IEEE MTT-V054-I08 (2006-08) [54, 8 ed.]

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AUGUST 2006

VOLUME 54

NUMBER 8

IETMAB

(ISSN 0018-9480)

MINI-SPECIAL ISSUE ON LARGE-SIGNAL CHARACTERIZATION AND MODELING OF NONLINEAR ANALOG DEVICES, CIRCUITS, AND SYSTEMS Guest Editorial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . N. Borges Carvalho, K. A. Remley, and D. E. Root

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MINI-SPECIAL ISSUE PAPERS

Envelope-Domain Time Series (ET) Behavioral Model of a Doherty RF Power Amplifier for System Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. Wood, M. LeFevre, D. Runton, J.-C. Nanan, B. H. Noori, and P. H. Aaen Calibration of Sampling Oscilloscopes With High-Speed Photodiodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T. S. Clement, P. D. Hale, D. F. Williams, C. M. Wang, A. Dienstfrey, and D. A. Keenan Noise Considerations When Determining Phase of Large-Signal Microwave Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P. S. Blockley, J. B. Scott, D. Gunyan, and A. E. Parker Large-Signal Behavioral Modeling of Nonlinear Amplifiers Based on Load–Pull AM–AM and AM–PM Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. Liu, L. P. Dunleavy, and H. Arslan Minimum-Phase Calibration of Sampling Oscilloscopes . . . A. Dienstfrey, P. D. Hale, D. A. Keenan, T. S. Clement, and D. F. Williams Experimental Characterization of the Nonlinear Behavior of RF Amplifiers . . . . Y. Rolain, W. Van Moer, R. Pintelon, and J. Schoukens Time-Domain Envelope Measurements for Characterization and Behavioral Modeling of Nonlinear Devices With Memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . F. Macraigne, T. Reveyrand, G. Neveux, D. Barataud, J.-M. Nebus, A. Soury, and E. NGoya In-Band Distortion of Multisines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. M. Gharaibeh, K. G. Gard, and M. B. Steer Amplitude and Phase Characterization of Nonlinear Mixing Products . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. C. Pedro and J. P. Martins

3163 3173 3182 3191 3197 3209 3219 3227 3237

CONTRIBUTED PAPERS

Linear and Nonlinear Device Modeling Design and Performance Analysis of Mismatched Doherty Amplifiers Using an Accurate Load–Pull-Based Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . O. Hammi, J. Sirois, S. Boumaiza, and F. M. Ghannouchi High-Efficiency Linear RF Amplifier—A Unified Circuit Approach to Achieving Compactness and Low Distortion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T. Y. Yum, L. Chiu, C. H. Chan, and Q. Xue

3246 3255

(Contents Continued on Back Cover)

(Contents Continued from Front Cover) Smart Antennas, Phased Arrays, and Radars Modified T-Shaped Planar Monopole Antennas for Multiband Operation . . . . . . . . . . . . S.-B. Chen, Y.-C. Jiao, W. Wang, and F.-S. Zhang

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Power-Efficient Switching-Based CMOS UWB Transmitters for UWB Communications and Radar Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. Xu, Y. Jin, and C. Nguyen

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Active Circuits, Semiconductor Devices, and ICs Low-Power-Consumption and High-Gain CMOS Distributed Amplifiers Using Cascade of Inductively Coupled Common-Source Gain Cells for UWB Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . X. Guan and C. Nguyen

3278

Signal Generation, Frequency Conversion, and Control Low Phase-Noise Microwave Oscillators With Interferometric Signal Processing . . . . . . . . . . . . . . . . . . . . . . . . E. N. Ivanov and M. E. Tobar A Low-Power Up-Conversion CMOS Mixer for 22–29-GHz Ultra-Wideband Applications . . . . . . . . . . . . . A. Verma, K. K. O, and J. Lin

3284 3295

Wireless Communication Systems On the Effects of Memoryless Nonlinearities on

M -QAM and DQPSK OFDM Signals . . . . . . . . . . . . . . . . . . . . . A. Chorti and M. Brookes

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Field Analysis and Guided Waves High-Order Runge–Kutta Multiresolution Time-Domain Methods for Computational Electromagnetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Q. Cao, R. Kanapady, and F. Reitich

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Filters and Multiplexers Design of a Dual-Band Bandpass Filter With Low-Temperature Co-Fired Ceramic Technology . . . . C.-W. Tang, S.-F. You, and I-C. Liu Design of Multiple-Stopband Filters for Interference Suppression in UWB Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. Rambabu, M. Y.-W. Chia, K. M. Chan, and J. Bornemann Adjoint Higher Order Sensitivities for Fast Full-Wave Optimization of Microwave Filters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. A. El Sabbagh, M. H. Bakr, and J. W. Bandler Broadband Quasi-Chebyshev Bandpass Filters With Multimode Stepped-Impedance Resonators (SIRs) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Y.-C. Chiou, J.-T. Kuo, and E. Cheng Novel Coplanar-Waveguide Bandpass Filters Using Loaded Air-Bridge Enhanced Capacitors and Broadside-Coupled Transition Structures for Wideband Spurious Suppression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S.-C. Lin, T.-N. Kuo, Y.-S. Lin, and C. H. Chen Miniature Broadband Bandpass Filters Using Double-Layer Coupled Stripline Resonators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Y. Zhang, K. A. Zaki, A. J. Piloto, and J. Tallo Packaging, Interconnects, MCMs, Hybrids, and Passive Circuit Elements Physics-Based Wideband Predictive Compact Model for Inductors With High Amounts of Dummy Metal Fill . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L. F. Tiemeijer, R. J. Havens, Y. Bouttement, and H. J. Pranger A Chip-Scale Packaging Technology for 60-GHz Wireless Chipsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . U. R. Pfeiffer, J. Grzyb, D. Liu, B. Gaucher, T. Beukema, B. A. Floyd, and S. K. Reynolds A Photonic Crystal Power/Ground Layer for Eliminating Simultaneously Switching Noise in High-Speed Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T.-L. Wu and S.-T. Chen Instrumentation and Measurement Techniques A Novel Technique for Deembedding the Unloaded Resonance Frequency From Measurements of Microwave Cavities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. J. Canós, J. M. Catalá-Civera, F. L. Peñaranda-Foix, and E. de los Reyes-Davó Investigation of Parylene-C on the Performance of Millimeter-Wave Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Kärnfelt, C. Tegnander, J. Rudnicki, J. P. Starski, and A. Emrich

3327 3333 3339 3352 3359 3370

3378 3387 3398

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Microwave Photonics Design of Low-Cost Multimode Fiber-Fed Indoor Wireless Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Das, A. Nkansah, N. J. Gomes, I. J. Garcia, J. C. Batchelor, and D. Wake

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MEMS and Acoustic Wave Components Micromachined Rectangular-Coaxial Transmission Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. R. Reid, E. D. Marsh, and R. T. Webster

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Biological, Imaging, and Medical Applications Estimation of Heating Performances of a Coaxial-Slot Antenna With Endoscope for Treatment of Bile Duct Carcinoma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K. Saito, A. Hiroe, S. Kikuchi, M. Takahashi, and K. Ito

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Information for Authors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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IEEE MICROWAVE THEORY AND TECHNIQUES SOCIETY The Microwave Theory and Techniques Society is an organization, within the framework of the IEEE, of members with principal professional interests in the field of microwave theory and techniques. All members of the IEEE are eligible for membership in the Society upon payment of the annual Society membership fee of $14.00, plus an annual subscription fee of $16.00 per year for electronic media only or $32.00 per year for electronic and print media. For information on joining, write to the IEEE at the address below. Member copies of Transactions/Journals are for personal use only. ADMINISTRATIVE COMMITTEE K. VARIAN, President S. M. EL-GHAZALY J. HAUSNER K. ITOH M. HARRIS D. HARVEY

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Honorary Life Members T. ITOH A. A. OLINER

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K. C. GUPTA (2005) R. J. TREW (2004) F. SCHINDLER (2003)

MTT-S Chapter Chairs Albuquerque: S. BIGELOW Atlanta: D. LEATHERWOOD Austria: R. WEIGEL Baltimore: A. D. BROWN Beijing: Z. FENG Beijing, Nanjing: W. X. ZHANG Belarus: A. GUSINSKY Benelux: D. V.-JANVIER Brasilia: A. KLAUTAU, JR. Buenaventura: C. SEABURY Buffalo: E. M. BALSER Bulgaria: K. ASPARUHOVA Cedar Rapids/Central Iowa: D. JOHNSON Central New England: K. ALAVI Central & South Italy: S. MACI Central No. Carolina: T. IVANOV Chicago: Z. LUBIN Cleveland: G. PONCHAK Columbus: F. TEIXEIRA Connecticut: C. BLAIR/R. ZEITLER Croatia: Z. SIPUS Czech/Slovakia: P. HAZDRA Dallas: R. EYE Dayton: A. TERZOUOLI, JR. Denver: M. JANEZIC Eastern No. Carolina: D. PALMER Egypt: I. A. SALEM Finland: T. KARTTAAVI Florida West Coast: K. O’CONNOR

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Editors-In-Chief DYLAN WILLIAMS NIST Boulder, CO 80305 USA Phone: +1 303 497 3138 Fax: +1 303 497 3970 email: [email protected] AMIR MORTAZAWI Univ. of Michigan Ann Arbor, MI 48109-2122 USA Phone: +1 734 936 2597 Fax: +1 734 647 2106 email: [email protected]

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Associate Editors

ANDREAS CANGELLARIS KENJI ITOH STEVEN MARSH Univ. of Illinois, Urbana Champaign Mitsubishi Electronics Midas Consulting USA Japan U.K. email:[email protected] email: [email protected] email: [email protected] YOSHIO NIKAWA DAVID LINTON TADEUSZ WYSOCKI Queen’s Univ. Belfast Univ. of Wollongong Kokushikan Univ. Northern Ireland Australia Japan email: [email protected] email: [email protected] email: [email protected] RUEY-BEEI WU MANH ANH DO JOSÉ PEDRO Univ. of Aveiro National Taiwan Univ. Nanyang Technological Univ. Portugal Taiwan, R.O.C. Singapore email: jcp.mtted.av.it.pt email: [email protected] email: [email protected] ZOYA POPOVIC ALESSANDRO CIDRONALI VITTORIO RIZZOLI Univ. of Colorado, Boulder Univ. of Florence Univ. of Bologna USA Italy Italy email: [email protected] email: [email protected] email: [email protected] M. GOLIO, Editor-in-Chief, IEEE Microwave Magazine G. E. PONCHAK, Editor-in-Chief, IEEE Microwave and Wireless Component Letters

SANJAY RAMAN Virginia Polytech. Inst. and State Univ. USA email: [email protected] JENSHAN LIN Univ. of Florida USA email: [email protected] RICHARD SNYDER RS Microwave Company USA email: [email protected] ALEXANDER YAKOVLEV Univ. of Mississippi USA email: [email protected] T. LEE, Web Master

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Digital Object Identifier 10.1109/TMTT.2006.882240

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 8, AUGUST 2006

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Guest Editorial S GUEST Editors, we are honored and excited to introduce this TRANSACTIONS’ Mini-Special Issue on “Measurements for Large-Signal Characterization and Modeling of Nonlinear Analog Devices, Circuits, and Systems.” Large-signal measurements reveal the detailed, often extremely rich, nonlinear dynamical responses exhibited by analog microwave devices, circuits, and systems under conditions of actual use in a way that static (dc) and linear ( -parameter) measurements simply cannot. In other words, large-signal measurements provide the quantitative basis for scientific understanding, as well as engineering exploitation of real-world nonlinear devices, circuits, and systems. Measurements of complex dynamic nonlinear phenomena can be made in the time, frequency, or mixed (e.g., envelope) domains. In many cases, new types of instrumentation (e.g., large-signal network analyzers) are necessary to capture this behavior. In some cases, new identification algorithms and techniques can be applied with existing hardware to obtain the needed data. The excellent papers that were submitted for this TRANSACTIONS’ Mini-Special Issue testify to the fact that large-signal measurement and modeling is an important and growing area of research within the microwave community. The techniques and approaches presented are indicative of the vibrant and interdisciplinary nature of this field. In fact, we have received papers from 12 different countries. Published papers were contributed by universities, industry, and government laboratories. Within this TRANSACTIONS’ Mini-Special Issue are papers that focus on several aspects of the large-signal problem: what we measure, how we measure it, and how we interpret and apply measured data in modeling applications. Papers can be categorized according to several key themes that address the current problems faced by the microwave community in largesignal measurement and modeling. These key themes include the following. Phase measurement: The accurate measurement of the phase relationships among frequency components has been and continues to be an important topic of research. For large-signal measurements, the magnitude and phase relationships among frequency components at the input of the system will typically affect the operating state of the nonlinear system. Since the excitation itself affects what signal and what level of distortion we see at the output of the system, knowledge of the phase relationships among the input signal frequency components is critical. While much recent research has focused on the measurement of the phase of harmonics with respect to a fundamental, papers in this TRANSACTIONS’ Mini-Special Issue demonstrate that measuring and interpreting phase relationships

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Digital Object Identifier 10.1109/TMTT.2006.879178

of signals within the modulation bandwidth itself has become an important topic of current research. No doubt this is because telecommunications applications typically utilize complex modulated-signal excitations. Instrumentation: Instruments that utilize time-domain acquisition are often used in large-signal measurement applications because they can provide the phase relationships among measured frequency components. In this TRANSACTIONS’ Mini-Special Issue, we see discussions on the use of the most ubiquitous of time-domain acquisition systems—the oscilloscope—in large-signal measurement applications. While the digital community has relied on this workhorse instrument to characterize system distortion for years, this TRANSACTIONS’ Mini-Special Issue illustrates its increasing prominence in large-signal analog applications. Calibration: The improvements in instrumentation for demanding time-domain measurements require new forms of calibration. Papers in this TRANSACTIONS’ Mini-Special Issue are devoted to the clarification of time-domain calibration methods and provide proposals for better time-domain measurements. This allows other researchers to improve their parameter-extraction capabilities and, thus, improve the understanding and modeling of nonlinear distortion behavior. Correlation Techniques: Separating the desired output signal from distortion products can be difficult in largesignal measurements. This is because in addition to ideal amplification, nonlinear elements also introduce distortion noise and generate intermodulation products that mix from other frequency bands into the frequency band of interest. As illustrated by papers within this TRANSACTIONS’ Mini-Special Issue, comparing input signals to a complex output signal by means of cross-correlation techniques can help us understand the sources and effects on distortion. Nonlinear Dynamical Measurement: Nonlinear systems and circuits demonstrate a richness of behavior that includes the static nonlinear point-of-view (generation of new spectral components and distortion of time-domain signals), but also goes beyond this to current dynamic phenomena that are quite complex and difficult to identify. Those dynamics can operate either at the RF (high values of frequency) or at the envelope (low values of frequency), or both, and these two time scales can even interact. These dynamics are also fundamentally nonlinear, which restricts the validity of any linear identification scheme. Thus, new methodologies to identify these dynamics are fundamental for a correct modeling of (e.g., wireless) system behavior. Some papers in this TRANSACTIONS’ Mini-Special Issue are devoted to nonlinear dynamic identification, and parameter extraction for nonlinear dynamical behavioral models, which will help the wireless design engineer properly account for nonlinear distortion.

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Specially Designed Stimulus for Nonlinear Identification: Nonlinear identification is fundamental when studying the signal distortion arising from nonlinear behavior. In order to identify and, thus, extract, a correct model of a nonlinear system, great care must be given to the design of the proper stimulus for the nonlinear black box model. From an instrumentation point-of-view, optimal performance exists when stimuli are periodic, but in this case, they should be precisely designed in order to excite most of the system states. One of those signals is the well-known multisine, and despite the long prominence of this theme, some new proposals are presented in this TRANSACTIONS’ Mini-Special Issue to give confidence bounds on nonlinear distortion using specially designed stimuli. While this TRANSACTIONS’ Mini-Special Issue has been devoted to new research-oriented results, we certainly see room for a future special issue or alternative focused publication on this topic that is more tutorial and also more retrospective in nature. This would provide pedagogical information to those entering the field and synthesize the great themes and major contributions from previously published material. Increased use of large-signal measurement methodologies and instrumentation will not only improve observation and characterization of nonlinear components and systems, but will also accelerate deployment of powerful modern mathematical approaches, both theoretical and those involving numerical simulation, to nonlinear problems. The end result will be improved design and optimization of wireless telecommunication and other important high-tech engineering systems.

This Guest Editorial would not be complete without expressing a special acknowledgment to the former Editor-inChief of this TRANSACTIONS, Prof. Michael B. Steer, for his unstinting help and advice in bringing this TRANSACTIONS’ Mini-Special Issue to fruition. We also welcome the new Editors-in-Chief, Dylan Williams and Amir Mortazawi. We express our gratitude to the expert reviewers who donated their time to make detailed reviews of these papers. Finally, this TRANSACTIONS’ Mini-Special Issue primarily represents the significant accomplishments and original work of the authors. We thank them for their contributions, and for their efficiency and flexibility working with the editors and reviewers both to improve their papers and simultaneously to meet the publication schedule.

NUNO BORGES CARVALHO, Guest Editor Instituto de Telecomunicações Universidade de Aveiro Aveiro, 3810-193 Portugal KATE A. REMLEY, Guest Editor National Institute of Standards and Technology Boulder, CO 80305 USA DAVID E. ROOT, Guest Editor Agilent Technologies Santa Rosa, CA 95403 USA

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Envelope-Domain Time Series (ET) Behavioral Model of a Doherty RF Power Amplifier for System Design John Wood, Senior Member, IEEE, Michael LeFevre, Member, IEEE, David Runton, Member, IEEE, Jean-Christophe Nanan, Basim H. Noori, Member, IEEE, and Peter H. Aaen, Member, IEEE

Abstract—In this paper, we present an envelope-domain behavioral model of a high-power RF amplifier. In this modeling approach, we use the signal envelope information, and the behavioral model is generated using an established nonlinear time-series approach to create a time-domain model that operates in the envelope or signal domain. We have generated a model of a 200-W Doherty amplifier from measured IQ data taken using a wideband code-division multiple-access excitation; the amplifier was driven from the linear regime into saturation. The time-series model was created using a time-delay embedding identified from auto-mutual information analysis, and an artificial neural network was used to fit the multivariate transfer function. The model has been validated using measured and simulated data, and it has been used in the development of a system-level design of a digital pre-distorter. Index Terms—Power amplifiers, modeling, nonlinear circuits, nonlinear systems.

I. INTRODUCTION HE behavioral modeling of RF and microwave amplifiers has attracted a great deal of attention in recent years (see, e.g., [1]). As microwave or RF systems become increasingly complex, it is becoming impossible to simulate their performance at the transistor level of circuit description. A typical module or subsystem of a microwave or RF communications system may contain several transistors or RF integrated circuit (RFIC) amplifiers, and even a relatively modest subsystem may contain too many components to allow a satisfactory simulation of the complete module—there are too many nonlinear equations to solve: the simulation may take too long to converge, may converge to an incorrect solution, or fail to converge altogether. By using a behavioral model of the nonlinear component or integrated circuit (IC), it is often possible to perform the system simulation successfully. Behavioral models are simplified models of the essential nonlinear behavior of the complex sub-circuits; this simplification means that these models will execute more quickly, and use much less memory than if an entire complex subsystem was simulated at the transistor level.

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Manuscript received December 15, 2005; revised March 15, 2006. J. Wood, M. LeFevre, D. Runton, B. H. Noori, and P. H. Aaen are with the RF Division, Freescale Semiconductor Inc., Tempe, AZ 85284 USA (e-mail: [email protected]). J.-C. Nanan is with the RF Division, Freescale Semiconductor Inc., Toulouse 31023, France. Digital Object Identifier 10.1109/TMTT.2006.879134

A variety of behavioral models for both wideband and relatively narrowband amplifiers have been reported [2]–[7] that mimic the RF performance of the amplifier, and that can be imported into the RF simulator, enabling RF system design to be carried out at the higher level of abstraction. The main focus of the behavioral modeling activity has thus far been on the development of models that can faithfully represent the nonlinear behavior of the amplifier, and yet can be simulated in a short time. This is particularly true for the power amplifiers used in wireless communications systems, wherein the accurate prediction of, and ultimately correction for, the nonlinear behavior of the amplifier is key to the overall system performance. Typically, a power amplifier will be operated with a signal drive sufficient to cause compression, to realize the highest operating efficiency. Clearly, the amplifier is operating in a nonlinear regime, and an accurate description of this behavior in the model is essential. It is not sufficient to design the amplifier using small-signal techniques alone, and the use of nonlinear simulation engines such as harmonic balance is virtually essential to capture the full RF behavior of the amplifier. Further, the measurement and characterization of the amplifiers’ RF performance is almost always carried out in the frequency domain using load–pull techniques [8], and large-signal network analysis (LSNA) [9], [10] where available. For these reasons, many behavioral models of RF and microwave amplifiers are identified using sinusoidal or multisine (periodic) signals, and have been targeted at the harmonic-balance simulator. Here, our focus is on high-power RF amplifiers that are capable of delivering hundreds of watts of RF power in wireless infrastructure applications. Both linearity and efficiency are significant design issues: they can be traded against each other for an optimal amplifier design for a given set of specifications or circumstances. Often the linearity is traded for higher efficiency, with the expectation that the nonlinear behavior can be suppressed at the system level, and the linear performance is recovered. As the trend is toward higher signal bandwidth and high peak-to-average ratios (PARs) for the digitally modulated signals, the requirement for an accurate dynamical model of the power amplifier at the system level is paramount. At the system level the designer is more concerned with the (digital) signal itself, and the RF carrier is of lesser importance: in other words, we are working in the envelope domain. This is particularly true for RF power amplifier design [11], where, because of the bandpass or low-pass filtering associated with the transistor itself, and its package and fixture, the measured RF

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signal is constrained to be sinusoidal. On the other hand, the nonlinear behavior, such as gain compression (AM-to-AM) and transfer phase (AM-to-PM) characteristics, occurs at the envelope of the signal. Thus, the behavioral model must be capable of describing these nonlinearities in the envelope-domain simulation using digitally modulated signals.

II. MODELING APPROACHES

A. Behavioral Model Structure A simple approach to modeling of (power) amplifiers is to fit the static AM-to-AM and AM-to-PM drive-up characteristics of the amplifier, using a polynomial or other curve-fitting method. This technique is the basis of several “system” or “behavioral” amplifier models that are supplied with commercial RF simulators. While straightforward to implement, this method suffers from the drawback that neither short-, nor long-term memory effects are described: these models are RF frequency-independent, and the polynomial fit yields an instantaneous function of the inputs. This modeling approach is generally inadequate for power amplifiers. A more rigorous description of nonlinear dynamical behavior can be found in Volterra series [12], [13], and several behavioral modeling approaches based on Volterra [14], [15] and modified Volterra series [16], [17] have been reported. Pedro and Maas [18] provide a detailed description and classification of these power amplifier models. The extraction of the (modified) Volterra kernels from RF measurements can be difficult, and often the model is derived from simulated data. The Volterrabased descriptions are often derived in the time domain, but can be expressed using analytical signals in the envelope domain. Time-series methods are also appropriate for nonlinear dynamical modeling [19], [20], and have been demonstrated for RF and microwave applications, showing good fidelity to the large-signal RF waveform when the RF circuit has been driven into nonlinear conditions [21], [22]. Since the digitally modulated envelope signal is essentially a discrete-time signal, the time-series approach seems a natural one for envelope-domain modeling of digital signals. Both Volterra and time-series methods can capture the signal history or memory effects. In the Volterra methodology, the signal history is included through multidimensional integrals over the memory time. In the time-series approach, the memory is captured naturally through the delay embedding [23], which will be described later. In the modeling technique described here, we use and adapt classic time-series analysis [19], [22] approaches to create the envelope-domain behavioral model of an RF power amplifier. In common with several reported nonlinear models for RF power amplifiers, we use an artificial neural network (ANN) for the multivariate function approximation because of the ease of use and accuracy that this method affords. Despite this apparent similarity with these other published envelope-domain models, the model presented here differs significantly in terms of the model identification techniques employed, and we believe this

leads to a worthwhile improvement in model performance. In particular, we use well-established techniques of system identification to determine the time-delay vectors that are used as the input data to our model; this is in contrast to the empirical methods employed to determine the number of delays, as found in previously reported models (see, e.g., [32]).

B. Simulation Environment Most of the reported behavioral models of the nonlinear amplifiers and other components are designed to be used directly in an RF simulator. Some commercial RF simulators (e.g., Agilent-EEof ADS, AWR Microwave Office) already include simple system-level models, as already outlined in Section II-A. The behavioral model may be implemented in the simulator using time- or frequency-domain constructs or descriptions. Nevertheless, the frequency-domain performance in harmonic-balance simulation is often used as a performance yardstick. The prediction of harmonic and intermodulation distortion products is used as a measure of the model’s ability to predict the nonlinear behavior of the amplifier. This is understandable when the model is to be used in a hierarchical simulation of a complete RF system, such as an RF receiver front end or broadband RF power source. Additionally, the models themselves are often identified using frequency-domain signals, either from measurement using an LSNA or simulation using harmonic balance. Such models are often also usable in Circuit Envelope simulation, and the RF behavior of the amplifier can then be predicted under the signal conditions appropriate to communication systems. However, when we adopt a systems perspective, we have a different set of model requirements. The design of the linearizer/ pre-distorter components of an RF transmitter is carried out in a digital signal domain: the system designers will typically work in a time-stepping simulation environment that includes digital signal processing (DSP) tools; Mathworks’ Simulink and MATLAB are popular choices. It would seem appropriate, therefore, to construct the behavioral model of the RF amplifier in such a way as to be usable directly in the DSP simulation environment. Accordingly, we have created our behavioral model in MATLAB; this also allows us to advantage of the various toolboxes that are available to assist in the construction of the model. Incidentally, we also avoid the often difficult step of converting a mathematically well-defined model into a component that can be used in the RF simulator; this is a step that can severely limit the performance of an otherwise sound model. By targeting MATLAB as the vehicle for our behavioral model, we are also able to use this model directly in the various system-level simulators available in the RF electronic design automation (EDA) tools. Time-stepping simulators such as Agilent-EEsof Ptolemy and AWR’s Visual System Simulator can read MATLAB functions directly at run time through a built-in component interface; no other modification is needed on our part. Of course, MATLAB also interfaces with Mathworks’ system-level simulator Simulink, and this approach has also been demonstrated for behavioral modeling of an RF amplifier [24].

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Fig. 1. Doherty amplifier module: the transistor package contains two separate transistors, which are configured as a Doherty amplifier. (Color version available online at: http://ieeexplore.ieee.org.)

Fig. 2. Experimental setup is an arbitrary signal generator and signal analyzer pair, the instruments and data manipulation all handled in MATLAB. (Color version available online at: http://ieeexplore.ieee.org.)

III. MEASUREMENTS AND SIMULATION FOR MODEL GENERATION The RF power amplifier under test is a Doherty amplifier using a Freescale MRF6P21190H transistor1; this component contains two separate power transistors that can be connected in either push–pull class AB configuration, or as a Doherty amplifier with class AB main and class C peaking transistor bias. The Doherty amplifier is capable of producing approximately 53 dBm of output power at 1-dB compression in the 2.1-GHz wireless band. The transistor is mounted in a fixture that provides the appropriate bias connections, quarter-wave transformers for the Doherty configuration, and impedance matching to 50 . An illustration of the fixture is shown in Fig. 1. The experimental test setup is shown in Fig. 2. MATLAB is used as the basis for controlling the complete system: it both manipulates the data and it controls the test equipment via the IEEE-488 interface bus. 1[Online].

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We use a complex digitally modulated RF signal for the model identification. The signal is characterized by in-phase (I) and quadrature (Q) components in the envelope that exhibit well-defined properties due to the modulation. An IQ signal is preferred over a continuous wave (CW) sinusoidal signal because it exercises the whole of the low frequency (memory) spectrum, and is convenient to generate and measure using a signal generator/analyzer pair. The test signal that we are using is the Test Model 1 3GPP complex modulation: this is a wideband code-division multipleaccess (WCDMA) signal, which has a channel bandwidth of 5 MHz and a data rate of 21.7 ns/bit, which is a sampling rate of 46.08 MSamples/s. The IQ data files are constructed in MATLAB and sent to the Rohde and Schwarz SMU arbitrary waveform generator, which uses a 16-bit DAC to reconstruct the signal. The SMU is used to modulate the data and up convert it to the RF frequency. We use two adjacent channels centered at 2.14 GHz. The composite input peak-to-average ratio (PAR) is 8.1 dB. This signal is amplified to the proper level using a very linear 10-W power amplifier. The average drive power from the SMU to the Doherty amplifier is measured using a power meter. The range of drive powers used was from approximately 22 to 35 dBm, which is sufficient to drive the Doherty amplifier from the linear region to 2-dB compression. The input power was increased in discrete steps over this range. This was done to allow the acquisition of the complete data signal, of some 50 000 samples, or approximately 2 ms, under conditions of constant drive. The received data is acquired using the Rohde and Schwarz FSQ spectrum analyzer in the IQ mode. In this mode, the FSQ samples the baseband data with a 14-bit ADC using the same sampling rate as the SMU. The receiver bandwidth is approximately 23 MHz, enough for capturing and compensating for third-order nonlinearities for the 10-MHz test signal. The input to the power amplifier is first captured, then the amplifier is placed into the circuit and its output is captured. The average output power from the amplifier is also measured using a power meter. The IQ baseband signal is read back into MATLAB to be time aligned and further processed as necessary for the model generation, as described below. Since the input and output waveforms are time aligned, the RF group-delay contribution of the amplifier is eliminated from the data, although the AM-to-PM information for the envelope is, of course, retained. This test setup can be replicated in the RF simulator. A wideband two-channel CDMA source was designed in ADS [25], and the Doherty amplifier circuit was simulated using the ADS Circuit Envelope for a range of input drive conditions up to 6-dB output/compression. The envelope data can then be read as a time series in to MATLAB for model generation. IV. ENVELOPE-DOMAIN TIME-SERIES MODEL EXTRACTION Following classic time-series analysis, the first thing to do is choose an appropriate embedding or phase-space reconstruction to describe the system [19]. The key idea is to represent the measured (or simulated) response data in a new phase or state space, which is a higher dimensional space, built not only from the measured data itself, but also transforms of this data. The embedding theorem states that, in this higher dimensional

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space, i.e., the “model” space, the trajectory of the reconstructed response and that of the original data will differ by no more than a smooth (differentiable) and invertible change of coordinates. In other words, the dynamical behavior of the original system is preserved by this mapping into the model space—the model is, therefore, an accurate representation of the dynamics of the actual system under test. This representation in the higher order space is called an embedding. The word “embedding” is often used for the procedure of creating the higher dimensional space, as well as the mathematical description of the mapping itself. A commonly used embedding in time-series analysis is to use time delays in the response data: the response data is then represented as a function of delayed values of itself as follows: (1) Thus, from the simple procedure of generating a delayed function, we are able to explicitly reveal the dynamics of the system [26]. The modeling tasks are to determine: 1) the value of the delay ( ); 2) how many delays we need ( ); and 3) find a function approximation ( ) to the trajectory in the model space. A description of how the embedding procedure can be applied to time-series analysis and modeling of RF and microwave amplifiers was presented in [22]. Naturally occurring time series are typically autonomous; i.e., they are a series of (discrete) data values. The time series model described by (1) is known as an autoregressive (AR) model. However, in the case of RF amplifiers, we have a driven system; using the embedding theorem of Takens [27] extended to the driven case by Stark [28], we can model this input–output system by including the drive signal in the embedding space

Fig. 3. Auto-mutual information for the output signal. The minimum value occurs at five time steps, indicating this is the characteristic delay time. (Color version available online at: http://ieeexplore.ieee.org.)

The delay time should, therefore, be chosen so that the vector or signal is effectively de-correlated from the delayed version. For periodic signals, this characteristic delay time can be found from the auto-correlation function; for example, a sinusoidal signal will be completely de-correlated from a delayed version of itself if the delay time is a quarter period. However, for nonlinear systems, the signals can have broadband power spectra, and correlation functions are not so useful; methods from information theory are used. For statistical information signals, such as the WCDMA signal used here, a technique known as auto mutual information is used. The mutual information is the amount of information that is shared between two data sets. It can be found from the difference in the information between two samples taken independently and taken together [26] as follows:

(2) This is known as an autoregressive-moving-average model (ARMA). The delay or memory depths and need not be identical. Again, the embedded model will mimic the dynamics of the original system, and we can predict the behavior of the system from the behavior of the driven model. In fact, from considerations of stability of the model in simulation, we have chosen to use delays in the drive signal only for the embedding. This may not result in the most compact model, in term of the number of parameters, but avoids any risk of instability with a recurrent model structure. A. Choosing the Delay As can be seen from (1), each delayed vector forms a dimension of the model space. The objective here is to choose a value for the delay that makes each dimension orthogonal to the others. This will result in the most compact model space. Whereas any (suitably small) value for the delay will work, the vectors will not necessarily be orthogonal, and will have projections from one onto another. This will result in a model with a larger number of parameters than is strictly necessary that may have stability or convergence problems.

(3) is the where is the mutual information for a delay of probability density of the data sample at times and , and is the joint probability of the sample and delayed sample. If the mutual information is zero, the two samples—the signal and delayed versions—are completely de-correlated. The characteristic delay can be found by sweeping over a number of time steps and finding when the mutual information falls to the first minimum value, indicating minimum correlation. For the two-channel WCDMA signal that was used for model identification, a characteristic delay time of five time steps was found from auto-mutual information analysis of the output signal, as shown in Fig. 3. Since the drive (input) and response (output) signals are very similar to each other, we use this delay for embedding the drive signal. B. Number of Delays The total number of delays represents the dimension of the model space. It is often associated with the “memory depth”

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of the system. The number of delays is often found using techniques of “false nearest neighbors” based on the work of Kennel et al. [29]. Essentially these techniques work by progressively increasing the number of the dimensions of the model space until some metric is satisfied that indicates that the time series follows a single-valued trajectory. In Kennel’s method, the metric is the number of data points in the set that lie close to the expected value for the next point in the time series. Since most of these data points are not the next point in the time series, they are “false neighbors” of the current time point. As the embedding dimension increases, then the number of values close to the true expected value for a given set of delay vectors will decrease. The true expected values will lie on a unique trajectory once the embedding dimension is sufficient: in this case, we have no “false neighbors,” only the correct next value in the time series. The metric is thus when the number of “false neighbors” falls close to zero. Kennel’s method was originally developed for autonomous time series. Cao et al. [30] extended this study in two ways: first, by extension to input–output systems; second, by implementing an averaging technique that avoided the use of a threshold parameter in Kennel’s original work. For a general delay-embedded time-series model of an inputoutput system, as described by (2), we have different memory depths for the drive and response signals: and . The total embedding dimension is given by the sum of these depths. In Cao et al.’s method, given some number of delays in the drive and response, a vector in can be constructed as follows: (4) where is some particular point in the data set. Another vector at some index is chosen so that the distance between and this new vector is minimized. This new vector is then the nearest neighbor to in this space. In this case, the distance between the outputs and should also be small. Cao et al. define the following ratio:

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Fig. 4. Minimum embedding dimension using Cao et al.’s method, applied to the output I signal from the Doherty amplifier. An embedding dimension of four delays is indicated as appropriate. (Color version available online at: http://ieeexplore.ieee.org.)

variables is fixed. In Fig. 4, we show the results of increasing the embedding dimension on the parameter for the I signal; the Q signal behaves identically, as expected—the dynamics of the amplifier are independent of the IQ signals themselves. Approximately four delays (i.e., five dimensions in total) are required to unfold the model space into enough dimensions to describe the system dynamics completely. For the identification of the time-series model parameters, we have used the auto-correlation, auto-mutual-information, and Cao nearest neighbor search functions found in OpenTSToolbox, which is shareware available for MATLAB: the software and user guide can be downloaded online.2 C. Multivariate Function Fitting The moving-average model for the embedded time-series model is expressed as

(7) (5) where is the maximum Euclidean distance for any delay, and then calculate the average value of over the whole data set. This is the “averaged false neighbors,” which is a function of the number of delays , and is denoted by . As the dimension of the model space is increased, say, from to , then the value of the averaged false neighbors will change, and this can be expressed as

(6) approaches unity, the embedding dimension is found. As We applied this method to the output signal of the Doherty amplifier. Since we are using delays only in the input drive signal, the procedure is simplified in that one of the embedding

is a single valued function relating the observable where output and the embedded drive variables. The relationship is a nonlinear one so we use a nonlinear function fitting method. ANNs are an established nonlinear function fitting technique, and as the dimensionality of the model space becomes larger because of the embedding, the ANN techniques become the natural method for fitting this multivariate discrete time data. We use a feed-forward neural network (FFNN), as this is straightforward to train and implement; we use the MATLAB Neural Network Toolbox. The input data IQ streams are embedded with the appropriate number of delays; this is done at each separate RF power level used in the measurement. Given that we take some 50 000 samples at each of 12 or so power levels, this yields a potentially huge dataset. We take a different subset of the embedded data 2[Online].

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Fig. 5. Convergence of the neural-network fitting; the black line is the training data, the solid gray line is the validation data, used for guiding the training, and the dotted gray line is the test data, which we observe as a check on generalization property of the network.

at each power level, thus every data point is unique in time. The subsets are then concatenated to form the training data. The complete training data set consists of three different sets of data, i.e.: 1) training; 2) validation; and 3) test data. Each of these three data sets is approximately 35 000 samples. The training and validation sets are used to control the gradient descent direction and to ensure that the network is not overtrained; the test data is used to ensure generalization, but not used in the actual training. The neural network is trained in batch mode. We have trained several neural networks having various hidden layer configurations, and found that a single hidden layer with a small number of neurons is suitable for fitting this data. The results presented below are for a single hidden layer network comprising 15 neurons. For a (semi) automated model extraction, we would employ pruning and neuron-addition techniques to size the hidden layer properly based on the training accuracy. The MATLAB neural-network function fitting was performed using Bayesian regularization techniques, which yielded smooth functions that were generally free from overtraining. The progress of the training over time for this model is shown in Fig. 5; here, the training was halted after 200 epochs. The normalized mean square error for this model is approximately 0.00144 for each of the IQ signals. The whole of the data analysis, embedding, neural network training, and verification is done in MATLAB. To compare with the results of the model fitting with published values, the measured and modeled IQ values were converted into envelope magnitude values, and the error recalculated. The normalized mean square error is then 3.706 10 , or 34.3 dB, comparable with the results of Isaksson et al. [31], where the magnitude of the envelope is modeled by a radial basis function neural network (RBFNN). For our application of the behavioral model to the development of a pre-distortion system, we have used the FFNN since the input data to the pre-distorter is acquired in batch mode. Another option would be to use a time-delay neural network (TDNN), wherein the raw I and Q data are delayed using a tapped delay line before being fed to the neural network [32]. From the perspective of training of a neural network, it is easier

Fig. 6. Comparison between the network output and measured data: I-channel data for the Doherty amplifier in 1-dB compression.



to train a regular FFNN, giving the network the complete set of delayed vectors and training in batch mode. In fact, once trained, the regular feed-forward network can be used in a TDNN format by providing it with the appropriate delay vectors from the delay line. Both feed-forward and TDNNs are standard features of the MATLAB neural-network toolbox. We are fitting IQ data at the same time in a single neural network—the I and Q are related, and only certain values are permitted by the modulation scheme. These restrictions are then learned by the network. It makes sense that generally for IQ digitally modulated signals this approach is used. The alternative to fitting the IQ data would be to use the magnitude and phase of the envelope signal: essentially fitting the AM-to-AM and AM-to-PM characteristics directly. The output signals are then derived from these functions. This is similar to the approach used by Isaksson et al. [31], where only the magnitude of the envelope is modeled, and the transfer phase is added linearly as bias to the output neurons. This is claimed to give a more compact network than a regular FFNN (as, indeed, only the magnitude function is being fitted). A time-delayed input was used to accommodate “memory” effects, though no details were given of how the number of delays had been chosen. V. MODEL VERIFICATION The first verification of the neural-network model is to see how well it predicts the “test” data set, which has not been used in the training. In Fig. 6, we show a small sample of the output I data with the amplifier at approximately 1-dB compression: the model predicts the measured data extremely well. The Q-channel data exhibits the same excellent agreement between the model and measured data. The dynamic AM-to-AM and AM-to-PM characteristics are shown in Figs. 7 and 8, respectively, comparing the measured amplifier and the envelope time-series neural-network model. Again, the “test” data set was used for this verification. It is clear that the new model predicts the amplifier’s dynamics very well, even when the amplifier is driven into compression. This range of powers includes the turn-on point of the peaking amplifier, at around 30-dBm input power, and thus any changes in the

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Fig. 7. Measured (grey circles) and modeled (black) AM-to-AM characteristics of the Doherty amplifier under two-channel WCDMA drive. This plot uses the “test” data set.

Fig. 8. Measured (grey circles) and modeled (black) AM-to-PM characteristics of the Doherty amplifier under two-channel WCDMA drive. This plot uses the “test” data set.

overall system dynamical behavior due to this are accommodated by this model. The memory effects associated with the amplifier operating in large-signal conditions are also captured accurately: the memory is associated with the width or spread of the curve. The large spread in data at low powers is attributable to the noise floor in the analyzer: this noise has been included in the model, but could be reduced by better averaging of the data. These dynamic AM-to-AM and AM-to-PM curves compare favorably with similar results presented in [32], wherein the authors, although using time-delayed IQ signals as inputs to a neural-network model, have determined the delays heuristically, whereas here we have used well-established techniques of system identification based on time-series analysis to determine the model dynamics properly. This has resulted in a more accurate capture of the breadth of these dynamic traces and, hence, a better model of the amplifier’s memory.

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Fig. 9. Measured (grey) and modeled (black) power spectra of the test data set, showing the IF bandwidth of the receiver, and the model accuracy in the spectral domain.

In Fig. 9, we show the output power spectrum of the envelope signal obtained from measurement, and compare with the model predictions, at approximately the 1-dB compression point for the amplifier. The measured data shows the limits of the IF bandwidth of the receiver. The model data tracks this spectrum well both in-band and in the adjacent channel with some small errors showing in the alternate channel. The model also attempts to predict the next adjacent channel (i.e., seventh order) intermodulation products, even though these signals are filtered from the input signal, indicating that we are modeling the amplifier’s dynamics and not merely mimicking the data. The above results indicate that the envelope time-series model describes the dynamical behavior of the amplifier accurately. Given the statistical nature of the drive signal, it is important that the signal statistics are also reproduced by the model. This can be verified by observing the complementary cumulative distribution function (CCDF) for the measured data and the model predictions. The CCDF is a measure of the amount of time the signal spends above a given power level. The shape of this curve is dependent upon signal modulation. As can be seen in Fig. 10, the measured and modeled output signals are very similar, indicating that the model has preserved the amplifier dynamics and memory behavior. In other words, the memory effects observed in the compression curves are not a result of noise or poor model accuracy. VI. DIGITAL PRE-DISTORTION The overall goal of this modeling exercise is to create a behavioral model that can be used in our development and understanding of digital pre-distorter (DPD) systems. As such, the model must be fast to simulate and accurate in response. A description of how DPD works is beyond the scope of this paper, but a background can be found in [33]. Our current DPD development setup is as shown in Fig. 1; this is the same arrangement as for the model extraction. The DPD algorithms are coded in MATLAB. A record of the amplifier output IQ data is captured by the spectrum analyzer, read into

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Fig. 10. CCDF curves for measured (grey) and modeled (black) output IQ signals.

MATLAB, and the DPD algorithm is applied. A typical record length may be 10 000 samples. Once the pre-distorter has been updated, the cycle begins again, a new record is captured, and the DPD system updates the pre-distorter again. After a few cycles, the DPD system converges. This can be a time-consuming process, as the measurement system needs to be set up, calibrated, and the data collected over a range of drive powers, and repeated for several cycles. This procedure can also be carried out in simulation. We use Agilent-EEsof Ptolemy to perform this system level simulation. Ptolemy includes WCDMA sources, sinks that mimic the signal analyzer, and it can co-simulate the Doherty amplifier circuit using the ADS Circuit Envelope simulator, and the DPD algorithms in MATLAB. Unfortunately, this procedure is also quite time consuming, the main culprit being the convergence of the Doherty amplifier circuit in Circuit Envelope, which results in an overall simulation time of approximately 4 h per RF power level. A behavioral model would clearly make this DPD development process more tractable. Further, as stated earlier, if the behavioral model were available in MATLAB, the co-simulation in the RF domain could be avoided. In fact, in the limit, the complete DPD development can be done in MATLAB. In Fig. 11, we show the power spectrum for the envelope time-series behavioral model of the Doherty amplifier, driven by the two-channel WCDMA signal to an average output power of 45 dBm with 8.1-dB PAR: this is equivalent to approximately 1-dB compression. This spectrum is in good agreement with measured data [34]. In the simulation, we have applied a simple predistortion using a seventh-order memory polynomial with five delay taps. The model predicts an improvement in ACLR of 15 dB, yielding an ACLR of approximately 45 dBc, which is quite close to the measured ACLR of approximately 50 dBc at this drive level. This result is obtained after only a few seconds of simulation in MATLAB, in contrast to needing up to 1 h for measurement, and several hours for simulation of the transistor-level circuit using Ptolemy and Circuit Envelope. This is an example of the tradeoff of speed versus accuracy in behavioral modeling.

Fig. 11. Simulated power spectrum of the envelope time-series model of the Doherty amplifier, showing the raw distortion performance and the improvement gained from using a simple pre-distortion algorithm.

VII. CONCLUSION We have developed an envelope-domain behavioral model of a Doherty high-power RF amplifier, using well-established model identification techniques founded in time-series analysis. The model reproduces the dynamical behavior of the amplifier accurately, capturing nonlinear and memory effects with high fidelity. The model is extracted from measured data in a simple and straightforward procedure, and it is implemented in MATLAB. System-level simulations of a pre-distorter using this model show generally good agreement with the measured data, and with orders of magnitude improvement in simulation time compared with the transistor-level circuit model. Such an improvement at almost no cost to accuracy can make model-based design of pre-distorter systems using a simulator a reality. ACKNOWLEDGMENT The authors would like to thank the management of the RF Division, Freescale Semiconductor Inc., Tempe, AZ, and their customers, for their encouragement of the authors’ pursuit of this modeling approach. REFERENCES [1] J. Wood and D. E. Root, Fundamentals of Nonlinear Behavioral Modeling for RF and Microwave Design. Norwood, MA: Artech House, 2005. [2] J. C. Pedro, N. B. Carvalho, and P. M. Lavrador, “Modeling nonlinear behavior of bandpass memoryless and dynamic systems,” in IEEE MTT-S Int. Microw. Symp. Dig., Philadelphia, PA, Jun. 2003, pp. 2133–2136. [3] J. Wood and D. E. Root, “The behavioral modeling of microwave/RF ICs using non-linear time series analysis,” in IEEE MTT-S Int. Microw. Symp. Dig., Philadelphia, PA, Jun. 2003, pp. 791–794. [4] A. Soury, E. Ngoya, and J. Rousset, “Behavioral modeling of RF and microwave circuit blocs for hierarchical simulation of modern transceivers,” in IEEE MTT-S Int. Microw. Symp. Dig., Long Beach, CA, Jun. 2005, pp. 975–978. [5] P. Asbeck, H. Kobayashi, M. Iwamoto, G. Nanington, S. Nam, and L. E. Larson, “Augmented behavioral characterization for modeling the nonlinear response of power amplifiers,” in IEEE MTT-S Int. Microw. Symp. Dig., Seattle, WA, Jun. 2002, pp. 135–138.

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[6] H. Ku and J. S. Kenney, “Behavioral modeling of RF power amplifiers considering IMD and spectral regrowth asymmetries,” in IEEE MTT-S Int. Microw. Symp. Dig., Philadelphia, PA, Jun. 2003, pp. 799–802. [7] J. J. Xu, M. Yagoub, R. Ding, and Q. J. Zhang, “Neural-based dynamic modeling of nonlinear microwave circuits,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 12, pp. 2769–2780, Dec. 2002. [8] J. F. Sevic, “Theory of high-power load–pull characterization for RF and microwave transistors,” in RF and Microwave Handbook, M. Golio, Ed. Boca Raton, FL: CRC, 2000. [9] J. Benedikt, R. Gaddi, P. J. Tasker, and M. Goss, “High-power timedomain measurement system with active harmonic load–pull for highefficiency base-station amplifier design,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 12, pp. 2617–2624, Dec. 2000. [10] D. Barataud, A. Mallet, M. Campovecchio, J. M. Nebus, J. P. Vilotte, and J. Verspecht, “Measurements of time-domain voltage/current waveforms at RF and microwave frequencies for the characterization of nonlinear devices,” in Instrum. Meas. Technol. Conf., 1998, vol. 2, pp. 1006–1010. [11] S. C. Cripps, RF Power Amplifiers for Wireless Communications. Norwood, MA: Artech House, 1999. [12] W. J. Rugh, Nonlinear System Theory—The Volterra-Wiener Approach. Baltimore, MD: The Johns Hopkins Univ. Press, 1981. [13] M. Schetzen, The Volterra and Wiener Theories of Nonlinear Systems. New York: Wiley, 1980. [14] E. Ngoya, N. LeGallou, J. M. Nebus, H. Buret, and P. Reig, “Accurate RF and microwave system modeling of wideband nonlinear circuits,” in IEEE MTT-S Int. Microw. Symp. Dig., Boston, MA, Jun. 2000, pp. 79–82. [15] T. Wang and T. Brazil, “Volterra-mapping-based behavioral modeling of nonlinear circuits and systems for high frequencies,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 3, pp. 1433–1440, May 2003. [16] F. Filicori, G. Vannini, and V. A. Monaco, “A nonlinear integral model of electron devices for HB circuit analysis,” IEEE Trans. Microw. Theory Tech., vol. 40, no. 7, pp. 1456–1465, Jul. 1992. [17] A. Soury, E. Ngoya, and J. M. Nebus, “A new behavioral model taking into account nonlinear memory effects and transient behaviors in wideband SSPAs,” in IEEE MTT-S Int. Microw. Symp. Dig., Seattle, WA, Jun. 2002, pp. 853–856. [18] J. C. Pedro and S. Maas, “A comparative review of microwave and wireless power amplifier behavioral modeling approaches,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 4, pp. 1150–1163, Apr. 2005. [19] H. Kantz and T. Schreiber, Nonlinear Time Series Analysis. Cambridge, U.K.: Cambridge Univ. Press, 1997. [20] D. M. Walker, R. Brown, and N. B. Tufillaro, “Constructing transportable behavioral models for nonlinear electronic devices,” Phys. Lett. A, vol. 255, pp. 236–242, 1999. [21] J. Wood, “Nonlinear time series analysis—Behavioral modeling of microwave amplifiers,” presented at the IEEE MTT-S Int. Microw. Symp. Workshop on Power Amplifiers, Long Beach, CA, 2005. [22] J. Wood, D. E. Root, and N. B. Tufillaro, “A behavioral modeling approach to nonlinear model order reduction for RF/microwave ICs and systems,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 9, pp. 2274–2284, Sep. 2004. [23] D. M. Walker, N. B. Tufillaro, and P. Gross, “Radial basis model for feedback systems with fading memory,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 48, no. 9, pp. 1147–51, Sep. 2001. [24] C. Warwick and M. Mulligan, “Using behavioral models to drive RF design and verify system performance,” RF Design, pp. 30–39, Mar. 2005. [25] F. Fernet, “An ADS bench for generating multi-carrier 3GPP WCDMA ACLR test signals,” High Freq. Electron., pp. 34–42, Nov. 2002. [26] N. Gershenfeld, The Nature of Mathematical Modeling. Cambridge, U.K.: Cambridge Univ. Press, 1999, ch. 16.3–16.4, pp. 208–220. [27] F. Takens, “Detecting strange attractors in fluid turbulence,” in Dynamical Systems and Turbulence, D. A. Rand and L.-S. Young, Eds. New York: Springer-Verlag, 1980. [28] J. Stark, “Delay embeddings and forced systems,” J. Nonlinear Sci., vol. 9, no. 3, pp. 255–332, Jun. 1999. [29] M. B. Kennel, R. Brown, and H. Abarbanel, “Determining embedding dimension for phase-space reconstruction using a geometrical construction,” Phys. Rev. A, Gen. Phys., vol. 45, no. 6, pp. 3403–3411, Mar. 1992.

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[30] L. Cao, A. I. Mees, K. Judd, and G. Froyland, “Determining the minimum embedding dimension of input–output time series data,” Int. J. Bifurcation and Chaos, vol. 8, no. 3, pp. 1490–1504, Aug. 1998. [31] M. Isaksson, D. Wisell, and D. Rönnow, “Nonlinear behavioral modeling of power amplifiers using radial basis function neural networks,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 11, pp. 3422–3428, Nov. 2005. [32] T. Liu, S. Boumaiza, and F. M. Ghannouchi, “Dynamic behavioral modeling of 3G power amplifiers using real-valued time-delay neural networks,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 3, pp. 1025–1033, Mar. 2004. [33] F. Zavosh, M. Thomas, C. Thron, T. Hall, D. Artusi, D. Anderson, D. Ngo, and D. Runton, “Digital predistortion techniques for RF power amplifiers with CDMA applications,” Microw. J. Oct. 1999. [Online]. Available: http://www.mwjournal.com/Archives/ [34] M. Lefevre, D. Runton, J. Kinney, J. Wright, J.-C. Nanan, and J.-J. Bouny, “Digital predistortion application of an LDMOS 200W Doherty amplifier,” presented at the IEEE Top. PA Workshop San Diego, CA, Jan. 2006.

John Wood (M’87–SM’03) received the B.Sc. and Ph.D. degrees in electrical and electronic engineering from The University of Leeds, Leeds, U.K., in 1976 and 1980, respectively. He is currently a Senior Technical Contributor responsible for RF computer-aided design (CAD) and modeling with the RF Division, Freescale Semiconductor Inc., Tempe, AZ. where he develops device and behavioral models for RF power transistors and ICs. From 1997 to 2005, he was with the Microwave Technology Center, Agilent Technologies (formerly Hewlett-Packard), Santa Rosa, CA, where his research included the investigation and development of large-signal and bias-dependent linear field-effect transistor (FET) models for millimeter-wave applications, and nonlinear behavioral modeling using large-signal network analyzer measurements and nonlinear system identification techniques. He has authored or coauthored over 80 papers.

Michael LeFevre (M’01) received the M.S.E.E. degree from Brigham Young University, Provo, UT, in 1997. Upon graduation he joined the Semiconductor Product Sector, Motorola (now Freescale Semiconductor Inc.), Tempe, AZ, where, in 1998, he transitioned to the Wireless Infrastructure System Division as a System Engineer. His current technical interests include digital pre-distortion techniques and system behavioral modeling.

David Runton (M’05) received the B.Sc. degree in applied physics from Jacksonville University, Jacksonville, FL, in 1993, the B.E.E. and M.S.E.E. degree in electrical engineering from the Georgia Institute of Technology, Atlanta, in 1993 and 1994, respectively, and the Master’s degree in business administration (as part of the High Technology Program) from Arizona State University, Tempe, in 1999. In 1994, he began as an RF Design Engineer with Motorola Semiconductor (now Freescale Semiconductor Inc.), Tempe, AZ, where he is currently the RF Systems Engineering Manager for the RF Division. His current interests include advanced linearization techniques and high-efficiency amplifier design.

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Jean-Christophe Nanan received the Eng. Diploma (with a specialty in electronics) from ENSEEIHT, Toulouse, France, in 1988. He spent seven years designing RF front-end circuits for fixed radio link prior to joining Motorola/SPS (now Freescale Semiconductor Inc.), Tempe, AZ, in 1998, as a RF Application Engineer. He is currently involved with RF system engineering for cellular network infrastructures with the RF Division, Freescale Semiconductor Inc., Toulouse, France.

Basim H. Noori (A’02–M’04) received the Electrical Engineering Diploma from the Higher Technical Institute, Nicosia, Cyprus, in 1991, and the M.Sc. degree in mobile communication systems from the University of Surrey, Guildford, U.K., in 2000. From 1991 to 2001, he was with Telectronics Ltd. (1991–1995), Limassol, Cyprus, Wood & Douglas Ltd. (1995–1996), Baughurst, U.K., and Nokia Telecommunications (1996–2001), Camberley, U.K. In 2001, joined Tropian Inc., Cupertino, CA. He then joined REMEC/Spectrian, prior to joining the

RF Division, Freescale Semiconductor Inc. (formerly Motorola), Tempe, AZ, in 2004, where he currently manages the Load–Pull Measurement Team. He holds two U.S. patents. His current research interests include power amplifier (PA) linearization techniques, high-efficiency PA design, and load–pull.

Peter H. Aaen (S’93–M’97) received the B.A.Sc. degree in engineering science and M.A.Sc. degree in electrical engineering from the University of Toronto, Toronto, ON, Canada, in 1995 and 1997, respectively, and the Ph.D. degree in electrical engineering from Arizona State University, Tempe, in 2005. In 1997, he joined the Wireless Infrastructure Systems Division, Semiconductor Products Sector, Motorola (now Freescale Semiconductor Inc.), Phoenix, AZ. His current research focuses on the development and validation of microwave transistor models and passive components. His technical interests include calibration techniques for microwave measurements and the development of package modeling techniques.

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Calibration of Sampling Oscilloscopes With High-Speed Photodiodes Tracy S. Clement, Senior Member, IEEE, Paul D. Hale, Senior Member, IEEE, Dylan F. Williams, Fellow, IEEE, C. M. Wang, Andrew Dienstfrey, and Darryl A. Keenan

Abstract—We calibrate the magnitude and phase response of equivalent-time sampling oscilloscopes to 110 GHz. We use a photodiode that has been calibrated with our electrooptic sampling system as a reference input pulse source to the sampling oscilloscope. We account for the impedance of the oscilloscope and the reference photodiode and correct for electrical reflections and distortions due to impedance mismatch. We also correct for time-base imperfections such as drift, time-base distortion, and jitter. We have performed a rigorous uncertainty analysis, which includes a Monte Carlo simulation of time-domain error sources combined with error sources from the deconvolution of the photodiode pulse, from the mismatch correction, and from the jitter correction. Index Terms—High-speed photodiode, impulse response, mismatch correction, oscilloscope calibration, sampling oscilloscope, uncertainty.

I. INTRODUCTION ROADBAND equivalent-time sampling oscilloscopes are routinely used to measure a variety of signals from data streams for communications to pulse time-of-flight measurements for biological applications. These instruments have the potential to be especially useful for the measurement of nonlinear microwave devices. In addition, techniques described here can be used to overcome the perceived problem of impedance mismatch in measuring microwave devices. At this time, there are several commercially available sampling oscilloscopes with bandwidths of 70–100 GHz. In the past, several methods have been described for measuring the impulse response or complex frequency response of broadband sampling oscilloscopes [1]–[16]. These methods fall into one of three categories, which are: 1) swept-sine calibrations; 2) “nose-to-nose” calibrations; and 3) calibration with a known pulse source. Our research extends previous techniques that use a known pulse source to higher frequencies and develops techniques for correcting significant sources of error including impedance mismatches and time-base distortions (TBDs). In addition, we provide a comprehensive uncertainty analysis for the complex frequency response of the oscilloscope. The swept-sine calibration [3], [4] compares the amplitude of sine waves measured with the oscilloscope to the measurement of the power of the sine waves measured with a calibrated power meter. This comparison determines the magnitude response of the oscilloscope at each frequency. The swept-sine calibration is traceable to the calibration of the power meter and is the most

B

Manuscript received December 15, 2005; revised March 25, 2006. The authors are with the National Institute of Standards and Technology, Boulder, CO 80305 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2006.879135

accurate oscilloscope amplitude calibration currently available. The main disadvantage to the swept-sine calibration method is that it cannot determine the phase of the oscilloscope frequency response. Phase calibration of broadband oscilloscopes, however, is required for many microwave applications. In addition, without the phase information, it is impossible to determine the impulse response of the oscilloscope in the time domain. Several groups have recently studied the nose-to-nose method as another possible oscilloscope calibration method [3]–[10]. The nose-to-nose technique uses as its pulse source a “kick-out” pulse that is generated by some oscilloscope sampler architectures. The nose-to-nose method relies on the assumption that this “kick-out” pulse is proportional to the impulse response of the oscilloscope. Several groups have studied the nose-to-nose assumption, and have quantified the error that this assumption introduces into the oscilloscope response determined from the nose-to-nose calibration [3], [4], [9], [10]. The concept of using a “known pulse source” to determine the response of the oscilloscope is relatively straightforward [11]. In an ideal system, the known pulse would be deconvolved from the measurement of the pulse source with the oscilloscope to determine the impulse response or complex frequency response of the oscilloscope. In practice, there can be many nonideal parts to the measurement of the known pulse source with the oscilloscope. In Section IV, we describe correction techniques for nonidealities such as drift, TBD, jitter, and impedance mismatch. One of the main problems with the known pulse source technique is finding a pulse source that has a response sufficiently fast that enough energy exists at high frequencies to carry out the deconvolution. In the past, both electrical and opto-electronic methods have been used to generate pulses with very wide bandwidths [11]–[16]. Another problem with the known pulse technique is accurately determining the response of the pulse source. In the 1990s, the National Institute of Standards and Technology (NIST), Boulder, CO, used a superconducting oscilloscope with a bandwidth greater than 60 GHz as an “ideal” measurement system to determine the response of electrical pulse generators [11]. However, this approach is limited by both the bandwidth of the pulse generator and the imperfect response of the superconducting oscilloscope, which deviates from ideal at higher frequencies. Other groups have used an electrooptic sampling (EOS) method to determine the response of high-speed pulses that are generated in coplanar waveguide (CPW) [13]–[16]. The main limitation of this approach is in characterizing the transition from the CPW to the coaxial geometry of the oscilloscope and accounting for errors in that transition, and accounting for the frequency-dependent impedance of the oscilloscope. Our research improves upon these techniques.

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Fig. 1. Portion of the time-domain waveform generated by the photodiode and measured on the oscilloscope. The inset shows a schematic diagram of the waveform measurement setup.

II. EXPERIMENTAL WAVEFORM MEASUREMENTS Fig. 1 shows a portion of a typical time-domain waveform acquired for the calibration of the sampling oscilloscope. The inset of Fig. 1 shows a schematic diagram of the waveform measurement setup. Our known pulse source consists of a commercially available high-speed photodiode with bandwidth greater than 50 GHz (XPDV2020R, u2t Photonics AG, Berlin, Germany)1 with a 1.85–1.0 mm coaxial adapter at the output. We calibrate the complex frequency response of the photodiode at the 1.0-mm coaxial plane from 0.2 to 110 GHz with our EOS system [17]–[19]. A mode-locked Er-doped fiber laser excites the calibrated photodiode with optical pulses that have a full-width at halfmaximum (FWHM) pulsewidth of less than 70 fs at a center wavelength of 1550 nm, and a repetition rate of 8.66 MHz. This generates a train of 6-ps duration electrical pulses at the photodiode output. We use a reflective neutral-density filter to attenuate the laser output prior to coupling to the fiber pigtail of the photodiode. In order to avoid nonlinearities in both the photodiode and oscilloscope, we ensure that the peak voltage of the impulse as measured by the oscilloscope is less than 150 mV. For the trigger signal, we use synchronous pulses from a pickoff in the Er-fiber laser. Another high-bandwidth photodiode converts the laser pulses into the electrical trigger signal. In order to improve the signal-to-noise ratio for the oscilloscope measurements, we acquire 100 waveforms. Time-domain averaging of 100 waveforms can increase the signal-to-noise ratio of the magnitude response by 20 dB. As Fig. 2 shows, the magnitude response can fall by as much as 40 dB at 110 GHz, thus the benefit from averaging is significant. We do not average the waveforms prior to acquisition from the oscilloscope because any drift during the measurements can broaden the measured pulse in time. From the individual waveforms, we also 1We use trade names only to specify our experimental conditions. This does not constitute an endorsement by NIST. Other products may perform as well or better.

Fig. 2. Solid lines show the oscilloscope response magnitude (in decibels). The dashed lines indicate the 95% confidence interval for the calibrated response. The inset figure shows details of the response up to 50 GHz.

obtain the noise statistics that we use for postprocessing algorithms that correct for drift and jitter. The time base of the oscilloscopes we use is known to have distortion [5], [20], [21], including discontinuities every 4 ns. By adjusting the delay between the photodiode pulse and trigger pulse, we can move the position of the photodiode pulse relative to the time-base discontinuities. We take care to position the pulses so that the discontinuities will not occur during the main part of the pulse or where there is significant energy in the pulse or its reflections. We discuss the characterization and correction of this TBD in Section IV. The waveform in Fig. 1 has a reflection that is caused by the impedance mismatch of the oscilloscope and photodiode at approximately 0.8 ns after the main pulse. Since we intentionally correct the measurements for the impedance mismatch of the source and oscilloscope (which can be manifest as reflections in the time domain), it is important to choose the time interval of the acquired waveform so that all of the significant energy due to reflections is captured. This is very different from many experiments where reflections are intentionally windowed out of the measured data, and where there is no accounting for impedance mismatch. To ensure that we capture all of the reflections, we set the vertical scale on the oscilloscope to it lowest value, average the displayed waveforms on the oscilloscope, and pick an end point for the window where the averaged signal shows only noise. For the measurements with our photodiode, we typically use a 5-ns window. To ensure that this window captures all of the pulse information, we have compared the results taken with a window that is twice as large (10 ns), and see no significant differences in the response of the oscilloscope. The 5-ns interval setting also provides us with a convenient spacing of points in the frequency domain of 200 MHz. It is also important to make sure that the spacing of points in the time domain is adequate for capturing the fast pulse events. When this is done, there is also no aliasing of energy in the frequency domain due to a sample rate that is too low. We sample 4096 points for each waveform, resulting in the spacing

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Fig. 3. Solid lines show the oscilloscope response phase in degrees. The dashed lines indicate the 95% confidence interval for the calibrated response. The inset figure shows details of the response up to 50 GHz.

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to a pure shift in time. The jump in the phase around 70 GHz corresponds to the notch in the magnitude response at the same frequency. The standard uncertainty for this detrended phase response is 1.6 at 50 GHz. The phase uncertainty rises significantly with the phase jump around 70 GHz, and stays high at the higher frequencies. This increase in phase uncertainty happens because the magnitude is becoming very small at high frequencies, thus any uncertainty in the vector response as the magnitude of the vector approaches the origin leads to a much larger uncertainty in the phase. The magnitude and phase response plotted in Figs. 2 and 3 were determined from multiple measurements of the same oscilloscope sampling head over more than two years time. The uncertainty includes the repeatability due to differences of many measurements made on the same day (within a set), and reproducibility due to differences between sets of data taken on different days, and with different experimental conditions. IV. CORRECTIONS TO WAVEFORM DATA

of points in the time domain approximately equal to 1.22 ps (5 ns/4096). For our measured waveforms, the rise and fall times are 8–9 ps, thus we have more than six points within the rising or falling edges. This spacing of points in the time domain is equivalent to a Nyquist frequency of 400 GHz. For our system, the waveform information drops below the noise floor between 150–200 GHz, thus this sampling rate is sufficient. In addition, if we drop the number of points by half (to 2048), we see no significant changes in the measured oscilloscope response below 110 GHz. III. OSCILLOSCOPE RESPONSE RESULTS In Section IV, we discuss in detail the corrections applied to the measured data. We correct for drift, TBD, jitter, impedance mismatch, and the known input pulse to obtain the complex frequency response at the 1.0-mm coaxial input of the oscilloscope. Figs. 2 and 3 show the complex frequency response results for a sampling oscilloscope that has a specified 3-dB bandwidth of greater than 50 GHz. The response is shown from 0.2 to 110 GHz on the larger plot. The smaller insets on each plot provide more detail of the response below 50 GHz. The dashed lines in both plots are calculated from the expanded uncertainty and indicate the 95% confidence interval of the results. Calculation of the uncertainty of the response is discussed in Section V. In Fig. 2, the oscilloscope magnitude response is given in decibel units on the vertical axis. The response is normalized to the value at the lowest frequency point and is found to be independent of the input pulse amplitude for peak voltage less than 150 mV. The response falls by more than 40 dB at 110 GHz, and there is a very large notch in the response around 70 GHz. From the inset, we can see that the 3-dB bandwidth of this sampling oscilloscope is greater than 50 GHz. The standard uncertainty for this response is 0.2 dB at 50 GHz and 2.0 dB at 100 GHz. Fig. 3 shows the phase response in degrees for the same sampling oscilloscope. For the phase response, we detrend the results by calculating a line with intercept equal to zero that best fits the phase data from 0.2 to 25 GHz in a least squares sense and subtracting that line. This detrending procedure corresponds

The waveform measured with the oscilloscope is two-dimensional (voltage versus time), and can have errors in either the time or voltage. A significant portion of the voltage errors are reduced by using the built-in dc calibration of the oscilloscope to correct for gain errors, offset errors, and “nonlinear distortion”—the manufacturer’s term for the voltage distortion that can change at different voltage levels. The nonlinear distortion is corrected within the oscilloscope with an internal lookup table. Even with those errors reduced, the measured voltage waveform still has errors due to additive noise and quantization, and errors due to the finite response time of the oscilloscope. In general, for our measurements, we empirically find that the additive noise level is approximately 1-mV rms, while the quantization interval is approximately 50 V. The quantization errors are small relative to the additive noise and are neglected for our results. The oscilloscope time-base measurement errors include errors due to drift, TBD, and jitter. Below we describe the methods used for correcting each of these errors. In addition to correcting for errors in the oscilloscope measurement of the waveforms, we also correct for the electrical mismatch between the photodiode pulse generator and oscilloscope, and we deconvolve the input electrical pulse. Fig. 4 shows a schematic flow diagram of the corrections that are applied to the measured data in order to obtain the frequencydomain impulse response for each sampling head. A. Oscilloscope Time-Base Corrections The first steps in our analysis are to correct the measured waveform data for errors due to the oscilloscope time base. We correct for drift, TBD, and jitter. In our correction procedures, we always correct for drift first, then TBD, and finally for jitter with the other frequency-domain magnitude corrections. Due to the nonlinear nature of the TBD errors, the order in the corrections can be important. To reduce the computation time, we first correct the individual waveforms for drift, then average and correct only the averaged waveform for the TBD. If, instead, we first correct for TBD of each individual waveform prior to correcting for drift and averaging, differences in the frequency-domain magnitude and phase responses are much smaller than the

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Fig. 4. Schematic flowchart of the corrections that are applied to the oscilloscope measurements to determine the oscilloscope response.

uncertainty in the response. Errors due to our order of processing the time-base errors is included in the uncertainty derived from the Monte Carlo analysis, which is described in Section V. 1) Drift: We define drift as a shift of the entire waveform in time. Since we acquire multiple waveforms, shifts in the waveforms in time will broaden the average of the pulses in time. One source of drift in sampling oscilloscope measurements is the temperature sensitivity of the trigger circuit [5]. Temperature stability in our laboratory during measurements is typically fairly good ( 1 C), and we generally see very small drift in the measurements. This corresponds to a standard deviation of the drift of 0.3 ps, and a 0.19-dB attenuation at 110 GHz if we assume a Gaussian drift distribution. To compensate for drift in the measurements, we align the individual waveforms before averaging, using an algorithm based on cross-correlation of all possible pairs of waveforms [22]. Each waveform is corrected for drift prior to averaging by Fourier transforming to the frequency domain, multiplying by , where is the estimated drift for the waveform, and inverse Fourier transforming back to the time domain. The remaining corrections are performed on the drift-corrected average of the data. 2) TBD: TBD is a deterministic deviation in the sample times of the oscilloscope from the ideal evenly spaced sample times predicted by the oscilloscope [20]. We estimate the TBD for our measurement window by acquiring multiple sine waves with the oscilloscope and analyzing the sine-wave data with an efficient least squares fit algorithm [20], [21]. For the TBD estimate, we acquire four sets of sine-wave data, each with frequency equal to 9.25 or 10.25 GHz, and with a relative phase shift of 0 or 90 (in-phase and quadrature). To best capture the TBD under the same laboratory conditions as the oscilloscope waveform

Fig. 5. TBD estimate for the sampling oscilloscope. The horizontal axis gives the sampling time expected from the oscilloscope, and the vertical axis gives the estimated error in that sampling time.

data, we determine the TBD of the oscilloscope for each set of experimental measurements. Fig. 5 shows a typical plot of the TBD for one of our oscilloscopes. In this figure, the horizontal axis represents the expected time at which the sample is taken (the number that the oscilloscope gives), and the vertical axis represents the difference between the expected sampling time and the actual sample time. The time axis of the averaged waveform is adjusted by the TBD estimate. After correcting the sampling time of each point in the waveform, the resulting data, which is unevenly spaced in time, is interpolated onto an evenly spaced time grid using a regression

CLEMENT et al.: CALIBRATION OF SAMPLING OSCILLOSCOPES WITH HIGH-SPEED PHOTODIODES

spline model [20], [23]. Finally, the averaged waveform is transformed to the frequency domain with a fast Fourier transform (FFT). Some groups have suggested taking extra data points at both ends of the pulse waveform in order to provide enough data for a true interpolation at the ends for cases where the TBD results in time compression. For our measurements, the TBD typically does not result in time compression, and it is sufficient to use the same time window for the waveform data as for the TBD data. Furthermore, if there were time compression, our interpolation method produces reasonable results for our data because we measure time intervals that are long enough to ensure that the signal is below the noise level on both ends of the time window. 3) Jitter: In equivalent-time sampling, each sample is taken at a given delay after a trigger signal, meaning that each voltage–time pair in a waveform is taken with a new trigger. Due to this, jitter in the time-base circuitry and between the trigger signal and measured signal can have a large effect. We obtain a jitter estimate separately from each of the sets of measurements since it can be influenced by the specific sampler being measured, the specific trigger signal, and the trigger level. We obtain the jitter estimate with a procedure that is based on the sample variance [24], [25]. Assuming that the jitter is small and has a symmetric probability density function, the variance of the measured signal due to jitter and additive noise can be expressed in a Taylor-series expansion as

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[17]–[19], [26]. The photodiode is calibrated at its 1.0-mm coaxial reference plane, and its response is determined from 0.2 to 110 GHz in 0.2-GHz steps. The known photodiode response is deconvolved by dividing it in the frequency domain from the complex frequency-domain oscilloscope data that have been corrected for the time-base errors. The photodiode response must have a significant signal level up to 110 GHz to obtain reasonable results from this division. The magnitude response of our high-speed photodiode falls off by only 12 dB at 110 GHz, providing a very good signal level for the deconvolution. 2) Laser Pulse: An estimate of the magnitude of the laser pulse envelope in the frequency domain is determined from a second-order autocorrelation measurement of the laser pulse. We fit the laser pulse magnitude (in decibels) up to 110 GHz with a fourth-order polynomial, and then correct the magnitude of the oscilloscope response with this quadratic function. Since the laser pulse is very short (less than 70-fs FWHM), corresponding to a very broad bandwidth in frequency (3-dB bandwidth 1 THz), the corrections to the oscilloscope response are very small (less than 0.019 dB at 110 GHz). One drawback to the use of the second-order autocorrelation is that all phase information about the laser pulse envelope is lost. However, following [27], we can conservatively estimate the maximum phase deviation due to asymmetry in the laser pulse envelope to be less than 0.0005 at 110 GHz. This is a negligible correction to the phase response of the oscilloscope. C. Impedance Mismatch Correction

(1) is the total measured signal variance, is the adwhere ditive noise variance, is the jitter variance, and is the derivative of the time-domain waveform. To determine an initial estimate of the jitter variance, we use only the time samples for which the magnitude of the estimated derivative of the waveform exceeds a selected threshold. A parametric bootstrap approach is then used to eliminate the inherent bias in this method [25]. Typically, is less than 1 ps for these measurements, and the jitter is assumed to be normally distributed. We then correct the frequency-domain magnitude for the estimated jitter by multiplying by , where is the estimated variance of the jitter. We note that the correction for jitter is an increasing function of frequency and can amplify noise at high frequencies, where the signal has rolled off to the noise floor.

Electrical mismatch between the photodiode and oscilloscope can cause both multiple reflections of the pulse and dispersion of the time-domain signal. Electrical reflection coefficients for both the photodiode and oscilloscope are measured from 200 MHz to 110 GHz with a vector network analyzer. These measurements are limited to a maximum frequency of 110 GHz by the 1.0-mm coaxial connectors of the devices, and this provides the upper frequency limit for the overall oscilloscope calibration. The reflection coefficients for the photodiode and the oscilloscope are frequency dependent. Although they are designed to have a good 50- match at low frequencies, the reflection coefficient is often as high as 0.5 at frequencies between 50–110 GHz. From the reflection-coefficient measurements, we correct for the mismatch by multiplying , the frequency-domain data after applying all of the corrections described above, by the mismatch correction factor [28], resulting in , the complex oscilloscope response

B. Known Photodiode Input Pulse The electrical waveform generated by the photodiode consists of the impulse response of the photodiode convolved with the laser pulse shape. Likewise, the measured data is a convolution of the waveform generated by the photodiode with the response of the oscilloscope sampling head. In the frequency domain, we correct the measured data to obtain the response of the sampling head by dividing the measured response by the complex frequency response of the photodiode and by the estimate of the laser excitation pulse. 1) Photodiode Response: The complex frequency response of the photodiode is determined with the NIST EOS system

(2) is the reflection coefficient of the photodiode and where is the reflection coefficient of the oscilloscope. Fig. 6 shows the magnitude of the frequency-domain response of the oscilloscope both with , and without , the mismatch correction up to 60 GHz. The ripple on the uncorrected signal comes from the multiple reflections between the photodiode and oscilloscope, and must be corrected with (2) to determine the oscilloscope response. The importance of the TBD correction is especially apparent when combined with

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multiplicative in the frequency domain, implying that these corrections are additive when we look at the magnitude in decibels and are additive in the phase. B. Components of Uncertainty

Fig. 6. Oscilloscope magnitude response without mismatch correction (grey line, offset by 1.5 dB for clarity) and with mismatch correction (black line).

0

the mismatch correction. Without TBD correction, the ripples due to electrical mismatch are only partially reduced because the spacing of the electrical reflections in the time domain is distorted by the TBD. V. UNCERTAINTY ANALYSIS A. Combining Uncertainties We perform a point-by-point uncertainty analysis in the frequency domain [29]. Given a scalar random variable with mean and variance , and variable , where is sufficiently differentiable, we have

(3) The multivariate analog involving the Jacobian of is straightforward; the details may be found in [29]. For most of the corrections described above, we use (3) to compute uncertainties. The main exception is for the determination of the uncertainty due to the time-base corrections described in Section IV. In this case, the functional forms are sufficiently complex that we instead use Monte Carlo analysis to determine the uncertainty of the frequency-domain results due to time-domain errors such as additive noise, correction of drift, and correction of TBD. The simulation includes both random variation in the observed time-domain waveform due to additive noise, jitter, and drift, and random variation in the TBD correction due to uncertainty in the TBD estimate. The results of this Monte Carlo simulation are time-domain waveforms, which are corrected for drift and TBD. These simulated waveforms are individually transformed to the frequency domain, and the variance in the frequency domain is used to obtain the uncertainty due to the combined effects of additive noise, drift, TBD, and jitter uncertainties. After the time-domain corrections, all of the remaining corrections (jitter, mismatch, photodiode response, laser pulse) are

Table I summarizes the components of the total uncertainty and gives the typical standard uncertainty in magnitude and phase for each component at 50 and 100 GHz. The combined uncertainty is calculated from the sum of squared standard deviations of the uncertainties in the frequency domain. The expanded uncertainty can be obtained by multiplying by a coverage factor of 2 to estimate the 95% confidence interval about the mean estimated value [29]. 1) Time-Domain Correction Uncertainties: We compute the variance due to the time-domain corrections from (typically ) simulation experiments. First, a reference waveform is calculated by correcting the measured waveforms for drift and TBD. For each of the simulation experiments, we carry out a Monte Carlo analysis by constructing (typically ) time-domain waveform realizations by perturbing the reference waveform with a smoothed version of TBD, and with random variations due to drift, jitter, and additive noise. In addition, we construct one realization of the TBD by perturbing the smoothed version of the TBD with random variations due to noise in the TBD measurement. The parameters for the random perturbation are derived from the sample waveforms obtained from the oscilloscope. The jitter, drift, and additive noise variances used in the Monte Carlo simulation are estimated from the original data described in Section IV. For each simulation experiment, this results in waveforms that are representative of the waveform data taken with the oscilloscope. These waveforms are then processed as if they were waveform data from the oscilloscope, correcting for drift and averaging, and estimating the jitter. We then correct this averaged simulation result for the perturbed TBD realization, and interpolate onto the evenly spaced grid. We repeat this process for each of the simulation experiments, resulting in averaged waveforms (analogous to laboratory experiments of waveforms). We transform these waveforms to the frequency domain and calculate the sample variances for both real and imaginary, or magnitude and phase, components due to the statistical deviations in the time-domain correction processes. 2) Mismatch Correction Uncertainty: The uncertainty in the mismatch correction is estimated from another Monte Carlo simulation that uses the uncertainties in the measurement of each scattering parameter. The simulation includes random variation in the reflection coefficients and scattering parameters due to uncertainty in the network analyzer measurements and the uncertainty in the coaxial standard definitions. For the reflection coefficients of the photodiode and oscilloscope, the network analyzer measurements are calibrated, and uncertainty in their calibration is determined using [30]. The standard uncertainty is determined from the manufacturer’s specified worst case uncertainty (from [30]) for the standard definitions by dividing by a coverage factor of 3. 3) Uncertainty in Laser Pulse Shape: Since the correction for the laser pulse is very small, the uncertainty due to this cor-

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TABLE I TYPICAL VALUES FOR THE COMPONENTS OF UNCERTAINTY

rection is also very small compared to all the other uncertainties. We conservatively estimate the uncertainty due to the laser pulse correction in magnitude as being equal to the half of the magnitude correction, giving an uncertainty of only 0.010 dB at 110 GHz. For the extremely small phase correction, the error is negligible compared to the other uncertainties and is neglected in the total phase uncertainty. 4) Photodiode Response Uncertainty: The uncertainty in the photodiode response is estimated from measurements of the response with the NIST EOS system. Systematic uncertainty in the EOS measurement of the photodiode response includes uncertainty in the mismatch correction due to uncertainties in the measurement of on-wafer scattering parameters, uncertainty in the measurement of coaxial scattering parameters, and uncertainty due to the finite impulse response of the EOS system [17], [18]. We do not include uncertainty due to the finite measurement time window in the EOS system or possible piezoelectric resonances. These effects may increase the uncertainty at the lowest frequencies, but are negligible at higher frequencies. 5) Repeatability: To estimate the statistically derived type-A uncertainty [29] in our measurements due to repeatability, we typically perform five sets of repeat measurements. The complex frequency response is obtained from each set of measurements as described above. Standard uncertainty in the mean of the magnitude and phase is obtained from the standard deviation of the five resulting responses. VI. DISCUSSION In order to check the oscilloscope complex frequency response results, we performed a swept-sine calibration of the magnitude of the complex frequency-domain response [31]. The top plot of Fig. 7 shows the magnitude response obtained from the known photodiode-based calibration described here (dark line) along with the response obtained from a swept-sine calibration (lighter line). The two responses agree very well over the region where we have swept-sine data (up to 40 GHz). The bottom plot in Fig 7 shows the difference in decibel units between the two calibrations. The difference is less than 0.3 dB from 0.2 to 40 GHz, and the difference is significantly less than the combined uncertainties of both measurements. The 0.2-GHz spacing of points in the oscilloscope calibration is limited by the data acquisition in the calibration of the photodiode and not by inherent limitations in the oscilloscope measurements. However, due to this limit, we are unable to capture ripples or features with a period of less than 0.2 GHz. We have calibrated the magnitude response with the swept-sine technique for various spacing of frequency points and find

Fig. 7. Comparison of photodiode-based known pulse calibration described here with a swept-sine calibration of the oscilloscope. The magnitude of the oscilloscope response is plotted in the top graph, and the difference between the two methods is plotted in the bottom graph.

no significant lower frequency features of the oscilloscope response [31]. The calibration of the photodiode with our EOS system has limited resolution at low frequencies due to the finite measurement window of the sampling system. This results in poor estimates of the response of the oscilloscope at low frequencies. Knowing the low-frequency response of the oscilloscopes can become very important, especially in determining the step response in the time domain [13]. It is possible to get a very good estimate of the magnitude of the frequency response at low frequencies with the swept-sine technique described in Section I. We are currently investigating ways to combine the magnitude response at low frequencies obtained from the swept-sine technique with the magnitude and phase response at higher frequencies obtained from the techniques presented here in order to obtain a more complete estimate of the complex frequency response, which can then be used to calculate the impulse or step response in the time domain [31]. VII. CONCLUSION We have calibrated the complex frequency response of highspeed oscilloscopes to 110 GHz using a calibrated photodiode as a reference pulse source. The results include corrections for impedance mismatch, drift, jitter, TBD, and the photodiode impulse. We have presented a complete uncertainty analysis for the complex frequency response calibration. We have verified

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the calibration by comparison with a swept-sine calibration of the oscilloscope. With our calibrated oscilloscope, we can now determine the response of a variety of sources (pulse sources, step inputs, communication signals, and circuits including harmonics and other distortion products generated by nonlinearities). We can deconvolve the response of the oscilloscope from the measurement of high-speed signals and improve the calibration of pulse sources.

APPENDIX I CALIBRATION WITH A COAXIAL ADAPTER BETWEEN THE PHOTODIODE AND OSCILLOSCOPE In order to calibrate the oscilloscope at a different coaxial reference plane (e.g., at the 2.4-mm coaxial connector on the front of the sampling head instead of at the 1.0-mm coaxial connector of the adapter), it is necessary to extend our mismatch correction to include the adapter’s -parameters. If we place an adapter with scattering parameters between the generator and oscilloscope, the equation for the oscilloscope response becomes

(4) This allows calibration through adapters and to different waveguide types and sizes. ACKNOWLEDGMENT The authors thank K. Coakley, National Institute of Standards and Technology (NIST), Boulder, CO, for his work in the development of the Monte Carlo simulations and H. Reader, University of Stellenbosch, Stellenbosch, South Africa, for his helpful comments on this paper’s manuscript. REFERENCES [1] K. A. Remley and D. F. Williams, “Sampling oscilloscope models and calibrations,” in IEEE MTT-S Int. Microw. Symp. Dig., 2003, pp. 1507–1510. [2] M. Kahrs, “50 years of RF and microwave sampling,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 6, pp. 1787–1805, Jun. 2003. [3] P. D. Hale, T. S. Clement, K. J. Coakley, C. M. Wang, D. C. DeGroot, and A. P. Verdoni, “Estimating magnitude and phase response of a 50 GHz sampling oscilloscope using the ‘nose-to-nose’ method,” in ARFTG Conf. Dig., Jun. 2000, vol. 55, pp. 335–342. [4] D. C. DeGroot, P. D. Hale, M. V. Bossche, F. Verbeyst, and J. Verspecht, “Analysis of interconnection networks and mismatch in the nose-to-nose calibration,” in ARFTG Conf. Dig., Jun. 2000, vol. 55, pp. 116–121. [5] J. Verspecht and K. Rush, “Individual characterization of broadband sampling oscilloscopes with a nose-to-nose calibration procedure,” IEEE Trans. Instrum. Meas., vol. 43, no. 2, pp. 347–354, Apr. 1994. [6] K. Rush, S. Draving, and J. Kerley, “Characterizing high-speed oscilloscopes,” IEEE Spectr., vol. 27, no. 9, pp. 38–39, Sep. 1990. [7] J. Verspecht, “Broadband sampling oscilloscope characterization with the ‘nose-to-nose’ calibration procedure: A theoretical and practical analysis,” IEEE Trans. Instrum. Meas, vol. 44, no. 6, pp. 991–997, Dec. 1995.

[8] N. G. Paulter and D. R. Larson, “An examination of the spectra of the ‘kick-out’ pulses for a proposed sampling oscilloscope calibration method,” IEEE Trans. Instrum. Meas, vol. 50, no. 5, pp. 1221–1223, Oct. 2001. [9] K. A. Remley, “The impact of internal sampling circuitry on the phase error of the nose-to-nose oscilloscope calibration,” NIST, Boulder, CO, Tech. Note 1528, Aug. 2003. [10] ——, “Nose-to-nose oscilloscope calibration phase error inherent in the sampling circuitry,” in ARFTG Conf. Dig., Dec. 2002, vol. 60, pp. 85–97. [11] W. L. Gans, “Dynamic calibration of waveform recorders and oscilloscopes using pulse standards,” IEEE Trans. Instrum. Meas, vol. 39, no. 6, pp. 952–957, Dec. 1990. [12] J. P. Deyst, N. G. Paulter, T. M. Souders, G. N. Stenbakken, and T. A. Daboczi, “Fast pulse oscilloscope calibration system,” IEEE Trans. Instrum. Meas., vol. 47, no. 5, pp. 1037–1041, Oct. 1998. [13] D. Henderson and A. G. Roddie, “Calibration of fast sampling oscilloscopes,” Meas. Sci. Technol., no. 1, pp. 673–679, 1990. [14] A. J. A. Smith, A. G. Roddie, and P. D. Woolliams, “Optoelectronic techniques for improved high speed electrical risetime,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2002, vol. 3, pp. 1501–1504. [15] D. Henderson, A. G. Roddie, and A. J. A. Smith, “Recent developments in the calibration of fast sampling oscilloscopes,” Proc. Inst. Elect. Eng.—Sci., Meas. Technol., vol. 139, no. 5, pp. 254–260, Sep. 1992. [16] S. Seitz, M. Bieler, M. Spitzer, K. Pierz, G. Hein, and U. Siegner, “Optoelectonic measurement of the transfer function and time response of a 70 GHz sampling oscilloscope,” Meas. Sci. Tech., vol. 16, no. 10, pp. L7–L9, Oct. 2005. [17] D. F. Williams, P. D. Hale, T. S. Clement, and J. M. Morgan, “Calibrating electro-optic sampling systems,” in IEEE MTT-S Int. Microw. Symp. Dig., May 2001, vol. 3, pp. 1527–1530. [18] D. F. Williams, P. D. Hale, T. S. Clement, and C. M. Wang, “Uncertainty of the NIST electrooptic sampling system,” NIST, Boulder, CO, Tech. Note 1535, Dec. 2004. [19] D. F. Williams, P. D. Hale, T. S. Clement, and J. M. Morgan, “Mismatch corrections for electro-optic sampling systems,” in ARFTG Conf. Dig., Nov. 2000, vol. 56, pp. 141–145. [20] C. M. Wang, P. D. Hale, and K. J. Coakley, “Least-squares estimation of time-base distortion of sampling oscilloscopes,” IEEE Trans. Instrum. Meas, vol. 48, no. 6, pp. 1324–1332, Dec. 1999. [21] C. M. Wang, P. D. Hale, K. J. Coakley, and T. S. Clement, “Uncertainty of oscilloscope timebase distortion estimate,” IEEE Trans. Instrum. Meas, vol. 51, no. 1, pp. 53–58, Feb. 2002. [22] K. J. Coakley and P. D. Hale, “Alignment of noisy signals,” IEEE Trans. Instrum. Meas, vol. 50, no. 1, pp. 141–149, Feb. 2001. [23] Y. Rolain, J. Schoukens, and G. Vandersteen, “Signal reconstruction for non-equidistant finite length sample sets: A ‘KIS’ approach,” IEEE Trans. Instrum. Meas., vol. 47, no. 5, pp. 1046–1052, Oct. 1998. [24] J. Verspecht, “Compensation of timing jitter-induced distortion of sampled waveforms,” IEEE Trans. Instrum. Meas, vol. 43, no. 5, pp. 726–732, Oct. 1994. [25] K. J. Coakley, C. M. Wang, P. D. Hale, and T. S. Clement, “Adaptive characterization of jitter noise in sampled high-speed signals,” IEEE Trans. Instrum. Meas, vol. 52, no. 5, pp. 1537–1547, Oct. 2003. [26] T. S. Clement, P. D. Hale, D. F. Williams, and J. M. Morgan, “Calibrating photoreceiver response to 110 GHz,” in 15th Annu. IEEE Lasers Electro-Optics Soc. Conf. Dig., Glasgow, U.K., Nov. 10–14, 2002, pp. 877–878. [27] J. Verspecht, “Quantifying the maximum phase-distortion error introduced by signal samplers,” IEEE Trans. Instrum. Meas, vol. 46, no. 3, pp. 660–666, Jun. 1997. [28] D. F. Williams, A. Lewandowski, T. S. Clement, C. M. Wang, P. D. Hale, J. M. Morgan, D. Keenan, and A. Dienstfrey, “Covariance based uncertainty analysis of the NIST electrooptic sampling system,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 1, pp. 481–491, Jan. 2006. [29] BIPM, IEC, IFCC, ISO, IUPAP, and OIML, “Guide to the expression of uncertainty in measurement,” Int. Org. Standard., pp. 1–101, 1993. [30] HP8510 Specifications and Performance Verification Analysis Software. Hewlett-Packard, Palo Alto, CA, part 08510-10033, program revision A.05.00, data revision A.05.00. [31] A. Dienstfrey, P. D. Hale, D. A. Keenan, T. S. Clement, and D. F. Williams, “Minimum-phase calibration of sampling oscilloscopes,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 8, pp. 3197–3208, Aug. 2006.

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Tracy S. Clement (S’89–M’92–SM’05) received the Ph.D. degree in electrical engineering from Rice University, Houston, TX, in 1993. Her Ph.D. research concerned the development and study of novel ultrashort pulse and very short wavelength lasers. Since 1998, she has been with the Optoelectronics Division, National Institute of Standards and Technology (NIST), Boulder, CO. Her current research interests include the development of measurement systems for high-speed electrooptic components, as well as ultrashort pulse laser measurements. Prior to joining the Optoelectronics Division, she was an Associate Fellow with JILA, in the Quantum Physics Division, NIST, and was an Assistant Professor Adjoint with the Department of Physics, University of Colorado at Boulder. From 1993 to 1995, she was a Director’s Post-Doctoral Fellow with the Los Alamos National Laboratory, Los Alamos, NM. Dr. Clement was the recipient of the Department of Commerce Silver Medal.

Paul D. Hale (M’01–SM’01) received the Ph.D. degree in applied physics from the Colorado School of Mines, Golden, CO, in 1989. Since 1989, he has been with the Optoelectronics Division, National Institute of Standards and Technology (NIST), Boulder, CO, where has conducted research in birefringent devices, mode-locked fiber lasers, fiber chromatic dispersion, broadband lasers, interferometry, polarization standards, and high-speed opto-electronic measurements. He is currently Leader of the High-Speed Measurements Project in the Sources and Detectors Group. His research interests include high-speed opto-electronic and microwave measurements and their calibration. Dr. Hale is currently an associate editor for the JOURNAL OF LIGHTWAVE TECHNOLOGY. He was the recipient of the Department of Commerce Bronze, Silver, and Gold Awards, two Automatic RF Techniques Group (ARFTG) Best Paper Awards, and the NIST Electrical Engineering Laboratory’s Outstanding Paper Award.

Dylan F. Williams (M’80–SM’90–F’02) received the Ph.D. degree in electrical engineering from the University of California at Berkeley, in 1986. In 1989, he joined the Electromagnetic Fields Division, National Institute of Standards and Technology (NIST), Boulder, CO, where he develops metrology for the characterization of monolithic microwave integrated circuits and electronic interconnects. He has authored or coauthored over 80 technical papers. Dr. Williams is currently Editor-in-Chief of the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES. He was the recipient of the Department of Commerce Bronze and Silver Medals, two Electrical Engineering Laboratory’s Outstanding Paper Awards, two Automatic RF Techniques Group (ARFTG) Best Paper Awards, the ARFTG Automated Measurements Technology Award, and the IEEE Morris E. Leeds Award.

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C. M. Wang received the Ph.D. degree in statistics from Colorado State University, Fort Collins, in 1978. He is currently a Mathematical Statistician with the Statistical Engineering Division, National Institute of Standards and Technology (NIST), Boulder, CO. His research interests include statistical metrology and the application of statistical methods to physical sciences. Dr. Wang is a Fellow of the American Statistical Association. He was the recipient of the Department of Commerce Bronze Medal.

Andrew Dienstfrey received the B.A. degree in mathematics from Harvard University, Cambridge, MA, in 1990, and the Ph.D. degree in mathematics from the Courant Institute of Mathematical Sciences, New York, NY, in 1998. From 1998 to 2000, he was a Post-Doctoral Scientist with the Courant Institute, where he investigated methods for remote sensing of dielectric properties of superconducting thin films. In 2000, he joined the Mathematical and Computational Sciences Division, National Institute of Standards and Technology (NIST), Boulder, CO. His research interests include theoretical and computational aspects of periodic scattering problems in acoustics and electromagnetics.

Darryl A. Keenan received the B.S. degree in physics from the University of Colorado at Boulder, in 1996. In 1989, he joined the National Institute of Standards and Technology (NIST) [then the National Bureau of Standards (NBS)], Boulder, CO, and has since been a member of the Sources and Detectors Group, Optoelectronics Division. He has run optical laser metrology laboratories including low-power continuous wave (CW) from the visible to near infrared, high-power CW at far infrared, -switched Nd : YAG at near infrared, and Excimer at ultraviolet to deep ultraviolet. He has worked with colleagues to develop optical fiber connector characterization and to develop a system for measuring detector nonlinearity at 193 nm. His current areas of research include optical laser metrology at 193 and 248 nm and time- and frequency-domain characterization of oscilloscopes using swept sine measurements and calibrated photodiodes.

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Noise Considerations When Determining Phase of Large-Signal Microwave Measurements Peter Stuart Blockley, Student Member, IEEE, Jonathan Brereton Scott, Senior Member, IEEE, Daniel Gunyan, Member, IEEE, and Anthony Edward Parker, Senior Member, IEEE

Abstract—Advances in microwave instrumentation now make it feasible to accurately measure not only the magnitude spectrum, but also the phase spectrum of wide-bandwidth signals. In a practical measurement, the spectrum is measured over a finite window of time. The phase spectrum is related to the position of this window, causing the spectrum to differ between measurements of an identical waveform. It is difficult to compare multiple measurements with different window positions or to incorporate them into a model. Several methods have been proposed for determining the phase spectrum such that multiple measurements can be effectively compared and utilized in models. The methods are reviewed in terms of the information required to determine the phase and compared in terms of their robustness in the presence of measurement noise. Index Terms—Intermodulation distortion, measurement uncertainty, nonlinear systems, phase measurement.

Fig. 1. Phase measurement repeated over a period of time (less than 1 h) for three tones in a multitone signal. Although the sources and measurement system are phase locked to a common reference, the measured phase varies slowly over time due to drift and imperfect phase locking in the system.

I. INTRODUCTION FTEN nonlinear systems are characterized only by their magnitude spectrum. For instance, verifying spectral mask compliance or determining the third-order intercept point of an amplifier only requires magnitude information. For advanced applications, such as device modeling and linearization, the phase spectrum provides important information and can be readily obtained over wide bandwidths with equivalent-time sampling oscilloscopes or nonlinear network analyzers [1], [2]. The magnitude spectrum of a periodic signal does not depend on the starting time of the time-domain window because the spectrum is independent of time shift. This property allows straightforward comparison of measurements made at different times. The phase spectrum is dependent on the relative starting time of the time-domain window, making it difficult to compare measurements, or incorporate several measurements made with different starting times into a model. The Fourier transform of a

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Manuscript received December 15, 2005; revised May 12, 2006. P. S. Blockley and A. E. Parker are with the Department of Electronics, Macquarie University, Sydney, N.S.W. 2109, Australia (e-mail: peterblockley@ieee. org). J. B. Scott was with the Microwave Technology Center, Agilent Technologies, Santa Rosa, CA 95404 USA. He is now with the School of Science and Engineering, University of Waikato, Hamilton, New Zealand (e-mail: [email protected]). D. Gunyan is with the Microwave Technology Center, Agilent Technologies, Santa Rosa, CA 95404 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2006.879136

time-shifted waveform, expressed in terms of magnitude and phase, is (1) where is a time-domain function with time shift of , is the corresponding frequency spectrum, is the angular frequency, and is the argument of (the angle associated with the complex quantity ). The difficulty is in finding the relative time shift between two measurements made at different times in the presence of measurement noise and without an absolute time-reference. Fig. 1 is an example of a phase measurement using the large-signal network analyzer (LSNA) system described in [2] repeated over a period of time. The measurements are before any processing and may differ from those obtained by other LSNA instruments, such as [1], published in the literature. The phase varies slowly over time due to drift and imperfect phase locking in the system. Note that while the measured phase appeared to follow a linear trend, consistent with a phase lock drift, this is not always the case. In linear systems, the issue is resolved by extracting a linear time-invariant model. For measurements of nonlinear systems, alignment methods and time-zero cancellation methods have been proposed for determining the phase. Alignment methods seek to align the measured signal to an explicit target signal, while time-zero cancellation methods seek to cancel in (1) through a linear transformation of the measured phase values.

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While a variety of phase processing methods have been reported in the literature, the consequences of measurement noise on the variance of the transformed phase is often not considered. However, it has been noted that different alignment methods can result in different uncertainties [3] and a study of time-domain methods for aligning noisy signals was conducted in [4]. This paper provides analysis of the uncertainty in the timezero cancellation methods and a comparison with the alignment methods using real measurements. It shows that the time-zero cancellation methods have a variance that can change significantly across the measurement bandwidth. The covariance matrix for the time-zero cancellation method is derived and can be used when the transformed measurements are used for developing models. Section II describes the extraction of linear time-invariant models for linear systems. Section III gives an overview of phase-determining methods that have been proposed for systems with energy at the fundamental frequency, and Section IV gives an overview of phase-determining methods that have been proposed for systems without energy at the fundamental frequency. The methods are presented in a historical order such that the reader may gain insight into the development of this theory. Section V derives the covariance matrix for the time-zero cancellation method. Section VI presents measurement examples using some of the methods described in Section IV.

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is the variance of the variable and is a where function that returns the argument of (the angle associated with the complex quantity of ). The model has the property that if the variance of and are statistically independent and do not change with frequency, then the variance of is statistically independent and does not change with frequency. Nonlinear system theory currently does not offer such a practical solution. Nonlinear time-series models, which are an extension of the linear moving average and autoregressive moving average models are possible, but a nonlinear system can have many states that depend on both time and amplitude. Unlike the linear case, it is difficult to excite every state in a nonlinear system and the large number of states can make the model computationally large. III. NONLINEAR SYSTEM-FUNDAMENTAL TONE PRESENT A simple case is one where the signals of interest contain energy at the fundamental frequency. The signal-alignment process is easier because the phase of the fundamental rotates through 360 once only, over one period of the signal. This section gives an overview of time-domain signal alignment, fundamental alignment, frequency-domain alignment, and time-zero cancellation methods when the fundamental tone is present. A. Time-Domain Signal Alignment

II. LINEAR SYSTEMS In linear systems, the phase-determination problem is often resolved by extracting a linear time-invariant model. The output of a linear time-invariant system as a function of input is given by

(2) where is the time and is the system’s impulse response. This definition is useful because does not depend on , even though the signals and do. Applying the Fourier transform to both sides of (2) gives

The goal of signal alignment is to calculate the relative time shift between signals. The simplest method is to calculate the time shift between a measured and a target signal. Ideally the measured signal is a time-delayed version of the target signal that has been corrupted by Gaussian noise. The first measurement can be used as the target signal (as is done in this paper) or estimated from multiple measurements [5]. When calculating relative time shift between three or more signals, the “complete cross-correlation” method has been shown to be the most accurate [4]. One method is to maximize the cross-correlation of the measured and target signal, i.e., to maximize

(5) (3) , and are the Fourier transforms of and respectively. A discrete version of is, therefore, easily extracted in the frequency domain using a vector network analyzer. A discrete frequency-domain model that uses the extracted data is a finite state moving average model in the time domain and has well-understood limitations. If the measurements of the phase and are independent, then the variance of is where

(4)

which is the cross-correlation of the target signal and measured signal . This is a nonlinear problem without an explicit solution and can be difficult to solve due to a large number of local maximums. The signal-alignment algorithm used in the comparative study [4] used a golden section search and parabolic interpolation to evaluate the local minimums of the error function in search of the global solution. For multiport systems, the signal-alignment method might be applied only to the incident wave and the other ports time shifted accordingly. This might be useful, for example, for observing changes in phase over bias in a transistor. The advantage of signal alignment is its good immunity to measurement noise. The method requires an explicit target

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signal. A distorted target signal (random noise, linear/nonlinear distortion, or systematic error) can have an effect on the variance and correctness of the assumptions. B. Fundamental Alignment In many situations, the target signal and corresponding phases may not be known. Fundamental alignment is a method that requires an arbitrary choice of target phase and has an explicit solution. The fundamental can be aligned to an arbitrary phase (this paper will consider the alignment of the fundamental phase to 0 ) without consideration of the harmonic frequencies. An explicit solution was formally considered by Jargon et al. [6], but had been previously implemented in LSNA software by Verspecht and Vanden Bossche for visualization purposes only [7]. Consider signal components at frequencies with phases , where is the fundamental frequency of the spectrum and is the measured phase at the fundamental frequency. The aligned phase of the component at , denoted , is given by

(6) where , is an integer , and the phases are modulo (i.e., ). Note that . It is interesting to note that an arbitrary does not affect the aligned phase and is referred to as “time-invariant phase” in [6]. This invariance to is the basis of time-zero cancellation discussed below. Now consider the variance (or combined uncertainty [8]) assuming that the frequency relationship is perfectly known and

of the measured and target signal. Frequency-domain alignment considered in [9] is performed by minimizing the least squared error between the measured and target phases. The choice of target signal/phases will depend on the application. In this paper, the target phases are chosen to be the first measurement, but various (often superior) approaches can be used for estimating the target signal from the measurement data [4]. Alternatively, the target phases could be those programed into the signal generator or from another port in a multiport system [9]. Care must be taken when selecting the target signal, as distortion can increase the observed variance. For aligning measurements that contain noise, a least squares or a weighted least squares problem is formulated. When the phase measurements have equal variance, the least squares problem is to minimize the error function

(8) where is the time shift, is a vector of phases with index , and is the vector of target phases with index . Frequency-domain alignment is closely related to time-domain signal alignment. In fact, the solutions converge to the same estimate when the signals perfectly align. To demonstrate this, consider that the real-valued time-domain signals to be aligned consist of a discrete set of tones. Noting the Fourier transform pair , the time-domain signal-alignment problem is that of maximizing a Fourier series

(9)

(10)

(11) (7) is the covariance of the variables and . where If the variance of the measurements of and are due to independent processes, then the covariance can be assumed zero. Therefore, for large values of , the aligned phase will have significantly greater variance than the measured phase. For multiport systems, the fundamental alignment method might be applied only to the incident wave and the other ports time shifted accordingly.

, , , and is the where amplitude of the target , is the phase of the target , is the amplitude of the measured signal , and is the phase of the measured signal . If the signals align perfectly, then at the solution, modulo must be zero. Expanding at the solution when the signals align perfectly in a Taylor series truncated to the second order and substituting gives

C. Frequency-Domain Alignment Frequency-domain alignment estimates a time when the phases align. This was formally considered in [9] and called “phase detrending.” The method is very similar to signal alignment and is referred to here as frequency-domain alignment. The signal-alignment method presented in Section III-A is performed in the time domain by maximizing the cross-correlation

(12)

(13) where

modulo

(i.e.,

).

BLOCKLEY et al.: NOISE CONSIDERATIONS WHEN DETERMINING PHASE OF LARGE-SIGNAL MICROWAVE MEASUREMENTS

Maximization of is equivalent to minimization of . Therefore, the solution of time-domain signal-alignment converges to the solution of the frequency-domain alignment problem (8) when the tones have equal magnitude and the signals are perfectly aligned. Frequency-domain alignment has the same advantage as signal alignment, i.e., that of good immunity to measurement noise. Both the time-domain signal-alignment and frequency-domain alignment methods require explicit target phases of the signal to be measured. The time-domain signal alignment implicitly weights the error function in proportion to the magnitude squared of the frequency components. This weighting may be correct for many systems where the variance of the phase is inversely proportional to the magnitude squared of the signal, but this is not the case for the LSNA presented in [2]. Frequency-domain alignment does not make any assumptions about the weights, hence, an appropriately weighted version of (8) would be expected to perform better in a broad range of measurement systems. D. Time-Zero Cancellation The high uncertainty in the fundamental alignment method described in Section III-B has lead to development of methods to reduce the propagation of errors for some measurement scenarios. The signal-alignment methods seek to estimate then align the signal to time zero (a point where ). Time-zero cancellation applies a linear transform to the phase such that the new phase values are not dependent on [10]. Consider expressing a frequency as (14) where and are the integer coefficients in a linear combination of the carrier frequency and frequency . The time-zero cancellation phase at frequency is defined as (15) is the measured phase at , is the measured where phase at the carrier frequency, and is the measured phase at the fundamental frequency. Assuming the variance of the measurement is not correlated across frequency, the variance of the time-zero cancellation phase is given by if if

(16) otherwise

where is the variance of the time-zero cancellation , is the variance of the measured phase , phase is the variance of the measured phase at the carrier and frequency. Therefore, when the carrier frequency is much greater than the frequency , the variance of the time-zero cancellation

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method is significantly less than the variance of the fundamental are typically much alignment method (7) because and less than . Similar to the fundamental alignment method, the time-zero cancellation method only requires arbitrary selected target phases. For a mixer application, the carrier frequency might exist at the local-oscillator port and the frequency at the IF port. The time-zero cancellation method might be applied to the RF port using from the local-oscillator port and from the IF port. For multiport applications, the variance will differ from that derived in (16). IV. NONLINEAR SYSTEM-FUNDAMENTAL SUPPRESSED Often a multitone signal has no energy at the fundamental frequency. The methods presented in Section III can be applied to nonlinear systems where the fundamental tone is not present with little or no modification. An overview of time-domain signal-alignment, frequency-domain alignment, two-tone envelope alignment (equivalent to fundamental alignment, but for the case when the fundamental is suppressed) and time-zero cancellation methods when the fundamental tone is suppressed or not present is presented here. A. Time/Frequency-Domain Alignment Time-domain signal alignment with a suppressed fundamental is the same as in Section III-A. The case is typically more difficult to solve due to the large number of local maximums with very close maximum values. This is because the tones of the signal traverse many 360 cycles over one period. Frequency-domain alignment with a suppressed fundamental tone is the same as in Section III-C. The case is typically more difficult to solve due to the large number of local minimums with very close minimum values. Consider the example of ten unity magnitude tones evenly spaced from 1 to 1.09 GHz. Each tone of the measured signal has a phase of 0 and the target signal is the same, but time shifted by 0.5 ns. A graphical example of the time-domain signal-alignment error function (inverted and with an offset) and frequencydomain alignment error function are shown in Fig. 2. The error functions have many local extreme points. The signals align perfectly at ns, the global solution to both error functions. B. Two-Tone Envelope Alignment Two-tone envelope alignment is an extension of the fundamental alignment method from Section III-B. When signals are considered that have no energy at the fundamental frequency , can be redefined as the difference in phase of two adjacent reference frequency components ( , , and ). The phase of the envelope of the two-tone signal is then . This effectively sets the phase difference of two adjacent frequencies to a fixed value. In this paper, the phase difference is set to zero, but could be set to an arbitrary target phase difference. Two-tone envelope alignment was first used as part of a derivation for the initial estimate of the frequency-domain alignment solution [9]. Letting , then

(17)

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be the difference in phase between two adjacent reference tones , , and ). This effectively sets the phase ( of two adjacent tones to a fixed value. In this paper, the phases of the two adjacent tones are set to zero, but could be set to any arbitrary target phases. This technique was first investigated for two-tone intermodulation distortion by [11] and further by [12] and [10]. Letting and expanding (15), for this case gives (19) Fig. 2. Example plot of the frequency-domain alignment error function E (t) and the time-domain alignment error function f ? g + 20. The error functions typically have many local extreme points, which makes evaluation of the global minimum difficult. This is an example of ten tones evenly spaced from 1 to 1.09 GHz with unity magnitude. There is no noise in this example, therefore, there is an exact solution E (t) = 0 and f ? g + 20 = 0 to both problems at t = 0:5 ns. The minimum value of the function log[E (t)] is much less than 1, but is not shown due to truncation of the y -axis.

0

The time-zero cancellation phase can be considered as the phase deviation from a “reference nonlinearity” excited with the adjacent reference tones [12]. For measurements, the “reference nonlinearity” might be the following:

0

0

where is the aligned phase at the frequency and is the measured phase at , is the measured phase at the carrier frequency and is the measured phase at the lower adjacent frequency. Consider the variance assuming that the frequency relationship is perfectly known and the phase measurements are independent as follows:

(20) Expanding the fundamental and

reference nonlinearity (20) for the distortion products, where gives

(21) where , , , and are the amplitudes of the refgenerated from the reference nonlinearity erence distortion . Subtracting the phase of the reference distortion from the measured phase for each corresponding frequency gives the time-zero cancellation phase. For the distortion example, the time-zero cancellation phases are

otherwise. (18) The resulting variance of the two-tone envelope alignment phase can be significantly greater than the variance of the measured phase in certain applications. Consider the measurement of the phase of the third-order intermodulation product ( ) in a two-tone test. The variance of the aligned phase would be significantly greater than the measured phase when the tone spacing is small. Consider a two-tone test with tones at 1 GHz [ ] and 1.001 GHz , where the variance of the measured phase is equal and independent across frequency. For this example, the variance of the aligned phase would be more than 60 dB greater than the variance of the measured phase . C. Time-Zero Cancellation Cancellation of time zero when the fundamental is suppressed is an extension of the time-zero cancellation method presented in Section III-D. The phase of the fundamental is chosen to

(22)

The time-zero cancellation method (19) is equivalent to signal alignment only when the following holds:

modulo

(23)

Therefore, time-zero cancellation cannot be regarded generally as a signal-alignment method and care must be taken when using data transformed in this way. Alternatively, an alignment method can be used to align the phases of two adjacent frequency components to their corresponding explicit target phases. This was considered in [9] as a starting estimate for the frequency-domain alignment problem.

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Assuming the variance of the measurement is not correlated across frequency, the variance of the time-zero cancellation phase is given by

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puted (which corresponds to the transfer function since the timezero cancellation phase is a linear transform)

otherwise. (24) Using the previous two-tone example from Section IV-B of a ] and a 1.001-GHz tone, the variance of the 1-GHz [ phase of the increases by less than 8 dB. Thus, this method is suitable for narrowband modulations, but is not suitable for wide-bandwidth applications. For the example shown in [13], a multitone signal was generated with 800-MHz bandwidth, carrier frequency of 3.66 GHz, and fundamental of 1.8 MHz. The resulting variance of the time-zero cancellation phase would be up to 48 dB greater than the measured phase for some frequency components in the signal. For a multiport system, and would typically be taken from the incident wave and the time-zero cancellation applied to all ports. References [11] and [12] give examples of a two-port excited by a two-tone excitation.

V. COVARIANCE MATRIX FOR TIME-ZERO CANCELLATION Phase determination using time-zero cancellation has been suggested for modeling applications [10]. At first glance, the method appears to simplify the fitting of models, but the method introduces correlation in the phase across frequency, which must be taken into consideration. Take an example of a four-tone signal . If the measurements of the phase are independent with equal variance, then the covariance matrix for the measured phase would be

(25)

(27) of quantities after the time-zero The covariance matrix cancellation transform is applied is given by (see [14] for a proof)

(28)

The time-zero cancellation transform is linear, therefore, this method provides an accurate covariance matrix for the timezero cancellation phase. From the covariance matrix, an increase in the variance is observed as well as a high correlation between the first and fourth tones. Typically the population covariance is not known, thus the sample covariance is used after transformation using (28). A confidence region for the mean vector can then be specified for the measurement as

(29) where is the number of samples, is the sample mean vector, is the transformed sample covariance matrix, and is the upper th percentile of the distribution [15]. VI. MEASUREMENTS

where is the variance of the measured phase. If the individual phases have Gaussian distribution, then the multivariate Gaussian (normal) distribution is given by

To verify the theoretical derivation for the propagation of measurement error, measurements were taken with the measurement system described in [2]. The signal source, phase reference clock, and the receiver hardware were phase locked via a 10-MHz reference. The phase varies slowly with time, as can be seen in Fig. 1, due to drift and imperfect phase locking. Two measurements were performed: a two-tone measurement and a multitone measurement.

(26) and is the where is the vector of phases vector of mean values corresponding to the elements in , and is the number of elements in . To compute the covariance matrix for the time-zero cancellation phase (19), the multivariate Jacobian matrix is first com-

A. Multitone Measurement The multitone consists of 41 pseudorandom phased tones spaced 1 MHz apart centered around 10 GHz (10-GHz carrier with 20 tones spaced either side), generated with an Agilent E8267C PSG. To calculate the sample variance of the

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Fig. 3. Time-zero cancellation phase and frequency-domain alignment phase for eight tones of the 41-tone (multitone) signal. The frequency-domain alignment phase and time-zero cancellation phase are represented with an error bar giving the minimum and maximum measured values around a mean. The frequency-domain alignment has fairly constant variance across the frequency range, resulting in fairly constant error bar size. The time-zero cancellation phase is set to zero with zero variance at k = 1, m = 0, and k = 1; m = 1, and has increasing sample spread for frequencies further away. The mean values of the frequency-domain alignment and time-zero cancellation phases have different values because the frequency-domain alignment is aligned to the first sample, while the time-zero cancellation phase is set to zero at m = 0 and m = 1.

0

Fig. 4. Comparison between the measured and calculated standard deviation for different tones in a multitone signal when using the time-zero cancellation method. The time-zero cancellation phase is set to zero with zero variance at k = 1; m = 0; and k = 1; m = 1, and has increasing standard deviation for frequencies further away. The standard deviation in the time-zero cancellation phase is calculated from (24) using the average variance of the frequency-domain alignment phase. This is contrasted with the standard deviation using the time- and frequency-domain alignment methods. The time-zero cancellation results in higher standard deviation compared to the alignment methods, but only requires arbitrary target phases.

0

0

time-zero cancellation (19) phase, time-domain signal-alignment (5) phase and frequency-domain alignment (8) phase, 172 measurements were performed. The frequency-domain alignment was implemented using MATLAB1 code from [9] and the time-domain signal-alignment method was implemented using a modified version of the code. The phase of the first measurement is used as the explicit target phase required to align the remaining 171 measurements. The time-zero cancellation phase and frequency-domain alignment phase for eight tones of the multitone signal are shown in Fig. 3. The frequency-domain alignment phase and time-zero cancellation phase are represented with an error bar giving the minimum and maximum measured values around a mean. The frequency-domain alignment phase has fairly constant variance across the frequency range, resulting in fairly constant error bar size. The time-zero cancellation phase is set to zero with zero variance at the reference frequencies ( , , and , ) and has increasing sample spread for frequencies further away. The mean values of the frequency-domain alignment and time-zero cancellation phases have different values because the frequency-domain alignment is aligned to the first sample, while the time-zero cancellation phase is set to zero at and . The measured standard deviation for the time-zero cancellation phase, calculated standard deviation (24) for the timezero cancellation phase, measured standard deviation for the frequency-domain alignment, and measured standard deviation for the time-domain signal-alignment are shown in Fig. 4. The calculated standard deviation in the time-zero cancellation phase (24) is evaluated where is the average variance of the frequency-domain alignment phase. 1MATLAB

is a registered trademark of The MathWorks Inc., Natick, MA.

Fig. 5. Phase deviation from the mean of the frequency-domain alignment phase and time-zero cancellation phase for samples of the 9.998-GHz tone against the 10.001-GHz tone. A 95% confidence region (29) for the mean of the time-zero cancellation phase is defined on the plot, where  is the average variance of the frequency-domain alignment phase. Analysis of multiple tones is possible, but it is difficult to graph more than three tones.

As the presented theory predicts, the time-zero cancellation method amplifies the variance of the underlying measurements used. This increase in variance was quite significant for the multitone measurement considered. The time-domain signal-alignment and frequency-domain alignment methods have significantly lower variance across the bandwidth than the time-zero cancellation method. Fig. 5 plots the deviation from the mean of the frequency-domain alignment phase and time-zero cancellation phase for samples of the 9.998-GHz tone against the 10.001-GHz tone. This plot clearly shows the high correlation between the tones of the time-zero cancellation phase. This is in contrast to the frequency-domain alignment phase, which has low correlation between the tones. A high correlation in the time-zero cancellation phase is observed between the 9.998- and 10.001-GHz tone because both tones share a component due to the tone at

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Fig. 6. Measured magnitude spectrum for two-tone test of a Mini-Circuits amplifier. The distortion products have smaller amplitude than the fundamental tones.

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Fig. 8. Comparison of the calculated standard deviation (24) and standard deviation in the measured phase of the time-zero cancellation method for the tones was in Fig. 6. The calculated standard deviation (24) was evaluated, where  the variance of the frequency-domain alignment phase.

at the output of the amplifier. The calculated standard deviation (24) and standard deviation in the measured phase for the time-zero cancellation method is shown in Fig. 8. The calculated standard deviation (24) was evaluated, where was the variance of the frequency-domain alignment phase. The difference in variance between the time-zero cancellation phase and alignment methods is not as significant as in the multitone example because the variance of the phase of the fundamental tones is less than that in the phase of the distortion products. Thus, two-tone distortion measurements are a possible candidate for time-zero cancellation methods, coupled with an appropriate covariance matrix. Fig. 7. Comparison of the standard deviation in the phase for the time-domain signal-alignment and frequency-domain alignment methods for the tones in Fig. 6. The frequency-domain alignment problem was weighted by an estimate of the variance of the measured phase.

and . A 95% confidence region (29) for the mean of the time-zero cancellation phase is defined in the plot, where is the average variance of the frequency-domain alignment phase. B. Two-Tone Measurement For the two-tone measurement, two equal-amplitude tones were generated with an Agilent E8267C PSG and used to drive a Mini-Circuits amplifier. The distortion at the output of the amplifier is shown in Fig. 6. Comparison of the standard deviation in the phase for the time-domain signal-alignment (5) and frequency-domain alignment (8) methods is shown in Fig. 7. The magnitude of the distortion products is less than the fundamental tones and, thus, the variance of the measured phase cannot be assumed equal. Therefore, the frequency-domain alignment problem was weighted by an estimate of the variance of the measured phase (the variance of the measured phase was estimated by first differencing sequential measurements to remove the trend in the measured data). The time-zero cancellation phase (19) was calculated with set to 992 MHz and to 960 MHz, the fundamental tones

VII. APPLICATIONS The alignment methods were shown to have good immunity to measurement noise, but require explicit target phase values. Alignment methods can be used to align narrow- or wide-bandwidth multitone signals. The fundamental alignment and time-zero cancellation methods are appropriate for weakly nonlinear distortion measurements where the variance of the phase for the distortion products is greater than the reference tones. Two-tone amplifier distortion measurements are a good candidate for time-zero cancellation methods, as the increase in variance tends to be small and good models can be extracted when coupled with the correct covariance matrix. VIII. CONCLUSION Several phase determination methods have been evaluated in terms of their performance in the presence of measurement noise. It was found that the alignment methods perform well in the presence of measurement noise, but require explicit target phases. The time-zero cancellation methods do not perform as well in the presence of measurement noise, but only require arbitrary target phase values. Covariance matrices have been derived that take into account the increased variance and the correlation between the phases at different frequencies that is introduced by the time-zero cancellation method.

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REFERENCES [1] J. Verspecht, F. Verbeyst, and M. V. Bossche, “Network analysis beyond S -parameters: Characterizing and modeling component behaviour under modulated large-signal operating conditions,” in 56th ARFTG Conf. Dig., Dec. 2000, pp. 9–12. [2] P. S. Blockley, D. Gunyan, and J. B. Scott, “Mixer-based, vector-corrected, vector signal/network analyzer offering 300 kHz–20 GHz,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2005, pp. 1497–1500. [3] J. Jargon, D. DeGroot, and D. Vecchia, “Repeatability study of commercial harmonic phase standards measured by a nonlinear vector network analyzer,” in 62nd ARFTG Microw. Meas. Conf., Dec. 2003, vol. 3, pp. 243–258. [4] K. J. Coakley and P. Hale, “Alignment of noisy signals,” IEEE Trans. Instrum. Meas., vol. 50, no. 1, pp. 141–149, Feb. 2001. [5] C. D. Woody, “Characterization of an adaptive filter for the analysis of variable latency neuroelectric signals,” IEE Med. Biol. Eng. Comput., vol. 5, pp. 539–553, 1967. [6] J. Jargon, D. DeGroot, K. Gupta, and A. Cidronali, “Calculating ratios of harmonically related, complex signals with application to nonlinear large-signal scattering parameters,” in 60th ARFTG Conf. Dig., Dec. 2002, pp. 113–122. [7] F. Verbeyst, 2006, Private communication. Note: F. Verbeyst and M. Vanden Bossche are with NMDG Engineering and J. Verspecht is with Jan Verspecht bvba.. [8] Guide to the Expression of Uncertainty in Measurement. Genve, Switzerland: Int. Org. Standard., 1995. [9] K. A. Remley, D. F. Williams, D. M. M.-P. Schreurs, G. Loglio, and A. Cidronali, “Phase detrending for measured multisine signals,” in 61th ARFTG Conf. Dig., Jun. 2003, pp. 73–83. [10] G. Loglio, J. Jargon, and D. C. DeGroot, “Phasor angle definition suitable for intermodulation measurements,” in 65th ARFTG Conf. Dig., Jun. 2005, pp. 83–89. [11] T. Nakatani, T. Matsuura, and K. Ogawa, “A simple method for measuring the IM3 components of multi-stage cascaded power amplifiers considering the phase characteristics,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2004, vol. 3, pp. 1731–1734. [12] J. P. M. Jose, C. Pedro, and P. M. Cabral, “New method for phase characterization of nonlinear distortion products,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2005, pp. 971–974. [13] P. Blockley, D. Gunyan, and J. B. Scott, “Wide-bandwidth, vector-corrected measurement with high spurious-free dynamic range,” in 65th ARFTG Conf. Dig., Jun. 2005, pp. 185–188. [14] S. J. Press, Applied Multivariate Analysis. New York: Holt, Rinehart and Winston, 1972, p. 63. [15] B. D. Hall, “Calculating measurement uncertainty for complex-valued quantities,” Meas. Sci. Technol., pp. 368–375, Feb. 2003.

Peter Stuart Blockley (S’03) received the B.Sc. degree from Macquarie University, Sydney, N.S.W., Australia, in 2002, and is currently working toward the Ph.D. degree at Macquarie University. He has authored or coauthored five publications. His research activities focus on device characterization and metrology for microwave and millimeterwave systems. Mr. Blockley is the chair of the IEEE student branch at Macquarie University. He was the recipient of the 1997 Merit Award in Electronics Technology, the 2005 Automatic RF Techniques Group (ARFTG) Best Interactive Session Paper Award, and the 2005 Macquarie University Innovation Award for work in partnership with Agilent Technologies.

Jonathan Brereton Scott (M’80–SM’99) received the B.Sc., B.E., M.Eng.Sc,. and Ph.D. degrees from The University of Sydney, Sydney, Australia in 1977, 1979, 1986, and 1997, respectively. Until 1997, he was with the Department of Electrical Engineering, University of Sydney. In 1995, he was a visitor with University College London, and subsequently a Visiting Lecturer with the University of Western Sydney. He was involved in establishing and subsequently teaching in the Graduate Program in Audio of the School of Architectural and Design Science, Sydney University. He was a founding member of the Collaborative Nonlinear Electronic Research Facility (CNERF) of the Electronics Department, Macquarie University. In 1997 and 1998, he was Chief Engineer with RF Technology, Sydney, Australia. He has served on committees of the Standards Association of Australia. He serves on the NRC Review Panel for the National Institute of Standards and Technology (NIST)’s Electronics and Electrical Engineering Laboratory. He is an Honorary Associate of Macquarie University. In 1998, he joined the Microwave Technology Center, Hewlett-Packard (now Agilent Technologies), Santa Rosa, CA, where he was responsible for advanced measurement systems. In 2006, he accepted the Foundation Professorship in Electrical Engineering with the University of Waikato, Hamilton, New Zealand. He has authored or coauthored numerous refereed publications, several book chapters, and a textbook. He holds a number of patents. Prof. Scott is a Fellow of the Institute of Engineers, Australia and a member of the Audio Engineering Society. In 1993, he held a British Telecom Research Fellowship. He was the recipient of the 1994 Electrical Engineering Foundation Medal for Excellence in Teaching.

Daniel Gunyan (M’97) received the B.S. degree in electrical and computer engineering and M.S. degree in electrical engineering from Brigham Young University, Provo, UT, in 1996 and 1997, respectively. Since 1997, he has been with Agilent Technologies (formerly Hewlett-Packard), Santa Rosa, CA, where he has been involved in both manufacturing and design of microwave and millimeter-wave components for measurement systems and radio transceivers. He is currently involved with the development of test systems and measurement methods for nonlinear characterization and modeling of high-frequency devices, components, and systems.

Anthony Edward Parker (S’84–M’90–SM’95) received the B.Sc., B.E., and Ph.D. degrees from The University of Sydney, Sydney, Australia, in 1983, 1985, and 1992, respectively. In 1990, he joined Macquarie University, Sydney, N.S.W., Australia, where he is a Professor of microwave device and circuit research within the Department of Electronics. He has a continuing project on characterization of microwave devices and design of low-distortion communications circuits. He has consulted with several companies including M/A-COM, Lowell, MA, and Agilent Technologies, Santa Rosa, CA. He is Director of the Collaborative Nonlinear Electronic Research Facility (CNERF), Macquarie University. He has developed accurate circuit simulation techniques, such as used in field-effect transistor (FET) and high electron-mobility transistor (HEMT) models. He has authored or coauthored over 120 publications. His recent research has been in the area of intermodulation in broadband circuits and systems, including a major project with Mimix Broadband Inc. Prof. Parker is a member of the Institution of Engineering and Technology, U.K., the Institution of Engineers, Australia, and the Information and Telecommunications and Electronic Engineers Society, Australia. He is also a committee member of the IEEE Antennas and Propagation (AP)/Microwave Theory and Techniques (MTT) N.S.W. Local Chapter.

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Large-Signal Behavioral Modeling of Nonlinear Amplifiers Based on Load–Pull AM–AM and AM–PM Measurements Jiang Liu, Lawrence P. Dunleavy, Senior Member, IEEE, and Huseyin Arslan, Senior Member, IEEE

Abstract—This paper presents an improved behavioral modeling technique that generates large-signal models for nonlinear amplifiers or devices based on load–pull AM–AM and AM–PM measurement datasets. The generated behavioral model characterizes the incident and scattering waveforms at two ports in the frequency domain based on the large-signal scattering function theory. The advantage of this technique is that it is derived entirely from load–pull measurements and provides an analytic method to utilize the load–pull measurements in practical designs. Examples are given to demonstrate the ability of the behavioral models to predict the load-related nonlinearities of the device-under-test. Index Terms—Large signal, microwave measurements, modeling, network analysis, nonlinear systems, scattering parameters.

I. INTRODUCTION EHAVIORAL modeling of RF components is receiving more and more interest recently. This is because of the increasing application of integrated circuits in wireless products, e.g., cellular phones and personal digital assistants (PDAs). Accurate behavioral models for these off-the-shelf components are very important for this practice to be successful. Probably the most successful behavioral model is the small-signal scattering parameter set that characterizes linear and mildly nonlinear devices and components. -parameter representation is a frequency-domain behavioral model for the network studied, characterizing the relationship between the incident and scattering waveforms at specific frequencies, one at a time. Since it deals with linear transfer relationship only, it cannot be used to model components [e.g., power amplifiers (PAs)] that present significant nonlinearities. However, with the advance of modern wireless communication systems, more and more demands are generated for nonlinear operation of devices and amplifiers to get better transmission efficiency and less power consumption. A large-signal scattering function theory is proposed to augment the limitation of the small-signal -parameter representation. This theory, which has been extensively studied, e.g., [1]–[6], extends small-signal theory to take into account not only the fundamental, but also harmonics at different ports. The contribution of all these spectral components is formulated into nonlinear functions, making it possible to characterize the nonlinearities. A specific measurement system, called a large-signal

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Manuscript received December 15, 2005; revised May 3, 2006. This work was supported by Modelithics Inc. The authors are with the Department of Electrical Engineering, University of South Florida, Tampa, FL 33620 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2006.879137

network analyzer (LSNA), is usually required to measure and derive this nonlinear behavioral model for the device-under-test (DUT). This theory has not been widely applied in real life thus far due to the limited access to the specialized LSNA systems. Alon [7] provides an interesting time-domain analysis/characterization of the large-signal -parameters. A modulated signal generator and vector signal analyzer are used to capture the memory effect of the nonlinear devices. However, this technique also requires specific instruments to perform the required measurement, which limits its adoption. Therefore, one question is asked: is it possible to derive the large-signal behavioral model through generally available measurement systems, e.g., a conventional network analyzer and load–pull measurement system? By looking closely at the LSNA, it is found that the measurement system can be considered as an active harmonic load–pull measurement system [2]. This analogy suggests the possibility to approximately create the large-signal model from a general load–pull measurement dataset. There are several existing techniques to utilize the load–pull dataset for modeling purposes [8], [9]. Some commercial computer-aided engineering (CAE) software provide the capabilities to read the load–pull data files into the simulator for linear or nonlinear simulation.1 2 These techniques create file-based models from the load–pull measurement dataset; the models created can accurately characterize the prescribed load points. The disadvantage of this technique is that it usually involves large file sizes and does not interpolate or extrapolate smoothly. What we will refer to here as the largetechnique is used in [10] to derive the behavioral model. This technique extends the traditional linear to be dependent on the input signal amplitude while keeping other -parameters constant. A typical swept power measurement will give enough information for deriving the largemodel. Therefore, it can capture the characteristics of small-signal load contours and compressions. However, due to the nature of the model derivation (only the gain compression curve at 50 is utilized), it does not take into account and fails to predict the load-related changes in the gain and phase compression curves. This paper presents a modeling technique that utilizes the load–pull AM–AM and AM–PM measurement datasets to derive the large-signal scattering function model. This is achieved 1Microwave Office, Applied Wave Res. Inc., El Segundo, CA. [Online]. Available: www.microwaveoffice.com 2Advanced Design System (ADS), Agilent Technol., Palo Alto, CA. [Online]. Available: www.agilent.com

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; it can be further Equation (6) is an implicit expression for transformed to an explicit function to simplify the model generation. Assume and are represented as

Fig. 1. Diagram of a two-port network with a voltage source of E and source impedance of Z . The load impedance is Z .

where are unknowns to be determined. Suppose and . and are the real and imaginary parts of , respectively. and are the real and imaginary parts of , respectively. Equation (6) can be rewritten as

through simplifying the original theory and omitting the influence from the harmonics. It is shown that a behavioral model can be created to characterize the load-related gain and phase compression properties accurately.

(7) where

II. ANALYSIS OF THE BEHAVIORAL MODELING TECHNIQUE A nonlinear amplifier can be treated as a two-port network shown in Fig. 1. A typical one-tone load–pull measurement gives information about the source impedance (or reflection coefficient ), load impedances (or reflection coefficient ), the input power , and the measured delivered power . To simplify the analysis, suppose the device is unilateral (i.e., ). This constraint can be removed if the input port reflected power is captured in the load–pull measurement. The incident and reflected waveforms at port 1 are calculated as (1)

(8) (9) Arrange the real and imaginary parts and we can get

(10) By solving the linear function (10), the real and imaginary parts of can be derived as

(2) (11) (3) where By adopting the large-signal scattering function proposed in [4]–[6] and considering only the fundamental tones, we get the incident and reflected waveforms at port 2 in (4) and (5). This model formulation is the result of the linearization around the large-signal operating point of a device and is valid for small . This extra conjugate term gives us additional flexibility to fit measurement datasets. Notice that, in this simplified model form, the phase normalization introduced in [4] and [5] is neglected. Therefore, this model does not guarantee time invariance; however, as will be demonstrated in the following equation, this deficiency has not led to significant problems, and the consequences will be further explored in the future: (4) (5) Combining (4) and (5) gives (6)

Obviously, in order to obtain and , the measurements for both the magnitude and phase are required. This is why it is important to obtain the load–pull AM–PM datasets. The load–pull AM–AM measurements provide the optimization criteria for the magnitude, while the load–pull AM–PM measurements set up the rule for the phase optimization. The magnitude can be derived from the delivered output power. The output power at port 2 is determined by and through (12)

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Fig. 2. AM–PM load–pull measurement system diagram.

Since the output power is known through the measurement, can be expressed as the magnitude of

(13) An optimization process can be applied to obtain the six unknown coefficients – . The least mean square (LMS) errors for the magnitude and phase can be represented by (14) and (15) as follows:

(14)

(15) where is the number of load points used in the optimization process. AM–PM is the phase compression data obtained through the load–pull AM–PM measurement. It is the phase difference between the voltages at the input and output ports. The input and output voltages are the sum of the incident and reflected waves at the port, respectively. To obtain the load–pull AM–PM datasets, an advanced measurement procedure is developed through the integration of the load–pull measurement system and a vector network analyzer (VNA). The setup diagram is shown in Fig. 2. The VNA is used as a vector receiver to obtain the relative AM–PM measurement dataset. The vector receiver captures the incoming signals and stores the corresponding complex values as an array in the instrument memory. The AM–PM curve is generated by normalizing the phases of these complex values against the phase of the first complex value in the array. The analysis given above has been implemented in a MATLAB program.3 Fig. 3 demonstrates the procedure to generate the behavioral model based on the load–pull datasets. The two error functions are normalized, respectively, and formulated into a weighted function in the MATLAB program. Notice that the load–pull datasets can come from either the measurements or from simulations, depending on the applications of this modeling technique. 3The MathWorks Inc., Natick, MA. [Online]. Available: www.mathworks.com

Fig. 3. Flowchart of the MATLAB program created for the behavioral model optimization based on the load–pull AM–AM and AM–PM datasets.

III. EXAMPLE MODEL OF A PA SAMPLE To illustrate this modeling technique, an example model is created based on the measurement datasets for a ISL3984 PA sample. This sample amplifier was characterized at 2.45 GHz. Load–pull gain and phase compression measurements were performed. Two-tone load–pull measurements were performed as well. The MATLAB modeling program was used to process the measurement data files and generate the model coefficients through the unconstraint nonlinear optimization procedure. The measurement condition is summarized as follows: • frequency: 2.45 GHz; • input power: 22–3.5 dBm; • two-tone frequency spacing: 100 kHz; • bias: 3.3 V. The model was implemented in ADS 2004A using the frequency-domain defined (FDD) device. The advantage of using this device is that it provides the ability to define the behavior of individual frequency components separately. Notice that although only the ADS implementation of the behavioral model is shown in this paper, this technique is general enough to be implemented in other CAE software. Fig. 4 compares the measured and simulated gain and phase compression performance of this PA at 50- condition. The model predicts the compression property correctly. Figs. 5 and 6 illustrate the optimized large-signal and coefficients versus input power levels. As you can see, the demonstrates strong compression at higher input power levels. The coefficient presents significant contributions at high input power. Therefore, it is important to characterize these effects instead of treating them as linear/constant. Fig. 7 shows the simulated output power contours compared with the measured result. The input power is at 20 dBm. Good agreement can be observed in this figure. In fact, the large-signal model reduces to a small-signal -parameter model when the input signal is low enough. The variation of the output power

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Fig. 4. Comparison of the measured and simulated gain and phase compression at 50 . Fig. 7. Comparison of the simulated output power contour with the measured dataset. TABLE I LIST OF THE SIX EXAMPLE LOAD REFLECTION COEFFICIENTS USED TO TEST THE PA MODEL

Fig. 5. Optimized S 22 versus P in.

Fig. 8. Illustration of the six load impedance examples on the Smith chart.

Fig. 6. Optimized T 22 versus P in.

with respect to the load can be characterized through the smallsignal -parameter. Detailed analysis can be found in [11]. Six load impedances are chosen as examples to test the large-signal model. The reflection coefficients of the six example loads are listed in Table I and plotted in a Smith chart, as

Fig. 9. Simulated fundamental tone at six loads are plotted. The markers of the curve are consistent with Fig. 8.

shown in Fig. 8. The load examples are chosen to spread over the Smith chart.

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the intermodulation products (which involves higher order harmonics) is limited. Therefore, to solve this issue, a file-based model is implemented for prediction of the third-order intermodulation (IM3) products. A contour interpolation algorithm is utilized during the generation of the data file. Fig. 11 illustrates the errors in the simulated IM3 at different loads. The new model has significantly better performance compared with the largemodel. IV. CONCLUSION

Fig. 10. Errors of the simulated fundamental tone at six loads are plotted. The black curves (enclosed by the rectangle) represent the errors associated with the newly developed model; the gray curves (enclosed by the ellipse) represent the errors associated with the large-S 21 model. The new model presents better performance, compared with the large-S 21 model.

A behavioral modeling technique has been presented that is directly derived on the load–pull gain and phase compression measurements. Simplified from the large-signal scattering function theory, the model formulation neglects the harmonic terms and phase normalization. An example model for a PA sample is given to demonstrate the possibility to generate the large-signal scattering function model using traditional load–pull measurement systems and exploiting advanced measurement procedures. Good agreement is observed between the simulated results and measurement datasets. More study is required in future research to study the consequences of neglecting the phase normalization. ACKNOWLEDGMENT The authors thank J. Verspecht, Jan Verspecht bvba, Steenhuffel, Belgium, for his valuable suggestions and insights. REFERENCES

Fig. 11. Errors of the simulated IM3 product at six loads are plotted. The black curves (enclosed by the rectangle) represent the errors associated with the newly developed model; the gray curves (enclosed by the ellipse) represent the errors associated with the large-S 21 model. The new model presents better performance compared with the large-S 21 model.

Fig. 9 shows the gain compression curves measured at the six loads. The markers of the curve are consistent with Fig. 8. As one can see, the compression rates of this device are different at different load points. Therefore, it is important to consider this effect when modeling a device. However, as mentioned in Section I, the largemodeling technique only considers the nonlinear effect of based on the gain compression measurement done at 50 . This limitation makes the largemodeling technique inaccurate when the load points are significantly deviated from 50 . As can be seen in Fig. 10, the proposed model presents a much better performance in predicting the gain compression effects of the device at different loads than the largemodel does. Since only the fundamental tone is considered and characterized during the model generation, its capability to predict

[1] J. Verspecht, M. V. Bossche, and F. Verbeyst, “Characterizing components under large signal excitation: Defining sensible ‘large signal S -parameters’,” in 49th ARFTG Conf. Dig., Jun. 1997, pp. 109–117. [2] J. Verspecht and P. V. Esch, “Accurately characterizing hard nonlinear behavior of microwave components with the nonlinear network measurement system: Introducing ‘nonlinear scattering functions’,” in Proc. 5th Int. Integr. Nonlinear Microw. Millimeterwave Circuits Workshop, Oct. 1998, pp. 17–26. [3] J. Verspecht, “Large-signal network analysis—Going beyond s-parameters,” presented at the 52nd ARFTG Conf., Dec. 2003, Short course notes. [4] J. Verspecht, D. Root, J. Wood, and A. Cognata, “Broad-band, multiharmonic frequency domain behavioral models from automated largesignal vectorial network measurements,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2005, 4 pp. [5] D. E. Root, J. Verspecht, D. Sharrit, J. Wood, and A. Cognata, “Broad-band poly-harmonic distortion (PHD) behavioral models from fast automated simulations and large-signal vectorial network measurements,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 11, pp. 3656–3664, Nov. 2005. [6] J. Verspecht, D. F. Williams, D. Schreurs, K. A. Remley, and M. D. McKinley, “Linearization of large-signal scattering functions,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 4, pp. 1369–1376, Apr. 2005. [7] Z. Alon, “Extraction and application of behavioral models in power amplifier simulation,” Master’s thesis, Dept. Elect. Comput. Eng., Univ. California at Santa Barbara, Santa Barbara, CA, 2003. [8] J. Olah and S. Gupta, “Power amplifier design using measured load–pull data,” in Microw. Eng. Europe, Aug. 2003, pp. 23–30. [9] R. L. Carlson, “Meld load–pull test with EDA tools,” Microw. RF, Apr. 2003 [Online]. Available: http://www.mwrf.com/Articles/ArticleID/ 5448/5448.html [10] W. Clausen, J. Capwell, L. Dunleavy, T. Weller, J. Verspecht, J. Liu, and H. Arslan, “Black-box modeling of RFIC amplifiers for linear and non-linear simulations,” Microw. Product Dig., p. 34, Oct. 2004. [11] G. Gonzalez, Microwave Transistor Amplifiers: Analysis and Design, 2nd ed. Englewood Cliffs, NJ: Prentice-Hall, 1996.

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Jiang Liu received the M.S.E.E. and Ph.D. degrees from University of South Florida, Tampa, in 2002 and 2005, respectively. He is currently a consultant providing services in the area of nonlinear modeling and customized measurement procedure developments. His main research interests are in advanced RF and microwave measurement procedure developments and behavioral modeling techniques for nonlinear RF devices or components.

Lawrence P. Dunleavy (S’80–M’82–SM’96) received the B.S.E.E. degree from Michigan Technological University, Houghton, in 1982, and the M.S.E.E. and Ph.D. degrees from The University of Michigan at Ann Arbor, in 1984 and 1988, respectively. Along with four faculty colleagues, he established the Center for Center for Wireless and Microwave Information Systems, University of South Florida, Tampa. In 2001, he co-founded Modelithics Inc., a USF spinoff company to provide a practical commercial outlet for developed modeling solutions and microwave measurement services. He has been involved in industry for E-Systems (1982–1983) and Hughes Aircraft Company (1984–1990), and was a Howard Hughes Doctoral Fellow (1984–1988). In 1990, he joined the Department of Electrical Engineering, University of South Florida, where he is currently a Professor. He guides a team of graduate students in various research projects related to microwave and millimeter-wave device, circuit, and system characterization and modeling. From 1997 to 1998, he spent a sabbatical year with the Noise Metrology Laboratory, National Institute of Standards and Technology (NIST), Boulder, CO. He has authored or coauthored over 75 technical papers. Dr. Dunleavy is very active in the IEEE Microwave Theory and Techniques Society (IEEE MTT-S) and the Automatic RF Techniques Group (ARFTG).

Huseyin Arslan (S’95–M’98–SM’04) received the Ph.D. degree from Southern Methodist University (SMU), Dallas, TX, in 1998. From January 1998 to August 2002, he was with the Research Group, Ericsson Inc., Research Triangle Park, NC, where he was involved with several project related to second-generation (2G) and third-generation (3G) wireless cellular communication systems. Since August 2002, he has been with the Department of Electrical Engineering, University of South Florida, Tampa. He was also a Visiting Professor (Summer 2005) and has been a Part-Time Consultant (since August 2005) with the Anritsu Company, Morgan Hill, CA. His research interests are related to advanced signal-processing techniques at the physical layer with cross-layer design for networking adaptivity and quality of service (QoS) control. More specifically, he is interested in signal-processing techniques for wireless communication systems including modulation and coding, interference cancellation and multiuser signal detection, channel estimation and tracking, equalization, soft information generation, adaptive receiver, transmission technologies, etc. He is interested in many forms of wireless technologies including cellular, wireless personal area networks (PANs)/local area networks (LANs)/metropolitan area networks (MANs), fixed wireless access, and specialized wireless data networks like wireless sensors networks and wireless telemetry. His current research interests are ultra-wideband (UWB), orthogonal frequency-division multiplexing (OFDM)-based wireless technologies with emphasis on WIMAX, and cognitive and software-defined radio. He is an Editorial Board member for the Wireless Communication and Mobile Computing Journal. Dr. Arslan has served as a Technical Program Committee member, session and symposium organizer for several IEEE conferences. He was Technical Program co-chair for the 2004 IEEE Wireless and Microwave Conference.

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Minimum-Phase Calibration of Sampling Oscilloscopes Andrew Dienstfrey, Paul D. Hale, Senior Member, IEEE, Darryl A. Keenan, Tracy S. Clement, Senior Member, IEEE, and Dylan F. Williams, Fellow, IEEE

Abstract—We describe an algorithm for determining the minimum phase of a linear time-invariant response function from its magnitude. The procedure is based on Kramers–Kronig relations in combination with auxiliary direct measurements of the desired phase response. We demonstrate that truncation of the Hilbert transform gives rise to large errors in estimated phase, but that these errors may be approximated using a small number of basis functions. As an example, we obtain a minimum-phase calibration of a sampling oscilloscope in the frequency domain. This result rests on data obtained by an electrooptic sampling (EOS) technique in combination with a swept-sine calibration procedure. The EOS technique yields magnitude and phase information over a broad bandwidth, yet has degraded uncertainty estimates from dc to approximately 1 GHz. The swept-sine procedure returns only the magnitude of the oscilloscope response function, yet may be performed on a fine frequency grid from about 1 MHz to several gigahertz. The resulting minimum-phase calibration spans frequencies from dc to 110 GHz, and is traceable to fundamental units. The validity of the minimum-phase character of the oscilloscope response function at frequencies common to both measurements is determined as part of our analysis. A full uncertainty analysis is provided. Index Terms—Hilbert transform, Kramers–Kronig relation, linear response functions, minimum phase, mismatch correction, oscilloscopes.

I. INTRODUCTION T THE National Institute of Standards and Technology (NIST), Boulder, CO, we are developing high-speed electrical time- and frequency-domain metrology that is based on an electrooptic sampling (EOS) system that is traceable to fundamental physical units. A fundamental component of this metrology is a photodiode (PD) whose phase and magnitude response is calibrated in the frequency domain up to 110 GHz using the NIST EOS system [1]–[3]. The PD calibration includes corrections for the complex characteristic impedances of the measurement system, as well as dispersion and multiple reflections. Once calibrated, the PD can be used to calibrate high-speed electrical test equipment, including sampling oscilloscopes [4]. We will refer to this as the EOS-PD calibration of the oscilloscope. The EOS system is designed for measurements requiring high bandwidths, 110 GHz or more at present, and has degraded un-

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Manuscript received December 15, 2005; revised March 14, 2006. The authors are with the National Institute of Standards and Technology, Boulder, CO 80305 USA (e-mail: [email protected]; hale@boulder. nist.gov). Digital Object Identifier 10.1109/TMTT.2006.879167

certainty below several hundred megahertz due to the maximum time interval the system is capable of measuring. This gives rise to an uncertainty in the PD response that propagates into an increased uncertainty in the EOS-PD oscilloscope calibration over this same low-frequency regime. This is a serious impediment for using the oscilloscope to obtain calibrated waveforms over time intervals larger than roughly 1 ns or, conversely, spectral information below 1 GHz. Furthermore, it is difficult to obtain accurate absolute scaling for the voltage pulse generated by the PD [1]. Thus, the EOS-PD calibration alone cannot be used to give accurate absolute voltages, as measured by the oscilloscope. As an alternative to the EOS-PD calibration, swept-sine (frequency-domain) measurements can be used to determine the magnitude of the frequency response of an oscilloscope, as described in [5] and [6]. The swept-sine calibration can be made at any frequency at which fundamental microwave power standards are available, typically from 0.1 MHz to greater than 50 GHz. However, since the swept-sine calibration does not give phase information, an oscilloscope calibrated using this technique alone is not adequately characterized for time-domain metrology. In this paper, we describe a procedure for reconstructing the minimum phase of the oscilloscope response function from its magnitude. Minimum-phase response functions have the property that, in principal, the phase can be recovered from the Hilbert transform of the logarithm of the magnitude [7]. In practice, naive attempts to apply the standard theory can yield extremely large absolute errors in the computed phase. This is due to the truncation of the required integral operators to the necessarily finite bandwidths attainable by measurement. Variations of this problem are well known and have been discussed previously by several authors, [8]–[11]. Our fundamental observation is that although the truncation error may be large in absolute scale, due to the localizing nature of the Hilbert transform, this error is inherently low rank in the sense that it can be approximated by a small number of customized basis functions. As we have independent and direct measurements of the phase response supplied by the EOS-PD calibration over a large bandwidth, we solve a linear least squares problem for the difference between our measured values of the phase and the values computed via a minimum-phase assumption as an expansion in our specialized basis. This expansion corrects the absolute size and coarse trends in the truncated minimum-phase approximation. We provide a complete uncertainty analysis of the procedure. Naturally, the final goal of a calibration is to determine the true frequency-domain response, which may or may not be min-

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imum phase. In our example, our analysis allows us to conclude that the oscilloscope response function is indeed minimum phase over the bandwidth attainable by direct EOS-PD measurements.1 Although the minimum-phase property would seem desirable for an oscilloscope, given that such response functions optimize a form of energy transfer (see, e.g., [12]), we have not seen such a claim made before in the literature. To the contrary, there exist several possible physical mechanisms [13] that could, in principle, lead to a nonminimum phase response. For theoretical reasons and based on experiments we have performed in the laboratory, we conjecture that the minimumphase character of the oscilloscope response extends into the low-frequency region. If so, our analysis results in a traceable extension of the oscilloscope calibration to frequencies below those unattainable by our present EOS techniques. Space does not permit these arguments to be made in their entirety below. To be clear, the results contained in this paper are as follows. 1) We describe an algorithm for correcting for truncation effects in minimum-phase analysis with a full description of error propagation, 2) We present a new uncertainty analysis for the swept-sine calibration procedure introduced in [5] and [6], 3) We demonstrate that the oscilloscope response function is given by the minimum-phase response from 1 to 100 GHz. A brief description of this paper follows. In Section II, we present the minimum-phase analysis and derive the basis functions suitable for correcting truncation effects. Section III contains our uncertainty analyses of the swept-sine calibration and the minimum-phase reconstruction. In Section IV, we implement the procedure to calibrate a high-speed equivalent-time sampling oscilloscope in the frequency domain. Finally, the assumption that the oscilloscope response function is minimum phase is fundamental to our analysis. We return to this in the conclusion presented in Section V. The swept-sine calibration of the oscilloscope magnitude was described previously in [5] and [6]. An abbreviated description is included here in Appendix A for completeness. The uncertainty analysis for this calibration appearing in Section III of this paper is new. We also include various numerical details in Appendix B, again for completeness. Throughout this paper, we use the “pseudowave” formalism of [14] to characterize signals in microwave networks and limit our discussion to frequencies below the cutoff frequency of the guiding structures. Furthermore, the desired instrument responses are assumed to be linear and time invariant, and also to satisfy the necessary technical constraints; e.g., they are finite square integrable, unless otherwise noted.

II. PHASE RECONSTRUCTION Here, we describe the theory and implementation of the minimum-phase analysis. 1Technically the minimum-phase property is a global one in the frequency domain. Phrases referring to the minimum-phase character of a frequency-domain response function over a finite bandwidth are to be understood as a lack of observable features attributable to Blaschke terms in the fully general factorization of (1).

A. Theory An arbitrary frequency-domain response function can be factored as [12]

(1) is the minimum-phase response, is a real time where offset, and is a Blaschke product or “all-pass filter.” One distinguishing characteristic of a minimum-phase function is that its phase is determined by its magnitude via a Hilbert transform relationship. There are various ways to express this; for our purposes, the most elementary version is sufficient. Writing , we have

(2)

One recognizes (2) as the usual Kramers–Kronig relations applied to the real and imaginary parts of the function . The analyticity of and consequent applicability of (2) is equivalent to the minimum-phase constraint. Note, by definition, a minimum-phase response has neither delay, nor all-pass components and . Currently we are unable to determine an absolute time origin. Therefore, we refer to response functions with possibly nonzero as minimum phase even though this is incorrect, strictly speaking. A pervasive problem in the application of any Kramers–Kronig analysis is to estimate the error due to the finite bandwidth of a measurement. As it is impossible to measure to infinite frequency, in practice, the integral in (2) is truncated. We define

(3)

where is the maximum frequency attainable by experiment. The problem is to estimate the effects of integrating only out to [8], [9], [11]. (We use as a dummy variable and it should not be confused with the usual Laplace transform argument.) Concerning this truncation effect, there is a significant distinction between phase recovery and the use of Kramers–Kronig relations to relate real and imaginary parts of response functions. As the time-domain impulse response functions of interest are real, standard parity arguments from Fourier analysis imply that the magnitudes of the frequency-domain response functions roll off as some even power, at large frequencies. Experimentally, the magnitudes of generic oscilloscope response functions have been observed to decay like , i.e., [15]. Regardless, as we will see, our results are independent of . We assume an asymptotic expansion of the form

(4)

DIENSTFREY et al.: MINIMUM-PHASE CALIBRATION OF SAMPLING OSCILLOSCOPES

where

and the ’s are constants. For fixed , such that , a simple estimate gives

(5) In this set of equations, the constant depends on and is allowed to change from line to line. The ellipses represent series terms in higher inverse powers of and, therefore, may be ignored asymptotically. Notice that the form of the estimate is independent of the rate of rolloff of the frequency-domain response function, i.e., . The point here is that the scaling converges so weakly as to not be useful. The implication is that even if the minimum phase hypothesis is valid, it is difficult to compute an estimate of the underlying minimum phase regardless of the experimental bandwidth. The situation is radically different when attempting to recover the imaginary from the real part. Using the same approach as in (5) yields an estimate of the truncation error (again assuming fixed ), which is

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TABLE I VALUE OF IMAGINARY PART AND PHASE OF RECONSTRUCTED FUNCTION h (1=3) FOR DIFFERENT TRUNCATION FREQUENCIES

If the real part is measured to twice the 3-dB frequency, we observe two-digit accuracy on the calculated imaginary part. By contrast, corresponding accuracy in the calculated phase requires magnitude measurements out to 1000 times the characteristic frequency. This slow convergence is consistent with the estimate given in (5). It is clear that estimation of the phase of a minimum-phase response function from (3) alone will exhibit large errors due to the truncation of the integral. However, we claim that although this error is large in magnitude, it is simultaneously low rank. By “low rank” we mean that it will be possible to expand the difference between the minimum phase and the phase computed by the truncated operator (3) in terms of a small number of functions that we define specially for this purpose. B. Definition of Basis Functions We assume that the response is minimum phase with the exception of a possible unknown time shift . In this case, (2) holds with a slight modification

(6) In this case, given a fixed interval of target frequencies, the size of the error due to truncation approaches zero rapidly as the measurement bandwidth increases. An example of these estimates may be instructive. Consider the second-order low-pass Butterworth filter given by

(8) is given by the truncated integral (3). The function where , defined implicitly by (8), is given by either of the equivalent expressions

(7) (9) As a low-pass filter with rolloff, (7) is a reasonable surrogate for an oscilloscope response function for which the 3-dB frequency point has been normalized to be (see [15]). We emphasize that our results rely neither on this class of filters, nor the order. The Butterworth filter is minimum phase, thus, (2) is valid. For fixed target frequency , we compute the functions

Consider the evaluation of and from and , respectively, using truncated integrals of the form (3). Computing the required Hilbert transforms for different truncation frequencies gives the results of Table I.

Note that encapsulates all unknown information of due to both finite bandwidth measurements and unknown time shift. As stated above, we know that admits an asymptotic expansion in even powers of . Assume that it is sufficient to keep only the first term of this expansion. Substituting this into the operator (9) gives

(10)

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as those of the magnitude measurements nor even as dense. All that is required is that

Fig. 1. Plot of orthonormal basis functions

where the constants depend on are

p (1= )9(f = ). and

. The functions

,

Informally, the frequencies for which we have magnitude data should “cover” the frequencies associated with the phase measurements. For technical reasons, we enforce that the second of the two inequalities is strict (see Appendix B). We assume that the desired phase response is minimum phase, i.e., it satisfies (8), and that is equal to a discrete sampling of . Using the magnitude measurements, we evaluate , the truncated minimum-phase function (3) at all frequencies for which we have measured phase data. For discretization of the singular integrals, we represent the logarithms of the measured magnitude data as a piecewise linear function of frequency. The singular integral (3) is applied analytically to each of the linear segments and then re-summed outside the integral. The operator is evaluated at all of the frequencies where the EOS-PD calibration returns direct measurements of the phase response, as described in Appendix B. The result is a dense matrix of order , where is the number of EOS-PD data points and is the total number of magnitude measurements. The fully discretized version of (3) reads

(12) (11) where is the Lerch transcendent. The equation for is derived by expressing under the integral as a power series in , integrating term by term, and rescaling the result to obtain the power series definition of [16]. This transcendent is a special function in the generalized Riemann–Zeta family and may be evaluated wherever needed either by the power series or its analytic continuation [16]. For conditioning purposes, it is preferable not to use these functions, but rather their orthogonalization given by a continuous version of the Gram–Schmidt procedure [17]. We denote the orthonormal functions by . They are shown for the case in Fig. 1. The case for general is obtained through a suitable rescaling. For a more technical analysis of truncation effects relevant to the application of Kramers–Kronig procedures in the classical setting—e.g., recovery of the imaginary part from the real—see [9]. C. Procedure We now have the tools necessary to describe our procedure. We assume that we have a reasonably dense sampling of magnitude measurements and direct measurements of the phase . The magnitude measurements are obtained from the EOS-PD measurement of [4] and the swept-sine measurement described in Appendix A. The magnitude data sets, acquired on different frequency grids, are merged using the method described in Appendix B. The frequencies at which the phase is measured need not be the same

where is the vector of logarithms of the magnitude response measurements. Next we form the difference (9) at the frequencies of the EOS measurements

(13) Following (10), we make the ansatz (14) in a least and solve for the undetermined coefficients squares sense. Our fundamental claim is that although is large relative to the true underlying phase , the functional form (14) will be sufficient to expand . This will be true if and only if the underlying response function is minimum phase. Thus, violations of this claim may be used as a test of the minimum-phase character of the measured response. We return to this below. Finally, given sufficiently low residual in the least squares fit, an indicator of the validity of the minimum-phase assumption, we compute the phase of the oscilloscope response function as the sum

(15) In this equation, the domain of the Kramers–Kronig operator consists of the same frequencies as the magnitude measurements ; the target frequencies are some arbitrary desired frequency grid.

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B. Minimum-Phase Computation

III. UNCERTAINTY ANALYSIS We perform a complete uncertainty analysis for the sweptsine measurement procedure and minimum-phase recovery algorithm. We recall the standard linearized propagation of errors. Given a scalar random variable with mean and variance , and variable , where is sufficiently differentiable, we assume that

We propagate errors through the minimum-phase reconstruction in a sequence of steps. First we evaluate via the truncated Hilbert transform (12). Using the rules for linear propagation of errors, we find that (18)

(16) The multivariate analog involving the Jacobian of is straightforward; details may be found in [18]. In most instances, we use the above formulas for computation of uncertainties. The one exception is the determination of the uncertainty in the mismatch factors in (29) or (30). In this case, the functional forms are sufficiently complex that we instead use Monte Carlo analysis to determine the uncertainties as functions of the constitutive reflection coefficients and -parameters. A. Swept-Sine Calibration We find the uncertainty in , as determined by the sweptsine calibration, by propagation of the uncertainties in (29) or (30) (of Appendix A), as appropriate. In linear units, we find the uncertainty in the magnitude of the impulse response spectrum as

(17) is the standard uncertainty of the quantity in where brackets, and is the mismatch term on the right-hand side of either (29) or (30). All other terms in (17) are defined in Appendix A. We estimate the uncertainty in each of the power measurements as the standard deviation of the mean of the measurements. The standard uncertainty in the calibration factor is taken from either the manufacturer’s specifications or the NIST calibration report for the power meter. The standard uncertainty in is determined from a Monte Carlo simulation that includes uncertainty in the magnitude and phase of all the scattering parameters. Uncertainty in the reflection coefficients in is determined from either: 1) the manufacturers specifications for the vector network analyzer used for the measurement or 2) NIST calibration reports. The scattering parameters of the adapter are determined from the Monte Carlo simulation of method described in [19], and includes uncertainties in the definitions of the open, short, and load terminations and the network analyzer manufacturer’s specifications. Applying this analysis, we obtain that the standard uncertainty [18] of our swept-sine data is typically between 0.05–0.08 dB.

where is the diagonal matrix containing the uncertainties (systematic and random) of the vector . (Note: this matrix could contain covariance estimates in off-diagonal terms if such information is available. These correlations are important for time-domain applications. See, e.g., [20].) Next we form , the difference between the EOS-PD determined phase and (12). Since the EOS and sweptsine measurements are independent, the uncertainties and are added in quadrature. The least squares problem (14) is then solved for the vector of coefficients . As with any linear least squares problem, the error in the solution vector will scale with the condition number of the underlying system matrix. In our case, this system matrix is given by (19) It is at this point that we benefit from having pre-orthogonalized the natural set of expansion functions (11) to instead form the orthonormal basis . Using the set to define in place of the naïve functions (11) reduces the condition number of from over 100 to around unity. Another way to view this is that, as the are orthonormal in a continuous sense, the discretized least squares solution of (14) effectively is given by a well-conditioned projection. Finally, we evaluate wherever desired as the sum (15). In principle, there exist correlations between the coefficients and the Kramers–Kronig operator applied to the magnitude measurements, given that the later appear in the evaluation of the initial phase estimate . We assume these to be negligible, as we require the values of at precisely those frequencies where the EOS-PD calibration yields unreliable phase measurements. For this reason, whatever correlations exist between the phase contributions of and the coefficients would have to be effected through the off-diagonal components of the Kramers–Kronig operator and subsequent least-squares analysis. We assume these to be insignificant. Therefore, in this final expression for the relevant random variables are and the set , which, by this argument, are independent. The error propagation through the integral operator and the three pointwise multiplications obey their respective linear propagation formalisms, and the resulting uncertainties of each are added in quadrature. IV. OSCILLOSCOPE CALIBRATION We implemented the procedures outlined above to obtain a minimum-phase calibration of our equivalent-time high-speed

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Fig. 2. EOS-PD calibration: (a) phase and (b) magnitude are shown in grey. Black lines show three separate sets of swept sine measurements. Together, measurements span a five decade frequency range.

sampling oscilloscope. This oscilloscope has a nominal bandwidth of approximately 50 GHz and has an adapter to convert from the oscilloscope’s usual 2.4-mm connector to a 1.0-mm coaxial input. It is at this 1.0-mm input reference plane that we calibrate. In our current configuration, the EOS-PD oscilloscope calibration gives the magnitude and phase of the oscilloscope response function at the 1.0-mm coaxial reference plane on an equi-spaced frequency grid from 200 MHz to 110 GHz in 200-MHz increments. These data are shown in Figs. 2 and 3. An uncertainty analysis, performed as part of that calibration, is given in [4]. In addition to the sources of error identified in that analysis, we also expect an increase in the uncertainty at low frequencies due to the limited time interval that is measured by the EOS platform currently used to calibrate the PD. As the oscilloscope is capable of taking measurements over large time windows, e.g., longer than 5 ns, the low frequency cutoff at 200 MHz is insufficient for utilizing the oscilloscope in the manner that it could potentially be used. To augment the EOS-PD calibration, swept-sine calibrations were performed at the oscilloscope’s 2.4-mm input connector plane, and the calibration was then embedded behind the 1.0-mm adapter. The measurements and uncertainty analysis were performed as discussed in Appendix A and Section III, respectively. After all mismatch corrections have been made to account for adapter transfer and device reflections, these experiments return the response magnitude on equi-spaced frequency grids. Separate calibrations were made with this technique over three overlapping frequency ranges. Plots of the resulting magnitudes are shown in Figs. 2 and 3, along with details of the measured frequency grids in the inset.

Fig. 3. Plot of the magnitude measurements from the EOS-PD calibration and three swept-sine experiments. The combined uncertainty in the swept-sine measurements is generally between 0.05–0.08 dB (the black error bar, shown for scale, is 0.08 dB.) Uncertainties for the EOS-PD calibration is shown as gray error bars. (Color version available online at: http://ieeexplore.ieee.org.)

6

In Fig. 3, we show the plots on both linear and logarithmic frequency scales so as to examine certain features in the data. The analysis of Section III-A yields a combined uncertainty in the swept-sine measurements of generally between 0.05–0.08 dB. A sample error bar is shown at the 0.20 GHz point in Fig. 3. We draw attention to two features in these plots. First, the EOS-PD calibration at the lowest two points (0.200 and 0.400 GHz) does not follow the response trend as measured by the swept-sine technique. In fact, the 0.200-GHz point differs from the swept-sine measurements by almost twice the combined measurement uncertainties as based on our current uncertainty analysis of the EOS-PD calibration [4]. Although less pronounced, a similar discrepancy appears at the 0.400-GHz point. Due to these discrepancies and additional time-domain measurements we have performed, we conclude that our current uncertainty analysis of the EOS-PD calibration does not accurately capture the uncertainty due to the limited time interval that is measured by the EOS system. For the purposes of our current minimum-phase analysis, we neglect these two points as outliers. Second, the notch feature in the magnitude response centered near 0.020 GHz, although small ( 0.1 dB), is both repeatable and large relative to the uncertainties as computed by our uncertainty analysis. Furthermore, this is a feature that would be missed entirely if one were to extrapolate the EOS-PD calibration to dc from the lowest EOS-PD frequency measurement of 0.200 GHz.

DIENSTFREY et al.: MINIMUM-PHASE CALIBRATION OF SAMPLING OSCILLOSCOPES

Fig. 4. Plot of the phase measured directly as part of the EOS-PD calibration  and the evaluation of the truncated Kramers–Kronig operator  calculated from (12).

We proceed to the minimum phase computation. For magnitude data, we used a synthesis of the three swept-sine data sets and the magnitude data of the EOS-PD calibration. Our procedure for combining data sets is described in Appendix B. The result is a single vector of magnitude measurements on an irregular frequency grid spanning five decades of frequencies (from 1 MHz to 110 GHz). This is the grid: . Next, the truncated singular integral operator was computed at all frequencies of the EOS-PD data set with the composite product rule described in Appendix B. This result is shown along with the EOS-PD measured phase in Fig. 4. We observe that the fine-scale structures of the two curves agree. Simultaneously, there are gross trends that have not been accounted for. The quantity is evaluated as the difference between these two curves and is shown in Fig. 5. The basis functions were tabulated at the same frequencies as and the system was inverted in a least squares sense for the coefficients . The resulting expansion of in the basis of is shown as the smooth curve in Fig. 5. We observe that this expansion captures all significant features of ; thus, we conclude that the minimum phase hypothesis is valid. We evaluate the minimum-phase response at all of the fundamental frequencies as the sum of the singular integral operator and computed as a linear combination of , as in (15). A plot over low frequencies of the measured phase and the minimum phase computed using our procedure is shown in Fig. 6. The standard uncertainties computed as in Section III-B are shown as the upper and lower dashed curves. The few EOS-PD measurements of phase lying below 2 GHz are included for comparison. The computed phase lies well within the EOS-PD measurement uncertainty. Note that the 0.075-dB dip in magnitude centered at 0.02 GHz (see Fig. 3) gives rise to the 0.4 rising region between 0.01–0.03 GHz. We have indeed observed these features (magnitude and phase) in measurements of comb generators performed with this oscilloscope.

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Fig. 5. Plot of the difference of the two curves in Fig. 4, in the orthogonalized basis functions .

9

1, and its expansion

Fig. 6. Phase response of our oscilloscope estimated using the ESO-PD calibration and the minimum-phase procedure described in this study. Combined uncertainties for the minimum-phase estimate are shown as the dashed lines, while uncertainties in the EOS-PD calibration are shown as gray error bars.

Finally, in Fig. 7, we plot the difference between the phase measured directly by the EOS-PD calibration and the computed minimum phase over all frequencies of the EOS-PD data set. The uncertainty of the difference (which is dominated by the EOS-PD measurements) is represented by the shaded gray region in this figure. The difference is less than a few degrees for GHz, and may be considered to be zero to within the

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directly connected to the signal source, the power meter reading is given by

(20)

Fig. 7. Difference between phase measured directly by the EOS-PD calibration and the computed minimum phase, plotted for all frequencies of the EOS-PD data set. The uncertainty in the difference is represented by the gray shaded region.

reported uncertainties. For frequencies above 70 GHz, the difference still remains small although the uncertainties returned by the EOS-PD calibration become quite large due to the small magnitude of the response function in this region. From this, we conclude that the oscilloscope response function is indistinguishable from the minimum-phase response function over this bandwidth. V. CONCLUSION We have described a procedure for systematically eliminating the large error due to truncation effects in the implementation of a minimum-phase analysis. A complete analysis of error propagation has been performed. As an example, we augmented swept-sine and EOS-PD data sets so as to obtain a minimum-phase calibration of our oscilloscope response function at frequencies unattainable by either technique individually. In the process of this analysis, we observed that the true oscilloscope response function as measured by the EOS-PD calibration is indistinguishable from the minimum-phase response over a very large bandwidth. The theoretical possibility that the true response function is not minimum phase cannot be ruled out from the results and arguments presented above. However, based on subsequent measurements and analysis, we conjecture that this oscilloscope response is, within practical limits, minimum phase to frequencies lower than we can attain using our EOS-PD calibration. We stress that this last assertion applies to our particular model oscilloscope and may not be valid for other models of oscilloscopes or oscilloscopes made by other manufacturers. We will report on these results in the future. APPENDIX A SWEPT SINE MAGNITUDE MEASUREMENT A. Theory A power sensor is commonly used to measure the power of a single-frequency microwave source, as described in [21]. When

Here, and [shown schematically in Fig. 8(a)] are the forward and reverse wave amplitudes of the “pseudowaves” at the junction between the source and the power meter (or oscilloscope) and are normalized to a 50- reference impedance (see [14]). These wave amplitudes have units of the square root of a watt.2 The quantity is the vector reflection coefficient of the power meter, and , known as the mismatch loss, accounts for the power reflected by the sensor. The sensor efficiency accounts for ohmic and radiation losses in the sensor interconnection and housing. Both factors are used to correct measured incident power in commercial power meters, and are usually combined in a calibration factor . From (20), we see that the calibrated power meter measures the power in the incident wave

(21) The net incident power can be related to the power the source by the relation would deliver to a 50- load

(22) is the vector reflection coefficient of the source. Comwhere bining (20) and (22) gives the standard expression for the power the source would deliver to a 50- load in terms of the power meter reading, the meter calibration factor, and the meter and generator reflection coefficients

(23) By analogy with (21), we calibrate the oscilloscope so that it measures the incident wave, as was used in earlier calibration methods [5]. That is, we calibrate the otherwise ideal3 oscilloscope to measure the convolution of the incident wave with the oscilloscope impulse response and use a scale factor to give the voltage reading. In the frequency domain, this becomes a product (24) where is the complex frequency response of the oscilloscope and is dimensionless, i.e., is the transform of the impulse response of the oscilloscope evaluated at our frequency of interest, 2References [5], [14], and [21] use a root-mean-square (rms) normalization. The rms-normalized pseudowaves a and b are related to those used here, with a = a= 2 and b = b= 2. 3By “ideal” we mean that the oscilloscope is free from timing errors, nonlinearity, etc. We use the methods described in this study to characterize the finite-impulse response of the otherwise ideal oscilloscope. Compensation for other nonidealities is described in [4] and the cited references therein.

p

p

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mode and is triggered asynchronously to measure the variance of the voltage at each frequency. We observe that this variance is the rms voltage measured by the oscilloscope. The power measured by the oscilloscope is calculated as

(28) is the measured signal variance and is the backwhere ground variance measured with the signal generator turned off. The frequency of the signal generator is then stepped over the same frequency range, and the response of the oscilloscope is calculated as

(29) Fig. 8. Schematic diagram showing pseudowave definitions for power meter and oscilloscope calibration in (a) and experimental configuration in (b). (Color version available online at: http://ieeexplore.ieee.org.)

and is the raw voltage measured by the oscilloscope. The scale factor converts the pseudowave description to a voltage. For a more detailed analysis of this scaling, see [20]. Using an analysis similar to that leading to (22), the oscilloscope measurement can be related to the wave and power the source would deliver to a 50- load

where is the monitor power meter reading while port 1 is connected to the calibrated meter and is the monitor reading when port 1 is connected to the oscilloscope. Using the ratio in (29) instead of (27) compensates for drift in the source power. When this type of ratio is used, the equivalent source reflection coefficient is found using the method described in [22]. In the case where the adapter is added to the power sensor, the response magnitude is calculated as

(30)

(25) where

and

are the scattering parameters of the adapter and .

(26) where is the vector reflection coefficient of the oscilloscope when in its quiescent state. Since is an invariant property of the signal generator, it can be used to relate the power meter and oscilloscope measurements. We equate (23) and (26) to give an expression for in terms of measured powers and reflection coefficients

(27) where

.

B. Implementation The configuration of the swept sine measurement system is shown in Fig. 8(b). The power at port 1 of the splitter is first measured by connecting directly to a calibrated power meter or, for low frequencies, a low-frequency power sensor is connected to port 1 through an adapter. The signal frequency is then stepped over the appropriate frequency range and, after allowing time for the equipment to settle, measurements are taken from the calibrated power meter and the monitor power meter. Port 1 of the splitter is then connected directly to the oscilloscope and the oscilloscope is placed in a vertical histogram

APPENDIX B NUMERICAL DETAILS Here we describe various numerical routines that were used for the computations above. We first present our algorithm for constructing a continuous description of tabulated noisy data. We refer to this procedure as “merging.” Next we derive the analytic expressions representing the application of the singular integral operator (3) applied to the result of the discrete merged data as in (12). We stress that the analytic basis of our argument—that the discrepancy between a minimum phase response and the truncated Kramers–Kronig operator is low rank—is independent of discretization. By contrast, the error analysis presented above does depend on the discretization. The routines that we employ are general in that they do not require function data on a regular grid, and are robust to the presence of experimental noise. They are also low order, a feature which simplifies the algebra and error analysis considerably. There exist natural higher order analogs to the procedures that we outline. For our current applications, these were deemed unnecessary. We may consider implementing them in the future. A. Merging The Hilbert transform operator is very sensitive to sharp changes or discontinuities in the function to which it is being

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applied. In our current application, such changes are inevitable as the magnitude data are obtained from different experiments over different frequency grids. Even when the discontinuities in magnitude are on the order of what could be expected given the experimental uncertainty of these measurements, they lead to sharp features in the reconstructed phase. Given some underlying assumption of smoothness on the true phase response, these features should be considered artifacts of the data collection procedure. Hence, it is desirable to merge the data sets in such a way that these effects are minimized. Synthesizing a (smooth) data curve from noisy overlapping samples is a topic in its own right. We employed a relatively simple procedure, which is adequate for the current purposes. As a first step, we combined all magnitude measurements—EOS-PD and the three swept-sine calibrations—into a single data set. To do this, we define the “fundamental” frequency grid as the union of all of the experimental frequencies. Next, for each experiment, we interpolated the magnitude data to the fundamental frequencies lying within the frequency interval of that individual experiment. We used piecewise-linear interpolation at this step, and propagated the experimental uncertainty accordingly. Finally, a global data set was formed by computing the average of the magnitude measurements at all frequencies where experiments overlapped. The average was formed by weighting each measurement by the inverse of its associated variance, i.e., the square of its standard uncertainty. In our example, this procedure yields interpolated averaged magnitude data at 800 irregularly spaced frequencies, as shown in Fig. 3. B. Kramers–Kronig Matrix Next we describe our implementation of the Kramers–Kronig operator and the entries of its matrix representation . We define the data vector and its separation into frequencies and values

The result of the merging procedure is a piecewise-linear function defined on an arbitrary frequency grid, i.e., it is a -spline interpolating the points . There are many ways to represent such curves, one being as a linear sum of “hat functions.” Given an arbitrary grid of abscissas , we define the th hat function as the piecewise linear function (see, e.g., [23, Ch. 3])

surement” about the origin and . Thereby, is defined. Finally, as there are no measurement frequencies beyond is given by the top line of (31) and is zero elsewhere. With these definitions, given tabulated data , the desired piecewise linear interpolating spline is given by

(32) is a simple function defined for all By construction, . As such, we may apply the Kramers–Kronig operator (32) analytically. Given a set of target frequencies, we define our discretization of the operator as the matrix that evaluates the singular integral operator against the interpolant formed from the discrete data at the desired target frequencies, e.g., , where the phase has been measured. From (32), we have

(33) Thus, the

th entry of the

matrix is given by (34)

is continuous for will likeAs wise be continuous. The discontinuity in at the top frequency has the consequence that the image will have a logarithmic divergence at . This divergence is mild and is not observable within our measurement error for . However, it is due to this logarithmic divergence that we enforce the strict inequality in Section II-B. It remains to evaluate in (34). For the sake of clarity, we drop the primes on in what follows. The operator applied to a general linear function is fundamental and is given by the identity

(35) For , application of to the hat function is a sum of terms of the form (35) with appropriate definitions of slopes and intercepts and . For , substituting (31) into (35) gives

(31) elsewhere This definition requires some modifications at the extreme frequencies. We assume that the lowest frequency is dc; . As we are interested in taking the Hilbert transform of , we know by standard Fourier arguments that this function is even with respect to frequency. Thus, we may augment the data vector to include a reflected frequency and “mea-

(36)

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Equation (36) makes sense for . At these three points, the logarithmic terms diverge. By construction, however, these divergences cancel and the limit of (35) is finite. For example, the limit as is given by

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[17] G. B. Arfken and H. J. Weber, Mathematical Methods for Physicists, 4th ed. Amsterdam, The Netherlands: Elsevier, 2000. [18] “Guide to the Expression of Uncertainty in Measurement,” BIPM, IEC, IFCC, ISO, IUPAC, IUPAP, and OIML, 1993, pp. 1–101. [19] R. F. Bauer and P. Penfield, “De-embedding and unterminating,” IEEE Trans. Microw. Theory Tech., vol. MTT-22, no. 3, pp. 282–288, Mar. 1974. [20] D. F. Williams, A. Lewandowski, T. S. Clement, C. M. Wang, P. D. Hale, J. M. Morgan, D. Keenan, and A. Dienstfrey, “Covariance-based uncertainty analysis of the NIST electro-optic sampling system,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 1, pp. 481–491, Jan. 2005. [21] “Agilent Fundamentals of RF and microwave power measurements (part 3),” Hewlett-Packard, Applicat. Note 1449-3, 2003. [22] J. R. Juroshek, “A direct calibration method for measuring equivalent source mismatch,” Microwave J., pp. 106–118, Oct. 1997. [23] C. de Boor, A Practical Guide to Splines, ser. App. Math. Sci.. New York: Springer-Verlag, 1978, vol. 27.

(37) We leave the other cases for the readers.

REFERENCES [1] D. Williams, P. Hale, T. Clement, and C.-M. Wang, “Uncertainty of the NIST electro-optic sampling system,” NIST, Boulder, CO, Tech. Note 1535, 2005. [2] T. S. Clement, D. F. Williams, P. D. Hale, and J. M. Morgan, “Calibrated photoreceiver response to 110 GHz,” in 15th Annu. IEEE Lasers Electro-Opt. Soc. Meeting, Glasgow, U.K., Nov. 10–14, 2002, pp. 877–878. [3] D. F. Williams, P. D. Hale, T. S. Clement, and J. M. Morgan, “Calibrating electro-optic sampling systems,” in IEEE MTT-S Int. Microw. Symp. Dig., May 2001, vol. 1473, pp. 1527–1530. [4] T. S. Clement, P. D. Hale, D. F. Williams, C. M. Wang, A. Dienstfrey, and D. A. Keenan, “Calibration of sampling oscilloscopes with highspeed photodiodes,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 8, pp. 3173–3181, Aug. 2006. [5] D. C. Degroot, P. D. Hale, M. Vanden Bossche, F. Verbyst, and J. Verspecht, “Analysis of interconnection networks and mismatch in the nose-to-nose calibration,” in ARFTG Conf. Dig., Jun. 2000, vol. 55, pp. 116–121. [6] P. D. Hale, T. S. Clement, K. J. Coakley, C. M. Wang, D. C. DeGroot, and A. P. Verdoni, “Estimating magnitude and phase response of a 50 GHz sampling oscilloscope using the ‘nose-to-nose’ method,” in 55th ARFTG Conf. Dig., Jun. 2000, pp. 335–342. [7] A. Papoulis, The Fourier Integral and Its Applications, ser. Electron. Sci.. New York: McGraw-Hill Book, 1962. [8] F. M. Tesche, “On the use of the Hilbert transform for processing measured CW data,” IEEE Trans. Electromagn. Compat., vol. 34, no. 3, pp. 259–266, Aug. 1992. [9] A. Dienstfrey and L. Greengard, “Analytic continuation, singular-value expansions, and Kramers–Kronig analysis,” Inverse Problems, vol. 17, pp. 1307–1320, 2001. [10] J. G. McDaniel and C. L. Clarke, “Interpretation and identification of minimum-phase reflection coefficients,” J. Acoust. Soc. Amer., vol. 110, no. 6, pp. 3003–3010, Dec. 2001. [11] J. Mobley, K. R. Waters, and J. G. Miller, “Finite-bandwidth effects on the causal prediction of ultrasound attenuation of the power law form,” J. Acoust. Soc. Amer., vol. 114, no. 5, pp. 2782–2790, Nov. 2003. [12] H. Dym and H. P. McKean, Fourier Series and Integrals. New York: Academic, 1972. [13] M. Efimchik and B. Levitas, “General properties of the transfer function of stroboscopic converters,” Sov. J. Commun. Technol. Electron., vol. 31, no. 4, pp. 110–119, 1986. [14] R. B. Marks and D. F. Williams, “A general waveguide theory,” J. Res. Nat. Inst. Stand. Technol., vol. 97, no. 5, pp. 533–562, 1992. [15] C. Mittermayer and A. Steininger, “On the determination of dynamic errors for rise time measurement with an oscilloscope,” IEEE Trans. Instrum. Meas., vol. 48, no. 6, pp. 1103–1107, Dec. 1999. [16] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. New York: Academic, 1994.

Andrew Dienstfrey received the B.A. degree in mathematics from Harvard University, Cambridge, MA, in 1990, and the Ph.D. degree in mathematics from the Courant Institute of Mathematical Sciences, New York, NY, in 1998. From 1998 to 2000, he was a Post-Doctoral Scientist with the Courant Institute, where he investigated methods for remote sensing of dielectric properties of superconducting thin films. In 2000, he joined the Mathematical and Computational Sciences Division, National Institute of Standards and Technology (NIST), Boulder, CO. His research interests include theoretical and computational aspects of periodic scattering problems in acoustics and electromagnetics.

Paul D. Hale (M’01–SM’01) received the Ph.D. degree in applied physics from the Colorado School of Mines, Golden, CO, in 1989. Since 1989, he has been with the Optoelectronics Division, National Institute of Standards and Technology (NIST), Boulder, CO, where has conducted research in birefringent devices, mode-locked fiber lasers, fiber chromatic dispersion, broadband lasers, interferometry, polarization standards, and high-speed opto-electronic measurements. He is currently Leader of the High-Speed Measurements Project in the Sources and Detectors Group. His research interests include high-speed opto-electronic and microwave measurements and their calibration. Dr. Hale is currently an associate editor for the JOURNAL OF LIGHTWAVE TECHNOLOGY. He was the recipient of the Department of Commerce Bronze, Silver, and Gold Awards, two Automatic RF Techniques Group (ARFTG) Best Paper Awards, and the NIST Electrical Engineering Laboratory’s Outstanding Paper Award.

Darryl A. Keenan received the B.S. degree in physics from the University of Colorado at Boulder, in 1996. In 1989, he joined the National Institute of Standards and Technology (NIST) [then the National Bureau of Standards (NBS)], Boulder, CO, and has since been a member of the Sources and Detectors Group, Optoelectronics Division. He has run optical laser metrology laboratories including low-power continuous wave (CW) from the visible to near infrared, high-power CW at far infrared, -switched Nd : YAG at near infrared, and Excimer at ultraviolet to deep ultraviolet. He has worked with colleagues to develop optical fiber connector characterization and to develop a system for measuring detector nonlinearity at 193 nm. His current areas of research include optical laser metrology at 193 and 248 nm and time- and frequency-domain characterization of oscilloscopes using swept sine measurements and calibrated PDs.

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Tracy S. Clement (S’89–M’92–SM’05) received the Ph.D. degree in electrical engineering from Rice University, Houston, TX, in 1993. Her Ph.D. research concerned the development and study of novel ultrashort pulse and very short wavelength lasers. Since 1998, she has been with the Optoelectronics Division, National Institute of Standards and Technology (NIST), Boulder, CO. Her current research interests include the development of measurement systems for high-speed electrooptic components, as well as ultrashort pulse laser measurements. Prior to joining the Optoelectronics Division, she was an Associate Fellow with JILA, in the Quantum Physics Division, NIST, and was an Assistant Professor Adjoint with the Department of Physics, University of Colorado at Boulder. From 1993 to 1995, she was a Director’s Post-Doctoral Fellow with the Los Alamos National Laboratory, Los Alamos, NM. Dr. Clement was the recipient of the Department of Commerce Silver Medal.

Dylan F. Williams (M’80–SM’90–F’02) received the Ph.D. degree in electrical engineering from the University of California at Berkeley, in 1986. In 1989, he joined the Electromagnetic Fields Division, National Institute of Standards and Technology (NIST), Boulder, CO, where he develops metrology for the characterization of monolithic microwave integrated circuits and electronic interconnects. He has authored or coauthored over 80 technical papers. Dr. Williams is currently Editor-in-Chief of the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES. He was the recipient of the Department of Commerce Bronze and Silver Medals, two Electrical Engineering Laboratory’s Outstanding Paper Awards, two Automatic RF Techniques Group (ARFTG) Best Paper Awards, the ARFTG Automated Measurements Technology Award, and the IEEE Morris E. Leeds Award.

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Experimental Characterization of the Nonlinear Behavior of RF Amplifiers Yves Rolain, Fellow, IEEE, Wendy Van Moer, Member, IEEE, Rik Pintelon, Fellow, IEEE, and Johan Schoukens, Fellow, IEEE

Abstract—Using specifically designed broadband periodic random excitation signals, the best linear approximation of RF amplifiers is measured. The proposed technique: 1) takes into account the measurement uncertainty and the nonlinear distortions and 2) detects, quantifies, and classifies the nonlinear distortions with confidence bounds. The approach is suitable for the experimental characterization of existing amplifiers. Index Terms—Linear approximation, linear characteristics, microwave amplifier, nonlinear distortions, system identification.

I. INTRODUCTION ONLINEAR distortion, or nonlinear system behavior in general, has become increasingly important for the RF and microwave community over the last few years. This movement has mainly been driven by the portable telecommunication market and its high demands on low power consumption and high RF efficiency. As a result, circuit operating points have migrated from the benign low-distortion region towards the compression region. As a result of this trend, the superposition principle is no longer applicable for these new designs. This simple statement, however, has very severe implications. For linear systems, the -parameter representation predicts the system response to an arbitrary signal, which can range from a simple sine wave to a multiple carrier in-phase and quadrature (IQ)-modulated signal such as an orthogonal frequency-domain modulation (OFDM) or a code-division multiple-access (CDMA) signal. This no longer holds for nonlinear systems. For “nonlinear” operation, the response signal will not only depend on the power of the input signal, but also on its spectral content. A system response in “nonlinear” operation is, therefore, only valid for a certain class of excitation signals. This means that it is pointless trying to extrapolate the response of, for example, an OFDM signal based on measurements performed with a sine wave or a two-tone only. If the response of the system to an OFDM signal is to be predicted, it has to be characterized using an OFDM-like signal, i.e., a signal with the same power spectrum and the same probability density function (pdf).

N

Manuscript received December 15, 2005; revised March 3, 2006. This work was supported under the Fund for Scientific Research (FWO-Vlaanderen), by the Flemish Government (GOA), by the Federal Government (IUAP 5), and by the Information Society Technologies Programme of the European Union under Contract IST-1-507893-NOE. The authors are with the Electrical Measurement Department, Vrije Universiteit Brussel, B-1050 Brussels, Belgium (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2006.879168

The problem with these new conditions is that there is no such thing as a “general theory” for nonlinear systems, or put in a more concrete form, -parameters no longer work for systems that are not linear and time invariant. This lack of clear theoretical support drastically increases the difficulty of the interpretation of the measured behavior. Previous attempts to describe a nonlinear system using a linear approximation and a perturbation source even date back from the late 1950s. These methods rely on a sinewave excitation or a pure noise excitation signal to be applied to the device (e.g., [1] and [2]). Unfortunately, these methods are of little practical interest in the context of the characterization of devices operated under complex modulated excitation signals: a sinewave is not acceptable as an approximation to a modulated signal, and the RF measurement of systems excited by noise is impossible with the current instrumentation. Measurements performed by a vectorial network analyzer also use sinwaves as an excitation signal and, therefore, are not really adequate either. In this paper, we will recycle the approach used for the characterization of systems in the linear time-invariant (LTI) framework to better understand the behavior of the nonlinear deviceunder-test (DUT). In a first step, spectral measurements will be used to answer the fundamental question: how big is the nonlinear contribution of the device given a certain class of excitation signals? Once confidence is built up here, a nonparametric best linear approximation (BLA) of the device is obtained in a second step. The BLA behaves as a local nonparametric -parameter representation, which is valid for a class of excitation signals that have their power spectrum (frequency band, power level, and pdf) in common. For the considered excitation signal class, it can be proven that the BLA indeed is the “best” approximation in the sense that it minimizes the difference between the output of the linear approximation and the actual system response in a least squares sense. The convergence between the BLA and the optimal filter that is defined in the context of the Wiener theory is then explicitly shown. In a third step, measurement-based statistical error bounds are obtained for the frequency response function (FRF). A statistical validation of the model, given the experimental uncertainty, can and will also be performed. Hence, the contributions of this paper are: 1) (simultaneous) measurement of the best linear two-port characteristics, the noise levels, and the levels of the in- and out-band nonlinear distortions using specially designed multisines; 2) classification of the nonlinear distortions of the device in odd and even degree contributions; and 3) nonparametric identification of the device

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characteristics taking into account the disturbing noise and the nonlinear distortions. II. INTRODUCTION TO THE EXPERIMENTAL FRAMEWORK To formalize the approach somewhat more, there is a need for a clear definition of the system class, excitation signal class, and model class that are going to be used. A. Class of Excitation Signals: Gaussian Noise Signals In this study, the signals that will be used belong to the class of Gaussian noise-like signals with an a priori fixed power spectral content. Remarkably enough, this class is wide enough to include some interesting periodic test signals, such as random phase multisines, besides the expected normally distributed noise and periodic Gaussian noise. Random phase multisines consist of a sum of commensurate tones that have a prescribed user-defined amplitude, but whose phase varies in a random way both over the different tones in one single signal and between different signals. One signal with a certain fixed phase realization drawn from this class is henceforth called a realization of the signal. In this case, the randomness does not change with time since it now lies in the random variation of the phase between different excitation signals that are sequentially applied to the DUT. To obtain different realizations of this noise-like signal, it will no longer be sufficient to measure successive periods of the same signal (as this will yield the same measurement), but it will be required to generate a new signal each time a new random realization is requested. As will be seen later on, this control of the realization of the periodic random signal comes in very handy for the data processing. Note that the proposed random phase multisine signal can be used as a test signal for any signal with a quasi-Gaussian pdf. Many practically used telecommunication signals with a complex modulation scheme, such as CDMA, OFDM, or quadrature amplitude modulation (QAM) signals, fall in this signal category [3]–[5]. The proposed class of excitation signals is hence certainly of practical interest. The normality assumption has been introduced here mainly for the sake of simplicity and because of its high practical relevance. If only a few carriers are present, the probability density of the modulated signal can no longer be assumed to be a Gaussian signal. However, it is then still possible to use a multisine excitation with a low number of lines that mimics the behavior of the “real-world” signal. The pdf of the multisine signal can then be tuned to approximate the real pdf, following the lines of [6]

Systems having bifurcation points, or what is even worse, chaotic systems, are not part of this class. C. Model Class: BLA Since there is no general framework equation for PISPO systems, and because of the huge potential complexity of the behavior of systems in this class, it is sensible to try an approximate modeling first. LTI models are a good starting point, as they can linearize the system around a certain operating condition. LTI system identification is well known, and many designed systems are “close” to being linear. To use this potential, a theory has been developed that shows that it is indeed possible to linearize a PISPO system. The BLA consists of an LTI model to describe the linearized dynamics of the system, and a noise source to describe the noise-like behavior of the nonlinearity when it is excited using a Gaussian noise signal. This model will be used in the rest of this paper. The BLA can be defined for single-input single-output (SISO) and multiple-input multiple-output (MIMO) systems. Put in a more familiar context, this means that the modeling can either be applied to one -parameter or to the -matrix as a whole. In the sequel, and to ease the notation, the SISO case will be considered, but please note that this is easily extended to the full MIMO case. If an excitation signal class that has non-Gaussian pdf would be selected, the BLA would also be changed. Again, one has to carefully select the class of excitation signals that is going to be used to match the real-world operating conditions of the DUT. D. Instrumentation: Large Signal Network Analyzer (LSNA) If a BLA is to be obtained, there is the need to measure the DUT when operated under modulated excitation using a large number of excitation lines. Besides a measurement on the excited lines, the waves must also be measured at nonexcited lines to assess the influence of the device nonlinearity. Hence, a vector network analyzer (VNA) is no longer adequate to obtain the measurements. Use of the LSNA [7], on the other hand, allows the measurement of the waveforms in the time domain and, hence, all the spectral components that are present in the signal will be measured simultaneously. The availability of an instrument such as the LSNA is, hence, an enabling technology for the estimation of the BLA. Now that the general framework has been put into place, the measurement procedure can be looked at next in some more detail.

B. Class of Considered Systems: PISPO Systems The class of allowed systems is not just the complement of the LTI class, as this would contain any nonlinear system. To focus a bit more, a class was selected that gently departs from the linear systems and contains those as a special case. The common feature of the systems that are considered in this paper is that the steady-state response to a periodic input signal is a periodic output (PISPO) signal having the same period. This class of systems encompasses systems with a saturating—or expanding—behavior that have a finite memory. A clipping device, for example, clearly belongs to this class.

III. MEASUREMENT PROCEDURE A. Introduction The amplifier characteristics are measured using random phase multisine excitation signals. This class of excitation signals has been proven to be asymptotically Gaussian distributed. Finite sample statistics prove that for a practical number of harmonic components, the Gaussian approximation is always valid. For a proof, the reader is referred to [11] or [12]. Random phase multisines are periodic signals consisting of the sum of

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harmonically related sine waves with user-defined amplitudes and random phases (see [9] and [10])

(1)

are randomly chosen such that , The phases where is the expected value operator; e.g., is uniformly distributed in . In [10]–[12], it has been shown that the FRF of a wide class of nonlinear systems, the PISPO systems, obtained using a random phase multisine as in (1) with sufficiently large, can be written as

(2) is the measured incident wave at port where is the measured reflected wave at port is the -parameter of the true underlying linear system, and is the measurement noise. The frequency and all quantities are defined in a 50- reference impedance. is the bias or deterministic nonlinear contribution, which depends on the odd degree nonlinear distortions and the power spectrum of the input only, and is the zero mean stochastic nonlinear contribution

for

(3)

where the overbar denotes the complex conjugate, and where the expected values are taken with respect to the different random phase realizations of the excitation defined in (1). Due to property (3), the stochastic nonlinear contributions act as a circular complex noise for sufficiently large. This means that the real and imaginary part of the spectrum have the same variance, and that their correlation is zero. Note also that the noise contributions at different frequencies are independent. The properties of the spectral measurement noise after a DFT also perfectly match these conditions. Hence, over different realizations of the random phase multisine, cannot be distinguished from the measurement noise . The sum

(4) is called the BLA to the nonlinear system for the class of Gaussian excitation signals (normally distributed noise, periodic Gaussian noise, and random phase multisines) with a given power spectrum (see [10]–[12] for the details). It can be

S f T

M P

T

Fig. 1. FRF ( ) measurement procedure: Apply different random phase realizations of the excitation, and measure each time periods of length after . a waiting time

approximated arbitrarily well by a rational form in the Laplace variable . Since the even degree nonlinear distortions do not affect the bias term while they increase the variance of , the variability of the FRF measurement (2) can be reduced by using random phase multisines, which excite the odd harmonics only [8], [9], i.e., (1) with . These so-called odd random phase multisines allow the detection of the presence and level of even degree nonlinear distortions by looking at the even harmonics in the output spectrum. To detect the presence and level of the odd degree nonlinear distortions, one should leave out some of the odd harmonics in the odd random phase multisine. The optimal strategy consists in splitting the odd harmonics in groups of equal number of consecutive lines, and eliminating randomly one line out of each group. This can be done for a linear, as well as a logarithmic frequency distribution (see [8] for the linear case). The resulting excitation, an odd random phase multisine with a random harmonic grid (linear or logarithmic frequency distribution), will be used throughout this paper. Two measurement strategies are proposed in the sequel of this section (see Fig. 1): the first uses one phase realization of the odd random phase multisine with a random harmonic grid, while the second uses multiple phase realizations (each time with the same random harmonic grid). B. First Measurement Strategy: Fast Method The first strategy for measuring the response , as defined in (2), and its uncertainty consists of the following steps. Step 1) Choose the amplitude spectrum and the frequency resolution of the odd random phase multisine (1) . This sets the period of the signal to a value of s. Step 2) Split the excited odd harmonics (linear or logarithmic frequency distribution) in groups of equal number of consecutive lines, and eliminate randomly one odd harmonic out of each group (e.g., 100 excited odd harmonics are split in 25 groups of four consecutive excited odd harmonics, and one out of the four odd harmonics is randomly eliminated in each group).

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subtracting the linear contribution of the DUT from the output at the nonexcited harmonics

excited harmonic in nonexcited harmonic in (6) Fig. 2. Basic scheme for measuring the characteristics of an amplifier using the LSNA.

Step 3) Make a random choice of the phases of the nonzero harmonics of the random phase multisine (1), and calculate the corresponding time signal . Step 4) Apply the excitation to the circuit (see Fig. 2) and measure consecutive periods1 of the steady-state input and output (see Fig. 1, one horizontal line). From the noisy input/output wave spectra , one can calculate the average -parameter and its sample variance

at the where, according to the frequency resolution, nonexcited frequencies is obtained through a linear or cubic interpolation of the values at the excited frequencies. Further, the sample mean and sample variance of the corrected output spectrum

(7) can be calculated. Within the measurement uncertainty , the presence and the level of the even and nonexcited odd harmonics in reveals the presence and the level of, respectively, the even and odd degree nonlinear distortions at the nonexcited frequencies [8], [9]. What about the nonlinear distortions at the excited odd harmonics in ? For odd random phase multisines with random harmonic grid (linear or logarithmic distribution), the level of the stochastic nonlinear contributions at the excited odd harmonics is obtained by a linear or cubic interpolation of the level at the nonexcited odd harmonics [8]. The bias contribution at the excited odd harmonics is bounded by (8)

with (5) is calculated over consecutive Since the sample variance periods of one particular realization of the random phase multisine, it is clear that it only contains the contribution of the measurement noise to the complex gain measurement . The presence and the level of the odd and even degree nonlinear distortions is revealed by analyzing the nonexcited frequencies [i.e., missing harmonics in ] in the output spec. trum Straightforward interpretation of the output spectrum is, however, sometimes impossible. Indeed, due to the possible presence of input impedance mismatch and reverse gain, a feedback loop in the set up for measuring is created: the input is also contaminated by the nonlinear wave spectrum distortions of the DUT by source pulling and, hence, the energy present at nonexcited frequencies in is partially due to the linear feed through of the distorted inputs. Since the DUT is dominantly linear, a first-order correction is obtained by 1At least six periods are needed to preserve the properties of the maximum likelihood estimator used in the parametric modeling step [10].

where is a heuristic factor depending on the power spectrum of the excitation and the system (see [8] and [9] for the linear frequency distribution). Typical values for lie between 2–10. Finally, calculating

and (9) at the excited odd harmonics gives the level of the stochastic and the bias contributions on the gain measurement . C. Second Measurement Strategy: The Robust Method The first three steps of the second strategy for measuring the amplifier characteristics are identical to the first strategy. In addition to Steps 1)–3) of Section III-B, we have the following steps. Step 5) Apply the excitation to the circuit in Fig. 2 and measure -parameter functions from consecutive periods of the steady-state response.2 2At

least two periods are needed to calculate a sample variance.

ROLAIN et al.: EXPERIMENTAL CHARACTERIZATION OF NONLINEAR BEHAVIOR OF RF AMPLIFIERS

Step 6) Repeat Steps 3) and 5) times3 (see Fig. 1, the horizontal lines). noisy FRFs and From the , one can calculate for each experiment the avand its sample variance erage -parameter

(10) An additional averaging over gives the final average rameter of the whole measurement procedure

-pa-

(11) . together with its sample variance From (2), (10), and (11), it follows that

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single experiment) and 2) the classification in odd and even degree nonlinear distortions. Its disadvantages are: 1) the variance of the -parameter measurement only accounts for the measurement noise and 2) an approximation (extrapolation) is needed to characterize the nonlinear stochastic contributions on the -parameter measurement. The advantages of the second measurement strategy are: 1) the contribution of the stochastic nonlinear distortions to the -parameter measurement are obtained without any approximation (extrapolation) and 2) the variance of the -parameter measurement accounts for the measurement noise, as well as the stochastic nonlinear distortions. Its disadvantages are: 1) the increased measurement time (several experiments are needed) and 2) no classification in odd and even degree nonlinear distortions can be made since, in this case, a general multisine without spectral gaps can equally well be used. If the input signal-to-noise ratio is smaller than 6 dB, then the relative bias on the -parameter measurement can no longer be neglected [10], [13]. It can be reduced by appropriate averaging noisy input/output spectra before calculating the of the -parameter. At the cost of exactly knowing a reference signal (typically the signal stored in the arbitrary waveform generator), the second measurement strategy can be generalized to handle input/output spectra (see [14] for the details).

and (12) , then should be Hence, if the system is linear divided by . approximately equal to the mean value of This quantity is labeled

, and is defined by

(13) The extra factor in (13) accounts for the fact that the variance of the mean value of the -parameter is calculated. If in in (13), then this is an indication that (11) is larger than , and the systems behaves nonlinearly

(14) otherwise is an estimate of linear bias contribution

. Using (8), (9), and (14), the nonin (2) can be bounded by

(15) where

is defined as in (8) (see [8] and [9] for the details).

D. Discussion The advantages of the first measurement strategy are: 1) the reduced measurement time (all information is gathered in one 3See

footnote 1.

E. Comparison With Gaussian Noise Excitations The concept of a BLA of a nonlinear system excited by Gaussian noise had been developed by Wiener in 1942 (see, e.g., [15]). It has formally been shown that the BLA (in least squares sense) for Gaussian excitations equals the ratio of the input–output crosspower spectrum by the input autopower spectrum [15], [16]. These results have for a long time been used in mechanics and acoustics [16], and recently also for the analysis of nonlinear power amplifiers in CDMA and OFDM systems [17], [18]. The latter is justified by the fact that CDMA and OFDM signals behave as band-limited Gaussian noise [3], [4]. The link between the BLA for Gaussian (like) signals and random phase multisines has recently been established in [10], [11], and Appendix A. Based on the fact that, for periodic signals, the division of crosspower by the autopower equals the ratio of the output and input spectra [10], it has formally been proven that the BLA [see (5) or (11)] obtained using random phase multisine excitations(1) is asymptotically (as the number of frequencies going to infinity) the same as the BLA obtained by band-limited Gaussian noise excitations with power spectral density , where is the th Fourier coefficient of the multisine, [see (1)], and with for . What are now the advantages and disadvantages of both approaches? The variance of the BLA obtained using Gaussian noise is calculated via the coherence function and contains the contribution of both the measurement noise and the nonlinear distortions [16]. Hence, no distinction can be made between both contributions. Moreover, since Gaussian noise is a random process, averaging is needed to guarantee a minimal excitation power at each frequency [10]. This is not the case for random

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phase multisines; hence, the minimal signal-to-total-distortion ratio with Gaussian noise excitations will be smaller than that with random phase multisines. At the price of requiring skilled users, the multisine approach allows one to distinguish between measurement noise and nonlinear distortions, and to classify the nonlinear distortions in odd and even contributions. IV. MEASUREMENT RESULTS A. Introduction The whole measurement/nonparametric modeling procedure is illustrated in the characterization of a global system for mobile communication (GSM)-band amplifier, i.e., the MRFIC 2006. The device is excited in a band ranging from 700 to 1100 MHz. The grid spacing of the multisine is chosen to be 500 kHz. The spacing between odd harmonics hence becomes 1 MHz. A nonparametric BLA of the device is then determined using the two methods proposed earlier. The whole process is repeated at different power levels; the results at a medium and high level are shown below. In this particular measurement, it was decided to characterize the device in a 50- reference impedance only. B. Instrumentation Setup To perform a BLA measurement, the instrumentation setup is required to be able to measure the undistorted time-domain waveforms at both ports of the device and, of course, needs to be able to generate a multisine, as described above. For the measurement of the waves, an LSNA is used. This device is fully calibrated to measure undistorted multiharmonic periodic wave signals. In this case, it measures the four waves at the ports of the DUT simultaneously. The multisine is generated by a Tektronix 4-Gsample arbitrary waveform generator, i.e., the AWG710. This generator has an analog bandwidth of 2 GHz, and can be used without an upconverter to generate the signal. The setup that was used is shown in Fig. 2. The LSNA is set to measure all the frequency lines between 600–1200 MHz with a frequency resolution of 500 kHz. This allows the simultaneous measurement of the in-band response and the adjacent channel distortion of the device. Since the DUT is used in a large-signal context, it is not possible to assume that the port match is perfect. When used in the forward mode, the device will, therefore, be excited by an incident wave at both ports resulting in

(16) The LSNA measures the three waves in this equation in each experiment. Since it is impossible to extract both and using only one equation, a second experiment will be required. As the DUT is a nonlinear device, it is not possible to swap the source between ports in general to get a second measurement. However, since the dominant nonlinearity is expected to be present in the forward gain path, a good first-order approximation of the output reflection can be obtained from a smallsignal measurement performed at port 2. This is then used to compensate the measurements of the output wave .

C. Generation of the Excitation Signal A multisine with a flat power spectrum is then constructed on a random odd grid. The frequency of the first line is chosen to be 700.5 MHz, and the successive odd harmonics are then spaced 1 MHz apart, up to 1099.5 MHz. To generate the random grid, the lines are grouped by three, and one randomly selected line is removed from each group. This results in 267 excited lines in the spectrum. This spectral grid defines the random multisine that is then fixed for the rest of the characterization of the system. To obtain a realization of the signal, the phases of the spectral lines are then randomly assigned using a uniform stochastic distribution in the interval . D. Data Collection For the experiments below, 20 realizations of the phase result in 20 signals that are, in turn, used to excite the device. Each signal is then measured during five successive periods . For the fast method, only one experiment is used for data processing. For the robust method, all the experiments are used simultaneously. The data are then first calibrated using a classical -parameter calibration. In this case, a simple SOLT calibration is performed. For the measurement of the BLA only, this is sufficient to ensure the integrity of the measurements. To obtain undistorted measurements of the power spectra, both at the excited lines, and the nonexcited lines, an additional power calibration is required. Next, if the phase spectrum of the waves is to be used, the calibration is to be extended with a phase calibration. This is not needed here, as the phase of spectral lines is only used for lines having the same frequency. The whole calibration process is explained in detail in [7]. E. Processing the Data With Both Methods As five successive periods of the random grid signal are used, it becomes possible to obtain the measured spectrum and its standard deviation at the excited lines and the nonlinear random contributions and their uncertainty at the detection lines in the measured spectrum. The measured wave spectra at a low excitation power level for one realization of the input phases are shown Fig. 3 (lefthand-side column of the plots). The input spectrum that is measured at the lines where an excitation signal has been applied is indeed almost flat, as was imposed by the signal design. The standard deviation on the measurements (full line) is approximately 30 dB lower than the excitation. This corresponds to a 63% confidence bound on the measurements. As approximately 57% of the distortion measurements lie within this bound, it can be concluded that the input spectrum is not significantly distorted. The output spectrum is shaped more with frequency, which shows the influence of the linearized dynamics of the system under test. The standard deviation of the spectrum is again approximately 30 dB down. The nonlinear contributions are still all down to the level of the standard deviation of the measured spectrum. This clearly indicates that the device is operating in a linear regime. Note that a direct measurement of the presence or absence, the level, and the possible shaping of the distortion in the passband of the device, is hereby

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Fig. 4. Compensated output wave spectrum (left-hand-side plot for a low input power and right-hand-side plot for a high input power). is the amplitude of the measured spectrum, full black line is the standard deviation on the spectrum, are the even nonlinearities and o are the odd nonlinearities.

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Fig. 3. (top) Measured input and (bottom) output wave spectra. Left-hand-side column is taken at a low input power and right column at a high input power. is the amplitude of the measured spectrum, full black line is the standard deviation on the spectrum, are the even nonlinearities, and o are the odd nonlinearities.

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obtained. Remember that all these values are directly measured using a single experiment! To show that the BLA is indeed dependent on the input power of the excitation signal, the previous experiment is repeated under identical experimental conditions, except for the input power level, which has now been increased by 20 dB. A striking difference in the right-hand-side column of Fig. 3 (high input power level), when compared to the left-hand-side column (low input power level) is that this time a significant distortion is present in the output wave. This is easily detected on the spectral plot, where the spectral contributions at the odd (circles) detection lines in are significantly larger than the standard deviation on this value (full black line under the ). As the even ( ) nonlinear contributions remain at the level of the measurement noise, this indicates that the nonlinearity is dominated by odd contributions. These contributions are now directly measured. A smooth transition is visible between the in-band and adjacent channel contributions, which are also directly measured. Based on the measurement of the input wave , there is no evidence of the presence of a significant distortion in the input wave : the detection lines all remain at the level of the measurement noise. Next, the distortion in the output spectrum is compensated for the presence of potential linear contributions coming from the input signal , as was discussed earlier. The result of the compensation is shown in the plots of Fig. 4. It could be argued that the compensation is not required since there is only a noise contribution at the detection lines of the input in this particular case. However, it is always sensible to perform the compensation, as this will allow one to properly take the input noise contribution into account in the standard deviation of the -parameter. Comparing the corrected output spectra between the plots in Fig. 4, the contribution in the nonlinearity to the noise-like character of the spectrum becomes evident: the excited lines show a more “noisy” behavior, but the standard deviation of the

Fig. 5. Measured S (left-hand-side plot for a low input power and right-handside plot for a high input power level), based on the fast method (top black line), measured experimental standard deviation (bottom light gray line), and measured total standard deviation (nonlinear measurement, central black line).

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measurement noise remains almost unchanged. The total standard deviation, on the other hand, increases by almost 15 dB, and clearly points towards the nonlinearity as the main source of variability in the device output. The fast method is then used to extract the nonparametric BLA for the forward gain of the device, and the results are shown in Fig. 5. The frequency rolloff of the forward gain is clearly visible in the measurement, as could be expected for a device that is optimized for the 900-MHz GSM band. For linear systems, it is common sense to assume that a noisy corresponds to a low excitation level, while here the opposite is true because of the presence of the stochastic contributions of the nonlinearity. On top of the linearized , an estimate of the standard deviation of the measurement noise is obtained at all the spectral lines using the sample variance taken over the five measured periods. An estimate of the nonlinear noise contribution is directly measured in the odd detection lines and is then extrapolated using the relation described in (8). Again, the nonlinear contribution to the standard deviation is much (15 dB) higher than the noise contribution for the high-power level. For the low-power level, no significant nonlinear contribution is detected (the standard deviation with and without the nonlinear contribution fall on top of each other). The measured phase of the BLA is shown in Fig. 6. Note that the phase is slightly more noisy for the high power experiment, also as a result of the presence of the nonlinear stochastic contributions. The robust method is used next. Remember that this time the 20 realizations of the random multisine are processed together, and the results of the left-hand-side plot of Fig. 7 display the behavior of the mean value. The BLA is very similar to the previous one, as was expected from a theoretical

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Fig. 6. Measured phase of S based on the robust method.

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for: (left) low and (right) high input power,

Fig. 9. Measured and modeled output spectrum (top lines), predicted 95% model uncertainty bound (bottom black line), together with the rms error between modeled and measured output, all obtained for four new realizations of the input signal at the high-power level (symbols).

Fig. 7. Measured S for: (left) low and (right) high input power levels, based on the robust method (top black line), measured experimental standard deviation (bottom light gray line), and measured total standard deviation (nonlinear measurement, central gray line).

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that the compression level of the device is indeed much higher. Note also that the measurement noise level has gone down while the total noise level has significantly increased. This can be explained by the increased nonlinear system contribution and the increased signal-to-noise ratio of the input spectrum, a consequence of the increased excitation signal power. V. VERIFICATION OF THE BLA

Fig. 8. Measured S for: (left) low and (right) high input power levels, based on both methods (top black line), measured experimental standard deviation of one experiment (fast: bottom line, robust: bottom crosses), and measured total standard deviation on one experiment (fast: central line, robust: central crosses).

point-of-view. Note that both the experimental and total standard deviation are lower than in the previous example. As these quantities are based on the mean value of the forward gain, it is expected that the noise levels would be smaller than with one single experiment. To cross-verify the results obtained by both approaches, the equivalent noise level on one experiment is calculated starting from the noise on the mean value using the law. These results are then plotted together with the results obtained by the fast method in Fig. 8. Note that the forward gain values, as obtained by both methods, fall on top of each other. The standard deviation of the measurement noise and total noise (including the contribution of the nonlinearity) for the robust method are denoted by a full line, while the values obtained by the fast method are indicated by crosses. Again, both set of values are almost identical. This validates the assumptions that were made in the fast method to allow the extrapolation of the nonlinear contributions from the odd detection lines to the odd excited lines. Note that the -parameter has gone down by approximately 4 dB over the band for the high power experiment, showing

Theoretically speaking, the BLA that was measured before is the BLA of the system under test when excited by Gaussian noise signals with a fixed power spectrum. During the measurement, a small number (20) of realizations of a random phase multisine have been used as a representative subset of this class. To prove that the BLA indeed captures the behavior of the system when excited with other members of the signal class, additional experiments were performed. The most extensive test is to use the nonparametric BLA and its confidence regions to predict the response to a real Gaussian noise signal, whose power spectral density matches the power spectrum of the random multisine, and then compare it to the measured response. This experiment cannot be performed using the LSNA, however, as harmonic mixing measurements require the use of periodic inputs. The BLA model is used to predict the output of four other multisine realizations instead. The upper traces in Fig. 9 show the measured and modeled output spectra that are obtained for these new signals. The norm of the complex error between the four measured and modeled outputs is represented by the gray symbols in Fig. 9, while the 95% confidence interval of the modeled BLA output is given by the full black line in the middle. Clearly, the majority of the residuals falls inside the confidence region. The experimental error falls outside the confidence region in 3% of the measured data points. Since only four realizations are taken into account, this shows that the model adequately predicts the system output for input signals that are members of the input signal class. To show the dependence of the BLA on the input power level, the power level of the high-power experiment is reduced by 3 dB, and the model for the higher power is used to predict the output of the system at the lower power level. The difference between the model and measurement now becomes large, and the spectral plot does not give much insight in the behavior of

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input and output spectra of consecutive periods and, hence, . The latter allows one to simplify (17) as

(18)

Fig. 10. BLA is shown for the high-power level (P ) as used before and a new experiment where the input power P is lowered by 3 dB. The magnitude of the BLA is shown on the left, while the phase difference (in degrees) of the BLAs is shown on the right.

which proves (5). Since for different realizations of the random phase multisine (1), exactly the same reasoning holds for (11). ACKNOWLEDGMENT

the error. If a BLA is also measured for the lower power level, it becomes easy to show in what sense the behavior of the system is influenced. Fig. 10 shows the magnitude of the BLA for both power levels. As was expected, the compression level increases for higher input power levels. The gain of the BLA is decreased by approximately 1 dB over the whole frequency band, while its phase difference between the two BLAs remains almost perfectly zero (right-hand-side plot). This means that the device exhibits an almost perfect static nonlinear behavior. It also shows the high dependence of the BLA on the power level of the measured signal when the device is operated under high compression levels. VI. CONCLUSION A method to measure the BLA of an RF amplifier has been proposed and verified experimentally. Using a single experiment, it is possible to measure the linearized frequency dynamics of the system, and the in-band and adjacent band nonlinear distortions for a signal with a fixed power spectrum. When experiments are repeated at different power levels, the dynamic compression characteristic of the device can be measured. This results in a very simple measurement-based way to obtain black-box models for systems that operate under a given class of random-like excitation signals.

APPENDIX FRF MEASUREMENT USING PERIODIC EXCITATIONS Assuming that the output measurements are disturbed by noise and that the (arbitrary) input is known exactly, the optimal (in least squares sense) FRF estimate is given by the ratio of the crosspower spectrum by the autopower spectrum [16]. For example, the FRF estimate of equals

(17)

where denotes the power spectrum estimated from meaand reflected waves, surements of the incident . If the excitation is periodic, then and are the

The research reported in this paper was performed in the context of the Top Amplifier Research Groups within a European Team (TARGET) network. The authors wish to thank the anonymous reviewers for their constructive comments, and want to express their special gratitude to the reviewer who did a wonderful job in improving the language. REFERENCES [1] J. F. Barrett and J. L. Coales, “An introduction to the analysis of nonlinear control systems with random inputs,” Inst. Elect. Eng. Monograph, no. 154, pp. 190–199, Nov. 1955. [2] J. L. Douce, “A note on the evaluation of the response of a non-linear element to sinusoidal and random signals,” Inst. Elect. Eng. Monograph, no. 257, pp. 1–5, Oct. 1955. [3] R. Dinis and A. Palhau, “A class of signal-processing schemes for reducing the envelope fluctuations of CDMA signals,” IEEE Trans. Commun., vol. 53, no. 5, pp. 882–889, May 2005. [4] P. Banelli, “Theoretical analysis and performance of OFDM signals in nonlinear fading channels,” IEEE Trans. Wireless Commun., vol. 2, no. 2, pp. 284–293, Mar. 2003. [5] J. Boutros and E. Viterbo, “Signal space diversity: A power- and bandwidth-efficient diversity technique for the Rayleigh fading channel,” IEEE Trans. Inf. Theory, vol. 44, no. 4, pp. 1453–1467, Apr. 1998. [6] J. Schoukens and T. Dobrowiecki, “Design of broadband excitation signals with a user imposed power spectrum and amplitude distribution,” in IEEE Instrum. Meas. Technol. Conf., 1998, pp. 1002–1005. [7] W. Van Moer and Y. Rolain, “A large signal network analyzer. What to do with it?,” IEEE Trans. Microw. Theory Tech., submitted for publication. [8] K. Vanhoenacker, T. Dobrowiecki, and J. Schoukens, “Design of multisine excitations to characterize the nonlinear distortions during FRF-measurements,” IEEE Trans. Instrum. Meas., vol. 50, no. 5, pp. 1097–1102, May 2001. [9] J. Schoukens, R. Pintelon, and T. Dobrowiecki, “Linear modeling in the presence of nonlinear distortions,” IEEE Trans. Instrum. Meas., vol. 51, no. 4, pp. 786–792, Apr. 2002. [10] R. Pintelon and J. Schoukens, System Identification: A Frequency Domain Approach. Piscataway, NJ: IEEE Press, 2001. [11] J. Schoukens, T. Dobrowiecki, and R. Pintelon, “Parametric identification of linear systems in the presence of nonlinear distortions. A frequency domain approach,” IEEE Trans. Autom. Contr., vol. 43, no. 2, pp. 176–190, Feb. 1998. [12] R. Pintelon and J. Schoukens, “Measurement and modeling of linear systems in the presence of non-linear distortions,” Mech. Syst. Signal Process., vol. 16, no. 5, pp. 785–801, 2002. [13] ——, “Measurement of frequency response functions using periodic excitations, corrupted by correlated input/output errors,” IEEE Trans. Instrum. Meas., vol. 50, no. 6, pp. 1753–1760, Jun. 2001. [14] R. Pintelon, P. Guillaume, S. Vanlanduit, K. De Belder, and Y. Rolain, “Identification of Young’s modulus from broadband modal analysis experiments,” Mech. Syst. Signal Process., vol. 18, no. 4, pp. 699–726, 2003. [15] M. Schetzen, The Volterra and Wiener Theories of Nonlinear Systems.. New York: Wiley, 1980. [16] J. S. Bendat and A. G. Piersol, Engineering Applications of Correlations and Spectral Analysis. New York: Wiley, 1980.

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[17] A. R. S. Bahai, M. Singh, A. J. Goldsmith, and B. R. Saltzberg, “A new approach for evaluating clipping distortion in multicarrier systems,” IEEE J. Sel. Areas Commun., vol. 20, no. 5, pp. 1037–1046, May 2002. [18] N. Ermolova and P. Vainikainen, “Analysis of nonlinear amplifiers with Gaussian input signals on the basis of complex gain measurements,” Eur. Trans. Telecommun., vol. 15, no. 5, pp. 501–505, 2004. [19] R. Pintelon, J. Schoukens, W. Van Moer, and Y. Rolain, “Identification of linear systems in the presence of nonlinear distortions,” IEEE Trans. Instrum. Meas., vol. 50, no. 4, pp. 855–863, Apr. 2001. [20] J. Schoukens, Y. Rolain, and R. Pintelon, “Modified AIC rule for model selection in combination with prior estimated noise models,” Automatica, vol. 38, no. 5, pp. 903–906, 2002. [21] I. Kollár, J. Schoukens, R. Pintelon, G. Simon, and G. Román, “Extension for the frequency domain system identification toolbox for MATLAB: Graphical user interface, objects, improved numerical stability,” in Proc. 12th IFAC Syst. Identification Symp., Santa Barbara, CA, Jun. 21–23, 2000, vol. 2, pp. 699–702.

Yves Rolain (M’90–SM’00–F’06) is currently with the Electrical Measurement Department (ELEC) Vrije Universiteit Brussel (VUB), Brussels, Belgium. His main research interests are nonlinear microwave measurement techniques, applied digital signal processing, parameter estimation/system identification, and biological agriculture. Dr. Rolain was the recipient of the 2004 IEEE Instrumentation and Measurement Society Award.

Wendy Van Moer (M’97) received the Engineer degree in telecommunication and Doctor degree in applied sciences from the Vrije Universiteit Brussel (VUB), Brussels, Belgium, in 1997 and 2001, respectively. She is currently a Post-Doctoral Researcher with the Electrical Measurement Department (ELEC), VUB. Her main research interests are nonlinear microwave measurement and modeling techniques.

Rik Pintelon (M’90–SM’96–F’98) was born in Gent, Belgium, on December 4, 1959. He received the Electrical Engineer (burgerlijk ingenieur) degree, Doctor degree in applied sciences, and the qualification to teach at the university level (geaggregeerde voor het hoger onderwijs) from the Vrije Universiteit Brussel (VUB), Brussels, Belgium, in 1982, 1988, and 1994, respectively. From October 1982 to September 2000, he was a Researcher with the Fund for Scientific Research–Flanders, VUB. Since October 2000, he has been a Professor with the Electrical Measurement Department (ELEC), VUB. His main research interests are in the field of parameter estimation/system identification, and signal processing.

Johan Schoukens (M’90–SM’95–F’97) received the Engineer degree and Doctor degree in applied sciences from the Vrije Universiteit Brussel (VUB), Brussels, Belgium, in 1980 and 1985, respectively. He is currently a Professor with the VUB. His research interests are in the field of system identification for linear and nonlinear systems. Dr. Schoukens was the recipient of the 2002 IEEE Instrumentation and Measurement society Best Paper Award and the 2003 Distinguished Service Award.

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Time-Domain Envelope Measurements for Characterization and Behavioral Modeling of Nonlinear Devices With Memory François Macraigne, Tibault Reveyrand, Guillaume Neveux, Denis Barataud, Jean-Michel Nebus, Arnaud Soury, and Edouard NGoya

Abstract—This paper presents a calibrated four-channel measurement system for the characterization of nonlinear RF devices such as power amplifiers. The main goal of this study is to perform the characterization of the bandpass response of a nonlinear device-under-test (DUT) driven by modulated carriers. The proposed setup enables the generation of - or -band (1–4 GHz) carriers with a modulation bandwidth up to 100 MHz. The carrier harmonics generated by the nonlinear DUT are ignored and considered to be sufficiently filtered. This characterization setup enables calibrated time-domain measurements of the complex envelopes of both incoming and outgoing RF waves at the input and output of the DUT. This means that the fundamental and harmonic frequencies of the envelope are measured and processed. A large set of modulation formats can be generated by using a computer-controlled arbitrary waveform generator. Complex envelopes are measured by using a four-channel sampling scope. The proposed calibrated setup can be used to study or to validate linearization techniques of power amplifiers. This characterization tool is also well suited for the extraction and validation of behavioral bilateral models of nonlinear RF analog equipment exhibiting memory effects. Index Terms—Behavioral third-order intermodulation measurements.

model, calibration, linearity, (IM3), time-domain envelope

I. INTRODUCTION NE METHOD for accurate characterization of nonlinear RF and microwave devices involves use of time-domain measurements. Therefore, absolute magnitude and phase of the spectral components of the signal at the input and output ports of the device-under-test (DUT) must be measured. As a consequence, measurements performed using a vector network analyzer or spectrum analyzer do not provide enough information.

O

Manuscript received December 15, 2005; revised May 10, 2006. This work was supported in part by the National French Space Agency under Contract 714/CNES/00/8167-IRCOM. F. Macraigne is with the Institut de Recherche en Communications Optiques et Microondes, University of Limoges, Unité Mixte de Recherche, 87060 Limoges Cedex, France, and also with Anritsu SA9, 91951 Les Ulis Cedex, France (e-mail: [email protected]). T. Reveyrand, G. Neveux, D. Barataud, J.-M. Nebus, and E. NGoya are with the Institut de Recherche en Communications Optiques et Microondes, University of Limoges, Unité Mixte de Recherche, 87060 Limoges Cedex, France, and also with the Faculté des Sciences et Techniques, XLIM, Unité Mixte de Recherche, Centre National de la Recherche Scientifique, 87060 Limoges Cedex, France (e-mail: [email protected]; [email protected]; [email protected]; [email protected]; [email protected]). A. Soury is with Xpedion Design Systems-24, 87000 Limoges, France (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2006.879169

A microwave transient analyzer (MTA) and large signal network analyzer (LSNA) enable the extraction of time-domain waveforms at the input and output of nonlinear devices. To get accurate time-domain waveform measurements, the following three main calibration procedures of the LSNA or MTA are required: • Short open load thru (SOLT), thru-reflect line (TRL), or line-reflect-reflect-match (LRRM) calibration for the correction of wave amplitudes ratios; • absolute calibration in magnitude by using a power meter; • absolute phase calibration by using a phase reference generator (this generator is a comb generator with well-known phase relationships between harmonic components of a continuous wave (CW) RF carrier [1]–[3]). Time-domain measurement principles and techniques are now well established for CW carriers and their harmonics [4]–[6]. However, accurate measurements of nonlinear devices driven by wideband modulated carriers are required to build behavioral models because modulated signals are necessary to probe memory effects in nonlinear solid-state RF devices [7]–[9]. Basically, using the current configuration of the LSNA [10], which can be considered as a reference instrument for calibrated measurements of time-domain waveforms, the modulation bandwidth of RF signals that can be measured is limited to 10 MHz [10]. This is due to the fact that the harmonic sub-sampling performed within the LSNA operates at a frequency of 25 MHz. Some solutions have been already proposed to perform wider bandwidth measurements, but are still under investigation [11]–[13]. Furthermore, the phase calibration of the LSNA for modulated carrier measurements remains a very difficult task because, to our knowledge, there is not, at the moment, any available solution to provide a wideband multitone phase calibrated reference source. This topic is discussed in [14] and [15]. Previously, a calibrated two-channel time-domain measurement system has been developed for the envelope characterization and behavioral modeling of nonlinear 50- amplifiers [16]. In this paper, we propose a calibrated four-channel time-domain measurement system that enables the measurement of the envelope response of mismatched nonlinear RF devices driven by modulated carriers having a modulation bandwidth as high as 100 MHz. In its current configuration, the setup operates at and -bands (1–4 GHz). In this paper, we will focus on the bandpass response of the DUT dedicated to the extraction of behavioral models, taking into account memory effects. The purpose of such models is to be implemented in system-level simulation

0018-9480/$20.00 © 2006 IEEE

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Fig. 2. Flowgraph used for the receiver section calibration procedure. (Color version available online at: http://ieeexplore.ieee.org.)

Fig. 1. Time-domain envelope measurement setup. (Color version available online at: http://ieeexplore.ieee.org.)

software like MATLAB/Simulink or Agilent/Ptolemy. Therefore, we will ignore the carrier harmonics and consider that they will be sufficiently filtered out. For that purpose, we have built a four-channel time-domain envelope measurement setup. This setup enables the characterization of the DUT in-band nonlinearities, as well as memory effects and impedance mismatches. The setup is based on the use of a 2-Gigasamples per second (GS/s)–8-bit digital sampling scope (DSO) that directly samples the complex envelopes of the incoming and outcoming RF fundamental carriers at the ports of the DUT. As mentioned in Section V, more powerful sampling scopes can be used without significantly modifying the principle of our study. The measurement setup is presented in Section II. The calibration procedure is explained in Section III. In Section IV-A, the application of the setup to the study of a possible linearization technique of a power transistor is shown. Finally, the application of our characterization tool to the behavioral modeling of a power amplifier is presented in Section IV-B. II. DESCRIPTION OF THE MEASUREMENT SETUP The block diagram of the measurement setup is given in Fig. 1. Modulation schemes are achieved by using a computer-controlled arbitrary waveform generator (AWG) (12 bit–250 MS/s). The AWG generates a modulated signal at an IF. This IF signal is then up-converted to - or -band by using a local oscillator (LO) and an in-phase/quadrature (I/Q) modulator. A specific arrangement, which consists of power dividers, a phase shifter, and a variable attenuator, is used to minimize the LO leakage at the RF output of the I/Q modulator. The microwave modulated signal is then amplified using a very linear amplifier. A bandpass filter is used to reject the signal present at the image frequency. A programmable step attenuator is used to enable power sweeps at the input of the DUT. The four measurement channels are built with two bi-directional couplers connected at the input and output of the DUT, calibrated step attenuators, isolators, mixers and a four-channel DSO (2 GS/s, 8 bit). A tuner is also used to make load–pull measurements. This setup must be calibrated in order to get error corrected time-

Fig. 3. Error coefficient matrix. (Color version available online at: http://ieeexplore.ieee.org.)

domain envelopes of the incident and reflected RF modulated signals at the DUT reference planes III. DESCRIPTION OF THE CALIBRATION PROCEDURE A. Receiver Section Calibration Procedure The calibration procedure of the receiver section of the setup consists of determining the frequency response of the four measurement channels at the envelope frequencies of the modulated signal that will be used for the DUT characterization. The flow-graph representation linking measured wave amplitudes and wave amplitudes at the DUT reference planes is given in Fig. 2. The associated error coefficient matrix is given in Fig. 3 [17]. In a first step, all the terms of this matrix, except the multiplying term , are determined using a SOLT, TRL, or LRRM calibration procedure. For that, a CW source is connected to the setup, as shown in Fig. 4. Calibration standards are connected to the DUT reference planes. The frequency of the source is swept over the bandwidth of interest (Basically, a frequency grid covering a 100-MHz bandwidth above the LO frequency). Error terms are extracted from the measurements of amplitude waves ratios made by the DSO at the IF: . In a second step, an absolute calibration in magnitude is performed to determine the magnitude of . For that purpose, a power meter is connected to the input reference plane of the DUT, as shown in Fig. 5. is determined by the following:

(1)

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Fig. 4. SOLT, TRL, or LRRM calibration procedure. (Color version available online at: http://ieeexplore.ieee.org.)

Fig. 6. Zoom of the unwrapped phase of the multisine measured at channel 1 without: (a) taking into account the time delay and (b) taking into account the time delay. (Color version available online at: http://ieeexplore.ieee.org.)

Fig. 5. Absolute power calibration procedure. (Color version available online at: http://ieeexplore.ieee.org.)

is the power measured by the power meter, while and are the waves measured by channels 1 and 2 of the DSO. In a third step, we would like to determine the phase of the error term . Unfortunately, we do not have a multitone RF reference generator with known phase relationships between tones. For that reason, we have to make the assumption that the group delay of the four measurement channels remains constant over the envelope frequency bandwidth of interest. We have validated this assumption by measuring on each channel of the setup a multitone phase aligned signal supplied by a Rhode & Schwarz generator (SMU 200 A). The signal supplied by this source contained 80 tones covering a 100-MHz bandwidth around a center frequency of 1.6 GHz. The phases of the tones were set to 0 . This source was connected in the place of the CW source, as depicted in Fig. 4. A thru connection was made between the reference planes of the DUT. A fast Fourier transform (FFT) was applied to the signal measured by the scope (respectively channels 1 and 3 and then channels 2 and 4). We then plotted the phase of the measured spectral lines versus frequency. We obtained a quasi-perfect line with a negative slope, as depicted in Fig. 6. This indicates that the group delay of the four channels is constant. We observed smooth phase ripples in

the order of three degrees, which gives an idea of the accuracy of the assumption. We also performed this measurement at a center frequency of 3 GHz and came to the same conclusion. By using our calibrated system, as depicted in Fig. 1, we have also measured the four -parameters of linear devices at - and -band. We have compared the -parameters measured by this setup with -parameters measured with a 360 B Anritsu vector network analyzer. The differences obtained are in the order of 3 for phase measurements and 0.1 dB for magnitude measurements B. Calibration of the Source Section of the System The complex envelope of the signal computed by the personal computer (PC) controlling the whole setup is not exactly the same as the one present in the input RF reference plane of the DUT. This is due to the linear distortion of the source section of the system (from the PC to the input RF reference plane). The chosen envelope model of the source section is represented by a nonideal I/Q modulator, as shown in Fig. 7. The goal of the calibration consists in determining the complex gains and linking the real and imaginary parts of the signal generated by the PC and the real and imaginary parts of the envelope of the RF signal present in the RF reference plane. The determination of these error terms is sketched in Fig. 7. A specific computed multitone excitation signal calculated by the PC is downloaded into the AWG. This specific signal, noted , is a band-limited multicarrier signal represented by the sum

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Fig. 8. Dynamic biasing modulation technique used for the linearization of a power transistor (illustrated here for a two-tone RF signal). (Color version available online at: http://ieeexplore.ieee.org.)

Fig. 7. Calibration of the source section. (Color version available online at: http://ieeexplore.ieee.org.)

The last step enables to characterize the phase of by applying at both and inputs of the I/Q modulator. is determined by the following relationship:

of a sufficient number of independent CW tones with the same magnitude and random relative phases [16]

(2) The random relative phases are chosen in order to minimize the peak to average ratio (PAR) of this stimulus ( dB typically). Using this low PAR signal improves the accuracy of the measurements performed with the scope. We used PAR minimization technique described in [18], but other techniques can be used. Three steps are then required to determine and . In a first step, the multitone signal is applied to the input of the I/Q modulator and no signal is applied to the input of the I/Q modulator. The RF envelope is determined from DSO measurements corresponding, respectively, to channels 1 and 2 via the error correction matrix of Fig. 3. The coefficient is then extracted at each frequency component of interest by calculating

(3) and are, respectively, the magnitude and phase where of each tone of the RF envelope at the DUT reference plane, and and are the magnitude and phase of the tones generated by the PC. In a second step, the same multitone signal is applied to the input of the I/Q modulator and no signal is applied to the input of the I/Q modulator. The RF envelope is determined from DSO measurements. The magnitude of the coefficient is then extracted at each frequency component of interest by calculating

(4)

(5) This source section calibration becomes of prime importance if one wants to couple the measurement setup with a system level simulator. In this calibration procedure, the dominant error comes from the 8-bit scope resolution. IV. SETUP CAPABILITIES A. Linearization of a Power Transistor The calibrated time-domain envelope measurement system has been used to validate a power transistor linearization technique [19]–[21]. It has been applied to a Fujitsu FLK10XM power transistor operating in class A ( V, mA, V) at 1.6 GHz. First, the load impedance of the transistor has been optimized for maximum output power at a CW carrier frequency of 1.6 GHz . The linearization technique tested here consists of applying to the drain bias circuit of the transistor the magnitude of the envelope of the modulated RF signal that is amplified by the transistor. Therefore, we perform a dynamic biasing technique at the drain port using a network that includes a 10- F capacitor and 1-mH inductor. In the case of a two-tone signal used for intermodulation measurements, the envelope is a sine wave with a frequency equal to the half of the tone spacing . The magnitude of this envelope is a positive rectified sine wave, as depicted in Fig. 8. This envelope signal used for drain modulation must be synchronized with the envelope of the modulated signal at the RF input of the transistor. It is achieved by using our setup because we use the two channels of the AWG. One is used to generate the baseband modulation format of the modulated input RF signal. The other one is used to generate the baseband signal for the dynamic drain bias modulation.

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Fig. 9. IM3 versus PAE and versus input power measured with and without dynamic bias modulation. (Color version available online at: http://ieeexplore. ieee.org.) Fig. 10. ACPR versus input power measured with and without dynamic bias modulation. (Color version available online at: http://ieeexplore.ieee.org.)

The principle of the linearity improvement is the following: when the instantaneous input power level of the modulated signal increases, the drain source voltage swing increases. Therefore, the drain source voltage can come close to the ohmic region (low ), which is a nonlinear region of the I/V characteristics. Thus, intermodulation performance decreases rapidly. By applying a drain bias modulation, the dc quiescent bias point can be pulled away from the ohmic region when the RF input power level increases, as sketched in Fig. 8. Therefore, the nonlinear ohmic region is not reached by the dynamic load line (as shown in Fig. 8), and the linearity of the transistor is enhanced. We have applied this linearization technique to two kinds of modulated signals: the first one is a two-tone signal with a tone spacing of 1 MHz. In this case, the signal used for drain bias modulation is a positive rectified sine wave having a frequency of 0.5 MHz. The magnitude and phase of the envelope signal generated for the drain bias modulation are tuned in order to minimize the third-order intermodulation (IM3) at the RF output of the transistor. This tuning is achieved for each input power level driving the device. Fig. 9 clearly shows the improvements of the linearity performance obtained with the dynamic bias modulation technique. By using the dynamic bias modulation, the IM3 product increases from 21 to 29 dBc (improvement of 8 dB) at an input power of 12 dBm. The dynamic bias modulation improves the power-added efficiency (PAE) from 14% to 31% for a constant equal to 31 dBc. This PAE is calculated without taking into account the consumption of the AWG to generate the drain bias modulation. The second kind of modulated signals applied to the transistor is a 3.84-Mbit/s quadrature phase-shift keying (QPSK) RF modulated signal. In this case, the signal used for drain-bias modulation has the shape of the magnitude of the baseband modulation format. The magnitude and phase of the envelope signal generated for the drain-bias modulation are tuned in order to minimize the adjacent channel power ratio (ACPR) criterion at the RF output of the transistor. This tuning is also achieved for each input power level driving the device. Fig. 10 shows the ACPR improvement obtained when the input RF signal is a QPSK 3.84-Mbit/s modulated signal. By using the dynamic bias modulation, the ACPR increases from 34 to 44 dBc at an input power of 12 dBm.

B. “Black Box” Modeling Application An interesting point is that this bench provides information about the complex envelope of the four wave amplitudes present at the access planes of the devices. To exploit this information, we build a black-box model of the device accounted for both memory effects and input/output impedance mismatches. This topic has received increasing attention from the international community since these problems become of prime importance in the design of new wireless communication applications. Among the interesting studies, we can quote the introduction of the “scattering functions” [22]–[25]. Recently, the introduction of the notion of the large-signal -parameters (LSSPs) enabled understanding of these problems in rigorous and powerful circuit envelope formalism [7]. Basically, without entering in the details, we can say that this concept is derived from the Volterra theory and enables us to write, to first order, the input/output relationships as follows:

(6) In the above, and represent, respectively, the incident and reflected wave amplitudes on port . The variables and stand for the frequency spectrum of and , respectively. Defining a time-varying spectrum , we can rewrite (6) so that with where

(7) (8)

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Fig. 11. Input/output transfer characteristic of the amplifier.

Fig. 12. First output matching characteristic of the amplifier.

with

and

Relation (8) reveals that the sub-system is characterized by two 2 2 matrices in a similar way that a linear two-port is characterized by only one 2 2 matrix. It is important to note that the matrix vanishes in small-signal conditions , and (7) reduces to the classical linear relationship. Thus, the different Volterra kernels in (8) defines the LSSPs, and (7) naturally bridges the extension of the classical -parameters notion towards nonlinear behaviors. As said in [2], the complete identification of the LSSPs can be stated driving the device with a two-tone signal and for three different output terminations. In this paper, we will focus on applications that do not generate low-frequency phenomena or on circuits that do not present spurious long-term memory effects. In such conditions, the influence of the terms and is minor and, thus, can be neglected. Hence, (8) becomes (9) with

and

These considerations simplify the identification procedure. Indeed, for extracting the six terms in (9), we just need to measure the response of the circuit for a single-tone stimulus and for three different output terminations. To fully characterize the circuit, the power and frequency of the single-tone signal have to be swept to cover the frequency bandwidth and power operating range. Fig. 11 presents the LSSPs of a 350-mW -band het-

Fig. 13. Second output matching characteristic of the amplifier.

erostructure field-effect transistor (HFET) amplifier extracted for a frequency ranging from 20 to 20 MHz and up to 4-dB compression gain. Figs. 11–13 illustrate the different transfer characteristics relating the reflected wave amplitude at port 2 to the incident wave vanamplitudes. We can notably verify that the parameter ishes at small-signal conditions, exhibiting the fact that (6) reduces to the classical linear relation. It is also interesting to note corresponds to the characteristic used that the parameter for building a model that does not take into account the output impedance mismatch (unilateral model). Equation (6) has been implemented in the circuit simulator GoldenGate [26] using a specific frequency-domain description (FDD) enabling the handling of nonlinear elements with memory in harmonic-balance (HB) or envelope-transient (ET) engines. Results of a single-tone analysis are presented in Fig. 14 where the results obtained by the model and measurements of the amplifier are superimposed. We can see that the model is able to predict the influence of the variations of the output impedance on the behavior and performance of the amplifier.

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study of linearization of power amplifiers using baseband predistortion techniques. For that, both calibrated measurements and behavioral models like those proposed in this paper are required. ACKNOWLEDGMENT The authors want to thank A. Mallet and F. Gizard, both with the National French Space Agency, Toulouse, France, for their helpful assistance. REFERENCES

Fig. 14. Performance of the amplifier for a single-tone signal predicted by the model (dots) and measured (lines) for two different output impedances (output reflection coefficient = 0 and 0:2) and for three input frequencies (freq = 20 MHz, 0 MHz, and 20 MHz).

0

V. CONCLUSION A calibrated four-channel time-domain envelope measurement system for the characterization of nonlinear devices such as power amplifiers has been presented. It enables the measurement of the complex envelope of both the incoming and outgoing RF carriers at the input and output of a DUT. The load impedance of the DUT can be controlled by using electromechanical tuners. Two main capabilities of the setup have been presented: the first one consists of applying a linearization technique to improve the linearity performance of a MESFET transistor. The performance of the transistor has been clearly improved by applying a dynamic drain bias modulation. The second main capability of this characterization tool is the validation or the extraction of behavioral bilateral models of nonlinear devices exhibiting memory effects. The behavioral model of an -band medium power amplifier has been extracted thanks to this new characterization tool and the LSSP’s formalism. The performance of the amplifier for a single-tone signal predicted by the model and measured for two different output impedances shows very good agreement and illustrates the interest in such a characterization tool for the system-level modeling of nonlinear devices. An interesting extension of the setup can be made by using a 40-GS/s-15-GHz sampling scope. This will eliminate the mixer stages of our measurement channels. A next step in the behavioral modeling will consist of taking into account both mismatch effects and long-term memory effects. Another important feature that is now under investigations is the experimental and theoretical

[1] J. Verspecht and K. Rush, “Individual characterization of broadband sampling oscilloscopes with a “nose to nose” calibration procedure,” IEEE Trans. Microw. Instrum. Meas., vol. 43, no. 2, pp. 347–354, Apr. 1994. [2] J. Verspecht, “Calibration of a measurement system for high frequency nonlinear devices,” Ph.D. dissertation, Dept. Electron., Vrije Univ. Brussel, Brussels, Belgium, Sep. 1995. [3] U. Lott, “Measurement of magnitude and phase of harmonics generated in nonlinear microwave two-ports,” IEEE Trans. Microw. Theory Tech., vol. 37, no. 10, pp. 1506–1511, Oct. 1989. [4] D. Barataud, A. Mallet, M. Campovecchio, J. M. Nebus, J. P. Villotte, and J. Verspecht, “Measurements of time domain voltage/current waveforms at R.F. and microwave frequencies for the characterization of nonlinear devices,” in IEEE Instrum. Meas. Technol. Conf., St. Paul, MN, 1998, pp. 1006–1010. [5] G. Kompa and F. van Raay, “Error-corrected large-signal waveform measurement system combining network analyzer and sampling oscilloscope capabilities,” IEEE Trans. Microw. Theory Tech., vol. 38, no. 4, pp. 358–365, Apr. 1990. [6] P. J. Tasker, “Non-linear vector network analyser (NLVNA),” in 14th IEE Microw. Meas. Training Course Conf., May 9–13, 2005, pp. 15–15/19, 2005/10870. [7] A. Soury, E. Ngoya, and J. Rousset, “Behavioral modeling of RF and microwave circuit blocs for hierarchical simulation of modern transceivers,” in IEEE MTT-S Int. Microw. Symp. Dig., Long Beach, CA, Jun. 2005, [CD ROM]. [8] C. J. Clark, G. Chrisikos, M. S. Muha, A. A. Moulthrop, and C. P. Silva, “Time-domain envelope measurement technique with application to wideband power amplifier modeling,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 12, pp. 2531–2540, Dec. 1998. [9] R. Hajji, F. Beauregard, and F. M. Ghannouchi, “Multitone power and intermodulation load-pull characterization of microwave transistors suitable for linear SSPA’s design,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 7, pp. 1093–1099, Jul. 1997. [10] M. Vanden Bossche, “Theoretical background on LSNA technology,” LSNA Technol. Library NMDG bvba, Brussels, Belgium, 1989 [Online]. Available: Available: http://www.nmdg.be [11] W. Van Moer and Y. Rolain, “An improved broadband conversion scheme for the large signal network analyzer,” in IEEE MTT-S Int. Microw. Symp. Dig., Long Beach, CA, Jun. 2005, pp. 1501–1504. [12] K. A. Remley, D. M. M.-P. Schreurs, D. F. Williams, and J. Wood, “Extended NVNA bandwidth for long-term memory measurements,” in IEEE MTT-S Int. Microw. Symp. Dig., Fort Worth, TX, Jun. 2004, vol. 3, pp. 1739–1742. [13] D. Schreurs and K. Remley, “Bandwidth extension using stitching approach,” in ARFTG Nonlinear Meas. Workshop, Washington, DC, Nov. 30, 2005, 6 pp. [14] S. Vandenplas, J. Verspecht, F. Verbeyst, E. Vandamme, and M. V. Bossche, “Calibration issues for the large signal network analyzer (LSNA),” in 60th ARFTG Conf. Dig., Washington, DC, Dec. 5-6, 2002, pp. 99–106. [15] K. A. Remley, P. D. Hale, D. I. Bergman, and D. Keenan, “Comparison of multisine measurements from instrumentation capable of nonlinear system characterization,” in 66th ARFTG Conf. Dig., Washington, DC, Dec. 2005, pp. 34–43. [16] T. Reveyrand, C. Maziere, J. M. Nébus, R. Quéré, A. Mallet, L. Lapierre, and J. Sombrin, “A calibrated time domain envelope measurement system for the behavioral modeling of power amplifiers,” in Eur. Microw. Week GaAs 2002, Milan, Italy, Sep. 2002, pp. 237–240. [17] J. V. Butler, Rytting, K. Douglas, M. F. Iskander, F. Magdy, R. D. Pollard, and M. Vanden Bossche, “16-term error model and calibration procedure for on-wafer network analysis measurements,” IEEE Trans. Microw. Theory Tech., vol. 39, no. 12, pp. 2211–2221, Dec. 1991. [18] P. Guillaume, J. Schoukens, R. Pintelon, and I. Kollar, “Crest factor minimization using nonlinear Chebyshev approximation methods,” IEEE Trans. Microw. Instrum. Meas., vol. 40, no. 6, pp. 982–989, Jun. 1991.

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[19] C. Duvanaud, F. Robin, S. Dardenne, F. Huin, and L. Dascalescu, “Effects of low-frequency drain termination and injection on nonlinear amplifier performances,” Int. J. RF Microw. Comput.-Aided Eng., vol. 15, no. 2, pp. 231–240, 2005. [20] A. Katz, “Linearization: Reducing distortion in power amplifiers,” IEEE Micro, pp. 37–39, Dec. 2001. [21] P. M. Asbeck, T. Itoh, Y. Qian, M. F. Chang, L. Milstein, G. Hanington, P. F. Chen, V. Schultz, D. W. Lee, and J. Arun, “Device and circuit approaches for improved linearity and efficiency in microwave transmitters,” in IEEE MTT-S Int. Microw. Symp. Dig., Denver, CO, Jun. 1998, pp. 327–330. [22] J. Verspecht, “Scattering functions for nonlinear behavioral modeling in the frequency domain,” in IEEE MTT-S Int. Microw. Symp. Fundamentals Nonlinear Behavioral Modeling: Foundations Applicat. Workshop, Philadelphia, Jun. 2003, [CD ROM]. [23] J. A. Jargon, K. C. Gupta, and D. C. DeGroot, “Nonlinear large-signal scattering parameters: Theory and applications,” in ARFTG Conf. Dig., Jun. 2004, pp. 157–174. [24] J. Verspecht, D. E. Root, J. Wood, and A. Cognata, “Broad-band multiharmonic frequency domain behavioral models from automated large signal vectorial network measurements,” in IEEE MTT-S Int. Microw. Symp. Microw. Symp. Fundamentals Nonlinear Behavioral Modeling: Foundations Applicat. Workshop, Long Beach, CA, Jun. 2005, [CD ROM]. [25] D. E. Root, J. Verspecht, D. Sharrit, J. Wood, and A. Cognata, “Broad-band poly-harmonic distortion (PHD) behavioral models from fast automated simulations and large-signal vectorial network measurements,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 11, pp. 3656–3664, Nov. 2005. [26] “GoldenGate Users Manual,” Xpedion Des. Syst., Santa Clara, 2005. [Online]. Available: Available: http://www.xpedion.com/ François Macraigne was born in Limoges, France, on November 6, 1977. He received the Ph.D. degree in electronics from the University of Limoges, Limoges, France, in 2005. He is currently a Sales Engineer with Anritsu SA9, Les Ulis, France.

Tibault Reveyrand was born in Paris, France, on September 20, 1974. He received the Ph.D. degree from the University of Limoges, Limoges, France, in 2002. From 2002 to 2004, he was a Post-Doctoral Scientist with the Centre National d’Etudes Spatiales (CNES) (French Space Agency). In 2005, he became a Contractual Center National de la Recherche Scientifique (CNRS) Engineer with XLIM [formerly the Institut de Recherche en Communications Optiques et Microondes (IRCOM)], Limoges, France. His research interests include the characterization and modeling of RF and microwave nonlinear components. Dr. Reveyrand was the recipient of the 2002 European GaAs Best Paper Award.

Guillaume Neveux was born in Civray, France, in 1976. He received the Diplôme d’Etudes Approfondies (DEA) degree from the Université Paris 11, Orsay, France, in 2000, and the Ph.D. degree in electronics and communications from the National Superior Institute of Telecommunications (ENST), Paris, France, in 2003. Since 2004, he has been with the Instrumentation Group, XLIM Laboratory, University of Limoges, Limoges, France. His research interests include nonlinear measurement with LSNA and the study of RF sampling systems.

Denis Barataud was born in Saint-Junien, France, in 1970. He graduated from the Ecole Nationale Superieure de Telecommunications de Bretagne, Bretagne, France, in 1994. He received the Ph.D. degree in electronics from the University of Limoges, Limoges, France, in 1998. From 1998 to 1999, he was a Post-Doctoral Scientist with the Microwave Laboratory, Centre National d’Etudes Spatiales (CNES), Toulouse, France. Since 2000, he has been with the XLIM [formerly the Institut de Recherche en Communications Optiques et Microondes (IRCOM)], University of Limoges, Limoges, France, where he became an Assistant Professor in 2001. His research interests include the development of time-domain equipment and techniques for the characterization of nonlinear devices.

Jean-Michel Nebus was born in Bourganeuf, France in 1963. He received the Ph.D. degree in electronics from the University of Limoges, Limoges, France in 1988. He was a Project Engineer with ALCATEL Space Industries France. He is currently a Professor with the XLIM [formerly the Institut de Recherche en Communications Optiques et Microondes (IRCOM)], University of Limoges. His main area of interest is nonlinear microwave device characterization and design.

Arnaud Soury was born in Confolens, France, in 1975. He received the Ph.D. degree in electrical engineering from the University of Limoges, Limoges, France, in 2002. From 2003 to 2004, he was a Post-Doctoral Scientist with the French Centre National de la Recherche Scientifique (CNRS), XLIM [formerly the Institut de Recherche en Communications Optiques et Microondes (IRCOM)], University of Limoges. In 2005, he joined Xpedion Design Systems, Limoges, France, where he is currently with the Development Team involved with the circuit-level simulation tool GoldenGate. His main research interests are in the field of computed-aided design techniques for RF and microwave circuits including behavioral modeling, analysis techniques, and stability of electronic devices.

Edouard NGoya received the Ph.D. degree in electronics from the University of Limoges, Limoges, France, in 1988. In 1990, he joined the French Centre National de la Recherche Scientifique (CNRS), XLIM [formerly the Institut de Recherche en Communications Optiques et Microondes (IRCOM)], University of Limoges. He is the inventor of key analog simulation and modeling technologies such as compressed transient, ET, and dynamic Volterra series. Since 1998, he has been the Consultant Chief Scientist with Xpedion Design Systems, Milpitas, CA. His current research activities focus on modeling and simulation techniques for large-scale integration RF circuits, RF sytem-on-chip (SOC) and system-in-package (SIP).

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In-Band Distortion of Multisines Khaled M. Gharaibeh, Member, IEEE, Kevin G. Gard, Member, IEEE, and Michael B. Steer, Fellow, IEEE

Abstract—Multisine signals are shown to be useful for estimating distortion of communication signals. In particular, a generalized approach for the evaluation of effective in-band distortion in a nonlinear amplifier using multisine excitation is presented. The output of the nonlinearity is represented as the sum of uncorrelated components by the transformation of a behavioral model. Simulated and measured results are presented for code-division multiple-access signals. Index Terms—Intermodulation distortion, multisine signals, nonlinear distortion, nonlinear systems, signal analysis, signal representations.

I. INTRODUCTION ULTISINE signals have been used to model the behavior of nonlinear systems because of the simplicity of the analysis and simulations. The design of multisines for modeling digitally modulated communication signals is usually based on the choice of the amplitudes, phases, and number of tones optimized for their suitability to capture adjacent channel power ratio (ACPR), in-band distortion, or other distortion metrics. The rationale for using multisines is that they require lower computational complexity than that required by direct use of the actual communication signals. Multisine representations of signals have direct application in harmonic-balance simulation [1] and in measurement characterization of nonlinear microwave circuits [2]. Moreover, multisine analysis leads to an analytic evaluation of distortion with simple expressions for nonlinear system figures-of-merit such as intermodulation ratio (IMR), signal-to-noise ratio (SNR), ACPR, etc. Transmitter signal quality in digital wireless communication systems is specified by the error vector magnitude (EVM) of the transmitted signal. EVM is inversely related to the signal-tonoise and distortion (SINAD) ratio, which measures the effective SNR including noise, distortion, and any other signals that degrade the effective SNR of the transmitter signal. There are many contributors to transmitter EVM degradation including thermal noise, phase noise, nonlinear distortion, spurious signals, etc. However, intermodulation distortion is a significant contributor to transmitter EVM degradation when operating at high output power levels in systems utilizing power efficient

M

Manuscript received December 15, 2005; revised May 12, 2006. This work was supported in part by the U.S. Army Research Office as a Multidisciplinary University Research Initiative on Multifunctional Adaptive Radio Radar and Sensors under Grant DAAD19-01-1-04 and under the William J. Pratt Assistant Professorship. K. M. Gharaibeh is with the Hijjawi Faculty for Engineering Technology, Yarmouk University, Irbid 21163, Jordan (e-mail: [email protected]). K. G. Gard and M. B. Steer are with the Electrical and Computer Engineering Department, North Carolina State University, Raleigh, NC 27695-7914 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2006.879170

nonlinear amplifiers. Here, the relationship between nonlinear transmitter distortion and transmitter SINAD degradation is investigated through analysis and measurements using multisine signals. In this context, we define effective in-band distortion as the component of the nonlinear output that shares the same frequency band as the input signal, but is “uncorrelated” with the ideal transmitter signal. From a communications point-of-view, the receiver is designed to distinguish between only two types of signals, which are: 1) the transmitted signal to which it is matched and 2) noise. Here, the term correlation refers not only to the statistical resemblance between the output and the input signals, but also to the ability of the receiver to recover useful information from the transmitter signal. Therefore, if part of the transmitted signal is uncorrelated with the expected waveform, that part of the signal is considered as uncorrelated distortion noise, which contributes to the degradation of transmitter SINAD. The problem with characterizing effective in-band distortion is the identification of the effective terms of the nonlinear output that are responsible for in-band distortion inside the main band of the input signal spectrum. The reason for this is that the nonlinear output is partially correlated with the input signal, which subtracts or adds to the desired output signal causing gain compression or expansion. The remaining part of the nonlinear output is the uncorrelated in-band distortion noise, which may be treated as an additional contributor to in-band noise. Thus, both the correlated and uncorrelated components of the nonlinear output contribute to the degradation of system SINAD in different ways. In [3], we presented an analysis of in-band distortion of multisines with random phases. In this paper, we provide an analytic approach to the identification of the effective in-band distortion of multisine signals using both deterministic and uniformly distributed random phases. We consider multisine signals with constant amplitudes and varying initial phases and derive the effective in-band distortion components. It is shown that the shape and level of the uncorrelated distortion spectrum depends on the initial phases, and this has different effects on in-band and out-of-band distortions. Estimated in-band distortion of multisine signals is verified by simulations and measured in-band distortion compared against an IS-95 forward-link code-division multiple-access (CDMA) signal using feed-forward cancellation. Multisine signals with 16 tones and random phase are shown to accurately predict the in-band distortion of a 64-user CDMA signal. This paper is organized as follows. Section II is a review of the existing approaches to multisine analysis. Section III presents an analysis of in-band distortion of multisines with fixed initial phases. In Section IV, we present the probabilistic view of distortion where multisines have random phases. Section V presents simulation and measurement results of distortion in

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multisine signals where we show how effective in-band distortion of a 64-user CDMA signal is estimated from that of multisines with random phases.

II. BACKGROUND Considerable research has been undertaken on using multisine signals for characterizing distortion in nonlinear systems. Interest has grown with the introduction of large-signal vector network analyzers (VNAs), which use multisine inputs for device characterization. Remley [4] used multisine signals with random and deterministic amplitude and phase for ACPR simulation. It was demonstrated that certain choices either overestimate or underestimate ACPR of real communication signals. Pedro and de Carvalho [5]–[7] used multisine signals to simulate the spectrum of communication signals subject to a nonlinear system with memory. Optimization criteria were used to design multisine signals that mimic the spectral characteristics of communication signals. Although the analysis was based on Volterra series, a simplification of the Volterra model was used to perform simulations. Boulejfen et al. [8] considered the estimation of in-band and out-of-band distortions of communication signals using multisine signals with random phases. The approach was analytic and enabled ACPR, noise power ratio (NPR) and co-channel power ratio (CCPR) to be estimated for a fifth-order nonlinearity. However, it was not shown that the approach can truly represent the in-band distortion in real communication signals. In [9] and [10], the design of multisine signals for a minimal crest factor was studied. The analysis was useful for constructing test signals to measure the frequency response of linear systems. It is questionable, however, that this approach could be used for estimating nonlinear system distortion because of the low correlation between the crest factor and distortion [11]. Geens et al. [12] used multisines with random phases to estimate NPR and CCPR. They showed that NPR is not always a valid metric for in-band distortion. A common characteristic of the above approaches is that they use the Gaussian approximation of multisine signals. The identification of the uncorrelated distortion noise is straightforward using the statistical properties of Gaussian processes. A theoretical analysis of the decomposition of the output spectrum into uncorrelated components without using the Gaussian assumption was studied in [13] and [14] where, based on the properties of the distribution function of the input signals, the output of a bandpass nonlinearity can be expressed as a sum of uncorrelated components. An approach for the identification of the effective in-band distortion of CDMA signals without using the Gaussian assumption was presented by the authors in [15]. The approach was based on the orthogonalization of the behavioral model and leads to exact identification of effective in-band distortion. The approach was used to accurately estimate system metrics such as SNR, EVM, and of CDMA signals. It is well established that the sum of a large number of sinusoids with random phases converges to a band-limited Gaussian noise process by the central limit theorem [2]. However, the approach is valid for any random process that satisfies the separability condition and,

Fig. 1. Geometrical interpretation of in-band distortion. (a) Distortion vector. (b) Orthogonal representations.

hence, it can be used with a small number of tones, as will be shown here. III. DISTORTION OF MULTISINES Effective in-band distortion of multisines can be understood by defining the uncorrelated component of the nonlinear output. Considering a memoryless nonlinearity, a geometrical representation of the nonlinear output is shown in Fig. 1 where the total output is the vector sum of the linear and third-order components. The third-order output can be partitioned into two components: one in the direction of the linear output and the other orthogonal to it. The uncorrelated distortion output can now be identified in terms of a canceling process where a scaled replica of the input signal is subtracted from the total nonlinear output. The orthogonal component cannot be canceled by a scaled replica of the input signal and, hence, it represents the effective uncorrelated distortion that contributes to the degradation of system SINAD performance, while the correlated component of the third-order output represents the correlated distortion that causes gain compression of the linear output. This definition complies with the definition of nonlinear distortion in communication theory [16]–[18] where the effective SNR is calculated as ratio of the effective signal component (which includes the correlated distortion or gain compression) to the effective nonlinear distortion component (which is the uncorrelated distortion). Therefore, the uncorrelated distortion

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Now consider a multisine input signal consisting of the sum of tones and applied to the nonlinear amplifier so that

(6) Here, the signal and phase

is the th tone with radian frequency

Fig. 2. (a) Instantaneous nonlinear model. (b) Instantaneous nonlinear model with uncorrelated outputs.

noise is treated as an additive noise component similar to the additive white Gaussian noise (AWGN). In the following, we consider a memoryless nonlinear system, which can be characterized by a power series model:

(1) The power series model represents the instantaneous relationship between the input and output waveforms. In order to identify the orthogonal component of the nonlinear output, we define a canceling signal as a replica of the input signal used to cancel the correlated component of the output

(7) The complex envelope of this signal is equivalent to the phasor form of sinusoids

(8) The response of the nonlinear system to a multisine signal consists of all intermodulation products of the input tones. Now using (1) and applying the multisine signal model in (7), the component of centered at frequency due to the intermodulation of input components (centered at the frequency vector ) can be expressed as [19]

(9) (2)

where

where

,

, and

(10) (3)

is a cross-correlation coefficient that represents the fraction of the cubic term that is correlated with the linear response. Using this representation, the nonlinear model in (1) depicted in Fig. 2(a), is converted into a model with orthogonal outputs, as shown in Fig. 2(b). Therefore, for a third-order nonlinearity, we define a new set of outputs

is the multinomial coefficient. In the following, we consider the special cases of single-, two-, and four-tone signals and we derive the output distortion components by which the effective in-band distortion is identified. A. Single Tone Let us first consider the case where the input consists of a single tone of frequency , therefore,

(11) (4) which is correlated to the input signal and can be canceled by , and

then using (9), the complex envelope of the output at the fundais mental frequency (12)

(5) where which is uncorrelated to the input signal. The new outputs and are orthogonal and represent a useful component and an uncorrelated distortion component and, hence, they define system SINAD.

(13)

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are the envelope coefficients and relate the complex envelope of the output to the complex envelope of the input signal [20]. Therefore, the total nonlinear output at the fundamental is

(14) and represent the well-known AM–AM and where AM–PM characteristics and, thus, the envelope coefficients can be obtained by polynomial fitting of the measured AM–AM and AM–PM characteristics. The nonlinear response to a single tone is, therefore, a replica of the input signal with modified amplitude and phase. A canceling signal in this case can be designed to cancel the whole output and is independent of the initial phase. The output phasor at any of the harmonics can also be derived in compact form in a similar way. Note that the response to a single tone is a single component that is correlated with the input and, hence, in-band distortion is absent. This is intuitive because the response of a nonlinear system to a single tone results in gain compression and not distortion. B. Two Tone Now consider a two-tone input with equal amplitudes, then

(15) where is the amplitude and is the phase of each of the input tones. Therefore, using (9) and the proper frequency vectors, the complex envelope of the output at the frequency of the first tone is

Fig. 3. Time-domain representation of a phase-aligned four-tone signal. Solid: output signal. Dotted: canceling signal. Dashed: in-band distortion.

respectively. Note that with a two-tone signal, the output tones that lie within the input band have the same phase change as the linear output regardless of the initial phase of the input tones and the nonlinear order. Therefore, effective in-band distortion is absent, as considering (19), the output signal has a phase that is totally correlated with the input signal. Thus, a canceling signal can be designed to cancel the output signal at the fundamental frequencies completely regardless of the initial phase. C. Four Tone For a four-tone input with each tone having an amplitude the input is

,

(16) (20) and for the second tone is

(17) where

is

The derivation of a closed form for the output for orders higher than 3 is more laborious than for the single- and two-tone cases. For illustrative purposes, we consider a third-order nonlinear system (one described by a third-order power series). Using (7), the output complex envelope at the fundamental frequencies is shown in (21) as follows:

(18) Therefore, the output and can be written as

at the fundamental frequencies

(19) and are real functions of the amplitudes where and which represent the AM–AM and AM–PM conversions,

(21)

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Fig. 5. In-band distortion for different phase combinations of a four-tone signal (solid) and zero initial phase (dashed).

For the special case where the initial phases of the input tones are zero, different terms in each of the output components in (21) add up in the following way:

(23) where (21) becomes

and

. Hence,

(24) Note that the amplitude of the output at each of the four tones is not equal and, therefore, a canceling signal will not cancel the whole signal at the fundamental frequencies and, hence, the remainder is in-band distortion. The coefficient for a four-tone signal can be computed from (3) as Fig. 4. (a) Output spectrum and (b) uncorrelated distortion spectrum for phase aligned four-tone signal (frequencies are offset from the carrier frequencies).

(25) where and . Therefore, the total output at the fundamental angular frequencies can then be written as

(22)

where and are real functions of the input amplitudes and the nonlinear coefficients and is a linear function of the initial phases . Note that a four-tone signal produces distortion components at the fundamental frequencies that can be either correlated or uncorrelated with the linear output depending on the initial phases of the input tones. The output components at the fundamental frequencies consist of the vector sum of different components and, therefore, depending on phase combinations.

and, hence, the in-band distortion component is

(26) Note that the in-band distortion components have equal powers at the four fundamental tones in the case of phase-aligned tones (zero initial phases). The same analysis can be performed when the initial phases are not zero. Fig. 3 shows a time-domain representation of the phase-aligned four-tone signal, a canceling signal, and the resulting in-band distortion. The frequency-domain representation is depicted in Fig. 4 where the total output spectrum and the total uncorrelated distortion spectrum are simulated. Fig. 5 shows the effective in-band distortion of the fourtone signal as a function of input power and for different initial phase combinations. Note that the minimum in-band dis-

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Fig. 7. In-band distortion of phase-aligned: (a) four- and (b) eight-tone signal. : measured. Solid: simulation.

Fig. 6. (a) Output spectrum and (b) uncorrelated distortion spectrum of a fourtone signal with random phase.

tortion occurs when the initial phases are zero (phase aligned multisines).

D. Discussion The above analysis provides a basis for the identification of effective in-band distortion of multisine signals. However, it is not adequate for identifying distortion when the input tones have random phases and/or amplitudes. Therefore, with random waveforms, the definition of in-band distortion needs a probabilistic model in order to provide a useful description of distortion in terms of signal statistics. A multisine signal with random amplitude and phase resembles a digitally modulated carrier where the signal does not have a deterministic representation. Therefore, multisines are a useful tool in understanding distortion in communication signals. In Section IV, we develop the

probabilistic view of in-band distortion using the same concepts developed here about the orthogonalization of a behavioral model, but with the probabilistic tools that suit the random nature of signals. IV. DISTORTION OF MULTISINES WITH RANDOM PHASES In [3], we presented a statistical analysis of in-band distortion of multisines with random phases. The orthogonalization of the behavioral model with random inputs is a probabilistic version of the one described in Section III for deterministic signals. The analysis of multisines with random phases assumes that multisines have uniformly distributed random phases and, hence, the estimated distortion is found as a statistical average. The statistical average is estimated in simulation by computing the average distortion of a large number of realizations of the multisines having phases generated using a random number generator for each realization, as will be shown in Section V. The orthogonalization of the behavioral model enables the uncorrelated output distortion to be treated as an additive noise similar to the AWGN and enables the SINAD to be determined since the effective noise-like component of the output nonlinear

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Fig. 8. Intermodulation distortion of phase aligned: (a) four- and (b) eight-tone signal.  measured. Solid: simulation.

power is determined. It is worth emphasizing here that the analysis presented in [3] does not require the input signal to have Gaussian distribution, which is a common assumption for a large number of tones with random phases. Therefore, it is valid with any number of tones provided that their phases are completely random. A complete derivation of the output autocorrelation function of different multisine signals with random phases can be found in [3]. It was shown that the response of the nonlinearity to a single-tone input with random phase is a single tone with a compressed amplitude regardless of the initial phase

(27) Therefore, the output is completely correlated with the input and, thus, a nonlinear system exhibits gain compression or expansion without generating effective in-band distortion. With two-tone excitation, the output consists of compressed outputs at the fundamental frequencies and uncorrelated out-of-band intermodulation components

(28)

Fig. 9. Simulated output spectrum of 16-tone signal and an IS-95 signal (solid line). (a) Phase aligned tones. (b) Random phases.

Note that a two-tone test does not predict the existence of uncorrelated in-band distortion since the only distortion terms that result as uncorrelated components are the intermodulation components, which lie outside the frequency band of the input tones. Therefore, single- and two-tone tests are inadequate for characterizing in-band uncorrelated distortion. In the case of tones where , the output distortion consists of uncorrelated in-band components (at the fundamental frequencies) in addition to uncorrelated out-of-band components. In particular, with a four-tone signal with random phases, the amount of in-band distortion is higher than if the phases are aligned, e.g., with zero degree initial phases

(29) This is because the output components add to the linear output, whereas with random phases the probability of having components that are orthogonal to the linear output is higher and,

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Fig. 10. Uncorrelated distortion spectrum of random-phase 16-tone signal and an IS-95 signal (solid line). (a) Simulated. (b) Measured.

Fig. 11. Uncorrelated distortion spectrum of phase aligned 16-tone signal and an IS-95 signal. (a) Simulated. (b) Measured.

hence, these components are considered as effective in-band distortion. Fig. 6 shows a simulated output spectrum and the uncorrelated distortion spectrum of a four-tone signal with uniformly distributed random phase.

model coefficients were then obtained using the development in Section II. Multisine signals were generated using an Agilent ESG 4438C vector signal generator. Uncorrelated distortion was obtained using the feed-forward cancellation scheme presented in [15] and the effective in-band distortion was measured within the signal bandwidth using an Agilent VSA. Simulated multisines were generated in MATLAB. The phases of the input tones were randomized using a uniform random number generator. Distortion was estimated by averaging a large number of phase realizations. The number of realizations required depends on the number of tones since, as the number of tones increases, the probability of having a uniformly distributed phases increases. Figs. 7 and 8 show measured and simulated effective in-band and out-of-band distortions of the phase-aligned multisine signals. A good agreement between predicted and simulated values of the in-band and out-of band distortions is shown. The analysis of multisine signals was used here to estimate distortion of CDMA signals. Nonlinear distortion of multisine

V. MEASUREMENTS AND SIMULATION RESULTS The analytical evaluation of effective in-band distortion obtained using the orthogonalization procedure was verified by measurements done with multisine signals. The measurements presented here were taken using an Agilent 8510 VNA, E4438C vector signal generator, E4445A spectrum analyzer, and 89600S vector signal analyzer (VSA). The amplifier considered has a gain of 21 dB, an output 1-dB compression point of 11 dBm, and an output third-order intercept (OIP3) of 18 dBm all at 2.0 GHz. The coefficients of the envelope model of the device-under-test were obtained by measuring the AM–AM and AM–PM characteristics at 2 GHz. A polynomial of order 5 was fitted to the complex data using classical least squares polynomial fitting and a set of envelope coefficients was obtained. The orthogonal

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CDMA signals and with measured data of a CDMA signal. The measured in-band distortion of fixed and constant phase multisine signals are both in good agreement with the simulated spectrums. The depth of the in-band notches observed at the edge of the signal bandwidth for the simulated fixed phase signal are not as deep for the measured in-band distortion due to the finite delay error in the feed-forward cancellation measurement setup. VI. CONCLUSION

3

Fig. 12. In-band distortion versus input power. : simulated random phase 16-tone signal. : simulated CDMA signal. : measured CDMA signal. Solid: simulated phase aligned 16-tone signal.

4



signals with random phases mimics that of CDMA signals. It was found that distortion of a 16-tone signal with random phases converges to that of a forward-link IS-95 CDMA signal, as shown in Fig. 9(a). This figure shows the output and the uncorrelated distortion spectra of a 16-tone signal with random phases compared to those of a forward-link IS-95 spectrum. The 16-tone signal was designed to have a total bandwidth equal to the bandwidth of an IS-95 CDMA signal (1.2288 MHz) and the tones were equally spaced. A maximum of ten runs for the randomization of the input phases was needed to reach convergence. Fig. 9(b) shows the spectrum of a phased-aligned 16-tone signal compared to that of a CDMA signal. Fig. 10(a) shows the simulated uncorrelated distortion spectrum of a 16-tone signal with random phases compared to that of a forward-link IS-95 signal (the solid line). While Fig. 10(b) shows measured uncorrelated distortion spectrum obtained using feed-forward cancellation. Fig. 11(a) and (b) shows the simulated and measured uncorrelated distortion spectrum of a phase-aligned 16-tone signal compared to that of a forward-link IS-95 signal (solid line). These figures also show that the shape of the effective in-band spectrum depends on the number of tones and their initial phases. It is clear that in-band distortion is minimum when the phases are aligned, while the out-of-band distortion is at its maximum. The uncorrelated distortion has an almost flat spectrum when the input phases are random. The in-band distortion in this case is at its maximum, while the out-of-band distortion is at its minimum. This is because a multisine signal with random phases approaches a Gaussian distribution, which has a flat spectrum for the uncorrelated distortion, as shown in Section III-D. It is clear that multisines with fixed phases overestimate the out-of-band distortion, while they underestimate in-band distortion of CDMA signals. Fig. 12 shows in-band distortion as a function of input power of multisine signals and a CDMA signal and compared to measured in-band distortion of the CDMA signal. This figure shows good agreement in the estimation of in-band distortion of the random multisine and

An analysis of multisine signals in a nonlinear system has been developed. The analysis is used for the estimation of effective in-band distortion of communication signals. Traditional two-tone tests have been shown to be inadequate for the estimation of in-band distortion. It was also shown that multisine signals (four tones and more) with random phases can be used to estimate in-band distortion in real communication signals. This is significant because multisine signals are easier and faster to simulate using harmonic-balance techniques. The accuracy of the simulations using multisine signals depends on the number of tones and the randomization of the initial phases. ACKNOWLEDGMENT The authors would like to thank the reviewers for their constructive suggestions and comments. REFERENCES [1] V. Rizzoli, N. Neri, F. Mastri, and A. Lipparini, “A Krylov-subspace technique for the simulation of integrated RF/microwave subsystems driven by digitally modulated carriers,” Int. J. RF Microw. Comput.Aided Eng., vol. 9, no. 6, pp. 490–505, 1999. [2] K. Vanhoenacker, T. Dobrowiecki, and J. Schoukens, “Design of multisine excitations to characterize the nonlinear distortions during FRF-measurements,” IEEE Trans. Instrum. Meas., vol. 50, no. 5, pp. 1097–1102, Oct. 2001. [3] K. M. Gharaibeh, K. G. Gard, and M. B. Steer, “Characterization of in-band distortion in RF front-ends using multisine excitation,” in IEEE Radio Wireless Symp., 2006, pp. 487–490. [4] K. A. Remley, “Multisine excitation for ACPR measurements,” in IEEE MTT-S Int. Microw. Symp., Jun. 2003, pp. 2141–2144. [5] J. C. Pedro and N. B. de Carvalho, “On the use of multitone techniques for assessing RF components’ intermodulation distortion,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 12, pp. 2393–2402, Dec. 1999. [6] ——, “Evaluating co-channel distortion ratio in microwave power amplifiers,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 10, pp. 1777–1784, Oct. 2001. [7] ——, “Characterizing nonlinear RF circuits for their in-band signal distortion,” IEEE Trans. Instrum. Meas., vol. 51, no. 3, pp. 420–426, Jun. 2003. [8] N. Boulejfen, A. Harguem, and F. M. Ghannouchi, “New closed-form expression for the prediction of multitone intermodulatiion distortion in fifth-order nonlinear RF circuits/systems,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 1, pp. 121–132, Jan. 2004. [9] S. Boyd, “Multitone signals with low crest factor,” IEEE Trans. Circuit Syst., vol. CAS-33, no. 10, pp. 1018–1022, Oct. 1986. [10] M. Friese, “Multitone signals with low crest factor,” IEEE Trans. Commun., vol. 45, no. 10, pp. 1338–1344, Oct. 1997. [11] J. F. Sevic and M. B. Steer, “On the significance of envelope peak-toaverage ratio for estimating the spectral regrowth of an RF/microwave power amplifier,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 6, pp. 1068–1071, Jun. 2000. [12] A. Geens, Y. Rolain, W. Van Moer, K. Vanhoenacker, and J. Schoukens, “Discussion on fundamental issues of NPR measurements,” IEEE Trans. Inst. Meas., vol. 52, no. 2, pp. 197–202, Feb. 2003. [13] N. Blachman, “The signal signal, noise noise, and signal noise output of a nonlinearity,” IEEE Trans. Inf. Theory, vol. IT-14, no. 1, pp. 21–27, Jan. 1968.

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[14] N. Blachman, “The uncorrelated output components of a nonlinearity,” IEEE Trans. Inf. Theory, vol. IT-14, no. 3, pp. 250–255, Mar. 1968. [15] K. M. Gharaibeh, K. G. Gard, and M. B. Steer, “Estimation of in-band distortion in digital communication system,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2005, pp. 1963–1966. [16] P. Banelli and S. Cacopardi, “Theoretical analysis and performance of OFDM signals in nonlinear AWGN channels,” IEEE Trans. Commun., vol. 48, no. 3, pp. 430–441, Mar. 2000. [17] A. Conti, D. Dardari, and V. Tralli, “An analytical framework for CDMA systems with a nonlinear amplifier and AWGN,” IEEE Trans. Commun., vol. 50, no. 7, pp. 1110–1120, Jul. 2002. [18] M. Eslami and H. Shafiee, “Performance of OFDM receivers in presence of nonlinear power amplifiers,” in Int. Comm. Syst. Conf., Nov. 2002, vol. 1, pp. 25–28. [19] J. W. Graham and L. Ehrmen, Nonlinear System Modeling and Analysis with Application to Communication Receivers. Rome, NY: Rome Air Develop. Ctr., Jun. 1973. [20] K. Gard, H. Gutierrez, and M. B. Steer, “Characterization of spectral regrowth in microwave amplifiers based on the nonlinear transformation of a complex Gaussian process,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 7, pp. 1059–1069, Jul. 1999. [21] K. M. Gharaibeh, K. Gard, and M. B. Steer, “The impact of nonlinear amplification on the performance of CDMA systems,” in IEEE Radio Wireless Conf., 2004, pp. 83–86.

Khaled M. Gharaibeh (S’01–M’04) received the B.S. and M.S. degrees in electrical engineering from the Jordan University of Science and Technology, Irbid, Jordan, in 1995 and 1998 respectively, and the Ph.D. degree in electrical engineering from North Carolina State University, Raleigh, in 2004. From 1996 to 2000, he was a Planning Engineer with Jordan Telecom, Amman, Jordan. From January 2004 to January 2005, he was a Post-Doctoral Research Associate with the Department Electrical and Computer Engineering, North Carolina State University. He is currently an Assistant Professor of electrical engineering with the Hijawi Faculty for Engineering Technology, Yarmouk University, Irbid, Jordan. His research interests are nonlinear system identification, behavioral modeling of nonlinear RF circuits, and wireless communications. Dr. Gharaibeh is a member of Eta Kappa Nu.

Kevin G. Gard (S’92–M’95) received the B.S. and M.S. degrees in electrical engineering from North Carolina State University, Raleigh, in 1994 and 1995, respectively, and the Ph.D. degree in electrical engineering from the University of California at San Diego, La Jolla, in 2003. He is currently the William J. Pratt Assistant Professor with the Electrical and Computer Engineering Department, North Carolina State University. From 1996 to 2003, he was with Qualcomm Inc., San Diego, CA, where he was a Staff Engineer and Manager responsible for the design and development of RF integrated circuits (RFICs) for code-division multiple-access (CDMA) wireless products. He has designed SiGe BiCMOS, Si BiCMOS, and GaAs metal–semiconductor field-effect transistor (MESFET) integrated circuits for cellular and personal communication systems (PCSs) CDMA, wideband code-division multiple-access (WCDMA), and AMPS transmitter applications. His research interests are in the areas of integrated circuit design for wireless applications and analysis and modeling of nonlinear microwave circuits with digitally modulated signals. Dr. Gard is a member of the IEEE Microwave Theory and Techniques and Solid-State Circuits Societies, Sigma Xi, Eta Kappa Nu, and Tau Beta Pi.

Michael B. Steer (S’76–M’82–SM’90–F’99) received the B.E. and Ph.D. degrees in electrical engineering from the University of Queensland, Brisbane, Australia, in 1976 and 1983, respectively. He is currently the Lampe Family Distinguished Professor of Electrical and Computer Engineering, North Carolina State University, Raleigh. In 1999 and 2000, he was a Professor with the School of Electronic and Electrical Engineering, The University of Leeds, where he held the Chair in microwave and millimeter-wave electronics. He was also Director of the Institute of Microwaves and Photonics, The University of Leeds. He has authored over 300 publications on topics related to RF, microwave and millimeter-wave systems, high-speed digital design, and RF and microwave design methodology and circuit simulation. He coauthored Foundations of Interconnect and Microstrip Design (Wiley, 2000). Prof. Steer is active in the IEEE Microwave Theory and Techniques Society (IEEE MTT-S). In 1997, he was secretary of the IEEE MTT-S. From 1998 to 2000, he was an elected member of its Administrative Committee. He was the Editor-in-Chief of the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES (2003–2006). He was a 1987 Presidential Young Investigator (USA). In 1994 and 1996, he was the recipient of the Bronze Medallion presented by the Army Research Office for “Outstanding Scientific Accomplishment.” He was also the recipient of the 2003 Alcoa Foundation Distinguished Research Award presented by North Carolina State University.

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Amplitude and Phase Characterization of Nonlinear Mixing Products José C. Pedro, Senior Member, IEEE, and João P. Martins, Student Member, IEEE

Abstract—This paper discusses the problem of amplitude and phase characterization of nonlinear mixing products arising from a broad class of nonlinear microwave devices. Realizing that the most difficult problem stands on the phase measurement, it starts by reviewing the most common approaches used for the phase characterization of spectral regrowth components. It is demonstrated that these methods can be viewed as correlation processes, and are so framed in a common theory. This allows a unique and universal definition of the input–output amplitude and phase relations, even for mixing products that fall on frequency positions for which there are no components present at the input, and for those arising from incommensurate excitations. Finally, it shows how that theory can be implemented in a laboratory measurement scheme, thus giving the introduced theoretical variables a practical engineering value. Index Terms—Behavioral science, nonlinear systems, waveform analysis.

I. INTRODUCTION CCURATE characterization of nonlinear mixing products has been drawing an increased attention in two different fields of microwave technology: power amplifier (PA) linearization [1]–[8] and nonlinear behavioral modeling [8]–[12]. In the first case, the objective consists of acquiring the necessary information of the PA so that an appropriate linearizer can be designed. This is simply a circuit that generates nonlinear distortion components, which can exactly cancel the ones produced by the nonlinear PA. Thus, since one cannot efficiently compensate what is not known, nonlinear distortion characterization plays a vital role here. Although that characterization has traditionally addressed static AM–AM and AM–PM distortion, the existence of PA long-term memory effects, i.e., dynamic behavior to the slowly varying modulating envelope, has demanded more detailed tests in which the previous continuous wave (CW) stimulus has been replaced by two tones [1]–[13]. In nonlinear behavioral modeling, this characterization aim is pushed even further until the complete identification of a certain nonlinear dynamic model structure is reached. The required tests depend on the type of desired model and have varied from former CW or two-tone measurements to multitones [12]–[16], RF carriers modulated under more or less complicated schemes [17], or even to pseudorandom [18], [19] or random stimuli [13]–[15].

A

Manuscript received December 15, 2005; revised April 27, 2006. This work was supported by the European Union, carried out by the Network of Excellence Top Amplifier Research Groups in a European Team under TARGET Contract IS-1–507893-NoE, and by the Instituto de Telecomunicacões under the ModEx Project. The authors are with the Instituto de Telecomunicacões, Universidade de Aveiro, 3810-193 Aveiro, Portugal (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2006.879171

Contrary to the PA linearization efforts, in which the distortion characterization is invariably carried out in the frequency domain (i.e., acquiring amplitudes and phases of spectral points), nonlinear model identification has been tried both in the frequency and time domains (in this case, extracting the required information from signal time samples) [9]–[12], [17]–[19]. Seen separately, these two fields have led to a set of disjoint approaches and laboratory benches, despite that they seem to be only two particular facets of the more general problem of nonlinear system identification. Hence, this paper aims at discussing the problem of amplitude and phase characterization of nonlinear mixing products arising from a broad class of nonlinear microwave devices. Thus, it is organized in the following way. In Section II, some of the existing measurement methods for finding the phase of intermodulation distortion (IMD) products are reviewed. These methods first define a certain phase reference for the IMD product in question, and then compute the IMD phase with respect to that reference. It is then demonstrated that such phase calculations can be viewed as a correlation process so that all of these methods become framed by a common theory. In Section III, we concentrate on defining a “universal” phase reference based on multidimensional, or higher order, inputs, and propose a method to measure IMD product phases with respect to it. This study, therefore, unites the previous methods, and subsumes into a unified treatment both commensurate and incommensurate cases. II. REVIEW OF THE MOST IMPORTANT METHODS FOR THE CHARACTERIZATION OF NONLINEAR DISTORTION PRODUCTS Although some time-domain approaches have already been tried [17]–[19], microwave nonlinear measurements tend to focus on the frequency-domain characterization of nonlinear distortion products.1 These tests have been carried out with CW [13], multitones [12]–[16] (from which the two-tone stimulus is a particular case of unarguable practical relevance), or even under arbitrarily stochastic signal excitations [13]–[15]. CW tests usually address static (from the point-of-view of long-term memory) AM–AM and AM–PM effects [8], [13], while multitone tests focus on the amplitude and phase characterization of spectral regrowth [1]–[12]. AM–AM and AM–PM measurements consist in acquiring the input–output relations of the device-under-test (DUT) in both amplitude and phase. Although we should intuitively expect multitone tests to do something similar, they tend to concentrate on the measurement of the input and output ampli1Since the referred time-domain tests were conducted with periodic excitations, even these have a direct frequency-domain counterpart as particular examples of multitone tests.

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tudes and phases, separately. This is due to the fact that, while CW tests have the input and output signals at the same frequency, the spectral regrowth produced by multitone or stochastic stimuli has naturally distinct frequency positions. In this case, the input–output relations lose their unique significance, and while the amplitudes of the input and output signals can be easily measured in a separate way, this same procedure cannot be extended to their phases. In fact, since phase is a relative entity (contrary to the amplitude, which has an absolute nature) finding a meaningful phase reference is not a trivial task, and has driven the efforts of many researchers [1]–[12]. Thus, in the following we will concentrate our attention on the phase measurement of mixing products. Due to the normal way multitone stimuli are synthesized, distortion tests involving more than two or three tones are usually performed with phase-correlated sines [12]. Two-tone tests can either be executed with phase-correlated [3]–[5], [9]–[12] or phase-uncorrelated sines [1], [2], [6]–[9] whether they are synthesized as a particular case of a multisine or generated from two independent (and relying on independent phase references) CW generators. Obviously, in the case of real telecommunication signals, which are stochastic in nature and involve continuous spectra, such phase properties are even more blurred. Being synthesized from the up-conversion of a baseband signal—created by scanning a lookup-table-based arbitrary waveform in a cyclic way—typical multitones, or pseudorandom modulated signals, have periodic envelopes. In addition, if the lookup-table sampling clock is derived from the same frequency reference of the RF carrier, then these modulated signals are periodic. Thus, their frequency-domain representation is composed of discrete lines in an evenly spaced frequency grid, being represented by the following Fourier series (truncated to the th harmonic):

its low-pass equivalent can still be represented by a similar Fourier series whose number of lines is now infinite, but can be truncated to some . Since these spectral lines have frequencies that are multiples of , one can still use the input fundamental (or, as we will see, one of its harmonics) as the phase reference, allowing in this way a straightforward and unique characterization of the input–output phase relations. When the tones are uncorrelated in phase, i.e., when they do not share the same phase reference, they have randomly varying (in time) phases between each other. Since frequency is the time derivative of phase, this implies that they are also separated by random frequency differences and, thus, they are no longer located in an evenly spaced frequency grid. The concept of a fundamental and its harmonics is lost, and, with it, the whole idea of a common phase reference. Using an example, let us see what happens with the IMD sidebands of a two-tone test whose excitation is

(3) , and and are two independent where stochastic variables. Assuming the system is described by the following dynamic polynomial [13]:

(4) are order-dependent time delays necessary to dein which scribe memory effects, its in-band distortion products can be given by [13]

(1) is the time-domain RF signal excitation and in which denotes the amplitude and phase of the th harmonic of the fundamental frequency . This implies that each of the RF tones is a higher order harmonic of the inverse of the baseband period . Thus, assuming the fundamental has zero phase ( taking on the role of the phase reference) all phases have a unique meaning that can be acquired when the fundamental passes through, e.g., zero with a positive slope. When this signal is applied to a “well behaved” nonlinear system,2 as a microwave PA, the RF output is again a multitone whose discrete lines are located in the same frequency grid as the input

(2) again denotes the amplitude and phase where of the th harmonic of the fundamental frequency . Thus, 2This

“well behaved” system will be precisely define in Section III.

(5a)

(5b)

(5c) . where each of the Contrary to the phase-correlated case, now we have output components that cannot be related in phase to any of the input excitations. For instance, although the phase of the output component at , , is correlated to the input at , the phase of the output component at, e.g., cannot be related to either or . In fact, it involves phase information ( and ) that is uncorrelated to any of them. Thus, any attempt to extrapolate the phase measurement methods of phase-correlated stimuli to this case would lead to a random result of zero mean value.

PEDRO AND MARTINS: AMPLITUDE AND PHASE CHARACTERIZATION OF NONLINEAR MIXING PRODUCTS

Fig. 1. IMD phase measurement technique based on a reference nonlinearity and an IMD cancellation loop.

Fig. 2. IMD phase measurement technique based on a reference nonlinearity and a general RF phase measurement instrument.

Nevertheless, it is intriguing to realize that these phases of the IMD products must have a precise physical meaning (i.e., the effects of these mathematical quantities can be measured in the laboratory) as they determine the performance of any PA linearizer. Being impossible to find, in the input signal, a common phase reference to these IMD sidebands, the characterization of the input–output phase relations of spectral regrowth components has relied on generating an auxiliary signal in a so-called reference nonlinearity. Such a signal is made to have the same frequency component as the tone to be characterized, for then being used as the phase reference. Examples of such setups can be seen in Figs. 1 and 2. They are both composed of two branches, one for the DUT, and another for the reference nonlinearity. Considering the example depicted in Fig. 1, the main branch carries the DUT and an attenuator, while the auxiliary branch includes the reference nonlinearity and a calibrated phase shifter. The signals of these two branches are then added, and the amplitude of this vector addition monitored in a spectrum analyzer. Adjusting both the attenuator and phase shifter to get a complete cancellation of the IMD product under test guarantees that the IMD phase of the DUT and reference nonlinearity must differ by exactly 180 . Thus, the sought IMD phase can be inferred from the phase-shifter reading. However, recognizing that the phase characteristics of the reference nonlinearity can be completely unknown in the microwave range, in an improvement to the original method proposed by Suematsu et al. [1], Yang et al. [2] suggested to first down-convert the DUT’s output to then operate the reference nonlinearity at the IF. A different IMD phase measurement technique was proposed by Martins and Carvalho [6], [7]. As seen in the sketch of Fig. 2, it no longer relies on the vector addition of the DUT’s mixing product with the one due to the reference nonlinearity. It first down-converts these two outputs to a common IF, selects the sought mixing product by a very narrow bandpass filter, and then

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Fig. 3. New “absolute” IMD phase measurement setup based on a dual-channel signal acquisition system such as the MTA.

measures their relative phases in a vector network analyzer, or a two-channel oscilloscope. One common feature that arises from this brief review is that, in both of these groups of methods, the underlying idea is always to generate an IMD phase reference in a reference nonlinearity, and then to measure the DUT’s IMD phase with respect to this signal. This has two important drawbacks. The first one is that we will never be able to know the exact phase of the DUT’s IMD unless the reference nonlinearity is calibrated using some other (today still nonexistent for uncorrelated tones) IMD phase measurement setup. Therefore, when, for instance, one is designing a PA linearizer, there is no guarantee that it is correctly adjusted, unless the PA and the linearizer are characterized against the same reference nonlinearity. The second disadvantage of these setups is their high complexity. Since they all rely on two parallel branches, they have associated important costs, face difficulties of calibration, and mostly, they depend on the matching of the components used in the two branches. More recently, a new phase measurement scheme conceived for correlated or uncorrelated two tones has been proposed [9]. Circumventing the need for the reference nonlinearity, it enabled “absolute” IMD phase measurements and in a much simpler way. As shown in Fig. 3, it has a single branch and is based in a dual channel time-domain waveform acquisition system, as a sampling oscilloscope, or a microwave transition analyzer (MTA) [20]. The idea behind this setup was to synchronously digitize the DUT’s input and output waveforms and then perform most of the signal operations—previously done by hardware—in the digital domain. To understand how it works, one first has to realize that the synchronous detectors provided by the vector network analyzer (VNA) of Fig. 2 (which are composed by the product between the DUT’s output and the VNA estimation of the reference nonlinearity signal, and with this latter signal and the reference nonlinearity signal itself, followed by low-pass filtering) are really correlators. Indeed, if the low-pass filters were substituted by ideal integrators and, for simplicity, we could assume that the VNA phase reference were synchronized with the reference nonlinearity signal with zero phase error, then the synchronous detector output result would be

(6)

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where, in this case, stands for the VNA local-oscillator estimated version of the reference nonlinearity signal (assumed of unity amplitude). Since the correlation is zero, except when and have exactly the same frequency, the phase reading would be

(7)

when the IF bandpass filter is tuned for selecting, e.g., the component. Similarly, as explained in [9], in the setup of Fig. 3, an ideal reference signal

III. BASIC NONLINEAR CHARACTERIZATION THEORY For developing a characterization theory capable of integrating the two fields of nonlinear network analysis mentioned in Section I, let us start by a mathematical model of a general “well behaved” nonlinear dynamic network. Such a system is one assumed to be causal, time-invariant, stable, and of finite memory (i.e., in which the output presents a vanishing dependence of the infinitely remote past). This is a class of nonlinear systems useful for describing many microwave devices that do not show instabilities, as transistors, PAs, mixers, and so on. As was shown by Boyd and Chua [21], the response of such a system to an arbitrary excitation can be approximated, with an error as low as desired, by the following polynomial filter (sometimes also called a Volterra filter):

(11)

(8) is first built from the sampled input of the DUT, and then the correlation of (6) is actually computed in the frequency domain via

in which the are the th-order system kernels. For a frequency-domain representation, we assume is an arbitrarily located multisine of incommensurate tones [i.e., : if and only if for any integers ] and randomly varying phases (from which the Fourier series of (1) would be the particular case of evenly separated and phase-correlated tones) [13], [22] (12)

(9) denotes the Fourier transform and and in which are the estimated spectra of the DUT’s input and output, respectively. As explained in [9], this frequency-domain version of the higher order cross-correlation between the DUT’s input and output can be estimated from many pairs of records of and , when and are uncorrelated, or simply from a single pair (obtained during exactly an integer number of periods of the signal) when they are phase correlated. In this latter case, is expressed by (2), the component at corresponds to one particular , say, , and (9) simply becomes

(10) i.e., the Fourier component of . As it will be seen in Section III, the coincidence of this result with the one previously obtained for phase-correlated multitones is far from being accidental, and will enable a precise definition of the general input–output amplitude and phase relations of a dynamic nonlinearity.

which leads to the following response [13], [22]:

(13) With such a model, the input–output relations of our nonlinear system can be defined as the magnitude and phase of the nonlinear transfer functions (NLTFs), that describe the way each of the output components is generated from the input. Note, however, that such a model is only used to relate the desired phase quantities to the input–output nonlinear mapping that represents the system. The NLTFs herein defined provide only a theoretical support to the multidimensional input signal references and the input–output cross-correlations. Hence, the system is not assumed to obey any particular model format. For example, considering our previous two-tone test, we can now define the phase of the input–output relation for the and tones from the first-order input–output cross-correlation, as is usually done in the transfer function identification of linear systems [23]. However, an immediate application of this reasoning to the nonlinear case would lead to a situation

PEDRO AND MARTINS: AMPLITUDE AND PHASE CHARACTERIZATION OF NONLINEAR MIXING PRODUCTS

in which not only the of the form

, but all odd-order NLTFs

would contribute to the output at . Fortunately, since all of these correspond to nonlinear mixing products involving plus pairs of and , they are all correlated in-phase. Thus, what we will do is to define the best linear approximator as the result of these combinations (in control theory, this would be the two sinusoidal input describing function for the output at [24]) to then relate the phase of the output component at to the input . Using this , we now have

(14) This enables the definition of the gain and phase of the output at as the magnitude and phase of calculated from the firstorder cross-spectral density and the input auto-spectral density (or the power spectral density) by

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Similar to what we saw for the first-order case, (17) shows that the input–output relations can be defined for the output component at considering as its signal reference the component of the “third-order input” , where and are the input components at and , respectively. Thus, this definition has a theoretical support and is consistent with the way the phase of the two-tone IMD products has been measured [9]. Obviously this idea can be extended to the input–output relations of any nonlinear mixing product with their correspondent higher order inputs or signal references [9]. Furthermore, its extension to the multitone excitation case is also immediate. In fact, since the tones were considered incommensurate, each output component will be determined by only one combination of input frequencies. In other words, this means that the output component at a certain mixing product will have null cross-correlation to any higher order input, except the one formed using the particular frequency vector . Thus, the amplitude and phase input–output relations can be defined as the magnitude and phase of the following th-order NLTF:

(15) As expected, (15) also shows that the signal reference for the output at is the input at that same frequency. Following the same reasoning, we can now precisely define the input–output amplitude and phase relations for the IMD sidebands from the computation of the third-order input–output cross-correlation and its correspondent third-order cross-spectral densities. Again, all products falling at, e.g., , will be of the form allowing a definition of a new best third-order approximator (the two-sinusoidal input describing function for the output at [24]). This NLTF can then be calculated from the third-order cross-correlation

(18) In summary, given a multitone test with incommensurate frequencies, the amplitude and phase relations of any linearly or nonlinearly generated component can be defined—via the higher order cross- and auto-correlations shown in (18)—with respect to a higher order input (the reference signal) built from the combination of the input frequency components used to generate that mixing product. The required auto- and cross-correlations can be estimated from the synchronous acquisition of a number of records of constant length of both the input and output, and then calculating their discrete Fourier transforms (DFTs) and so that [9], [25]

(19) and (16) using (20)

(17)

Note that, because each of the input tones uses its own phase reference, there is no single phase reference available at either the input or output to trigger the acquisitions. Also, note that, since the frequency components of (12) were considered as incommensurate, and are necessarily aperiodic. There-

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Thus, there is no unique way to define the input signal reference, as we did before. However, we could still apply the concepts previously proposed, and thus, determine an unique IMD phase measurement, if we assume that this wanted input signal reference is the combined input products of minimum order that gives nonnull correlation with the desired output frequency component

(21) Fig. 4. Illustrative DFTs Y (! ) obtained from the output records microwave low-power HEMT subject to a two-tone test.

y (t) of a

Fig. 5. Magnitude and phase moving averages of the estimated H (! ; ! ; ! ) obtained from (18) via the third-order cross- and auto-correlation of (19) and (20) of a microwave HEMT subject to a two-tone test. The sideband tone considered was the one referred as the “IMD Tone” in Fig. 4.

0

fore, the DFTs of each of the acquired records will necessarily present spectral leakage, with varying amplitudes and randomized phases, as is illustrated from the two tone-test results shown in Fig. 4. Thus, the amplitude and phase measurements obtained from (18) will only tend to their correct values when is made sufficiently large. This is shown in Fig. 5 where two-tone test results were used to build the magnitude and phase moving averages of from (18) to (20). In case of commensurate tones, i.e., when every tone of the input and the output is correlated in phase to a unique phase reference, and the frequencies are evenly located in a uniform grid, there is a large number of ways the input tones can be combined to generate a particular output mixing product. For example, the fourth harmonic of the fundamental can be generated from combining four times that fundamental, from combining the third harmonic with the fundamental or even from a linear transposition of the fourth harmonic possibly present at the input. Thus, there will be a much smaller number of independent output mixing products and, thus, of available observations, than the number of distinct NLTFs to be identified. This demonstrates that, contrary to a widely adopted thought, a harmonically related multisine cannot be used to extract the system model because it is unable to expose all system’s properties [26], [27].

Since both the input and output can now be described by the Fourier series of (1) and (2), we could use, for example, the fundamental and its harmonics as the higher order input signal references, as is usually done for this type of test signal [12]. In fact, as the calculation of the th-order coefficient of the Fourier series of the output signal can be understood as the cross-correlation of with the unit amplitude th harmonic of the input fundamental, it works similarly to the uncorrelated case. Since all input and output tones are correlated to the fundamental , we can use that fundamental as the single tone base and still define the amplitude and phase of any output component, say, at , as the cross-correlation of it to the correspondent higher order input signal reference whose frequency is exactly . That is, we can still define a certain and compute it by

(22) as we did before. Unfortunately, in practical RF and microwave nonlinear measurements, the most common situation is to use bandpass signals that do not have the component readily available. That is, and the direct application of (22) would lead to an impossibility. The way this has been circumvented for nonlinear phase measurements is to estimate the phase of via a phase detrending procedure [12]. Conversely, this cannot provide the phase of the spectral regrowth components in case of multitones of incommensurate frequencies. Indeed, that procedure is restricted to tests made with commensurate tones and applied to measure the absolute phase change introduced by the system [10]–[12] provided the phases of two consecutive input tones are known [12] or to find only the phase deviation if this knowledge of the input excitation is unavailable. In summary, using the presented method, the main difference we will face with respect to the phase uncorrelated case is that, because both the input and output now depend on a single phase reference, we can use that phase reference to trigger the acquisition of the records and . Also, since they are both periodic, we can make the acquisition during an integer number of periods. Doing that implies that the obtained DFTs of each

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Fig. 6. Nonlinear dynamic system used in the simulated results.

of the records will now have deterministic and constant amplitudes and phases, and the higher order cross- and auto-spectral densities of (19) and (20) will converge to their final value in only one acquisition (except, perhaps, for random measurement noise, which will need to be averaged out). IV. SIMULATED EXAMPLE To validate the proposed theory, we first applied it to a controlled environment: the simulation of the dynamic nonlinear system shown in Fig. 6. As seen, it is composed of a static nonlinearity and a linear filter in a cascade arrangement known as the Hammerstein model. The nonlinearity is a hyperbolic tangent (e.g., an amplifier with saturation), and the filter is a Butterworth design of 1.9-GHz center frequency and 10-MHz bandwidth. As the nonlinearity is memoryless, it does not introduce any phase delay in the fundamentals, but contributes with 180 for the distortion products (gain compression). The phase added by the filter can be known from the filter response and the frequency position of the desired output component. Hence, it can be easily concluded that the th-order NLTFs of this Hammerstein model must be given by

Fig. 7. Simulated evolution of the phase of the fundamental tone f with the number of considered multisine realizations.

(23) where is a real coefficient dependent on the nonlinearity and, thus, whose phase can only be 0 or 180 , and and are the amplitude and phase response functions of the linear filter at the frequency position . To show the ability of the current method in determining the phase of output components, the system of Fig. 6 was excited with nine commensurate tones centered at GHz and with MHz frequency spacing. The tested tones were one fundamental, located at , and one IMD component at . The first test assumed all input fundamentals had and served as a control experiment. In this case, all distortion components falling at any frequency component share the same phase, which is either the one imposed by the filter, for the fundamentals, or that plus 180 , for the distortion. These are represented by the dashed horizontal lines shown in Figs. 7 and 8, respectively. The next test used a series of realizations of our nine-tone multisine in which the amplitudes were kept constant, but the phases were randomized. As shown in [23], the ensemble of a large number of tests of this random multisine is equivalent (in a statistical sense) to a single test made of incommensurate frequencies. Hence, it emulates the wanted test made with an

Fig. 8. Simulated evolution of the phase of the distortion tone f number of considered multisine realizations.

with the

incommensurate tone excitation, allowing an independent determination of all kernels of different phase. The obtained results are shown in Fig. 7 for (one of the fundamentals) and in Fig. 8 for (the distortion product generated by mixing and such that ). Note that the phase estimator is unbiased and consistent since it tends to the desired result with a steadily decreasing variance. In fact, it is noticed that, although the error is decreasing in a long-term perspective, it does not always decrease in a short period. That was confirmed to be due to an imperfection of the random number generator used to create the randomized phases, not to the tested phase measurement algorithm. Another interesting observation is that the phase of the fundamental converges to the final value in a significantly faster manner than the distortion tone. That is due to the fact that, while in we have a large number of distortion components with different phases, but of comparable amplitude, in case of the fundamental, any component having a different phase must necessarily be a distortion product and, thus, having a much smaller amplitude.

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(a) Fig. 10. Moving average of the measured phase of the spectral regrowth tone indicated in Fig. 9(b) as “IMD Tone.”

(b) Fig. 9. Illustrative example of the: (a) waveform and (b) DFT of one of the records y (t) and Y (! ) acquired at the output of our DUT. (a)

In conclusion, these simulation results fully validate the measurement procedure, as they prove that the obtained results indeed converge to the predicted theoretical ones. V. EXPERIMENTAL RESULTS In order to illustrate the practical application of this new phase measurement technique, we will now present some experimental results obtained with a measurement setup similar to the one depicted in Fig. 3, but now using a ten-tone stimulus. The used DUT was again the low-power microwave high electron-mobility transistor (HEMT) used to obtain the measurement results reported in Figs. 4 and 5. Since we had not ten independent generators, we had to rely on a single multitone signal generator (HP ESG E4433B). Unfortunately, these multitone generators produce periodic phase-correlated tones. Thus, to emulate the general case of incommensurate tones, we followed the procedure adopted in the simulation example, performing the test with a random multisine [23]. In addition, the MTA sampler trigger was set to operate in a free-running mode. Fig. 9(a) and (b) shows an example of the acquired output record waveforms and DFTs, respectively. Fig. 10 shows the moving average of the measured phase corresponding to the spectral regrowth tone indicated in Fig. 9(b). Finally, as an illustration of the characterization capabilities of the presented theory, Fig. 11 shows typical measurement results of the input–output amplitude [see Fig. 11(a)] and phase

(b) Fig. 11. Typical: (a) input–output amplitude and (b) phase relations of the fundamentals and sidebands, as defined in (18), resulting from a two-tone input power sweep test of a class-C amplifier.

[see Fig. 11(b)] relations of the fundamentals (1.8 GHz 50 kHz and 1.8 GHz 50 kHz) and sidebands (1.8 GHz 150 kHz and 1.8 GHz 150 kHz), as defined in (18), when a class-C biased commercial drop-in amplifier was subject to a two-tone input power sweep. Note the amplitude and phase asymmetry of the two IMD tones and the expected change of the sideband phases when the amplifier crosses the transition between the smalland large-signal operating regimes (where a smooth large-signal IMD sweet spot is visible).

PEDRO AND MARTINS: AMPLITUDE AND PHASE CHARACTERIZATION OF NONLINEAR MIXING PRODUCTS

VI. CONCLUSION This paper has introduced a theoretically supported way to precisely define the input–output phase relations of microwave nonlinear dynamic systems. Beyond proposing an input signal reference for the phase of each output tone, both on frequency components shared by the input and spectral regrowth, it provided a way to characterize the phase of distortion products independently of any arbitrarily selected reference nonlinearity. The proposed theory—supported by an ideal simulation example and some illustrative laboratory measurements to give it a physical meaning—is believed to play a paramount role on the future extraction of nonlinear behavioral models and the design of PA linearizers. ACKNOWLEDGMENT The authors would like to acknowledge P. M. Lavrador, Instituto de Telecomunicações, Universidade de Aveiro, Aveiro, Portugal, for performing the simulations presented in Section IV. The authors would also like to express their gratitude to Dr. D. Root, Guest Editor of this TRANSACTIONS’ Special Issue, and the reviewers for their comments and suggestions. REFERENCES [1] N. Suematsu, Y. Iyama, and O. Ishida, “Transfer characteristics of IM3 relative phase for a GaAs FET amplifier,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 12, pp. 2509–2514, Dec. 1997. [2] Y. Tang, J. Yi, J. Nam, B. Kim, and M. Park, “Measurement of two-tone transfer characteristics of high-power amplifiers,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 3, pp. 568–571, Mar. 2001. [3] J. Vuolevi, T. Rahkonen, and J. Manninen, “Measurement technique for characterizing memory effects in RF power amplifiers,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 8, pp. 1383–1389, Aug. 2001. [4] J. Vuolevi and T. Rahkonen, Distortion in RF Power Amplifiers. Norwood, MA: Artech House, 2003. [5] J. Dunsmore and D. Goldberg, “Novel two-tone intermodulation phase measurement for evaluating amplifier memory effects,” in 33rd Eur. Microw. Conf. Dig., Munich, Germany, Oct. 2003, pp. 235–238. [6] J. Martins and N. Carvalho, “Spectral filtering setup for uncorrelated multi-tone phase and amplitude measurement,” in Proc. 34th Eur. Microw. Conf., Amsterdam, The Netherlands, Oct. 2004, pp. 201–204. [7] J. Martins and N. Carvalho, “Multitone phase and amplitude measurement for nonlinear device characterization,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 6, pp. 1982–1989, Jun. 2005. [8] H. Ku and J. Kenney, “Behavioral modeling of nonlinear RF power amplifiers considering memory effects,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 12, pp. 2495–2504, Dec. 2003. [9] J. Pedro, J. Martins, and P. Cabral, “New method for phase characterization of nonlinear distortion products,” in IEEE MTT-S Int. Microw. Symp. Dig., Long Beach, CA, Jun. 2005, pp. 971–974. [10] K. Remley, D. Schreurs, D. Williams, and J. Wood, “Extended NVNA bandwidth for long-term memory measurements,” in IEEE MTT-S Int. Microw. Symp. Dig., Fort Worth, TX, Jun. 2004, pp. 1739–1742. [11] ——, “Simplifying and interpreting two-tone measurements,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 11, pp. 2576–2584, Nov. 2004. [12] K. Remley, D. Williams, D. Schreurs, G. Loglio, and A. Cidronali, “Phase detrending for measured multisine signals,” in 61st ARFTG Conf. Dig., Philadelphia, PA, Jun. 2003, pp. 73–83. [13] J. Pedro and N. Carvalho, Intermodulation Distortion in Microwave and Wireless Circuits. Norwood, MA: Artech House, 2003. [14] J. Pedro and N. Carvalho, “On the use of multi-tone techniques for assessing RF components’ intermodulation distortion,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 12, pp. 2393–2402, Dec. 1999.

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[15] J. C. Pedro and N. B. Carvalho, “Characterizing nonlinear RF circuits for their in-band signal distortion,” IEEE Trans. Instrum. Meas., vol. 51, no. 6, pp. 420–426, Jun. 2002. [16] G. Loglio, J. Jargon, and D. DeGroot, “Phasor angle definition suitable for intermodulation measurements,” in 65th ARFTG Conf. Dig., Long Beach, CA, Jun. 2005, pp. 83–89. [17] A. Zhu, M. Wren, and T. Brazil, “An efficient Volterra-based behavioral model for wideband RF power amplifiers,” in IEEE MTT-S Int. Microw. Symp. Dig, Philadelphia, PA, Jun. 2003, pp. 787–790. [18] C. Silva, “Time-domain measurement and modeling techniques for wideband communication components and systems,” Int. J. RF Microw. Comput.-Aided Eng., vol. 13, pp. 5–31, Jan. 2003. [19] C. Silva, A. Moulthrop, and M. Muha, “Introduction to polyspectral modeling and compensation techniques for wideband communications systems,” in 58th ARFTG Conf. Dig., San Diego, CA, Nov. 2001, pp. 1–15. [20] “Microwave Transition Analyzer User’s Guide,” Agilent Technol., Palo Alto, CA, Nov. 1992. [21] S. Boyd and L. Chua, “Fading memory and the problem of approximating nonlinear operators with Volterra series,” IEEE Trans. Circuits Syst., vol. CAS-32, no. 11, pp. 1150–1161, Nov. 1985. [22] S. A. Maas, Nonlinear RF and Microwave Circuits. Norwood, MA: Artech House, 2003. [23] J. Schouckens and R. Pintelon, System Identification—A Frequency Domain Approach. Piscataway, NJ: IEEE Press, 2001. [24] A. Gelb and W. Vander Velde, Multiple-Input Describing Functions and Nonlinear System Design. New York: McGraw-Hill, 1968. [25] J. Bendat and A. Piersol, Engineering Applications of Correlation and Spectral Analysis. New York: Wiley, 1993. [26] S. Boyd, Y. Tang, and L. Chua, “Measuring Volterra kernels,” IEEE Trans. Circuits Syst., vol. CAS-30, no. 8, pp. 571–577, Aug. 1983. [27] L. Chua and Y. Liao, “Measuring Volterra kernels—II,” Int. J. Circuit Theory Applicat., vol. 17, pp. 151–190, Apr. 1989. José C. Pedro (S’90–M’95–SM’99) was born in Espinho, Portugal, in 1962. He received the Diploma and Doctoral degrees in electronics and telecommunications engineering from the Universidade de Aveiro, Aveiro, Portugal, in 1985 and 1993, respectively. From 1985 to 1993, he was an Assistant Lecturer with the Universidade de Aveiro, and a Professor since 1993. He is currently a Senior Research Scientist with the Instituto de Telecomunicações, Universidade de Aveiro, as well as a Full Professor. He has authored or coauthored several papers appearing in international journals and symposia. He coauthored Intermodulation Distortion in Microwave and Wireless Circuits (Artech House, 2003). His main scientific interests include active device modeling and the analysis and design of various nonlinear microwave and opto-electronics circuits, in particular, the design of highly linear multicarrier PAs and mixers. Dr. Pedro is an associate editor for the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES and is also a reviewer for the IEEE Microwave Theory and Techniques Society (IEEE MTT-S) International Microwave Symposium (IMS). He was the recipient of the 1993 Marconi Young Scientist Award and the 2000 Institution of Electrical Engineers (IEE) Measurement Prize.

João P. Martins (S’06) was born in Sever do Vouga, Portugal, on May 13, 1973. He received the B.Sc. and M.Sc. degrees from the Universidade de Aveiro, Aveiro, Portugal, in 2001 and 2004, respectively. Since 2001, he was a Researcher with the Instituto de Telecomunicações, Universidade de Aveiro. His main interests are wireless systems and nonlinear microwave circuit design.

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Design and Performance Analysis of Mismatched Doherty Amplifiers Using an Accurate Load–Pull-Based Model Oualid Hammi, Student Member, IEEE, Jérôme Sirois, Member, IEEE, Slim Boumaiza, Member, IEEE, and Fadhel M. Ghannouchi, Senior Member, IEEE

Abstract—An active load–pull-based large-signal modeling approach, suitable for designing and optimizing load modulated amplifiers such as Doherty or linear amplification using nonlinear components based amplifiers, is proposed. A Doherty amplifier was designed by optimizing the dynamic loads seen by the amplifier’s transistor using a large-signal load–pull-based behavior model built into computer-aided-design software. Simulation and measurement results showed good agreement, while results obtained using an empirical model of this transistor demonstrated discrepancies. The load–pull-based model was then used to study performance degradation of the Doherty amplifier when load impedance was moved out from the perfect 50 . It has been shown that the load mismatch can greatly affect the linearity and efficiency performance of the amplifier unless its phase is controlled and kept within a specific range. A load mismatch system level compensator scheme, capable of restituting the linearity loss and maintaining the power-added efficiency close to its maximum range, is proposed.



Index Terms—Active load modulation, Doherty amplifier, high efficiency, load–pull measurement.

I. INTRODUCTION HE constraints imposed by the new wireless communications access technologies, such as multicarrier wideband code division multiple access (MC-WCDMA) and orthogonal frequency-division multiplexing (OFDM), are becoming more and more stringent. In fact, linearity and power efficiency are both crucial factors in designing communication transmitters since they permit signal integrity upholding, base-station deployment and operating cost reduction, and an increase in battery life for mobile stations. Many techniques have been proposed during the last few decades to enhance the transmitters’ linearity and power efficiency by using class-AB amplifiers along with a linearization technique [1]. However, they have shown relatively limited performances when dealing with the new standards’ signals that have highly varying envelopes.

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Manuscript received January 11, 2006; revised April 29, 2006. This work was supported by the Natural Sciences and Engineering Research Council of Canada, by the Canadian Space Agency, by TRLAbs, and by the Informatics Circle of Research Excellence. O. Hammi, S. Boumaiza, and F. M. Ghannouchi are with the iRadio Laboratory, Electrical and Computer Engineering Department, University of Calgary, Calgary, AB, Canada T2N 1N4 (e-mail: [email protected]; [email protected]; [email protected]). J. Sirois is with the Poly-Grames Research Center, Département de Génie Électrique, École Polytechnique de Montréal, Montréal, QC, Canada H3V 1A2, and also with Freescale Semiconductor, Tempe, AZ 85284 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2006.877810

Recently, new approaches, based on load modulation [2]–[18] and switching-mode amplifiers [19]–[22] have been introduced to overcome the limitations of the older techniques. These new techniques increase the power efficiency of the power amplifier (PA), even for high output-power backoff (OPBO) values, by changing the amplifier’s load according to the input signal amplitude. Doherty amplifiers [2]–[14] and linear amplification using nonlinear components (LINC) transmitters [15]–[18], which are the two most common techniques that fall into this category, are currently the research focus of many laboratories. However, they still face implementation difficulties due to imprecise characterization and modeling of the RF transistors that either keep such approaches in the conceptual phase or limit their outcomes. Hence, accurate and comprehensive transistor models are essential for running realistic simulations for the optimization of the RF front-end stage performance early in the design step. Fundamentally, Doherty amplifiers are composed of two amplifiers, denoted as carrier and peaking amplifiers, that operate in classes B and C, respectively. The use of a nonisolated power combiner, which ensures an instantaneous summation of the two amplifiers outputs, initiates an active load modulation. Hence, the transistor model is required for an accurate prediction of its complex gain (AM/AM and AM/PM curves) and drain current characteristics to properly simulate and design the Doherty amplifier. In order to obtain an accurate model of a given device, this latter must be characterized in linear and nonlinear operating classes, under realistic conditions that are very similar to those that will be encountered during regular operations. This requires the use of load–pull measurement that allows for the extraction of precise and reliable data when parameters such as input power, source and load impedances, are varied. Load–pull measurement results are composed of a complete set of information that can be used in the design of an actively load modulated amplifier. Recently, highly integrated Doherty amplifier implementations suitable for handset devices were proposed [9]–[14]. These implementations addressed the critical issue of the transformers that were substituted with discrete element equivalent transformers using more integrated and/or networks [9], [10]. Thus, even though the use of Doherty amplifiers in handset devices is not widespread for the time being, it is perceived as an enabling high-efficiency amplification technique for the near future and is gaining more and more research and development attention.

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HAMMI et al.: DESIGN AND PERFORMANCE ANALYSIS OF MISMATCHED DOHERTY AMPLIFIERS USING ACCURATE LOAD–PULL-BASED MODEL

In the literature, the design of Doherty amplifiers has always been limited to optimal matched loads (50 ). However, in practical situations, especially for handset terminals, load mismatch may occur due to variations in the antenna impedance. This problem is often encountered in handset transceivers where no isolators are used, in order to keep designs cost effective and highly integrated. The load mismatch degrades the linearity performance of the amplifier, as well as its power efficiency. The above-mentioned sensitivity of the amplifier’s performances to the load mismatch is even more pronounced for amplifiers that are designed with and utilize beneficially active load modulation principles such as Doherty or LINC amplifiers. Accordingly, the effect of the load mismatch on PA performance is a critical issue that has been recently studied [23]–[26]. These studies primarily considered continuously driven single-branch PA’s (classes A and AB) linearity degradation under load mismatch. To overcome the observed linearity degradation, three linearity preservation techniques that adaptively control the output power of the amplifier, its load line, or supply voltage have been proposed [24]. However, the power-efficiency degradation of the amplifier was not considered. A dynamic output phase technique was proposed to control the phase of the reflection coefficient at the output of the PA [25]. This circuit tuned the reflection coefficient’s phase to an optimal phase, which alleviated the degradation in the amplifier performances. In this paper, the reported results did not investigate the linearity characteristics over a power sweep, in terms of AM/AM and AM/PM characteristics, and focused only on the empirical and parameters. Moreover, the efficiency was only given for a fixed power level. The limitation of this approach is that the adjustment of the reflection coefficient phase to its optimal value keeps the performance of the amplifier, in terms of , , as well as its gain and power-added efficiency (PAE), at a given power level. However, there is no information on how these performances can be maintained over a power level range and a certain frequency bandwidth. Qiao et al. [26] proposed a real-time technique that corrects for load mismatch by measuring the voltages at three points of a 50- transmission line at the PA output. According to the measured voltages, the parameters of a tunable matching network are adjusted iteratively. Thus, this technique is not suitable to compensate for rapid and sudden variations of load mismatch in handset devices and is very sensitive to the possible local minima of the searching algorithm. In this paper, a large-signal characterization and modeling approach suitable for the design of actively load modulated amplifiers is presented, evaluated, and applied to the design of an -band Doherty amplifier prototype. The proposed model is then used to investigate the sensitivity of the Doherty amplifier to the output load mismatch. Section II describes the extraction and calculation of a load–pull table-based model. Next, the design and optimization process of a Doherty amplifier, based on the previously established model, is presented in Section III. Measured performances are compared to those obtained by simulation, using the load–pull-based model and an empirical model extracted from the literature [27]. Once the model accuracy is proven and its robustness is assessed in a Doherty amplifier configuration, the load–pull based model is utilized to study performance variations experienced when

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the amplifier’s load moves away from perfect 50 , as shown in Section IV. This study is confirmed in Section V by experimental results showing the behavior of the Doherty amplifier under load mismatch conditions. This establishes the acceptable impedance variation before the performance degradation reaches an unacceptable level. Finally, a system level scheme that aims at correcting for the performance degradation due to load mismatch is proposed in Section VI. II. EXTRACTION OF A TABLE-BASED MODEL The design and simulation of a load modulated amplifier, using the Doherty or LINC techniques, requires a robust and accurate nonlinear model of the selected transistors. The load–pull measurements technique is by far the most precise means to extract the nonlinear behavior of a transistor as a function of source and load impedances at a quiescent biasing condition . Some design tools, such as AmpLoadPull in Agilent Technologies’ Advanced Design System (ADS) software, are now available to use load–pull data in the design software. This simulation tool is based on the following function: (1) where is the input reflection coefficient of the device, is the complex gain, is the power available from source and is the load reflection coefficient. These parameters are extracted from load–pull measurement results [7]. Output power and actual drain and gate currents were also measured in order to calculate the PAE of the transistor as follows: (2) Once the measurement data is collected, it has to be manipulated and converted to be compatible with AmpLoadPull. As defined in (1), this model requires the knowledge of and , which can be deduced from the measurement results. The AmpLoadPull model assumes that the input source has an impedance of 50 , which implies that is equal to the device’s input power . To generalize the use of this model, actual power available from the source is used instead of power delivered to transistor. To get from , the following relationship has to be used: (3) The measurement results are treated using The MathWorks’ MATLAB software procedure that allows for the extraction of the required data needed to construct the model. A CITI file is then generated based on these results and imported into ADS for simulations. A different sub-model has to be generated for every quiescent biasing point. The model is excellent in simulating dynamic load matching conditions; however, it suffers from two main limitations. First, it is only valid for a single frequency, and secondly, it requires

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Fig. 2. Optimal (measured) and model predicted (simulated) load impedance displacements at drain reference plane for carrier amplifier in 10-dB dynamic range from P sat.

Fig. 1. Flowchart of the Doherty amplifier combining network optimization process.

interpolation to estimate measurement data away from the actual measurement points. The desired simulation conditions must remain inside the validity range of the model at all times to ensure accuracy of the results. This means that a critical number of measurements are required to obtain an accurate and flexible model. III. DOHERTY AMPLIFIER DESIGN Developing the layout of a Doherty amplifier is not a straightforward process. The load impedance interaction between carrier and peaking amplifiers, which is required for observing the Doherty effect, relies on a complex impedance-matching network. Fortunately, the design process can be started from existing class AB and class C initial designs. In this study, a model of the transistor operating in class AB was used to design the carrier amplifier. The same process was applied to design the peaking amplifier, but this time using a model of the transistor operating in class C. Once these two layouts were available, the Doherty amplifier design process could be initiated. This process [5]–[7] is described in Fig. 1 and requires optimization. The optimization step is fundamental since the addition of a theoretical impedance transforming network will not automatically lead to an optimal Doherty amplifier for many reasons. First, the Doherty amplifier theory often assumes that, at maximum input power, both amplifiers deliver the same output power and see the same virtual impedance at the power combiner reference plane. This assumption is not valid when using the same device for both amplifiers since one is operated in deep class AB or class B, while the other one is operated in

Fig. 3. Simulated reflection coefficients at the drain of the peaking amplifier versus the theta parameter.

class C. This means that they cannot achieve the same output power at saturation. The same virtual load can practically never be reached for both amplifiers at the power combiner reference plane. To compensate for this effect, the impedances values of the Doherty power-combining network can be readjusted to present 50 to both amplifiers at maximum output power. Alternatively, the carrier and peaking amplifiers load transforming network can be readjusted slightly and optimized by using segment of a 50- transmission line. The electrical length of the transmission-line segment, which is denoted as theta, is calculated so that the amplifier delivers its maximum output power into a resistive load that is slightly different from 50 . Second, considering that we are using the same device to build both carrier and peaking amplifiers, the phase compression experienced in both amplifiers may not be exactly the same. The phi offset line at the input of the device allows the designer to compensate for eventual differences in the propagation delay and in the shape of the two curves for high input power levels. Finally, the active load modulation at the transistor’s drain reference plane may be optimal only at maximum output power. The variation of the previously introduced theta parameter allows the designer to obtain a better correlation between the two curves at all input power levels (Fig. 2). Moreover, the optimization of the theta parameter makes possible the adjustment of the locus of load displacements to fit those derived from the load–pull measurements. Fig. 3 shows the load displacement at the output of the peaking amplifier as a function of theta.

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Fig. 4. Designed Doherty amplifier prototype. (Color version available online at: http://ieeexplore.ieee.org.) Fig. 6. Simulatedand measured AM/PM curves of the Doherty amplifier. (Color version available online at: http://ieeexplore.ieee.org.)

Fig. 5. Simulated and measured AM/AM curves of the Doherty amplifier. (Color version available online at: http://ieeexplore.ieee.org.) Fig. 7. Simulated and measured PAE curves of the Doherty amplifier. (Color version available online at: http://ieeexplore.ieee.org.)

IV. ACCURACY ASSESSMENT OF THE PROPOSED APPROACH The design approach described in Section III was used to build the Doherty amplifier prototype shown in Fig. 4. In order to assess the accuracy of the proposed design methodology, the AM/AM and AM/PM characteristics, as well as the PAE of the simulated Doherty amplifier were compared to those measured using the designed prototype. As demonstrated in Figs. 5–7, good agreement was obtained between simulation and measurement results. These figures also present the characteristics (complex gain and PAE) of the simulated Doherty amplifier when the proposed table-based model is substituted by an empirical model [27]. In this case, the empirical model was unable to accurately predict the PA’s behavior. Indeed, even though the empirical-model-based approach leads to relatively good results in modeling and designing single-ended amplifiers such as class

AB, as reported in [27], it does not show the same accuracy when the transistors are used in designing Doherty amplifiers, especially in the 1.9–2-GHz frequency range. This might be attributed to the fact that Doherty amplifiers design requires an accurate nonlinear transistor model at both low- and high-drive levels to take into account the saturation of the carrier amplifier when the peaking amplifier turns on. In addition, it requires an accurate transistor model from classes AB to B and class C operation modes. The observed discrepancy between the analytical model, the load–pull-based model, and the measurement results, shown in Figs. 5–7, might be attributed to the difference of the class of operation (classes AB, B, and C) and the difference in the power-domain validation range. Indeed, the power-domain large-signal model validation, as proposed in [27], was carried

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V. ANALYSIS OF THE DOHERTY AMPLIFIER’S SENSITIVITY TO LOAD MISMATCH

Fig. 8. Impedances seen at the drains of the carrier and peaking amplifiers versus input power. (proposed model: PM, empirical model: EM). (Color version available online at: http://ieeexplore.ieee.org.)

Fig. 9. Simulated trajectories of the impedances seen by the transistors versus input power level. (Color version available online at: http://ieeexplore.ieee.org.)

out up to 20 dBm as input power, which is approximately 10 dB lower than the saturation point. To further investigate the origin of the observed discrepancy between the results obtained using both models, the active load modulation presented to both the peaking and carrier amplifiers’ drain reference planes were derived. Fig. 8 shows that the impedances predicted by the empirical model are quite different from those obtained when the proposed model is used. According to the previous results, the implemented model is accurate enough and can be used to carry out transistor-level investigations of the Doherty amplifier’s behavior for which measurements are difficult or even impossible to perform. As an application, the dynamic trajectories of the impedances seen by both the carrier and peaking amplifiers at the combiner level were simulated. These trajectories correspond to the variation of the impedances seen by both amplifiers versus the input power level. Fig. 9 shows that both trajectories plotted in the Smith chart converge toward the chart center (50 ) without reaching it for the reasons given in Section III. Theoretically, the impedance seen by the carrier amplifier should vary from to , whereas the impedance seen by the peaking amplifier should move from that of an open circuit for low input power levels to as the driving level increases.

In Section IV, the accuracy of the proposed Doherty amplifier’s model was assessed. Moreover, it has been shown that this model can be used to simulate the behavior of the Doherty amplifier at the transistor level and observe parameters that can be very difficult or require nonstandard and complex measurement setups such as the active impedances seen by each of the two transistors. Here, the model implemented on Agilent Technologies’ ADS software is used to investigate the sensitivity of the Doherty amplifier to load mismatch. Indeed, the load impedance seen by the Doherty amplifier may vary considerably following a variation in the transmitting antenna’s impedance, which can be caused by changes in the antenna’s vicinity. Such variations in the impedance seen by the Doherty amplifier involve important changes in the amplifier’s characteristics, namely, its gain compression characteristics and its power-efficiency performance. This is a major concern in the case of Doherty amplifiers where the correct operation closely relies on the active load modulation concept. To evaluate the sensitivity of the Doherty amplifiers to the load mismatch, the designed amplifier was simulated under load mismatch conditions by varying the magnitude of the load reflection coefficient from 0- to 0.2-in steps of 0.05. For each of these values, the phase of the load reflection coefficient was swept over 360 in increments of 45 . The obtained PAE values for various input power backoff (IPBO) levels are presented in Fig. 10. According to this figure, one can distinguish two ranges for the phase of the load reflection coefficient. Indeed, for phases ranging around 270 , the PAE of the Doherty amplifier is slightly affected by the load mismatch over the whole sweep range of the load reflection-coefficient magnitude. The PAE is even increased by a few percents. However, out of this phase range, the Doherty amplifier’s PAE decreases as the magnitude of the load reflection coefficient increases. This degradation is more pronounced for a phase span around 135 . To better point out the behavior of the Doherty amplifier under load mismatch conditions, the PAE curves versus the input power levels were derived for various load reflection coefficients having the same magnitude and different phases. These results, presented in Fig. 11, show that for 270 , the predicted PAE characteristic is quite similar to that obtained with a matched load. However, as this phase value increases from 0 to 180 , the discrepancy between the PAE obtained using a matched load and that obtained with the mismatched load becomes more pronounced. Finally, one can notice that as long as the magnitude of the load reflection coefficient is kept below 0.05 (approximately 26-dB return loss), no significant degradation in the PAE is observed over the whole phase sweep range. Such requirement is difficult to constantly achieve in practical designs. The output power of the Doherty amplifier, also plotted for the same IPBO level, shows similar dependency upon the variation of the load reflection-coefficient magnitude and phase, as shown in Fig. 12. The gain of the Doherty amplifier is also sensitive to the load mismatch. In order to investigate its dependency upon the magnitude and phase of the load reflection coefficient, several sim-

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Fig. 11. Simulated PAE of the Doherty amplifier versus the phase of the load reflection coefficient. (mag(0 ) = 0:15). (Color version available online at: http://ieeexplore.ieee.org.)

Fig. 10. Simulated PAE of the Doherty amplifier versus the magnitude and phase of the load reflection coefficient. (a) At 2-, 4-, and 6-dB IPBO. (b) At 6-dB IPBO. (Color version available online at: http://ieeexplore.ieee.org.)

ulations were carried out. Fig. 13 illustrates this dependency. In Fig. 13(a), the AM/AM curves are plotted versus the phase of the load reflection coefficient, the magnitude of which is set to 0.15. Additionally, the same curves are plotted in Figs. 13(b) and (c) for different load reflection coefficients with a magnitude sweep from 0 to 0.2 and with the phase set to 180 and to the optimal value of 270 , respectively. Accordingly, it appears that the AM/AM characteristics of the Doherty amplifier depend on both the magnitude and phase of the reflection coefficient seen at the load side. However, as observed for the PAE and output power, the dependency of the PA’s nonlinear characteristics (AM/AM and AM/PM) is less important when the phase of the load reflection coefficient is kept around 270 . To confirm the simulation results, an experimental validation of the mismatch effect was carried out on the designed Doherty amplifier. For this purpose, the load reflection coefficient was set to the desired values using an impedance tuner. For every magnitude of , the phase was swept from 0 to 360 in steps of 45 . The obtained results are plotted in Fig. 14. This figure shows that for all the values of , the PAE sensitivity to the phase of is

Fig. 12. Simulated output power variation of the Doherty amplifier at 6-dB backoff versus the magnitude and phase of the load reflection coefficient. (Color version available online at: http://ieeexplore.ieee.org.)

similar to what was predicted by the simulation results. Indeed, the PAE greatly decreases around 135 and reaches its highest values around 270 and 315 . Moreover, this figure shows that as the magnitude of is increased, the efficiency drop at 135 gets more significant. VI. PROPOSED SYSTEM FOR LOAD MISMATCH COMPENSATION As illustrated above, the load mismatch at the output of the Doherty amplifier affects both the linearity and power efficiency of the PA over the whole input power range. It has been shown that the power efficiency is mainly affected by the phase of the load reflection coefficient, whereas the linearity depends on the complex value of the load reflection coefficient. Herein, a first attempt for the compensation of the load mismatch effects

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Fig. 14. Measured PAE of the Doherty amplifier versus the magnitude and phase of the load reflection coefficient at 6-dB IPBO. (Color version available online at: http://ieeexplore.ieee.org.)

Fig. 15. Proposed load mismatch effects compensation.

at the output of the PA to control the phase of the load reflection coefficient, the proposed approach takes into account the linearity degradation due to the changes in the complex load reflection coefficient. Indeed, once the phase adjustment is performed, a multiple lookup table predistorter is applied to compensate for the nonlinear complex gain variation as function of the magnitude of reflection coefficient. This noniterative approach makes it possible to track rapid variations of the load reflection coefficient. VII. CONCLUSION Fig. 13. Simulated AM/AM curves of the Doherty amplifier. (a) AM/AM versus phase of the load reflection coefficient for mag(0 ) = 0:15. (b) AM/AM versus magnitude of the load reflection coefficient for phase(0 ) = 180 . (c) AM/AM versus magnitude of the load reflection coefficient for phase(0 ) = 270 . (Color version available online at: http://ieeexplore.ieee.org.)

is proposed. The compensator, presented in Fig. 15, is mainly composed of two parts. The first is used to maintain the power efficiency of the amplifier at its maximum value by adjusting the phase of the load reflection coefficient in real time. Unlike the technique proposed in [25], which only uses phase detection

In this paper, an accurate design approach suitable for designing active load-modulated amplifiers, such as Doherty or LINC amplifiers, has been proposed. The design methodology is based on the use of a tabulated transistor load–pull-based model capable of precisely predicting the large-signal performance of an -band Doherty amplifier in terms of AM/AM and AM/PM characteristics and PAE over a wide range of drive levels. This model was compared to a published empirical-based model, and it was found that it is a better predictor of the RF large-signal response of the Doherty amplifier system. The developed model was used to study and investigate, for the first time, the sensitivity of the Doherty amplifiers to load mismatch. It was found

HAMMI et al.: DESIGN AND PERFORMANCE ANALYSIS OF MISMATCHED DOHERTY AMPLIFIERS USING ACCURATE LOAD–PULL-BASED MODEL

that this load mismatch seriously affects the linearity, as well as the efficiency of the Doherty amplifiers. Therefore, one can state that Doherty amplifiers are vulnerable to load mismatch effects. Consequently, they need to include extra circuitry and means to correct for these undesirable effects. For this purpose, a system-level load-mismatch compensator scheme, which is capable of recovering the linearity and maintaining the PAE close to its maximum range, has been proposed. ACKNOWLEDGMENT The authors would like to thank C. Simon, Department of Electrical and Computer Engineering, Schulich School of Engineering, University of Calgary, Calgary, AB, Canada, for providing technical support during the measurements. The authors would also like to acknowledge Agilent Technologies for software donation. REFERENCES [1] P. B. Kenington, “Linearized transmitters: An enabling technology for software defined radio,” IEEE Commun. Mag., pp. 156–162, Feb. 2002. [2] W. H. Doherty, “A new high efficiency power amplifier for modulated waves,” Proc. IRE, vol. 24, no. 9, pp. 1163–1182, Sep. 1936. [3] F. H. Raab, “Efficiency of Doherty power-amplifier systems,” IEEE Trans. Broadcast., vol. BC-33, no. 3, pp. 77–83, Sep. 1987. [4] M. Iwamoto, A. Williams, P.-F. Chen, A. G. Metzger, L. E. Larson, and P. M. Asbeck, “An extended Doherty amplifier with high efficiency over a wide power range,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 12, pp. 2472–2479, Dec. 2001. [5] Y. Yang, J. Yi, Y. Y. Woo, and B. Kim, “Optimum design for linearity and efficiency of a microwaves Doherty amplifier using a new load matching technique,” Microw. J., vol. 44, no. 12, pp. 20–36, Dec. 2001. [6] N. Srirattana, A. Raghavan, D. Heo, P. E. Allen, and J. Laskar, “Analysis and design of a high-efficiency multistage Doherty power amplifier for wireless communications,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 3, pt. 1, pp. 852–860, Mar. 2005. [7] J. Sirois, S. Boumaiza, M. Helaoui, G. Brassard, and F. M. Ghannouchi, “A robust modeling and design approach for dynamically loaded and digitally linearized Doherty amplifiers,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 9, pp. 2875–2883, Sep. 2005. [8] J. Sirois, S. Boumaiza, and F. M. Ghannouchi, “Large-signal characterization and modeling approach suitable for the design of actively load-modulated amplifiers,” in IEEE Radio Wireless Symp., San Diego, CA, Jan. 2006, accepted for publication. [9] J. Nam, J. H. Shin, and B. Kim, “A handset power amplifier with high efficiency at a low level using load-modulation technique,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 8, pp. 2639–2644, Aug. 2005. [10] J. Kim, S. Bae, J. Jeong, J. Jeon, and Y. Kwon, “A highly integrated Doherty amplifier for CDMA handset applications using an active phase splitter,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 5, pp. 333–335, May 2005. [11] J. Jung, U. Kim, J. Jeon, J. Kim, K. Kang, and Y. Kwon, “A new “series-type” Doherty amplifier for miniaturization,” in IEEE Radio Freq. Integr. Circuits Symp. Dig., Long Beach, CA, Jun. 2005, pp. 259–262. [12] D. W. Ferwalt and A. Weisshaar, “A base control Doherty power amplifier for improved efficiency in GSM handsets,” in IEEE MTT-S Int. Microw. Symp. Dig., Fort Worth, TX, Jun. 2004, vol. 2, pp. 895–898. [13] S. M. Wood, R. S. Pengelly, and M. Suto, “A high power, high efficiency UMTS amplifier using novel Doherty configuration,” in Proc. IEEE Radio Wireless Conf., Boston, MA, Aug. 2003, pp. 329–332. [14] S. Bae, J. Kim, I. Nam, and Y. Kwon, “Bias-switching Quasi-Doherty type amplifier for CDMA handset applications,” in IEEE Radio Freq. Integr. Circuits Symp. Dig., Philadelphia, PA, Jun. 2003, pp. 137–140.

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[15] H. Chireix, “High power outphasing modulation,” Proc. IRE, vol. 23, no. 11, pp. 1370–1392, Nov. 1935. [16] D. C. Cox, “Linear amplification with nonlinear components,” IEEE Trans. Commun., vol. COM-22, no. 12, pp. 1942–1945, Dec. 1974. [17] F. H. Raab, “Efficiency of outphasing RF power-amplifier systems,” IEEE Trans. Commun., vol. COM-33, no. 10, pp. 1094–1099, Oct. 1985. [18] A. Birafane and A. B. Kouki, “On the linearity and efficiency of outphasing microwave amplifiers,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 7, pp. 1702–1708, Jul. 2004. [19] F. Lepine, A. Adahl, and H. Zirathe, “A high efficient LDMOS power amplifier based on inverse class F architecture,” in 34th Eur. Microw. Conf., Oct. 2004, pp. 1181–1184. [20] H. Tsai-Pi, A. G. Metzger, P. J. Zampardi, M. Iwamoto, and P. M. Asbeck, “Design of high efficiency current-mode class-D amplifiers for wireless handsets,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 1, pp. 144–151, Jan. 2005. [21] H. Jager, A. Grebennikov, E. Heany, and R. Weigel, “Broadband highefficiency monolithic InGaP/GaAs HBT power amplifiers for wireless applications,” Int. J. RF Microw. Comput.-Aided Eng., vol. 13, no. 6, pp. 496–510, Nov. 2003. [22] N. Legallou, J. Y. Touchais, J. F. Villemazet, A. Darbandi, J. L. Cazaux, A. Mallet, and L. Lapierre, “High efficiency 45 W HPA for space application,” in 34th Eur. Microw. Conf., Oct. 2004, pp. 181–184. [23] A. V. Bezooijen, C. Chanlo, and A. H. M. van Roermund, “Adaptively preserving power amplifier linearity under antenna mismatch,” in IEEE MTT-S Int. Microw. Symp. Dig., Fort Worth, TX, Jun. 2004, vol. 3, pp. 1515–1518. [24] ——, “Adaptive methods to preserve power amplifier linearity under antenna mismatch conditions,” IEEE Trans. Circuits Syst. I, Reg. Papers, vol. 52, no. 10, pp. 2101–2108, Oct. 2005. [25] A. Keerti and A. Pham, “Dynamic output phase to adaptively improve the linearity of power amplifier under antenna mismatch,” in IEEE RFIC Symp. Dig., Long Beach, CA, Jun. 2005, pp. 675–678. [26] D. Qiao, D. Choi, Y. Zhao, D. Kelly, T. Hung, D. Kimball, M. Li, and P. Asbeck, “Real-time adaptation to antenna impedance mismatch for CDMA transceivers,” in IEEE Power Amplifiers for Wireless Commun. Dig. Top. Workshop, San Diego, CA, Jan. 2006. [27] Y. Yang, Y. Y. Woo, J. Yi, and B. Kim, “A new empirical large-signal model of Si LDMOSFETs for high-power amplifier design,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 9, pp. 1626–1633, Sep. 2001.

Oualid Hammi (S’03) received the B.Eng. degree in electrical engineering from the École Nationale d’Ingénieurs de Tunis, Tunis, Tunisia, in 2001, the M.Sc. degree from the École Polytechnique de Montréal, Montréal, QC, Canada, in 2004, and is currently working toward the Ph.D. degree at the University of Calgary, Calgary, AB, Canada. His current research interests are in the area of microwave and millimeter-wave engineering in general. His particular research activities are related to the design of intelligent and highly efficient linear transmitters for wireless communications, the development of DSP techniques for PAs linearization purposes.

Jérôme Sirois (M’05) received the B.Eng. degree in electrical engineering from the Université de Sherbrooke, Sherbrooke, QC, Canada, in 2001, and the M.A.Sc. degree from the École Polytechnique de Montréal, Montréal, QC, Canada, in 2005. In 2005, he was an RF Application and Support Engineer with Focus Microwaves, Dollard-des-Ormeaux, QC, Canada. He is currently with Freescale Semiconductor, Tempe, AZ, where he is involved with the enhancement of load–pull measurement techniques and accuracy. His current research interests are characterization and modeling of RF transistors, load–pull measurement, PA design, and efficiency enhancement techniques.

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Slim Boumaiza (S’00–M’04) received the B.Eng. degree in electrical engineering from the École Nationale d’Ingénieurs de Tunis, Tunis, Tunisia, in 1997, and the M.S. and Ph.D. degrees from the École Polytechnique de Montréal, Montréal, QC, Canada, in 1999 and 2004. In May 2005, he joined the Electrical Engineering Department, University of Calgary, Calgary, AB, Canada, as an Assistant Professor and Faculty Member with the iRadio Laboratory. His research interests are in the areas of design of RF/microwave and millimeter-wave components and systems for wireless communications. His current interests include RF/DSP mixed design of intelligent transmitters, design, characterization, modeling, and linearization of high-power amplifiers, reconfigurable and multiband RF transceivers, and adaptive DSP.

Fadhel M. Ghannouchi (S’84–M’88–SM’93) received the Ph.D. degree in electrical engineering from the University of Montréal, Montréal, QC, Canada, in 1987. He is currently an iCORE Professor, a Canada Research Chair, and the Director of the iRadio Laboratory, Department of Electrical and Computer Engineering, University of Calgary, Calgary, AB, Canada. He has held invited positions at several academic and research institutions in Europe, North America, and Japan. He has authored or coauthored over 300 publications. He holds seven patents. His research interests are in the areas of microwave instrumentation, modeling of microwave devices and communications systems, design and linearization of RF amplifiers and SDR radio systems.

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High-Efficiency Linear RF Amplifier—A Unified Circuit Approach to Achieving Compactness and Low Distortion Tsz Yin Yum, Member, IEEE, Leung Chiu, Student Member, IEEE, Chi Hou Chan, Fellow, IEEE, and Quan Xue, Senior Member, IEEE

Abstract—The realization of a highly efficient linearized amplifier has emerged as a paramount issue in the design of advanced mobile handsets. In this paper, a new RF amplifier linearization scheme using a compensating transistor combination is proposed. The devised approach ably utilizes all terminals of an additional transistor that offers a unified pre–post-distortion and cubic distortion characteristic for performance improvement. Meticulous modeling along with a power-dependent Volterra series is performed to identify contributions on each mechanism under various power levels. An experimental four-tone test reveals a maximum 28-dB reduction for the intermodulation distortion at 1.95 GHz, which outperforms typical pre-distortions of 5–10 dB. A single-ended two-stage amplifier module demonstrates a state-of-the-art power efficiency of 55% with 27-dB transducer gain at 24-dBm output power. Meanwhile, the adjacent channel power ratio ( 1 ) is maintained with good margins of 35 dBc for a four-channel wideband code-division multiple-access signal under all output dynamics. Graceful degradations on modulation bandwidths, tone spacing, bias, and gain variations are also discussed, showing superb performance with virtually no dedicated retuning circuit parameters for multicarrier applications. By combining a bias control along with the proposed linearization technique, the average efficiency (12%) is 3 higher than that of the fixed bias (3.94%), demonstrating the potential utility on further prolonging battery lifetime in practical scenarios.

ACPR

ACPR

Index Terms—Adjacent channel power ratio ( 1 ), cubic distortion, intermodulation distortion (IMD), linearization, postdistortion, power efficiency and pre-distortion.

I. INTRODUCTION HE tremendous growth of the wireless market, coupled with the fierce competition over the last few years, has spurred unprecedented interest on low-cost and high-performance RF amplifiers. Concern for performance is motivated by the significant impact the amplifier has on the talk time

T

Manuscript received March 8, 2006; revised April 13, 2006. This work was supported by the City University of Hong Kong under Strategic Research Grant 7001586 and by the Hong Kong Research Grant Council under Competitive Earmarked Research Grant CityU110605. T. Y. Yum was with the Wireless Communications Research Center, City University of Hong Kong, Kowloon, Hong Kong. He is now with Integrated Display Technology International Limited, Hong Kong (e-mail: kenji.yum@gmail. com). L. Chiu, C. H. Chan, and Q. Xue are with the Wireless Communications Research Center, City University of Hong Kong, Kowloon, Hong Kong (e-mail: [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.1109/TMTT.2006.877815

and spectral distortions of mobile terminals, which can represent a substantial marketing advantage. New developments in linearity-efficiency boosting techniques, such as signals injection [1], envelope elimination and restoration (EER) [2], out-phasing, or Doherty amplifiers [3], [4] have been proposed to allow optimal tradeoffs. Despite their conceptual appeals, none have found widespread use in portable handsets. The suspected reasons rest with their intrinsic bandwidth (BW) limitations, as well as lingering doubts about complexity and cost. By far the most popular approach in handheld mobiles is the well-known RF pre-distortion. The approach tends to be stable and provides modest efficiency enhancement. It has advantages historically in terms of miniaturized size and low complexity, making it attractive for monolithic implementations. Various RF pre-distorters have been proposed over the years with either series [5], [6] or shunt [7] and active [8], [9] or passive [10] configurations. Improvements on linearity [adjacent channel power ratios (ACPRs)] are, however, marginal and, at best, can provide a few critical extra decibels [5]–[10]. With the widespread use of sophisticated multicarrier formats, it is important that the intrinsic linearity of amplifiers be made as high as possible. Digital adaptive pre-distortion [11], [12] has been gaining wide favor. This added feature is acquired, albeit at the expense of increased complexity and digital storage overhead, thus preventing RF designers to embrace the concept in battery-operated applications. An altogether different approach to linearization derived from a simple resonant cell concept was reported in [13]. The main prerequisite for correcting distortions is the availability of the well-designed stopband structure controlling numerous harmonics and phases simultaneously. While the new design improves the performance, the size is inevitably increased due to the resonant cell. Table I summarizes some representative linearization techniques with their associated linearity—efficiency—complexity tradeoffs. Obviously, the challenge has been to devise a low-cost and power-efficient RF amplifying concept that inherits the simplicity of earlier circuit techniques, yet exhibits a superior distortion-correcting capability. Recent research efforts focus on maximizing average efficiency at low-power drives. Power control like dual-bias control [14], dc–dc voltage converters [15], and parallel-chain high–low power amplifiers [16] were proposed with great success. However, none of the efforts have been paid on combining simple RF pre-distorters with adaptive bias control in practice. Technological constraints on this end are probably confined by either

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TABLE I COMPARISONS OF REPRESENTATIVE DISTORTION COMPENSATION TECHNOLOGIES

Fig. 1. Schematic diagram and simplified small-signal modeling of the compensating transistor pair module.

overcome is to place a compensatory nonlinear element alongside the target junction such that the combination of both is linear over the entire dynamic range [18], [19]. the marginal improvement on preserving linearity or the suitable match of compensatory nonlinear transfer functions over different biasing conditions. An adaptive feedback can be introduced, though rather complex. This paper presents for the first time a detailed strategy on the design and performance of a power amplifier module (PAM) based on the compensating transistor combination [17] under adaptive power management. A rigorous Volterra series is employed to analyze the design concept. The dominant compensation mechanism under different power levels is investigated as a highlight. Such an approach possesses many advantages. First, the scheme offers intact benefits of the ACPRs’ improvement and simplicity of both pre- and post-distortion. It also ably integrates the cubic distorter in a parallel configuration and unifies three techniques into one versatile framework for superior improvement. Second, the proposed method eliminates complex and expensive circuitries and, hence, reduces size, cost, and power consumption. Third, adopting an adaptive biasing along with our linearization method further reduces dc consumption and fully exploits the average efficiency in a much-relaxed linearity under wide power range. Last, but not least, this method takes full advantages of the unified pre–post-distortion feature, making it transparent to the modulation BW and less sensitive to memory effects for emerging multicarriers usages. II. GENERAL PRINCIPLE AND COMPENSATING TRANSISTOR COMBINATION Under a large-signal drive, the base–emitter (BE) voltage ( ) drops with the decrease of trans-conductance ( ), resulting in severe signal clippings (AM–AM). Design of the biasing circuit, which sources sufficient base dc capability, becomes the prime issue at hand. Nonlinear components (e.g., , and ) are a function of input power, and also affect the linearity. Distortions are largely attributed to their dynamic variations over the device load line. The simplest method to

A. Pre-Distortion Fig. 1 illustrates the main core of the devised approach that comprises two identical SiGe bipolar transistors, one for amplification and the other for distortion compensation. The transistor amplifies the RF signal, while the forward-biased base–collector (BC) diode of , resistor , and capacitor form the biasing circuit for dc boosting and simultaneously a pre-distorter for the input . The low impedance of a tuned capacitor decreases the impedance looking into the biasing circuit, directing an RF coupling path to the . The pre-distorted junction relies on its second-order response to boost the base voltage of at large signal. Being a current sharing path, the incident RF signal causes increase voltage drop in the BC junction of and, thus, compensates the voltage drop in the BE junction of . Depicted in the inset of Fig. 1, these two input diodes are connected in an opposite polarity. Consequently, coefficients for them in Volterra expansions are also opposite in sign, which, in turn, compensate the dynamic base voltage. To confirm the pre-distortion, Fig. 2 plots the ADS simulated AM–AM and AM–PM characteristics versus input powers at BE and BC junctions of and , respectively. Their variations are seen to be opposite, which in total are almost constant for the normal operating range. B. Post-Distortion As shown in Fig. 1, the BE junction of is utilized and connected in series with a capacitor ( 0.5 pF) to the output. This capacitor has interesting usages. First, it serves as a dc block capacitor that separates the base ( ) and collector ( ) bias. Second, in conjunction with the parasitic capacitance of , this capacitor offers a deliberate design flexibility to match with the nonlinear capacitance variation of for linearization. For the sake of illustration, of is transformed by Miller theorem at the output, referred in the inset of Fig. 1. Table II indicates

YUM et al.: HIGH-EFFICIENCY LINEAR RF AMPLIFIER—UNIFIED CIRCUIT APPROACH TO ACHIEVING COMPACTNESS AND LOW DISTORTION

Fig. 2. Simulated AM–AM and AM–PM characteristics versus input powers in BE and BC junctions of BJT and BJT , respectively.

TABLE II TYPICAL CAPACITIVE VALUES OF C

,

C

, AND

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Fig. 4. Simulated AM–AM and AM–PM characteristics versus input powers in CE junction of BJT under different source impedances.

C

Fig. 5. Equivalent-circuit model for the computation of the second- and third-order intermodulation products.

C. Cubic-Distortion

Fig. 3. Simulated AM–AM and AM–PM characteristics versus input powers in BC and BE junctions of BJT and BJT , respectively.

the 2-V-biased capacitance values of , and are all in the same order that fulfill the task. Third, the added small capacitor subtly acts as an impedance transformation chain to prevent excessive output power from leaking to the compensation branch. Referring to Fig. 1, the ADS simulated input impedance is transformed from 25 (at point A) to approximately 262 (at point B) by using this series capacitor . An optional inductor can also be inserted for proper phase alignments. Fig. 3 depicts the ADS simulated AM–AM and AM–PM at the BC junction of and the BE junction of , which infers an opposite characteristic. In fact, the value of the resistor acts as an intended design parameter to determine the operation of the BE junction.

One should notice that the (lower path) serves as a distortion contributor by means of its nonlinear collector–emitter (CE) terminal. The RF signal coupled to the lower path will launch a distorted component (say, ) with a fortuitous phasing at the output for distortion cancellation. Building on published works [20], [21], the junction simply resorts to the use of a variable resistor to exhibit gain and phase deviations under different signal drives. Due to the fact that the common emitter delivers the phase deviation out-of-phase while the common base does the same in-phase, their AM–PM conversions are expected to be opposite at saturation. Recall the input impedance at point B (Fig. 1) is high enough to prevent signal leakage; the signal content subsequently flows from the input to output rather than feeding back. Instability issues are thus essentially absent. In order to illustrate how the lower path works, a two-port CE junction with fixed bias is simulated by disconnecting the output. Fig. 4 depicts its simulated AM–AM and AM–PM characteristics of the CE junction alone versus input powers as a function of input impedances. The lower path virtually possesses opposite slopes for both gain and phase deviations against that in the for all cases. By tuning the input and output impedances (in Fig. 4), both the degree of gain and phase deviation can be manipulated.

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Fig. 6. (a) and (b) Simulated normalized parameters of the BJT transistor used in the power stage.

III. DEVICE MODELING AND VOLTERRA ANALYSIS To show how the compensating transistor combination works, we analyze its equivalent nonlinear model according to Fig. 5, where and denote the input and load impedances, respectively. For ease of analysis, the intrinsic composite transistor pair is modeled by the combined resistance ( ) and capacitance ( ) at the input, nonlinear capacitance ( ) and trans-conductance ( ) of and nonlinear capacitance ( ) and resistance ( ) of . Each nonlinear source is modeled by their linear element with a parallel distortion current source , in which values are calculated based on polynomial coefficients and voltage phasors. To that end, a table-based nonlinear component value was first extracted from simulated multiple-bias -parameters under various RF powers, which was then reconverted to -parameters. The component values of the composite transistor pair are then easily extracted as follows: (1)

also possesses a varistor charat the output nonlinearity. acteristic that feeds the distortion current forward, and should contribute itself to distortion as well. For a two-tone excitation with or , a narrow tone separation, i.e., and , the BE dc voltage (second-order) and third-order intermodulation distortion ( ) at collector can be derived based on the “method of nonlinear currents” [22] shown in (7) and (8) as follows:

(7)

(2) (3) (4)

Current feeds forward to output

(5)

(8)

(6)

is the trans-impedance transfer where function from a distortion current source between nodes and to the voltage in node . Expressions are described in the Appendix and can be referred to [18] and [23]. The values of and calculated from Volterra expressions (7) and (8) are superimposed with ADS simulation. Results of a typical amplifier in the same bias are also shown for comparisons. Excellent agreements throughout the entire dynamic range are obtained for all calculations, verifying the extraction method, the completeness of our simplified model, as

Fig. 6(a) and (b) shows the power dependency of all nonlinear parameters, which can then be subsequently fitted into a thirdorder truncated power series for Volterra analysis. As anticipated, variations on and as well as their effects on were “linearized” over the entire dynamic range, attributed by the pre-distortion effect. It can also be noted that, at an output power of 22 dBm or smaller, the variation on and are oppositely equal, which, in turn, compensates the distortion

YUM et al.: HIGH-EFFICIENCY LINEAR RF AMPLIFIER—UNIFIED CIRCUIT APPROACH TO ACHIEVING COMPACTNESS AND LOW DISTORTION

Fig. 7. Comparisons of simulated and calculated: (a) dc BE voltages and (b) third-order IMD ratios versus output powers.

well as the superior improvement of the compensating transistor combination. While Fig. 7(a) proves the calculated voltage in the typical design decreases with the increasing input power, of in the devised approach remains unchanged up to the compression point, showing a 9-dB improvement in the constant power range of and demonstrating its dc boosting capability. Noteworthy to see is the superior in-band third-order intermodulation distortion ratio ( ) improvement in Fig. 7(b). An null of more 30-dB difference is achieved at an approximately 21-dBm turning point, and the improvement is sustained as much as 10 dB over a wide enough dynamic range until dB. Moreover, the curve exhibits a nearly 7 : 1 slope beginning at an output power of 22.5 dBm, showing all ’s and ’s are totally suppressed for the power below that value. Calculated from (7), Fig. 8 plots the contributions of each mechanism to the output for the following three cases. Case 1) Contributions of the pre-distortion effect. (By setting all nonlinear current sources to zero, except

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IMD

Fig. 8. (a) Calculated contributions to the output from the pre- and postdistortion nonlinearities for both the conventional and linearized amplifiers in our case. (b) Calculated amplitude and phase contributions of cubic distortion nonlinearity (r ) in the linearized amplifier.

and in (7). The contributions from only and in the conventional design are also shown in Fig. 8(a) for comparison.) Case 2) Contributions of the post-distortion effect. (By setting all nonlinear current sources to zero, except and in (7). The contribution from only in the conventional design is also shown in Fig. 8(a) for comparison.) Case 3) Contributions of the cubic distortion effect. (By setting all nonlinear current sources to zero, except in (7). The contribution by setting only in (7) to zero is also shown in Fig. 8(b) for comparison.) As depicted, though nonlinearities of input and output BC junction are quite large in typical designs, they are compensated in our case by pre–post-distortion effect throughout most power levels. Fig. 8(b) compares the magnitude and phase contribution

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by setting the nonlinear source “on” and then by setting all remaining nonlinear sources “on” in the compensating transistor combination case. Their combined result, as shown in Fig. 7(b), is also plotted in the same graph for better illustration. As seen, the contribution of the cubic distortion is exactly equal in magnitude and out-of-phase with the combination of all remaining nonlinear current sources at exactly the output power null of 21 dBm, which confirms the appearance of the as alluded to earlier and shown in this same figure by a nearly 8-dB suppression. Below 14 dBm, the effect of cubic distortion is small compared with that in other current sources, which has no effect on its compensation. Luckily, as magnitudes of all nonlinear current sources in such a region is not so large and their resultant [“All” in Fig. 8(b)] is still within acceptable margins. IV. EXPERIMENTAL RESULTS The design methodology is demonstrated for a two-stage PAM operating at 1.95 GHz. A commercial NPN silicon–germanium (SiGe) bipolar junction transistor (BJT) from Infineon Technologies, Munich, Germany, was used as the power stage. With a dc supply voltage of 4.4 V and a collector current of 60 mA, the device is capable to produce 25-dBm saturated power. Another fully matched driver stage was used to provide an overall transducer gain of more than 25 dB. For comparisons, three power amplifiers, one with the proposed and the other with a diode-based pre-distortion and also a reference amplifier, were designed and fabricated. All designs were then tested under various biases and power levels using AM–AM and AM–PM characteristics, two/four-tone, and real time 3GPP wideband code-divison multiple-access (W-CDMA) vector modulation.

Fig. 9. Measured transducer gains and phase distortions as a function of output powers.

A. Gain (AM–AM) and Phase (AM–PM) Distortion The measured transducer gain of the working prototype is 27.4 dB, which is similar to the diode-based pre-distortion with a negligible 1.3-dB insertion loss compared with the reference. Using a single-tone signal, Fig. 9 plots the gain and phase deviations in three cases. The output power of dB has been improved significantly by 7 and 3.7 dB compared with the reference and the diode-based pre-distortion, respectively. While linearization aims to provide the amplifier a harder saturation, the gain curve in our proposed technique reveals a near ideal limiter property, showing a dB of 24 dBm located near the device saturation. Our approach is also capable of improving the phase distortion. A peak-to-peak phase deviation of less than 16.5 was achieved over the entire output dynamic range, improvements of 10 and 6.5 compared with the reference and the diode-based pre-distortion amplifiers at 23-dBm output power, respectively. B. Four-Tone and Vector Signal Test Performance A more stringent four-tone test is performed to illustrate the improvement for less output power backoff. Four tones from a single generator centred at 1.95 GHz with frequency spacing of 1 MHz were used. At an output power of 14.5 dBm per tone, the designed amplifiers were optimized, resulted in a 28-dB reduction of IMDs compared with the reference, as shown in Fig. 10.

Fig. 10. Measured four-tone test of the amplifiers centered at 1.95 GHz with 1-MHz frequency offset.

The improvement is only limited by the noise floor of the equipment and is at least 15 dB better than that in the simple diodebased pre-distortion. Fig. 11 illustrates the measured spectra of three cases at a (maximum) four-channel mode W-CDMA power of around 22 dBm. A substantial reduction of 18 and 12 dB, respectively, in (power ratio of 60-kHz BW at 5-MHz offset from the center frequency to that in 3.84-MHz channel BW) were achieved when compared with the reference and the simple diode-based pre-distortion. Note that with the spectrum emission mask in the 3GPP class-3 standard, only the devised method can meet the requirement with good margins. Without further retuning circuit parameters, Fig. 12 shows the of designed amplifies in a 3GPP W-CDMA vector signal test over various power levels. The proposed circuit operates very well over a wide output power up to dB. These results also imply that no adaptation to different power levels is required. It is worth noting that by using the proposed method, the is further reduced by almost 5–10 dB over the entire dynamic range, even compared to the diode-based pre-distortion. In theory, linearization allows the amplifier to deliver more output power and operate at a higher efficiency under

YUM et al.: HIGH-EFFICIENCY LINEAR RF AMPLIFIER—UNIFIED CIRCUIT APPROACH TO ACHIEVING COMPACTNESS AND LOW DISTORTION

Fig. 11. Comparisons of the spectral re-growth with four-channel mode 4.096-Mb/s 3GPP W-CDMA modulation signal. (Average output power = 22 dBm).

Fig. 12. Measured ACPR ’s of three amplifier modules (4.096-Mb/s 3GPP W-CDMA modulation signal). Measurement was taken as the power ratio of 60-kHz BW at 5-MHz offset from the center frequency to that in 3.84-MHz channel BW.

a prescribed distortion level. Fig. 13 plots the collector efficiency and of a standard two-tone test (1-MHz frequency offset) over the entire output power range. Under the condition of dBc, 22-dBm output power and 39% efficiency was obtained, which translates to a 5.3-dBm output power increase and a substantial 26% efficiency improvement compared with the reference. Results of should be compared to that in Fig. 7(b), which shows good agreement. C. Performance Degradation on Modulation BWs, Tone Spacing, and Bias Variations Results of some performance corners are presented here. For a multicarrier W-CDMA signal, one should consider additional nonlinear effects caused by electrical memories. For example, the BW and envelope frequency of a four-carrier W-CDMA signal extend up to 20 MHz, which degrades the performance of typical RF pre-distorters. To investigate the cancellation capability, particularly for an RF signal with large modulation

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Fig. 13. Measured collector efficiencies and third-order IMD ratios (IMR in two-tone test) of three designed amplifier modules.

Fig. 14. Dependence of measured transducer gain and IMD improvement of the proposed linearized module on the tone spacing.

BW, the system has been tested experimentally by sweeping the two-tone spacing over wide modulation frequencies, as seen in Fig. 14. The diagram clearly indicates that the reduction factor is nearly transparent to the tone spacing up to 60 MHz, which exceeds far enough for the entire W-CDMA application and is superior to any narrowband linearization techniques reported [24], [25] in the order of kilohertz. Transducer gain is maintained over 27.4 1 dB for the whole measurement range. Fig. 15 shows the measured power densities of three amplifiers at 20.8-dBm average output power under a four-carrier WCDMA signal. An of 47.6 dBc is obtained symmetrically for both the upper and lower carrier. Obvious improvement of 16 and 10 dB is achieved when compared with the reference and diode-based pre-distortion, respectively. In addition, handset terminals are always affected by environmental factors such as temperature and supply voltage over time. To show the current linearization scheme is also well behaved in this regard, the dependence of the measured output power and on the collector biasing voltage was investigated.

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TABLE III SUMMARY TABLE OF POWER MANAGEMENT

Fig. 15. Linearized spectrums of three designed amplifier modules under a four-carrier 3GPP W-CDMA signal (Average output power = 18 dBm).

Fig. 16. Conceptual block diagram of the linearized PAM under adaptive power management.

Generally, the linearity of an amplifier decreases with reduced saturation power. Interestingly, we found that the improvement range of is retained nearly unchanged when the supply voltage decreases, thereby confirming the devised approach can be operated properly under various biasing. The result also illustrates the potential power controllability of the working prototype facilitated by the collector voltage to improve average efficiency in a much-relaxed linearity margin. V. EFFECT OF NEW LINEARIZATION TECHNIQUE UNDER ADAPTIVE BIAS CONTROL New linearization approaches should be able to compensate distortions under various biasing, rather than fix conditions that normally reported [20]. Though little can demonstrate their viability on this end, improvements on average efficiency are still confined by marginal distortion improvements [26]. Neither are the complex control circuitries, due to numerous parameters [20]. The dynamic bias control introduced in the following illustrates how these impediments can be overcome by the superior improvement of the devised linearization approach. The conceptual block diagram of the amplifier module is given in Fig. 16. A simple digital control is attempted with three stepwise settings designated as the low, medium, and high

Fig. 17. PAM with adaptive power control unit (Color version available online at: http://ieeexplore.ieee.org).

power region. Each range has a predetermined threshold, which can be best traded with the linearity. During transmission, the amplifier adjusts the collector bias between preset limits, depending upon signal strength detected. When the power limit of one range is reached in response to the required linearity, the collector voltage is sensed and then switched to the next higher mode, raising the level of transmitted power. Table III summarizes the power management defined with the intent to maintain a minimum of 40 dBc (7-dB margin from the 3GPP W-CDMA standard in both low- and medium-power range). An experimental realization thereof is depicted in Fig. 17. Full investigations have been validated in Fig. 18 for the reference and the proposed PAM, both with dynamic-biased supply. Thanks to improvement in three power modes, only linearized amplifiers can keep below 35 dBc over the whole dynamic range, as intended. The dynamic-biased amplifier incorporated with the devised linearization technique establishes

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Fig. 18. ACPR comparison of the dynamic-supply amplifier with proposed linearization technique and the dynamic-supply amplifier without linearization.

at least 10-dB improvement on ’s when compared to that without linearization. This potentially permits the former to work at a lower dc consumption to trade the linearity, which further enhances average efficiencies at deep power backoffs. In addition, the dynamic-biased control with our linearization technique, working in unison, possesses an extra benefit on improving the light-load linearity when compared with that using fix biasing. The direct consequence is a gain from the reduced dc power consumption. As shown in Fig. 19(a), the dc power drained from the battery dramatically decreases from 450 mW (a full twostage module) to 60 and 155 mW in low- and medium-power regions, respectively. To estimate the battery lifetime improvement, weighted dc power profiles are acquired by timing the probability density function of W-CDMA in an urban area, and the result is shown in Fig. 19(b). Clearly, area under the curve is the average dc power consumed in the amplifier, which shows a substantially decrease in our case and translates into a talktime improvement. With the amplifier functioning at switching points of 18 (refers as 6-dB backoff) and 8 dBm (refers as 16-dB backoff), collector efficiencies of 29% and 7% can be achieved, respectively. The calculated average efficiency of our dynamically linearized system is 12%, which is 3 higher than that in the fixed supply of 3.94%. Gain variations in three modes are within 5 dB, which is much smaller than reported literatures of 8–15 dB [16], [26], attributed by only collector–voltage variations in our case. VI. SUMMARY AND DISCUSSION The performance summary of the measured PAM together with other “circuit-level” linearization techniques are shown in Tables IV and V, respectively. Comparing with state-of-the-art circuit-level solutions that are normally reported by 5–10 dB improvement (either ), our measured 30 dB in and 20 dB in should be especially highlighted despite the utilization of SiGe BJT technology. While this good linearity result may be overkill for the W-CDMA system, it indeed demonstrates the uniqueness of our proposed linearization scheme. Undoubtedly, it will become an even more significant advantage when the architecture is attempted for a

Fig. 19. Comparisons of: (a) input dc supply power and (b) weighted input dc supply power of the fixed-bias supply and adaptive-bias supply PAMs after adopting proposed linearization approach.

TABLE IV SUMMARY OF THE MEASURED PAM PERFORMANCE

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TABLE V COMPARISON OF PUBLISHED LINEARIZED AND DYNAMIC-BIASED POWER AMPLIFIERS

system [such as the orthogonal frequency-division multiplexing (OFDM)-based wireless local area network (WLAN)] where more stringent linearity and error vector magnitude (EVM) requirements are demanded. In terms of the simplicity on the whole implementation, this study demands no extra and external circuitries and requires only one additional composite transistor with a few circuit components consumed by no dc power. Consequently, this work can be seen as among the most power efficient in terms of peak efficiency (55%). It also features one of the most aggressive, yet simple adaptive power-reduction schemes for the backoff mode, and can be further improved if today’s advanced adapted biasing schemes are applied. The improvement is transparent to memory effects, bias variations, and maintained, respectively, at least 30 dB over 300-MHz BW, which provide an extra insight on covering multimode and multistandard systems.

comparable to the feed-forward design, which is bulky, expensive, and more complicated. Meanwhile, a significant improvement in the spectral re-growth is obtained well with wide output powers. With high power efficiency, the devised approach exactly meets today’s demands, which is expected to be used in a wide range of wireless applications. APPENDIX We now derive (7) and (8) using the method of nonlinear currents [22], [23]. By applying Kirchoff’s current law at nodes B to C and ignoring nonlinear current sources in Fig. 5, the fundamental linear voltages at the base and collector are given by (9) (10)

VII. CONCLUSION Spectral efficiency is becoming a dominant requirement in modern wireless systems, leading to tight IMD specifications for all transmitter components, especially power amplifiers. In this paper, a simple, low-cost, and high-performance linearized PAM with adaptive power control has been realized using a compensating transistor combination. The distortion behavior was rigorously studied by Volterra analysis. Very good agreement is observed among theories, simulations, and experiments. Measured maximum IMD improvement is superior, and is even

where

(11)

YUM et al.: HIGH-EFFICIENCY LINEAR RF AMPLIFIER—UNIFIED CIRCUIT APPROACH TO ACHIEVING COMPACTNESS AND LOW DISTORTION

in which and are evaluated at the fundamental frequency . Similarly, the responses of the base and collector voltages at are given by

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which can then be used for calculating the BE dc voltage ) and the third-order IMD ratio ( ) as shown in ( Fig. 7(a) and (b), respectively. ACKNOWLEDGMENT

(12) (13) and are evaluated at the frequency . in which As in our case and for simplicity, we neglect the up-conversion effect between the fundamental and secondorder ( ) base voltage to the third-order terms, which then limit the complexity of nonlinear voltage analysis to their own nonlinear current sources only. As subsequent results indicate, this simplifying assumption is valid over the range of signal power and bias conditions that the amplifier operates on. From [22], we find (14) (15) and (16) (17) , , and and , , and , respecwhere tively. By setting the signal source ( ) to zero and applying Kirchoff’s current law in the circuit of Fig. 5 for the second and the third-order voltage, respectively, we can obtain

(18)

(19)

The authors wish to acknowledge Dr. A. Katz, Linearizer Technology Inc., Hamilton, NJ, for his valuable discussions at the 2004 IEEE Microwave Theory and Techniques Society (IEEE MTT-S) International Microwave Symposium (IMS) Student Paper Competition and Dr. C. Wang, GoldRadio Communications Company Ltd., ShenZhen, China, for his help with the adaptive bias control. REFERENCES [1] C. S. Aitchison, M. Mbabele, M. R. Moazzam, D. Budimir, and F. Ali, “Improvement of third-order intermodulation product of RF and microwave amplifiers by injection,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 6, pp. 1148–1153, Jun. 2001. [2] F. H. Raab, P. Asbeck, S. Cripps, P. B. Kenington, Z. B. Popovic´ , N. Pothecary, J. F. Sevic, and N. O. Sokal, “Power amplifiers and transmitters for RF and microwave,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 3, pp. 814–826, Mar. 2002. [3] X. Zhang, L. E. Larson, and P. M. Asbeck, Design of Linear RF OutPhasing Power Amplifiers. Norwell, MA: Artech House, 2003. [4] Y. Yang, J. Yi, Y. Y. Woo, and B. Kim, “Optimum design for linearity and efficiency of microwave Doherty amplifier using a new load matching technique,” Microw. J., vol. 44, no. 12, pp. 20–36, Dec. 2001. [5] C. Haskins, T. Winslow, and S. Raman, “FET diode linearizer optimization for amplifier predistortion in digital radios,” IEEE Microw. Guided Wave Lett., vol. 10, no. 1, pp. 21–23, Jan. 2000. [6] J. Sun, B. Li, and M. Y. W. Chia, “Linearized and highly efficient CDMA power amplifier,” Electron. Lett., vol. 35, no. 10, pp. 786–787, May 1999. [7] K. Yamauchi, K. Mori, M. Nakayama, Y. Mitsui, and T. Takagi, “A microwave miniaturized linearizer using a parallel diode with a bias feed resistance,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 12, pp. 2431–2435, Dec. 1997. [8] H. Kawamura, K. Sakuno, T. Hasegawa, M. Hasegawa, H. Koh, and H. Sato, “A miniature 44% efficiency GaAs HBT power amplifier MMIC for the W-CDMA application,” in 22nd Annu. GaAs IC Symp., Nov. 2000, pp. 25–28. [9] Y. S. Noh and C. S. Park, “PCS/W-CDMA dual-band MMIC power amplifier with a newly proposed linearizing bias circuit,” IEEE J. SolidState Circuits, vol. 37, no. 9, pp. 1096–1099, Sep. 2002. [10] T. Yoshimasu, M. Akagi, N. Tanba, and S. Hara, “An HBT MMIC power amplifier with an integrated diode linearizer for low voltage portable phone applications,” IEEE J. Solid-State Circuits, vol. 33, no. 9, pp. 1290–1296, Sep. 1998. [11] S. Kusunoki, K. Yamamoto, T. Hatsugai, H. Nagaoka, K. Tagami, N. Tominaga, K. Osawa, K. Tanabe, S. Sakurai, and T. I. Iida, “Power-amplifier module with digital adaptive predistortion for cellular phones,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 12, pp. 2979–2986, Dec. 2002. [12] E. G. Jeckeln, F. M. Ghannouchi, and M. A. Sawan, “A new adaptive predistortion technique using software-defined radio and DSP technologies suitable for base-station 3G power amplifiers,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 9, pp. 2139–2147, Sep. 2004. [13] T. Y. Yum, Q. Xue, and C. H. Chan, “Amplifier linearization using compact microstrip resonant cell (CMRC) structure—Theory and experiment,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 3, pp. 927–934, Mar. 2004. [14] K. Yang, G. I. Haddad, and J. R. East, “High-efficiency class-A power amplifiers with a dual-bias-control scheme,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 8, pp. 1426–1432, Aug. 1999. [15] G. Hanington, P.-F. Chen, P. M. Asbeck, and L. E. Larson, “Highefficiency power amplifier using dynamic power-supply voltage for CDMA applications,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 8, pp. 1471–1476, Aug. 1999. [16] J. H. Kim, J. H. Kim, Y. S. Noh, and C. S. Park, “An InGaP–GaAs HBT MMIC smart power amplifiers for W-CDMA mobile handsets,” IEEE J. Solid-State Circuits, vol. 38, no. 6, pp. 905–910, Jun. 2003.

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[17] T. Y. Yum, Q. Xue, and C. H. Chan, “A novel amplifier linearization technique using anti-parallel re-configurable transistor (ART) pair,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2004, pp. 685–688. [18] C. Wang, M. Vaidyanathan, and L. E. Larson, “A capacitance-compensation technique for improved linearity in CMOS class-AB power amplifiers,” IEEE J. Solid-State Circuits, vol. 39, no. 11, pp. 1927–1937, Nov. 2004. [19] C. S. Yu, W. S. Chan, and W. L. Chan, “1.9 GHz low loss varactor diode pre-distorter,” Electron. Lett., vol. 35, no. 20, pp. 1681–1682, Sep. 1999. [20] G. Hau, T. Nishimura, and N. Iwata, “A highly efficient linearized wideband CDMA handset power amplifier based on predistortion under various bias conditions,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 6, pp. 1194–1201, Jun. 2001. [21] A. Katz, S. Moochalla, and J. Klatskin, “Passive FET MMIC linearizers for C; X; and Ku-band satellite applications,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 1993, pp. 353–356. [22] S. A. Maas, Nonlinear Microwave Circuits. Norwood, MA: Artech House, 1988. [23] J. Vuolevi and T. Rahkonen, “Analysis of third-order intermodulation distortion in common-emitter BJT and HBT amplifiers,” IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., vol. 50, no. 12, pp. 994–1001, Dec. 2003. [24] K.-K. M. Cheng and C.-S. Leung, “A novel generalized low-frequency signal injection method for multistage amplifier linearization,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 2, pp. 553–559, Feb. 2003. [25] Y. Yang and B. Kim, “A new linear amplifier using low-frequency second-order intermodulation component feedforwarding,” IEEE Microw. Guided Wave Lett., vol. 9, no. 10, pp. 419–421, Oct. 1999. [26] Y. S. Noh and C. S. Park, “An intelligent power amplifier MMIC using a new adaptive bias control circuit for W-CDMA applications,” IEEE J. Solid-State Circuits, vol. 39, no. 6, pp. 967–970, Jun. 2004. [27] C.-C. Yen and H.-R. Chuang, “A 0.25-m 20-dBm 2.4-GHz CMOS power amplifier with an integrated diode linearizer,” IEEE Microw. Wireless Compon. Lett., vol. 13, no. 2, pp. 45–47, Feb. 2003.

Tsz Yin Yum (S’02–M’05) was born in Hong Kong. He received the B.Eng. degree (with first-class honor) and Ph.D. degree in electronic engineering from the City University of Hong Kong, Kowloon, Hong Kong, in 2002 and 2005, respectively. In 2005, he joined Integrated Display Technology (IDT) Ltd., Hong Kong, as the Group RF Design Consultant, where he is involved with wireless product and business development. His research interests include both RF/microwave active and passive circuit and antenna designs. Mr. Yum served as the President of the IEEE Student Branch (Hong Kong Section) at the City University of Hong Kong from 2003 to 2005. He was the

recipient of 25 scholarships, fellowships and prize awards, including the VTech Scholarship, the 2002 and 2003 Sir Edward Youde Memorial Fellowship, the 2004 IEEE Microwave Theory and Techniques Society (IEEE MTT-S) Graduate Fellowship, third and first place in the Student Paper Competition at the 2003 and 2004 IEEE MTTT-S International Microwave Symposium (IMS), first place in the 2004 IEEE Region 10 (Asia–Pacific Region) Student Paper Contest (Postgraduate Level) and the Hong Kong Young Scientist Excellence Award.

Leung Chiu (S’05) received the B.Eng. degree in electronic engineering and M.Eng. degree in electronic engineering with business management from the City University of Hong Kong, Kowloon, Hong Kong, in 2004, and is currently working toward Ph.D. degree at the City University of Hong Kong. His research interests include microwave circuits and antenna arrays. Mr. Chiu was the recipient of second place of the 2004 IEEE Region 10 (Asia–Pacific Region) Student Paper Contest (Undergraduate Level).

Chi Hou Chan (S’86–M’86–SM’00–F’02) received the Ph.D. degree in electrical engineering from the University of Illinois at Urbana-Champaign, in 1987. Since April 1996, he has been with the Department of Electronic Engineering, City University of Hong Kong, Kowloon, Hong Kong. He is currently Dean of the Faculty of Science and Chair Professor of Electronic Engineering. He is also an Adjunct Professor with the University of Electronic Science and Technology of China, Peking University, and Zhejiang University. His research interests include computational electromagnetics, electronic packaging, antennas design, and microwave and millimeter-wave communications systems. Prof. Chan was a recipient of the 1991 U.S. National Science Foundation (NSF) President Young Investigator (PYI) Award.

Quan Xue (M’02–SM’05) received the B.S., M.S., and Ph.D. degrees in electronic engineering from the University of Electronic Science and Technology of China, Chengdu, China, in 1988, 1990, and 1993, respectively. In 1993, he joined the Institute of Applied Physics, University of Electronic Science and Technology of China, as a Lecturer. He became an Associate Professor in 1995 and a Professor in 1997. From October 1997 to October 1998, he was a Research Associate and then a Research Fellow with the Chinese University of Hong Kong. In 1999, he joined the City University of Hong Kong, Kowloon, Hong Kong, where he is currently an Associate Professor, the Director of the Applied Electromagnetics, and the Director of Industrial Technology Center. His research interests include microwave circuits, antennas and wireless communications.

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Modified T-Shaped Planar Monopole Antennas for Multiband Operation Sheng-Bing Chen, Yong-Chang Jiao, Wei Wang, and Fu-Shun Zhang

Abstract—In this paper, we propose a novel modified T-shaped planar monopole antenna in that two asymmetric horizontal strips are used as additional resonators to produce the lower and upper resonant modes. As a result, a dual-band antenna for covering 2.4and 5-GHz wireless local area network (WLAN) bands is implemented. In order to expand the lower band, a multiband antenna for covering the digital communications systems, personal communications systems, Universal Mobile Telecommunications Systems, and 2.4/5-GHz WLAN bands is also developed. Prototypes of the multiband antenna have been successfully implemented. Good omnidirectional radiation in the desired frequency bands has been achieved. The proposed multiband antenna with relatively low profile is very suitable for multiband mobile communication systems. Index Terms—Monopole antenna, multiband antenna, wireless communication.

I. INTRODUCTION IRELESS communications have been developed widely and rapidly in the modern world, which leads to a great demand in designing compact, low-profile, and multiband antennas for mobile terminals. To meet these requirements, compact high-performance multiband planar antennas with good radiation characteristics are needed. Recently many antennas with multiband and wideband characteristics have been successfully designed for wireless applications [1]–[8]. In these designs, they can provide a dual-band operation for the application in the wireless local area network (WLAN) communication systems. However, a multiband system is becoming necessary to provide more services including the digital communication system (DCS), Universal Mobile Telecommunications System (UMTS) and personal communications system (PCS), and yet a multiband antenna covering DCS1800, PCS1900, UMTS2000, and 2.4- and 5-GHz WLAN bands is very scant in the literature. In this paper, we first propose a novel modified T-shaped planar monopole antenna. Two asymmetric horizontal strips are used to provide two broadband dual-resonance modes. As a result, a dual-band antenna for covering the 2.4- and 5-GHz WLAN systems is achieved, which was initially presented in [9]. By widening the right horizontal strip and using an L-shaped notch in the right horizontal strip, a multiband antenna covering DCS1800, PCS1900, UMTS2000, and 2.4-

W

Manuscript received December 24, 2005; revised March 23, 2006. This work was supported by the National Natural Science Foundation of China under Grant 60171045. S.-B. Chen, Y.-C. Jiao, and F.-S. Zhang are with the National Laboratory of Antennas and Microwave Technology, Xidian University, Xi’an, Shaanxi 710071, China (e-mail: [email protected]; ychjiao@xidian. edu.cn; [email protected]). W. Wang is with the East China Research Institute of Electronic Engineering, Hefei, Anhui 230031, China (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2006.877811

Fig. 1. Geometry of the modified T-shaped monopole antenna. The dimensions (in millimeters) shown in this figure are not to scale.

and 5-GHz WLAN bands is then implemented. The proposed antenna is both compact in size and multiband operation suitable for the DCS (1710–1880 MHz), PCS (1850–1990 MHz), UMTS (1920–2170 MHz), 2.4-GHz WLAN (IEEE 802.11b in the U.S.: 2400–2484 MHz), and 5-GHz WLAN (HIPERLAN/ 2 in Europe: 5150–5350/ 5470–5725 MHz and IEEE 802.11a in the U.S.: 5150–5350/ 5725–5825 MHz) applications. This paper is organized as follows. As a starting point, Section II presents the dual-band antenna and its measured, as well as simulated results. After that, the multiband antenna design and parameter study are described in Section III, and a prototype of the antenna is also constructed and tested. Finally, the entire study is summarized in Section IV. II. DUAL-BAND ANTENNA A. Antenna Geometry Fig. 1 shows the geometry and dimensions of the initial proposed modified T-shaped planar monopole antenna for multiband operation. The proposed antenna is excited using a 50microstrip feed line. The radiating element has compact dimensions of 10 26 mm and is printed on the front of an inexpensive FR4 substrate with a thickness of 1 mm and a relative permittivity of 4.4. The ground plane is selected to be 40 50 mm and is printed on the back of the substrate. In this design, the proposed antenna consists of a vertical strip and two asymmetric horizontal strips on the top that can produce two different surface current paths and result in a dual-resonance mode. The right horizontal strip provides the lower mode covering the 2.4-GHz WLAN band, while the left one controls

0018-9480/$20.00 © 2006 IEEE

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Fig. 2. Simulated and measured VSWRs for the proposed antenna. Standards bandwidth requirement in gray. (Color version available online at: http://ieeexplore.ieee.org.)

Fig. 3. Geometry of the multiband monopole antenna. The dimensions (in millimeters) shown in this figure are not to scale.

L

Fig. 5. Simulated VSWRs with variation of for the antenna in Fig. 3 with = 25 mm. (Color version available online at: http://ieeex= 0 mm and plore.ieee.org.)

t

L

Fig. 6. Prototype of the proposed multiband antenna. (Color version available online at: http://ieeexplore.ieee.org.)

H for the antenna in Fig. 3 with

Fig. 7. Simulated and measured VSWRs for the multiband antenna in Fig. 6. Standards bandwidth requirement in gray. (Color version available online at: http://ieeexplore.ieee.org.)

the upper mode including 5-GHz WLAN bands. A conducting triangular section is also added into the vertical strip of the T-shaped monopole, as shown in Fig. 1, which improves the

impedance matching for both bands. With this structure, it is very easy to achieve the desired bandwidth for the lower and upper modes.

Fig. 4. Simulated VSWRs with variation of = 0 mm and = = 25 mm.

t

L

L

CHEN et al.: MODIFIED T-SHAPED PLANAR MONOPOLE ANTENNAS FOR MULTIBAND OPERATION

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Fig. 8. Measured radiation patterns for the proposed multiband antenna. (a) 1.9 GHz. (b) 2.4 GHz. (c) 5.2 GHz. (d) 5.8 GHz. (Color version available online at: http://ieeexplore.ieee.org.)

B. Measured and Simulated Results The proposed antenna was simulated using two commercial electromagnetic solvers, Ansoft’s High Frequency Structure Simulator (HFSS) and CST’s Microwave Studio, and a prototype of the antenna was constructed and tested. Fig. 2 shows the measured and two simulated voltage standing-wave ratios (VSWRs) of the proposed antenna in Fig. 1. The measured data in general agree with the simulated results obtained from Ansoft’s HFSS and CST’s Microwave Studio. Two separate broadband resonant modes are clearly excited at 2.4 and 5.5 GHz simultaneously with a good matching condition. With the definition of 2 : 1 VSWR, the impedance bandwidth of the lower mode covers the 2.4-GHz WLAN band (2400–2484 MHz). For the upper mode, a wide impedance bandwidth is obtained, which is sufficient to encompass the 5-GHz WLAN bands (5150–5350/5470–5825 MHz). III. MULTIBAND ANTENNA

Here, effects of two key antenna parameters, i.e., the width of the right horizontal strip and the length of the lower right strip, are considered on the antenna bandwidth. The geometry of the improved antenna is shown in Fig. 3. For the simulations here, and are fixed at 0 and 25 mm, respectively. Fig. 4 illustrates the simulated VSWRs versus width of the right horizontal strip ( mm) when mm. It can be seen from Fig. 4 that with increasing width , the bandwidths for the lower band increase and the resonant frequencies of the lower band shift down, while the bandwidths of the upper band change slightly. As a matter of fact, the use of a rectangular corner notch leads to a closer dual-resonant response and, therefore, extends the impedance bandwidth of the lower resonant mode. The simulated VSWR curves with optimal width of 7 mm and different length of the lower branch in the right horizontal strip are plotted in Fig. 5. It can be observed that the impedance matching is improved with decreasing .

A. Antenna Design In order to make the antenna cover simultaneously the DCS, PCS, and UMTS bands, we widen the right horizontal strip and use an L-shaped notch in the right horizontal strip. The parametric studies of the antenna are conducted to obtain the influence of the dimensions on the antenna performance and to find a set of optimal design parameters for the desired operating frequency bands. The simulations are performed using CST’s Microwave Studio package, which utilizes the finite integration technique for electromagnetic computation.

B. Measured and Simulated Results After the parametric study, an optimal design of the antenna is suggested, as shown in Fig. 3. Moreover, the use of an additional rectangular notch of the right horizontal strip is helpful for extending the lower mode. The L-shaped notch has two different removed sections with 9 3 mm and 1 1 mm areas at the lower right corner. A prototype of the proposed multiband antenna was fabricated according to the aforementioned design result, as shown

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in Fig. 6. A 50- microstrip line is used to feed this proposed antenna and is etched on the FR-4 substrate. The dimensions of the multiband antenna with mm, mm, and mm are depicted in Fig. 3. The measured and simulated VSWRs of the proposed multiband antenna are shown in Fig. 7. The measured data agree in general with the simulated results. Two separate broadband resonant modes are clearly excited at 2.15 and 5.47 GHz simultaneously with a good matching condition. With the definition of a 2 : 1 VSWR, the impedance bandwidth of the lower mode is 930 MHz (1.66–2.59 GHz) covering the DCS, PCS, UMTS, and 2.4-GHz WLAN bands. For the upper mode, an impedance bandwidth of 1410 MHz (4.48–5.89 GHz) is obtained, which is sufficient to encompass the 5-GHz WLAN bands. The measured radiation patterns at 1.9, 2.4, 5.2, and 5.8 GHz for the proposed antenna are plotted in Fig. 8. It can be seen that the radiation patterns in – -plane are nearly omnidirectional for four frequencies, and those in the – - plane, as expected, are very monopole-like. Thanks to the low signal level and complex environments for the measurement of an omnidirectional antenna, some measured patterns look rough in spots. IV. CONCLUSION A compact modified T-shaped planar monopole antenna has been proposed. The proposed antenna can provide sufficient impedance bandwidths and suitable radiation patterns for wireless applications. A dual-band antenna for the 2.4- and 5-GHz WLAN systems is addressed. As a further design, a multiband antenna prototype is constructed for applications in the DCS, PCS, UMTS, and 2.4- and 5-GHz WLAN systems. The measured impedance bandwidths are 0.93 GHz at the lower frequency band of 2.4 GHz, and 1.41 GHz at the upper frequency band of 5 GHz. The measured radiation patterns in the – -plane are of a nearly omnidirectional characteristic at operating frequencies. Therefore, the antennas are good candidates for multiband communication applications. REFERENCES [1] P. Ciais, R. Staraj, G. Kossiavas, and C. Luxey, “Compact internal multiband antenna for mobile phone and WLAN standards,” Electron. Lett., vol. 40, pp. 920–921, Jul. 2004. [2] H. C. Go and Y. W. Jang, “Multi-band modified fork-shaped microstrip monopole antenna with ground plane including dual-triangle portion,” Electron. Lett., vol. 40, pp. 575–577, May 2004. [3] S. Y. Lin, “Multiband folded planar monopole antenna for mobile handset,” IEEE Trans. Antennas Propag., vol. 52, no. 7, pp. 1790–1794, Jul. 2004. [4] H. D. Chen, J. S. Chen, and Y. T. Cheng, “Modified inverted-L monopole antenna for 2.4/5 GHz dual-band operations,” Electron. Lett., vol. 39, pp. 1567–1568, Oct. 2003. [5] C.-M. Su, H.-T. Chen, and K.-L. Wong, “Printed dual-band dipole antenna with U-slotted arms for 2.4/5.2 GHz WLAN operation,” Electron. Lett., vol. 38, pp. 1308–1309, Oct. 2002. [6] Y.-L. Kuo and K.-L. Wong, “Printed double-T monopole antenna for 2.4/5.2 GHz dual-band WLAN operations,” IEEE Trans. Antennas Propag., vol. 51, no. 9, pp. 2187–2192, Sep. 2003. [7] J.-Y. Jan and L.-C. Tseng, “Small planar monopole antenna with a shorted parasitic inverted-L wire for wireless communications in the 2.4-, 5.2-, and 5.8-GHz bands,” IEEE Trans. Antennas Propag., vol. 52, no. 7, pp. 1903–1905, Jul. 2004. [8] T. H. Kim and D. C. Park, “CPW-fed compact monopole antenna for dual-band WLAN applications,” Electron. Lett., vol. 41, pp. 291–293, Mar. 2005.

[9] S.-B. Chen, Y.-C. Jiao, F.-S. Zhang, and Q.-Z. Liu, “Modified T-shaped planar monopole antenna for 2.4/5 GHz WLAN applications,” in Proc. Asia–Pacific Microw. Conf., Suzhou, China, Dec. 4–7, 2005, vol. 4, pp. 2714–2716. Sheng-Bing Chen was born in Anhui, China. He received the B.S., M.S., and Ph.D. degrees in electrical engineering from Xidian University, Xi’an, China, in 1998, 2001, and 2006, respectively. His current research interests include antenna theory and design, antenna measurements, microwave engineering, computational electromagnetics, and optimization methods.

Yong-Chang Jiao received the B.S. degree in mathematics from Shanxi University, Taiyuan, China, in 1984, and the M.S. degree in applied mathematics and Ph.D. degree in electrical engineering from Xidian University, Xi’an, China, in 1987 and 1990, respectively. In 1990, he joined the Institute of Antennas and Electromagnetic Scattering, Xidian University, where he is currently a Professor. He was supported by the Japan Society for the Promotion of Science (JSPS) and, in 1996, was a Visiting Priority-Area Research Fellow with the University of Tsukuba, Japan. From October 1997 to January 1998 and from July 1999 to April 2000, he was also a Research Associate with the Chinese University of Hong Kong. From March to September in 2002, he was a Research Fellow with the City University of Hong Kong. He has authored or coauthored over 60 papers in technical journals and conference proceedings. His current research interests include antenna design, antenna engineering, antenna measurements, optimization algorithms and applications, and evolutionary computation. Dr. Jiao is a Senior Member of the Chinese Institute of Electronics (CIE). He is a member of the Youth Committee of the CIE and a member of the Antenna Committee of CIE. He was elected deputy to the 9th and 10th Shaanxi Provincial People’s Congress. He is a member of the Standing Committee of the 10th Shaanxi Provincial People’s Congress.

Wei Wang received the B.S. degree in physics from Anhui University, Hefei, China, in 1993, the M.S. degree from Xidian University, Xi’an, China, in 2001, and the Ph.D. degree from Shanghai University, Shanghai, China, in 2005, all in electrical engineering. From 1993 to 1998 and 2001 to 2002, he was with the East China Research Institute of Electronic Engineering (ECRIEE), as an Assistant Engineer and Engineer, respectively. He is currently a Senior Engineer with the ECRIEE. He has authored or coauthored over 20 journal papers and over ten conference papers. His research interests include waveguide slot antennas, microstrip antennas for radar, ultra-wideband (UWB) and small antennas for wireless communications, microwave passive devices and circuits, and microwave/millimeter systems. Dr. Wang is a Senior Member of the Chinese Institute of Electronics (CIE).

Fu-Shun Zhang received the Ph.D. degree in electrical engineering from Xidian University, Xi’an, China, in 2000. Since 1981, he has been with the Institute of Antennas and Electromagnetic Scattering, Xidian University, where he is currently a Professor. He has authored or coauthored over 40 papers in technical journals and conference proceedings. He coauthored Antenna Measurements (Xidian Univ. Press, 1995, in Chinese), and has coauthored one book chapter. His current research interests include antenna theory, engineering, and measurements, as well as microwave technology. Dr. Zhang is a Senior Member of the Chinese Institute of Electronics (CIE).

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Power-Efficient Switching-Based CMOS UWB Transmitters for UWB Communications and Radar Systems Rui Xu, Student Member, IEEE, Yalin Jin, and Cam Nguyen, Fellow, IEEE

Abstract—This paper presents a new carrier-based ultra-wideband (UWB) transmitter architecture. The new UWB transmitter implements a double-stage switching to enhance RF-power efficiency, reduce dc-power consumption, and increase switching speed and isolation, while reducing circuit complexity. In addition, this paper also demonstrates a new carrier-based UWB transmitting module implemented using a 0.18- m CMOS integrated pulse generator-switch chip. The design of a UWB sub-nanosecond-switching 0.18- m CMOS single-pole single-throw (SPST) switch, operating from 0.45 MHz to 15 GHz, is discussed. The design of a 0.18- m CMOS tunable impulse generator is also presented. The edge-compression phenomenon of the impulse signal controlling the SPST switch, which makes the generated UWB signal narrower than the impulse, is described. Measurement results show that the generated UWB signal can vary from 2 V peak-to-peak with 3-dB 4-ns pulsewidth to 1 V with 0.5 ns, covering 10-dB signal bandwidths from 0.5 to 4 GHz, respectively. The generated UWB signal can be tuned to cover the entire UWB frequency range of 3.1–10.6 GHz. The sidelobe suppression in the measured spectrums is more than 15 dB. The entire CMOS module works under a 1.8-V supply voltage and consumes less than 1 mA of dc current. The proposed carrier-based UWB transmitter and the demonstrated module provide an attractive means for UWB signal generation for both UWB communications and radar applications. Index Terms—CMOS RF integrated circuit (RFIC), pulse generator, single-pulse single-throw (SPST) switch, ultra-wideband (UWB) communications, UWB radar, UWB system, UWB transmitter.

I. INTRODUCTION LTRA-WIDEBAND (UWB) technology has received significant interests, particularly after the Federal Communications Commission (FCC)’s Notice of Inquiry in 1998 [1] and Report and Order in 2002 [2] for unlicensed uses of UWB devices within the 3.1–10.6-GHz frequency band. UWB techniques are promising technology, capable of both accurate position location and high-rate short-range ad hoc networking, as well as high-resolution sensing. Carrier-based UWB signals have been widely used in various radar and communication applications [3], [4]. Compared with carrier-free impulse/monopulse signals, carrier-based UWB

U

Manuscript received February 27, 2006; revised May 3, 2006. This work was supported in part by the National Science Foundation and in part by the U.S. Army Corps of Engineers. The authors are with the Electrical and Computer Engineering Department, Texas A&M University, College Station, TX 77843-3128 USA (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TMTT.2006.877830

Fig. 1. Block diagram of a typical carrier-based UWB transmitter.

signals hold the advantage of more convenient spectrum management and less distortion through antennas [5]. Moreover, the use of carrier-based signals facilitates the design of all components including antennas in the system, due to the signals’ much narrower bandwidth. Essentially, a carrier-based UWB signal is generated by multiplying an impulse signal with a single-tone carrier signal. Therefore, the bandwidth and central frequency of the generated signal can be manipulated by adjusting the pulsewidth of the impulse signal and the frequency of the single tone. Fig. 1 shows a typical carrier-based UWB transmitter. The clock generator is usually composed of timing-control digital circuits and sends out a clock signal that can include modulation information. The impulse generator produces an impulse signal whose pulsewidth is inversely proportional to the bandwidth of the required signal. The mixer performs a multiplication between the impulse and a single-tone signal generated by the oscillator. The up-converted signal is then sent to the wideband power amplifier (PA) to achieve a required amplitude. In this approach, the oscillator is required to generate multiple tones if the transmitter needs to operate over multiple frequency bands. This carrier-based UWB technique suffers from two major disadvantages. First, the PA design in the last stage is a challenging issue. This PA needs to supply enough gain and has reasonable power efficiency and good output matching over a wide frequency range. Second, in UWB radar applications, low pulse repetition frequency (PRF) is often utilized. Since the transmitted signal duration is usually very short, the resultant peak-to-average power ratio is extremely high. This means no signal needs to be transmitted during most of the time. However, because the PA and other circuits of the transmitter are “on” all the time, a large amount of power is wasted, rendering the approach power inefficient. Various carrier-based UWB transmitters using CMOS and BiCMOS SiGe processes have been developed [6]–[8].

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Fig. 2. Proposed carrier-based UWB transmitter topology. Fig. 3. Waveforms of the building blocks in the proposed UWB transmitter.

In this paper, we propose a new architecture for carrier-based UWB transmitters that is not only power efficient, but also reduces power consumption, enhances switching speed and isolation, and reduces circuit complexity. In addition, we also demonstrate the workability and performance of a new transmitting module, covering the entire 3.1–10.6-GHz UWB band, using a CMOS chip, consisting of an impulse generator integrated with a single-pole single-throw (SPST) switch, and an external synthesizer. The CMOS chip was designed and fabricated using a standard low-cost 0.18- m CMOS process. It consumes less than 1 mA of dc current. II. CARRIER-BASED UWB TRANSMITTER ARCHITECTURE Fig. 2 shows the block diagram of the newly proposed carrier-based UWB transmitter, consisting of a voltage-control oscillator (VCO), a buffer, an SPST switch, and two pulse generators. In this approach, power switching is used to perform the signal multiplication, instead of mixing, as used in the more typical UWB transmitter structure shown in Fig. 1 and those in [6]–[8]. The transmitter’s principle is based upon the concept of generating a carrier-based UWB signal by gating a single-tone signal with a small time window, thereby only producing a signal during a small time period. A double-stage switching procedure, using two pulse generators of wide and narrow pulses, and two switches, is adopted in the proposed transmitter to remedy the switching speed limitation of the buffer, inherent in CMOS circuits, to achieve sub-nanosecond gating required in UWB signal generation. The VCO generates carrier signals that define the center frequencies of the UWB signal to feed the buffer to realize sufficient transmitted power and proper output impedance matching. The buffer is gated through its internal switch (first-stage switching) using a wide pulse produced by pulse generator 1, which should be wide enough to allow the buffer to start and reach stabilization. The second-stage switching, performed by the SPST switch and pulse generator 2 generating narrower pulses, is then used to reduce the pulsewidth of the generated signal, making it a UWB signal having spectrum bandwidth of at least 500 MHz, as defined by the FCC [2]. Pulse generators 1 and 2 are synchronized using a common clock generator, as shown in Fig. 2. Fig. 3 illustrates the time-domain waveforms at the output of different blocks of the transmitter. It is noted that the SPST’s gating signal from pulse generator 2 needs to fall behind the rising edge of the buffer’s gating signal from pulse generator 2

Fig. 4. Effects of power leakage using: (a) single- and (b) double-stage switching.

to accommodate the slow switching time of the buffer. Using the first-stage switching to turn on/off the buffer not only saves power, particularly useful for battery-operated UWB devices, but is also needed to relax the isolation requirement for secondstage switching. In low PRF UWB applications, the level of power leakage needs to be very small, imposing a very strict isolation requirement on the gating components, thus necessitating the use of two switching stages. Fig. 4 shows the effects of power leakage and demonstrates the need for two switching stages. In Fig. 4(a), only the secondstage switching is used. stands for the pulsewidth of the UWB signal, while is the interval between two consecutive pulses. During the time the second-stage switching is off, the local oscillator (LO) signal (i.e., the VCO’s signal) still manages to arrive at the transmit antenna due to limited SPST switch isolation. Although this LO leakage has much smaller amplitude than the transmitted UWB signal, it can still accumulate sufficiently large power over the duration to overdrive the UWB signal on the transmitted spectrum. When this happens, a high-power single tone would be observed at the carrier frequency above the UWB signal spectrum. To avoid this problem, the signal-to-leakage ratio should be much higher than . For instance, for a 1-ns UWB signal pulse to be transmitted at 10-kHz PRF with negligible power leakage, the ratio is roughly approximately 10 and an isolation of much more than 50 dB is thus required to satisfy the leakage requirement if only one switching stage was used (i.e., the second stage). This level of isolation is very difficult to achieve in CMOS switches. In Fig. 4(b), both switching stages are applied. The LO leakage

XU et al.: POWER-EFFICIENT SWITCHING-BASED CMOS UWB TRANSMITTERS FOR UWB COMMUNICATIONS AND RADAR SYSTEMS

Fig. 5. Schematic of the CMOS SPST switch.

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Fig. 6. Insertion loss comparison of two 0.18-m MOSFETs having different widths.

only appears during the time the buffer is on; i.e., within the time widow , which is usually no longer than 10 ns. The isolation of the second-stage switching is only required to be larger than . In this case, 30-dB isolation is sufficient to ensure small LO leakage regardless of the PRF used. For CMOS switching, 30-dB isolation is a modest requirement and can be achieved by careful design. (a)

III. DEMONSTRATION OF A CARRIER-BASED UWB TRANSMITTER USING A PULSE GENERATOR–SPST SWITCH CMOS CHIP AND EXTERNAL SYNTHESIZER A new carrier-based UWB transmitting module covering the entire UWB band of 3.1–10.6 GHz is realized using a pulse generator–SPST switch CMOS chip, designed and fabricated using the TSMC 0.18- m CMOS process [9], and an external frequency synthesizer. The module is based on the transmitter concept proposed in Section II, but implemented without pulse generator 1 and the VCO and buffer replaced with an external frequency synthesizer. A. CMOS SPST Switch Design Fig. 5 shows the schematic of the CMOS SPST switch. One series and two shunt MOSFETs are used to provide compromise between insertion loss and isolation. Biasing resistors are used in lieu of RF chokes to minimize the chip area. In order to achieve an ultra-wide bandwidth, on-chip inductors between adjacent transistors are combined with the transistors’ capacitances to form a synthetic transmission line between the input and output of the SPST switch. The bulk (or substrate) terminals of the transistors are floated to improve the power-handling ability and insertion loss [10]. The SPST’s on and off states are obtained when the control signals , are set to , 0 and 0, , respectively. To ensure wideband performance, the sizes of the series and shunt transistors need to be carefully determined. The series transistor particularly plays an important role in the switch’s insertion loss, while the shunt transistors enhance the isolation when the switch is off. Shunt devices, however, inadvertently aggravate the insertion loss due to their parasitics. For the series-connected transistor, as the gatewidth is increased, the on resistance reduces, resulting in low insertion loss in the low-frequency region. The gate–source capacitance, however, increases and, hence, degrading the insertion loss at high frequencies. Fig. 6 compares the insertion loss of two 0.18- m

(b)

Fig. 7. (a) MOSFET with a step function applied to the gate. (b) Behavior of the gate voltage V for different gate bias resistances.

MOSFETs, each in series configuration, having two different gatewidths (64 and 192 m). As can be seen, the smaller size transistor has higher loss at lower frequencies, but its insertion loss maintains relatively constant over a wide frequency range. For shunt-connected MOSFETs, a larger size provides higher isolation at lower frequencies. Typical SPST transistor switch structures with gate biasing resistors are only suitable for slow switching due to the fact that large resistors are normally used to make the gate open at RF so that the switch performance is not affected. The large gate biasing resistor, however, leads to a large RC constant, which effectively slows down the control signal applied to the control terminal connecting to the gate via the resistor. Fig. 7 demonstrates the effect of the gate biasing resistor, assuming an ideal step signal is applied to the control terminal. Due to the gate resistor and the total gate-to-ground capacitance , which includes the parasitic and intrinsic capacitances seen at the gate, the rising edge of the resultant gate voltage is slowed down. As can be seen, a larger gate resistor gives a slower rising edge, leading to slower switching speed. From Fig. 7, it is apparent that in order to maintain the fidelity of the control sub-nanosecond pulse signals, the gate resistance should be less than a few hundred ohms. Although this may aggravate the insertion loss and return loss of the switch, the switch is expected to have reasonably good performance by optimizing its other components, while maintaining the rising/falling edge of the control pulse signal. The gate-to-ground capacitance is estimated as 0.5 pF, whereas the calculated gate–source ( ) and gate–drain ( ) capacitances are each between 0.2–0.3 pF. Numerically, is thus approximately equal to and in parallel, which is actually expected from the MOSFET’s equivalent circuit. Using the

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TABLE I SUMMARY OF THE DESIGNED SPST’S COMPONENTS

Fig. 9. CMOS impulse generation using digital logics. (Color version available online at: http://ieeexplore.ieee.org.)

Fig. 8. Measured insertion loss (S 21: on), input return loss (S 11), output return loss (S 22) and isolation (S 21: off) of the 0.18-m CMOS SPST switch. (Color version available online at: http://ieeexplore.ieee.org.)

estimated value of and the input control signal, the optimal value for the gate resistor , that produces the required 300-ps rising/falling time for the SPST’s output pulse, can thus be determined as 100 . The SPST switch was designed and fabricated using the TSMC 0.18- m CMOS process [9]. Table I summarizes the designed circuit elements. The SPST switch has been measured on-wafer using a probe station and an automatic network analyzer. Fig. 8 shows the measured results with 10-dBm input power. The SPST switch exhibits measured insertion loss less than 2 dB, return loss greater than 15 dB, and isolation more than 40 dB across the entire UWB band of 3.1–10.6 GHz. B. CMOS Pulse Generator Design The pulse generator generates an impulse signal to control the SPST switch. It is preferred that the impulse duration be adjustable so that different bandwidths can be attained for the UWB signal. The pulse generator is implemented based on digital logics [11]. This method of pulse generation is especially attractive for the proposed transmitter because it can provide full voltage swing required for the SPST switch operation. Fig. 9 shows the designed CMOS pulse generator. An inverter chain is used to sharpen the rising/falling edge of the clock generator’s signal needed for subsequent generation of narrow impulse-like signals. The sharpened clock signal is split into two branches. In one of the branch, the clock is delayed and inverted with respect to the other. The NAND gate then combines the rising and falling edges of these clocks to form an impulse. This pulse and its inversion are used to control the shunt and

Fig. 10. (a) Current starving inverter structure. (b) Its delay tuning range.

Fig. 11. Different simulated pulsewidths controlled by the bias voltage.

serial transistors in the SPST switch, respectively. The duration of the generated impulse is determined by the delay between the rising and falling edges. The current starved inverter, indicated in Fig. 9 and elaborated in Fig. 10, is used to flip the clock and generate tunable delay. The delay of an inverter is determined by the time used by the current to charge and discharge the load capacitance. Consequently, varying the inverter charging/discharging current through biasing can change the inverter delay, hence, producing tunable impulse, which consequently produces variable UWB signal bandwidths. Fig. 11 displays different durations of the impulse reaching the gate of the SPST switch at different biasing voltages. It is a common assumption that the pulsewidth of an impulse, used as the gate control signal for an SPST transistor switch, is equal to the duration of the output signal emerging from the switch. However, we find that, in practical circuits, these durations are not the same. This is mainly due to the fact that the amplitude of the output signal of the switch does not have a linear relationship

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Fig. 13. Microphotograph of the 0.18-m CMOS chip integrating the pulse generator and the SPST switch.

Fig. 12. Rising-edge compression of the SPST switch.

with the control voltage, which results in different time dependence between the switch’s output signal and the control signal. Fig. 12 briefly explains this phenomenon. Suppose an impulse with 400-ps rising edge, as shown in Fig. 12(a), is used to control the switch. Fig. 12(b) displays the calculated normalized output amplitude of the switch versus the control voltage, showing that the switch output begins to appear for control voltages higher than 0.7 V. Fig. 12(c) shows the switch’s output amplitude as a function of time, obtained by mapping the output amplitude into the time domain using the data from Fig. 12(a) and (b). As can be seen, the rising edge of the output signal is compressed to 200 ps. This resultant sharpened edge is expected to result in an UWB signal having narrower duration than that of the control impulse. It is expected that the gating impulse signal will couple to the transmitter’s output port through the transistors’ gate–source capacitances. To minimize this undesirable effect, a slower rising/ falling edge for the control signal is preferred. By exploiting the edge compression property of the switch, as explained earlier, we can afford to relax the rising/falling edge restriction for the control signal while still meeting the minimum pulsewidth requirement for the UWB signal. As will be seen in the measurement results of the new transmitting module, the gate coupling is negligible as compared with the generated UWB signal. The pulse generator circuit is not fabricated and measured separately. Nevertheless, its workability is demonstrated through the measurement of the entire CMOS chip integrating the pulse generator and SPST switch. C. Measurement Results The 0.18- m CMOS pulse generator-SPST switch chip has been used in conjunction with an external frequency synthesizer to demonstrate a new carrier-based UWB transmitter. The CMOS chip’s microphotograph is displayed in Fig. 13. The die area of the whole circuit is 850 m 700 m including input

Fig. 14. Measured UWB signal having 500-MHz bandwidth. (a) Time-domain waveform. (b) Spectrum.

and output on-wafer pads. The measurement was conducted on-wafer. The frequency synthesizer supplied the LO signal feeding the input port of the pulse generator–SPST switch chip. A 15-MHz clock was used to drive the on-chip pulse generator. The whole circuit consumes less than 1-mA dc current. Fig. 14 displays the time-domain waveform and the spectrum of a measured UWB signal. The UWB signal has a 3-dB pulsewidth of 4 ns and a 10-dB bandwidth of 500 MHz at 5-GHz center frequency, which conforms to the FCC’s minimum UWB bandwidth requirement. The UWB signal amplitude is 2 V (peak-to-peak) and its amplitude at 5 GHz is 20 dBm. The 2-V peak-to-peak voltage is the maximum voltage level corresponding to the SPST being turned on completely. The sidelobe level is below 15 dBc. By increasing the external biasing voltage for the pulse generator, the pulsewidth of the UWB signal is reduced. When the pulsewidth is reduced

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Fig. 16. Measured spectrums of UWB signals covering the 3.1–10.6-GHz UWB band. (Color version available online at: http://ieeexplore.ieee.org.)

Fig. 15. Measured UWB signal having 4-GHz bandwidth. (a) Time-domain waveform. (b) Spectrum.

to a certain value, the amplitude of the UWB signal, however, diminishes due to the fact that the SPST switch cannot be completely turned on any more, which results in partial reflections of the carrier signal. When the amplitude of the UWB signal decreases to half of its maximum value (2 V), the corresponding duration is defined as the minimum pulsewidth. This minimum pulsewidth is measured as 0.5 ns. Fig. 15 displays the UWB signal with this minimum pulsewidth. The peak-to-peak voltage amplitude is around 1 V. The 10-dB bandwidth is around 4 GHz. The LO leakage overshooting can be clearly seen from the spectrum of this signal because only a single-stage switching is used here. This LO leakage confirms experimentally the need of a double-stage switching for the carrier-based UWB transmitters (Fig. 2), as explained in Section II. With another stage of switching, the LO leakage can be reduced to a negligible level. With both levels of switching, the efficiency of the transmitter will be reduced due to the added power consumptions of the VCO and buffer, with the latter particularly dominating the efficiency owing to its larger power consumption. It is, however, relatively easy to achieve high power efficiency for a single-tone buffer. The pulsewidth of the obtained UWB signal can be further reduced by increasing the bias voltage of the pulse generator, hence achieving wider signal bandwidth than 4 GHz. This, however, is obtained at the expense of its signal amplitude. Fig. 16 shows the measured spectrums of different UWB signals, obtained by varying the carrier frequency at a certain bias voltage for the pulse generator, demonstrating that the entire UWB band of 3.1–10.6 GHz can be achieved with the developed CMOS chip by using a multiband signal source. To evaluate effects of coupling from the impulse signal applied to the SPST’s control terminals, we remove the LO input

Fig. 17. (a) Measured output signal coupled from the impulse control signal. (b) Its spectrum.

signal and directly observe the coupled signal at the output. The measured coupled signal and its spectrum are displayed in Fig. 17. The observed positive and negative portions correspond to the rising and falling edges of the control impulse, respectively. When the pulsewidth of the control impulse is increased, the positive and negative parts move away from each other. On the contrary, when the pulsewidth is reduced, they overlap and cancel each other. The time-domain results show that the amplitude of the coupled signal is negligible as compared to the UWB signals shown in Figs. 14 and 15. Furthermore, the energy of this coupled signal is mainly distributed below 1 GHz, as seen in Fig. 17(b), signifying that the signal is out of the UWB band and can be easily removed using a filter. IV. CONCLUSION We have introduced a new carrier-based UWB transmitter architecture that is suitable for both UWB radar and communications. The newly proposed UWB transmitter is simple and implements two switching stages to conserve RF power, reduce power consumption, and enhance switching and isolation. We have also demonstrated a new carrier-based UWB transmitting module implemented using a fully integrated 0.18- m CMOS chip that combines a tunable pulse generator and an UWB SPST switch and an external frequency synthesizer. The

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0.18- m CMOS SPST switch shows very good performance over the entire UWB band. The 0.18- m CMOS tunable pulse generator generates an impulse signal to control the SPST switch. The transmitting module produces UWB signals of different bandwidths ranging from 0.5 to 4 GHz. By varying the LO frequency, the signal spectrum’s center frequency can be shifted to any frequency within the UWB band of 3.1–10.6 GHz. The coupling of the gate control signal is found to be negligible and can be filtered out easily. Measurement results confirm the workability of the new transmitter as another simplified approach for carrier-based UWB signal generation. Given a particular UWB application and corresponding requirements, we can expect improvement on the transmitter’s performance by optimizing its individual components for that application specifically.

Yalin Jin was born in Baoding, China, in 1974. He received the B.S. degree in electrical engineering from Xi’an Jiaotong University, Xi’an, China, in 1995, the M.S. degree in electrical engineering from the Academy of Sciences, Beijing, China, in 1998, and is currently working toward the Ph.D. degree in electrical engineering at Texas A&M University, College Station. From 1998 to 2001, he was with XinWei Telecom Inc., Beijing, China, where he was involved in designing RF circuits for synchronous code division multiple access base stations and mobile phones. From 2001 to 2002, he was a Senior Design Engineer involved with RF circuits on global system for mobile communication (GSM) cellular phones with Motorola Electronics Inc., Beijing, China. During the summer of 2006, he interned as an RF Integrated Circuit (RFIC) Designer for orthogonal-frequency division-multiplexing (OFDM) UWB integrated transceivers in BiCMOS with Alereon Inc., Austin, TX. His research interest are high-speed integrated circuits and high-frequency RFICs.

REFERENCES

Cam Nguyen (F’05) received the B.S. degree from the California State Polytechnic University, Pomona, in 1980, the M.S. degree from the California State University, Northridge, in 1983, and the Ph.D. degree from the University of Central Florida, Orlando, in 1990, all in electrical engineering, while working full time at nearby industries. Following over 12 years in industry, in 1991 he joined the Department of Electrical Engineering, Texas A&M University, where he is currently the Texas Instruments Endowed Professor. From 2003 to 2004, he was Program Director with the National Science Foundation (NSF), where he was responsible for research programs in RF electronics and wireless technologies. From 1979 to 1990, he had various engineering positions in industry, including being a Microwave Engineer with the ITT Gilfillan Company, a Member of Technical Staff with the Hughes Aircraft Company (now Raytheon), a Technical Specialist with the Aeroject ElectroSystems Company, a Member of Professional Staff with the Martin Marietta Company (now Lockheed-Martin), and a Senior Staff Engineer and Program Manager with TRW (now Northrop Grumman). While in industry, he led numerous microwave and millimeter-wave activities and developed many microwave and millimeter-wave hybrid and monolithic ICs and systems up to 220 GHz for communications, radar and remote sensing. At Texas A&M University, he has developed and taught different courses in microwave electronics—from theoretical EM field analysis to practical design of RF, microwave ICs and systems, and established a prominent diversified and interdisciplinary research program in microwave electronics. His research group at Texas A&M University currently focuses on CMOS RF ICs and systems, microwave and millimeter-wave ICs and systems, and UWB devices and systems for wireless communications, radar, and sensing applications—developing not only individual components, but also complete systems including design, signal processing, integration, and test. His research group has particularly been at the forefront of developing UWB ICs and systems for subsurface sensing and wireless communications and pioneered the development of microwave IC systems for sensing applications. His group developed some of the first complete all-MIC planar time-domain and frequency-domain UWB ground-penetrating radars, incorporating antennas, transmitters, and receivers completely on single packages. His group also developed some of the first millimeter-wave planar IC Doppler velocimetry for low-velocity measurement and interferometric sensors for displacement sensing with an unprecedented resolution of =840. He has authored or coauthored over 150 refereed papers, one book, and several book chapters, and given more than 80 conference presentations and numerous invited lectures. He has served as a member of the Editorial Boards and Technical Committees and a reviewer for various journals and conferences. He is the founding Editor-in-Chief of Sensing and Imaging: An International Journal. Dr. Nguyen was the chairman of the International Conference on Subsurface Sensing Technologies and Applications from 1999 to 2001.

[1] “Revision of part 15 of the Commission’s rules regarding ultra-wideband transmission systems,” FCC, Washington, DC, Sep. 1, 1998 [Online]. Available: http://www.fcc.gov/oet/dockets/et98-153, FCC notice of inquiry, adopted Aug. 20, 1998, released Sep. 1, 1998 [2] “Revision of part 15 of the Commission’s rules regarding ultra-wideband transmission systems,” FCC, Washington, DC, 2002, FCC report and order, adopted Feb. 14, 2002, released Jul. 15, 2002. [3] R. J. Fontana, “Recent system applications of short-pulse ultra-wideband (UWB) technology,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 9, pp. 2087–2104, Sep. 2004. [4] G. R. Aiello, “Challenges for ultra-wideband (UWB) CMOS integration,” in Proc. IEEE RFIC Symp., Jun. 2003, pp. 497–500. [5] R. J. Fontana and J. F. Larrick, “Waveform adaptive ultra-wideband transmitter,” U.S. Patent 6026 125, Feb. 15, 2000. [6] J. Ryckaert, C. Desset, A. Fort, M. Badaroglu, V. De Heyn, P. Wambacq, G. Van der plas, S. Gonnay, B. Van Poucke, and B. Gyselinckx, “Ultra-wide-band transmitter for low-power wireless body area networks: Design and evaluation,” IEEE Trans. Circuits Syst. I, Fundam. Theory Appl., vol. 52, no. 12, pp. 2515–2525, Dec. 2005. [7] J. Zhao, C. Maxey, A. Narayanan, and S. Raman, “A SiGe BiCMOS ultra wide band RFIC transmitter design for wireless sensor networks,” in Proc. Radio Wireless Conf., Sep. 2004, pp. 215–218. [8] D. D. Wentzloff and A. P. Chandrakasan, “A 3.1 GHz-10.6 GHz ultrawideband pulse-shaping mixer,” in Proc. IEEE RFIC Symp., Jun. 2005, pp. 83–86. [9] “TSMC 0.18-m CMOS process,” MOSIS Foundry, Marina del Rey, CA, 2005. [10] M.-C. Yeh, Z.-M. Tsai, R.-C. Liu, K.-Y. Lin, Y.-T. Chang, and H. Wang, “Design and analysis for a miniature CMOS SPDT switch using body-floating technique to improve power performance,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 1, pp. 31–39, Jan. 2005. [11] K. Marsden, H. Lee, D. S. Ha, and H. Lee, “Low power CMOS re-programmable pulse generator for UWB systems,” in Proc. IEEE Ultra Wideband Syst. Technol. Conf., Nov. 2003, pp. 443–447. Rui Xu (S’06) was born in Bengbu, China, in 1976. He received the B.S. and M.S. degrees in electrical engineering from Southeast University, Nanjing, China, in 1998 and 2001, respectively, and is currently working toward the Ph.D. degree at Texas A&M University, College Station. In 2002, he joined the Sensing, Imaging and Communication Systems Laboratory, Texas A&M University, as a Research Assistant. His research interests include RF, and microwave and millimeter-wave integrated circuits for wireless and sensor systems.

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Low-Power-Consumption and High-Gain CMOS Distributed Amplifiers Using Cascade of Inductively Coupled Common-Source Gain Cells for UWB Systems Xin Guan and Cam Nguyen, Fellow, IEEE

Abstract—A distributed amplifier with new cascade inductively coupled common-source gain-cell configuration is presented. Compared with other existing gain-cell configurations, the proposed cascade common-source gain cell can provide much higher transconductance and, hence, gain. The new distributed amplifier using the proposed gain-cell configuration, fabricated via a TSMC 0.18- m CMOS process, achieves an average power gain of around 10 dB, input match of less than 20 dB, and noise figure of 3.3–6.1 dB with a power consumption of only 19.6 mW over the entire ultra-wideband (UWB) band of 3.1–10.6 GHz. This is the lowest power consumption ever reported for fabricated CMOS distributed amplifiers operating over the whole UWB band. In the high-gain operating mode that consumes 100 mW, the new CMOS distributed amplifier provides an unprecedented power gain of 16 dB with 3.2–6-dB noise figure over the UWB range. Index Terms—CMOS RF integrated circuit (IC), distributed amplifiers, low-noise amplifiers (LNAs), low power-consumption amplifiers, RF IC, ultra-wideband (UWB) systems.

I. INTRODUCTION LTRA-WIDEBAND (UWB) has received significant interests, particularly after the Federal Communications Commission (FCC)’s Notice of Inquiry in 1998 [1] and Report and Order in 2002 [2] for unlicensed uses of UWB devices within the 3.1–10.6-GHz frequency band. UWB techniques are promising technology capable of both accurate position location and highrate short-range ad hoc networking. UWB systems transmit and receive information using many narrow pulses each second with extremely low-power spectral densities across extremely wide bandwidths. This effectively produces very small interference to other radio signals while maintaining excellent immunity to interference from these signals. UWB devices can, therefore, work within frequencies already allocated for other radio services, thus helping to maximize this dwindling resource. Distributed amplifiers are attractive candidates for UWB systems due to its inherently ultra-wide bandwidth. It first appeared in a patent filed by Percival in 1937 [3]. Since then, it has been used for decades in technologies ranging from vacuum tubes to

U

Manuscript received February 10, 2006; revised April 25, 2006. This work was supported in part by the National Science Foundation and in part by the U.S. Army Corps of Engineers. The authors are with the Electrical and Computer Engineering Department, Texas A&M University, College Station, TX 77843-3128 USA (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TMTT.2006.877812

hybrid and monolithic microwave integrated circuits (MMICs), e.g., [4]–[7]. The recent emergence of silicon as a viable alternative for realization of low-cost and highly integrated RF integrated circuits (RFICs) has created great interest in the implementation of distributed amplifiers in silicon-based CMOS (and related BiCMOS, SiGe, etc.) With the advances of CMOS technology and availability of accurate CMOS device and passiveelement models, many CMOS distributed amplifiers having relatively flat gains over very wide bands within the dc to 40-GHz range have been reported, e.g., [9]–[18]. A major drawback of distributed amplifiers for UWB applications is their large dc power consumption, severely limiting their usage in wireless portable devices. This is due to the fact that several parallel transistors, with each transistor draining current from the source, are needed to form the required artificial transmission lines and to achieve a reasonable gain on a 50- load. Recently, the design of low-power distributed amplifier in CMOS has been addressed [16]–[18]. Most of these designs are based on the gaincell topology presented in [11] and [12] and do not provide enough gain and bandwidth at very low power consumption. In this paper, we report on the development of a new CMOS distributed amplifier having very low power consumption for UWB applications, particularly suitable for portable UWB wireless devices. Significant performance advantages of the new low-power distributed amplifier design are demonstrated. The new UWB distributed amplifier is fabricated using a standard TSMC 0.18- m CMOS process [19]. It achieves the highest gain (in high-gain mode) and lowest power consumption (in low-power mode) ever reported across the UWB frequency range of 3.1-10.6 GHz. The measured noise figure is similar to the best published noise figure. II. CIRCUIT ANALYSIS The major challenge in designing a low power-consumption distributed amplifier is the tradeoff between power consumption and gain. In order to lower the power consumption, each transistor has to be biased at a very low overdrive voltage. This, however, leads to insufficient gain for the whole amplifier. To address this issue, we propose a new distributed amplifier topology, as shown in Fig. 1. New gain cells, each consisting of two cascade common-source transistors with peaking inductor, are used. This new gain-cell configuration improves the gain significantly with similar power consumption or similar gain with substantially reduced power consumption, over the entire

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Fig. 1. Simplified schematic of the new low-power distributed amplifier.

Fig. 2. Lumped-element network representing one element of the artificial transmission line.

Fig. 3. (a) Simple amplifier with input and output capacitors. (b) Distributed amplifier implementing the simple amplifiers as gain cells.

B. Gain and Bandwidth UWB band of 3.1-10.6 GHz, as compared to the conventional distributed amplifiers. A. Basic Design Principle The basic design principle of distributed amplifiers is based on forming two artificial transmission lines at the input and output ports of the constituting gain cells by periodically combining serial inductive components with the parasitic capacitors of the gain cells. Essentially, the input and output capacitances are absorbed into the artificial transmission lines, resulting in an extremely wideband performance. These transmission lines consist of multiple elements, each approximately represented by a frequencydependent lumped-element network, as shown in Fig. 2. The characteristic impedance of the artificial transmission line and its phase velocity can be found, respectively, as (1) (2) , where is the cutoff frequency of the artificial for transmission line, which normally determines the bandwidth of the distributed amplifier, given as (3) of the transmission If the desired characteristic impedance line is fixed, the cutoff frequency can be expressed as (4) Equation (4) shows that the bandwidth of a distributed amplifier decreases as value of the (parasitic) capacitance increases. Since this capacitance is proportional to the dimension of the employed MOSFET transistors and the gain of the distributed amplifier, (4) also shows possible tradeoff between gain and bandwidth in distributed amplifier design.

We consider a simple amplifier, as shown in Fig. 3(a). The frequency response of this amplifier is affected by the poles at both the input and output nodes. As several of these amplifiers, acting as gain cells, are parallel-connected in distributed amplifiers, as shown in Fig. 3(b), their input and output capacitors are absorbed into the artificial transmission lines of the distributed amplifiers. The two poles are then pushed far away from their original locations at very low frequencies to the cutoff frequency of the two artificial transmission lines. As a result, the distributed amplifier provides a constant gain over a very wide frequency band. The gain of the conventional distributed amplifier has been analyzed in [6] and can be estimated, in a loss-free case, by (5) is the transconductance where is the number of the stages, of the each gain cell, and and are the characteristic impedance of the input line and output line, respectively. Equation (5) shows that, if and are fixed to a certain value, the gain of the distributed amplifier is only affected by and . Although the gain can be increased by simply increasing the number of stages before it reaches a maximum number, as given in [8], limited by the increasing losses of the artificial transmission lines, this increase is obtained at the cost of larger die area and higher power consumption, which are not very desirable, particularly for commercial CMOS-based portable devices. Apparently, the transconductance of each gain cell is the most important issue for the gain. It should be noted that the intrinsic gain of each gain cell is no longer important because each gain cell in the distributed amplifier is driving a small load. Furthermore, these gain cells cannot take advantages of their large output impedance for gain enhancement because of the small load. The capability to provide sufficiently large current at the output of the gain cells will determine the gain of the distributed amplifier. Fig. 4(a)–(c) shows several gain-cell configurations used in CMOS distributed amplifiers [3]–[17]. Fig. 4(a) is a common-source stage, which has been used in distributed amplifier designs for a very long time. It provides a decent gain

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Fig. 4. Gain cell configurations used in CMOS distributed amplifiers. (a) Traditional common source gain cell. (b) Cascode gain cell. (c) Current reuse gain cell. (d) Proposed cascade common-source gain cell.

Fig. 5. (a) Small-signal equivalent circuit of the proposed cascade gain cell. (b) Equivalent-circuit model for the gain and bandwidth analysis.

and very large bandwidth. Fig. 4(b) is a cascode structure used to enhance reverse isolation. This structure does not provide significantly higher than a single common-source transistor and, thus, does not have considerable gain advantage over that in Fig. 4(a). In Fig. 4(c), a current reuse gain cell is formed by nMOS and pMOS transistors. This configuration is supposed to have higher transconductance at proper bias voltages and transistor dimensions. However, because of the mobility difference of nMOS and pMOS devices, the bias and dimensions for this structure are very difficult to be determined. Fig. 4(d) shows the newly proposed cascade common-source gain cell. Two common-source transistors are connected with each other through a peaking inductor. A resistor is connected to the gate of the second transistor. Passive inductors are used, instead of active inductors, to avoid possible variation of inductance and increase in noise over extremely wide bandwidths such as the UWB band of 3.1–10.6 GHz. As will be seen, this new structure enhances the transconductance significantly and, hence, the amplifier’s gain. Fig. 5(a) shows a small-signal equivalent circuit of the new cascade common-source gain cell, which is used to determine its equivalent transconductance. , , and

Fig. 6. Calculated transconductance of the proposed gain cell and a commonsource gain cell stage.

are the respective gate-to-source, drain-to-bulk, and gate-to-drain capacitances of the nMOS transistor at the first and second stage in each gain cell. , and , are the transconductances and output resistances of the two nMOS devices, respectively. and are the resistance and inductance used in each gain cell, as and are shown in Fig. 4(d), respectively. Typically, relatively large and, therefore, can be neglected. Neglecting and and combining , with , and , to form and , which are the Miller equivalent capacitance observed at the gate and drain of the first transistor, we can derive the transconductance shown in (6) at the bottom of this page. Usually, and, thus, the transconductance expression, can be further simplified as

(7) From (7), three poles can be observed. One pole is formed by the input capacitor of the upper transistor. This pole normally dominates the low-frequency response due to the fact that the value of the gate-to-source capacitance of a transistor is usually large. The other two complex conjugate poles are created by the inductance and output capacitance of the lower transistor. In practice, these two poles are located on the left -plane instead of exactly on the complex axis because of the loss incurred in real circuits. The presence of these two poles will boost up the transconductance of this gain cell at frequency (8) which can also be considered as the cutoff frequency of the proposed gain cell’s transconductance.

(6)

GUAN AND NGUYEN: LOW-POWER-CONSUMPTION AND HIGH-GAIN CMOS DISTRIBUTED AMPLIFIERS

Fig. 7. Calculated frequency response of the transconductance of the proposed gain cell for different values of inductance.

Fig. 8. Microphotograph of the low power distributed LNA (1.6

2 0.9 mm )

By properly choosing the inductance and resistance in the new cascade common-source gain cell, a nearly constant transconductance can be obtained between the first-pole frequency and the cutoff frequency in (8). Unlike a singletransistor gain cell, whose transconductance remains fairly constant over a particular frequency range, the proposed gain cell exhibits more frequency variation for its transconductance (see Fig. 7 for various designed cells). Considering this and the fact that the gain of distributed amplifiers is proportional to the transconductance of each gain cell, the cutoff frequency of a distributed amplifier employing the proposed gain cell is determined by the cutoff frequency of each gain cell’s transconductance, instead of that of the artificial transmission lines at the input and output of the amplifier, provided that the former is lower than the latter. Fig. 6 shows the calculated transconductance of the new cascade gain cell and that of a single common-source transistor. Comparing with the single common-source transistor, this cascade common-source gain cell provides a significantly higher transconductance, which is expected to lead to a significantly higher gain. Exploiting this resultant unique capability of providing high transconductance, the cascade common-source gain cells can be biased at a very low voltage to consume a very small amount of power, while still providing enough transconductance to obtain a decent gain for the entire amplifier. The new distributed amplifier topology facilitates the enhancement of stability. The cascade structure used in each gain cell stage results in higher reverse isolation as compared to a single-transistor cell, while maintaining good input and output matching, which may help improve the stability of the distributed amplifier. Although the inductors used in the

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Fig. 9. Measured and simulated power gain (S ) and input return loss (S ) at low power consumption mode (measured P = 19:6 mW).

Fig. 10. Measured and simulated power gain and input return loss at high gain mode (measured P dc = 100 mW).

Fig. 11. Measured and simulated output return loss (S ) and isolation (S ).

Fig. 12. Measured and simulated noise figure at high-gain and low-power modes.

gain cells may resonate with the input gate capacitors of some transistors, no negative resistance is expected to occur at the

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TABLE I PERFORMANCE SUMMARY OF CMOS DISTRIBUTED AMPLIFIERS

gates of these transistors due to the facts that all the transistors are grounded at the source and an 80- resistor is used in each gain cell. Our simulations of the designed distributed amplifier show that the stability factor is greater than 1 across dc to 18 GHz, demonstrating its expected unconditional stability. III. CIRCUIT DESIGN The new distributed amplifier is designed and fabricated using the TSMC 0.18- m CMOS process. The gatewidth of each transistor is optimized for gain and power consumption. The inductance in each gain cell determines the bandwidth of the distributed amplifier. The calculated frequency response with different values of inductance is shown in Fig. 7. 1 nH is chosen for the inductance to obtain a reasonably flat gain over the considered UWB frequency range of 3.1–10.6 GHz. On-chip inductors are used for the inductive elements of the artificial transmission lines in the distributed amplifier. Very small inductance values are used in order to have high- resulting from less eddy current loss in the silicon substrate. The full-wave electromagnetic (EM) simulator IE3D [20] is used for simulations of all passive components including bends, interconnects, and spiral inductors. The two artificial transmission lines are terminated with 50- resistors and ac-coupling capacitors to minimize reflections. The new CMOS distributed amplifier design uses three stages to ensure a decent gain, reasonable die area, and small power consumption. The designed amplifier has a chip size of 1.6 0.9 mm . Fig. 8 shows its microphotograph. IV. MEASUREMENT RESULTS The fabricated CMOS distributed amplifier was measured using on-wafer probes. Bias tees and dc-biasing probes were used to bias the circuit. Fig. 9 shows the calculation and measurement results for the power gain and input return loss. The amplifier exhibits a measured gain of around 10 dB and input return loss less than 20 dB across the entire UWB of 3.1-10.6 GHz, which agree reasonable with those calculated. The power consumption was measured at 19.6 mW with the entire circuit biased at an extremely low voltage of 0.7 V. Fig. 10 shows the calculated and measured gain and input return loss at a high-gain mode. As can

be seen, the distributed amplifier demonstrates a high gain of around 16 dB and return loss below 20 dB, measured across the UWB 3.1-10.6-GHz range, with a power consumption of 100 mW. The measured and calculated results also agree fairly well. A very high gain near dc is observed from both Figs. 9 and 10. This is due to the and network that appears between the two transistors in each gain cell, as expected from the simulations. The measured gain shows a lower cutoff frequency than that of simulation. This may be attributed to the parasitics resulting from the layout that were not taken into account in the simulations. Fig. 11 shows the measured and simulated results for the isolation and return loss at the output. Both calculated and measured results show that there is only a very small difference between the high-gain mode and low-power mode. It is also noted that use of the new cascade gain cells results in high isolation of more than 30 dB for the distributed amplifier. The noise observed in the measured and data shown in Figs. 9–11 were due to calibration of the vector network analyzer. Nevertheless, multiple measurements for different chips have been performed and, despite this noise, the results are very consistent. The abnormal peak in the measured near 6.5 GHz seen in Fig. 11 may be due to some unexpected parasitics at the output port of the fabricated amplifier. The noise figure of the new CMOS distributed amplifier at both high-gain and low-power modes has also been measured using an Agilent noise-figure analyzer. The measured and simulated results are plotted in Fig. 12. The noise figure in the low-power mode is slightly higher than that in the high-gain mode both in simulation and measurement. Since sufficient gain is provided by the lower transistor in each gain cell, the new distributed amplifier with cascade gain cell configuration does not give higher noise figure than that of traditional distributed amplifiers. Table I compares the performance of the new CMOS distributed amplifier with those of recently published CMOS distributed amplifiers. The new distributed amplifier demonstrates both the highest gain (in high-gain mode) and lowest power consumption (in low-power mode) with gain comparable to the best reported gain across the UWB frequency range of 3.1–10.6 GHz. The measured noise figure is also similar to the best published noise figure.

GUAN AND NGUYEN: LOW-POWER-CONSUMPTION AND HIGH-GAIN CMOS DISTRIBUTED AMPLIFIERS

V. CONCLUSION A new low power-consumption design approach along with its analysis for CMOS distributed amplifiers has been presented. The new design employs cascade of common-source gain cells with peaking inductance to provide substantially enhanced transconductance and gain over the entire UWB frequency of 3.1–10.6 GHz. The new 0.18- m CMOS distributed amplifier implementing these gain cells achieves the lowest power consumption, with decent gain comparable to the best gain reported, over the entire UWB band, good input match, and good noise figure similar to the best published noise figure. The new CMOS distributed amplifier also exhibits the highest gain ever reported across the UWB range with good noise figure when operated in the high-gain mode. The performance confirms the suitability of the developed CMOS amplifier for UWB systems. REFERENCES [1] “Revision of part 15 of the Commission’s rules regarding ultra-wideband transmission systems,” FCC, Washington, DC [Online]. Available: http://www.fcc.gov/oet/dockets/et98-153, notice of inquiry adopted August 20, 1998; released September 1, 1998 [2] “Revision of part 15 of the Commission’s rules regarding ultra-wideband transmission systems,” FCC, Washington, DC, Report and order, adopted February 14, 2002; released July 15, 2002. [3] W. S. Percival, “Thermionic valve circuits,” U.K. British Patent 460 562, Jan. 29, 1937. [4] E. W. Strid and K. R. Gleason, “A DC–12 GHz monolithic GaAs FET distributed amplifier,” IEEE Trans. Microw. Theory Tech., vol. MTT-30, no. 7, pp. 969–975, Jul. 1982. [5] B. Agarwal, A. E. Schmitz, J. J. Brown, M. Matloubian, M. G. Case, M. Le, M. Lui, and M. J. W. Rodwell, “112-GHz, 157-GHz, and 180-GHz InP HEMT traveling-wave amplifiers,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 12, pp. 2553–2559, Dec. 1998. [6] C. S. Aitchison, “The intrinsic noise figure of the MESFET distributed amplifier,” IEEE Trans. Microw. Theory Tech., vol. MTT-33, no. 6, pp. 460–466, Jun. 1985. [7] Y. Ayasli, R. L. Mozzi, J. L. Vorhaus, L. D. Reynolds, and R. A. Pucel, “A monolithic GaAs 1–13-GHz traveling-wave amplifier,” IEEE Trans. Microw. Theory Tech., vol. MTT-30, no. 7, pp. 976–981, Jul. 1982. [8] J. B. Beyer, S. N. Prasad, J. E. Nordman, R. C. Becker, G. K. Hohenwarter, and Y. Chen, “Wideband monolithic microwave amplifier study,” Office Naval Res., Washington, DC, ONR Rep. NR243-033-02, Jul. 1982.

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[9] B. M. Ballweber, R. Gpta, and D. J. Allstot, “A fully integrated 0.5–5.5-GHz CMOS distributed amplifier,” IEEE J. Solid-State Circuits, vol. 35, no. 2, pp. 231–239, Feb. 2000. [10] H. Ahn and D. J. Allstot, “A 0.5–8.5-GHz fully differential CMOS distributed amplifier,” IEEE J. Solid-State Circuits, vol. 37, no. 8, pp. 985–993, Aug. 2002. [11] R.-C. Liu, K.-L. Deng, and H. Wang, “A 0.6–22-GHz broadband CMOS distributed amplifier,” in IEEE RFIC Symp. Dig., 2003, pp. 103–106. [12] R.-C. Liu, C.-S. Lin, K.-L. Deng, and H. Wang, “A 0.5–14 GHz 10.6-dB CMOS cascade distributed amplifier,” in IEEE VLSI Circuits Symp. Dig., 2003, pp. 139–140. [13] R. E. Amaya, N. G. Tarr, and C. Plett, “A 27 GHz fully integrated CMOS distributed amplifier using coplanar waveguide,” in Proc. IEEE RFIC Symp., 2004, pp. 193–196. [14] H. Shigematsu, M. Sato, T. Hirose, F. Brewer, and M. Rodwell, “40 GB/s CMOS distributed amplifier for fiber-optic communication systems,” in Proc. IEEE Int. Solid-State Circuits Conf., 2004, pp. 476–477. [15] X. Guan and C. Nguyen, “A 0.25-m CMOS ultra-wideband amplifier for time-domain UWB applications,” in IEEE Radio Freq. Integr. Circuits Symp., Long Beach, CA, Jun. 2005, pp. 339–342. [16] H.-L. Huang, M.-F. Chou, W.-S. Wuen, K. A. Wen, and C-Y. Chang, “A low power CMOS distributed amplifier,” in IEEE Annu. Wireless Microw. Tech. Conf., 2005, p. 3. [17] S. Arekapudi, E. Iroaga, and B. Murmann, “A low power distributed wideband LNA in 0.18 m CMOS,” in Proc. IEEE Circuits Syst. Dig., 2005, pp. 5055–5058. [18] F. Zhang and P. Kinget, “Low power programmable-gain CMOS distributed LNA for ultra-wideband applications,” in Proc. VLSI Circuits Dig., 2005, pp. 78–81. [19] “TSMC 0.18-m CMOS process,” MOSIS Foundry, Marina del Rey, CA, 2005. [20] IE3D. Zeland Software Inc., Fremont, CA, 2005. Xin Guan was born in Beijing, China in February 1976. He received the B.S. and M.S. degrees in electronic engineering from Tsinghua University, Beijing, China, in 1999 and 2002, respectively, and is currently working toward the Ph.D. degree at Texas A&M University, College Station. In August 2002, he joined the Sensing, Imaging, and Communication Systems Laboratory, Texas A&M University, as a Research Assistant. His current research interests include RF, microwave, and millimeter-wave integrated circuits for wires sensors and systems. Cam Nguyen (F’05), photograph and biography not available at time of publication.

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Low Phase-Noise Microwave Oscillators With Interferometric Signal Processing Eugene N. Ivanov and Michael E. Tobar, Senior Member, IEEE

Abstract—Phase-noise spectral density of a 9-GHz oscillator has been reduced to 160 dBc/Hz at 1-kHz offset frequency, which is the lowest phase noise ever measured at microwave frequencies. This performance was achieved by frequency locking a conventional loop oscillator to a high- sapphire dielectric resonator operating at the elevated level of dissipated power ( 0.4 W). Principles of interferometric microwave signal processing were applied to generate the error signal for the frequency control loop. No cryogenics were used. Two almost identical oscillators were constructed to perform classical two-oscillator phase-noise measurements where one oscillator was phase locked to another. The phase locking was implemented by electronically controlling the level of microwave power dissipated in the sapphire dielectric resonator. Index Terms—Microwave interferometry, oscillator frequency stabilization, phase locking.

I. INTRODUCTION PPLICATION of interferometric signal processing to frequency stabilization of microwave oscillators resulted in a dramatic improvement in their phase-noise performance [1]. Subsequent progress in this field was rather slow due to the fundamental limitations imposed by thermal fluctuations (Nyquist noise) on the resolution of frequency measurements. During this time, most of the research efforts were focused on improving the short-term frequency stability of low phase-noise microwave oscillators and developing the methods of their electronic frequency tuning [2]–[4]. Arrival of high-power solid-state microwave amplifiers offered an opportunity of lowering the thermal noise limit by increasing the power dissipated in the high- resonator. By following this approach, we have constructed two 9-GHz oscillators with the single-sideband (SSB) phase-noise spectral density 160 dBc/Hz at a Fourier frequency of 1 kHz, which represents an order of magnitude improvement relative to the current state-of-the-art. In this paper, we: 1) summarize the basic stages of the design of high-power microwave oscillators with interferometric signal processing; 2) describe the two-oscillator phase-noise measurement system; 3) analyze the methods of evaluating the contributions of various noise mechanisms to the oscillator phase noise; 4) discuss the phase-noise performance of the high-power oscillators developed; and 5) describe the excess phase-noise phenomena in high-power oscillators.

A

Manuscript received February 17, 2006; revised May 12, 2006. This work was supported by the Australian Research Council and by Poseidon Scientific Instruments Pty. Ltd. The authors are with the School of Physics, University of Western Australia, Crawley 6009 W.A., Australia (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2006.879172

II. DESIGN, NOISE PROPERTIES, AND PHASE LOCKING OF MICROWAVE OSCILLATORS WITH INTERFEROMETRIC SIGNAL PROCESSING Fig. 1 shows a schematic diagram of a low phase-noise microwave oscillator with interferometric signal processing. It is based on a high- microwave resonator, which serves as both a bandpass filter of a self-sustaining loop oscillator and a dispersive element of a frequency discriminator (FD). The latter consists of a microwave interferometer and a phase sensitive readout system featuring a low-noise microwave amplifier and a mixing stage. Microwave signal reflected from the high- resonator interferes destructively with a fraction of incident signal at the “dark port” of the interferometer. This cancels the carrier of the difference signal while preserving the noise sidebands caused by frequency fluctuations of the loop oscillator. The noise sidebands are amplified (low-noise amplification is a key factor in reducing the FD effective noise temperature, which otherwise would have been limited by fluctuations of the nonlinear mixing stage) and demodulated to dc producing an error voltage varying synchronously with the oscillator frequency. The filtered error voltage from the mixer’s output is applied, in an appropriate phase, to the voltage controlled phaseshifter (VCP) in the loop oscillator steering its frequency to the given resonant mode of a high- resonator. The FD, filter, and VCP form a frequency control system (FCS), which detects and cancels the oscillator frequency fluctuations. Assuming a frequency control loop of sufficiently high gain, the quality of oscillator frequency stabilization would be entirely determined by the noise properties of the FD [1]. In this respect, the interferometric measurement systems are greatly superior to their conventional counterparts. The main reasons for this are the low effective noise temperature of the interferometric systems and their ability to handle high levels of signal power without saturating the readout system, provided that the carrier at the interferometer “dark port” is adequately suppressed. We consider a typical interferometric FD with the effective noise temperature K based on a 9-GHz room-temperature stabilized sapphire loaded cavity (SLC) resonator with a factor 2 10 . Assuming the level of dissipated microwave power mW, the SSB phase-noise floor of such a discriminator at Fourier frequency kHz would be close to 165 dBc/Hz. This is almost four orders of magnitude better than has been ever achieved with the conventional FDs based on the SLC resonators. The phase-noise floor in the above example is set by intrinsic fluctuations in the microwave electronics of the frequency discriminator. This limit scales inversely with the microwave power dissipated in the resonator [2]. However, there exists detremental

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Fig. 1. Two-oscillator phase-noise measurement system. (Color version available online at: http://ieeexplore.ieee.org.)

noise mechanisms of which the resolution of spectral measurements (and, therefore, the spectral purity of oscillator signal) is either independent or increases with signal power. One of the noise mechanisms involves a direct conversion of oscillator power fluctuations into the phase noise due to the dependence of the SLC resonant frequency on power. For a typical -band SLC resonator operating at room temperature, the magnitude of the power-to-frequency conversion is approximately 0.4 kHz/mW [6]. Fluctuations of signal power are largely responsible for the excess noise in the spectrum of oscillator phase fluctuations at the relatively low Fourier frequencies Hz , where thermal inertia of the sapphire resonator does not strongly manifest itself. In such a regime, increasing the resonator dissipated power by a factor of would result in an increase in the intensity of oscillator phase fluctuations. Another noise mechanism due to which oscillator power fluctuations could affect its phase noise is associated with the residual amplitude sensitivity of the FD. In principle, by utilizing a source of a pure amplitude modulated signal, one could completely cancel the amplitude sensitivity of an FD by adjusting the reference phase shift in the mixer local oscillator (LO) arm (see Fig. 1). Alternatively (as it is not clear whether a source of a pure amplitude modulation can be constructed), one could fine tune the FD by monitoring the voltage noise at the output of two-oscillator noise measurement system and minimizing it by adjusting the reference phase shifts in both oscillators. This may be a time-consuming procedure, as both phase shifts must be set with a relatively high accuracy given by

where and are spectral densities of oscillator phase and amplitude fluctuations, respectively. Assuming a typical

microwave oscillator with a spectral density of amplitude fluctuations (1 kHz) 130 dBc/Hz, the error in setting the reference phase shift needs to be less than a degree if the phase noise of kHz dBc/Hz is to be achieved. It should be noted that the spectral density of the excess phase noise resulting from the FD residual amplitude sensitivity is independent of signal power [1]. A serious attention must be paid to the sources of nonthermal fluctuations inside the interferometer associated with the voltage-controlled and ferrite devices. Such components are known to exhibit a flicker phase noise, which can prevent one from reaching the oscillator phase-noise performance set by the FD effective noise temperature. Setting the resonator effective coupling close to critical is an effective way of reducing the oscillator frequency flicker noise. In our experiments, coupling coefficients of both resonators were set close to unity (carrier suppression by each resonator was 20 dB). The classical technique of oscillator phase-noise measurements involves phase locking of one oscillator (“slave”) to another (“master”). This means making the “slave” oscillator frequency tunable while preserving its low phase-noise operation. It is not a trivial task, as those two requirements are contradictory. In the case of the oscillator with interferometric signal processing, the problem of its frequency tuning is usually solved by introducing a VCP into the interferometer’s compensating arm. This enables fast (in a sense that it is not limited by the SLC thermal time constant ) tuning of the oscillator frequency by varying the phase mismatch of the microwave interferometer. The main problem here is the increased level of oscillator phase noise caused by the intrinsic phase fluctuations of the VCP. Setting the resonator coupling coefficient close to unity overcomes the excess phase-noise problem while sacrificing the frequency tunability.

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Another approach to oscillator frequency tuning relies on changing the operating temperature of the SLC resonator, say, with a thermal-electric cooler [7]. This type of frequency tuning is characterized by a large dynamic range (tens of megahertz), but is very slow. Another problem is that the temperature-to-frequency transfer function may look similar to that of a high-order low-pass filter due to the imperfect thermal joints in the path of the heat flow. This greatly complicates the design of the phase-locked loop (PLL). Power-to-frequency conversion in the SLC oscillator offers the most convenient way of oscillator frequency tuning by changing the level of microwave power dissipated in the resonator. This technique combines a relatively wide tuning range with a relatively quick response time equal to the thermal time s for a room-temperaconstant of a sapphire cylinder ( ture -band SLC resonator) [6]. Most importantly, it preserves the oscillator phase noise outside the lock-in range of the PLL. We followed this approach while designing the two-oscillator noise measurement system. III. TWO-OSCILLATOR NOISE MEASUREMENT SYSTEM An electronic frequency tuning of a “slave” oscillator was implemented by varying the level of microwave power dissipated in the SLC resonator with a voltage-controlled attenuator (VCA) placed before the microwave loop amplifier (Fig. 1). Two main reasons were considered when deciding about the location of the VCA. First, there is a risk of damaging the VCA by the high microwave power when placing it after the loop amplifier (power rating of a typical VCA is 100 mW). Second, the need to avoid unnecessary power loss in the path from the amplifier to resonator (this may not be the main reason anymore since the amplifiers capable of delivering powers in excess of 1 W are becoming available). Ignoring the practical limitations associated with the power rating of a VCA, we calculated the power incident on the resonator as a function of the VCA bias voltage for two different locations of the VCA within the microwave loop. The simulations were conducted assuming the following set of parameters: amplifier maximum output power W, the insertion loss from the output of the loop amplifier to the SLC resonator dB, and the oscillator gain margin (amplifier compression level) dB. The results of these simulations, as well as the experimental data, are shown in Fig. 2. Here, curves 1 and 2 correspond to the situations when the VCA was placed, respectively, after and before the loop amplifier. Curve 3 shows the experimental dependence of the incident power on the VCA bias (with the VCA located before the loop amplifier). In the latter case, power incident on the resonator varied from 520 to 380 mW following the variations of the VCA bias voltage from 0.5 to 1.5 V. The corresponding range of oscillator frequency tuning was equal to mW kHz/mW kHz. This was large enough to keep the “slave” oscillator locked to the “master” for 10 15 giving us sufficient time to accumulate at least 20 30 averages from the fast Fourier transform (FFT) spectrum analyzer even when taking data in the frequency range from 1 to 10 Hz. The relatively short

Fig. 2. Power incident on resonator as a function of VCA bias voltage: VCA is placed after the loop amplifier (curve 1, simulations); VCA is placed in front of the loop amplifier (curve 2, simulations); VCA is in front of the loop amplifier (curve 3, experiment) (experiments were conducted with commercial amplifiers JCA0911 from JCA Technology, Sunnyvale, CA). (Color version available online at: http://ieeexplore.ieee.org.)

phase-locking time was a consequence of poor temperature control of one of the resonators, which caused the difference frequency between two oscillators to vary (with the amplitude 20 kHz) following the room-temperature variations induced by the air conditioner. To ensure a stable phase locking of the “slave” oscillator to the “master,” one needs to know its complex frequency tuning coefficient. The analytical expression for such a coefficient could be derived by introducing the variation of the VCA bias voltage and solving the loop oscillator characteristic equation with respect to the corresponding variation of amplifier input power

(1)

are the VCA insertion loss and its voltage where and derivative , and are the gain of the loop amplifier and its power derivative , respectively. In case of a VCA located after the loop amplifier, the oscillator and the frequency tuning coefficient is derived from (1) and is given by

(2)

is the power-to-frequency conversion coefwhere ficient of the SLC resonator and is the loss in the path from the amplifier to the resonator (additional to the VCA loss). In deriving (2), it was assumed that the cavity coupling was close to critical.

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Fig. 4. Structure of the PLL active filter.

Fig. 3. Amplitude of frequency tuning coefficient as a function of bias voltage applied to VCA: VCA is in front of the loop amplifier (curve 1); VCA is behind the loop amplifier (curve 2). (Color version available online at: http://ieeexplore. ieee.org.)

In case of a VCA located before the loop amplifier, the frequency tuning coefficient is

(3)

The dependencies of the frequency tuning coefficient on the VCA bias voltage calculated in accordance with (2) and (3) are shown in Fig. 3 (curves 1 and 2, respectively). In both cases, the oscillator gain margin was chosen to be 4 dB at the VCA mean bias voltage V. As follows from the data in Fig. 3 at low bias voltages ( V), it is better (from the viewpoint of the efficient frequency tuning) to have the VCA located after the loop amplifier, while at higher bias voltages, the more efficient frequency tuning is achieved with the VCA placed before the loop amplifier. In the latter case, the frequency tuning coefficient was calculated to be 30 kHz/V at V, which was 20% less than the measured value. Parameter in (2) and (3) is a complex function of Fourier frequency due to the thermal inertia of the sapphire resonator. The frequency dependence of is similar to that of a voltage transfer function of a first-order low-pass filter with the time constant equal to . When designing a PLL for referencing the “slave” to “master,” one has to compensate for the relatively large phase lag introduced by the oscillator frequency actuator with a phase lead of the loop filter. This could be achieved with an active filter shown in Fig. 4. It contains a quasi-differentiator with a flat–ramp–flat (FRF) response, a “leaking” integrator (with unity ac gain), and a sum device. The quasi-differentiator provides the phase lead required for making the control loop stable, while the integrator ensures that PLL has enough gain to cope with the temperature-induced oscillator phase fluctuations (spectral density of which varies as

Fig. 5. Voltage noise suppression factor of the PLL based on the filter with parameters shown on a circuit diagram in Fig. 4. (Color version available online at: http://ieeexplore.ieee.org.)

). A small resistor at the input of the quasi-differentiator is needed to prevent the “short circuiting” of the preceding stage by the large input capacitance of the quasi-differentiator. Finally, the sum device combines the bias voltage and correction signal from the phase detector of the PLL. It also limits the dynamic range of the correction signal in order to avoid the negative bias on the VCA. The PLL filter shown in Fig. 4 permitted a stable phase locking of the “slave” oscillator to the “master.” The quality of the phase locking was characterized by the PLL voltage noise suppression factor . The latter was determined with the FFT network analyzer by measuring the transfer function of the PLL sum device in a closed-loop configuration. The results of such measurements, as well as the computed dependence of , are shown in Fig. 5. As follows from the above data, the bandwidth of the PLL was approximately 10 Hz. A “hump” at the dependence of the was observed at frequencies around 20 Hz. It corresponded to voltage noise enhancement due to the low phase margin of the feedback loop. This noise enhancement could be minimized by increasing the upper corner frequency of the FRF filter or, in other words, by making the quasi-differentiator closer to the ideal one. PLL transfer function , along with the sensitivity of the PLL phase detector, are the key parameters required for inferring the oscillator phase noise from the spectrum of voltage fluctuations at the output of the noise measurement system (see Section V).

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is the loaded half-bandwidth of the SLC resonator where and is the total phase shift around the loop consisting of three separate terms (5) and are phase shifts induced by the loop where amplifier, VCP, and VCA, respectively. The last two terms in (5) depend on voltages and , which are given by and , where is the bias voltage of the VCP, is the correction voltage produced by the FCS, is the bias voltage of the VCA, and is the correction voltage produced by the PLL. Introducing the differential operator , one can express the FCS correction voltage as Fig. 6. Phase-noise floors of 9-GHz two-oscillator phase-noise measurement systems (mixer RF power 1 mW). (Color version available online at: http:// ieeexplore.ieee.org.)



Measurements of the oscillator phase noise were carried out with a dual-channel readout system featuring a cross-correlation signal processing. This enabled the improved spectral resolution of the phase-noise measurements due to the cancellation of uncorrelated fluctuations of the each channel [8]–[10]. The SSB phase-noise floors of both single- and dual-channel readout systems are shown in Fig. 6. The phase-noise floor of the dualchannel noise readout system was approximately 168 dBc/Hz at kHz, representing an 12 dB advantage over its single-channel counterpart. In principle, the oscillator phase-noise measurements could be conducted with much better accuracy by making use of the interferometric signal processing (the phase-noise floor of the interferometric phase detector was measured to be 180 dBc/Hz at kHz when comparing phases of two 10-mW signals). Unfortunately, a relatively poor temperature control of one of the SLC resonators prevented us from utilizing such a readout system. Variations of the difference frequency between two oscillators kept upsetting the phase balance of the interferometer, making the readout system susceptible to oscillator amplitude noise. With the intensity of amplitude noise being more than two orders of magnitude higher than that of a phase noise (at 1 kHz kHz), the experimental data obtained with the interferometric system were largely corrupted by the AM noise.

(6) is the Laplace transform of the FCS loop filter where transfer function and is the frequency-dependent voltage at the output of the FD. The PLL correction voltage is given by

(7) is the Laplace transform of the PLL loop filter where transfer function and is the phase detector phase-to-voltage conversion efficiency. Parameter is a phase difference between “slave” and “master” oscillators, which, in general case, is a function of time . Assuming a stationary regime of the PLL and introducing small fluctuations of the bias voltages around their mean values and , the “slave” oscillator frequency fluctuations can be expressed from (4) as (8) is the fluctuations of the SLC resonant frequency where and corresponds to intrinsic phase fluctuations of the microwave loop amplifier. Two other fluctuating terms in (8) are given by

IV. NOISE ANALYSIS OF A PHASE-LOCKED OSCILLATOR Here, we analyze the noise properties of a “slave” oscillator based on a real VCA whose insertion loss and phase shift are both functions of the applied voltage. For the loop oscillator in a stationary regime, the relationship between frequency of oscillations and frequency of the resonator is given by (4)

where and are fluctuations of the correction voltages produced, respectively, by the frequency and phase control loops. The SLC resonant frequency depends on both ambient temperature and microwave power dissipated, which, in turn, depends on the bias voltages on the VCA and VCP. As a result,

IVANOV AND TOBAR: LOW PHASE-NOISE MICROWAVE OSCILLATORS WITH INTERFEROMETRIC SIGNAL PROCESSING

fluctuations of the SLC resonant frequency fluctuations in (8) can be expressed as

where is the term representing the temperature-induced fluctuations of the SLC resonant frequency. Parameters and are complex functions of Fourier frequency characterizing the SLC resonant frequency sensitivity to variations of VCP and VCA bias voltages, respectively. At this stage, it is useful to introduce frequency fluctuations of a free-running “slave” oscillator as (9) The first term of the sum (9) corresponds to frequency fluctuations associated with the noise in the microwave electronics of the loop oscillator (10) The second term in (9) corresponds to fluctuations of the SLC resonant frequency

characterizes the FD residual sena real FD. Parameter sitivity to amplitude fluctuations of the “slave” oscillator (see Section II). Parameter corresponds to the intrinsic voltage noise at the output of the FD. By solving the system of linear differential equations (12)–(14), one can obtain analytical expressions for various components of “slave” oscillator phase noise. To make these expressions less cumbersome, we introduce the special notations for the gains of different control systems of a “slave” oscillator. In particular, gain of the FCS can be presented as a sum of “fast” and “slow” terms: , where and . Term reflects the fast action of the FCS via instantaneous change of oscillator frequency due to variation of the VCP phase describes the slow action of the FCS via the delay. Term power induced change in the SLC resonant frequency. For most of the VCPs, one can safely assume that . Gain of the PLL can be expressed in a sim, where ilar fashion: and , where . By making use of these new notations, the relationship between frequency fluctuations of the “slave” and “master” oscillator is obtained as follows:

(15)

(11) . where In case of a perfect voltage-controlled element (15) takes more familiar form

Substituting (10) and (11) in (8) results in

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,

(12) The latter equation along with (6) and (7) forms the necessary basis for the phase noise analysis of a “slave” oscillator. When searching for voltage variations and in (12), we assume that the phase shift between two oscillators is a sum of a mean value and fluctuating term . In such a case, from (7), it follows that

(13) where is the intrinsic voltage noise a the output of the PLL phase detector. By analogy, voltage fluctuations are obtained from (6) as follows: (14) where is the frequency sensitivity of the FD. The last two terms in (14) are due to the imperfections of

which shows that within the PLL bandwidth , the frequency of the “slave” oscillator varies synchronously with the frequency of the “master” . Fluctuations of bias voltages and result in frequency fluctuations of the “slave” oscillator (16) and are the frequency fluctuations of the where free-running “slave” oscillator given by (10) and (11). Once again, assuming that , (16) is simplified as follows:

The latter equation indicates that “slave” oscillator frequency fluctuations due to microwave electronics (10) are suppressed by both frequency control and PLLs, while frequency fluctuations due to instability of the SLC resonant frequency (11) are suppressed by the PLL only.

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Intrinsic voltage noise of the phase detector imposes the following limit on “slave” oscillator frequency noise

(17) is the PLL frequency noise floor

where . At

, (17) becomes

which shows that noise process does not affect “slave” oscillator frequency fluctuations outside the PLL bandwidth (noise limits the resolution of phase noise measurements outside the PLL bandwidth). Finally, spurious amplitude sensitivity of the FD, along with its intrinsic voltage noise, is responsible for the oscillator excess frequency noise given by

Fig. 7. Phase-noise spectra of a 9-GHz high-power oscillator with interferometric signal processing. (Color version available online at: http://ieeexplore. ieee.org.)

(18)

and correspond, respectively, to the frewhere quency fluctuations of the phase-unlocked “slave” oscillator and voltage noise suppression factor of the PLL

Equations (15)–(18) show how various noise mechanisms affect frequency stability of a phase-locked “slave” oscillator. These expressions will be used for interpreting the data from the twooscillator noise measurement system.

Equation (20) relates voltage noise at the output of the measurement system and phase fluctuations of two oscillators under test. Assuming that both oscillators are similar and the measurement system noise floor is low , the phase noise of the individual oscillator can be derived from , where is the spectral density of voltage noise . Fig. 7 illustrates the phase-noise performance of the highpower oscillators developed. The top trace corresponds to the free-running oscillator (FCS is disabled), the bottom trace gives the measurement system phase noise floor, and the intermediate trace shows the phase noise spectrum of a frequency stabilized oscillator. These measurements were performed with a dual-channel readout system. The increased “fuzziness” of the phase-noise trace (especially near the measurement system noise floor) is an “artifact” of the cross-correlation signal processing, which requires long data acquisition times for averaging out the uncorrelated noise processes of both channels. At Fourier frequencies ( Hz kHz), the spectral density of oscillator phase noise falls as reaching 160 dBc/Hz at kHz. This phase-noise performance is more than an order of magnitude better than the previous state-of-the-art [2]. It was achieved by lowering the noise floor of the FD, which, in turn, resulted from the increase of signal power and setting of the resonator coupling closer to critical.

where At

is the FD frequency noise floor . , the latter equation is transformed into

V. NOISE MEASUREMENTS AND THEIR INTERPRETATION With a “slave” oscillator phase referenced to the “master,” voltage noise at the output of the two-oscillator measurement system is given by

(19) where corresponds to intrinsic voltage fluctuations of the measurement system (measured by driving both inputs of the measurement system with the signals derived from the same oscillator and adjusted to be in quadrature). The first fluctuating term in (19) is a symbolic sum of noise components given by (15)–(18) . Substituting this sum in (19) results in (20)

IVANOV AND TOBAR: LOW PHASE-NOISE MICROWAVE OSCILLATORS WITH INTERFEROMETRIC SIGNAL PROCESSING

Fig. 8. Phase and amplitude noise spectra of a 9-GHz high-power frequencystabilized oscillator. (Color version available online at: http://ieeexplore.ieee. org.)

At low Fourier frequencies Hz, the phase noise of a frequency stabilized oscillator varies as . This, we believe, is primarily due to the ambient temperature fluctuations influencing the SLC resonant frequency. In such a case, the spectral density of oscillator phase noise is given by , where is the resonator frequency-temperature coefficient and is the spectral density of the ambient temperature fluctuations. The latter parameter was measured to be a rapidly diverging function at low Fourier frequencies . This explains the observed 40-dB/dec increase in the spectral density of oscillator phase noise (corresponding to the so-called random walk of oscillator frequency) at . It is worth mentioning that not only classical oscillators, but lasers and some atomic clocks as well, also exhibit the random walk-type noise at low Fourier frequencies [11]. With the temperature fluctuations being the primary source of oscillator frequency instability, it is not surprising that the phase-noise performance of a high-power oscillator at low Fourier frequencies is not very different from that of its low-power counterpart described in [2]. The phase-noise spectrum in Fig. 7 was inferred by assuming that both “slave” and “master” oscillators were identical. In reality, the oscillators were slightly different. This was established by degrading the noise of each oscillator by the controlled amount and measuring the effect on the overall phase noise. We found that the SSB phase noise of the “slave” oscillator was close to 158 dBc/Hz at Fourier frequency kHz, while the phase noise of the “master” was 161 162 dBc/Hz at kHz. The SSB spectra of phase and amplitude fluctuations of the developed oscillators are shown in Fig. 8. Within the frequency range 1 kHz kHz spectral density of AM fluctuations is almost three orders of magnitude higher than that of phase fluctuations. This comparison clearly demonstrates the need for AM-insensitive tuning of the oscillator FCS. We followed a purely empirical approach to tuning of the oscillators,

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Fig. 9. Inferred voltage noise spectrum at the output of two-oscillator readout system due to intrinsic fluctuations of the FCS of the “slave” oscillator (curve 1). Intrinsic voltage noise at the output of FCS low-pass filter measured at the output of 40-dB gain amplifier (curve 2). Transfer function (curve 3). (Color version available online at: http://ieeexplore.ieee.org.)

H

! PD

which involved in situ optimization of the reference phase shifts in both FDs (Section II). A set of equations from Section IV could be used to model the phase-noise performance of a frequency-stabilized oscillator and reveal the noise mechanisms responsible for its phase fluctuations at different Fourier frequencies. This approach is similar to that taken in [1]. As an alternative to predicting the oscillator phase noise, one could infer it by measuring the intrinsic voltage fluctuations of various control systems, along with the transfer functions describing the contribution of these fluctuations to the overall voltage noise at the output of the two-oscillator readout system. To illustrate how this technique works, we consider the limit imposed on “slave” oscillator phase noise by intrinsic voltage fluctuations of the FCS. In such a case, one measures a transfer function from the bias port of the VCP to the output of the PLL phase detector . This is done by injecting voltage noise (from a network analyzer) at the input of the sum device in the frequency control loop and observing the response of the readout system (output of mixer 2 or 3 in Fig. 1). During these measurements, the “slave” oscillator remains phase locked to the “master.” At the next stage, one measures the intrinsic voltage noise of the FCS. This is done by terminating the input of the low-noise microwave amplifier (LNA in Fig. 1) and measuring voltage noise at the output of the FCS filter . Contribution of the FCS intrinsic fluctuations to the overall voltage noise at the output of the two-oscillator measurement system is found from . The spectrum of the inferred voltage noise is shown by curve 1 in Fig. 9. Curves 2 and 3 show the voltage noise spectrum and transfer function , respectively. By following the same logic, one can estimate the voltage noise floor of the two-oscillator measurement system due to the active filter of the PLL. This noise may limit the resolution of

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system. The comparison of these noise spectra shows that intrinsic fluctuations of the FCS are the main factor affecting the oscillator phase noise at Fourier frequencies around 1 kHz. The above transfer functions and can be calculated by using the results of noise analysis from Section IV. Thus, by making use of (12)–(14), one can derive

Fig. 10. Inferred voltage noise spectrum at the output of two-oscillator readout system due to intrinsic fluctuations of the PLL (curve 1). Intrinsic voltage noise at the output of the PLL active filter measured at the output of 40-dB gain am(curve 3). (Color version plifier (curve 2). Transfer function available online at: http://ieeexplore.ieee.org.)

H

! PD

One can also derive the expression for the voltage noise suppres, which is sion factor of the FCS required for analyzing the feedback stability of the phase-locked “slave” oscillator. VI. FEEDBACK STABILITY AND EXCESS NOISE IN HIGH-POWER MICROWAVE OSCILLATORS

Fig. 11. Spectrum of voltage fluctuations measured at the output of two-oscillator readout system (curve 1). Spectra of the inferred voltage noise at the output of the two-oscillator readout system due to intrinsic fluctuations of the PLL (curve 2) and the FCS (curve 3). (Color version available online at: http://ieeexplore.ieee.org.)

spectral measurements at Fourier frequencies within the bandwidth of the PLL. First, one measures voltage transfer function from the VCA bias port to the output of the PLL phase detector with the “slave” oscillator referenced to “master.” At the next stage, voltage noise at the output of the PLL filter is measured with its input terminated. Voltage noise induced at the output of the two-oscillator readout system by noise of the PLL electronics is found from . It is shown by curve 1 in Fig. 10. Curves 2 and 3 show the voltage noise spectrum and transfer function , respectively. Fig. 11 shows the spectra of the inferred voltage fluctuations ( and ) along with the spectrum of voltage noise measured directly at the output of the two-oscillator readout

Ensuring a stable operation of a high-power oscillator is one of the major problems of its design. This is due to the relatively high level of phase fluctuations of a free-running loop oscillator and the corresponding need to adequately suppress them over the broadest possible range of frequencies impose. The first prototypes of the high-power oscillators featured feedback loops based on the relatively narrowband operational amplifiers (gain-bandwidth product: 8 MHz). Phase noise spectral density of such oscillators at Fourier frequencies of interest (around 1 kHz was a nonmonotonic function of the loop gain: the initial suppression of the phase noise with the loop gain was followed by the noise enhancement, when the loop gain exceeded a certain threshold ( 60 65 dB). This phenomenon was linked to the onset of instability of the frequency control loop and could be observed by viewing the voltage noise spectrum at the output of the FD (there is no need of setting up a two-oscillator noise measurement system). For example, a narrow peak appeared in the spectrum of voltage noise at the FD output (“master” oscillator) when the FCS dc gain exceeded 60 dB. Appearance of this peak was accompanied by the excess voltage noise with the spectral density independent on frequency. The height of the peak and spectral density of the excess voltage noise were increasing with the loop gain. As a result, the spectrum of voltage noise at the output of the FD was almost entirely “white” at dB, which corresponded to the spectrum of oscillator phase fluctuations. The problem of the excess phase noise has been solved (to some extent) by redesigning the filter of the FCS and making use of the broadband operational amplifiers (gain-bandwidth product: 120 MHz). This enabled us to increase the gain of the FCS and construct the oscillators with the phase-noise performance shown in Fig. 7. Still, we could not claim that the problem of the excess phase noise was solved completely or that the

IVANOV AND TOBAR: LOW PHASE-NOISE MICROWAVE OSCILLATORS WITH INTERFEROMETRIC SIGNAL PROCESSING

origin of the excess noise was well understood. We are yet to explain some anomalous behavior exhibited by the high power oscillators at high levels of phase-noise suppression. For example, in a “well-behaved” oscillator, the spectral density of voltage must be independent fluctuations at the output of the FD on FD conversion efficiency provided that . In our experiments (with “master” oscillator), the spectral density of voltage fluctuations remained independent on only at the relatively low values of the loop gain dB. At higher loop gains, intensity of voltage noise was increasing rapidly with . Voltage noise was also dependent on oscillator gain margin. A possible way of circumventing the excess noise problem in the ultra-low phase-noise oscillators could rely on a better frequency stabilization of a free-running oscillator. In such a case, gain of the FCS does not have to be too high for achieving the fundamental noise floor set by effective noise temperature of the FD. In this respect, the methods of amplifier phase-noise reduction described in [1] look quite promising. VII. CONCLUSION Two 9-GHz oscillators with the SSB phase-noise spectral density close to 158 and 162 dBc/Hz at Fourier frequency kHz have been constructed. Each oscillator features a room-temperature stabilized SLC resonator used both as a narrowband filter of a loop oscillator and highly dispersive component of the interferometric FD. These oscillators are currently the lowest phase-noise sources at microwave frequencies. Construction of the ultra-low phase-noise oscillators turned out to be a relatively simple part of this project, as compared to their phase-noise measurements. Such measurements required the design of a special PLL capable of keeping two oscillators in a tight synchronism for a prolonged time without degrading their phase noise. In this study, we have described the technique of oscillator phase locking based on controlling the microwave power dissipated in the SLC resonator. The noise analysis of a phase-locked oscillator enabled us to “visualize” contributions of various noise mechanisms to oscillator overall phase noise and justify the method of transfer functions as a way of predicting the oscillator phase noise from a set of some auxiliary measurements. It also enabled us to find the relationship between the phase and voltage noise spectra, which is required for processing the experimental data. Further research in the field of SLC-based oscillators is yet to uncover their true potential and find out whether they can meet the challenge imposed by photonic devices [12]–[15]. In this respect, we believe that a new family of extremely low phase-noise oscillators could be created on the basis of cryogenically cooled SLC resonators. The renewed interest in cryogenic SLC resonators is explained by the better performance and affordability of closed cycle refrigerators. The latter eliminate the need for cryogenic fluids allowing uninterrupted operation of the oscillator for many years. By making use of cryogenic SLC resonators with a factor of 10 30 million, phase noise of an -band oscillator could be reduced to 180 dBc/Hz at 1-kHz Fourier frequency with only 10 mW of signal power dissipated inside the resonator. A microwave oscillator based on a cryogenically cooled SLC resonator will serve as an “ideal”

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reference with respect to which any noncryogenic microwave oscillator can be characterized. ACKNOWLEDGMENT The authors are very grateful to J. H. Searls and C. McNeilage, both with Poseidon Scientific Instruments Pty. Ltd., Fremantle, Australia, for technical assistance with temperature stabilized microwave resonators and servo electronics. The authors also appreciate J. Torrealba, University of Western Australia, Perth, Australia, for his help with various types of measurements. REFERENCES [1] E. N. Ivanov, M. E. Tobar, and R. A. Woode, “Applications of interferometric signal processing to phase noise reduction in microwave oscillators,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 10, pp. 537–1545, Oct. 1998. [2] E. N. Ivanov, M. E. Tobar, and R. A. Woode, “Microwave interferometry: Application to precision measurements and noise reduction techniques,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control, vol. 45, no. 6, pp. 1526–1537, Nov. 1998. [3] M. E. Tobar, E. N. Ivanov, C. R. Locke, and J. G. Hartnett, “Difference frequency technique to achieve frequency-temperature compensation in whispering-gallery sapphire resonator-oscillator,” Electron. Lett., vol. 38, no. 17, pp. 948–950, Aug. 2002. [4] M. E. Tobar, G. L. Hamilton, E. N. Ivanov, and J. G. Hartnett, “New method to build a high stability sapphire oscillator from the temperature compensation of the difference frequency between modes of orthogonal polarization,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control, vol. 50, no. 3, pp. 214–219, Mar. 2003. [5] E. N. Ivanov and M. E. Tobar, “Real time noise measurement system with sensitivity exceeding the standard thermal noise limit,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control, vol. 49, no. 8, pp. 1160–1161, Aug. 2002. [6] E. N. Ivanov, “Noise properties of microwave oscillators based on sapphire loaded cavity resonators,” in Proc. 16th Int. Noise in Phys. Syst. 1=f Fluctuations Conf., Gainsville, FL, Oct. 2001, pp. 473–478. [7] C. McNeilage, J. H. Searls, E. N. Ivanov, P. R. Stockwell, D. M. Green, and M. Mossamaparast, “A review of sapphire whispering gallery-mode oscillators including technical progress and future potential of the technology,” in Proc. IEEE Freq. Control Symp., Montreal, QC, Canada, Aug. 2004, pp. 210–218. [8] F. L. Walls, A. J. D. Clements, C. M. Felton, M. A. Lombardi, and M. D. Vanek, “Extending the range and accuracy of phase noise measurements,” in Proc. 42nd Freq. Control Symp., Baltimore, MD, Jun. 1988, pp. 432–441. [9] E. Rubiola and V. Giordano, “Correlation-based phase noise measurements,” Rev. Sci. Instrum., vol. 71, no. 8, pp. 3085–3091, Aug. 2000. [10] E. N. Ivanov and F. L. Walls, “Interpreting anomalously low voltage noise in two-channel measurement systems,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control, vol. 49, no. 1, pp. 11–19, Jan. 2002. [11] E. N. Ivanov, S. A. Diddams, and L. Hollberg, “Experimental study of noise properties of a Ti : sapphire femtosecond laser,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control, vol. 50, no. 4, pp. 355–360, Apr. 2003. [12] R. J. Jones and J. C. Diels, “Stabilization of femtosecond lasers for optical frequency metrology and direct optical to radio frequency synthesis,” Phys. Rev. Lett., vol. 86, no. 15, pp. 3288–3291, Apr. 2001. [13] L. Hollberg, C. W. Oates, E. A. Curtis, E. N. Ivanov, S. A. Diddams, T. Udem, H. G. Robinson, J. C. Bergquist, R. J. Rafac, W. M. Itano, R. E. Drullinger, and D. J. Wineland, “Optical frequency standards and measurements,” IEEE J. Quantum Electron., vol. 37, no. 12, pp. 1502–1513, Dec. 2001. [14] J. L. Hall, J. Ye, S. A. Diddams, L.-S. Ma, S. T. Cundiff, and D. J. Jones, “Ultrasensitive spectroscopy, the ultrastable lasers, the ultrafast lasers, and the seriously nonlinear fiber: A new alliance for physics and metrology,” IEEE J. Quantum Electron., vol. 37, no. 12, pp. 1482–1492, Dec. 2001. [15] J. J. McFerran, E. N. Ivanov, A. Bartels, G. Wilpers, C. W. Oates, S. A. Diddams, and L. Hollberg, “Low-noise synthesis of microwave signals from an optical source,” Electron. Lett., vol. 41, no. 11, pp. 650–651, May 2005.

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Eugene N. Ivanov was born in Moscow, Russia, in 1956. He received the Ph.D. degree in radio science from the Moscow Power Engineering Institute, Moscow, Russia, in 1987. From 1980 to 1990, he was involved with applications of low-loss dielectric resonators to frequency stabilization of microwave oscillators. In 1991, he joined the Physics Department, University of Western Australia, Crawley, W.A., Australia. From 1992 to 1994, he was involved in the development and operation of the cryogenic resonant-mass gravitational wave antenna Niobe. Since 1994, he has been concerned with the applications of interferometric signal processing to precision noise measurements. This research resulted in the development of a new generation of ultra-low phase-noise microwave oscillators and “real-time” measurement systems with sensitivity surpassing the standard thermal noise limit. From 1999 to 2005, he has been a Visiting Scientist with the Time and Frequency Division, National Institute of Standards and Technology (NIST), Boulder, CO. He was involved in the study of noise properties of femtosecond lasers and development of the coherent link between the optical and microwave domains. Dr. Ivanov was the recipient of the 1994 Japan Microwave Prize and the 2002 W. G. Cady Award presented by the IEEE Ultrasonics, Ferroelectrics, and Frequency Control Society.

Michael E. Tobar (S’87–A’90–M’96–SM’01) was born in Maffra, Vic., Australia, on January 3, 1964. He received the B.Sc. degree in theoretical physics and mathematics and B.E. degree (with honors) in electrical and computer systems engineering from Monash University, Melbourne, Vic., Australia, in 1985 and 1988, respectively, and the Ph.D. degree in physics from the University of Western Australia, Perth, W.A., Australia, in 1992. His doctoral dissertation concerned gravitational wave detection and low noise sapphire oscillators. From 1989 to 1992, he was with the Department of Physics, University of Western Australia. From 1992 to 1993, he was a Research Associate with the University of Western Australia. From 1994 to 1996, he was an Australian PostDoctoral Research Fellow with the University of Western Australia. During 1997, he was a Senior Research Associate with the University of Western Australia. From 1997 to 1998, he was a Research Fellow of the Japan Society for the Promotion of Science, University of Tokyo, Tsukuba, Japan. During 1998, he was a Visiting Professor with the Institut de Recherche en Communications Optiques et Microondes, University of Limoges, Limoges, France. From 1999 to 2000 (for of 11 months), he was a Research Director of the Centre National de la Recherche Scientifique (CNRS). He is currently a Research Professor with the School of Physics, University of Western Australia, Crawley, W.A., Australia. His research interests encompass the broad discipline of frequency metrology, precision measurements, and precision tests of the fundamental of physics. Prof. Tobar was the recipient of an Australian Professorial Fellowship presented by the Australian Research Council. He was also the recipient of the 1999 Best Paper Award presented by the Institute of Physics Measurement Science and Technology, the 1999 European Frequency and Time Forum Young Scientist Award, the 1997 Australian Telecommunications and Electronics Research Board (ATERB) Medal, the 1996 Union of Radio Science International (URSI) Young Scientist Award, and the 1994 Japan Microwave Prize.

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A Low-Power Up-Conversion CMOS Mixer for 22–29-GHz Ultra-Wideband Applications Ashok Verma, Student Member, IEEE, Kenneth K. O, Senior Member, IEEE, and Jenshan Lin, Senior Member, IEEE

Abstract—A double-balanced, low-power, and low-voltage -band is designed dual-gate up-conversion mixer working at and fabricated in the UMC 130-nm logic CMOS process. The mixer achieves a 3-dB conversion-gain bandwidth of 1.8 GHz at the input IF port and a 3-dB conversion-gain bandwidth of 10 GHz at the output RF port. The mixer achieves an output referred 1-dB compression point as high as 5.8 dBm and an output referred third-order intercept point as high as 5.8 dBm, while consuming 8.0 mW from a 1.2-V supply. This study demonstrates that the implementation of low-power mixers operating in the 22–29-GHz band for ultra-wideband automotive radar applications is possible in low-cost and low-voltage logic CMOS technology. Index Terms—Broadband, CMOS, dual-gate, ultra-wideband.

-band, mixer,

I. INTRODUCTION T HAS been suggested that with the improvement of CMOS technologies, RF CMOS circuits operating in the 22–29-GHz range will be possible [1]–[3]. Such circuits can be used in radars, wireless local area networks, local multipoint distribution services (LMDSs), and other industrial–scientific–medical (ISM) band applications. Since many wireless applications run off batteries, low power consumption is important. Furthermore, the RF-CMOS circuits have to operate with supply voltage of 1.2 V or less in order to be compatible with the supply voltage requirement of deep submicrometer transistors [6]. CMOS low-noise amplifiers operating at such frequencies have received a great deal of attention [7]–[11], while mixers have received little attention. To have a compatible fabrication process for future single-chip integration with digital logic circuits, mixers designed and fabricated using the logic CMOS process is of interest [4], [5]. In particular, very little has been done for up-conversion CMOS mixers for such a frequency range. This paper presents the design and measurement results of a low-power -band up-conversion dual-gate mixer with improved linearity. The mixer is fabricated via the UMC 130-nm logic CMOS process. Although passive mixers have the inherent advantage of better frequency and linearity response, they have conversion loss and also require a high local oscillator (LO) drive for operation, which can significantly

I

Manuscript received February 23, 2006; revised April 28, 2006. This work was supported in part by the National Science Foundation under Grant 0421218. A. Verma and J. Lin are with the RF System on a Chip Research Group, Department of Electrical and Computer Engineering, University of Florida, Gainesville, FL 32611 USA (e-mail: [email protected]). K. K. O is with the Silicon Microwave Integrated Circuits and Systems Laboratory, Department of Electrical and Computer Engineering, University of Florida, Gainesville, FL 32611 USA. Digital Object Identifier 10.1109/TMTT.2006.879173

Fig. 1. (a) Schematics of Gilbert cell. (b) Dual-gate mixer.

increase power consumption. A smaller LO drive is crucial for low-voltage/low-power integrated-circuit (IC) design, as it is becoming more difficult to generate high LO drive in low-voltage circuits, especially as those operating in -band. Due to lower available LO drive, a dual-gate active mixer topology has been chosen. Although the dual-gate topology shown in Fig. 1 looks similar to the conventional double-balanced Gilbert cell [4], the mixing process is quite different. In a dual-gate mixer, a finite LO signal must be present at the drains of lower field-effect transistors (FETs) ( – ),

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which causes the drain-to-source voltages of the transistors to change. This causes the switching of devices between linear and saturation regions [21]. Due to this, the transconductance of lower FETs ( ) is modulated at the LO frequency. In a Gilbert cell mixer, the LO signal at the drain of differential pair is ideally zero and the mixing is caused by switching action of the LO pair between the cutoff and saturation region. In a Gilbert cell mixer, the lower (transconductance) stage is primarily biased in saturation region, whereas in a dual-gate mixer, the lower FETs operates in linear region during most part of the LO cycle, resulting in lower transconductance than an FET biased in the saturation region [16]–[23]. Due to this, the conversion gain of the Gilbert cell design is normally higher than that of dual-gate designs, whereas linearity performance of dual-gate mixers is moderately better than the Gilbert cell. Dual-gate mixers, like Gilbert cell mixers, also have the inherent property of separation of the LO and RF ports, which makes them more amenable for integration. The up-conversion dual-gate mixer has been realized using a cascode connection of two single-gate FETs [24] for which relatively accurate simulation models are available. This topology also gives the freedom to utilize different device geometry for the two transistors. This paper is organized as follows. Design considerations of the mixer are presented in Section II. The measured performance is described in Section III. Conclusions are then drawn in Section IV. II. DESIGN METHODOLOGY The design of up-conversion mixers involves tradeoffs among conversion-gain, linearity, LO power, port-to-port isolation, and power consumption. The dual-gate FET behavior has been realized with a cascode connection of two single-gate FETs with different gatewidths. As the IF frequency for up-conversion mixer is low ( 7 lower than the LO frequency), in order to match the input port and to achieve conversion gain, the lower transistors ( – ) should be larger than the LO transistors. If LO transistors are chosen to be the same as that for the lower stage, the parasitic capacitance at floating nodes ( – in Fig. 1) increases, which decreases the voltage swing at the nodes and degrades the linearity and gain performance. A larger width for the LO transistors can improve noise figure. However, since noise is not an important design criterion for an up-converter design due to high signal-to-noise ratio of baseband signal, the size of the LO transistor ( – ) gatewidth is chosen to be 20 m that presents a small load to the LO drivers. This is 1/8 of the IF transistor gate width of 165 m. To provide broadband impedance matching and tolerance to component variations at the IF and RF ports, quality factors of the input and output matching networks ( and ), as defined later in this paper, in (2) and (4), are designed to have low values. A higher quality factor can reduce the power consumption, but the circuit becomes more narrowband and sensitive to process variations [25]. As shown in Fig. 1, gate inductor ( or ) and source degeneration inductor ( or ) are used to match the input to 50 . Having source degeneration also helps to improve the linearity at some expense of conversion gain. Planar spiral structures have been used to implement on-chip gate inductors

Fig. 2. Simplified schematic of input matching network.

(6.9 nH), which are large and has considerable parasitic series resistance ( in this case) associated with them. For simplification, the effect of parasitic shunt capacitances of the inductor structure has been neglected in calculation of input impedance. lowers the resonant frequency of the input matching network. With this assumption, a simplified approximate expression for the impedance looking through the inductor into the gate of transistor pairs – and – can be written as follows, which is equal to when matched at the resonance frequency [15]:

(1) is much smaller than , it can be neglected in (2). Since In matched condition at resonance, the input matching network can be simplified as shown in Fig. 2, and the quality factor of the input matching network including the source impedance can be expressed in terms of and capacitances (2) as follows:

(2)

Using the large low- inductor, low- matching was determined. The simulated 3-dB conversion-gain bandwidth for input port (IF) is 2 GHz from 1.6 to 3.6 GHz. For this particular value of and center resonant frequency of 2.6 GHz, should be 6.7 nH. Due to its large value, this inductor occupies a significant portion of the chip area. By matching the real part to 50 , is determined. Transconductance of lower FETs is set by biasing them at the edge of the linear and saturation region, and is limited by maximum tolerable current consumption. Assuming the use of minimum channel length devices, once is set, the widths of transistors – are determined by (2). The size of transistors depends on the requirements for and , and is fixed for the chosen .

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Fig. 3. Simplified schematic of output matching network.

For output matching, an -matching network consisting of a shunt ( or ) and a series ( or ) has been used. The drains of coupled transistors – and – are connected to the power supply through the inductors and (0.34 nH). The quality factor of output matching network is kept low to cover the 22–29-GHz band for ultra-wideband applications. The quality factor was determined to be 2.2 at the center frequency of 25 GHz, which is related to the quality factor of drain inductor and resistance looking into drain of LO transistors ( in Fig. 1), as expressed in (4), [25]. To achieve of 2.2, inductor only needs to be 8.9 at 25 GHz (3). One terminal of the drain inductors and is connected to , which is an ac ground. For each output side, the spiral inductor ( or ) along with its parasitic series resistance can be equivalently represented with a parallel combination of and . For of , is . The output matching network can be simplified as shown in Fig. 3 and as follows:

(3) To match the impedance at the output port, should be equal to shunt resistance of in Fig. 3. When this is satisfied, at the resonant frequency is given by (4) as follows:

Fig. 4. Die photograph of the dual-gate mixer. (Color version available online at: http://ieeexplore.ieee.org.)

Fig. 5. Isolation between LO-to-RF and LO-to-IF ports.

layout, shielded pads are employed at RF, LO, and IF ports [26]. The ground shield reduces the signal power loss and noise generation associated with the substrate resistance. Wide ground rings are placed around all the transistors at minimum distances allowed to reduce the substrate losses. To mitigate the LO feed-through problem, the layout was made symmetric. III. EXPERIMENTAL RESULTS

(4) Choice of a lower quality factor value for the output matching network ensures wideband impedance matching and, hence, broadband response at the RF port. The LO ports are terminated with 50- resistors. This helps achieve broadband matching for the LO port and better control of LO signal amplitude during testing. The effects of the parasitic associated with the spiral inductor and bond pads were taken into account during simulation/layout. The chip occupies an area of 680 690 m . The die photograph is shown in Fig. 4. A large inductor (7.0 nH) is placed between the ground and the common node of the degeneration inductors ( and ). At resonance frequency of 2.6 GHz, this provides an impedance of 115 . This helps reduce the effects of common-mode signals in the circuit. An on-chip bypass capacitor has been included between the ground and . In the

The mixer has been characterized on wafer using differential [ground–signal–signal–ground (GSSG)] probes. The measurements were performed at LO power of 3.0 dBm. Optimum performance has been achieved at RF of 23 GHz. The measured LO-to-RF isolation is better than 30 dB, LO-to-IF isolation is better than 55 dB, and IF-to-RF and IF-to-LO isolations are both better than 50 dB. Isolation between LO-to-RF and LO-to-IF was better than 30 and 50 dB for LO frequency variation from 14 to 26 GHz, as is shown in Fig. 5. Good impedance matching has been achieved at RF and IF ports, as shown in Fig. 6. Input return loss is better than 10 dB from 2.1 to 3.2 GHz and output return loss is better than 10 dB from 19.8 to 25.8 GHz. The 10-dB frequency range for output return loss was shifted down in the measurements. Unaccounted parasitic capacitances in the device models and passive component variations are the likely cause of such a result. To evaluate the broadband frequency response of the circuit, conversion gain was measured by varying IF with constant LO

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Fig. 8. Measured and simulated 1-dB compression, OIP3, and conversion gain versus RF. Fig. 6. Measured and simulated input and output return loss.

Fig. 7. Measured and simulated 1-dB compression, OIP3, and conversion gain versus IF.

frequency of 21 GHz, which results in different RF signals at the output of the mixer. Measurements were performed at a current consumption of 6.8 mA from a 1.2-V supply. Conversion-gain variation of the mixer is 3 dB over an input bandwidth of 1.8 GHz from 1.5 to 3.5 GHz. Measured output 1-dB compression and output third-order intermodulation intercept point (OIP3) points are also flat over more than 2 GHz of input frequencies. Fig. 7 shows the plot of conversion gain and linearity characteristics versus IF. The circuit also exhibits broadband performance at the RF port. The simulated 3-dB gain bandwidth of the circuit is 10 GHz between 19–29 GHz. For this measurement, IF was kept fixed at 2.3 GHz and LO was varied to achieve different RF values. Measurements were performed at a current consumption of 6.8 mA with 3 dBm of LO drive. The measured 3-dB conversion-gain bandwidth is 10 GHz from 18 to 28 GHz. OIP3 value as high as 5.8 dBm and output 1-dB compression value as high as 4.6 dBm have been achieved from 17 to 29 GHz, which is very wide. Fig. 8 shows the plots of conversion gain and linearity characteristics versus RF. Frequency response measurements determining the 3-dB input and output

Fig. 9. Measured 1-dB compression and conversion gain versus LO power for constant bias current and versus bias current for constant LO.

bandwidth show that the measured results are in close agreement with simulated values for output and OIP3. However, the measured conversion gain is lower than the predicted value by 2 dB, which is likely due to degradation of transconductance of the devices from actual simulated values and unaccounted parasitics. The behavior of 1-dB compression point and conversion gain as a function of LO drive was also characterized. At a fixed bias point, as the LO drive is increased, the lower FETs operate in the linear region for a larger fraction of an LO cycle, and overall linearity is improved. Conversion gain is also improved as the amplitude of modulating transconductance also increases with increased LO drive [21]. A plot of measured 1-dB compression point and conversion gain versus LO drive at a constant current consumption of 6.8mA is shown in Fig. 9. An increase of LO power beyond 6.0 dBm does not further improve as the nonlinearities of LO transistors become dominant. Measured conversion gain and 1-dB compression points were also characterized as a function of current consumption

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TABLE I PERFORMANCE SUMMARY OF MIXERS OPERATING AROUND 20 GHZ

of the circuit while keeping LO power fixed at 3 dBm. Current consumption was varied by changing the bias voltages of LO and IF ports. At current consumption of 9 mA, the circuit can achieve a gain of 1.7 dB and an output 1-dB compression point of 3.4 dBm. Fig. 9 shows the measured performance versus varying current consumption. A list of performance of current design and previously published CMOS [3]–[5] and SiGe [12]–[14] mixers operating at similar frequency ranges are shown in Table I. The dual-gate mixer presented in this paper achieves very good isolation between all the ports and flat linearity response over a bandwidth of more than 10 GHz at output RF port, while consuming only 8 mW from a 1.2-V supply. Compared to the 20-GHz Gilbert cell up-conversion mixer [4], is 4–6 dB higher while consuming the same power. IV. CONCLUSION This paper has presented the design and measured characteristics of a broadband CMOS up-conversion mixer working at -band. By carefully choosing the quality factors of the input and output matching networks, a wideband response was achieved at IF and RF ports for a simple dual-gate configuration. The mixer achieved 1.8-GHz 3-dB gain bandwidth at input and 10-GHz 3-dB gain bandwidth at output. Good linearity was achieved at a low power-consumption level and supply voltage of 1.2 V. The measured LO-to-RF isolation is better than 30 dB, LO-to-IF isolation is better than 55 dB, and IF-to-RF and IF-to-LO isolations are both better than 50 dB. These results successfully show that an up-conversion mixer for ultra-wideband radars operating in the 22–29-GHz range [27] can be implemented in low-cost and low-voltage deep submicrometer logic CMOS technology.

ACKNOWLEDGMENT The authors would like to thank United Microelectronics Corporation (UMC), Hsinchu, Taiwan, R.O.C., for fabricating the chip. REFERENCES [1] B.-A. Floyd, C.-M. Hung, and K. K. O, “Intra-chip wireless interconnect for clock distribution implemented with integrated antenna, receivers and transmitters,” IEEE J. Solid-State Circuits, vol. 37, no. 5, pp. 543–552, May 2002. [2] C.-M. Hung, L. Shi, I. Lagnado, and K. K. O, “A 25.9 GHz voltage controlled oscillator fabricated in a CMOS process,” in VLSI Circuits Symp., Honolulu, HI, Jun. 2000, pp. 100–101. [3] X. Guan and A. Hajimiri, “A 24 GHz CMOS front end,” IEEE J. SolidState Circuits, vol. 39, no. 2, pp. 368–373, Feb. 2004. [4] A. Verma, L. Gao, K. K. O, and J. Lin, “A K -band up-conversion mixer in 0.13-m CMOS technology,” in Proc. Asia–Pacific Microw. Conf., Dec. 2004. [5] M. Tiebout, C. Kinemeyer, R. Thuringer, C. Sandner, H. D. Wohlmuth, M. Berry, and A. L. Scholtz, “A 17-GHz transceiver design in 0.13 m CMOS,” in Radio Freq. Integr. Circuits Symp., 2005, pp. 101–104. [6] Q. Huang, F. Piazza, P. Orasatti, and T. Ohgura, “The impact of scaling down to deep submicron on CMOS RF circuits,” IEEE J. Solid-State Circuits, vol. 33, no. 7, pp. 1023–1036, Jul. 1998. [7] B. A. Floyd, L. Shi, Y. Taur, I. Lagnado, and K. K. O, “A 23.8-GHz SOI CMOS tuned amplifier,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 9, pp. 2193–2195, Sep. 2002. [8] K.-W. Yu, Y.-L. Lu, D.-C. Chang, V. Liang, and M. F. Chang, “K -band low-noise amplifiers using 0.18 m CMOS technology,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 3, pp. 106–108, Mar. 2004. [9] S.-C. Shin, M.-D. Tsai, R.-C. Liu, K.-Y. Lin, and H. Wang, “A 24-GHz 3.9-dB NF low-noise amplifier using 0.18 m CMOS technology,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 7, pp. 448–450, Jul. 2005. [10] Y. Su, J.-J. Lin, and K. K. O, “A 20 GHz CMOS RF down-converter with an on-chip antenna,” in Int. Solid-State Circuits Conf. Tech. Dig., Feb. 2005, pp. 270–271. [11] X. Guo and K. K. O, “A power efficient differential 20-GHz low noise amplifier with 5.3-GHz 3-dB bandwidth,” IEEE Microw. Wireless. Compon. Lett., vol. 15, no. 9, pp. 603–605, Sep. 2005.

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[12] K. B. Schad, H. Schemacher, and A. Schuppen, “Low-power active -band applications using SiGe HBT MMIC technology,” mixer for in IEEE Radio Freq. Integr. Circuits Symp., 2000, pp. 263–266. [13] S. Hackl, J. Bock, J. Bobk, and G. Ritzberger, “28 GHz active IQ mixer with integrated QVCO in SiGe bipolar technology,” in Eur. Solid-State Circuits Conf., 2002, pp. 159–162. [14] M. W. Lynch, C. D. Holdecried, and J. W. Haslett, “A 17-GHz direct down-conversion mixer in a 47 GHz SiGe process,” in IEEE Radio Freq. Integr. Circuits Symp., Jun. 2003, pp. 461–464. [15] A. Ismail and A. A. Abidi, “A 3–10 GHz low noise amplifier with wideband LC-ladder matching network,” IEEE J. Solid-State Circuits, vol. 39, no. 12, pp. 2269–2277, Dec. 2004. [16] C. Tsironis, R. Meierer, and R. Stahlmann, “Dual-gate MESFET mixers,” IEEE Trans. Microw. Theory Tech., vol. MTT-32, no. 3, pp. 248–255, Mar. 1984. [17] P. J. Sullivan, B. A. Xavier, and W. H. Ku, “Doubly balanced dualgate CMOS mixer,” IEEE J. Solid-State Circuits, vol. 34, no. 6, pp. 878–881, Jun. 1999. [18] S.-I. Liu and Y.-S. Hwang, “CMOS four-quadrant multiplier using bias feedback techniques,” IEEE J. Solid-State Circuits, vol. 29, pp. 750–752, Jun. 1994. [19] P. J. Sullivan, B. A. Xavier, and W. H. Ku, “Low voltage performance of a microwave CMOS Gilbert cell mixer,” IEEE J. Solid-State Circuits, vol. 32, pp. 1151–1155, Jul. 1997. [20] G. Han and E. S. Sinencio, “CMOS transconductance multipliers: A tutorial,” IEEE Trans. Circuits Syst. II, Analog Digit. Signal Process., vol. 45, no. 12, pp. 1550–1563, Dec. 1998. [21] S. A. Mass, Microwave Mixer. Norwood, MA: Artech House, 1986. [22] S. Ye and C. A. T. Salama, “A 1-V 1.9 GHz low distortion dual-gate CMOS on SOI mixer,” in IEEE Int. SOI Conf., Oct. 2000, pp. 104–105. [23] S. Ye and Y. Lu, “A 1-V 1.9-GHz folded dual gate mixer in CMOS,” in Int. Microw. Millimeter Wave Tech. Conf., Aug. 2002, pp. 175–178. [24] H. I. Kang, J. H. Kim, Y. W. Kwon, and J. E. Oh, “An asymmetric GaAs MMIC dual-gate mixer with improved intermodulation characteristics,” IEEE MTT-S Int. Microw. Symp. Dig., vol. 2, pp. 795–798, Jun. 1999. [25] K. K. O, “Design and analysis of tuned RF circuits including the effects of process and component variations,” Asia–Pacific Microw. Conf., Nov. 2003, tutorial. [26] J. T. Colvin, S. S. Bhatia, and K. K. O, “Effects of substrate resistances on LNA performance and a bond pad structure for reducing the effects in a silicon bipolar technology,” IEEE J. Solid-State Circuits, vol. 34, no. 9, pp. 1339–1344, Sep. 1999. [27] “Technical requirements for vehicular radar systems,” FCC, Washington, DC, Code Fed. Reg., Title 47, part 15, sec. 15. 515, Sep. 19, 2005.

Ku

Ashok Verma (S’04) was born in Rajasthan, India. He received the B.S. degree in electronics and communication engineering from the Indian Institute of Technology, Guwahati, Assam, India, in 2002, the M.S. degree in electrical and computer engineering from the University of Florida, Gainesville, in 2004, and is currently working toward the Ph.D. degree in electrical and computer engineering at the University of Florida. Since 2002, he has been with the Radio Frequency System on a Chip (RFSOC) Research Group, Department of Electrical and Computer Engineering, University of Florida . His research interests include RF and mixed-signal IC design in CMOS technology, wireless communications, and high power RF-to-dc rectifiers using GaN technology for wireless power transmission systems.

Kenneth K. O (S’86–M’89–SM’04) received the S.B., S.M., and Ph.D. degrees in electrical engineering and computer science from the Massachusetts Institute of Technology (MIT), Cambridge, in 1984, 1984, and 1989, respectively. From 1989 to 1994, he was with Analog Devices Inc., where he developed submicrometer CMOS processes for mixed-signal applications and high-speed bipolar and BiCMOS processes for RF and mixed-signal applications. He is currently a Professor with the Department of Electrical and Computer Engineering, University of Florida, Gainesville. He has authored or coauthored approximately 130 journal and conference publications. He holds four patents. His research group (Silicon Microwave Integrated Circuits and Systems Research Group) develops circuits and components required to implement analog and digital systems operating from 1 GHz to 1 THz using silicon IC technologies. Dr. O was the general chair of the 2001 IEEE Bipolar/BiCMOS Circuits and Technology Meeting (BCTM). He was an associate editor for the IEEE TRANSACTIONS ON ELECTRON DEVICES from 1999 to 2001. He also served as the publication chairman of the 1999 International Electron Device Meeting. He was the recipient of the 1995, 1997, and 2000 IBM Faculty Development Awards and the 1996 National Science Foundation (NSF) Early CAREER Development Award.

Jenshan Lin (S’91–M’94–SM’00) received the B.S. degree from National Chiao Tung University, Hsinchu, Taiwan, R.O.C., in 1987, and the M.S. and Ph.D. degrees in electrical engineering from the University of California at Los Angeles (UCLA), in 1991 and 1994, respectively. In 1994, he joined AT&T Bell Laboratories (later Lucent Bell Laboratories), Murray Hill, NJ, as a Member of Technical Staff, and became the Technical Manager of RF and High Speed Circuit Design Research in 2000. Since joining Bell Laboratories, he has been involved with ICs using different technologies for different applications, He led the Bell Laboratories Base Station RF Integrated Circuits (RFIC) Team to demonstrate the first low-cost high-performance silicon CMOS RFIC solution for wireless base stations, which was press released at the 2001 International Solid-State Circuits Conference (ISSCC). In September 2001, he joined Agere Systems, a spin-off from Lucent, and was involved with high-speed CMOS circuit design for 10G/40G broadband communications. In July 2003, he joined the University of Florida, Gainesville, as an Associate Professor. He has authored or coauthored over 90 technical publications in referred journals and conferences proceedings. He holds five patents. His current research interests include RF system-on-chip integration, high-speed broadband circuits, high-efficiency transmitters, wireless sensors, biomedical applications of microwave and millimeter-wave technologies, and software-configurable radios. Dr. Lin has been active in the IEEE Microwave Theory and Techniques Society (IEEE MTT-S). He is an elected Administrative Committee (AdCom) member serving the term of 2006–2008, and a member of the Wireless Technology Technical Committee. He also serves on several conference Steering Committees and Technical Program Committees, including the IEEE MTT-S International Microwave Symposium (IMS), Radio Frequency Integrated Circuits Symposium (RFIC), Radio and Wireless Symposium (RWS), and Wireless and Microwave Technology Conference (WAMICON). He is currently technical program co-chair of the 2006 RFIC Symposium and workshop co-chair of the 2006 RWS. He was the recipient of the 1994 UCLA Outstanding Ph.D. Award and the 1997 Eta Kappa Nu Outstanding Young Electrical Engineer Honorable Mention Award. He coauthored/advised on several IEEE MTT-S IMS Best Student Paper Awards (Dawson: second place 1997, Droitcour: honorable mention 2001, Droitcour: first place 2003) and advised on an IEEE MTT-S Undergraduate/Pre-Graduate Scholarship Award (2004).

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On the Effects of Memoryless Nonlinearities on -QAM and DQPSK OFDM Signals

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Arsenia Chorti and Mike Brookes, Member, IEEE

Abstract—In the design of RF up-conversion and down-conversion communication links, an issue of special interest is presented by the nonlinear characteristic of analog devices. In this paper, we deal with the effect of memoryless nonlinear distortion on orthogonal frequency-division multiplexing (OFDM) transceivers. We tackle the issue of calculation of the number of intermodulation products with methods from combinatorics theory and derive closed-form expressions for the signal-to-noise ratio (SNR). We deal with third-order nonlinearities alone though the methodology used can be extended to cover higher order nonlinear phenomena. We then proceed to deriving SNR expressions in the presence of a high adjacent channel of the same service and predict the generation of in-band tonal interference. Finally, we generalize to the case of a multichannel OFDM transceiver. In each case, bit-error-rate estimations for differential quadrature phase-shift -quadrature keying and symbol-error-rate estimations for amplitude-modulation constellations are presented and a mapping between circuit characteristics and OFDM performance is outlined. Index Terms—Adjacent channel, bit error rate (BER), combinatorics, digital audio broadcasting (DAB), digital video broadcasting (DVB), discrete multitone (DMT), intermodulation, multicarrier system, nonlinearity, orthogonal frequency-division multiplexing (OFDM), RF circuits, signal-to-noise ratio (SNR), symbol error rate (SER).

I. INTRODUCTION ULTICARRIER systems have emerged over recent years as preferred candidates for a variety of wired and wireless applications, allowing higher symbol rates to become attainable. Amongst various implementation approaches, orthogonal frequency-division multiplexing (OFDM) systems [1] stand out owing to their superior performance over frequency-selective channels, their resilience to inter-symbol interference, and their ease in hardware realization by means of fast Fourier transform (FFT). As a result, OFDM has been employed in various applications such as digital video broadcasting (DVB), digital audio broadcasting (DAB), asynchronous digital subscriber line (ADSL) and wireless local area network (LAN). Nonetheless, because OFDM signals result from the superposition of a high number of modulated carrier signals, they exhibit a high peak-to-meanratioorcrestfactor.Itisthuslikelythatsomeof the RF devices in the up- and down-conversion stages of the communication link will have to cope with high power levels and may

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Manuscript received January 6, 2006; revised April 29, 2006. This work was supported by Panasonic System LSI Design Europe. A. Chorti is with the Electronic Systems Design Group, Department of Electronics and Computer Science, University of Southampton, Southampton SO17 1BJ, U.K. (e-mail: [email protected]; [email protected]). M. Brookes is with the Communications and Signal Processing Group, Department of Electrical and Electronic Engineering, Imperial College London, London SW7 2AZ, U.K. (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2006.879129

be driven into the unwanted nonlinear operating region of their input/output characteristic [2]. The effect manifests itself through a severe bit-error-rate (BER) degradation of the OFDM system. Simulation results are presented in [3] and [4]. In [5], Shimbo examined the effects of intermodulation in multicarrier systems. More recently, many authors, [6]–[8] amongst them, have dealt with the impact in OFDM of traveling-wave tube (TWT) [9] or other high power amplifiers (HPAs) that exhibit AM/AM and AM/PM characteristics. The noise due to the nonlinearity was shown in these analyses to be approximately uncorrelated with the OFDM signal that is modeled as a Gaussian random process. However, these results do not apply directly in the case of transistor circuits that are normally modeled as weakly nonlinear memoryless systems [10] because a different nonlinearity model has to be used. In that context, many authors, amongst them [11] and [12], have tackled the problem by assuming the nonlinearity is memoryless. The OFDM signal is represented as a Gaussian random process and the analysis is based on the derivation of the nonlinearity output autocorrelation function as a function of the autocorrelation of the Gaussian input signal based on statistical signal-processing analyses included in [13]. It is worth noting that Blachman in [14] has given closed-from evaluations of the relevant autocorrelation functions and examined important limiting cases. In order to derive an approximate signal-to-distortion ratio (SDR), van den Bos et al. in [12] assumes that the nonlinear intermodulation noise is dominantly uncorrelated with the OFDM signal. The results presented in this study lend themselves as a useful rough estimation of the system performance degradation. In our analysis, we intend to lift the restrictions of the approximate analysis in the case of memoryless weakly nonlinear circuits and extract accurate SNR expressions. We represent the OFDM spectrum as a sum of carrier tones missing the central carrier (application in DAB), taking into account their spectral broadening because of phase noise. It is demonstrated that, in the case of weakly nonlinear memoryless systems, phase noise and intermodulation distortion (IMD) can be treated as uncorrelated noise sources, but that their joint effect is more complex than a mere addition of the relevant variances, as presented in [15]. Applying techniques from combinatorics theory, we determine the exact number of IMD products on any given in-band frequency bin and extract closed-form expressions for the carrier signal-to-noise ratio (SNR) and OFDM symbol-error-rate (SER) or BER according to the modulation scheme. Our approach offers an insight into the way the IMD products are generated and into the degree of phase correlation between the OFDM signal and the IMD. To demonstrate the benefit from following a more accurate analysis, we compare our results with the results of [12].

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The use of combinatorics was introduced quite early and, in [16], Wescott discusses the case of multicarrier FM signals through TWT amplifiers. Pedro and de Carvalho in [17] have calculated many important IMD figures-of-merit in regard to third-order nonlinear circuits. In [18], Liu presents an approximate SNR evaluation. In [19], Boulejfen et al. applied the methodology in multitone signals through fifth-order nonlinear distortion and derived new closed-form expressions for the relevant figures-of-merit. In this study, we are mainly interested in deriving closed-form expressions for the carrier SNR to allow for using the results for BER and SER estimations. Calculations of the number of IMD products due to thirdorder nonlinearities already exist in the literature (e.g., [16] and [19]). Extending the previous use of combinatorics, we examine the effect of IMD distortion in the presence of phase noise and of additive Gaussian noise channel. Subsequently, we present an analysis for the case of a multitone channel received along with a high adjacent channel at any spacing and the generalization to multichannel multitone excitation. Although our derivations follow the DAB specifications (suppressed central carrier), the results practically apply in the general case as well. It is worth noting that the proposed combinatorics approach leads to much simpler SNR expressions if the OFDM envelope is continuous. This paper is organized as follows. In Section II, we present the nonlinearity model we will use in our analysis and discuss the suitability of OFDM signals for the methodology derived from combinatorics. In Section III, we deal with a single OFDM channel through a third-order nonlinearity, including band-limited flat spectral density noise and common to all carriers phase noise. A comparison between our results and the results assuming uncorrelated IMD distortion [12] is included. In Section IV, we consider the case of reception of an OFDM channel along with a high power adjacent channel of the same service and discuss issues concerning the reduction of the receiver effective dynamic range. In Section V, we generalize for the case of multiple adjacent channels and highlight the restrictions imposed on the RF circuit designer by a given OFDM service layout. Conclusions are presented in Section VI. II. WEAKLY NONLINEAR CIRCUITS AND THE USE OF COMBINATORICS According to the theory of weakly nonlinear systems, we can represent a component’s input/output characteristic by a truncated power series around the dc operating point as long as a frequency-independent model is appropriate [20]. On the other hand, if the component’s nonlinearities are not memoryless, Volterra analysis should be used instead for the representation of the component’s characteristic [21]. In the following, we will assume that: 1) the circuit is weakly nonlinear and 2) the circuit input/output characteristic can be approximated by the first four terms of its Taylor series expansion

where

The above approximation is realistic for a wide range of lownoise amplifiers, mixers, and baseband amplifiers [22], [23]. The use of combinatorics and of the generating function technique in particular [24], [25] to calculate the number of IMD products presents, in the general case, clear limitations. The problem to be solved is particularly complex if the multitone excitation of the nonlinearity have an irregular frequency spacing or if the tones are not of the same amplitude. However, in OFDM systems, we do indeed have a regular carrier spacing due to the inverse fast Fourier transform (IFFT) in the transmitter. Furthermore, since we are interested in SNR expressions, the requirement of carriers of the same amplitude can actually be reduced to the following. The random variables (RVs) representing the carrier amplitudes should be independent or at least uncorrelated up to the sixth order and the modulation phases need to be independent RVs with known moments up to the sixth order. The above requirements are reasonable approximations in many cases of real broadcasting systems and are presumed valid in the rest of this paper. The combinatorial approach is well suited to DAB as the carriers are designed to have the same amplitude. The effect of frequency-selective broadcasting channels is not examined in this study. The main difficulty in using combinatorics for IMD distortion analysis is in the evaluation of the coefficients of products of infinite-length polynomials in a parametric closed form. In the remainder of this paper, these coefficients have been readily calculated for all the examined cases. III. OFDM SIGNALS THROUGH THIRD-ORDER NONLINEARITY In Section III-A, we calculate the number of in-band IMD products in the case of a third-order nonlinear circuit with OFDM input degraded with common to all carriers phase noise. In Section III-B, we derive the SNR expression including the effect of phase noise, while in Section III-C, we also account for the presence of band-limited white-like noise. Finally, in Section III-D, we present the mapping of the circuit nonlinearities to BER and SER estimations. A. OFDM With Phase Noise Through Third-Order Nonlinearity To begin our analysis with, we model the nonlinear RF component characteristic as a third-order polynomial. Modeling the circuit nonlinearity with such a low-order polynomial is actually an efficient approximation in many real systems since the third-order nonlinearity is responsible for most of the IMD distortion [16]. Furthermore, we neglect even-order nonlinearities because the resulting IMD products will be out of the OFDM band. Without loss of generality, we may neglect the dc term and express the output of the nonlinear circuit as (1) where and are the linear gain and third-order nonlinearity coefficient, respectively, while is the input OFDM voltage. Commonly, for the assessment of a circuit’s nonlinear characteristics, circuit designers make use of the 1-dB compression point or second- and third-order intercept points. In the following, we will characterize the nonlinearity through the output-inferred

CHORTI AND BROOKES: ON THE EFFECTS OF MEMORYLESS NONLINEARITIES ON

third-order intercept point (OIP3) [26]. Its dependence on the linear gain is cubic, while it is inversely proportional to :

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TABLE I -QAM AMPLITUDES

(2) The analysis can be summarized as follows. If we represent the OFDM signal as a sum of carrier tones, then the third-order term resulting from substitution in (1) is expressed as two triple sums; one centered in the OFDM band and the second at thrice the frequency. Neglecting out-of-band terms, the evaluation of the output SNR requires the determination of the number of in-band IMD components and of their phase relation to the given carrier. For the evaluation of the number of IMD terms, we can either perform a threefold convolution of the sum of the Dirac deltas representing the OFDM spectrum or use counting methods from combinatorics theory. Subsequently, further use of combinatorics enables us to count the number of IMD products that are in-phase with a given OFDM carrier and, hence, add with it in amplitude. The effect of the remaining IMD components depends on the statistics of the modulation phases. As long as the RVs that represent the OFDM carrier phases can be assumed independent and the IMD products have null averages, the problem is simplified and the out-of-phase IMD tones will add in power with each other and with the OFDM carrier. We start the analysis by assuming that represents an OFDM signal corrupted only by phase noise (common to all carriers1) (3)

The random process accounts for the OFDM signal phase noise from oscillators in earlier stages and denotes the real value. The lower frequency of the OFDM channel is , the carrier spacing is , and and are the th carrier amplitude and phase. We assume that the RVs are independent and that the process is zero mean, which is the case for quadrature amplitude modulation ( -QAM). For differential quadrature phase-shift keying (DQPSK), to ensure that the above is valid, we need to further assume that the transmitted symbols are uncorrelated. Finally, we consider OFDM signals missing the central carrier and of an even number of carriers (application in DAB). The output voltage can be expressed as in (4), obtained by substitution of (3) into (1) as follows:

Equation (4) highlights two important conclusions. Firstly, the phase noise is not altered by the nonlinearity for in-band products [the third term in (4)] while it is tripled in the out-ofband products [the second term in (4)]. Secondly, we need to evaluate up to the sixth-order moments of the modulation amplitudes RVs for the calculation of the nonlinearity output power. If the modulation scheme is either -QAM or DQPSK, then the amplitudes are generated by the general expression [27]

where The RVs and are independent and the number of constellation points equals . We only need to calculate second-, fourth-, and sixth-order moments. In Table I, we have evaluated for various -QAM constellations the fourth- and sixth-order moments as a function of the second-order moment . The case of DQPSK is covered by 4-QAM. We also assume the use of a rectangular time window, which is typical in OFDM systems. As shown in (4), the in-band intermodulation products are at frequencies . Therefore, to calculate their number, we have to compute the number of possible ways for an IMD product to appear at a frequency

in the bandpass . This problem is equivalent to computing the number of integer solutions of the equation with conditions

(5) (6)

We denote the set of the possible combinations on any given frequency as . Making use of combinatorics theory [24], [25], we can identify the size of from the generating function of (5)

(7) (4) 1Phase noise generated by mixing with a noisy oscillator is common to all carriers.

The number of in-band IMD products is derived in Appendix I. We distinguish four subsets of according to

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TABLE II IMD SUBSETS

the phase relation of the above IMD products in regard to the carrier , as described in Table II. In the subsets and , two terms arise with the same frequency and phase by interchanging and or and , respectively, resulting in a term with double the amplitude. The sizes , , , and of the respective subsets are included in Appendix I. We note that is of the order , is the unit, and are of the order , and is of the order . In the case of DAB where , it follows that the number of IMD products is 1 767 167 on the carrier next to the central frequency, of which, 1536 are in-phase with it. B. Derivation of the SNR at the Output of the Nonlinearity We now have to identify how the various subsets of IMD products affect the useful signal. We begin by noticing that subsets and are in-phase with the carrier at and will, therefore, add with it in amplitude. According to the sign of the third-order nonlinearity coefficient , those two subsets will provide either gain enhancement or gain compression to the wanted signal, evaluated as

(8) It follows that the gain enhancement/compression expressed in (8) depends on , , , , and . It increases with decreasing and increasing . The useful output power on a carrier at a frequency bin is

Fig. 1. Output power as estimated analytically and through simulation.

proof can be found in Appendix IV. This criteria is fulfilled for -QAM and DQPSK so that the intermodulation noise can be expressed as

(10) In Fig. 1, we plot the estimated overall output power along with simulation results in the absence of phase noise . The simulated OFDM output power was evaluated by realizing threefold convolution of QPSK OFDM with carriers and was averaged over 1000 OFDM symbols. The third-order nonlinearity was chosen such that dBm, while and . The good agreement of the results validates our output power estimator . In the absence of phase noise, the orthogonality of the carriers is not affected by the nonlinearity and we can use (9) and (10) for the SNR evaluation at the output of the FFT. However, in the presence of phase noise, we need to further account for energy spillage between consecutive carriers. Assuming phase noise is small, it is shown in [28] that the effect of phase noise on OFDM can be decomposed into two terms, i.e., the common phase error (CPE) and the inter-carrier interference (ICI). Their variances and , respectively, as a function of the phase noise spectrum , are given as follows:

(9) denotes the binomial coefficient. where On the other hand, subsets and correspond to intermodulation noise. In order to derive the intermodulation noise power, we need to examine whether the IMD products included in subsets and will add in amplitude or in power. As long as the modulation phases’ probability density function is symmetrical in regard to both axes of the constellation diagram, IMD terms in and are mutually uncorrelated. The

(11)

(12) . where The CPE demonstrates the effect of the broadening of the carrier spectrum, while the ICI results from the effect of energy

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spillage from neighboring carriers. Therefore, IMD terms on the same frequency as the carrier will contribute in terms of CPE, while IMD terms on the remaining carriers will contribute in terms of ICI. As a result, the SNR of the carrier at frequency , due to both phase noise and third-order nonlinearity, can be expressed as (13) where (14) (15) The CPE is not Gaussian and, therefore, the SNR expression (13) cannot be directly used for SER or BER estimation. An important conclusion of this analysis is that the CPE is not “common” to all carriers in the presence of nonlinearities. Unlike in a previous study [15], we have readily shown that the amount of CPE depends on the carrier index and on the number of OFDM carriers . This effect should be taken into consideration in compensation strategies.

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Therefore, the noise power per carrier due to the terms is evaluated as

(18) where is the variance of . In order to calculate , we need to express the power of the baseband components generated by . is centered on dc ( takes values in the range ) and it is worth noting that phase noise is null in those IMD terms. We define the set of baseband IMD products generated by as and use the generating-function method to calculate its size

(19) add in power, except for , The components in in which case, they are all in-phase and add in amplitude. The distortion power in the set is

C. OFDM and Noise Through Third-Order Nonlinearity We now proceed by assuming that the input signal is expressed as , where is given in (3) and is band-limited additive noise of flat power spectral density . The OFDM band is included in the band-limited white-like noise band. We note that

(20) Therefore, the variance

is expressed as

(21) By substitution of (21) in (18), we get (16) The terms in set represent the amplified OFDM signal, while set contains the third-order IMD products previously derived. Set contains the terms resulting from the intermodulation of the additive white noise with the OFDM channel. To account for the noise power of terms in , we identify as the dominant term within this set. We assume that the bandwidth of , on either side of the central OFDM frequency, is at least equal to the OFDM bandwidth, so as to ensure that the spectrum of is flat in the OFDM band. In that case, exploiting the independence of the processes and , we can calculate the resulting noise power per carrier as the product of the variances of the respective noise terms. Assuming a rectangular time window is employed at the OFDM receiver, the variance of in a carrier spacing bandwidth is given by

(17)

(22) Finally, we need to take into account the nonlinear circuit internal noise sources. We assume there is a noise contribution per carrier so that the output carrier SNR can be expressed as

(23) D. SNR Mapping to SER and BER We have obtained an expression that maps the quantities , , , , and (the distance in carrier spacing from the lower edge of the OFDM channel) to the carrier SNR. Neglecting the effect of phase noise and setting , we can isolate the effect of IMD distortion on the output SNR. In that case, the central limit theorem ensures that the distortion components can

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Fig. 2. Comparison of exact and approximate SER evaluation. QPSK OFDM, N = 1024, a = 1.

Fig. 3. Absolute value of the difference between the exact and approximate SER expressions as a function of the number of constellation points M and ja j. We have set N = 1024 and a = 1.

be approximated as Gaussian noise. We can as a result relate the carrier SNR to average SER for -QAM and BER for DQPSK through the formulas provided in [27]. For the SER, we will use the formula (24) is the symbol SNR, is the number of constellawhere tion points of the -QAM, with being the complementary error function. For the DQPSK BER, we have the expression

(25) where

is Marcum’s

function defined as Fig. 4. Analytic and simulation BER for DQPSK OFDM, ! = 2 krad/s, OIP3 = 8 dBm.

with , is . Finally, is the modified the bit SNR, Bessel function of the first kind and order . In Fig. 2, we compare the results of our analysis to: 1) the approximate results in [12] denoted in the graph as “SER approximate (Bos)” and 2) the simulated results in [12] denoted in the graph as “Simulation results from Bos.” In the same graph, we plot the SER evaluated for QPSK OFDM with and unit input power, both for negative (denoted as “SER with negative ”) and positive (denoted as “SER with positive ”), following our accurate SNR estimator. We note that the results produced by our accurate SNR estimator match very well with the simulated results presented in [12], thus offering a degree of validation. Our exact analysis substantially increases the degree of accuracy in SER estimation, as

N = 1536,

compared to the approximate, in the case of negative . Moreover, in the case of positive , the approximate analysis does not account for the gain enhancement and the approximate SER prediction is rather pessimistic. In Fig. 3, we plot the absolute value of the difference between the exact and approximate SER expressions as a function of the number of constellation points and of the absolute value of the nonlinearity coefficient , assuming . The input power is unit, and . The underestimation or overestimation of the SER is shown to be more important for constellations with either a small or a large number of constellation points. Therefore, in such OFDM systems, the exact SNR expressions should be used in IMD performance analysis. Furthermore, in Fig. 4, we compare BER results obtained from our analysis in the case of negative to simulation results

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Fig. 6. BER of DQPSK OFDM with N = 96 versus hP i=OIP3. Fig. 5. SER of 4-QAM for different values of hP i=OIP3 versus the number of carriers.

obtained from a DAB OFDM simulator developed in Simulink/ MATLAB. The output BER was evaluated for DQPSK OFDM over 100 symbols in our simulator. The OIP3 was set to 8 dBm, while the linear gain was assumed unit and the simulator generated carriers at a carrier spacing krad/s. The results present good match and further validate our analysis. To continue, we define as “linear output power” the output power in the absence of nonlinearities

(26) and plot in Fig. 5 the SER of a 4-QAM OFDM versus the number of carriers for different values of . For sufficiently high values of (around 100), the SER depends only on the ratio of the linear term to the OIP3. For values of below 100, the SER is slightly improved. It is thus inferred that the proportion of correlated and uncorrelated to the carrier intermodulation terms tends to be insensitive to the OFDM service layout. This result is in accordance with (9) and (10) as the dominant terms in both expressions are of the order of so that their ratio is asymptotically independent of . Finally, in Figs. 6 and 7, we investigate the performance of OFDM with for different constellations versus . As long as the target BER in the case of DQPSK or SER for -QAM is defined, we can identify the requirement on . From Fig. 7 stems that a decrease of approximately 1.4 dB in is necessary for each doubling of the number of constellation points. This arises because, for large , the SER is approximately a function of [27] and, thus, a doubling of may be compensated by a 3-dB increase in SNR. From (10), it follows that the SNR is proportional to or, equivalently, from (2) to .

Fig. 7. SER of M -QAM OFDM with N = 96 versus hP i=OIP3.

IV. OFDM WITH AN ADJACENT CHANNEL OF DIFFERENT AMPLITUDE A. Adjacent Channel at Integer Spacing In Section III, we examined the performance of an OFDM service under third-order nonlinearities. Here, we will further assume an adjacent OFDM channel of different amplitude. The adjacent channel amplitude relative to the carrier is , and is the spacing between adjacent channels, in units of carrier spacing. The input signal is expressed as

(27)

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TABLE III SUB-SUBSETS ACCORDING RELATIVE TO THEIR CARRIER AMPLITUDE

In order to calculate the size of the set of in-band IMD products, we identify the generating function for this problem as

Fig. 8. Overall in-band output power of a QPSK OFDM channel with an adjacent channel 60 dB higher. N = 96, g = 10, P = 30 dBm, OIP3 = 18:24 dBm.

h i 0

(28) where (29) (30) At present, we have assumed that is a multiple of the carrier spacing to simplify calculations. Later on, we will disengage from that requirement and examine the case of a noninteger inter-channel spacing. The case presenting the greatest interest is that of an immediate adjacent channel, where normally . The size of the set of in-band IMD products is expressed in (61), shown at the bottom of page 3313 when . To get the output SNR, we have to identify the subsets , , , and , as defined in Table II. However, within each of these subsets, we have to further identify sub-subsets of amplitudes. Use of the generating-function technique provides the sizes of the sub-subsets defined in Table III. All amplitudes are relative to the carrier amplitude. The sizes of the various sub-subsets are calculated in Appendix II. The useful in-band output power is thus expressed as

(31) With respect to intermodulation noise, using similar reasoning to the case of a single OFDM channel, the various sub-subsets of subsets and are mutually uncorrelated and resulting cross-terms have null averages. Therefore, all sub-subsets add in power. The corresponding intermodulation noise is now expressed as

(32) In Fig. 8, we plot the total output power and compare the results of our estimation with simulation results averaged over 1000 symbols. The analytic and simulation curves are practically indistinguishable. The OFDM channel has carriers, the adjacent channel is at a distance of carriers, the expected value of the output linear power, as expressed in (26), is dBm and dBm. In the specific example, the adjacent channel is chosen dB higher than the OFDM channel of interest. As expected from a closer inspection of (71)–(73), shown at the bottom of page 3314, the output power on carriers with indices 0 to contains intermodulation noise from set with relative power 0, 60, and 120 dB. From (74), shown at the bottom of page 3314, it becomes clear that IMD products purely generated from intermodulation of the adjacent channel

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Fig. 9. Simulated versus analytic BER for DQPSK BER with an adjacent channel 40 dB higher than the channel of interest. N = 1536, g = 200, ! = 2 krad/s.

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Fig. 10. SER versus hP i=OIP3 for M -QAM OFDM with N = 96 and g = 12. The relative power of the adjacent channel is 20 dB.

with itself and of relative power 180 dB affect only carriers that are at least carriers away from the lower edge of the OFDM channel. We conclude that, in the presence of a significantly higher immediate adjacent channel, OFDM carriers in its vicinity are much more severely affected. The carrier SNR is expressed as

(33) is given in (31), is expressed in (32), and is where the circuit noise. The estimation of BER for DQPSK OFDM and SER for -QAM OFDM are produced based on (24) and (25). The term contains IMD products of the adjacent channel with itself, thus we expect a severe degradation of the carrier SNR in the presence of a high adjacent channel compared to the case of single OFDM channel through similar nonlinear circuits. The result is a reduction in the effective dynamic range of the OFDM receiver. It is thus inferred that, in the receiver downconversion chain, either channel selection has to be performed at an early stage or a strict requirement on RF circuits linearity has to be adopted. In order to validate the derived SNR expression, in Fig. 9 we compare simulation results using the Simulink/MATLAB simulator with the estimated BER based on (31)–(33). The DQPSK OFDM simulator generates carriers, the inter-channel spacing is , and the carrier spacing is krad/s while the adjacent channel is at dB. Theoretical and simulation results match well. In Figs. 10 and 11, we investigate the impact of an adjacent channel on the SER of an -QAM OFDM. In Fig. 10, we have , and an adjacent channel of relative amplitude dB and highlight the impact of the number of constellation points on the output SER. Our previous remark

Fig. 11. SER versus hP i=OIP3 for 16-QAM OFDM with N = 1536, ! = 2 krad/s, g = 200, and various adjacent channel relative amplitudes.

about the reduction in receiver dynamic range is confirmed. In order to achieve a target SER for a given constellation, we need to decrease the nonlinear circuit output power level. The limitation in decreasing the output power is placed by the nonlinear circuit internal noise sources that at low output power levels will have a nonnegligible effect on the carrier SNR. The effect is investigated in Fig. 11, where, we assume a circuit with noise figure dB2 and plot the SER of 16-QAM OFDM with , krad/s, and for various adjacent channel power levels. From this graph, we can identify in each case the optimal reception region in regard to the ratio . As long as the circuit operates in that region, the system SER can be kept under the occasional target. When is large, (32) is dominated by the term in and 2An

NF = 15 dB is a pessimistic value used for illustrative purposes.

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for every 20-dB increase in adjacent channel power, a 30-dB decrease in is necessary to prevent performance degradation. This follows from (2) and the quadratic dependency of the noise on . As an interesting limiting case, when the adjacent channel amplitude becomes dB higher than the OFDM channel of interest, we can identify the optimal value of the in order to minimize the SER. In the quantity receiver design, extra care should be put into meeting the compromise between circuit NF and nonlinearities, as well as in the detection of adjacent channels and the adjustment of automatic gain control (AGC) stages accordingly. As a concluding comment, previous analyses of the effects of adjacent channel on multicarrier systems [19], [29] were based on the evaluation of the spectral regrowth of the adjacent channel alone. According to the terminology we introduced, this implies that only the subset was taken into account. There is an underestimation of the adjacent channel effect, as subsets , , , , , and were completely disregarded. Unless the joint effect of the channel of interest and of the adjacent is investigated, the estimation of the intermodulation power is underestimated.

the subsequent FFT. In the frequency domain, this is equiva. The lent to convolving the power spectrum with a overall effect of the time windowing of the sampler of the ADC and of the FFT block is to project part of the in-band IMD distortion power onto all OFDM carriers. The projected IMD power onto an individual carrier is scaled by

offset

(34)

The intermodulation noise power on a carrier at a distance from the lower edge of the OFDM band is expressed as offset

B. Adjacent Channel at Noninteger Spacing We will now consider the case of an adjacent channel at a noninteger spacing in units of OFDM carrier spacing so is now a rational number instead of an integer. The issue is to identify which of the previously defined subsets fall exactly on carrier frequencies in the band of interest or in between two carriers. We express offset, where and offset , being the remainder of the integer division. We examine the various subsets separately. Subsets , , , and are generated from combinations of the channel of interest and fall on carrier frequencies. Subsets , , , and are generated from combinations of two carriers of the wanted channel and one carrier from the adjacent. The IMD products in the above sets fall on frequencies offset, where is the frequency the products would fall on if the adjacent channel was at a spacing . Subsets , , , and are generated from the combination of two carriers of the adjacent channel and one of the wanted and fall on frequencies offset offset . Therefore, these sets fall exactly on the frequencies defined in the analysis involving an adjacent channel at an integer spacing. Subset is generated exclusively from combinations of carriers of the adjacent channel and the IMD products fall on frequencies offset offset offset offset. Therefore, all IMD products in this set fall on the frequencies defined from the previous analysis plus offset. The IMD products that fall in between two carriers affect all the OFDM carriers to a varying extend. This phenomenon is the result of the time windowing used to isolate individual OFDM symbols. In the following, we assume a rectangular time window of length is used before the ADC converter and

(35) where offset offset

(36) (37) (38)

, the carrier power is not altered and the As carrier SNR is expressed as

offset

(39)

is expressed in (31) and is the circuit noise. The where overall intermodulation noise power is larger compared to the case of integer and OFDM will perform worse under a high power adjacent channel. Validation of the presented SNR estimator is provided by comparison of the analytically evaluated BER for DQPSK OFDM and simulated results. The Simulink/MATLAB simulator generated carriers and an adjacent channel at 40 dB higher at a frequency of with a carrier spacing of krad/s. The results match well and are depicted in Fig. 12. The SNR degradation strongly depends on the number of OFDM carriers. For a high adjacent channel, the main contribution in comes from set . For , we can approx-

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of the SER as a function of the offset are determined by two tendencies: the decrease in the SER as the inter-channel gap increases and the expression in (40) becomes maximal when the offset is half the carrier spacing. From the previous analysis, we conclude that although theoretically there is a slight gain in carrier SNR if the system designer chooses an integer inter-channel spacing, the effect is almost imperceivable in realistic terms. As a result, the analysis for an adjacent channel at an integer spacing lends itself as a quite precise approximation for the case of noninteger spacing and can, therefore, safely be adopted as the general case. V. OFDM WITH AN ARBITRARY NUMBER OF ADJACENT CHANNELS

Fig. 12. Simulated versus analytic BER for DQPSK OFDM with an adjacent channel at 40 dB higher than wanted, at an offset of half the carrier spacing. N = 1536, g = 200:5, ! = 2 krad/s.

Finally, in this section, we examine the case of an OFDM channel along with an arbitrary number of adjacent channels of the same amplitude. In the following, we have included the central carrier in order to avoid overcomplicating the derived expressions. The input signal to the nonlinear device can be expressed as

(41) where is the number of adjacent OFDM channels. The generating function to calculate the size of the set of in-band IMD products is

(42) where Fig. 13. SER of 4-QAM OFDM versus offset in adjacent channel spacing for OIP3 = 18:24 dBm and H = 40 dB.

imate the number of IMD as a constant term in (35) can be approximated by

(43) (44) (45)

. Therefore, the final with

(46) (40) The expression in (40) is an increasing function that asymptotically reaches as . In Fig. 13, we plot the output SER of a 4-QAM OFDM with an adjacent channel at a distance , denoting the closest integer rounding function, while dB and , and overall input power 20 dBm. For a variety of values of , the offset increases from 0 to . Two conclusions can be drawn. Firstly, that the larger , the less varied is the SER, as we previously predicted. Secondly, the local minima

Finally, (47) (48) (49) The size of the set nents is given in Appendix III.

of the in-band compo-

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Fig. 15. Output SER of 64-QAM OFDM with r adjacent channels.

Fig. 14. Output power as estimated and through simulation.

Again, we will distinguish between the four subsets , , , and whose phase relation is already described in Table II. The sizes of the relevant sets are calculated in Appendix III. The useful power on a carrier is expressed as

(50) and . The intermodulation noise results from subsets Based on the fact that the cross-correlation terms have null averages, we express in closed form the intermodulation noise as (51) In Fig. 14, we plot the estimated output power on a carrier and the simulated output power for QPSK OFDM with , , averaged over 1000 symbols. We have adjacent channels, and the overall linear output power

(52) is dBm and dBm. The output power slightly increases with the carrier index. The carrier SNR if we ignore all other noise sources is simply given by

(53) In Fig. 15, we plot the SER of a 64-QAM OFDM as a function of with and without adjacent channels. It is shown

Fig. 16. SER of 64-QAM OFDM with ten adjacent channels as a function of the inter-channel spacing.

that for every 10-dB increase in the total power of the adjacent channels, an approximately 10-dB increase in the is necessary to maintain performance. The effect of multiple adjacent channels can be mitigated by an increase of the inter-channel spacing. However, a limiting factor in the BER or SER improvement is placed by the fact that a number of IMD products is independent of . The effect is investigated in Fig. 16 where we examine the effect of inter-channel spacing on the SER of a 64-QAM OFDM with ten adjacent channels. As the inter-channel spacing becomes equal to the OFDM channel bandwidth, the SER is still higher than in the case with no adjacent channels. Intermodulation products generated from mixing of the adjacent channels without the participation of the channel of interest places a limit on the achievable system performance. The effect should be taken into account, especially in the case of “congestive” broadcasting services.

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-QAM AND DQPSK OFDM SIGNALS

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VI. CONCLUSIONS

(57)

The use of combinatorics in nonlinear distortion analysis of OFDM signals can provide precise SNR evaluations. As long as the dominant circuit nonlinearities can be described by the theory of weakly nonlinear circuits, our SNR formulas are precise. We have derived in closed-form expressions for the carrier SNR in the cases of single OFDM channel, OFDM along with an adjacent channel of different amplitude and an OFDM channel with an arbitrary number of adjacent channels. A mapping to -QAM SER and DQPSK BER was obtained by analyses of the occurring cross-term correlations. Comparison with other published studies shows that the new formulas notably increase the accuracy in SNR prediction [12]. Our results can provide important guidelines both to the OFDM service regulator and to the RF transceiver engineer and save valuable simulation time. Exploiting either a priori knowledge of the circuit nonlinear characteristics or an ad hoc estimation, our carrier SNR prediction can help improve the performance of subsequent soft decision stages of the decoder. More importantly, the resilience of specific OFDM receiver architectures to nonlinearities can be precisely evaluated. The receiver designer can reliably achieve the necessary compromise between performing channel selection filtering at late stages of the tuner and using highly linear analog circuits. Finally, our SNR estimators in the presence of single or multiple adjacent channels can help identify how the AGC levels should be chosen in order to avoid unacceptable levels of intermodulation noise. APPENDIX I SET SIZES IN THE CASE OF SINGLE OFDM CHANNEL In the case of an OFDM channel through a third-order nonlinearity, we have calculated the sizes of sets , , , , and as follows:

and (58) ,

with ,

,

,

,

,

, . By

value of If

modulo ,

. ,

, , and , we denote the

simplifies to

(59) We note that, in the calculation of expansion (assume )

, we have used the

(60) with

denoting the floor function. APPENDIX II SETS SIZES IN THE CASE OF AN ADJACENT CHANNEL OF DIFFERENT AMPLITUDE

In the case of an OFDM channel along with an adjacent channel of different amplitude, we have evaluated the sizes of the various sets. The sizes of sets , , , , , and are given in (61) and (71)–(74), while the sizes of all other sets are given as follows: (62) (63)

(54) where

,

(64)

,

,

(65) (55)

(66) (56)

(67) (68)

(61)

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where

(69) (70) ,

,

and

(84)

, ,

, , , and . The number of carriers is assumed to satisfy and . APPENDIX III SET SIZES IN THE CASE OF AN ARBITRARY NUMBER OF ADJACENT CHANNELS We have evaluated the sizes of the various sets in the case of an arbitrary number of adjacent OFDM channels of the same amplitude as follows:

(75) where

We define

while is defined in (47). APPENDIX IV CROSS-CORRELATION OF IMD SETS

and In order to calculate the intermodulation noise in sets , we need to evaluate the cross-correlations of the following. (i) Any two distinct terms in set . (ii) Any two distinct terms in set . (iii) Any two terms in sets and . Lemma 1: We assume that the RVs satisfy the following. (a) They are independent and identically distributed. (b) Their probability density function is symmetrical about and . We construct the random processes , where

(76) (77) (78)

(85) and with

odd, we define (86)

(79) (80) (81) where

(87) The thus generated random processes

are zero mean.

Proof of Lemma 1: (82) (83)

Remark 1: Due to condition (b), the RVs 1) In the case of , and ensures that .

are zero mean. and Remark 1

(71)

(72)

(73)

(74)

CHORTI AND BROOKES: ON THE EFFECTS OF MEMORYLESS NONLINEARITIES ON

2) For

, either or or will include a variable Due to condition (a),

is odd so that either with unit multiplicity.

The criteria of Lemma 1 are satisfied in the case of -QAM and DQPSK modulations. As a result, in both and , the IMD products are zero mean. Furthermore, for any distinct pair of IMD products, the relevant angles are uncorrelated. Therefore, the cross-correlations in cases (i)–(iii) are estimated as null. ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers for their valuable comments. REFERENCES [1] W. Zou and Y. Wu, “COFDM: An overview,” IEEE Trans. Broadcast., vol. 41, no. 1, pp. 1–8, Mar. 1995. [2] T. Keller and L. Hanzo, “Adaptive multicarrier modulation: A convenient framework for time-frequency processing in wireless communications,” Proc. IEEE, vol. 88, no. 5, pp. 611–640, May 2000. [3] X. Li and L. J. Cimini, “Effects of clipping and filtering on the performance of OFDM,” IEEE Commun. Lett., vol. 2, no. 5, pp. 131–133, May 1998. [4] S. Merchan, A. G. Armada, and J. Garcia, “OFDM performance in amplifier nonlinearity,” IEEE Trans. Broadcast., vol. 44, no. 1, pp. 106–114, Mar. 1998. [5] O. Shimbo, “Effects of intermodulation, AM–PM conversion, and additive noise in multicarrier TWT systems,” Proc. IEEE, vol. 59, no. 2, pp. 230–238, Feb. 1971. [6] D. Dardari, V. Tralli, and A. Vaccari, “A theoretical characterization of nonlinear distortion effects in OFDM systems,” IEEE Trans. Commun., vol. 48, no. 10, pp. 1755–1764, Oct. 2000. [7] G. Santella and F. Mazzenga, “A hybrid analytical-simulation procedure for performance evaluation in -QAM-OFDM schemes in presence of nonlinear distortions,” IEEE Trans. Veh. Technol., vol. 47, no. 1, pp. 142–151, Feb. 1998. [8] P. Banelli, G. Baruffa, and S. Cacopardi, “Effects of HPA non linearity on frequency multiplexed OFDM signals,” IEEE Trans. Broadcast., vol. 47, no. 2, pp. 123–136, Jun. 2001. [9] A. M. Saleh, “Frequency independent and frequency dependent nonlinear models of TWT amplifiers,” IEEE Trans. Commun., vol. COM-29, no. 11, pp. 1715–1720, Nov. 1981. [10] S. A. Maas, “How to model intermodulation distortion,” in IEEE MTT-S Int. Microw. Symp. Dig., 1991, pp. 149–151. [11] Q. Shi, “OFDM in bandpass nonlinearity,” IEEE Trans. Consum. Electron., vol. 42, no. 3, pp. 253–258, Aug. 1996. [12] C. van den Bos, M. H. L. Ksuwenhoven, and W. A. Serdijn, “Effect of smooth nonlinear distortion on OFDM symbol error rate,” IEEE Trans. Commun., vol. 49, no. 9, pp. 1510–1514, Sep. 2001. [13] W. B. Davenport and W. L. Root, An Introduction to the Theory of Random Signals and Noise. New York: McGraw-Hill, 1958. [14] N. M. Blachman, “The signal signal, noise noise, and signal noise output of a nonlinearity,” IEEE Trans. Inf. Theory, vol. IT-14, no. 1, pp. 21–27, Jan. 1968. [15] E. Costa and S. Pupolin, “ -QAM–OFDM performance in the presence of a nonlinear amplifier and phase noise,” IEEE Trans. Commun., vol. 50, no. 3, pp. 462–472, Mar. 2002.

M

x

M

x

x

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[16] R. J. Westcott, “Investigation of multiple FM/FDM carriers through a satellite TWT operating near to saturation,” Proc. IEEE, vol. 114, no. 6, pp. 726–740, Jun. 1967. [17] J. C. Pedro and N. B. de Carvalho, “On the use of multitone techniques for assessing RF component’s intermodulation distortion,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 12, pp. 2393–2402, Dec. 1999. [18] C. Liu, “The effect of nonlinearity on a QPSK–OFDM–QAM signal,” IEEE Trans. Consum. Electron., vol. 43, no. 3, pp. 443–447, Aug. 1997. [19] N. Boulejfen, A. Harguem, and F. A. Channouchi, “New closed-form expressions for the prediction of multitone intermodulation distortion in fifth-order nonlinear RF circuits/systems,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 1, pp. 121–132, Jan. 2004. [20] R. Deutsch, Non-Linear Transformations of Random Processes. New York: Prentice-Hall, 1962. [21] A. Heiskanen and T. Rahkonen, “Fifth-order multi-tone Volterra simulator with component-level output,” in Proc. Int. Circuits Syst. Symp., May 2002, vol. 3, 2, pp. 591–594. [22] C. Chien, Digital Radio on a Chip. Norwell, MA: Kluwer, 2001. [23] C. Fager, J. C. Pedro, N. B. de Carvalho, H. Zirath, F. Fortes, and M. J. Rosario, “A comprehensive analysis of IMD behavior in RF CMOS power amplifiers,” IEEE J. Solid-State Circuits, vol. 39, no. 1, pp. 24–34, Jan. 2004. [24] J. Bradley, Applied Combinatorics With Problem Solving. Reading, MA: Addison-Wesley, 1990. [25] A. Tucker, Applied Combinatorics, 3rd ed. New York: Wiley, 1994. [26] U. Rodhe, RF/Microwave Circuit Design for Wireless Applications. New York: Wiley, 2000. [27] J. G. Proakis, Digital Communications, 4th ed. New York: McGrawHill, 2001. [28] J. Stott, “The effects of phase noise in COFDM,” EBU Tech. Rev.,, vol. 276, pp. 1–19, Jul. 1998. [29] J. C. Pedro and N. B. de Carvalho, “Characterizing nonlinear RF circuits for their in-band distortion,” IEEE Trans. Instrum. Meas., vol. 51, no. 3, pp. 420–426, Jun. 2002. Arsenia Chorti received the M. Eng. degree in electrical engineering from the University of Patras, Patras, Greece, in 1998, the D.E.A. degree in electronics from the University Pierre et Marie Curie–Paris VI, Paris, France, in 2000, and the Ph.D. degree in communications and signal processing from Imperial College London, London, U.K., in 2005. She is currently a Research Fellow with the Electronic Systems Design group, Electronics and Computer Science Department, University of Southampton, Southampton, U.K., where she is involved in the area of intelligent sensing. Her research interests include multicarrier communication systems—OFDM, stochastic signal processing, density estimation, novelty detection, and prediction/estimation algorithms.

Mike Brookes (M’88) received the B.A. degree in mathematics from Cambridge University, Cambridge U.K., in 1972. Upon graduation, he spent four years with the Massachusetts Institute of Technology (MIT), Cambridge, where he was involved with astronomical instrumentation and telescope control systems. Since 1979, he has been with the Electrical and Electronic Engineering Department, Imperial College London, London, U.K., where he is a Reader with the Communications and Signal Processing Research Group. His main areas of research are speech processing where he has been involved with speech production modeling, speaker recognition algorithms, and techniques for speech enhancement using both single microphones and microphone arrays. He currently applies techniques from speech processing to RADAR target identification and is also actively involved in computer vision research.

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High-Order Runge–Kutta Multiresolution Time-Domain Methods for Computational Electromagnetics Qunsheng Cao, Ramdev Kanapady, and Fernando Reitich

Abstract—In this paper we introduce a class of Runge–Kutta multiresolution time-domain (RK-MRTD) methods for problems of electromagnetic wave propagation that can attain an arbitrarily high order of convergence in both space and time. The methods capitalize on the high-order nature of spatial multiresolution approximations by incorporating time integrators with convergence properties that are commensurate with these. More precisely, the classical MRTD approach is adapted here to incorporate th-order -stage low-storage Runge–Kutta methods for the time integration. As we show, if compactly supported wavelets are used (e.g., the Daubechies functions) and of order , then the RK-MRTD methods deliver solutions that converge with this overall order; a variety of examples illustrate these properties. Moreover, we further show that the resulting algorithms are well suited to parallel implementations, as we present results that demonstrate their near-optimal scaling.

=

Index Terms—High-order accuracy, multiresolution time domain (MRTD), Runge–Kutta methods, wavelets.

I. INTRODUCTION HE multiresolution time-domain (MRTD) method for the numerical simulation of solutions to Maxwell’s electromagnetic equations was initially introduced in [1] as an alternative to the popular and, by now, classical finite-difference time-domain (FDTD) [2] approach. The basic idea behind MRTD is rather simple, as it reduces to a method of moments [3] (MoM) wherein the spatial basis functions are chosen from a multiresolution analysis (MRA) [4]. This choice naturally results in the potential for highly resolved spatial variations of the fields. However, in typical implementations, the realization of this potential is hindered by a low-order (leap-frog) time-stepping procedure. In this paper, we show how this limitation can be overcome by introducing a new class of MRTD schemes that incorporate high-order Runge–Kutta time integrators. We show that, with this addition, the resulting algorithms can be made to

T

Manuscript received January 23, 2006; revised May 5, 2006. This work was supported in part by the Army High Performance Computing Research Center under Army Research Laboratory Cooperative Agreement DAAD19-01-2-0014. The work of F. Reitich was supported by the Air Force Office of Scientific Research under Contract FA9550-05-1-0019 and by the National Science Foundation under Grant DMS-0311763. Q. Cao is with the College of Information Science and Technology, Nanjing University of Aeronautics and Astronautics, 210016 Nanjing, China. R. Kanapady is with the Department of Mechanical Engineering, University of Minnesota, Minneapolis, MN 55455 USA. F. Reitich is with the School of Mathematics, University of Minnesota, Minneapolis, MN 55455 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2006.879130

converge with arbitrarily high order in space and time, and that they consequently lead to more efficient simulations. Moreover, we demonstrate that these new numerical procedures retain the highly parallelizable characteristics of standard MRTD methods [5], [6] as we present a variety of results from a fully three-dimensional parallel implementation. The original developments of MRTD [1] highlighted an improvement in the observed dispersion characteristics over classical FDTD. Since then, a variety of studies have concentrated on the further analysis and application of these techniques, including the investigation of their stability, accuracy, and dispersion properties (see, e.g., [7]–[13]), the design of schemes for the incorporation of boundary and radiation conditions (e.g., [12], [14]–[16]), and the development of implementations for use in a variety of specific configurations [17]–[20]; the recent book by Chen et al. [21] provides a good introduction to the subject. The recent development of higher order finite-difference schemes for electromagnetic applications (see, e.g., [22]–[26]), on the other hand, has provided potential options over the classical FDTD scheme that are alternative to MRTD-based methodologies. Indeed, for instance, as noted in [27], the improved dispersion characteristics of MRTD techniques are attained at the cost of a larger stencil when compared to that of the Yee scheme [2] and, in fact, it results in solutions of a comparable quality to those produced by higher order differencing schemes. The simplicity of implementation of standard, as well as adaptive forms of MRTD procedures [13], [28]–[30], on the other hand, have continued to provide further impetus for the development of these methodologies. In fact, it is arguably these characteristics that account for the current popularity of the use of MRTD methods (e.g., over higher order finite-difference schemes) for tackling problems of practical relevance. As we mentioned, our research provides a further step toward the attainment of “optimal” and widely applicable MRTD procedures for computational electromagnetics. Our results, originally announced in [31], are related to the concurrent work communicated in [32]. Both our and this latter study begin with the basic realization that the low-order time-integration strategy inherent in standard MRTD is rather contrary to their high spatial resolution. The work in [32], however, further advocates the use of an MRA in time to circumvent this limitation. In contrast, our current study views the equations for the coefficients in the spatial multiresolution expansion as satisfying a system of linear ordinary differential equations to which some recently derived th-order, -stage, low-storage Runge–Kutta methods for the

0018-9480/$20.00 © 2006 IEEE

CAO et al.: HIGH-ORDER RUNGE–KUTTA MRTD METHODS FOR COMPUTATIONAL ELECTROMAGNETICS

time integration [33], [34] can be readily applied. We contend that this latter choice may be preferable over a space–time MRA [32] for a number of reasons. Most importantly, perhaps, this approach automatically chooses a minimal stencil to attain an accuracy that is commensurate with the spatial resolution, and it does so with minimal storage requirements. In addition, as we explain in Section III-C, the implementation of the resulting scheme is rather straightforward, necessitating only the repeated application of “forward Euler” steps and, thus, it can be readily used to upgrade existing MRTD implementations. The remainder of this paper is organized as follows. First, in Section II, we review the basic equations and concepts related to MRTD schemes. The classical method is introduced in Section II-A and its convergence properties are reviewed in Section II-B. Our new algorithms are then presented in Section III. The spatial discretization and its convergence properties are described in Sections III-A and B, respectively. The time-integrator, in turn, is described in Section III-C. Numerical results that demonstrate the overall high-order accuracy of the resulting algorithm are presented in Section III-D. Section III-E is devoted to the stability and dispersion characteristics of the RK-MRTD procedures, and Section III-F to their computational cost and memory requirements. In both cases, the properties of the new schemes are contrasted with those of standard MRTD methods, a comparison that further highlights the advantages of the proposed approach. Next, the details of the parallel implementation for the treatment of three-dimensional configurations are discussed in Section IV-A and results from this follow in Section IV-B. Finally, our conclusions are summarized in Section V.

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in such a way that the total field

satisfies (1). At material interfaces, the tangential components of the (total) electromagnetic field must be continuous

where denotes the normal vector to the interface and “ ” denotes the limiting values from either side of the corresponding quantity. In the limiting case of a perfect electric conductor (PEC), these conditions reduce to

A. MRTD Method The original MRTD scheme [1] is based on a spatial multiresolution representation of the fields at each instant in time. For instance if, for simplicity, only scaling functions are used (“S-MRTD” [1]), this representation takes on the form

(2a)

II. ELECTROMAGNETICS AND MRTD As we said, we shall be concerned with the numerical solution of the (time-domain) Maxwell equations

(2b)

(1a) (2c)

(1b) (1c) (1d)

(2d)

where , , , , and are the electric field, magnetic field, electric displacement, magnetic induction and current, respectively. We shall assume that the media are linear

(2e)

and will consider both initial-value problems, wherein initial conditions

are prescribed, and scattering problems where an unknown scattered field arises in response to incident radiation

(2f) where and , are the (constant, unknown) field expansion coefficients. Here, the function is the scaling function

for

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TABLE I NONZERO COEFFICIENTS a( ) FOR THE DAUBECHIES SCALING FUNCTIONS D –D

Fig. 1. Daubechies scaling functions D , N = 1; 2; 3; and 4.

is the “impulse function” if if if and (3) The update equations then follow from the relations (4a)

Fig. 2. Geometrical representation of the classical MRTD discretization.

(4b)

(4c)

(5b)

(4d)

upon substituting the expansions (2) into (1) and integrating against , , and . The nonzero coefficients are shown in Table I for the cases of the Daubechies scaling functions through (see Fig. 1). Explicitly, in a region of constant material properties, the resulting scheme takes on the form

(5a)

and similarly for the remaining components of the electromagnetic field. A geometrical interpretation of the staggered approximation (5) is given in Fig. 2. B. Convergence of MRTD Solutions As it follows from the decomposition (2), the standard MRTD scheme is highly accurate in the spatial description of the fields. The use of the impulse functions (3) and the staggered approximations (see Fig. 2), however, limit its accuracy to second order, much as in the classical FDTD scheme. An example of the effect of these limitations is provided in Fig. 3. There we display the results corresponding to one- and three-dimensional examples wherein a compactly supported

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Similar considerations apply to a spherical pulse propagating in three dimensions. The results in Fig. 3 correspond to the mean squared error over the support of the pulse at time . Note that the order of convergence is approximately equal to two, regardless of the specific wavelet basis chosen for the simulation.

III. RK-MRTD METHOD Fig. 3. Convergence of the classical MRTD and FDTD schemes. (a) One-dimensional example. (b) Three-dimensional example.

Fig. 4. Solution to the one-dimensional problem corresponding to the results in Fig. 3(a).

pulse propagates in free space. More precisely, in one space dimension, Maxwell’s equations reduce to [cf. (1)] (6a) (6b) where are the transverse components of the total electromagnetic field. The propagation of a pulse in free space ( in dimensionless form) can be simulated, in this context, by imposing exact radiation conditions at the boundary of the computational domain. Indeed, in one space dimension, these conditions take on the simple (local) form

Here, we present the details of our new algorithm. First, in Section III-A, we describe the wavelet expansion that we use for the spatial discretization. In Section III-B, we recall the convergence properties of these expansions and we motivate the need to incorporate a higher order time integrator. One such scheme is presented in Section III-C, and numerical results that confirm the attainment of an overall order of convergence that respects the spatial resolution are shown in Section III-D. Moreover, the stability and dispersive characteristics of the methods are discussed in Section III-E and, finally, their computational cost is analyzed in Section III-F. In both cases, as we mentioned, we contrast these characteristics with those of standard MRTD implementations, and we show that the enhanced accuracy of the new procedures translates, in fact, in higher quality solutions and lower computational times.

A. Spatial Discretization The expansion of the fields that we shall use is largely as that shown in (2), except for the use of impulse functions in time. Instead, we let the actual wavelet coefficients be functions of this variable, which are to be determined. More precisely, we let

(11a)

(7a) (7b) The example in Fig. 3 corresponds to a smooth compactly supported pulse with an initial form

(11b)

(8a) (8b)

(11c)

(9)

(11d)

where

for otherwise

(10)

, , and (see Fig. 4). and In this case, of course, the exact solution is simply given by

(11e)

(11f)

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and derive the equations

(12a)

(12b) analogous to (5). B. Convergence in Space As we stated, the use of a spatial MRA (or simply that of appropriate scaling functions) provides the potential for accurate spatial representations, which may converge with high order. The basic reason can be traced to a rather fundamental result (see, e.g., [35]), which states that the following are equivalent properties for any integer . • Smooth functions can be approximated with error at every scale , i.e.,

for suitable for orthonormal families [cf. (4c)]). • The polynomials are linear combinations of the translates . • The first moments of the wavelet vanish for • The wavelet coefficients of a smooth function decay as

For the Daubechies scaling functions , for instance, we have [35]. As a result, the spatial representation of any (smooth) function in terms of these [cf. (11)] converges with order (see Fig. 5). This high-order convergence, however, is limited in standard MRTD implementations by the low (second) order time integration strategy and the choice (13) to satisfy the Courant–Friedrichs–Lewy (CFL) stability condition. Indeed, with this choice, the overall error is bounded by Error (14)

Fig. 5. Convergence of the projection onto the span of the scaling functions at . The order of convergence of the projection is two, three, and varying scales four when using , , and , respectively.

1x D D

D

(1x)

Fig. 6. Convergence of the MRTD approach for time steps of order and . Note that the order of convergence for the MRTD results on the left and right panels are approximately four and six, respectively.

(1x)

and, thus, limited to second order. This estimate is confirmed in Fig. 6, where we further show that the higher spatial order can be recovered in the overall solution by simply setting

with . More precisely, in this figure, we show that fourth and sixth order can be attained in space time using, for instance, the Battle–Lemarie scaling functions and time steps proportional to and , respectively. C. Higher Order Time Integration The system of ordinary differential equations (12) (and its version for the remaining components of the field) can be symbolically written as

CAO et al.: HIGH-ORDER RUNGE–KUTTA MRTD METHODS FOR COMPUTATIONAL ELECTROMAGNETICS

If sources are included, then the system can be put into the form

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TABLE II CONVERGENCE OF RK-MRTD FOR DIFFERENT VALUES OF

m AND N

(15) where

and

To achieve the same order of convergence as that in Fig. 6 without resorting to unduly small time steps, we propose to discretize the system (15) with an th-order stage strong stability preserving Runge–Kutta (SSP-RK) method with low storage requirements. The scheme, originally introduced in [33] for homogeneous systems and extended to systems such as (15) in [34], has the form

(16) where

and denotes the identity operator. The coefficients given by [33], [34]

are

D. Convergence in Space Time Here, we present results that confirm the expected rate of convergence of the RK-MRTD approach. To this end, we evaluate the errors incurred by the approximation of the problem described in (6)–(10). The expansion coefficients and for the initial data in (8) are computed by projecting onto the corresponding set of scaling functions ( , ), i.e.,

These errors and respective estimates of convergence orders are presented in Table II. The results demonstrate that a uniform overall order can be attained with the use of the proposed scheme. This, of course, results from the easily derivable estimate error (17) which, for reasons of stability, holds provided (13) is satisfied for a suitable CFL constant (see Section III-E). We also note that these results, together with (13), demonstrate that high-order convergence can be achieved with time steps whose size is just linearly proportional to that of the spatial discretization (compare with Fig. 6). Moreover, the last subtable provides further evidence that a refinement in space (e.g., using the fourth-order scaling function ) does not translate into significant gains if not accompanied by a commensurate treatment of the time variable. In fact, as it follows from (17), if (13) holds, then the overall order of the algorithm can be summarized as

In particular, if

, then error

(see Section III-B), and the error is measured in the mean square error

(18)

E. Dispersion and Stability As we have argued, the use of RK-MRTD schemes can be extremely beneficial from an accuracy standpoint. This, of course, translates, in particular, into a more accurate satisfaction of the

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series of the fields needs to be stored in both cases. If, however, only the fields at a final time are of interest, the memory needs of the new schemes are twice that of the standard MRTD. Indeed, while for the latter only the values of and [cf. (5)] need to be stored at each time step, the RK-MRTD procedure demands that both and [cf. (16)] be stored as the latter is evaluated. The issue of computational cost, on the other hand, is a bit more subtle. For a single time step, the cost (number of operations) of standard MRTD is [cf. (5)] Cost Per Time Step Number of grid points

Size of stencil (19)

Fig. 7. Long-time evolution of the pulse in Fig. 4 using the RK-MRTD scheme with N = m = 2; 3; 4. For reference, the solid line depicts the exact solution.

if the scaling function is used. Here is the number of dimensions (1, 2, or 3) and denotes the number of points in a single direction, i.e.,

(20)

TABLE III CFL NUMBERS FOR RK-MRTD

denotes the grid spacing. for a domain of unit size, where The RK-MRTD scheme, on the other hand, demands that, for each time step, we perform stages in which we must: 1) evaluate and 2) add the result . Clearly, 1) costs operations, while 2) involves , where exact dispersion relation by the corresponding numerical solutions (see, e.g., [36]). An example of this is presented in Fig. 7 where we have plotted the result of a long-time propagation experiment for the pulse in Fig. 4. While almost no dissipation is visible in any of the implementations, the effects of dispersion are rather pronounced for . These effects get mollified when we set ; when , the approximate solution is almost indistinguishable from the exact solution. However, these significant accuracy gains are obtained at a cost of a more stringent stability requirement [37]. Indeed, the result of some simple numerical experiments demonstrate that, for the RK-MRTD methods introduced here, the maximal time step for which the method remains stable is inversely proportional to (see Table III). Within a typical simulation, these requirements, of course, translate into the need to effect a larger number of time steps. As we explain below, however, these additional demands are largely compensated for by the higher space–time accuracy, resulting in an overall computational effort to attain any given accuracy that can be substantially lower than that required by standard MRTD implementations. F. Computational Cost and Memory Requirements Here, we discuss the computational cost and memory requirements of our new numerical procedures, and we contrast them with those of classical MRTD. The issue of memory, in fact, is rather simple: if the values of the fields are needed for a period of time, then the memory requirements are the same, as the time

(21) is the grid spacing for the order- method. As a result, and a single time step of the RK-MRTD procedure calls for Cost Per Time Step In particular, if

(cf. Table II), we have

Cost Per Time Step

(22)

On the other hand, as described in Section III-E, if denotes the maximal stable time step for MRTD, the time step for RK-MRTD must be chosen as

Collecting these results, we obtain Cost to time

(23)

and Cost to time (24) (25)

CAO et al.: HIGH-ORDER RUNGE–KUTTA MRTD METHODS FOR COMPUTATIONAL ELECTROMAGNETICS

Fig. 8. Log–log plot of the errors from Table II, and best fits to lines with slopes given by the order of the method.

In order to compare the efficiency of these alternative schemes, we must now choose appropriate values of and . It is here then where the higher order convergence properties of RK-MRTD manifest themselves as, for any given accuracy, a larger grid spacing can be taken when compared to that required by standard MRTD. Indeed, the estimates (14) and (18) suggest that similar accuracies between standard (second-order accurate) MRTD and higher order RK-MRTD evaluations should be attainable provided

(26) for some constants , depending only on the dimension and on the order of approximation (i.e., independent of the grid spacing). Using (20), (21), and (26), we conclude that comparable errors can be obtained provided

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Fig. 9. (a) Log–log plot of the size of the spatial grid as a function of the desired accuracy (“L Error” from Fig. 8). (b) Values of the quotients x = x and x = x as functions of the desired accuracy, as derived from the left plot.

1 (1 )

1 (1 )

the data that have slopes prescribed by the order of convergence . These plots clearly convey the effect of higher order convergence and, moreover, the linear fits allow us to extrapolate to a plot of the necessary grid spacing to attain any given accuracy, by simply interchanging the axes; see Fig. 9(a). From this, in turn, the constants in (26) can be estimated by equating the error in standard MRTD to that of and evaluating the quotients in the left-hand side of (26). The results of this exercise are displayed in Fig. 9(b), which shows that, for ,

(28) In these cases, (27) then translates to if

From (24), we then get

and Cost for a given accuracy if Letting , we obtain that the cost of RK-MRTD is less than that of standard MRTD provided if if

Cost for the same accuracy provided

Similarly, if standard MRTD if

, (27) implies that the cost is half that of

(27) The key feature of (27) is, of course, that the first ratio on the left-hand side can be made arbitrarily small as the number of points in a single direction is increased. Moreover, this ratio is raised to a power that increases with increasing dimension, providing a further mechanism for the reduction of the magnitude of the left-hand side. To provide a numerical example, we consider the most unfavorable case of the one-dimensional example of Table II. In Fig. 8, we have plotted the results in this table in logarithmic scales, together with the best linear fits to

if if

IV. PARALLEL IMPLEMENTATION AND RESULTS Here, we present performance results for a parallel implementation of the RK-MRTD formulation. The details of the implementation are first reviewed in Section IV-A. In Section IV-B, we then present some speed-up results that highlight the almost optimal scalability of the algorithm.

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10 m

30 m 10 m, is illuminated by a Gaussian pulse at , , and with increasing center frequencies from 5 to 10 MHz. A perfectly matched layer (PML) is used to truncate the computational domain, as shown in Fig. 11. Fig. 12 shows a parallel speed-up, which is, again, close to ideal as the number of degrees of freedom is increased. V. CONCLUSIONS Fig. 10. Parallel speedup for a one-dimensional example of application of the RK-MRTD formulation.

Fig. 11. Parallel speedup for a three-dimensional example (see Fig. 12) of application of the RK-MRTD formulation.

Fig. 12. Problem description for the parallel performance study of the RK-MRTD formulation, and schematic of subdomain partitioning.

A. Parallel Implementation The implementation is based on “coarse grained parallelization” using the message passing interface (MPI). We recall that coarse grained parallelization is based on a division of the computational domain into subdomains, which are mapped onto individual processors. In general, the parallel performance of the algorithm depends on the parallel overhead, which is mainly a function of the collective communication overhead between processors and the communication between neighboring processors. Since the method is based on explicit time discretizations, the communication overhead is restricted to nearest neighbors. B. Numerical Examples The parallel performance of the implementation on an IBM-SP parallel machine is illustrated in Figs. 10 and 11. Fig. 10 corresponds to parallel implementations of the one-dimensional example in (6)–(10), showing almost optimal speed-up with decreasing grid spacing. The three-dimensional example in Fig. 11, on the other hand, corresponds to a simple scattering problem for a hexahedral PEC, as depicted in Fig. 12. The scatterer, with dimensions

In this paper, we have presented a new method for the numerical simulation of electromagnetic wave propagation that is based on the classical MRTD procedure. The basic premise relies on the observation that this scheme produces high-order accurate approximations in space, to which, for instance, its improved dispersion characteristics over the standard FDTD method can actually be attributed. As we have shown, however, the high quality of the spatial approximations in standard MRTD is severely impaired by the typical leap-frog time integrator associated with it. A possible remedy could rely on the use of much reduced time steps to bring the time-integration error in line with that of the spatial approximations (cf. Fig. 6) or on the use of space–time MRA [32]. Here, in contrast, we propose a significantly more efficient strategy that relies on the replacement of the second-order leap-frog scheme with a high-order Runge–Kutta procedure to approximate the time evolution while retaining the accurate multiresolution treatment in space. We demonstrated that the resulting method provides solutions wherein the errors from both discretizations are comparable and, thus, that it realizes the full potential of spatial multiresolution decompositions. Moreover, we have also shown that the enhanced accuracy of the new numerical schemes results in correspondingly reduced numerical dispersion, and that it also leads (beyond a relatively low threshold) to a significant decrease in computational effort to attain a prescribed accuracy. Finally, we have also demonstrated that this new approach can be easily and efficiently parallelized, and we have presented a number of fully three-dimensional calculations that illustrate its almost optimal scalability. ACKNOWLEDGMENT The computations for this project were carried out at the Minnesota Supercomputing Institute, Minneapolis, MN, whose support is gratefully acknowledged. The U.S. Government is authorized to reproduce and distribute reprints for governmental purposes notwithstanding any copyright notation thereon. The views and conclusions contained herein are those of the author and should not be interpreted as necessarily representing the official policies or endorsements, either expressed or implied, of the Air Force Office of Scientific Research, the Army Research Laboratory, or the U.S. Government. REFERENCES [1] M. Krumpholz and L. Katehi, “MRTD: New time-domain schemes based on multiresolution analysis,” IEEE Trans. Microw. Theory Tech., vol. 44, no. 4, pp. 555–571, Apr. 1996. [2] K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag., vol. AP-14, no. 3, pp. 302–307, May 1966.

CAO et al.: HIGH-ORDER RUNGE–KUTTA MRTD METHODS FOR COMPUTATIONAL ELECTROMAGNETICS

[3] R. F. Harrington, Field Computation by Moment Methods. New York: Macmillian, 1968. [4] I. Daubechies, Ten Lectures on Wavelets, ser. CBMS-NSF Lecture Notes. Philadelphia, PA: SIAM, 1992, vol. 61. [5] X. Zhu, L. Carin, and T. Dogaru, “Parallel implementation of the biorthogonal multiresolution time-domain method,” J. Opt. Soc. Amer. A, Opt. Image Sci., vol. 20, no. 5, pp. 844–855, May 2003. [6] C. D. S. K. Tomko, P. Czarnul, S.-H. Hung, R. L. Robertson, D. Chun, E. S. L. Davidson, and L. P. B. Katehi, “Multiresolution time domain modeling for large scale wireless communication problems,” in IEEE AP-S Int. Symp., Boston, MA, Jul. 2001, vol. 3, pp. 557–560. [7] E. M. Tentzeris, R. L. Robertson, J. F. Harvey, and L. P. B. Katehi, “Stability and dispersion analysis of Battle–Lemarie-based MRTD schemes,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 7, pp. 1004–1013, Jul. 1999. [8] M. Fujii and W. J. R. Hoefer, “Numerical dispersion in Haar-wavelet based MRTD scheme—Comparison between analytical and numerical results,” in Proc. 15th Annu. Rev. Progr. Appl. Comput. Electromagn., Monterey, CA, Mar. 1999, vol. 1, pp. 602–607. [9] S. Grivet-Talocia, “On the accuracy of Haar-based multiresolution time-domain schemes,” IEEE Microw. Guided Wave Lett., vol. 10, no. 10, pp. 397–399, Oct. 2000. [10] T. Dogaru and L. Carin, “Multiresolution time-domain using CDF biorthogonal wavelets,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 5, pp. 902–912, May 2001. [11] C. D. Sarris and L. P. B. Katehi, “Fundamental gridding-related dispersion effects in multiresolution time-domain schemes,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 12, pp. 2248–2257, Dec. 2001. [12] T. Dogaru and L. Carin, “Scattering analysis by the multiresolution time-domain method using compactly supported wavelet systems,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 7, pp. 1752–1760, Jul. 2002. [13] Y. A. Hussein and S. M. El-Ghazaly, “Extending multiresolution timedomain (MRTD) technique to the simulation of high-frequency active devices,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 7, pp. 1842–1851, Jul. 2003. [14] E. M. Tentzeris, R. L. Robertson, and L. P. B. Katehi, “PML implementation for the Battle–Lemarie multiresolution time-domain schemes,” in Proc. 14th Annu. Rev. Progr. Appl. Comput. Electromagn., Monterey, CA, Mar. 1998, vol. 2, pp. 647–654. [15] Y. W. Cheong, Y. M. Lee, K. H. Ra, J. G. Kang, and C. C. Shin, “Wavelet-Galerkin scheme of time-dependent inhomogeneous electromagnetic problems,” IEEE Microw. Guided Wave Lett., vol. 9, no. 8, pp. 297–299, Aug. 1999. [16] N. Kovvali, W. Lin, and L. Carin, “Image technique for multiresolution time-domain using nonsymmetric basis functions,” Microw. Opt. Technol. Lett., vol. 47, no. 1, pp. 44–47, Oct. 2005. [17] Z. Chen and J. Zhang, “Efficient eigen-based spatial-MRTD method for computing resonant structures,” IEEE Microw. Guided Wave Lett., vol. 9, no. 9, pp. 333–335, Sep. 1999. [18] T. Dogaru and L. Carin, “Time-domain sensing of targets buried under a Gaussian, exponential, or fractal rough interface,” IEEE Trans. Geosci. Remote Sens., vol. 39, no. 8, pp. 1807–1819, Sep. 2001. [19] N. Bushyager, E. Dalton, and E. M. Tentzeris, “Modelling of complex RF/wireless structures using computationally optimized time-domain techniques,” Int. J. Numer. Model., vol. 17, no. 3, pp. 223–236, May/ Jun. 2004. [20] X. Zhu and L. Carin, “Application of the biorthogonal multiresolution time-domain method to the analysis of elastic-wave interactions with buried targets,” IEEE Trans. Geosci. Remote Sens., vol. 42, no. 7, pp. 1502–1511, Jul. 2004. [21] Y. Chen, Q. Cao, and R. Mittra, Multiresolution Time Domain Scheme for Electromagnetic Engineering, ser. Microw. Opt. Eng. Hoboken, NJ: Wiley, 2005, vol. 1. [22] J. Zhang and Z. Chen, “Low-dispersive super high-order FDTD schemes,” in Proc. IEEE AP-S Int. Symp., Salt Lake City, UT, Jul. 2000, vol. 3, pp. 1510–1513. [23] D. W. Zingg, “Comparison of high-accuracy finite-difference methods for linear wave propagation,” SIAM J. Sci. Comput., vol. 22, no. 2, pp. 476–502, 2000. [24] M. D. White and M. R. Visbal, “Implicit high-order generalized coordinate solution of Maxwell’s equations,” in Proc. IEEE AP-S Int. Symp., San Antonio, TX, Jun. 2002, vol. 3, pp. 256–259. [25] S. Zhao and G. W. Wei, “High-order FDTD methods via derivative matching for Maxwell’s equations with material interfaces,” J. Comput. Phys., vol. 200, no. 1, pp. 60–103, Oct. 2004.

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[26] S. E. Sherer and M. R. Visbal, “Time-domain scattering simulations using a high-order overset-grid approach,” in Proc. Comput. Electromagn. in Time Domain, Sep. 2005, pp. 44–47. [27] K. L. Shlager and J. B. Schneider, “Comparison of the dispersion properties of higher order FDTD schemes and equivalent-sized MRTD schemes,” IEEE Trans. Antennas Propag., vol. 52, no. 4, pp. 1095–1104, Apr. 2004. [28] E. M. Tentzeris, R. L. Robertson, L. P. B. Katehi, and A. Cangellaris, “Space- and time-adaptive gridding using MRTD technique,” in IEEE MTT-S Int. Microw. Symp. Dig., Denver, CO, Jun. 1997, vol. 1, pp. 337–340. [29] E. M. Tentzeris, A. Cangellaris, L. P. B. Katehi, and J. Harvey, “Multiresolution time-domain (MRTD) adaptive schemes using arbitrary resolutions of wavelets,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 2, pp. 501–516, Feb. 2002. [30] N. Bushyager and E. M. Tentzeris, “Haar-MRTD time and space adaptive grid techniques for practical RF structures,” in IEEE MTT-S Int. Microw. Symp. Dig., Long Beach, CA, Jun. 2005, pp. 1123–1126. [31] F. Reitich, “Signature modeling for advanced hardware assessment and design,” Bull. Army High Performance Comput. Res. Center, vol. 15, no. 1, pp. 15–18, 2005. [32] C. D. Sarris, “New concepts for the Multiresolution Time Domain (MRTD) analysis of microwave structures,” in Proc. 34th Eur. Microw. Conf., London, U.K., Oct. 2004, vol. 2, pp. 881–884. [33] S. Gottlieb, C.-W. Shu, and E. Tadmor, “Strong stability-preserving high-order time discretization methods,” SIAM Rev., vol. 43, no. 1, pp. 89–112, 2001. [34] M.-H. Chen, B. Cockburn, and F. Reitich, “High-order RKDG methods for computational electromagnetics,” J. Sci. Comput., vol. 22/23, no. 1–3, pp. 205–226, Jun. 2005. [35] G. Strang, “Wavelets and dilation equations: A brief introduction,” SIAM Rev., vol. 31, no. 4, pp. 614–627, Dec. 1989. [36] B. Cockburn, “Discontinuous Galerkin methods for convection-dominated problems,” in High-Order Methods for Computational Physics, ser. Lecture Notes in Comput. Sci. Eng., T. Barth and H. Deconink, Eds. Berlin, Germany: Springer-Verlag, 1999, vol. 9, pp. 69–224. [37] B. Cockburn and C.-W. Shu, “Runge–Kutta discontinuous Galerkin methods for convection-dominated problems,” J. Sci. Comput., vol. 16, no. 3, pp. 173–261, Sep. 2001.

Qunsheng Cao received the Ph.D. degree in electrical engineering from The Hong Kong Polytechnic University, Hong Kong, in 2001. From 2001 to 2005, he was a Research Associate with the Department of Electrical Engineering, University of Illinois at Urbana-Champaign, and with the Army High Performance Computing Research Center (AHPCRC), University of Minnesota. In 2005, he joined the University of Aeronautics and Astronautics (NUAA), Nanjing, China, as a Professor of electrical engineering. He has authored or coauthored over 30 papers in refereed journals and conference proceedings. He coauthored Multiresolution Time Domain Scheme for Electromagnetic Engineering (Wiley, 2005, Microw. Opt. Eng. ser., vol. 1). His current research interests are in computational electromagnetics, and particularly in time-domain numerical techniques [FDTD, MRTD, and time-domain finite-element method (TDFEM)] for the study of microwave devices and scattering applications.

Ramdev Kanapady received the Ph.D. degree in mechanical engineering from the University of Minnesota, Twin Cities, in 2001. From 2001 to 2003, he continued his tenure as a Research Associate with the Department of Mechanical Engineering, University of Minnesota. From 2003 to 2005, he was Research Scientist and Technology Transfer Scientist with the Army High Performance Computing Research Center (AHPCRC), and continued with the Affiliate Graduate Faculty, Department of Mechanical Engineering, University of Minnesota, Minneapolis. His research interests are in the areas of computational mechanics and high-performance computing in multidisciplinary/interdisciplinary fields.

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Fernando Reitich received the Ph.D. degree in mathematics from the University of Minnesota, Twin Cities, in 1991. From 1991 to 1994, he was a Zeev Nehari Assistant Professor with the Department of Mathematics, Carnegie–Mellon University, Pittsburgh, PA. From 1994 to 1997, he was an Assistant and then Associate Professor with North Carolina State University. In 1997, he joined the School of Mathematics, University of Minnesota, Minneapolis, where he is currently a Professor. He is currently the Associate Director of

the Minnesota Center for Industrial Mathematics (MCIM), and he also serves as Portfolio Coordinator for the Battlefield Environment Portfolio with the Army High Performance Computing Research Center (AHPCRC). His current research interests are broadly in the area of computational electromagnetics.

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Design of a Dual-Band Bandpass Filter With Low-Temperature Co-Fired Ceramic Technology Ching-Wen Tang, Member, IEEE, Sheng-Fu You, and I-Chung Liu

Abstract—A novel dual-band bandpass filter with low-temperature co-fired ceramic technology is proposed in this paper. By adopting the structure of an asymmetrical short-circuit coupled line, two open-stub lines, and two reactances, the transmission zeros, which increase the degree of isolation between the passbands, will easily appear at the higher side of two passband’s skirts. The analysis of theorem and the procedure of design are described. The measurements of fabricated units match well with the electromagnetic simulation results, which is an evidence of the feasibility of the proposed filter configuration.

Fig. 1. Two types of short-circuit coupled lines. (a) Asymmetrical feeding. (b) Symmetrical feeding.

Index Terms—Coupled line, dual-band bandpass filter, lowtemperature co-fired ceramic (LTCC), open-stub line.

I. INTRODUCTION ROADER AND multiple bandwidths are essential in modern wireless communication systems. Recently, many studies on the dual-band transceivers with the functionality of switching between two different bands have been conducted. Wu and Razavi [1] and Tham et al. [2] used the parallel architecture in mobile phones. Such parallel architecture can integrate both receivers of wideband code division multiple access (WCDMA) and global system for mobile communication (GSM) [3] into a single path. A newly concurrent dual-band receiver architecture capable of simultaneous operating at two different frequencies and dissipating power less than twofold, but significantly increases both of cost and footprint, was introduced [4]. The wireless communication systems can create the dual-band operation on the RF front-end. Such dual-band devices include the antenna, filter, low-noise amplifier (LNA), and power amplifier (PA) [5], [6]. Dual-band bandpass filters play the most important roles in the above circuits. They can be classified into several categories such as the wideband Zolotarev filter [7], the two different filters set in parallel [8], the frequency-selective resonator filter [9]–[11], the stepped-impedance resonator (SIR) filter [12], [13], the multiple capacitively loaded coupled-line filter [14], and the coupled resonator pairs [15].

B

Manuscript received November 25, 2005; revised May 5, 2006. This work was supported in part by the National Science Council, Taiwan, R.O.C., under Grant NSC 94-2213-E-194-028. C.-W. Tang is with the Department of Communication Engineering, Department of Electrical Engineering, Center for Telecommunication Research, National Chung Cheng University, Chiayi 621, Taiwan, R.O.C. (e-mail: [email protected]). S.-F. You was with the Department of Electrical Engineering, National Chung Cheng University, Chiayi 621, Taiwan, R.O.C. He is now with the Arima Communications Corporation, Taipei 242, Taiwan, R.O.C. I-C. Liu is with the Department of Electrical Engineering, National Chung Cheng University, Chiayi 621, Taiwan, R.O.C. Digital Object Identifier 10.1109/TMTT.2006.879174

Fig. 2. Simulated responses of two types of short-circuit coupled lines and an open-stub line.

In this paper, a new method to design a dual-band bandpass filter has been proposed. We first use the asymmetrical short-circuit coupled line, as shown in Fig. 1(a), to construct a broadband bandpass filter, and then incorporate some open-stub lines into this broadband filter with the notch effect. This way will divide a broadband bandwidth into two separate passbands. With asymmetrical feeding, the short-circuit coupled line proposed in this paper is more appropriate for the design of dual-band filter. Fig. 2 shows that the transmission zero of the asymmetrical short-circuit coupled line appears at two times the normalized frequency when the electric length is 90 . On the contrary, the transmission zero of the short-circuit coupled line with the symmetrical feeding, which is shown as Fig. 1(b) and proposed for the design of the narrowband bandpass filter [16], shows up at the normalized frequency when the electric length is 90 . Fig. 2 also indicates that the transmission

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Fig. 3. Structure of proposed dual-band bandpass filter.

zero of the open-stub line appears at the normalized frequency when the electric length is 90 . Moreover, by using the open-stub lines, the notch effect can make circuit size more compact, and also generate two transmission zeros between two separated passbands to greatly enhance the degree of isolation, which will reduce the image signal in the receiver link. Although the higher order method, adding more sections of the asymmetrical short-circuit coupled lines, can also increase the degree of isolation, it will result in larger circuit size. In Section II, a theory required for developing dualband filters is presented. The design and measured results are provided in Section III. Section IV will describe the influence of changing the electric lengths. The conclusions are given in Section V.

Fig. 4. Transformed circuit of asymmetrical short-circuit coupled-line section. (a) Equivalent circuit. (b) Impedance form. (c) Admittance form.

II. DESIGN THEORY Fig. 3 shows the architecture of the proposed dual-band bandpass filter. The main structure of this dual-band bandpass filter is composed of an asymmetrical short-circuit coupled line, two open-stub lines, and two reactances ( and ). Each line of the coupled line is connected to one end of the opposite direction to ground. Both the open-stub line and reactance are shunted to each side of the input and output ports. A. Coupled Transmission Line The equivalent circuit of the asymmetrical short-circuit coupled line, as shown in Fig. 4(a), is provided by [17]. Using the impedance inverter, such as in Fig. 4(b), this equivalent circuit can be transferred more generally. In order to adopt the technique of the immittance inverter provided in [16], we modify the transformed circuit with the admittance inverter, as shown in Fig. 4(c). The even- and odd-mode line admittances and can then be derived as

Fig. 5. Equivalent circuit of the proposed dual-band bandpass filter.

can be shown as in Fig. 5. Thus, we can apply the immittance inverter [16]–[18] to analyze and design the dual-band bandpass filter. Compared to the generalized bandpass filter, the parameters of susceptances and susceptance slopes can be obtained from

(1)

(3)

(2)

(4)

where is the corresponding electric length of the asymmetrical short-circuit coupled line.

(5)

B. Design Procedure Substituting the transformed circuit of the asymmetrical short-circuit coupled-line section, as shown in Fig. 4(c), the equivalent circuit of the proposed dual-band bandpass filter

(6) (7) (8)

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where ’s are the element values of the prototype of the lowpass filter, is the fractional bandwidth, and and are the impedances of source and load transmission lines, respectively [17]. ’s (where or ) are resonated at the central frequencies, which will result in . The reactances are then derived as in (9) and (10). When the reactances have positive values, they can be substituted with inductors; on the contrary, the capacitors are adopted for the negative values Fig. 6. Structure of 2.4/5.2-GHz dual-band bandpass filter.

(9) (10) In general, the left-hand sides of both (9) and (10) are very small so that

In the proposed dual-band bandpass filter, the central frequencies of two passbands are correlated to the electric length of asymmetrical short-circuit coupled line. Their relation can be expressed as

(11) (18) (12) and are the central frequencies of the first and where second passbands, respectively.

C. Analysis of Transmission Zeros In order to increase the isolation between two passbands, our proposed dual-band bandpass filter can generate three transmission zeros. First, the asymmetrical short-circuit coupled line can generate the transmission zero at the electric length of 180 . Second, the transmission zeros of two open-stub lines can appear at the electric length of 90 . In the design of the dual-band bandpass filter, two open-stub lines are shunted with both ends of the asymmetrical short-circuit coupled line, separately, to split the original passband generated by the asymmetrical shortcircuit coupled line into two parts. Moreover, tuning two reactances can also shift the frequencies of the two passbands. The frequencies of three transmission zeros can be represented as (13) (14) (15) where is the resonance frequency of the asymmetrical shortcircuit coupled line, and and are the resonance frequencies of the two open-stub lines. For simplicity, we assume that the admittances of and are equal. It will result in . Thus,

(16) We can also get the result

(17)

III. FILTER DESIGN The procedures of developing the multilayered dual-band bandpass filter are as follows. First, utilizing the derived equations (1)–(12) to obtain the circuit parameters. Secondly, translating these parameters into the circuit models with a circuit simulator such as ADS or other software. Thirdly, converting these circuit models into a three-dimensional (3-D) structure with the assistant of the full-wave electromagnetic (EM) simulator Sonnet from Sonnet Software, North Syracuse, NY. A. Circuit Model The 2.4/5.2-GHz dual-band bandpass filter is used as a design example. The two frequencies of 2.45 and 5.25 GHz are the central frequencies of the two passbands, and their fractional bandwidths are the same as 10%. In the initial design, we choose the electric lengths and of two open-stub lines that are equal to , and set the ripple and impedance of the asymmetrical short-circuit coupled line to be 0.01 dB and 15.5 , respectively, for the Chebyshev low-pass prototype filter [17]. The electric length , the impedances and of the two open-stub lines, and even- and odd-mode impedances and of the asymmetrical short-circuit coupled line can be obtained from (1)–(12) as 57.27 and 39.7, 39.7, 21.34, and 12.17 , respectively. Since the two reactances and are negative, we can substitute them with the capacitors shown in Fig. 6. Two calculated capacitances and are found to be the same, i.e., 0.148 pF. These calculated parameters are substituted into the circuit simulator to carry out the circuit simulation, and the simulated responses of the dual-band bandpass filter are shown in Fig. 7 (dotted line). The transmission zeros are located at 3.85- and 7.7-GHz frequencies. If three electric lengths ( and )

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Fig. 7. Theoretical responses of 2.4/5.2-GHz dual-band bandpass filter.

and two capacitances ( and ) are modified as 55 , 59.27 and 55.27 and 0.06 and 0.29 pF, respectively, their responses can be as shown in Fig. 7 (solid line). The frequencies of the three transmission zeros can then also be calculated using (13)–(15) as 8.02, 3.72, and 3.99 GHz. B. EM Simulation and Measurement The modularized concept [19] that each part of a circuit can be designed independently is adopted for the design of the multilayered dual-band bandpass filter. The proposed modularized circuit with vertical allocation can reduce the mutual coupling effects and increase the fabricated yield rate as well. We utilize the physical model of the circuit simulator to obtain the initial dimension of the coupled line, while it takes too much time to employ the full-wave EM simulator to design the whole 3-D structure. Subsequently, the EM simulator is used to obtain the precise dimension of the rest of the proposed structure. These circuits are composed of two pairs of the open-stub line and the shunt-connected capacitor, which are placed on the upper and lower layers separately to fine tune the return losses of two passbands. The multilayered dual-band bandpass filters have been fabricated with low-temperature co-fired ceramic (LTCC) technology [20]–[24] on a substrate of Dupont 951. Its dielectric constant and loss tangent are 7.8 and 0.0045, respectively. This LTCC dual-band bandpass filter is designed on four upper layers with the 1.57-mil sheet, six middle layers with the 3.6-mil sheet, four following layers with the 1.57-mil sheet, and two lowest layers with the 3.6-mil sheet. Fig. 8(a) and (b) shows the 3-D architecture and photograph of the modified dual-band bandpass filter with the overall circuit size of 120 mil 100 mil 41.4 mil. Three electric lengths and are fabricated with different lengths, as indicated in Fig. 7. Two capacitors and

Fig. 8. Fabricated 2.4/5.2-GHz dual-band bandpass filter. (a) 3-D structure. (b) Photograph. (c) EM simulated and measured responses.

are inserted at the ends of the input and output ports, as shown in Fig. 8(a), which can also be used to fine tune the responses

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spectively. The on-wafer tester has been employed in order to improve the accuracy of measurement. The network analyzer, Agilent N5230A PNA L, is used for measurement, whereas the short-open-load-thru (SOLT) is adopted for calibration. The measured and EM simulated results are shown in Fig. 8(c). For the first passband (between 2.3–2.7 GHz), the measured insertion loss is less than 1.4 dB, and the return loss is greater than 14 dB. In the second passband (between 5–5.55 GHz), the measured insertion loss is less than 1.8 dB, and the return losses are greater than 14 dB. The transmission zero frequencies are located at 3.74, 3.96, and 8.14 GHz for simulation, and at 3.73, 4.05, and 8.02 GHz for measurement. As shown in Fig. 8(c), the measured results match well with the EM simulation results. IV. EFFECT OF TUNING THE ELECTRIC LENGTHS Here, we still use the frequency band of 2.4/5.2 GHz as a design example to discuss the factor that may influence the dualband bandpass filter. The electric length and even- and oddmode impedances and of the asymmetrical short-circuit coupled line are chosen as 54 and 18.3 and 10.8 , respectively, for the resonance frequency at 2.4 GHz. The impedances and and electric lengths and of the two open-stub lines are also selected to be 31.4 and 31.8 and 61 and 52.5 , respectively, for the resonance frequency at 2.4 GHz, and two capacitances and are 0.1 and 1.2 pF, respectively. According to (13)–(15), the frequencies of the three transmission zeros are derived as 3.54, 4.11, and 8 GHz, as indicated in Fig. 9(c) with the dotted line. In order to further describe the effect of unequal electric lengths of and , two capacitors and are placed at different layers with varying thicknesses, as shown in Fig. 9(a). This filter is also designed with the substrate of Dupont 951, and with the size of 132 mil 100 mil 41.4 mil, as shown in Fig. 9(b). Here, the smaller capacitor (such as ) is connected to the longer electric length (open-stub line), and the larger capacitor (such as ) is connected to the shorter electric length (open-stub line). As shown in Fig. 9(c), the measured frequencies of transmission zeros are located at 3.55, 4.2, and 8.09 GHz, respectively. Other measured results such as the insertion and return losses for the first passband (between 2.3–2.64 GHz) are less than 1.5 dB and greater than 13 dB, respectively. In the second passband (between 5.05–5.51 GHz), the measured insertion loss is less than 2.1 dB and the return loss is greater than 13 dB. V. CONCLUSION

Fig. 9. Developed 2.4/5.2-GHz dual-band bandpass filter. (a) 3-D structure. (b) Photograph. (c) Simulated and measured responses.

of dual passbands. The simulated responses demonstrate that this filter has two passbands, centered at 2.45 and 5.25 GHz, re-

The newly proposed dual-band bandpass filter has been disclosed in this paper. The dual-band filter with the capability of high integration and small size is very suitable for implementation in the multichip module. The approaches that use the structure of an asymmetrical short-circuit coupled line, two openstub lines, and two reactances can easily generate the transmission zeros on the higher side of the two passband’s skirts. The split frequencies in the region of higher side of the lower passband’s skirt can increase the isolation between the passbands. Two reactances and connected to the input and output ports are used to fine tune the responses of the passbands.

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The LTCC dual-band bandpass filter was designed and measured at the two frequency bands of 2.4 and 5.2 GHz. In the realization of the physical 3-D circuit, the parasitic effect among capacitors may cause their model to be slightly different from the ideal values provided by the circuit simulator. The agreement between experimental and theoretical results shows the feasibility of the proposed filter. ACKNOWLEDGMENT The authors would like to thanks F.-L. Lin, Indiana University, Bloomington, and the reviewers of this paper’s manuscript for their helpful comments. REFERENCES [1] S. Wu and B. Razavi, “A 900-MHz/1.8-GHz CMOS receiver for dualband applications,” IEEE J. Solid-State Circuits, vol. 33, no. 12, pp. 2178–2185, Dec. 1998. [2] J. L. Tham, M. A. Margarit, B. Pregardier, C. D. Hull, R. Magoon, and F. Carr, “A 2.7-V 900-MHz/1.9-GHz dual-band transceiver IC for digital wireless communication,” IEEE J. Solid-State Circuits, vol. 34, no. 3, pp. 282–291, Dec. 1999. [3] J. Ryynanen, K. Kivekas, J. Jussila, A. Parssinen, and K. A. I. Halonen, “A dual-band RF front-end for WCDMA and GSM applications,” IEEE J. Solid-State Circuits, vol. 36, no. 8, pp. 1198–1204, Aug. 2001. [4] H. Hashemi and A. Hajimiri, “Concurrent dual-band CMOS low noise amplifiers and receiver architectures,” in VLSI Circuits Symp. Dig., Jun. 2001, pp. 247–250. [5] K. Kunihiro, S. Yamanouchi, T. Miyazaki, Y. Aoki, K. Ikuina, T. Ohtsuka, and H. Hida, “A diplexer-matching dual-band power amplifier LTCC module for IEEE 802.11a/b/g wireless LANs,” in IEEE Radio Freq. Integr. Circuits Symp. Dig., 2004, pp. 303–306. [6] S. F. R. Chang, W. L. Chen, S. C. Chang, C. K. Tu, C. L. Wei, C. H. Chien, C. H. Tsai, J. Chen, and A. Chen, “A dual-band RF transceiver for multistandard WLAN applications,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 3, pp. 1048–1055, Mar. 2005. [7] H. Clark Bell, “Zolotarev bandpass filters,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 12, pp. 2357–2362, Dec. 2001. [8] H. Miyake, S. Kitazawa, T. Ishizaki, T. Yamada, and Y. Nagatomi, “A miniaturized monolithic dual band filter using ceramic lamination technique for dual mode portable telephones,” in IEEE MTT-S Int. Microw. Symp. Dig., 1997, pp. 789–792. [9] C. Quendo, E. Rius, and C. Person, “An original topology of dual-band filter with transmission zeros,” in IEEE MTT-S Int. Microw. Symp. Dig., 2003, pp. 1093–1096. [10] C. H. Chang, H. S. Wu, H. J. Yang, and C. K. C. Tzuang, “Coalesced single-input single-output dual-band filter,” in IEEE MTT-S Int. Microw. Symp. Dig., 2003, pp. 511–514. [11] C. M. Tsai, H. M. Lee, and C. C. Tsai, “Planar filter design with fully controllable second passband,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 11, pp. 3429–3439, Nov. 2005. [12] S. F. Chang, Y. H. Jeng, and J. L. Chen, “Dual-band step-impedance bandpass filter for multimode wireless LANs,” Electron. Lett., vol. 40, no. 1, pp. 38–39, Jan. 2004. [13] J. T. Kuo, T. H. Yeh, and C. C. Yeh, “Design of microstrip bandpass filters with a dual-passband response,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 4, pp. 1331–1337, Apr. 2005. [14] C. W. Tang and S. F. You, “Design the duplexer and dual-band filter with multiple capacitively loaded coupled lines,” in Int. Antennas, Radar, Wave Propag. Conf., Banff, AB, Canada, Jul. 8–10, 2004, pp. 161–165. [15] C. C. Chen, “Dual-band bandpass filter using coupled resonator pairs,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 4, pp. 259–261, Apr. 2005. [16] C. W. Tang and S. F. You, “Design methodologies of LTCC bandpass filters, diplexer and triplexer with transmission zeros,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 2, pp. 717–723, Feb. 2006.

[17] G. L. Matthaei, L. Young, and E. M. Jones, Microwave Filters, Impedance-Matching Network, and Coupling Structures. Norwood, MA: Artech House, 1980. [18] J. S. G. Hong and M. J. Lancaster, Microstrip Filters for RF/Microwave Applications. New York, NY: Wiley, 2001. [19] C. W. Tang, “Harmonic-suppression LTCC filter with the step impedance quarter-wavelength open stub,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 2, pp. 617–624, Feb. 2004. [20] J. Muller and H. Thust, “3D-integration of passive RF-components in LTCC,” in Pan Pacific Microelectron. Symp. Dig., 1997, pp. 211–216. [21] C. Q. Scrantom and J. C. Lawson, “LTCC technology: Where we are and where we’re going—II,” in IEEE MTT-S Int. Microw. Symp. Dig., 1999, pp. 193–200. [22] Y. Rong, K. A. Zaki, M. Hageman, D. Stevens, and J. Gipprich, “Low temperature cofired ceramic (LTCC) ridge waveguide bandpass filters,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 1999, pp. 1147–1150. [23] D. Heo, A. Sutono, E. Chen, Y. Suh, and J. Laskar, “A 1.9 GHz DECT CMOS power amplifier with fully integrated multilayer LTCC passives,” IEEE Microw. Wireless Compon. Lett., vol. 11, pp. 249–251, Jun. 2001. [24] W. Y. Leung, K. K. M. Cheng, and K. L. Wu, “Design and implementation of LTCC filters with enhanced stop-band characteristics for Bluetooth applications,” in Proc. Asia–Pacific Microw. Conf., Dec. 2001, pp. 1008–1011. Ching-Wen Tang (S’02–M’03) received the B.S. degree in electronic engineering from Chung Yuan Christian University, Chungli, Taiwan, R.O.C., in 1991, and the M.S. and Ph.D. degree in communication engineering from National Chiao Tung University, Hsinchu, Taiwan, R.O.C., in 1996 and 2002, respectively. In 1997, he joined the RF Communication Systems Technology Department, Computer and Communication Laboratories, Industrial Technology Research Institute (ITRI), Hsinchu, Taiwan, R.O.C., as an RF Engineer, where he developed LTCC multilayer-circuit (MLC) RF components. In 2001, he joined Phycomp Taiwan Ltd., Kaohsiung, Taiwan, R.O.C., as a Project Manager, where he continues to develop LTCC components and modules. Since February 2003, he has been with the Department of Communication Engineering and Department of Electrical Engineering, Center for Telecommunication Research, National Chung Cheng University, Chiayi, Taiwan, R.O.C., where he is currently an Assistant Professor. His research interests include microwave and millimeter-wave planar-type and multilayered circuit design and the analysis and design of thin-film components.

Sheng-Fu You was born in Changhua, Taiwan, R.O.C., on December 12, 1980. He received the B.S. degree in electronic engineering from Feng Chia University, Taichung, Taiwan, R.O.C., in 2003, and the M.S. degree in electrical engineering from National Chung Cheng University, Chiayi, Taiwan, R.O.C., in 2005. He is currently an RF Engineer with the Arima Communications Corporation, Taipei, Taiwan, R.O.C. His research interests include the design of microwave planar and multilayered filers and associated RF components.

I-Chung Liu was born in Keelung, Taiwan, R.O.C., on September 6, 1981. He received the B.S. degree in electrical engineering from Chung Yuan Christian University, Chungli, Taiwan, R.O.C., in 2005, and the M.S. degree in electrical engineering from National Chung Cheng University, Chiayi, Taiwan, R.O.C., in 2006. His research interests include the design of microwave passive and multilayered circuits for RF applications.

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Design of Multiple-Stopband Filters for Interference Suppression in UWB Applications K. Rambabu, Michael Yan-Wah Chia, Member, IEEE, Khee Meng Chan, and Jens Bornemann, Fellow, IEEE

Abstract—A design of multiple-stopband filters is presented for the suppression of interfering signals in UWB applications. Since possible interferers can be located at fixed frequencies or within a defined frequency band, the design of both fixed and tunable narrow stopband filter sections is addressed. For multiple fixed stopband filters, bent resonators, coupled to the main line, are introduced in order to more effectively suppress harmonics. A new tunable tapped stopband section is proposed, which allows the simultaneous control of stopband frequency and bandwidth. The final multiple-stopband design combines fixed and tunable sections and simultaneously suppresses interferences from global system for mobile communication, wireless local area network, worldwide interoperability for microwave access, and industrial–scientific–medical applications. Measurements verify the design process. Index Terms—Microstrip filters, multiband filters, stopband filters, ultra-wideband (UWB) applications.

I. INTRODUCTION LTRA-WIDEBAND (UWB) technology started in 1960 as a time-domain study of electromagnetic wave propagation [1]. It finds applications in low-probability radar and in data communications. From a radar perspective, short-pulse UWB techniques have many advantages over conventional radar approaches [2]. In communication applications, short-pulse UWB techniques provide increased immunity to multipath cancellation and improve operational security. Moreover, low pulserate UWB technologies are ideally suited for battery-operated equipment. Due to the broad definition of UWB transmission technologies, various frequency bands are in use for different applications. UWB radar for monitoring the motion of objects at short distance has been designed [3]. It operates between very low frequencies and 5 GHz and has also been applied to measure the heart and respiratory beats and other vital activities of patients. Through-wall UWB radars in the frequency range of 3.1–10.6 GHz are designed to detect people buried under debris [4]. UWB radar system designers have the option to utilize the wide frequency spectrum from very low frequencies up to 10.6 GHz. However, the power level allowed for UWB emitters in the frequency range of 1.61–10.6 GHz is lower than 41 dBm/MHz. Outside this band, emission should be reduced by another 10 dB. Measured interferences due to coexisting

U

Manuscript received December 25, 2005; revised April 20, 2006. K. Rambabu, M. Y.-W. Chia, and K. M. Chan are with the Institute for Infocomm Research, Singapore 117674. J. Bornemann is with the Department of Electrical and Computer Engineering, University of Victoria, Victoria, BC, Canada V8W 3P6. Digital Object Identifier 10.1109/TMTT.2006.877813

Fig. 1. Coupled resonator bandstop filter sections with: (a) straight and (b) bent resonators.

narrowband applications such as global system for mobile communication (GSM), wireless local area network (WLAN), worldwide interoperability for microwave access (WIMAX), and industrial–scientific–medical (ISM) are 20 dB stronger than the maximally permitted UWB emission. To operate UWB systems in the presence of such strong narrowband interferers is difficult at current power levels. In this paper, we propose multiple-stopband filters to suppress narrowband interferences. Except for ISM band interference, which varies in frequency from 5 to 6 GHz with a bandwidth of 100 MHz, all other interferences are located at fixed frequencies. Therefore, Section II presents the design of narrow stopband filters with harmonic suppression. Section III introduces a tunable resonator with varying bandwidth for narrow stopband filters. In Section IV, both circuits are integrated into a multiple-stopband filter for UWB interference suppression. II. FIXED STOPBAND SECTIONS Different topologies are available for the design of fixed bandstop filters, e.g., open-circuit quarter-wavelength stubs, short-circuit half-wavelength stubs, and coupled resonators [5]. Generally, bandstop filters are concerned only with narrow frequency bands and bandwidths up to a few percent. Microwave stopband filters can be designed based on low-pass prototypes with suitable frequency transformation [6]. Coupled resonator bandstop filters are useful when relatively narrow stopbands are required. The resonator lengths are a quarter-wavelength at the center frequency of the stopband with a short circuit at one end. Resonators are capacitively coupled to the main line and spaced a quarter-wavelength apart at the stopband’s center frequency [7]. In this paper, we use parallel-coupled resonator filters for suitable narrow stopbands. Fig. 1 shows coupled resonator bandstop filter sections with straight and bent resonators. It is well known that maximum coupling occurs between coupled transmission lines, when the coupled line length is an odd multiple of a quarter-wavelength. If the coupled line length is an even multiple of a

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Fig. 3. Principal layout of the new tunable narrow bandstop section.

at 7.32 GHz) are effectively suppressed using bent resonators (solid line). Fig. 2. Measured jS j performances of five-stage multiple-stopband filter for GSM, WLAN, WIMAX, and ISM suppression using straight (dashed line) and bent resonators (solid line).

quarter-wavelength, then energy couples from the main arm to the coupled arm in the first quarter-wavelength section and back to the main arm in the following one. Therefore, even multiple quarter-wavelength coupling sections have minimum coupling. In the configuration with straight resonators depicted in Fig. 1(a), the length amounts to a quarter-wavelength including fringing fields and introduces stopbands at all odd harmonics. If a stopband at one of the harmonics needs to be suppressed, then the above concept is used in a bent configuration according to Fig. 1(b). Within the prototype design presented in Section IV, let us consider that we want to eliminate the fifth harmonic of 1.84 GHz (9.2 GHz). A quarter-wavelength at 1.84 GHz results mm [see Fig. 1(b)] in the resonator length of including fringing capacitances on a substrate material with . To avoid the stopband at 9.2 GHz, this resonator is mm now bent such that the coupling section length (including fringing fields) is a full wavelength at 9.2 GHz. Similarly, in order to eliminate the third harmonic of 2.44 GHz (7.32 GHz), the quarter-wavelength at 2.44 GHz is mm on the same substrate (including fringing-field lengths). A half-wavelength at 7.32 GHz is mm [see Fig. 1(b)]. In order to effectively suppress GSM, WLAN, and WIMAX interference in UWB applications, the required attenuation levels are expected to be 20 dB (GSM and WLAN) and 15 dB (WIMAX). Since a single coupled-resonator stopband section will realistically produce an attenuation of approximately 16 dB, the sections to suppress GSM and WLAN have been doubled in the design presented in Section IV. Here, in Fig. 2, we compare the measured results of a design using five straight resonators—two of them each doubled at two fundamental frequencies—with those of a similar design using bent resonators. According to the above examples, note that the harmonic stopbands labeled 5R1 (fifth harmonic of 1.84 GHz at 9.2 GHz) and 3R2 (third harmonic of 2.44 GHz

III. TUNABLE STOPBAND SECTION Tunable bandpass or bandstop filters can be designed by loading the resonators (stubs) with variable capacitors. In general, for a stopband resonance to occur, an open-circuit quarterwavelength resonator or a short-circuited half-wavelength resonator can be employed. The design of a narrow bandstop filter with conventional tunable stubs is difficult due to their low quality factor and the fact that maintaining a constant bandwidth over the entire range of frequencies is not possible. In this paper, we propose a new tunable resonator (Fig. 3), which is different from conventional tapped resonators [8], [9] for the following reasons. First, the center frequency of the stopband can be tuned by tuning , and the bandwidth can be controlled through . Secondly, the resonator in Fig. 3 inherently possesses a high quality factor due to longer stub lengths required for its bandstop resonance. Thirdly, the bandwidth of this resonator is much narrower than that of the circuit in [9] due to the fact that the tuning capacitor is not located at the end of the stub. The resonator proposed here has an open-circuited stub , whose length is close to a half-wavelength, a capacitor and a very short stub . Although are close to a half-wavelength, the total electrical length of the resonator (including ) is a quarterwavelength at the center frequency of the stopband. Since the input impedance of an open-circuit quarter-wavelength resonator is zero, capacitor in parallel to the resonator will not affect the frequency of the stopband. A. Theory The resonance condition is obtained from the imaginary part of the input impedance (1) For a given center frequency of the stopband, capacitance can be calculated as (2)

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The electrical length of the resonator decides the nature of the resonance. For a stopband, the phase should be (3) where is an integer. The total node admittance of the resonator including bandwidth-controlling capacitor can be written as (4), shown at the bottom of this page, and defines the quality factor of the resonator via (5) and, thus, (6), shown at the bottom of this page, where Fig. 4. Variation of center frequency and bandwidth with of C .

and is the speed of light. From (6), it follows that the quality factor of the circuit will increase with . B. Results Fig. 4 shows the change in resonant frequency and bandwidth ( 10 dB) of the resonator with tuning capacitor in the absence of and mm, mm. The solid line without diamonds or circles shows the capacitance calculated from (2) to change the resonant frequency from 4.8 to 6.2 GHz. The line with solid diamonds shows the capacitance variation calculated from ADS software and includes the junction effects and capacitor pad effects. The line with hollow diamonds is obtained from measuring the capacitance for different resonant

C

in the absence

frequencies. It is in good agreement with the capacitance from (2) over a wide frequency range. Of course, the bandwidth (line with circles) decreases with increasing resonant frequency due to less capacitive loading. For lower resonant frequencies and fixed and , the required capacitive load is large compared to that required at higher resonant frequencies. Thus, the quality factor at lower resonant frequencies will be low. Fig. 5 displays the effect of on the resonant frequency and bandwidth for fixed quantities pF, mm, and mm. Equation (1) predicts no effect of on the resonant frequency, and (6) shows the increase in quality factor with . Both facts are essentially confirmed by ADS simulations and measurements presented in Fig. 5. The differences between simulations and measurements are attributed to two facts: first, ADS is a circuit based, not a full-wave simulation tool and, second, the capacitance added by the solder material is not considered in the simulation. It is possible to obtain a constant bandwidth throughout the tuning band by simultaneously adjusting and . For mm and mm, Fig. 6 shows the required variation of and to achieve a constant 10-dB bandwidth of 4.5% while varying the center frequency from 5 to 6 GHz. is reduced to shift the resonant frequency from 5 to 6 GHz, and is reduced from 1.1 to 0.05 pF to maintain the constant bandwidth of 4.5% throughout the band.

(4)

(6)

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Fig. 5. Effect of C on resonant frequency and bandwidth. Fig. 7. Variation of reactance of input impedance for open-circuit resonator.

Fig. 6. Variation of C and C for constant bandwidth (ADS simulation).

For practical microstrip lines, the impedance of an open circuit including fringing capacitances is in the order of a few hundred to a thousand ohms. The sequence of resonances (stopband and passband) can be estimated by calculating the imaginary part of the input impedance, as shown in Fig. 7. Stopbands and passbands occur at frequencies where the imaginary part of the input impedance vanishes. For case 1, mm, pF, mm, and pF, the first stopband occurs at 5.4 GHz, the first passband at 5.75 GHz. The second stopband and passband are located at 10.87 and 11.35 GHz, respectively. This sequence is similar for case 2 ( mm, pF, mm, pF) and case 3 ( mm, pF, mm, pF). However, it can be seen from Fig. 7 that the harmonics of the stopband can be pushed upwards in frequency by reducing and increasing . The ADS simulation in Fig. 8 (case 1 only) confirms the sequence of stopbands and passbands. It is an essential requirement for UWB applications that harmonic stopbands be pushed out of the frequency region where pulse energy is concentrated.

Fig. 8. Scattering parameters of tunable narrow bandstop section (case 1): simulated with ADS.

IV. COMBINED FIXED/TUNABLE MULTIPLE-STOPBAND FILTER In full UWB band operation, harmonics of stopbands severely distort the pulse due to excessive loss at higher frequencies. Therefore, energy loss due to harmonics should be reduced to retain the pulse’s shape. In order to achieve a design for the fixed frequency stopbands for GSM, WLAN, WIMAX (Section II), and a variable stopband for ISM interference, the tunable tapped resonator section (Section III) is integrated with fixed coupled resonator filters sections presented in Section II. The final circuit is shown in Fig. 9. Note that compactness is achieved by structural folding and by varying the positions of individual resonators. Following the microstrip line from the bottom left to top right, the bent resonator sections 1 and 3 as well as 2 and 5 are identical to satisfy the attenuation requirements in the GSM and WLAN bands (see Section II). They also suppress the undesired third harmonic of the WLAN and the fifth harmonic of

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them as follows: first, losses of the microstrip lines increase with frequency; second, the prototype circuit is fabricated using simple means for circuit etching, the creation of the via-hole for and the placement and soldering of capacitors. Therefore, it is expected that utilizing suitable fabrication techniques can reduce insertion losses at higher frequencies. V. CONCLUSION

Fig. 9. Multiple-stopband filter for GSM, WLAN, WIMAX, and tunable ISM suppression. (Color version available online at: http://ieeexplore.ieee.org.)

The multiple-stopband printed-circuit filter scheme presented in this paper offers attractive solutions for the suppression of interferences in UWB applications. Bent resonators, which are coupled to the main line, allow the designer to control the stopband location and reduce the number of harmonics. Therefore, they are superior to hitherto known straight stopband resonators. The new tunable stopband section offers high design flexibility and presents a viable option to suppress interference from varying sources. The prototype design can be used to suppress interference from GSM, WLAN, and WIMAX applications, and additionally, allows tuning out signals in varying sections of the ISM band. Practical realization and prototype measurements validate the design process. REFERENCES

Fig. 10. Measured performances of the multiple-stopband filter for GSM, WLAN, WIMAX, and tunable ISM suppression.

GSM according to Section II. The main reason for not bending the small (fourth) resonator operating at 3.5 GHz (WIMAX) is that its harmonic is at 10.3 GHz, which is outside the frequency band of interest for UWB applications. Fig. 10 displays the measured performances of the integrated filter (Fig. 9) over the entire UWB frequency range and for two different sets of capacitances. It is obvious that this prototype circuit meets requirements encountered in UWB applications. Moreover, tunability over the ISM frequency range is demonstrated through two different sets of capacitances. However, it is also observed that the insertion loss increases markedly towards higher frequencies. Although losses at frequencies higher than 9 GHz will have only a small effect on UWB pulses due to the pulse’s low energy content at those frequencies, the losses below 9 GHz are a concern. We attribute

[1] C. L. Bennet and G. F. Ross, “Time-domain electromagnetics and its applications,” Proc. IEEE, vol. 66, no. 3, pp. 299–318, Mar. 1978. [2] R. J. Fontana, “Recent system applications of short-pulse ultra-wideband (UWB) technology,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 9, pp. 2087–2104, Sep. 2004. [3] I. Y. Immoreev, S. Samkov, and T. H. Tao, “Short-distance ultrawideband radars,” IEEE Aerosp. Electron. Syst. Mag., vol. 20, pp. 9–14, Jun. 2005. [4] M. Y. W. Chia, S. W. Leong, C. K. Sim, and K. M. Chan, “Throughwall UWB radar operating within FCC’s mask for sensing heart beat and breathing rate,” in Proc. 35th Eur. Microw. Conf., Paris, France, Oct. 2005, pp. 267–270. [5] G. Matthaei, L. Young, and E. M. T. Jones, Microwave Filters, Impedance Matching Networks, and Coupling Structures. Boston, MA: Artech House, 1980, ch. 12. [6] L. Young, G. L. Matthaei, and E. M. T. Jones, “Microwave bandstop filters with narrow stopbands,” IRE Trans. Microw. Theory Tech, vol. MTT-10, no. 11, pp. 416–427, Nov. 1962. [7] B. M. Schiffman and G. L. Matthaei, “Exact design of band-stop microwave filters,” IEEE Trans. Microw. Theory Tech., vol. MTT-12, no. 1, pp. 6–15, Jan. 1964. [8] P. Rizzi, Microwave Engineering-Passive Circuits. Englewood Cliffs, NJ: Prentice-Hall, 1998. [9] J. M. Drozd and T. Joines, “A capacitively loaded half-wavelength tapped-stub resonator,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 7, pp. 1100–1104, Jul. 1997.

K. Rambabu received the Ph.D. degree from the University of Victoria, Victoria, BC, Canada, in 2004. He is currently a Research Staff Member with the Institute for Infocomm Research, Singapore. He has authored or coauthored over 40 papers published in refereed journals and conferences. He holds a patent for beam shaping of a cellular base station antenna. His research interests include design and development of miniaturized passive microwave components and antennas for various applications.

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Michael Yan-Wah Chia (M’94) was born in Singapore. He received the B.Sc. (first-class honors) and Ph.D. degrees from Loughborough University, U.K., in 1990 and 1994, respectively. In 1994, he joined the Center for Wireless Communications (CWC), Singapore, as a Member of Technical Staff (MTS), and was then promoted to Senior MTS, then Principal MTS, and finally Senior Principal MTS. He is currently a Principal Scientist and Division Director with the Communications Division, Institute for Infocomm Research, A-STAR. He holds adjunct positions with the National University of Singapore and Nanyang Technological University of Singapore. In 1999, he started fundamental work on UWB research at I2R. Since then, his team has reported UWB transmission at a data rate of 500 Mb/s in April 2003 and 1 Gb/s in June 2004 conforming to FCC’s mask. In 2002, he also led the development of a direct conversion transceiver design for wireless local area network (LAN) in collaboration with IBM. Since April 2004, his team has been invited into the IBM Business Partner Program for UWB-MBOA silicon design. He has authored or coauthored over 120 publications in international journals and conferences. He holds ten patents, some of which have been commercialized. His main research interests are UWB, beamsteering, wireless broadband, RF identification (RFID), antennas, transceivers, radio over fiber, RF integrated circuits (RFICs), amplifier linearization, and communication and radar system architecture. Dr. Chia is a member of the Radio Standards (IDA), Telecommunications Standards Advisory Committee (IDA) and Technical Advisory Member of Rhode & Schwartz Communications & Measurements (Asia). He has been an active member of organizing committees in various international conferences and was program cochair of IWAT 2005. He was a keynote speaker at the International Conference of UWB in 2005. He is general chair of ICUWB 2007. He was the recipient of the Overseas Research Student Award and Studentship from British Aerospace, U.K.

Khee Meng Chan received the B.Eng. degree in electrical and electronic engineering from the University of Queensland, Brisbane, Qld., Australia, in 1997. In 2001, he joined the Institute for Infocomm Research, Singapore, as a Research Engineer and was involved in projects developing RFID and UWB systems. He is currently a Senior Research Engineer with the Institute for Infocomm Research, where he is involved with passive and active circuits for a UWB radar project. His research interests are microwave filters, UWB systems, and circuit design.

Jens Bornemann (M’87–SM’90–F’02) received the Dipl.-Ing. and Dr.-Ing. degrees in electrical engineering from the University of Bremen, Bremen, Germany, in 1980 and 1984, respectively. From 1984 to 1985, he was a Consulting Engineer. In 1985, he joined the University of Bremen, as an Assistant Professor. Since April 1988, he has been with the Department of Electrical and Computer Engineering, University of Victoria, Victoria, BC, Canada, where he became a Professor in 1992. From 1992 to 1995, he was a Fellow of the British Columbia Advanced Systems Institute. In 1996, he was a Visiting Scientist with Spar Aerospace Limited (now the MDA Corporation), Ste-Anne-de-Bellevue, QC, Canada, and a Visiting Professor with the Microwave Department, University of Ulm, Ulm, Germany. From 1997 to 2002, he was a Co-Director of the Center for Advanced Materials and Related Technology (CAMTEC), University of Victoria. In 2003, he was a Visiting Professor with the Laboratory for Electromagnetic Fields and Microwave Electronics, Eidgenössische Technische Hochschule (ETH) Zürich, Zürich, Switzerland. He coauthored Waveguide Components for Antenna Feed Systems. Theory and Design (Artech House, 1993) and has authored/coauthored over 200 technical papers. His research activities include RF/wireless/microwave/millimeter-wave components and systems design, and problems involving electromagnetic-field theory in integrated circuits, feed networks, and radiating structures. He serves on the Editorial Advisory Board of the International Journal of Numerical Modeling. Dr. Bornemann is a Registered Professional Engineer in the Province of British Columbia, Canada. He serves on the Technical Program Committee of the IEEE Microwave Theory and Techniques Society (IEEE MTT-S) International Microwave Symposium (IMS).

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 8, AUGUST 2006

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Adjoint Higher Order Sensitivities for Fast Full-Wave Optimization of Microwave Filters Mahmoud A. El Sabbagh, Member, IEEE, Mohamed H. Bakr, Member, IEEE, and John W. Bandler, Fellow, IEEE

Abstract—For the first time, full-wave optimization exploiting adjoint Hessian matrices is applied to the design of microwave filters and transitions. The first- and second-order sensitivities of the scattering parameters are computed analytically using the adjoint network method (ANM). The mode-matching-based ANM is applied to the generalized scattering matrices of the different filter/transition components. Analytical gradient and Hessian matrices of differentiable objective functions are expressed in terms of the first- and second-order response adjoint sensitivities. Optimization techniques exploiting second-order information such as the Levenberg–Marquardt method are applied using the adjoint first- and second-order information. Significant acceleration is achieved using these techniques over gradient-based optimization techniques such as the Broyden–Fletcher–Goldfarb–Shanno method. The adjoint-based sensitivities are also exploited in efficient tolerance analysis of microwave filters. Index Terms—Adjoint networks, computer-aided design (CAD), mode-matching (MM) methods, optimization methods, sensitivity analysis.

I. INTRODUCTION HE DESIGN process of microwave filters or transitions involves imposing design specifications on the scattering parameters in certain frequency bands. An optimization algorithm is utilized to obtain a feasible optimal design. This algorithm carries out a number of iterations starting from an initial design. The optimizer invokes the electromagnetic (EM) simulator in each iteration to check the design feasibility and to obtain derivative information necessary to generate the next design. Derivative-based optimization algorithms can exploit first- or second-order derivatives [1]. Algorithms exploiting first-order derivatives when properly modified to exploit available secondorder derivative information are expected to deliver faster convergence. Second-order derivatives, however, are expensive to calculate using central difference (CD) approximations. The adjoint network method (ANM) is an efficient technique for sensitivity (derivative) estimation [2]–[8]. Using only the original simulation, the first-order sensitivities of the network functions with respect to (w.r.t.) all the design variables are ob-

T

Manuscript received January 8, 2006; revised April 20, 2006. M. A. El Sabbagh was with the Department of Electrical and Computer Engineering, McMaster University, Hamilton, ON, Canada L8S 4K1. He is now with the Faculty of Engineering, Department of Electronics and Communication Engineering, Ain Shams University, Abbassia, Cairo 11517, Egypt, and also with the Electrical and Computer Engineering Department, University of Waterloo, Waterloo, ON, Canada N2L 3G1 (e-mail: [email protected]). M. H. Bakr is with the Department of Electrical and Computer Engineering, McMaster University, Hamilton, ON, Canada L8S 4K1. J. W. Bandler is with the Department of Electrical and Computer Engineering, McMaster University, Hamilton, ON, Canada L8S 4K1, and also with Bandler Corporation, Dundas, ON, Canada L9H 5E7. Digital Object Identifier 10.1109/TMTT.2006.877814

tained. These sensitivities give an indication of how critically the designed circuit depends on the design parameters. They are essential in optimization, yield analysis, and tolerance analysis. Formulations of first- and second-order sensitivities w.r.t. network parameters in terms of wave variables based on the ANM were presented in [5]. Formulations of first- and second-order sensitivities of voltage in terms of circuit parameters were presented in several papers, e.g., [6]–[9]. In [7], third-order sensitivity of voltage in terms of circuit admittance matrix was obtained. ANM-based full-wave optimization of microwave circuits exploiting admittance matrices was introduced in [10]. The design of corrugated feed horn antennas using ANM was discussed in [11]. In [10] and [11], gradient-based optimization techniques were utilized and the derivatives of individual network components were determined by numerical differences. In this paper, the ANM is applied to calculate first- and second-order sensitivities of the scattering parameters obtained with the full-wave mode-matching (MM) technique. Using only the MM simulation of the original network, first- and second-order sensitivities of the complex scattering parameters and their magnitudes w.r.t. all designable parameters are obtained. Closed-form formulation for the sensitivity analysis of microwave filters and transitions is derived. The formulation exploits generalized scattering matrices (GSMs) of individual components obtained with the full-wave MM technique. GSMs contain entries for both propagating and evanescent modes. Including higher order modes improves the accuracy of both the scattering parameters and their estimated sensitivities. The ANM-based first- and second-order sensitivities are integrated with the Levenberg–Marquardt (LM) optimization method. This method is then exploited in the design of microwave filters. Results show that convergence of the design process using the ANM-based second-order information is significantly faster than gradient-based algorithms such as the quasi-Newton Broyden–Fletcher–Goldfarb–Shanno (BFGS) method. We start by briefly reviewing the ANM-based approach for estimating first-order sensitivities in Section II. In Section III, it is shown how second-order derivatives are obtained using the ANM. In Section IV, the ANM-based sensitivities are exploited to estimate the first- and second-order sensitivities of differentiable objective functions. The numerical results and discussions are presented in Section V. Finally, conclusions are drawn in Section VI. II. ADJOINT FIRST-ORDER DERIVATIVES The ANM enables efficient estimation of the sensitivities of the scattering parameters w.r.t. the designable parameters. In

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Using the relationship [13]

(7)

Fig. 1. Network representation of circuit.

N

and with simple manipulations, (6) becomes two-port-components of a microwave

[5], [12], and [13], it was shown how to get the first-order sensitivity (derivative) of the scattering parameters. In [14], the first-order sensitivity of ridge waveguide scattering parameters w.r.t. the relative dielectric constant was formulated and applied. Here, the ANM is briefly reviewed. Consider the network representation of a microwave circuit shown in Fig. 1. It is composed of a cascade of two-port components and independent generators at the input and output. The first-order derivative of the incident wave vector, w.r.t. a design parameter , is [12]

(8) Similar to (4), the second-order derivative of the ingoing wave variable at the th port of the network is

(9) Equation (9) is reformulated using (1) and (5) as

(1) where is the incident wave vector. The connection scattering matrix is given by

(10) The scattering parameter

is given by (11)

(2) where is the connection matrix describing the network topology and is the block diagonal system scattering matrix. Its submatrices along the diagonal are the GSMs of the various components of the network. The sensitivity of the ingoing wave variable at the th port of the network is computed by multiplying the left-hand side of (1) by a row vector [12]

is the ingoing wave variable at the th port with a In (11), matched load terminating the th port of the network. The condition is imposed by the matched generator with connected to port (see Fig. 1). The adjoint network is excited at port by a matched generator with impressed wave [12]. Thus,

(3) to obtain (4)

(12) w.r.t. design The derivatives of the system scattering matrix parameters and/or are obtained analytically. For example, consider one of its blocks to be the GSM of a transmission line of length with modes in each port

where (13)

(5) is the adjoint system and

is the adjoint incident wave vector.

where given by

, and

is a diagonal matrix

III. ADJOINT SECOND-ORDER DERIVATIVES Taking the derivative of (1) w.r.t. another design parameter , the second-order derivative of the incident wave vector is obtained as follows:

(6)

.. .

.. .

.. .

.. .

(14)

is the propagation constant of the th and mode in each port. Here, it is assumed that all the connected

EL SABBAGH et al.: ADJOINT HIGHER ORDER SENSITIVITIES FOR FAST FULL-WAVE OPTIMIZATION OF MICROWAVE FILTERS

ports have the same number of modes. The first-order derivative of the GSMs w.r.t. the length is given by

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cond 2 otherwise (21)

(15) where

.. .

.. .

where is the passband weighting factor and is the stopband weighting factor. is the absolute value of the desired return loss in the passband. is the absolute value of the desired attenuation in the stopband. The switching conditions in (20) and (21) are cond 1

(16)

.. .

cond 2

The second-order derivative of the GSM is given by

w.r.t. the length

The error function is minimized subject to simple lower and upper bounds on the design variables

subject to

(17)

(22)

This error function has the gradient IV. OPTIMIZATION As described in Sections II and III, the ANM is used to obtain the first- and second-order derivatives of the scattering parameters. The Jacobian and Hessian matrices of differentiable objective functions are expressed in terms of these derivatives. Here, how the analytical Hessian matrices are obtained and implemented in optimization is presented. The error function to be minimized is defined as a weighted norm of the difference between the actual and desired response of the circuit (sum of squares of nonlinear functions [16]–[18]). The following objective function was successfully used in filter optimization [19]–[21]:

(23) and the Hessian matrix

(18) where is the th frequency point in the passband and is the th frequency point in the stopband. and are the number of frequency points in the passband and stopband, respectively. and are the residual vectors corresponding to and , respectively. They are expressed as

(24) where

is the Jacobian of

defined by

(25) (19) Here, vectors

and

and are given by

. The elements of the

The expressions of the gradient in (23) and the Hessian in (24) require the first- and second-order derivatives of the magnitude of the scattering parameters. The first-order derivative of the magnitude, w.r.t. a variable , is given as [13]

cond 1 otherwise (20)

(26)

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Fig. 2. Top view of the five-pole ridge waveguide filter and its initial frequency response. MM response compared to HFSS [25] response. Waveguide cross section: 0.24 in 0.071 in, ridge width w = 0:08 in, ridge gap = 0:0138 in; " = 5:9; L to L dimensions are as indicated in Table II.

2

Fig. 3. Real part of the dominant mode scattering parameters secondorder derivatives w.r.t. L of the five-pole filter with center frequency f = 7:66 GHz, obtained using the ANM and finite CDs. (Color version available online at: http://ieeexplore.ieee.org.)

TABLE I OPTIMIZATION GOALS

TABLE II DIMENSIONS OF x (IN INCHES) BEFORE AND AFTER OPTIMIZATION L = L ;L = L ;L = L ;L = L ;L = L

Taking the derivative of (26) w.r.t. and after simple algebraic manipulations, the second-order derivative of the magnitude is given as (details of the derivation are shown in the Appendix)

(27) The first-order derivatives that form the Jacobian matrix are computed using (26). The Jacobian matrix is used to calculate the error function gradient in (23). A distinctive feature of the least squares formulation is that by knowing the Jacobian, the first part of the Hessian is computed for free. Many optimization techniques such as Gauss–Newton, Newton–Raphson, quasi-Newton, gradient methods, and LM method can be applied to solve the minimization problem in

Fig. 4. Real part of the dominant mode scattering parameters second-order derivatives w.r.t. L and L of the five-pole filter with center frequency f = 7:66 GHz, obtained using the ANM and finite CDs. (Color version available online at: http://ieeexplore.ieee.org.)

(22). In these techniques, the methods of LM and quasi-Newton prove to be the best ones in most cases in terms of convergence properties. The BFGS method, regarded as the most efficient algorithm among various quasi-Newton methods, is more often adopted to solve unconstrained optimization problems. It presents good performance and does not require the Hessian matrix. However, the line search in the BFGS when applied to complex systems may be time consuming. Two optimization procedures are compared to each other to show the advantage of having analytical Hessian matrices. These optimization procedures are briefly described below. A. BFGS BFGS is a quasi-Newton method used to solve nonlinear problems with simple bounds. It is based on the gradient

EL SABBAGH et al.: ADJOINT HIGHER ORDER SENSITIVITIES FOR FAST FULL-WAVE OPTIMIZATION OF MICROWAVE FILTERS

Fig. 5. Real part of the dominant mode scattering parameters second-order derivatives w.r.t. L and L of the five-pole filter with center frequency f = 7:66 GHz, obtained using the ANM and finite CDs. (Color version available online at: http://ieeexplore.ieee.org.)

Fig. 6. Real part of the dominant mode scattering parameters secondorder derivatives w.r.t. L of the five-pole filter with center frequency f = 7:66 GHz, obtained using the ANM and finite CDs. (Color version available online at: http://ieeexplore.ieee.org.)

projection method and uses the BFGS update formula to approximate the Hessian matrix of the objective function. The BFGS formula is given as [16]

(28) is the current optimization step and . More details about the algorithm can be found in [22] and [23].

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Fig. 7. Real part of the dominant mode scattering parameters secondorder derivatives w.r.t. L of the five-pole filter with center frequency f = 7:66 GHz, obtained using the ANM and finite CDs. (Color version available online at: http://ieeexplore.ieee.org.)

Fig. 8. Imaginary part of the dominant mode scattering parameters secondorder derivatives w.r.t. L of the five-pole filter with center frequency f = 7:66 GHz, obtained using the ANM and finite CDs. (Color version available online at: http://ieeexplore.ieee.org.)

duces a sufficient decrease in the residuals of at the new point . It then sets and begins a new iteration with replacing . A sufficient decrease in the residuals implies that . The correction depends upon a damping parameter, the residual and its Jacobian at . The optimization step is defined as [1]

where

B. LM Method In this method, the user provides an initial approximation of the vector of design parameters to the solution of the problem. Usually, the initial solution is obtained from synthesis. The algorithm then determines a correction to that pro-

(29) where is a diagonal matrix of damping parameters. During iterations, the value of is controlled by the gain ratio

(30)

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Fig. 9. Imaginary part of the dominant mode scattering parameters secondorder derivatives w.r.t. L and L of the five-pole filter with center frequency f = 7:66 GHz, obtained using the ANM and finite CDs. (Color version available online at: http://ieeexplore.ieee.org.)

Fig. 10. Imaginary part of the dominant mode scattering parameters secondorder derivatives w.r.t. L and L of the five-pole filter with center frequency f = 7:66 GHz, obtained using the ANM and finite CDs. (Color version available online at: http://ieeexplore.ieee.org.)

Fig. 11. Imaginary part of the dominant mode scattering parameters secondorder derivatives w.r.t. L of the five-pole filter with center frequency f = 7:66 GHz, obtained using the ANM and finite CDs. (Color version available online at: http://ieeexplore.ieee.org.)

Fig. 12. Imaginary part of the dominant mode scattering parameters secondorder derivatives w.r.t. L of the five-pole filter with center frequency f = 7:66 GHz, obtained using the ANM and finite CDs. (Color version available online at: http://ieeexplore.ieee.org.)

as follows: (31) and (32) Although the value of is controlled by the gain ratio, its initial value can be critical in determining the number of iterations and the optimization time. V. NUMERICAL RESULTS AND DISCUSSION A five-pole ridge waveguide filter (see Fig. 2) is first designed, using full-wave modeling, to have a center frequency GHz and a bandwidth GHz. The filter has

the initial response shown in Fig. 2. The passband return loss (12.6 dB), the center frequency (7.8425 GHz), and the bandwidth (0.863 GHz) do not satisfy the specifications as required in Table I. The filter response is obtained using the rigorous MM technique. The number of modes used to characterize each ridge to rectangular waveguide discontinuity is 12 (eight TE modes and four TM modes). This number of modes satisfies convergence, as indicated in [24]. Results obtained from Ansoft’s High Frequency Structure Simulator (HFSS) [25] are compared to those computed using the MM technique. Both results are in very good agreement, which validates the accuracy of the developed in-house MM simulator. The vector of designable parameters is . The initial value of is indicated in Table II. Figs. 3–7 and Figs. 8–12 show the real and

EL SABBAGH et al.: ADJOINT HIGHER ORDER SENSITIVITIES FOR FAST FULL-WAVE OPTIMIZATION OF MICROWAVE FILTERS

Fig. 13. Real part of the dominant mode scattering parameters first- and second-order derivatives w.r.t. L of the five-pole filter with center frequency f = 7:66 GHz, obtained using the ANM. The displayed values of the real part of the second-order derivatives of S and S were divided by 75 and 20, respectively, for better display of the curves. (Color version available online at: http://ieeexplore.ieee.org.)

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Fig. 15. Optimized responses of the five-pole filter with center frequency f = 7:66 GHz obtained using BFGS and LM methods compared to HFSS (using the LM dimensions). The optimized dimensions using BFGS and LM are shown in Table II. (Color version available online at: http://ieeexplore.ieee.org.)

Thus, has no other peak within the passband and it remains below the specified return loss in the entire passband. Similarly, the minimum of (in absolute value) occurs at the edges of the passband and then alternates between maximum and minimum due to ripples in the transmission within the passband. The results in Figs. 13 and 14 are typical for a good designed filter where the main objective is to minimize and to maximize in the passband. In Figs. 3–14, the utilized values of the design parameters are those obtained from the LM optimization, as indicated in Table II. All results obtained using the ANM are compared to those obtained from CDs, and they are in excellent agreement. The formulas used for CDs are based on function calls only and are given by [15]

(34) Fig. 14. Imaginary part of the dominant mode scattering parameters first- and second-order derivatives w.r.t. L of the five-pole filter with center frequency f = 7:66 GHz, obtained using the ANM. The displayed values of the real part of the second-order derivatives of S and S were divided by 20 for better display of the curves. (Color version available online at: http://ieeexplore.ieee. org.)

imaginary parts, respectively, of the second-order derivatives of the dominant mode scattering parameters w.r.t. the design parameters. It should be noted that in Figs. 7 and 12,

(33) Figs. 13 and 14 show the real and imaginary parts, respectively, of the dominant mode scattering parameters first- and secondorder derivatives w.r.t. . These figures show the following. and at the passband edges so the minimum of (in absolute value) occurs at the edges of the passband. The first order derivative has no other zeros.

and

(35) , subscripts is the vector of design parameters, and . The optimization goals for the implemented optimization routines (BFGS and LM) are as shown in Table I. The BFGS method gives the following results: where

of iterations of function evaluations of segments explored during Cauchy searching Optimization time

s (36)

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6

Fig. 16. Tolerance analysis assuming 1-mil statistical variation of the dimensions L to L of the five-pole filter with center frequency f = 7:66 GHz. (a) Responses are obtained using first- and second-order derivatives. (b) Responses are obtained from direct MM simulation.

The LM method (initial value of each element of is the norm of the corresponding column of the initial Hessian matrix) gives the following results: of iterations of function evaluations of gradient evaluations of Hessian evaluations Optimization time

s

(37)

The optimized vector of design parameters has the dimensions shown in Table II. The filter is symmetric around . The computation time for the BFGS method is approximately 5 that of the LM method. The simulations, obtained from a developed FORTRAN code, were run on a dual 750-MHz UltraSPARC III processors with 1 GB of memory and running SunOS 5.8 (i.e., Solaris 8). The savings in time using the LM method is

6

Fig. 17. Tolerance analysis assuming 2-mil statistical variation of the dimensions L to L of the five-pole filter with center frequency f = 7:66 GHz. (a) Responses are obtained using first- and second-order derivatives. (b) Responses are obtained from direct MM simulation.

due to the sparsity of the matrices in (10). This makes the computation of the Hessian matrix in (24) very fast. Comparison between the responses obtained from both methods and HFSS using the LM optimized lengths is shown in Fig. 15. The results are similar, which shows the advantage of the LM method over the BFGS method regarding computation time. The return loss obtained from the BFGS optimization is better than the one obtained from the LM routine. However, the selectivity obtained from the LM is better than the one obtained from the BFGS algorithm and the bandwidth obtained from the LM fits exactly the specifications. Tolerance analysis for 1 mil (25.4 m), 2 mil (50.8 m), and 5 mil (127 m) statistical variation of the dimensions to are shown in Figs. 16–18, respectively. Figs. 16(a), 17(a), and 18(a) show the tolerance analysis obtained using the firstand second-order derivative information (38)

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Fig. 19. Optimized responses obtained using BFGS and LM methods compared to the nonoptimized response for the five-pole filter with center frequency 0.071 in, ridge width f = 9:89 GHz. Waveguide cross section: 0.18 in w = 0:08 in, ridge gap = 0:0138 in, " = 5:9; L to L initial and optimized using BFGS and LM are shown in Table III. (Color version available online at: http://ieeexplore.ieee.org.)

2

6

Fig. 18. Tolerance analysis assuming 5-mil statistical variation of the dimensions L to L of the five-pole filter with center frequency f = 7:66 GHz. (a) Responses are obtained using first- and second-order derivatives. (b) Responses are obtained from direct MM simulation.

TABLE III DIMENSIONS OF x (IN INCHES) BEFORE AND AFTER OPTIMIZATION FOR THE RIDGE WAVEGUIDE FILTER WITH CENTER FREQUENCY f = 9:89 GHZ L = L ;L = L ;L = L ;L = L ;L = L

Fig. 20. Top view of the transition from the input ridge waveguide to a 50- stripline. The inset shows the cross section of the stripline. a = 0:24 in; b = 0:071 in; g = 0:0138 in; w = 0:08 in; w = 0:04 in; w = 0:082 in; w = 0:0135 in. (Color version available online at: http://ieeexplore.ieee.org.)

% varies from 0.26% to 1.32%. The maximum relative perturbation is defined as

(39) where is the perturbation of from its nominal value. Figs. 16(b), 17(b), and 18(b) show the tolerance analysis obtained using direct MM simulation. These figures show that up to a manufacturing tolerance of 2 mil, the responses are good. The responses obtained from the Taylor expansion (38) also agree very well with the responses obtained from direct MM simulations. For a maximum tolerance varying from 1 to 5 mil, the corresponding maximum relative perturbation percentage

where it is assumed that the design variables are all perturbed by the maximum manufacturing tolerance . Actually, this tolerance variation corresponds from 2.8% to 14% of the middle resonator length , while it represents from 0.41% to 2% of the middle evanescent sections length . In Figs. 16–18, and 20 perturbed responses are run for each manufacturing tolerance. Each response has 100 frequency points.

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Fig. 21. Real part of the dominant mode scattering parameters sensitivity w.r.t. L of the transition, obtained using the ANM and finite CDs. (Color version available online at: http://ieeexplore.ieee.org.)

Fig. 23. Real part of the dominant mode scattering parameters second-order derivatives w.r.t. L of the transition, obtained using the ANM and finite CDs. (Color version available online at: http://ieeexplore.ieee.org.)

Fig. 22. Imaginary part of the dominant mode scattering parameters sensitivity w.r.t. L of the transition, obtained using the ANM and finite CDs. (Color version available online at: http://ieeexplore.ieee.org.)

The time required by direct MM simulation is 286 s. The Taylor approximation, on the other hand, required only 103 s. A second optimization example for a five-pole ridge waveguide filter with a center frequency GHz, passband ripple 0.01 dB, and equiripple bandwidth from 9.23 to 10.63 GHz is carried out. The BFGS method gives the following results: of iterations Optimization time

s

(40)

The LM method gives the following results: of iterations of gradient evaluations of Hessian evaluations Optimization time

s

(41)

Fig. 24. Real part of the dominant mode scattering parameters second-order derivatives w.r.t. L and L of the transition, obtained using the ANM and finite CDs. (Color version available online at: http://ieeexplore.ieee.org.)

The optimized vector of design parameters has the dimensions shown in Table III. The computation time for the BFGS method is more than 4 that of the LM method. The optimized responses compared to the nonoptimized response are shown in Fig. 19. The developed optimization technique is also applied to a third example. The transition from the input/output ridge waveguide to a 50- stripline, shown in Fig. 20, is optimized w.r.t. and . The number of modes used to charthe lengths acterize each stripline to stripline discontinuity is 15 (ten TE modes and five TM modes). The real and imaginary parts of the first- and second-order derivatives are shown in Figs. 21–26. Using the obtained first- and second-order derivatives, the transition is optimized over the frequency range of 7–10 GHz to

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TABLE IV DIMENSIONS OF x (IN INCHES) BEFORE AND AFTER OPTIMIZATION FOR THE RIDGE WAVEGUIDE TO 50- STRIPLINE TRANSITION

Fig. 25. Imaginary part of the dominant mode scattering parameters secondorder derivatives w.r.t. L of the transition, obtained using the ANM and finite CDs. (Color version available online at: http://ieeexplore.ieee.org.)

Fig. 27. Optimized responses of the transition obtained using BFGS and LM methods compared to the nonoptimized. The optimized dimensions using BFGS and LM are shown in Table IV. (Color version available online at: http://ieeexplore.ieee.org.)

Fig. 26. Imaginary part of the dominant mode scattering parameters secondorder derivatives w.r.t. L and L of the transition, obtained using the ANM and finite CDs. (Color version available online at: http://ieeexplore.ieee.org.)

The optimized vector of design parameters for the transition has the dimensions shown in Table IV. The computation time for the BFGS method is more than 3 that of the LM method. Comparison between the initial response and those obtained from both methods is shown in Fig. 27. The results are similar, which again shows the advantage of the LM method over the BFGS method regarding computation time. VI. CONCLUSION

satisfy a return loss of 0.01. The BFGS method gives the following results: of function evaluations of segments explored during Cauchy searching Optimization time

s (42)

The LM method gives the following results: of function evaluations of gradient evaluations of Hessian evaluations Optimization time

s

(43)

The ANM has been applied to estimate the sensitivities of scattering parameters of microwave filters and transitions, obtained with the full-wave MM technique. Using only the MM simulation of the original network, the first- and second-order sensitivities of the scattering parameters w.r.t. all the designable parameters have been obtained. The higher order modes (propagating and evanescent) characterizing the discontinuities have been considered for better accuracy. The obtained sensitivities (first- and second-order derivatives) have been used to compute the gradient and the Hessian matrices of differentiable objective functions. The formulation has been applied to the sensitivity analysis of ridge waveguide filters and transitions from ridge waveguides to 50- striplines. Our optimization routine utilizing analytical Hessian matrices is faster than the BFGS routine using only the analytical gradient. Statistical analysis obtained using an adjoint-based approximate model is accurate and much faster than direct MM simulation.

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APPENDIX DERIVATION OF (27) Let the magnitude of the complex scattering parameter be represented as

(A-1) where and are the real and imaginary parts of the complex scattering parameter, respectively. Taking the derivatives of both sides of (A-1), the derivative of the magnitude is given as

(A-2) Taking the derivative of (A-2) w.r.t. ,

(A-3) The following relations can also be easily proven:

(A-4) where denotes the conjugate. Substituting from (A-4) into (A-3) and doing simple manipulations, (27) is obtained.

[10] F. Alessandri, M. Mongiardo, and R. Sorrentino, “New efficient full wave optimization of microwave circuits by the adjoint network method,” IEEE Microw. Guided Wave Lett., vol. 3, no. 11, pp. 414–416, Nov. 1993. [11] M. Mongiardo and R. Ravanelli, “Automated design of corrugated feeds by the adjoint network method,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 5, pp. 787–793, May 1997. [12] J. Dobrowolski, Computer-Aided Analysis, Modeling, and Design of Microwave Networks. The Wave Approach. Norwood, MA: Artech House, 1996. [13] K. C. Gupta, R. Garg, and R. Chadha, Computer Aided Design of Microwave Circuits. Norwood, MA: Artech House, 1981. [14] M. A. El Sabbagh and M. H. Bakr, “Analytical dielectric constant sensitivity of ridge waveguide filters,” J. Electromagn. Waves Applicat., vol. 20, no. 3, pp. 363–374, 2006. [15] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions. New York: Dover, 1972, p. 884. [16] R. Fletcher, Practical Methods of Optimization. New York: Wiley, 1980, vol. 1, Unconstrained Optimization. [17] P. E. Gill, W. Murray, and M. H. Wright, Practical Optimization. London, U.K.: Academic, 1981. [18] J. Nocedal and S. J. Wright, Numerical Optimization. New York: Springer-Verlag, 1999. [19] Y. Rong, “Modeling of combline coaxial, ridge waveguide filters and multiplexers,” Ph.D. dissertation, Dept. Elect. Comput. Eng., Univ. Maryland at College Park, College Park, MD, 1999. [20] J. W. Bandler, “Optimization methods for computer-aided design,” IEEE Trans. Microw. Theory Tech., vol. MTT-17, no. 8, pp. 533–552, Aug. 1969. [21] J. W. Bandler and C. Charalambous, “Practical least pth optimization of networks,” IEEE Trans. Microw. Theory Tech., vol. MTT-20, no. 12, pp. 834–840, Dec. 1972. [22] R. H. Byrd, P. Lu, and J. Nocedal, “A limited memory algorithm for bound constrained optimization,” SIAM J. Sci. Stat. Comput., vol. 16, no. 5, pp. 1190–1208, 1995. [23] C. Zhu, R. H. Byrd, and J. Nocedal, “L-BFGS-B: Algorithm 778: L-BFGS-B, FORTRAN routines for large scale bound constrained optimization,” ACM Trans. Math. Softw., vol. 23, no. 4, pp. 550–560, Dec. 1997. [24] M. A. El Sabbagh, H. T. Hsu, and K. A. Zaki, “Stripline transition to ridge waveguide bandpass filters,” Progr. Electromagn. Res., vol. 40, pp. 29–53, 2003. [25] HFSS. ver. 9.2.1, Ansoft, Pittsburgh, PA, 2004.

REFERENCES [1] J. E. Dennis, Jr. and R. B. Schnabel, Numerical Methods for Unconstrained Optimization and Nonlinear Equations. Englewood Cliffs, NJ: Prentice-Hall, 1983. [2] E. S. Kuh and C. G. Lau, “Sensitivity invariants of continuously equivalent networks,” IEEE Trans. Circuit Theory, vol. CT-15, no. 9, pp. 175–177, Sep. 1968. [3] S. W. Director and R. A. Rohrer, “The generalized adjoint network and network sensitivities,” IEEE Trans. Circuit Theory, vol. CT-16, no. 8, pp. 318–323, Aug. 1969. [4] M. Sablatash and R. Seviora, “Sensitivity invariants for scattering matrices,” IEEE Trans. Circuit Theory, vol. CT-18, no. 3, pp. 282–284, Mar. 1971. [5] J. W. Bandler and R. E. Seviora, “Wave sensitivities of networks,” IEEE Trans. Microw. Theory Tech., vol. MTT-20, no. 2, pp. 138–147, Feb. 1972. [6] G. I. Vasilescu and T. Redon, “A new approach to sensitivity computation of microwave circuits,” in Proc. IEEE Int. Circuits Syst. Symp., Helsinki, Finland, Jun. 1988, pp. 1167–1170. [7] T. Redon and G. I. Vasilescu, “Second- and third-order sensitivities of microwave circuits,” Electron. Lett., vol. 25, no. 9, pp. 607–609, Apr. 1989. [8] V. A. Monaco and P. Tiberio, “Computer-aided analysis of microwave circuits,” IEEE Trans. Microw. Theory Tech., vol. MTT-22, no. 3, pp. 249–263, Mar. 1974. [9] O. Wing and J. V. Behar, “Circuit design by minimization using the Hessian matrix,” IEEE Trans. Circuits Syst., vol. CAS-21, no. 5, pp. 643–649, Sep. 1974.

Mahmoud A. El Sabbagh (S’93–M’02) received the B.S. (with honors) and M.S. degrees in electrical engineering from Ain Shams University, Cairo, Egypt, in 1994 and 1997, respectively, and the Ph.D. degree from the University of Maryland at College Park, in 2002. From 1994 to 1998, he was a Lecturer and Research Assistant with the Department of Electrical Engineering, Ain Shams University, where his research dealt with applications of superconductors in microwave circuits. In June 1998, he joined the Microwave Group, University of Maryland at College Park. From May 2001 to June 2002, he was a Guest Researcher with the National Institute of Standards and Technology (NIST) Gaithersburg, MD. From September 2002 to July 2003, he was a Visiting Scientist with the United Stated Department of Agriculture—Agricultural Research Service (USDA-ARS). Since October 2003, he has been an Assistant Professor with the Faculty of Engineering, Electronics and Communications Department, Ain Shams University. From December 2003 to January 2005, he led the microwave team of the National Authority for Remote Sensing and Space Sciences, Cairo, Egypt. In 2005, he was a Post-Doctoral Fellow with the Electrical and Computer Engineering Department, McMaster University. He is currently a Post-Doctoral Fellow with the Electrical and Computer Engineering Department, University of Waterloo, Waterloo, ON, Canada. His research interests include computer-aided design of microwave devices, microwave filters modeling and design, dielectric characterization, microwave remote sensing, and EM theory.

EL SABBAGH et al.: ADJOINT HIGHER ORDER SENSITIVITIES FOR FAST FULL-WAVE OPTIMIZATION OF MICROWAVE FILTERS

Mohamed H. Bakr (S’98–M’00) received the B.Sc. degree in electronics and communications engineering [with distinction (honors)] and M.Sc. degree in engineering mathematics from Cairo University, Cairo, Egypt, in 1992 and 1996, respectively, and the Ph.D. degree from McMaster University, Hamilton, ON, Canada, in 2000. In 1997, he was a Student Intern with Optimization Systems Associates Inc. (OSA), Dundas, ON, Canada. From 1998 to 2000, he was a Research Assistant with the Simulation Optimization Systems (SOS) Research Laboratory, McMaster University. In November 2000, he joined the Computational Electromagnetics Research Laboratory (CERL), University of Victoria, Victoria, BC, Canada, as a Natural Sciences and Engineering Research Council of Canada (NSERC) Post-Doctoral Fellow. He is currently an Assistant Professor with the Department of Electrical and Computer Engineering, McMaster University. His research areas of interest include optimization methods, computer-aided design and modeling of microwave circuits, neural-network applications, smart analysis of microwave circuits, and efficient optimization using time-/frequency-domain methods. Dr. Bakr was a recipient of the Premier’s Research Excellence Award (PREA) presented by the Province of Ontario, Canada, in 2003.

John W. Bandler (S’66–M’66–SM’74–F’78) was born in Jerusalem, on November 9, 1941. He received the B.Sc.(Eng.), Ph.D., and D.Sc.(Eng.) degrees from the University of London, London, U.K., in 1963, 1967, and 1976, respectively. He joined Mullard Research Laboratories, Redhill, Surrey, U.K., in 1966. From 1967 to 1969, he was a Postdoctorate Fellow and Sessional Lecturer at the University of Manitoba, Winnipeg, Canada. He joined McMaster University, Hamilton, ON, Canada, in 1969. He was Chairman of the Department of Electrical Engineering and Dean of the Faculty of Engineering. He is currently Professor Emeritus in Electrical and Computer Engineering, directing research

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in the Simulation Optimization Systems Research Laboratory. He has authored or coauthored over 385 papers. He was a member of the Micronet Network of Centres of Excellence. He was President of Optimization Systems Associates Inc. (OSA), which he founded in 1983, until November 20, 1997, the date of acquisition of OSA by the Hewlett-Packard Company. OSA implemented a first-generation yield-driven microwave computer-aided design (CAD) capability for Raytheon in 1985, followed by further innovations in linear and nonlinear microwave CAD technology for the Raytheon/Texas Instruments Joint Venture MIMIC Program. OSA introduced the CAE systems RoMPE in 1988, HarPE in 1989, OSA90 and OSA90/hope in 1991, Empipe in 1992, and Empipe3D and EmpipeExpress in 1996. OSA created the product empath in 1996 which was marketed by Sonnet Software Inc. He is President of Bandler Corporation, which he founded in 1997. He joined the Editorial Boards of the International Journal of Numerical Modelling in 1987, the International Journal of Microwave and Millimeterwave Computer-Aided Engineering in 1989, and Optimization and Engineering in 1998. He was a Guest Editor of the International Journal of Microwave and Millimeter-Wave Computer-Aided Engineering Special Issue on “Optimization-Oriented Microwave CAD” (1997). He was Guest Coeditor of the Optimization and Engineering Special Issue on “Surrogate Modelling and Space Mapping for Engineering Optimization” (2001). Dr. Bandler is a Fellow of the Canadian Academy of Engineering, the Royal Society of Canada, the Institution of Electrical Engineers, and the Engineering Institute of Canada. He is a member of the Association of Professional Engineers of the Province of Ontario, Canada, and the MIT Electromagnetics Academy. He was an associate editor of the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES (1969–1974) and has continued serving as a member of the Editorial Board. He was guest editor of the IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES Special Issue on “Computer-Oriented Microwave Practices” (1974) and on “Automated Circuit Design Using Electromagnetic Simulators” (1997) and guest coeditor of the Special Issue on “Process-Oriented Microwave CAD and Modeling” (1992) and on “Electromagnetics-Based Optimization of Microwave Components and Circuits” (2004). He was chair of the MTT-1 Technical Committee on Computer-Aided Design. He was the recipient of the 1994 Automatic Radio Frequency Techniques Group Automated Measurements Career Award and the 2004 IEEE Microwave Theory and Techniques Society (MTT-S) Microwave Application Award.

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Broadband Quasi-Chebyshev Bandpass Filters With Multimode Stepped-Impedance Resonators (SIRs) Yi-Chyun Chiou, Student Member, IEEE, Jen-Tsai Kuo, Senior Member, IEEE, and Eisenhower Cheng

Abstract—Planar broadband bandpass filters of order up to 9 are synthesized based on the multimode property of stepped-impedance resonators (SIRs). Based on the transmission line theory, the modal frequencies of the SIRs are calculated based on the impedance and length ratios of its hi- and lowsegments. In the synthesis, the SIR coupling schemes are determined by the split mode graphs. Using one, two, two, three, and three dual- or triple-mode SIRs, quasi-Chebyshev filters with four, six, six, eight, and nine transmission poles, respectively, are . Emsynthesized with a fractional bandwidth (BW) phasis is placed not only on designing the I/O coupling structures for matching the external ext and the circuit BW, but also on matching the resonant peaks of the circuit with the nominal Chebyshev poles. Measured results of experimental circuits show good agreement with simulated responses.

(

)

1 = 50%

Index Terms—Bandpass filter (BPF), broadband, external multimode, stepped-impedance resonator (SIR).

,

I. INTRODUCTION N THE RF front end of a modern communication system, especially for satellite and mobile applications, high-performance RF/microwave bandpass filters (BPFs) are essential devices. The multimode BPFs have many attractive features such as a simple design procedure, high selectivity, and low cost. Recently, many researches on multimode resonator filters have been published for innovative design and analysis methods [1]–[5]. In [1], a narrowband third-order Chebyshev filter is realized by a triple-mode cavity. A shorted waveguide stub is used to set up couplings between the degenerate modes. The method developed in [2] significantly extends the design possibilities for multimode cavity filters. Several degenerate resonances of a cavity are simultaneously coupled to the same I/O port. A third-order elliptic function filter is realized with one triple-mode cavity. In [3], a six-pole pseudoelliptic function BPF based on the sixfold degeneracy of a single cavity is presented. The design achieves significant savings in mass and volume when compared to six-pole dual- and/or triple-mode filters of equivalent performance. In [4], new common multimode cavity multiplexing and double-band filter methods are introduced. In [5], dual- and triple-mode resonators are used to implement miniaturized dielectric resonator filters.

I

Manuscript received February 11, 2006; revised April 10, 2006. This work was supported under the MoE ATU Program and by the National Science Council, Taiwan, R.O.C., under Grant NSC 94-2213-E-009-073 and Grant NSC 94-2752-E-009-003-PAE. The authors are with the Department of Communication Engineering, National Chiao Tung University, Hsinchu, 300 Taiwan, R.O.C. (e-mail: jtkuo@cc. nctu.edu.tw). Digital Object Identifier 10.1109/TMTT.2006.879131

It is noted that all of above-mentioned multimode resonator filters are three-dimensional and have a relatively narrow band. They are designed based on degenerated modes, i.e., modes with identical resonant frequencies. The stepped-impedance resonators (SIRs), on the other hand, have a planar structure and are advantageous in designing BPFs due to its versatile resonant characteristics [6]–[10]. One of the key features of an SIR is that its resonant frequencies can be easily altered by tuning its geometric parameters. For example, in [6], the SIR filters are designed to have a wide stopband by locating the first higher order resonance as far away from the fundamental frequency as possible. Recently, periodic stepped-impedance ring resonators (PSIRRs) [7] have been proposed to design dual-mode filters. The two modes are orthogonal and have the same frequencies, and perturbation is required to split up the resonances and create a narrow passband. The filters in [8]–[10], on the other hand, have a broadband characteristic by incorporating the first three SIR modes at distinct frequencies into the passband. Five-pole filters can be built up with a single triple-mode SIR and an excitation structure with strong coupling. The impedance junctions between the excitation and the SIR are capable of creating two additional transmission poles. In this paper, we aim at developing a systematic procedure for synthesizing broadband BPFs based on single or plural multimode SIRs. When only a single SIR is involved, the design is similar to those in [8]–[10]. Each SIR is treated as a multimode cavity and contributes two or three resonances at different frequencies to the circuit. The resonances of coupled SIRs are tuned to match the transmission poles of a Chebyshev passband. The BPFs will have a quasi-Chebyshev response since the final poles deviate slightly from the designated positions due to the I/O coupling. To this end, the impedance and length ratios of the SIRs are properly chosen based on resonant spectrum in readiness. Based on the external formulas, proper I/O coupling is devised to meet the bandwidth (BW) specification. For demonstration, four-, six-, eight-, and nine-pole BPFs are fabricated and measured. II. DUAL-MODE AND TRIPLE-MODE SIRs The resonant frequencies of an SIR in Fig. 1 can be calculated by the following two transcendental equations [6], [11]: odd mode even mode

(1) (2)

is the impedance ratio and and are where electrical lengths of the microstrip sections with characteristic

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edges, respectively. The transmission poles in (4), or the zeros of , are given as

(5) Fig. 1. Geometry of an SIR. Z > Z in this study.

For a th-order Chebyshev BPF with a fractional BW pole frequency can be easily calculated by

, the

(6)

Fig. 2. Resonant spectrum of SIRs with R = 5; 7:25; and 9:5. All resonances are normalized with respect to the fundament frequency f . The horizontal axis u =  =( +  ). (Color version available online at: http://ieeexplore.ieee. org.)

impedances and , respectively. The fundamental and consecutive higher order resonances occur alternatively in the odd and even modes. Fig. 2 plots through normalized with respect to the fundamental frequency for and . The horizontal axis is the length ratio defined as

(3) The data shown in Fig. 2 form an important basis of our design for determining the SIR geometry. The SIRs used here are either dual- or triple-mode elements. The center frequency of a dual-mode element is defined as the arithmetic mean of the two resonances, and that of a triple-mode SIR is at its second resonance, provided that the three resonances are equally spaced. In the following demonstration for four-, six-, and nine-pole filters, either dual- or triple-mode SIRs with identical resonances are used; while for an eighth-order BPF, both dual- and triple-mode elements with distinct resonant frequencies are used.

where is the center frequency. The idea of our design is to match the resonant frequencies of each BPF circuit with these transmission poles. In a BPF, the couplings between the I/O feeder and end resonator and between adjacent resonators have to be properly designed. When the BPF has only one SIR, only the I/O coupling has to be considered. If there are two or more SIRs, the couplings among the SIRs are then chosen to match split-off resonant frequencies with the Chebyshev poles in (6). It is assumed that the I/O feeders do not shift the split-off frequencies significantly. In this paper, all circuits are designed on a substrate with and thickness mm and have with a 0.1-dB ripple level. Before the I/O feeders are equipped, the split-off poles are detected by a loose coupling scheme [6]. All simulation are done via IE3D [13]. A. I/O Coupling For the broadband BPFs in [8]–[10], strong couplings are essentially required for the I/O structures. The couplings, however, seem to be lack of quantitative description. In this study, coupled-line stages are used to implement the required I/O couplings. The modal characteristic impedances of the stage and can be determined by the formulas in [14, Secs. 8.09 and 10.02] when the circuit BW is small and wide, respectively. Once and are known, the linewidth and gap size can be found. The linewidth, however, is one of geometric parameters of the SIR and has been fixed by the designated resonant frequencies. Thus, the I/O feeders need redesigning as follows. A coupled-line stage can be modeled as two quarter-wave lines with an admittance inverter in between [12]. The admittance value can be expressed in terms of the characteristic impedances of the stage

III. CIRCUIT SYNTHESIS AND MEASUREMENT The insertion loss function of a th-order Chebyshev filter can be expressed as [12]

(7) where

(4) is the th-order Chebyshev polynomial of the first where kind and specifies the passband ripple level. The frequencies corresponding to and are center frequency and band

is the reference port impedance. The external of the stage and the admittance inverter is related

by

(8)

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Fig. 3. Simulated loaded Q (Q ) for coupled-line stages versus gap size G . (Color version available online at: http://ieeexplore.ieee.org.)

Fig. 4. Simulated and measured results of the dual-mode BPF. W = 3:76, W = W = 0:2, ` = 3:19, ` = 26:45, ` = 22:95, and G = 0:13 (all in millimeters). (Color version available online at: http://ieeexplore.ieee.org.)

The loaded by

and the 3-dB BW

of a stage is related

(9) is specified by the stage rather than the passNote that band response, i.e., the stage is required to have a fulfilling for lossless cases. Fig. 3 plots the simulated values against gap size for three sets of linewidths and . It can be seen that the smaller the gap size or the linewidths, the smaller the values. When mm, the values of the three cases are nearly the same. In simulation, for the particular substrate with linewidth of 0.15 0.2 mm, the stage shows two extra transmission poles when mm and no poles when mm [8]. B. BPFs With a Single SIR

Fig. 5. Design and results of two six-pole BPFs. (a) Circuit layout and mode graph of the two coupled dual-mode SIRs. (b) Circuit layout and mode graph of the two coupled triple-mode SIRs. (c) Simulated and measured results. Dualmode BPF: W = 3:76, W = 0:2, ` = 4:18, ` = 26:1, W = 0:2, ` = 23:5, G = 0:165, and S = 0:265. Triple-mode BPF: W = 7:24, W = 0:15, ` = 21:5, ` = 23:8, W = 0:2, ` = 24:2, G = 0:215, and S = 0:371 (all in millimeters). (Color version available online at: http:// ieeexplore.ieee.org.)

Suppose we are designing a dual-mode filter based on two resonances at and , which are designated by (5) and (6). Fig. 4 plots the simulated and measured responses of the dualmode BPF designed at GHz with . The geometric parameters of the dual-mode SIR are chosen by locating GHz and GHz at the two transmission poles of . For , there are two possible solutions in

Fig. 2, i.e., points and , to have . Obviously, is preferable to since the next resonance of the former at is much higher than that of the latter at . At point , . The response shows that the circuit has four transmission poles. The two extra poles are from the I/O stage [8]. The length

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TABLE I COMPARISON OF FREQUENCIES IN TRANSITION BANDS WITH SPECIFIC ATTENUATION LEVELS FOR CIRCUITS IN FIG. 5

Fig. 6. Group delays for the six-pole BPFs in Fig. 5. (Color version available online at: http://ieeexplore.ieee.org.)

of the coupled-line stage mm is a quarter-wavelength long at . The measurement shows and agree well with the design. The poles at and shift to 1.85 and 3.03 GHz, respectively, after the I/O structures are applied. The shifts of these two poles are less than 9% and slightly increase the circuit BW. The transmission zero at 5 GHz is an inherent property of a parallel-coupled stage [15]. When the SIR is used as a triple-mode element with resonances at , , and , it will be convenient to choose , i.e., , since is the arithmetic average of and . It is because from (1), and at and , respectively, and from (2), at , which is the midpoint of the values at and . Points and in Fig. 2 show a good example. Note that this equal-space property is independent of the value. The absolute BW of a third-order circuit based on this triple-mode resonator is close to , and it can be controlled by . There have been many good BPF examples designed with a single triple-mode SIR in [8]–[10], and they will be not repeated here. C. BPFs With Two and Three Triple-Mode SIRs Two dual-mode SIRs are used to construct a six-pole BPF. The four resonant peaks of the coupled dual-mode SIRs, without the I/O stage, are symmetric about GHz. Based on (6), the geometric parameters of each SIR are chosen to have two resonances at and . Two triple-mode SIRs are also used to design a six-pole BPF with the same . The width of the low- section is 7.24 mm to realize . The possible spurious corresponding to its transverse resonance is approximately GHz . The split mode graphs in Fig. 5(a) and (b), respectively, investigate the split off of the resonant peaks of the two dual- and two triple-mode SIRs with respect to coupling gap size . The split

Fig. 7 Synthesis and design of a ninth-order BPF with three triple-mode SIRs. (a) Half of the circuit layout. (b) Filter performance in a band from 5 to 11 GHz. (c) Simulation and measured responses in a broadband. (d) Circuit photograph. W = 7:60, W = 0:17, W = 8:35, W = 0:15, ` = 6:29, ` = 6:11, ` = 0:70, W = 0:15, ` = 6:76, G = 0:225, and S = 0:375 mm (all in millimeters). (Color version available online at: http://ieeexplore.ieee.org.)

off increase as is decreased, as expected. mm and mm are then chosen for the dual- and triple-mode cases, respectively, to match the frequencies with the nominal poles of the fourth- and the sixth-order BPF. The maximal relative deviation of the values from those given in (6) is no larger than 1%. Fig. 5(c) shows the simulated and measured results of the two BPFs. For the sake of clarity, only measured data are plotted for . Measured data indicate that the in-band insertion loss is 0.9 dB and the return loss is 15 dB. Note that both circuits

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TABLE II CHEBYSHEV POLES (THEORY) AND TUNED RESONANT PEAKS (IN GIGAHERTZ) OF THE NINTH-ORDER BPF

have six transmission poles. Of the dual-mode BPF, again, two extra poles are resulted from the I/O stages with a gap size of 0.165 mm. In the triple-mode case, however, there is no extra pole since gap size of the I/O stages mm corresponds to a relatively weak coupling. Although both of the circuits in Fig. 5 have the same number of poles, one can easily identify that the two circuits have different attenuation rates in their transition bands by comparing responses in Fig. 5(c). Table I compares the frequenthe cies of the two circuits at specific attenuation levels with those obtained by theoretical calculations, i.e., Chebyshev function of the th-order ( or ) with a 0.1-dB ripple and [14]. The frequencies of the triple-mode circuit match well with those of , while those of the dual mode agree better with those of a fourth-order Chebyshev function. This means that the two extra poles created by the I/O stages in the dual-mode BPF have little contribution to rejection levels in the transition bands. The layouts of the circuits in Fig. 5 involve asymmetric coupled lines. Rigorous analysis for the coupling and based on the transmission line theory would be favorable for circuit design. The circuits, nevertheless, involve strong couplings and discontinuities, which need further simulation for final confirmation anyway. Thus, from the design point of view, the couplings and for the structures are directly characterized by an electromagnetic software package. For a broadband filter, variations of group delay within the passband can be of important concern. Fig. 6 plots group delay responses for the fabricated circuits in Fig. 5. The maximum group delay variations for the dual- and triple-mode SIR circuits are 1 and 2.5 ns, respectively. The latter shows a worse phase linearity in the passband. Synthesis of a ninth-order BPF with three triple-mode SIRs is also studied. The circuit is designed at GHz and the SIRs have and . Fig. 7(a) shows half of the circuit layout on the left-hand side of its plane of symmetry . Fig. 7(b) illustrates the performance of the BPF from 5 to 11 GHz. Detailed data indicate that the measurement has in-band insertion loss dB and for a 15-dB return loss. Before the I/O coupling is imposed, the chosen circuit poles together with the Chebyshev function poles are listed in Table II. The maximal relative deviation between the theory and tuned values is only 1.8%. In simulation of the entire circuit, linewidths and and the parameter are finely adjusted for the in-band level. The poles in the measured response are also listed and show reasonably good agreement with the chosen values. Fig. 7(c) plots the simulated and measured responses up to 25 GHz. From Fig. 2, since are used for the passband, the first spurious may occur at or . The possible resonances in transverse direction corresponding

Fig. 8. Synthesis and design of an eighth-order BPF with two triple-mode and a dual-mode SIRs. (a) Layout and the split mode graph. (b) Filter performance from 5 to 11 GHz. (c) Simulation and measured responses in broadband. (d) Photograph of the circuit. W = 7:72, W = 0:15, W = 3:86, = 0:15, ` = 6:29, ` = 6:07, ` = 1:02, ` = 7:42, ` = 0:65, W = 0:15, ` = 6:52, G = 0:22, and S = 0:358 (all in millimeters). W (Color version available online at: http://ieeexplore.ieee.org.)

to widths of the low- sections, and , are also close to 16 GHz or . They are, however, suppressed by the zero created by the I/O coupled stage [14]. As a result, the 25-dB peak at 22.5 GHz is the next spurious or of the SIR. Fig. 7(d) shows a photograph of the fabricated circuit. One practical important point of this study is as follows. Broadband BPFs are usually hardly realizable by the parallel-coupled configuration using the standard PCB process

CHIOU et al.: BROADBAND QUASI-CHEBYSHEV BPFs WITH MULTIMODE SIRs

TABLE III CHEBYSHEV POLES (THEORY) AND TUNED RESONANT PEAKS (IN GIGAHERTZ) OF THE EIGHTH-ORDER BPF

since the gap sizes are usually too small to fabricate. For example, for a ninth-order BPF with and 0.1-dB ripple on a substrate of and thickness mm, the gap sizes range from 5 to 66 m. Based on the proposed approach, however, the required gap sizes have no such difficulty, as shown in Fig. 7. D. BPF With Both Dual- and Triple-Mode SIRs Two triple-mode SIRs, with , and one dual-mode SIR, with , in the middle are organized to synthesize an eighth-order BPF. Half the circuit layout is shown in Fig. 8(a). In the mode split-off graph (not shown), the resonances and are from the dual-mode element, and they have negligible changes when is varied. The triple-mode SIRs have , , and when is large. Note that the dual- and triple-mode SIRs have identical center frequencies, but distinct individual resonances. Table III lists the theoretical pole positions and tuned resonant peaks when mm. The maximal deviation is only 2.2% and occurs at values of . Fig. 8(b) shows the filter performance from 5 to 11 GHz. The measurement shows that in-band insertion loss is 1.6 dB and for a 15-dB return loss. Fig. 8(c) plots the simulation and measured data up to 25 GHz. The circuit has a wide upper stopband. For example, if a 25-dB rejection is used, the stopband extends to over 25 GHz. This could be attributed to the fact that the circuit consists of SIRs with staggered higher order resonances [16]. Fig. 8(d) presents a photograph of the circuit. IV. CONCLUSION Quasi-Chebyshev BPFs of order up to 9 have been synthesized and implemented by either dual- or triple-mode SIRs. An eighth-order BPF has been realized by both of them, where they have distinct resonant frequencies. Mode graphs have been presented to depict the split off of the resonant frequencies of the coupled SIRs before the I/O feeders are applied. The resonant peaks of the multielement structure have been chosen to match the transmission poles of a Chebyshev bandpass function. It has been validated that their positions are not shifted significantly by the I/O couplings for such multielement circuits with . A particular example has shown that the extra transmission poles created by the strong I/O couplings have little contributions to the rejection levels of the BPF in the transition bands. The measured responses show good agreement with the simulated results. REFERENCES [1] W. Steyn and P. Meyer, “A shorted waveguide-stub coupling mechanism for narrowband multimode coupled resonator filters,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 6, pp. 1622–1625, Jun. 2004.

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[2] U. Rosenberg and W. Hagele, “Advanced multimode cavity filter design using source/load-resonance circuit cross couplings,” IEEE Microw. Guided Wave Lett., vol. 2, no. 12, pp. 508–510, Dec. 1992. [3] R. R. Bonetti and A. E. Williams, “A hexa-mode bandpass filter,” in IEEE MTT-S Int. Microw. Symp. Dig., 1990, pp. 207–210. [4] U. Rosenberg, “Multiplexing and double band filtering with commonmultimode cavities,” IEEE Trans. Microw. Theory Tech., vol. 38, no. 12, pp. 1862–1871, Dec. 1990. [5] K. Wakino, T. Nishikawa, and Y. Ishikawa, “Miniaturization technologies of dielectric resonator filters for mobile communications,” IEEE Trans. Microw. Theory Tech., vol. 42, no. 7, pp. 1295–1230, Jul. 1994. [6] J.-T. Kuo and E. Shih, “Microstrip stepped impedance resonator bandpass filter with an extended optimal rejection bandwidth,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 5, pp. 1554–1559, May 2003. [7] J.-T. Kuo and C.-Y. Tsai, “Periodic stepped-impedance (PSIRR) bandpass filter with a miniaturized area and desirable upper stopband characteristics,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 3, pp. 1107–1112, Mar. 2006. [8] L. Zhu, H. Bu, and K. Wu, “Aperture compensation technique for innovative design of ultra-broadband microstrip bandpass filter,” in IEEE MTT-S Int. Microw. Symp. Dig., 2000, pp. 315–318. [9] W. Menzel, L. Zhu, K. Wu, and F. Bogelsack, “On the design of novel compact broadband planar filters,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 2, pp. 364–370, Feb. 2003. [10] L. Zhu, C. Sun, and W. Menzel, “Ultra-wideband (UWB) bandpass filters using multiple-mode resonators,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 11, pp. 796–798, Nov. 2005. [11] M. Makimoto and S. Yamashita, “Bandpass filters using parallel coupled stripline stepped impedance resonators,” IEEE Trans. Microw. Theory Tech., vol. MTT-28, no. 12, pp. 1413–1417, Dec. 1980. [12] D. M. Pozar, Microwave Engineering, 2nd ed. New York: Wiley, 1998. [13] IE3D Simulator. Zeland Software Inc., Fremont, CA, Jan. 1997. [14] G. L. Matthaei, L. Young, and E. M. T. Jones, Microwave Filters, Impedance-Matching Network, and Coupling Structures. Norwood, MA: Artech House, 1980. [15] J.-T. Kuo, S.-P. Chen, and M. Jiang, “Parallel-coupled microstrip filters with over-coupled end stages for suppression of spurious responses,” IEEE Microw. Wireless Compon. Lett., vol. 13, no. 10, pp. 440–442, Oct. 2003. [16] C.-F. Chen, T.-Y. Huang, and R.-B. Wu, “Design of microstrip bandpass filters with multiorder spurious-mode suppression,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 12, pp. 3788–3793, Dec. 2005. Yi-Chyun Chiou (S’04) was born in Taoyuan, Taiwan, R.O.C., on September 25, 1979. He received the B.S. and M.S. degrees in the electronic engineering from Feng Chia University (FCU), Taichung, Taiwan, R.O.C., in 2001 and 2003, respectively, and is currently working toward the Ph.D. degree in communication engineering at National Chiao Tung University (NCTU), Hsinchu, Taiwan, R.O.C. From 2003 to 2004, he was a Lecturer with the Department of Electronic Engineering, Nan-Kai Institute of Technology, Nanto, Taiwan, R.O.C. His research interests include the design of microwave devices and associated RF modules for microwave and millimeter-wave applications.

Jen-Tsai Kuo (S’88–M’92–SM’04) received the Ph.D. degree from the Institute of Electronics, National Chiao Tung University (NCTU), Hsinchu, Taiwan, R.O.C., in 1992. Since 1984, he was with the Department of Communication Engineering, NCTU, where he is currently a Professor. From 1995 to 1996, he was a Visiting Scholar with the Electrical Engineering Department, University of California at Los Angeles (UCLA). From Aug. 2001 to July 2003, he was the Deputy Department Chair, from Aug. 2003 to July 2005, he was the Director of the Degree Program, and from February 2005 to March 2006, he was the Director of the Industrial Degree Program of the Electrical Engineering and Computer Science Colleges. His research interests include analysis and design of microwave integrated circuits (MICs) and numerical techniques in electromagnetics.

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Dr. Kuo is currently an Editorial Board member of the IEEE TRANSACTIONS and the IEEE MICROWAVE AND a corecipient of the Best Paper Award of the 2002 National Telecommunication Conference, Taiwan, R.O.C.

ON MICROWAVE THEORY AND TECHNIQUES WIRELESS COMPONENTS LETTERS. He was

Eisenhower Cheng was born in Hualien, Taiwan, R.O.C., on January 16, 1981. He received the B.S. degree in the physics from National Tsing Hua University (NTHU), Hsinchu, Taiwan, R.O.C., in 2004, and the M.S. degree in communication engineering from National Chiao Tung University (NCTU), Hsinchu, Taiwan, R.O.C., in 2006. His research interests include the design of microwave planar filters and RF modules for microwave and millimeter-wave applications.

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Novel Coplanar-Waveguide Bandpass Filters Using Loaded Air-Bridge Enhanced Capacitors and Broadside-Coupled Transition Structures for Wideband Spurious Suppression Shih-Cheng Lin, Tsung-Nan Kuo, Yo-Shen Lin, Member, IEEE, and Chun Hsiung Chen, Fellow, IEEE

Abstract—Novel inline coplanar-waveguide (CPW) bandpass filters composed of quarter-wavelength stepped-impedance resonators are proposed, using loaded air-bridge enhanced capacitors and broadside-coupled microstrip-to-CPW transition structures for both wideband spurious suppression and size miniaturization. First, by suitably designing the loaded capacitor implemented by enhancing the air bridges printed over the CPW structure and the resonator parameters, the lower order spurious passbands of the proposed filter may effectively be suppressed. Next, by adopting the broadside-coupled microstrip-to-CPW transitions as the fed structures to provide required input and output coupling capacitances and high attenuation level in the upper stopband, the filter with suppressed higher order spurious responses may be achieved. In this study, two second- and fourth-order inline bandpass filters with wide rejection band are implemented and thoughtfully examined. Specifically, the proposed second-order filter has its stopband extended up to 13 3 0 , where 0 stands for the passband center frequency, and the fourth-order filter even possesses better stopband up to 19 04 0 with a satisfactory rejection greater than 30 dB. Index Terms—Air bridge, bandpass filter, coplanar waveguide (CPW), microstrip, quarter-wavelength resonator, spurious suppression, stepped-impedance resonator (SIR).

I. INTRODUCTION OPLANAR WAVEGUIDE (CPW) as one type of popular guided structures [1] has received much attention in the field of bandpass-filter design. Comparing with other planar transmission-line configurations, such as microstrip lines, the CPW permits easier integration of both series and shunt components since its grounds are located on the same surface as the signal line. It also possesses the features such as low dispersion, easy fabrication of short-circuited elements, and insensitivity to the substrate thickness, etc. With such advantages, the CPW turns into an admirable candidate for microwave filter implementation.

C

Manuscript received March 1, 2006; revised May 2, 2006. This work was supported by the National Science Council of Taiwan under Grant NSC 94-2752-E002-001-PAE, Grant NSC 94-2219-E-002-008, and Grant NSC 94-2213-E-002055. S.-C. Lin, T.-N. Kuo, and C. H. Chen are with the Department of Electrical Engineering and Graduate Institute of Communication Engineering, National Taiwan University, Taipei 106, Taiwan, R.O.C. (e-mail: [email protected]. tw). Y.-S. Lin is with the Department of Electrical Engineering, National Central University, Chungli 320, Taiwan, R.O.C. Digital Object Identifier 10.1109/TMTT.2006.879175

Over the past few years, a considerable amount of studies have been made on the development of CPW bandpass filters. Capacitively end-coupled half-wavelength resonator filters built in the CPW by excavating gaps in the center conductor were investigated in [2]. To circumvent the high radiation loss produced from the gaps, inductively direct-coupled bandpass filters [3], [4] were demonstrated by means of patterning a thin strip across the center conductor to join the grounds. The CPW filters with attenuation poles for improved skirt selectivity were proposed both in [5] by using tapped feeds for input/output and interstage couplings and in [6] by rearranging the meander-line interval and shape to introduce cross coupling. Since these filters are principally based on uniform-impedance resonators, their spurious passbands are observed around multiples of the passband center frequency . For the purposes of size miniaturization and pushing the first spurious passband higher, quarter-wavelength resonators are adopted in CPW filter design. The tapped-feed combline-type bandpass filters using the CPW resonators were presented in 1995 [7]. Distinct from the conventional combline-type configuration, the CPW resonators with two ends driven by series-capacitive and shunt-inductive coupling elements [8] were designed into hairpin shapes to implement a miniaturized end-coupled filter owning multiple passes. Furthermore, originated from the transmission-line stepped-impedance resonators (SIRs), which are composed of transmission lines (TLs) with different characteristic impedances [9], the SIR inline CPW filter was proposed in [10] on the basis of a conventional filter designed with impedance inverters. As regards the spurious passbands, the filters consisting of resonators have the higher order resonances occurring around , instead of for the ones made of resonators. Since the filters composed of distributed elements inevitably come into existence of unwanted spurious responses in the upper frequency band, which may degrade the filter performance in rejecting the out-of-band interference, numerous attempts have been made using various techniques [11]–[14] to break away from those annoying spurious passbands. By means of proper tappings at both input and output resonators, two independent notches can be created at specified frequencies, thereby neutralizing the spurious passbands [11]. Only those bandpass filters with tapped input/output feeds may take advan-

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Fig. 2. Equivalent transmission-line model of the proposed =4 SIR loaded by a lumped-element capacitor (Y = 1=Z ; Y = 1=Z ; R = Z =Z = Y =Y ).

Fig. 1. Three-dimensional physical layout of the proposed second-order inline CPW bandpass filter loaded by air-bridge enhanced capacitors and fed by microstrip-to-CPW transition structures for wideband rejection.

tage of these notches to suppress the spurious responses. For the sake of removing the second harmonic response of conventional microstrip parallel coupled-line filters, omnigenous notions to equalize the even- and odd-mode velocities were reported [12]–[14] such as perturbing the shapes of couple-line sections, employing the suspended coupled microstrips on a substrate with a proper suspension height, etc. By using various dissimilar SIRs, wide-stopband filters may be accomplished [15]. However, the utilization of dissimilar resonators for spurious suppression demands long tuning period and destructs the symmetry of the whole filter structure. Up to now, the focus of the published literature was mainly on the improvement of the out-of-band attenuation performance of microstrip-type filters. Little attention has been drawn to the spurious-suppression methods suitable for filters of CPW type [16], [17], especially to those adequate for the inline CPW bandpass filter. The particular pattern of via-holes was used in the double-surface CPW filter [16] to reduce the spurious responses. Another four-pole bandpass filter was introduced in [17] using the two asymmetric parallel-coupled CPW stages at the input/output terminated ends to generate the two separated transmission zeros for cancelling the harmonic passband occurring at . As mentioned above, very few published documents were reported to achieve wide rejection band and effective suppression of the spurious passbands, especially for the inline CPW bandpass filter design. In this paper, novel inline CPW bandpass filters with very wide rejection band are proposed. The proposed filter configuration is composed of CPW SIRs, which are loaded by the air-bridge enhanced capacitors for lower order spurious suppression, as well as size reduction, and are coupled to the input/output ports through the microstrip-to-CPW transition structures for higher order spurious attenuation. Fig. 1 shows one demonstration of a proposed second-order inline CPW filter for wide stopband performance. The loaded lumped-element capacitors are implemented by means of enhancing the air-bridge strips printed over the CPW structure. By properly adjusting the loaded capacitors, the odd-number spurious passbands of proposed filters may essentially be destructed. Moreover, the transmission zero

inherently associated with the air-bridge enhanced capacitors may suitably be allocated to cancel the second spurious passband, thereby effectively suppressing the lower order spurious passbands. In addition, by adopting the broadside-coupled microstrip-to-CPW transitions as the input/output fed structures so as to provide required input/output capacitive couplings at the passband and also to give a high attenuation level in the upper stopband, the remaining unsuppressed higher order spurious passbands of proposed filter may effectively be attenuated. In this study, the concepts of spurious destruction, lower order spurious cancellation, and higher order spurious attenuation are demonstrated by the full-wave simulation of a proposed second-order filter. By making full use of the above-mentioned spurious-suppression mechanisms, second- and fourth-order inline CPW bandpass filters are implemented with the stopbands extended up to and . II. SIRS LOADED BY AIR-BRIDGE ENHANCED CAPACITORS A. Resonance Condition The equivalent transmission-line model of the proposed SIR loaded by a lumped-element capacitor is shown in Fig. 2. The resonator is composed of two transmission-line sections with unequal characteristic impedances. The section with open-circuited termination has a characteristic impedance and electrical length , while the section with short-circuited termination owns characteristic impedance and electrical length . A lumped-element capacitor with capacitance is attached between the two sections. For this proposed resonator, the input admittance seen from the open-end side and the input impedance seen form the short-end side are simply given by (1) (2) denotes the susceptance of the loaded capacwhere itor. The proposed resonator (Fig. 2) exhibits parallel or series resonances when or , both leading to the resonance condition (3) represents the impedance ratio where between low- and high-impedance sections. With the aid of the

LIN et al.: NOVEL CPW BANDPASS FILTERS

Fig. 3. Normalized length versus the length ratio with parameters.

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C

(in picofarads) as

simple (3), the fundamental and higher order resonant frequencies may easily be calculated. Note that these results reduce to those in [9] when , as expected. To demonstrate the effect of the loaded capacitor, Fig. 3 shows the relationship between the length ratio and the normalized resonator length for . Here, the value of has been fixed to be 50 and the center frequency is set to be 1.5 GHz to facilitate the calculation. One can easily observe from Fig. 3 that the larger the capacitance, the more evident the capacitor reduces the normalized length. Equivalently, the size of the proposed resonator may be reduced by simply increasing the loaded capacitor without requiring large impedance difference between low- and high-impedance sections. It is known that a very low impedance ratio sometimes is not achievable under the fabrication process limit since low requires extremely wide or narrow metal-strip width. Wide metal-strip width may cause transverse resonance, while narrow metal-strip width may increase the conductor loss. With the aid of our proposed resonator, the requirement of enforcing low for resonator miniaturization may be prevented. B. Physical Realization and Model Extraction Bond wires or air bridges are common in CPW design for suppressing the unwanted modes such as the slot-line mode. Thin wires are usually introduced to keep original circuits unperturbed. The loaded capacitor for the proposed resonator can simply be implemented by enhancing the conventional air bridges. The three-dimensional physical realization and the top-/bottom-plane layouts of the proposed SIR loaded by air-bridge enhanced capacitor are shown in Fig. 4(a) and (b). The physical widths and are determined by the impedance ratio of the resonator, while the lengths and are governed by the length ratio. By employing a metal strip printed over the CPW structure and connecting two ends of the strip to CPW grounds through via-holes, an air-bridge enhanced capacitor can be constructed. The strip not only plays the role of suppressing the unwanted modes, but also serves as

Fig. 4. (a) Three-dimensional physical layout. (b) Top-/bottom-plane layouts of the proposed CPW =4 SIR loaded by air-bridge enhanced capacitor with dimensions labeled (W = 1:02; G = 1:27; W = 6:35; G = 0:51; W = 11:56; D = 1:78).

a shunt capacitor across the middle of SIR. As observed from Fig. 4, the region formed by the CPW center conductor and the metal strip of the air-bridge span provides the metal–insulator–metal capacitance. However, due to the fringing effect, step discontinuity, and some parasitic effects, this structure becomes too complicated to be analyzed. Although the use of via-holes is somewhat of a drawback, industry may build large number of via-holes without adding too much cost and complexity in the modern fabrication process. Alternatively, the use of via-holes provides the advantage that facilitates the designers to extend microwave components from planar structure to three-dimensional topology. Therefore, the occupied size of components may become miniaturized, as expected. In this study, by making good use of via-holes, one may build loaded capacitors in the CPW structure with satisfactory large capacitance. Actually, the enhanced air-bridge structure behaves capacitive at lower frequency band, but resonates at higher frequency. The lumped-element model shown in Fig. 5(a) may be adopted to characterize this structure by setting two ports at low- and high-impedance lines separately. The inductors , , and capacitors are contributed by discontinuity effects. The

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Fig. 5. (a) Lumped-element model. (b) Simplified RLC resonator model of the proposed air-bridge enhanced capacitor.

kernel portion of the enhanced air bridge is modeled by the couand the LC resonant tank ( and ). Each pling capacitor element value for this model may be extracted from full-wave simulated scattering parameters through optimization. Since the enhanced air bridge behaves like a series resonator, it can approximately be modeled as a series RLC resonator, as shown in Fig. 5(b). As a matter of fact, the model in Fig. 5(a) may show better agreement with the real structure, but its application will be restricted. Accordingly, the simplified model [see Fig. 5(b)] is used in the following discussion. In the beginning, the approximate parameters for the simplified series resonator model may be extracted using the equations given in [18]. The relationship between the transmission coefficient and and may be expressed as (4) From (4), one may evaluate the values of given by

and

Fig. 6. Comparison of the results, for the structure shown in Fig. 4(a), from full-wave simulation and a simplified model. Here, C represents one of the ABCD -matrix parameters of the structure.

Fig. 7. Top-/bottom-plane layouts of the broadside-coupled microstrip-toCPW transition for higher order spurious attenuation with W to be adjusted. (W = 1:12; D = 1:1; W = 6:35; G = 0:51, the substrate thickness h = 0:508, the dielectric constant " = 3:38).

, as

(5a)

the equivalent loaded capacitance of this structure approximately equals . The related simulated result of using only is also included in Fig. 6 for comparison. III. BROADSIDE-COUPLED MICROSTRIP-TO-CPW TRANSITION STRUCTURE

(5b) where

represents the , and and

series resonant frequency, are the frequencies at which

. Fig. 6 illustrates the comparison between full-wave simulated and simplified model results including the transmission coefficient and the imaginary part of “ ,” which is one of the -matrix parameters. The air-bridge enhanced structure is fabricated on a Rogers RO4003 substrate of thickness 0.508 mm, dielectric constant 3.38, metal thickness 17 m, and loss tangent 0.0027. The diameter of via-holes is 1 mm. The dimensions for the simulated structure are annotated in Fig. 4. The extracted element values of the simplified model are nH and pF. Note that the responses for full-wave simulation and a simplified model coincide well over a wide frequency range. In the lower frequency band, this simplified model may be reduced to only a capacitor left since the inductor will be of minor significance. Therefore,

A broadside-coupled microstrip-to-CPW transition structure [19], [20] is adopted to feed the proposed inline CPW bandpass filter, as shown in Fig. 1. As a transition, this structure should provide a broadband transmission behavior. However, as a fed structure for the proposed CPW filter, it is only required to implement the predetermined capacitive coupling level. Therefore, the condition that the longitudinal coupled-strip length should approximately equal half-wavelength is no longer of necessity. For spurious suppression, the broadside-coupled microstrip-to-CPW transition structure should properly be designed so as to provide required feeding capacitance for the filter at the center frequency and to give a large insertion loss in the higher frequency band. Shown in Fig. 7 is the corresponding circuit layout fabricated on the 0.508-mm-thick RO4003 substrate for higher order spurious suppression and 1.5-GHz filter application. Here, the conductor and slot widths ( and ) of the CPW portion are specified by the required impedance for the SIR. The required metal–insulator–metal capacitance essentially determines the coupled-strip length between the microstrip and CPW after assigning

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shown in Fig. 9 are presented in Fig. 10(a) and (b). Defining and with and given by (1) and (2), the susceptance and reactance slope parameters can be obtained from their definitions as follows: (6) With the derived susceptance and reactance slope parameters, one may express the filter design parameters as

W

Fig. 8. Simulated frequency responses of the structure in Fig. 7 for various values of . (Unit: millimeters).

. The remaining dimension left to be adjusted is with mm being fixed in characterizing this structure. Fig. 8 depicts the four sets of full-wave simulated . frequency results under the selected dimensions for When just equals , a deep transmission notch occurs at 26.64 GHz, which is the frequency at which becomes a full-wavelength-long microstrip counterpart. Equivalently, the length measured from the open end of the upper microstrip to the feed-line center is equal to a half-wavelength. Physically, this notch may be associated with the fundamental resonance of a strip structure for which the oppositely directed vertical-coupled electric-field distributions between the overlapped portion of microstrip and CPW would be cancelled along the half-wavelength resonator; thus, no signal would pass through this transition. As increases, the attenuation ability degrades gradually in the higher frequency band and the notch gets unobvious. Note that the response around 1.5 GHz remains unaltered, implying that the required capacitive coupling is almost unchanged as is changed. Note that a critical value of 11.4 mm of resulting in unfavorable rejection at high frequency over 25 GHz is observed in Fig. 8. In other words, needs to be adjusted for our specification of the rejection band. In this demonstrated case, the optimal value of for a high rejection level at high frequency ranges from 9.2 to 10.42 mm. IV. FILTER DESIGN For the proposed CPW filter with loaded air-bridge enhanced capacitors, its design procedure needs to be clearly specified. Based on the filter synthesis technique developed by Matthaei et al. [21], the general design equations for the proposed th-order filter, which is composed of SIRs together with loaded capacitors, will systematically be established here. Given the desired filter specification of center frequency and factional bandwidth , Fig. 9 reveals the equivalent circuit of our proposed bandpass filter of order loaded by lumped-element capacitors. The filter is made of SIRs, which are coupled alternatively to - and -inverters. Each resonator is comprised of two joined transmission-line sections with impedances and , and electrical lengths and . The loaded capacitor for the resonator has a capacitance . The equivalent circuits for the - and -inverter blocks

(7) where are the element values of the low-pass prototype filter. With the - and -inverter values calculated from (7), the corresponding capacitances, inductances, and electrical lengths may be found using the formulas for the conventional impedance and admittance inverters comprised of lumped and transmission-line elements [22]. As a result, the complete design formulas for the inline bandpass filter with loaded capacitors are established. In this study, the -inverters at input and output stages will be implemented by the broadside-coupled microstrip-to-CPW transitions, as shown in Fig. 7, to provide suitable capacitive coupling at low frequency and spurious attenuation at high frequency. For the other values, they are all realized by using the gap capacitors between the open ends of CPW SIRs, as shown in Fig. 11(a). The -inverters are realized by the shunt inductors implemented between the short ends of CPW SIRs, as shown in Fig. 11(b). Note that the electrical length and may be converted to physical CPW lengths denoted as and through the propagation constants for different CPW impedances and of resonators. V. MECHANISMS OF WIDEBAND SPURIOUS-SUPPRESSION A. Spurious Destruction Very wide rejection band may potentially be achieved for the inline coplanar filter when appropriate spurious-suppression mechanisms are used. Based on the second-order filter with the equivalent circuit shown in Fig. 12, the concept for spurious destruction will qualitatively be discussed. At center frequency, the proposed filter (Fig. 12) is simply the cascade of two SIRs loaded by capacitors. On the foundation of the theory established by Matthaei et al. [21], an inline bandpass filter

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Fig. 9. Equivalent circuit of the proposed N th-order bandpass filter using =4 SIRs with loaded lumped-element capacitors.

TABLE I REQUIRED PARAMETERS FOR SECOND-ORDER BANDPASS FILTER SHOWN IN FIG. 12 WITH ABSENCE OF ODD-NUMBER SPURIOUS PASSBANDS

Fig. 10. (a) J -inverter network realized by series capacitor C . (b) K -inverter network realized by shunt inductor L between resonators i and (i + 1).

Fig. 11. Realizations of J - and K -inverters (capacitor and inductor) adopted in the proposed CPW filter. (White region: slot. Gray region: conductor.)

Fig. 12. Equivalent circuit of the proposed second-order bandpass filter with absence of odd-number spurious passbands.

using TLs as its distributed elements requires - and -inverters placed alternatively between the resonant elements to produce a passband at a designated frequency. It implies that once the placement of - and -inverters does not follow the above-mentioned rule, the filter passband may be destructed. Instead of constructing the center passband, our proposed filter does exactly the opposite to destruct the spurious passband at . Our effort is to arrange an improper placement of - and -inverters at first spurious passband on the basis of the secondorder filter. One can easily observe that there are four sections of

TL in the second-order filter using capacitor-loaded SIRs. The two loaded capacitors may be considered as another two -inverters realized by shunt capacitors. By properly choosing the values of , impedance ratio , and length ratio , the first spurious resonance can be pushed up to the frequency around . The frequency-dependent electrical lengths and at will be times their electrical lengths at , i.e., , under the assumption that the electrical lengths are proportional to the frequency. Moreover, the capacitor-loaded SIR must appropriately be designed to give and , which are simultaneously close to 90 . As a result, after the absorption of electrical lengths from - and -inverters at , the second-order filter configuration designed at behaves approximately like a cascade of -90 TL- -90 TL- -90 TL- -90 TL- , equivalently a fourth-order filter with resonators at . Since the - and -inverters are improperly arranged as expected, the spurious passband at eventually will not appear. A circuit-level simulation is conducted to demonstrate the spurious-destruction technique. To this end, a second-order filter is designed with center frequency GHz, and 3-dB fractional bandwidth of 10% for Chebyshev response. Choosing pF, , and , the corresponding electrical lengths of resonators will be . The required design parameters for the second-order filter evaluated from (7) are listed in Table I. The calculated first five higher order spurious resonances are at 6.44, 11.07, 16.2, 21.57, and 26.45 GHz ( and ), respectively, as shown in Fig. 13 (bottom half). Since the and at are 108.4 , which would be close to 90 after the absorption of electrical lengths from the - and -inverters at , the condition for resonance destruction is met. The circuit simulation is carried out by AWR Microwave Office to get the frequency responses presented in Fig. 13 (upper half). Apparently, over the frequency range of 1.0–30 GHz, the filter only exhibits the spurious passbands around and , while the passbands at and are completely absent. In conclusion, the odd-number spurious resonances ( etc.) will entirely be destructed due to the notion of improper inverter arrangement.

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C. Higher Order Spurious Attenuation

i

Fig. 13. Frequency responses of the proposed second-order filter (top half) described in Table I and the corresponding th spurious (harmonic) frequencies (bottom half). denoted as

f

Thus far, three spurious passbands have been either destructed or suppressed with the innovative use of the air-bridge enhanced capacitors. The next effort is focused on the suppression of the rest of higher order resonances. The broadside-coupled microstrip-to-CPW transition described in Section III holds the possibility to attenuate the passed signal in the higher frequency band exceeding . As illustrated in Fig. 8, the microstrip-to-CPW transition structure resembles a low-pass structure when the dimensional sizes are optimally tuned. By designing the strip length (Fig. 7) ranging from 9.2 to 10.42 mm, the higher order spurious passbands not lower than may be attenuated. Of course, when fabricating this structure in a different substrate, this range might be different. Eventually, the key steps to realize the proposed filter with a wide stopband are worthy of mention and are concluded as follows. First of all, the values of and should properly be chosen to make the utilized SIR with and approximately equal to 90 after the electrical-length absorption from inverters at the first spurious frequency so that the odd-number spurious resonances may entirely be destructed. The capacitor width (Fig. 4) is then suitably adjusted such that the inherent transmission zeros associated with the air-bridge enhanced capacitor is allocated around , thereby canceling the second spurious passband around . Finally, the spurious passbands not lower than is attenuated by optimally tuning the dimension (Fig. 7) of the microstrip-to-CPW transition structures. By combining the abovementioned spurious-suppression schemes, a very-wide rejection band can be accomplished in the proposed filter configuration. VI. FILTER IMPLEMENTATION AND RESULTS

W

Fig. 14. Curve to relate transmission-zero frequencies, inherently associated (Fig. 4). with the air-bridge enhanced capacitor, to capacitor strip width

B. Spurious Cancellation A transmission zero is inherently created by the air-bridge enhanced capacitor structures discussed in Section II. Adjusting the dimensional sizes of strip, the transmission-zero position can be moved downward or upward to the specified frequency point. Based on the structure in Fig. 4 with all dimensions identical, except for , the relation between the transmission-zero frequencies inherently associated with the air-bridge enhanced capacitor and the capacitor strip width is plotted in Fig. 14. One can easily observe that the wider the strip, the lower the transmission-zero frequency. Intuitively, the related element values of the simplified model would be altered by adjusting the width , but the would only slightly be changed. To predict the generation of the transmission zero, the inductor presented in the simplified model for air-bridge enhanced capacitor needs to be included into the equivalent circuit of the proposed filter. Ensuring that the series resonance of the LC tank occurs exactly at , the spurious passband arising there will be suppressed. Once this spurious is cancelled, the next repeated passband for the demonstrated second-order filter listed in Table I will appear at 21.57 GHz .

Sufficient information for designing the inline CPW bandpass filters loaded by air-bridge enhanced capacitors with very-wide rejection bandwidth has been provided. To validate the design procedures, several filters based on the proposed structures are fabricated and thoughtfully examined. A. Second-Order Filter Spurious-destruction technique has been detailed qualitatively by simulating a second-order filter as an example in Section V. Further verification is done by fabricating the second-order filter with center frequency of 1.5 GHz, 3 dB-fractional bandwidth of 10% on the FR4 substrate ( mm, ). The SIRs possess the first four predicted spurious resonances at 6.69, 10.24, 16.7, and 19.77 GHz with the given parameters , and pF. Accordingly, the corresponding electrical lengths of SIRs are and . Since and at are 123.45 and 109.45 , which are close to 90 after the electrical-length absorption from inverters, the resonance-destruction condition is met. The filter design parameters evaluated from (7) are listed in Table II. The air-bridge enhanced capacitor is designed with the extracted nH and pF to result in a transmission zero around 9.31 GHz for cancelling the spurious passbands at 9.56 GHz . Note that the second spurious passband of the

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No spurious passbands appear at 6.69 and 16.7 GHz because of spurious-destruction condition. The spurious resonance at 19.77 GHz has also been effectively suppressed using the broadside-coupled microstrip-to-CPW transition structures. Note that the simulated result of the conventional inline CPW filter with identical without loaded is also presented in Fig. 16. It is found that the rejection level has been significantly improved in our proposed second-order filter.

TABLE II REQUIRED PARAMETERS FOR FILTERS IN SECTION VI

B. Fourth-Order Filter

W

Fig. 15. Top-/bottom-plane layouts of the proposed second-order bandpass = filter loaded by air-bridge enhanced capacitors described in Table II. ( 1:02; G = 1:27; W = 6:35; G = 0:51; D 2:9; d = 0:76; W = 10:74; D = 1:78; D

= 5:9;D = 0:79; W

= 6:93; s = 9:02).

=

Fig. 16. Measured/simulated frequency responses of the proposed second-order filter fabricated on the 0.508-mm-thick FR4 substrate. (The second-order filter constructed by SIRs with identical R without loaded C is also included as “SIR-NoC” for comparison.)

fabricated filter shifts approximately 0.68 GHz lower than the calculated one. It is due to some parasitic effects of the CPW resonators using air-bridge enhanced capacitors not taken into consideration when calculating by (3). However, the resonance frequencies are still predicted accurately enough for our design. The physical layout is presented in Fig. 15 with dimensions labeled. The full-wave simulated result with the dielectric loss excluded is shown in Fig. 16 together with the measured result for comparison. During the simulation, it is found that the loss obviously influences the in-band insertion loss, but is of minor significance on the out-of-band rejection. The fabricated filter has a center frequency at 1.549 GHz, measured 3-dB fractional bandwidth of approximately 11.27%, minimum insertion loss of 2.723 dB, and minimum passband return loss of 15.5 dB. The filter has a miniature size of only , where stands for the wavelength of 50- CPW at center frequency. This second-order bandpass filter possesses a stopband extended up to 20 GHz with a rejection level better than 25.19 dB.

Intuitively, the attenuation level of the stopband can further be reinforced by increasing the filter order. Another fourth-order bandpass filter loaded by air-bridge enhanced capacitors is implemented on a 0.508-mm-thick RO4003 substrate. Since the fourth-order filter is symmetric with respect to [see Fig. 17(a)], only the design parameters on the left-hand side of the equivalent circuit are presented in Table II. This filter is designed with GHz, 3-dB fractional bandwidth % for the Chebyshev response. The equivalent circuit of this filter is illustrated in Fig. 17(a). Shown in Fig. 17(b) are the layouts of the fourth-order filter with dimensions labeled. The extracted and of the simplified equivalent model related to the dimensions of the air-bridge enhanced capacitor are 0.23 nH and 1.14 pF with respect to mm and mm. The SIRs loaded by capacitors in this design have parameters, and pF to satisfy the spurious-destruction condition. Accordingly, the predicted first four spurious resonances of utilized resonators are 6.58, 11.37, 16.85, and 22.22 GHz ( and ). The loaded air-bridge enhanced capacitors introduce an inherent transmission zero in the proximity of 9.83 GHz to cancel the second spurious passband at 9.92 GHz . The offset of the second spurious passband is also due to the parasitic effects of the utilized SIRs not taken into consideration. The broadside-coupled microstrip-to-CPW transition has been optimally adjusted to possess mm to increase the attenuation level exceeding 20 GHz. Consequently, a very wide-stopband inline CPW filter has been fabricated. Its measured and simulated results are shown in Fig. 18. This fourth-order filter has a center frequency of 1.48 GHz, measured 3-dB fractional bandwidth of approximately 11.27%, minimum insertion loss of 1.73 dB, and minimum passband return loss of 15.5 dB. The proposed filter has a miniature size of only . The stopband is extended up to 28.18 GHz with a rejection level better than 30 dB. Note that the simulated results of a conventional inline CPW filter using uniform-impedance resonators with the same specification is also included in Fig. 18 as a curve “Inline UIR” for reference. Remarkably, more than eight spurious passbands associated with the filter using a uniform-impedance resonator are suppressed in our proposed filter. Note that there is some discrepancy between measured and simulated results from 5 to 13 GHz due to the inaccuracy of aligning the top- and bottom-plane layouts in the printed-circuit board fabrication. However, the level and trend of the measured out-of-band response are still predicted by the simulated one,

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W

Fig. 17. (a) Equivalent circuit. (b) Top-/bottom-plane layouts of the proposed fourth-order CPW bandpass filter loaded by air-bridge enhanced capacitors de= 1:02; G = 1:27; W = 6:35; G = 0:51; D = 5:05; D = 7:95; s = 3:3; d = 0:76; D = 7:95; D = 10:8; scribed in Table II. ( = 1:63;W = 1:12; D = 9:4; W = 11:56; D = 1:78). d

Fig. 19. Rectangular housing structure for the second-order filter in Fig. 15. (W = 30:48 mm, L = 41:66 mm, H = 15:24 mm). Note that the bold black lines represent the housing boundary and “O” indicates the coordinate origin. (a)

VII. HOUSING EFFECTS

(b) Fig. 18. (a) Narrowband and (b) wideband measured/simulated frequency responses of the proposed fourth-order filter described in Table II fabricated on the 0.508-mm-thick RO4003 substrate.

thereby verifying the three spurious suppression mechanisms, as mentioned in Section V.

For system application, the filter is usually surrounded by the metallic housing for preventing unnecessary interferences. Physically, the housing may cause unwanted cavity resonances to degrade the wideband spurious-suppression characteristic. It is very complicated to conduct the simultaneous design of the housing and filter. Therefore, a separate design of housing is essential from the practical point-of-view. Note that the cavity resonances would eventually appear no matter how the housing is arranged and reshaped. An approach to effectively suppress the unwanted cavity resonances is to attach the absorbing material inside the housing walls. Shown in Fig. 19 is a typical rectangular housing structure for the proposed second-order filter in Fig. 15. Curve 1 of Fig. 20 exhibits the simulated result of the filter with housing (but without the absorbing material). Obviously, the wideband spurious-suppression characteristic has been destructed due to the cavity resonances and the first spurious passband occurs at the resonant frequency of the cavity associated with the mode. By attaching the absorbing material on the top

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Fig. 20. Results of the second-order filter with and without the housing. (Curve 1: housing only, Curve 2: with absorber, Curve 3: without housing.)

and bottom walls of the housing, those unwanted resonances disappear, as shown in the simulated Curve 2 in Fig. 20. The ability of spurious suppression is recovered as well. Note that the measured result of the second-order filter (Curve 3) is also included in Fig. 20 for comparison. VIII. CONCLUSION To the authors’ knowledge, the inline CPW bandpass filters loaded by capacitors with wideband spurious suppression have never been presented. The filters composed of SIRs loaded by lumped-element capacitors with an absence of an odd number of spurious passbands are originally investigated in this study. Based on the mechanisms of the spurious destruction, the spurious cancellation due to the transmission zero inherently associated with the air-bridge enhanced capacitor structure, and the higher order spurious attenuation provided by broadside-coupled microstrip-to-CPW transition, the implemented fourth-order inline CPW filter has its stopband significantly extended up to 19.04 with a rejection level better than 30 dB. The implemented filter occupies a size also smaller than that of the conventional one due to the use of resonators loaded by lumped-element capacitors. Since all occupied capacitors are realized in a dual-plane vertical configuration, the proposed inline CPW filters are especially suitable for the multilayer fabrication process. ACKNOWLEDGMENT The authors would like to thank Dr. C.-H. Wang, National Taiwan University, Taipei, Taiwan, R.O.C., for his valuable discussions and suggestions on this topic. REFERENCES [1] C. P. Wen, “Coplanar waveguide: A surface strip transmission line suitable for nonreciprocal gyromagnetic device applications,” IEEE Trans. Microw. Theory Tech., vol. MTT-17, no. 12, pp. 1087–1090, Dec. 1969. [2] D. F. Williams and S. E. Schwarz, “Design and performance of coplanar waveguide bandpass filters,” IEEE Trans. Microw. Theory Tech., vol. MTT-31, no. 7, pp. 558–566, Jul. 1983. [3] J. K. A. Everard and K. K. M. Cheng, “High performance direct coupled bandpass filters on coplanar waveguide,” IEEE Trans. Microw. Theory Tech., vol. 41, no. 9, pp. 1568–1573, Sep. 1993.

[4] A. Vogt and W. Jutzi, “An HTS narrow bandwidth coplanar shunt inductively coupled microwave bandpass filter on LaAl ,” IEEE Trans. Appl. Supercond., vol. 45, no. 4, pp. 492–497, Apr. 1997. [5] K. Wada and I. Awai, “Heuristic models of half-wavelength resonator bandpass filter with attenuation poles,” Electron. Lett., vol. 35, no. 5, pp. 401–402, Mar. 1999. [6] H. Kanaya, J. Fujiyama, R. Oba, and K. Yoshida, “Design method of miniaturized HTS coplanar waveguide bandpass filters using cross coupling,” IEEE Trans. Appl. Supercond., vol. 13, no. 2, pp. 265–268, Jun. 2003. [7] K. Wada, Y. Noguchi, E. Higashino, and J. Ishii, “Tapped-feed combline-type coplanar waveguide resonator bandpass filters,” in IEEE MTT-S Int. Microw. Symp. Dig., Feb. 1995, pp. 141–146. [8] T. Tsujiguchi, H. Matsumoto, and T. Nishikawa, “A miniaturized endcoupled bandpass filter using =4 hair-pin coplanar resonators,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 1998, pp. 829–832. [9] M. Sagawa, M. Makimoto, and S. Yamashita, “Geometrical structures and fundamental characteristics of microwave stepped-impedance resonators,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 7, pp. 1078–1085, Apr. 1997. [10] A. Sanada, H. Takehara, and I. Awai, “Design of the CPW in-line =4 stepped-impedance resonator bandpass filter,” in Proc. Asia–Pacific Microw. Conf., Dec. 2001, pp. 633–636. [11] J.-T. Kuo and E. Shih, “Microstrip stepped impedance resonator bandpass filter with an extended optimal rejection bandwidth,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 5, pp. 1554–1559, May 2003. [12] T. Lopetegi, M. A. G. Laso, J. Hernández, M. Bacaicoa, D. Benito, M. J. Garde, M. Sorolla, and M. Guglielmi, “New microstrip ‘wigglyline’ filters with spurious passband suppression,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 9, pp. 1593–1598, Sep. 2001. [13] T. Lopetegi, M. A. G. Laso, F. Falcone, F. Martin, J. Bonache, J. Garcia, L. Perev-Cuevas, M. Sorolla, and M. Guglielmi, “Microstrip ‘wiggly-line’ bandpass filters with multispurious rejection,” IEEE Microwave Wireless Compon. Lett., vol. 14, no. 11, pp. 531–533, Nov. 2004. [14] J.-T. Kuo, M. Jiang, and H.-J. Chang, “Design of parallel-coupled microstrip filters with suppression of spurious resonances using substrate suspension,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 1, pp. 83–89, Jan. 2004. [15] S.-C. Lin, P.-H. Den, Y.-S. Lin, C.-H. Wang, and C. H. Chen, “Wide-stopband microstrip bandpass filters using dissimilar quarter-wavelength stepped impedance resonators,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 3, pp. 1011–1018, Mar. 2006. [16] T. Tsujiguchi, H. Matsumoto, and T. Nishikawa, “A miniaturized double-surface CPW bandpass filter improved spurious responses,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 5, pp. 879–885, May 2001. [17] J. Gao and L. Zhu, “Asymmetric parallel-coupled CPW stages for harmonic suppressed =4 bandpass filters,” Electron. Lett., vol. 40, no. 18, pp. 1122–1123, Sep. 2004. [18] C.-C. Chang, C. Caloz, and T. Itoh, “Analysis of a compact slot resonator in the ground plane for microstrip structures,” in Proc. Asia–Pacific Microw. Conf., Dec. 2001, pp. 1100–1103. [19] J. J. Burke and R. W. Jackson, “Surface-to-surface transition via electromagnetic coupling of microstrip and coplanar waveguide,” IEEE Trans. Microw. Theory Tech., vol. 37, no. 3, pp. 519–525, Mar. 1989. [20] L. Zhu and W. Menzel, “Broad-band microstrip-to-CPW transition via frequency-dependent electromagnetic coupling,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 5, pp. 1517–1522, May 2004. [21] G. Matthaei, L. Young, and E. Jones, Microwave Filters, Impedance Matching Networks, and Coupling Structures. New York: McGrawHill, 1964, pp. 427–440, 464–471. [22] J. S. Hong and M. J. Lancaster, Microstrip Filter for RF/Microwave Applications. New York: Wiley, 2001, pp. 56–63.

Shih-Cheng Lin was born in Taitung, Taiwan, R.O.C., in 1981. He received the B.S. degree in electrical engineering from National Sun Yet-Sen University, Kaohsiung, Taiwan, R.O.C., in 2003, and is currently working toward the Ph.D. degree in communication engineering at National Taiwan University, Taipei, Taiwan, R.O.C. His research interests include the design and analysis of microwave filter circuits and passive components.

LIN et al.: NOVEL CPW BANDPASS FILTERS

Tsung-Nan Kuo was born in Taoyuan, Taiwan, R.O.C., in 1981. He received the B.S. degree in electrical engineering from National Dong Hwa University, Hualien, Taiwan, R.O.C., in 2003, the M.S.E.E. degree from National Taiwan University, Taipei, Taiwan, R.O.C., in 2005, and is currently working toward the Ph.D. degree at National Taiwan University. His research interests include the design and analysis of microwave filter circuits.

Yo-Shen Lin (M’04) was born in Taipei, Taiwan, R.O.C. in 1973. He received the B.S. and M.S.E.E. degrees in electrical engineering and Ph.D. degree in communication engineering from National Taiwan University, Taipei, Taiwan, R.O.C., in 1996, 1998, and 2003, respectively. From 1998 to 2001, he was an RF Engineer with the Acer Communication and Multimedia Inc., Taipei, Taiwan, R.O.C., where he designed global system for mobile communication (GSM) mobile phones. From 2001 to 2003, he was with the Chi-Mei Communication System Inc., Taipei, Taiwan, R.O.C., where he was involved with the design of low-temperature co-fired ceramic (LTCC) RF transceiver modules for GSM mobile applications. In August 2003, he joined the Graduate Institute of Communication Engineering, National Taiwan University, as a Post-Doctoral Research Fellow, and became an Assistant Professor in August 2004. Since August 2005, he has been with the Department of Electrical Engineering, National Central University, Chungli, Taiwan, R.O.C., where he is currently an Assistant Professor. His research interests include the design

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and analysis of miniature planar microwave circuits and RF transceiver module for wireless communication systems. Dr. Lin was the recipient of the Best Paper Award presented at the 2001 Asia–Pacific Microwave Conference (APMC), Taipei, Taiwan, R.O.C. and the 2005 Young Scientist Award presented at the URSI General Assembly, New Delhi, India.

Chun Hsiung Chen (SM’88–F’96) was born in Taipei, Taiwan, R.O.C., on March 7, 1937. He received the B.S.E.E. and Ph.D. degrees in electrical engineering from National Taiwan University, Taipei, Taiwan, R.O.C., in 1960 and 1972, respectively, and the M.S.E.E. degree from National Chiao Tung University, Hsinchu, Taiwan, R.O.C., in 1962. In 1963, he joined the Faculty of the Department of Electrical Engineering, National Taiwan University, where he is currently a Professor. From August 1982 to July 1985, he was Chairman of the Department of Electrical Engineering, National Taiwan University. From August 1992 to July 1996, he was the Director of the University Computer Center, National Taiwan University. In 1974, he was a Visiting Scholar with the Department of Electrical Engineering and Computer Sciences, University of California at Berkeley. From August 1986 to July 1987, he was a Visiting Professor with the Department of Electrical Engineering, University of Houston, Houston, TX. In 1989, 1990, and 1994, he visited the Microwave Department, Technical University of Munich, Munich, Germany, the Laboratoire d’Optique Electromagnetique, Faculte des Sciences et Techniques de Saint-Jerome, Universite d’Aix-Marseille III, Marseille, France, and the Department of Electrical Engineering, Michigan State University, East Lansing, respectively. His areas of interest include microwave circuits and computational electromagnetics.

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Miniature Broadband Bandpass Filters Using Double-Layer Coupled Stripline Resonators Yunchi Zhang, Student Member, IEEE, Kawthar A. Zaki, Fellow, IEEE, Andrew J. Piloto, and Joseph Tallo

Abstract—A novel double-layer coupled stripline resonator structure is introduced to realize miniature broadband bandpass filters. Filters with relative bandwidth up to 60% and size less 8 ( is wavelength at the midband frequency; than 8 is the substrate height, which is much smaller than 8) can be fulfilled using such resonators. Two possible filter configurations are proposed in this paper: combline and interdigital. The filter synthesis procedure follows the classical coupling matrix approach that generates very good initial responses. Optimization by the mode-matching method and fine tuning in Ansoft’s High Frequency Structure Simulator are combined to improve the filter performance. Two filter design examples are given to validate the feasibility. Low temperature co-fired ceramic (LTCC) technology is employed to manufacture the filters. Experimental results of the two manufactured filters are presented. The effects of LTCC manufacturing procedure on the filter performance are also discussed. Index Terms—Bandpass filter, broadband, combline, compact, interdigital, low-temperature co-fired ceramic (LTCC), miniature, resonator, stripline.

I. INTRODUCTION INIATURE broadband filters compatible with printed circuit board (PCB) and monolithic-microwave integrated-circuit (MMIC) fabrication technologies are required in many communication systems. The filter size is usually constrained by the size of the employed resonator structures, and the filter bandwidth is limited by the achievable maximum couplings between these resonators. Many available compact resonator structures can be found in literatures. Some of them, such as a stripline resonator with one grounded end [1], hairpin resonator [2], etc., have the size constraint of a quarter-wavelength. Others, such as folded quarter-wavelength resonator [3], ring resonator [4], spiral resonator [5], etc., have a smaller size than a quarter-wavelength, but are usually not applicable for broadband filter designs due to the difficulty in realizing the strong couplings. The goal of this paper is to design broadband filters with compact size less than (filter height is usually very small and approximately equal to the substrate height of the stripline). A novel double-layer coupled stripline resonator structure is, therefore, proposed to fulfill the purpose. The size of the proposed resonator structure can be less than , and the coupling between two resonators can be realized strong enough

M

Manuscript received April 6, 2006; revised May 15, 2006. Y. Zhang and K. A. Zaki are with the Department of Electrical and Computer Engineering, University of Maryland at College Park, College Park, MD 20742 USA (e-mail: [email protected]). A. J. Piloto and J. Tallo are with Kyocera America, San Diego, CA 92123 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2006.879176

Fig. 1. Double-layer coupled stripline resonator structure. (a) Threedimensional view. (b) Cross section. (c) Side view. The structure is filled with a homogeneous dielectric material.

for filter designs up to 60% bandwidth by proper mechanisms. Physical realization of the resonator structure can be easily performed in low-temperature co-fired ceramic (LTCC) technology that is a suitable manufacturing choice for a high-integration level of multiple-layer structures. Two types of filter configurations can be implemented using the proposed resonator structures: combline and interdigital [6]. In this paper, two design examples of interdigital filters are presented to validate the theory. The two filters having the same specifications, but employing different resonator dimensions are manufactured to investigate the LTCC manufacturing effects on the performance. II. FILTER CONFIGURATION A. Proposed Resonator Structure The proposed double-layer coupled stripline resonator structure is shown in Fig. 1. The idea is to introduce a strong capacitive loading effect inside the resonator to reduce its physical length [7]. The resonator structure consists of two strongly coupled strips, as shown in Fig. 1(b). The opposite ends of these two strips are shorted to ground, as shown in Fig. 1(c), so the coupling between these two strips will behave like a capacitance that will lower the resonant frequency. Due to this capacitive coupling effect, the total physical length of the resonator will be much shorter than a quarter-wavelength at the desired resonant frequency. The physical length of the resonator is deterof the two mined by three factors, which are: 1) the width strips; 2) the coupled (overlapped) length ; and 3) the vertical distance between them because these three factors will affect the capacitive coupling between the two strips. Actually,

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III. FILTER DESIGN AND MODELING

Fig. 2. Proposed filter configurations using double-layer coupled stripline resonators. (a) Interdigital filter configuration. (b) Combline filter configuration.

To design a filter with either of the proposed configurations in Fig. 2, the following three steps are implemented. Step 1) Initial filter dimensions are determined according to the given filter specifications. Step 2) Optimization using the mode-matching method (MMM) is performed to find the optimum filter dimensions. Step 3) Ansoft’s High Frequency Structure Simulator (HFSS) is employed to check the optimum design from (2) and fine tune the filter dimensions if needed. The detailed information is illustrated below. A. Initial Design

Fig. 3. Inter-coupling curves of interdigital and combline configurations. Identical resonators are used. S is the separation between two resonators.

resonator with physical length less than at the desired resonant frequency can be realized by such a configuration with properly selected dimensions of these three factors. It must be noted that the vertical distance between the two strips must be an integer multiple of the thickness of one ceramic layer in LTCC technology, and so is the whole height of the resonator. B. Possible Filter Configurations Shown in Fig. 2 are two possible filter configurations using the proposed double-layer coupled stripline resonators: interdigital as in Fig. 2(a), and combline as in Fig. 2(b). These two filter structures are more compact compared with conventional one-layer microstrip/stripline interdigital and combline filters. The input/output external couplings are realized by the tapped-in 50- striplines. The inter-coupling between resonators is achieved by the fringing fields between two resonator lines. Usually the coupling between two interdigital resonators is stronger than the coupling between two combline resonators having the same spacing [8]. Shown in Fig. 3 are the coupling curves for both cases with identical resonator dimensions. The smaller the separation between two resonators, the more noticeable the coupling difference between the two configurations. Therefore, interdigital configuration is more appropriate for broadband filter designs, while combline configuration is a proper choice for some relatively narrower bandwidth filter designs since it has more compact size than the interdigital one.

Given the specifications of a desired filter, the filter design starts with synthesizing a circuit model prototype. Physical realization is then performed according to such ideal model. For the cases presented in this paper, the initial design procedure is given as follows. 1) An ideal circuit model is generated according to the filter requirements [9]. 2) The dimensions of the double-layer coupled stripline resonator are determined in terms of the center frequency of the filter. 3) Determine the tapped-in stripline position to achieve the external coupling [6], [10]. 4) Determine the separations between resonators to provide the required inter-couplings [6], [11]. 5) Assemble the tapped lines and resonators together according to the calculated dimensions. Initial filter responses can be obtained by full-wave electromagnetic simulation in MMM or Ansoft’s HFSS. One of the advantages of combline and interdigital filters is that the resonant frequency of the resonators will not be changed much by the loading effects of the couplings. Therefore, the initial filter response is usually a good starting point (return loss is typically below 10 dB) for further optimization. B. Optimization by MMM The initial design procedure given above does not take into account the higher order modes and the nonadjacent couplings between resonators, which have more effects on broadband filter designs than narrowband ones. To achieve the desired filter performance, optimization by MMM can be employed. To demonstrate the optimization procedure by MMM, an interdigital filter configuration is used (the combline case is similar). Shown in Fig. 4(a) is an interdigital filter with two stripline ports along the -axis. To model this filter configuration, the MMM should be applied along the -direction. The cross sections involved are a stripline (I), an asymmetric stripline (II), side view of a double-layer coupled stripline resonator (III), a rectangular waveguide (IV), etc. [as in Fig. 4(b)]. Eigenmodes and eigenfields of these cross sections can be calculated as in [12] and [13]. Basically, a given cross section is decomposed into many parallel-plate regions and the eigenfields in each region are expressed as a summation of Fourier series with unknown coefficients. By applying the boundary conditions

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Fig. 5. End view along x-direction of the filter configuration in Fig. 4(a). Fig. 4. (a) Tenth-order interdigital filter using double-layer coupled stripline resonators. (b) Involved cross sections along z -axis.

and the field-continuity conditions between the parallel-plate regions, a linear system is obtained and the solution of this system will lead to the cutoff frequencies of eigenmodes and the coefficients of Fourier series. Accordingly, the eigenfields of the total cross section can be solved. The discontinuities between these cross sections are modeled by generalized scattering matrices (GSMs) solved according to the matching boundary conditions [13]. Finally, the overall response can be calculated by cascading all the GSMs together. In principle, this MMM approach is very rigorous, but the convergence and simulation speed must be considered. The rectangular waveguides separating the discontinuities are very short, and the fundamental resonating modes of the interdigital filter are mainly operating with fields, while the fields in cross sections of III, IV, etc. are presented as a summation of and modes in the MMM analysis. As a consequence, a very large number of modes are needed for convergence, which would result in large computation time and numerical errors. In order to do the optimization with this MMM approach, the appropriate number of modes should be selected to have the fast optimization speed and acceptable simulation results. Usually the simulation durations and responses using different number of modes are examined, and a tradeoff between simulation speed and accuracy is made to select the number of modes for optimization. The tapped-line position, widths of resonators, and separations between resonators can be optimized to improve the filter performance. The error function to be minimized is constructed depending on the locations of the poles and equal-ripple points (1) and are the number of poles and equal-ripple where points, respectively. and are the optimization weights. represents the equal-ripple return loss. C. Fine-Tuning in HFSS (If Needed) The optimized filter response by the MMM with the selected number of modes might be different from the converged one. Ansoft’s HFSS is then applied to verify the design and fine tune the filter if needed. The parameter extraction method [14] can be employed to guide the fine tuning in HFSS to speed up the

procedure. Basically, only the tapped-line position and the width of the first resonator are needed to be tuned in this step. HFSS is not used in the previous optimization step because of the slow simulation speed. For the design examples in this paper, it takes MMM approximately 5 h and 1000 iterations to generate the desired performance after the eigenmodes and eigenfields have been calculated (150 modes are used), while it takes HFSS more than 6 h to obtain the converged response for one single filter structure. For an optimization procedure of 1000 iterations, it will take HFSS approximately 6000 h to acquire the optimum design, which is not acceptable. A desktop PC using 3.0-GHz Pentium 4 processor and 4-GB memory is employed to perform the designs. IV. DESIGN EXAMPLES Two design examples of interdigital filters are performed to show the feasibility. They have the same specifications, but different resonator dimensions, and are manufactured to investigate the effects of the LTCC manufacturing procedure. A. Design Example I A design of a ten-pole Chebyshev filter with a center frequency of 1.125 GHz and 500-MHz bandwidth is performed. The relative bandwidth is approximately 45%. The desired stopband rejection level below 0.75 GHz and above 1.5 GHz must be larger than 50 dB. The external couplings and the normalized inter-couplings are

(2) The interdigital filter configuration to be employed is shown in Fig. 4(a). This filter will be embedded inside a PCB system using LTCC technology, and the stack-up options with other components set many constraints on the vertical dimensions. Shown in Fig. 5 is the end view of the filter along the -direction. The height of the whole filter box is mil, the vertical position of the lower strip is mil, and the vertical position of the upper strip is mil. Only one ceramic layer (thickness is approximately 3.74 mil) exists between the two strips. The metallization thickness of the strip is mil

ZHANG et al.: MINIATURE BROADBAND BANDPASS FILTERS USING DOUBLE-LAYER COUPLED STRIPLINE RESONATORS

Fig. 7. External coupling curve: normalized htapin.

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R

versus tapped-in position

Fig. 6. (a) Configuration to decide the dimensions of the resonator. (b) Typical frequency response of S for configuration (a).

and the relative permittivity of the ceramic is , which are decided by the selected LTCC technology. Fig. 6(a) is the configuration to determine the resonator dimensions to have the fundamental resonating mode at the center frequency of 1.125 GHz. Two ports are weakly coupled to the resonator and the peak frequency of the simulated response, as shown in Fig. 6(b), is the resonant frequency. The length of the resonator is selected as mil. The coupled length and width of the two strips (two strips in one resonator have the same dimensions) are being swept until the required resonant frequency of 1.125 GHz is achieved. The found values are mil and mil. The external coupling curve is presented in Fig. 7, which shows the external coupling is linearly proportional to the tapped-in position. The value of htapin to have is approximately 432 mil. The computed inter-coupling curve is shown in Fig. 8 and the separations are calculated by interpolation to have the desired inter-couplings. The found separations for the five desired adjacent couplings in (2) are mil, mil, mil, mil, and mil, respectively. The initial filter response is presented in Fig. 9. The simulated frequency responses by MMM and HFSS of the final filter design are both shown in Fig. 10, which demonstrates a good agreement between them. The minimum return loss over the passband is approximately 15 dB because the nonadjacent couplings for such broadband filter are not avoidable and make it very difficult

Fig. 8. Inter-coupling curve: normalized coupling m versus separation S .

Fig. 9. Frequency response of filter example I with initial dimensions.

to achieve a return loss better than 15 dB. The upper stopband of this filter has clean spurious response up to the third

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Fig. 10. Simulated responses of the optimized filter I by MMM and HFSS.

harmonic. The final dimension of the whole filter box is approximately 500 mil 515 mil 59.28 mil, which is less than mil 540 mil 59.28 mil at the center frequency of 1.125 GHz. This filter is manufactured in LTCC technology for testing. Fig. 11(a) shows the filter prototype and the arrangement of the measurement. Input and output ports of the filter are bent toward the same direction for the convenience to connect with other components. Transitions from 50- microstrip lines to the tapped-in striplines are also added on the filter. -probe launches and a Cascade Microwave Probe Station are used in the measurement. The measured response is shown in Fig. 11(b). The insertion loss at the center frequency is approximately 3 dB and at the higher band edge is approximately 6 dB. The wideband response shows that the spurious response that starts around 3.6 GHz is approximately three harmonics. A simulation by HFSS with lossy materials is also performed to investigate the response difference between the designed and manufactured filters. The employed parameters for loss are: finite conductivity S/m for conductor and loss tangent of 0.001 for ceramics. Both the simulated and measured responses are shown in Fig. 11(c). A good agreement is noticed, except a frequency shift between them. The frequency shift is caused by the effects of the LTCC manufacturing procedure such as vias, inhomogeneous ceramic layers, etc. (in HFSS simulation, solid walls and homogeneous materials are assumed), which will be discussed later. B. Design Example II In design example I, one ceramic layer of thickness mil exists between the two strips of a resonator. To investigate the sensitivity of the filter with respect to and the effects of the LTCC manufacturing procedure on filter performance, a second design with larger is carried out for an odd-order interdigital filter. The filter requirements are identical with design example I, but a design of an eleven-pole filter is performed for this example. The stack-up option is different from example I. For this example, mil, mil, and mil. Definitions of , and

Fig. 11. (a) Built filter I and the measurement arrangement. (b) Measured frequency response of filter I. (c) Comparison between the measurement and simulated response by HFSS. (Color version available online at: http://ieeexplore. ieee.org.)

are given as shown in Fig. 5. The separation between the two strips is mil, which means that the physical length of the resonator will be longer than example I due to the relatively weaker coupling between two strips. The selected dimensions of the resonator are mil, mil, and mil. The same design procedure as example I is followed for this design. The final dimension of the whole filter box is approximately 700 mil 540 mil 63.02 mil, which is larger than design example I. Fig. 12 shows both manufactured filters of examples I and II. The measured response is shown in Fig. 13(a). The spurious response starts around 2.5 GHz, which is about

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Fig. 12. Manufactured filters (design example I and design example II). Filter II is slightly larger than filter I. (Color version available online at: http://ieeexplore. ieee.org.)

Fig. 14. Draft of the physical realization of filters in LTCC technology. (Color version available online at: http://ieeexplore.ieee.org.)

frequency, which causes the spurious performance to be worse than filter I. The response comparison between the measurement and HFSS simulation is shown in Fig. 13(b). The measured bandwidth is slightly narrower than the simulated one, which is also due to the manufacturing effects. In both design examples, the measured insertion loss is slightly larger than the simulated one. Two main reasons are responsible for that, which are: 1) vias in actual structures might introduce more loss than the solid wall model in HFSS and 2) the loss tangent of the actual ceramic material is larger than 0.001. V. LTCC MANUFACTURING EFFECTS ON FILTER PERFORMANCE Fig. 13. (a) Measured response of filter II. (b) Comparison between the measurement and the simulated response by HFSS.

two harmonics. The reason why filter II has worse spurious performance than filter I is related to the resonator structure and dimensions. The first higher order resonating mode of the resonator is controlled by the introduced capacitive coupling between two strips. The stronger the coupling, the further the first higher order mode. Resonators in filter II has larger than filter I and, thus, smaller capacitive coupling between strips; therefore, the first higher order resonating mode is closer to the center

The measured responses of the two filters are slightly different from the simulated ones, which is usually caused by the LTCC manufacturing effects. Shown in Fig. 14 is a draft of the physical realization of the designed filters in LTCC technology. Basically, the filled ceramic is placed layer by layer with fixed thickness of each layer. The horizontal walls and resonator striplines inside the ceramic are implemented by metallization of gold. The vertical walls are realized by a via-fence of closely placed vias. If the signal is communicating between different layers, vias are also applied to connect the signal lines. Such a manufacturing procedure will affect the filter performance from several points-of-views.

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First, the via-fence to realize the vertical walls of the filter will affect the resonator length, which can be observed from the zoom-in view of Fig. 14. Diameter of vias is approximately 6 mil. The distance between two vias should be selected to be as small as possible to decrease the parasitic resistance and also create an effective “pure resistance” environment (i.e., parasitic inductance and capacitance are counteracted), which is usually decided experimentally. The via-fence position relative to the ideal vertical wall position ( in Fig. 14) should be determined to have the manufactured resonator resonating at the same frequency as the ideal resonator, otherwise the actual filter response will be shifted from the designed one. The optimum via-fence position is approximately . Second, the thickness of gold metallization usually varies from 0.4 to 0.6 mil. This effect will cause the variation of the vertical distance (as in Fig. 1) between two strips in a resonator and, thus, the frequency shift of the resonators. The filters will be influenced more by this effect designed with smaller than those with larger since the variation occupies more percentage in smaller . This might be one of the reasons that the measured response of filter I is shifted from the desired center frequency. Third, assembling the filled ceramic layer-by-layer causes the variation of the permittivity of different layers, i.e., the filled ceramic is not perfectly homogeneous. The relative permittivity can be from 4.7 to 6.3. This effect will cause the frequency shift and mainly influence all the couplings existing in the filters. The filters with larger will be affected more since larger means greater variation of the permittivity. This could be one of the reasons that filter II has narrower bandwidth than the design bandwidth. In the actual manufacture, several test pieces are usually manufactured and measured first. The proper processing conditions to reduce the aforementioned effects are then determined according to the comparison between the measured response and the designed one. Once the processing conditions are found, the mass production of components can be performed with more than 95% yield. VI. CONCLUSION A novel double-layer coupled stripline resonator has been introduced for compact size and broadband bandpass filter designs. Two interdigital filter examples using such resonators have been presented. LTCC technology has been employed to manufacture the filters and the effects of LTCC manufacturing procedure on the filter performance have been discussed. The experimental results are in good agreement with the designed responses, validating the theory and design method. ACKNOWLEDGMENT The authors would like to acknowledge Dr. J. A. Ruiz-Cruz, Universidad Autónoma de Madrid, Madrid, Spain, for his helpful discussions. REFERENCES [1] G. L. Matthaei, L. Young, and E. M. T. Jones, Microwave Filters Impedance-Matching Networks and Coupling Structures. Norwood, MA: Artech House, 1980.

[2] E. Cristal and S. Frankel, “Hairpin-line and hybrid hairpin-line/halfwave parallel-coupled-line filters,” IEEE Trans. Microw. Theory Tech., vol. MTT-20, no. 11, pp. 719–728, Nov. 1972. [3] C.-Y. Chang, C.-C. Chen, and H.-J. Huang, “Folded quarter-wave resonator filters with Chebyshev, flat group delay, or quasi-elliptical function response,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2–7, 2002, vol. 3, pp. 1609–1612. [4] K. Chang, Microwave Ring Circuits and Antennas. New York: Wiley, 1996, ch. 3, 7, and 12. [5] F. Huang, “Ultra-compact superconducting narrowband filters using single- and twin-spiral resonators,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 2, pp. 487–491, Feb. 2003. [6] Y. Zhang and K. A. Zaki, “Compact, coupled strip-line broadband bandpass filters,” presented at the IEEE MTT-S Int. Microw. Symp., Jun. 2006. [7] L. A. Robinson, “Wideband interdigital filters with capacitively loaded resonators,” in IEEE MTT-S Symp. Dig., May 1965, vol. 65, pp. 33–38. [8] R. Levy, R. V. Snyder, and G. Matthaei, “Design of microwave filters,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 3, pp. 783–793, Mar. 2002. [9] A. E. Atia, A. E. Williams, and R. W. Newcomb, “Narrow-band multiple-coupled cavity synthesis,” IEEE Trans. Circuits Syst., vol. CAS-21, no. 9, pp. 649–655, Sep. 1974. [10] M. El Sabbagh, H. T. Hsu, and K. A. Zaki, “Full-wave optimization of stripline tapped-in ridge waveguide bandpass filters,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2002, vol. 3, pp. 1805–1808. [11] J.-S. Hong and M. J. Lancaster, Microstrip Filters for RF/Microwave Applications. New York, NY: Wiley, 2001. [12] Y. Rong and K. A. Zaki, “Characteristics of generalized rectangular and circular ridge waveguides,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 2, pp. 258–265, Feb. 2000. [13] J. A. Ruiz-Cruz, M. A. El Sabbagh, and K. A. Zaki, “Canonical ridge waveguide filters in LTCC or metallic resonators,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 1, pp. 174–182, Jan. 2005. [14] H.-T. Hsu, Z. Zhang, K. A. Zaki, and A. E. Atia, “Parameter extraction for symmetric coupled-resonator filters,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 12, pp. 2971–2978, Dec. 2002.

Yunchi Zhang (S’04) received the B.S. and M.S. degrees in physics from Peking University, Beijing, China, in 1997 and 2000, respectively, the M.S. degree in electrical and computer engineering from North Carolina State University, Raleigh, in 2002, and is currently working toward the Ph.D. degree at the University of Maryland at College Park. He is currently a Research Assistant with the Microwave Laboratory, Department of Electrical and Computer Engineering, University of Maryland at College Park. His current research interests include analysis, modeling, and design of microwave and millimeter-wave devices and circuits.

Kawthar A. Zaki (SM’85–F’91) received the B.S. degree (with honors) from Ain Shams University, Cairo, Egypt, in 1962, and the M.S. and Ph.D. degrees from the University of California at Berkeley, in 1966 and 1969, respectively, all in electrical engineering. From 1962 to 1964, she was a Lecturer with the Department of Electrical Engineering, Ain Shams University. From 1965 to 1969, she was a Research Assistant with the Electronics Research Laboratory, University of California at Berkeley. In 1970, she joined the Electrical Engineering Department, University of Maryland at College Park, where she is currently a Professor of electrical engineering. She has authored or coauthored over 200 publications. She holds five patents on filters and dielectric resonators. Her research interests are in the areas of electromagnetics, microwave circuits, simulation, optimization, and computer-aided design of advanced microwave and millimeter-wave systems and devices. Prof. Zaki was the recipient of several academic honors and awards for teaching, research, and inventions.

ZHANG et al.: MINIATURE BROADBAND BANDPASS FILTERS USING DOUBLE-LAYER COUPLED STRIPLINE RESONATORS

Andrew J. Piloto received the B.S. degree in mechanical engineering and B.S. degree in solid-state physics from from the University of Texas, Austin, in 1984 and 1985, respectively. From 1994 to 1996, he attended the Graduate School of The Johns Hopkins University, Baltimore, MD, under the graduate educational program of Westinghouse Electronics Systems. He is currently a Member of the Technical Staff with the Product Technology Center, Kyocera America, San Diego, CA. He is currently involved in the development of next-generation transmit/receive (T/R) module and radar support electronics packaging for active phased-array radar systems. He was a Process Engineer with the Hybrid Microelectronics Laboratory, and a Design Engineer with the Advanced Missile Systems Department, Texas Instruments Incorporated. He participated in the design of ATF T/R modules for the F-22 and YF-23, as well as several classified programs. He was also with Northrop Grumman Electronic Systems (formerly Westinghouse Electronic Systems), where he was involved with the development of advanced T/R modules. While with Northrop Grumman Electronic Systems, he was granted 44 U.S. patents, two of which include both the JSF (F-35) T/R module and the stealthly ferrite used in the JSF circulator assembly. Upon leaving Northrop Grumman Electronic Systems, he co-founded the Design Integrated Circuits Division, Nurlogic, where he led opto-electronic (OE) integrated-circuit (IC)

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development. Subsequently sold to Artisan Components, Nurlogic led the OE IC industry producing the first 72-channel parallel optical transceiver currently deployed in Cisco routers. He is currently involved with the development of highly integrated T/R modules and frequency converters for both space and terrestrial applications as both a contractor to the U.S. Government, as well as the original equipment manufacturers (OEMs).

Joseph Tallo received the B.S. degree in electrical engineering from the University of California at San Diego, La Jolla, in 2000. He has been a Research and Design Engineer with Kyocera America, San Diego, CA. His research involves modeling and measurement of ceramic packages.

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Physics-Based Wideband Predictive Compact Model for Inductors With High Amounts of Dummy Metal Fill Luuk F. Tiemeijer, Ramon J. Havens, Yann Bouttement, and Henk Jan Pranger

Abstract—A computationally efficient physics-based wideband predictive inductor compact model is presented that is capable of evaluating the impact on inductance and quality factor of dummy metal fill-cells. In a modern integrated-circuit process, high amounts of these fill-cells are required to meet metal density rules and guarantee adjacent circuit integrity. The predictions made with this model are shown to be in agreement with data measured on symmetrical octagonal inductors realized in a 90-nm CMOS process with varying amounts of dummy metal fill-cells. We further provide all inductance equations, list additional sources of eddy current losses, and discuss layout modifications, which would further improve the performance of the integrated inductors. Index Terms—Dummy metal fill-cells, integrated circuits (ICs), on-chip inductors, on-wafer microwave measurements, tiling.

I. INTRODUCTION NTEGRATED spiral inductors, despite their significant area consumption, are essential to get the best performance out of RF circuits such as (voltage-controlled) oscillators and (low-noise) amplifiers, and to realize (narrowband) impedance matching, filter, and decoupling functionality. Many studies on spiral inductor performance in bulk (Bi)CMOS processes have been published over the past decade [1], [2] starting with issues such as optimum trace width and gap [3], or the impact of metal thickness [4], [5], and later on reporting on other items such as current crowding [6], [7], substrate modifications [8], [9], or the benefits of a ground shield [10], [11]. Still other issues, such as the need for a considerable amount of dummy metal fill-cells to meet the metal density rules of the modern integrated-circuit (IC) process have not yet received adequate attention [12]–[15]. Even more publications have appeared on efforts to develop compact and computationally efficient physics-based wideband equivalent-circuit models, which would allow one to optimize the inductor within the circuit design process. Although initially in these models the focus was on predicting the correct inductance value [16]–[22], to date, parasitic capacitances and various losses associated with the skin effect, proximity effect, and

I

Manuscript received December 22, 2005; revised February 20, 2006. This work was supported by the Crolles2 Alliance. L. F. Tiemeijer is with Philips Research Europe, 5656 AE Eindhoven, The Netherlands (e-mail: [email protected]). R. J. Havens and H. J. Pranger are with Philips Semiconductors, 6534 AE Nijmegen, The Netherlands. Y. Bouttement is with Philips Semiconductors, 14079 Caen Cedex 5, France. Digital Object Identifier 10.1109/TMTT.2006.877831

Fig. 1. (left) 20-GHz single-loop inductor in the conventional test environment and (right) with the high amounts of dummy metal fill-cells required to meet metal density rules in today’s advanced IC processes.

currents induced in the substrate are also captured quite well [23]–[29], although sometimes phenomenological fitting factors have to be introduced to achieve good accuracy. In today’s advanced IC processes, high amounts of dummy metal fill-cells are required to meet metal density rules and guarantee adjacent circuit integrity. In this paper, we present the first predictive physics-based wideband compact inductor model capable of evaluating the impact on the inductance and quality factor of these dummy metal fill-cells (Fig. 1). We will demonstrate that employing only well-known process parameters as resistivity’s and dielectric constants as input, inductances, and quality factors observed on a set of symmetrical octagonal inductors realized in a 90-nm node CMOS process can be predicted down to the few percent level, while the impact of different dummy metal fill schemes can be accurately reproduced. The model presented here is an extension of earlier work reported in [26], where we now are able to provide all inductance equations, and include eddy-current loss in the substrate, patterned ground shield (PGS), and dummy metal fill-cells.

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Fig. 3. Simplified circuit to illustrate the impact of the ground shield.

Fig. 2. Nested inductor loops can be series connected in different ways. (left) A single spiral provides the highest resonance frequency; however, for many applications (right) a symmetrical scheme suitable for differential excitation is preferred.

II. DESIGN CONSIDERATIONS To date, on-chip inductors have been realized in circular, square, and octagonal layouts. In this study, octagonal layouts are used since they provide quality ( ) factors close to that of circular layouts, but can be more easily realized with the current layout tools. To achieve higher inductance values, either size has to be increased or several (narrower) inductor loops must be series connected. Within a given outer and inner diameter and available metal stack thickness, both the inductance, as well as the series resistance, increase quadratically with the number of turns [19] so the remains roughly the same at a given frequency. Since the increases with frequency until about one-half of the inductor’s self resonance frequency, and this resonance frequency comes down quickly when the number of turns is increased, for the same footprint, practically available peak quality factors decrease for the larger inductance values [26]. To achieve a high self-resonance frequency, the capacitance between the different inductor loops should be minimized, and nested inductor loops seem preferable over stacked inductor loops [30]. These nested inductor loops can be series connected in different ways (Fig. 2). On an isolating substrate, a single spiral (left) provides the highest resonance frequency since it minimizes the energy stored in the capacitances between adjacent inductor windings. However, for many bulk (Bi)CMOS applications, a symmetrical layout suitable to enhance the [31] through differential excitation is preferred (right). When integrated in a bulk (Bi)CMOS process, the spiral inductor is only a few micrometer away from the conductive substrate. For the typical substrate resistivity’s of 10 cm and higher, encountered in these processes, the losses due to eddy currents is small. Losses from the capacitive coupling to the substrate can be eliminated using a proper PGS [11], [26], as illustrated in Fig. 3. Here, and represent the inductance and series resistance, and , , and represent the inter-loop, loop to shield, and shield to substrate capacitances, respectively. For a proper ground shield, the effective resistance is much less than the resistance seen in the substrate. This shield, therefore, effectively short circuits , thereby reducing the inductor resonance frequency, but improving the peak factor considerably, particularly for single-loop inductors, where is small compared to and [11].

Fig. 4. Circuit to capture the increase of the inductor series resistance at high frequency.

In modern CMOS processes, design rules require inclusion of dummy metal fill-cells inside the inductor loops to meet certain minimum metal density levels. A reduction of the inductor self-resonance frequency [15] is avoided when these dummy metal fill-cells are kept approximately three times the distance to the ground shield away from the inductor loops. This may require some stretching of the process design rules, but since minimum rule features are not required for the inductor loops, this can usually be tolerated. Inductive effects, however, have a longer reach, and eddy-current losses in these dummy metal fill-cells should be considered when optimizing inductor performance [12], [13]. III. INDUCTOR MODEL BASICS In the inductor, at low frequency, the current flow across the available inductor loop cross section adjusts itself for minimum resistance, whereas at high frequency, the skin and proximity effects redistribute the current flow in an effort to minimize overall inductance. To capture the skin effect, the inductor series resistance (Fig. 3) should be evaluated using the exact expression for the complex internal impedance of a current sheet [32], [33]

(1) , , and and are the conwhere ductivity and thickness of the conductive material, respectively. However, for the typical loop width to thickness ratios encountered in today’s integrated spiral inductors, the series resistance increase due to this ”classical” skin effect is overwhelmed by that coming from lateral current crowding [6], [7]. To first order, this current crowding can be modeled [25], [27], [34], [35] by the second-order network shown in Fig. 4, where and represent the low-frequency resistance and inductance, respectively, and and represent their high-frequency counterparts. However, after the initial transition, the resistance increase with frequency is typically too steep, and also the high-frequency limit of the skin effect is not captured very well in this approach and, therefore, the use of higher order networks [26], [29], [36] is desirable to improve model accuracy. In our previous study [26], we showed that by introducing only one new inductance parameter , in addition to and

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, we were able to model the increase of the series resistance at high frequency observed on many different inductors taken from various IC processes with good accuracy using

(2) where represents the corner frequency for a square root of frequency resistance increase. Using a fixed fourth-order network, this frequency-dependent complex impedance ((2)) can be approximated with good accuracy [26] over a large ) with conventional resistances frequency range (up to and inductances. We also demonstrated that we could evaluate the onset of current crowding and calculate the corner frequency by splitting each inductor loop in a limited set of sub-loops of equal resistance and evaluating their inductances. A single-turn inductor, for example, can be decomposed into a parallel connection of nested sub-loops with resistance and inductance , where

Fig. 5. Impact of a dissipative eddy-current loop L =R on an inductor L to first order can be modeled by a parallel conductance G.

need to identify all relevant eddy-current patterns in the vicinity of the inductor and evaluate their impact on the impedance seen at the terminals of the inductor. As illustrated in Fig. 5, we may model each eddy-current pattern as a current loop with an impedance and mutual inductance with the main inductor L and approximate the impedance seen at the terminals of this device as

(6) (3) and represents the mutual inductance matrix between the nested sub-loops (appendix ). The impedance of this inductor is

Neglecting the higher order terms for the moment, the eddycurrent loss yields an increase of the resistance proportional to the square of the frequency, which we can model by adding a small conductance in parallel with the main inductor L. As long as the total amount of eddy-current losses is sufficiently small that the above perturbation result is valid, we are able to include all of them in the model using

(4) (7) where represents the unity matrix. After Taylor matrix expansion in , summation, and a second scalar Taylor expansion in , this impedance could be written as

where the summation is taken over all relevant eddy-current patterns. A. Substrate

(5) showing an increase of the series resistance at high frequency proportional to the variance of the normalized sub-loop inductances . Taylor expanding (2) up to and equating the first three terms enabled us to identify and as the average and the standard deviation of the sub-loop inductances . Considering five parallel sub-loops is enough to find and with sufficient accuracy [26]. Generalization of the above approach to multiturn and multiloop [7] inductors only requires adding the results obtained for each individual inductor winding.

For the mirror currents induced in the high resistive silicon substrate and the moderately doped p-well located in the top few micrometers of the substrate, we thus use (8) vskip4pt (9) represent the mutual inductance between the inductor where and a nested filamentary current loop of radius at depth (see the Appendix ) and and represent the sheet and bulk resistivity of the well and substrate, respectively.

IV. MODELING EDDY-CURRENT LOSS For the well optimized inductors reported here, dissipative losses from eddy currents induced in the substrate, ground shield, and dummy metal fill-cells should only appear at frequencies well beyond . To include them in the model, we

B. Dummy Metal Fill-Cells For dummy metal fill-cells of size (Fig. 6), we first need to add the contributions from all the (rectangular) eddycurrent loops induced in such a metal fill-cell. Introducing a

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Fig. 6. Eddy-current loops assumed in a rectangular dummy metal fill-cell.

normalized radius , the resistance loop is

seen by the current

(10) is the sheet resistivity of the metal fill-cell. Simiwhere larly, the mutual inductance with the main inductor is (11) is the mutual inductance per unit of area, which we where calculate from

(12) Integration over the fill-cell radius provides the eddy-current loss for a single dummy metal fill-cell (13)

Fig. 7. Equivalent-circuit model of the symmetrical inductors reported in this . paper. New with respect to [26] are the two conductances

G

has been described above. Fig. 7. The calculation of and The inductor resistance follows in a straightforward way from the sheet resistance of the inductor loops. To include a first-order account of the eddy-current losses in the substrate, the shield and dummy metal fill-cells, the two conductances of value (16)

covered with a A second integration over the entire area density of these dummy metal fill-cells then gives

(14) C. Ground Shield Despite the fact that the parallel bars (of width ) of the PGS are grounded and varies along the bar length, similar eddycurrents loops, as shown in Fig. 6, are expected in these bars. Simplifying (14) for , and substituting , the eddy-current loss in the PGS is thus accounted for using

(15) where we integrate over the area covered by the shield, where is the shield density, and is the sheet resistivity of the shield. Numerical approximations for the integrals in these formulas are presented in the Appendix . V. FULL INDUCTOR EQUIVALENT CIRCUIT The full five terminal equivalent circuit of the symmetrical inductors reported in this paper is depicted in Fig. 7. To approximate the frequency-dependent equation (2) for both differential and common mode excitation (see the Appendix), 12 conventional inductances and eight conventional resistances, which are directly derived from the inductance parameters , , and , are used. For differential excitation, these ratios are listed in

have been added, where represents the inductance of a single-loop inductor with the same inner and outer diameter (see the Appendix ). Finally, six equivalent capacitances [37] are needed to model the energy stored in the inter-loop ( ) and loop to shield ( ) capacitances (Fig. 3) for the three different excitation modes of the three inductor terminals with respect to the ground shield, assuming (Fig. 3) to be sufficiently low to be neglected. The local capacitances and are obtained from simple analytical expressions fitted to numerical interconnect capacitance simulations [38]. Further detailing the calculation of these circuit elements would be beyond the scope of this paper. This is also the case for the shield to substrate capacitance, and the shield, center tap, and substrate resistances seen in Fig. 7. VI. VERIFICATION Differentially driven octagonal symmetric inductors [31] (with a PGS as described in [26]) employing polysilicon bars perpendicular to the symmetry plane of the inductor were used in this study. These inductors were fabricated in an industrial 90-nm node CMOS process with a copper back-end. Using our predictive inductor model, a set of five inductors in metal 5 (330-nm Cu), and metal 6 (900-nm Cu) with low-frequency inductances of 0.5, 1, 2, 4, and 8 nH, was defined with parameters listed in Table I [5]. To verify whether the excellent quality factors reported in [5] could be realized while simultaneously meeting the strict active area, poly silicon and

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TABLE I OUTER AND INNER DIAMETERS ( , , IN MICROMETERS), NUMBER OF TURNS ( ), LOOP WIDTH AND SPACE ( , , IN MICROMETERS), ) INDUCTANCE ( ) AND INDUCTANCE ERROR ( OF THE OCTAGONAL INDUCTORS

N

L

D D

W S

1L

TABLE II SHEET RESISTANCES AND METAL DENSITIES USED FOR EVALUATING THE LOSSES OF THE INDUCTORS REPORTED IN THIS STUDY

Fig. 8. Typical inductance modeling error observed in this study, our previous work [26] and reported by other groups: Mohan et al. [19], Jenei et al. [20], Asgaran [21], Burghartz and Rejaei [2], Leduc et al. [22], and Rotella et al. [29].

metal density rules of a modern CMOS process, versions with dummy active area fill-cells below the polysilicon patterned shield, additional metal 1 shield bars, and various amounts of dummy metal fill-cells inside the inductor loops were also fabricated. All inductors were characterized up to 50 GHz in a two-port ground–signal–ground (GSG) configuration (Fig. 1). Deembedding was performed using the “open–short–load” and “pad–open–short” techniques described in [39]. First the “pad–open–short” deembedding was used to find the actual resistance (targeted at 50 ) and verify the capacitance (5 fF) of the load standard embedded in the load dummy, and then “open–short–load” deembedding was applied to correct the inductor measurements. The resistance parameters used in the inductor model are listed in Table II. First, measured and simulated inductances were compared. The differences at approximately one-tenth of the resonance frequency appeared to be only a few tenths of a percent, as listed in the last column of Table I, for inductors with additional metal 1 shield bars. Given that the minute amount of inductance systematically lost in the measurement could well be caused by the inductor magnetic field seeing other inductors, or the RF probes used in the measurements, we conclude that due to our analytical inductance formulation, we are able to predict the inductance with unprecedented accuracy. This is illustrated in the benchmark Fig. 8 where the 0.5% inductance modeling error observed for 14 different 90-nm node inductors and 1.3% average inductance modeling error seen on 52 devices taken from three older technologies compares favorably to the typical inductance modeling errors reported by other groups. The impact of the shield modifications and the dummy metal fill-cells on inductance remains below our detection limit of 3 pH, in line with the model predicting there should be none. To continue with the factors, in Fig. 9, measured and simulated single-ended and differential peak factors [11] for the 90-nm node inductors with a combined poly and metal 1 shield are compared in a similar plot as in [26]. Generally the agreement is within a few percent. Only for the two-turn inductor is

Q

Fig. 9. Measured peak factor versus simulated peak shown for both single-ended and differential factor.

Q

Q factor. Data are

the measured factor somewhat larger than simulated. This is typical for two-turn inductors and comes from the fact that both the proximity and skin effects are modeled by a single parameter. In Fig. 10, differential factor data taken on four different versions of the same 20-GHz 0.5-nH single-loop inductor is collected to demonstrate the impact of eddy-current losses. Here, inductor A is the 20-GHz 0.5-nH single-loop inductor of Fig. 1 (left) in a conventional test environment where the inclusion of dummy metal fill-cells is blocked up to 50- m distance of the inductor loops. For inductor B, 3- m squared dummy metal fill-cells were added outside the inductor up to 10 m away from the inductor loops. For inductor C, the polysilicon shield is additionally replaced by a combined polysilicon and metal 1 shield. For inductor D, Fig. 1 (right) 3- m squared dummy metal fill-cells were additionally added inside the inductor up to 10- m distance from the inductor loops. As shown in Fig. 10,

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Fig. 10. Measured (symbols) and simulated (lines) differential Q factor for four types of single-turn 0.5-nH inductors (see text) versus frequency.

Fig. 12. Measured (symbols) and simulated (lines) ac resistance increase versus frequency seen when 3-m dummy metal fill-cells are inserted up to 10m from the inner inductor loops.

VII. INDUCTOR OPTIMIZATION

Fig. 11. Measured (symbols) and simulated (lines) ac resistance increase versus frequency seen when a metal 1 shield is added to inductors with a shield. The dashed line was simulated without eddy-current losses.

each step going from layout A to the final layout D reduces the peak -factor by approximately 5%, a trend well reproduced by the predictive inductor model. It should be noted that due to the test-structure parasitics, it is much more difficult to obtain reliable -factor data at 20 GHz, than for instance at 3 GHz, where the of the 8-nH inductor peaks. Unfortunately at these low frequencies, the impact of eddy-current losses in the dummy metal fill-cells is only approximately 0.5%, less than the uncertainty in the inductor resistance due to process spread. Therefore, to further verify the modeling of eddy-current losses, in Fig. 11, the ac differential resistance increase seen when a metal 1 shield is added is plotted versus frequency. The dashed line represents the increase in resistance simulated when only the increase in capacitance to ground is accounted for, whereas the solid line represents the same simulation, but now with the eddy-current losses in the patterned metal 1 shield included. Similarly, in Fig. 12, the ac resistance increase versus frequency seen when 3- m dummy metal fill-cells are inserted up to 10 m from the inner inductor loops is plotted versus frequency. Again, it is seen that the model predicts the additional losses correctly.

A breakdown of the inductor losses for the five inductors of this study is given in Table III. The losses were calculated using the predictive inductor model at the frequency where the peak differential factor is obtained. As seen, the “traditional” eddy-current losses in the substrate and p-well, although increasing linearly with frequency, even for the smaller inductances are almost negligible at typical levels of only 1% and 3%. The impact of the eddy-current losses in the dummy metal fill-cells and the metal 1 shield shows a steeper increase with frequency and reaches the 20% level for the 0.5-nH inductor. For all inductors however, the loop resistance, which can be split into dc resistance and in excess resistance due to the skin and proximity effects, remains the predominant determinant of the factor, accounting for 72%–97% of the inductor losses. When employing processes with thicker metal layers and lower values of (Table II) to reduce these resistive losses, it should be noted that the eddy-current losses in dummy metal fill-cells may increase considerably since inevitably will also be lower in these processes, while furthermore, the process may require the use of larger dummy metal fill-cells. As a result of this, the characteristic frequencies where eddy-current loss in dummy metal fill-cells may become a point of concern could be reduced considerably compared to what is presented here in Table III for a standard 90-nm node CMOS process. Taking this loss analysis of our current inductors as a starting point, a first step to further improve the performance of the 20-GHz 0.5-nH type-D inductor would be to reduce the metal 1 finger width of the PGS from the current value of 3 m down to the minimum design rule of 0.12 m. This provides a 600-fold reduction in the eddy-current losses in the metal 1 shield, and increases the peak from 16 to 18, as shown in Fig. 13. A similar improvement is obtained when a polysilicon shield is used instead. The next step would be to reduce the size of the dummy metal fill-cells from the current 3 3 m down to the minimum design rule of approximately 0.6 0.6 m, providing a 20-fold

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TABLE III BREAKDOWN OF THE INDUCTOR LOSSES (IN PERCENTAGE) CALCULATED USING THE PREDICTIVE INDUCTOR MODEL AT THE FREQUENCY WHERE THE PEAK DIFFERENTIAL FACTOR IS OBTAINED

Q

f

demonstrated that whereas for 90-nm node CMOS inductors of several nanohenrys operating at frequencies of a few gigahertz, eddy-current losses in the substrate, the ground shield and dummy metal fill-cells tend to be negligible compared to the resistive losses of the inductor loops, for sub nanohenry inductors operating at tens of gigahertz, this is no longer the case, and there minimizing these losses requires using the smallest possible PGS finger widths and dummy metal fill-cell sizes. APPENDIX A. Mutual Inductance of Nested Circular Loops Generally, assuming a uniform current distribution across fixed conductor cross sections, the mutual inductance between two current loops can be calculated using (17)

Q

Fig. 13. Improvements in differential -factor of the 0.5-nH inductor simulated for various layout and process modifications.

where , is the distance between two points in the conductors, while the integration is taken over their crosssectional areas and and along their directions of current flow and . To use this in an inductor model, the complex inductor layout has to be decomposed into smaller elements for which this multiple integral can be evaluated either analytically or numerically within an acceptable amount of computation time. An accurate analytical expression for the mutual inductance of nested filamentary circular current loops for evaluating spiral inductors was first published in [17]. To cover our modeling needs, we extended this expression to provide the mutual inductance and self-inductance of nested current loops of radii and finite width and thickness (18)

reduction in the eddy-current losses in the tiles, and further increasing the peak to 22. After that, locally blocking the well implant would further enhance the peak to 25, whereas ultimately a peak of 26 would be expected when the current 15 cm substrate would be replaced by a high-resistivity substrate (HRS) of 100 cm. In these virtual optimization steps, the strongest contributors to eddy-current losses have been dealt with first. It may be clear from this analysis that just further enhancing the substrate resistivity without using an optimal PGS, minimum dummy metal fill-cell sizes, and locally blocking all well implants, in an effort to improve the overall quality factors, is not effective. VIII. CONCLUSIONS We have provided additional details on our physics-based wideband predictive inductor model, and documented how eddy-current losses in the substrate, ground shield, and dummy metal fill-cells required in a 90-nm node CMOS process can be included. With these extensions, our compact inductor model is not only capable of modeling inductance to within a few tenths of a percent, but also to accurately predict the factor and differences in expected for different shield and dummy metal fill-cell design choices. We have further

(19) denotes their vertical height difference, and and represent the complete elliptic integrals of the first and second kinds [17], [40]. When the two current loops coincide with each other, this mutual inductance equation gives the selfinductance of this loop. Furthermore, this expression is now sufficiently accurate that when the loop is split into a number of sub-loops, as is required to find , the self-inductance, as obtained from the average normalized sub-loop inductance, does not depend on the number of sub-loops. where

B. Inductance Corrections After numerical evaluation of (17), we find that only a small correction needs to be applied to this result for octagonal inductor layouts

(20) . Since the loop length and footprint increase where by 5.5% for the same minimum inner diameter for octagonal

TIEMEIJER et al.: PHYSICS-BASED WIDEBAND PREDICTIVE COMPACT MODEL FOR INDUCTORS WITH HIGH AMOUNTS OF DUMMY METAL FILL

inductors, whereas the inductance only increases by approximately 3%, compared to circular inductors, the octagonal inductors are slightly less area efficient. For inductors having a center tap, numerical calculations show that the common mode mutual inductance [26], where the two inductor halves are connected in parallel instead of in series, can be derived from the differential mutual inductance through (21) II. EDDY-CURRENT LOSS NUMERICAL RESULTS To obtain simple expressions for the eddy-current losses, we replace the actual inductor by a single loop of equal inner and and inductance . Numerical calculaouter radius and tions then yield

(22) where we have split the area covered with dummy metal fillcells in an inner and an outer region and and denote their end and start radii, respectively. ACKNOWLEDGMENT The authors wish to acknowledge the Crolles2 Alliance for providing the silicon, and particularly D. Gloria, ST Microelectronics, Crolles, France, and D. Monk and M. Petras, Freescale Semiconductor Inc., Austin, TX, for valuable discussions regarding the definition of the experiments reported in this study. REFERENCES [1] N. M. Nguyen and R. G. Meyer, “Si IC-compatible inductors and LC passive filters,” IEEE J. Solid-State Circuits, vol. 25, no. 8, pp. 1028–1031, Aug. 1990. [2] J. N. Burghartz and B. Rejaei, “On the design of RF spiral inductors on silicon,” IEEE Trans. Electron Devices, vol. 50, no. 3, pp. 718–729, Mar. 2003. [3] J. M. Lopez-Villegas, J. Samitier, C. Cane, P. Losantos, and J. Bausells, “Improvement of the quality factor of RF integrated inductors by layout optimization,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 1, pp. 76–83, Jan. 2000. [4] Y.-S. Choi and J.-B. Yoon, “Experimental analysis of the effect of metal thickness on the quality factor in integrated spiral inductors for RF ICs,” IEEE Electron Device Lett., vol. 25, no. 2, pp. 76–78, Feb. 2004. [5] L. F. Tiemeijer, R. J. Havens, Y. Bouttement, and H. J. Pranger, “The impact of an aluminum top layer on inductors integrated in an advanced CMOS copper backend,” IEEE Electron Device Lett., vol. 52, no. 11, pp. 722–724, Nov. 2004. [6] W. B. Kuhn and N. M. Ibrahim, “Analysis of current crowding effects in multiturn spiral inductors,” IEEE Trans. Microw. Theory Tech., vol. 49, no. 1, pp. 31–38, Jan. 2001.

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[29] F. Rotella, B. K. Bhattacharya, V. Blaschke, M. Matloubian, A. Brotman, Y. Cheng, R. Divecha, D. Howard, K. Lampaert, P. Miliozzi, M. Racanelli, P. Singh, and P. J. Zamapardi, “A broadband lumped element analytic model incorporating skin effect and substrate loss for inductors and inductor like components for silicon technology performance assessment and RFIC design,” IEEE Trans. Electron Devices, vol. 52, no. 7, pp. 1429–1441, Jul. 2005. [30] C.-C. Tang, C.-H. Wu, and S.-I. Liu, “Miniature 3-D inductors in standard CMOS process,” IEEE J. Solid-State Circuits, vol. 37, no. 4, pp. 471–480, Apr. 2002. [31] M. Danesh and J. R. Long, “Differentially driven symmetric microstrip inductors,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 1, pp. 332–341, Jan. 2002. [32] R. Horton, B. Easter, and A. Gopinath, “Variation of microstrip losses with thickness on strip,” Electron. Lett., vol. 7, no. 17, pp. 490–491, Aug. 1971. [33] J. C. Rautio and V. Demir, “Microstrip conductor loss models for electromagnetic analysis,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 3, pp. 915–921, Mar. 2003. [34] J. Sieiro, J. M. Lopez-Villegas, J. Cabanillas, J. A. Osorio, and J. Samitier, “A physical frequency-dependent compact model for RF integrated inductors,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 1, pp. 384–392, Jan. 2002. [35] J. Gil and H. Shin, “A simple wideband on-chip inductor model for silicon-based RF ICs,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 9, pp. 2023–2027, Sep. 2003. [36] S. Kim and D. Neikirk, “A compact equivalent circuit model for the skin effect,” in IEEE MTT-S Int. Microw. Symp. Dig., 1996, pp. 1815–1818. [37] C.-H. Wu, C.-C. Tang, and S.-I. Liu, “Analysis of on-chip spiral inductors using the distributed capacitance model,” IEEE J. Solid-State Circuits, vol. 38, no. 6, pp. 1040–1044, Jun. 2003. [38] U. Choudhury and A. Sangiovanni-Vinticelli, “Automatic generation of analytical models for interconnect capacitances,” IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst., vol. 14, no. 4, pp. 470–480, Apr. 1995. [39] L. F. Tiemeijer, R. J. Havens, A. B. M. Jansman, and Y. Bouttement, “Comparison of the ‘pad-open-short’ and ‘open-short-load’ de-embedding techniques for accurate on-wafer RF characterization of high quality passives,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 2, pp. 723–729, Feb. 2005. [40] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions. New York: Dover, 1970, ch. 9.

Luuk F. Tiemeijer was born in Son en Breugel, The Netherlands, in 1961. He received the M.S. degree in experimental physics from the State University of Utrecht, Utrecht, The Netherlands, in 1986, and the Ph.D. degree in electronics from the Technical University of Delft, Delft, The Netherlands, in 1992. In 1986, he joined Philips Research Laboratories, Eindhoven, The Netherlands, where he has conducted research on InGaAsP semiconductor lasers and optical amplifiers. Since 1996, he has been involved in the RF characterization and modeling of advanced IC processes.

Ramon J. Havens was born in Nijmegen, The Netherlands, in 1972. He received the Bachelor’s degree from Eindhoven Polytechnic, Eindhoven, The Netherlands, in 1995. He subsequently joined Philips Research Laboratories, Eindhoven, The Netherlands, where he became involved in the on-wafer RF characterization of the various active and passive devices found in advanced IC processes. In the end of 2005 he joined Philips Semiconductors, Nijmegen, The Netherlands, where he is currently active in the characterization of RF microelectromechanical systems (MEMS).

Yann Bouttement was born in Berlin, Germany, in 1976. He received the Engineer’s degree from the Ecole Polytechnique Universitaire de Lille (EUDIL) Lille, France, in 2000. In 2001, he joined Philips Semiconductor, Caen, France, where he is currently involved with the characterization and modeling of RFIC processes.

Henk Jan Pranger was born in Assen, The Netherlands, in 1963. He received the Electrical Engineering degree and Ph.D. degree in electronics from the University of Twente, Enschede, The Netherlands, in 1988 and 1993, respectively. In 1994, he joined Philips Research Laboratories, Eindhoven, The Netherlands, where he conducted research on analog design automation. In 2002, he joined Philips Semiconductors, Nijmegen, The Netherlands, where he is involved in library and flow development for RF design in advanced CMOS processes.

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A Chip-Scale Packaging Technology for 60-GHz Wireless Chipsets Ullrich R. Pfeiffer, Member, IEEE, Janusz Grzyb, Member, IEEE, Duixian Liu, Senior Member, IEEE, Brian Gaucher, Member, IEEE, Troy Beukema, Brian A. Floyd, Member, IEEE, and Scott K. Reynolds

Abstract—In this paper, we present a cost-effective chip-scale packaging solution for a 60-GHz industrial–scientific–medical band receiver (Rx) and transmitter (Tx) chipset capable of gigabit-per-second wireless communications. Envisioned applications of the packaged chipset include 1–3-Gb/s directional links using amplitude shift-keying or phase shift-keying modulation and 500-Mb/s–1-Gb/s omni-directional links using orthogonal frequency-division multiplexing modulation. This paper demonstrates the first fully package-integrated 60-GHz chipset including receive and transmit antennas in a cost-effective plastic package. A direct-chip-attach (DCA) and surface mountable land-grid-array (LGA) package technology is presented. The size of the DCA package is 7 11 mm2 and the LGA package size is 6 13 mm2 . Optionally, the Tx and Rx chip can be packaged together with Tx and Rx antennas in a combined 13 13 mm2 LGA transceiver package. Index Terms—Chip-scale packaging, low cost, millimeter wave, silicon germanium, wireless communication.

I. INTRODUCTION ACKAGING OF electronic devices plays a key role in the proper functioning of any semiconductor product. Electronic packages conduct signals through a circuit by metal in the form of wires, contacts, foils, plating, and solders. The package further insulates circuits from others and provides an environmental protection and physical support of the circuit. Packaging of millimeter-wave components is particularly challenging because of the associated complexity both in design and fabrication. The small wavelength involved often demands high-precision machining, accurate alignment, or high-resolution photolithography. Furthermore, millimeter-wave circuits usually exhibit low integration levels and are often assembled using expensive and bulky waveguides [1]–[5]. Common are monolithic-microwave integrated-circuit (MMIC) packages [6], [7], which are primarily an outgrowth of the hybrid microwave integrated-circuit (MIC) technology and the discrete device packaging technology. The use of conventional low-cost chip-scale packaging (CSP), plastic ball grid arrays (PBGAs), or direct-chip-attach (DCA) technologies is limited and has only been reported at lower frequencies [8]. This is in part because the typical 1-nH/mm lead and wire-bond inductances

P

Manuscript received January 21, 2006; revised April 10, 2006. This work was supported in part by the National Aeronautics and Space Administration under Grant NAS3-03070 and by the Defense Advanced Research Projects Agency under Grant N66001-02-C-8014 and Grant N66001-05-C-8013. The authors are with the IBM T. J. Watson Research Center, Yorktown Heights, NY 10598 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2006.877832

are prohibitively high at millimeter-wave frequencies [9] and plastic packaging materials are quite lossy. Moreover, standard mold materials have not generally been characterized at millimeter-wave frequencies [10]. Historically, high-quality passive devices were needed inside millimeter-wave packages to complement millimeter-wave circuitry. The only option was to use highly complex and costly packaging technologies, e.g., like low-temperature co-fired ceramic (LTCC) [11], [12]. With the recent progress in the semiconductor technologies like BiCMOS SiGe and CMOS, however, the integration level has made significant progress and passive device functions can be eliminated from millimeter-wave packages. This opens up new avenues for low-cost packaging technologies. A millimeter-wave radio architecture with baseband modulated interface can further relax the frequency requirements. Packages of highly integrated I/Q radios, for example, do not have to conduct frequency higher than the baseband if the antenna can be integrated together with the chip in a single package. This represents an inherent advantage compared to other III/V semiconductor and MMIC technologies, which exhibit lower integration levels [13], [14]. In this paper, we present a cost-effective CSP solution for 60-GHz wireless chipsets capable of multigigabit per second wireless communication. We have integrated a single-chip transmitter and single-chip receiver together with 7-dBi cavity-backed folded dipole antennas in a DCA or direct surface mountable land-grid array (LGA). The area of the LGA package is 2.8 the area of its components (chip plus antenna). The size of the DCA package is 7 11 mm . The package buildup provides a well-controlled electromagnetic (EM) environment, which makes the antenna performance less sensitive to the surrounding package-level and printed circuit board (PCB)-level dielectric and metal structures—an important design feature simplifying the simulation and modeling complexity typically involved in millimeter-wave package designs [15]. Although there has been some successful early work in integrating antennas with simple mixers directly in silicon [16], the LGA package represents an individual component with a formal package for environmental protection around it. See [17] for more details on the LGA fabrication technique. This paper begins with a short description of the millimeter-wave chipset design (see Section II). Next, the packaging concept is outlined in Section III. Simulated and measured antenna performance is given in Section IV. The packaged transmitter performance is covered in Section V, including anechoic chamber measurements showing its radiation pattern, large-signal compression, total conversion gain, and the interconnect modeling approach being taken. This

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Fig. 2. (a) DCA package encapsulated with a standard low-cost glob-top material. (b) Encapsulated LGA package including a Tx and Rx chip. Note: the antenna efficiency in both packages is above 90%. The efficiency will degraded by approximately 5% if the antenna opening is omitted.

III. PACKAGING CONCEPT

Fig. 1. Block diagrams of the 60-GHz Tx and Rx chips.

TABLE I 60-GHz CHIPSET PERFORMANCE

includes reliability and system link tests, which demonstrate the feasibility of the packaged radio system. II. 60-GHz CHIPSET The two chips [receiver (Rx) and transmitter (Tx)] are implemented in a 0.13- m SiGe BiCMOS technology with GHz and GHz [18]. The radio design uses a superheterodyne architecture with a variable IF for operation in the 59–64-GHz band. Fig. 1 shows a block diagram of the Rx and Tx chips. Image rejection and IF filters are integrated on chip. The frequency of a voltage-controlled oscillator (VCO) is either tripled or halved to generate the local oscillator for the millimeter-wave and IF mixers. The frequency plan results in an IF of 8.4–9.1 GHz, a VCO frequency of 16.8–18.3 GHz, and an image frequency of 42–46 GHz. The bandwidth of the in-phase/quadrature (I/Q) baseband signal is currently limited by the IF bandwidth of 700 MHz. On-wafer measurements for the Rx and Tx chips have previously been published [19] and are repeated in Table I for comparison with their packaged performance. The die sizes are 3.4 1.7 mm and 4.0 1.6 mm , respectively.

An inherent advantage of the highly integrated radio architecture with respect to packaging is that the maximum frequency required for a package lead is given by the maximum bandwidth of the I/Q baseband signals. The VCO, phase-locked loop (PLL), and IF signals are confined within the chip and do not have to be connected from the outside. This represents an inherent advantage compared to other III/V semiconductor and MMIC technologies, which exhibit lower integration levels [14]. However, this assumes that a low-loss interconnection of the 60-GHz RF signal to the antenna exists. We have solved this by integrating the antenna together with the chip into the same package. Alternative approaches include integrating the antenna directly on the chip; however, due to the increase in chip area, we have not considered this as being a cost-effective solution. Fig. 2 shows two package encapsulation options that currently exist. A DCA is shown in Fig. 2(a) and a direct surface mountable LGA is shown in Fig. 2(b). The size of the DCA package is 7 11 mm and the LGA package size is 13 13 mm . The Tx and Rx chip can also be packaged individually in separate Tx and Rx LGA packages, each having a size of 6 13 mm (not shown). Fig. 3 shows a conceptual drawing of the package buildup. The antenna is constructed from a fused silica (SiO ) substrate, which is bonded to a covar metal frame using a thermosetting adhesive. A covar alloy was chosen because of its superior thermal expansion (3.7 ppm C) characteristic, which is matched closely to that of fused silica (0.55 ppm C). A chemical etching and photolithography process was used for frame fabrication to comply with tight tolerances and yet high volume fabrication. The antenna is flipped to the millimeter-wave circuit chip using a thermal compression flip-chip bonding technique. Simultaneously, the antenna frame is attached to the package base using a silver-filled adhesive. This way, the antenna is suspended in air below the silica superstrate and the metallic base plate of the package acts as a reflecting ground. A uniform ball height of 30 m could be achieved across the flip-chip ball interconnect. The spacing between the radiating element and the ground influences the bandwidth of the antenna and is defined by the thickness of the covar frame. This buildup leads to a direct connection to the antenna through the 30- m–high flip-chip ball interconnect, without the

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Fig. 3. Conceptional drawing for the buildup of the LGA. Standard wire bonding is used, except for the 60-GHz RF signal. A fused silica antenna is flipped to the chip providing the shortest possible interconnect. Silver-filled adhesive is used for chip and antenna attachment. An under-fill material is dispensed locally between the antenna substrate and the die surface, as indicated. The mold material in the antenna feed-line opening, as well as the package encapsulation material, are omitted for better clarity.

need for vias or EM coupled substrate interconnects [20], [21]. Flip-chip interconnects have shown excellent performance at millimeter-wave frequencies and have been characterized and modeled up to 110 GHz [22]. Unlike other suspended antennas [23]–[26], the covar frame serves two purposes. First, it provides the mechanical support for the antenna, and second, it provides a well-controlled EM environment, making the antenna performance less sensitive to the surrounding package- and PCB-level dielectric and metal layers. The frame decouples the design of the antenna from the exact physical properties of the package such as dimension and material parameters—an important design feature simplifying simulation and modeling complexity. Use of the frame avoids any unknown shapes or encapsulants in close proximity to the radiating element. Special care, however, is required for the antenna feed line that interfaces the active circuit to the antenna. A shift in the dielectric constant or loss tangent of any feed-line encapsulation will de-tune the antenna impedance match and reduce receiver sensitivity and transmitter efficiency. The feed line, therefore, passes through a dedicated opening in the frame, which was encapsulated prior to the mounting of the antenna. This pre-molding step ensures the hermeticity of the cavity and allows the antenna design process to be greatly independent of the electrical properties of the final package encapsulation material. In this configuration, the folded-dipole antenna achieves 7-dBi gain and 30 3-dB beamwidth. The simulated interconnect shows a large impedance bandwidth of over 30% defined by a 10-dB return loss and a radiation efficiency above 90% over the entire 59–64-GHz band. Fig. 4 shows a photograph of the PCB used for testing of the transmitter chip. The board uses a multilayer stack-up with an FR4 core and a Teflon top-layer dielectric for low-loss and impedance-controlled interconnects. Although the Teflon adds

Fig. 4. Test board with packaged transmitter chip including its antenna. The size of the Tx board is 6 5 cm and the package has a size of approximately 4 7 mm . A similar sized board was used for receiver tests.

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little to the total cost of the board, FR4 could have been used exclusively due to the relatively low-frequency requirements. Fig. 5 shows a close-up side view of the DCA package before encapsulation. Both packages exhibit a low operating junction temperature and low temperature-activated failure rate. This is achieved due to use of both a low thermal resistance from the junction to the board-level heat-sink and coefficient of thermal expansion (CTE) matched materials combined with low-stress molding compound and die-attach adhesives. The DCA package has an inherent low thermal resistance since the package ground connects directly to the board-level ground plane. The LGA

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Fig. 5. DCA package before encapsulation. The fused silica is transparent, thus allowing one to see through the antenna superstrate. The antenna feed is flip-chip connected to the antenna and the antenna is radiating through the silica with the main radiation direction being orthogonal to the board surface.

package uses a thin (250- m thick) interposer layer with thermal vias that add little to its thermal resistance. Results on the package-device performance and reliability are given in Section V. The most critical point in terms of CTE mismatch is the gap between the chip and the covar frame that is bridged by the fused silica antenna substrate. The different CTEs between the package ground material, typically copper, and the antenna substrate material can cause a strain on the flip-chip joints. However, this gap is small enough (500 m) that the stress can be absorbed by an under-fill material, dispensed locally between the antenna substrate and the die surface. The length expansion at that point is only on the order of a half micrometer. A low-viscosity mold material was dispensed on the chip and the antenna to provide a final void-free hermetical seal. Deliberately, air is only enclosed within the cavity of the frame. The enclosed air is not of concern since additional stress due to the thermal expansion of air can be absorbed without any delamination, blistering, or cracking. Unlike other packaging technologies, the power of the chip is not dissipated within the air-filled cavity, leaving the air closer to the package heat-sink temperature. This greatly relaxes any temperature-activated component failures. Furthermore, glass is used in high-reliability applications and is known for its excellent hermeticity and strength [27]. According to Gay–Lussac’s Law (constant volume enclosed by the frame), the elevated pressure is only 28.8% for a change in ambient temperature from 25 C to 85 C. At sea level, this corresponds to a force equivalent to approximately 77 g applied to the area of the fused silica substrate opening. The stiffness of the covar metal frame and fused silica can withstand such forces, as will be further discussed in Section V. However, it is to be understood that such packages should not be used in other than terrestrial (e.g., space) conditions, which have low or no external ambient pressure. IV. INTEGRATED ANTENNA DESIGN Another major challenge is the development of antennas that can be integrated into low-cost packaging solutions at millimeter-wave frequencies. The main requirements for the antenna, beyond low-cost and small size, are wide bandwidth (e.g., approximately 10%–15% for 59–64-GHz coverage) and high efficiency [28], [29]. This is usually achieved by choosing a substrate with the lowest possible dielectric constant [28],

Fig. 6. Simulated and measured antenna gain.

suspending the antenna in air [29], or adding a superstrate over a planar antenna [24]–[26]. The antenna used in this paper combines a metal cavity with a suspended folded dipole antenna [23] to provide a mechanically stable and reliable solution for high-volume manufacturing. Moreover, the antenna is compatible with standard plastic packaging technologies and provides higher bandwidth than standard planar antennas while maintaining high antenna efficiency. The free-space characteristics of folded dipole antennas are well known [30] in the literature. This includes folded dipole antennas parallel to a metal plate [31] or printed on a PCBs [32]. Folded dipole antennas printed on a circuit board that is backed with a metal cavity, however, has not been reported. The metal frame, previously described in Fig. 3, is crucial for the presented packaging approach. It provides a barrier that protects the inside of the antenna cavity from the mold material. The metal walls of the frame further provide a well-controlled shield for an easier design process. However, approximately 500 m of the antenna feed is exposed by the frame and will be covered with a lossy mold material that needs to be considered during the feed-line design. See [17] for a sensitivity analysis of the antenna in respect to the electrical properties of mold materials. The shape and size of the applied cavity was optimized according to the antenna input impedance bandwidth. Other important parameters for impedance bandwidth were the strip width of the dipole and spacing between its front and rear parts. Two additional metal bars (reflectors) were placed on top of the fused silica (see Fig. 5) to minimize radiation in the antenna plane. The antenna with its surrounding cavity was measured in an anechoic chamber as a standalone part using a probe-tip-based measurement setup described in [33] with the maximum measured frequency limited to 65 GHz. For performance characterization, the antenna was mounted in a specially designed sample holder and connected with a coplanar probe. To emulate the package environment, its feed line outside of the cavity up to the probe tips was covered with the mold material used in the package. Fig. 6 shows the measured and simulated antenna gain over frequency. Very good agreement between the measurement and

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Fig. 9. Setup used for radiation pattern measurements of the packaged transmitter in an anechoic chamber. The Rx horn is spaced 38 cm apart from the Tx chip antenna. Fig. 7. Simulated and measured antenna pattern along  in the  = 0 direction. Note:  and  are defined as spherical polar coordinates throughout this paper. See Fig. 3 for their definition in respect to the package and antenna orientation.

Fig. 8. Simulated and measured antenna pattern along  in the  = 90 direction.

simulation was found. The simulated antenna efficiency is above 90% within the entire frequency range of 55–65 GHz. The good agreement between the measured and simulated gains indicate that the radiation efficiency is very close to its simulated value. Simulations indicate that the lossy encapsulant reduces the antenna efficiency by only 4% from the initial 95%. The measured return loss is better than 15 dB for 56–65 GHz. Both vertical co-polar radiation patterns for and are presented in Figs. 7 and 8. Very good correlation between the measured and simulated curves exist. Note, the packaged DCA antenna has improved back radiation due to the larger board-level ground plane, which will be shown in Section V. A high-resolution gold deposition process on fused silica was selected for its tight metal feature tolerances required at millimeter-wave frequencies. Furthermore, fused silica has a very

low loss tangent, below 0.001 at 60 GHz [23], and a relatively low dielectric constant of 3.8. A half-wavelength folded dipole topology was chosen as the radiating element due to its small dimensions and the fact that it matches to the coplanar stripline (CPS) used for connecting the antenna to the transmitter chip. It shows higher input impedance when compared to a regular dipole, which is a desirable feature when considering the range of feasible characteristic impedance values for the CPS lines on fused silica. A dual dipole configuration was chosen for the Rx chip taking into account that coplanar waveguide (CPW) line results in the most straightforward interconnect scheme between both parts. The Tx and Rx antennas are matched to 100 and 50 , respectively. A 250- m-thick antenna substrate was used, being a compromise between the mechanical stability and manufacturability on one side, and the deteriorating influence of surface waves on the other. The metal frame defines a cavity depth of 500 m, which is a tradeoff between the available chip thicknesses and the required antenna bandwidth. The internal cavity size is 3 4.2 mm and its metal walls are 0.4-mm thick. The 1.46-mm-long dipole is fed by a quarter-wavelength transformer line followed by a uniform 100- CPS line within the cavity, and a 100- tapered CPS line. Tapering was required to connect to the chip bonding pads. The lateral dimensions (strip and slot width) of the tapered line were adjusted so that it could accommodate the influence of encapsulant material covering this part of the feed structure. The encapsulant dielectric constant and loss tangent were measured to be 4.4 and 0.03 at 60 GHz, respectively [23]. The radiation window above the antenna substrate is used to improve the radiation efficiency, although the simulated antenna efficiency is only degraded by less than 5% if a 200- m-thick mold layer is applied. V. PACKAGED CHIP PERFORMANCE The packaged chip performance was measured at various steps during the packaging process to monitor any package-related performance degradation. First, initial on-wafer measurements are required to provide a reference for the conversion gain, noise figure, or maximum available output power that

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Fig. 10. Radiation pattern along  (0 –120 ) for 61 GHz. (a) Before molding ( = 0 ). (b) After molding ( = 0 ). (c)  = 90 direction. The cross-polarization in each direction is 20 dB down (not shown). (Color version available online at: http://ieeexplore.ieee.org.)

is achievable by the receiver or transmitter chip, respectively (see Section II for details). Next, such on-wafer results can be verified on the board-level if the input pads to the LNA or output pads from the power amplifier (PA) are not obstructed by any objects in close proximity to the probe pads. A final test, however, is required that shows the overall encapsulated performance of the packaged chip including its antenna. Since the antenna cannot be measured in situ, the exact interconnect losses are unknown and can only be inferred by accurately calibrated gain measurements in an anechoic chamber. To do this, a similar measurement setup like the one described in [33] was used here to provide accurate gain calibration and radiation pattern measurement capabilities. Unlike passive antennas, however, an active radio circuit includes a frequency conversion. A network-analyzer-based gain calibration and measurement technique, therefore, cannot be used. Fig. 9 shows the measurement setup that was used for the transmitter chip. The transmitter anechoic chamber measurements achieve a better sensitivity than the receiver measurements due to the high transmit power that is available from the Tx. Three calibrations are required, which are: 1) a gain calibration; 2) an I/Q input power calibration; and 3) a loss calibration. The gain calibration was done across the industrial–scientific–medical (ISM) band with a standard gain horn antenna that was used instead of the Tx chip. The standard gain horn was driven by a high-power synthesizer and the power that was received by the receiver horn at the opposite side of the chamber was measured with a power sensor (including an extra gain stage). A loss calibration is required to provide absolute power numbers during the measurement; hence, the frequencydependent loss from the synthesizer to the waveguide connector of the standard gain horn was calibrated. The I/Q input power delivered to the on-chip differential 100- terminations was calibrated from 50 to 5 dBm with a 100-MHz continuous wave (CW) tone. In the main radiation direction, this leads to an overall gain measurement that includes the chip’s conversion gain, the antenna gain, and the interconnect and packaging related losses. As mentioned previously, the chip’s conversion gain was measured accurately on wafer prior to packaging and the antenna gain was simulated using a full-wave EM simu-

lator, as described in Section IV. From such data, one can finally infer the interconnect losses. Due to the calibration complexity, the measurement accuracy is limited to an estimate of 0.5 dB. The anechoic chamber setup was used to measure the radiation pattern, the interconnect losses, as well as the large-signal gain compression. The following data was taken at 61 GHz with a 10-dB backoff from the 1-dB compression point of the Tx. The transmitter on-wafer conversion gain at that point was 35 dB. Fig. 10 shows the vertical radiation pattern along to 120 for 61 GHz. Fig. 10(a) and (b) shows the package radiation pattern before and after molding with a standard low-cost mold material in the direction. The direction of the molded package is assumed to be symmetrical around and is shown in Fig. 10(c). The cross-polarization is 20 dB down (not shown). The overall gain in the main radiation direction is approximately 41 dB with a 3-dB beamwidth of 30 . The pattern is smooth, implying little package and board-level interference. The molded package shows a slightly higher disturbance at 30 due to interference with, and refraction at, the package boundary. Fig. 11 shows the to 120 pattern from 59 to 64 GHz in a waterfall diagram. The pattern changes only slightly across frequency due to a small shift in the radiation wavelength. The location of the ripples changes across frequency without any major distortion or notches. The observed spatial variation is within 1 dB. A. Interconnect Modeling Approach The quality of the electrical interconnect between the PA [or low-noise amplifier (LNA)] to the antenna is a critical measure of the packaged chipset performance since it reduces transmitter efficiency and receiver sensitivity. A well-modeled interconnect is an important design features since it enables detailed chip-package co-simulations. The electrical interconnect model consists of a two-port lumped equivalent circuit for the flip-chip interconnect in series with a one-port scattering parameter model of the antenna. The flip-chip interconnect model was extracted based on a direct ball-probing technique [34] and the antenna model was extracted from a full-wave simulation of the

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Fig. 11. Waterfall diagram of the  = 0 to 120 radiation pattern from 59 to 64 GHz.

physical three-dimensional (3-D) antenna, cavity, and feed-line structure. Although the impedance at the on-chip pads can be arbitrarily assigned within the limits given by the antenna impedance and the optimum LNA or PA load, it is convenient to use a 50- characteristic impedance, which is compatible with standard test equipment. The LNA input match, therefore, is designed for a 50- ground–signal–ground (GSG) coplanar input impedance, whereas the PA output match is designed for a differential 100- ground–signal–ground–signal–ground (GSGSG) load impedance. The differential 100- impedance was chosen to utilize an additional 3-dB power pick up at the antenna port. Any on-chip microstrips, contact pads, or matching elements have been modeled as part of the circuit and are not considered as part of the interconnect model. Although the interconnect and antenna are highly efficient and have little resistance, they have associated reactance that will change the optimum load impedance seen by the PA and, likewise, the LNA. The PA, for example, is designed for maximum power delivery into a 100- load while providing good linearity and high 1-dB compression point (CP1 dB). A shift in its optimum load impedance causes an lower voltage or current compression that will change the gain and compression characteristic, respectively. A low insertion and return loss are, therefore, important design features that need to be modeled and co-simulated with the active circuits. From 59 to 64 GHz, the simulated return loss at the antenna feed line is better than 10 dB and the return loss seen at the PA chip pads is better than 6 dB. The simulated frequency response of the packaged transmitter is less broadband and has a slightly higher gain peak shifted upwards by approximately 1 GHz compared with an ideal 100- load. The simulated CP1 dB at 61.5 GHz is approximately 1 dB lower. Fig. 12 shows the interconnect loss versus frequency that was calculated from the total measured gain of the transmitter. This was done directly after attaching the antenna, under-filling the interconnect, molding ( ), and after thermal cycling tests had been completed (see Section V-B). Note that the antenna is designed for encapsulation and, therefore, shows a higher interconnect loss before molding. The interconnect loss

Fig. 12. Interconnect loss measured at various steps during the assembly. Data was taken after attaching the antenna (Plain), after under-filling the interconnect (Ufill), after molding  = 0 ; 90 (Enc,Enc90), and after thermal cycling tests (Temp) have been completed. (Color version available online at: http://ieeexplore.ieee.org.)

Fig. 13. Large-signal compression of the transmitter chip at 61 GHz. Output power and power gain is shown before (on-wafer) and after packaging. The packaged part exhibits a 5-dB higher net gain due to the 7-dBi antenna gain pickup (2-dB loss). (Color version available online at: http://ieeexplore.ieee. org.)

is around 2 dB across the ISM band. Although the accuracy of this measurement is limited to approximately 1 dB, it is consistent with the link budget described in Section V-C. Fig. 13 shows a comparison of the 61-GHz large-signal compression characteristic of the transmitter chip measured on wafer and after packaging in the anechoic chamber. The input-referred 1-dB compression point is slightly lower than on wafer, indicating a slight shift in the PA load impedance, as mentioned before. This effect and the slight dip in the packaged transmitter gain characteristic can be simulated and are due to the packaged PA load impedance. The packaged transmitter has approximately 5 dB more gain due to the additional antenna

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gain pickup. This is equivalent to a 2-dB interconnect loss since a 7-dBi antenna was used. B. Reliability Tests The reliability of an electronic system comprising a group of components is the probability, expressed in percent, of operating continuously over a specified period of time. The more components or chips that are contained within a package, the higher the failure rate will be. A higher circuit-level integration, therefore, is inherently more reliable than any multichip module or MMIC technology can be. However, the fact that the antenna is integrated into the package might cause some increased temperature-activated failures due to higher stress at the antenna interconnect [35] so this was tested. A similar setup like the one shown in Fig. 9 was used for thermal cycling tests in a Tenney temperature chamber. Due to the size of the chamber, the radiated power from the Tx board was simply captured with a horn antenna that was positioned above the Tx package at a distance of only about 2 in. Such a setup cannot be calibrated accurately since the location of the horn will vary with temperature, but it can be used qualitatively to monitor the Tx performance. The Tx board was cooled down to 5 C and heated up to 85 C within 30 min including 10-min soak time at the extrema. The board was tested over 35 h (70 temperature cycles) and showed failure-free operation while it was operated at the 1-dB compression point with approximately 10-dBm output power. The receiving horn antenna is located in the temperature chamber and, hence, cooled and heated likewise, which dislocates the horn slightly over time. This leads to an additional variation of the received signal strength on top of a 3-dB Tx gain variation. Most of the Tx circuitry uses a proportional-to-absolute-temperature (PTAT) bias that will compensate for some temperature degradation. Constant output power, however, needs to be regulating externally with the IF variable gain amplifier gain settings, which is not being anticipated in this test. Fig. 14 shows a total 4-dB gain variation over temperature, recorded at the end of the cycling period. After thermal cycling, the board was remeasured in the anechoic chamber at room temperature and no performance degradation was observed (see Fig. 12 for the remeasured interconnect loss).

Fig. 14. Thermal reliability test results. The Tx board was cycled from 5 C to 85 C with a 4-dB gain variation. No failure or performance degradation was observed over 70 cycles (35 h).

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Fig. 15. Expected coverage range versus data rate. The measured range at 630 Mb/s (limited by test equipment) is included. TABLE II LINK BUDGET

C. System Link Tests The chipset is intended primarily for half-duplex communications links to avoid the need for high- frequency-domain duplex filtering in the front-end antenna system. In the half-duplex system architecture, the Tx chip is powered down, while the Rx is active and vice versa. This avoids degrading receiver performance from transmitter power leakage into the receiver band. The receiver and transmitter boards have been mounted on separate carts in order to demonstrate the system link performance. A 700-Ms/s arbitrary waveform generator (ARB) was used for the Tx baseband I/Q modulation and a 700-Ms/s 8-bit PCI ADC including a software demodulator was used at the Rx I/Q output. The expected coverage range versus data rate is shown in Fig. 15 based on the link budget given in Table II. An IEEE 802.11a-based orthogonal frequency-division multiplexing (OFDM)-quadrature phase-shift keying (QPSK) modulation was used at 630 Mb/s for bit error rate (BER) and cov-

erage range testing. Note, the 630-Mb/s speed is a test equipment limitation, not related to this chip or package. At 630 Mb/s, the link ran up to a maximum separation of 10 m with less than

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10% frame error rate (FER). This lines up very well with the expected coverage range shown in Fig. 15 and validates the inferred package interconnect loss and antenna gain. Further system performance evaluation will include high-gain (20 dBi) antennas with amplitude shift-keying (ASK)/phase-shift keying (PSK) modulation for directional 1–3-Gb/s applications. Highgain antennas will extend the coverage range into the kilometer range. Other packaging options are being developed that provide a waveguide interface for external high-gain antennas. VI. SUMMARY A cost-effective CSP solution for a millimeter-wave chipset capable of multigigabit per second wireless communications in the 60-GHz ISM band has been presented. Envisioned applications of the packaged chipset include 1–3 Gb/s or greater directional links using ASK or PSK modulation and 500 Mb/s–1 Gb/s omni-directional links using OFDM modulation. The wireless chipset was packaged together with 7-dBi cavity-backed foldeddipole antennas using a DCA or direct surface mountable LGA. The LGA area of 13 13 mm is no more than 2.8 the original die plus antenna size. The cavity-backed antenna design provides a well-controlled EM environment, simplifying antenna simulation and modeling complexity. The antenna interconnect shows an impedance bandwidth of over 30% defined by a 10-dB return loss and a radiation efficiency above 90% over the entire 59–64-GHz ISM band. The inferred interconnect loss is 2 dB. The transmitter chip achieves 3-dB beamwidth of 30 with a 41-dB total conversion gain in the main radiation direction. The pattern is smooth with little package and board-level interference. The packaged Tx chip has a saturated output power of 15 dBm with a 10-dBm 1-dB compression point. The packaged chips achieve a level of integration unique for millimeter-wave applications, with a complexity approaching that seen in chips for the cellular bands. The chipset is flexible enough to work in a variety of wireless systems. Link tests using a IEEE 802.11abased OFDM–QPSK modulation have shown 630 Mb/s over a 10-m coverage range with less than 10% FER. ACKNOWLEDGMENT The authors thank T. Zwick for early antenna developments, R. John for wire-bonding, D. Beisser for layout support, and M. Soyuer and M. Oprysko for management support, all with the IBM T. J. Watson Research Center, Yorktown Heights, NY. The authors would also like to thank all who contributed to the fabrication and mounting of the chips, especially the IBM SiGe Technology Group, IBM Burlington, Essex Junction, VT, for chip fabrication. REFERENCES [1] D. Parker, “Microwave industry outlook—Defense applications,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 3, pp. 1039–1041, Mar. 2002. [2] J. Schepps and A. Rosen, “Microwave industry outlook—Wireless communications in healthcare,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 3, pp. 1044–1045, Mar. 2002. [3] K. Kitazawa, S. Koriyama, H. Minamiue, and M. Fujii, “77-GHz-band surface mountable ceramic package,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 9, pp. 1488–1491, Sep. 2000. [4] N. Jain, “Designing commercially viable mm-wave modules,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2000, pp. 565–568.

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[5] M. Oppermann, “Multichip-modules for micro- and millimeter-wave applications—A challenge?,” in Int. Multi-Chip Modules and HighDensity Packag. Conf., Apr. 1998, pp. 279–284. [6] T. Midford, J. Wooldridge, and R. Sturdivant, “The evolution of packages for monolithic microwave and millimeter-wave circuits,” IEEE Trans. Antennas Propag., vol. 43, no. 9, pp. 983–991, Sep. 1995. [7] M. Hauhe and J. Wooldridge, “High density packaging of -band active array modules,” IEEE Trans. Manufact. Technol., vol. 20, no. 3, pp. 279–291, Aug. 1997. [8] A. Bessemoulin, M. Parisot, and M. Camiade, “1-watt -band power amplifier MMICs using low-cost quad-flat plastic package,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2004, vol. 2, pp. 473–476. [9] J.-Y. Kim, H.-Y. Lee, J.-H. Lee, and D.-P. Chang, “Wideband characterization of multiple bondwires for millimeter-wave applications,” in Asia–Pacific Microw. Conf., Dec. 2000, pp. 1265–1268. [10] T. Zwick, A. Chandrasekhar, C. Baks, U. R. Pfeiffer, S. Brebels, and B. P. Gaucher, “Determination of the complex permittivity of packaging materials at millimeter-wave frequencies,” IEEE Trans. Microw. Theory Tech., vol. 54, no. 3, pp. 1001–1010, Jan. 2006. [11] J.-H. Lee, G. DeJean, S. Sarkar, S. Pinel, K. Lim, J. Papapolymerou, J. Laskar, and M. Tentzeris, “Highly integrated millimeter-wave passive components using 3-D LTCC system-on-package (SOP) technology,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 6, pp. 2220–2229, Jun. 2005. [12] J. Heyen, T. von Kerssenbrock, A. Chernyakov, P. Heide, and A. Jacob, “Novel LTCC/BGA modules for highly integrated millimeter-wave transceivers,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 12, pp. 2589–2596, Dec. 2003. [13] K. Ohata, K. Maruhashi, M. Ito, S. Kishimoto, K. Ikuina, T. Hashiguchi, K. Ikeda, and N. Takahashi, “1.25 Gb/s wireless gigabit ethernet link at 60 GHz-band,” in Radio Freq. Integr. Circuits Symp., Jun. 2003, pp. 509–512. [14] S. E. Gunnarsson, C. Kärnfelt, H. Zirath, R. Kozhuharov, D. Kuylenstierna, A. Alping, and C. Fager, “Highly integrated 60 GHz transmitter and receiver MMICs in a GaAs pHEMT technology,” IEEE J. Solid-State Circuits, vol. 40, no. 11, pp. 2186–2174, Nov. 2005. [15] D. Griffin and A. Parfitt, “Electromagnetic design aspects of packages for monolithic microwave integrated circuit-based arrays with integrated antenna elements,” IEEE Trans. Antennas Propag., vol. 43, no. 9, pp. 927–931, Sep. 1995. [16] R. Carrillo-Ramirez and R. Jackson, “A highly integrated millimeter-wave active antenna array using BCB and silicon substrate,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 6, pp. 1648–1653, Jun. 2004. [17] U. Pfeiffer, J. Grzyb, D. Liu, B. Gaucher, T. Beukema, B. Floyd, and S. Reynolds, “A 60-GHz radio chipset fully-integrated in a low-cost packaging technology,” in 56th Electron. Compon. Technol. Conf., Jun. 2006, pp. 1343–1346. [18] B. Jagannathan, “Self-aligned SiGe NPN transistors with 285 GHz and 207 GHz in a manufacturable technology,” IEEE Electron Device Lett., vol. 23, no. 5, pp. 258–260, May 2002. [19] B. Floyd, S. Reynolds, U. R. Pfeiffer, T. Beukema, J. Grzyb, and C. Haymes, “A silicon 60 GHz receiver and transmitter chipset for broadband communications,” in IEEE Int. Solid-State Circuits Conf., Feb. 2006, pp. 184–185. [20] G. Strauss and W. Menzel, “Millimeter-wave monolithic integrated circuit interconnects using electromagnetic field coupling,” IEEE Trans. Compon. Packag. Technol., vol. 19, no. 2, pp. 278–282, May 1996. [21] W. M. L. Zhu, “Broad-band microstrip-to-CPW transition via frequency-dependent electromagnetic coupling,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 5, pp. 1517–1521, May 2004. [22] U. Pfeiffer and A. Chandrasekhar, “Characterization of flip-chip interconnects up to millimeter-wave frequencies based on a non-destructive in situ approach,” IEEE Trans. Advanced Packag., vol. 28, no. 5, pp. 160–167, May 2005. [23] T. Zwick, D. Liu, B. P. Gaucher, and U. R. Pfeiffer, “Apparatus and methods for constructing and packaging printed antenna devices,” U.S. Patent Applicat. 10/881 104, Jun. 2004. [24] W. Choi, C. Pyo, Y. H. Cho, J. Choi, and J. Chae, “High gain and broadband microstrip patch antenna using a superstrate layer,” in IEEE Antennas Propag. Symp., Jun. 2003, vol. 2, pp. 292–295. [25] N. Alexopoulos and D. Jackson, “Fundamental superstrate (cover) effects on printed circuit antennas,” IEEE Trans. Antennas Propag., vol. AP-32, no. 8, pp. 807–816, Aug. 1984. [26] C. Gürel and E. Yazgan, “Bandwidth widening in an annular ring microstrip antenna with superstrate,” in IEEE Antennas Propag. Symp., Jun. 1995, pp. 692–695.

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[27] R. Traeger, “Hermeticity of polymeric lid sealants,” in Proc. 25th Electron. Compon. Conf., 1976, pp. 361–367. [28] N. Herscovici, “A wideband single-layer patch antenna,” IEEE Trans. Antennas Propag., vol. 46, no. 4, pp. 471–474, Apr. 1999. [29] M. Faiz and P. Wahid, “A high efficiency L-band microstrip antenna,” in IEEE Int. URSI Conf., Jul. 1999, pp. 272–275. [30] W. L. Stutzman and G. A. Thiele, Antenna Theory and Design, 2nd ed. New York: Wiley, 1998, vol. 2. [31] P. Raumonen, L. Sydanheimo, L. Ukkonen, M. Keskilnmmi, and M. Kivikoski, “Folded dipole antenna near metal plate,” in Proc. IEEE AP-S Int. USNC/URSI Nat. Radio Sci. Meeting Symp., Jun. 2003, vol. 1, pp. 848–851. [32] C. Lin, C. Su, F. Hsiao, and K. Wong, “Printed folded dipole array antenna with directional radiation for 2.4/5 GHz WLAN operation,” Electron. Lett., vol. 39, no. 24, pp. 1698–1699, Nov. 2003. [33] T. Zwick, C. Baks, U. Pfeiffer, D. Liu, and B. Gaucher, “Probe based MMW antenna measurement setup,” in IEEE Int. Antennas Propag. Symp., Jun. 2004, pp. 747–750. [34] U. Pfeiffer and B. Welch, “Equivalent circuit model extraction of flipchip ball interconnects based on direct probing techniques,” IEEE Microw. Wireless Compon. Lett., vol. 15, no. 9, pp. 594–596, Sep. 2005. [35] Y. Lin, W. Liu, Y. Guo, and F. Shi, “Reliability issues of low-cost overmolded flip-chip packages,” IEEE Trans. Adv. Packag., vol. 28, no. 1, pp. 79–88, Feb. 2005.

Ullrich R. Pfeiffer (M’02) received the Diploma degree in physics and Ph.D. degree in physics from the University of Heidelberg, Heidelberg, Germany, in 1996 and 1999, respectively. In 1997, he was a Research Fellow with the Rutherford Appleton Laboratory, Oxfordshire, U.K., where he developed high-speed multichip modules. In 2000, his research was based on high-integrated real-time electronics for a particle physics experiment with the European Organization for Nuclear Research (CERN), Geneva, Switzerland. In 2001, he joined IBM and is currently a Research Staff Member with the IBM T. J. Watson Research Center, Yorktown Heights, NY. His current research involves RF circuit design, power-amplifier design at 60 and 77 GHz, high-frequency modeling, and packaging for millimeter-wave communication systems. Dr. Pfeiffer is a member of the German Physical Society (DPG). He was the recipient of the 2004 Lewis Winner Award for Outstanding Paper presented at the IEEE International Solid-State Circuit Conference.

Janusz Grzyb (M’01) received the M.Sc. degree in electrical engineering (with honors) from the Technical University of Gdansk, Gdansk, Poland, in 1997, and the Ph.D. degree from the Swiss Federal Institute of Technology Zürich, Zürich, Switzerland, in 2004. From 1997 to 1999, he was a Research Assistant with the Technical University of Gdansk, where he was involved in the area of CMOS and BiCMOS integrated circuits. While with the Swiss Federal Institute of Technology, he was mainly involved in the development of multichip-module (MCM)-based packaging and system-on-package solutions for 60- and 77-GHz applications. Since joining the IBM T. J. Watson Research Center, Yorktown Heights, NY, in 2004, his primary responsibilities has been antenna and package design for 60-GHz wireless systems. His main interests include the development of packaging and subsystem integration solutions for millimeter-wave applications, EM modeling, analysis and design of antennas and passive structures at millimeter-wave frequencies, and CMOS and BiCMOS analog and mixed-signal integrated circuits.

Duixian Liu (S’85–M’90–SM’98) received the B.S. degree in electrical engineering from XiDian University, Xi’an, China, in 1982, and the M.S. and Ph.D. degrees in electrical engineering from The Ohio State University, Columbus, in 1986 and 1990, respectively. From 1990 to 1996, he was with Valor Enterprises Inc. Piqua, OH, initially as an Electrical Engineer and then as Chief Engineer, during which time he designed an antenna product line ranging from 3 MHz to 2.4 GHz . Since April 1996, he has been with the IBM T. J. Watson Research Center, Yorktown Heights, NY, as a Research Staff Member. He is also an external Ph.D. examiner for several universities and external examiner for some government organizations on research funding proposals. He has authored or coauthored approximately 50 journal and conference papers. He holds 17 patents with 20 pending. His research interests are antenna design, EM modeling, digital signal processing, and communications technology. Dr. Liu is an associate editor for the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION. He has been the organizer or chair for numerous international conference sessions or special sessions and also a technical program committee member. He was the general chair of the 2006 IEEE International Workshop on Antenna Technology: Small Antennas and Novel Metamaterials, White Plains, NY. He was the recipient of the prestigious Presidential “E” Award for Excellence in Exporting in 1994. He was a two-time recipient of the 2001 and 2002 IBM Outstanding Technical Achievement Award. He was also the recipient of the 2003 IBM Corporate Award (IBM’s highest technical award) for contributions to the integrated antenna subsystems for laptop computers.

Brian Gaucher (M’81) received the B.S. degree from the University of Massachusetts at Lowell, in 1982, and the M.S. degree from Northeastern University, Boston, MA, in 1993. From 1982 to 1983, he was with the Research and Development Laboratory, Alpha Industries, where he designed microwave GaAs field-effect transistor (FET) amplifiers, switches detectors, limiters, filters, and supercomponents. In 1984, he joined the Communication Systems Division, GTE, where he was involved with research and development of secure spread-spectrum communication and radar systems for the military across the 900-MHz–60-GHz frequency bands. In 1993, he joined IBM. He is currently a Research Staff Member with the IBM T. J. Watson Research Center, Yorktown Heights, NY, where he manages a communication system design and characterization group. His group has helped more than five products come to market. He is an IBM Master Inventor. His current research interests include 60-GHz multigigabit-per-second wireless communication design and 77- and 94-GHz radar and biomedical applications of wireless technology. Mr. Gaucher was the recipient of two Outstanding Technical Achievement Awards and one corporate award.

Troy Beukema received the B.S.E.E. and M.S.E.E. degrees from the Michigan Technological University, Houghton, in 1984 and 1988, respectively. From 1984 to 1988, he was a Research and Development Engineer with the Hewlett-Packard Company, where he was involved in the communications test equipment area. In 1989, he joined the Communications Sector, Motorola, Schaumburg, IL, where he contributed to the development of digital cellular wireless systems. In 1996, he joined the IBM T. J. Watson Research Center, Yorktown Heights, NY, where he is currently a Research Staff Member with a concentration on radio architecture and modulation for 60-GHz wireless systems and high-speed serial I/O core architecture for adaptive equalization of 6–12 –Gb/s backplane/wire-line links. His research interests include communication link system design and simulation with an emphasis on modulation, equalization, and synchronization for wireless and high-speed wire-line channels.

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Brian A. Floyd (S’98–M’01) received the B.S. degree (with highest honors), M. Eng., and Ph.D. degrees in electrical and computer engineering from the University of Florida, Gainesville, in 1996, 1998, and 2001, respectively. While with the University of Florida, he was involved with wireless clock distribution for multigigahertz microprocessors. In 2001, he joined IBM. He is currently a Research Staff Member with the IBM T. J. Watson Research Center, Yorktown Heights, NY. His research interests include millimeter-wave and RF integrated circuit design. Dr. Floyd was the recipient of the Intersil/Scientific Research Council (SRC) Graduate Fellowship and the Pittman Fellowship while with the University of Florida. He was the recipient of the Best Paper Award presented at the 2004 International Solid-State Circuits Conference. He was also the recipient of the 2000 SRC Copper Design Challenge.

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Scott K. Reynolds received the B.S.E.E. degree from The University of Michigan at Ann Arbor, in 1983, and the M.S.E.E. and Ph.D. degree in electrical engineering from Stanford University, Stanford, CA, in 1984 and 1987, respectively. In 1998, he joined IBM. He is currently a Research Staff Member with the IBM Thomas J. Watson Research Center, Yorktown Heights, NY. His job responsibilities have involved analog- and mixed-signal circuit design for a wide variety of high-speed communication systems, both IBM products and research projects. He is currently engaged in the development of CMOS and BiCMOS integrated circuits for high-data-rate wired, RF wireless, and optical communication links. He has authored and coauthored numerous technical publications. Dr. Reynolds was the recipient of the Lewis Winner Outstanding Paper Award presented at the 2004 International Solid-State Circuits Conference for his paper on 60-GHz wireless transceiver circuits.

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A Photonic Crystal Power/Ground Layer for Eliminating Simultaneously Switching Noise in High-Speed Circuit Tzong-Lin Wu, Senior Member, IEEE, and Sin-Ting Chen Abstract—A novel photonic crystal power/ground layer (PCPL) is proposed to efficiently suppress the power/ground bounce noise (P/GBN) or simultaneously switching noise (SSN) in high-speed digital circuits. The PCPL is designed by periodically embedding high dielectric-constant rods into the substrate between the power and ground planes with a small area filling ratio less than 10%. The PCPL can efficiently eliminate the SSN (over 60 dB) with broad stopband bandwidth (totally over 4 GHz) below the 10-GHz range, and in the time domain, the P/GBN can be significantly reduced over 90%. The PCPL not only performs good power integrity, but also keeps good signal quality with significant improvement on eye patterns for high-speed signals with via transitions. In addition, the proposed designs perform low radiation (or electromagnetic interference) caused by the SSN within the stopbands. These extinctive behaviors both in signal integrity and electromagnetic compatibility are demonstrated numerically and experimentally. Good agreements are seen. The bandgap maps to help design the PCPL structure are also demonstrated based on the two-dimensional finite-difference time-domain method. Index Terms—Electromagnetic bandgap (EBG), electromagnetic interference (EMI), high-speed digital circuits, photonic crystal, power/ground bounce noise (P/GBN), power integrity (PI), signal integrity (SI), simultaneously switching noises (SSNs).

I. INTRODUCTION ITH THE trend of fast edge rates, high clock frequencies, and low voltage in high-speed circuits, power/ground bounce noise (P/GBN) or simultaneously switching noise (SSN) has become one of the major concerns for the power/ground plane’s design in the high-speed circuit package. In a traditional lead frame package, the SSN is resulted from the fast transient switching current passing through the equivalent inductance of the power or ground leads. As the electronic packages progressed from the lead frame package to the package with power/ground planes due to the high-speed trend, the SSN problem is moved from the inductance effect to the cavity resonance between the power and ground planes on the package. Several previous research has showed the resonance noise propagating between the power and ground planes could cause serious signal integrity (SI) or power integrity (PI)

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Manuscript received February 19, 2006; revised April 24, 2006. This work was supported by the National Science Council, Taiwan, R.O.C., under Grant NSC 93-2213-E-110-010. T.-L. Wu is with the Department of Electrical of Engineering and Graduate Institute of Communication Engineering, National Taiwan University, Taipei, Taiwan 10617, R.O.C. (e-mail: [email protected]). S.-T. Chen is with the Department of Electrical of Engineering, National Sun Yat-Sen University, Kaohsiung, Taiwan 80424, R.O.C. Digital Object Identifier 10.1109/TMTT.2006.879132

problems for high-speed circuits [1]–[3]. Moreover, due to the cavity resonance effect between the power/ground planes, the P/GBN also results in significant radiated emissions or electromagnetic interference (EMI) issues [3]. Several studies have contributed to the suppression of the SSN. Adding decoupling capacitors between power and ground planes is a typical way to eliminate the SSN and EMI, but they are not effective above several megahertz due to the effective series inductance of the capacitors. The embedded capacitance with a very thin dielectric layer between the power and ground planes are another solution to suppress the SSN [4]. However, the electromagnetic waves still propagate between the planes with resonance at specific frequencies. Recently, several electromagnetic bandgap (EBG) structures employing either the high-impedance surface (HIS) concept [5]–[7] or the long-period coplanar (LPC) EBG design [8], [9] on the power or ground planes are proposed to eliminate the P/GBN in high-speed circuits. The basic idea of these EBG structure is periodically cascading the designed unit cell with specific L and C characteristics to form a bandstop filter. These structures are realized on the power or ground planes with several etched slits on the metals. These EBG power/ground planes’ design behaves like broadband SSN isolation and EMI elimination, but the SI issue could arise for the signal traces referring to the imperfect ground or power planes with slits [10]. This paper proposes a photonic crystal power/ground layer (PCPL) to eliminate the SSN and corresponding radiated emission without etching the power or ground metal planes. The main idea of the PCPL is periodically embedding the relatively high dielectric-constant material (rods) into the original isolation layer between the solid power and ground planes. The twodimensional (2-D) photonic crystal layer with a periodic dielectric contrast will form a stopband and can be used to suppress the resonant modes excited by the SSN inside the power/ground parallel plate. By suitably choosing the pitch and dimension of the embedded rods, the stopband of the PCPL can be designed at frequencies below 10 GHz, where the P/GBN is dominantly distributed in high-speed circuits [5] with the rejection bandwidth totally above 4 GHz. Based on the 2-D finite-difference time-domain (FDTD) approach, the design diagram for the frequency range and the bandwidth of the stopband for the PCPL is calculated. The advantages of the proposed PCPL are broad stopband for SSN with a very low filling ratio of the high dielectric-constant rods, good SI with continuous power and ground planes, and an easy fabrication process compatible with standard package [or printed circuit board (PCB)] substrate manufacturing. This idea was recently proven for the capability of

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Fig. 1. PCPL concept in a typical four layer high-speed PCB circuits.

noise suppression [11] in a chip package substrate with small area, but the performance is not very satisfied. This study not only extends the idea to the PCB-like substrate (larger area) with significantly improved noise suppression performance, but also provides a complete theoretical and experimental investigation of the PCPL behavior both in the time and frequency domains, the impact of the PCPL for SI and electromagnetic compatibility (EMC), and the design strategies for the proposed structure. Photonic crystal structures have been widely used in the fiber communication and microwave circuit design, but the application in the SSN suppression for high-speed circuits, to the best of our knowledge, has not been seen in the previous literature. This paper is organized as follows. Section II describes a modified 2-D FDTD method to solve the dispersion diagram of the PCPL. The design and fabrication concept for the PCPL is also discussed. In Section III, the distinctive behavior of the P/GBN elimination both in the frequency and time domains is measured and compared with simulation. The broadband EMI suppression performance is also presented in Section III. The design diagram relating the geometrical parameters with stopband behavior will be discussed in Section IV. Conclusions are drawn in Section V. II. PCPL MODEL AND DESIGN BY 2-D FDTD METHOD A. PCPL Concept Fig. 1 shows a PCPL concept in typical four-layer high-speed PCB circuits. Top and bottom layers are for routing signal traces between the circuit components such as integrated circuits and passive elements. The inner two layers, i.e., the second and third layer, are typically for power and ground planes. In typical packages, the substrate thickness is very thin and it can be assumed that only TM modes propagate between the parallel-plate cavity formed by the power and ground planes. The PCPL concept is periodically embedding high dielectric-constant rods between the power and ground planes, as shown in Fig. 1, to eliminate the propagation of the TM modes. From the SI and EMI point-of-view, keeping the reference planes continuous is important to have a good return path for the high-speed signals. Therefore, consistent with the layout strategy for the high-speed circuit package, there are no additional etching slots or partitioning on the metal power or ground planes in our proposed design. The P/GBN and their corresponding EMI are omni-directionally and efficiently suppressed by the EBG of the PCPLs.

Fig. 2. Schematic diagram of proposed test boards. (a) SL PCPL board. (b) TL PCPL board. (Color version available online at: http://ieeexplore.ieee.org.)

B. PCPL Design and Fabrication Fig. 2(a) and (b) shows two PCPL designs with a square lattice (SL) of 40 (8 5) embedded rods and triangular lattice (TL) of 39 embedded rods on a two-layer Rogers/Duroid 5870 substrate. The dimension of the substrate for the SL PCPL and TL PCPL is 62.5 mm 100 mm and 60 mm 60 mm, respectively, with 0.8-mm thickness. The dielectric constant of the substrate is . The radius of the circular rod and the pitch between adjacent rods are denoted as and , respectively. is designed as 0.16 and 0.2, respectively, for the SL PCPL and TL PCPL. The high dielectric-constant rod with 2-mm radius and 0.8-mm height is fabricated with the mixing of BaCO and TiO under the typical ceramic fabrication process. The dielectric constant of the rods is approximately 102. The top and bottom faces of the circular rods are coated with silver metal. The PCPL is simply fabricated by drilling the 2-mm-radius holes on the substrate at designed positions and embedding the circular rods into the holes. As shown in Fig. 2, two measurement ports, i.e., port1 and port2, are located at 25 and 12.5 mm and 47 and 23.5 mm, respectively, for the SL PCPL and 50 and 15 mm and 42.5 and 40 mm, respectively, for the TL PCPL. It is noted that the filling ratio , defined as the total area of the all embedded rods to the area of the substrate, is only approximately 8% and 13% for the SL PCPL and TL PCPL, respectively. C. Bandgap Modeling by 2-D FDTD Method As shown in Fig. 2(a) and (b), the unit cell of the SL PCPL and TL PCPL is chosen as the computational domain due to the periodic structure. One of the plane waves from the initial

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excitation source for the TM polarization can be given as follows [12], [13]:

(1a)

(1b)

(1c) where (2) where is the wave vector, of the reciprocal lattice vectors

and , and

are primitive vectors are integers. The

is usually small and the important renecessary number of quirement for the initial field is the nonzero projections on the interested normal modes. Each plane wave would be different both in phase and amplitude for different , i.e., a different mode pair . The mode pairs that satisfy the criterion of are chosen as the excitation sources, where is the maximum frequency we consider and is the light velocity in free space. The total number of the chosen is denoted as . The plane wave summation of the mode pairs are used as the initial source excitation for following time stepping. In the FDTD calculation, the boundary conditions that satisfy the Bloch theory are applied on boundaries of the unit cell as Fig. 3. Dispersion diagram of proposed PCPL. (a) SL PCPL board. (b) TL PCPL board. (Color version available online at: http://ieeexplore.ieee.org.)

(3a) (3b) where is the lattice vector. For each mode pair , the time-domain response of the field at meshed position is denoted as . In general, 60–80 points of different are evenly chosen in the unit-cell calculation space. The eigenfrequencies of the normal modes for a given wave vector are then identified by the peaks of the spectral intensity , which is the double summation of the Fourier transform of the recorded and is derived as

(4) Fig. 3(a) and (b) shows the band structure for the designed SL PCPL and TL PCPL for the TM polarization. The dots denote the band structure obtained from the 2-D FDTD method and the solid lines are obtained by the Massachusetts Institute of Technology (MIT) photonic band tool based on the planar wave expansion (PWE) method [14]. Good agreement is seen. As shown in Fig. 3, there are two bandgaps below 8 GHz. They are in the frequency range from 2.8 to 5 GHz and from 5.6 to 6.5 GHz for

the SL PCPL, and from 3 to 5 GHz and from 6 to 8 GHz for the TL PCPL. Within these stopband, the eigenmodes at any propagation direction can not exist inside the PCPL, i.e., the resonant modes caused by the SSN can be omni-directionally suppressed in the PCPL. III. PI/SI PERFORMANCE A. PI Performance 1) Frequency Domain: We first see the bandstop behavior for eliminating the P/GBN in the frequency domain. Fig. 4(a) and (b) shows both the measured and simulated for the SL PCPL and TL PCPL, respectively. The behaviors of the reference board without embedding the high dielectric-constant rods are also included in these two figures for comparison. Anosft’s High Frequency Structure Simulator (HFSS) is used to simulate the -parameters for all the power/ground-plane structures. As shown in Fig. 4(a) and (b), good agreement between measurement and modeling is obtained. A slight discrepancy at frequencies above 5 GHz is seen for both PCPL boards. The reasons could be that the fabrication accuracy for drilling holes and their alignment is not good enough for the prototype boards. As shown in Fig. 4(a) and (b), there are two

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the performance of the SL PCPL. For the second stopband, the noise peak at 6.2 GHz is reduced approximately 50 dB, and on average, there is over 40-dB noise suppression. It is also found that TL PCPL has almost the same bandwidth for the first stopband, but significantly larger bandwidth for the second band, which is similar to the band structure prediction shown in Fig. 3. It is noted that there are additional resonant peaks, which are not seen in the reference boards, which appear below the first stopband (from dc to approximately 2.8 GHz) for both PCPL boards. However, as shown in Fig. 4(a) and (b), these peaks would not worsen the P/GBN coupling because the effective dielectric constants of the PCPL are increased significantly. The effective dielectric constant is defined as

Fig. 4. Comparison of jS j obtained by HFSS and measurement for: (a) SL PCPL board and (b) TL PCPL board. (Color version available online at: http://ieeexplore.ieee.org.)

stopbands for both the SL PCPL and TL PCPL below 8 GHz. The bandwidth of the SL PCPL for the first stopband is approximately 2.6 GHz (from 2.6 to 5.2 GHz) and approximately 1.3 GHz (from 5.8 to 7.1 GHz) for the second band. The bandwidth for the TL PCPL is approximately 2.6 GHz (from 2.8 to 5.4 GHz) and 1.6 GHz (from 6 to 7.6 GHz) for the first and second stopbands, respectively. The bandwidth is defined by the insertion loss less than 30 dB. There are other stopbands above 8 GHz, but it is not discussed in this study because we are interested in the P/GPN dominantly distributed below 10 GHz. For the SL PCPL, it is seen the power plane resonance noise on the reference board at 2.9, 3.2, 3.9, and 5.0 GHz are efficiently reduced over 70 dB by the SL PCPL. The P/GBN is reduced over 60 dB on average within the first stopband, which is an excellent performance for suppressing the SSN coupling. For the second stopband, the noise peak at 6.2 GHz is reduced approximately 35 dB, and on average, there is over 25-dB noise elimination capability. For the TL PCPL, it is found the power plane noise on the reference board at 3.2, 3.7, and 5.0 GHz are efficiently eliminated over 70 dB. The P/GBN is reduced over 60 dB on average within the first stopband, which is similar to

which is 10.348 for the SL PCPL and 15.899 for the TL PCPL. The equivalent parallel-plate capacitance for the SL PCPL pF and pF, respectively. and TL PCPL is The insertion loss of the shunt capacitor is also plotted in Fig. 4(a) and (b), and is quite consistent with both the measured and simulated results at low frequency below the first resonant peak. It can be clearly seen that the increase of the effective dielectric constant not only significantly reduces the power noise coupling between the two ports at frequency below 500 MHz, but also decreases the coupling strength at those resonant peaks below approximately 2.8 GHz. 2) Time Domain: Next, we try to understand the P/GBN suppression capability in the time domain for the proposed PCPL. The power/ground planes of those test boards are excited by a pulse pattern generator (Anritsu MP1763C) to emulate the noise on the power plane, and the coupling noise at the receiving port is measured in the time domain by a signal analyzer (Agilent DCA 86100A). All test boards including two reference boards, TL PCPL, and SL PCPL boards are measured. Fig. 5(a) shows the waveform of the excitation waveform launched from the pattern generator. It is a periodic squarelike wave with a frequency of 10 Gb/s and an amplitude of 125 mV. Port1 and port2 are the exciting and receiving ports, respectively. Fig. 5(b) and (c) shows the measured P/GBN at the receiving port for the SL PCPL and the corresponding reference board, respectively. Fig. 5(d) and (e) shows the measured P/GBN for the TL PCPL and the corresponding reference board, respectively. It is seen that peak-to-peak amplitude of the coupling noise is approximately 16.1 and 171.2 mV for the SL PCPL and the corresponding reference board, respectively, and 21.4 and 107 mV for the TL PCPL and the corresponding reference board, respectively. Compared with the reference boards, the P/GBN can be reduced approximately 91% and 80% for the SL PCPL and TL PCPL, respectively. It is clearly seen that the P/GBN can be efficiently suppressed by the proposed PCPL structure. B. SI Performance Fig. 6 shows a typical high-speed signal trace routing on a four-layer PCB two via-transitions between the top and bottom

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Fig. 5. Measured P/GBN suppression behavior in time domain for the proposed power plane. (a) Waveform of the excitation source launched from a pattern generator. (b) Coupling P/GBN at the receiving port for the reference board of SL PCPL. (c) Coupling P/GBN at the receiving port for SL PCPL board. (d) Coupling GBN at the receiving port for reference board of TL PCPL. (e) Coupling P/GBN at the receiving port for the TL PCPL board. (Color version available online at: http://ieeexplore.ieee.org.)

layers. The PCPL is designed in the inner two layers. Since the power and ground metal planes are perfect (or continuous), the influence of the PCPL on the signal quality could only result from the via-transitions. It is well known that through-hole viatransition will excite the noise between the power and ground

planes and degrades the SI. The SI performance of the proposed PCPL is discussed here. Fig. 7(a) and (b) shows the simulated eye patterns for TL PCPL and the reference board, respectively. The input signal is 10 Gb/s with 500-mV amplitude and 35-ps edge rate. The traces

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Fig. 6. Four-layer structure with transmission line transient between the PCPL power plane and solid ground plane for single-ended trace. (Color version available online at: http://ieeexplore.ieee.org.)

Fig. 8. Measurement setup for EMI in 3-m fully anechoic chamber.

quality. It is seen that for the TL PCPL, mV and ps, and for the reference board, mV and ps. Compared with the reference board, the MEO and MEW is significantly improved approximately 19% and 18% for the TL PCPL. Since the power and ground plane noise excited by the via transition is efficiently suppressed in the PCPL, the signal quality can be well enhanced. The SL PCPL has a similar SI improvement capability as the TL PCPL does and is not shown here. IV. EMI PERFORMANCE

Fig. 7. Eye patterns for signal trace with two via transitions in: (a) reference board of TL PCPL and (b) TL PCPL board. (Color version available online at: http://ieeexplore.ieee.org.)

are designed as 50 with a total length of 100 mm. Eye patterns for evaluating the signal quality are obtained via the following three steps. First, the two-port -parameters for the signal trace are simulated by the commercial tool HFSS. Incorporating with the simulated two-port -parameters, the eye patterns at the output side are finally generated in the Ansoft Designer environment via launching a pattern source of 2 1 pseudorandom bit sequence (PRBS), nonreturn to zero (NRZ), coded at 10 GHz. Two parameters, maximum eye open (MEO) and maximum eye width (MEW), are used as metrics of the eye pattern

It is known that the P/GBN can cause a significant EMI issue in high-speed circuits because of the resonance effect in the cavity formed between the power and ground planes. Low radiated emission or EMI is important in high-speed circuits for the compliance of the strict EMC regulations. Here, the EMI behavior of the proposed PCPL by comparing with the reference board is numerically and experimentally investigated. Fig. 8 shows the EMI measurement setup in an EMC fully anechoic chamber. The test board is put on a wooden table, and the RF signal of 0 dBm generated by the signal source (HP 8324) is launched into the power plane of the board through port1. The height of the receiving antenna and test board is fixed at 1-m height from the floor of chamber, and the distance between them is 3 m. The radiated -field from 1 to 8 GHz is measured by the horn antenna (R&S HF906). The wooden table with a test board is rotated in 360 at the speed of 4.5 /s for each excited frequency point, and the maximum radiated -field is recorded by the spectrum analyzer (R&S FSP) with 100-kHz resolution bandwidth. The radiated -field in the 3-m test distance is also modeled by HFSS. Fig. 9(a) and (b) shows the simulated and measured EMI radiation at 3-m distance for the SL PCPL and TL PCPL board, respectively. The reference board with both the power and ground plane being solid is also shown in this figure for comparison. Although the EMI performance below 1 GHz is mandatory in the EMC compliance regulations, it is not discussed here because the stopbands in the proposed design are all above the

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Fig. 9. Simulated and measured EMI radiation at 3 m for: (a) SL board and (b) TL board. (Color version available online at: http://ieeexplore.ieee.org.)

gigahertz range. The agreement between the measurement and simulation is reasonably good. As shown in Fig. 9(a) for the SL PCPL, the radiation peaks of approximately 90 dB V/m within the first stopband are significantly reduced to approximately 55 dB V/m with over 30-dB EMI improvement. At the second stopband, the radiation strength can be also reduced on average by approximately 30 dB. Similar efficient radiation suppression behavior can be seen for the TL PCPL, as shown in Fig. 9(b). There is approximately 35-dB EMI reduction on maximum and approximately 30-dB improvement on average within two stopbands. The proposed PCPL designs not only have good PI, but also keep good EMI performance caused by the power/ground plane noise. V. DESIGN DIAGRAM Fig. 10(a) and (b) shows the bandgap map for the SL PCPL and TL PCPL, respectively, by the 2-D FDTD method. The dependence of the normalized stopband distributions on the normalized radius of the high dielectric-constant rods for the first three bands are presented, where is the speed of light in free space. As shown in Fig. 10, the first stopband for both PCPL appear for small radius at approximately with narrow bandwidth. The bandwidth gradually increases with center frequency decreased as is increased. It is found

Fig. 10. Simulated gap map for: (a) SL and (b) TL. (Color version available online at: http://ieeexplore.ieee.org.)

that there is maximum bandwidth of for the SL PCPL and for the TL PCPL, as is designed between 0.08–0.1. The bandwidth of the first stopband decreases gradually with the center frequency decreased as is increased from 0.1 for both PCPL. The bands will disappear as is larger than approximately 0.45, where the substrate is almost filled with high dielectric-constant rods. Similar phenomena are seen for the second and third stopbands, where the second stopband starts from and for the SL PCPL and TL PCPL, respectively. For the application of suppressing the P/GBN in this study, we need the broad stopband at a low-frequency range below 10 GHz. Therefore, there is design tradeoff between the bandwidth and center frequency for the PCPL structure. As described in Sections III and IV, we design the SL PCPL with and the TL PCPL with . These parameters do not provide maximum bandwidth, but has a low center frequency as needed. Fig. 11(a) and (b) shows the influence of the dielectric constant of the embedded rods ( and ) on the gap map for the SL PCPL and TL PCPL, respectively. Only the first bands are shown. As shown in Fig. 11(a), the bandgap appears at smaller for a larger dielectric constant of the rods and

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can efficiently eliminate the SSN with approximately 90% reduction. In addition, the EMI caused by the P/GBN is also efficiently suppressed within the stopband with 30-dB reduction on average. All the excellent SI/PI and EMI performance are verified both by the numerical simulation and experimental measurement. Good agreement is seen. The gap map for helping design the SL PCPL and TL PCPL structure have been demonstrated based on a 2-D FDTD method. REFERENCES

Fig. 11. Influence of the dielectric constant of the embedded rods on the gap map. (a) SL. (b) TL. (Color version available online at: http://ieeexplore.ieee. org.)

disappears at the same . The center frequency of the band at the same is higher for the smaller dielectric constant. It is also found that the maximum bandwidth is approximately the same for these three cases, but the corresponding center frequency is significantly higher for the lower dielectric constant case. Similar phenomena are seen for the TL PCPL shown in Fig. 11(b), but the maximum bandwidth is approximately for these three cases.

VI. CONCLUSIONS A novel PCPL has been proposed to suppress the P/GBN or SSN in a high-speed circuit package. The PCPL has been fabricated by periodically embedding the high dielectric-constant material (rods) into the original isolation layer between the solid power and ground planes. Two types of PCPL lattice structures have been discussed: one is the SL PCPL and the other is the TL PCPL. It is found that both PCPLs provide excellent P/GBN elimination capability with over 60 dB on average reduction of the power noise and over totally 4-GHz bandwidth below the 10-GHz range. In the time-domain measurement, both PCPLs

[1] T. L. Wu, Y. H. Lin, J. N. Hwang, and J. J. Lin, “The effect of test system impedance on measurements of ground bounce in printed circuit boards,” IEEE Trans. Electromagn. Compat., vol. 43, no. 4, pp. 600–607, May 2001. [2] G.-T. Lei, R. W. Techentin, and B. K. Gilbert, “High frequency characterization of power/ground-plane structures,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 5, pp. 562–569, May 1999. [3] S. Radu and D. Hockanson, “An investigation of PCB radiated emissions from simultaneous switching noise,” in Proc. IEEE Int. Electromagn. Compat. Symp., 1999, pp. 893–898. [4] M. Xu, T. H. Hubing, J. Chen, T. P. Van Doren, J. L. Drewniak, and R. E. DuBroff, “Power-bus decoupling with embedded capacitance in printed circuit board design,” IEEE Trans. Electromagn. Compat., vol. 45, no. 1, pp. 22–30, Feb. 2003. [5] R. Abhari and G. V. Eleftheriades, “Metallo-dielectric electromagnetic bandgap structures for suppression and isolation of the parallel-plate noise in high-speed circuits,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 6, pp. 1629–1639, Jun. 2003. [6] T. Kamgaing and O. M. Ramahi, “A novel power plane with integrated simultaneous switching noise mitigation capability using high impedance surface,” IEEE Microw. Wireless Compon. Lett., vol. 13, no. 1, pp. 21–23, Jan. 2003. [7] S. Shahparnia and O. M. Ramahi, “Electromagnetic interference (EMI) reduction from printed circuit boards (PCB) using electromagnetic bandgap structures,” IEEE Trans. Electromagn. Compat., vol. 46, no. 4, pp. 580–587, Nov. 2004. [8] T. L. Wu, Y. H. Lin, and S. T. Chen, “A novel power planes with low radiation and broadband suppression of ground bounce noise using photonic bandgap structures,” IEEE Microw. Wireless Compon. Lett., vol. 14, no. 7, pp. 337–339, Jul. 2004. [9] T. L. Wu, Y. H. Lin, T. K. Wang, C. C. Wang, and S. T. Chen, “Electromagnetic bandgap power/ground planes for wideband suppression of ground bounce noise and radiated emission in high-speed circuits,” IEEE Trans. Microw. Theory Tech., vol. 53, no. 9, pp. 2935–2942, Sep. 2005. [10] Y. H. Lin and T. L. Wu, “Investigation of signal quality and radiated emission of microstrip line on imperfect ground plane: FDTD analysis and measurement,” in Proc. IEEE Int. Electromagn. Compat. Symp., Montreal, QC, Canada, Aug. 2001, pp. 319–324. [11] T. L. Wu and S. T. Chen, “An electromagnetic crystal power substrate with efficient suppression of power/ground plane noise on high-speed circuits,” IEEE Microw. Wireless Compon. Lett., to be published. [12] C. T. Chan, Q. L. Yu, and K. M. Ho, “Order- spectral method for electromagnetic waves,” Phys. Rev. B, Condens. Matter, vol. 51, pp. 16 635–16 642, 1995. [13] M. Qiu and S. He, “A nonorthogonal finite-difference time-domain method for computing the band structure of a two-dimensional photonic crystal with dielectric and metallic inclusions,” J. Appl. Phys., vol. 87, pp. 8268–8275, 2000. [14] S. G. Johnson and J. D. Joannopoulos, “Block-iterative frequency-domain methods for Maxwell’s equations in a planewave basis,” Opt. Exp., vol. 8, no. 3, pp. 173–190, 2001.

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Tzong-Lin Wu (S’91–M’93–SM’04) received the B.S.E.E. and Ph.D. degrees from National Taiwan University (NTU), Taipei, Taiwan, R.O.C., in 1991 and 1995, respectively. From 1995 to 1996, he was a Senior Engineer with Microelectronics Technology Inc., Hsinchu, Taiwan, R.O.C. From 1996 to 1998, he joined the Central Research Institute, Tatung Company, Taipei, Taiwan, R.O.C., where he was involved with the analysis and measurement of EMC/EMI problems of high-speed digital systems. From 1998 to 2005,

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he was with the Electrical Engineering Department, National Sun Yat-Sen University (NSYSU), Kaohsiung, Taiwan, R.O.C. He is currently a Professor with the Department of Electrical Engineering and Graduate Institute of Communication Engineering, NTU. His research interests include design and analysis of fiber-optic components, EMC and SI design, and measurement for high-speed digital/optical systems. He was listed in Marquis’ Who’s Who in the World in 2001. Dr. Wu is a member of the Chinese Institute of Electrical Engineers. He was the recipient of the 2000 Excellent Research Award and 2003 Excellent Advisor Award presented by NSYSU, the 2002 Outstanding Young Engineers Award presented by the Chinese Institute of Electrical Engineers, and the 2005 Wu Ta-You Memorial Award presented by the National Science Council (NSC).

Sin-Ting Chen was born in Pingtung, Taiwan, R.O.C., in 1980. He received the B.S.E.E. degree from National Sun Yat-Sen University (NSYSU), Kaohsiung, Taiwan, R.O.C., in 2002, and is currently working toward the Ph.D. degree in electrical engineering at the NSYSU. His research interests are modeling and measurement for PI of high-speed package and PCBs.

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A Novel Technique for Deembedding the Unloaded Resonance Frequency From Measurements of Microwave Cavities Antoni J. Canós, José M. Catalá-Civera, Member, IEEE, Felipe L. Peñaranda-Foix, Member, IEEE, and Elías de los Reyes-Davó

Abstract—A novel technique to extract the influence of coupling networks on the resonant frequency of cavities in one-port measurements is presented. The determination of the unloaded resonant frequency is performed directly from measurements without either the need to obtain the electromagnetic fields in the resonator or to deembed the delay of transmission lines from the measuring equipment to the resonator. The importance of the Foster’s form on the modeling of the frequency detuning of the resonators is also discussed and a criterion for the choice of the appropriate Foster’s form is suggested. The procedure is validated with simulations and experimental measurements of manufactured cavities. Index Terms—Cavity resonators, microwave resonators, one-port measurements, factor, resonance frequency.

I. INTRODUCTION HE PRECISE characterization of microwave resonators (determination of unloaded parameters and ) is a very important task for many microwave applications, such as filters, oscillators, dielectric resonators, frequency meters, the dielectric and magnetic characterization of materials, etc. Measuring a resonator involves using feeding mechanisms (e.g., apertures, slits, probes, current loops, microstrip gaps) that modify its original unloaded response: resonance frequency is shifted to , and the quality factor is lowered to . Therefore, when the unloaded parameters of the isolated resonator have to be determined, it is necessary to deembed the effect of the coupling networks from measurements [1]. Different methods have been described in the literature to obtain unloaded parameters from one-port measurements of resonators [1]–[7]. Most of them are based on equivalent circuits derived from Foster’s forms, and they model the resonators and coupling networks in the resonance vicinity. By using these procedures, the unloaded quality factor of the resonator and the external quality factor related to the coupling elements can be extracted from measurements independently of the mechanisms used for exciting the resonators. However, papers in the literature mainly focus on the extraction of , but not on the unloaded resonance frequency , which is the scope of the work presented here.

T

Manuscript received January 21, 2005; revised April 8, 2006. The authors are with the Department of Communications, Institute for the Applications and Advanced Communication Technologies, Technical University of Valencia, Valencia E-46022, Spain (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2006.877833

In the case of cavity measurements taken under very weak coupling conditions (e.g., very short probes or small apertures), the values are high, and the resonances are slightly altered by coupling networks; thus, . In many cases, however, strong coupling conditions are needed to perform accurate measurements, which implies low values, thus invalidating the previous assumption. In principle, the change of frequency due to the feeding mechanisms can be determined once their reactive effects and are known [1], [3]. From the measurements point-of-view, the main effect of the reactive character of coupling networks is a phase shifting in the response, (i.e., a rotation of the reflection coefficient on the Smith chart). The length of the transmission line connecting the measuring equipment to the resonator also adds a phase to the measurement, which means that it is difficult to quantify the reactive character of the coupling networks from the measurement of the reflection coefficient. Consequently, it is not possible to extract the unloaded frequency of the resonator directly from measurements unless the delay of the line from the reflectometer to the resonator is accurately known, something that is not easy to determine [1]. In addition to this, even when the delay of the line and, therefore, the reactive effects of coupling networks are known, the choice of one or another Foster’s form in the model of the resonator yields to different values of , as explained below, so the appropriate Foster’s form must be known a priori to determine correctly the detuning due to the feeding mechanisms from equivalent circuits. On the other hand, the coupling networks and their influence on the resonance frequency may be evaluated from exhaustive electromagnetic analyses of the structure, but sometimes this cannot be easily performed, especially for irregular shapes. An empirical expression of the detuning of microwave cavities due to the coupling mechanisms was given in [8], where new equivalent circuits were suggested for the modeling of resonators. Those circuits add some elements, which may be not necessary, to the traditional circuit representations. In this study, a novel technique for deembedding the unloaded resonance frequency from measurements of microwave cavities is presented, thus eliminating the influence of coupling networks on the resonance frequency. The method is derived from traditional equivalent circuits without the addition of new elements. The characterization of coupling structures is performed directly from measurements without either the need to obtain the delay of lines from the measurement equipment or the knowledge of electromagnetic fields in the resonators, which is very

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taking these considerations into account, the total impedance at plane 1 in Fig. 1(b) can be expressed as (1) where

(2) (3) (4) (5) The coupling factor representing the ratio of the power dissipated in the external circuit to the power dissipated in the unloaded resonator can be written as (6)

Fig. 1. Equivalent circuits of a microwave resonator in the resonance vicinity including the measurement system. (a) First Foster’s form. (b) Second Foster’s form.

interesting from a practical point-of-view. The importance of the Foster’s form in the equivalent circuits of resonators is also discussed in order to model the frequency detuning due to the coupling networks, and a criterion for the use of the appropriate form is suggested.

The input reflection coefficient at plane pressed as

can be then ex-

(7) where the detuned reflection coefficient representing the reflection coefficient at a frequency far away from the resonance is (8)

II. EQUIVALENT-CIRCUIT MODEL

It is usual to use the following representation around

:

A. Circuit Representations A one-port microwave resonator, connected to a vector network analyzer (VNA) by means of a transmission line, can be represented in the vicinity of a resonance by the equivalent circuits represented in Fig. 1(a) and (b), which are referred to as the first and second Foster’s form, respectively [6]. In Fig. 1, elements , , and are interior parameters since they model the behavior of the isolated (unloaded) resonator. The coupling networks are modeled by an ideal transformer of ratio , and by the impedance or admittance , in the first and second Foster’s form, respectively. The ohmic losses within the coupling mechanisms are represented by and . Elements and represent the extra energy storage introduced by the coupling structures. All the parameters have been normalized to the transmissionline characteristic impedance, which is usually real and equal to the resistance source of the VNA. The values of and can be considered constant with the frequency near the resonance for high- microwave resonators [1]. For simplicity purposes, ohmic losses in the coupling networks will be neglected. By

(9) where

is the frequency detuning parameter defined as

(10) The loaded resonance frequency is defined as the frequency at which the imaginary part of vanishes. If we consider the approximation of (9), this occurs at

Thus, the loaded frequency frequency as

(11) can be related to the unloaded

(12)

CANÓS et al.: NOVEL TECHNIQUE FOR DEEMBEDDING UNLOADED RESONANCE FREQUENCY FROM MEASUREMENTS OF MICROWAVE CAVITIES

With the approximation of (9) and by defining the input reflection coefficient can be expressed as

,

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By using the approximation of (9) for this last circuit representation, the detuning of the resonator due to the measurement setup becomes

published results have been found concerning the effect it has on the determination of . Once and are known, the natural frequency of the resonator can be calculated from (12) or (19). However, it presents the following two basic difficulties. • The equivalent susceptance in (12), or the equivalent reactance in (19), can be obtained from by using (8) or (18), respectively. However, as stated from (21), the position of on the Smith chart cannot be determined unless the length of the transmission line and connectors is known with accuracy, something uncommon in real measurements. Some procedures to obtain the group delay due to the transmission line and connectors length have been published [7], [9], but they present certain limitations. The method described in [7] requires the measurement of an undercoupled resonance in a wide span that is free of neighboring resonances, which is not always possible in real measurements. To determine in [9], two identical transmission lines of different and accurately known lengths are needed, as is an accurate value of the electrical delay due to the connector, which may be unpractical. • Even if the transmission line and connectors length and, therefore, the actual position of on the Smith chart were known, the resonator must be modeled by using the appropriate Foster’s form to correctly determine the relation between and . A wrong choice of the Foster’s form in the model would yield an error in the calculation of from the measured , as discussed in detail in Section II-B.

(19)

B. Importance of Choice of the Foster’s Form in the Modeling of Frequency Detuning

(13) It should be remarked that the same expression is obtained for the input reflection coefficient at plane of the circuit in Fig. 1(a) [1], where the following expressions are satisfied: (14) (15) (16) with

(17) (18)

The reflection coefficient given by (13) represents the input reflection coefficient at the detuned plane (plane in Fig. 1), while the measurements are performed at the calibration plane of the VNA (plane in Fig. 1). These planes rarely coincide due to the transmission line and connectors length , thus, the measured input reflection coefficient at plane is given by (20) where is the propagation constant in the feeding line. In a narrow band around , can be considered constant with the frequency, thus, can be expressed as (21) The representation of both (13) and (21) on the Smith chart describes a circle with a diameter of , [1], [3], [4], and the values of are rotated with respect to around the origin of the chart. From this circle, referred to as the circle, the procedures described in [1], [3] allow , , , and to be directly calculated from measurements, by neglecting , and they are calculated independently of the Foster’s form chosen to model the resonator since they are based upon (13). The effect of on the determination of of high- resonators was analyzed in [1], and a slight influence was shown, but no

Once the response of a resonator is measured, a Foster-type equivalent circuit is usually employed to obtain the relevant parameters of the resonance. From the literature, it may be considered that the first and second Foster’s forms are equivalent representations since different forms have been used to model the same resonant structure. For instance, a rectangular cavity coupled by an electrical probe is modeled by the circuit in [7, Fig 1(a)], [10] (first Foster’s form) and also by the circuit in [6, Fig 1(b)] (second Foster’s form). On the other hand, a rectangular cavity coupled by a circular iris is modeled by the circuit in [11, p. 523, Fig 1b] (second Foster’s form), and by the circuit in [6, Fig 1(a)], [12], and [13] (first Foster’s form), even though the model in [12] and [13] does not include the series reactance . Both Foster’s forms yield to the expression (13) so they can be indifferently used, as mentioned above, to obtain the parameters , , , and from measurements. However, the correct determination of the detuning caused by the coupling network and, therefore, the calculation of , directly depends on the Foster’s form chosen to model the resonator. This is graphically shown in Fig. 2(a) and (b), where an hypothetical measurement of the reflection coefficient of a resonator is represented on the Smith chart in the vicinity of a resonance. The shift due to the connectors and transmission lines has been already deembedded in the reflection coefficient plotted in both Fig. 2(a) and (b), referred to as . For simplicity, coupling losses are neglected here. The reflection coefficient plotted

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frequency, the isolated resonator becomes a pure resistive element, and then the input impedance is equal to . This frequency point is referred as in Fig. 2(a). It can also be followed from Fig. 2(a) and from (19) that , thus, if the resonator was represented as a first Foster’s form circuit, the deviation of resonance frequency due to the coupling mechanism would be inferred as positive. On the other hand, if the resonator was modeled by the second Foster’s form [see Fig. 1(b)], the position of the circle that describes in Fig. 2(b) would be seen as the response of a series RLC circuit with a parallel coupling susceptance , which shifts the circle along lines of constant conductance. The position of on the Smith chart is not the open point (as it would be is above the real in a zero coupling susceptance circuit), but axis so according to (8). This movement due to is indicated in Fig. 2(b) with a dashed arrow in the counterclockwise sense. Now the input admittance of the resonator and coupling network at the unloaded resonance frequency is equal to . This frequency point is referred as in Fig. 2(b). From Fig. 2(b) and (12), it is followed that , thus, if the resonator was considered to behave as a second Foster’s form, the deviation of resonance frequency due to the coupling mechanism would be inferred as negative. Similar conclusions can be obtained if was located in the capacitive half of the Smith chart. In fact, from (13), one may conclude that the position of the circle on the Smith chart is determined by (assuming ). Any location of on the Smith chart can be explained from both first and second Foster’s forms, with their respective reactive elements and related by (22)

Fig. 2. Input reflection coefficient of a resonator measured at the calibration plane. (a) Resonator modeled by a first Foster’s form. (b) Resonator modeled by a second Foster’s form.

in Fig. 2(a) and (b) may correspond to the measurement of a rectangular cavity coupled by a lossless very thin circular iris located at the calibration plane ( ). Increasing frequencies are indicated by solid arrows in the circles. If the resonator is assumed to behave as the first Foster’s form circuit [see Fig. 1(a)], the position of the circle that describes in Fig. 2(a) is interpreted as the response of a parallel RLC circuit with , but shifted due to the series coupling reactance . The detuned point is not located at the short point of the Smith chart (as it would be in a zero coupling reactance circuit), but it is in the inductive half of the Smith chart (above the real axis), thus, according to (18). This effect is indicated in Fig. 2(a) via a dashed arrow in clockwise sense. The diameter of the circle is decreased because the points are shifted along lines of constant resistance. At the unloaded resonance

Therefore, it is stated that both Foster’s forms are not equivalent representations from the point-of-view of the detuning caused by the coupling networks because they imply frequency deviations of different sign. Thus, although the delay of lines and connectors may be determined and, consequently, the actual position of on the Smith chart was known, the natural resonance frequency of a resonator would not be correctly determined from equivalent circuits, unless the appropriate Foster’s form was chosen. Conversely, Foster’s form must be chosen according to the location of the detuned point on the Smith chart and to the sign of the detuning produced by coupling networks as well, as summarized in Table I. III. DETERMINATION OF THE UNLOADED FREQUENCY A. Description of the Method If the first Foster’s form is used to model the resonator, from (14),

(23) since

is in practice.

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TABLE I CRITERION FOR FOSTER’S FORM CHOICE

Fig. 3. Empty rectangular cavity. (a) Coupled by a circular iris. (b) Coupled by an electrical probe.

From (19), the detuning of the system due to the coupling network can be expressed as

feeding mechanism so this takes a dependence on parameter . From (24) and (26), and according to previous considerations, the following expression is put forward to model the detuning of a resonator due to the coupling elements

(24) If the second Foster’s form is used in the modeling, from (6),

(25) is in practice, thus, the loaded and unloaded fresince quencies can be related as

(26) The external quality factor represents the influence of the resonator’s external elements on the resonance so it may be considered an indicator of the perturbation caused by excitation networks on the electromagnetic fields pattern of the isolated resonator. As a consequence of the field pattern modification, the resonance frequency is shifted, so the detuning is somehow related with . It is observed in practice that under very weak coupling conditions, (e.g., when very short electrical probes or very small irises are being used), reaches very high values and the field pattern into the resonator remains almost unaltered. In this case, the resonator is hardly detuned from its natural resonance frequency. In the limit case of , should be the natural frequency of the cavity since the perturbation of electromagnetic fields into the resonator becomes negligible. The and parameters may be considered independent of the coupling networks and constants in a narrow frequency range around . From (6) and (14), one can conclude that the manipulation of the coupling network affects and (or ), thus yielding a change in and, therefore, in . The transformer ratio can be known for some well-defined resonator geometries, examples of such are cylindrical cavity, rectangular cavity, and coaxial cavity, with coupling mechanisms that can be fully analyzed. In some structures, the term may be considered constant in the resonance vicinity, as shown in Section III-B, where a rectangular cavity coupled by a circular iris is analyzed. In general, also depends on the

(27) Parameters and depend on electromagnetic fields into the cavity (physical dimensions, resonant mode, etc.), the coupling network (type, position, etc.), and on the changes in the feeding network modifying (location, dimensions, orientation, etc.), and they can be obtained from an electromagnetic analysis of the structure, as described in Section III-B. Nevertheless, since depends on , they can also be determined directly from measurements of the input reflection coefficient, by conveniently modifying the feeding structure (e.g., by using probes of different lengths, circular irises of different radii, by changing its location, etc.), thus producing different coupling factors, and so the following system can be solved as: (28) where the external factor , and the loaded frequency produced by a coupling network can be obtained, for instance, by means of the procedure described in [1] and [4]. The solution of the system provides the unknowns and and the unloaded resonant frequency . Therefore, the unloaded resonance frequency can be determined with this procedure without any knowledge of the delay of lines or without or . Additionally, this method can be used to appropriately select the Foster’s form in the modeling by analyzing and , as discussed in Section II (see Table I). With the use of (12) and (19), the values of the equivalent parallel susceptance of the coupling network (for the second Foster’s form) or the equivalent series reactance (for the first Foster’s form) can be obtained. B. Validation With a Rectangular Cavity Excited by a Circular Iris We considered the case of a rectangular cavity formed by a small circular iris of zero thickness and a radius , centered in a transverse wall of a rectangular waveguide at a distance from the short-circuited end, as shown in Fig. 3(a). This cavity can be modeled by the equivalent circuit in Fig. 1(b), as described in Section II. The inductance is

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the sum of that for cavity and arising from the surface impedance of the cavity walls [11, p. 523]. Element is the equivalent parallel susceptance of the iris. According to the incident mode, it may be small-hole theory and for the expressed as (29) is the phase constant in the wavewhere . guide, and From this equivalent circuit, the input admittance can be expressed as

choice is to take then it can be written

[11, p. 525], and

(35) From (25), the external

factor is expressed as

(36) As appointed above, the total impedance at plane 1 in Fig. 1(b) at the loaded resonance frequency is real, (37) It should be appointed that the loaded resonance frequency of , the cavity is derived in [11, p. 526] by imposing and then

(30)

with as the loss-free cavity resonant frequency and as the factor of the mode. Both the unloaded resonance frequency and the unloaded factor defined in Section II-A can be expressed as

(38) Equation (38) is an equivalent to (37), but only when , which occurs in a critical coupling. Thus, the following is obtained from (33):

(31)

(32) respectively. By assuming resonances in the cavity, and by following a similar procedure to that indicated in [11], the input admittance of the structure satisfies (33)

where . By comparing (30) and (33), the transformer ratio can be written as

(39)

since normally . With the use of (36),

(40)

where . By comparing this expression with (27), it follows that

(34) (41) and and is thus independent of the coupling iris. The term can be considered constant with the coupling network at the resonance. In this case, parameter is arbitrary, which corresponds to the arbitrary choice of the impedance level of the equivalent circuit representing the cavity. In practice, a useful

(42)

thus confirming the validity of the expression (27).

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TABLE II PARAMETERS OF THE RESPONSES OF A SIMULATED PROBE-COUPLED RECTANGULAR CAVITY. f (ref:) = 1:99600867 GHz

Fig. 5. Loaded resonance frequencies versus external quality factors of a rectangular cavity fed by electrical probes of different length.

Fig. 4. Simulated responses of a rectangular cavity coupled with several probes of different length t. (a) Magnitude of the reflection coefficient. (b) Reflection coefficient on the Smith chart.

IV. EXPERIMENTAL METHOD AND RESULTS The above-described procedure was used to determine from the reflection coefficient of several cavities excited by different coupling networks. A. Numerical Results The electromagnetic simulator Concerto [14] was used to generate the reflection coefficient ( ) of some cavities coupled by coupling networks with different dimensions. The -Prony method was applied to obtain more accurate responses, especially for high- resonances [14]. The response of a rectangular cavity ( mm, mm, mm) was firstly simulated. Several electrical probes of a different length located at the top wall, as illustrated in Fig. 3(b), were used to excite the resonant mode. Fig. 4 shows the simulated responses of the cavity for the different probes. The values of and the factors of these

responses calculated with the QZERO program [1] are given in Table II. A completely closed cavity excited by a lumped element was also simulated, thus obtaining a resonance frequency of 1.99600867 GHz. This value may be taken as reference for since the cavity can be considered isolated under these conditions. Longer probes yield higher frequency deviations from the reference value, as seen in Fig. 4 and Table II. The loaded resonance frequency tends to the reference value in the case of a high excitation (very short probes) since the perturbation into the resonator then becomes negligible. The new approach described in Section III-A was applied here. The and values from Table II were used to solve the equation system written in (28), and then , , and GHz were obtained. Fig. 5 shows the values of as a function of and the curve given by (27). It can be observed that tends to the unloaded frequency when . The value obtained with this approach is well in accordance with the reference value. Since the detuned point is located below the real axis on the Smith chart and the detuning due to the coupling networks is negative, the equivalent circuit of this resonator should be a parallel RLC resonant circuit (first Foster’s form) with a series reactance according to Table I. The values of for each of the probes were calculated from (19), and they are also provided in Table II.

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Fig. 6. Sectorial cavity with a dielectric rod fed by an electrical probe.

Fig. 7. Loaded resonance frequencies versus external quality factors of an irregular cavity fed by electrical probes of a different length.

The procedure has also been applied to determine the unloaded frequency of an irregular structure, as the loaded sectorial cavity illustrated in Fig. 6, where electromagnetic fields do not follow such canonical shapes. Fig. 7 shows the and values obtained when the cavity is excited with probes of a different length and the curve derived from (27) after the fitting procedure. A cavity excited by a lumped element was also simulated and, thus, a resonance frequency of 2.28707519 GHz was obtained. Values of , , and GHz were obtained, which represents a deviation of 0.009% with respect to the reference, confirming the validity of the approach for irregular-shaped cavities. B. Measurements of Real Cavities The above-presented procedure was also used to determine the unloaded resonance frequency from measurements. A rectangular cavity ( mm, mm, mm) was manufactured and excited with 0.5-mm-thick irises of a different radius centered on a transverse wall, as schematically shown in Fig. 3(a). The theoretical resonance frequency of the mode from the physical dimensions of the closed resonator is 1.88413760 GHz. A range of frequencies

Fig. 8. Measured responses of a rectangular cavity iris-coupled. (a) Magnitude of the reflection coefficient. (b) Reflection coefficient on the Smith chart.

around this value was selected to perform the measurements. Fig. 8 shows the measured reflection coefficient of the cavity ( ) for different iris radii. Calculated and values are provided in Table III. This time only the responses with irises of radii 10, 15, and 17.5 mm were used in the fitting procedure, and values of , , and GHz were thus obtained. The values of as a function of are plotted in Fig. 9. This alternative representation may be useful to illustrate the fitting procedure. The unloaded resonant frequency is the crosspoint of the fitted straight line with the vertical axis ( ). The value of the extracted is well in accordance with the theoretical value (0.004% of deviation) despite the uncertainty that is always present in the physical dimensions of a real cavity, basically associated to mechanical tolerances. It is also observed that the values of and are very similar to those predicted by (41) and (42) in the analysis of Section III-B, which yield values of and , respectively. Once the resonator

CANÓS et al.: NOVEL TECHNIQUE FOR DEEMBEDDING UNLOADED RESONANCE FREQUENCY FROM MEASUREMENTS OF MICROWAVE CAVITIES

Fig. 9. Loaded resonance frequencies versus external quality factors of a manufactured iris-coupled rectangular cavity.

TABLE III PARAMETERS OF THE MEASURED RESPONSES OF A REAL IRIS-COUPLED RECTANGULAR CAVITY. f (theor:) = 1:88413760 GHz

has been characterized, the frequency deviation caused by another similar coupling network can be accurately predicted, given its value. This is also shown in Fig. 9, where the predicted and measured values of the rest of irises are in good accordance. This can be used to predict the resonance frequency of the cavity under desired coupling conditions. The procedure was also applied to the responses obtained with all the irises by obtaining values of , , and GHz. Therefore, the dependence of calculations with selected responses is not significant. The delay of noncalibrated lines may now be neglected since very thin irises were used and they were located at the calibration plane of the VNA. The detuned point is located above the real axis on the Smith chart (see Fig. 8) and , thus, according to Table I, the equivalent circuit of this resonator should be a series RLC (second Foster’s form) with a parallel susceptance , which is in concordance with the model suggested for this structure in Section III-A. The values for each of the irises were calculated from (12), and they are compared with the values obtained with the small-hole theory (29) in Table III, and similar behaviors were presented. Another rectangular cavity ( mm, mm, mm) was manufactured. Electrical probes from a subminiature A (SMA) coaxial line were located at the top wall

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Fig. 10. Loaded resonance frequencies versus external quality factors of a real rectangular cavity coupled with electrical probes.

to excite the resonant mode. Two sets of measurements were performed with different strategies to modify the coupling factor. Firstly, the probe length penetrating the cavity was varied while maintaining its location constant. A probe with a length of 6.8 mm that penetrated the cavity was then located at different positions. Fig. 10 shows the values where the coordinates are the locations of the probes in mm, as indicated schematically in Fig. 3(b). After applying the procedure to both sets of measurements, values of 2.01025069 and 2.00978561 GHz were obtained, respectively. These values differ from the theoretical value obtained from physical dimensions of the cavity (2.00959963 GHz) by less than 0.033%, independently of the procedure used to modify . However, as Fig. 10 illustrates, parameters and depend on the strategy used to modify the coupling factor. V. CONCLUSION A novel technique for deembedding the influence of coupling networks on resonant cavities has been presented. The characterization of the coupling structure is performed directly from measurements so the unloaded resonance frequency can be calculated without obtaining the electromagnetic fields inside the cavity, and without knowing the proper equivalent circuit of the resonator or the delay of noncalibrated lines, which is very interesting from a practical point-of-view. This approach allows for accurate predictions of the frequency detuning of resonators due to the coupling networks so the performance of some cavity applications, such as dielectric and magnetic characterization techniques, can be directly improved with this approach. The method has been validated with simulated results and experimental measurements of some manufactured cavities. In all studied cases, the derived values show a very good agreement with the unloaded resonant frequencies of the ideally isolated resonators. As stated, the appropriate Foster’s form must be chosen in the equivalent circuit of a resonator to correctly model the frequency detuning due to the coupling networks. A criterion for

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the choice of the Foster’s form is provided. Further work is necessary to extend the application of this technique to another type of resonator, such as microstrip or dielectric resonators. REFERENCES

Q

[1] D. Kajfez, Factor. Oxford, MS: Vector Fields Ltd., 1994. [2] E. L. Ginzton, Microwave Measurements. New York: McGraw-Hill, 1957. [3] D. Kajfez and E. J. Hwan, “ -factor measurement with network analyzer,” IEEE Trans. Microw. Theory Tech., vol. MTT-32, no. 7, pp. 666–670, Jul. 1984. [4] D. Kajfez, “Linear fractional curve fitting for measurement of high -factors,” IEEE Trans. Microw. Theory Tech., vol. 42, no. 7, pp. 1149–1153, Jul. 1994. [5] ——, “ -factor measurement with a scalar network analyser,” Proc. Inst. Elect. Eng.—Microw. Antennas Propag., vol. 142, no. 5, pp. 369–372, Oct. 1995. [6] E.-Y. Sun and S.-H. Chao, “Unloaded measurement – The criticalpoints method,” IEEE Trans. Microw. Theory Tech., vol. 43, no. 8, pp. 1983–1986, Aug. 1995. [7] K. Leong and J. Mazierska, “Precise measurements of the factor of dielectric resonators in the transmission mode—Accounting for noise, crosstalk, delay of uncalibrated lines, coupling loss, and coupling reactance,” IEEE Trans. Microw. Theory Tech., vol. 50, no. 9, pp. 2115–2127, Sep. 2002. [8] A. J. Canós, J. M. Catalá-Civera, F. L. Peñaranda-Foix, J. MonzóCabrera, and E. De los Reyes, “A new empirical method for extracting unloaded resonant frequencies from microwave resonant cavities,” in IEEE MTT-S Int. Microw. Symp. Dig., 2003, pp. 1823–1825. [9] L. H. Chua and D. Mirshekar-Syahkal, “Accurate and direct characterization of high- microwave resonators using one-port measurement,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 3, pp. 978–985, Mar. 2003. [10] D. Kajfez, “Correction for measured resonant frequency of unloaded cavity,” Electron. Lett., vol. 20, no. 2, pp. 81–82, Jan. 1984. [11] R. E. Collin, Field Theory of Guided Waves. New York: IEEE Press, 1991. [12] D. M. Pozar, Microwave Engineering. Reading, MA: Addison-Wesley, 1990. [13] R. E. Collin, Foundations for Microwave Engineering. New York: McGraw-Hill, 1992, ch. 7. [14] Concerto User Guide. Oxford, U.K., Vector Fields Ltd., 1999.

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of Valencia, where he received a Readership in 2000. He is currently Head of the Microwave Applications Research Group, Institute for the Applications and Advanced Communication Technologies (ITACA), Technical University of Valencia. He has coauthored approximately 60 papers in referred journals and conference proceedings and over 50 engineering reports for companies. He holds five patents. His research interests encompass the design and application of microwave theory and applications, the use of microwaves for electromagnetic heating, microwave cavities and resonators, measurement of dielectric and magnetic properties of materials, and development of microwave sensors for nondestructive testing. Dr. Catalá-Civera is a member of the International Microwave Power Institute (IMPI). He is currently a Board member of the Association of Microwave Power in Europe for Research and Education (AMPERE), a European-based organization devoted to the promotion of RF and microwave energy.

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Antoni J. Canós was born in Almenara (Castelló de la Plana), Spain, in 1973. He received the Dipl. Eng. and M.S. degrees in electrical engineering from University of Valencia, Valencia, Spain, in 1999 and 2003, respectively, and is currently working toward the Ph.D. degree at the University of Valencia. In 2001, he joined the Institute for the Applications and Advanced Communication Technologies (ITACA), Technical University of Valencia, as a Research and Development Engineer. Since 2005 he has been an Assistant Professor with the Communications Department, University of Valencia. His current research interests include numerical analysis and design of waveguide components, microwave measurement techniques and devices for the electromagnetic characterization of materials, and noninvasive monitoring of processes involving dielectric changes.

José M. Catalá-Civera (M’04) received the Dipl.Ing. and Ph.D. degrees in telecommunications engineering from the Technical University of Valencia, Valencia, Spain, in 1993 and 2000, respectively. From 1993 to 1996, he was a Research Assistant with the Microwave Heating Group, Technical University of Valencia, where he was involved with microwave equipment design for industrial applications. Since 1996, he has been with the Communications Department, Technical University

Felipe L. Peñaranda-Foix (M’92) was born in Benicarló, Spain, in 1967. He received the M.S. degree in electrical engineering from Polytechnical University of Madrid, Madrid, Spain, in 1992, and the Ph.D. degree in electrical engineering from the Technical University of Valencia (UPV), Valencia, Spain, in 2001. In 1992, he joined the Department of Communications, UPV, where he is currently a Senior Lecturer. He has coauthored approximately 40 papers in referred journals and conference proceedings and over 40 engineering reports for companies. He is a Reviewer for several international journals. His current research interests include electromagnetic scattering, microwave circuits and cavities, sensors, and microwave heating applications. Dr. Peñaranda-Foix is a member of the Association of Microwave Power in Europe for Research and Education (AMPERE), a European-based organization devoted to the promotion of RF and microwave energy.

Elías de los Reyes-Davó was born in Albatera (Alicante), Spain, in 1950. He received the Dipl.Ing. degree from the Polytechnical University of Madrid, Madrid, Spain, in 1974, and the Ph.D. degree in telecommunications from the Polytechnic University of Catalunya, Catalunya, Spain, in 1978. In 1986, he became a Tenured Professor of radar technology with the Polytechnical University of Barcelona. In 1988, he joined the Technical University of Valencia, Valencia, Spain. He is currently the Dean of the Telecommunications School and Acting Director of the Institute for the Applications and Advanced Communication Technologies (ITACA), where he is the Leader of the Applied Electromagnetic Group (GEA). He has authored or coauthored over 100 technical papers and three technical books. His research interests are high-power microwave systems and their industrial applications, together with the electromagnetic compatibility and the environmental radio-electric control. Dr. de los Reyes-Davó was chairman of the 7th International Conference on Microwave Heating in 1999. He has been or is member of the Commission for the Scientific and Technological Research (ICT) of the Spanish Ministry for Science and Technology, president of the Electromagnetism Commission and Scientific Advisor to the Ècole Supérieure d’Electricité (SUPLELEC), Paris, France, and is currently president of the Association of Microwave Power in Europe for Research and Education (AMPERE), a European-based organization devoted to the promotion of RF and microwave energy. He was the recipient of a 1976 grant by the Government of France for a long-term stay at the National Centre for Telecommunications Studies (CNET), Bretagne, France. He was the recipient of an Outstanding Thesis of the Year Award for his doctoral dissertation.

IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. 54, NO. 8, AUGUST 2006

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Investigation of Parylene-C on the Performance of Millimeter-Wave Circuits Camilla Kärnfelt, Member, IEEE, Christina Tegnander, Janusz Rudnicki, J. Piotr Starski, Senior Member, IEEE, and Anders Emrich

Abstract—Parylene-C is a polymeric material primarily used in hybrid manufacturing for humidity protection and dielectric isolation. In this study, the influence of Parylene-C on passive millimeter-wave circuits such as transmission lines and resonators is investigated in electromagnetic simulations up to 100 GHz and measurements up to 67 GHz. It is demonstrated that when applying 5- m Parylene-C, the resonance frequency of a resonator is shifted 0.4% and the value is changed slightly. The dissipation factor of the Parylene-C versus frequency has been calculated from measured data. The flip-chip mounted broadband traveling-wave monolithic-microwave integrated-circuit (MMIC) amplifier is also investigated. A 5- m-thick Parylene-C coat results in a total loss of 1.04 dB. A positive side effect of the Parylene-C is that it allows heat, dissipated in the amplifier, to spread over a larger area, consequently lowering the backside temperature of the flipped MMIC with as much as 10 C. The results from this study demonstrate that, concerning the electrical performance, Parylene-C is very well suited as protective coating in millimeter-wave applications and can be used as an alternative to a hermetic package in the frequency range from dc to 67 GHz to reduce weight and cost. Index Terms—Coatings, dielectric loss, dissipation factor, flip chip, humidity protection, monolithic microwave integrated circuits (MMICs), Parylene, poly-para-xylylene, resonators.

I. INTRODUCTION ARYLENE-C is a polymeric material based on the polymonochloro-para-xylylene molecule, which has been used for over 30 years in hybrid manufacturing [1], [2], high-power low-frequency circuits, and as antireflective coating for terahertz optics. The primary use in these applications is as humidity protection and dielectric isolation. Parylene-C has also been proposed for electromagnetic (EM) shielding instead of metal shielding, as presented in [3]. One of the advantages of Parylene is the coating process; since it is vapor phase deposited, the coating is truly conformal and covers every surface, crack, void, or recess in every direction and the protection is complete. The Parylene coating grows

P

Manuscript received January 12, 2006; revised March 31, 2006. This work was supported by Västra Götalandsregionen. C. Kärnfelt, J. Rudnicki, and J. P. Starski are with the Department of Microtechnology and Nanoscience, Microwave Electronics Laboratory, Chalmers University of Technology, SE 412 96 Göteborg, Sweden (e-mail: [email protected]). C. Tegnander and A. Emrich are with Omnisys Instruments AB, SE 421 30 Västra Frölunda, Sweden. Digital Object Identifier 10.1109/TMTT.2006.877834

in the direction normal to any surface with a speed of approximately 0.2 m per minute, thus the thickness can be controlled with good accuracy. Another characteristic of Parylene-C is that no shrinkage occurs during the polymerization process. Even though Parylene-C has been used for several years in the microwave hybrid industry, it is hard to find any publications on the influence on the microwave performance, although some electrical data can be found in data sheets.1 The relative dielectric constant of Parylene-C is 2.95 at 1 MHz and the dissipation factor is 0.013. The addition of Parylene-C coating on an microwave circuit will change the effective dielectric constant and, thus, influence the circuits’ performance. By knowing how large the perturbation is, the design engineer can compensate for the effect to be able to fulfill the design requirements. For the active circuits, hermetic packages are often used to protect the components from humidity to guarantee long life of the system. The intent of this study is to investigate the possibility to use Parylene-C, as means to exclude the hermetic metal package. The weight and cost reduction can be considerable. Since the interest for flip-chip assembly is increasing due to benefits compared to chip and wire technology (such as the possibility of miniaturization, better reproducibility, and wideband performance due to the small parasitics of the carrier-to-chip transition), the investigation of the influence of Parylene on a broadband flip-chip test assembly was thought to be suitable. In a previous study [4], we have examined the effect of flip-chip assembly on an 8-dB 40–60-GHz microstrip (MS) monolithic microwave integrated circuit (MMIC) amplifier. In this study, the MMIC subjected to flip-chip mount is a commercially available traveling-wave amplifier with gain larger than 8 dB from 10 MHz to 67 GHz. The amplifier is an MS design on 100- m-thick GaAs with gold pads originally intended to be connected by wire bonding. II. PASSIVE CIRCUITRY DESIGN AND MEASUREMENT The thickness recommended for Parylene-C in different standards is not unambiguous. The MIL standard [5] states that the thickness should be 15 2.5 m, whereas the IPC-2221 recommends 10 m as a minimum thickness for adequate protection [6]. The moisture vapor permeability value was measured at different thicknesses below 0.1 m.2 The conclusion from this study is that the environmental protection is proportional to the thickness of the Parylene-C. In order to investigate the influence of Parylene-C, different circuits were designed such as a coplanar transmission line, an 1[Online]. 2[Online].

Available: www.vp-scientific.com/parylene_properties.htm Available: www.scscookson.com/library/

0018-9480/$20.00 © 2006 IEEE

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Fig. 1. MS to CPW transition with Parylene coating after removal of the solder mask. (Color version available online at: http://ieeexplore.ieee.org.)

MS transmission line, and a resonant circuit. The coplanar and MS lines are used to evaluate the additional loss caused by the Parylene-C coating. The resonator is used to evaluate the change in the value, dielectric loss, and dissipation factor. The passive circuits were subjected to full-wave EM simulation in the Quickwave three-dimensional (3-D) finite-difference time-domain (FDTD) simulator [7]. The properties of the 99.5% alumina substrate are and (at 10 GHz). The alumina substrate thickness is 254 m with a polished surface, and on top there is a 4- m-thick gold metallization. The Parylene-C properties at 1 MHz are and according to [4]. These values were used in the EM simulations. The main task was to assess the influence of 5- or 10- m-thick Parylene-C coating on the circuits’ performances. Six samples each of the three designs were fabricated. Out of the six samples, two samples of each were used as reference samples without any coating, two were coated with 2 m, and two samples were coated with 5 m of Parylene-C. To protect the probe pads from being covered with Parylene, a solder mask was used that could easily be removed after the coating process. The solder mask did only cover as much area as was needed to place the probes properly. The probe pad after Parylene coating and removal of the solder mask is shown in Fig. 1. The solder mask was removed gently by tearing it off and the Parylene edge had to be torn as well. Measurements were performed using an E8361A network analyzer from Agilent Technologies, Palo Alto, CA. The frequency range of this instrument is from 10 MHz to 67 GHz. A. Coplanar Transmission Line The coplanar transmission line has 70- m-wide signal trace and a 40- m gap to the ground traces. The ground traces are 100 m. There is no plating on the backside of this carrier. The dimensions are chosen so that the coplanar line impedance is 50 at 60 GHz. The full length of the CPW transmission line is 9800 m and the carrier size is 4000 10 000 m (see Fig. 2). A cross section of the CPW transmission line covered with Parylene-C is outlined in Fig. 3. Fig. 4 shows the simulated response of the coplanar transmission line with different coating. At 60 GHz, the loss is 0.18, 0.18,

Fig. 2. Coplanar transmission-line test structure. The inset shows the details of the transmission line. There is no backside metallization. (Color version available online at: http://ieeexplore.ieee.org.)

Fig. 3. Cross section of the CPW transmission line covered with Parylene-C, not to scale. (Color version available online at: http://ieeexplore.ieee.org.)

Fig. 4. Response of EM simulations of the CPW transmission line using 0-, 5-, and 10-m Parylene coating. (Color version available online at: http://ieeexplore.ieee.org.)

and 0.20 dB for the 0-, 5-, and 10- m-thick Parylene coating, respectively. At 100 GHz, the transmission loss is 0.30, 0.31, and 0.33 dB. In the coplanar case, the losses increases linearly with frequency. Fig. 5 shows the measurement of the coplanar transmission line. The effect of the Parylene coating is negligible. Comparing these results with the EM simulated results, it is obvious that the losses in the coplanar case are not accounted for properly in EM simulations. B. MS Transmission Line The MS transmission line is designed on a 4000 10 000 m carrier as well. To enable probe measurements, an MS to CPW transition is placed at each end of the transmission line. The transition is designed in momentum simulator to ensure good performance. The transmission loss is below 0.2 dB up to 65 GHz.

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Fig. 5. Measurement of coplanar transmission line with 0-m (circles), 2-m (squares), and 5-m (diamonds) Parylene coating. The effect of the Parylene coating is negligible. (Color version available online at: http://ieeexplore.ieee. org.)

Fig. 6. MS transmission-line test structure protected with Parylene-C. (Color version available online at: http://ieeexplore.ieee.org.)

The MS line is 290- m wide and 9020- m long, as measured from probe pad to probe pad (see Fig. 6). Fig. 7 shows the simulated performance for the MS line with 0, 5, and 10 m of coating. From simulation, it is clear that the thickness of the Parylene has no effect on the losses. The plots follow each other all the way up to 100 GHz. After 60 GHz, the forward transmission falls off rapidly, which indicates that other modes are present in addition to the MS mode. The measured data from the MS transmission line are presented in Fig. 8. Comparing the results of 0- and 5- m thickness in Fig. 8 with the EM simulated data of Fig. 7, the agreement is excellent. The Parylene coating has a very small effect on the performance of the transmission line. C. Ring Resonator The ring resonator has an outer diameter of 3688 m and an inner diameter of 3110 m (see Fig. 9). The gap to the connecting transmission lines is 30 m. The carrier dimensions are 7200 5600 m. The resonance frequency is 10 GHz and resonances appear at every multiple of this frequency. This resonator has the same MS to CPW transitions for probing as the MS transmission line, attached at both ends. The results from the EM simulations of the ring resonator with 0-, 5-, and 10- m Parylene-C coating are shown in Fig. 10. The uncoated circuit resonates at 59.1 GHz. When applying 5 m of Parylene-C, the resonance frequency shifts to 59.0 GHz, i.e., a frequency shift of approximately 100 MHz takes place. The simulations with 10 m of Parylene-C show that no additional frequency shift occurs. This indicates that the entire electric field is contained within the first 5 m of the Parylene

Fig. 7. EM simulations of the MS transmission line with 0-, 5-, and 10-m -thick Parylene. (a) From 0 to 100 GHz. (b) Enlargement around 60 GHz. (Color version available online at: http://ieeexplore.ieee.org.)

coating and, thus, the effective dielectric constant of the MS line does not change any further. The ring resonator measurements are shown in Fig. 11. The measured result is again very similar to the simulated result, although the loss level is 2.5 dB at the peak in reality and only 0.5 dB in the simulation. The simulations show a 100-MHz frequency shift when changing from 0- to 5- m-thick Parylene, Fig. 10, as compared to the measured shift of 240 MHz, Fig. 11. The loaded ( ) and the unloaded ( ) of the th peak ( ) were calculated from the reflection coefficient factor according to [8]. The values are presented in Table I. The effective dielectric constant of the material is calculated from

(1) where is the th resonance, is the speed of light in vacuum, is the mean radius of the resonator ring, and equals the th resonance frequency. The values found by using (1) are presented in Table II. The variation of the effective dielectric constant due to different thickness of Parylene-C is less than 1% and, thus, has very limited influence on the circuit performance. The total loss per unit length was calculated from (2)

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Fig. 8. Measured results of the MS transmission line with 0-m (circles), 2-m (squares), and 5-m (diamonds) Parylene coating. The full frequency range is presented in (a) and a close-up around 60 GHz is presented in (b). (Color version available online at: http://ieeexplore.ieee.org.) Fig. 10. 3-D full-wave simulation of the ring resonator. (a) Simulations from 0 to 70 GHz of all three cases. (b) Close-up on the peak at 59 GHz. (Color version available online at: http://ieeexplore.ieee.org.)

60 GHz. The dielectric loss , also expressed in decibels/centimeter, was then calculated from (5) as follows:

(5) Knowing the dielectric losses, the dissipation factor be extracted from

can

Fig. 9. Ring resonator test structure. (Color version available online at: http:// ieeexplore.ieee.org.)

(6) where

(3) The conductor loss was calculated using the equation

(4) from [9], where denotes the normalized series distributed resistance of the MS and denotes the normalized series resistance of the ground plane. The conductor loss was found to vary linearly from 0.169 dB/cm at 10 GHz to 0.414 dB/cm at

where is the dielectric constant of the thin film alumina (9.8) and is the resonance frequency expressed in gigahertz. Using (1)–(6) on the measured data from the resonator circuit without Parylene-C, the dissipation factor of the thin-film substrate could be calculated (see Fig. 12). For the resonators covered with Parylene-C, the loss from the Parylene adds to the total loss. Thus, the difference in dielectric loss compared to the dielectric loss of the noncovered circuit can be calculated from

(7) where the dielectric loss from the resonator with 0 m of Parylene-C is used as , and is again the conductor loss, as pre-

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Fig. 12. Extracted dissipation factor of the resonator without Parylene-C, i.e., the dissipation factor of the substrate including top and bottom metal. (Color version available online at: http://ieeexplore.ieee.org.)

Fig. 11. Measurements of ring resonator with 0-m (circles), 2-m (squares), and 5-m (diamonds) Parylene coating. (a) Entire measured frequency range. (b) Close-up on one of the peaks. (Color version available online at: http://ieeexplore.ieee.org.)

TABLE I

Q VALUE OF THE RESONATOR

Fig. 13. Dielectric loss of the thin-film substrate and the loss due to 2- and 5-m Parylene-C coating. (Color version available online at: http://ieeexplore. ieee.org.)

TABLE II EFFECTIVE DIELECTRIC CONSTANT

Fig. 14. Dissipation factor of 2- and 5-m Parylene-C. (Color version available online at: http://ieeexplore.ieee.org.)

viously calculated from (4). Solving this equation for the resonances of the resonators covered with 2 and 5 m leads to the results plotted in Fig. 13, where of the thin-film substrate is plotted against frequency together with the loss of the Parylene-C, i.e., . Using (6) while replacing with will yield the dissipation factor of the Parylene-C itself. These results are presented in Fig. 14.

From Fig. 14, we can conclude that the dissipation factor of the Parylene-C itself does not change much compared to the dominant changes of dissipation factor of the thin-film alumina in Fig. 12. III. ACTIVE TEST CIRCUIT DESIGN AND ASSEMBLY The MMIC chosen for the active circuit study is a commercially available medium-power traveling-wave amplifier [10].

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Fig. 16. Forward transmission measurement of the 12 MMICs before assembly. (Color version available online at: http://ieeexplore.ieee.org.)

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Fig. 15. AUH 312 broadband amplifier. The die size is 1.2 mm 1.0 mm. Courtesy of Velocium Products, Redondo Beach, CA. (Color version available online at: http://ieeexplore.ieee.org.)

This particular amplifier was chosen from approximately ten possible candidates, primarily because of its flat gain response, evenly distributed pads, and the surface passivation that would make it a good object for flip-chip assembly. The technology is a 0.1- m GaAs pseudomorphic high electron-mobility transistor (pHEMT) using MS design. The chip is 100- m thick with gold plating on the backside. The pads are quadratic with a length of 100 m and the pitch is 200 m. The total dissipated power is typically 360 mW at 5-V drain bias. The layout of the amplifier is seen in Fig. 15. To design the transition from the carriers’ coplanar transmission line to the chip via the bumps, the FDTD solver Quickwave was used. The coplanar waveguide (CPW) linewidth is 70 m and the gapwidth is 40 m. The length of the taper of 100 m was chosen. Our simulations also indicate that a good transition can be achieved with the pillar’s diameter of 75 m and height of 50 m. This study was thoroughly described in [11]. When using GaAs MMICs, the most common way of applying bumps is to form stud bumps in a regular ball bonding machine. In this project, we used electroplated cylindrical gold pillars on the carrier itself. This pillar technology was earlier presented in [12]. The benefit is mainly that the pillars are uniform with a very low variation in height and diameter. Thermo-compression flip-chip bonding was made using a Finetech LAMBDA. A special tilt fixture was manufactured as a precaution to ensure that the air bridges of the MMIC were not damaged while turning the chips upside down prior to picking them up in the vacuum chuck. The substrate, as well as the chip were heated to 250 C prior to applying the force. The force was 30 N, which is quite high for only 12 pillars; however, this is the lowest force that could be applied with the equipment at hand. The bonding time was 20 s. The pillar height after assembly is approximately 45 m. The measured result of the MMIC before and after assembly was presented earlier [12]. IV. MEASUREMENT RESULTS Measurements before and after Parylene coating were performed using the same network analyzer as before. The probes used are 67-GHz ground–signal–ground (GSG) probes with a pitch of 150 m.

Fig. 17. Measurement of a flip-chip assembly. The solid line indicates that no rigid foam is used. The circles indicate the result when rigid foam is used. (Color version available online at: http://ieeexplore.ieee.org.)

An initial -parameter test of the MMICs, to investigate the parameter spread before assembly and Parylene-C coating, was performed with results according to Fig. 16. At 60 GHz, the gain is 10.4 0.2 dB for all samples but one, which has a 9.55-dB gain. The MMICs were marked with an individual number and the same number was used for the assemblies to see the influence of the test. Throughout all the following measurements, the drain bias was set at 5 V and the drain current was maintained at 60 mA. Starting this study with 12 MMICs, 11 circuits worked electrically after flip-chip assembly. The circuits were divided into three subgroups, which were subjected to 0, 2, and 5 m of Parylene-C coating, respectively. Once again, the probe pads were protected with a solder mask to enable measurement after the coating process. In the probe station, the chuck is normally grounded and, in this case, since the chuck ground is not connected with the ground traces of the carriers CPW transmission line, unwanted parallel-plate modes occur and propagate through the carrier. To avoid this problem, the devices-under-test was placed on top of a 10-mm-thick board of rigid foam, i.e., Rohacell 31F, with a dielectric constant of 1.041 and a loss tangent of 0.0106 at 26.5 GHz. Thus, the test device was moved away from the ground plane of the chuck and the parallel-plate modes did not occur. In Fig. 17, the forward transmission and input reflection of a measured flip-chip assembly with and without the rigid foam is presented. It can be seen that when using the rigid foam, the gain curve is much smoother. Also there is approximately 1-dB gain loss penalty to be paid.

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Fig. 18. Cross-section of the assembly covered with Parylene-C, not to scale. (Color version available online at: http://ieeexplore.ieee.org.)

Fig. 19. Cross section through one flip-chip mounted assembly covered with 5-m Parylene-C. The Parylene-C is seen as the dark material formed around the geometry of the assembly.

When comparing the same MMIC results before and after assembly, the gain loss due to the flip-chip assembly is found to be 0.5 dB, after compensation for the losses in the transmission lines, which were measured to be 0.48 dB in total. This loss is accounted for in the two bump transitions, the proximity effect of having the chip surface close to the carrier surface and also due to heating of the chip, which will deteriorate its performance. A. Parylene-C Impact on Scattering Parameters A schematic cross section of the chip assembly covered with Parylene-C is outlined in Fig. 18 for clarity. The scanning electron microscope (SEM) picture in Fig. 19 shows the cross section of one circuit covered with Parylene-C. The measured results of flip-chip assembly with 0-, 2-, and 5- m Parylene coating are presented in Fig. 20. To enhance readability of the plots, only one sample from each subgroup (0, 2, and 5 m) is presented since the deviation between individuals is very small. The three individual MMICs had a maximum gain difference of 0.15 dB over the whole frequency band before assembly. The results from this test show that 2- m coating results in less than 0.9 dB gain loss over most of the frequency band

Fig. 20. Scattering parameters of the flip chip assembly with 0-m (circles), 2-m (squares), and 5-m (triangles) Parylene-C coating. The gain drop of 2-m Parylene-C coating is 0.9 dB at 60 GHz. (Color version available online at: http://ieeexplore.ieee.org.)

(0-60 GHz). The gain difference attained between the 2- and 5- m assembly is less than 0.14 dB. Comparing these results with what was obtained with the passive circuits, the loss due to the Parylene-C coating on the active circuit is quite large and could not be anticipated. One possible reason is that by placing Parylene-C directly on top of the amplifier, the characteristics impedance and effective dielectric constant is perturbed. Thus, the line lengths between the stages of the traveling-wave amplifier and the matching circuitry is slightly erroneous, which leads to phase error, and the consequence is gain loss. The phase error and gain loss in the

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The overall loss due to flip-chip assembly of the amplifier is found to be 0.5 dB taking the double transitions into account. The losses are increased by less than 0.9 dB when coating the circuit with 2- m Parylene-C and 1.04 dB when coating the circuit with 5- m Parylene-C. The Parylene-C helps to spread the dissipated heat in the assembly so that the backside temperature of the flipped MMIC is decreased by 10 C. Fig. 21. Temperature measurement at position T and T was made to evaluate the heat transfer capability of the assembly. (Color version available online at: http://ieeexplore.ieee.org.)

TABLE III TEMPERATURE MEASUREMENTS

first stage affects the following stage, and this way the gain loss multiplies throughout the circuit. The Parylene has little effect on the matching, but a more pronounced effect on the forward transmission. The other reason is the penetration of Parylene between the active circuit and the carrier, which makes the proximity effect more pronounced with Parylene than with air in the gap. B. Thermal Properties The Parylene-C has a moderate thermal conductivity of 0.08368 W/mK. In a flip-chip assembly, the heat is transported away through the pillars and by convection from the chip backside. To evaluate the heat transfer in the assemblies, we measured the temperature on the backside of the MMIC ( ) and on the alumina carrier ( ) with a thermometer probe placed at positions and (Fig. 21). The assembly was biased at 5-V drain voltage and 60-mA drain current. In this case, the flip-chip assembly was placed directly onto to probe station chuck held at room temperature. The result is seen in Table III. In spite of the rough measurement method, there is a clear difference between the coated and noncoated samples, which indicates that the Parylene-C contributes to lowering the temperature of the flipped circuits. V. CONCLUSION The results in this study show that Parylene-C is very well suited as a protective coating in millimeter-wave applications. Simulations up to 100 GHz indicate that the performance is not deteriorated using a 5- m layer. By doubling the Parylene thickness from 5 to 10 m, the time of humidity protection is extended twice without any further change in electric performance. The EM simulated results have been confirmed by measurements up to 67 GHz. The dielectric loss and dissipation factor of the thin-film substrate and the Parylene-C itself has been calculated from resonator measurements.

ACKNOWLEDGMENT This research was carried out within the cluster organization Microwave Road as part of the Ceramic Substrate Technologies for High Frequency Applications Project. M. Ericson, Omnisys Instruments AB, Västra Frölunda, Sweden, is acknowledged for the photographs. C. Andersson and P. Rundqvist, both with the Chalmers University of Technology, Göteborg, Sweden, are acknowledged for preparing the SEM sample and running the SEM microscope. REFERENCES [1] V. S. Kale and T. J. Riley, “A production Parylene coating process for hybrid microcircuits,” IEEE Trans. Parts, Hybrids, Packag., vol. PHP-13, no. 3, pp. 273–279, Sep. 1977. [2] E. A. Noga, “Evaluation of Parylene as a barrier coating for semiconductors,” in Proc. Int. Hybrid Microelectron. Society Microelectron. Symp., 1975, pp. 273–279. [3] J. Noordegraaf and H. Hull, “C-SHIELD Parylene allows major weight saving for EM shielding of microelectronics,” in Proc. 1st IEEE Int. Polymeric Packag. Symp., Oct. 26–30, 1997, pp. 189–196. [4] C. Kärnfelt, H. Zirath, J. P. Starski, and J. Rudnicki, “Flip chip assembly of a 40–60 GaAs microstrip amplifier,” in Proc. 34th Eur. Microw. Conf., Amsterdam, The Netherlands, Oct. 11–15, 2004, vol. 1, pp. 89–92. [5] Military Specification, Insulating Compound, Electrical (for Coating Printed Circuit Assemblies), MIL-I-46058C, MIL, 1972, p. 3. [6] Generic Standard on Printed Circuit Board Design, IPC-2221, IPC, 1998, p. 23. [7] “QuickWave 3-D Software Manual,” 5.0 ed. QWED, Warsaw, Poland, 2005. [8] D. R. Conn, H. M. Naguib, and C. M. Anderson, “Mid-film for microwave integrated circuits,” IEEE Trans. Compon., Hybrids, Manuf. Technol., vol. CHMT-5, no. 1, pp. 185–191, Mar. 1992. [9] R. E. Collins, Foundations of Microwave Engineering, 2nd ed. New York: McGraw-Hill, 1992. [10] “65 GHz broadband amplifier—AUH 312,” Velocium Products, Redondo Beach, CA, Datasheet, 2005. [11] C. Kärnfelt, J. Rudnicki, J. P. Starski, H. Zirath, and K. Boustedt, “GaAs flip chip evaluation in the 3 to 110 GHz range,” in Proc. 3rd Eur. Microelectron. Packag. Symp., Prague, Czech Republic, Jun. 16–18, 2004, pp. 181–186. [12] C. Kärnfelt, C. Tegnander, J. P. Starski, J. Rudnicki, and A. Emrich, “Flip chip assembly of a commercial MMIC amplifier on thin film alumina with electroplated pillars,” in Proc. Gigahertz 2005, Uppsala, Sweden, Nov. 8–9, 2005, pp. 234–237. Camilla Kärnfelt (M’06) was born in Dragsmark, Sweden, in 1965. She received the M.Sc. degree in engineering physics from Chalmers University of Technology, Göteborg, Sweden, in 2001. From 1987 to 2001, she was with Ericsson Microwave Systems as a Preproduction Engineer specializing in microwave hybrid manufacturing. In September 2001, she joined the startup company Optillion, as a Research Engineer. Since October 2002, she has been with the Microwave Electronics Laboratory, Chalmers University of Technology, as a Research Engineer. Her research interests lie in millimeter-wave MMIC design and packaging, especially flip-chip assembly of millimeter-wave MMICs.

KÄRNFELT et al.: INVESTIGATION OF PARYLENE-C ON PERFORMANCE OF MILLIMETER-WAVE CIRCUITS

Christina Tegnander was born in Uddevalla, Sweden, 1959. She received the M.Sc. degree in electrical engineering from Chalmers University of Technology, Göteborg, Sweden, in 1983. From 1983 to 1995, she was a Design Engineer with Ericsson Microwave Systems AB. Since 1995, she has been with Omnisys Instruments AB, Västra Frölunda, Sweden, a company that has mainly been engaged in custom design projects, incorporating analog, microwave, application-specific integrated circuits (ASICs), and power electronics. Her main interests include microwave design.

Janusz Rudnicki was born in Minsk Mazowiecki, Poland, in 1974. He received the M.S. degree from the Warsaw University of Technology, Warsaw, Poland, in 2000, and is currently working toward the Ph.D. degree at the Warsaw University of Technology. His special interests are computer programming and MMIC design and packaging.

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vision of Network Theory and Division of Microwave Technology, Chalmers University of Technology. He is currently a Docent with the Microwave Electronics Laboratory, Chalmers University of Technology. His current research activities are in the area of microwave circuits and devices, as well as in interconnections for RF applications. Dr. Starski was chairman of the IEEE Swedish section from 1987 to 1992 and vice-chairman from 1992 to 1999. He was the recipient of a 1978 Fellowship of the Sweden–America Foundation.

Anders Emrich was born in Uddevalla, Sweden, 1962. He received the M.Sc. degree in electrical engineering and Ph.D. degree from the Chalmers University of Technology, Göteborg, Sweden, in 1985 and 1992, respectively. Since 1992, he has been President of Omnisys Instruments AB, Västra Frölunda, Sweden, and served as a Specialist and Project Manager for several high-speed, microwave, and terahertz development projects. This includes several radiometer subsystems for the ODIN satellite, as well as five generations of autocorrelation spectrometers. From 2004 to 2005, he was Project Manager for the ESA GeoMS study, an aperture synthesis instrument for operation in GEO orbit with a continuous coverage of 400 400 pixels over 50-500 GHz. His main interest is in optimized development and implementation of advanced microwave and terahertz systems.

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J. Piotr Starski (M’78–SM’92) was born in Lodz, Poland, in 1947. He received the M.Sc. and Ph.D. degrees in electrical engineering from the Chalmers University of Technology, Göteborg, Sweden, in 1973 and 1978, respectively. In 1983, he became an Associate Professor with the Chalmers University of Technology. From 1972 to 1978, he was with the division of Network Theory, Chalmers. From 1978 to 1979, he was a Design Engineer with Anaren Microwave Inc., Syracuse, NY. From 1979 to 1997, he was a Researcher with the Di-

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Design of Low-Cost Multimode Fiber-Fed Indoor Wireless Networks Anjali Das, Anthony Nkansah, Nathan J. Gomes, Member, IEEE, Ignacio J. Garcia, John C. Batchelor, Member, IEEE, and David Wake, Member, IEEE

Abstract—A low-cost option for transporting global system for mobile communication, Universal Mobile Telecommunication System and wideband local area network (WLAN) signals using multimode fiber (MMF) with 850-nm vertical-cavity surface-emitting lasers (VCSELs) is investigated through range predictions from a link budget analysis. These predictions are experimentally verified for WLAN signal transmission in an office environment, using a commercial access point and a 300-m (OM1/OM2) MMF link with low-cost 850-nm VCSEL transmitters. The analysis indicates that good performance and signal coverage is possible with optimum design of indoor fiber-fed wireless systems, even when using such inexpensive components. Index Terms—Distributed antenna systems (DASs), multimode fiber (MMF), radio-over-fiber, vertical-cavity surface-emitting lasers (VCSELs), wireless local area networks (WLANs).

I. INTRODUCTION ADIO-OVER-FIBER-BASED distributed antenna systems (DASs) can enable the deployment of picocellular access networks, providing high-quality mobile/wireless services for dense in-building user populations [1]. The use of multimode fiber (MMF) in such systems is attractive as it continues to be deployed in greater volumes than single-mode fiber, with typical installation lengths of up to 300 m [2]. Further, it has been shown that the MMF may be used for the transmission of RF carriers beyond the modal dispersion limited 3-dB bandwidth [3], [4]. For the DAS to be implemented with low incremental cost for the overall wireless network, inexpensive components such as vertical-cavity surface-emitting laser (VCSEL) transmitters can be used. Indeed, it has been shown that short-wavelength VCSELs can provide low distortion performance close to that of Fabry–Perot and distributed feedback (DFB) lasers for analog links [5]. For a DAS, the transmission of different radio signals over the same fiber-optic infrastructure would be of interest to neutral host providers. At short wavelengths where component costs are less, a 300-m link comprising 850-nm multimode VCSEL and high-bandwidth MMF has been demonstrated with a spurious-free dynamic range (SFDR) of 94 dB Hz in the frequency range of 1–8 GHz [6]. Good signal performance has also been reported for the transmission of emulated WLAN signals

R

Manuscript received December 3, 2005; revised March 24, 2006. This work was supported in part by the European Union under the ROSETTE Interreg project. The authors are with the Broadband and Wireless Communications Group, Department of Electronics, University of Kent, Canterbury CT2 7NT, U.K. (e-mail: [email protected]). Digital Object Identifier 10.1109/TMTT.2006.877835

Fig. 1. Basic fiber-fed DAS.

at 2.4 and 5 GHz over a VCSEL–MMF link [7]. Error-free transmission of GSM1800 and emulated Universal Mobile Telecommunication System (UMTS) signals using 850-nm VCSEL and 50/125- m MMF has also been shown in [8]. However, the experiments in [6]–[8] characterize only the optical link performance. Wireless path characterization is important as it has an impact on the link performance due to multipath signal fading. Moreover, bi-directional operation using real wireless signals over a VCSEL–MMF link has additional restrictions compared to emulated signal transmission. In this paper, we analyze the link performance for a DAS fed by VCSEL–MMF-based optical links. A detailed link budget analysis is verified by experimental measurements on a wireless local area network (WLAN) demonstrator. The link budget analysis is then used to optimize remote antenna unit (RAU) design and predict maximum coverage ranges for WLANs, GSM900, GSM1800, and UMTS. II. SYSTEM LINK BUDGET The basic DAS may be divided into four main parts, which are: 1) the central unit (CU); 2) the optical link; 3) the RAU; and 4) the mobile unit (MU), as shown in Fig. 1. The RF signal fed to the downlink (DL) path modulates the CU laser; the RF modulated optical signal is sent over the 300-m MMF link to the RAU. At the RAU, the incoming DL signal is detected by a photodiode (PD) and the retrieved RF signal is amplified and transmitted over the wireless path. For the uplink (UL), the received RF signal is first amplified and then fed to a laser in the RAU. The modulated optical signal is again sent over 300-m MMF and detected by a PD at the CU. For the RAU, a two-antenna design (using separate transmit and receive antennas), as shown in Fig. 2, is considered in this paper. Although a single antenna RAU may offer reduced size, for broadband operation, the greater isolation offered by using separate antennas enables the use of higher amplification levels and, hence, greater radio ranges. The effect of the isolation on the level of amplification possible will become apparent below. The link budget analysis that follows allows radio range predictions for the two-antenna RAU. The analysis can be divided

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Using , the maximum tolerable propagation loss may be calculated as

(4) where is the DL receiver sensitivity specified for each standard and is the mobile terminal antenna gain. The above calculated loss can be matched against the propagation loss values calculated using (1) to find the maximum achievable range for the DL.

Fig. 2. Two-antenna RAU design.

C. UL Optical Path into: 1) wireless path; 2) DL optical path; and 3) UL optical path calculations. A. Wireless Path The Keenan–Motley propagation model [9] is frequently used for calculating path loss (PL) in indoor environments. However, when the effects of transmission loss through walls and floors are ignored, the Keenan–Motley equation reduces to the openspace PL equation with a modified PL exponent [10]

The UL radio-range calculations depend ultimately on the receiver equipment sensitivity at the CU, defined in terms of a minimum detectable signal ( ), the value of which is obtained from the standards (or may be specified by the equipment manufacturer). This leads to a minimum RF power required to drive the UL laser, and a minimum UL power at the receive antenna ( ). However, additional noise from the optical link increases the total UL noise floor, which, in turn, causes an increase in . The final value of the UL minimum detectable signal ( ) may then be written as

(1) Here, is the frequency of the RF signal, is the velocity of light, and is the distance between the transmitter and receiver. The PL exponent is a characteristic of the propagation environment and typically takes on values from 3 to 4 for indoor environments.

(5) is the required carrier-to-noise ratio at the receiver where specified in the relevant documents for each standard and is the total output UL noise power defined as (6)

B. DL Optical Path The input RF power ( ) to the CU laser needs to be chosen such that distortion is avoided. This is especially important for systems using OFDM modulation (e.g., IEEE802.11g at 54 Mb/s) [11]. Once has been set, the DL transmit power ( ) is calculated as

where is Boltzmann’s constant, is the ambient room temperature (290 K), and and are the cascaded gain and cascaded noise figure of the overall UL subsystem, respectively, using the standard cascade relationships [12]. For calculating , the noise figure of the optical link needs to be determined as

(2)

(7)

is the DL amplifier gain and is the RAU antenna where gain. is defined as the DL optical link insertion loss and is calculated as

where link and

is the total noise power at the output of the optical is the UL optical insertion loss. is defined as (8)

(3)

is the slope efficiency of the CU laser, is the where gain of the DL MMF (at any particular frequency), is the RAU PD impedance, and is the CU laser impedance, and the parameter takes into account laser frequency response variations, which would not be modeled using the static slope efficiency.

is the RIN noise power, is the shot noise power, where and is the thermal noise power measured at the output of the CU PD. The other noise powers are calculated as (9a) (9b) where is the electronic charge, is the dc photocurrent, and

is the CU PD impedance, is the relative intensity noise

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of the laser. Both and demonstrate frequency dependence and were, therefore, experimentally determined for each of the standards. As the UL and DL paths are similar, may be calculated in the same way as (6)

increased significantly, thus having little effect on the capping limit discussed below. 2) Capping: In order to obtain low distortion performance for the UL, a maximum is set for the amplified RF power fed to the RAU laser ( ). This maximum must not be exceeded even when the MU is at a minimum distance ( ) from the receive antenna. Thus,

(10) (14) is the RAU laser slope efficiency, is the gain of where the UL MMF (determined from the MMF response), is the CU PD responsivity, and is the RAU laser impedance. Once is calculated and is determined, is calculated as

is the UL

where is the PL at . It should be noted that the results presented in this paper are capping limited. 3) Transmitted Noise: Equations similar to (7)–(9) were also used to calculate transmitted DL noise power components. This power must be within acceptable limits as defined by IEEE or ETSI [13] standards. The above calculations were automated and carried out iteratively such that the restrictions posed limits on the amplifier gains that could be used in the link budget/range calculations.

) is then

III. SYSTEM PARAMETERS

(11) where is the UL optical insertion loss and amplifier gain. The maximum tolerable UL propagation loss ( calculated as

(12) is the mobile transmit power (and includes ) where specified in the European Telecommunications Standards Institute (ETSI) documents for each standard. Using , the maximum achievable UL range may be determined by comparison with (1) in the same way as for the DL. D. Additional Restrictions Although the above equations represent the link budget for each direction of operation, further restrictions are posed in a real system. 1) Loop Gain: The RAU subsystem forms a loop with its DL and UL branches, as shown in Fig. 2, due to imperfect isolation between the two antennas and due to RF coupling between the RF drive to the UL VCSEL and DL PD circuit. This results in DL bleed and UL bleed signals being coupled to the UL and DL, respectively. To avoid oscillation (with some margin), the maximum loop gain was set to be less than 10 dB. The loop-gain condition may be written as

(13) where is the isolation between the two antennas and is the isolation between the DL and UL paths at the VCSEL-PD end of the RAU. With such a margin for the loop gain, it has been calculated that the signal coupled back into the DL causes little interference to the actual DL signal, and that the increase in the total signal level for the UL (the signal applied to the RAU laser) is not

The component parameters used as inputs to the link budget for the range calculations were experimentally determined and are listed in Table I. The transmission of different standards was possible as the RAU contains no filters and uses multiband antennas [14]. The parameters of the standards evaluated are listed in Table II. These system parameters have been taken from or calculated using 3GPP standards for GSM900 [15], GSM1800 (DCS) [15], and UMTS [16]–[19] and the IEEE802.11b/g [20], [21] standard. The data rate assumed for IEEE 802.11g is 54 Mb/s and for IEEE 802.11b, 11 Mb/s, and 12.2 kb/s users are assumed for UMTS. IV. SYSTEM DEMONSTRATOR A system demonstrator for transmission of WLAN signals has been used to practically verify the link budget calculations. The calculations were carried out for both IEEE 802.11b and 802.11g for the achievement of a radio range (in both UL and DL) of 5 m, to cover a target room of size 6 m 5 m, containing typical office furniture. The results are shown in Table III. A. Experimental Setup Fig. 3 shows the complete experimental setup of the WLAN system demonstrator. The CU and the RAU are placed in separate rooms and connected by 300-m MMF. The WLAN signal is generated using a commercial access point (AP) (D-Link DWL-AP2000+). ULM VCSELs operating at 850 nm are used as optical transmitters for both the DL and UL paths. An RF attenuator attenuates the WLAN signal before it is fed to the DL VCSEL to avoid distortion. Standard 50/125 m (OM1) and 62.5/125 m (OM2) MMF, with lengths of 300 m, are used for the DL and UL, respectively. The amplifier gains are adjusted to the values given in Table III through bias control.

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TABLE I INPUT PARAMETERS TO THE LINK BUDGET

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TABLE III CALCULATIONS FOR THE SYSTEM DEMONSTRATOR

Fig. 3. Experimental setup of the system demonstrator.

less card (D-Link AirPlus G DWL-G650 ) installed. The MU had line-of-sight at all times with the RAU, although multipath effects were apparent.

B. Measurements and Results

TABLE II SYSTEM PARAMETERS

For the measurements, the RAU was mounted on the ceiling towards one end of the room. The MU is a laptop with a wire-

The received signal strength at the MU was measured using commercial software (Wireless Network Ignition), which itself was checked against measurements with an electrical spectrum analyzer (Agilent E4407B) and known antenna. Fig. 4(a) and (b) shows the variation of signal strength with distance from the RAU for both IEEE 802.11b and IEEE 802.11g signals. As expected, the received signal strength is seen to generally decrease as the MU moves away from the RAU. The predicted signal strength from the link budget calculations is also plotted and, as can be seen, there is a close match between the two for both standards. The link quality was tested by streaming a video signal over the link. It was possible to obtain high-quality signal reception throughout the room. In order to quantify the link performance, throughput measurements were carried out using Networx commercial software during the data transfer of a large file over the link. The average throughput against distance from the RAU is plotted in Fig. 5 for both the 802.11b and 802.11g standards. Throughputs as high as 5.2 Mb/s for the IEEE 802.11b and 20 Mb/s for the IEEE 802.11g (at 54 Mb/s) were achieved closer to the RAU. These values are very close to the approximate maximum throughput values of 6 Mb/s for IEEE 802.11b and 22 Mb/s for IEEE 802.11g (at 54 Mb/s) reported in the literature [22]. From Fig. 5(a), it is seen that for IEEE 802.11b, the throughput values are relatively stable over the 5-m range. However, during the measurements for the IEEE 802.11g system, it was observed that on approaching the edge of the 5-m range, the data rate of the signal fluctuated between 54 and 36 Mb/s, due to fluctuations in signal strength (Fig. 4 shows average values). This leads to the decrease in the average

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Fig. 4. Received signal strength for: (a) IEEE 802.11b and (b) IEEE 802.11g. Fig. 5. Throughput measurement for: (a) IEEE 802.11b and (b) IEEE 802.11g.

throughput, even though the measured average signal strengths match predictions.

TABLE IV POTENTIAL RANGE CALCULATIONS FOR DIFFERENT SYSTEMS

V. FUTURE SYSTEM OPTIMIZATION The system demonstrator performance is limited by the available component specifications. We now compute the ranges possible for all systems presented in Section III when using the better available system components. For example, referring to Table I, it is assumed the lower RIN laser (with 0.38-W/A slope efficiency), the higher 0.44-A/W responsivity PD, and the 300-m OM1 MMF with gain 3.67 dB are available for both UL and DL. The remaining parameters are assumed the same as in Table I. The resulting radio range predictions for the different systems are then listed in Table IV. The range predictions are based only on the assumed system parameters of Table II. Within each system’s bandwidth, the respective access protocol may be used to support multiple MUs depending on service requirements. The results in Table IV show that reasonable ranges can be obtained for all systems. In fact, it is unlikely that 30 m radius cells would be required for indoor coverage when the fiber link lengths are themselves typically no more than 300 m. System architectures can be envisaged where many RAUs provide the coverage for high-speed WLANs, whereas only subsets of these (simultaneously [23]) provide coverage for the less demanding systems.

The DL transmitted noise power was found to be in accordance with the spurious emission limit stated in each of the standards (the required noise power column in Table IV gives this value). However, the standards impose a much stricter limit on the spurious emission from a base station in another base-station’s frequency band when co-locating different types of base stations or, in other words, where there is simultaneous transmission of different systems. In such a case, bandpass filters may be employed to further attenuate the noise power outside the wanted frequency band.

DAS et al.: DESIGN OF LOW-COST MMF-FED INDOOR WIRELESS NETWORKS

TABLE V RIN VALUES FOR THE DIFFERENT STANDARDS

VI. CONCLUSION A link budget analysis has been developed for the bidirectional operation of a fiber-fed indoor wireless network. The analysis allows for the optimization of component values to obtain the best possible range, while taking into account restrictions such as crosstalk and noise emissions. A WLAN system demonstrator using low-cost components (VCSELs, OM1/OM2 MMF) has been used to verify the link budget predictions. The demonstrator also verifies the operation of the whole fiber-fed WLAN system, giving throughput values similar to those normally expected. Theoretical range predictions for different radio systems such as GSM900, GSM1800, UMTS [both frequency division duplex (FDD) and time division duplex (TDD)], as well as WLANs have been carried out for a system using the better measured component parameters. The results show that reasonable cell sizes may be achieved for all systems. APPENDIX RIN VALUES FOR THE DIFFERENT STANDARD FREQUENCIES RIN values for the different standard frequencies for each of the DL and UL lasers were experimentally determined, and the values are listed in Table V. ACKNOWLEDGMENT The authors acknowledge many useful discussions with their colleagues in the virtual center Transmanche Telecom, and particularly J.-P. Vilcot, B. Sanz-Izquierdo, and J. Assaouré. The authors are also grateful to R. Davis, N. Simpson, T. Rockhill, and C. Birch for their assistance during the setup of the experiments. REFERENCES [1] D. Wake, “Trends and prospects for radio over fibre picocells,” in Int. Microw. Photon. Top. Meeting , Awaji, Japan, Nov. 2002, pp. 21–24. [2] A. Flatman, “In-premises optical fibre installed base analysis to 2007,” presented at the IEEE 802.310GBE Over FDDI Grade Fiber Study Group, Orlando, FL, Mar. 2004. [3] D. Wake, S. Dupont, J.-P. Vilcot, and A. J. Seeds, “32-QAM radio transmission over multimode fibre beyond the fibre bandwidth,” in Int. Microw. Photon. Top. Meeting, Long Beach, CA, Jan. 2002, vol. supp., 4 pp. [4] E. J. Tyler, M. Webster, R. V. Penty, and I. H. White, “Penalty free subcarrier modulated multimode fiber links for datacomm applications beyond the bandwidth limit,” IEEE Photon. Technol. Lett., vol. 14, no. 1, pp. 110–112, Jan. 2002.

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[5] R. V. Dalal, R. J. Ram, R. Helkey, H. Rousell, and K. D. Choquette, “Low distortion analog signal transmission using vertical cavity lasers,” Electron. Lett., vol. 34, pp. 1590–1591, Aug. 1998. [6] C. Carlsson, H. Martinsson, A. Larsson, and A. Alping, “High performance microwave link using a multimode VCSEL and high-bandwidth multimode fiber,” in Int. Top. Microw. Photon. Meeting, Long Beach, CA, Jan. 2002, pp. 81–84. [7] M. Y. W. Chia, B. Luo, M. L. Yee, and E. J. Z. Hao, “Radio over multimode fibre transmission for wireless LAN using VCSELs,” Electron. Lett., vol. 39, pp. 1143–1144, Jul. 2003. [8] R. E. Schuh, A. Alping, and E. Sundberg, “Penalty-free GSM-1800 and WCDMA radio-over-fibre transmission using multimode fibre and 850 nm VCSEL,” Electron. Lett., vol. 39, pp. 512–514, Mar. 2003. [9] T. S. Rapapport and S. Sandhu, “Radio-wave propagation for emerging wireless personal-communication systems,” IEEE Antennas Propag. Mag., vol. 36, pp. 14–23, Oct. 1994. [10] ASH Transceiver Designer’s Guide. Dallas, TX: RF Monolithic Inc., 2004, p. 51. [11] H. Sasai, T. Niiho, K. Tanaka, and K. Utsumi, “Radio over fiber transmission performance of OFDM signal for dual-band wireless LAN systems,” in Int. Microw. Photon. Topical Meeting, Budapest, Hungary, Sep. 2003, pp. 139–142. [12] T. S. Rappaport, Wireless communications— Principles and practice. Upper Saddle River, NJ: Prentice-Hall, 1996, pp. 565–568, Appendix B. [13] Radio Equipment and Systems (RES); Wideband Transmission Systems; Technical Characteristics and Test Conditions for Data Transmission Equipment Operating in the 2,4 GHz ISM Band and Using Spread Spectrum Modulation Techniques, ETSI 300 328, Nov. 1996. [14] B. Sanz-Izquierdo, J. Batchelor, and R. Langley, “Broadband multifunction planar PIFA antenna,” in Loughborough Antennas Propag. Conf., Loughborough, U.K., Apr. 2005, pp. 209–212. [15] 3rd Generation Partnership Project; Technical Specification Group GSM/EDGE Radio Access Network; Radio Transmission and Reception, 3GPP TS 45.005, 2004-11, ver. 6.7.0. [16] Universal Mobile Telecommunications System (UMTS); User Equipment (UE) Radio Transmission and Reception (FDD), 3GPP TS 25.101, 2003-03, ver. 6.7.0, rel. 6. [17] Universal Mobile Telecommunications system (UMTS); User Equipment (UE) Radio Transmission and Reception (TDD), 3GPP TS 25.102, 2003-12, ver. 6.0.0, rel. 6. [18] Universal Mobile Telecommunications system (UMTS); Base Station (BS) Radio Transmission and Reception (FDD), 3GPP TS 25.104, 2004-12, ver. 6.8.0, rel. 6. [19] Universal Mobile Telecommunications system (UMTS); Base Station (BS) Radio Transmission and Reception (TDD), 3GPP TS 25.105, 2004-12, ver. 6.2.0, rel. 6. [20] Supplement to IEEE Standard for Information Technology—Telecommunications and Information Exchange Between Systems—Local and Metropolitan Area Networks—Specific Requirements—Part 11: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications: Higher-Speed Physical Layer Extension in the 2.4 GHz Band, IEEE Standard 802.11b-1999, 1999. [21] Supplement to IEEE Standard for Information Technology—Telecommunications and Information Exchange Between Systems—Local and Metropolitan Area Networks—Specific Requirements Part 11: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications Amendment 4: Further Higher Data Rate Extension in the 2.4 GHz Band, IEEE Standard 802.11g-2003, 2003. [22] R. Seide, “Capacity, coverage, and deployment considerations for IEEE 802.11g, white paper,” Cisco Syst., San Jose, CA, 2003. [23] A. Nkansah, A. Das, I. J. Garcia, C. Lethien, J.-P. Vilcot, N. J. Gomes, J. C. Batchelor, and D. Wake, “Simultaneous transmission of dual-band radio signals over a multimode fibre fed indoor wireless network,” IEEE Microw. Wireless Compon. Lett., submitted for publication. Anjali Das received the B.Sc. (with honors) and M.Sc. degrees in electronic science from Delhi University, Delhi, India, in 1999 and 2001, respectively, the M.Sc. degree in information and communication engineering from the University of Leicester, Leicester, U.K., in 2003, and is currently working toward the Ph.D. degree in electronic engineering at the University of Kent, Canterbury, Kent, U.K. Her research interests include low-cost radio-overfiber systems and their deployment within buildings for improving coverage.

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Anthony Nkansah received the B.Eng. (with honors) degree in electronic engineering and the M.Sc. degree in broadband and mobile communication networks from the University of Kent, Canterbury, U.K., in 2000 and 2001, respectively, and is currently working toward the Ph.D. degree in electronic engineering at the University of Kent. His research interests include low-cost microwave and millimeter-wave radio-over-fiber networks and their deployment within premises.

Nathan J. Gomes (M’92) received the B.Sc. degree from the University of Sussex, Sussex, U.K., in 1984, and the Ph.D. degree from University College London, London, U.K., in 1988, both in electronic engineering. From 1988 to 1989, he held a Royal Society European Exchange Fellowship with the Ecole Nationale Supérieure des Télécommunications (ENST), Paris, France. Since late 1989, he has been a Lecturer, and since 1999, a Senior Lecturer, with the Electronics Department, University of Kent. His current research interests include radio-over-fiber systems and networks, the photonic generation and transport of millimeter-wave signals, and photoreceivers for such applications.

Ignacio J. Garcia received the B.Eng (with honors) degree in telecommunications engineering from Queen Mary University of London, London, U.K., in 2003, and is currently working toward the Ph.D. degree in electronic engineering at the University of Kent, Canterbury, U.K. From March 2004 to March 2005, he was a Research Associate with the University of Kent, where he was involved with the ROSETTE project, which concerned the design of fiber-fed antenna units for pico-cellular systems. He is currently a Research En-

gineer with the University of Wales, Swansea, U.K., where he is involved with the IRIS project, which concerns the implementation of optical wireless systems. His research interests are the miniaturization and design of multiband antennas for mobile wireless communication systems.

John C. Batchelor (M’94) received the B.Sc. and Ph.D. degrees from the University of Kent, Canterbury, U.K., in 1991 and 1995, respectively. From 1994 to 1996, he was a Research Assistant with the Electronics Department, University of Kent, and in 1997, became a Lecturer of electronic engineering. His current research interests include printed antennas, compact multiband antennas, electromagnetic-bandgap structures, and low-frequency frequency-selective surfaces.

David Wake (M’02) received the B.Sc. degree in applied physics from the University of Wales, Cardiff, U.K., in 1979, and the Ph.D. degree from the University of Surrey, Surrey, U.K., in 1987. He is currently a Visiting Senior Research Fellow with the University of Kent, Canterbury, U.K. From May 2003 to February 2005, he was Director of Research and Development and Chief Scientist with Microwave Photonics Inc., a startup company based in Los Angeles, CA, which was formed to develop a product set for the mobile communications industry based on novel radio-over-fiber technology. In 2002, he cofounded Zinwave Ltd., a startup company aimed at exploiting innovative radio-over-fiber technology for the mobile communications industry. He has been involved in the radio-over-fiber research topic for approximately 15 years, initially with BT Laboratories, where he was the Program Manager for the microwave photonics research domain and then with University College London as a Senior Research Fellow.

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Micromachined Rectangular-Coaxial Transmission Lines J. Robert Reid, Member, IEEE, Eric D. Marsh, Member, IEEE, and Richard T. Webster, Senior Member, IEEE

Abstract—Rectangular-coaxial (recta-coax) transmission lines fabricated through a three-dimensional micromachining process are presented. These lines are shown to have significant advantages over competing integrated transmission lines such as microstrip and coplanar waveguides. Design equations are presented for impedance, loss, and frequency range. The equations are confirmed with simulations and measurements. The quality factor of 4 resonators is measured to be 156 at 60 GHz. This shorted corresponds to a line loss of 0.353 dB/cm. Advantages of these lines for passive millimeter-wave circuits including ease of signal routing, high isolation, and signal crossovers are demonstrated with realized lines and couplers. Index Terms—Coplanar waveguides (CPWs), couplers, filters, microstrip, millimeter-wave circuits, resonators.

I. INTRODUCTION NCREASING use of the microwave spectrum combined with the demand for high bandwidth secure communications is increasingly pushing users towards millimeter-wave frequencies. However, the design of circuits above 30 GHz is complicated by signal crosstalk and the tight fabrication tolerances required to ensure accurate reproduction of simulated circuit designs. Over the past several years, the development of multilayer electroplated metal micromachining processes has made it possible to realize integrated rectangular-coaxial (recta-coax) transmission lines such as that illustrated in Fig. 1. These lines have the potential to greatly simplify the design of passive millimeter-wave circuits. Recta-coax lines consist of a rectangular center line suspended in air and enclosed on all sides by a ground plane. They are similar to microstrip and coplanar waveguide (CPW) lines in that they are fabricated using thin-film processing on top of a flat substrate. It is, therefore, possible to route the lines along any two-dimensional path. Further, the lines can be integrated on top of substrates such as silicon, gallium arsenide, and alumina. However, recta-coax lines offer significant advantages over both CPW and microstrip transmission lines. First, they are TEM and not quasi-TEM, mitigating dispersion, thus allowing operation over a very broad frequency range (dc to over 200 GHz). Second, the fields are completely enclosed

I

Manuscript received December 15, 2005; revised April 2, 2006. This work was supported in part by the Air Force Office of Scientific Research under LRIR 92SN04COR and in part by the Center for Advanced Sensor and Communications Antennas. The authors are with the Antenna Technology Branch, Air Force Research Laboratory, Hanscom AFB, MA 01731 USA (e-mail: [email protected]. mil). Digital Object Identifier 10.1109/TMTT.2006.879133

Fig. 1. Schematic illustration of a recta-coax line labeling the design dimensions.

by the ground. Therefore, two lines can be routed very close to each other. More importantly, the lines can easily cross paths without significant signal interference. Third, the design and performance of the lines are independent of the substrate. This eliminates the need for substrate thinning, and allows the substrate to be chosen based on parameters such as cost and the requirements of active devices. It also allows designs to be implemented on multiple substrates without any changes. The earliest investigations of micromachined transmission lines utilized bulk etching of silicon to form lines on thin membranes [1], [2]. In that study, the lines were referred to as microshield lines. The lines were implemented as coplanar transmission lines on a thin membrane with an air-filled metal cavity under the lines. In some cases, a second metal cavity was added above the line as well to create an entirely enclosed structure. The design of the CPW had to be modified to account for the additional capacitive loading of the added cavity ground planes. In addition, removing nearly all of the dielectric allowed the lines to operate in an effectively pure TEM mode [2]. The lines were fabricated by bulk etching of the back side of multiple silicon wafers. Metalizations were then added, and two or three wafers were bonded together. In subsequent studies, the microshield lines were designed to take on more of a microstrip style and then used in a large variety of millimeter-wave circuit demonstrations [3]–[9]. Of particular note is the exceptional millimeter-wave filter performance achieved [5]–[7]. Unfortunately, the complexity of the fabrication has limited the

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commercial acceptance of the microshield lines. One attempt to address this is the work of Melanovic´ et al. [10], [11], in which CPWs were also formed on membranes. However, the lines were formed using post processing of CMOS wafers. The final bulk etch was done after all of the lines were patterned. That approach does not provide a cavity around the CPW, but does eliminate dielectric losses. Neither of these approaches allows the flexibility and complex routing provided by the recta-coax lines described in this paper. Approaches utilizing SU-8 and polymer films to create microstrip on a thin film [12], CPW on thin film [13], and membrane-supported CPW lines [14], [15] have also been demonstrated. These polymer approaches typically offer simpler fabrication, but for microstrip lines offer high losses. Very low losses (0.18 dB/cm at 20 GHz) have been reported for large CPW lines on SU-8 films by etching out the SU-8 in the gap region between the lines. However, this technique is not suitable for multiple layers of transmission lines. In the case of membrane-supported CPW lines, relatively low losses (0.7 dB/cm) have been reported at -band when the line is fully enclosed [14]; however, the enclosure process is not detailed and is not inherent to the line as fabricated, and the design of multiple conductor layers would be difficult for the process as described. Over the past several years, work has progressed on the fabrication of transmission lines using electroplated films on top of the substrate, leading to the realization of suspended CPW transmission lines [16]–[20] and recta-coax lines [18]–[26]. Demonstrations of couplers and filters at microwave frequencies have shown the potential for these lines to simplify the design of millimeter-wave systems [21], [23]–[26]. This paper provides a detailed description of integrated recta-coax lines including design equations for impedance, loss, and resonator quality factor. Calculations from these equations are compared with Ansoft’s High Frequency Structure Simulator (HFSS) [27] simulations and measured values. In addition, examples of the transmission lines are provided, including lines of different lengths, lines with bends, and 60-GHz couplers.

II. RECTA-COAX DESIGN EQUATIONS

A. Characteristic Impedance The characteristic impedance of a transmission line can be calculated as

(1) where m/s is the phase velocity of the line, which is equal to the speed of light in vacuum for an air core TEM transmission line, and is the capacitance per unit length of the transmission line. It is, therefore, only necessary to solve for the capacitance per unit length between the center line and the ground. Chen [28] provides equations for solving this capacitance over a range of possible line dimensions. For lines

TABLE I POSSIBLE LINE CONFIGURATIONS FOR INTEGRATED RECTA-COAX

where the width and thickness are larger than the gaps ( and ), the capacitance is calculated as

(2) where all dimensions are defined in Fig. 1 and pF/m is the permittivity of the dielectric. Without loss of generality, it is assumed that . For lines that are thin, i.e., , the corner capacitances cannot be considered independently and the capacitance is calculated as

(3) Chen does not provide an estimate as to the accuracy of these equations. Weil provides a review of the methods for calculating this capacitance [29], and notes that Chen’s work is based on the coupled stripline equations developed by Cohn [30], and these equations are only valid when , which can be approximated by the condition . In order to confirm the validity of Chen’s equations and get an approximation of their accuracy for the types of lines that can be fabricated with integrated processes, calculations using (2) and (3) were compared with finite-element simulations performed using Ansoft’s HFSS. The line dimensions used for this comparison are shown in Table I. Calculations and simulations were done by fixing , , and , and then varying over the range shown in the column in Table I. When using the equations as defined above, the calculated value was within 4% of the simulation in all cases. While this is not a rigorous analysis, it does give us confidence that Chen’s equations are accurate for the lines that can be realized. More exact methods of calculating this capacitance for these cases have been detailed by other authors, but are not considered here [31]–[37]. B. Line Loss The loss of a transmission line is evaluated as the sum of conductor and dielectric losses. For an air core TEM transmission line, it is reasonable to ignore the dielectric loss and calculate

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only the conductor loss . The conductor loss is calculated using the Wheeler incremental inductance rule as [38]

(4) is the power loss in the conductor per unit length, and is the reference power. The power loss can be calculated as

where

C. Frequency Range A transmission line is typically used at frequencies where only one mode is not cutoff. For TEM transmission lines, the lower cutoff frequency is 0 Hz, or dc. The upper frequency cutoff is set by the next lowest mode that will propagate on the transmission line. Gruner has shown that the mode with the lowest cutoff frequency for a recta-coax line will always be either the or mode [41]. Further, because of symmetry, these modes are the same as the modes in ridged waveguide. The cutoff frequency of the mode in a ridged waveguide occurs when [42]

(5) (10) is the change in inductance where is the line resistance, when all conductor walls are receded by an amount , and , , and are the conductivity, permeability, and skin depth of the metal, respectively. Substituting values for and into (4) and simplifying results in

m/s is the speed of light in vacuum and is a shunt susceptance associated with the discontinuity created by the ridge. An approximation for the shunt susceptance can be calculated using [42]

where

(6) Since the inductance is related to the characteristic impedance as , the incremental inductance is related as , and (6) can be rewritten in terms of the change in the characteristic impedance to get

(7) where is the impedance calculated as described in Section II-A, and is the change in the impedance when the conductor walls are receded by half the skin depth of the conductor . This equation is not the same as that reported by Lau [39] because Lau uses the approximation . While this approximation is generally accurate for good conductors, it is not necessary given the widespread availability of digital computers to directly evaluate . This has the added advantage that any method for calculating including simulation can be used. The conductor loss must also account for metal roughness. This is done by adding a correction factor as given by Edwards [40]

(8) where is the line loss as calculated using (7) and is the rms surface roughness of the conductor. Since line loss can be difficult to measure, it is helpful to extract it from the measured quality factor of a resonator, which can be measured accurately. The line loss is related to the quality factor for shorted resonators by [38]

(9)

(11) . The accuracy of this equation will where degrade when . However, in this range, the susceptance is not the primary factor in (10) and, thus, the net effect on accuracy is negligible. To find the lowest cutoff frequency, it is necessary to calculate both the and cutoff frequencies. The cutoff frequency can be found using the same equations, but with the widths and thicknesses swapped. A good practice to ensure that operation is well below the cutoff frequency is to include a 15% buffer in the final calculation of the maximum operating frequency , resulting in

(12)

III. FABRICATION All of the devices presented in this paper were fabricated through a commercially available three-dimensional (3-D) microfabrication process called EFAB. The authors fabricated devices using a foundry model where design and simulation were done in-house. Design files were then submitted to the vendor, Microfabrica Inc., for fabrication under contract and the devices were returned to the authors for testing. A detailed description of the process is not within the scope of this paper. However, a basic understanding of the process is essential as it impacts the design of the transmission lines. Additional details on EFAB are available in [43] and is also available online.1 The EFAB process begins with a planar surface that is suitable for electroplating. On top of this surface, the following three steps are repeated for each desired layer. Step 1) A patterned sacrificial layer is deposited onto the substrate. 1[Online].

Available: http://www.microfabrica.com

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Fig. 3. SEM picture showing the transition from the CPW probe pads to rectacoax line of type A. The line was fabricated with 12 layers of nickel. Scalloping can be seen between each of the layers.

Fig. 2. Schematic illustration of the fabrication of an AFRL logo using the EFAB process. (0) Process begins with planar substrate. (1) Patterned layer is then deposited followed by (2), blanket deposition of a second layer. (3) Top surface is then planarized. Steps (1)–(3) are repeated to create the desired number of layers, as shown in (4). (5) Finally, the sacrificial layer is removed resulting in the desired structure. A scanning electron micrograph (SEM) picture of the realized structure is shown. (Color version available online at: http://ieeexplore.ieee.org.)

Step 2) A second layer, i.e., the structural layer, is blanket deposited over the entire substrate. Step 3) The top surface is planarized to create a single layer of uniform thickness with two separate materials. After all of the layers are completed, the sacrificial layer is removed via a chemical etch leaving the structural layers to form the desired structure. A schematic illustration of this process flow is provided in Fig. 2. The process is fundamentally the same as a sacrificial surface micromachining process, with two critical distinctions. First, the layers in this process are typically much thicker 2–40 m than in surface micromachining 0.5–4 m . Second, the planarization allows significantly more layers ( 10) than are common in surface micromachining. In the study reported here, three separate fabrication builds were done. The first two builds had 12 layers, while the third had 26 layers. Each run had different layer thicknesses so

that the final realized lines had different dimensions. Lines of type A and B from Table I could be realized in build 1, lines of type C could be realized in build 2, and lines of type D and E could be realized in build 3. In addition to EFAB, these transmission lines can be fabricated in a number of other processes. Custom processes are available either through in-house development or at several universities [16]–[20], [44]–[46]. In addition, the Defense Advanced Research Projects Agency (DARPA) 3-D microelectromagnetic RF systems (MERFS) program aims to develop a low-cost multilayer process utilizing copper as the structural material. This process uses layers that are even thicker than the EFAB layers to minimize the number of masking steps and, thus, reduce the costs for recta-coax lines.2 IV. RESULTS A. Impedance Designing recta-coax lines using a layered sacrificial process such as EFAB requires certain considerations. The layered nature of these processes can result in scalloping at the layer interfaces, as is visible in Fig. 3. In order to mitigate any effects this might have on the impedance of the lines, all of our lines are designed so the ratio of linewidth to gap thickness is always greater than the ratio of thickness to gapwidth or . Thus, the capacitance from the signal line is always dominated by the capacitances associated with the top and bottom ground planes and the capacitance to the sidewalls is reduced. In addition to the scalloping, there is some concern with layer-to-layer misalignment. To date, neither of these concerns has had a measurable effect on our devices. In addition to these concerns, sacrificial etching requires the regular, but not necessarily periodic, spacing of etch access holes. These holes are visible in Fig. 4. However, the design constraint mentioned above also mitigates this problem. Preliminary electromagnetic simulations indicate 2[Online]. 3dmerfs.html

Available:

http://www.darpa.mil/MTO/people/pms/evans/

REID et al.: MICROMACHINED RECTA-COAX TRANSMISSION LINES

Fig. 4. SEM picture showing the ports for two 60-GHz bandpass filters. Etch access holes are visible on the sidewalls of the filters.

that etch holes have only a negligible effect when the transmission lines are used for signal routing and nonresonant devices. However, in resonator designs, the etch holes may play a more significant role, and further investigations are required. Recta-coax lines are designed by first selecting , and based on selected layer thicknesses. Next, is fixed so that the sidewall capacitance is lower than the top and bottom wall capacitances. Based on this, calculations of the characteristic impedance were compared to HFSS simulations for lines of type A, D, and E in Table I. These geometries were specifically chosen to cover a wide range of possible aspect ratios. The results of these calculations are shown in Fig. 5. The calculations based on (1)–(3) agree well with the simulated port impedances with the only significant ( 2%) variations occurring for line type E at high impedances over 115 . This is not surprising, as these lines violate the accuracy criteria for the capacitance model. The width required to achieve a 50- line for types A–E is reported in Table II in the column. As can be seen, lines of type B are not realistic due to the very narrow width required to achieve 50 . Lines of the other four types have been realized in three separate fabrication runs. Fig. 6 shows a comparison of the measured and simulated characteristic impedance of recta-coax lines. The impedance was calculated from both the measured and simulated -parameters using the conventional single-line technique described in [47]. The cross-sectional dimensions of the line are those of Line A in Table I, with the center conductor width at 47 m, corresponding to a simulated port impedance of 50 . The measured lines are nominally identical with a length of 1.1 mm. The vector network analyzer was calibrated to a reference impedance of 50 at the probe tips using the short, open, load, thru (SOLT) method with the probe manufacturers calibration substrate. The measured -parameters include effects of the coplanar-to-rectacoax transition, while the electromagnetic simulation does not include transitions. The mean value of the measured over the measured frequency range is . This compares well with the mean value of the simulated at .

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Fig. 5. Impedance of three different lines plotted as a function of linewidth. The continuous lines are modeled values, while the discrete points are simulated values.

TABLE II

50

LINEWIDTHS AND MAXIMUM OPERATING FREQUENCIES FOR

DIFFERENT RECTA-COAX LINES

Fig. 6. Characteristic impedance Z of measured recta-coax lines is compared to Z of an electromagnetic simulation of the lines. The line dimensions are those of Line A in Table I, with a center conductor width of 47 m, corresponding to a simulated port impedance of 50 . The measured lines are nominally identical with a length of 1.1 mm. Coplanar to recta-coax transitions are included in the measurements.

B. Line Loss The loss for lines A, D, and E was calculated using (7) and (8). The loss for these lines fabricated in nickel s/m is presented in Fig. 7 in terms decibels/centimeter. For all of the loss calculations, a surface roughness of m is used. This value was measured from the top surface of the build

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Fig. 7. Modeled line loss for 50- nickel lines of types A, D, and E.

1 recta-coax signal lines using a optical profilometer. Based on (8), this increases the loss of the nickel lines by less than 3% over the entire frequency range. At the time of fabrication, nickel was the only metal available through EFAB. Metals such as gold, copper, and silver offer significantly lower losses, and Fig. 8 provides a comparison between line type D in nickel, gold, and copper to circuit simulations of 50- microstrip on 50- m-thick GaAs and 50- CPW on sapphire. The microstrip line has a width of 35 m and is simulated with a 2- m-thick gold metallization. This line was chosen because 50- m GaAs is an industry standard. The CPW line dimensions are those used in the design of microwave microelectromechanical systems (MEMS) switches at the Air Force Research Laboratory (AFRL), Hanscom AFB, MA [48], [49]. These lines are also simulated with a 2- m-thick gold metallization. Recta-coax line type D was chosen because it yields lines that are generally comparable in size. A 50- line of type D has a total width of 195 m including the line, gaps, and ground walls. This is significantly more than the 35- m-wide microstip. However, the microstrip line generally needs a buffer of at least 1.5–2 linewidths on each side, resulting in a total path width that is 140–175 m. As shown in the comparison, recta coax is relatively low loss in terms of decibels/centimeter even when it is fabricated with nickel. While the loss per unit length is important for signal routing across a die, many circuits are more concerned with loss as a function of electrical length. In this case, loss per wavelength or resonator quality factor are more appropriate measures of line performance. From (9), it is clear that quality factor is inversely proportional to the loss per wavelength. Since resonator quality factor can be measured accurately, independent of the calibration, quality factor is presented here. The quality factor of shorted resonators is presented in Fig. 9 for line type E with nickel, gold, and copper metallizations. For comparison, the calculated loss of 50- microstrip lines on a 15- m-thick fused silica substrate are also provided. On 150- m-thick silica, the microstrip lines are calculated to be 324- m wide. The microstrip line loss is calculated using Agilent ADS with a 2- m-thick gold line. Recta-coax resonators

Fig. 8. Comparison of the modeled line loss for recta-coax line type D with 50- microstrip line on a 50-m-thick GaAs substrate, and a 50- CPW on a sapphire substrate with a linewidth of 80 m and a gap of 40 m. Both the microstrip and CPW lines are simulated using 2-m-thick gold.

Fig. 9. Measured quality factor of nickel =4 resonators compared with modeled quality factors for lines of different metallizations and a 50- microstrip line on 150-m-thick fused silica with a 2-m-thick gold metallization.

fabricated with nickel have lower quality factors than the microstrip lines. However, when gold or copper are used, the rectacoax compares well with the microstrip line. The final choice of a metal must also consider the mechanical properties of the material and environmental factors such as corrosion. While nickel has relatively low conductivity, it has a relatively high Young’s modulus, leading to mechanical strength and rigidity. This allows nickel lines to be suspended for several millimeters without significant deviation from the design parameters. We are currently investigating plating the nickel lines with gold to provide a mechanically strong line with low electrical loss. At millimeter-wave frequencies, a nickel line plated with 1–2 m of gold would provide good mechanical strength, low electrical loss, and a corrosion resistant surface. The loss calculations are verified by measuring the quality factors of , line type E, shorted resonators at 44 and 60 GHz. Measurements were taken

REID et al.: MICROMACHINED RECTA-COAX TRANSMISSION LINES

Fig. 10. Measured phase of S versus frequency for four recta-coax lines and the deviation from linear phase indicate low dispersion. The line dimensions are those of line A in Table I, with a center conductor width of 47 m, corresponding to a simulated port impedance of 50 . The line lengths are 0.2, 1.1, 1.6, and 3.2 mm. Coplanar to recta-coax transitions are included in the measurements.

on nine 44-GHz resonators, and ten 60-GHz resonators located on five different die. The average measured quality factors were 142.8 and 155.8 at 44 and 60 GHz, respectively. These values are slightly below the HFSS simulations (145 and 169, respectively) and the calculated values (150.9 and 175.7, respectively), but are generally in good agreement. This corresponds to a line loss of 0.282 dB/cm at 44 GHz and 0.353 dB/cm at 60 GHz. C. Dispersion and Group Velocity The measured transmission phase for four 50- lines shown in Fig. 10. The line lengths are 0.2, 1.1, 1.6, and 3.2 mm and the corresponding group delays obtained from a linear fit to the phase curves are 2.68, 5.76, 7.43, and 12.83 ps. These measurements were taken using a SOLT calibration and, therefore, do include coplanar-to-recta-coax transitions. The deviation from linear phase remains within 0.02 rad, except for a few spikes above 50 GHz. A linear regression of the delays on the line lengths gives a group velocity of 2.958 10 m/s. This value is within 1.5% of the speed of light expected for an air dielectric recta-coax line. D. Operation Frequency The operational frequency ranges for 50- recta-coax lines of types A–E calculated using (10)–(12) are presented in Table II. It can be seen that for types A–D, the maximum operating frequencies are well in excess of 500 GHz, while for type E, it is over 200 GHz. E. Thru-Reflect-Line (TRL) Calibration Standard In order to facilitate measurements, a TRL calibration set is implemented on chip with each die. The TRL from our first build was implemented using 50- lines of type A. This cal set is shown in Fig. 11. Measurements are taken using ground–signal–ground probes with a 0.15-mm center-to-center spacing. Since the coax can not be directly probed, it was necessary to terminate the lines in a thick CPW section that is

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Fig. 11. TRL calibration set. The thru is the short line in the lower left. The lines are 1.1- and 1.6-mm long. The four reflect standards are implemented to facilitate easy probing.

designed to have a 50- characteristic impedance. This CPW section extends 0.2 mm from the beginning of the recta-coax. It is designed so that the center conductor does not have to change height or width, but there is a transition in the ground at the entrance to the recta-coax. The TRL set includes a 0.2-mm thru, multiple reflect standards that have 0.1 mm of line length, 1.1- and 1.6-mm lines. As a result, the TRL calibration sets the measurement reference plane 0.1 mm in from the entrance to the recta-coax. Using the TRL set will then exclude any effects caused by the transitions from probe to CPW to recta coax. The accuracy of the TRL calibration is generally best when the electrical length of the lines is 30 –150 Therefore, this TRL set is good from 16.7 to 125 GHz. Typically, when the TRL calibration set is used, measurements are taken from 30 to 67 GHz. F. Line With Bends One of the reported advantages of these lines is their ability to turn corners without significant design issues. As such, a demonstration line was fabricated with four 90 bends. This line is shown in Fig. 12. The total length of the bent line is 3.2 mm measured on a path running along the center of the line with cross-sectional dimensions of line type A. Fig. 13 shows the measured scattering parameters for the 3.2-mm line with four bends compared to a 3.2-mm straight line. The two lines appear almost identical with the average loss always within 0.3 dB. Measurements from multiple lines typically show the bent lines to have 5 –10 less phase shift at 60 GHz. However, an accurate characterization of the bend will require implementing it in a resonator type structure. As is clear in Fig. 12, the line is routed in an S-configuration with all three sections close to each other. The ground lines are spaced 54 m apart for a center to center spacing of 302 m. This is only possible because of the signal isolation achieved by having and enclosed TEM line, and this feature of the transmission lines can be used to make devices like branch-line couplers significantly more compact, as demonstrated by Chen et al. [21].

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Fig. 12. 3.2-mm line (50- type A) with four 90 bends.

Fig. 14. SEM picture showing a 60-GHz branch-line coupler.

TABLE III LINE DIMENSIONS FOR A 60-GHz BRANCH-LINE COUPLER USING LINE TYPE A

H. Cross Overs In the third build, a six-port coupler was implemented. This coupler is shown in Fig. 15. Detailed design and testing of this coupler are provided in [25]; however, the coupler is shown here because the physical implementation requires lines that cross over. These cross overs are shown in Fig. 15. Simulations of the coupler showed no significant crosstalk between the two line layers. As with the branch-line coupler, the measured coupler performs exactly as designed. Fig. 13. Measured scattering parameters for two 3.2-mm-long lines. One of the lines is straight while the other has four 90 bends.

G. 60-GHz Branch-Line Coupler In order to demonstrate the performance of these transmission lines, a branch-line coupler was realized at 60 GHz [23]. The coupler can be seen in Fig. 14, and Table III provides the widths and characteristic impedances for the two arms. Over a 10% bandwidth (57–63 GHz), the two coupled lines have measured average insertion loss of 3.625 and 3.870 dB. The measured insertion loss of these paths is within 0.25 dB of the design simulation, while the measured phase is within 5 of the original simulation. In order to save space on the die, a second test structure was inserted inside the coupler. The inserted test structure places metal walls 50 m from the cross arms of the branch-line coupler. This structure, visible in Fig. 14, is an unrelated test structure and was not included in any of the coupler simulations.

V. DISCUSSION Design equations for recta-coax lines have been presented for impedance, loss, and frequency range. Calculations of impedance, loss, and resonator quality factor are shown to be in good agreement with full-wave electromagnetic simulations over a range of design parameters. Measured line impedance is shown to agree very well with simulated line impedance over a wide frequency range. Measured lines showed losses consistent with the calculated values. However, for better accuracy, resonators were measured at two frequencies and are shown to be in good agreement with predictions. In addition, the group velocity of the lines was measured and the line dispersion was shown to be very low up to 67 GHz. Integrated recta-coax lines offer significant advantages over microstrip and CPW lines for millimeter-wave circuit design. We have presented many of these advantages here and in previous work including low-loss 180 bends, signal cross overs, and isolation of signals. Recta-coax lines have also been shown

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on the substrate considering only the need to physically probe the devices. This greatly simplifies the design issues involved in millimeter-wave circuits, and can possibly lead to a much higher level of integration. ACKNOWLEDGMENT This paper has been cleared for public release: Distribution Unlimited (ESC06-0062). REFERENCES [1] N. Dib, W. Harokopus, L. Katehi, C. Ling, and G. Rebeiz, “Study of a novel planar transmission line,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 1991, pp. 623–626. [2] N. Dib and L. Katehi, “Impedance calculation for the microshield line,” IEEE Trans. Microw. Guided Wave Lett., vol. 2, no. 10, pp. 406–408, Oct. 1992. [3] L. Katehi, G. Rebeiz, T. Weller, R. Drayton, H.-J. Cheng, and J. Whitaker, “Micromachined circuits for millimeter- and sub-millimeter-wave applications,” IEEE Antennas Propag. Mag., vol. 35, no. 5, pp. 9–17, Oct. 1993. [4] T. Weller, L. Katehi, and G. Rebeiz, “High performance microshield line components,” IEEE Trans. Microw. Theory Tech., vol. 43, no. 3, pp. 534–543, Mar. 1995. [5] C.-Y. Chi and G. Rebeiz, “Conductor-loss limited stripline resonators and filters,” IEEE Trans. Microw. Theory Tech., vol. 44, no. 4, pp. 626–630, Apr. 1996. [6] S. Robertson, L. Katehi, and G. Rebeiz, “Micromachined -band filters,” IEEE Trans. Microw. Theory Tech., vol. 44, no. 4, pp. 598–606, Apr. 1996. [7] P. Blondy, A. Brown, D. Cros, and G. Rebeiz, “Low-loss micromachined filters for millimeter-wave communication systems,” IEEE Trans. Microw. Theory Tech., vol. 46, no. 12, pp. 2283–2288, Dec. 1998. [8] A. Brown and G. Rebeiz, “A high-performance integrated -band diplexer,” IEEE Trans. Microw. Theory Tech., vol. 47, no. 8, pp. 1477–1481, Aug. 1999. [9] A. Margomenos, K. Herrick, M. Herman, S. Valas, and L. Katehi, “Isolation in three-dimensional integrated circuits,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 1, pp. 25–32, Jan. 2003. [10] V. Milanovic´ , M. Gaitan, E. Bowen, and M. Zaghloul, “Micromachined coplanar waveguides in CMOS technology,” IEEE Trans. Microw. Guided Wave Lett., vol. 6, no. 10, pp. 380–382, Oct. 1996. [11] ——, “Micromachined microwave transmission lines in CMOS technology,” IEEE Trans. Microw. Theory Tech., vol. 45, no. 5, pp. 630–635, May 1997. [12] J. R. Thorpe, D. P. Steenson, and R. E. Miles, “High frequency transmission line using micromachined polymer dielectric,” Electron. Lett., vol. 34, no. 12, pp. 1237–1238, Jun. 1998. [13] D. Newlin, A.-V. H. Pham, and J. Harriss, “Development of low loss organic-micromachined interconnects on silicon at microwave frequencies,” IEEE Trans. Compon. Packag. Technol., vol. 25, no. 3, pp. 506–510, Sep. 2002. [14] W. Y. Liu, D. P. Steenson, and M. B. Steer, “Membrane-supported CPW with mounted active devices,” IEEE Microw. Wireless Compon. Lett., vol. 11, no. 4, pp. 167–169, Apr. 2001. [15] F. D. Mbairi and H. Hesselbom, “High frequency design and characterization of SU-8 based conductor backed coplanar waveguide transmission lines,” in Proc. Int. Adv. Packag.: Processes, Properties, Interfaces Symp., Mar. 2005, pp. 243–248. [16] J.-B. Yoon, B.-I. Kim, Y.-S. Choi, and E. Yoon, “3-D lithography and metal surface micromachining for RF and microwave MEMS,” in IEEE Int. Microelectromech. Syst. Conf., Jan. 2002, pp. 20–24. [17] E.-C. Park, Y.-S. Choi, B.-I. Kim, J.-B. Yoon, and E. Yoon, “A low loss MEMS transmission line with shielded ground,” in IEEE Int. Microelectromech. Syst. Conf., Jan. 2003, pp. 136–139. [18] J.-B. Yoon, B.-I. Kim, Y.-S. Choi, and E. Yoon, “3-D construction of monolithic passive components for RF and microwave ICs using thick-metal surface micromachining technology,” IEEE Trans. Microw. Theory Tech., vol. 51, no. 1, pp. 279–288, Jan. 2003. [19] E.-C. Park, T.-S. Song, S.-H. Baek, and E. Yoon, “Low phase noise CMOS distributed oscillators using MEMS low loss transmission lines,” in IEEE Int. Microelectromech. Syst. Conf., Jan. 2004, pp. 645–648.

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Fig. 15. (a) SEM picture of a six-port coupler. The six circular structures are probe ports used for testing. The couplers ports are located at the junctions. The probe ports are connected the coupler ports by transmission lines running under the coupler and then rising up to the coupler signal lines, as illustrated in the – cutaway shown in (b). (Color version available online at: http://ieeexplore. ieee.org.)

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to have relatively low losses compared to both microstrip and CPW lines. This is true when considering loss over a given distance (decibels/centimeters), loss per wavelength, or resonator quality factor. Due to the air dielectric, recta-coax resonators are physically longer than either microstrip or CPW resonators at the same frequency, but even when the recta-coax lines are fabricated with nickel, the resulting quality factor is comparable. The eventual use of either plated lines or copper lines will dramatically improve the quality factor of recta-coax resonators. The ability to repeatably realize resonators with quality factors above 100 will make this technology suitable to applications including filters and oscillators. Recta-coax lines do require a more complicated fabrication process than either CPW or microstrip lines. At this time, it is not possible to do an accurate cost comparison between the technologies because the recta-coax processes are still relatively immature. However, as this technology develops, this comparison will need to take into account the reduced design cost that should be possible with recta-coax lines. Reduced design costs are anticipated because recta-coax lines can be designed independent of the substrate and independent of the surrounding environment. As an example, we have presented a branch-line coupler and noted that, after the design was completed, a structure was added to the center area of the coupler with no effect on simulated performance. In fact, at this point, all of the designs we have completed have been done independently, and then placed

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[20] H.-S. Lee, D.-H. Shin, S.-C. Kim, B.-O. Lim, T.-J. Baek, B.-S. Ko, Y.-H. Chun, S.-K. Kim, H.-C. Park, and J.-K. Rhee, “Fabrication of new micromachined transmission line with dielectric posts for millimeter-wave applications,” J. Micromech. Microeng., vol. 14, pp. 746–749, 2004. -band filter [21] R. Chen, E. Brown, and C. Bang, “A compact low-loss using 3-dimensional micromachined integrated coax,” in IEEE Int. Microelectromech. Syst. Conf., Jan. 2004, pp. 801–804. [22] E. Brown, A. Cohen, C. Bang, M. Lockard, G. Byrne, N. Vendelli, D. McPherson, and G. Zhang, “Characteristics of microfabricated rectanband,” Microw. Opt. Technol. Lett., vol. 40, pp. gular coax in the 365–804, Mar. 2004. [23] J. Reid and R. Webster, “A 60 GHz branch line coupler fabricated using integrated rectangular coaxial lines,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2004, pp. 441–444. [24] ——, “A compact integrated coaxial -band bandpass filter,” in IEEE AP-S Int. Symp. Dig., Jun. 2004, pp. 990–993. beam former,” in IEEE [25] ——, “A six-port 60 GHz coupler for an AP-S Int. Symp. Dig., Jul. 2006. [26] ——, “A 55 GHz bandpass filter realized with integrated TEM transmission lines,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2006, pp. 132–135. [27] High Frequency Structure Simulator (HFSS) ver. 9.0, Ansoft Corporation, Palo Alto, CA [Online]. Available: http://www.ansoft.com [28] T.-S. Chen, “Determination of the capacitance, inductance and characteristic impedance of rectangular lines,” IRE Trans. Microw. Theory Tech., vol. MTT-8, no. 9, pp. 510–519, Sep. 1960. [29] C. Weil, “The characteristic impedance of rectangular transmission lines with thin center conductor and air dielectric,” IEEE Trans. Microw. Theory Tech., vol. MTT-26, no. 4, pp. 238–242, Apr. 1978. [30] S. Cohn, “Thickness corrections for capacitive obstacles and strip conductors,” IRE Trans. Microw. Theory Tech., vol. MTT-9, no. 11, pp. 638–644, Nov. 1961. [31] W. Getsinger, “Coupled rectangular bars between parallel plates,” IRE Trans. Microw. Theory Tech., vol. MTT-10, no. 9, pp. 65–72, Sep. 1962. [32] R. Garver, “ of rectangular coax,” IRE Trans. Microw. Theory Tech., vol. MTT-9, no. 5, pp. 262–263, May 1961. [33] O. Cruzan and R. Garver, “Characteristic impedance of rectangular coaxial transmission lines,” IRE Trans. Microw. Theory Tech., vol. MTT-12, no. 9, pp. 488–495, Sep. 1964. [34] H. Riblet, “The exact dimensions of a family of rectangular coaxial lines with given impedance,” IEEE Trans. Microw. Theory Tech., vol. MTT-20, no. 8, pp. 538–541, Aug. 1972. [35] ——, “An approximation for the characteristic impedance of shieldedslab line,” IEEE Trans. Microw. Theory Tech., vol. MTT-27, no. 6, pp. 557–559, Aug. 1979. [36] ——, “Two limiting values of the capacitance of symmetrical rectangular coaxial strip transmission line,” IEEE Trans. Microw. Theory Tech., vol. MTT-29, no. 7, pp. 661–666, Aug. 1981. [37] J. Tippet and D. Chang, “Characteristic impedance of a rectangular coaxial line with offset inner conductor,” IEEE Trans. Microw. Theory Tech., vol. MTT-26, no. 11, pp. 876–883, Nov. 1978. [38] D. Pozar, Microwave Engineering. Reading, MA: Addison-Wesley, 1990. [39] K. Lau, “Loss calculations for rectangular coaxial lines,” Proc. Inst. Elect. Eng., vol. 135, no. 3, pp. 207–209, Jun. 1988. [40] T. Edwards, Foundations for Microstrip Circuit Design, 2nd ed. New York: Wiley, 1992. [41] L. Gruner, “Higher order modes in rectangular coaxial waveguides,” IEEE Trans. Microw. Theory Tech., vol. MTT-15, no. 8, pp. 483–485, Aug. 1967. [42] R. Collin, Foundations for Microwave Engineering, 2nd ed. New York: McGraw-Hill, 1992. [43] A. Cohen, G. Zhang, F. Tseng, F. Mansfield, U. Frodis, and P. Will, “EFAB: Rapid low-cost desktop micromachining of high aspect ratio true 3-D MEMS,” in IEEE Int. Microelectromech. Syst. Conf., Jan. 1999, pp. 244–251. [44] Y.-J. Kim and M. Allen, “Surface micromachined solenoid inductors for high frequency applications,” IEEE Trans. Compon., Packag., Manuf. Technol., vol. 21, no. 1, pp. 26–33, Jan. 1998.

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[45] Y.-H. Joung, S. Nuttinck, S.-W. Yoon, M. Allen, and J. Laskar, “Integrated inductors in the chip-to-board interconnect layer fabricated using solderless electroplating bonding,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2002, pp. 1409–1412. [46] S. Pinel, F. Cros, S. Nuttinck, S.-W. Yoon, M. Allen, and J. Laskar, “Very high- inductors using RF-MEMS technology for system-onpackage wireless communication integration module,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2003, pp. 1497–1500. [47] W. Eisenstadt and Y. Eo, “ -parameter-based IC interconnect transmission line characterization,” IEEE Trans. Compon., Hybrids, Manuf. Technol., vol. 15, no. 4, pp. 483–490, Aug. 1992. [48] J. Reid, L. Starman, and R. Webster, “RF actuation of capacitive MEMS switches,” in IEEE MTT-S Int. Microw. Symp. Dig., Jun. 2003, pp. 1919–1922. [49] J. Ebel, R. Corez, K. Leedy, and R. Strawser, “Hermetic thin-film encapsulation for RF MEMS switches,” in Proc. Gomactech, 2005, pp. 327–330.

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J. Robert Reid (S’94–M’97) received the B.S.E.E. degree from Duke University, Durham, NC, in 1992, and the M.S.E.E. and Ph.D. degrees from the Air Force Institute of Technology (AFIT), Dayton, OH, in 1993 and 1996, respectively. He is currently an Electronics Engineer with the Antenna Technology Branch, Electromagnetics Technology Division, Air Force Research Laboratory (AFRL), Hanscom AFB, MA. In 1992, he was commissioned a second lieutenant with the US Air Force and sent to the AFIT. Since 1997, he has been with the AFRL, where he conducts research into the application of micromachining and MEMS to front-end antenna technology. In 2002, he separated from the Air Force and joined the AFRL as a civilian. His current research includes design, fabrication, and testing of micromachined transmission lines, RF MEMS switches and varactors, millimeter-wave phase shifters, and switching networks. Dr. Reid is a member of Sigma Xi.

Eric D. Marsh (S’03–M’04) received the Bachelor’s degree in electrical engineering from Iowa State University, Ames, in 1999, and the Master’s degree in electrical engineering from the Air Force Institute of Technology, Dayton, OH, in 2004. In 1999, he joined the United States Air Force as a Developmental Engineer. Since 2004, he has been with the Sensor’s Directorate, Air Force Research Laboratory, Hanscom AFB, MA, where he performs basic research on micromachined millimeter-wave transmission lines and components.

Richard T. Webster (S’76–M’76–SM’02) received the B.S. and M.E. degree in electrical engineering from the Rensselaer Polytechnic Institute, Troy, NY, in 1973 and 1976, respectively. He is currently an Electronics Engineer with the Antenna Technology Branch, Electromagnetics Technology Division, Air Force Research Laboratory (AFRL), Hanscom AFB, MA. Since 1980, he has been with the AFRL, where he plans and conducts programs to develop monolithic microwave and millimeter-wave integrated circuits for amplification and signal control in phased-array antenna systems. His current research is focused on the development of amplifiers, switches, phase shifters, and related components using a variety of technologies including semiconductor, microelectromechanical, and electromagnetic. Mr. Webster is a member of Sigma Xi.

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Estimation of Heating Performances of a Coaxial-Slot Antenna With Endoscope for Treatment of Bile Duct Carcinoma Kazuyuki Saito, Member, IEEE, Atsushi Hiroe, Satoru Kikuchi, Masaharu Takahashi, Senior Member, IEEE, and Koichi Ito, Fellow, IEEE

Abstract—Hyperthermia is one of the modalities for cancer treatment, utilizing the difference of thermal sensitivity between tumor and normal tissue. Up to now, the authors have been studying various coaxial-slot antennas for microwave hyperthermia. A coaxial-slot antenna aiming at intracavitary heating for bile duct carcinoma has been especially developed. In this paper, heating performances of such an antenna are estimated for the actual treatments. As a result of the investigations, the possibility of this treatment by use of the proposed coaxial-slot antenna could be confirmed. Index Terms—Coaxial-slot antenna, finite-difference timedomain (FDTD) method, intracavitary microwave hyperthermia, specific absorption rate (SAR) distribution, temperature distribution.

I. INTRODUCTION N RECENT years, various types of medical applications of microwaves have widely been investigated and reported [1]. They are microwave hyperthermia [2], [3] and microwave coagulation therapy (MCT) [4], [5] for medical treatment of cancer, cardiac catheter ablation for ventricular arrhythmia treatment [6], [7], thermal treatment of benign prostatic hypertrophy (BPH) [8], [9], etc. Until now, the authors have been studying antennas for microwave hyperthermia. Hyperthermia is one of the modalities for cancer treatment, utilizing the difference of thermal sensitivity between tumor and normal tissue. In this treatment, the tumor is heated up to the therapeutic temperature from 42 C and 45 C without overheating the surrounding normal tissues. There are a few methods for heating the cancer cells inside the body. The authors have been especially studying the coaxial-slot antenna [10], [11], which is one of the thin microwave antennas, for the interstitial microwave hyperthermia. As a result of these investigations, some cases of actual treatments could be realized by use of our developed antenna, and the effectiveness of these treatments could be confirmed [11]. This time we have developed a coaxial-slot antenna aiming at intracavitary heating for a bile duct carcinoma. The bile duct

I

Manuscript received January 19, 2006; revised March 31, 2006. K. Saito and M. Takahashi are with the Research Center for Frontier Medical Engineering, Chiba University, Chiba 263-8522, Japan (e-mail: [email protected]). A. Hiroe and S. Kikuchi are with the Graduate School of Science and Technology, Chiba University, Chiba 263-8522, Japan. K. Ito is with the Faculty of Engineering, Chiba University, Chiba 263-8522, Japan. Digital Object Identifier 10.1109/TMTT.2006.879177

Fig. 1. Scheme of the treatment (the calculated region explained in Section III is included).

is seated in deep region of the body and some thick blood vessels lay near the bile duct. Therefore, the bile duct carcinoma is difficult to treat via the surgical operation. In such cases, a multidisciplinary approach, which combines radiotherapy, chemotherapy, thermal therapy, etc. is expected to improve the quality of life (QOL) of the patient. For these reasons, the investigations of the antenna for such a treatment are important. Previously, heating performances of the coaxial-slot antenna were considered by numerical calculations using a simple model [12]. In this paper, in order to check the possibility of heating the bile duct carcinoma, we numerically calculated the temperature distribution around the tip of the antenna by use of a calculation model based on the realistic human model developed at Brooks Air Force Laboratories, San Antonio, TX.1 In Section II, the scheme of the treatment and the structure of the antenna are explained. In Section III, the procedure of numerical calculation and the calculation model based on the realistic human model are introduced. In Section IV, the heating performances of the antenna for the treatment are described. Finally, conclusions are presented in Section V. II. SCHEME OF THE TREATMENT AND STRUCTURE OF THE ANTENNA Fig. 1 shows the scheme of the treatment. In this treatment, the endoscope is first inserted into the duodenum and a long and flexible coaxial-slot antenna is then inserted into the forceps channel of the endoscope, which is used to insert the tool for surgical treatment. Finally, the antenna is guided to the bile duct 1Brooks Air Force Base Homepage. [Online]. Available: http://www. brooks.af.mil/AFRL/HED/hedr/hedr.html

0018-9480/$20.00 © 2006 IEEE

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TABLE I PARAMETERS FOR FDTD CALCULATIONS

Fig. 2. Structure of the coaxial-slot antenna for treatment of bile duct carcinoma. (a) Structure of the coaxial-slot antenna. (b) Tip of the endoscope with the coaxial-slot antenna.

through the papilla of Vater, which is located in the duodenum, and is inserted in the bile duct. Fig. 2(a) shows the structure of the proposed coaxial-slot antenna for this treatment. The basic structure of the antenna is the same as in [10] and [11], except that this antenna is fabricated using a commercially available flexible coaxial cable. We confirmed that the fabricated prototype antenna can be inserted into the forceps channel without any problems [see Fig. 2(b)]. The antenna is bent at the outlet of the channel by a barb whose bend angle is controllable by the operator. The heating pattern around the antenna is controllable by changing the parameters of the slots. In [11], the antenna with two slots was employed for generating a localized heating region only around the tip of the antenna. However, in this treatment, a relatively long heating region in the antenna axial direction is required. Therefore, the antenna with a single slot, which generates a relatively long heating region, is employed.

Fig. 3. Heating patterns around the tip of the antenna (experimental results).

III. PROCEDURE OF NUMERICAL CALCULATIONS AND CALCULATION MODEL A. Procedure of Calculations In the numerical calculation, we first analyze the electric field around the antenna by the finite-difference time-domain (FDTD) method and calculate the specific absorption rate (SAR) from the following equation: W/kg

(1)

where is the conductivity of the tissue (S/m), is the density of the tissue (kg/m ), and is the electric field (rms) (V/m). The SAR takes a value proportional to the square of the electric field generated around the antenna and is equivalent to the heating source created by the electric field in the tissue. The SAR distribution is one of the most important characteristics of antennas for heating. Table I shows the parameters for FDTD calculations.

Fig. 4. Calculation model.

Next, we calculate the temperature distribution around the antenna. In order to obtain the temperature distribution in the tissue, we numerically analyze the bioheat transfer equation [13] including the obtained SAR values using the finite-difference method (FDM). The bioheat transfer equation is given by (2) where is the temperature ( C), is the time (s), is the density of the tissue (kg/m ), is the specific heat of the tissue (J/kg K), is the thermal conductivity of the tissue (W/m K), is the

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TABLE II PHYSICAL PROPERTIES OF THE BIOLOGICAL TISSUES , [16], [17]

Fig. 5. Organs in the observation planes. (a) Observation plane #1. (b) Observation plane #2.

density of the blood (kg/m ), is the specific heat of the blood (J/kg K), is the temperature of the blood ( C), and is the blood flow rate (m /kg s). The finite-difference approximation and details for calculation are the same as [14]. In addition, in order to calculate the SAR and the temperature distributions during the microwave heating exactly, the temperature-dependent physical properties of the tissues should be considered. However, in our previous study [14], we could observe an almost good agreement between the calculated and experimental temperature distributions without any dependency. Therefore, we consider that the calculations without the dependency give us results that are satisfactory enough for finding the possibilities of the treatment.

Fig. 6. Calculated SAR distributions. (a) Observation plane #1. (b) Observation plane #2.

B. Calculation Model As shown in Fig. 2, the coaxial-slot antenna for the treatment of bile duct carcinoma has a thin and long structure. Therefore, it is difficult to analyze the whole structure of the antenna. In this paper, numerical calculations will be performed only around the tip of the antenna. Fig. 3 shows the measured SAR distributions by use of the tissue equivalent solid phantom and the fabricated antenna (thermographic method [15]). In the measurements, the fabricated antenna with a bend was employed because it will be strongly bent at the outlet of the forceps channel shown in Fig. 2(b) during the actual treatment. In addition, the heating patterns around the antenna are observed by the infrared camera. In the

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Fig. 7. Calculated temperature transitions. (a) Temperature transitions by feeding parameter (A). (b) Temperature transitions by feeding parameter (B). (c) Temperature transitions by feeding parameter (C). (d) Observation points.

experiments, irradiation times of microwave were set as 10 s to neglect the heat conduction inside the phantom for SAR measurement. The radiation power from the antennas can be estiand 11.8 W by conmated as 11.6 W sidering the transmission loss of the cable and the reflection power from the antenna. In addition, large differences are not observed depending on the bent angle. Details of these losses are explained in Section IV-B. From Fig. 3, we can observe a localized heating pattern only around the tip of the antenna without any undesirable hot spots at the bent portion. Therefore, we can construct a calculation model, which only includes the tip of the antenna. Fig. 4 shows the calculation model based on the realistic human model developed at Brooks Air Force Laboratories. In order to construct this model, a region including the bile duct is extracted. The position of the calculation model is illustrated in Fig. 1. In Fig. 4, only the bile duct is indicated inside the calculated region though there are a few other organs such as the liver, stomach, duodenum, small intestine, etc. The physical properties of the biological tissues are listed in Table II. In addition, in Fig. 4, two observation planes are shown for consideration in Section IV. Here, observation mm (including the slot) and plane #1 is the -plane at observation plane #2 is the -plane at . Fig. 5 shows the organs that are included in two observation planes. Note that the walls of the large intestine, small intestine, bile duct, etc. are expressed as small isolated squares in the longitudinal plane in Fig. 5(b). In addition, although all the tissues listed in

Table II are included in the calculation model, we cannot find all of them in Fig. 5. The FDTD calculation model of the coaxial-slot antenna is constructed by staircasing approximation and is placed inside the bile duct. We employed nonuniform grids and used smallsize grids only for the antenna. In addition, the same grids as those of the electromagnetic analysis are employed for thermal calculations. The initial temperature of the tissue is 37 C. In addition, the temperature of the thick blood vessels and the abdominal cavities were set to 37 C to consider the cooling effect, and the validity of this assumption has been confirmed. IV. HEATING PERFORMANCES OF THE ANTENNA A. Calculated SAR Distribution Fig. 6 shows the calculated SAR distributions around the antenna. The observation planes of the distributions are defined in Fig. 4. Here, the values of the SAR are normalized by 1.0-W net input (incident-reflection) power to the antenna. From Fig. 6(b), we can observe the high SAR value around the slot ( mm in this plane). In addition, complicated SAR distributions are observed around the antenna because of the complex electrical constants distribution. The high SAR region is especially observed not only at the region close to the antenna ( in observation plane #1), but also in the border of the small intestine ( mm, in observation plane #1 and mm, – mm in observation plane #2).

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B. Calculated Temperature Distribution In hyperthermic treatment, it is important to heat the tumor more than the therapeutic temperature. Therefore, the temperature measurement in and around the tumor is needed for a reliable treatment. In general, some thin thermosensors such as fiber-optic thermosensors are employed for the measurement [11]. However, in this case, it is difficult to place any sensors noninvasively, especially inside the tumor. In addition, it is clear that the region close to the antenna is heated up at a very high temperature. This high temperature will cause lesions of the bile duct. In order to avoid the high temperature, if we choose low input power of the antenna, the size of the therapeutic heating region will not be satisfied. In order to solve these problems, an on–off feeding control is employed. We defined the feeding parameters as follows: • output power of the generator: 15.0 W; • allowed maximum temperature: 60 C. The antenna cable has large transmission loss because of its thin, long, and flexible structure. In addition, although it is not so large, miss matching loss of the antenna also exists. In order to estimate the situation of practical treatment, we included the influence of these losses by adjusting the input power of the antenna in the calculations. According to our preliminary investigations, the transmission loss of the cable and the mismatching loss due to the antenna structure were 65.3% and 11.0% to the total input power, respectively. Therefore, in this case, the radiation power from the antenna to the tissue is approximately 4.7 W. Thus, in the calculation, we employed a lossless antenna model and set 4.7-W input power as the 15.0-W generator output. In addition, a temperature exceeding 60 C causes a tissue coagulation. Thus, we kept the maximum temperature below 60 C. Since this temperature is higher than the therapeutic temperature of the hyperthermia, a lesion may be caused around the surface of the antenna. However, it is considered that if this high-temperature region is generated in the targeted tumor, it contributes to a sure treatment. In order to heat the tumor effectively without any lesion, the following three types of intervals for the on–off feeding control are employed to find the effective feeding parameters: A) on-feeding 2 s, off-feeding 2 s; B) on-feeding 5 s, off-feeding 5 s; C) on-feeding 10 s, off-feeding 10 s. Fig. 7(a)–(c) shows the calculated temperature transitions. Here, the temperature observation points are indicated in Fig. 7(d). In addition, the calculated temperature transition without on–off feeding control is also shown for comparison. From Fig. 7(a) and (b), we can observe that the maximum temperature at observation point #1 is less than 60 C with on-off feeding control, although the temperature of this point exceeds 70 C without control. Moreover, the temperature at observation point #2, which is placed 5.0 mm from the antenna axis, exceeds the lowest therapeutic temperature (42 C) in both cases. On the other hand, in Fig. 7(c), the maximum temperature of observation point #1 exceeds 60 C by feeding parameter (C). Fig. 8(a) and (b) shows the calculated temperature distributions of feeding parameters (A) and (C) in the observation planes

Fig. 8. Calculated temperature distributions. (a) Feeding parameter (A). (b) Feeding parameter (C).

defined in Fig. 4 [we confirmed that distributions by the feeding parameter (B) are the almost same as the parameter (A)]. In this figure, the white dotted lines indicate 42 C, which is the lowest temperature for the treatment. These temperature distributions, for which the minimum size of the heating region by the on–off feeding control is chosen, are the results at the steady state. Although the SAR distributions are not uniform, especially around the border of the small intestine, from Fig. 8, we can find an almost uniform heating region because of the heat transfer. Moreover, we confirmed that the temperature distribution in the -plane (observation plane #2) and -plane were almost the same. Therefore, we may say that the temperature distributions are similar to the results obtained inside a homogeneous medium. Here, the diameter of the effective heating region (the region higher than 42 C) is approximately 15 mm in the -plane and the length in the axial direction (in the -plane) is approximately 30 mm (heating pattern in the axial direction of the antenna can be controlled by shifting the antenna) in both parameters (A) and (C). By comparison of Fig. 8(a) and (b), we can observe almost the same size of effective heating regions in both parameters (A) and (C). However, in Fig. 7(c), the maximum temperature of observation plane #1 by parameter (C) exceeds 60 C. Therefore, it is considered that the quick switching [parameter (A)] suppresses the high temperature region well around the slot while keeping the relatively large heating region.

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V. CONCLUSION In this paper, the possibility of treatment for bile duct carcinoma by the intracavitary microwave hyperthermia using the coaxial-slot antenna has been described. First, the scheme of the numerical calculation and the construction of the calculation model were introduced. The calculated SAR and temperature distributions were then explained. From the results of these investigations, we could find the possibility of the treatment. As a further study, we will fabricate the antenna for practical use. Moreover, some animal experiments are needed before clinical trials.

[14] K. Saito, Y. Hayashi, H. Yoshimura, and K. Ito, “Heating characteristics of array applicator composed of two coaxial-slot antennas for microwave coagulation therapy,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 11, pp. 1800–1806, Nov. 2000. [15] Y. Okano, K. Ito, I. Ida, and M. Takahashi, “The SAR evaluation method by a combination of thermographic experiments and biological tissue-equivalent phantoms,” IEEE Trans. Microw. Theory Tech., vol. 48, no. 11, pp. 2094–2103, Nov. 2000. [16] F. A. Duck, Physical Properties of Tissue. New York: Academic, 1990. [17] P. M. Van Den Berg, A. T. De Hoop, A. Segal, and N. Praagman, “A computational model of the electromagnetic heating of biological tissue with application to hyperthermic cancer therapy,” IEEE Trans. Biomed. Eng., vol. BME-30, no. 12, pp. 797–805, Dec. 1983.

ACKNOWLEDGMENT The authors would like to thank Dr. T. Tsuyuguchi, Chiba University Hospital, Chiba, Japan, for his valuable comments from the clinical side. REFERENCES [1] F. Sterzer, “Microwave medical devices,” IEEE Micro, vol. 3, no. 1, pp. 65–70, Mar. 2002. [2] M. H. Seegenschmiedt, P. Fessenden, and C. C. Vernon, Eds., Thermoradiotherapy and Thermochemotherapy. Berlin, Germany: Springer-Verlag, 1995. [3] M. Converse, E. J. Bond, S. C. Hagness, and B. D. Van Veen, “Ultrawide-band microwave space-time beamforming for hyperthermia treatment of breast cancer: A computational feasibility study,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 8, pp. 1876–1889, Aug. 2004. [4] T. Seki, M. Wakabayashi, T. Nakagawa, T. Itoh, T. Shiro, K. Kunieda, M. Sato, S. Uchiyama, and K. Inoue, “Ultrasonically guided percutaneous microwave coagulation therapy for small carcinoma,” Cancer, vol. 74, no. 3, pp. 817–825, Aug. 1994. [5] P. Liang, B. Dong, X. Yu, D. Yu, Z. Cheng, L. Su, J. Peng, Q. Nan, and H. Wang, “Computer-aided dynamic simulation of microwave-induced thermal distribution in coagulation of liver cancer,” IEEE Trans. Biomed. Eng., vol. 48, no. 7, pp. 821–829, Jul. 2001. [6] R. D. Nevels, G. D. Arndt, G. W. Raffoul, J. R. Carl, and A. Pacifico, “Microwave catheter design,” IEEE Trans. Biomed. Eng., vol. 45, no. 7, pp. 885–890, Jul. 1998. [7] P. Bernardi, M. Cavagnaro, J. C. Lin, S. Pisa, and E. Piuzzi, “Distribution of SAR and temperature elevation induced in a phantom by a microwave cardiac ablation catheter,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 8, pp. 1978–1986, Aug. 2004. [8] D. Despretz, J.-C. Camart, C. Michel, J.-J. Fabre, B. Prevost, J.-P. Sozanski, and M. Chivé, “Microwave prostatic hyperthermia: Interest of urethral and rectal applicators combination—Theoretical study and animal experimental results,” IEEE Trans. Microw. Theory Tech., vol. 44, no. 10, pp. 1762–1768, Oct. 1996. [9] A. Dietsch, J.-C. Camart, J. P. Sozanski, B. Prevost, B. Mauroy, and M. Chivé, “Microwave thermochemotherapy in the treatment of the bladder carcinoma—Electromagnetic and dielectric studies—Clinical protocol,” IEEE Trans. Biomed. Eng., vol. 47, no. 5, pp. 633–641, May 2000. [10] K. Ito, K. Ueno, M. Hyodo, and H. Kasai, “Interstitial applicator composed of coaxial ring slots for microwave hyperthermia,” in Proc. Int. IEEE AP-S Symp., Tokyo, Japan, Aug. 1989, pp. 253–256. [11] K. Saito, H. Yoshimura, K. Ito, Y. Aoyagi, and H. Horita, “Clinical trials of interstitial microwave hyperthermia by use of coaxial-slot antenna with two slots,” IEEE Trans. Microw. Theory Tech., vol. 52, no. 8, pp. 1987–1991, Aug. 2004. [12] K. Saito, K. Ito, M. Takahashi, and A. Hiroe, “Numerical calculations of heating pattern around a coaxial-slot antenna with endoscope aiming at intracavitary microwave hyperthermia for treatment of bile duct carcinoma,” in Proc. Int. Union of Radio Sci. Gen. Assembly, New Delhi, India, Oct. 2005, [CD ROM]. [13] H. H. Pennes, “Analysis of tissue and arterial blood temperatures in the resting human forearm,” J. Appl. Physiol., vol. 1, pp. 93–122, Aug. 1948.

Kazuyuki Saito (S’99–M’01) was born in Nagano, Japan, in May 1973. He received the B.E., M.E., and D.E. degrees in electronic engineering from Chiba University, Chiba, Japan, in 1996, 1998 and 2001, respectively. He is currently a Research Associate with the Research Center for Frontier Medical Engineering, Chiba University. His main interest is in the area of medical applications of the microwaves including microwave hyperthermia. Dr. Saito is a member of the Institute of Electrical, Information and Communication Engineers (IEICE), Japan, the Institute of Image Information and Television Engineers of Japan (ITE), and the Japanese Society of Hyperthermic Oncology. He was the recipient of the IEICE Antennas and Propagation Society (AP-S) Freshman Award, the Award for Young Scientist of URSI General Assembly, the IEEE AP-S Japan Chapter Young Engineer Award, the IEICE Young Researchers’ Award, and the International Symposium on Antennas and Propagation (ISAP) Paper Award in 1997, 1999, 2000, 2004, and 2005, respectively.

Atsushi Hiroe was born in Hyogo, Japan, in March 1983. He received the B.E. degree in electrical engineering from Chiba University, Chiba, Japan, in 2005, and is currently working toward the M.E. degree at Chiba University. His main interests include analysis and design of antennas for hyperthermia and the research on evaluation of the interaction between electromagnetic (EM) fields and the human body by use of numerical and experimental phantoms. Mr. Hiroe is a member of the Institute of Electrical, Information and Communication Engineers (IEICE), Japan, and the Japanese Society of Hyperthermic Oncology.

Satoru Kikuchi was born in Aomori, Japan, in March 1983. He received the B.E. degree in electrical engineering from Chiba University, Chiba, Japan, in 2005, and is currently working toward the M.E. degree at Chiba University. His main interests include numerical calculation of thin antennas for hyperthermia using the high-resolution numerical human model. Mr. Kikuchi is a member of the Institute of Electrical, Information and Communication Engineers (IEICE), Japan, and the Japanese Society of Hyperthermic Oncology.

SAITO et al.: ESTIMATION OF HEATING PERFORMANCES OF COAXIAL-SLOT ANTENNA WITH ENDOSCOPE

Masaharu Takahashi (M’95–SM’02) was born in Chiba, Japan, on December, 1965. He received the B.E. degree in electrical engineering from Tohoku University, Miyagi, Japan, in 1989, and the M.E. and D.E. degrees in electrical engineering from the Tokyo Institute of Technology, Tokyo, Japan, in 1991 and 1994, respectively. From 1994 to 1996, he was a Research Associate, and from 1996 to 2000, he was an Assistant Professor with the Musashi Institute of Technology, Tokyo, Japan. From 2000 to 2004, he was an Associate Professor with the Tokyo University of Agriculture and Technology, Tokyo, Japan. He is currently an Associate Professor with the Research Center for Frontier Medical Engineering, Chiba University, Chiba, Japan. His main interests are electrically small antennas, planar array antennas, and electromagnetic compatibility. Dr. Takahashi is a member of the Institute of Electrical, Information and Communication Engineers (IEICE), Japan. He was the recipient of the 1994 IEEE Antennas and Propagation Society (IEEE AP-S) Tokyo Chapter Young Engineer Award.

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Koichi Ito (M’81–SM’02–F’05) was born in Nagoya, Japan, in June 1950. He received the B.S. and M.S. degrees from Chiba University, Chiba, Japan, in 1974 and 1976, respectively, and the D.E. degree from the Tokyo Institute of Technology, Tokyo, Japan, in 1985, all in electrical engineering. From 1976 to 1979, he was a Research Associate with the Tokyo Institute of Technology. From 1979 to 1989, he was a Research Associate with Chiba University. From 1989 to 1997, he was an Associate Professor with the Department of Electrical and Electronics Engineering, Chiba University, and is currently a Professor with the Faculty of Engineering. Since April 2005, he has been one of the Deputy Vice-Presidents for Research, as well as Director, Office of Research Administration, Chiba University. In 1989, 1994, and 1998, he was with the University of Rennes I, Rennes, France, as an Invited Professor. Since 2004, he has been an Adjunct Professor with the Institute of Technology Bandung (ITB), Bandung, Indonesia. His main research interests include analysis and design of printed antennas and small antennas for mobile communications, research on evaluation of the interaction between electromagnetic fields and the human body by use of numerical and experimental phantoms, and microwave antennas for medical applications such as cancer treatment. Dr. Ito is a member of the American Association for the Advancement of Science (AAAS), the Institute of Electrical, Information and Communication Engineers (IEICE), Japan, the Institute of Image Information and Television Engineers of Japan (ITE), and the Japanese Society of Hyperthermic Oncology. He served as chair of Technical Group on Radio and Optical Transmissions of the ITE from 1997 to 2001 and chair of the Technical Group on Human Phantoms for Electromagnetics of the IEICE from 1998 to 2006. He also served as chair of the IEEE Antennas and Propagation Society (AP-S) Japan Chapter from 2001 to 2002 and Technical Program Committee (TPC) co-chair of the International Workshop on Antenna Technology (IWAT) 2006, New York, NY. He is currently vice-chair of the 2007 International Symposium on Antennas and Propagation (ISAP2007). He is an associate editor for the IEEE TRANSACTIONS ON ANTENNAS AND PROPAGATION.

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Digital Object Identifier 10.1109/TMTT.2006.882243

Digital Object Identifier 10.1109/TMTT.2006.882245

EDITORIAL BOARD Editors: D. WILLIAMS and A. Mortazawi Associate Editors: A. CANGELLARIS, A. CIDRONALI, M. DO, K. ITOH, J. LIN, D. LINTON, S. MARSH, Y. NIKAWA, J. PEDRO, Z. POPOVIC, S. RAMAN, V. RIZZOLI, R. SNYDER, R. WU, T. WYSOCKI, A. YAKOVLEV REVIEWERS A. Abbaspour-Tamijani D. Abbott M. Abdulla M. Abe M. Abedin M. Abouzahra M. Abramowicz R. Achar E. Ackerman D. Adam E. Adle M. Adlerstein M. Afsar K. Agarwal K. Agawa K. Ahmed D. Ahn H.-R. Ahn M. Aikawa M. Akaike Y. Akaiwa E. Akmansoy S. Aksoy A. Akyurtlu F. Alessandri C. Algani F. Ali M. Ali W. Ali-Ahmad F. Alimenti C. Alippi B. Alpert A. Alphones S. Al-Sarawi A. Altintas A. Alvarez-Melcom S. Amari C. Anastasiou U. Andersson Y. Ando P. Andreani K.-S. Ang I. Angelov S. Anlage O. Anwar I. Aoki R. Aparicio V. Aparin F. Arndt U. Arz M. Asai P. Asbeck H. Ashok H. Ashoka A. Atalar A. Atia N. Audeh S. Auster P. Auxemery I. Awai A. Aydiner K. Aygun R. Azadegan A. Babakhani I. Bahl D. Baillargeat S. Bajpai W. Bakalski J. Baker-Jarvis B. Bakkaloglu K. Balmain Q. Balzano S. Banba J. Bandler R. Bansal F. Bardati I. Bardi S. Barker D. Barlage J. Barr J. Bartolic D. Bates G. Baudoin Q. Balzano B. Beker G. Belenky D. Belot C. Bell P. Bell J. Benedikt J. Bernhard G. Bertin H. Bertoni E. Bertran W. Beyene A. Beyer M. Bialkowski E. Biebl P. Bienstman S. Bila M. Bilakowski A.-L. Billabert F. Bilotti H. Bilzer O. Biro R. Bisiso B. Bisla D. Blackham M. Blank P. Blondy D. Boccoli F. Bohn B. Boeck L. Boglione R. Boix J. Booske N. Borges de Carvalho V. Boria O. Boric-Lubecke A. Borji J. Bornemann W. Bosch R. Bosisio S. Boumaiza M. Bozzi E. Bracken R. Bradley V. Bratman T. Brazil G. Brehm K. Breuer B. Bridges J. Brinkoff S. Broschat S. Brozovich D. Budimir D. Buechler M. Buff C. Buntschuh J. Bunton J. Burghartz P. Burghignoli Y. Bykov A. Caballero B. Cabon J. Calame

M. Calcatera C. Caloz C. Camacho-Penalosa E. Camargo R. Cameron S. Cammer C. Campbell M. Campovecchio F. Canavero J. Cao J. Capmany F. Capolino G. Carchon R. Carter N. Carvalho F. Casas J. Catala R. Caverly J. Cavers Z. Cendes B. Cetiner R. Chair H. Chaloupka A. Chambarel B. Chambers C.-H. Chan Y.-J. Chan C.-Y. Chang F. Chang G. Chang H.-C. Chang H.-R. Chang K. Chang E. Channabasappa H. Chapell W. Chappell M. Chatras S. Chaudhuri S. Chebolu C.-C. Chen C.-H. Chen H.-H. Chen J. Chen R. Chen W.-K. Chen Y.-J. Chen K.-K. Cheng Y.-C. Cheng W.-C. Chew C.-Y. Chi Y.-C. Chiang C.-F. Chiasserini I.-T. Chiang J. C. Chiao I. Chiba D. Chigrin A. Chin C.-C. Chiu Y. Cho C. Choi J. Choi M.-J. Choi C.-K. Chou Y.-H. Chou D. Choudhury K. Choumei Y. Chow C. Christodoulou C. Christopoulos H.-R. Chuang Y. Chung B. Chye R. Cicchetti C. Cismaru D. Citrin P. Civalleri A. Ciubotaru T. Clark R. Clarke J. Cloete E. Cohen F. Colomb B. Colpitts M. Condon D. Consonni J. Corral A. Constanzo I. Corbella E. Costamagna A. Coustou J. Craninckx J. Crescenzi S. Cripps D. Cros T. Crowe M. Cryan J. Culver C. Curry W. Curtice M. da Cunha W.-L. Dai T. Dahm G. Dambrine B. Danly F. Danneville N. Das M. Davidovich A. Davis C. Davis L. Davis H. Dayal F. De Flaviis H. De Los Santos A. De Lustrac P. De Maagt J. de Mingo R. De Roo L. de Vreede D. De Zutter B. Deal A. Dearn P. Debicki J. Deen A. Deleniv M. DeLisio S. Demir A. Deutsch V. Devabhaktuni Y. Deval A. Diet L. Ding A. Djermoun T. Djordjevic J. Dobrowolski D. Dolfi W. Dou M. Douglas P. Draxler A. Dreher F. Drewniak J. Drewniak D. Dubuc S. Dudorov L. Dunleavy V. Dunn A. Duzdar

S. Dvorak L. Dworsky M. Dydyk M. Edwards R. Ehlers H. Eisele G. Eisenstein G. Eleftheriades M. Elliott T. Ellis A. Elsherbeni R. Emrick N. Engheta A. Enokihara Y. Eo H. Eom C. Ernst M. Esashi L. Escotte I. Eshrah V. Esposti M. Essaaidi K. Esselle H. Estaban J. Esteban C. Fager J. Fan D.-G. Fang M. Farina W. Fathelbab A. Fathy J. Favennec A. Fazal E. Fear M. Feldman A. Fernandez A. Ferrero T. Fickenscher J. Fiedziuszko D. Filipovic A. Fliflet B. Floyd P. Focardi N. Fong K. Foster P. Foster B. Frank C. Free J. Freire M. Freire R. Freund F. Frezza I. Frigyes C. Froehly J. Fu R. Fujimoto T. Fujioka O. Fujiwara H. Fukuyama V. Fusco D. Gabbay N. Gagnon J. Gallego B. Galwas O. Gandhi B.-Q. Gao J. Gao J. Garcia R. Garver A. Gasiewski B. Geelen B. Geller V. Gelnovatch W. Geppert F. Gerecht J. Gering M. Gerken S. Gevorgian R. Geyer O. Ghandi F. Ghannouchi K. Gharaibeh G. Ghione D. Ghodgaonkar F. Giannini J. Gilb A. Glisson M. Goano E. Godshalk M. Goldfarb P. Goldsmith M. Golio N. Gomez X. Gong R. Gonzalo S. Gopalsami A. Gopinath R. Gordon A. Gorur K. Goverdhanam W. Grabherr L. Gragnani J. Grahn G. Grau A. Grebennikov T. Gregorzyk I. Gresham A. Griol D. R. Grischowsky C. Grossman E. Grossman T. Grzegorczyk A. Gupta K. Gupta M. Gupta R. Gutmann W. Gwarek J. Hacker M. Hafizi S. Hadjiloucas S. Hagness D. Haigh P. Hale D. Ham K. Hamaguchi S. Hamedi-Hagh J. Hand K. Hashimoto Q. Han T. Hancock A. Hanke V. Hanna Z. Hao S. Hara L. Harle A. Harish P. Harrison H. Hartnagel J. Haslett G. Hau R. Haupt S. Hay H. Hayashi J. Hayashi L. Hayden J. Heaton

P. Hedekvist W. Heinrich G. Heiter M. Helier R. Henderson F. Henkel J. Herren P. Herczfeld F. Herzel J. Hessler A. Hiatala C. Hicks M. Hieda A. Higgins M. Hikita W. Hioe Y. Hirachi T. Hiraota A. Hirata T. Hiratsuka Y.-C. Ho W. Hoefer K. Hoffmann R. Hoffmann J. Hong J.-S. Hong K. Horiguchi Y. Horii J. Horng J. Horton K. Hosoya R. Howald H. Howe H.-M. Hsu H.-T. Hsu J.-P. Hsu C.-W. Hsue C.-C. Huang C. Huang F. Huang H. Huang H.-C. Huang J. Huang T.-W. Huang P. Huggard H.-T. Hui D. Humphreys A. Hung C.-M. Hung H. Hung J.-J. Hung I. Hunter H.-Y. Hwang T. Idehara S. Iezekiel J.-Y. Ihm Y. Iida H. Iizuka P. Ikalainen Y. Ikeda P. Ikonen K. Ikossi M. Ilic J. Inatani K. Iniewski H. Inokawa A. Inoue M. Ishida A. Ishimaru T. Ishizaki S. Islam Y. Ismail Y. Isota M. Ito T. Itoh Y. Itoh T. Ivanov C. Iversen D. Iverson M. Iwamoto Y. Iyama H. Izumi D. Jachowski C. Jackson D. Jackson R. Jackson M. Jacob S. Jacobsen D. Jaeger B. Jagannathan N. Jain R. Jakoby G. James V. Jandhyala M. Janezic H. Jantunen B. Jarry P. Jarry A. Jastrzbeski E. Jeckein W. Jemison Y. Jeon J. Jeong Y.-H. Jeong G. Jerinic A. Jerng T. Jerse D. Jiao J.-M. Jin J. Joe L. Johansson T. Johnson A. Joseph K. Joshin J. Joubert P. Juodawlkis P. Kabos S.-T. Kahng T. Kaho D. Kajfez T. Kamel Y. Kamimura H. Kamitsuna K. Kamogawa S. Kanamaluru H. Kanaya M. Kanda P. Kangaslahtii V. Kaper M. Kärkkäinen A. Karpov U. Karthaus A. Karwowski T. Kashiwa R. Kaul K. Kawakami A. Kawalec T. Kawanishi S. Kawasaki H. Kayano M. Kazimierczuk R. Keam L. Kempel P. Kenington K. Kenneth S. Kenny

Digital Object Identifier 10.1109/TMTT.2006.882242

A. Kerr A. Khalil A. Khanifar J. Kiang Y.-W. Kiang P.-S. Kildal O. Kilic B. Kim H. Kim I. Kim J.-P. Kim M. Kim W. Kim B. Kimm K. Kimura S. Kimura A. Kirilenko V. Kisel S. Kishimoto A. Kishk T. Kitamura K. Kitayama T. Kitazawa W. Klaus E. Klumprink R. Knerr R. Knöchel L. Knockaert K. Kobayashi Y. Kogami B. Kolner S. Komaki M. Komaru J. Komiak A. Komijani G. Kompa A. Konczykowska Y. Konishi A. Koonen B. Kopp K. Kornegay M. Koshiba T. Kosmanis J. Kot Y. Kotsuka S. Koul V. Kourkoulos A. B. Kozyrev A. Krenitskiy N. Kriplani K. Krishnamurthy V. Krishnamurthy A. Kroenig C. Kromer C. Krowne V. Krozer W. Kruppa R. Kshetrimayum H. Ku H. Kubo E. Kuester Y. Kuga W. Kuhn T. Kuki M. Kumar M. Kunert J. Kuno M. Kunst C.-N. Kuo J.-T. Kuo H. Kurebayashi T. Kuri F. Kuroki S. Kusunoki D. Kuylenstierna M. Kuzuhara I. Kwon Y.-W. Kwon R. Lai Y.-L. Lai P. Lampariello M. Lanagan M. Lancaster P. Lane U. Langmann Z. Lao G. Lapin L. Larson J. Laskar A. Lauer G. Lazzi Y. Le Coz Y. Le Guennec S. Le Maguer B. Lee C. Lee J.-F. Lee J.-W. Lee K. Lee R. Lee S.-G. Lee T. Lee Y.-C. Leong R. Leoni K.-W. Leung P. Leuchtmann G. Leuzzi A. Leven A. Levi R. Levy A. Lewandowski M. Lewis K. Li L.-W. Li X. Li Y. Li Y.-M. Li M. Liberti L. Ligthart S. Lim E. Limiti C. Lin J. Lin Y.-D. Lin Y.-S. Lin L. Lind S. Lindenmeier F. Ling A. Lipparini D. Lippens V. Litvinov C.-P. Liu Q.-H. Liu S.-I. Liu W. Liu O. Llopis D. Lo A. Loayssa R. Loison J. Long K. Lorincz U. Lott J.-H. Loui H.-C. Lu L.-H. Lu S. Lu

W.-T. Lu V. Lubecke G. Lucca S. Lucyszyn R. Luebbers L. Lunardi J. Luy S. Lyshevski J.-G. Ma Z. Ma S. Maas P. Maccarini G. Macchiarella P. Macchiarella J. Machac S. Maci J. Maciel M. Madihian B. Madhavan V. Madrangeas M. Magana S. Mahmoud S. Mahon I. Maio A. Majedi M. Majewski M. Makimoto J. Malherbe D. Malocha T. Manabe G. Manganaro T. Maniwa C. Mann H. Manohara R. Mansour D. Manstretta J. Mao S.-G. Mao S. Marchetti R. Marques J. Martens J. Marti F. Martin E. Martinez K. Maruhashi D. Masotti A. Massa S. Masuda A. Materka B. Matinpour M. Matsuo A. Matsushima A. Matsuzawa S. Matsuzawa G. Matthaei D. Matthews J.-P. Mattia J. Maurer J. Mayock J. Mazierska S. Mazumder G. Mazzarella K. McCarthy T. McKay J. McKinney R. McMillan D. McQuiddy P. Meany F. Medina S. Melle F. Mena C. Meng H.-K. Meng W. Menzel F. Mesa A. Metzger P. Meyer C. Mias K. Michalski G. Michel E. Michielssen A. Mickelson R. Miles D. Miller R. Minasian B. Minnis D. Mirshekar J. Mitchell O. Mitomi R. Mittra M. Miyakawa R. Miyamoto M. Miyazaki K. Mizuno S. Mizushina M. Mohamed S. Mohammadi A. Mohammadian M. Mongiardo J. Morente M. Morgan K. Mori A. Morini N. Morita E. Moros A. Morris J. Morsey H. Mosallaei M. Mrozowski J.-E. Mueller M. Muraguchi K. Murata H. Muthali T. Nagatsuma P. Nagel K. Naishadham T. Nakagawa M. Nakajima N. Nakajima J. Nakayama M. Nakayama M. Nakhla J. Nallatamby S. Nam S. Narahashi A. Natarajan J. Nath B. Nauwelaers J. Navarro I. Nefedovlgor H.-C. Neitzert B. Nelson S. Nelson A. Neri H. Newman D. Ngo E. Ngoya C. Nguyen K. Niclas E. Niehenke P. Nikitin A. Niknejad N. Nikolova T. Nirmalathas K. Nishikawa T. Nishikawa

K. Nishimura T. Nishino K. Nishizawa G. Niu W. Ng S. Nogi K. Noguchi T. Nojima A. Nosich B. Notaros K. Noujeim D. Novak T. Nozokido T. Nurgaliev D. Oates J. Obregon J. O’Callahan M. O’Droma M. Odyneic I. Ogawa M. Ogusu K. Oh M. Ohawa T. Ohira I. Ohta M. Ohtsuka S. Oikawa K. Okada Y. Okano H. Okazaki V. Okhmatovski A. Oki M. Okoniewski A. Oliner J. Olsson F. Olyslager A. Omar M. Omiya K. Onodera B.-L. Ooi I. Oppermann R. Orta S. Ortiz J. Ou T. Owada M. Ozkar J. Page de la Pega W. Palmer G.-W. Pan A. Paolella C. Papanicolopoulos J. Papapolymerou B.-K. Park C.-S. Park W. Park A. Parker D. Parker T. Parker J. Pearce B. Pejcinovic S.-T. Peng R. Pengelly R. Penty J. Pereda B. Perlman L. Perregrini M. Petelin R. Petersen W. Petersen A. Peterson A. Petosa A.-V. Pham J. Phillips H. Pickett M. Pieraccini L. Pierce B. Piernas J. Pierro P. Pieters M. Piket-May L. Pileggi Z.-Y. Ping M. Pirola A. Platzker C. Plett C. Pobanz R. Pogorzelski R. Pokharel R. Pollard G. Ponchak M. Popovic J. Portilla M. Pospieszalski V. Postoyalko A. Pothier S. Prasad D. Prather D. Prescott A. Priou D. Purdy Y. Qian T. Quach C. Quendo R. Quere F. Raab V. Radisic K. Radhakrishnan T. Rahkonen C. Railton A. Raisanen K. Rajab O. Ramahi J. Randa R. Ranson T. Rappaport J. Rathmell C. Rauscher J. Rautio B. Rawat J. Rayas-Sanchez R. Reano G. Rebeiz J. Rebollar B. Redman-White M. Reddy R. Reid H.-M. Rein J. Reinert R. Remis K. Remley C. Rey L. Reynolds A. Rezazadeh E. Rezek A. Riddle B. Riddle J.-S. Rieh E. Rius I. Robertson R. Robertson A. Rodriguez R. Rogers H. Rogier U. Rohde N. Rolland R. Romanofsky

A. Rong Y. Rong D. Root L. Roselli A. Rosen U. Rosenberg L. Roy M. Royer J. Roychowdury T. Rozzi B. Rubin M. Rudolph P. Russer D. Rutledge T. Ruttan A. Rydberg T. Rylander D. Rytting C. Saavedra A. Safavi-Naeini A. Safwat M. Sagawa B. Sahu A. Saitou I. Sakagami K. Sakaguchi K. Sakakibara K. Sakamoto K. Sakoda M. Salazar-Palma C. Samori L. Samoska A. Sanada Y. Sanada M. Sanagi P. Sandhiva U. Sangawa A. Sangster K. Sano K. Sarabandi T. Sarkar C. Sarris H. Sato M. Sato S. Sato H. Sawada H. Sawaya A. Sawicki A. Sayed I. Scherbatko J. Schellenberg G. Schettini F. Schettino B. Schiek M. Schindler E. Schlecht E. Schmidhammer D. Schmitt J. Schneider J. Schoukens A. Schuchinsky R. Schuhmann J. Schultz J. Schutt-Aine A. Seeds Y. Segawa T. Seki S. Selberherr G. Semouchkin E. Semouchkina Y.-K. Seng R. Settaluri J. Sevic O. Sevimli Y. Segawa Z. Shao M. Shapiro A. Sharma S. Sharma T. Shen Z.-X. Shen Y. Shestopalov H. Shigesawa Y.-C. Shih H. Shimasaki S. Shinjo N. Shino N. Shinohara T. Shimozuma W. Shiroma K. Shogen N. Shuley M. Shur D. Sievenpiper A. Sihvola C. Silva M. Silveira M. Silveirinha M. Silveirinhao K. Silvonen G. Simin R. Simons B. Sinha F. Sinnesbichler J. Sinsky J. Sitch H.-J. Siweris R. Sloan A. Smith D. Smith G. Smith P. Smith R. Snyder H. Sobol A. Sochava M. Solano K. Solbach M. Solomon M. Sorolla Ayza R. Sorrentino C. Soukoulis N. Soveiko E. Sovero J. Sowers M. Soyuer R. Sparks P. Staecker D. Staiculescu S. Stapleton J. Staudinger P. Stauffer P. Steenson K. Stephan M. Steyaert S. Stitzer A. Stoehr B. Strassner M. Stubbs M. Stuchly A. Suarez G. Subramanyam R. Sudbury N. Suematsu M. Sugiyama D. Sullivan L. Sundstrom

Y. Suzuki J. Svacina D. Swanson D. Sweeney R. Syms B. Szendrenyi W. Tabbara M. Tabib-Azar A. Taflove M. Taghivand N. Taguchi Y. Tahara G. Tait Y. Tajima T. Takagi K. Takahashi S. Takayama Y. Takayama S. Takeda I. Takenaka M. Taki K. Takizawa S. Talisa N. Talwalkar B.-T. Tan C.-Y. Tan J. Tan C.-W. Tang W.-C. Tang S. Tanaka T. Tanaka Y. Tanaka M. Tani E. Taniguchi H. Tanimoto R. Tascone J. Taub J. Tauritz R. Tayrani D. Teeter F. Teixeira R. Temkin M. Tentzeris K. Thakur H. Thal W. Thiel H.-W. Thim B. Thompson D. Thompson M. Tiebout L. Tiemeijer H. Toda M.-R. Tofighi M. Togashi T. Tokumitsu R. Tomasiunas A. Tombak K. Tomiyasu I. Toyoda S. Tretyakov R. Trew A. Trifiletti C. Trueman A. Truitt C.-M. Tsai E. Tsai L. Tsang H.-Q. Tserng T. Tsiboukis J. Tsui M. Tsuji T. Tsujiguchi T. Tsukahara K. Tsukamoto K. Tsunoda H. Tsurumi S. Tu R. Tucker M. Tur C.-K. Tzuang H. Uchida S. Uebayashi T. Ueda S. Ueno J. Uher F. Uhlmann T. Ulrich T. Umeda Y. Umeda F. Urbani T. Uwano P. Vainikainen P. Valanju F. Van de Water P. van den Berg D. Van der Weide G. Vandenbosch A. Vander Vorst D. Vanhoenacker-Janvie J. Vankka F. Van Straten K. Varian G. Vasilecu A. Vegas-Garcia L. Vegni A. Verma R. Vernon J. Verspecht B. Vidal L. Vietzorreck A. Viitanen A. Vilches C. Vittoria S. Vitusevich D. Viveiros V. Volman K. Wada K. Wakino D. Walker R. Walker M. Wallis C. Walsh C. Wan S. Wane B.-Z. Wang C. Wang D. Wang E. Wang H. Wang J. Wang K.-C. Wang S. Wang T.-H. Wang W. Wang X. Wang K. Warnick P. Warr S. Wartenberg O. Watanabe S. Watanabe R. Waugh D. Webb K. Webb R. Webster S. Wedge C.-J. Wei

J. Weirt R. Weigel G. Weihs R. Weikle C. Weil D. Weile A. Weily S. Weinreb J. Weiss C. Weitzel T. Weller C.-P. Wen M.-H. Weng R.-M. Weng S. Wentworth J. Whelehan L. Whicker J. Whitaker N. Whitbread D. White I. White S. Whiteley A. Whittneben B. Widrow G. Wilkins J. Williams T. Williams A. Williamson B. Willen B. Wilson J. Wiltse T. Winslow J. Winters A. Wittneben M. Wnuk M.-F. Wong S. Wong W. Woo J. Wood R. C. Wood G. Woods D. Woolard B.-L. Wu C. Wu H. Wu K. Wu K.-L. Wu Q. Wu Y.-S. Wu J. Wuerfl M. Wurzer J. Wustenberg G. Xiao C. Xie H. Xin Y.-Z. Xiong J. Xu Y. Xu Q. Xue T. Yakabe K. Yamamo S. Yamamoto S. Yamashita K. Yamauchi F. Yang H.-Y. Yang K. Yang Y. Yang Y.-J. Yang Z. Yang S. Yanagawa F. Yanovsky H. Yao J. Yao J. Yap B. Yarman K. Yashiro H. Yasser K. Yasumoto S. Ye J. Yeo S.-P. Yeo A. Yilmaz W.-Y. Yin S. Yngvesson N. Yoneda T. Yoneyama C.-K. Yong J.-G. Yook J.-B. Yoon R. York I. Yoshida S. Yoshikado L. Young M. Yousefi J.-W. Yu M. Yu P.-K. Yu W. Yu S.-W. Yun P. Yue A. Zaghoul A. Zaghloul A. Zajic K. Zaki P. Zampardi J. Zapata L. Zappelli J. Zehentner L. Zhang Q.-J. Zhang R. Zhang S. Zhang W. Zhang Y. P. Zhang A. Zhao L. Zhao Y. Zhao F. Zhenghe W. Zhou A. Zhu L. Zhu N.-H. Zhu Y.-S. Zhu Z. Zhu R. Zhukavin D. Zimmermann R. Ziolkowski H. Zirath J. Zmuidzinas A. Zozaya