IEEE Transaction on communications (October) [Volume 53, Number 10]

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OCTOBER 2005

VOLUME 53

NUMBER 10

IECMBT

(ISSN 0090-6778)

TRANSACTIONS LETTERS

Coding Iterative Decoding and Channel Parameter Estimation Algorithms for Repeat–Accumulate Codes . . . . . . . . . . . W. Oh and K. Cheun Fading/Equalization A Soft Decoding Scheme for Vector Quantization Over a CDMA Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .H. H. Nguyen Combined Equalization for Uplink MC-CDMA in Rayleigh Fading Channels . . . . . . . . . . . I. Cosovic, M. Schnell, and A. Springer Modulation/Detection Generalized APP Detection of Continuous Phase Modulation Over Unknown ISI Channels . . . . . . . . . . . . A. Hansson and T. Aulin Transmission Systems Capacity and Coverage Increase With Repeaters in UMTS Urban Cellular Mobile Communication Environment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. N. Patwary, P. B. Rapajic, and I. Oppermann Noncooperative Power-Control Game and Throughput Game Over Wireless Networks . . . . . . . . . . . . . . . . Z. Han and K. J. R. Liu

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TRANSACTIONS PAPERS

Coding Turbo Codes With Rate-m=(m + 1) Constituent Convolutional Codes . . . . . . . . . . . . . . . . . . . . . . . . C. Douillard and C. Berrou DC-Free Error-Control Block Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .F. Zhai, Y. Xin, and I. J. Fair Digital Communications Joint Source/Channel Coding for Multiple Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Z. Wu, A. Bilgin, and M. W. Marcellin Diversity Combining With Imperfect Channel Estimation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. You, H. Li, and Y. Bar-Ness Fading/Equalization Rate-Adaptive Transmission Over Correlated Fading Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . T. Ji and W. Stark The Expectation-Maximization Viterbi Algorithm for Blind Adaptive Channel Equalization . . . . . . . . . . . H. Nguyen and B. C. Levy Modulation/Detection Multirate Modulation: A Bandwidth- and Power-Efficient Modulation Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H. B. Peek A New Performance Bound for PAM-Based CPM Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. Perrins and M. Rice Design of Turbo-Coded Modulation for the AWGN Channel With Tikhonov Phase Error . . . . . . . . . . . Y. Zhu, L. Ni, and B. J. Belzer Multiple Access Adaptive Opportunistic Fair Scheduling Over Multiuser Spatial Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Li and X. Wang Systolic Array Implementation of a Real-Time Symbol-Optimum Multiuser Detection Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .C.-C. Lu, J.-Y. Hsu, and C.-C. Cheng

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(Contents Continued on Back Cover)

(Contents Continued from Front Cover)

Optical Communication An Overlay Architecture for Managing Lightpaths in Optically Routed Networks . . . . . . . . . . . V. A. Vaishampayan and M. D. Feuer Multiple-Subcarrier Optical Communication Systems With Subcarrier Signal Point Sequence. . . . . . . . . S. Teramoto and T. Ohtsuki Spread Spectrum On the Tradeoff Between Two Types of Processing Gain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. Fishler and H. V. Poor Transmission Systems An Iterative Extension of BLAST Decoding Algorithm for Layered Space–Time Signals . . . . . . . . . . . . . . K. Liu and A. M. Sayeed Blind Carrier Frequency Tracking for Filterbank Multicarrier Wireless Communications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .V. Lottici, M. Luise, C. Saccomando, and F. Spalla

1729 1738 1744 1754 1762

Abstracts of Forthcoming Manuscripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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

1776

PAPERS SCHEDULED TO BE IN THE NEXT ISSUE NOVEMBER 2005

Transactions Letters

Improving the Performance of Turbo Codes With a Simple Protection Scheme for Error-Prone Bit Positions. . . . W. Oh, Y. Kim, and K. Cheun Infinite Series Representations Associated With the Bivariate Rician Distribution and Their Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. A. Zogas and G. K. Karagiannidis Bit-Error Probability for Orthogonal Space–Time Block Codes With Differential Detection . . . . . . . . . . . . T. P. Soh, P. Y. Kam, and C. S. Ng On the Probability of Error of Antenna Subset Selection With Space-Time Block Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. J. Love Cooperative Space–Time Coding for Wireless Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Stefanov and E. Erkip Adaptive Resource Allocation for Multiaccess MIMO/OFDM Systems With Matched Filtering . . . . . . . . . . . . .Y. J. Zhang and K. B. Letaief Low-Complexity Maximum-Likelihood Decoder for Four-Transmit-Antenna Quasi-Orthogonal Space–Time Block Code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M.-T. Le, V.-S. Pham, L. Mai, and G. Yoon

Transactions Papers

Nonsystematic Turbo Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .A. Banerjee, F. Vatta, B. Scanavino, and D. J. Costello, Jr. A Subband Approach to Channel Estimation and Equalization for DMT and OFDM Systems . . . . . . . . . . . . . . . . . . . .D. Marelli and M. Fu Optimized Decision Feedback Equalization for Convolutional Coding With Reduced Delay . . . . . . . . . . . . . . . . . J.-T. Liu and S. B. Gelfand Iterative Design and Detection of a DFE in the Frequency Domain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . N. Benvenuto and S. Tomasin Simulation of Rayleigh Faded Mobile-to-Mobile Communication Channels . . . . . . . . . . . . . . . . . . . C. S. Patel, G. L. Stüber, and T. G. Pratt Capacity-Approaching Turbo Coding and Iterative Decoding for Relay Channels . . . . . . . . . . . . . . . . . . . . . . . . Z. Zhang and T. M. Duman Wireless Packet Scheduling Based on the Cumulative Distribution Function of User Transmission Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . D. Park, H. Seo, H. Kwon, and B. G. Lee Expanding the Switching Capabilities of Optical Crossconnects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.-T. Lea On the Combined Input-Crosspoint Buffered Switch With Round-Robin Arbitration . . . . . . . . . . . . . R. Rojas-Cessa, E. Oki, and H. J. Chao Timing Ultra-Wideband Signals With Dirty Templates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . L. Yang and G. B. Giannakis Abstracts of Forthcoming Manuscripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Modulation & Signal Design M. BRANDT-PEARCE, CDMA Systems Univ. of Virginia, Charlottesville, VA G. CHERUBINI, CDMA Systems IBM Research Lab., Zurich, Switzerland G. E. CORAZZA, Spread Spectrum DEIS, Univ. of Bologna, Bologna, Italy A. HOST-MADSEN, Multiuser Communications Univ. of Hawaii, Honolulu, HI R. A. KENNEDY, Data Commun. Res. School Info. Sci. Eng. Australian Nat’l. Univ., Canberra, Australia R. KOHNO, Spread Spectrum Theory & Appl. Yokohama Univ., Yokohama, Japan H. LEIB, Commun. & Inform. Theory McGill Univ., Montreal, QC, Canada L. RASMUSSEN, Iterative Detection Decoding & ARQ Univ. of South Australia, Adelaide, Australia R. SCHOBER, Detection, Equalization, & MIMO Univ. of British Columbia, Vancouver, BC, Canada C. TELLAMBURA, Multicarrier Systems Univ. of Alberta, Edmonton, AB, Canada S. ULUKUS, CDMA Systems Univ. of Maryland, College Park, MD M. Z. WIN, Area Editor Lab. for Inform. & Decision Syst. (LIDS) Massachusetts Inst. of Technol., Cambridge, MA X. WANG, Multiuser Detection & Equalization Columbia Univ., New York, NY A. ZANELLA, Wireless Systems DEIS, Univ. of Bologna, Bologna, Italy

S. JAFAR, Wireless Commun. Theory & CDMA Univ. of California, Irvine, CA H. JAFARKHANI, Space–Time Coding Univ. California, Irvine, CA P. Y. KAM, Modulation & Detection National Univ. of Singapore, Singapore D. KIM, Spread Spectrum Transmission & Access Simon Fraser Univ., Burnaby, BC, Canada Z. KOSTIC, Wireless Systems AT&T Shannon Labs.–Research, Middletown, NJ I. LEE, Wireless Commun. Theory Korea Univ., Seoul, Korea K. K. LEUNG, Wireless Network Access & Performance Lucent Technol., Bell Labs, Holmdel, NJ Y. LI, Wireless Communication Theory Georgia Inst. of Technol., Atlanta, GA A. LOZANO, Wireless Network Access & Performance Lucent Technologies, Bell Labs., Holmdel, NJ P. T. MATHIOPOULOS, Wireless Personal Commun. Inst. Space Applicat. & Remote Sensing Nat. Observatory of Athens, Athens, Greece A. NAGUIB, Space–Time and CDMA Qualcomm, Inc., San Diego, CA R. A. VALENZUELA, Wireless Systems Lucent Technologies, Bell Labs., Holmdel NJ J. WANG, Wireless Spread Spectrum Univ. of Hong Kong, Hong Kong T. WONG, Wideband & Multiple Access Wireless Systems Univ. of Florida, Gainesville, FL

Wireless Communication N. AL-DHAHIR, Space–Time, OFDM & Equalization Univ. of Texas, Dallas, TX A. ANASTASOPOULOS, Iterative Detection, Estimation, & Coding Univ. of Michigan, Ann Arbor, MI S. ARIYAVISITAKUL, Area Editor Texas Instruments, Alpharetta, GA S. BATALAMA, Spread Spectrum & Estimation SUNY Buffalo, Buffalo, NY N. C. BEAULIEU, Wireless Commun. Theory Univ. of Alberta, Edmonton, AB, Canada Y. FANG, Wireless Networks Univ. of Florida, Gainesville, FL R. HEATH, MIMO Techniques Univ. of Texas, Austin, TX B. JABBARI, Wireless Multiple Access George Mason Univ., Fairfax, VA

Optical Communication I. ANDONOVIC, Optical Networks & Devices Univ. of Strathclyde, Glasgow, U.K. R. HUI, Optical Transmission & Switching Univ. of Kansas, Lawrence, KS D. K. HUNTER, Photonic Networks Univ. of Essex, Colchester, U.K. K. KITAYAMA, Photonic Networks & Fiber-Optic Wireless Osaka Univ., Osaka, Japan W. C. KWONG, Optical Networks Hofstra Univ., Hempstead, NY P. PRUCNAL, Area Editor & Light. Networks Princeton Univ., Princeton, NJ J. SALEHI, Optical CDMA Sharif Univ., Tehran, Iran

Editors Speech, Image, Video, & Signal Process. F. ALAJAJI, Source & Source/Channel Coding Queen’s Univ., Kingston, ON, Canada K.-K. MA, Video & Signal Processing Nanyang Tech. Univ., Singapore C. S. RAVISHANKAR, Speech Processing Hughes Network Systems, Germantown, MD K. ROSE, Source/Channel Coding Area Editor Univ. of California, Santa Barbara, CA M. SKOGLUND, Source/Channel Coding Royal Inst. of Technol., Stockholm, Sweden Transmission Systems V. AALO, Diversity & Fading Channel Theory Florida Atlantic Univ., Boca Raton, FL A. ABU-DAYYA, Diversity & Modulation AT&T Wireless, Redmond, WA M.-S. ALOUINI, Modulation & Diversity Syst. Univ. of Minnesota, Minneapolis, MN M. CHIANI, Wireless Commun. Univ. of Bologna, Bologna, Italy X. DONG, Modulation & Signal Design Univ. of Alberta, Edmonton, AB, Canada H. EL GAMAL, Space–Time Coding & Spread Spectrum Ohio State Univ., Columbus, OH G.-H IM, Equalization & Multicarrier Techniques Pohang Univ. Sci. & Technol., Pohang, Korea H. MINN, Synchronization & Equalization Univ. of Texas, Dallas, TX R. RAHELI, Detection, Equalization, & Coding Univ. of Parma, Parma, Italy R. REGGIANNINI, Synchronization & Wireless Appl. Univ. of Pisa, Pisa, Italy F. SANTUCCI, Wireless Syst. Performance Univ. of L’Aquila, L’Aquila, Italy E. SERPEDIN, Synchronization & Sensor Networks Texas A&M Univ., College Station, TX C. TEPEDELENLIOGLU, Synchronization & Equalization Arizona State Univ., Tempe, AZ G. VITETTA, Equalization & Fading Channels Univ. of Modena, Modena, Italy C.-L. WANG, Equalization National Tsing Hua Univ., Taiwan K. WILSON, Multicarrier Modulation Royal Inst. of Technology, Stockholm, Sweden

I. JACOBS, Senior Advisor Dept. Elec. Eng. Virginia Polytechnic Inst. Blacksburg, VA 24061 M. Z. WIN, Equalization & Diversity Lab. for Inform. & Decision Syst. (LIDS) Massachusetts Inst. of Technol., Cambridge, MA J. WINTERS, Area Editor & Equalization Motia, Inc., Middletown, NJ Network Architecture R. FANTACCI, Wireless Networks & Systems Univ. Firenze, Firenze, Italy M. HAMDI, Switching, Routing, & Optical Networks HKUST, Hong Kong G. S. KUO, Area Editor & Commun. Architect. National Central Univ., Chung-Li, Taiwan T. LEE, Switching Architecture Performance Chinese Univ. of Hong Kong, Hong Kong A. PATTAVINA, Switching Arch. Perform. Politecnico di Milano, Italy T.-S. P. YUM, Packet Access & Switching Chinese Univ. of Hong Kong, Hong Kong M. ZORZI, Wireless Multiple Access University of Padova, Padova, Italy Coding & Commun. Theory E. AYANOGLU, Commun. Theory & Coding Appl. Univ. of California, Irvine, CA A. BANIHASHEMI, Coding & Commun. Carleton Univ., Ottawa, ON, Canada F. FEKRI, LDPC Codes & Applications Georgia Inst. of Technol., Atlanta, GA M. FOSSORIER, Coding & Commun. Theory Univ. of Hawaii, Honolulu, HI B. HOCHWALD, MIMO Techniques Lucent Technol. Bell Labs., Murray Hill, NJ P. HOEHER, Coding & Iterative Processing Kiel Univ., Kiel, Germany A. K. KHANDANI, Coding & Information Theory Univ. of Waterloo, Waterloo, ON, Canada K. NARAYANAN, Modulation, Coding, & Equalization Texas A&M Univ., College Station, TX W. E. RYAN, Modulation, Coding & Equalization Univ. of Arizona, Tucson, AZ C. SCHLEGEL, Coding Theory & Techniques Univ. of Alberta, Edmonton, AB, Canada TRIEU-KIEN TRUONG, Coding Theory & Techniques I-Shou Univ., Taiwan R. D. WESEL, Coding & Coded Modulation Univ. of California, Los Angeles, CA S. G. WILSON, Area Editor & Coding Theory & Appl. Univ. of Virginia, Charlottesville, VA

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Digital Object Identifier 10.1109/TCOMM.2005.859279

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Transactions Letters________________________________________________________________ Iterative Decoding and Channel Parameter Estimation Algorithms for Repeat–Accumulate Codes Wangrok Oh, Member, IEEE, and Kyungwhoon Cheun, Member, IEEE

Abstract—The sensitivity of the iterative decoder for repeat–accumulate (RA) codes to carrier phase and channel signal-to-noise ratio estimation errors is investigated, and efficient algorithms to estimate and correct these errors are developed. The behavior of RA codes with imperfect channel estimation is different from that of turbo codes, and correction algorithms specific to RA codes must be formulated. The proposed algorithms use the soft information generated within the iterative decoder, and thus, are not only hardware-efficient, but also offer excellent performance. Index Terms—Iterative decoding, phase synchronization, repeat–accumulate (RA) codes, signal-to-noise ratio (SNR) estimation.

I. INTRODUCTION REVIOUS works [1]–[3] have demonstrated that repeat–accumulate (RA) codes, one of the simplest serial concatenated codes, can achieve remarkable performance over additive white Gaussian noise channels with iterative decoding. These results were based on the assumption of perfect estimation of channel parameters at the decoder, such as the channel signal-to-noise ratio (SNR) and carrier phase offset. This assumption may be unrealistic in practical implementations, due to the extremely low operating SNR range of RA codes. Simulation studies indicate that the performance of RA codes is very sensitive to errors in channel SNR and carrier phase offset estimates. Also, the behavior of RA codes as a function of these measurement errors may be quite different from that corresponding to turbo codes. Hence, in order to fully realize the exceptional performance of RA codes, it is crucial to formulate proper estimation and correction algorithms specific to RA codes, giving satisfying performance even at very low SNRs. Naturally, it would also be desirable to achieve this with minimum hardware overhead. In this letter, we investigate the sensitivity of the iterative decoder for RA codes to channel SNR and carrier phase offset estimation errors, and propose very simple correction algorithms.

P

Paper approved by W. E. Ryan, the Editor for Modulation, Coding, and Equalization of the IEEE Communications Society. Manuscript received April 30, 2004; revised January 27, 2005 and May 4, 2005. This work was supported by the Center for Broadband OFDM Mobile Access (BrOMA) at POSTECH, supported by the ITRC Program of the Ministry of Information and Communication, Korea, supervised by the Institute of Information Technology Assessment (IITA). The authors are with the Division of Electrical and Computer Engineering, Pohang University of Science and Technology (POSTECH), Pohang 790-784, Korea (email: [email protected]; [email protected]). Digital Object Identifier 10.1109/TCOMM.2005.857150

The algorithms exploit the properties of the log-likelihood ratios (LLRs) generated within the iterative decoder which are similar to the algorithms proposed for turbo codes [4], [5]. The carrier phase offset correction algorithm also exploits the differential encoding characteristic specific to RA codes. Since the estimation is performed using the soft information already available within the iterative decoder, the required additional hardware complexity is minimal. There are two notable differences in the behaviors of RA and turbo codes under channel parameter estimation errors. A more subtle difference is that unlike turbo codes [4], [6], RA codes are much less lenient to overestimation of the channel SNR. A more significant difference is that due to inherent differential inner encoding, RA codes exhibit relatively good performance under carrier phase offset in the vicinity of 180 . This inhibits straightforward application of stochastic gradient-type carrier phase offset correction algorithms, due to the presence of a local maximum. Hence, in order to fully realize the exceptional performance of RA codes, it is crucial to develop channel parameter estimation and correction algorithms specific to RA codes. The remainder of the letter is organized as follows. In Section II, we present the system model and investigate the sensitivity of RA codes to channel SNR and carrier phase offset estimation errors. In Section III, channel parameter estimation and correction algorithms appropriate for RA codes are described and their performances are evaluated. Finally, conclusions are drawn in Section IV. II. SYSTEM MODEL The system model under consideration is shown in Fig. 1. Binary information bits are grouped into frames of size and encoded by an RA encoder. The RA encoder consists of a serial concatenation of a –times repeater and a rate–1 accu. The encoded mulator, separated by an interleaver of size are binary phase-shift keying modulated, symbols with a double–sided power specand white Gaussian noise tral density of is added before being presented to the demodulator. The corresponding demodulator output with a car, is prerier phase offset of , denoted is a zero-mean comsented to the iterative decoder, where per diplex Gaussian random variable (r.v.) with variance mension. Let be the channel SNR, where is the code rate and is the received energy per information bit. The resulting LLRs for at the end of the th decoder

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Fig. 1. System model. u are the input information bits and c are the RA encoded symbols, y is the demodulator output with a carrier phase offset  , u ^ denotes the decoder output for u , L (u ) is the resulting LLR for u at the end of the lth decoder iteration, ~ is the initial carrier phase offset estimate, and ^ and ^ are the channel SNR and carrier phase offset estimates used in the lth decoder iteration.

^ and ^ are the channel reliability value and carrier phase offset estimates used in the lth decoder iteration. Fig. 2. Factor graph of an RA code. L

iteration, denoted , are used to generate the estimates and of and to be used in the th decoder iteration. The th iteration of the iterative decoder is carried out using the demodulator outputs compensated by , and using on the factor graph [7] shown in Fig. 2, where the dependence of the messages on was suppressed to simplify the figure. The iterative decoder computes the messages and using the following recursions:

(3)

(4) Here

(1)

(2)

[7], denotes the real part of , is the estimated channel reliability value [8], and denotes the th output position of the interleaver (Fig. 2). Also, are the messages

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Fig. 3. BER performance of the iterative decoder for an RA code versus the

dB, assumed to be fixed for all l. Frame ^ channel SNR estimation error size is 512, q = 3, l = 10, and perfect carrier phase offset estimation is assumed.

Fig. 4. BER performance of the iterative decoder for an RA code versus the carrier phase offset estimation error  , assumed to be fixed for all l. Frame size is 512, q = 3 , l = 10, and perfect channel SNR estimation is assumed.

0

passed from the information nodes to the check nodes, initialized to zero and computed as (5) where denotes the integer part of , and denotes the th output position of the deinterleaver (Fig. 2). The messages are the messages originating from the check nodes destined for and as the information nodes, computed using (6) with . Finally, the LLRs for the information bits the end of the th iteration are then given as

at

with turbo codes [4], [6], a slight underestimation of the channel SNR optimizes the BER performance of the iterative decoder for RA codes.2 In Fig. 4, the performance of the standard iterative RA dewith , versus , assumed to be coder with and 2 dB. This graph fixed for all , is shown for clearly shows the detrimental effect of carrier phase offset estimation errors. However, note that the iterative decoder for RA codes shows relatively good performance, with carrier phase offset estimation errors in the vicinity of 180 . This is due to the differential encoding of the accumulator inner code. This characteristic is unique to RA codes, and must be taken into account when designing carrier phase offset correction algorithms. III. PROPOSED CHANNEL PARAMETERESTIMATION ALGORITHMS

(7) The decoder iteration stops when it reaches a preset maximum number , or when all bits are successfully decoded, which can be verified using a cyclic redundancy check [9]. It is clear that the estimated channel reliability value plays a key role in the decoding of RA codes. The bit-error rate (BER) for performance of a standard iterative decoder with and versus the channel SNRan RA code with estimation error is shown in Fig. 3 for and 2 dB.1 Here, the assumed channel SNR-estimation error was held fixed throughout the decoder iterations, and the carrier phase offset , is assumed to be zero estimation error, denoted for all decoder iterations. We observe that, unlike turbo codes, both over- and underestimation of the channel SNR results in a sizable degradation. However, it is interesting to note that as 1For all numerical results, we assume that a random interleaver pattern is generated pseudorandomly for each frame, i.e., the uniform interleaver. However, proper operation of the proposed algorithms was verified under various fixed interleavers. Also, 1000 frame errors were observed for each BER simulation point.

A. Channel SNR Estimation Similar to the channel SNR estimation algorithms for turbo codes developed in [4], we investigate the behavior of the variance of the absolute values of the LLRs, computed as follows: (8) where denotes the LLR for the th bit after the th iteragiven by (8) may be tion of the th frame. Also, as in [4], replaced with the following simplified version: (9) The ensemble averaged behavior of simulation runs given by

and

using 100 ,

2This trend may be slightly changed by employing decoder-scheduling schemes different from the one described in Section II, e.g., full flooding [10]. However, the proposed algorithms are directly applicable to iterative decoders employing any decoder-scheduling scheme.

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Fig. 5. Behavior of v , i = 1; 2 versus the channel SNR estimation error for l = 10. Frame size is 512 and q = 3. The channel SNR estimation error is

BER performance of the iterative decoder using ^ with v and  = = 10, and perfect carrier phase offset estimation is assumed.

assume to be fixed for all decoder iterations, and perfect carrier phase offset estimation is assumed.

versus the channel SNR estimation error is shown in Fig. 5 for . Here, we assume perfect carrier phase offset estimation and the fixed channel SNR estimation error for all decoder iterations. Comparing this with Fig. 3, we observe that the BER of the iterative decoder for RA codes may be minimized by minior . Based on this observation, we propose the mizing following stochastic gradient-type recursive update equation for : (10) is the estimated channel SNR for the th iteration of where the th frame, and is the update gain. Also, (10) can be further simplified by replacing with its polarity, resulting in (11) if , and , otherwise. The BER where performance of the iterative decoder using the proposed channel given by (9), is shown SNR estimate given by (11), with in Fig. 6 with perfect carrier phase offset estimation. We observe that, as with turbo codes [4], the iterative decoder using the proposed channel SNR estimate gives BER performance slightly better than that using the exact channel SNR. B. Carrier Phase Estimation The presence of a carrier phase offset estimation error results in an effective reduction in the received signal power, as seen by the matched-filter receiver. This will, in effect, reduce the average power of the LLRs of the information bits within the iterative decoder. This indicates that the measured power of the LLRs of the information bits may be used for carrier phase offset estimation and correction. As in [5], we consider the following

Fig. 6.

2:5. Frame size is 512, and q = 3, l

simplified measure in place of the more complex power estimator: (12) We may also consider the following much simpler measure [5] that may further reduce the hardware complexity, with minimal loss in performance: (13) In Fig. 7, the ensemble averaged behavior of and , using 100 simulation runs given by , , versus the carrier phase offset estimation error are shown for . Here, we assume perfect channel SNR estimation and the fixed carrier phase offset estimation error for all decoder iterations. Note the existence of a prominent local maximum at 180 carrier phase offset estimation error. This indicates that unlike turbo codes, straightforward application of a stochastic gradient update algorithm of the following type may converge to the undesirable local maximum at 180 , depending on the initial carrier phase offset:3

(14) Here, is the estimated carrier phase offset for the th iteration of the th frame. However, if a coarse initial carrier phase recovery is available which guarantees carrier phase offset in the vicinity of 0 , (14) may successfully be used to compensate and track the carrier phase offset. If the coarse initial carrier phase 3The update term in (14) has polarity opposite to that of (11), due to the fact that s , j = 1; 2 are convex functions of the carrier phase offset estimation error, while v , j = 1; 2 are concave function of the channel SNR-estimation error.

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Fig. 7. Behavior of s , i = 1; 2 versus the carrier phase offset estimation error for l = 10. Frame size is 512 and q = 3. The carrier phase offset estimation error is assumed to be fixed for all decoder iterations, and perfect channel SNR estimation is assumed.

Fig. 8. BER performance of the proposed iterative decoder using ~ and ^ with s and  = 0:5, versus the channel SNRs for p = 1; 5. Frame size is ^ = 4 . 512, q = 3, l = 10, and L

recovery algorithm is nondata aided and thus possesses a 180 phase ambiguity [11], we may use the following observation to easily detect and compensate for the 180 phase ambiguity within the iterative decoder before the tracking operation with (14). generated under a carrier phase Note that the messages offset of 180 are identical to those generated under a zero car, which has a reversed polarity. rier phase offset, except for This is due to the differential encoding inherent in RA codes. is alHence, neglecting the effect of noise, the polarity of ways opposite to that of under a 180 carrier phase offset. Based on this observation, we may modify the iterative decoder to detect and compensate for carrier phase offsets near 180 by comparing the polarity of the messages and after the first decoder iteration. Let denote the initial carrier phase offset estimation with a 180 resolution, given as if otherwise.

(15)

If is 0 , the proposed iterative decoder moves on with the remaining decoder iterations. Otherwise, the decoder negates and the message , and recalthe demodulator outputs before culates the messages affected by the negation of performing the remaining decoder iterations. Note that only out of , ’s need be recalculated as a result of negating . Hence, the additional complexity required to detect and compensate for carrier phase offset near 180 is much less than of that required for one decoder iteration. We have also verified that this trend is maintained and reinforced even after iterations. Hence, in order to increase the reliability of the estimate of , the detection operation described above may be performed after decoder iterations. The BER performance of the proposed iterative decoder and 180 with versus the channel SNR under

is shown in Fig. 8, assuming perfect channel SNR estimation.4 Note that the proposed iterative decoder under a carrier phase offset of 0 and 180 offers performance within 0.03 and 0.1 at BER dB of the standard iterative decoder, with , respectively. The BER performance curve of the with proposed iterative decoder under an initial carrier phase offset of 45 is also included in Fig. 8, and lies between those correand 180 . Although not shown in Fig. 8, sponding to we have carried out extensive simulations, and verified that the initial carrier phase offset of 180 results in the largest SNR loss for the proposed iterative decoder. We have also verified that the increase in the required average number of decoder iterations, due to the increase in , is slightly less than . In most packet-based networks, a preamble is usually available, and a crude initial carrier phase recovery may be performed using the preamble. In continuous transmission systems, the stochastic gradient update algorithm of (14) may first be used to let the residual carrier phase offset converge to the vicinity of either 0 or 180 . A decision on may then be made using (15), after which the tracking operation may be resumed. IV. CONCLUSIONS In this letter, we investigated the sensitivity of the iterative decoder for RA codes to carrier phase and channel SNR estimation errors, and developed algorithms to accurately estimate and correct these errors. The characteristics specific to RA codes were identified and fully exploited. The proposed algorithms, using the soft information generated within the iterative decoder, are not only very hardware-efficient, but also provide excellent performance.

4Note that due to the fact that s takes on values much larger than those of v , the update gain used for the channel SNR estimation is much larger than

that used for the carrier phase offset compensation.

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REFERENCES [1] D. Divsalar, H. Jin, and R. McEliece, “Coding theorems for “turbo-like” codes,” in Proc. 36th Allerton Conf. Commun., Control, Comput., Sep. 1998, pp. 201–210. [2] H. Jin and R. J. McEliece, “RA codes achieve AWGN channel capacity,” in Proc. 13th Symp. Appl. Algebra, Algebraic Algorithms, Error Correcting Codes, 1999, pp. 10–18. [3] H. Jin, A. Khandekar, and R. McEliece, “Irregular repeat-accumulate codes,” in Proc. 2nd Int. Symp. Turbo Codes, Sep. 2000, pp. 1–8. [4] W. Oh and K. Cheun, “Adaptive channel SNR estimation algorithm for turbo decoder,” IEEE Commun. Lett., vol. 4, no. 8, pp. 255–257, Aug. 2000. , “Joint decoding and carrier phase recovery algorithm for turbo [5] codes,” IEEE Commun. Lett., vol. 5, no. 9, pp. 375–377, Sep. 2001. [6] T. A. Summers and S. G. Wilson, “SNR mismatch and online estimation in turbo decoding,” IEEE Trans. Commun., vol. 46, no. 4, pp. 421–423, Apr. 1998.

[7] F. R. Kschischang, B. J. Frey, and H. Loeliger, “Factor graphs and the sum-product algorithm,” IEEE Trans. Inf. Theory, vol. 47, no. 2, pp. 498–519, Feb. 2001. [8] J. Hagenauer, E. Offer, and L. Papke, “Iterative decoding of binary block and convolutional codes,” IEEE Trans. Inf. Theory, vol. 42, no. 3, pp. 429–445, Mar. 1996. [9] A. Shibutani, H. Suda, and F. Adachi, “Reducing average number of turbo decoding iterations,” Electron. Lett., vol. 35, pp. 701–702, Apr. 1999. [10] F. R. Kschischang and B. J. Frey, “Iterative decoding of compound codes by probability propagation in graphical models,” IEEE J. Sel. Areas Commun., vol. 16, no. 2, pp. 219–230, Feb. 1998. [11] H. Meyr, M. Moeneclaey, and S. A. Fechtel, Digital Communication Receivers. New York: Wiley, 1998.

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A Soft Decoding Scheme for Vector Quantization Over a CDMA Channel Ha H. Nguyen, Senior Member, IEEE

Abstract—The optimal decoding of vector quantization (VQ) over a code-division multiple-access (CDMA) channel is too complicated for systems with a medium-to-large number of users. This paper presents a low-complexity, suboptimal decoder for VQ over a CDMA channel. The proposed decoder is built from a soft-output multiuser detector, a soft bit estimator, and the optimal soft VQ decoding of an individual user. Simulation results obtained over both additive white Gaussian noise and flat Rayleigh fading channels show that with a lower complexity and good performance, the proposed decoding scheme is an attractive alternative to the more complicated optimal decoder.

The soft-output MUDs considered include the jointly optimal MUD (OPT-MUD), the minimum mean-squared error MUD (MMSE-MUD), and the decorrelating MUD (DC-MUD) [9]. For each type of MUD, the soft-bit estimates are calculated and fed into the soft VQ decoders. Due to its lower complexity and good performance, the proposed decoding scheme is an attractive alternative to the complicated optimal decoder.

Index Terms—Code-division multiple access (CDMA), combined source and channel coding, multiuser detection, soft decoding, vector quantization (VQ).

The system model considered in this letter is the same as users in a synchronous CDMA the one in [6]. There are system, where each user transmits its source vectors by means of VQ. The th user produces a -dimensional random vector . The vector is then encoded into an index , where for some integer . The bits per source transmission rate of the system is thus dimension. The th encoder is described by a partition

I. INTRODUCTION HE source and channel coding of a communication system are often designed and implemented separately. This common practice is mainly due to the work by Shannon [1], where it was shown that such a separation can perform optimally. However, the positive coding theorems of information theory only show such separability in the limit of infinite codeword length, and hence, infinite delay [1], [2]. Furthermore, there exist channels for which the separation theorem is not valid, even asymptotically. One important class of such channels is the class of multiple-access channels [2]. It is, therefore, important to study the combined source-channel coding for this type of channel. The combined source-channel coding considered in this letter is restricted to block coding, where the code is defined by a robust vector quantization (VQ). In robust VQ, the channel imperfections are taken into account when assigning VQ codevectors to the transmission codewords. The codeword-assignment problem is generally referred to as the index-assignment (IA) problem [3]–[5]. The optimal soft decoding of such VQ over code-division multiple-access (CDMA) channels was recently considered in [6]–[8]. Such an optimal decoder is, however, too complicated for CDMA systems with a medium or large number of users. In this letter, a suboptimal approach to VQ decoding over CDMA channels is developed. The proposed decoder is built from a soft-output multiuser detector (MUD), a soft bit estimator, and the optimal soft VQ decoders of individual users.

T

Paper approved by X. Wang, the Editor for Modulation, Detection, and Equalization of the IEEE Communications Society. Manuscript received March 15, 2004; revised November 10, 2004; February 8, 2005; May 8, 2005. This work was supported by NSERC under a Discovery Grant. This paper was presented in part at the IASTED International Conference on Communications Systems and Applications (CSA), Banff, AB, Canada, July 2004. The author is with the Department of Electrical Engineering, University of Saskatchewan, Saskatoon, SK S7N 5A9, Canada (e-mail: [email protected]). Digital Object Identifier 10.1109/TCOMM.2005.857151

II. SYSTEM MODEL

of the Euclidean source space . Let

, such that if

, then . Also, de-

fine the th encoder centroid of user as . For a VQ with a mean squared error (MSE) distortion measure, are the optimal reconstruction vectors for a the centroids noiseless channel. For a noisy channel, the optimal hard-decision reconstruction vectors are formed as linear combinations of the centroids [18]. With binary phase-shift keying (BPSK) modulation, the index is converted into a block of bits, where . For simplicity, it is assumed that all users have the same block length . The bits of the th user’s indexes are transmitted over a synchronous CDMA channel by modulating the th user’s distinct signature waveform. Let be the bit duration, and , be the signature waveform of the th user whose energy is normalized to unity. Because the system is synchronous, it suffices to consider the transmission of a single index of every user. Two channel models, namely, the additive white Gaussian noise (AWGN) and the slow flat Rayleigh fading channels, are considered in this letter. For an AWGN channel, the received ) can baseband signal over the first index interval ( be expressed as (1) is the received amplitude of the th users and where real AWGN of power spectral density (PSD) W/Hz.

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is

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Fig. 1. Structure of the proposed decoder.

For a slow flat Rayleigh fading channel, the parameters ’s in (1) are changed to ’s. These parameters are modeled as independent zero-mean complex Gaussian random variables with independent real and imaginary parts. Furthermore, the is zero-mean, complex white Gaussian noise of PSD noise per dimension. For simplicity, an AWGN channel shall be used in presenting the proposed decoding scheme. Application of the proposed decoder to a flat Rayleigh fading channel is straightforward. It is well known (see [9]) that the sufficient statistics for the channel model in (1) can be obtained by a bank of matched filters (sampled at the bit rate), and are given by (2) is the correlation matrix of the signature wavewhere forms with , is a Gaussian vector of zero mean and covariance matrix and independent of the transmitted . bits, and , the decoder needs Based on the sufficient statistics to make a decision on the transmitted source vectors of all users. Of course, different processing algorithms on yield different decoders. In the remainder of this section, two such decoders shall be reviewed, as they will be used as benchmark decoding schemes to compare with the decoder proposed in this letter.

is the sample received data, where readily follows from the and the exact expression for CDMA channel model in (2) (see [6]). In [6], the implementation of the optimal decoder in (3) based on Hadamard matrix description of the VQs is presented. Such an optimal decoder is named the Hadamard-based multiuser decoder (HMD). The structure of HMD shows how to use the a priori and channel information in an optimal fashion to counteract channel noise and multiple-access interference (MAI). The total decoding complexity of HMD is about operations, which is clearly prohibitive for a CDMA system with a medium-to-large number of users. B. Suboptimal Decoder Based on Table Lookup An alternative decoding approach is based on a combination of separate multiuser detection and table-lookup (or hard) VQ decoding. The multiuser detection can be, for example, the OPT or the MMSE receiver [9]. Such a tandem approach first gives the hard decision for the transmitted vector of bits . For each user , the bits are then converted to the corresponding estimated index . The VQ decoder of the th user then finds and outputs the centroid for VQ decoding. If the OPT-MUD is used, then the complexity of the suboptimal hard decoder is about operations per user. On the other hand, the decoding operations if the MMSE-MUD complexity is about is employed. III. A SUBOPTIMAL SOFT DECODER

A. Jointly Optimal Multiuser-VQ Decoder denote the vector Similar to [6], let consisting of all users’ indexes having sample values . . Also define Let and

Assume that the sources of the different users are statistically independent, which implies that . The jointly optimal decoder minimizes the distortion for each user . From estimation theory, the optimal estimate of is the conditional mean and is given by [6] (3)

The suboptimal decoder proposed in this letter is also based on separate multiuser detection and VQ decoding, as illustrated in Fig. 1. However, the key difference, compared with the decoder described in Section II-B, is that the soft bit estimates from the soft-output MUD are first generated and then fed into the individual soft VQ decoders. It should also be pointed out here that unlike the receiver in [10] for (channel) coded CDMA, there are no iterations between the soft-output MUD and the individual soft VQ decoders in Fig. 1. Although such iterations might also be implemented (similar to the single-user systems [11]) by appropriately modeling the VQ encoders with Markov sources, they are not considered here for two main reasons. First, the emphasis of the letter is on the low complexity of the receiver, which does not favor iterations. Second, as will be seen in Section IV, the performance of the proposed decoder can approach that of the op-

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timum decoder quite closely, which suggests that there might be only a little room for performance improvement with iterations. Furthermore, the iterations are only helpful for VQ encoders that have a large amount of redundancy.1 This is because VQ’s redundancy is essential to produce enhanced a priori probabilities of the binary bits for the soft-input soft-output MUDs through iterations. To see what are the soft bit estimates needed for individual soft VQ decoders in Fig. 1, it is appropriate to consider the optimal decoding of VQ over a single-user channel first.

ated from the soft-output MUD in Fig. 1. To this end, define the following soft bit estimate for a given MUD: MUD (8) where of the bit

MUD MUD

A. Optimal Soft VQ Decoder for a Single-User Channel If there is only the user signaling over an AWGN channel, the discrete channel output is simply (4) is a zero-mean Gaussian random variable with variwhere . The optimal decoder that minimizes the MSE comance putes the following conditional expectation [7]: (5) where

is the a posteriori log-likelihood ratio (LLR) at the output of the MUD, defined as

is the sample value of

. A detailed treatment of the above decoder based on a Hadamard matrix and the related Hadamard transform is given in [7]. Such a decoder is useful, since it provides a description of the optimal decoder in terms of the estimates of the individual bits of the transmitted index. The main operation of the Hadamard-based decoder can be summarized as follows. Let be the encoder matrix that satisfies , where is Sylvester-type Hadamard matrix the th column of an . Then (5) can be computed as [7]

(6)

where (7)

Computation of the a posteriori LLR for each type of MUD is described below. For the optimal MUD, the a posteriori LLR is readily given in [10, eq. (28)]. Furthermore, it should be pointed out that with the use of the optimal MUD, the resulting decoder is the same as the user-separated HMD (US-HMD) in [6]. The complexity , which is still exponential of this decoder is about in the number of users . To further reduce the computational complexity of the US-HMD, an approximation to the US-HMD (called US-HMD) is presented in [6]. In essence, the US-HMD is obtained by reducing the number of terms in the summations needed to compute the LLR to be . For example, the sums nearest neighbors of , a hard can be limited to the decision on the transmitted value of obtained by applying a hard-decision DC-MUD. Such an approach, however, requires determining and storing nearest neighbors for each of all possible vectors of . The total decoding complexity of US-HMD is reduced to about . The choice of depends on the number of users , but generally, it has to grows to maintain a particular performance be increased as level [6]. Here we take a different and simpler approach to reduce the computational complexity of the US-HMD. Instead of the optimal MUD, a soft-output MMSE-MUD or a soft-output DC-MUD with a much lower complexity is applied. The soft bit estimates are then generated accordingly to be the inputs to the individual soft VQ decoders. denote a -vector of all zeros, except for the th eleLet ment, which is one. Then the output of the linear MMSE-MUD is2 [9], [10] for user that corresponds to bit

and

denotes the Kronecker matrix product. The quantity is the MMSE soft estimate of the bit . For the channel model in (4), this soft bit estimate is computed as [7] . B. Proposed Suboptimal Soft Decoder Observe that the optimal Hadamard-based soft VQ decoder for a single-user channel can also be employed for an individual user in a CDMA channel, if the soft bit estimates can be gener1Note that because there is a much larger amount of redundancy typically introduced by channel coding, the use of interleavers/deinterleavers and iterative processing is very effective for channel-coded CDMA systems, as demonstrated in [10].

(9) It is shown in [12] that the distribution of the residual interference-plus-noise at the output of the linear MMSE-MUD is well approximated by a Gaussian distribution. Thus, it can be in (9) represents the output of the following assumed that as its input symbol: equivalent AWGN channel having (10) in (10) is the equivalent amplitude of the th The parameter user’s signal, and is a zero-mean Gaussian noise sample 2There

is a typo in [10, eq. (42)]. The term A should be changed to A

.

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Fig. 2. Performance comparison of different decoding schemes in a CDMA system with two users (the first user is illustrated). Sources are 256 256 monochrome images, and the channel is AWGN.

formance of different decoders is measured at different values , where of the channel signal-to-noise ratio (CSNR) is the energy per bit. The simulation results were obtained for the case where the two users’ amplitudes are equal, i.e., . Note that also shown in Fig. 2 is the performance of the Hadamard-based optimal decoder to serve as the upper bound. The advantage of the proposed decoder with soft-output MUD and soft VQ decoding over the table-lookup decoder can be clearly observed from Fig. 2 for each type of MUD, especially at the low-to-medium CSNR region. Such an advantage can be achieved with essentially the same decoder complexity (as a function of the number of users). Another observation is that there seems to be a little performance improvement by the use of MMSE-MUD over DC-MUD in this case. This is due to the fact that the level of MAI in a two-user system is typically small, and the two MMSE-MUD and DC-MUD perform fairly close. The advantage of using MMSE-MUD over DC-MUD becomes more evident when a system with a larger number of users (i.e., a higher level of MAI) is considered next. Here the system has four users. In order to make the results more general, different decoders are tested for “standard” synthetic data produced by a Gauss–Markov source. The Gauss–Markov source has been widely used in VQ research, because the statistical properties of the source can be easily adjusted [5]–[8], [13]–[18]. Specifically, the individual user’s source is modeled as a zero-mean, unit-variance, stationary, and first-order Gauss–Markov random process with correlation , coefficient . The source is described by is an independent and identically distributed where . The parameters of the Gaussian process with variance VQ used in the simulations are . The VQ was trained for the Gauss–Markov source having and a noiseless channel. The VQ codevectors are then given good IAs based on the LISA algorithm [13]. For this VQ, the entropy is 2.88 bits, and the signal-to-distortion ratio, which is the highest achievable value of the SNR, is 9.4 dB. Also, a simple channel model of four users and the following cross correlation matrix [6] is considered:

2

with variance . The expressions for shown in [10] to be

and

are

and . It is then simple to show that the a posteriori LLR delivered by the soft MMSE-MUD is given by . On the other hand, the soft output of the DC-MUD for the th user is , where is a Gaussian random variable with zero mean and variance . The a posteriori LLR provided by the soft DC-MUD is thus . Finally, the computational complexity of the proposed decoder based on either the soft-output MMSE-MUD or the soft, which is clearly lower output DC-MUD is about than that of the US-HMD. IV. NUMERICAL RESULTS AND COMPARISON Simulation results using real images are first presented to compare the proposed decoder with other decoders described in Section II. In designing the VQ codebook, 20 different 512 512 monochrome images, including “baboon,” “bridge,” “pepper,” and “f16” images, are used as the training data. The pixels of all images are represented by 8 bits. The codevector dimension and and , the number of codevectors are set to respectively. The compression ratio is thus bits/pixel. The IAs of the codevectors are based on the LISA algorithm [13]. A two-user system with a correlation coefficient of 0.7 between the two users’ signature waveforms is simulated over an AWGN channel. User 1 transmits the “Lena” image, while user 2 sends the “Zelda” image. Both images are not included in the training data. Fig. 2 presents the peak signal-to-noise ratio (PSNR) of the received image data for user 1, which is de. The perfined as PSNR

(11)

The performance of VQ decoders is measured in terms of the , versus the output SNR, SNR CSNR . All the simulations for this system were obtained with 120 000 source samples. Also, for simplicity, it is assumed that all users’ amplitudes are equal. Fig. 3 compares the performance of the proposed soft decoder with that of the table-lookup decoder and the optimal HMD over an AWGN channel. Observe that all the decoders can offer the highest achievable value of the SNR (which is 9.4 dB for the selected VQ) at a high CSNR region (more than 10 dB). The superiority of the proposed soft decoder over the table-lookup hard decoder can also be clearly seen. Moreover, there is a clear advantage of using a more complicated soft-output MUD in the proposed decoder at a medium range of the CSNR (about 0–8 dB) for the system under consideration. More specifically,

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Fig. 3. Performance comparison of different decoding schemes in a CDMA system with four users (the first user is illustrated). The source is modeled by the first-order Gauss–Markov process, the channel is AWGN.

Fig. 5. Performance comparison of the proposed decoding schemes with the US-HMD. The source is modeled by the first-order Gauss–Markov process, the channel is AWGN.

a given type of MUD, the proposed soft decoder always outperforms the table-lookup decoder at no extra complexity. Due to the fading effects, it requires a much higher CSNR to achieve the same SNR level, compared with the case of an AWGN channel. Finally, Fig. 5 compares the performance of the proposed decoder with different types of soft-output MUD with that of the suboptimal US-HMD suggested in [6] over an AWGN channel. Note that with the four-user system under consideration, the complexity of the US-HMD is about the same as that of the proposed decoder (using either soft-output MMSE-MUD . The superiority of the proposed deor DC-MUD) if coder is thus clear from Fig. 5, where it is seen that US-HMD with produces the worst performance. A closer look at Fig. 5 reveals that the proposed decoder with soft-output MMSE-MUD still performs better than US-HMD, with at low CSNR and almost the same at high CSNR. Also observe that US-HMD with performs almost identically to the ). true US-HMD (i.e., Fig. 4. Performance comparison of different decoding schemes in a CDMA system with four users (the first user is illustrated). The source is modeled by the first-order Gauss–Markov process, the channel is flat Rayleigh fading.

with the optimal MUD, the proposed decoder (which is the same as the US-HMD in [6] in this case) can perform very close to that of the optimal HMD. At a low CSNR region (less than 0 dB), the performance degradation due to a lower complexity soft-output MUD becomes smaller. In particular, it is interesting to point out that the performance of the proposed decoder using a soft-output MMSE-MUD closely approaches that of the US-HMD at a low CSNR region. Note that the former decoding scheme has a much lower computational complexity, compared with that of the latter one. Simulations of the four-user system (with the same average received power for all users) employing different decoders are also carried out over a flat Rayleigh fading channel. The results are reported in Fig. 4. Again, the general observation is that for

V. CONCLUSIONS A suboptimal approach to VQ decoding over CDMA channels has been presented. The proposed decoder is built from a soft-output MUD, a soft bit estimator, and the optimal soft VQ decoders of individual users. The soft-output MUDs can be the OPT-MUD, the MMSE-MUD, or the DC-MUD. It has been demonstrated that the proposed decoding scheme offers a great flexibility to trade performance for receiver complexity over both AWGN and flat Rayleigh fading channels. The extension of the technique to a multipath fading channel is also being investigated. ACKNOWLEDGMENT The author would like to thank the anonymous reviewers and the Editor for their comments and suggestions, which improved the presentation of this letter.

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REFERENCES [1] C. E. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J., vol. 27, pp. 379–423, Jul./Oct. 1948. [2] T. M. Cover and J. A. Thomas, Elements of Information Theory. New York: Wiley, 1991. [3] N. Rydbeck and C. E. Sundberg, “Analysis of digital errors in nonlinear PCM systems,” IEEE Trans. Commun., vol. COM-24, no. 1, pp. 59–65, Jan. 1976. [4] K. A. Zeger and A. Gersho, “Zero redundancy channel coding in vector quantization,” Electron. Lett., vol. 23, pp. 654–655, Jun. 1987. [5] N. Farvardin, “A study of vector quantization for noisy channels,” IEEE Trans. Inf. Theory, vol. 36, no. 7, pp. 799–809, Jul. 1990. [6] M. Skoglund and T. Ottosson, “Soft multiuser decoding for vector quantization over a CDMA channel,” IEEE Trans. Commun., pp. 327–337, Mar. 1998. [7] M. Skoglund and P. Hedelin, “Hadamard-based soft decoding for vector quantization over noisy channels,” IEEE Trans. Inf. Theory, vol. 45, no. 3, pp. 515–532, Mar. 1999. [8] T. Ottosson and M. Skoglund, “Joint source-channel multiuser decoding for Rayleigh fading CDMA channels,” IEEE Trans. Commun., vol. 48, no. 1, pp. 13–16, Jan. 2000. [9] S. Verdú, Multiuser Detection. Cambridge, U.K.: Cambridge Univ. Press, 1998. [10] X. Wang and V. Poor, “Iterative (turbo) soft interference cancellation and decoding for coded CDMA,” IEEE Trans. Commun., vol. 47, no. 7, pp. 1046–1061, Jul. 1999.

[11] N. Görtz, “On the iterative approximation of optimal joint sourcechannel decoding,” IEEE J. Sel. Areas Commun., vol. 19, no. 9, pp. 1662–1670, Sep. 2001. [12] H. V. Poor and S. Verdú, “Probability of error in MMSE multiuser detection,” IEEE Trans. Inf. Theory, vol. 43, no. 5, pp. 858–871, May 1997. [13] P. Knagenhjelm and E. Agrell, “The Hadamard transform—A tool for index assignment,” IEEE Trans. Inf. Theory, vol. 42, no. 7, pp. 1139–1151, Jul. 1996. [14] J. Han and H. Kim, “Joint optimization of VQ codebooks and QAM signal constellations for AWGN channels,” IEEE Trans. Commun., vol. 49, no. 5, pp. 816–825, May 2001. [15] V. Vaishampayan and N. Farvardin, “Joint design of block source codes and modulation signal sets,” IEEE Trans. Inf. Theory, vol. 38, no. 7, pp. 1230–1248, Jul. 1992. [16] F. H. Liu, P. Ho, and V. Cuperman, “Joint source and channel coding using a nonlinear receiver,” in Proc. IEEE Int. Conf. Commun., Geneva, Switzerland, 1993, pp. 1502–1507. [17] , “Sequential reconstruction of vector quantized signals transmitted over Rayleigh fading channels,” in Proc. IEEE Int. Conf. Commun., New Orleans, LA, 1994, pp. 23–27. [18] N. Farvardin and V. Vaishampayan, “On the performance and complexity of channel-optimized vector quantizers,” IEEE Trans. Inf. Theory, vol. 37, no. 11, pp. 155–159, Nov. 1991.

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Combined Equalization for Uplink MC-CDMA in Rayleigh Fading Channels Ivan Cosovic, Student Member, IEEE, Michael Schnell, Senior Member, IEEE, and Andreas Springer, Member, IEEE

Abstract—In this letter, we propose the concept of combined equalization for uplink multicarrier code-division multiple access (MC-CDMA) and perform a theoretical analysis which shows that better single-user bounds than the classical matched-filter bounds are achieved with this new concept. Moreover, we illustrate how to properly design an uplink MC-CDMA transmitter and receiver for combined equalization, and show by Monte Carlo simulations that the improved single-user bounds are closely approached, even in the case of a fully loaded system. Index Terms—Combined equalization, multicarrier code-division multiple access (MC-CDMA), single-user bounds, uplink.

I. INTRODUCTION ULTICARRIER code-division multiple access (MC-CDMA) is a transmission technique which combines advantages of both code-division multiple access (CDMA) and orthogonal frequency-division multiplexing (OFDM) [1]. A considerable part of the research on MC-CDMA deals with bit-error rate (BER) minimization, the corresponding performance bounds, and how to improve MC-CDMA performance in order to approach these bounds more closely. All previous research has one thing in common: the investigated techniques are lower bounded by the classical matched-filter (MF) bound. Note, the classical MF bound is identical to the single-user performance in MC-CDMA with maximum ratio combining (MRC) at the receiver (Rx). In this letter, we propose the concept of combined equalization for uplink MC-CDMA, which enables us to go beyond the MF performance. The proposed concept operates on the premise that channel state information (CSI) is available at both transmitter (Tx) and Rx, and performs pre- and postequalization. The knowledge about the transmission channel can be made available, for example, by exploiting time-division duplex (TDD) to gather CSI at Tx needed for pre-equalization, and by performing channel estimation at Rx in order to obtain CSI needed for post-equalization.

M

Paper approved by V. A. Aalo, the Editor for Diversity and Fading Channel Theory of the IEEE Communications Society. Manuscript received March 29, 2004; revised November 29, 2004 and April 21, 2005. This work was carried out in part within the projects 4MORE (4G MC-CDMA Multiple-Antenna Systems On Chip For Radio Enhancements) and WINNER (Wireless World Initiative New Radio), supported by the European Commission under the framework of FP6 under Contracts IST-2002-507039 and IST-2003-507581. This paper was presented in part at the International Workshop on Multicarrier Spread-Spectrum, Wessling, Germany, September 2003, and in part at the IST Mobile and Wireless Communications Summit, Lyon, France, June 2004. I. Cosovic and M. Schnell are with the German Aerospace Center (DLR), Institute of Communications and Navigation, 82234 Wessling, Germany (e-mail: [email protected]; [email protected]). A. Springer is with the Institute for Communications and Information Engineering, University of Linz, 4040 Linz, Austria (e-mail: [email protected]). Digital Object Identifier 10.1109/TCOMM.2005.857143

For uplink MC-CDMA, both different post- and pre-equalization techniques have been proposed to increase performance. Since some of these techniques form the basis for combined equalization, a short literature overview is given in the following. Considering postequalization, it is well known that single-user detection (SUD) techniques for fully loaded uplink MC-CDMA result in poor performance [1]. However, multiuser detection (MUD) techniques improve uplink MC-CDMA performance significantly. In [2], it has been shown that the maximum-likelihood detector (MLD) for fully loaded uplink MC-CDMA is capable of almost reaching the classical MF bound. The remaining performance degradation for uncoded uplink MC-CDMA is only around 1 dB in signal-to-noise ratio (SNR). In addition, suboptimum MUD techniques have been investigated by several authors. In [3] and [4], joint detection is investigated, in [5] and [6], parallel interference cancellation (PIC), and finally, in [7], a more superior cancellation technique, i.e., successive interference cancellation (SIC). Pre-equalization for uplink MC-CDMA has also been addressed in the literature recently. In [8], a pre-equalization technique that maximizes the signal-to-interference-plus-noise ratio (SINR) is introduced. Moreover, several suboptimum pre-equalization techniques have been developed by several authors, e.g., [9], [10]. Analysis of the single-user case of the post- and pre-equalization techniques proposed in [1]–[10] reveals that their performance is lower bounded by the classical MF bound. The contributions of this letter are summarized in the following. 1) We analytically show that for MC-CDMA, a considerably better single-user performance than the classical MF performance can be achieved by employing the concept of combined equalization. 2) We propose a flexible pre-equalization technique, termed generalized pre-equalization (G-pre-eq), which comprises both several well known and a class of new pre-equalization techniques, and is especially suited for use within combined equalization. 3) We develop a combined-equalization approach for coded uplink MC-CDMA which is based on the combination of G-pre-eq at Tx and an advanced Rx structure, and is capable of closely approaching the improved single-user bounds, even in the case of a fully loaded system. 4) To illustrate the practical significance of the combined-equalization approach for coded uplink MC-CDMA, we give numerical results obtained from Monte Carlo simulations. The results are compared with the classical MF bound and corresponding post- and pre-equalization techniques. It turns out that the classical MF bound, as well as the results obtained with the

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represents the diagonal channel matrix for where user , with the diagonal elements , . A diagonal element represents the fading coefficient on the th represubcarrier for user . The vector sents the AWGN with variance in each component. As shown in Fig. 1(b), the received signal is detected with a generalized detector which comprises detection, despreading, demapping, deinterleaving, and decoding.

Fig. 1. Block diagram of the uplink MC-CDMA transmission system. (a) Tx. (b) Rx.

corresponding post- and pre-equalization techniques, are outperformed significantly by the proposed approach. II. UPLINK MC-CDMA TRANSMISSION SYSTEM We consider a synchronous low-mobility uplink TDD/MCof subcarriers. The CDMA system with a total number subsystems of subcarriers. system is divided into This flexible scheme is better known as MC-CDMA with -modification [1]. Note that denotes the number of parallel data symbols transmitted per OFDM symbol for each denotes the number of different, parallel user user, while groups per OFDM symbol. Without loss of generality, we users ( ). The concentrate on a single subsystem of users are separated by orthogonal Walsh–Hadamard spreading sequences of length . Moreover, we assume that the corresubcarriers are uniformly spread over the signal sponding bandwidth so as to better exploit frequency diversity of the channel, i.e., they are frequency-interleaved with the remaining subcarriers. We denote with , , the subcarriers in the considered subsystem. The block diagram of the th, , uplink MC-CDMA Tx of the considered subsystem is shown in Fig. 1(a). After channel coding, outer interleaving , and symbol mapping, the complex-valued is spread by a unit-energy spreading sequence symbol . The spreading process results in given by the sequence (1) where denotes transposition. Then, pre-equalization is applied, resulting in a new sequence (2) where is a diagonal pre-equalization matrix with diagonal elements , . Finally, the pre-equalized sequence is OFDM modulated onto the corresponding out of subcarriers and transmitted. OFDM comprises the inverse fast Fourier transform and addition of a guard interval by cyclic extension of the OFDM symbol. The block diagram of an uplink MC-CDMA Rx of the considered subsystem is shown in Fig. 1(b). After the inverse OFDM (IOFDM) operation, the received signal results in (3)

III. COMBINED EQUALIZATION SINGLE-USER BOUNDS FOR MC-CDMA In order to explore the potential benefits of combined equalization, the maximum achievable performance improvements are evaluated by considering the single-user case. As a result, new, considerably improved single-user bounds are obtained, proving the concept of combined equalization and making available a quantitative measure of possibly achievable performance improvements. In this letter, two different combined-equalization techniques are considered. The first technique is called selection diversity (SD) combined equalization, and is the optimal single-user combined-equalization technique. The second technique is actually a class of techniques, which represents a certain tradeoff between the classical MF and the SD combined-equalization technique. This class of techniques is obtained by combining G-pre-eq with postequalization MRC (post-eq MRC), and is named G-pre-eq-MRC combined equalization. Note, for convenience, the indexes that mark different users are omitted in this section, since the single-user case is assumed. The received signal is processed in the generalized detector at Rx. After equalization and despreading, the soft estimate of the transmission symbol is given by (4) For postequalization, a SUD technique is applied, which is represented by the elements , , of the diagonal postequalization matrix . In the following, it is shown that in the case of combined , i.e., the transmitted enequalization, the SNR given by over the one-sided noise spectral density ergy per bit at Rx, differs from the SNR at Rx after postequalization. The SNR is improved, compared with , leading to an improved single-user performance. Note that in the singleuser case for MC-CDMA with post- or pre-equalization applied alone, both SNRs and single-user bounds are identical. Before we proceed with the SNR analysis, from (4), we calculate the generalized expression for the SNR which equals

(5)

and is the basis for the SNR analysis performed in the following. A. SD Single-User Bound In the single-user case, the optimal combined-equalization technique is based on the application of the SD criterion at both

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Tx and Rx. SD applied solely for post- or pre-equalization is introduced in [11]. In these two cases, the SNR obtained with SD is worse than the SNR obtained with MRC at Rx. However, SD is the optimal technique, in the sense of SNR maximization, for the case of combined equalization. The pre-equalization coefficients for SD (pre-eq SD) are chosen according to

same optimal subcarrier for transmission. Such a situation instantaneously leads to an unacceptable high BER caused by multiple-access interference (MAI).

otherwise

(6)

while the SD postequalization coefficients are given by

otherwise.

(7) returns the index or position of the specific The function returns the maximal value out of a specelement, while ified set. Note, phase correction of the fading coefficient can be performed on any of the two sides, Tx or Rx. Here, phase correction is performed at Rx. From (5)–(7), the SNR can be easily obtained, and is equal to (8) In the limit and under the assumption of uncorrelated Rayleigh fading on each subcarrier, (8) transforms to (9) Moreover, in the case of finite and uncoded binary or quarternary phase-shift keying (BPSK/QPSK) transmission, the coris equal to responding BER

(10) Due to the space limitations of this letter, we skip derivation details which lead to (10). It is obvious that allocating the whole transmit power to the best subcarrier ensures the maximization of the SNR of the received symbol when there is only one symbol to be transmitted over a plurality of subcarriers. Note, this statement is valid if sole power loading is considered without bit loading. Techniques which take into account power loading together with bit loading and aim at SNR maximization for a certain target data rate have been thoroughly investigated in the literature, e.g., [12] and [13]. Such approaches are beyond the scope of this letter, since bit-loading techniques are difficult to apply within MC-CDMA due to its CDMA component. Thus, this letter concentrates on “standard” MC-CDMA with a fixed symbol-mapping scheme [1]. Combined equalization based on SD is mainly of theoretical importance and has no significant practical value for MC-CDMA, since the considered system reduces to a single-user, single-carrier system with no MC-CDMA-specific features. Moreover, in the multiuser case, SD does not lead to optimal performance, since several users might choose the

B. G-pre-eq-MRC Single-User Bounds In the following, the single-user bounds for combined equalization based on the combination of the proposed G-pre-eq technique with post-eq MRC are considered. G-pre-eq corrects the phase of the fading channel and weights the transmission signal with a coefficient proportional , where is any real number. The pre-equalization to coefficients are given by (11)

and are normalized such that the transmit power is the same as in the case without pre-equalization, i.e., power-constrained preequalization is applied. G-pre-eq is a unified approach to preequalization and comprises several well-known pre-equalization techniques, such as pre-equalization MRC (pre-eq MRC) , pre-equalization equal gain combining (pre-eq EGC) for , and pre-equalization zero forcing (pre-eq ZF) for for . For more information about pre-eq MRC, pre-eq ZF, and pre-eq EGC, please refer to [9]. G-pre-eq is especially designed for combined equalization, since it allows going beyond the well-known pre-equalization techniques. For application of are of G-pre-eq within combined equalization, values special interest, since then G-pre-eq-MRC leads to improved single-user bounds, which are the better the larger is chosen, as will be shown in the following. At Rx, we apply the optimal SUD technique for the singleuser case, i.e., post-eq MRC. Its coefficients are given by (12) It is noteworthy that postequalization coefficients in (12) are adapted not only to distortions caused by the fading channel, but also to distortions caused by pre-equalization. can be From (5), (11), and (12), the corresponding SNR calculated

(13)

This SNR depends on the chosen parameter of the G-pre-eq , (13) becomes equal to technique at Tx. In the case the SNR of post-eq MRC, which is identical to the classical MF , the SNR becomes better, leading to performance. For improved single-user bounds. Moreover, it can be easily shown that for any , , i.e., we can conclude that the SNR is a monotonously nondecreasing function in dependence of the parameter . Note, the equality sign is valid only in the special case of flat fading over all subcarriers. Finally, , (13) reduces to (8). for

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TABLE I REQUIRED SNR FOR SEVERAL VALUES p TO ACHIEVE BER OF P = 10 FOR CODED MC-CDMA APPLYING COMBINED EQUALIZATION. QPSK, L = 16, K = 1, R = 1=2

Assuming and using (16) and (17) from the Appendix, it is easily shown that under the assumption of uncorrelated Rayleigh fading on each subcarrier, (13) reduces in the to limit

(14) is the gamma function [14]. Thus, the SNR is linearly where . Note that the dependent on the parameter in the limit (i.e., classical MF bound) and (i.e., two cases SD combined equalization bound) are comprised in (14). Moreover, for uncoded BPSK/QPSK transmission, the corresponding is given by BER (15) denotes the complementary error function. Note, where to the best of our knowledge, a closed-form solution of the corresponding BER formula for finite is not possible to obtain or . except for some special cases, e.g., Table I summarizes the SNR given in required to for different values of for conachieve a BER of volutionally coded single-user MC-CDMA in an uncorrelated Rayleigh fading channel. The results are obtained by Monte Carlo simulations. A standard convolutional code with memory is applied for coding, and spreading is per6 and rate . In this case, formed by a spreading sequence of length the optimal theoretical SD combined equalization single-user . bound is approached relatively closely already for Thus, by setting , only relatively small additional improvements can be achieved. Moreover, considering a multiuser environment, MAI will be increased as grows, making it more difficult to cancel MAI at Rx, as will be shown in the following section. Note, G-pre-eq is not the only solution capable of redistributing transmission power so as to improve single-user bounds. There are many possible solutions which can achieve similar effects. Nevertheless, G-pre-eq is an interesting pre-equalization technique, because it achieves the desired effect, unifies several well-known pre-equalization techniques, and has a simple mathematical interpretation. IV. CONCEPT OF COMBINED EQUALIZATION FOR UPLINK MC-CDMA In Section III, it has been shown that the single-user bounds for combined equalization, applying SD or G-pre-eq-MRC (already for ), are considerably better than the conventional MF bound, proving the potential benefit of combined equalization. To address the significance of these bounds in a multiuser environment, we propose an appropriate concept of combined

equalization for uplink MC-CDMA in this section. A theoretical approach toward optimal combined equalization with respect to BER performance is difficult to follow, since the optimal power assignment at Tx depends on the MAI-suppression capabilities at Rx and, thus, requires a combined optimization of Tx and Rx. Moreover, the optimal power assignment also depends on several system parameters, which include channel coding, symbol mapping, length of spreading sequence, and system load. The proposed pragmatic approach is to use G-pre-eq, which introduces the parameter for optimizing the power assignment, with the goal of achieving the minimal BER for a given Rx structure and for a specific set of system parameters. By enlarging , the power is redistributed over the available subcarriers in a more and more unequal way. Thus, the single-user bound is further improved, since the corresponding SNR at Rx is improved. However, MAI is increased as grows, since pre-equalization no longer compensates the channel influence, but aims at improving the SNR. If the Rx structure is able to eliminate almost all MAI, the corresponding combined equalization single-user bound will be closely approached. According to the above statements, there is an optimal value , for which BER is minimized at a given SNR or, alternatively, for which the required SNR is minimized at a given BER. The optimal value of the parameter can be determined during the development of a system for the corresponding Rx structure and the different sets of system parameters, and saved in a lookup table. Afterwards, in an operational system, the parameter can be read from the precalculated lookup table. If the system load is not taken into account, the parameter can be adjusted to the worst case, i.e., full system load. All other parameters of interest, i.e., channel coding, symbol mapping, and length of spreading sequence, are always known at the uplink Tx, as they are required to properly modulate the transmission signal. The simplicity of the proposed combined-equalization approach is reflected in the fact that the very complex problem of joint Tx-Rx optimization is reduced to a simplified problem, in which a single optimization parameter needs to be determined. Note, the goal of this section is not the investigation of novel uplink MC-CDMA Rx structures, but the demonstration of the proposed combined-equalization concept. Thus, at Rx we apply a “standard” PIC detector, similar to the one described in [5] and [6]. The considered PIC detector operates with soft bits and consists of a minimum mean-square error MUD in the initial cancellation stage, and post-eq MRC in the following cancellation stages. Moreover, we note that in the case of combined equalization, detection is performed with respect to a pre-equalized channel, which takes into account both the fading channel and the pre-equalization influence, similar to what was done in (12). Due to the fact that a nonlinear detection technique, namely PIC, is applied at Rx, a closed-form solution for the optimization of the parameter with the goal to minimize BER is difficult to obtain. However, as we show in Section V, the parameter can be optimized by Monte Carlo simulations. V. NUMERICAL RESULTS In this section, performance results of various coded uplink MC-CDMA systems with pre-, post-, or combined equalization

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Fig. 2. Performance of coded uplink MC-CDMA for combined equalization, applying G-pre-eq together with PIC in dependence of the G-pre-eq parameter p; 1, 3, or 5 PIC iterations, QPSK, L = 16, K = 16, R = 1=2.

are compared within a mobile radio environment. The performance results are obtained by Monte Carlo simulations. The . The considered system parameters are SNR is given in as follows. The transmission bandwidth is 60 MHz and the carrier frequency is fixed to 5 GHz. The number of subcarriers is , while sequences of length are used modification. The guardfor spreading, thus allowing interval duration exceeds the maximum delay of the mobile radio channel. The MC-CDMA frame consists of 24 subsequent MC-CDMA symbols. For channel coding, a standard convoluis used. tional code with memory 6 and code rate The coded data symbols are QPSK mapped. We assume that the fading coefficients within a subsystem are Rayleigh distributed and uncorrelated due to appropriate frequency inter) leaving. Moreover, only fully loaded systems ( are considered. Fig. 2 shows the BER of coded uplink MC-CDMA with combined equalization, applying G-pre-eq together with PIC at Rx. Simulation results are given for 1, 3, and 5 PIC iterations and for a fixed SNR, while the parameter for G-pre-eq is varied in order to find the optimal value which minimizes the BER. The corresponding SNR values are chosen to lead to an approxand . They are set to 1.5, 0.0, imate BER between and 0.0 dB, respectively. It can be seen that in all three cases, the minimal BER is achieved by setting . In Fig. 3, the performance of uplink MC-CDMA with combined equalization, applying G-pre-eq with together with PIC at Rx is shown. PIC with three iterations is considered. As references, the AWGN performance, the classical MF , and the SD combound, the G-pre-eq-MRC bound for bined-equalization bound are given. As additional references, performance results of uplink MC-CDMA applying PIC [5], and uplink MC-CDMA applying pre-equalization which maximizes SINR [8] are given. With the combined-equalization con, and cept, the corresponding G-pre-eq-MRC bound for even the SD combined equalization bound, are approached quite

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Fig. 3. Performance of coded uplink MC-CDMA with pre-, post-, or combined equalization; QPSK, L = 16, K = 16, R = 1=2.

closely. Moreover, the proposed combined-equalization concept outperforms the classical MF bound, as well as corresponding post- and pre-equalization references, significantly. This reflects the uniqueness of the proposed concept, in which the classical MF bound is no longer valid as the ultimate MC-CDMA performance bound, as it is for other known concepts [1]–[10]. Finally, we note that the combined-equalization concept is not limited to the combination of G-pre-eq and PIC. Instead of PIC, we can, for example, employ SIC at Rx, and expect some further improvements. However, as we concentrate on the principle of combined equalization in this letter, we leave such investigations for possible future work. VI. CONCLUSIONS The concept of combined equalization is proposed and investigated for uplink MC-CDMA in this letter. The potential benefits of combined equalization are proven by a theoretical analysis, which states that new, considerably improved MC-CDMA single-user bounds are valid for combined equalization. Moreover, numerical results show that these improved single-user bounds are closely approached, even for fully loaded uplink MC-CDMA. As a consequence, the proposed uplink MC-CDMA concept based on combined equalization significantly outperforms the classical MF bound, as well as corresponding pre- and postequalization concepts. APPENDIX Assuming uncorrelated Rayleigh fading on each subcarrier which, on average, does not amplify or attenuate the transmission signal, and bearing in mind that in such a case (16) is valid [14], where (13) reduces in the limit

denotes mathematical expectation, to (14). Moreover, to obtain

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(14), we have implicitly used Kolmogorov’s strong law of large numbers [15] in order to conclude that (17) REFERENCES [1] K. Fazel and S. Kaiser, Multi-Carrier and Spread Spectrum Systems. New York: Wiley, 2003. [2] M. Schnell, Systeminhärente Störungen bei “Spread-Spectrum”—Vielfachzugriffsverfahren für die Mobilfunkübertragung, ser. 10. Düsseldorf, Germany: VDI Verlag, Fortschritt-Berichte VDI, 1997. [3] S. Verdu, Multiuser Detection. Cambridge, U.K.: Cambridge Univ. Press, 1998. [4] F. Bader, S. Zazo, and J. M. Borrallo, “Decorrelation MUD for uplink MC-CDMA in an uplink transmission mode,” in Proc. Int. Workshop Multi-Carrier Spread-Spectrum, Related Topics, Sep. 2001, pp. 173–180. [5] V. Kühn, “Combined MMSE-PIC in coded OFDM-CDMA systems,” in Proc. IEEE Global Telecommun. Conf., Nov. 2001, pp. 231–235. [6] L. Sanguinetti, M. Morelli, and U. Mengali, “Channel estimation and tracking for MC-CDMA signals,” Eur. Trans. Telecommun., vol. 3, pp. 249–258, May/Jun. 2004.

[7] J. G. Andrews and T. H. Meng, “Performance of multicarrier CDMA with successive interference cancellation in multipath fading channel,” IEEE Trans. Commun., vol. 52, no. 5, pp. 811–822, May 2004. [8] D. Mottier and D. Castelain, “SINR-based channel pre-equalization for uplink multi-carrier CDMA systems,” in Proc. IEEE Int. Symp. Pers., Indoor, Mobile Radio Commun., Sep. 2002, pp. 1488–1492. [9] I. Cosovic, M. Schnell, and A. Springer, “On the performance of different channel pre-compensation techniques for uplink time division duplex MC-CDMA,” in Proc. IEEE Veh. Technol. Conf., Oct. 2003, pp. 857–861. [10] P. Bisaglia, N. Benvenuto, and S. Quitadamo, “Performance comparison of single-user pre-equalization techniques for uplink MC-CDMA systems,” in Proc. IEEE Global Telecommun. Conf., Dec. 2003, pp. 3402–3406. [11] D. G. Brennan, “Linear diversity combining techniques,” Proc. IRE, vol. 47, pp. 1075–1102, Jun. 1959. [12] P. S. Chow, J. M. Cioffi, and J. A. C. Bingham, “A practical discrete multitone transceiver loading algorithm for data transmission over spectrally shaped channels,” IEEE Trans. Commun., vol. 43, no. 2-4, pp. 773–775, Feb.-Apr. 1995. [13] R. Fischer and J. Huber, “A new loading algorithm for discrete multitone transmission,” in Proc. IEEE Global Telecommun. Conf., Nov. 1996, pp. 724–728. [14] J. G. Proakis, Digital Communications. New York: McGraw-Hill, 1995. [15] W. Feller, An Introduction to Probability Theory and its Applications, 3rd ed. New York: Wiley, 1968.

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Generalized APP Detection of Continuous Phase Modulation Over Unknown ISI Channels Anders Hansson, Member, IEEE, and Tor Aulin, Fellow, IEEE

Abstract—Techniques for computing soft information in the presence of unknown intersymbol interference are presented, with a particular focus on iterative detection of serially concatenated continuous phase modulation. The techniques are centered around the recursive least-squares algorithm, thus enabling unsupervised detection. In particular, we employ bidirectional estimation. Index Terms—Channel estimation, continuous phase modulation (CPM), intersymbol interference (ISI), iterative decoding, maximum a posteriori (MAP) detection.

I. INTRODUCTION N THE WAKE of the tremendous success of iterative detection over additive white Gaussian noise (AWGN) channels [1], similar techniques have been proposed for detection over channels characterized by, for example, frequency-flat Rayleigh fading [2]. In contrast to [2], where the channel is described in statistical terms, most papers are devoted to the somewhat artificial problem of iterative detection over known channels.1 In particular, many of the iterative schemes that have been suggested for communication in the presence of known intersymbol interference (ISI) may be sorted under the rubrics of turbo equalization [3]. If the receiver instead lacks perfect knowledge of the channel, it somehow needs to estimate (explicitly or implicitly) a number of unknown parameters. Anastasopoulos and Chugg proposed bidirectional estimation as a means of achieving a desired estimation diversity [4]. In this letter, we employ their ideas in the context of iterative detection of serially concatenated continuous phase modulation (CPM) and unknown ISI.

I

Consider a block of data symbols , linearly . Assume that the modulated and transmitted at the rate transmission is subject to ISI. The received waveform, observed in AWGN, can then be expressed as (1)

Paper approved by W. E. Ryan, the Editor for Modulation, Coding, and Equalization of the IEEE Communications Society. Manuscript received December 9, 2003; revised November 20, 2004 and April 2, 2005. This work was supported in part by the Swedish Foundation for Strategic Research under Personal Computing and Communication Grant PCC-9706-01. A. Hansson is with Discrete Simulation Sciences, Los Alamos National Laboratory, Los Alamos, NM 87545 USA (e-mail: [email protected]). T. Aulin is with the Department of Computer Engineering, Chalmers University of Technology, SE-412 96 Göteborg, Sweden (e-mail: [email protected]). Digital Object Identifier 10.1109/TCOMM.2005.857164 thermal noise is, of course, unknown.

(2) is an observation vector comprising samples in which , is a matrix of data from symbol interval 0 up to consymbols, is an unknown parameter vector, and sists of independent noise samples with variance , is the cutoff rate of the lowpass filter (which is aswhere sumed to be ideal).3 The reader is referred to [8] for a detailed derivation of (2). We would like to employ an algorithm that functions as a soft inverse, i.e., an algorithm that computes soft information on the modulated symbols. In order to handle the parameter vector, we choose to explore the generalized likelihood principle [7], in which the receiver performs estimation and detection jointly. Moreover, let us employ the concept of bidirectional estimation/detection, which has been suggested by Anastasopoulos and Chugg [4]. Given the state-transition diagram of the encoder/modulator, it is straightforward to arrange the data hypotheses in a search tree. The generalized a posteriori probability (GAPP) of an edge in this tree may be calculated by averaging over all consistent data hypotheses4 GAPP

II. JOINT BIDIRECTIONAL CHANNEL ESTIMATION/ SOFT SYMBOL DETECTION

1The

be comMoreover, let the (noiseless) channel response , and let the noise be characterized pactly supported on by the covariance function . If the receiver front end employs lowpass filtering followed by (uniform, possibly fractionally spaced) sampling at the Nyquist rate, it becomes straightforward to derive a discrete representation of (1)2

(3) Here, is the hypothesized maximum-likelihood (ML) estimate of the parameter vector, which is readily found

ht ht

2If ( ) is known, it is straightforward to devise an optimal front end that discretizes the received, continuous-time waveform into a sufficient statistic [5]. Here, ( ) is unknown, and the issue of designing an optimal front end is nontrivial [6]. It should be noted that sampling leads to suboptimal detection; bandlimited pulses are not time-limited, and require an infinite number of samples (taken over the whole time axis) to attain statistical sufficiency, whereas time-limited pulses are not bandlimited and require infinitely dense sampling [7]. 3In practice, if is greater than the practical bandwidth of the modulation pulse (e.g., indirectly defined by the 99% inband power measure), and (loosely speaking) if each decision is based on a great number of samples, the performance loss due to sampling should not dominate the overall performance of the receiver. On the other hand, if the decisions are based on short data blocks, sampling may very well be significantly inferior to other expansion schemes. This is demonstrated in [8]. 4A proportionality constant, identical for all data hypotheses, is omitted. The GAPPs must sum up to one at each time index, and normalization can thus be done at a later stage.

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by the standard method of normal equations, , where denotes the Moore–Penrose pseudoinverse of [9]. Following [4], the the hypothesized data matrix least-squares (LS) metric is partitioned into three terms (4) The first term is associated with a forward estimate of the parameter vector, whereas the second term is associated with a backward estimate of the same quantity. Both these terms are efficiently computed by means of the recursive least-squares (RLS) algorithm [10]. Finally, the third term can be interpreted as a penalty term, which raises the total LS metric in case of mismatching forward/backward estimation processes. For brevity, we do not present the explicit expression for this penalty term, and the reader is instead referred to [4] and [8] for details. It should be noted, however, that the complexity of this term increases with the dimension of the unknown parameter vector , as it involves three matrix inverses of size , is the sampling rate. where A search tree that starts at time index 0 is grown, where the incremental metric of each emerging edge is computed by a forward-directed RLS algorithm. From a given search depth, only a subset of the nodes are expanded, preferably nodes with low metrics. Also, in a per-survivor processing fashion [11], each survivor node keeps and updates its own individual vector of estimated parameters, based on its own hypothesized data sequence. In a similar manner, a time-reversed search tree is grown . At every time epoch, and pruned, starting from time index we complete sequences in this backward-directed tree with sequences from the previously grown forward-directed tree. Note that every completed sequence can be associated with an exact ML estimate (on a given hypothesis), which is an advantage, compared with force-folding the tree search into a trellis search. It is convenient to employ a breadth-first search algorithm [12], since a time-regular search front facilitates the completion step. In this letter, we elaborate on the Viterbi algorithm. A. Details of the Tree Search in the Case of CPM For simplicity, we discuss the forward-directed tree search in the case of minimum-shift keying (MSK) [13]. To comply with (1), MSK is represented as offset quadrature phase-shift keying, with a half-cycle sinusoid having period as modulation pulse, i.e., the transmission format is with data symbols

for

and

for

and modulation pulse for . Here, is the th bit (the bits are differentially encoded), while denotes energy per bit. Using this description of MSK, Fig. 1 illustrates how two parent nodes, “Node 0” and “Node 4,” are expanded into four

Fig. 1. phase.

Illustration of how the forward tree is grown during the expansion

children nodes, “Node a,” “Node b,” “Node A,” and “Node B.” Each node is associated with a set of hypothesized data symbols. These symbols have been encircled in Fig. 1. The numbers below the circles represent the corresponding bits. Moreover, the metrics of the children nodes are refinements of their parent’s metrics, where the refinements are provided by the RLS algorithm, and this computation resembles the add step of the Viterbi algorithm. Next, the children nodes can be grouped in pairs of two, such that the last differential bits are identical within each pair. In our example, we choose , and “Node a” and “Node A” are then grouped, as well as “Node b” and “Node B.” The add-compare-select unit subsequently compares the LS metrics of the two nodes in each pair, and declares the node having the lowest metric to be the survivor node of that pair. Just like in the Viterbi algorithm, the survivor sequences are thus forced to have different recent (differential) bits. Put differently, the survivor sequences comprise all combinations of the last bits. Note that only one path enters each node. This means that each child node in Fig. 1 could have been built up by four data symbols instead of five, because the very first symbol can be found by a traceback operation, i.e., by backward following the branch that enters the child node. Now, look at “Node a” and “Node A.” Apart from the first data symbol, we see that the two nodes correspond to sign-reversed data words. One could then ask whether it is somehow possible to always continue the search from, say, “Node a”? To answer this question, imagine that the metrics of “Node a” and “Node A” have already been computed. If “Node a” is selected to be the survivor node, there is, of course, no problem in continuing the search from “Node a.” When “Node A” is selected to be the survivor node, we would still like to continue from “Node a.” Since an imaginary transition from “Node A” to “Node a” involves a sign switch of the hypothesized sequence, we must also switch the sign of the associated channel estimate. Thus, if “Node A” is selected to be the survivor node, we store its metric in “Node a,” together with

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Fig. 2. Illustration of how a vector of eight nodes is updated as the forward tree is selectively expanded. Overlapping symbols have been marked with double circles, and a dashed transition implies a need for sign-reversing the associated parameter estimate.

a sign-switched copy of the estimate calculated for “Node A.”5 The search is then continued from “Node a” as if the sign-reversed sequence had been the original candidate. Fig. 1 is finally redrawn in Fig. 2 such that all its nodes are shown. Before explaining the details of Fig. 2, it should be stressed that the concept of mergers is invalid; the figure merely illustrates how a regular data structure may account for the metric recursion. In spite of the similarity to one recursion step of the Viterbi algorithm, one must remember that our search is performed in a tree. Now, the time index in Fig. 2 is an odd number, and we have marked all transitions involving a sign switch with a dashed arrow. The node vector at time is built up by the eight (differential) bit hypotheses . Each node (at time ) may also be associated with a combina. tion of four hypothesized data symbols More importantly, each node represents two sequences, where the two sequences are sign-reversed replicas of each other. The backward tree is searched in a similar manner, and we therefore omit details.6 Regarding the sequence completion, it is important to notice that there will be constraints on how we are allowed to combine partial sequences in the forward tree with partial sequences in the backward tree. This is due to the channel memory, and is clear from observing how the 5There is also a Hessian matrix associated with the RLS algorithm, and whenever we change the sign of the parameter estimate, we must not forget to change the sign of the associated Hessian, as well. See [8] for details. 6The backward recursion is not a mirrored replica of the forward recursion, due to the direction of the channel memory (which leads to minor differences). See [8] for details.

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Fig. 3. Illustration of the completion phase when L = 3 and eight nodes are kept in each node vector. The L 1 = 2 overlapping symbols have been marked with double circles, and a dashed completion implies a need for sign-reversing the associated backward survivor sequence.

0

backward LS metric turn, depends on

depends on

, which, in [8]. The symbols are also part of the forward LS metric [8], i.e., any given forward sequence must have the symbols in common with a backward sequence, if the two partial sequences are to be united into a complete sequence. As an example, Fig. 3 illustrates the , and on the addicompletion phase for the case when tional assumption that we employ a search strategy in which eight nodes are kept in each search tree. The eight nodes of the forward search are stored in a forward-search node vector, while the eight nodes of the backward search are similarly kept in a backward-search node vector. It should also be mentioned that we have assumed the completion time index to be an odd number. Since , there are 2 ( ) overlapping data symbols, namely and , and these two symbols have been marked with double circles in Fig. 3. If we instead turn our attention to the corresponding differential bits, we will find bit that needs to be identical for the that there is only forward/backward sequences. This differential bit is denoted in Fig. 3. Now, consider the forward survivor sequence that ends in “Node 0” of the node vector associated with the forward-directed tree search. It can be seen how this forward sequence may be completed with backward survivor sequences ending in “Node 0,” “Node 2,” “Node 4,” and “Node 6” of the backward-search node vector. Recall that each node represents two sequences, which are sign-reversed replicas of each other. This means that we can switch the sign of any backward survivor sequence, if we at the same time switch the sign of its associated parameter estimate. By switching the sign of a backward

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survivor sequence, we simply assume that the MSK termination state is phase-shifted 180 degrees. If the transmitter and receiver only agree to communicate an even (or odd) number of data symbols, there is an inherent phase ambiguity of the termination state. Returning to Fig. 3 and the forward sequence ending in “Node 0,” we see how the backward sequences ending in “Node 2” and “Node 4” would have to be sign-reversed, and the associated completions have consequently been marked with dashed lines. In order to reduce the computational complexity, we may digroups, where vide the backward survivor sequences into each group corresponds to a particular differential hypothesis . Let us call these groups backward survivor classes. In Fig. 3, there are two backward survivor classes: the nodes {0, 2, , and the nodes {1, 3, 5, 7} corre4, 6} corresponding to . Complexity reduction is obtained by comsponding to pleting each forward survivor with only one backward survivor, namely, the backward survivor having the lowest LS metric of all candidates in its survivor class. If a better GAPP approximation is desired, we could instead consider the two best metrics within each class, etc.

Fig. 4. Influence of the number of sequences used to approximate each GAPP. The error rate is plotted for the first three decoding iterations of a rate-1/2 repetition code in series with a 512-bit interleaver and MSK. 32 nodes and 25 pilot symbols were used in the forward/backward search. The performance of the standard forward-backward algorithm over a known channel is shown for reference.

III. NUMERICAL EXAMPLE In this section, we study the error performance of a serially concatenated system in which the inner code is the continuous phase encoder (CPE) that is inherent in MSK. Given our particular avatar of MSK, the minimum weight of input sequences generating error events is two. In accordance with Benedetto et al. [14], we should then expect an interleaver gain over the AWGN channel. Also, based on an upper AWGN bound on the average ML bit-error probability, Benedetto et al. [14] proved that it is preferable to choose an outer nonrecursive encoder with (a large and) odd free distance. A simple choice is then the standard feedforward convolutional code with octal generators (7, 5) and free distance 5. However, the given rules of thumb for designing good serially concatenated codes are valid for the AWGN channel. Our channel has memory, and it can, hence, be thought of as an additional code, in series with the inner CPE. Had the (noiseless) channel impulse response been perfectly known, this joint inner code would have reduced to MSK followed by a finite impulse response (FIR) filter. The joint inner code would then have been recursive, just as the CPE of MSK, and we should have expected an interleaver gain. In this letter, we investigate a case in which we lack perfect knowledge of the channel. However, for an unlimited block length in combination with an unlimited tree search, we would be able to perfectly estimate the unknown channel parameters, and in this limit, the impulse response can be regarded as a known FIR filter, and the recursive nature of the inner code is preserved. For this reason, it seems motivated to also employ a recursive inner encoder for our channel. When it comes to the outer code, however, extrinsic information transfer (EXIT) chart analysis [15] reveals how a (7, 5) code is a particularly bad choice. Instead, a simple rate-1/2 repetition code opens up the tunnel in the EXIT chart, and is thus a better choice. Fig. 4 shows the bit-error rate (BER) for the first three decoding iterations of a rate-1/2 repetition code in series with a

512-bit interleaver and MSK. 32 nodes were used in the forward/backward search, and 25 pilot symbols were introduced to improve the convergence of the estimation processes. Here, the (time-invariant) channel parameter vector is

The error rate clearly decreases as more than one sequence is used for computing soft information on each survivor edge. At in BER, the discrepancy between two-sequence GAPP detection and the standard forward–backward algorithm (assuming known channel parameter vector) is less than 1 dB in signal-to-noise ratio. IV. CONCLUSIONS We studied bidirectional channel estimation and soft symbol detection jointly in the context of serially concatenated CPM and unknown ISI. The algorithm in [4] was slightly modified, and its performance was investigated by iteratively decoding a rate-1/2 repetition code in series with a 512-bit interleaver and MSK. REFERENCES [1] C. Berrou, A. Glavieux, and P. Thitimajshima, “Near-Shannon-limit error-correcting coding and decoding: Turbo codes,” in Proc. IEEE Int. Conf. Commun., Geneva, Switzerland, May 1993, pp. 1064–1070. [2] A. Å. Hansson and T. M. Aulin, “Iterative diversity detection for correlated continuous-time Rayleigh fading channels,” IEEE Trans. Commun., vol. 51, no. 2, pp. 240–246, Feb. 2003. [3] M. Tüchler, R. Koetter, and A. C. Singer, “Turbo equalization: Principles and new results,” IEEE Trans. Commun., vol. 50, no. 5, pp. 754–767, May 2002.

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[4] A. Anastasopoulos and K. M. Chugg, “Adaptive soft-input soft-output algorithms for iterative detection with parametric uncertainty,” IEEE Trans. Commun., vol. 48, no. 10, pp. 1638–1649, Oct. 2000. [5] G. D. Forney, Jr., “Maximum-likelihood sequence estimation of digital sequences in the presence of intersymbol interference,” IEEE Trans. Inf. Theory, vol. IT-18, no. 5, pp. 363–378, May 1972. [6] K. M. Chugg and A. Polydoros, “MLSE for an unknown channel—Part I: Optimality considerations,” IEEE Trans. Commun., vol. 44, no. 7, pp. 836–846, Jul. 1996. [7] H. L. Van Trees, Detection, Estimation, and Modulation Theory, Part I. New York: Wiley, 1968. [8] A. Å. Hansson, “Generalized APP detection for communication over unknown time-dispersive waveform channels,” Ph.D. dissertation, Chalmers Univ. Technology, Göteborg, Sweden, 2003. [9] G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed. Baltimore, MD: The Johns Hopkins Univ. Press, 1996.

[10] S. Haykin, Adaptive Filter Theory, 3rd ed. Upper Saddle River, NJ: Prentice-Hall, 1996. [11] R. Raheli, A. Polydoros, and C. Tzou, “Per-survivor processing: A general approach to MLSE in uncertain environments,” IEEE Trans. Commun., vol. 43, no. 2-4, pp. 354–364, Feb.-Apr. 1995. [12] T. M. Aulin, “Breadth-first maximum-likelihood sequence detection: Basics,” IEEE Trans. Commun., vol. 47, no. 2, pp. 208–216, Feb. 1999. [13] P. A. Laurent, “Exact and approximate construction of digital modulations by superposition of amplitude modulated pulses (AMP),” IEEE Trans. Commun., vol. COM-34, no. 2, pp. 150–160, Feb. 1986. [14] S. Benedetto, D. Divsalar, G. Montorsi, and F. Pollara, “Serial concatenation of interleaved codes: Performance analysis, design, and iterative decoding,” IEEE Trans. Inf. Theory, vol. 44, no. 5, pp. 909–926, May 1998. [15] S. ten Brink, “Convergence of iterative decoding,” IEE Electron. Lett., vol. 35, pp. 1117–1118, Jun. 1999.

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Capacity and Coverage Increase With Repeaters in UMTS Urban Cellular Mobile Communication Environment Mohammad N. Patwary, Student Member, IEEE, Predrag B. Rapajic, Senior Member, IEEE, and Ian Oppermann, Senior Member, IEEE

Abstract—In this letter, we propose to use repeaters to increase system information capacity in urban areas, where path loss exponents are higher (as much as 3.4 or more). There is at least a 10% increase in coverage area when repeaters are placed in every cell within the network to increase system capacity. The overall system information capacity is doubled at propagation exponent 3.7–3.9. Index Terms—Line of sight (LOS), quality of service (QoS), universal mobile telecommunication systems (UMTS).

I. INTRODUCTION N UNIVERSAL mobile telecommunication systems (UMTS), cells are designed with a layered structure, such as picocell, microcell, macrocell. The available radio resources also vary from layer to layer, but keep the same quality of service (QoS) demanded [1]. One of the benefits of UMTS is an improved and continuous QoS guarantee with extended capacity and coverage, compared with the existing systems, such as GSM. Due to the terrain variety in rural areas, suburban areas, and the dense urban structure, there are places that cannot have as good coverage as expected according to the network design. This leads to a call-dropping situation, even though the user remains within the designed coverage area. From a QoS perspective, call dropping is more problematic than call blocking. One of the most cost-effective engineering solutions for this situation is to insert repeaters [2]–[8], which may incur only 15%–20% of the cost of a new basestation. To extend the coverage of the cellular system, repeaters were proposed originally for rural and suburban environments in [2]–[5]. Different aspects of using repeaters in code-division multiple-access (CDMA) cellular networks have been analyzed in [4] and [5]. Even though it is a common belief that the repeater will reduce system capacity in the cellular network, in [6] and [7], it has been shown that cellular system capacity can be significantly improved by inserting repeaters. Capacity increase within a hotspot has been shown in [7], exploiting repeaters. An intelligent switching scheme was proposed in [8] that optimizes unnecessary interference from repeaters. Repeaters have also been recommended for UMTS cellular networks in [9] and [10]. If the basestation-originated signal

I

Paper approved by K. K. Leung, the Editor for Wireless Network Access and Performance of the IEEE Communications Society. Manuscript received December 3, 2003; revised November 11, 2004 and April 27, 2005. This paper was presented in part at the Personal Wireless Communications Conference, Singapore, October 2002. M. N. Patwary and P. B. Rapajic are with the School of Electrical Engineering and Telecommunications, University of New South Wales, Sydney NSW 2052, Australia (e-mail: [email protected]; [email protected]). I. Oppermann is with the Centre for Wireless Communications, University of Oulu, Oulu, FIN-90014, Finland (e-mail: [email protected]). Digital Object Identifier 10.1109/TCOMM.2005.857144

faces severe fading, there will be at least another reliable signal reflected through the repeater, when there are repeaters that have been inserted within the wireless communication system environment. In this letter, we show that by taking advantage of the closed-loop power-control algorithm proposed in UMTS, cellular radio link capacity can be significantly increased. We found that for urban areas, where the propagation exponent is beyond 3.4, the amount of signal-to-noise ratio (SNR) degradation due to repeater noise is not significant, compared with the propagation loss. A significant amount of capacity increase has been found when repeaters are placed in the dense urban environment (with propagation exponent above 3.4). We propose to insert repeaters in dense urban areas to increase the coverage, as well as system capacity. The network coverage and capacity with and without repeaters in different propagation conditions have been analyzed. The analytical and simulation results show the following. 1) The CDMA system capacity in systems with repeaters is the tradeoff with coverage, up to the path-loss exponent of 3.4. 2) Beyond the path-loss exponent 3.4, the presence of repeaters increases system capacity and coverage significantly. 3) Doubled system capacity can be achieved by inserting repeaters in the propagation environment with the propagation exponent between 3.7–3.8 for the International Telecommunications Union (ITU) Pedestrian A channel and between 3.8–3.9 for the ITU Vehicular A channel. 4) There is at least a 10% increase in terrain coverage when the repeaters are placed within the network to increase the overall system capacity. The rest of the paper has been organized as follows. A system model is introduced in Section II to analyze the scenarios. Capacity analysis is discussed in Section III. Some realistic powercontrolled scenarios of the network with repeater are also discussed in Section III. Simulation results are shown and discussed in Section IV, and we conclude in Section V. II. SYSTEM MODEL A seven-cell cluster cellular mobile system has been considered, and the central cell serves the test mobile. It has been assumed that the mobiles within the network are randomly distributed, and there are regular grids of basestations [1]. In this analysis, we have considered different scenarios in urban environments within a UMTS network, where a fast power-control technique has been used for both links (uplink and downlink). To increase the coverage and capacity, three repeaters are placed in every cell, as shown in Fig. 1. Repeaters are selected in such

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high. In other words, the overall received signal at the receiver within the blind spot is mainly the reflected multipaths served from the repeater. For each scenario, the total power received by the mobile is the vector sum of all considerable multipath signals, arriving from the basestation, repeaters, and different scatterers or reflectors within the environment (1) is the total number of multipaths. For this analysis, where we have considered only four multipaths for uplink and three multipaths for downlink. The repeater parameters considered in our analysis have been followed as in [9]–[11]. The analytical model for capacity analysis has been used as in [12]. The channel-delay profile and multipaths model have been developed as in [13]–[16] for our analysis. III. CAPACITY ANALYSIS

Fig. 1. Single cell with three sectors and three repeaters.

a way that the downlink sensitivity level of the repeater’s donor antenna is equal to the sensitivity level of the mobile receiver. For uplink, repeater service antenna sensitivity level is the same as the basestation receiver sensitivity level. The repeaters are placed within the cell coverage, from where the mobiles can receive signals from the basestation with enough strength. The repeaters will provide an additional strong multipath to the user. However, in both uplink and downlink, the receivers are dealing with two strong sources of signals. For each link, the instantaneous signal power received at the receiver is the vector sum of the entire multipath signals. There are three different propagation scenarios considered in our analysis. These are given below, and denote the received power where the notations from the basestation antenna and repeater service antenna, respectively. A. Scenario I The probability of the signal received from the repeater being ) within the basestation service equal to zero (i.e., antenna coverage is very high, if the proper antenna isolation is preferred for the maximum repeater gain. B. Scenario II The received signal at the mobile arrives from the basestation and repeater with approximately the same signal strength (i.e., ), which is difficult to detect without an equalizer if the delay between two signals is very large, or as low as one chip duration. Maximum tolerable delay is 20 s [13], [14]. C. Scenario III In a blind spot (conventionally known as a dead spot), the probability of the signal being received from the basestation close to null (i.e., ) at the mobile receiver is very

to maintain a radio link for any speThe necessary cific service in UMTS [Wideband CDMA (WCDMA)] can be determined by the following equation from [12]: (2) is the chip rate, is the transmitted power from the where basestation for a specific service radio link, denotes bit rate, is the path loss between the serving basestation and the test mobile, is total interference and noise power that is (3) where , , and are thermal noise, basestation noise, and repeater noise, respectively. Note that the repeater noise ( ) will be added only when the system is considered with a repeater. The interference power is given by (4) where (5) (6) is the link budget transmit power from the basestation. is the same-cell interference, and is other-cell interference. In this letter, the interference from the other cell is considered only for the cells within the active set. The other cell interference for the systems with a repeater consists of signals from the adjacent basestations along with the repeater gain. For the advanced reader, we are referring to W. C. Y. Lee et al. [6, p. 1765], where the authors have mentioned that the interference due to repeater gain of the adjacent cell is negligible, in comparison with the interference signal itself. is the orthogonality factor (orthogonality factor is the uniqueness of the transmitted signal from each basestation), which is 90% for ITU Pedestrian A and 60% for ITU Vehicular A ( is 100% for a single propagation path, we assume another 10% decrement in orthogonality factor when

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the system with a repeater is considered) [1], [4]–[7], [11]. However, this is an assumption for our analysis, and it will vary from scenario to scenario. The amount of this decrement is a function of the size of the search window, associated delay, and Doppler is the path loss from the neighboring basestations frequency. within the active set to the test mobile. For simplicity, we can rewrite (3) as follows:

are nothing but a special case of assumed user distribution, the so-called fringe user distribution, as shown in [17]. Another generalized user distribution can be obtained from the , , and , above-mentioned Gaussian parameters called uniform random distribution. For uniform random user distribution, the Gaussian parameters are set to and . It has also been considered that the link budgeted transmit power from every basestation is constant. As a closed-loop power-control scheme has been exploited as recommended in UMTS, explained in the following section, the assumed user distribution holds for a generalized system model. The total number of radio links supported with that , is given as follows: amount of power, denoted as

(7) represents overall noise, and represents the total where interference. From (2), the power required to maintain a radio link for a specific service can be rewritten as as the following equation: (8) Substituting the values from (4)–(7) in (8), it can be rewritten as follows: (9)

A. Other Cell Interference Calculation The users are assumed to follow a random distribution within the cell of interest. The central cell in the cluster, as assumed in the system model, is the cell of interest. We have assumed this user distribution within a uniform grid of basestations within an active set (seven cells in UMTS [9]). To calculate the other cell interference in a seven-cell cluster, the position of the central cell is assumed as the origin for the Cartesian coordinate. To realize the user distribution in urban areas, we have assumed a random uniform distribution of user groups within the active set, and Gaussian user distribution within each of these groups. The location of the th mobile user distributed in Gaussian manner within a user group, denoted by , where Cartesian coordinates and , where denotes the Gaussian distribution with mean and variance , and denotes the minor (smaller) radius of the hexagonal cell. The Euclidian distances between the test mobile within the cell of interest and the neighboring active set cells within the cluster gives the instantaneous distances between them. More specifically, these distances in this system model are the Euclidian distances between the coordinate of the test mobile and the coordinates of the cells within the active set, i.e., ( ), and ). These distances are denoted as with , where is the number of cells within the active set. The instantaneous distance between the origin and the location of the test mobile has been of (9) is determined with denoted as . The value of the following equation: (10) Average cell capacity is largely dominated by users at the cell boundaries. Most users located in the cell boundaries

(11) VAF where VAF is the voice activity factor. The multipaths from the repeater have the same delay properties, identical to the other multipaths, those reflected from the different nongenerative sources (obstacles, such as buildings) within the network. Maximum sustainable limit of this delay is 20 s in UMTS (wideband channel) [14] to detect the signal without an equalizer. The cell sizes in urban areas are very small. Subsequently, the delay is very small, which causes negligible phase deviation to the reflected signal, in reference to the signal received from the basestation. So the resultant signal can be considered as additive. In contradiction with [2] and consistent with [6], there would not be any significant QoS degradation when the signals from the primary source and the reflected (secondary) source are equal, as there is some delay between them, and these signals are within the multipaths search window. The minimum size for the , is determined by the following search window for UMTS, equation: (12) is the repeater group delay in s, is the distance where, is between the basestation and the repeater in miles, and the distance between the repeater and the mobile in miles. B. Power-Controlled Scenarios The instantaneous power received at the receiver is approximately equal to the vector sum of two signals received from two different sources. All multipaths within the search window is the total received signal power have been considered. If is the total signal received from from the basestation, and the reflected paths through the repeater, then the total received can be written as power (13) the minimum required power at any If we consider time instant, then with the closed-loop power-control technique, it is always expected to adjust the transmitted power to reach the required level. A regulating factor has been introduced to maintain as constant, which has been used to decide the transmit power command (TPC), as in UMTS. So is the controlled received power (14)

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TABLE I TRANSMITTER–RECEIVER PARAMETERS USED IN THIS ANALYSIS

Fig. 3.

Fig. 2. Radio-link support capacity with and without repeaters (12.2 kb/s). (a) Pedestrian A channel. (b) Vehicular A channel.

It is possible when there is at least one strong signal path. Repeaters provide this service for the shadowed region to survive against deep fading without degradation of QoS. This letter is concerned with urban areas, and we propose using repeaters to increase capacity only in an urban environment. Besides, in UMTS, there is 1500 Hz closed-loop power-control tracking recommended, which continuously monitors power every 667 s, and ensures the lower probability of the network being unstable due to the feedback conditions. IV. SIMULATION RESULTS The transmitter and receiver parameters given in Table I have been used for computer simulation. Simulation results are given

Comparison of coverage cell radius with and without repeaters.

in Figs. 2 and 3. Fig. 2(a) shows that for the ITU Pedestrian A channel within the environment of propagation exponent 2–3.4, increasing coverage with a repeater is the tradeoff with the numbers of radio link capacity. The same results have been found for the ITU Vehicular A channel, as shown in Fig. 2(b) within the propagation exponent 2–3.3. But within these ranges of propagation exponents, operators are more concerned with minimizing coverage cost, as there are a smaller number of users within this region. On the other hand, beyond these propagation exponents, the system capacity of the network with repeaters starts to dominate the capacity of the network without repeaters. A doubled system capacity can be obtained for the path-loss exponent between 3.7–3.8 for ITU pedestrian A channel, as shown in Fig. 2(a). For the ITU Vehicular A channel, doubled system capacity has been obtained between the propagation exponents 3.8–3.9, which are very common in dense urban areas. This gain is due to the capability of having the basestations be considerably apart from each other, and thus, reducing intercell interference significantly. This lowered interference allows the mobile receiver to reach its target SNR with a low-budgeted power requirement from the basestation to maintain the radio link. Hence, with the same budgeted transmit power, the basestations are able to support more radio links than that of the system without repeaters. From the results shown in Fig. 2, it could be concluded that for rural and suburban areas, where the path-loss exponent is low and coverage is the most considerable issue, repeaters can be a significant solution. On the other hand, for urban areas, where the path-loss exponent is high, repeaters can dominate the capacity of the network without repeaters, as well as the coverage, significantly. It should be noted that even though we have considered a seven-cell hexagonal cluster, the behavior of the number of links supported with respect to path-loss exponents will remain the same, regardless of the cell geometry. The network with repeaters has to be designed in such a way that the extended radius remains uniform in each direction. Fig. 3 shows that the extension of coverage with the repeater is more efficient in rural and suburban areas than in urban areas. The coverage radius can be extended 60%–80% in different areas, but doing that causes a cutoff in the capacity of

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about 20%–50%. This cutoff in capacity is found as a noticeable effect of repeater noise. We found that a basestation and three sets of duplex repeaters can serve a coverage area for which at least three basestations have to be installed. V. CONCLUSION Both the system capacity and the coverage in the UMTS environment have been analyzed, where fast power control has been adopted in both the uplink and the downlink. From the simulation results, it could be concluded that the repeaters, which are usually used in extension of coverage in rural and suburban areas, can also be exploited to increase the system capacity in dense urban areas. Due to the dynamic control in each radio-link power of different services and environment to meet the target SNR, it is possible to design the UMTS network with repeaters, implementing intelligent control to the repeaters. The algorithm proposed in [8] optimizes the capacity cutoff in rural and suburban areas. In urban areas, one of the most common optimization issues that the operators have faced from the beginning of the cellular system is to serve the network with the least percentage of blocking probability. One of the major measures of the network quality is the percentage of blocking probability, which is comparatively higher in urban areas than in the rural and suburban areas. We proposed to solve this problem by using repeaters within the cellular network to provide additional multipaths, which not only ensures the minimization of the blocking probability, but also increases the instantaneous capacity, as well as extension of the coverage. REFERENCES [1] H. Holma and A. Taskala, WCDMA for UMTS. New York: Wiley, 2001. [2] E. H. Drucker, “Development and application of cellular repeater,” in Proc. Veh. Technol. Conf., 1988, pp. 321–325. [3] W. T. Slingsby and J. P. Mcgeehan, “A high gain cell enhancer,” in Proc. Veh. Technol. Conf., 1992, pp. 756–758. [4] M. R. Bavafa and H. H. Xia, “Repeaters for CDMA system,” in Proc. Veh. Technol. Conf., 1998, pp. 1161–1165. [5] S. J. Park, W. W. Kim, and B. Kwon, “An analysis of effect of wireless network by a repeater in CDMA system,” in Proc. Veh. Technol. Conf., vol. 4, 2001, pp. 2781–2785. [6] W. C. Y. Lee and D. J. Y. Lee, “The impact of repeaters on CDMA system,” in Proc. Veh. Technol. Conf., 2000, pp. 1763–1767. [7] M. Rahman and P. Ernstrom, “Repeaters for hotspot capacity in DS-CDMA networks,” IEEE Trans. Veh. Technol., vol. 53, no. 3, pp. 626–633, May 2004. [8] W. Choi, B. Y. Cho, and T. W. Ban, “Automatic on-off switch repeater for DS/CDMA reverse link capacity improvement,” IEEE Commun. Lett., vol. 4, no. 5, pp. 138–141, Apr. 2001. [9] “UTRA Repeater: Radio Transmission and Reception,”, 3GPP TS25.106.

[10] “UTRA Repeater Conformance Testing,”, 3GPP TS25.143. [11] “Repeater site survey – Highway coverage with repeater,” Allgon Repeater Syst. Equip.. [12] K. Sipila, Z. C. Honkasalo, J. L. Steffens, and A. Wacker, “Estimation of capacity and required transmission power of WCDMA downlink based on a downlink pole equation,” in Proc. Veh. Technol. Conf., 2000, pp. 1002–1005. [13] V. Erceg, D. G. Michelson, S. S. Ghassemzadeh, L. J. Greenstein, A. J. Rustako, Jr., P. B. Guerlain, M. K. Dennison, R. S. Roman, D. J. Barnickel, S. C. Wang, and R. R. Miller, “A model for the multipath delay profile of fixed wireless channels,” IEEE J. Sel. Areas Commun., vol. 17, no. 3, pp. 399–410, Mar. 1999. [14] K. H. Tsioumparakis, T. L. Doumi, and J. G. Gardiner, “Delay-spread considerations of same frequency repeaters in wideband channels,” IEEE Trans. Veh. Technol., vol. 46, no. 3, pp. 664–675, Aug. 1997. , “Delay spread statistics in a two-path radio environment,” in Proc. [15] Veh. Technol. Conf., 1996, pp. 642–646. [16] J. P. Decruyenaere and D. Falconer, “A shadowing model for prediction of coverage in fixed terrestrial wireless systems,” in Proc. Veh. Technol. Conf., 1999, pp. 1427–1433. [17] T. S. Rappaport and L. B. Milstein, “Effects of path loss and fringe user distribution on celluler frequency reuse efficiency,” in Proc. IEEE Globecom, vol. 1, Dec. 1990, pp. 500–506.

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Noncooperative Power-Control Game and Throughput Game Over Wireless Networks Zhu Han and K. J. Ray Liu

Abstract—Resource allocation is an important means to increase system performance in wireless networks. In this letter, a gametheory approach for distributed resource allocation is proposed. Observing the bilinear matrix inequality nature of resource allocation, we construct two interrelated games: a power-control game at the user level, and a throughput game at the system level, respectively, to avoid local optima. An optimal complex centralized algorithm is developed as a performance bound. The simulations show that the proposed games have near-optimal system performance. Index Terms—Adaptive modulation, game theory, power control, rate adaptation.

I. INTRODUCTION NE OF THE major challenges in wireless networks is to efficiently use the limited radio resources, which are restrained by the cochannel interference (CCI) and time-varying nature of channels. Resource allocation [1], such as power control [2] and adaptive modulation, is an important means to combat these detrimental effects and increase the spectrum efficiency in the interference-limited wireless networks. Joint consideration of power control and rate adaptation can further improve the system performance [3]–[5]. Since individual mobile users do not have the knowledge of other users’ conditions and cannot cooperate with each other, they act selfishly to maximize their own performances in a distributed fashion. Such a fact motivates us to adopt the game theory. The resource allocation can be modeled as a noncooperative game that deals largely with how rational and intelligent individuals interact with each other in an effort to achieve their own goals. In the resource-allocation game, each mobile user is self-interested and trying to maximize his/her utility function, where the utility function represents the user’s performance and controls the outcomes of the game. Some related work can be found in [6]–[10]. In most of the previous work, only one utility function is defined for both power and throughput, which can result in local optima because of nonlinearity. We can show that joint power control and adaptive rate-allocation problem can be formulated as a bilinear matrix inequality (BMI) [11] constraint if the bit-error rate (BER) is fixed, i.e., the throughput is linearly constrained if the power is fixed, and the power is linearly constrained if the throughput is fixed. So this gives us motivation to

O

Paper approved by K. K. Leung, the Editor for Wireless Network Access and Performance of the IEEE Communications Society. Manuscript received June 24, 2003; revised October 3, 2004 and April 19, 2005. This work was supported in part by MURI under AFOSR F496200210217. The authors are with the Electrical and Computer Engineering Department and Institute for Systems Research, University of Maryland, College Park, MD 20742 USA (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TCOMM.2005.857136

design two interconnected games for power and throughput, respectively, so that higher system performance can be more likely to be achieved. In [8], the idea of two-level optimization was first proposed, while only one utility was used and no adaptive modulation was applied. In order to achieve better system performance, our primary concern is to design the utility functions and the rules of the games. One of the goals is to motive individual users to adopt a social behavior and enhance the system performance by sharing the resources. Consequently, we can make the distributed selfoptimizing decisions compatible with the demand for a higher overall system performance. In doing so, we link both power control and adaptive modulation by designing games at both the user level and the system level. A noncooperative powercontrol game (NCPCG) is designed at the user level. At the system level, the optimization goal is to maximize the overall system throughput under the maximal transmitted power constraint. A noncooperative throughput game (NCTG) is designed. There may be multiple Nash equilibriums in this game. A distributed algorithm is constructed to achieve the better Nash equilibrium by employing a proposed game rule and an initialization method. An optimal but complex centralized algorithm that achieves the optimal system performance is developed as a performance upper bound. From simulations, the proposed games are optimal for the power at the user level, and can be optimal or near-optimal for network throughput at the system level. II. SYSTEM MODEL AND BILINEAR MATRIX INEQUALITY cochannel uplinks that may exist in disConsider tinct cells of wireless networks. Each link consists of a mobile and its assigned base station (BS). We assume the average transmitted power for different modulation conas the noise level. The stellations is normalized. Define th user’s signal-to-interference-plus-noise ratio (SINR) is , where is the th user’s transmitted power, and is the channel gain from the th user to the th BS. Adaptive modulation provides the links with abilities to match the effective bit rates (throughput), according to interference and channel conditions. -ary quadrature amplitude modulation (MQAM) is a modulation method that has high spectrum efficiency. We assume each user has a unit bandof MQAM, the th width. In [4], for a desired throughput , user’s BER can be approximated as BER where and when BER is small. For a specific desired BER , the th link’s required SINR for the desired can be expressed as , throughput where BER . If the users’ throughput is too large, CCI is severe, and it is possible that there exists no feasible power allocation for the

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desired throughput and BER. In order to prevent the system from not being feasible, we need to analyze the feasibility condition. First, we use the targeted SINR and require that the received SINR be larger than or equal to this targeted SINR, , in order to ensure the desired BER for the i.e., throughput . Rewriting these inequalities in a matrix form, , where is an identity matrix, we have with , , , if ; , if . The and above inequality is a BMI [10], i.e., the power vector is linearly constrained if the targeted SINR vector (throughput) is fixed, and vice versa. Since linearity can achieve global optimum, this motivates us using the two-game approach developed later. By the Perron–Frobenius theorem [1], there exists a feasible solution with positive power and rate allocation only if the maximum [spectrum radius ] is inside the unit eigenvalue of circle. III. CONSTRUCTING GAMES FOR DISTRIBUTIVE RESOURCE ALLOCATION In wireless communication networks, because of the bandwidth limitation, it is impractical for the mobile users to communicate, thus cooperate with each other, so as to optimally use the wireless resources. Each individual mobile user tries to maximize his/her performance, based only on his/her perceived self-interest. However, this will cause the system to be balanced in some undesired nonoptimal equilibriums. We design the game rules for the users’ competitions such that the system will be balanced in the desired optimal and efficient resource allocation. Because power and throughput are bilinearly constrained, it is natural to divide the optimization efforts into the system level and the user level. We define value function as the connection between two levels. The goals for both levels are given by the following. 1) User Level: The goal is to define a utility function , and then each user can compete with other users in an NCPCG to maximize his/her utility function. There are some practical constraints, such as the max. The proposed NCPCG imum transmitted power , where is formulated as , and is the assigned value function that is related to throughput . is optiAt the user level, the transmitted power is assigned mized by the proposed NCPCG, while is equal to zero, by the BS. When the throughput no transmitted power is needed and should be zero. Otherwise, we define the value function as a function of , the desired throughput as ; , if , where is a function if of only throughput and . is related to the desired BER, and is usually predefined and fixed. When the CCI is high, from a system optimization point of should be increased view, the cost for the desired to reduce CCI. We represent this cost as , where reflects the severity of the CCI and can be fed back from the BS to the mobile. We define the utility function . NCPCG is played iteratively as

satisfies the until convergence. It can be shown that three requirements of the “standard function” in [2] for iterative function. Consequently, NCPCG converges to the unique optimum that achieves the minimal SNR for the desired BER. 2) System Level: The goal is to assign a user his/her value by an NCTG, such that the overall system function throughput is maximized, under the constraint . When the system is balanced, and are functions of , where . The overall network throughput is optimized by NCTG, and the corare assigned to the users for NCPCG. responding , subThe problem can be formulated as . ject to A. Noncooperative Throughput Game at the System Level At the system level, NCTG is constructed for the users to compete distributively, while the system maintains feasibility. We define as an indication function for system feasibility. When the BS detects that all required transmitted power for the , desired BER and throughput are less than or equal to equals one, otherwise, it equals zero. Since the users compete with each other for the throughput, we define each user’s utility function for NCTG as a product of his/her throughput and , i.e., (NCTG) . The game starts from any feasible initial values and is balanced when no user can increase his/her throughput. The existence of the Nash equilibrium can be shown by a similar proof in [6]. However, there might be multiple Nash equilibriums, which will be shown in the simulations later. If the users with bad channels get high throughput, they will produce large CCI to other users. Consequently, the system’s overall throughput will be reduced. So how to initialize the proposed game and how to design the game rule for each user to compete his/her throughput play a critical role on finding the better optimum. The idea for initialization comes from the following theorem. when Theorem 1: Define maximal achievable SINR as . Then the value is feasible for both games, where is floor function for . the maximal integer smaller. For example, . Proof: Define Since is an increasing function, . Because and , we have , since any component in , , and is nonnegative. So all components in are nondecreasing , any functions of . When we select the targeted SINR component of the power vector must be smaller than or equal , so the value functions satisfy the maximum power to constraint and the system must be feasible. First, every user transmits the maximal power. The BS detects the received SINR. Using the above theorem, the BS decides what is the largest achievable throughput and sends back the corresponding . The system is sure to be feasible, but not necessarily optimal. By doing this, the users with good channels will get higher throughput. Each user will then play NCPCG until convergence. The convergence is achieved when each user detects that the NCPCG utility is stable, because this means the interfering users’ power and utilities are also stable. After

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TABLE I DISTRIBUTED SYSTEM ALGORITHM

NCPCG convergence, the users decide if they can increase their throughput, while the system is still feasible. We need to find the criteria for the users to decide when to send the requests to as the required the BS for throughput increase. We define , if we assume SINR for the desired throughput . When the interferences, noise, and channel gains are fixed, the required will be , power for throughput where is the current power. We compare this desired power , where is a constant and . If the with desired power is larger than , the user can send a request to the BS to increase his/her throughput by one. When this user increases his/her power, he/she causes interference to the others. Others have to increase their power, which causes interference to this user, as well. So this user has to increase his/her power more. is such a factor that takes into consid, all reeration this “mutual interfering” effect. When ceived power is the interference plus noise power, defined as . The channel gain can be estimated during the initialization when this user transmits the maximal power. This user can calculate estimated received SINR by trans. If the received SINR is larger than , mitting power the user will send the throughput increase request. Let us define as the throughput request factor, the criteria for the users to , request their throughput to increase by one is when where , if and , if . The above game rule for NCTG and the distributed adaptive algorithm are summarized in Table I. After initialization, two games are played iteratively. First, users play NCPCG. After convergence, users play NCTG, , and send requests if necessary. If the request is calculate granted, the selected user will increase the throughput, and then NCPCG is played again. Both games are distributive, because only local information is necessary to play. It is worth mentioning that in order to determine the largest , adjacent BSs should exchange the values of , which can be implemented with limited signaling. B. Centralized Scheme as a Performance Upper Bound The distributed algorithm may not be optimal. First, there is a probability that the users do not send requests, while the system might be feasible if users sent requests. Second, there exist Nash equilibriums that are not global optimum. To understand the performance loss, we need to find the optimal solution as a performance bound. The most straightforward idea is to let the system centrally decide how to allocate throughput to users, with the assumption that all channel responses are known. This is not im-

Fig. 1. Nash equilibriums of NCPCG.

plementable, since the channel conditions from the users to the BS in other cells are hard to obtain. The problem becomes a constrained optimization: to maximize overall throughput under the maximum power and maximum eigenvalue constraints, which can be solved by standard nonlinear programming s.t.

IV. SIMULATION RESULTS We evaluate the performances of the proposed algorithms by two simulation setups. First, we consider a two-user case. , , , Here we assume , BER , and . In Fig. 1, we show the Nash equilibriums of NCPCG when the different gets the throughput allocations are given. On any solid line, maxima. On any dotted line, has the optima. Starting from any feasible power allocation, each user tries to maximize his utility function by controlling his power, such that the power allocation is closer to the corresponding lines. When the system is balanced, any intersection is a Nash equilibrium, where we denote the throughput as (user1’s throughput, user2’s throughput). obtained from will inWe can see that the maxima for crease with increasing . This is because the CCI increases. In Table II, we list the strategic form of NCTG at the system level for all the nonzero throughput allocations. Each row lists user1’s

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TABLE II STRATEGIC FORM FOR TWO-USER NCTG EXAMPLE

Fig. 3.

Fig. 2. System throughput versus R .

throughput, and each column lists user2’s throughput. The bold numbers are the overall throughput. If the system is not feasible, the overall throughput is zero. We can see that (4,2), (2,3), and (1,4) are Nash equilibriums, because no user can improve his throughput alone. However, (2,3) and (1,4) are not desired Nash equilibriums for the optimal overall network throughput. The proposed distributed algorithm in Table I will be initialized at ), because (3,2). If is properly selected (in this case, , the algorithm will increase user1’s throughput first and converge to the optimal Nash equilibrium (4,2). So, from this example, we can see that we can achieve both power optimum and throughput optimum by playing the NCPCG at the user level and NCTG at the system level. We set up another simulation to test the proposed algorithms. A network is constructed with one cell at the center and the other six at the degrees of [0, 60, 120, 180, 240, 300], respectively. One BS is located at the cell’s center, and one user is randomly m, the located within each cell. The cell radius is m, minimal distance between the user and the BS is and the distance between the centers of two adjacent cells is m, where is the reuse factor. W, , , BER , and W. The simulations times. run with Fig. 2 compares overall network throughput versus different . When is small, CCI is severe. After initializing by sending the maximal transmitted power, most users get throughput of zero. The overall network throughput is increased by the users’ throughput-increasing requests. If is large enough, the proposed games can achieve the optimal

Tradeoff between rejection and throughput loss for .

system performance when is small enough. The overall network throughput is minimal when , because different BSs and users are mixed together, and CCI is most is large, CCI is minor. severe under this condition. When After the initialization, most users get the desired throughput. The overall network throughput is refined when is large. The optimal system performance can be achieved when both and are large enough. When is in the middle range, the proposed games may fall into some local minima and produce . The overall throughput suboptimal solutions, even when can be improved by increasing . However, the overall throughput improvement by increasing is at the expense of a possible high request-rejected probability defined as the ratio of the number of rejected requests over the total number of requests. Fig. 3 shows the throughput loss compared with the optimal solution and rejection probability versus for different . When , the rejection probability is always zero, and the throughput loss is monotonically decreasing with . This is because the optimal solution is that only the user with the best channel condition transmits,and there is no CCI from other users. So there is no penalty from other users if the transmitting user increases and aggressively . When sends the request. Therefore, it is optimal to select , the rejection probability monotonically increases with . There is a tradeoff between the throughput loss and rejection probability. The higher the , the lower the throughput loss, but the higher the rejection probability. If the system wants a very with a perforlow rejection probability, we can select , the rejection probamance loss of 2.35 b/s/Hz. When bility is almost zero when , and the overall throughput . The tradeoff loss is approximately 0.44 b/s/Hz when only occurs when is large. The reason is that the users get almost optimal throughput after initialization. Consequently, the refinement only happens when the users are more aggressive for throughput requests. We define the fairness factor as

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Fig. 4. Fairness and average transmitted power versus reuse distance.

where is the maximal throughput if user is the only transaverage . The physical meaning mitting user, and of is the normalized variance of users’ throughput compared with that of the single-user case. The higher the , the more unfair among users, i.e., the users throughput is more affected by CCI. is one possible definition to measure the fairness. Fig. 4 shows the fairness and average transmitted power versus with . When is small and CCI is severe, is large, and the users with the better channel condition occupy most of the resources. The average transmitted power is also low, because becomes large and CCI most users cannot transmit. When is reduced, the users with worse channel conditions can compete for their transmissions, while the users with better channel conditions are not so dominant. Consequently, is reduced and users transmit more fairly, like the single-user case. The average transmitted power is increased and saturated with increase , because most users can transmit according to their own of channel conditions, regardless of the low CCI from others. for Fig. 5 shows the average throughput per user versus different with . We can see that the average throughput is large. This is because the increases more slowly when is small. When is deCCI is increasing, especially when creasing, the point where the average throughput per user saturates moves to the lower . There is no need for higher if the performance curve is saturated already. So when is deaccordingly. creasing, we can reduce V. CONCLUSIONS In order to achieve better overall network throughput, we construct NCPCG and NCTG at the user level and at the system level, respectively. From the simulations, the proposed games

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Fig. 5.

Average throughput per user versus P

.

converge to the near-optimal solutions, compared with the optimal solutions obtained from the centralized scheme. The proposed two-game approach explores the BMI nature of the resource allocation to avoid local optima, and consequently, has high performance in a distributed implementation. REFERENCES [1] J. Zander and S. L. Kim, Radio Resource Managment for Wireless Networks. Norwood, MA: Artech House, 2001. [2] R. Yates, “A framework for uplink power control in cellular radio systems,” IEEE J. Sel. Areas Commun., vol. 13, no. 7, pp. 1341–1348, Sep. 1995. [3] K. K. Leung and L. C. Wang, “Controlling QoS by integrated power control and link adaptation in broadband wireless networks,” Eur. Trans. Telecommun., no. 4, pp. 383–394, Jul. 2000. [4] X. Qiu and K. Chawla, “On the performance of adaptive modulation in cellular systems,” IEEE Trans. Commun., vol. 47, no. 6, pp. 884–895, Jun. 1999. [5] E. Armanious, D. D. Falconer, and H. Yanikomeroglu, “Adaptive modulation, adaptive coding, and power control for fixed cellular broadband wireless systems: Some new insights,” in Proc. Wireless Commun. Netw. Conf., vol. 1, Mar. 2003, pp. 238–242. [6] C. U. Saraydar, N. B. Mandayam, and D. J. Goodman, “Efficient power control via pricing in wireless data networks,” IEEE Trans. Commun., vol. 50, no. 2, pp. 291–303, Feb. 2002. [7] A. B. MacKenzie and S. B. Wicker, “Game theory in communications: Motivation, explanation, and application to power control,” in Proc. GLOBECOM, vol. 2, Nov. 2001, pp. 821–826. [8] N. Feng, S. Mau, and N. Mandayam, “Pricing and power control for joint network-centric and user-centric radio resource management,” in Proc. 39th Allerton Conf. Commun., Control, Comput., Allerton, IL, Oct. 2001, pp. 202–211. [9] P. Zhang et al., “Power control of voice users using pricing in wireless networks,” in Proc. SPIE ITcom, Aug. 2001, pp. 155–165. [10] Z. Han, Z. Ji, and K. J. Ray Liu, “Fair multiuser channel allocation for OFDMA networks using Nash bargaining and coalitions,” IEEE Trans. Commun., vol. 53, no. 8, pp. 1366–1376, Aug. 2005. [11] M. Mesbahi, M. G. Safonov, and G. P. Papavassilopoulos, “Bilinearity and complementarity in robust control,” in Advances in Linear Matrix Inequality Methods in Control. Philadelphia, PA: SIAM, 2000.

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Transactions Papers Turbo Codes With Rate-m=(m + 1) Constituent Convolutional Codes Catherine Douillard, Member, IEEE, and Claude Berrou, Member, IEEE

Abstract—The original turbo codes (TCs), presented in 1993 by Berrou et al., consist of the parallel concatenation of two rate-1/2 binary recursive systematic convolutional (RSC) codes. This paper explains how replacing rate-1/2 binary component codes by ( + 1) binary RSC codes can lead to better global perrateformance. The encoding scheme can be designed so that decoding can be achieved closer to the theoretical limit, while showing better performance in the region of low error rates. These results are illustrated with some examples based on double-binary ( = 2) 8-state and 16-state TCs, easily adaptable to a large range of data block sizes and coding rates. The double-binary 8-state code has already been adopted in several telecommunication standards. Index Terms—Iterative decoding, permutation, rate( + 1) recursive systematic convolutional (RSC) code, tailbiting code, turbo code (TC).

I. INTRODUCTION CLASSICAL turbo code (TC) [1] is a parallel concatenation of two binary recursive systematic convolutional (RSC) codes based on single-input linear feedback shift registers (LFSRs). The use of multiple-input LFSRs, which allows several information bits to be encoded or decoded at the same time, offers several advantages compared with classical TCs. In the past, the parallel concatenation of multiple-input LFSRs had mainly been investigated for the construction of turbo trellis-coded modulation schemes [2]–[4], based on Ungerboeck trellis codes. Actually, the combination of such codes, providing high natural coding rates with high-order modulations, leads to very powerful coded modulation schemes. In this paper, we propose the construction of a family of TCs calling for RSC constituent codes based on -input LFSRs, that outperforms classical TCs. We provide two examples of TCs with reasonable decoding complexity that allow decoding to be achieved very close to the theoretical limit, and at the same time, show good performance in the region of low error rates.

A

Paper approved by R. D. Wesel, the Editor for Coding and Communication Theory of the IEEE Communications Society. Manuscript received May 1, 2003; revised August 6, 2004. The authors are with the Electronics Department, GET/École Nationale Supérieure des Télécommunications de Bretagne, CS 83818 – 29238 Brest Cedex 3, France (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TCOMM.2005.857165

Section II describes the structure adopted for -input RSC component encoders, along with some conditions that guarantee large free distances, regardless of the value of . In Section III, we describe the turbo-encoding scheme, and the advantages of this construction compared with classical TCs. Section IV presents some practical examples of TCs with and their simulated performance. The 8-state family has already been adopted in the digital video broadcasting (DVB) standards for return channel via satellite (DVB-RCS) [5] and the terrestrial distribution system (DVB-RCT) [6], and also in the 802.16a standard for local and metropolitan area networks [7]. Combined with the powerful technique of circular trellises, TC offers good performance and versatility for enthis coding blocks with various sizes and rates, while keeping reasonable decoding complexity. Replacing the 8-state component encoder by a 16-state encoder allows better performance at low error rates, at the price of a doubled decoding complexity. Minimum Hamming distances are increased by 30%–50%, with regard to 8-state TCs, and allow frame-error rate (FER) curves to without any noticeable change in the slope decrease below (the so-called flattening effect). Finally, conclusions and perspectives are summarized in Section V. II. RATEBASED ON

RSC ENCODERS -INPUT LFSRS

In this section, we will define the constituent RSC codes to be used in the design of the proposed TCs. Fig. 1 depicts the general structure of the RSC encoder under study. It involves a single -stage LFSR, whose -row and -column generator matrix is denoted . At time , the -component input vector is connected to the possible taps via a connection grid represented by a -row and -column binary matrix denoted . The tap column vector at time , , is then given by (1) In order to avoid parallel transitions in the corresponding trellis, the condition has to be satisfied, and matrix has to be full rank.

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DOUILLARD AND BERROU: TURBO CODES WITH RATE-

CONSTITUENT CONVOLUTIONAL CODES

Fig. 1. General structure of a rate-m=(m + 1) RSC encoder with code memory  .

Except for very particular cases, this encoder is not equivalent to a single-input encoder fed successively by , that is, the -input encoder is not generally decomposable. The redundant output of the machine, not represented in Fig. 1, is calculated at time as (2) where denotes the -component column vector describing the encoder state at time , and is the -component row-redundancy vector. The th component of is equal to one if the th component of is present in the calculation of , and zero, otherwise. The code being linear, we assume that the “all zero” sequence is encoded. Let us define a return to zero (RTZ) sequence as an input sequence of a recursive encoder, that makes the encoder leave state 0 and return to it again. Calculating the minimum free distance of such a code involves finding the RTZ sequence path with minimum output Hamming weight. and at least Leaving the null path at time implies that is equal to one. In this case, (2) ensures one component of that the Hamming weight of is at least two when leaving the reference path, since the inversion of one component of implies the inversion of . Moreover, since (3) it can be shown that

may also be written as (4)

on the condition that (5) Consequently, if the code is devised to verify condition (5), (4) ensures that if the RTZ sequence retrieves the all-zero reference path at time ( ) because one of the information bits is equal to one, is also equal to one. Hence, relations (2) and (4) together guarantee that the minimum free distance of the unpunctured code, whose rate is , is at least four, whatever . The two codes proposed in this paper for meet this requirement. The minimum Hamming distance of a concatenated code being larger than that of its constituent codes, provided that the

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permutation function is carefully devised, we can imagine that large minimum distances may be obtained for TCs, for low as well as for high coding rates. implies high decoding comChoosing large values of plexity, because also has to be large, and paths have to be processed for each trellis node. For this reason, only low values of can be contemplated for practical applications for the time being (typically, 2, possibly 3). Up to now, we have only investigated the case in order to construct practical coding and decoding schemes in this new RSC or family of TCs. In the following, we call such turbo encoders double binary. Double-binary RSC codes have a natural rate of 2/3. When higher coding rates are required, a simple regular or quasi-regular puncturing pattern is applied. The decoding solution calling for the application of the maximum a posteriori (MAP) algorithm based on the dual code [8] can also be considered for large values of , since it requires fewer edge computations than the classical MAP algorithm for high coding rates. However, for practical implementations, its application in the log-domain means the computation of transition metrics with a far greater number of terms, and the compu(Jacobian) function requires great precision. tation of the Consequently, the relevance of this method for practical use is not so obvious when the rate of the mother code is not close to 1.

III. BLOCK TURBO CODING WITH RATECONSTITUENT RSC CODES The TC family proposed in this paper calls for the parallel concatenation of two identical rateRSC encoders , are enwith -bit word interleaving. Blocks of bits, coded twice by this bidimensional code, whose rate is . A. Circular RSC Codes Among the different techniques aiming at transforming a convolutional code into a block code, the best way is to allow any state of the encoder as the initial state, and to encode the sequence so that the final state of the encoder is equal to the initial state. The code trellis can then be viewed as a circle, without any state discontinuity. This termination technique, called tailbiting [9], [10] or circular, presents three advantages in comparison with the classical trellis-termination technique using tail bits to drive the encoder to the all-zero state. First, no extra bits have to be added and transmitted; thus, there is no rate loss, and the spectral efficiency of the transmission is not reduced. Next, when classical trellis termination is applied for TCs, a few codewords with input Hamming weight one may appear at the end of the block (in both coding dimensions), and can be the cause of a marked decrease in the minimum Hamming distance of the composite code. With tailbiting RSC codes, only codewords with minimum input weight two have to be considered. In other words, tailbiting encoding avoids any side effects. Moreover, in a tailbiting or circular trellis, the past is also the future and vice versa. This means that a non-RTZ sequence produces effects on

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the whole set of redundant symbols stemming from the encoder, around the whole circle. Consequently, the output weights associated with non-RTZ sequences are large, and do not contribute to the minimum Hamming distance of the code. In practice, the circular encoding of a data block consists of a two-step process [9]. At the first step, the information sequence is encoded from state 0 and the final state is memorized. During this first step, the outputs bits are ignored. The second step is the actual encoding, whose initial state is a function of the final state previously memorized. The double encoding operation represents the main drawback of this method, but in most cases, it can be performed at a frequency much higher than the data rate. The iterative decoding of such codes involves repeated and continuous loops around the circular trellis. The number of loops performed is equal to the required number of iterations. The state probabilities or metrics, according to the chosen decoding algorithm, computed at the end of each turn are used as initial values for the next turn. With this method, the initial, or final, state depends on the encoded information block and is a priori unknown to the decoder at the beginning of the first iteration. If all the states are assumed to be equiprobable at the beginning of the decoding process, some side errors may be produced by the decoders at the beginning of the first iteration. These errors are removed at the subsequent iterations, since final state probabilities or metrics computed at the end of the previous iteration are used as initial values.

Among the numerous permutation models that have been suggested up to now, the apparently most promising ones, in terms of minimum Hamming distances, are based on regular permutation calling for circular shifting [11] or the co-prime [12] principle. After writing the data in a linear memory, with address ( ), the information block is likened to a circle, both extremities of the block ( and ) then being contiguous. The data are read out such that the th datum read was written at the position , given by (6) where the skip value is an integer, relatively prime with , and is the starting index. This permutation does not require the block to be seen as rectangular; that is, may be any integer. In [13] and [14], two very similar modifications of (6) were proposed, which generalize the permutation principle adopted in the DVB-RCS/RCT or IEEE802.16a TCs. In the following, we will consider the almost regular permutation (ARP) model detailed in [14], which changes relation (6) into (7) is an integer, whose value is taken in a limited set , in a cyclic way. , called the cycle of the permutation, must be a divider of and has a typical value , the permutation law is of four or eight. For instance, if defined by if if if

(8)

and must be a multiple of four, which is not a very restricting condition, with respect to flexibility. In order to ensure the bijection property of , the values are not just any values. A straightforward way to satisfy the bijection condition is to choose all ’s as multiples of . The regular permutation law expressed by (6) is appropriate for error patterns which are simple RTZ sequences for both encoders; that is, RTZ sequences which are not decomposable as a sum of shorter RTZ sequences. A particular and important case of a simple RTZ sequence is the two-symbol RTZ sequence, which may dominate in the asymptotic characteristics of a TC ). A two-symbol sequence is a se(see [15] for TCs with quence with two nonzero -bit input symbols, which may contain more than one nonzero bit. Let us define the total spatial as the sum of the two spatial distance (or total span) distances, before and after permutation, according to (6), for a given pair of positions and (9) where (10) Finally, we denote by for all possible pairs and

the minimum value of

, (11)

B. Permutation

where

if

It was demonstrated in [16] that the maximum possible value for , when using regular interleaving, is (12) If any two-symbol RTZ sequence for one component encoder into another two-symbol RTZ seis transformed by or quence for the other encoder, the upper bound given by (12) is amply sufficient to guarantee a large weight for parity bits, and thus, a large minimum binary Hamming distance. This is the same for any number of symbols, on the condition that both RTZ sequences, before and after permutation, are simple RTZ sequences. On the other hand, ARP aims at combating error patterns which are not simple RTZ sequences, but are combinations of simple RTZ sequences for both encoders. Instilling some convalues in (7), tends to break trolled disorder, through the most of the composite RTZ sequences. Meanwhile, because the value of cycle is small, the good property of regular permutation for simple RTZ sequences is not lost, and a total span close can be achieved. [14] describes a procedure to obtain to appropriate values for and for the set of parameters. The algorithmic permutation model described by (7) is simple to implement, does not require any ROM, and the parameters can be changed on-the-fly for adaptive encoding and decoding. Moreover, as explained in [14], massive parallelism, allowing several processors to run at the same time without increasing the memory size, can be exploited. In addition to the ARP principle and the advantages develcomponent code adds one more oped above, the rate-

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Fig. 3. Periodicities of the double-binary encoder of Fig. 2(b). Input couples (0,0), (0,1), (1,0), and (1,1) are denoted 0, 1, 2, and 3, respectively.

Fig. 2. Possible rectangular error patterns. For (a) binary and (b) double-binary TCs with regular permutations.

degree of freedom in the design of permutations: intrasymbol permutation, which enables some controlled disorder still to be added into the permutation without altering its global quasi-regularity. Intrasymbol permutation means modifying the contents of the -bit symbols periodically, before the second encoding, in such a way that a large proportion of composite RTZ sequences for both codes can no longer subsist. Let us develop . this idea in the simplest case of Fig. 2(a) depicts the minimal rectangular error pattern (input ) for a parallel concatenation of two identical biweight nary RSC encoders, involving a regular permutation (linewise writing, columnwise reading). This error pattern is a combination of two input weight-two RTZ sequences in each dimension, leading to a composite RTZ pattern with distance 16, for coding rate 1/2. If the component encoder is replaced by a double-binary encoder, as illustrated in Fig. 2(b), RTZ sequences and error patterns involve couples of bits, instead of binary values. Fig. 2(b) gives two examples of rectangular error patterns, corresponding to distance 18, still for coding rate 1/2 (i.e., no puncturing). Data couples are numbered from 0 to 3, with the following notation: (0,0):0; (0,1):1; (1,0):2; (1,1):3. The periodicities of the double-binary RSC encoder, depicting all the combinations of pairs of input couples different from 0 that are RTZ sequences, are summarized in the diagram of Fig. 3. For instance, if the encoder, starting from state 0, is fed up with successive couples 1 and 3, it retrieves state 0. The same behavior can be observed with sequences 201, 2003, 30002, 3000001, or 30000003, for example. The change from binary to double-binary code, though leading to a slight improvement in the distance (18 instead of 16), is not sufficient to ensure very good performance at low error rates. Let us suppose now that couples are inverted (1 becomes 2 and vice versa) once every other time before second (vertical) encoding, as depicted in Fig. 4. In this way, the error patterns displayed in Fig. 2(b) no longer remain error patterns. For instance, 20000002 is still an RTZ sequence for the second (vertical) encoder, but 10000002 is no longer RTZ. Thus, many error patterns, especially short patterns, are eliminated, thanks to the disorder introduced inside input symbols. The right-hand side of Fig. 4 shows two examples of rectangle error patterns

Fig. 4. Couples in gray spaces are inverted before second (vertical) encoding. 1 becomes 2; 2 becomes 1; 0 and 3 remain unaltered. The three patterns on the left-hand side are no longer error patterns. Those on the right-hand side remain possible error patterns, with distances 24 and 26 for coding rate 1/2.

that remain possible error patterns after the periodic inversion. The resulting minimal distances, 24 and 26, are large enough for the transmission of short data blocks [17]. For longer data blocks (a few thousand bits), combining this intrasymbol permutation with intersymbol ARP, as described above, can lead to even larger minimum distances, at least with respect to the rectangular error patterns with low input weights we gave as examples. C. Advantages of TCs With RateCodes

RSC Constituent

Parallel concatenation of -input binary RSC codes offers several advantages in comparison with classical (one input) binary TCs, which have already been partly commented on in [18]. 1) Better Convergence of the Iterative Process: This point was first observed in [19] and commented on in [20]. The better convergence of the bidimensional iterative process is explained by a lower error density in each code dimension, which leads to a decrease in the correlation effect between the component decoders. Let us consider again (12), which gives the maximum total span achievable when using regular or quasi-regular permutation. For a given coding rate , the number of parity bits involved all along the total span, and used by either one decoder or the other, is (13) Thus, replacing a classical binary ( ) with a double) TC multiplies this number of parity bits by , binary ( though dividing the total span by the same value. Because the

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Performance in BER of three single 16-state RSC codes with m = 1, m = 2, and m = 3. Encoder polynomials: 23 (feedback) and 35 (redundancy)

Fig. 6.

m

Fig. 5. Performance in BER of single 8-state RSC codes with = 1 and = 2. Encoder polynomials: 15 (feedback) and 13 (redundancy) in octal form (DVB-RCS constituent encoder for = 2). Coding rates are 2/3 and 6/7 (regular puncturing). Binary/quaternary phase-shift keying (BPSK/QPSK) modulation, additive white Gaussian noise (AWGN) channel, and MAP decoding. No quantization.

m

m

parity bits are not a matter of information exchange between the two decoders (they are just used locally), the more numerous they are, with respect to a given possible error pattern (here, the weight-two patterns), the fewer correlation effects between the component decoders. Raising beyond two still improves the turbo algorithm, regarding correlation, but the gains get smaller and smaller as increases. 2) Larger Minimum Distances: As explained above, the number of parity bits involved in simple two-symbol RTZ sequences for both encoders is increased when using ratecomponent codes. The number of parity bits involved in any simple RTZ sequence, before and after permu, regardless of the number tation, is at least equal to of nonzero symbols in the sequence. The binary Hamming distances corresponding to all simple RTZ sequences are then high, and do not pose any problem with respect to the minimum Hamming distance of the TC. This comes from error patterns made up of several (typically, two or three) short simple RTZ sequences on both dimensions of the TC. Different techniques can be used to break most of these patterns, one of them (ARP) having been presented in Section III-B. 3) Less Puncturing for a Given Rate: In order to obtain , from the RSC encoder of coding rates higher than Fig. 1, fewer redundant symbols have to be discarded, compared binary encoder. Consequently, the correcting with an ability of the constituent code is less degraded. In order to illustrate this assertion, Fig. 5 compares the performance, in

in octal notation. Coding rate is 3/4, regular puncturing. BPSK/QPSK modulation, AWGN channel and MAP decoding. No quantization.

terms of bit-error rate (BER), of two 8-state RSC codes with the same generator polynomials (15, 13) in octal notation, for

The two-input RSC code displays better performance than the one-input code, for both simulated coding rates. Raising to three and beyond for this code is not of interest, since there is no full-rank three-column matrix that satisfies (5). The choice of a constituent code with parallel transitions in the trellis would lead to a TC with very low minimum Hamming distance. curve has been introduced for the (23,35) In Fig. 6, an 16-state RSC code with coding rate 3/4. The connection matrices are equal to for for for As expected, we observe that the performance gain between and is smaller than the gain between and . Raising to four and beyond for this 16-state code is not of interest, since there is no full-rank four-column matrix that satisfies (5). 4) Higher Throughput and Reduced Latency: The decoder convolutional code provides bits at each of an decoding step. Thus, once the data block is received, and for a

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nonbinary turbo decoding actually makes the full MAP decoder unnecessary (or the Max-Log-MAP decoder with Jacobian logarithm correction [24]), which requires more operations than the Max-Log-MAP decoder. Also, the latter does not need the knowledge of the noise variance on Gaussian channels, which is a nice advantage. The rigorous explanation for this quasi-equivalence of the MAP and the Max-Log-MAP algorithms, when decoding -input TCs, has still to be found. IV. PERFORMANCE OF DOUBLE-BINARY TCS This section describes two examples of double-binary TCs, with memory 3 and 4, whose reasonable decoding complexity allows them to be implemented in actual hardware devices for practical applications. Simulation results for transmissions over an AWGN channel with QPSK modulation are provided. A. Eight-State Double-Binary TC Fig. 7. Comparison of performance in FER of two TCs based on both 8-state RSC codes of Fig. 5, with k = 1504, R = 4=5, for MAP and Max-Log-MAP decoding. AWGN channel, QPSK modulation, eight iterations. Scaling factor for Max-Log-MAP decoding: 0.7 for iterations 1–7, 1.0 for iteration 8. No quantization.

given processing clock, the decoding throughput of a hardware decoder is multiplied by . The latency, i.e., the number of clock periods required to decode a data block, is then divided ). by , compared with the classical case ( However, the critical path of the decoder being the Add–Comhas a lower pare–Select (ACS) unit, the decoder with maximum clock frequency than with . For instance, for , the Compare–Select operation has to be done on four metrics instead of two, thus with an increased propagation delay. The use of specialized look-ahead operators and/or the introduction of parallelism, in particular, the multistreaming method [21], makes it possible to significantly increase the maximum frequency of the decoder, and even to reach that of the decoder . with 5) Robustness of the Decoder: Fig. 7 represents the sim, of four ulated performance in FER, as a function of coding/decoding schemes dealing with blocks of 1504 information bits and coding rate 4/5: binary and double-binary 8-state TCs exhibited in Fig. 2, both with the full MAP decoding algorithm [22], [23], and with the simplified Max-Log-MAP version [24]. In the latter case, the extrinsic information is less reliable, especially at the beginning of the iterative process. To compensate for this, a scaling factor, lower than 1.0, is applied to extrinsic information [25]. The best observed performance was obtained when a scaling coefficient of 0.7 for all the iterations, except for the last one, was applied. In Fig. 7, both codes have ARP internal permutation with optimized span. We can observe that the double-binary TC per, and the steeper slope forms better, at both low and high for the double-binary TC indicates a larger minimum binary Hamming distance. These characteristics were justified by 1) and 2) of this section. What is also noteworthy is the very slight difference between the decoding performance of the double-binary TC when using the MAP or the Max-Log-MAP algorithms. This property of

The parameters of the component codes are

(14) The diagram of the encoder is described in Fig. 8. Redunis only used for coding rates less than 1/2. For dancy vector coding rates higher than 1/2, puncturing is performed on redundancy bits in a regular periodical way, following patterns that are described in [5]. These patterns are identical for both constituent encoders. is performed on two The permutation function levels, as explained in Section III-B. , we have the following. For • Level 1: inversion of and in the data couple, if . • Level 2: this permutation level is described by a particular form of (8) with if if if if

(15)

Value is added to the incremental relation in order to comply with the odd–even rule [26]. The disorder is instilled in the permutation function, according to the ARP principle, in two ways. • A shift by is added for odd values of . This is done because the lowest subperiod of the code generator is one (see Fig. 3). The role of this additional increment is thus to spread to the full the possible errors associated with the shortest error patterns. , , and act as local additional pseudorandom fluc• tuations. Notice that the permutation equations and parameters do not depend on the coding rate considered. The parameters can be

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Fig. 8. Structure of the 8-state encoder. Redundancy y is only used for turbo-coding rates less than 1/2.

Fig. 9.

Structure of the proposed 16-state double-binary turbo encoder.

optimized to provide good behavior, on average, at low error rates for all coding rates, but seeking parameters for a particular coding rate could lead to slightly better performance.

TABLE I OF ESTIMATED VALUES OF MINIMUM BINARY HAMMING DISTANCES d PROPOSED 8-STATE AND 16-STATE DOUBLE-BINARY TCS FOR 188-B DATA BLOCKS. DISTANCES WERE ESTIMATED WITH THE ALL-ZERO ITERATIVE DECODING ALGORITHM [27]

B. Sixteen-State Double-Binary TC The parameters of the best component code we have found are

(16) The diagram of the encoder is described in Fig. 9. Puncturing is performed on redundancy in a periodic way, with identical patterns for both constituent encoders. It is usually regular, except when the puncturing period is a divisor of the LFSR period. For example, for coding rate 3/4, the puncturing period is chosen equal to six, with puncturing pattern [101000]. For this code, the permutation parameters have been carefully chosen, following the procedure described in [14], in order to guarantee a large minimum Hamming distance, even for high rates. The level-1 permutation is identical to the intrapermutation of the 8-state code. The level-2 intersymbol permutation is given by For with if if if if

(17)

The spirit in which this permutation was designed is the same as that already explained for the 8-state TC. The only difference is that the lowest subperiod of the 16-state generator is two, in) is applied stead of one. That is why the additional shift (by consecutively, twice every four values of . Table I compares the minimum binary Hamming distances of the proposed 8-state and 16-state TCs, for 188-B data blocks and four different coding rates. The distance values were estimated with the so-called all-zero iterative algorithm, a fast computational method described in [27], which provides distance values with very high reliability for block sizes larger than a few hundred bits. We can observe a significant increase in the minimum distance when using 16-state component codes; the gain varies from 30%–50% depending on the case considered. With this code, we were also able to define permutation param, eters leading to minimum distances as large as 33 for 22 for , and 16 for for ( )-B blocks.

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Fig. 10. Performance in FER of 8-state and 16-state double-binary TCs for ATM (53 B) blocks and rates 1/2, 2/3, and 3/4. QPSK modulation and AWGN channel. Max-Log-MAP decoding with 4-b input samples and 8 iterations. The theoretical limits on FER are derived from [28].

Fig. 11. Performance in FER of 8-state and 16-state double-binary TCs for MPEG (188 B) blocks and rates 1/2, 2/3, and 3/4. QPSK modulation and AWGN channel. Max-Log-MAP decoding with 4-b input samples and 8 iterations. The theoretical limits on FER are derived from [28].

From an implemention point of view, the complexity of the corresponding decoder is about twice the complexity of the 8-state decoder.

performance improves predictably with block size and coding rate in relation to the theoretical limit. The reported limits given in Figs. 10 and 11, as well as in Fig. 7, take the block size and the target FER into account. They are derived from the Gallager random coding bound on the error probability for binary-input , the simulated channels, as described in [28]. At FER curves lie within 0.6–0.8 dB from the limit, regardless of block size and coding rate. To improve the performance of this , the more powerful 16-state code family at FER below component code has to be selected so as to increase the overall minimum Hamming distance of the composite code. Dotted-line curves in Figs. 10 and 11 show the 16-state TC performance for the same simulation conditions as for the 8-state code. Similar to this code, the permutation parameters are related to the block size, not to the coding rate. We can observe that the selected code does not lead to a convergence-threshold shift of the iterative decoding process in comparison with the previous 8-state code. For FERs above , 16-state and 8-state codes behave similarly. For lower error rates, thanks to the increase in distance, there is no noticeable floor effect for the simulated signal-to-noise ratio (SNR) . Again, performance imranges, that is, down to a FER of proves predictably with block size and coding rate in relation to , the simulated curves lie the theoretical limit. At FER within 0.7–1.0 dB from the limits, regardless of block size and coding rate, even with the simplified Max-Log-MAP algorithm.

C. Simulation Results We have simulated and compared these two codes for two block sizes and three coding rates for transmissions over an AWGN channel with QPSK modulation. The simulation results take actual implementation constraints into account. In particular, the decoder inputs are quantized for hardware complexity considerations. According to our experience, the performance degradation due to input quantization is not significant beyond 5 b. The observed loss is less than 0.15 dB for 4-b quantization, and about 0.4 dB for 3-b quantization. When quantization is applied, clipping extrinsic information at a threshold around twice the maximum range of the input samples does not degrade the performance, while limiting the amount of required memory. Solid line curves in Figs. 10 and 11 show the FER as a for the transmission of ATM (53 B) and function of MPEG (188 B) packets for three different values of coding rate of the 8-state double-binary TC. The component decoders use the Max-Log-MAP algorithm, with input samples quantized on 4 b. Eight iterations were simulated and at least 100 erroneous frames were considered for each point indicated, except for the lowest points, where approximately 30 erroneous frames were simulated. We can observe good average performance for this code whose decoding complexity is very reasonable. For a hardware implementation, less than 20 000 logical gates are necessary to implement one iteration of the decoding process when decoding is performed at the system clock frequency, plus the memory required for extrinsic information and input data. Its

V. CONCLUSION Searching for perfect channel coding presents two challenges: encoding in such a way that large minimum distances can be reached; and achieving decoding as close to the theoretical limit as possible. In this paper, we have explained why

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-input binary TCs combined with a two-level permutation can represent a better answer to these challenges than classical one-input binary TCs. , we have been able to design coding In practice, with schemes with moderate decoding complexity, and whose performance approaches the theoretical limit by less than 1 dB at . The 8-state TC with has already found FER practical applications through several international standards. Furthermore, the parallel concatenation of RSC circular codes leads to flexible composite codes, easily adaptable to a large range of data block sizes and coding rates. Consequently, as -input binary codes are well suited for association with high-order modulations, in particular -ary quadrature -ary PSK, TCs based on these amplitude modulation and constituent codes appear to be good candidates for most future digital communication systems based on block transmission. ACKNOWLEDGMENT The authors are grateful to the Associate Editor and the reviewers for their valuable comments and suggestions for improving this paper. REFERENCES [1] C. Berrou and A. Glavieux, “Near-optimum error-correcting coding and decoding: Turbo codes,” IEEE Trans. Commun., vol. 44, no. 10, pp. 1261–1271, Oct. 1996. [2] P. Robertson and T. Wörz, “Bandwidth-efficient turbo trellis-coded modulation using punctured component codes,” IEEE J. Sel. Areas Commun., vol. 16, no. 2, pp. 206–218, Feb. 1998. [3] S. Benedetto, D. Divsalar, G. Montorsi, and F. Pollara, “Parallel concatenated trellis-coded modulation,” in Proc. IEEE Int. Conf. Commun., vol. 2, Dallas, TX, Jun. 1996, pp. 974–978. [4] C. Fragouli and R. D. Wesel, “Turbo-encoder design for symbol-interleaved parallel concatenated trellis-coded modulation,” IEEE Trans. Commun., vol. 49, no. 3, pp. 425–435, Mar. 2001. [5] “Interaction Channel for Satellite Distribution Systems,” DVB, ETSI EN 301 790, vol. 1.2.2, 2000. [6] “Interaction Channel for Digital Terrestrial Television,” DVB, ETSI EN 301 958, vol. 1.1.1, 2002. [7] IEEE Standard for Local and Metropolitan Area Networks, IEEE 802.16a, 2003. [8] S. Riedel, “Symbol-by-symbol MAP decoding algorithm for high-rate convolutional codes that use reciprocal dual codes,” IEEE J. Sel. Areas Commun., vol. 16, no. 2, pp. 175–185, Feb. 1998. [9] C. Weiss, C. Bettstetter, S. Riedel, and D. J. Costello, “Turbo decoding with tailbiting trellises,” in Proc. IEEE Int. Symp. Signals, Syst., Electron., Pisa, Italy, Oct. 1998, pp. 343–348. [10] R. Johannesson and K. S. Zigangirov, Fundamentals of Convolutional Coding, ser. Digital, Mobile Commun.. New York: IEEE Press, 1999, ch. 4. [11] S. Dolinar and D. Divsalar, “Weight distribution of turbo codes using random and nonrandom permutations,” JPL, NASA, TDA Prog. Rep. 42–122, 1995. [12] C. Heegard and S. B. Wicker, Turbo Coding. Norwell, MA: Kluwer, 1999, ch. 3. [13] S. Crozier, J. Lodge, P. Guinand, and A. Hunt, “Performance of turbo codes with relatively prime and golden interleaving strategies,” in Proc. 6th Int. Mobile Satellite Conf., Ottawa, ON, Canada, Jun. 1999, pp. 268–275. [14] C. Berrou, Y. Saouter, C. Douillard, S. Kerouédan, and M. Jézéquel, “Designing good permutations for turbo codes: Toward a single model,” in Proc. IEEE Int. Conf. Commun., Paris, France, Jun. 2004, pp. 341–345. [15] S. Benedetto and G. Montorsi, “Design of parallel concatenated convolutional codes,” IEEE Trans. Commun., vol. 44, no. 5, pp. 591–600, May 1996.

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[16] E. Boutillon and D. Gnaedig, “Maximum spread of -dimensional multiple turbo codes,” IEEE Trans. Commun., vol. 53, no. 8, pp. 1237–1242, Aug. 2005. [17] C. Berrou, E. A. Maury, and H. Gonzalez, “Which minimum Hamming distance do we really need?,” in Proc. 3rd Symp. Turbo Codes, Brest, France, Sep. 2003, pp. 141–148. [18] C. Berrou, M. Jézéquel, C. Douillard, and S. Kerouédan, “The advantages of nonbinary turbo codes,” in Proc. Inf. Theory Workshop, Cairns, Australia, Sep. 2001, pp. 61–63. [19] C. Berrou, “Some clinical aspects of turbo codes,” in Proc. Int. Symp. Turbo Codes Related Top., Brest, France, Sep. 1997, pp. 26–31. [20] C. Berrou and M. Jézéquel, “Nonbinary convolutional codes for turbo coding,” Electron. Lett., vol. 35, no. 1, pp. 39–40, Jan. 1999. [21] H. Lin and D. G. Messerschmitt, “Algorithms and architectures for concurrent Viterbi decoding,” in Proc. IEEE Int. Conf. Commun., Boston, MA, Jun. 1989, pp. 836–840. [22] L. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal decoding of linear codes for minimizing symbol-error rate,” IEEE Trans. Inf. Theory, vol. IT-20, no. 3, pp. 284–287, Mar. 1974. [23] S. Benedetto, D. Divsalar, G. Montorsi, and F. Pollara, “A soft-input soft-output APP module for iterative decoding of concatenated codes,” IEEE Commun. Lett., vol. 1, pp. 22–24, Jan. 1997. [24] P. Robertson, P. Hoeher, and E. Villebrun, “Optimal and suboptimal maximum a posteriori algorithms suitable for turbo decoding,” Eur. Trans. Telecommun., vol. 8, pp. 119–125, Mar./Apr. 1997. [25] J. Vogt and A. Finger, “Improving the max-log-MAP turbo decoder,” Electron. Lett., vol. 36, no. 23, pp. 1937–1939, Nov. 2000. [26] A. S. Barbulescu, “Iterative decoding of turbo codes and other concatenated codes,” Ph.D. dissertation, Univ. South Australia, 1996. [27] R. Garello and A. V. Casado, “The all-zero iterative decoding algorithm for turbo code minimum distance computation,” in Proc. Int. Conf. Commun., Paris, France, Jun. 2004, pp. 361–364. [28] R. G. Gallager, Information Theory and Reliable Communication. New York: Wiley, 1968, sec. 5.6.

Catherine Douillard (M’01) received the engineering degree in telecommunications from the École Nationale Supérieure des Télécommunications (ENST) de Bretagne, Brest, France, in 1988, and the Ph.D. degree in electrical engineering from the Université de Bretagne Occidentale, Brest, France, in 1992. In 1991 she joined ENST Bretagne, where she is currently a Professor in the Electronics Department. Her main interests are turbo codes and iterative decoding, iterative detection, and the efficient combination of high spectral efficiency modulation and turbo-coding schemes.

Claude Berrou (M’87) was born in Penmarc’h, France, in 1951. In 1978, he joined the Ecole Nationale Supérieure des Télécommunications (ENST) de Bretagne, where he is currently a Professor in the Electronics Department. In the early 80’s, he started up the training and research activities in VLSI technology and design, to meet the growing demand from industry for microelectronics engineers. Some years later, he took an active interest in the field of algorithm/silicon interaction for digital communications. In collaboration with Prof. A. Glavieux, he introduced the concept of probabilistic feedback into error-correcting decoders, and developed a new family of quasi-optimal error-correction codes that he named “turbo codes.” He also pioneered the extension of the turbo principle to joint detection and decoding processing, known today as turbo detection and turbo equalization. His current research topics, besides algorithm/silicon interaction, are electronics and digital communications at large, error-correction codes, turbo codes and iterative processing, soft-in/soft-out (probabilistic) decoders, etc. He is the author or co-author of 8 registered patents and about 60 publications in the field of digital communications and electronics. Dr. Berrou has received several distinctions (with Prof. Glavieux), amongst which are the 1997 SEE Médaille Ampère, one of the 1998 IEEE (Information Theory) Golden Jubilee Awards for Technological Innovation, the 2003 IEEE Richard W. Hamminag medal, the 2003 French Grand Prix France Télécom de l’Académie des sciences, and the 2005 Marconi Prize.

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DC-Free Error-Control Block Codes Fengqin Zhai, Student Member, IEEE, Yan Xin, Member, IEEE, and Ivan J. Fair, Member, IEEE

Abstract—DC-free codes and error-control (EC) codes are widely used in digital transmission and storage systems. To improve system performance in terms of code rate, bit-error rate (BER), and low-frequency suppression, and to provide a flexible tradeoff between these parameters, this paper introduces a new class of codes with both dc-control and EC capability. The new codes integrate dc-free encoding and EC encoding, and are decoded by first applying standard EC decoding techniques prior to dc-free decoding, thereby avoiding the drawbacks that arise when dc-free decoding precedes EC decoding. The dc-free code property is introduced into standard EC codes through multimode coding techniques, at the cost of minor loss in BER performance on the additive white Gaussian noise channel, and some increase in implementation complexity, particularly at the encoder. This paper demonstrates that a wide variety of EC block codes can be integrated into this dc-free coding structure, including binary cyclic codes, binary primitive BCH codes, Reed–Solomon codes, Reed–Muller codes, and some capacity-approaching EC block codes, such as low-density parity-check codes and product codes with iterative decoding. Performance of the new dc-free EC block codes is presented. Index Terms—Complementary codeword pair, dc-free codes, error-control (EC) block codes, guided scrambling (GS), multimode coding.

I. INTRODUCTION RROR-CONTROL (EC) codes and constrained sequence (CS) codes are channel codes that are used to improve the performance of digital communication systems. EC codes enable detection and/or correction of errors when the coded sequence is corrupted due to an imperfect channel. CS codes make it possible for the channel-input sequences to avoid some patterns that are susceptible to corruption in practical systems. CS codes include runlength-limited (RLL) codes and dc-free codes, in which the coded sequences have explicit constraints in the time domain and in the frequency domain, respectively. In an RLL-coded sequence, the lengths of consecutive like-valued symbols are limited to a prescribed range. In a dc-free coded sequence, the power spectral density (PSD) function has value zero at zero frequency. DC-free codes are commonly used in wired transmission systems to improve performance that is degraded due to the use

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Paper approved by G. M. Vitetta, the Editor for Equalization and Fading Channels of the IEEE Communications Society. Manuscript received May 24, 2004; revised December 3, 2004 and April 1, 2005. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada, in part by the Alberta Ingenuity Fund, and in part by iCORE. This paper was presented in part at the IEEE International Symposium on Information Theory, Yokohama, Japan, June 29-July 4, 2003, and in part at the IEEE Pacific Rim Conference on Communications, Computers, and Signal Processing, Victoria, BC, Canada, August 2005. The authors are with Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB T6G 2V4, Canada (e-mail: [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.1109/TCOMM.2005.857160

of coupling components and/or isolating transformers [1], are widely employed in optical recording to allow low-frequency noise caused by dust and fingerprints to be filtered out with minimal loss of signal [2], and have been proposed in wireless systems to assist with the insertion of pilot tones [3]. The conventional method of incorporating both EC and CS coding into a digital communication or storage system is through concatenation of an EC code as an outer code, and a CS code as an inner code [4]. In general, CS decoders exhibit error extension, expect binary input, and output hard decisions. To avoid the impact of error extension during CS decoding on the subsequent EC decoder, and to enable the use of soft-decision information during EC decoding, it is desired that EC decoding precede CS decoding [4]. Concatenation schemes with a partially reversed order of conventional CS and EC coding have been proposed in [5] and [6], and the performance of these schemes has been evaluated in [4]. Constructions for some integrated binary dc-free EC codes have been proposed in [3] and [7]–[11]; however, these codes have limited EC ability, and/or have limited flexibility regarding tradeoffs between bit-error rate (BER) and spectral performance. For example, in [7], one specific dc-free block code with minimum distance 4 was synthesized with a lookup table. In [8], dc-free block coset codes were proposed, where construction was limited to binary dc-free coset Bose–Chaudhuri–Hocquengem (BCH) codes, and in general, the number of augmenting bits for dc control is limited by the manner in which the codeword length can be factored. It remains to be determined how this coset coding technique can be applied to nonbinary codes, product codes, and low-density parity-check (LDPC) codes in order to generate dc-free EC block codes. dc-free -ary error-correcting codes were introduced in [3]; howeve, the spectral performance of these codes diminishes when the number of parity-check bits in the EC code is relatively large. In [9]–[11], consideration focused on dc-free convolutional codes. This paper introduces a new general approach for integrating dc-free codes and EC block codes for highly efficient and reliable digital transmission and storage systems. The main idea is to map a source word to one or multiple complementary pairs, and to encode these pairs with an EC encoder that preserves the complementary nature of words after EC encoding. Multimode selection techniques are then used to generate a dc-balanced EC sequence. In the new coding scheme, well-established EC block codes, such as binary cyclic codes, binary primitive BCH codes, Reed–Solomon (RS) codes, Reed–Muller codes, and some capacity-approaching EC block codes, including LDPC and product codes, can be employed, and the BER performance is determined mainly by the EC codes. The remainder of the paper is organized as follows. In Section II, we briefly review the required background. In Sec-

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tions III and IV, we present the proposed new coding scheme, and discuss some EC block codes that are suitable for this new scheme. In Section V, we present performance evaluation of the new coding scheme, and in Section VI, we give concluding remarks.

to form a set of quotients, and selecting an appropriate encoded word from the quotient set [14]. Its decoding involves unscrambling the received word and discarding the augmenting bits. The scrambling and unscrambling operations can be interpreted as division and multiplication processes, respectively, from the ring of polynomials over the Galois field of order two [GF(2)] [19]. Both the scrambling polynomial and selection criteria affect the power spectrum of the coded sequence. Polynomial representations are convenient for the description of the GS coding process. Let the components of a code vector of length be coefficients of a code polynomial , where the term with the highest degree represents the first bit in time. We use the terms “code vector” and “code polynomial” interchangeably in this paper. In each GS encoding of length is preceded by all interval, the source word -bit binary patterns , , to obtain augmented words of length . A quotient set is obtained by scrambling these augmented of length corresponding to the words. The quotient is augmented word

II. PRELIMINARIES A. DC-Free Codes Let binary and bipolar symbols be from the alphabets {0,1} , respectively. Let denote a bipolar coded seand at the th time quence. The running digital sum (RDS) of . It has been shown that instant, , is defined as is bounded, the sequence has a spectral null at dc if [12]. DC-free codes can be classified as monomode, bimode, and multimode codes [13]. When both code rate and spectrum performance are considered, dc-free multimode codes have been shown to have advantages over the other two classes of dc-free codes [13]. In a multimode code, each source word is mapped into a set of multiple representations, and a selection criterion is employed to select the “best” representation from the selection set. In the literature, three approaches have been developed to map source words to sets of potential codewords in multimode codes: dc-free coset coding [8], guided scrambling (GS) [14], and the scrambling of an RS code [15]. GS has received particular attention, since it is easy to implement and to integrate with other codes, and is well-developed [13]. The disparity of a binary codeword is defined as the difference between the number of ones and the number of zeros in the codeword. Note that the disparity value equals the word-end RDS value evaluated over the length of the codeword. In order to ensure generation of a balanced sequence in dc-free multimode codes, it is required that there exist at least one codeword with zero disparity, or that there exist codewords with opposite polarity of disparity in each selection set. Furthermore, to improve spectrum performance, it is desirable that each selection set have as many such codewords as possible. Since a pair of complementary codewords have equal and opposite disparity, in general, in a dc-free multimode code, it is advantageous if each selection set consists of complementary codeword pairs. In practice, it is desired that the spectral components around zero frequency in the dc-free sequence be significantly suppressed. A performance metric that indicates the width of the spectral null at low frequencies is the cutoff frequency , where represents the variance of the RDS (sum variance in brief) [16], [17]. Another performance metric, called low-frequency spectrum weight (LFSW) [18], indicates the depth of the spectral null at low frequencies. , where denotes the At low frequencies, PSD of a dc-free sequence, and denotes the LFSW which equals the zero-frequency content of the continuous PSD of the corresponding RDS sequence . In this paper, all frequency values are normalized by the symbol rate. B. Guided Scrambling GS encoding involves augmenting a source word with bits, scrambling the augmented words with a scrambling polynomial

(1) where the operator denotes the generation of a quotient through modulo-2 division of its argument by the scrambling polynomial of degree ( ) where 1 or 0, and . Based on a given is selected from the quotient selection criterion, a quotient denote the set as a codeword to be sent to the channel. Let augmented word that generated the selected quotient . In GS decoding, the decoded augmented word is ob, which is the tained by multiplying the received word hard-decision output of the demodulator, by the scrambling . and discarding the least significant bits of polynomial (2) If no errors are present at the output of the demodulator, the source word can be recovered correctly by removing the augmenting bits from . To take errors into account, let , where is an error pattern at the output of the in indicates channel, and a coefficient of 1 for the term an error at position . From (2), it follows that: (3) which demonstrates that at the output of the GS decoder, one error at the input of the decoder may be extended to multiple errors during GS decoding, and that this extension is upper . When GS is used in a noisy bounded by the weight of channel, to prevent large error extension, it is desirable to choose a scrambling polynomial with small weight. Note that the minimum weight of is two. Let denote an even-weight scrambling polynomial with denote the weight-two degree not greater than , and let scrambling polynomial of degree (i.e., ). It

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has been shown that use of ensures the generation of complementary codeword pairs in the quotient selection set [20]. Since, in general, a scrambling polynomial with a higher degree (up to degree ) results in larger suppression of low frequencies, and scrambling polynomials of the same degree do not result in a significant difference in spectral performance [21], we proin the new scheme introduced in this paper pose using to yield good spectral performance and minimum error extension in GS decoding. C. Error-Control Block Codes [19] Let and denote the generator matrix and parity-check matrix for an ( , ) linear EC block code, respectively. A code, word of length can be generated according to where is the message word of length . Also, a codeword satisfies , where is the transpose of . Let be a symbol from a Galois field GF , where is a positive integer. Then the set includes the elements of this field. Note that . distinct The nonzero elements can be represented by the or less. nonzero polynomials of over GF(2) with degree These polynomials are obtained by setting , where is a primitive polynomial of degree over GF(2). Addition of elements is carried out using the polynomial representations of the elements. , and let be a polynomial Let be an element in GF , then is with coefficients from GF(2). If is a root of , where is a positive integer. The element also a root of is called a conjugate of . The elements of GF form all the roots of . may be a root of a polynomial over GF(2) with a degree less than . Let be the polynomial . This of smallest degree over GF(2), such that is called the minimal polynomial of ; it is irreducible, and it divides . III. NEW CODING SCHEME It has been shown that in binary multimode encoding, the existence of at least one complementary codeword pair in each codeword selection set guarantees the ability to generate a dc-free-coded sequence [14], and that use of scrambling polynomials in GS ensures the generation of complementary GS codeword pair(s) [20]. It is also straightforward to show that it is possible to use linear EC block codes to generate complementary EC codewords when input words to the EC encoder are complementary. Note that the sum of complementary words is the all-one word, and that a linear systematic EC code which contains the all-one codeword will map the all-one input word to the all-one codeword (note that this is also possible with some nonsystematic codes). Owing to the linear nature of this EC code, it will also map complementary input words to complementary codewords. Therefore, it is sufficient for the all-one word to be an EC codeword, in order to ensure the presence of complementary EC codewords for selection when the inputs to an EC encoder are complementary words. For example, a systematic (7, 4) Hamming code has an all-one codeword. This code has eight pairs of complementary

Fig. 1. (a). Block diagram of the new dc-free EC block code. (b). Block diagram of the equivalent new dc-free EC block code.

codewords, and the encoder will map a pair of complementary input words to a pair of complementary codewords. We consider integration of a GS encoder, which uses ( ) augmenting bits and , with an EC block code which contains the all-one codeword. Fig. 1(a) shows our new coding structure for dc-free EC block codes. To encode, a source word is mapped to a set of pairs of complementary EC codewords through concatenation of a GS encoder with identical EC encoders. The GS encoder generates pairs of comple, and the EC mentary words and , encoders encode these complementary words to form the com. Based plementary EC words and , on a predetermined selection criterion, an EC codeword from the selection set is selected to ensure that the RDS of the coded sequence is bounded. Therefore, the input to the channel is a fully EC-protected dc-free sequence. Several selection criteria have been developed for multimode codes. They include minimum word-end running digital sum (MRDS), which selects the word from the selection set which results in the minimum absolute word-end RDS value [14], and minimum squared weight (MSW), which selects the word with the minimum sum of the squared RDS values at each bit position within the word [17]. It has been shown that the MSW criterion yields excellent spectral performance for dc-free multi, and provides approximately the same mode codes when performance as that of MRDS when [13]. Since MRDS is simpler than MSW, in this paper, we consider use of MRDS and MSW when . when It can be shown that the all-one binary codeword exists in most linear EC block codes, including binary cyclic codes,

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binary primitive BCH codes, the Golay code, RS codes, and Reed–Muller codes. For LDPC codes [22], if each row of the parity-check matrix has even weight, the all-one word is a codeword. Furthermore, product codes [23] with the above codes as component codes also have the all-one codeword. We consider appropriate EC codes in more detail in Section IV. Note that the use of EC codes that map the all-one input word to the all-one codeword is not strictly necessary in order to ensure valid operation of this new encoding approach. In general, in order to ensure that the RDS of the coded sequence can be bounded, we must ensure that in each selection set, there exist words of opposing disparity, or at least one word with zero disparity. However, it is not clear how to construct a dc-free EC code with this general requirement. Instead of considering this general construction of dc-free EC codes, we propose selecting GS and EC code parameters, as outlined above, to ensure the presence of complementary words in each selection set. It is well known how to generate complementary words at the output of , and that an EC encoder the GS encoder through use of with an all-one codeword will preserve this complementary nature. Alternatively, if the EC encoder maps some input word rather than the all-one word to the all-one codeword, the GS encoder could be designed to guarantee that words in the GS selection set are related through elementwise modulo-2 addition with . This would involve construction of the appropriate GS scrambling polynomial whose weight would be greater than two [24]. An equivalent form of the new coding scheme is shown in Fig. 1(b). The difference between these two structures is the manner in which encoding is implemented. Since both the GS encoder and the EC encoders in Fig. 1(a) are linear, there are predetermined additive patterns between the codewhich depend only on words the GS scrambling polynomial and the EC encoder. In order to determine these additive patterns, one can augment the all-zero source word with all patterns of the augmenting bits , , scramble these augmented words with the scrambling polynomial, and encode these scrambled words to EC codewords . Then the in each codeword candidates encoding interval are related by and , . Fig. 1(b) demonstrates that the same codeword set as that generated in Fig. 1(a) can be constructed by generating , and then forming the rest of the codeword alternatives through addition with the predetermined additive is the patterns stored within the encoder. Note that since all-zero sequence, it is not indicated in Fig. 1(b), and that since the predetermined additive patterns are complementary, only half of them need to be stored. Fig. 1(a) and (b) show that decoding of the new coding scheme is completed first through EC decoding and then through GS decoding, and that the EC decoding is independent of the GS decoding. Therefore, 1) soft-decision information available at the output of the demodulator can be used by the EC decoder, resulting in improved BER performance, and 2) error extension that occurs during GS decoding follows the EC decoder, and therefore, has no detrimental impact on the performance of the EC decoder.

The BER performance of this new technique is governed largely by the characteristics of the EC code. When is used for GS, subsequent error extension in GS decoding is upper bounded by two. IV. ERROR-CONTROL BLOCK CODES SUITABLE FOR THE NEW CODING SCHEME denote the all-one word of length . If is a codeword, nonsystematic codes, in general, map . The constructed by a nonall-one input word to using this nonall-one word as the relationship sequence likely has weight greater than two. In systematic codes, however, the , and all-one input word is mapped to will generate this all-one relationship sequence. Since low weight is preferred to decrease the error extension during GS decoding, systematic EC codes are considered in the remainder of this paper. in a number of difWe now show the existence of ferent EC block codes, in order to demonstrate that these EC codes can be integrated into our new dc-free EC block coding scheme. Let

A. Binary Cyclic Codes In an ( , ) binary cyclic code, a word is a codeword if is divisible by the generator polynomial and only if of degree ( ) [19]. Also, divides . It is straightforward to verify that . If is except (which is often the selected from factors of case), the all-one word is a codeword. B. Binary Primitive BCH Codes BCH codes are cyclic codes. For an ( , ) binary primitive -error-correcting BCH code, the generator polynomial is the least common multiple of the minimal polynomial of , . Since is a multiple of all the minimum polynomials except for those of elements is a multiple of , and is, zero and one [19], therefore, a codeword. For shortened BCH codes, unfortunately, the all-one codeword is not available in BCH codes shortened in the conventional manner [19]. Instead, we develop a new approach to shorten the code. Consider the codewords in the ( , ) . Note code that have all-one parity check of length , there are of them. that if These codewords are called shortening patterns. For example, is such a codeword in the (15, ). This codeword indicates 11) BCH code ( that the (15, 11) code can be shortened to a (12, 8) code in a different manner. Zeros in indicate the positions in which data bits in all words must be ensured to be zero. To construct the codewords, insert three zeros in the 8-b mesto obtain the 11-b word sage word . This word is processed as the input word of the (15, 11) encoder, and the three zeros are removed after encoding to create a (12, 8) shortened code. Based on the shortening pattern, it is straightforward to construct the shortened GS-BCH scheme by modifying Fig. 1(a) or

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(b) to include appropriate insertion of zeros before the BCH encoder and removal of these zeros after the BCH encoder, and insertion of zeros (or appropriate soft-decision values) before the BCH decoder and removal of these zeros after the BCH decoder. Shortening patterns can be constructed by simply adding ap, which is equivalent propriately left-shifted versions of , and therefore, constructing a to evaluating a multiple of valid codeword in the code. Alternatively, to find the shortening pattern for the (12, 8) code, appending the all-zero vector of yields the codeword . The complement of length 10 to , the desired pattern , is also a codeword, because the all-one word of length is a codeword. Based on these approaches, , ) codes, the (15, 11) code can be shortened to ( . The most typical extension of a binary code is the inclusion of an overall parity check, which is the modulo-2 sum of all other bits in the word. Since the codeword length is odd in BCH codes [19], it is straightforward to verify that the all-one codeword exists in extended BCH codes.

C. Reed–Solomon Codes RS codes are cyclic codes. For ( , ) primitive -error-correcting RS codes with and , the generator polynomial has as all its roots. from GF , , form all the roots of Since , is a multiple of . However, the biis not all-one when . Let nary representation of ( ) from GF be the element whose binary vector representation is an -bit all-one vector. Since in a linear code over GF , the product of any codeword by a field element is a codeword [25], in RS codes, there exists , whose corresponding binary representation is the all-one word. As with shortened BCH codes, the all-one codeword is not available in RS codes shortened in the usual manner. The method used for BCH codes can be employed to find shortening patterns for RS codes, however, it is not as straightforward because of the nonbinary elements. As an example, Fig. 2 shows how to obtain the shortening patterns for codes shortened from a (7, 5) RS code. From Fig. 2, the shortening pattern for the (3, 1) code, when and omitting the first written with coefficient 1 instead of , which is the of the (7, three zeros, is 4) binary BCH code. This observation provides an easier approach to construct the shortening patterns for RS codes based , which are given in many EC text books. Consider on binary ( , ) BCH codes and ( , ) RS codes, such that both and have the same . Since codes are constructed over GF has and their conjugates as all its roots, has as all its roots [19], is a and multiple of and is a codeword of the RS code. Therefore, the shortening patterns for the RS codes can be obtained through summation of shifted versions of the corresponding , and subsequent replacement of the coefficients of value 1 by .

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Fig. 2. Construction of the shortening patterns c (x) and c (x) for a (7, 5) RS code shortened to (3, 1) and (4, 2) RS codes, respectively.

D. Reed–Muller Codes It follows from observation of the generator matrix of binary th-order Reed–Muller codes, which are nonsystematic, that the all-one codeword exists [25]. The codes can be converted to a systematic form through elementary row operations and/or column permutations on the generator matrix [25]. If there exists the all-one codeword, it remains a codeword after any of the above transformations of the generator matrix. E. LDPC Codes Length- LDPC codes are specified by the sparse nonsystematic parity-check matrix [22]. For regular LDPC codes, there and fixed column weight . These is a fixed row weight codes are usually denoted as ( , , ). If all rows of have even weight, the all-one word is a codeword. As an example, all rows of in the (12, 3, 6) LDPC code constructed in [26] have even weight, and it is straightforward to verify that the all-one word is a codeword. F. Product Codes An ( , ) product code is constructed using two component codes, an ( , ) inner block code and an ( , ) outer block code [19]. If all-one binary codewords exist for both component codes, it can be shown that an all-one message word of length results in the all-one codeword of length . All codes discussed above can be used as component codes in these product codes. V. PERFORMANCE EVALUATION In this section, we evaluate the performance of our new scheme, and compare it with dc-free coset coding and the conventional approach of concatenating EC and constrained codes. Note that for the simulation results reported in this section, independent and equiprobable information bits are assumed unless otherwise specified. Due to the complexity of GS dc-free multimode codes, it is not yet known how to fully analyze their dc-free performance in a practical manner [13]. Instead, the spectral performance of these codes is obtained through simulation or approximations. In both cases, metrics such as sum variance and LFSW are used to assess the performance of these codes. A random-drawing

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block codes to accurately estimate sum variance and LFSW values. In [8], a dc-free coset code is constructed from a binary BCH code through modification of the nonsystematic generator matrix of the BCH code. We call this approach a dc-free coset BCH code. For a binary -error-correcting BCH code of length , let and be two odd integers such that and . As described in [8], the dc-free augcoset BCH code of length can be generated with menting bits. Due to the similarity of the approach to dc control in the dc-free coset BCH code [8] of length and a polarity switch code of length , we propose estimating the sum variance of the dc-free coset BCH code with the analytical method developed for the polarity-switch code [17]. By extending the approach outlined in [17], the sum variance of the dc-free coset BCH code can be approximated as (6) Also, based on a closed-form expression for the approximate PSD of the polarity-switch code [26], it follows that the LFSW of the polarity-switch code can be approximated as (7)

Fig. 3. (a) Sum variance of GS dc-free multimode codes (calculation and simulation), and simulated sum variance of three different dc-free GS-EC codes. (b) LFSW of GS dc-free multimode codes (calculation and simulation), and simulated LFSW of three different dc-free GS-EC codes.

model was proposed to estimate the sum variance of multimode codes which used the MRDS selection criterion [13], and a method has been developed to estimate the sum variance and the LFSW of GS dc-free multimode codes which use and the MSW criterion [21]. As described in [21], the sum variance and LFSW of GS dc-free multimode codes which use the MSW criterion and can be approximated as (4) (5) is the binary codeword length and and are where factors which are dependent only on the number of augmenting and are reported in [21]. bits . Values of Fig. 3(a) and (b) compare the sum variance and LFSW calculated from (4) and (5) with simulated sum variance and LFSW for GS dc-free multimode codes. These figures also show simulated sum variance and LFSW for our new dc-free GS-EC block codes, where the (255, 239) BCH code, (504, 252) LDPC code, and (255, 239) RS codes are used as EC codes. These figures show that relations (4) and (5) can also be applied to our GS-EC

Therefore, the LFSW of the dc-free coset BCH code can be estimated from (7). In order to compare the performance of our GS-BCH code with the dc-free coset BCH code, consider the (255, 239) BCH (in octal) [19] and code with augmenting bits in both the GS-BCH code and the dc-free coset BCH code; in this coset BCH code, . Note that the BCH code used in our GS-BCH code is a standard systematic BCH code, and the code used in the dc-free coset BCH code is a nonsystematic modified BCH code [8]. From (4) through (7), and when [21], noting that and for the GS-BCH code, we obtain and for the dc-free coset BCH and and values equal 15.58 and 912.15 code. Our simulated for the GS-BCH code, and 33.67 and 2905.22 for dc-free coset BCH code, respectively. Fig. 4 shows simulated PSDs for both these codes. These results demonstrate that with the same number of augmenting bits, the GS-BCH code results in significant improvement in spectral performance, compared with the dc-free coset BCH code. Fig. 5 illustrates the simulated distribution of RDS values for these two codes. From [8], the RDS of this dc-free coset BCH code can . Fig. 5 shows be upper bounded by RDS are still far that for this code, RDS values with probability from this bound. We also note that if the Berlekamp–Massey algorithm [19] is used to decode the dc-free coset BCH code, error extension cannot be avoided, since the source word is recovered from the decoded codeword by solving equations based on the nonsystematic generator matrix. In the remainder of this section, we show other advantages of our new coding scheme. We report performance results when an RS code is used as the component EC code to show that our new coding scheme yields coding gain when compared with the conventional concatenation of RS and GS codes. We also present results when an LDPC code is used as the component

ZHAI et al.: DC-FREE ERROR-CONTROL BLOCK CODES

Fig. 4. Simulated PSDs (solid curves) and approximate PSDs (dashed lines, based on the simulated LFSW) of the dc-free coset BCH code and the GS-BCH code employing the (255, 239) BCH code with A = A = 5.

Fig. 5. Simulated distribution of RDS values for the dc-free coset BCH code and the GS-BCH code employing the (255, 239) BCH code with A = A = 5.

EC code to illustrate integration of soft-decision EC decoding, which is not possible, in general, in the conventional scheme. and the MSW Note that in the following results, we use selection criterion in our codes. The performance of the proposed new GS and RS integrated coding scheme (GS-RS scheme) is shown in Figs. 6–8. Our sim; ulation parameters include: (1912, 1904) GS code with (255, 239) RS code with (in octal) for generating . For comparison, we also give the BER performance GF for uncoded signaling for (255, 239) RS coding, and for conventional concatenation of RS and GS codes. In this conventional code, the (255, 239) RS code was used as the outer code, and a (2048, 2040) GS code was used as the inner code. Fig. 6 depicts the PSD performance of the GS-RS scheme. The corresponding conventional scheme provides almost the same low-frequency suppression as that of the new scheme; RS coding alone does not result in suppression of low frequencies. Fig. 7 presents the BER performance of this new GS-RS scheme over the additive white Gaussian noise (AWGN) channel, and compares this with the performance of uncoded

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Fig. 6. Simulated PSD (solid curve) and approximate PSD (dashed line, based on the simulated LFSW) for a GS-RS coding scheme.

Fig. 7.

BER performance over the AWGN channel.

signaling, the conventional scheme, and RS coding without denotes the average energy per informaspectrum control. is the single-sided PSD of the white Gaussian noise, tion bit, and matched filtering is used. RS decoding is implemented using the hard-decision Berlekamp–Massey algorithm [19]. As indicated in this figure, under these conditions, our GS-RS when compared with scheme offers 1-dB gain at BER conventional concatenation of RS and GS codes. In order to obtain a general indication of error performance that can be expected on a channel with low-frequency constraints, we model the noisy dc-constrained channel using a simple first-order RC high-pass filter (HPF) and AWGN (HP-AWGN channel). The HPF can be used to model a specific dc constraint through selection of the value of the normalized , where denotes symbol duration. time constant Fig. 8 illustrates the BER performance of our new GS-RS coding scheme compared with the other three schemes for different source-logic probabilities when and a rectangular pulse shape and a rectangular receiving filter are used. This figure presents results when the probability of a logic 0 in the source sequence, , is 0.5, 0.45, and 0.4. It can be seen that the

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Fig. 8. BER performance with different source-symbol probabilities over the HP-AWGN channel.

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Fig. 10. BER performance of the GS-LDPC coding scheme over the AWGN channel.

the case with . This is in agreement with the results obtained with increasing in GS dc-free multimode codes without provision for EC [13]. Fig. 10 shows the BER performance in and the AWGN channel for the GS-LDPC scheme with , and the BER performance for LDPC codes without the dc constraint. It is evident that when the BER is less than , the BER performance of the new GS-LDPC scheme is within 0.15–0.2 dB of that of the LDPC code without the dc constraint. This loss in performance in the AWGN channel is due to the error extension in GS decoding and the rate penalty associated with including GS augmenting bits. As with the GS-RS scheme, however, the performance of the GS-LDPC scheme will be superior on channels with a dc constraint. VI. CONCLUSION

Fig. 9. Simulated PSDs (solid curves) and approximate PSDs (dashed lines, based on the simulated LFSW) of two GS-LDPC codes.

change of this probability does not affect the performance of the GS-RS scheme and the conventional scheme significantly, but has a large, negative effect on the performance of the uncoded and RS codes, which justifies the use of dc-constrained codes on this channel. While these results are for the simple first-order HP-AWGN channel model, it is expected that these trends would also be apparent in more realistic noisy dc-free constrained systems. The performance of the proposed new GS and LDPC inteand , in grated coding scheme (GS-LDPC) for terms of PSD and BER, is shown in Figs. 9 and 10. The rate-1/2 LDPC code uses a parity-check matrix from [28] with column weight 3 and row weight 6, and LDPC decoding is performed using the iterative sum-product algorithm [22] with a maximum of 1000 iterations. Fig. 9 illustrates the PSDs for this GS-LDPC scheme with and . Clearly, a 10-dB improvement in performance is obtained at when , compared with

A new general technique for integrating dc-free CS codes and EC block codes has been reported. Our new coding method proposes concatenation of GS multimode coding and EC block coding. In this new scheme, the codeword selection set consists of EC block codewords. Selection criteria previously developed for dc-free multimode codes can be used. Well-established standard EC block decoders can be employed to complete the first stage of decoding of these new codes, allowing for the use of soft-decision information and iterative decoding. Many well-known block codes can be used in this new coding scheme. We have introduced a method for estimating the spectral performance of the new codes, and have shown that this new coding scheme demonstrates excellent performance in terms of lowfrequency suppression. We have also demonstrated that, compared with conventional concatenation of EC and dc-free codes, this scheme provides superior BER performance over a simple HP-AWGN channel. ACKNOWLEDGMENT The authors would like to thank the anonymous reviewers, whose comments resulted in significant improvement in this paper.

ZHAI et al.: DC-FREE ERROR-CONTROL BLOCK CODES

REFERENCES [1] K. W. Cattermole, “Principles of digital line coding,” Int. J. Electron., vol. 55, pp. 3–33, Jul. 1983. [2] K. A. S. Immink, P. H. Siegel, and J. K. Wolf, “Codes for digital recorders,” IEEE Trans. Inf. Theory, vol. 44, no. 6, pp. 2260–2299, Oct. 1998. [3] A. Kokkos, A. Popplewell, and J. J. O’Reilly, “A power efficient coding scheme for low-frequency spectral suppression,” IEEE Trans. Commun., vol. 41, no. 11, pp. 1598–1601, Nov. 1993. [4] J. L. Fan and A. R. Calderbank, “A modified concatenated coding scheme with applications to magnetic data storage,” IEEE Trans. Inf. Theory, vol. 44, no. 4, pp. 1565–1574, Jul. 1998. [5] W. G. Bliss, “Circuitry for performance error correction calculations on baseband encoded data to eliminate error propagation,” IBM Tech. Discl. Bull., vol. 23, pp. 4633–4634, 1981. [6] K. A. S. Immink, “A practical method for approaching the channel capacity of constrained channels,” IEEE Trans. Inf. Theory, vol. 43, no. 5, pp. 1389–1399, Sep. 1997. [7] H. C. Ferreira, “Lower bounds on the minimum Hamming distance achievable with runlength constrained or dc-free block codes and the synthesis of a (16, 8) d = 4 dc-free block code,” IEEE Trans. Magn., vol. MAG-20, no. 9, pp. 881–883, Sep. 1984. [8] R. H. Deng and M. A. Herro, “DC-free coset codes,” IEEE Trans. Inf. Theory, vol. 34, no. 4, pp. 786–792, Jul. 1988. [9] M. C. Chiu, “DC-free error-correcting codes based on convolutional codes,” IEEE Trans. Commun., vol. 49, no. 4, pp. 609–619, Apr. 2001. [10] T. Wadayama and A. J. H. Vinck, “DC-free binary convolutional coding,” IEEE Trans. Inf. Theory, vol. 48, no. 1, pp. 162–173, Jan. 2002. [11] I. J. Fair and D. R. Bull, “DC-free error control coding through guided convolutional coding,” in Proc. IEEE Int. Symp. Inf. Theory, 2002, p. 297. [12] G. L. Pierobon, “Codes for zero spectral density at zero frequency,” IEEE Trans. Inf. Theory, vol. IT-30, no. 2, pp. 435–439, Mar. 1984. [13] K. A. S. Immink and L. Patrovics, “Performance assessment of dc-free multimode codes,” IEEE Trans. Commun., vol. 45, no. 3, pp. 293–299, Mar. 1997. [14] I. J. Fair, W. D. Grover, W. A. Krzymien, and R. I. MacDonald, “Guided scrambling: A new line coding technique for high-bit-rate fiber optic transmission systems,” IEEE Trans. Commun., vol. 39, no. 2, pp. 289–297, Feb. 1991. [15] A. Kunisa, S. Takahasi, and N. Itoh, “Digital modulation method for recordable digital video disc,” IEEE Trans. Consum. Electron., vol. 42, no. 3, pp. 820–825, Aug. 1996. [16] J. Justesen, “Information rates and power spectra of digital codes,” IEEE Trans. Inf. Theory, vol. IT-28, no. 3, pp. 457–472, May 1982. [17] K. A. S. Immink, Codes for Mass Data Storage Systems. Rotterdam, The Netherlands: Shannon Foundation, 1999. [18] Y. Xin and I. J. Fair, “A performance metric for codes with a high-order spectral null at zero frequency,” IEEE Trans. Inf. Theory, vol. 50, no. 2, pp. 385–394, Feb. 2004. [19] S. Lin and D. J. Costello, Jr., Error Control Coding: Fundamentals and Applications. Englewood Cliffs, NJ: Prentice-Hall, 1983. [20] Y. Xin and I. J. Fair, “Polynomials for generating selection sets with complementary quotients in guided scrambling line coding,” Electron. Lett., vol. 37, pp. 365–366, Mar. 2001. , “Low-frequency performance of guided scrambling dc-free [21] codes,” IEEE Commun. Lett., vol. 9, no. 6, pp. 537–539, Jun. 2005. [22] D. J. C. MacKay, “Good error-correcting codes based on very sparse matrices,” IEEE Trans. Inf. Theory, vol. 45, no. 2, pp. 399–431, Mar. 1999. [23] R. M. Pyndiah, “Near-optimum decoding of product codes: Block turbo codes,” IEEE Trans. Commun., vol. 46, no. 8, pp. 1003–1010, Aug. 1998.

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[24] I. J. Fair, Q. Wang, and V. K. Bhargava, “Polynomials for guided scrambling line codes,” IEEE J. Sel. Areas Commun., vol. 13, no. 4, pp. 499–509, Apr. 1995. [25] R. E. Blahut, Theory and Practice of Error Control Codes. Reading, MA: Addison-Wesley, 1983. [26] P. Sweeney, Error Control Coding: From Theory to Practice. Chichester, U.K.: Wiley, 2002. [27] L. J. Greenstein, “Spectrum of a binary signal block coded for dc suppression,” Bell Syst. Tech. J., vol. 53, pp. 1103–1126, Jul.–Aug. 1974. [28] D. J. C. MacKay. Online code resources of low-density paritycheck codes. [Online]. Available: http://www.inference.phy.cam. ac.uk/mackay/codes/data.html

Fengqin Zhai (S’98) was born in Yangquan, Shanxi, China. She received the M.Sc. degree in electrical and computer engineering in 2001 from the University of Alberta, Edmonton, AB, Canada, where she is currently working toward the Ph.D. degree. Her current research interests include error-control coding and constrained coding in wireless communication, digital transmission, and data-storage systems.

Yan Xin (S’97–M’03) was born in Shenyang, Liaoning, China. He received the Ph.D. degree in electrical and computer engineering from the University of Alberta, Edmonton, AB, Canada, in 2002. Since 2002, he has been a Postdoctoral Fellow, holding an Alberta Ingenuity Associateship at the University of Alberta. His current research interests include error-control coding and constrained coding in digital transmission, data storage, and multicarrier communications systems.

Ivan J. Fair (S’91–M’95) received the B.Sc. and M.Sc. degrees from the University of Alberta, Edmonton, AB, Canada, in 1985 and 1989, respectively, and the Ph.D. degree from the University of Victoria, Victoria, BC, Canada, in 1995. He was employed with Bell Northern Research Ltd. (now Nortel Networks) from 1985 to 1987, and with MPR TelTech Ltd. from 1989 to 1991, working on various aspects of communication system design and implementation. In 1995, he joined the Technical University of Nova Scotia (since amalgamated with Dalhousie University), Halifax, NS, Canada, as an Assistant Professor, and was promoted to Associate Professor before joining the University of Alberta in 1998, where he is now a Professor. From 2001 to 2004, he was the Associate Chair for Undergraduate Studies and Acting Director of Computer Engineering in the Department of Electrical and Computer Engineering at the University of Alberta. He is presently an Associate Editor for IEEE COMMUNICATIONS LETTERS.

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Joint Source/Channel Coding for Multiple Images Zhenyu Wu, Member, IEEE, Ali Bilgin, Member, IEEE, and Michael W. Marcellin, Fellow, IEEE

Abstract—A joint source/channel coding algorithm is proposed for the transmission of multiple image sources over memoryless channels. The proposed algorithm uses a quality scalable image coder to optimally allocate a limited bit budget among all the sources to achieve the optimal overall distortion reduction for the multiple reconstructed images. In addition to the conventional un gain, it provides channel multiplexing gain, which can be much more significant. Furthermore, an extended scheme is proposed to provide flexibility between the optimization performance and complexity. Index Terms—Joint source/channel coding, scalable source coding, un (UEP).

I. INTRODUCTION ODAY’S multimedia applications often require images and video to be transmitted over noisy channels. State-of-the-art image coders, such as set partitioning in hierarchical trees (SPIHT) [1] and JPEG2000 [2], can provide high compression efficiency and scalability. With the quality scalable property, a bitstream obtained by compressing an image at a high bit rate can provide many embedded subsets. Each subset represents an efficient compression of the original image, as if it were compressed at the corresponding lower rate. This property makes it possible for a source coder to perform compression only once, but still be able to satisfy a wide range of application requirements, based on access bandwidth, user interests, etc. Together with the advance of error-control coding, joint source/channel coding has attracted a lot of research effort recently. Various schemes have been proposed for the transmission of scalable images over different channels. Rate-compatible punctured convolutional (RCPC) codes were used to protect SPIHT bitstreams in [3]. In that paper, a SPIHT bitstream is protected equally by a channel code rate for a given channel. The unequal importance of different segments of a scalable bitstream can be exploited to provide performance improvement by applying the proper amount of channel protection to each segment. An un (UEP) scheme from [4] provides about 0.3 dB improvement over [3] with the same source and channel coders.

T

Paper approved by M. Skoglund, the Editor for Source/Channel Coding of the IEEE Communications Society. Manuscript received September 24, 2004; revised March 20, 2005. This work was supported in part by the National Science Foundation under Grants CCR-9979310 and ANI-0325979. This paper was presented in part at the 47th Annual IEEE Global Telecommunication Conference (Globecom), Dallas, TX, USA, November 29–December 3, 2004. Z. Wu was with the Department of Electrical and Computer Engineering, University of Arizona, Tucson, AZ 85721 USA. He is now with Thomson Inc., Princeton, NJ 08540 USA (e-mail: [email protected]). A. Bilgin and M. W. Marcellin are with the Department of Electrical and Computer Engineering, University of Arizona, Tucson, AZ 85721-0104 USA (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TCOMM.2005.857142

A dynamic-programming-based rate allocation was proposed in [5] using optimization criteria based on mean-square error (MSE), peak signal-to-noise ratio (PSNR), and available source rate. The last criterion was suggested as the best approach, since it reduces the complexity and allows optimal transmission at many rates lower than the optimized target rate. The authors of [6] used a Viterbi-algorithm (VA)-based rate allocation for UEP with turbo codes and JPEG2000. A similar problem was solved by a local search in [7]. The algorithm discussed there starts from a rate-optimal solution and converges to a locally distortion optimal solution, and gives comparable performance to [6], but with reduced complexity. A scheme employing the same rate allocation but more powerful irregular repeat–accumulate (IRA) channel codes was presented in [8]. While the above schemes deal mostly with binary symmetric channels (BSCs), image transmission over other types of channels has also been studied. For example, product channel codes with different component codes and decoding schemes were employed in [9] and [10] for fading channels. Joint rate allocation over fading channels with and without feedback was investigated in [11], based on an empirical model of decoded bit-error rate as a function of the channel code rate. Rate allocation for image transmissions over packet-erasure channels was explored in [12] and [13]. All the above schemes consider the case in which only one image is transmitted. However, with current state-of-the-art source-coding technology, many communication channels are now capable of delivering several compressed images or video sequences concurrently. This makes it possible for some applications to transmit multiple images together sharing a common channel, such as multimedia client/server systems, wireless sensor networks, and so forth. We note that the problem of multiplexing variable bit-rate (VBR) videos has been studied previously (for example, [14] and [15]). It has been shown that such multiplexing can lead to more efficient use of the channel capacity, and also more uniform picture quality [14]. However, these schemes only involve source coding, and do not take joint source/channel coding into consideration. In this paper, a joint source/channel coding algorithm is proposed to code multiple image sources jointly. It uses the quality scalable property of image coders to dynamically allocate limited bandwidth among all the sources, in order to achieve optimal distortion reduction over all the reconstructed images. It provides both UEP and channel-multiplexing gains, and shows the advantage of performing rate allocation jointly for the multiple image sources. The paper is organized as follows. Section II develops the proposed joint rate-allocation algorithm. Experimental results are given in Section III, while Section IV draws conclusions.

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WU et al.: JOINT SOURCE/CHANNEL CODING FOR MULTIPLE IMAGES

II. JOINT RATE ALLOCATION Consider that multiple images are to be transmitted over a noisy channel within a given total bit rate. At the transmitter end, each image is first progressively encoded by a source coder. For each image, its resulting bitstream is partitioned into several segments, and each segment is then channel encoded to form a fixed-length channel packet to be sent over the channel. Each packet is assigned a specific channel code rate, and the code rate, in turn, determines the number of source and parity bytes in the packet. At the receiver end, for each image, channel decoding is first performed to recover source bytes from its received channel packets. Channel decoding stops for a given image whenever a channel decoding failure occurs, or all the channel packets for the image have been correctly decoded. The recovered source bytes are then decoded by a source decoder to reconstruct the image. This practice may result in decoding only a prefix of an original source bitstream. However, it prevents the otherwise possibly catastrophic reconstruction effects due to loss of synchronization during source decoding. With the aforementioned source/channel decoding practice, the goal of the proposed joint rate allocation is to optimally assign a channel code rate for each channel packet such that the overall expected distortion reduction (measured in MSE) for the reconstructed image(s) is minimized. A. Single-Source Case Let be the bitstream length (corresponding to a given be bit rate) including both source and parity bytes, and the corresponding number of channel packets allocated to a source . Because each channel packet has a fixed length, can be obtained by dividing with the packet length. Let be a vector representing the channel code channel packets. Therefore, deterrates assigned to the mines, for source , the rate allocation between its source and channel coders, and the protection level each source segment receives from the channel codes. The expected distortion reduction (as compared with the zero-rate case) for a reconstructed image can be written as

(1) is the expected distortion reduction when where is employed, and is the distortion reduction brought by the source bytes included in the th channel packet with code denotes the probability of channel-packet derate . coding failure when code rate is used to protect the th packet (note that is defined to be 1 to indicate the end of the bitstream.) The joint rate-allocation problem for the single-source case is . The to find a code-rate vector which maximizes optimal solution can be obtained by a brute-force search. However, when the number of channel packets is large, the search is computationally prohibitive. The forward dynamic-programming-(VA)-based algorithm from [6] can be used to solve this problem efficiently. With this algorithm, each VA stage corresponds to a channel packet, and each trellis state corresponds

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to an available channel code rate. As noted in [6], the VA is suboptimal in this case. However, it has low complexity, and its solutions are close to optimal most of the time. Compared with using one channel code rate to protect a source bitstream, the scheme discussed above can provide a performance improvement by applying unequal amounts of channel protection to different segments of the source bitstream, according to . The gain obtained in this manner for a single source is called the UEP gain hereafter. B. Multiple-Sources Case Let be the number of images to be transmitted through a as the total bitstream length (corcommon channel. Denote responding to a given total bit rate), including both source and sources. parity bytes, to be shared by the One possibility is to divide the total bit rate equally into pieces and statically assign each piece to a source. Each source can perform the joint rate allocation as described in Section II-A to obtain a UEP gain. However, images from different sources can be drastically different in terms of their rate-distortion characteristics. By taking this factor into account and dynamically allocating the total bit rate among multiple sources, the overall distortion reduction for the multiple reconstructed images could be improved compared with the static rate-allocation case. This gain is called the channel multiplexing gain hereafter. Therefore, the goal of the proposed dynamic rate-allocation algorithm for the multiple sources case is s.t.

(2)

Lagrange-multiplier methods can be employed to transform this constrained optimization into an unconstrained version (3) Similar to [16], for a given , the solution to (3) can be obtained by solving each term (corresponding to each source) independently. By sweeping from zero to infinity, sets and can be created for each source . If for some , equals , then an optimal solution has been found. , minimizing each term in (3) corresponds For a given to an optimization task for each source. Combining the result from (1), the objective function to be minimized for source can be written as

(4) where is the fixed channel-packet length. Notice that unlike (1) where is fixed, to minimize (4), and must be optimized jointly according to the value. and over their search By enumerating all permissible space, an optimal solution can be found for (4). But when the

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number of channel packets or the number of channel code rates is large, this approach quickly becomes infeasible. A forward dynamic-programming-(VA)-based approach is therefore proposed to reduce the complexity in solving (4). The notation is based on that in [17]. The derivation in the remainder of this subsection is for a single source , and thus the subscript is dropped for simplicity. By proper rearrangement and modification, (4) can be rewritten as

multiple-sources case, it is easy to observe that when and is fixed (this also leads to the exclusion of the zero-value state from the code-rate space for each stage), the algorithm becomes the VA optimization for the single-source case, as given in [6]. So single-source optimization is a special case of the mulis specified and is set to zero. tiple-sources case when 1) Rate-Allocation Adjustment: Most practical scalable image coders can provide only a finite number of points on their operational distortion-rate curve for a given image. Specifically, a scalable bitstream cannot generally be truncated at any arbitrary point without incurring some truncation loss [2].2 Furthermore, UEP is provided mainly based on a finite set of channel codes. Because of the discrete nature of the problem, with the use of Lagrangian optimization, for any given total bit , it may be impossible to find such that rate is exactly equal to . Consequently, the optimal solution cannot always be achieved by the above Lagrangian-optimization-based method. A rate-allocation-adjustment algorithm is proposed to address this problem. Suppose with Lagrangian optimization, a value is found that is closest to . Denote as which gives and , which can be positive the gap between or negative. Because the goal of the algorithm is to maximize the overall distortion reduction for the multiple images, when , it is desirable to include more from the source which gives the greatest distortion reduction. On the other hand, when , some bits should be trimmed from the source which brings the smallest distortion reduction. Suppose after the Lagrangian optimization, a total given bit rate cannot be achieved. A value can be chosen which gives . Each source is allocated channel the smallest associated packets, and there is a distortion reduction with its last included packet. Based on the information, a source can be chosen according to the criteria of the previous paragraph.3 This source is then augmented or trimmed to . One channel packet is added to or subtracted from decrease the source. Adding or subtracting a packet to/from the chosen source (or requires the the reoptimization of source with ) channel packets. Because the number of channel packets is then specified, the problem can be readily solved using the single-source optimization. Recall this is just a special case of the multiple-sources case. To be more specific, the optimization can be done by setting and specifying a new fixed or ) in the corresponding number of packets (

(5) is the maximum number of channel packets that can where is defined possibly be allocated to the source, and function as

In light of (5), for the proposed VA algorithm, the cost funcis defined as tion at stage , state (6) where stage corresponds to the th channel packet ( ), and state belongs to the set of available channel code . rates at stage denoted by The cost-to-arrive function at stage , state is defined as when and as shown in (7) at the bottom of the page, where (8) The quality scalable property of the source bitstream implies a nondecreasing channel code-rate assignment. This can be im( ). At each stage of the posed as cost-to-arrive function , in addition to the states corresponding to the available channel code rates, a zero-value entry ) is added to the state space to indicate that only the first ( channel packets are required to minimize (4). Therefore, for each source and a given , the proposed VA algorithm proceeds from the first channel packet to the last one that can be possibly included by the source, computing the cost-to-arrive using (7) for each packet (stage) and the available states at that stage. The minimal cost associated with a state in the last stage becomes the optimal cost, and the path which leads to this state from the first stage given by (8) determines the optimal rate allocation for source .1 From the derivation for the 1The VA is not always strictly guaranteed to find the optimal solution. This point is discussed carefully in Section II-C.

2For example, most embedded bitstreams can only be truncated without truncation loss at the end of a “coding pass” (e.g., SPIHT). 3Note that when adding more data, we decide which source to augment based . This reduces complexity with negon d , even though we should use d ligible suboptimality.

1

1

when (7)

WU et al.: JOINT SOURCE/CHANNEL CODING FOR MULTIPLE IMAGES

multiple-sources algorithm. This selection/optimization procedure can be carried out one channel packet at a time until the total bit-rate requirement is satisfied. 2) Algorithm Summary: The proposed dynamic rate allocation for multiple sources can be divided into two levels. At the lower level, each source performs joint rate allocation with a dynamic programming algorithm for a given to achieve UEP gain. At the higher level, the value of is adjusted to achieve the desired total bit rate. The higher level is aimed to give channel multiplexing gain in addition to the UEP gain provided by the lower level. Note that other optimization schemes which solve (4) can be used instead of dynamic programming. When the algorithm of the previous paragraph cannot provide a solution that exactly satisfies a given total bit rate, the rateadjustment part of the algorithm is called for. It executes (in an iterative fashion) to augment or trim one channel packet for one source at a time. This procedure is repeated until the total desired rate is reached. Because all the lower-level source optimizations are independent of each other, they can be done in parallel, which leads to a total algorithm-processing time possibly independent of the number of sources. In this case, a controller is needed to coordinate the rate allocation for the multiple sources. Its job is simply to assign choices of to all the source processors, and to collect and add the returned bitstream lengths from each source, and compare with the given total bit rate. Because of the monotonic relationship between and the resulting total length, a bisection strategy can be adopted by the controller to find the proper . Therefore, the controller has very low complexity.

C. An Optimality Improvement When the dynamic-programming approach is employed in the above optimization, a term associated with all the previous stages is multiplied with the adjusted cost function in the current stage, as shown in (7). With the presence of this term, the cost of each stage is not simply additive, and hence, dynamic programming cannot guarantee an optimal solution [6]. Depending on the actual optimization space and the number of stages involved, in some cases, the solution given by the dynamic-programming approach could be significantly worse than the optimal one. When this happens, some sources may yield substantially suboptimal rate allocations which affect their UEP gains and also the channel multiplexing gain. To that end, an improvement over the dynamic-programming optimization is proposed. It is based on a variation of the list Viterbi decoding algorithm (LVA) [18]. With conventional dynamic programming, at each state of a stage, only the minimal cost and the associated path are stored. In contrast, with the proposed LVA-based approach, each state keeps a list of the minimal costs and their paths. For each state, the new trellis search involves the computation of all candidate states from the previous stage, and the resulting best costs (and their paths) are selected and ordered to form the list associated with the current state. With conventional LVA, at the final stage, its best metric is the same as the VA. If the path associated with this metric

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cannot satisfy some constraint (such as an inner cyclic redundancy check (CRC) code), the LVA searches for the path corresponding to the next best metric, until all stored metrics and their paths are exhausted. However, with the proposed LVA, the minimal cost (and its path) at the final stage can be equal to or better than the VA. And only the minimal cost and its associated path of all the states at the final stage is chosen to give the desired solution. Because the VA is a special case of the proposed LVA with its list depth being 1, the proposed approach gives a consistent but flexible method to adjust the optimality and complexity for both single- and multiple-source optimization. Since the lower level optimization is independent for each source, when a source has enough computational resources, it can choose a LVA with large depth to achieve most of the UEP gain; while with a resource-limited source, a faster but possibly more suboptimal VA optimization can be called with little impact on the other sources. III. EXPERIMENTAL RESULTS In the following experiments, multiple images are transmitted through BSCs with crossover probability and at different total bit rates. Rate-compatible punctured turbo (RCPT) codes [19] are employed as our channel codes. A rate-1/3 parallel concatenated is convolutional coder with generator polynomial chosen as the encoder, and a Bahl–Cocke–Jelinek–Raviv maximum a posteriori (BCJR-MAP) decoder is used with a maximum of 20 iterations. An S-random interleaver [20] is used , where is the number of message bits with plus memory flush bits in a codeword. Puncturing patterns are chosen from [19] with a puncturing period of eight to provide a set of codes for each channel, with tradeoffs between rates and protection levels. Each channel packet has a fixed length of 500 bytes. Inside each packet, 4 bytes are used for CRC-32 as an inner code with generator polynomial to indicate a turbo-decoding failure, and 1 byte is reserved for transmitting side information (such as the turbo-decoding rate for the next channel packet and the source ownership of the received channel packet). , code rates {8/9, 4/5, 8/11, 2/3} are For BSC chosen with {439, 394, 358, 327} source bytes included in the corresponding channel packets, according to the packet structure adopted. Their probabilities of packet-decoding ] for the channel are {0.8730, 1.1675 , failure [ 1.2037 , 1.30 }, obtained based on 2 sim, code ulations for each rate. Similarly, for BSC rates {4/9, 8/19, 2/5, 8/21, 4/11, 1/3} are chosen with {216, 206, 194, 185, 176, 161} source bytes included within each corresponding channel packet. Their probabilities of packet-de, 1.125 , coding failure for the channel are {5.119 , 5.15 , 2.10 , 5.00 }. 8.650 Five images, Lenna, Peppers, Goldhill, Baboon, and Whitehouse, are chosen as our test images. Each is a 512 512, 8-bit, gray-level image. These images are transmitted with total bit rates 1.00, 0.50, and 0.25 bpp/source over the two channels. JPEG2000-based Kakadu software is selected as the source

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TABLE I PERFORMANCE COMPARISON FOR BSC 0.01

Fig. 1. Differences in optimization results between different schemes for the Lenna image and BSC 0.01.

coder, and layering functionality is employed during the source encoding to generate quality scalable bitstreams. To show the utility of the LVA approach, optimization is performed with the Lenna image for BSC 0.01. A small number of states and stages are used to make a manageable comparison with exhaustive search. Fig. 1 shows the difference of values [see (4)] between the exhaustive search (EX), the VA, and the proposed LVA with list depths of 10 and 20. Although the figure shows only the single-source case, the results are similar for multiple sources. From the figure, when the list depth is large enough, the proposed LVA approach always achieves a smaller margin than the VA. In most cases, the LVA approach achieves the optimum given by the EX. On the other hand, depending on the characteristics of specific source and channel codes, the VA can sometimes result in substantial suboptimality in early stages, as shown in the figure. The rate-allocation results based on the proposed LVA algorithm with a list depth of 25 are further compared with those of VA by simulations with multiple images and at least 2000 trials for each case. Up to 0.2 dB per image (0.09 dB, on average) improvement has been observed for BSC 0.01. For BSC 0.1, up to 0.46 dB per image improvement is observed (0.17 dB, on average). Joint source/channel coding is performed for both the static and dynamic rate allocations, which are based on the singlesource and multiple-sources cases in Sections II-A and II-B, respectively. In the single-source case, the UEP results are compared with those in the corresponding equal error protection (EEP) cases. The EEP scheme is realized by protecting an entire bitstream with code rate 2/3 for BSC 0.1 and code rate 1/3 for BSC 0.01. These code rates are the highest that provide close to error-free decoding with their corresponding channels, consistent with the criterion used in [3]. In the multiple-sources case, additional channel multiplexing gains are compared with the UEP gains provided by the single-source case. For each case, at least 2000 simulations are conducted. The mean PSNR values

are calculated by averaging the decoded MSE values, and then converting the mean MSE to the corresponding PSNR value. In every case, both static and dynamic rate allocations are performed based on the LVA algorithm with a list depth of 25. Corresponding to 1.00, 0.50, and 0.25 bpp/source and 500 bytes/packet, there are a total of 330, 165, and 80 channel packets to be allocated among 5 images. With static rate allocation, each image gets an equal number of channel packets. In contrast, the number of channel packets allocated to each image can be very different in the dynamic rate-allocation case. The rate-allocation results for the static and dynamic cases for the two test channels are listed in Tables I and II. Along with them are the simulation results for the average MSE and PSNR values for the two cases. From the tables, it can be observed that in the multiple-sources case, packets are effectively taken from “easier-to-code” images (such as Lenna and Peppers), and given to “harder-to-code” images (such as Baboon and Whitehouse). The resulting distortion increase for the “easy” images is more than compensated by the distortion decrease in the “hard” images, which leads to an overall distortion reduction. Also from these tables, it can be seen that UEP can provide a performance improvement in the static bit-allocation case. However, a substantial channel multiplexing gain can be achieved, in addition, by coding these images jointly in the corresponding dynamic case. For the cases with BSC 0.01, channel multiplexing gains are 6.60, 3.14, and 2.90 times as much as the corresponding UEP gains. For BSC 0.1, the ratios are 2.11, 1.28, and 0.55, respectively. Note that the channel multiplexing gain decreases along with the total bit rate for both channels. This is because with a reduced total bit rate, the corresponding source rate is effectively

WU et al.: JOINT SOURCE/CHANNEL CODING FOR MULTIPLE IMAGES

TABLE II PERFORMANCE COMPARISON FOR BSC 0.1

reduced as well, which limits the rate-distortion diversity that can be exploited by the proposed algorithm. It also explains the gain drop in BSC 0.1, compared with BSC 0.01. For the same total bit rate, more bits are dedicated to the channel coding for BSC 0.1, and hence, the source rate becomes smaller. IV. CONCLUSION A joint source/channel coding algorithm is proposed for the transmission of multiple image sources. It exploits the quality scalable property of image coders to provide UEP and channel multiplexing gains, where the latter gain could be substantially greater. The main part of the algorithm can be executed in parallel for each source which gives a processing time possibly independent of the number of sources. It includes the singlesource optimization as a special case. The LVA offers the flexibility to trade off optimality and complexity.

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[3] P. G. Sherwood and K. Zeger, “Progressive image coding for noisy channels,” IEEE Signal Process. Lett., vol. 4, no. 7, pp. 189–191, Jul. 1997. [4] M. Zhao and A. N. Akansu, “Optimization of dynamic UEP schemes for embedded image sources in noisy channels,” in Proc. Int. Conf. Image Process., vol. 1, 2000, pp. 383–386. [5] V. Chande and N. Farvardin, “Progressive transmission of images over memoryless channels,” IEEE J. Sel. Areas Commun., vol. 18, no. 6, pp. 127–131, Jun. 2000. [6] B. A. Banister, B. Belzer, and T. R. Fischer, “Robust image transmission using JPEG2000 and turbo-codes,” IEEE Signal Process. Lett., vol. 9, no. 4, pp. 117–119, Apr. 2002. [7] R. Hamzaoui, V. Stankovic, and Z. Xiong, “Rate-based versus distortionbased optimal joint source-channel coding,” in Proc. Data Compress. Conf., 2002, pp. 63–72. [8] C. Lan, R. Narayanan, and Z. Xiong, “Scalable image transmission using rate-compatible irregular repeat–accumulate (IRA) codes,” in Proc. Int. Conf. Image Process., 2002, pp. 717–720. [9] P. G. Sherwood and K. Zeger, “Error protection for progressive image transmission over memoryless and fading channels,” IEEE Trans. Commun., vol. 46, no. 12, pp. 1555–1559, Dec. 1998. [10] L. Cao and C. W. Chen, “A novel product coding and recurrent alternate decoding scheme for image tranmission over noisy channels,” IEEE Trans. Commun., vol. 51, no. 9, pp. 1426–1431, Sep. 2003. [11] A. Nosratinia, J. Liu, and B. A. Aazhang, “Source-channel rate allocation for progressive transmission of images,” IEEE Trans. Commun., vol. 51, no. 2, pp. 186–196, Feb. 2003. [12] A. Mohr, E. A. Riskin, and R. E. Ladner, “Unequal loss protection: Graceful degradation of image quality over packet erasure channels through forward error correction,” IEEE J. Sel. Areas Commun.., vol. 18, no. 6, pp. 819–828, Jun. 2000. [13] J. Kim, R. M. Mersereau, and Y. Altunbasak, “Error-resilient image and video transmission over the internet using unequal error protection,” IEEE Trans. Image Process., vol. 12, no. 2, pp. 121–131, Feb. 2003. [14] L. Wang and A. Vincent, “Bit allocation and constraints for joint coding of multiple video programs,” IEEE Trans. Circuits Syst. Video Technol., vol. 9, no. 9, pp. 949–959, Sep. 1999. [15] M. Balakrishnan and R. Cohen, “Global optimization of multiplexed video encoders,” in Proc. Int. Conf. Image Process., vol. 1, 1997, pp. 377–380. [16] Y. Shoham and A. Gersho, “Efficient bit allocation for an arbitrary set of quantizers,” IEEE Trans. Acoust., Speech, Signal Process., vol. 36, no. 9, pp. 1445–1453, Sep. 1988. [17] D. P. Bertsekas, Dynamic Programming and Optimal Control. Belmont, MA: Athena Scientific, 1995. [18] N. Seshadri and C. Sunderg, “List Viterbi decoding algorithm with applications,” IEEE Trans. Commun., vol. 42, no. 2, pp. 313–323, Feb. 1994. [19] D. N. Rowitch and L. B. Milstein, “On the performance of hybrid FEC/ARQ systems using rate compatible punctured turbo (RCPT) codes,” IEEE Trans. Commun., vol. 48, no. 6, pp. 948–959, Jun. 2000. [20] D. Divsalar and F. Pollara, “Multiple turbo codes for deep-space communication,” JPL TDA Prog. Rep., pp. 42–121, May 1995.

ACKNOWLEDGMENT Zhenyu Wu (S’00–M’05) received the B.S. degree in telecommunication engineering and the M.S. degree in Signal and information processing engineering from Shanghai University, Shanghai, China, in 1996 and 1999, respectively, and the Ph.D. degree in electrical engineering from the University of Arizona, Tucson, in 2005. He is currently a Technical Staff Member with Thomson Inc. Corporate Research, Princeton, NJ. His research interests include multimedia communications, source and channel coding, and signal

The authors would like to thank the Editor and the anonymous reviewers for their valuable comments. REFERENCES [1] A. Said and W. A. Pearlman, “A new, fast, and efficient image codec based on set partitioning in hierarchical trees,” IEEE Trans. Circuits Syst. Video Technol., vol. 6, no. 6, pp. 243–250, Jun. 1996. [2] D. S. Taubman and M. W. Marcellin, JPEG2000: Image Compression Fundamentals, Practice and Standards. Norwell, MA: Kluwer, 2002.

processing.

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Ali Bilgin (S’94–M’03) received the B.S. degree in electronics and telecommunications engineering from Istanbul Technical University, Istanbul, Turkey, the M.S. degree in electrical engineering from San Diego State University, San Diego, CA, and the Ph.D. degree in electrical engineering from the University of Arizona, Tucson. He is currently a Research Assistant Professor with the Department of Electrical and Computer Engineering, University of Arizona. He has served as a program committee member for international conferences and was a member of the organizing committee for the 2000 IEEE International Conference on Acoustics, Speech, and Signal Processing (ICASSP 2000). His current research interests are in the areas of signal and image processing, and include image and video coding, data compression, and magnetic resonance imaging. He has co-authored more than 75 technical publications in these areas.

Michael W. Marcellin (S’81–M’82–SM’93–F’02) was born in Bishop, CA, on July 1, 1959. He received the B.S. degree (summa cum laude (highest honors)) in electrical engineering from San Diego State University (SDSU), San Diego, CA, in 1983, and the M.S. and Ph.D. degrees in electrical engineering from Texas A&M University, College Station, in 1985 and 1987, respectively. Since 1988, Dr. Marcellin has been with the University of Arizona, Tucson, where he is a Professor of Electrical and Computer Engineering. His research interests include digital communication and data storage systems, data compression, and signal processing. He has authored or coauthored more than 150 papers in these areas. He is a major contributor of technology to JPEG2000, the emerging second-generation standard for image compression. Throughout the standardization process, he chaired the JPEG2000 Verification Model Ad Hoc Group, which was responsible for the software implementation and documentation of the JPEG2000 algorithm. He is coauthor, with D. S. Taubman, of the book, JPEG2000: Image Compression Fundamentals, Standards and Practice (Norwell, MA: Kluwer, 2002), intended to serve as a graduate-level textbook on image-compression fundamentals, as well as the definitive reference on JPEG2000. Dr. Marcellin was named the most outstanding student in the College of Engineering of SDSU in 1983, and is a member of Tau Beta Pi, Eta Kappa Nu, and Phi Kappa Phi. He is a 1992 recipient of the National Science Foundation Young Investigator Award, and a corecipient of the 1993 IEEE Signal Processing Society Senior (Best Paper) Award. He has received teaching awards from NTU (1990, 2001), IEEE/Eta Kappa Nu student sections (1997), and the University of Arizona College of Engineering (2000). In 2001, Dr. Marcellin was named the Litton Industries John M. Leonis Distinguished Professor of Engineering at the University of Arizona.

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Diversity Combining With Imperfect Channel Estimation Roy You, Member, IEEE, Hong Li, Student Member, IEEE, and Yeheskel (Zeke) Bar-Ness, Fellow, IEEE

Abstract—The optimal diversity-combining technique is investigated for a multipath Rayleigh fading channel with imperfect channel state information at the receiver. Applying minimum mean-square error channel estimation, the channel state can be decomposed into the channel estimator spanned by channel observation, and the estimation error orthogonal to channel observation. The optimal combining weight is obtained from the first principle of maximum a posteriori detection, taking into consideration the imperfect channel estimation. The bit-error performance using the optimal diversity combining is derived and compared with that of the suboptimal application of maximal ratio combining. Numerical results are presented for specific channel models and estimation methods to illustrate the combined effect of channel estimation and detection on bit-error rate performance. Index Terms—Diversity combining, imperfect channel estimation, Rayleigh fading.

I. INTRODUCTION IVERSITY-combining techniques have often been used to combat the deleterious effect of channel fading [1], [2]. If the channel state is known perfectly at the receiver, maximum ratio combining (MRC) can be applied to minimize system biterror rate (BER) [2]. However, in practice, since the channel estimation at receiver is often imperfect, the estimation error will degrade the BER performance. While such a problem has long been studied [3], [4], the current development of high-data-rate, multiple-input multiple-output mobile communication systems has renewed the interest in understanding the impact of imperfect channel estimation on diversity techniques [5]–[7]. In [4] and [5], the BER performance of MRC of independent and identically distributed (i.i.d.) diversity branches with Rayleigh fading is studied. In [6], the distribution of signal-to-noise ratio (SNR) is given for similar scenario. In [7], the BER performance of MRC with independent but not identically distributed (i.n.d.) branches is studied. In this paper, we apply the techniques developed in [8] and study the impact of imperfect channel estimation on a general correlated Rayleigh fading channel. We assume that an imperfect channel observation is available at the receiver through a pilot scheme, which is jointly Gaussian distributed with the channel state. With the imperfect channel observation,

D

Paper approved by A. Abu-Dayya, the Editor for Modulation and Diversity of the IEEE Communications Society. Manuscript received October 8, 2004; revised March 9, 2005. This paper was presented in part at the Conference on Information Sciences and Systems, Princeton, NJ, March 2004. The authors are with the Center for Communications and Signal Processing Research, Electrical and Computer Engineering Department, New Jersey Institute of Technology, Newark, NJ 07102 USA (e-mail: [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.1109/TCOMM.2005.857156

Fig. 1.

Diversity combiner with imperfect channel state information.

minimum mean-square error (MMSE) channel estimation is performed where the channel state is decomposed into the MMSE channel estimator spanned by channel observation, and the estimation error orthogonal to channel observation. The optimal detection rule is derived based on maximum a posteriori (MAP) detection, given the channel observation and the decision variable. The detection error performance is analyzed, and we show that the channel-estimation error should be treated as an additional source of noise, which might not be white. Therefore, MMSE combining with reliable channel estimation should be used, instead of MRC with imperfect channel observation. Furthermore, since the combiner output SNR is quadratic, the error performance can be calculated analytically as a function of the eigenvalues of certain covariance matrices. As examples of the theory put forth, BER performance of frequency- and space-diversity systems with pilot-symbol-aided channel estimation [9] are analyzed and simulated to verify the combined effect of channel estimation and diversity combining. II. SYSTEM MODEL AND OPTIMAL DETECTION RULE A. System Model The system model is shown in Fig. 1. is the transmitted signal and is, in general, complex-valued. In this paper, we use binary phase-shift keying (BPSK) modulation, where is drawn from an i.i.d. source with equal symbol probability. The fading channel is modeled with a correlated Rayleigh fading model, where the channel state is a proper complex Gaussian random vector [10] with zero mean and covariance matrix (1) denotes the Hermitian of , which is the complex-conHere, jugate transpose for complex vectors, and reduces to transpose operator for real vectors. We denote the channel state as . The channel noise is also zero-mean proper complex white Gaussian with covariance matrix

0090-6778/$20.00 © 2005 IEEE

(2)

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and is denoted as vector

. At the receiver, a random is received, where

Given and , has a complex Gaussian distribution with mean and variance

(3)

(12) (13)

,

Using a combiner with weight vector we create a combined decision random variable , where

We write the conditional probability density function (pdf) as (4)

If the channel state is known at the receiver, MRC can be applied to maximize the combiner output SNR, thus minimizing BER. However, the channel state is usually not known perfectly, and only an imperfect channel observation is available at the receiver through the pilot scheme. The imperfect channel obis also assumed to be proper servation complex Gaussian and zero mean with covariance matrix

(14) Assuming that the transmitted symbol has equal probability and the channel observation is independent of the transmitted symbols, the decision rule is simply the minimum-distance decision rule, where

(5)

(15)

and is denoted as . Furthermore, we assume that the channel observation and the channel state are jointly proper and have the cross-correlation

With BPSK modulation, the decision rule reduces to a threshold test

(6) In this paper, while we do not assume the perfect knowledge of the channel state, we do assume that perfect channel statistics are available at the receiver. Such an assumption is made since the channel statistics vary much more slowly than the channel state itself, and thus can be obtained at the initialization stage using a long training sequence, and continuously improved during the whole communication period. The channel statistics depend on the channel-state model and pilot scheme, which is kept generic here. Numerical results with specific models will be given as examples at the end of this paper. B. Optimal Detection Rule

(16) denotes the real part of a complex number. The comwhere bining weight vector is determined such that the average BER can be minimized, which we derive in the next section. III. OPTIMAL COMBINING AND CORRESPONDING ERROR PERFORMANCE The system error performance is measured by the average BER over additive noise , channel state , channel observation , and transmitted symbol

Given the channel observation , the MMSE estimation of the channel state is [11] (7) where with covariance matrix

is zero-mean Gaussian distributed

(17) Define the average error probability conditioned on the transmitted symbol and channel observation as (18)

(8)

and assuming that the channel observation is independent of the transmitted symbol , the average bit error can be expressed as

The channel state is the combination of the linear estimator and the estimation error (9) where

(hence,

) and

(19)

are orthogonal, and (10)

Since the channel state is not available at the receiver, the MAP detection is performed conditioning on the channel observation and combiner output (11)

A. Optimal Combining With Perfect Channel Knowledge When channel state information is perfect ( (19) becomes

),

(20)

YOU et al.: DIVERSITY COMBINING WITH IMPERFECT CHANNEL ESTIMATION

where is the average error due to the additive Gaussian noise , given the perfect channel knowledge. When the MAP detection rule is applied, it can be shown that the is conditional error probability

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portion of the channel state is known to the receiver, and the estimation error is independent of , thus, not observable. as additive noise on top of . Therefore, we need to treat The total noise, taking into account the imperfect observation, with zero mean and covariance is (28)

(21) We also get due to the symmetry in BPSK. The conditional error probability is minimized when . After averaging over , we get

Since is often not white, MMSE combining should be applied with as the observable channel state, and as the total noise to get the appropriate weighting coefficients . As before, we can define the combiner output SNR with optimal combining as (29)

(22) then This is just MRC, which is optimal when the channel state is known perfectly at the receiver, and the noise is white. If we define the performance parameter as the combiner output SNR with perfect channel knowledge (23) then (24)

(30) C. Evaluation of Optimal Combining Performance Once the conditional detection error probability is is evaluated by averaging determined, the error performance the conditional error over all possible channel observations, as in (30). The approach used traditionally [4], [5] is to use (29) to and then calculate derive the distribution of SNR (31)

B. Optimal Combining With Imperfect Channel Knowledge When the channel state is not perfectly known, but the channel statistics (i.e., , , ) are known, MMSE estimation can be applied to the channel observation to obtain the . Then, the channel state can channel estimate be decomposed as , where is the estimation error independent of . Applying MAP detection as before, the conditional error probability can be expressed as

However, as shown in [8], when the combiner output SNR is can be alternatively calculated by using in quadratic form, (32) as in [12], for optimal Using this analytical expression of combining with imperfect channel knowledge, of (30) can be written as shown in (33) at the bottom of the page, where we use the fact that is zero-mean proper complex Gaussian. Denote

(25) Again,

, and the

that minimizes

is

(34) denote the set of eigenvalues of and let integration with respect to , we get

, then, after

(26) After averaging over (27) It can be seen that when performing MAP detection conditioned on the channel observation , only the MMSE estimator

(35)

(33)

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The last step follows from the partial-fraction expansion is the number of distinct eigenvalues, method, where is the eigenvalue’s multiplicity, and is the th residue associated with the th power in the partial-fraction expansion. can be calculated using the residue theorem, and The term the bit-error probability (BEP) can be further simplified using the following closed-form integral [8]:

It is interesting to compare the the suboptimal-combining error performance with the optimal case. Following the previous development, it can be shown that the suboptimal BER is (43) where

(36)

(44) In this case, the combiner output SNR can be defined as

where

(45)

(37) In the case where all the eigenvalues are equal to , partial expansion is not necessary, and directly from (35), we have

For such combining, the error performance is difficult to evalis not itself a uate, since the effective combiner output SNR quadratic form, but rather a quotient of quadratic forms. However, using the Cauchy–Schwartz inequality, we can find a lower bound on error performance

(38) (46)

In the case where the eigenvalues are distinct

Therefore, the suboptimal BEP is lower bounded by the optimal performance. (39) (40) Other cases of eigenvalues with multiplicity can be calculated accordingly [8]. Similarly, it can be shown [12] that the error performance of (24) under perfect channel knowledge is in the same form as should be replaced by the (35), except the parameters eigenvalues of (41) IV. SUBOPTIMAL COMBINING AND CORRESPONDING ERROR PERFORMANCE Due to implementation simplicity or other constraints, sometimes suboptimal combining is used instead of the optimal weights. For example, one common case of suboptimal combining is to treat as the true channel state and use it for MRC combining. In this case, the combining weight is set to , is and the real part of the decision variable used for a threshold test (42)

V. EXAMPLES AND NUMERICAL RESULTS In the previous sections, we have considered the detection error probability with imperfect channel estimation and perfect channel statistics. However, the correlation between channel state and channel observation is determined by the pilot scheme, and is, in general, complicated. In this section, a simple example of pilot-symbol-aided channel estimation [9] is given to illustrate the concept of analyzing BER while taking into consideration the channel-estimation uncertainty. A. Pilot-Symbol-Aided Channel Estimation The pilot-symbol-aided channel estimation refers to the general scheme where predetermined special symbols are transmitted over the same channel as the data symbols, and the received signals are used for channel estimation. For example, in frequency-diversity systems, such as multicarrier code-division multiple access (MC-CDMA), it could be sending pilot symbols over different frequency subcarriers; or in a space-diversity system, it could simply be transmitting pilot symbols over multiple antennas. Using pilot-symbol-aided channel estimation, we assume that the pilot symbol has energy (47) The corresponding received signal is (48)

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where is the additive noise at the receiver. The channel observation can be expressed as

It is interesting to compare this result with combining using perfect channel knowledge, where

(49)

(64) (65)

It can be seen that, under the pilot-symbol scheme, the covariand can be diagonalized by the same ance matrices unitary matrix as

Applying a similar technique, we get the eigenvalues of (41) as

(50)

(66)

(51)

Comparing with , we notice that can be factored in two is the same as , indicating parts. The first term the SNR gain of the decorrelated subchannel. The second term shows the loss due to channel estimation. Notice that at low SNR, this loss could be significant, due to the poor quality of channel estimation. Furthermore, when is large, we get

We define (52) (53) then one can show that (54)

(67)

where

(55)

which shows that for high transmission SNR, the case with imperfect channel knowledge still has dB power loss, compared with the case with perfect channel knowledge.

where

(56)

C. Suboptimal Combining

(57) (58)

For suboptimal combining with pilot-symbol-aided channel estimation, it can be shown that

Furthermore, we define (59) then (60) B. Optimal Combining In the case with imperfect channel knowledge, the combiner output SNR can be written as

(61) where

(68)

Then, it can be shown that (62) For the two channel models of frequency and space diversity that we will consider, the eigenvalues of the channel-state covariance matrix are distinct. Therefore, we can calculate as in (39) and (40) with the eigenvalues of (34) as

(63)

Although the above expression of is complicated, it is interesting to compare it with the expression of in (61). It can be seen that when the eigenvalues of are identical, the suboptimal combining SNR is the same as . This implies that if the multipath channel can be diagonalized into parallel subchannels with i.i.d. (as in rich scattering environment), then there is no difference between optimal and suboptimal com) rises, bining. Furthermore, as the transmission SNR ( also approaches the optimal combining SNR . This should not be surprising, since at high SNR, the channel observation is very close to the channel state and the MMSE channel estimation, thus making the optimal and suboptimal combining weights

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similar to each other. Although the optimal and suboptimal combining performances are similar under high SNR or identical eigenvalue conditions, the performances are indeed different for practical situations. Compared with combining with perfect channel knowledge, there are also performance losses due to pilot-symbol power. D. Channel Models and Simulation Results Aside from detection and estimation methods, the system BER is fundamentally determined by the underlying channels, where correlation between channel state can be over time, frequency, or space, or their combinations. In this paper, we consider two examples of correlated Rayleigh fading channels, where the correlation is either between subcarriers at different frequencies or between antennas at different space locations. The first example we consider is to combine over frequency diversity, such as in MC-CDMA systems. The correlation for the channel state is modeled as [13] (69) is the channel delay spread and is set to 25 ns. is the frequency separation between two adjacent subcarriers. We allocate a total bandwidth of 20 MHz and divide the bandwidth MHz for into either 8 or 16 subcarriers. Thus, and MHz for . From Fig. 2(a), we can see that when perfect channel knowledge is available, using more subcarriers increases BER performance because of the increased frequency diversity. However, when the channel knowledge is not perfect, the system designer can choose to improve BER performance by either improving the channel observation quality (e.g., increasing the ratio of pilot power ) in this case) or increasing system to symbol power ( diversity (e.g., increasing the number of subcarriers from 8 to 16). When SNR is low or the subcarriers are highly correlated, the diversity gain is not as effective as improved channel observation. Therefore, even with more combining branches, the BER performance could be worse than system with fewer combining branches but better channel observation. In this situation, to improve system BER, it would be more cost effective and less complex to implement systems with better channel observation and fewer combining branches. As shown by Fig. 2(b), compared with the case with perfect channel knowledge, imperfect channel knowledge degrades system performance, even with optimal combining, and BER is further reduced when suboptimal combining is used. In this case, the loss due to suboptimal combining is significant, and can be attributed to inaccurate estimation of channel state. The second example we consider is to combine over space diversity. In this example, we allocate a fixed length of 0.05 m and ) or three ( ) anassign this space to either two ( tennas. The antennas are assumed to be placed in line and spaced equally. The normalized covariance matrix for the channel state is modeled as [14] (70)

Fig. 2. Example of frequency-diversity model ( = 25 ns). (a) Comparing the analytical error performance of optimal combining with perfect channel knowledge and imperfect channel knowledge, using different estimation power for N = 8 and N = 16. (b) Analytical error performance and Monte Carlo simulation results for the perfect channel knowledge, optimal combining with imperfect channel knowledge and the suboptimal combining performance with N = 16 and E = E .

where is the distance between two adjacent antennas, GHz is the carrier frequency, and is the light speed. is the zeroth-order Bessel function of the first kind. In the case , cm, and the two antennas are uncorrelated. of In the case of , cm, and the three antennas are correlated. From Fig. 3(a), we can see a similar tradeoff between combining diversity and channel observation quality. More antennas provide more space diversity, which improves system BER performance. However, such improvement can be achieved by better channel observation, which could be more cost-effective to implement. In Fig. 3(b), the analytical error performance of optimal combining with perfect and imperfect channel knowledge are compared with Monte Carlo simulation and are shown to be in good agreement. It is somewhat surprising that simulation also shows that the suboptimal combining in this example has only slightly worse performance than

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case, the channel can be approximated by two parallel subchannels with i.i.d. distributions of channel states. However, in general, the optimal combining has better performance and should be applied. VI. CONCLUSION In this paper, we investigate the effect of imperfect channel estimation on diversity combining. For Rayleigh fading channel, when the channel observation at the receiver is imperfect, MMSE channel estimation should be performed. The channel state is the sum of the channel estimator and estimation error, where the estimation error mixed with transmitted signal contributes an additional source of noise. Since this noise is usually not white, MMSE combining should be used for optimal diversity combining, with the adjusted channel state and noise model. The optimal combining error performance is calculated and compared with MRC with perfect channel knowledge and suboptimal combining with imperfect channel knowledge cases. Numerical results with specific models show good agreement with Monte Carlo simulations. REFERENCES

Fig. 3. Example of spatial diversity model (f = 2:4 GHz). (a) Comparing the analytical error performance of optimal combining with perfect channel knowledge and imperfect channel knowledge using different estimation power for N = 2 and N = 3. (b) Analytical error performance and the Monte Carlo simulation results for the perfect channel knowledge, optimal combining with imperfect channel knowledge, and the suboptimal combining performance with =E . N = 3 and E

optimal combining. However, by applying the corresponding parameters into (70), we get the eigenvalues , , . It can be seen that the eigenvalue is either “off” ( ) or “on.” For the eigenvalues ). Applying that are “on,” their values are quite close ( these eigenvalues in (61) and (68), we can see that the combiner output SNR and are very close. Hence, the suboptimal performance is very close to that of the optimal combining. In this

[1] W. C. Jakes, Microwave Mobile Communications. New York: Wiley, 1974. [2] J. G. Proakis, Digital Communications, 4 ed. New York: McGrawHill, 2000. , “Probabilities of error for adaptive reception of M -phase signals,” [3] IEEE Trans. Commun., vol. COM-16, no. 2, pp. 71–81, Feb. 1968. [4] M. Gans, “The effect of Gaussian error in maximal ratio combiners,” IEEE Trans. Commun., vol. COM-19, no. 8, pp. 492–500, Aug. 1971. [5] B. R. Tomiuk, N. C. Beaulieu, and A. A. Abu-Dayya, “General forms for maximal ratio diversity with weighting errors,” IEEE Trans. Commun., vol. 47, no. 4, pp. 488–492, Apr. 1999. [6] S. Roy and P. Portier, “Maximal-ratio combining architectures and performance with channel estimation based on a training sequence,” IEEE Trans. Wireless Commun., vol. 3, no. 7, pp. 1154–1164, Jul. 2004. [7] J. S. Thompson, “Antenna array performance with channel estimation errors,” in Proc. ITG Workshop Smart Antennas, Munich, Germany, Mar. 2004, pp. 75–78. [8] M. K. Simon and M.-S. Alouini, Digital Communication Over Fading Channel: A Unified Approach to Performance Analysis. New York: Wiley, 2000. [9] J. K. Cavers, “An analysis of pilot symbol assisted modulation for Rayleigh fading channels,” IEEE Trans. Veh. Technol., vol. 40, no. 6, pp. 686–693, Nov. 1991. [10] F. D. Neeser and J. L. Massey, “Proper complex random processes with applications to information theory,” IEEE Trans. Inf. Theory, vol. 39, no. 7, pp. 1293–1302, Jul. 1993. [11] S. M. Kay, Fundamentals of Statistical Signal Processing. Englewood Cliffs, NJ: Prentice-Hall PTR, 1993, vol. I, Estimation Theory. [12] V. V. Veeravalli, “On performance analysis of signaling on correlated fading channels,” IEEE Trans. Commun., vol. 49, no. 11, pp. 1879–1883, Nov. 2001. [13] M. D. Yacoub, Foundations of Mobile Radio Engineering. Boca Raton, FL: CRC, 1993. [14] G. Stuber, Principles of Mobile Communication. Norwell, MA: Kluwer, 1996.

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Roy You (S’98–M’03) received the B.S. and Ph.D. degrees in electrical engineering from the Department of Electrical Engineering and Computer Sciences, University of California, Berkeley, in 1997 and 2003, respectively. Since 2003, he has been an Assistant Professor of Electrical and Computer Engineering with the New Jersey Institute of Technology, Newark. In his previous research, he applied communication techniques to analyze an optical wireless channel’s information capacity, developed new techniques to reduce power requirements of optical wireless systems, and investigated the application of trellis-coded modulation for fiber-optic systems with optical amplifiers. His current research includes analysis and design of multicarrier CDMA systems, study of diversity combining performance under channel uncertainty, and application of hybrid-ARQ to fading channels. Dr. You received the UC Regents Fellowship in 1997 and NASA Faculty Fellowship in 2004. He serves as faculty advisor for the IEEE student branch at NJIT.

Yeheskel Bar-Ness (M’69–SM’78–F’89) received the B.Sc. and M.Sc. degrees in electrical engineering from the Technion, Haifa, Israel, and the Ph.D. degree in applied mathematics from Brown University, Providence, RI. Currently, he is a Distinguished Professor of Electrical and Computer Engineering, and the Foundation Chair of the Center for Communication and Signal Processing Research at the New Jersey Institute of Technology (NJIT), Newark. Previously, he was with the Rafael Armament Development Authority, Israel, working in the field of communications and control; and with the Nuclear Medicine Department, Elscint Ltd., Haifa, Israel, as a Chief Engineer in the field of control, and image and data processing. In 1973, he joined the School of Engineering, Tel-Aviv University, Tel Aviv, Israel, where he was an Associate Professor of Control and Communications. Between September 1978 and September 1979, he was a Visiting Professor with the Department of Applied Mathematics, Brown University. He was on leave with the University of Pennsylvania and Drexel University, both in Philadelphia, PA. He came to NJIT from AT&T Bell Laboratories in 1985. Between September 1993 and August 1994, he was on sabbatical with the Telecommunications and Traffic Control Systems Group, Faculty of Electrical Engineering, Delft University of Technology, Delft, The Netherlands. Between September 2000 and August 2001, he was on sabbatical at Stanford University, Stanford, CA. His current research interests include adaptive multiuser detection, array processing and interference cancellation, and wireless mobile and personal communications. Dr. Bar-Ness was an Area Editor for the IEEE TRANSACTIONS ON COMMUNICATIONS (Transmission Systems), and Editor for Adaptive Processing Systems. He is the Founder and Editor-in Chief for IEEE COMMUNICATIONS LETTERS. He is also Editor for the journal Wireless Personal Communications. He was Chairman of the Communication Systems Committee, and currently is the Vice Chair of the Communications Theory Committee of the IEEE Communication Society. He served as the General Chair of the 1994 and 1999 Communication Theory Mini-Conference. He was also the Technical Chair for the IEEE Sixth International Symposium on Spread Spectrum Techniques and Applications. He is a recipient of the Kaplan Prize (1973), which is awarded annually by the government of Israel to the ten best technical contributors.

Hong Li (S’04) received the B.S. degree in electrical engineering from Shanghai Jiao Tong University, Shanghai, China, in 1997, and the M.S. degree in communication and electronic systems from the Second Academy of China Aerospace Science and Industry Corp., Beijing, China, in 2000. He is currently working toward the Ph.D. degree at the New Jersey Institute of Technology (NJIT), Newark. From 2000 to 2001, he was a R&D Engineer with Alcatel Shanghai Bell, Shanghai, China. Since 2001, he has been a Research Assistant with the Center for Communication and Signal Processing Research at the Department of Electrical and Computer Engineering, NJIT. His research interests include channel estimation, signal detection, and modulation classification.

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Rate-Adaptive Transmission Over Correlated Fading Channels Tingfang Ji and Wayne Stark, Fellow, IEEE

Abstract—In this paper, we investigate link adaptation and incremental redundancy (IR) retransmission schemes over correlated wireless channels. While computer simulations have been used to study the performance of these techniques, a numerically tractable analytical approach is more desirable to analyze generic protocols, and to reveal insights into the performance tradeoffs. An error-recursion approach is developed in this paper to mathematically analyze the throughput, delay, and energy efficiency of rate-adaptation techniques over fading channels with arbitrary correlations between retransmissions. Using Reed–Solomon codes as an example, we quantatitively predict the performance tradeoff of throughput and latency for IR schemes and the performance dependency on the channel correlation. Numerical results also show that reactive rate-adaptation schemes with IR retransmission outperform proactive rate-adaptive schemes, even with perfect channel side information, in terms of throughput and energy efficiency. Index Terms—Automatic repeat request (ARQ), channel knowledge, fading channels, forward error correction (FEC), incremental redundancy (IR) retransmission, link adaptation, rate adaptation, Reed–Solomon (RS) code, Suzuki process, wireless communications systems.

I. INTRODUCTION ELIABLE communications over unreliable channels require error-control mechanisms such as forward error correction (FEC) codes and/or automatic repeat request (ARQ). FEC codes introduce redundancy into a transmission so that the receiver can exploit the redundancy in the received signal to correct the channel-corrupted message. Usually, more redundancy allows more reliable communications at the expense of bandwidth efficiency. Redundancy can also be used to detect errors in a transmission and to trigger ARQ. In a traditional ARQ protocol, when error(s) are detected in the received message, the receiver will request a retransmission until the message is successfully decoded or a stopping criterion is met. FEC codes have often been used for applications where residual errors are tolerable, such as voice, and ARQ protocols have been used for applications where error-free communication is required, such as

R

Paper approved by M. Hamdi, the Editor for Network Architecture of the IEEE Communications Society. Manuscript received December 19, 2003; revised March 3, 2005. This paper was presented in part at the 37th Allerton Conference on Communications, Control, and Computing, Pacific Grove, CA, September 1999. T. Ji is with Qualcomm Inc., San Diego, CA 92121 USA (e-mail: [email protected]). W. Stark is with the Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, MI 48109 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TCOMM.2005.857147

file transfer protocol (FTP). In a hybrid FEC/ARQ scheme, retransmissions are used to correct residual errors after error correction with FEC. Rate-adaptation schemes have been designed for channels with large signal-to-noise ratio (SNR) dynamic ranges, such that low-rate robust FEC codes are used under poor channel conditions, and high-rate efficient FEC codes are used under good channel conditions. In a rate-adaptive type-I hybrid ARQ system, the transmitter proactively varies the code rate, according to the predicted channel quality for each transmission. In an incremental redundancy (IR) retransmission type-II ARQ scheme, the transmitter starts with a high-rate FEC code for the first transmission, then responds to any retransmission request(s) from the receiver by transmitting additional redundancy blocks. When the receiver combines the redundancy blocks with previously received code blocks, the effective code rate is lowered to match the channel condition. Both proactive rate-adaptation techniques, such as modulation and coding scheme (MCS) selection, and reactive rate-adaptation techniques, such as synchronous IR, have been adopted in 3G wireless standards like CDMA2000-1xEVDO and UMTS. Rate-adaptive ARQ protocols have been studied in many aspects, such as asymptotic capacity [1], simulation-based performance evaluation [2] and rate-compatible code design [3]. In [1], IR protocols are shown to achieve similar performance as a random binary code for Gaussian block collision channels. In [2], a link-adaptive scheme and an IR scheme are compared via computer simulations in a practical cellular network. In this paper, the exact throughput, delay, and energy efficiency of rate-adaptive hybrid ARQ schemes are derived using error-recursion techniques for communication channels with arbitrary correlation between transmissions. This approach is then applied to Reed–Solomon (RS) codes to demonstrate the effect of channel correlation, code selection, channel side information, decoding algorithm, and proactive versus reactive mechanism on system performance. The analytical methodology proposed in this paper is suitable for most ARQ schemes based on block codes with bounded distance decoding. II. RELATED WORK The performance of RS-code-based type-II hybrid ARQ schemes has been analyzed over a variety of memoryless channels. Generating function method can be used to capture the state transition of an ARQ protocol [4] for simple protocols with a small number of redundancy blocks. This approach, however, is not scalable for either complicated ARQ protocols or complicated channel models.

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codeword is divided into a length- information block and -length redundancy blocks for successive transmissions. Coded RS symbols are then interleaved and modulated onto BFSK signals. The modulator maps each RS symbol onto BFSK signals, according to the natural binary mapping. BFSK is assumed in this paper because of its known robustness over fading channels [7].

Fig. 1.

Block diagram of an RS-coded ARQ system.

In [5] and [6], the performance of the type-II hybrid ARQ scheme, traditional type-I hybrid ARQ scheme, and rate-adaptive type-I hybrid ARQ schemes were analyzed for meteor-burst communications. The meteor-burst channel has a deterministic exponentially decaying SNR over time. The analysis has a similar flavor to the work presented here in the sense that it also employs error-recursion techniques. Our approach, however, differs significantly from [5] and [6] in two aspects: individual code blocks are not necessarily decodable, and the underlying channel is a finite-state channel.

B. Channel Model In this paper, the fading process is modeled by the Suzuki process, where the fading process is represented by the product of two independent fast and slow fading processes [8]. For simplicity, we assume the channel experienced by each modulation symbol is frequency-nonselective, e.g., each tone in a properly designed orthogonal frequency-division multiple access (OFDMA) system. The error-recursion method can also be applied to frequency-selective fading with a different , the symbol-error model. Given the transmitted signal is given by received signal (1)

A. Coding and ARQ Schemes

where is the path loss due to the distance between the transmitter and the receiver, is the slow shadowing process, is the fast fading process, and is an additive white Gaussian noise process with double-sided power spectrum den. sity (PSD) The path loss is assumed to be constant over a packet duration, which may include multiple frame transmissions. The shadowing process is modeled as a zero-mean log-normal random process with exponential decaying correlation as a function of distance [9]. Since the shadowing process is very slow, compared with the data rate, the shadowing is assumed to be constant over a frame duration and correlated between frames. Suppose a mobile’s velocity is m/s, the normalized correlation of the shadowing levels measured at seconds apart is given by

In the system studied in this paper, the information sequence is encoded with a CRC code. The CRC-encoded sequence is then grouped into symbols of size , and each length- symbol sequence is encoded into an extended RS codeword of length symbols. This codeword can be punctured to form a family of rate-compatible punctured RS codes with rates from to 1. As most well-designed CRC codes can usually guarantee a very small undetected error probability with only a small number of parity-check bits (16 b in ANSI and CCITT standards), we assume that the undetected error probability and overhead of the CRC code is negligible. Since the point of the paper is regarding reactive and proactive rate-adaptation schemes, we assume an error-free feedback channel, selective repeat retransmission, and infinite buffer size. In a rate-adaptive type-I hybrid ARQ protocol, a punctured code with an appropriate rate is selected from a family of codes for each transmission, based on channel quality information. In an IR type-II hybrid ARQ protocol, the RS

where is the standard deviation of the shadowing level, and is the correlation of measurements between two points separated by distance . Under the assumption that the acknowledgment delay is constant and much longer than the packet transmission time, as it is in most practical systems, the distance that a mobile travels between the transmission of a packet and its immediate retransmission is roughly constant. Hence, the correlation of the shadowing process between retransmissions can be modeled as constant, regardless of the redundancy block size. into levels If we quantize the shadowing process , we can approximate the shadowing channel with an -state first-order Markov model. Since follows a Gaussian distribution in the log domain, a Lloyd–Max minimum mean-square error quantizer for Gaussian source can be , and the employed to determine the reconstruction levels [10]. The steady-state probability of the decision regions channel to be in state , , can be obtained by integrating

III. SYSTEM AND MODELS In this section, we describe the system of interests and the underlying channel models assumptions. An RS-coded system is used as an example to illustrate the error-recursion method for analyzing hybrid ARQ schemes with bounded distance decoders. As shown in Fig. 1, the transmitter is composed of a cyclic redundancy check (CRC) encoder, a rate-compatible RS encoder, an interleaver, and a binary frequency-shift keying (BFSK) modulator. Note that a symbol interleaver that matches the size of the RS code symbol is used instead of a bit interleaver. The receiver consists of a (noncoherent) BFSK demodulator, a deinterleaver, an RS decoder, and a CRC decoder.

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the Gaussian density over . The state-transition probability can then be easily obtained using (2) where , follows a joint Gaussian distribution with covariance , as given by (3) is modeled as a Rayleigh fading The fast fading process process, constant over each symbol and independent from symbol to symbol. The independence of fading between symbols could be achieved through interleaving and time/frequency diversity of the channel. For the channel model described in this section, the average SNR can be written as (4) Note that the finite-state channel correlation could also be used to model fast fading, such as the Doppler effect. Shadowing is used as an example to illustrate the methodology and to generate numerical results. C. Demodulation and Symbol-Error Rate The BFSK noncoherent demodulation bit-error probability and the RS symbol-error probability (SEP) for a Rayleigh fading channel can be easily obtained for hard-decision demodulation. RS codes, however, provide additional error-correcting capability when used in conjunction with soft-decision erasure demodulation. An RS code of minimum distance can correct symbol errors and erasures, as long as . In this paper, a parity-check method is used to erase unreliable symbols. For each RS code symbol ( bits), an even parity-check bit is appended to it, then the symbol is modulated with BFSK signals. If parity check of a demodulated symbol fails, it is erased [11]. The following performance analysis can also be easily extended for other erasure-generating techniques, such as Viterbi’s ratio threshold test (RTT) and the Baysian detection method. IV. PERFORMANCE ANALYSIS In this section, we will analyze the throughput, delay, and energy-efficiency performance of three ARQ protocols based on RS codes. We first derive the performance for hard-decision decoders, then extend the results to soft-decision error-and-erasure decoders. A. Rate-Adaptive Type-I Hybrid ARQ Consider a rate-adaptive type-I hybrid ARQ scheme where a genie tells the transmitter the channel condition for each transmission. The transmitter then chooses an appropriate code to maximize the instantaneous throughput, based on the channel condition. The performance of this scheme provides an upper bound for practical systems with imperfect channel estimation. We will first describe the code-selection algorithm, then derive the performance over correlated shadowing channels.

Fig. 2. Code length of optimal (N ; 16) punctured RS codes GF(2 ) over Rayleigh fading channels.

1) Optimal Code Selection: The code-selection algorithm is designed to maximize the throughput for each transmission. The code candidates are chosen from a family of codes with rate compatibility, in the sense that every high-rate code is a prefix of lower rate codes in the family. Suppose the information packet , then the extended RS code family has symbols over GF , where is an punctured is given by , RS code, and . For the convenience of analysis, is assumed to be an even number. code over a Consider the transmission of an channel in state . The throughput is defined as the expected number of correctly received information bits per channel use, as given by (5) where

is given by

where and denote the SEP over a channel in state . Note that is the probability of correctly decoding a codeword of code over a channel in state . The opfor channel state is given by timal code (6) For a given channel state , the code-selection algorithm is described by (6). To illustrate the code-selection algorithm, the code lengths of RS codes over GF are shown in Fig. 2 the optimal for a range of channel SNRs. For channel SNR above 27 dB, the rate-1 code is selected, since no error protection is needed, as shown in Fig. 2. As the channel quality degrades, more and more parity checks are added to the codeword to protect the transmission from channel corruption. When the channel SNR drops to 6.5 dB, the lowest rate code becomes the optimal code. When

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the channel SNR goes below 5 dB, the additional error proteccode, over tion provided by the lowest rate code, i.e., the code, can no longer the next lowest rate code, i.e., the offset the smaller code rate of the code. Thus, the optimal code rate starts to increase. When the channel SNR goes dB, the rate-1 code again becomes the optimal code below selection, because codes of all rates result in equally poor error performance. 2) Error Recursion: Throughput, delay, and energy efficiency of ARQ protocols can be easily derived from the decoding error probabilities after each (re)transmission. Let denote the event of decoding failures on each of the first transmissions. Let denote the joint probability of denote the event , and the channel be in state , and let indicator function. The average number of retransmissions is given by

The proof of (10) can be obtained as a special case of the proof in the next section. Interested readers are encouraged to refer to [12] for a detailed derivation. Note that the initial value is equal to . For fixed-rate type-I hybrid ARQ, the error-recursion algois the same as given in (10), with the optimal rithm for for state replaced by a fixed . code length B. IR Type-II Hybrid ARQ As described in Section III-A, for type-II hybrid ARQ, the lowest rate RS codeword is divided into blocks, where the first block is of size , and the rest of size . After all blocks are transmitted, the ARQ protocol restarts denote the probability that each from the first block. Let transmissions fail, and the th transmission of the first leads to a successful decoding. The expected number of retransmissions for IR is given by

(7) (12) The throughput for selective-repeat ARQ schemes, , is defined as the average number of information bits successfully received per channel symbol transmitted, as given by

and the throughput

is given by (13)

(8)

is the number of code symbols transmitted during the where th transmission. For the first transmission, , is the optimal code length in channel state . For where , can be written as for where the outer summation is over the channel states of the th transmission, and the inner summation is over the channel states of the th transmission. The energy efficiency is defined as the reciprocal of the the total energy dissipated for each corJoule/bit. To generalize the results for any rectly received bit, , the normalized energy efficiency is background noise PSD defined as (9) The probability , which is defined earlier in this section, is recursively computed for all transmissions associated with the same information sequence, as given by

where the state-transition probability and is given by

where is the number of code symbols in the th transmitted block, as given by mod mod Let denote the joint probability of decoding failures, the th transmission occurs in channel state , and the th redundancy block has symbol errors. can be written as (14) The recursive algorithm for calculating can be described by the following theorem. Theorem: For IR retransmission schemes with incremental block size , the probability is given by (15), shown at the bottom of the next page, where is the error-correcting capability at the th transmission, and (16)

(10)

(17)

is given by (2),

(18)

(11)

Proof: When the th (re)transmission packet is not the information packet, i.e., mod , the probability of success of the th decoding trial is a function of the accumulated symbol

JI AND STARK: RATE-ADAPTIVE TRANSMISSION OVER CORRELATED FADING CHANNELS

errors in previous transmissions and the number of symbol errors in current redundancy block, . See (19) at the bottom of is independent of previous transmisthe page. Observe that sion errors, given current channel state . Previous transmisand , given the last sion errors are also independent of . Hence, (19) can be rewritten as channel state

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TABLE I PARAMETERS USED IN NUMERICAL EVALUATION: CHANNEL CORRELATION , SHADOWING STANDARD DEVIATION  , SPATIAL CORRELATION OF DISTANCE  , MAXIMUM NUMBER OF RECURSION N , AND NUMBER OF STATES IN FINITE-STATE CHANNEL L

(20) deNote that (20) is the first half of the theorem, where . notes When mod , say, , the IR protocol restarts. Based on the same conditional independent argument as above, can be written as

(21) is Note that the joint probability independent of and is always equal to . Although can be calculated from by , it is computationally effisumming over cient. The purpose of computing is to derive , as given by (14), where is only required to be comto , which is usually much smaller than puted for from , the RS codeword length. The following algorithm for comis based on in the same puting range:

(22) where the term in summation can be rewritten as

(23) Combining (21)–(23), we obtain the second half of the theorem.

C. Soft-Decision Decoding The error-counting method for the soft-decision decoding algorithm is very similar to the methods described above for harddecision decoding. Instead of counting the number of errors in a codeword, the new algorithm keeps track of the sum of the number of erasures and twice the number of errors, as denoted by . In addition, the error-correcting capability is redefined . After the first transmissions, the RS code can be as . correctly decoded as long as V. NUMERICAL RESULTS The performance of a genie-aided rate-adaptive type-I hybrid ARQ scheme and IR type-II hybrid ARQ schemes are numerically evaluated over fading channels with log-normal shadowing. In Table I, we list the parameters of the shadowing process ( , , ), the maximum number of recursions of the , and the number of states in the Markov fading algorithms corresponds to a model . The channel correlation of space separation of .08 m in an urban environment, which also corresponds to a mobile speed of 30 km/h with retransmission delay of 10 ms. A. Rate-Adaptive Type-I Hybrid ARQ and Fixed-Rate Type-I Hybrid ARQ In this section, a genie-aided optimal rate-adaptive type-I hybrid ARQ scheme and fixed-rate type-I hybrid ARQ schemes are compared in terms of throughput. The results presented in RS code family over GF Fig. 3 are based on the and channel models with correlation 0.99. In Fig. 3, the throughput performance is plotted versus the average channel SNR. For fixed-rate protocols, higher rate codes are shown to have better throughput at high average channel SNRs, while lower rate codes are shown to have

for mod (15) for mod

(19)

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Fig. 3. Throughput of a genie-aided rate-adaptive type-I hybrid ARQ scheme and fixed-rate type-I hybrid ARQ schemes. Channel correlation  = 0:99, RS code family (N ; 16) over GF(2 ).

Fig. 5. Delay comparison of IR retransmission type-II hybrid ARQ schemes and rate-adaptive type-I hybrid ARQ schemes. Channel correlation  = 0:99, RS code family (N ; 16) over GF(2 ).

Fig. 4. Throughput comparison of IR retransmission type-II hybrid ARQ schemes and rate-adaptive type-I hybrid ARQ schemes. Channel correlation  = 0:99, RS code family (N ; 16) over GF(2 ).

Fig. 6. Energy-efficiency comparison of IR retransmission type-II hybrid ARQ schemes and rate-adaptive type-I hybrid ARQ schemes. Channel correlation  = 0:99, RS code family (N ; 16) over GF(2 ).

better throughput at low average channel SNRs. The rate-adaptive protocol is shown to achieve higher throughput than all fixed-rate codes for all SNRs. One interesting observation is that the throughput of the genie-aided rate-adaptive protocol is not a tight upper bound of the throughput of fixed-rate protocols. This is because an optimal code rate is selected for each transmission in the rate-adaptive protocol based on the SNR of a particular shadowing realization, instead of the average SNR. Hence, for any given average SNR, the rate-adaptive protocol outperforms all fixed-rate protocols.

RS code family over GF and channel models with retransmission channel correlation 0.99. In Figs. 4 and 6, the throughput and energy efficiency of IR type-II ARQ schemes are shown to be significantly higher than the rate-adaptive type-I ARQ scheme with perfect rate prediction over a highly correlated channel. There are two fundamental advantages of IR over rate-adaptive type-I hybrid ARQ that lead to the superior performance of IR. First of all, IR schemes accumulate channel capacities through retransmissions, and type-I ARQ schemes discard erroneous transmissions which contain some valuable information. Second, the perfect rate prediction of type-I ARQ chooses the optimal code rate based on average error performance of a given channel SNR; however, a code of rate higher than the optimal rate may still decode for a particular realization of fading of the given SNR. The second advantage will be minimal for codes with a steep

B. IR and Rate-Adaptive Type-I Hybrid ARQ Figs. 4–6 illustrate the throughput, delay, and energy-efficiency performance of a genie-aided rate-adaptive type-I ARQ scheme and IR type-II hybrid ARQ schemes with redundancy blocks of size 2, 4, and 8 symbols/block. All results assume a

JI AND STARK: RATE-ADAPTIVE TRANSMISSION OVER CORRELATED FADING CHANNELS

Fig. 7. Throughput of rate-adaptive type-I hybrid ARQ scheme over channels with different correlations. RS code family (N ; 16) over GF(2 ).

SNR waterfall region in packet-error performance, e.g., long turbo codes. In Fig. 5, the average delay performance of IR schemes is shown to be inferior to that of the rate-adaptive scheme at low SNR. This is mainly due to the larger number of retransmissions required to reach the desired code rate. Among the type-II schemes, the schemes with smaller redundancy block sizes are shown to achieve better throughput and energy efficiency than the schemes with larger block sizes, due to a more refined rate increase profile. On the other hand, small redundancy block sizes also results in long delay. Note that the rate-adaptive type-I ARQ scheme considered here is a genie-aided scheme with perfect channel information and optimal code selection. In practice, rate-adaptive systems usually require additional hardware for channel quality measurements, additional feedback signaling overhead, and rate prediction algorithm to select the appropriate MCS. In reality, imperfect channel quality measurement, limited feedback information, and rate-prediction inaccuracy for a high-speed channel may significantly degrade the system performance. More significant throughput difference between IR and link adaptation was reported in [2] through computer simulations of convolutional coded systems in enhanced data rate for global evolution (EDGE) cellular networks. C. Effect of Channel Correlation and Decoding Algorithms The throughput curves of ARQ schemes over channels with three correlation values, 0.01, 0.50, and 0.99, are plotted in Fig. 7. The throughput is shown to degrade with the increase of channel correlation in the low SNR region. This complies to our intuition of better time diversity for channels with smaller correlation. One interesting observation is the slight performance gain of a channel with high correlation at very high SNR. This phenomenon can be explained with the convexity of throughput as a function of the effective SNR, say , collected over all transmissions of an ARQ protocol realization. has a smaller spread for channels with higher interpacket correlation, and a larger spread for channels with lower interpacket correlation.

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Fig. 8. Throughput of IR retransmission type-II hybrid ARQ scheme with errors-only and error-and-erasure decoding algorithms. Channel correlation  = 0:99, RS code family (N ; 64) over GF(2 ).

In the region where the throughput is a convex cap function of the SNR, a smaller spread in effective SNR will result in a larger average throughput. Hence, at high SNR, the channel has a higher throughput than the channel with with . The same argument can also be used to explain the loss of throughput due to high correlation at low SNR, where the throughput is a convex cup function of . The impact of the decoding algorithm on the throughput performance is demonstrated in Fig. 8. The error-and-erasure dedB gain when SNR coding algorithm is shown to have dB, and to lose 11% in throughput at higher SNR compared with the errors-only decoding algorithm. At low SNR, the error-and-erasure decoding algorithm benefits from the additional reliability information about each symbol provided by the erasure mechanism. At high SNR, however, the gain cannot , due to the redundancy offset the throughput loss of of the additional parity-check bits for every -bit RS symbol. VI. CONCLUSIONS The main contribution of this paper is the analytical approach for performance evaluation of rate-adaptation algorithms over correlated fading channels. As both IR and link adaptation schemes are adopted in many commercial systems, it is of great interest to develop an analytical method for performance evaluation beyond computer simulations, which could be time consuming and do not always yield insight into a problem. The proposed error recursion differs from previous analysis of type-II ARQ protocols in that it is not limited to a small number of independently decodable redundancy blocks, and it is applicable to arbitrary channel correlations. Using a punctured RS code family as an example, the proposed error-recursion method shows that IR type-II hybrid ARQ schemes achieve notably better throughput and energy efficiency than a genie-aided rate-adaptive type-I hybrid ARQ scheme at the expense of larger delay. This is confirmed by previous simulation studies of similar protocols with different

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codes and channels. High channel correlation is shown to degrade the system performance at moderate SNR for ARQ systems. The recursive methodology proposed in this paper can also be used to analyze other rate-adaptive block codes over general finite-state machine channels. The analytical approach developed in this paper is limited to block codes and bounded distance decoding. ARQ systems with quasi-capacity-achieving codes, such as turbo codes, are hard to analyze, due to the lack of mathematically tractable error-performance expression. However, a capacity-based approach could be used to approximate the error performance. For instance, the packet-error probability after each retransmission could be computed as the probability of the accumulated channel capacity (with some reasonable backoff to account for coding and channel estimation loss) being less than the information contained in the transmitted message. Comparison of type-I and type-II ARQ protocols performances with the capacity-based approach could be of great interest to readers in this field.

[9] M. Gudmundson, “Correlation model for shadow fading in mobile radio systems,” Electron. Lett., vol. 27, no. 23, pp. 2145–2146, Nov. 1991. [10] K. Sayood, Introduction to Data Compression. San Francisco, CA: Morgan Kaufmann, 1996. [11] M. B. Pursley and C. S. Wilkins, “Adaptive-rate coding for frequency-hop communications over Rayleigh fading channels,” IEEE J. Sel. Areas Commun., vol. 17, no. 7, pp. 1224–1232, Jul. 1999. [12] T. Ji, “Rate adaptive coding for wireless data networks,” Univ. Michigan, Commun., Signal Process. Lab., Ann Arbor, MI, Tech. Rep. 329, 2001.

Tingfang Ji received the Ph.D. degree from the University of Michigan, Ann Arbor, in 2001, the M.S. degree from the University of Toledo, Toledo, OH, in 1997, and the B.E. degree from Tsinghua University, Beijing, China, in 1995, all in electrical engineering. Since 2003, he has been with Qualcomm Inc. Corporate R&D, San Diego, CA. His responsibilities include designing and evaluating PHY and MAC algorithms for next-generation wireless systems. Prior to joining Qualcomm, he was with Bell Labs, where he was responsible for supporting 3G standards and drafting research initiatives related to UWB and sensor networks as a Member of Technical Staff.

REFERENCES [1] G. Caire and D. Tuninetti, “The throughput of hybird-ARQ protocols for the Gaussian collision channel,” IEEE Trans. Inf. Theory, vol. 47, no. 7, pp. 1971–1988, Jul. 2001. [2] X. Qiu, J. Chuang, K. Chawla, and J. Whitehead, “Performance comparison of link adaptation and incremental redundancy in wireless data networks,” in Proc. IEEE Wireless Commun. Netw. Conf., vol. 2, 1999, pp. 771–775. [3] J. Hagenauer, “Rate-compatible punctured convolutional codes (RCPC codes) and their applications,” IEEE Trans. Commun., vol. 36, no. 4, pp. 389–400, Apr. 1988. [4] S. B. Wicker and M. J. Bartz, “Type-II hybrid ARQ protocols using punctured MDS codes,” IEEE Trans. Commun., vol. 42, no. 2–4, pp. 1431–1440, Feb.–Apr. 1994. [5] M. B. Pursley and S. D. Sandberg, “Incremental-redundancy transmission for meteor-burst communications,” IEEE Trans. Commun., vol. 39, no. 5, pp. 689–702, May 1991. , “Variable-rate hybrid ARQ for meteor-burst communications,” [6] IEEE Trans. Commun., vol. 40, no. 1, pp. 60–73, Jan. 1992. [7] J. G. Proakis, Digital Communications, 3rd ed. New York: McGrawHill, 1995. [8] H. Suzuki, “A statistical model for urban radio propagation,” IEEE Trans. Commun., vol. COM-25, no. 7, pp. 673–680, Jul. 1977.

Wayne E. Stark (S’77–M’78–SM’94–F’98) received the B.S. (with highest honors), M.S., and Ph.D. degrees in electrical engineering from the University of Illinois, Urbana-Champaign, in 1978, 1979, and 1982, respectively. Since September 1982, he has been a Faculty Member in the Department of Electrical Engineering and Computer Science, University of Michigan, Ann Arbor, where he is currently a Professor. His research interests include the areas of coding and communication theory, especially for spread-spectrum and wireless communications networks. Dr. Stark is a member of Eta Kappa Nu, Phi Kappa Phi, and Tau Beta Phi. He was involved in the planning and organization of the 1986 International Symposium on Information Theory, held in Ann Arbor, MI. From 1984 to 1989, he was Editor for Communication Theory of the IEEE TRANSACTIONS ON COMMUNICATIONS in the area of spread-spectrum communications. He was selected by the National Science Foundation as a 1985 Presidential Young Investigator. He is Principal Investigator of an Army Research Office Multidisciplinary University Research Initiative (MURI) Project on low-energy mobile communications.

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The Expectation-Maximization Viterbi Algorithm for Blind Adaptive Channel Equalization Hoang Nguyen, Member, IEEE, and Bernard C. Levy, Fellow, IEEE

Abstract—A blind maximum-likelihood equalization algorithm is described and its convergence behavior is analyzed. Since the algorithm employs the Viterbi algorithm (VA) to execute the expectation step of the expectation-maximization (EM) iteration, we call it the expectation-maximization Viterbi algorithm (EMVA). An EMVA-based blind channel-acquisition technique which achieves a high global convergence probability is developed. The performance of the method is evaluated via numerical simulations under static and fading channel conditions. Index Terms—Blind equalization, expectation-maximization (EM) algorithm, expectation-maximization Viterbi algorithm (EMVA), Rayleigh fading channel, Ricean fading channel, Viterbi algorithm (VA).

I. INTRODUCTION digital communication system comprised of a modulator, transmit filter, physical channel, receive filter, and demodulator can be modeled as a discrete time finite-state machine (FSM). Let , where the ’s belong to a finite alphabet of size , denote the data sequence transmitted by such a system. For the highest level of generality, suppose is the response of the FSM to the input , where the vector represents all the unknown system parameters, such as the channel impulse response and the channel noise variance. In the presence of an additive circular complex white Gaussian noise (WGN) , the observed signal takes the form

A

(1.1) For constant channels, the transformation is assumed to be known, or the equivalent, to admit an exact model. For fading channels, it is assumed to have a statistical description, such as a Ricean or Rayleigh fading model. For reasons including the need for higher data throughput and easier management of multipoint networks, considerable attention has focused over the last several decades on the problem are estiof blind equalization [2]–[4], wherein both and mated based just on alone. Even when training data is periodically available, such as in GSM wireless systems, the use of a semiblind equalization approach to track the channel changes Paper approved by G. M. Vitetta, the Editor for Equalization and Fading Channels of the IEEE Communications Society. Manuscript received August 3, 2003; revised February 18, 2005 and April 12, 2005. This work was supported in part by the National Science Foundation (NSF) under a Graduate Fellowship, and in part by the NSF under Grant ECS-0121469. The authors are with the Department of Electrical and Computer Engineering, University of California, Davis, CA 95616 USA (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TCOMM.2005.857162

between training periods can significantly enhance the system performance for rapidly varying channels [5], [6]. The expectation-maximization Viterbi algorithm (EMVA) that we consider falls within the same class as the expectation-maximization (EM)-based blind technique proposed by Kaleh and Vallet [7]. The method of [7] relies on the forward–backward algorithm [8] for symbol-by-symbol maximum a posteriori (MAP) probability estimation of the data , whereas the EMVA uses the Viterbi algorithm (VA) to execute the E-step of the EM algorithm [1], [9]. In both cases, the channel is described by a hidden Markov model (HMM), and the maximization (M-step) of the likelihood function for the complete data yields an estimate of . Due to the generality of the HMM formulation, this approach is applicable to linear, as well as nonlinear, modulation schemes such as continuous phase modulation (CPM) [10]. However, the MAP algorithm has several disadvantages. First, it is relatively costly, since for a Markov model with states and possible transitions operations to process from each state, it requires a data block of length , whereas the VA requires operations to accomplish the same task, i.e., it is times faster than the MAP-based approach. Also, as noted in [7, Sec. IV], it allows forbidden state transitions in an estimated sequence, an event which tends to occur at low signal-to-noise ratios (SNRs) or with inaccurate parameter estimates. The latter case is important, since during the startup, only coarse parameter estimates are available, which means that the algorithm could get locked in permanently on a bad maximizer. By comparison, the survivors of the VA never contain forbidden state transitions, even with inaccurate channel information or under high noise conditions. The application of the EMVA to the blind and semiblind equalization of CPM signals is discussed in [11] and [12]. A general description of the EMVA and its relation to other methods are given in [11]. In addition, [11] contains a general description of the EMVA, and of a channel order estimation method which can be useful in applications where the channel delay spread is unknown. The focus of this paper is on the channel acquisition and adaptive properties of the algorithm for frequency-selective fading channels, so without loss of generality, we restrict our attention to linearly modulated signals and refer the reader to [11] for nonlinear modulation examples. The implementation of the EMVA for constant frequency-selective channels is described in Section II. A convergence analysis is given in Section III, where the existence of a capture set is established, and a blind channel-acquisition (BCA) method is devised which ensures that global convergence occurs with a high probability. In Section IV, we introduce blockwise and

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online adaptive forms of the EMVA, suitable for tracking of Ricean and Rayleigh channels with moderate fading rates. Due to space limitations, a version of the EMVA applicable to fast-fading Ricean channels is presented separately in [13] and [14]. Simulation results are presented in Section V, and concluding remarks are given in Section VI.

is a WGN process, the posterior probabilities of the Since VA survivors are (2.9) where (2.10)

II. EMVA FOR CONSTANT CHANNELS A. Signal Model Consider the linear signal model where the observed sequence is given by (2.1) where (2.2) represents the channel response, with denoting the transmitted sequence of independent and identically distributed denoting the channel tap values, and (i.i.d.) symbols, being a circularly complex zero-mean white Gaussian process of variance . Also, denotes the channel order which represents the channel memory in symbol periods. For a signal extending over , the observed sequence can be expressed as (2.3) where

(2.11) where is independent of

. Maximizing (2.11) over

yields

(2.12) (2.13)

(2.4) (2.5) (2.6) and

is the path metric (the squared Euclidean distance) for the th survivor path, is an arbitrary constant, and is a normalizing factor satisfying . To prevent underflow, one should select . Note that the path metric is an output of the VA. The is subtracted from the cumulative metric pruning constant metric of every survivor path of the VA to prevent the path metrics from becoming too large. The expected log-likelihood function can then be expressed in the form (see [11])

represents the convolution matrix

.. . Note that the superscript

.. .

.. .

(2.7)

denotes matrix transposition.

B. Brief Description of the EMVA The signal model (2.1) can be described as the output of an FSM observed in WGN. The state of the FSM at time is defined as (2.8) With this state definition, a complete trellis can be constructed. and the current input , the current Given the current state , given by (2.2), and the next state output of the FSM is is uniquely determined. The trellis thus has states with transitions into each state. In the following, denotes the iteration index, and repreat the th iteration. The sents the estimate of complete data used to formulate the EM iteration is .

where the superscript denotes the Hermitian operator. beObserve that the approximation (2.11) for comes an equality if the set of survivors includes all paths in the Markov chain trellis extending from to . Including all paths in this sum is costly and unnecessary, since most paths in the trellis have a negligible probability. Hence, can be evaluated by considering only the paths with a significant a posteriori probability. Typically, these paths are among the survivor paths of the VA, but if it becomes necessary to consider a larger set of paths, a list VA [15] can be substituted for the VA. Alternatively, if it is desired to consider fewer of states, a beam VA [16, Ch. 5] paths than the number can be implemented to retain only survivor paths whose metric is sufficiently close to the minimum metric. In the extreme case, where only the most likely path of the VA is retained, the EMVA reduces to the method of [17] and [18]. Note that for certain applications, such as turbo equalization, it is possible to produce soft-output information by employing the soft-output VA [19]; a discussion on this topic is given in [11]. Since the posterior probabilities in (2.9) are computed from byproducts of the VA, the computational complexity of the EMVA is essentially that of the VA plus the computations required to solve the linear matrix equation (2.12). To evaluate the EMVA, we will compare its performance with that of the VA-based method proposed in [17] and [18]. As pointed out in [11], this method, which employs the VA within

NGUYEN AND LEVY: THE EXPECTATION-MAXIMIZATION VITERBI ALGORITHM FOR BLIND ADAPTIVE CHANNEL EQUALIZATION

the EM framework, differs from the EMVA by the fact that it does not evaluate the path probabilities, and hence, ignores the E-step. Instead, it maximizes

(2.14) where is the most likely path of the VA, based on the pa. Comparing (2.11) with (2.14), we see that rameter vector , instead the algorithm of [17] and [18] maximizes of , alternately over and . This technique therefore implements a Gauss–Seidel alternating coordinate-ascent algorithm, which converges to a local maximum of , instead of a local maximum of , as the EM method would be expected to do. More fundamentally, this approach treats the transmitted sequence as a fixed unknown parameter vector, instead of random data. For brevity, in the following, we refer to this method as the maximization-maximization Viterbi algorithm (MMVA). III. CONVERGENCE ANALYSIS AND CHANNEL ACQUISITION A. Convergence and Existence of a Capture Set The EMVA can be interpreted as a quasi-Newton or a gradenote the vector dient-ascent method. Let of system parameters, where is complex, with real part and imaginary part . Consider the scaled likelihood function (3.1) It is shown in [11] that the EMVA iteration can be expressed as (3.2) (3.3) where (3.4) (3.5) In the above expressions (3.6) Note that we have used the Wirtinger calculus [20, pp. 287–288] in computing the gradient with respect to the complex vector . If we introduce the real-valued parameter vector (3.7) which will be used interchangeably with ment, by using the operator identity

as a function argu-

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the EMVA iteration (2.12) and (2.13) can be shown to have the real-domain equivalent (3.9) where (3.10) In the above expression, the operators and return the real and imaginary parts of their argument, respectively. Thus, the iteration (3.9) can be interpreted as a quasi-Newton or gradient-ascent method of the type discussed in [21]. This property allows a more precise characterization of the EMVA convergence than is available for the general EM iteration. Specifically, observe that the incomplete-data likelihood is multimodal in the parameter space, and the convergence reconverge to a sult of [22] only ensures that the iterates local maximum. Consequently, if we initialize an EM iteration in the domain of attraction of a local (or global) at some point of the likelihood, there is no guarantee that the almaximum gorithm will converge to . Although this is rather uncommon, the EM iteration can conceivably jump to another domain of attraction and ultimately converge to a local maximum other than . Fortunately, for the case of the EMVA, it can be shown that under mild conditions, any local maximizer of the likelihood admits a capture set. This is of practical interest, since in situations where some prior knowledge is available, the initial estican be properly chosen so that global convergence is mate guaranteed, or highly probable. When prior knowledge is available in the form of training data, it can be fully exploited by a semiblind EMVA implementation [5], [11]. In the next subsecsuch that tion, we will describe a blind method for finding a convergence to a global maximum of the likelihood function is guaranteed with a high probability, even when no information is available for obtaining the initial estimate . Lemma: Let be a local maximum of , and be is the only stationary point of any instance of the EMVA. If in some open set, and if there exists a constant such for all , where denotes the maximum that singular value of , then there exists an open set containing , such that if for some , then for all and . Also, given any scalar , the , which can be set can be chosen as a solid hypersphere centered at . The proof of this lemma can be sketched as follows. Since is continuous in both and , any limit point of an instance of the EMVA is a stationary point of (see [22, Th. 2]). We also have that as a general property of the EM algorithm. From (3.9), the EMVA according to , generates the sequence . Further, we have where (3.11)

(3.8)

The result then follows from [21, Prop. 1.2.5].

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B. BCA Method

is the squared Euclidean distance (2.10) of the th Note that survivor evaluated at the last iteration of the th trial. Observe also that (3.12) is numerically stable, since is on the order of unity. The final parameter estimate is selected as

It turns out that the capture set for the global maximizer is reasonably large, since for the vast majority of all possible channel realizations, the channel-acquisition probability is contains quite high if the initializer only one nonzero real-valued tap and one nonzero purely imaginary tap, of correct sign and unit magnitude, located at the proper position. For an arbitrary complex channel of order , there are possibilities for such an initializer. For a real channel, the imaginary tap is not needed, and the . Note that even though the number of possibilities is channel-identification problem has a sign ambiguity (replacing by and by leaves the output vector invariant), when the FSM initial state is fixed, say, to the zero state, a very small symbols is known, which can be used to resolve block of the sign ambiguity by tap flipping. For certain signal alphabets, however, differential encoding is possible, so that it is unnecessary to resolve the channel sign ambiguity by flipping the sign of each initializer tap. In such cases, the number of possible for complex-valued channels, initializers is reduced to or for real-valued channels. For example, it is well known that differential modulation can be easily implemented with binary phase-shift keying (BPSK) modulation. It is also possible to employ differential encoding with quaternary phase-shift keying (QPSK), provided the mapping is defined in a proper manner. To see this, note that a sequence of QPSK symbols can be represented by a sequence of 2-b pairs , . Then, if the QPSK alphabet symbols where are, respectively, represented by the couples {(0, 0), (0, 1), (1, 1), (1, 0)}, the sign ambiguity in the demodulated binary and sequences is circumvented if the binary sequences have been individually differentially encoded. Note that this mapping rule is consistent with the Gray code mapping, where any two adjacent signal points differ by only one bit. For higher order alphabets, however, differential encoding cannot completely preserve the Gray mapping rule [23]. In light of this observation, a BCA technique can be devised trial runs of the EMVA for complex based on at most runs for real channels. Let the trials channels, or as few as be indexed by . Suppose that at trial , the EMVA converges to . Evaluating the log of (3.1) at yields

(3.16) where (3.17) To evaluate the acquisition probability, we need a quantitative definition of acquisition. To measure the channel-estimation error, we employ the per-tap squared error (3.18) where (3.19) (3.20) and we define the acquisition probability as (3.21) so that for some threshold . In this paper, we select the channel is considered acquired if the per-tap squared error is smaller than the noise variance. If the noise is low and the error is large, the algorithm is considered to have misconverged. In cases where the noise is high, the error may still be reasonably large, even though the algorithm has indeed converged to the true maximum-likelihood (ML) estimate. Note that the lesser of and is used as error measure in order to account for the sign ambiguity inherent in the blind-estimation problem. The sign ambiguity in the estimated bits can be resolved by employing simple error-control techniques such as differential encoding, as discussed above. Note that the MMVA-based BCA algorithm is the same as , with indexing the most described above, except likely path of the VA. IV. EMVA FOR SLOWLY FADING CHANNELS A. Channel Model In this section, we consider the fading channel model [24] (4.1)

(3.12)

where represents the channel response at time , due to a unit impulse transmitted at time . A commonly used model has the form for

where

(4.2)

(3.13) (3.14) (3.15)

where nience, let

is a zero-mean random process in . For conve(4.3)

NGUYEN AND LEVY: THE EXPECTATION-MAXIMIZATION VITERBI ALGORITHM FOR BLIND ADAPTIVE CHANNEL EQUALIZATION

(4.4) (4.5) where is usually called the specular part. We examine the case is a stationary Gaussian process with autocorrelation where matrix (4.6) In this case, (4.2) specifies a Rayleigh fading channel if , or a Ricean fading channel, otherwise. Two important parameand the spread ters characterizing the model are the -factor factor , given by (4.7) (4.8) where denotes the coherence time. If , for some , the coherence time is given by

for (4.9)

. For practical which is essentially the overall support of as the range of values of for which all purposes, we define elements of remain above 10% of their respective maximum values. Slow-fading channels are characterized by small (large ), and spread factors greater than 0.001 typically indicate fast fading. Assuming the signal alphabet is power-normalized, the receive SNR is defined as SNR

(4.10)

B. Block EMVA For sufficiently slowly fading channels, the EMVA as described previously can be implemented to equalize the channel in a block-by-block manner. In such an implementation, the block length is selected to be a small fraction of the fading memory , so that the channel remains approximately constant over each block. To describe the procedure, suppose and , respectively, denote the block index and the block length. The th block of the transmitted data is represented by (4.11) Employing the EMVA, we seek the estimate the observations in the th block

of

based on

(4.12) denotes the overlap time between two consecutive where , where can be blocks. Note that the length of is thought of as a decoding delay, usually assigned a value equal to several times the channel order . Consequently, each survivor of the EMVA has length , and the estimate of results from the first symbols of the best survivor. For each block, the quantities that need to be initialized can be obtained as follows.

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), the Channel Initialization: For the first block ( channel is estimated via the BCA algorithm described in Sec, the EMVA can be initialized with the tion III-B. For th block. channel estimate obtained for the Initial State: The state of the FSM at the beginning of block is defined by (2.8). With a “reasonable” decision delay , the state can be estimated from the decoded th block by data belonging to the (4.13) Since the block EMVA is simply the EMVA operating on the blockwise basis, it is obvious that the per-block complexity of the block EMVA is similar to that of the EMVA. Since the per EMVA has complexity is due to the compublock, where the term is incurred by tation of the data autocorrelation matrix and the maximization, the block EMVA has complexity per length- block. The multiplier is due to periodic updates. As , the complexity of the block EMVA approaches that of the EMVA. C. Online EMVA In addition to the blockwise implementation, the EMVA can also operate in an adaptive online mode. The development of the online EMVA is based on the following observation. Suppose , the channel estimate is . Since the that at time channel does not change significantly between time and , for some window width , the VA can operate for , with based on denoting the update period. To track the channel variations, is updated via the channel estimate at time one iteration of the EMVA based on the newest observations. The online EMVA is implemented as follows. samples of the 1) Channel Acquisition: Based on the first observed sequence, the channel is acquired via the BCA algorithm of Section III-B. 2) Channel Tracking: The channel estimate is updated every time epochs. Each update is based on the most recent symbols of the VA survivors, using (2.12) and (2.13). The computational complexity of the online EMVA is similar to that of the VA. More specifically, the number of computations required in addition to the operation of the VA are mainly those incurred by the M-step (2.12), which is per update. Therefore, the online EMVA has complexity per block of length , where the multiplier is due to one channel estimate update per time steps.

V. SIMULATION RESULTS A. Constant Channel To illustrate the performance of the EMVA, we consider several examples.

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Fig. 1. SER performance of the EMVA-based and MMVA-based BCA algorithm as a function of SNR. (a) BPSK. (b) QPSK.

Example 1: A BPSK signal ( the channel

) is transmitted through

(5.14) considered in [25]. The simulated bit-error rate (BER) performance of the BCA algorithm based on the EMVA and MMVA is plotted in Fig. 1(a), for which the channel initializer taps are assigned a positive value in every trial. We see that for a small b), the MMVA exhibits a noise floor block length ( and is outperformed by the EMVA by a significant amount. At b, the MMVA performs close to the EMVA until it dB. When is increased starts to “floor” out at SNR to 57 b, the MMVA achieves the performance of the EMVA. For the 3 BER curves of the EMVA above, the 8 mostly likely paths of the 16-state trellis, identified via the list VA [15], are included in the E-step. For comparison, we include the BER of b. In addition, the EMVA-based BCA algorithm for we show the EMVA performance subject to the fixed channel initializer (5.15)

used in the first block of the simulations. For each subsequent block, the channel initializer is obtained from the final channel estimate of the previous block. Finally, the performance of the VA based on the true channel is included for reference. When the block length is sufficiently long, the survivors of the VA merge at a point several times the channel order away from their front end. In this case, the sum (2.11) is dominated by the common part of the survivors, so that (2.14) closely approximates (2.11), and thus the MMVA approaches the performance of the EMVA. When the block length is short, however, (2.14) cannot approximate (2.11) well enough, and the MMVA is outperformed by the EMVA, as expected. ) is transmitted through Example 2: A QPSK signal ( channel (5.14). The simulated symbol-error rate (SER) curves of the BCA algorithm based on the EMVA and the MMVA are symplotted in Fig. 1(b), where the block length is bols. To save some complexity, the trial channel taps used in the BCA algorithm are assigned only a positive value. To avoid the sign (polarity) ambiguity in the detected sequence, the differential encoding scheme described in Section III-B is employed. In addition, we show the EMVA performance subject to the fixed channel initializer (5.15) used in the first block of the simulations. The performance of the VA based on the true channel is also included for reference. It is interesting to examine briefly the error patterns of the EMVA and MMVA at a high SNR in this example. At SNR dB, we ran the EMVA until 25 symbol errors were accumulated, which occurred out of 936 blocks. No instance of misconvergence was observed. With the same simulation configuration and random generator state, the MMVA committed several misdetections which could be a result of misconvergence. As seen in simulation Example 1, the performance of the MMVA approaches that of the EMVA as the block length increases. Hence, without repeating the simulations, we can conclude that the same holds for QPSK. B. Time-Varying Channel In Fig. 2(a), we show the simulation results for the five-tap Ricean fading channel with specular part given by (5.14), , and . The fading taps , , are uncorrelated Gaussian processes with autocorrelation matrix (5.16) denotes the standard triangular pulse of width 2 where , and has been chosen to achieve the and height 1, value. The triangular shape of the autocorrelation desired function has been chosen only for simulation convenience. It is worth noting that the Ricean channel model above can be viewed as the discrete-time equivalent of a continuous-time Ricean channel. The specular part of the channel corresponds to the sampled version of the transmit–receive filter impulse response dispersed by fixed multipaths. Even if there is only one fixed path, imperfect timing phase results in an intersymbol interference (ISI) specular part having more than one tap. Similarly, the scattering part corresponds to fading paths. In fact, the general form of this channel model was developed in 1960 by

NGUYEN AND LEVY: THE EXPECTATION-MAXIMIZATION VITERBI ALGORITHM FOR BLIND ADAPTIVE CHANNEL EQUALIZATION

PERFORMANCE

OF THE

TABLE I EMVA-BASED AND MMVA-BASED BCA METHOD ESTIMATED FROM 2000 MONTE CARLO RUNS RAYLEIGH FADING CHANNEL WITH BLOCK LENGTH L = 60 b, BPSK, AND S = N

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BLOCKWISE

In Fig. 2(b) we show the BER curves of the EMVA and MMVA for a four-tap unit-energy Rayleigh fading channel and ) with . The fading taps ( , , are uncorrelated Gaussian processes whose autocorrelation matrix is given by (5.16) with and . For comparison purposes, we plot also the performance of the training-based VA employing the channel estimate obtained from the training data contained in every block, each consisting of 40 training bits and 160 information bits, with an overlap time of b. It is seen that the training-based VA is outperformed by both EMVA and MMVA. In this respect, one should note that the EMVA has a semiblind implementation that fully exploits all training data available. The reader is referred to [5] and [11] for a performance comparison between the semiblind EMVA and a training-based VA. C. Acquisition Probability of the BCA Algorithm To illustrate the performance of the BCA method, we show in Table I the acquisition probability for a five-tap, blockwise Rayleigh fading channel. That is, the channel is constant over each block, and for each block, a new random channel is generated, where all taps are circularly complex independent variables having zero mean and the same variance. The Rayleigh channel realized in this manner has all possible shapes, thereby representing a demanding acquisition scenario. Some attention should be paid to the fact that in cases with a large SNR, the algorithm may still have converged to the correct maximum, even though the error is only slightly larger than the threshold . Also shown in the table is the normalized mean-squared error (NMSE) NMSE

Fig. 2. Performance of the EMVA and MMVA for differentially modulated BPSK signal transmitted over (a) a Ricean fading channel and (b) a Rayleigh fading channel. For the trained VA, each block of L bits contains L training bits and L L information bits.

0

Kailath [24]. In conclusion, the discrete-time taps are not to be confused with the continuous-time multipaths. We observe from the figure that the block EMVA outperforms the online EMVA by about 2 dB, and both of them are superior to a training-based VA. The block MMVA is outperformed by the block EMVA and clearly exhibits a noise floor before the online EMVA. For comparison, we also show the performance of the VA based on the specular part of the channel, i.e., the scattering part of the channel is assumed to be absent.

(5.17)

estimated from 2000 Monte Carlo runs. As an example, examine the results for the EMVA. Suppose the transmit power is normalized, as is the case in the simulation. For SNRs of 8 and 16 dB, the error is less than 16% and 2.5% of the channel energy for 98.30% and 99.10% of the time, respectively. For the MMVA, these probabilities decrease to 97.70% and 98.90%, respectively. By extrapolation, the error is less than 1% of the dB. channel energy for nearly 100% of the time at SNR Note also that the NMSE is quite small, indicating global convergence and the acquisition of a highly accurate channel estidB, the NMSE is about 2.20% mate. For example, at SNR for the EMVA and about 3.03% for the MMVA. VI. CONCLUSIONS In this paper, we have analyzed the convergence behavior of the EMV algorithm for blind equalization of communications

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channels. The algorithm has a broad range of applications, including equalization of constant, Rayleigh, and Ricean channels. This approach is computationally faster than other HMM blind-equalization methods based on the forward–backward MAP algorithm. When the block length is sufficiently long, it is desirable to use only the most likely survivor of the EMVA in the E-step, resulting in the MMVA. However, under conditions such as fast fading that prevent the use of large block lengths, the EMVA achieves better performance than the MMVA. A BCA method has also been developed, which employs multiple EMVA or MMVA runs to find the global maximum of incomplete-data likelihood. It is noteworthy that even with a very short observation record (e.g., 60 b long) the BCA algorithm is able to attain near 100% acquisition probability. With the BCA method, the EMVA can thus be viewed as a completely blind equalizer, since global convergence is almost guaranteed due to the existence of a capture set associated with the global maximum. Hence, the algorithm can achieve a highly reliable channel estimate without training or prior channel knowledge, and may prove to be a powerful tool for blind equalization of wireline or wireless communication links. The blind-acquisition feature represents a robust capability of the EMVA not usually found in other blind techniques.

REFERENCES [1] A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum-likelihood from incomplete data via the EM algorithm,” J. Roy. Stat. Soc., ser. B, vol. 39, pp. 1–38, 1977. [2] Y. Sato, “A method of self-recovering equalization for multi-level amplitude modulation,” IEEE Trans. Commun., vol. COM-28, no. 6, pp. 678–682, Jun. 1975. [3] D. N. Godard, “Self-recovering equalization and carrier tracking in twodimensonal data communication system,” IEEE Trans. Commun., vol. COM-28, no. 11, pp. 1867–1875, Nov. 1980. [4] A. Benveniste and G. Goursat, “Blind equalizers,” IEEE Trans. Commun., vol. COM-32, no. 8, pp. 871–883, Aug. 1984. [5] H. Nguyen and B. C. Levy, “A semi-blind EMVA for maximum-likelihood equalization of GMSK signal in ISI fading channels,” in Proc. 36th Asilomar Conf. Signals, Syst., Computers, Pacific Grove, CA, Nov. 2002, pp. 1905–1908. [6] D. Boss, K.-D. Kammeyer, and T. Petermann, “Is blind channel estimation feasible in mobile communication systems?,” IEEE J. Sel. Areas Commun., vol. 16, no. 10, pp. 1479–1492, Oct. 1998. [7] G. K. Kaleh and R. Vallet, “Joint parameter estimation and symbol detection for linear or nonlinear unknown channels,” IEEE Trans. Commun., vol. 42, no. 7, pp. 2406–2413, Jul. 1994. [8] L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal decoding of linear codes for minimizing symbol error rate,” IEEE Trans. Inf. Theory, vol. IT-20, no. 3, pp. 284–287, Mar. 1974. [9] G. J. McLachlan and T. Krishnan, The EM Algorithm and Extensions. New York: Wiley, 1997. [10] H. A. Cirpan and M. Tsatsanis, “Blind receivers for nonlinearly modulated signals in multipath,” IEEE Trans. Signal Process., vol. 47, no. 2, pp. 583–586, Feb. 1999. [11] H. Nguyen and B. C. Levy, “Blind and semi-blind equalization of CPM signals with the EMV algorithm,” IEEE Trans. Signal Process., vol. 51, no. 10, pp. 2650–2664, Oct. 2003. , “Blind ML detection of CPM signals via the EMV algorithm,” in [12] Proc. IEEE Int. Conf. Acoust., Speech, Signal Process., vol. 3, Orlando, FL, May 2002, pp. 2457–2460. , “Blind equalization of fast fading Ricean channels via the EMV [13] algorithm,” in Proc. IEEE Veh. Technol. Conf., vol. 2, Orlando, FL, Oct. 2003, pp. 1157–1161. , “Blind equalization of dispersive fast fading Ricean channels via [14] the EMV algorithm,” IEEE Trans. Veh. Technol., vol. 54, no. 5, Sep. 2005. [15] N. Seshadri and C.-E. W. Sundberg, “List Viterbi decoding algorithms with applications,” IEEE Trans. Commun., vol. 42, no. 2–4, pp. 313–323, Feb.–Apr. 1994.

[16] F. Jelinek, Statistical Methods for Speech Recognition. Cambridge, MA: MIT Press, 1998. [17] M. Feder and J. A. Catipovic, “Algorithms for joint channel estimation and data recovery—Application to equalization in underwater communications,” IEEE J. Ocean. Eng., vol. 16, no. 1, pp. 42–55, Jan. 1991. [18] K. H. Chang, W. S. Yuan, and C. N. Georghiades, “Block-by-block channel and sequence estimation for ISI/fading channels,” in Proc. 7th Thyrrhenian Workshop Digital Commun., E. Biglieri and M. Luise, Eds., Viareggio, Italy, Sep. 10–14, 1995, pp. 153–170. [19] J. Hagenauer and P. Hoeher, “A Viterbi algorithm with soft-decision outputs and its applications,” in Proc. IEEE Globecom, Dallas, TX, Nov. 1989, pp. 1680–1686. [20] P. Henrici, Applied and Computational Complex Analysis. New York: Wiley, 1986, vol. III. [21] D. P. Bertsekas, Nonlinear Programming. Belmont, MA: Athena Scientific, 1999. [22] C. F. J. Wu, “On the convergence properties of the EM algorithm,” Ann. Statist., vol. 11, pp. 95–103, 1983. [23] W. J. Weber, III, “Differential encoding for multiple amplitude and phase-shift-keying systems,” IEEE Trans. Commun., vol. COM-26, no. 5, pp. 385–391, May 1978. [24] T. Kailath, “Correlation detection of signals perturbed by a random channel,” IEEE Trans. Inf. Theory, vol. IT-6, no. 6, pp. 361–366, Jun. 1960. [25] B. Porat and B. Friedlander, “Blind equalization of digital channels using high-order moments,” IEEE Trans. Signal Process., vol. 39, no. 2, pp. 522–526, Feb. 1991. Hoang Nguyen (S’97–M’05) received the B.S. degree (summa cum laude) in 1999 from the University of Missouri-Columbia, Columbia, MO, and the M.S. and Ph.D. degrees in 2002 and 2003, respectively, from the University of California-Davis (UC Davis), Davis, CA, all in electrical engineering. From November 1995 to August 1999, he was a member of the Mathematics Tutoring Staff at Penn Valley Community College, Kansas City, MO. From August to December 2002, he was a Research Intern with Nokia Research Center, Dallas, TX, where he developed efficient equalization algorithms for downlink CDMA receivers. From January to September 2003, he held a Research Assistant position in electrical engineering at UC Davis. He is currently a Research Engineer with Nokia Research Center, San Diego, CA, where he performs radio systems research and participates in transceiver design. His research interests include statistical signal processing, estimation, detection, equalization, diversity techniques, and coding for communications. Dr. Nguyen is a member of the Phi Kappa Phi, Tau Beta Pi, and Eta Kappa Nu honor societies. He was the 1998 recipient of the Outstanding Junior Award within the Electrical and Computer Engineering Department, University of Missouri, and was a 1997 Barry M. Goldwater scholarship nominee. He holds a Missouri Registered Engineer-in-Training license. He was a graduate Research Fellow of the U.S. National Science Foundation, and spent the three-year fellowship tenure at UC Davis. Bernard C. Levy (F’94) received the diploma of Ingénieur Civil des Mines from the Ecole Nationale Supérieure des Mines, Paris, France in 1974, and the Ph.D. in electrical engineering from Stanford University, Stanford, CA, in 1979. From July 1979 to June 1987, he was Assistant and then Associate Professor in the Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge. Since July 1987, he has been with the University of California-Davis (UC Davis), Davis, CA, where he is Professor of Electrical Engineering and a member of the Graduate Group in Applied Mathematics. He served as Chair of the Department of Electrical and Computer Engineering at UC Davis from 1996 to 2000. He was a Visiting Scientist at the Institut de Recherche en Informatique et Systèmes Aléatoires (IRISA), Rennes, France, from January to July 1993, and at the Institut National de Recherche en Informatique et Automatique (INRIA), Rocquencourt, France, from September to December 2001. His research interests are in statistical signal processing, estimation, detection, and multidimensional signal processing. Dr. Levy currently serves as Associate Editor of the IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS II, and of the EURASIP Journal on Applied Signal Processing. He was a Member of the Image and Multidimensional Signal Processing Technical Committee of the IEEE Signal Processing Society from 1992 to 1998. He is a Member of SIAM and the Acoustical Society of America.

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Multirate Modulation: A Bandwidth- and Power-Efficient Modulation Scheme Hans B. Peek, Fellow, IEEE

Abstract—Multirate (MR) modulation resembles block-coded modulation (BCM), since matrices are being used to transform binary input vectors to multilevel output vectors (blocks) of length . Unlike BCM, attention is given to the spectral shaping of the signal to be transmitted. Hence, the encoding matrices are designed to provide simultaneous spectral shaping and Euclidean distance. The encoding matrices can be implemented by using MR digital filters of low complexity. MR modulation also resembles partial response (PR) modulation since, in both cases, a transmitter and receiver filter is used with an overall duobinary impulse response. It will be shown that MR modulation has a number of significant advantages compared with PR modulation. Thus, for example, with MR modulation, loss of synchronization or gain control, as can occur with PR modulation, cannot happen in the receiver. Furthermore, computer simulations for an additive white Gaussian noise channel demonstrate that, for a bit-error rate of = 10) gives a gain of 1.5 dB, 10 6 , MR modulation (with compared with PR modulation and symbol-by-symbol detection. However, MR modulation requires a slightly higher bandwidth. It is also explained how, for block lengths 10, MR modulation gives a larger bandwidth efficiency than -ary pulse-amplitude modulation with raised-cosine pulses and a rolloff factor 0 1. Index Terms—Bandlimited communication, decoding, digital communication, discrete-time filters, modulation.

I. INTRODUCTION HIRD-GENERATION cellular exemplifies wireless systems where the realization of sharp spectral characteristics is traded off against processing complexity in signal generation and detection. In universal mobile telecommunications systems (UTMSs), a square-root raised cosine pulse with a rolloff factor of 0.22 is used. This corresponds with excess bandwidth of 22% and, therefore, the bandwidth efficiency is low in this case. The technique described here takes a new approach that realizes excellent bandwidth efficiency at a minimal cost in power efficiency and processing complexity. be a discrete-time signal, representing a data Let and stream and consisting of an arbitrary sequence of samples with a sampling period of s that has to be transmitted over an additive Gaussian noise channel. Assume a strict Hz that could theoretically be bandwidth limitation of as the weights of a series of achieved by applying the delta pulses (with period ) fed to an ideal low-pass analog Hz. However, because filter with a bandwidth

T

Paper approved by X. Dong, the Editor for Modulation and Signal Design of the IEEE Communications Society. Manuscript received April 30, 2003; revised August 3, 2004, January 27, 2005, and April 27, 2005. The author is at Dennenlaan 10, 5671 BX Nuenen, The Netherlands (e-mail: [email protected]). Digital Object Identifier 10.1109/TCOMM.2005.857145

of the brick-wall-shaped frequency characteristic of an ideal low-pass analog filter, such a filter cannot be realized. Partial response (PR) filters [1, Sec. 4.6], in particular, the duobinary and modified duobinary impulse responses, are realizable filters meeting the same strict bandwidth limitation. These can be implemented as the combination of a digital filter, with system function or , respectively, , and an ideal analog low-pass filter, where resulting in a realizable analog low-pass filter with a cosineor sine-shaped frequency-response characteristic, respectively. , correEach has a spectral null at the Nyquist frequency . sponding with Although PR modulation is very bandwidth-efficient, the output of these filters can become zero for long input strings of plus-ones and minus-ones or alternating sign samples. Zero transmitted energy over a long time can cause loss of synchronization or gain control in the receiver. Besides, in order to avoid error propagation, the digital signal has to be precoded [1, Sec. 4.5]. To obtain maximum power efficiency for PR-modulated signals, maximum-likelihood (ML) sequence detection must be used in the receiver. This makes the receiver complicated, in particular, for PR signaling with many levels. However, it is also possible to use simple symbol-by-symbol detection. This will, however, give a loss in power efficiency. In this paper, an extension of PR modulation is described that does not have the disadvantages of PR modulation mentioned above. This extension of PR modulation makes use of multirate (MR) digital filters [1, Sec. 4.7], [2], [3]. “Multirate” refers to the property that the output sampling rate of such a digital filter can be larger or smaller than the input sampling rate. It will be explained how, with such an MR filter, a spectral null at the Nyquist frequency of the output signal can be realized. The times larger output sampling rate will, however, be than the input sampling rate, where is a positive integer such . The operation of the digital MR filter can also be that regarded as transforming an input block of samples taken from the set into an output block (codeword) of samples taken from the set . The system to be described combines a digital MR filter with an ideal low-pass analog filter. To make it realizable, the part of the digital MR filter that is responsible for the Nyquist null can be merged with the ideal low-pass analog filter resulting in a realizable analog filter. This filter will have a duobinary impulse response. Because of the higher output sampling rate with , the bandwidth efficiency of MR modulation a factor will be smaller by this factor, compared with normal PR modcan be chosen, say 20, the loss in bandwidth ulation. Since efficiency is only 5% in that case.

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TABLE I CODEBOOK

It will be shown that the creation of a Nyquist null results in a larger minimum Euclidean distance between the output codewords of the MR filter. This larger Euclidean distance can be exploited in the receiver by using an easy-to-implement Wagner decoding [6], [7]. MR modulation over an additive white Gaussian noise (AWGN) channel with Wagner decoding and, for in the receiver gives, for a bit-error rate (BER) of , a gain in power efficiency of 1.5 dB, compared with standard duobinary PR signaling. A compact and useful comparison of digital modulation (b/s/Hz methods is based on the normalized data rate of bandwidth) versus the signal-to-noise ratio (SNR) per bit ) required to achieve a given error probability. In [5, ( Fig. 5, 2–17, p. 282], such a comparison is made of several modsymbol-error probability. Proakis ulation methods for a distinguishes in [5, Fig. 5, 2–17, p. 282] a bandwidth-limited and a power-limited region. MR modulation belongs to the bandwidth-limited region, but intends to be power-efficient in that region, like coded modulation. Just like block-coded modulation (BCM) [7, pp. 528–529], MR modulation is power-efficient in the sense that it provides Euclidean distance at the transmitter and exploits it in the receiver.

Fig. 1. MR filter structure for realizing the matrix transformation given by (1). Input block length 3; output block length 4.

demonstrates that the Nyquist frequency or is absent. Note further that there is no output block consisting of only zero samples. This implies that a concatenation of output blocks will never result in a loss of synchronization or gain control in the receiver. This is an advantage over PR modulation, in which certain data sequences can cause a loss of synchronization and gain control in the receiver. or samples is 1/4, The frequency of occurrence of whereas that of a zero sample is 1/2. This is similar to duobinary signaling, which also has a Nyquist null and where and . The transformation from an input block to an output block is given by the matrix multiplication

II. EXAMPLE OF MR MODULATION The main idea in MR modulation is based on a transformation of an input block to a different output block such that the spectrum of the output block possesses a spectral null. The creation of a spectral null can best be explained by giving an example. Then it will be easier to understand the theory of MR modulation. A. Example , consisting of a series of The discrete-time signal plus-one and minus-one samples, can be regarded as a concatenation of blocks (vectors) of length 3. Now let each input block be transformed into an output block of length 4, where every consists of samples from . For this example, this transformation is given by the codebook in Table I. be equal to that of Let the duration of every input block , where is the output samthe corresponding output block and, hence, the output sampling pling period. Thus, rate will be 4/3 times the input sampling rate. Application of the Fourier transform to an arbitrary output block

where (1) Fig. 1 shows an MR filter implementation of this matrix transformation [3]. This MR filter contains three finite impulse re, , sponse (FIR) digital filters with system functions , where . It is important to note that and the coefficients of the first, second, and third polynomials correspond, respectively, to the first, second, and third columns of . The two delay elements, each of s, and the threefold downsampling in every branch, can be regarded as a serial-to-parallel conversion. A fourfold upsampling in every branch results in a fourfold increase of the sampling frequency. In general, an -fold downsampling means that only the input sample numbers equal to a multiple of are retained [4]. A -fold upzero-valued samples are inserted sampling means that between adjacent input samples [4]. The three FIR filters, with coefficients or , and their short impulse responses are of low complexity. An advantage of representing the matrix transformation (1) in real time as given in Fig. 1 is that it can be extended to represent the required signal-processing operation

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give a Nyquist null. Thus, the loss in bandwidth efficiency given can be made small. by the ratio The minimum Hamming distance of the code given in Table I , whereas the minimum Euclidean distance is is . This is the same minimum Euclidean distance as between two PR sequences of 2, 2, and zero samples. If the noise samples at the output of the matched filter in the receiver were uncorrelated, a 3-dB gain in power efficiency using ML sequence detection could be obtained, compared with symbol-by-symbol detection. It will be shown, however, that the noise samples are correlated and that, with Wagner decoding, an asymptotic gain of 1.76 dB can be achieved, compared with PR modulation using and , symbol-by-symbol detection. For a BER of however, the gain is 1.5 dB. Fig. 2. Cascade of an MR filter and an analog filter that gives a spectrum null at the Nyquist frequency. Input block length 3; output block length 4.

for any matrix. In Section IV, matrix transformations, for various values of and , are described. All of these transformations can be implemented by generalizing the scheme given in Fig. 1 by using parallel branches and, in each branch, an -fold downsampling, followed by a -fold upsampling and simple FIR filters. However, there are various ways to implement a digital MR filter [8], and the final choice depends on the preference of the designer. Analysis of the MR filter operation, given in Fig. 1, will show that it causes a delay of one s between input and output. block, i.e., The three system functions are zero at the Nyquist frequency or at . Thus, ( ) can be factored out from all three system functions, and made a common . When multiplier without changing the output signal ) is merged with an ideal low-pass filter with the factor ( a bandwidth , the result is the realizable shaped-spectrum low-pass filter of Fig. 2. The frequency response of the cascade of the two filters can be determined as follows. An ideal , has an impulse relow-pass filter, with a bandwidth sponse

where . The cascade of a filter with a system function ideal low-pass filter thus has the impulse response

III. BLOCK TRANSFORMATION USING AN MR DIGITAL FILTER In this section, we explain how a binary signal can be transformed into a digital output signal by using a is divided into digital MR filter. The binary input signal blocks of length , and the MR filter output signal is divided into blocks of length . The transformation of the th input block to the corresponding output block is linear and is given by (3) matrix with entries and where and where is a are column vectors of length and , respectively. The dus equals the time duration ration of every input block s of every output block. Thus (4) This can be implemented with an MR digital filter similar to that given in Fig. 1. , the input and output rates differ, hence, the word If “multirate.” The filter is realized by a parallel connection of different FIR filters that are preceded by an -fold downsampling and a -fold upsampling. The impulse response of the th FIR filter is ( ) (5)

and the where

This is called a duobinary pulse [1, Sec. 4.6.1]. The Fourier transform of gives the frequency response of the total filter (2) The magnitude of this frequency response is depicted in Fig. 2. For this example, MR modulation would require a 4/3 times larger bandwidth than PR modulation. In the next section, it will be shown, however, that it is easy to create matrices with rows columns, where can be arbitrary large, that also and

Thus, the impulse response of the th FIR filter corresponds with the coefficients of the th column of matrix . and are relatively prime, the order of downsampling If and upsampling can be changed [4]. At the instant the th input block enters the MR filter, the th output block starts to emerge from the filter. Thus, the MR filter causes a time delay of one block. In Section I, it was shown that this implementation is ideally suited for combining with an ideal low-pass filter to obtain a realizable analog filter with a Nyquist null and, thus, limit the bandwidth of the ). output signal to the Nyquist frequency bandwidth (

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IV. NYQUIST NULL IN THE OUTPUT SIGNAL SPECTRUM OF AN MR FILTER FOR AN -ARY PAM INPUT SIGNAL

An example of such a matrix

is given by

The Fourier transform of an output block is (6) is the relative frequency. If the row vector where denotes the th row of the matrix , then (6) can be written as (7) where is the input vector corresponding with the output vector . Now, define the system functions

(15) All entries not indicated have zero value and are left out for ( ) rows and clarity. The matrix has columns. The system functions are

and the system row vector (8) Then, (7) can be written as

(16) (9)

All system functions are zero for

, and thus (17)

If for

or i.e.,

then (10) . The regardless of , i.e., regardless of the input signal , in the fundamental interval shape of the spectrum , depends, of course, on and . ( ) A spectral null at the Nyquist frequency is obtained when (11) From (6), it follows that in that case (12) The code defined by the matrix given by (1) is an example where all output codewords fulfill (12). In certain applications such as baseband data transmission [PCM codecs and digital subscriber line (DSL)], a spectral null at dc (zero frequency) is desired. This can be obtained when (13) In that case, it follows from (6) that (14) A matrix that gives either a dc null or a Nyquist null can be converted into the other type by multiplying all entries in the even rows by minus one. A matrix generates codewords with a Nyquist null if according to (11)

A few words on how the matrix [given by (15)] was found might be useful. Since each component of an output block is an element from , each row of the matrix must exist of zero the set entries except two entries taken from the set . Next, every column of the matrix must correspond with a . By trying, one can system function that is zero for yields the smallest matrix, as given by (1). show that Once this result is obtained, it is not difficult to discover the reggiven by (15). Furthermore, the matrix ular structure for transformation must give a one-to-one correspondence between each input vector and its corresponding output vector [3]. Matrix and matrix give a one-to-one correspondence between and . The ratio of the output sample rate and the input sample rate for an MR filter that is described by matrix is and approaches unity for large values of . Because gives codewords with a Nyquist null, it follows from (12) that (18) This property will be used later in the decoding (error correction). After error correction in the receiver, the original input block must be recovered. This step will be called data reof the vector covery. For data recovery, the last component can be deleted and the last row of . Now, define the reduced vector (19) and let be the square by deleting the last row of . Clearly, vector can be recovered by

matrix that is obtained is invertible, and the (20)

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Fig. 3.

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K 0 1); output block length K .

Cascade of an MR filter and an analog filter that gives a spectrum null at the Nyquist frequency. Input block length (

The inverse matrix operation can be implemented with an MR filter. is an invertible matrix, it follows that a vector Because with arbitrary components can also be recovered from the output vector by using (20). is an -ary (digital) pulse-amAssume now that is a vector plitude modulation (PAM) vector, i.e., with PAM components taken from the set , and 2, 4, 8, etc. This set contains where distinct integers. The case has already been discussed. , the set is . For If the input to an MR filter, given by the matrix , is an -ary , with a Nyquist PAM block , then the output block null, has components taken from the set

( ). Since both and have the difference vector also has a Nyquist null. In order that a Nyquist null, has a Nyquist null, it must have at least two components different from zero. Hence, the minimum Hamming distance is . These two components of have a minimum absolute value of 2, and thus the minimum Euclidean distance . between two output codewords is Now, the system functions given by (16) can be written as

(21)

In the same manner as was done for the MR filter that implemented the matrix (Section II), the digital filter with system and an ideal analog low-pass filter can be function merged into a realizable analog low-pass filter (Fig. 3) with a . magnitude response For certain applications, such as in wireless systems, it is advantageous to have both a dc null and a Nyquist null. If the transmitted signal spectrum has a null at the carrier frequency, a synchrodyne receiver can be applied, which is much easier to realize on a chip than a heterodyne receiver with its intermediate frequency (IF) filter [9]. In [2], a matrix is given, with rows and columns, where , and even that generates codewords with a dc null and a Nyquist null. Implementing this matrix with an MR filter and merging the common system functions, either or with an ideal low-pass filter will give a realizable analog filter that will either have a duobinary or modified duobinary impulse response. For the sake of brevity, this case will not be considered further.

, the set is . These For equidistant levels are identical with those at the output of a PR duobinary filter with an -ary PAM signal as input [5, Sec. is 9.2.3]. Furthermore, a reduced vector of length defined by (19). The components of are taken from the set is invertible. Thus, (21). In that case, (20) also holds, since the -ary PAM vector can be recovered. It is important to note that, if is an -ary PAM vector, then , and consequently, also . Thus, no loss of synchronization or gain control, as can occur with PR, can happen in the receiver. Moreover, error propagation, as occurs with PR without precoding, is impossible with MR. All output blocks have a Nyquist null. After having used (18) for error correction (decoding) in the receiver (Section V), and assuming no errors are left in the reduced block, the original -ary PAM block is recovered by using (20). Consider, in order to determine the minimum Hamming and minimum Euclidean distance of the code given by matrix ,

(22)

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Fig. 4. Bandwidth efficiency of M -ary PAM,with raised cosine pulses,as a a function of the rolloff factor . The bandwidth efficiencies of M -ary MR modulation for a block length of K = 5, K = 10, and K = 20 are also indicated.

The bandwidth efficiency of tion with a Nyquist null is

-ary PAM using MR modula-

b s Hz

(23)

which is times smaller than for standard duobinary , which is easy to implement, the PR modulation. For loss in bandwidth efficiency is 5%. For -ary PAM with raised cosine pulses and a rolloff factor , the bandwidth efficiency is b s Hz

(24)

From (23) and (24), it can be concluded that -ary MR modulation for gives a higher bandwidth efficiency than -ary PAM and . In Fig. 4, (24) is plotted as a function of , and the bandwidth efficiencies as given by (23) are indi, , and . cated for V. DECODING OF AN MR-MODULATED SIGNAL Let an MR-modulated signal with a Nyquist null be transmitted over an additive Gaussian noise channel using duobinary pulses,as explained earlier. It is well known [5, Sec. 9.2.3] that the frequency response (2), i.e., the Fourier transform of a duobinary pulse, has to be split evenly between the frequency responses and of the transmitter and receiver filters. Thus (25) is given by (2). where in the receiver (Fig. 5) The output of the matched filter is sampled with a period . The resulting samples, belonging to a transmitted block , can be written as (26) where is a noise vector with correlated Gaussian noise components. As indicated in Section IV, is a block of samples with values taken from a set of equidistant levels.

Fig. 5.

Structure of an MR modulation receiver.

Next, the components of are rounded to the nearest of the levels, which gives

We now apply a decoding algorithm attributed to Wagner [6], [7]. Define

and the error vector

If is a codeword with a Nyquist null, then , then the decoded vector is to (18). Thus, if

according

If, however, , then we determine the largest magnitude error component of , say , and modify to so that

This guarantees that the corrected sum is zero. Thus

Implementation of Wagner decoding is very simple. In successively rounding samples, one only has to keep track of the largest rounding error, and to mark at which position in the block this occurred. Initially, the largest error is at the first position in the block. Assume, for example, that the second rounding error is smaller than the first, but that the third rounding error is larger than the first. In that case, one deletes the value of the first rounding error and retains the value of the third rounding error, and marks that this error occurred at the third position in the block. This process continues until, at the end of the block, the position of the largest rounding error is determined.

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Since Wagner decoding is not a recursive algorithm as Viterbi decoding is, it can be executed at very high speeds. In the Appendix , it is shown that, after Wagner decoding and is upperfor large SNRs, the probability of a block error bounded by (27) where is the average energy per bit that is related to the distance between two adjacent levels (see the Appendix), and is the power spectral density of the AWGN. A block where error occurs if two or more symbols in a block are erroneous (see the Appendix). Proakis [5, Sec. 9.3.2] showed that, for -ary duobinary PR modulation and symbol-by-symbol detection, the average probability of a symbol error is upper bounded as (28) where

is the average energy per bit, and where (29)

M

Fig. 6. Performance of MR modulation for = 2 and for various values of the block length . The theoretical and simulated performances of PR modulation for = 2 and for symbol-by-symbol detection are also indicated.

M

K

In order to compare the bounds given by (27) and (28), we use as a bound for the function (30) However, this bound is only tight for large values of . Therefore, this comparison of PR and MR is only relevant for high SNRs. We thus obtain for (28) (31) For large SNRs, the power efficiency of a modulation scheme is determined by the argument of the exponential function. Thus, the energy per bit required by PR with symbol-by-symbol detection to achieve the same small value for the error probability as MR is times greater, or 1.76 dB. The exponent in the BER expression for MR is the same as that of the block-error probability. Only the coefficient in front of the exponent is different. This coefficient can be made smaller by using an additional error correction after the inverse matrix operation and after recovering the bits from the PAM vector. For , the error-correction operation is as follows. Correct the symbols 3 and 2, after the inverse matrix processing by the and by the symbol symbol 1, and correct the symbols . Finally, correct the symbol 0 by the symbol 1. Ciacci made computer simulations for MR modulation with Wagner decoding and the additional error correction for and , each for block lengths , , , and . The BER curves he obtained for these cases as are given in Figs. 6 and 7. Here, dea function of notes the energy per bit in each case. His results were obtained

M

Fig. 7. Performance of MR modulation for = 4 and for various values of the block length . The theoretical upper bound for the BER of PR modulation for = 4 is also indicated.

M

K

by using statistically independent and symbols for generating PAM symbols. He also simulated PR modulation for . The result of these simulations, as well as the theoretical upper bound on the BER, are also shown in Fig. 6. From Figs. 6 and 7, one can conclude the following.

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1) For and with and a BER of , the gain of MR with Wagner decoding is 1.5 and 1.3 dB, respectively, compared with PR and symbol-by-symbol detection. , the BER curves for the various 2) For increasing converge. This is in agreement with (27) where, for , the difference between the block-error increasing probabilities, for any two values of but for the same , will go to zero. and , the difference between the power 3) For efficiencies, at a BER of , for both MR (same ) and PR is about 4 dB. This is in accordance with (27) and (28). is a good compromise between power effiA value of ciency and bandwidth efficiency. Finally, we compare the power efficiencies of MR modulation with Wagner decoding and -ary PAM with raised cosine pulses. In [1, Sec. 4.4], the probability of a symbol (sample) error for -ary PAM with raised cosine pulses is given by

Compared with -ary PAM, MR modulation for gives a loss of 0.65 dB. a BER of

and

APPENDIX Before deriving an upper bound for the block-error probability after Wagner decoding, we first determine the correlation between the noise components of the noise vector . The power spectral density (PSD) function of the noise at the output of the matched filter in the receiver is

where is given by (2), and where is the PSD of the AWGN. The autocorrelation function of the output noise is

(34) Thus, the noise variance is

(32) where is the average energy per bit. With Gray coding, the times the symbol-error BER for -ary PAM is rate given by (32). Multiplying (32) by the factor and solving this equation for a BER of and for the cases and yields an of, respectively, 10.6 and 14.41 dB. Comparison of these two results with the correvalues for MR, at a BER of , shows a loss sponding and , and in power efficiency of MR for of approximately 0.65 dB. Using the bound (30) in (32) and comparing this result with the bound (27) yields that the loss in asymptotic power efficiency of MR modulation and Wagner decoding compared with -ary PAM and raised cosine pulses is dB

(33)

(35) and the normalized autocorrelation coefficients are where Note that the correlation is largest for two adjacent noise components . The probability of the occurrence of a zero symbol is larger than for any other symbol. Therefore, the probability of a block can be upper bounded by assuming that a block conerror sisting of only zero symbols is transmitted. After Wagner decoding, there remain two types of errors. The first type of error is a double or triple sample error. Of this type, the dominant term for high SNRs is a double sample (symbol) ) error. Since two adjacent samples (noise correlation are more likely to be in error than two nonadjacent samples, the of a double or triple error is approximated by probability (36)

VI. CONCLUSION It has been shown that MR modulation is bandwidth- and power-efficient, and is simple to implement. It was also shown that MR modulation has a number of advantages, compared with PR modulation. Thus, with MR modulation, loss of synchronization or gain control, as can occur with PR modulation, cannot happen in the receiver. Precoding, as needed in PR modulation, is not required for MR modulation, since error propagation is impossible for MR modulation. Furthermore, compared with PR modulation and symbol-by-symbol detection, , MR modulation with simple Wagner decoding gives for , and a gain of 1.5 and 1.3 dB, respectively, at a BER of . Finally, compared with -ary PAM and raised-cosine pulses, MR modulation, for , gives a greater bandwidth efficiency for rolloff factors . However, the bandwidth efficiency of MR modulation de, compared with PR modulation. creases by a factor

where

is the bivariate normal probability density function with a nor, is the malized correlation coefficient number of adjacent sample combinations in a block of length , and is the distance between two levels. The second type of error occurs when Wagner decoding incorrectly determines the position where a sample has to be corrected. This event occurs when the roundoff error, at the position where a sample error did indeed happen, is smaller than roundoff errors. In that case, the value one of the other of the sample, which is different from the erroneous sample, is changed, resulting in a second error. The probability of such an event is called . For this type of decoding error, the dominant

PEEK: MULTIRATE MODULATION: A BANDWIDTH- AND POWER-EFFICIENT MODULATION SCHEME

term for high SNRs is that only one of the roundoff errors is larger than that which occurred at the position where the error did indeed happen. Furthermore, if after rounding only one sample (symbol) has an error, it is likely that, because two adja), the cent noise samples have the largest correlation ( erroneously determined position of this error is adjacent to the can, for high SNRs, be approximated true position. Hence, by

Since

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Substituting (41) and (35) into (39) gives as an upper bound for the block-error probability after Wagner decoding [(27)]. ACKNOWLEDGMENT The author would like to thank M. Ciacci for making the computer simulations and for the discussions with him. The author would also like to thank S. Baggen, J. Bergmans, L. Tolhuizen, and S. Weinstein for reading the manuscript and providing their valuable comments. REFERENCES

, we have

(37) Using (29) and (30), we find after tedious but straightforward computation that (38) into (37) and (38), we have as an After substituting upper bound for the block-error probability (39) Proakis [5, Sec. 9.3.2] derived for PR modulation a relation belevels and the average transtween the distance of the mitted signal power. This relation also holds for MR modulaequidistant levels, tion, with a distance between the and is given by (40) is the average transmitted signal power for MR modwhere ulation. , where is the average energy Since per bit, (40) can also be written as (41)

[1] R. D. Gitlin, J. F. Hayes, and S. B. Weinstein, Data Communications Principles. New York: Plenum, 1992. [2] J. B. H. Peek, “Multirate block codes,” in Proc. 20th Symp. Inf. Theory Benelux, Haasrode, Belgium, 1999, pp. 205–213. [3] J. B. H. Peek and L. F. P. M. Lakeman, “Generating block line codes with spectrum nulls using multirate digital filters,” in Proc. IEEE Int. Conf. Commun., Denver, CO, pp. 1098–1102. [4] P. P. Vaidyanathan, “Multirate digital filters, filter bank, polyphase networks and applications: A tutorial,” Proc. IEEE, vol. 78, no. 1, pp. 56–93, Jan. 1990. [5] J. G. Proakis, Digital Communications, 4th ed. New York: McGrawHill, 2001. [6] R. A. Silverman and M. Balser, “Coding for constant data rate systems,” IRE Trans. Inf. Theory, vol. 4, pp. 50–63, Sep. 1954. [7] S. G. Wilson, Digital Modulation and Coding. Englewood Cliffs, NJ: Prentice-Hall, 1996, p. 518. [8] A. W. M. van den Enden, Efficiency in Multirate and Complex Digital Signal Processing. New York: Delta, 2001. [9] S. Mirabbasi and K. Martin, “Classical and modern receiver architectures,” IEEE Commun. Mag., no. 11, pp. 132–139, Nov. 2000.

Hans B. Peek (SM’75–F’87) received the degree in electrical engineering from the Technical University of Delft, Delft, The Netherlands, in 1959, and the Ph.D. degree from the Technical University of Eindhoven, Eindhoven, The Netherlands, in 1967. In 1961, he joined the Philips Research Laboratories, Eindhoven, where he was involved in the areas of digital signal processing and communications, and where he was a Department Head from 1969 to 1987. From 1987 to 1991, he was a Chief Scientist with the same laboratories and, from 1987 to 2001, a part-time Professor with Nijmegen University, Nijmegen, The Netherlands. Dr. Peek was the recipient of various awards, including the Gold Medal from the Dutch Veder Foundation in 1967, the IEEE Communications Magazine Best Paper Award in 1985, and the IEEE Millenium Medal 2000. He is a member of Eta Kappa Nu.

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A New Performance Bound for PAM-Based CPM Detectors Erik Perrins, Member, IEEE, and Michael Rice, Senior Member, IEEE

Abstract—It is well understood that the pulse amplitude modulation (PAM) representation of continuous phase modulation (CPM) can lead to reduced-complexity detectors with near optimum performance. It has recently been shown that the PAM representation also extends to CPM schemes with multiple modulation indexes (multi- CPM). In this paper, we present a detector for multiCPM which is based on the PAM representation. We also give an exact expression for the pairwise error probability for the entire class of PAM-based CPM detectors (single- and multi- , optimal, and reduced-complexity) over the additive white Gaussian noise (AWGN) channel and show that this bound is tighter than the previously published bound for approximate PAM-based detectors. In arriving at this expression, we show that PAM-based detectors for CPM are a special case of the broad class of mismatched CPM detectors. We also show that the metrics for PAM-based detectors accumulate distance in a different manner than metrics for other CPM detectors. These distance properties are especially useful in applications with greatly reduced trellis sizes. We give thorough examples of the analysis for different single- and multi- signaling schemes. We also apply the new bound in comparing the performance of PAM-based detectors with other reduced-complexity detectors for CPM. Index Terms—Continuous phase modulation (CPM), mismatched detector, pairwise error probability, pulse amplitude modulation (PAM), reduced-complexity detector, union bound.

I. INTRODUCTION ONTINUOUS phase modulation (CPM) is advantageous for its efficient use of power and bandwidth. It also has a constant signal envelope, which is essential in applications using nonlinear amplifiers. However, the optimal maximumlikelihood sequence detection (MLSD) scheme, which is implemented via the Viterbi algorithm (VA), often suffers from high complexity in terms of the required number of correlators (matched filters or MFs) and trellis states. A number of techniques have been proposed to reduce the number of MFs, e.g., [1]–[4], and similarly to reduce the number of trellis states, e.g., [1], [5]–[7]. Of particular interest here is the PAM representation of CPM, which was introduced by Laurent in 1986 [8]. In his paper, Laurent showed that any binary single- CPM scheme can be ex-

C

Paper approved by C. Tellambura, the Editor for Modulation and Signal Design of the IEEE Communications Society. Manuscript received October 12, 2004; revised March 8, 2005. This paper was presented in part at the IEEE Military Communications Conference, Monterey, CA, November 2004. E. Perrins was with the Department of Electrical and Computer Engineering, Brigham Young University, Provo, UT 84602 USA. He is now with the Department of Electrical Engineering and Computer Science, University of Kansas, Lawrence, KS 66045 USA (e-mail: [email protected]). M. Rice is with the Department of Electrical and Computer Engineering, Brigham Young University, Provo, UT 84602 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TCOMM.2005.857133

actly represented by a superposition of PAM waveforms. He also showed that the binary single- CPM signal is often well approximated by a reduced number of PAM pulses or even by the main pulse alone. Kaleh followed in 1989 [2] by deriving the MLSD structure for PAM-based CPM detectors. He also showed that suboptimal PAM-based detectors require an appreciably reduced number of MFs (since they are based on a limited number of pulses) and that they simultaneously achieve a reduction in the number of trellis states. Kaleh also provided a simple performance bound, which is of little use beyond the binary single- CPM schemes considered in his paper. The PAM representation of CPM has since been extended to -ary signaling by Mengali and Morelli [9] and for the special case of CPM schemes with integer modulation index by Huang and Li [10]. It has also been confirmed in these cases that reduced-complexity PAM-based detectors achieve a simultaneous reduction in the number of MFs and trellis states with manageable performance tradeoffs [10], [11]. These performance assessments have been made using computer simulations. There is a need for analysis and explanation of the performance of PAM-based detectors in general, since the computer simulations do not reveal the reasons for the strong performance of PAM-based detectors. The PAM representation has also been extended to -ary multi- CPM very recently in [12]. In this paper, we take this recent extension of the PAM-based CPM model and apply it to the problem of detecting -ary multi- CPM signals. We generalize Kaleh’s results from [2] and arrive at an optimal MLSD structure for multi- CPM that is based on the PAM representation. We confirm that the MF and trellis reduction properties of suboptimal PAM-based detectors also hold for the multi- case. One important facet of PAM-based detectors for CPM which is missing from [2], [11] is an adequate performance analysis of these detectors. We study the problem of performance in this paper and derive the exact expression for the pairwise error probability for PAM-based CPM detectors in AWGN. This pairwise error probability, though given in multi- terms, is also applicable to the entire class of PAM-based detectors for CPM in [2], [11] (both optimal and approximate). In carrying out this analysis, we show that detectors based on the PAM approximation can be viewed as mismatched CPM detectors. Schemes of this type, where the internal signal model of the detector is mismatched (different) with respect to the signal produced by the transmitter, were first analyzed in [1]. The class of mismatched detectors is quite broad and includes the schemes in e.g., [3], [4]. As such, the analysis presented here is a special case of that given in [1]. In order for this viewpoint to yield correct results, however, proper consideration is given to the precise

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PERRINS AND RICE: NEW PERFORMANCE BOUND FOR PAM-BASED CPM DETECTORS

manner in which metrics are computed in PAM-based detectors. We demonstrate that, like other mismatched detectors, the pairwise error probability is a function of specific pairs of data sequences, in contrast with the optimal detector where only the difference between pairs of data sequences is needed. This in turn means that the error performance for reduced-complexity PAM detectors is not dominated by a single distance parameter, i.e., the minimum distance. We apply the new bound in a performance comparison between PAM-based detectors and the reduced complexity detectors given in [1] and [6]. In this comparison, we demonstrate explicitly how metrics within PAM-based detectors accumulate distance in a different manner than metrics in other CPM detectors. For instance, if two CPM signals are different from each other for a brief interval, the optimal detector observes the distance between these signals over this entire event. By contrast, a detector with a reduced trellis often observes the distance between these signals over some fraction of this interval (thus forfeiting some portion of the optimal distance). The reduced complexity scheme in [1] observes the (mismatched) distance during the center of the interval, splitting the omitted portion evenly between the beginning and the ending tails. The decision feedback scheme in [6] observes the distance from the beginning of the event to a certain point, discarding whatever remaining distance there is on the ending tail of the event. On the other hand, PAM-based detectors observe the (mismatched) distance from the beginning of the event to a certain point, after which they continue to observe the distance to a lesser degree up to the completion of the event. This behavior allows PAM-based detectors to be used with relatively small performance losses in spite of aggressively reduced trellis sizes. This study gives a theoretical basis to confirm the simulation results which have been reported for single- schemes in e.g., [2], [11]. In the next section, we review the traditional and PAM-based signal models for multi- CPM and give the structure of PAMbased detectors. In Section III, we analyze these detectors and obtain a new performance bound. In Section IV, we use the bound to characterize the performance of several single- and multi- schemes. We also apply the bound in a performance comparison in Section V and give conclusions in Section VI. II. MULTI- CPM SIGNAL MODEL A. Traditional Model The complex-baseband multi- CPM signal is given by

(1) (2) where is the symbol energy, is the symbol duration, is the set of modulation indexes, are the information , and symbols in the -ary alphabet is the phase pulse. In this paper, the underlined subscript . notation in (2) is defined as modulo- , i.e.,

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We assume the modulation indexes are rational numbers of the form [13] (3) We determine by expressing all of the modulation indexes as a fraction and taking as the value of the smallest common denominator. is the integral of the frequency pulse The phase pulse . The frequency pulse is zero outside the time interval and is scaled such that . In light of these constraints on and , and considering the -ary digits , (2) can be written as (4) where (5) (6) and is a data-independent phase is a function of the symbols being tilt [13]. The term modulated by the phase pulse. The substitution of is confined to the phase state , which takes on only distinct values. Therefore, the phase signal in (4) is described by a trellis containing states, with branches -tuple at each state. Each branch is defined by the (7)

B. PAM Model for Multi- CPM In [12] it is shown that the right-hand side of (1) can be exactly written as (8) , , and is the integer that where satisfies the conditions . The signal is the scaled by pseudo-symbols . superposition of pulses This is a generalization to the multi- case of the results in [8], [9] which apply to binary and -ary single- CPM, respectively. The details of the construction of and are found in [12]. For this paper, it is important to know that the vary in amplitude and duration, with the longest pulses pulses also having the largest amplitude. This is shown in Fig. 1 for a quaternary CPM scheme with a raised-cosine frequency (3RC) and . Each pulse pulse of duration symbol times, where is an integer in has a duration of the set . We can group the pulses into sets with common duration, i.e., (9)

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C. PAM-Based Detectors for Multi- CPM The optimal detector for the equivalent PAM representation of single- CPM was derived by Kaleh [2]. We now show the extension needed to accommodate the multi- case. The complex-baseband received signal model is (11) where is a complex valued AWGN process with one-sided . Due to the AWGN assumption, the power spectral density detector selects as its output the information sequence which minimizes the Euclidian distance (12) is constant envelope, minimizing (12) is equivalent Since to maximizing the correlation Fig. 1. Signal pulses g (t) for M = 4, 3RC, with h = f4=16; 5=16g. The 48 pulses for n-even are on top and the 48 for n-odd are on bottom. In each case there is one pulse of duration 4T , two of duration 3T , nine of duration 2T , and 36 of duration T (too small to view in the figure).

(13) which can be computed in a trellis using the recursive metric (14)

Another important characteristic of the PAM representation is that not all the pseudo-symbols require the full -tuple in (7) to describe them. We obtain a reduced trellis simply by discarding those signal terms that require the most states. In fact, it can be shown that the number of states required is , for by the pseudo-symbols in each set [14]. In order for this rule to be complete, we must also account , i.e., the set , which we do by for the case where grouping with , as was also done in [2] and [9]. In terms and ) require a of Fig. 1, the first three pulses (those in . If the nine lengthtrellis with only pulses in are included then a trellis of states is required. An additional -fold increase in states is required if the 36 length- pulses are included. By simply discarding the smaller pulses and their costly pseudo-symbols, we see that the PAM representation simultaneously reduces the number of MFs and the number of trellis states. We stress that the principle behind the trellis reduction is a function of the pseudo-symbols and not the pulses. It just so happens that the structure is conveniently summarized in terms of the pulse duration. The PAM model we will consider from this point on is (10) is an approximation of the exact CPM signal in (8). The summation in (10) is over an arbitrary subset , of signal terms , which is a proper subset of and is usually chosen in terms of . The number of ele, and the shortest pulse duration is ments in the set is . The pulses and pseudo-symbols are related to the original pulses and pseudo-symbols by some averaging scheme. One such example is the minimum mean-squared error approximation in [8], [9], [12]. Another pulses in Fig. 1, which are example is averaging the lengthall very similar, to produce a single pulse [11]. where

where corresponds to the -tuple associated with the th branch in the trellis and is the cumulative metric associated with at index . We refer to (14) as the traditional detector metric and its performance is well understood [15]. To arrive at the PAM detector metric, we insert (10) into (12) and with some simple manipulations arrive at the recursion (15) where (16) is the output of a filter matched to and

sampled at

(17) is a bias term that is a consequence of the approximation (10) no longer being constant envelope. The hypothesis along the th branch, , is associated with a set of branch pseudo-sym. A slightly different version of the branch pseudo-symbols is used (17). These are modified by decision feedbols back, as explained shortly. We pause to discuss an important special case of (15) where the exact PAM representation in (8) is used. In other words, no , , approximations are made and and . Here the detector is based on a constant envelope signal. Therefore, the bias term is no longer necessary, since is a constant for all values of . The correlation in (13) is computed exactly, the difference is that the computation is made using PAM pulses and pseudo-symbols rather than the traditional data-dependent CPM matched filters in (14). This special case represents an alternate form of MLSD and has equivalent performance to that of (14).

PERRINS AND RICE: NEW PERFORMANCE BOUND FOR PAM-BASED CPM DETECTORS

The trellis for the PAM-based detector has where

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states, (18)

We note that this reduced trellis is identical to those obtained from [1] and [6]1; the former reduces the trellis by basing the detector on a simpler (mismatched) CPM scheme, while the latter reduces the trellis with the use of decision feedback. In the present case, the trellis is reduced as a natural consequence of the PAM approximation. Each branch in the trellis has an -tuple (19) pseudo-symassociated with it, and a corresponding set of bols . Since the interconnections (i.e., branches) in the fashion, we have added multi- trellis vary in a moduloa moduloindex to the branch pseudo-symbols . Practically speaking, the limits of integration in (16) mean are computed with a delay of the sampled MF outputs symbol intervals, since . This delay is directly related to the different manner in which the PAM metrics accumulate distance, as we shall see in the next section. The bias term provides compensation for the signal energy variations present in the approximation. It presents a minor difficulty since it remains a function of the original -tuple in (7). To cope with this, we use decision feedback where each state in the reduced trellis maintains a record of recent merge decisions which are used to fill out the original -tuple. This minor use of decision feedback results in no performance loss (as we shall see in Section IV), which is consistent with other cases where decision feedback has been used, e.g., [6]. The record of recent decisions along the -th branch is denoted by and the in (17) forms an -tuple. The bias concatenation of term is not present in the detector configuration in [2], and can be ignored if (10) is very close to being constant envelope (as was the case in [2]); however, for more coarse PAM approximations, such as the multi- scheme discussed in Section IV, the removal results in a surprisingly large performance penalty. of The structure of the detector is shown in Fig. 2. The received is fed into the bank of MFs. The sampled filter signal are the inputs to the VA, which computes branch outputs metrics, determines the surviving path at each merging node, and outputs a decision. The figure also shows an expanded view of the th filter in the bank. Each filter actually consists of a filters whose sampled outputs are cyclically selected set of using a commutator and then delayed by the amount needed to have an overall filter delay of symbol times.

h

Fig. 2. PAM-based detector structure for multi- CPM with expanded view of matched filter and delay. modulo-

N

where

(21) While the limits of integration in (20) are infinite, the integrands are identical to each other (and thus cancel one another) except for the finite interval when the trellis paths taken by the two data sequences are different. If the data are different over a span of symbol times, then the two paths can merge together in the trellis after (22) symbol times. We examine the branch metric in (15) over the where is arbitrary and interval . We concentrate on the first term in (15), which is a function of the received signal and ignore for the moment. The PAM-based metric is (23) where the signal

is defined as

(24)

III. PERFORMANCE ANALYSIS A. Pairwise Error Probability for PAM-Based Detectors , which is the probability of We seek the quantity the detector outputting the sequence given is transmitted. From (12), this pairwise error probability is (20) 1We emphasize that the expressions for the branch metrics for each of these three detectors are each different from one another. The three detectors merely operate on an identical trellis structure.

for . This signal is key to understanding the performance of PAM-based detectors for CPM. Conceptually speaking, it is obtained by summing the filters with impulse response , outputs of a set of . These filters are idle (i.e., their output is zero) for . During the symbol times where the two trellis paths are different, these filters are fed with the pseudo-symbols that correspond to the trellis path. After the paths merge, the filters are starved of input and their collective output returns

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to zero for . The cumbersome indexes on the inner sum in (24) ensure that pseudo-symbols are not clocked into the pulses for values of outside the range . We now include in the analysis. For convenience, we refer as and as . We expand the terms to in (20) in the same manner in which (15) was obtained from (12). By ,this we mean that the expansion includes a mix of and terms, just as there are two terms found in (15). The pairwise error probability becomes

times later. As such, distance not only continues to accumulate to a lesser degree, but this extra distance is applied retroactively at the time of the merge. As mentioned earlier, this attribute of the PAM-based CPM detector is distinct from those in [1] and [6]. We now make a final comment on (29). Although this is given as a squared quantity (for conceptual and historical reasons, e.g., [1]), there is nothing to prevent this quantity from assuming a negative value, especially as the PAM approximation (mismatch) becomes more coarse. In light of this, the final distance is correctly used in unsquared form in the pairmeasure wise error probability (26) to preserve the sign. The case where is negative corresponds to the detector outputting errors (i.e., an error floor). for arbitrarily large values of

(25) which simplifies to

B. Probability of Bit Error (26)

where

is the energy per bit (

) and (27) (28) (29) (30)

We stress that (29) and (30) contain two forms of the approximate PAM signal. The received signal and noise are correlated from (24). The branch bias with the modified signal and are from the traditional form of the PAM apterms and proximation in (10). Some discussion of (28)–(30) is in order. While these expressions were derived from basic principles, they match the general form of the distance measure for mismatched CPM detectors in [1]. Here the transmitter/detector mismatch is due to the PAM approximation (i.e., discarding the less significant pulses and averaging the pulses). Other examples of mismatched signals are found in [1] and [4]. In light of this, the above analysis is a special case of the results in [1]. However, there are certain nuances regarding the PAM-based detector which must be given proper attention in order to obtain the correct result; namely, the mismatched signal must take the nonobvious form in (24) in order to to compute the distance measure correctly instead of its original form in (10). It can be shown that it is ), with its shortactually the reduced trellis itself (i.e., ened merger duration in (22), which motivates the need for the modified signal in (24). It is also (24) which shows the unique manner by which distance is accumulated in the PAM-based detector. While the competing paths in the trellis are different for symbol intervals, the true CPM signals in (1) differ only for symbol intervals (a difference of ). The limits on the integral in (16) show that, at index , the distance measure has contributions from the received signal as far in ad. Therefore, while (24) begins to decay after vance as index symbol times, it does not return to zero until symbol

Some additional steps are needed to convert the pairwise error probability into a probability of bit error. It is well known that, as grows large in the AWGN environment, a pairwise error probability term (26) corresponding to the minimum-distance becomes dominant [15]. This results from the nature of (27). large enough for The only question is at what point is this approximation to be accurate?2 For the optimal detector, there are many pairs of data sethat have the minimum distance. Therefore, quences the pairwise error probability associated with the minimum disand tends to tance is independent of specific pairs of of practical interest. In the dominate the union bound for case of the PAM-based detector (and other mismatched detecpair has its own distance. Therefore, tors), each specific the pairwise error probability associated with the minimum disof practical interest, tance is not typically dominant for since there are many near-minimum-distance terms also present in the union bound. In the case of the PAM-based detector, the probability of bit error is given by a sum of near-minimum-distance terms, as shown below. Examples illustrating this point are given in Section IV. Based on the above arguments, the probability of bit error for the optimal detector is well approximated by (31) We define the terms in (31) below. We start with the concept pairs with a common of a merger, which is a set of difference such that (32) where is the th coordinate of , and and are from the in (3). Equation (32) simply states that if modulation index two data sequences deviate from each other and their signals are to merge together at some later point, it must be that the summation of the difference in their phase is zero (modulo- ) when 2In this paper, we study the answer to this question in the context of uncoded CPM. The answer is different when considering the case of coded CPM, e.g., [16].

PERRINS AND RICE: NEW PERFORMANCE BOUND FOR PAM-BASED CPM DETECTORS

properly scaled by the modulation indexes. In general, the difference sequence has nonzero coordinates that span a limited number of symbol times ( ) and we arbitrarily assign the first of these coordinates to be . We can easily count the number with a common difference of pairs of data sequences as of

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TABLE I MINIMUM-DISTANCE MERGER PARAMETERS FOR THE EXAMPLES

(33) Identifying the difference sequence corresponding to the minimum-distance is a straightforward task for the optimal multi- CPM detector [15]. It is essentially to find the sequence (34) times For multi- CPM, the search in (34) must be repeated to allow each modulation index to coincide with . Also, since CPM is constant envelope, this distance does not vary across the individual pairs in this merge. in (31) to be the difference in bits between We define a pair (i.e., the bit error weight), which is a function of the mapping from bits to symbols (typically a Gray code), is the number of bits transmitted per use of the and channel. The other terms in the denominator of (31) constitute , which is the probability that a given -ary length- data sequence is transmitted with a particular alignment to the modulation indexes (we assume the uniform distribution). For the PAM case, the minimum-distance merger itself is as the optimal detector, although there is often the same nothing to prevent another merger from becoming dominant as the PAM approximation (mismatch) becomes more coarse. Therefore, the search over all mergers in (34) must be performed explicitly for the PAM distance measure in (28). Another difference from the optimal case, as was mentioned earlier, is that the distance in (28) is a function of specific sequence pairs and is different (in general) for each of the pairs in the merger. [This variation in distance is a general attribute of all mismatched CPM detectors; in the PAM case, it is caused by the missing or averaged pseudo-symbols in (10) that are still present in the exact signal in (8).] Modifying (31) accordingly produces (35) in (31) is replaced by a summaThe scale factor of tion over all sequence pairs for which . There will that produces ; howbe a combination of ever, since this is only one of (potentially) many terms in the sum whose distances are relatively close to each other, it genis outside the erally does not become dominant until range of practical interest. Therefore, PAM-based CPM detectors, generally speaking, are not well characterized by just one minimum-distance parameter as in (31); instead, the summation in (35) must be used.

Fig. 3. Performance of M = 4, 3RC, with h = f4=16; 5=16g. For the PAM detector, the P curve shows strong agreement with the simulated data points. In this instance, the performance bound from Kaleh [2] essentially fails.

IV. EXAMPLES A. Quaternary 3RC With The first example we consider is the multi- scheme , . This is the 3RC (raised cosine), and Advanced Range Telemetry (ARTM) CPM waveform for aeronautical telemetry defined in IRIG 106–04 [17], [18]. The MLSD trellis has 256 states in this case, which is a considerable number. The performance of the optimal detector is given by (31) using the parameters in the first entry in Table I and is plotted in Fig. 3. The exact PAM representation requires the large set of 48 2 pulses shown in Fig. 1. We obtain a much smaller set of three pulses by taking the following steps: 1) we apply the minimum mean-squared error approximation in [12] to obtain one aver) aged set of 12 pulses (this results in an effective value of and 2) we further reduce the number of MFs by combining the two length- pulses to form one averaged pulse and repeat this for the nine lengthpulses. This gives the final filter bank of three MFs (these steps are explained in detail in [14]). With , the 64-state trellis has branches defined by (36) pairs within We must now compute (28) for all of the each merger . The search in (34) finds the same minimum-distance merger for the PAM case as was found for the MLSD case, namely, the first entry in Table I. Technically speaking, , the distance metric in (29) is a function of with the two symbols preceding the merge and the three symbols following the merge; however, the dependence is strongest only on the symbol immediately preceding and immediately following the merge. The result is that we must consider all of the

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pairs with , where we have explicwhich may not itly padded a zero coordinate on each end of be ignored when counting the number of terms in (33) (there are 1152 sequence pairs of this type). The squared distances range from 1.03 to 1.52 over these sequence pairs, which is a distance sequences. By conloss of 1.02 dB for certain transmitted trast, the bounding technique in [2] essentially fails by upper for the bounding the distance loss at 27.24 dB. Fig. 3 shows PAM-based detector where we see that the bound in (35) becomes indistinguishable from data points obtained by computer simulation as grows large. We also point out that Fig. 3 shows an actual loss of only , which is 0.17 dB for the PAM-based detector at much less than the 1.02 dB loss of the minimum-distance term in (35); this underscores the point that the actual minimum distance for PAM-based detectors (or all mismatched CPM detectors, for that matter) typically does not have enough weight to well for of practical interest. approximate

Fig. 4. Performance of 4-ary 2RC with h = 1=4. The four-state PAM detector has a negligible loss with respect to MLSD. The Kaleh bound [2] is also shown for reference (dashed line).

B. Quaternary 2RC With The second scheme we consider is , 2RC with , where the optimal trellis has 16 states. The merger parameters for the MLSD scheme are given in the second entry in Table I. We consider the PAM configuration for this scheme given in , which has only two pulses and a four-state [11] with trellis with branches defined by (37) Since , the dependence of on the symbols preceding and following the merger is small and can be ignored in pairs of interest this case. We evaluate (28) for the 18 and obtain squared distances ranging from 1.16 to 1.51, which is a maximum distance loss of 0.59 dB (though the loss is much ). By contrast, the distance bounding techsmaller at nique in [2] yields a less useful upper bound of 1.38-dB loss. Fig. 4 shows for this PAM-based detector, which again shows grows strong agreement with computer simulations as large. C. Binary GMSK With

Fig. 5. Performance of binary GMSK with L = 4 and BT = 1=4. The P curves show strong agreement with the simulated data points for the two PAM-based detectors (four- and two-states). The bound from Kaleh [2] is also shown for both cases (dashed lines) but it is not tight for the two-state case.

MF. This gives branches defined by

and produces a two-state trellis with (39)

and

The last example we consider is binary Gaussian minimumand , which was the shift keying (GMSK) with central example in [2]. The optimal trellis has 16 states, and the last entry in Table I gives the merger parameters for the optimal detector. The exact PAM representation for this scheme contains eight down to . We select approxipulses with durations of mate PAM-based detectors with two different configurations. and , as The first uses the two most significant pulses, and produces a four-state trellis with MFs. This makes branches given by

As with the first example, we must pad a zero coordinate on each end of the sequences when performing the search in (34). For the four-state detector, the squared distances range from 1.66 to 1.72. The analysis in [2] upper bounds this distance loss at 0.24 dB. For the second detector configuration (two states), the squared distances range from 1.29 to 2.07 and the distance curves bound from [2] is less helpful at 3.18-dB loss. The generated by (35) for the two PAM configurations are shown in Fig. 5 along with data from computer simulations.

(38)

Given that the reduced-complexity detectors in [1], [6], and Section II each gather distance in their own unique manner while using the exact same trellis, an interesting question to ask is: which method yields the best performance for a given trellis

This configuration was considered at length in [2]. The second , as an configuration uses only the most significant pulse,

V. APPLICATIONS

PERRINS AND RICE: NEW PERFORMANCE BOUND FOR PAM-BASED CPM DETECTORS

TABLE II PERFORMANCE OF REDUCED-COMPLEXITY DETECTORS FOR M = 4, 3RC, h

= f4=16; 5=16g

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on a 1RC approximation, and the detector from [6] uses deciand that sion feedback to approximate the symbols are found in the original 4-tuple in (7). The PAM-based detector uses the 3 2 pulses from the minimum mean-squared error approximation in [12]. The performance of these three detectors is shown in the second grouping in Table II. Here the decision feedback approach from [6] and the PAM-based detector perform close to one another, while the detector from [1] suffers a large loss due to the coarseness of the 1RC approximation. For the third reduced trellis example, we use decision feedto back to reduce the number of phase states from , cf. e.g., [6]. The branches in this 16-state trellis are defined by (41)

complexity? It comes as no surprise that the answer depends on the CPM scheme and the trellis in question. In this section we demonstrate the typical characteristics of each detec, 3RC, tion scheme, using the multi- scheme as a case study. We will show that the PAM-based detector performs well in cases where the trellis size is aggressively reduced. This is due primarily to the different manner in which distance accumulates in PAM-based detectors. In Table II we consider three reduced trellis configurations. There are three groupings in the table, one for each configuration. The first entry in the table is for the MLSD scheme which serves as the reference detector for what follows. The first reduced trellis is the 64-state configuration with branches defined by (36). This is a relatively minor state reduction of a factor of 4. The first grouping in Table II shows that the three reduced complexity detectors perform relatively close to one another, with the PAM-based detector being slightly the worst of the three both in terms of minimum distance and loss . In all of these detectors, the minimum-distance at merger is the one given for this CPM scheme in Table I. The detector from [1] is based on a 2RC approximation. Its minor losses are the result of this transmitter/receiver misis shortened from seven match and from the fact that to six symbol times. As mentioned before, the one symbol time worth of distance loss is divided equally between the beginning and ending tail of the error event. The detector from [6] uses decision feedback to approximate the symbol which is found in the original 4-tuple in (7). Its minor loss is . In this case, it forfeits entirely due to the shortening of the distance increment that might have come in the seventh symbol time of the error event, namely, the small increment . The PAM-based detector is the same as given in Section IV-A, which uses three pulses to approximate the original set of 48 2. Its third-place ranking is balanced by the fact that its filtering requirements are much less than the other two detectors. We can pursue more aggressive trellis approximations. The second reduced trellis we consider has branches defined by (40) which is a trellis of 16 states (a state reduction by a factor of 16). To achieve this reduction, the detector from [1] is based

where the modulo-4 operation in (41) is understood to apply to the modulo- operation in (6). We use the same three detector approximations as in the 64-state case, namely the 2RC misdecision feedback and the three-pulse PAM match, the trellis is an overall reapproximation. The benefit of the duction in states by a factor of 16, which gives the same number of states as the previous trellis example, although it is an entirely different trellis. The downside of the approximation is that this new trellis definition permits mergers according to (32) using instead of . The most potentially catastrophic of these is , where symbol times. The last grouping in Table II shows the performance of the three detectors using this trellis approximation. In this example merger is the detector from [6] suffers greatly because the dominant, where squared distance is only 0.31 after two symbol times. This is unfortunate, because after the second symbol time the distance grows rather quickly (and indefinitely) with a linear slope, since the data sequences represented by produce CPM signals which remain different forever. On the other hand, the mismatched detector from [1] benefits greatly from the extra merger. The squared half symbol time it gets to observe the distances for this merger range from 1.32 to 1.51, with a somewhat uniform distribution. Therefore, the minimum distance for this detector in Table II is unchanged from the 64-state example. However, these near-minimum-distance mergers have a high probability of transmission and produce a 1.2-dB loss for this detector at . The PAM-based detector also benefits from its different observation of the merger. Here the squared distances range from 1.20 to 5.29, which is a large spread with most of the terms at the high end. The minimum distance for the PAM-based detector is also unchanged from the 64-state exis due to the ample. Its smaller loss of 0.9 dB at terms. larger distance of most of the Although we have focused on only one CPM scheme in this study, the results given are typical for these detectors (see also [1], [2], [6], and [11]). As is decreased, the detector from [1] suffers the most due to the coarseness of its mismatch with the transmitter. As is decreased, the decision feedback detector from [6] suffers the most due to the rogue mergers permitted by the reduced trellis. In either case, PAM-based detectors sustain relatively manageable losses.

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VI. CONCLUSION We have presented the form of the optimal and approximate detectors for multi- CPM which are based on the recently extended PAM representation of multi- CPM. We have also found an exact expression for the pairwise error probability for all PAM-based CPM detectors. This expression was used to evaluate the performance of detectors for single- and multiCPM, where there was strong agreement with computer simulations and where a previously reported performance bounding technique was found to be less useful. It was shown that the key performance characteristic of these detectors is that their metrics observe the received signal for longer durations than those of other reduced-complexity detectors for CPM. This permits a very aggressive application of trellis reduction techniques, parameterized by and , with relatively minor performance losses. This result unifies and confirms those reported by other authors which were obtained by computer simulations. It was also shown that detectors with traditional metrics can suffer great losses when similarly aggressive trellis reductions were applied. REFERENCES [1] A. Svensson, C.-E. Sundberg, and T. Aulin, “A class of reduced-complexity Viterbi detectors for partial response continuous phase modulation,” IEEE Trans. Commun., vol. 32, no. 10, pp. 1079–1087, Oct. 1984. [2] G. K. Kaleh, “Simple coherent receivers for partial response continuous phase modulation,” IEEE J. Sel. Areas Commun., vol. 7, no. 12, pp. 1427–1436, Dec. 1989. [3] J. Huber and W. Liu, “An alternative approach to reduced complexity CPM receivers,” IEEE J. Sel. Areas Commun., vol. 7, no. 12, pp. 1427–1436, Dec. 1989. [4] P. Moqvist and T. Aulin, “Orthogonalization by principal components applied to CPM,” IEEE Trans. Commun., vol. 51, no. 11, pp. 1838–1845, Nov. 2003. [5] T. Aulin, “Study of a New Trellis Decoding Algorithm and its Applications,” European Space Agency, Noordwijk, The Netherlands, ESTEC Contract 6039/84/NL/DG, 1985. [6] A. Svensson, “Reduced state sequence detection of partial responce continuous phase modulation,” Proc. Inst. Elect. Eng., pt. I, vol. 138, pp. 256–268, Aug. 1991. [7] S. J. Simmons and P. H. Wittke, “Low complexity decoders for constant envelope digital modulations,” IEEE Trans. Commun., vol. COM-31, no. 12, pp. 1273–1280, Dec. 1983. [8] P. A. Laurent, “Exact and approximate construction of digital phase modulations by superposition of amplitude modulated pulses (AMP),” IEEE Trans. Commun., vol. COM-34, no. 2, pp. 150–160, Feb. 1986. -ary CPM signals [9] U. Mengali and M. Morelli, “Decomposition of into PAM waveforms,” IEEE Trans. Inf. Theory, vol. 41, no. 9, pp. 1265–1275, Sep. 1995. [10] X. Huang and Y. Li, “The PAM decomposition of CPM signals with integer modulation index,” IEEE Trans. Commun., vol. 51, no. 4, pp. 543–546, Apr. 2003. [11] G. Colavolpe and R. Raheli, “Reduced-complexity detection and phase synchronization of CPM signals,” IEEE Trans. Commun., vol. 45, no. 9, pp. 1070–1079, Sep. 1997.

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[12] E. Perrins and M. Rice, “PAM decomposition of -ary multi- CPM,” IEEE Trans. Commun., to be published. [13] B. E. Rimoldi, “A decomposition approach to CPM,” IEEE Trans. Inf. Theory, vol. 34, no. 2, pp. 260–270, Mar. 1988. [14] E. Perrins and M. Rice, “Optimal and reduced complexity receivers for -ary multi- CPM,” in Proc. IEEE Wireless Commun. Netw. Conf., Atlanta, GA, Mar. 2004, pp. 1165–1170. [15] J. B. Anderson, T. Aulin, and C.-E. Sundberg, Digital Phase Modulation. New York: Plenum, 1986. [16] P. Moqvist and T. Aulin, “Serially concatenated continuous phase modulation with iterative decoding,” IEEE Trans. Commun., vol. 49, no. 11, pp. 1901–1915, Nov. 2001. [17] M. Geoghegan, “Description and performance results for a multi- CPM telemetry waveform,” in Proc. IEEE MILCOM, vol. 1, Oct. 2000, pp. 353–357. [18] (2004) IRIG Standard 106–04: Telemetry Standards. Range Commanders Council Telemetry Group, Range Commanders Council, White Sands Missile Range, White Sands, NM. [Online]. Available: http://jcs.mil/RCC/manuals/106–04

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Erik Perrins (S’96–M’05) received the B.S. (magna cum laude), M.S., and Ph.D. degrees from Brigham Young University, Provo, UT, in 1997, 1998, and 2005, respectively. From 1998 to 2004, he was with Motorola, Inc., Schaumburg, IL, where he was involved with advanced development of land mobile radio products. Since 2004, he has been an industry consultant on receiver design problems such as synchronization and complexity reduction. He joined the faculty of the Department of Electrical Engineering and Computer Science, University of Kansas, Lawrence, in August 2005. His research interests are in digital communication theory, synchronization, channel coding, and complexity reduction in receivers. Prof. Perrins is a member of the IEEE Communications Society.

Michael Rice (M’82–SM’98) received the B.S.E.E. degree from Louisiana Tech University, Ruston, in 1987 and the Ph.D. degree from the Georgia Institute of Technology, Atlanta, in 1991. He was with Digital Transmission Systems, Inc., Atlanta, and joined the faculty at Brigham Young University, Provo, UT, in 1991 where he is currently the Jim Abrams Professor with the Department of Electrical and Computer Engineering. He was a NASA/ASEE Summer Faculty Fellow with the Jet Propulsion Laboratory during 1994 and 1995, where he was involved with land mobile satellite systems. During the 1999–2000 academic year, he was a Visiting Scholar with the Communication Systems and Signal Processing Institute, San Diego State University, San Diego, CA. His research interests are in the area of digital communication theory and error control coding, with a special interest in applications to telemetering and software radio design. He has been a consultant to both government and industry on telemetry related issues. Prof. Rice is a member of the IEEE Communications Society. He was Chair of the Utah Section of the IEEE from 1997 to 1999 and Chair of the Signal Processing and Communications Society Chapter of the Utah Section from 2002 to 2003.

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Design of Turbo-Coded Modulation for the AWGN Channel With Tikhonov Phase Error Ying Zhu, Li Ni, and Benjamin J. Belzer, Member, IEEE

Abstract—We design 1-b/symbol/Hz parallel concatenated turbo-coded modulation (PCTCM) for the additive white Gaussian noise (AWGN) channel with Tikhonov phase error. Constituent recursive convolutional codes are optimized so that the turbo codes have low error floors and low convergence thresholds. The pairwise error probability based on the maximum-likelihood decoding metric is used to select codes with low error floors. We also present a Gaussian approximation method that accurately predicts convergence thresholds for PCTCM codes on the AWGN/Tikhonov channel. Simulation results show that the selected codes perform within 0.6 dB of constellation constrained capacity, and have no detectable error floor down to bit-error rates of 10 6 . Index Terms—Additive white Gaussian noise (AWGN), Tikhonov phase error, turbo-coded modulation.

I. INTRODUCTION HE invention of turbo codes [1], [2] has enabled coherent communication at lower signal-to-noise ratios (SNRs) than previously possible; at these lower SNRs, receiver phase-estimation errors can be significant. This paper addresses the design and performance analysis of phase-robust turbo-coded modulation. In coherent receivers, the phase-locked loop (PLL) is commonly used to track carrier phase. Assuming pilot-tone-PLL phase estimation, our objective is to design codes that minimize the SNR required for reliable communication, at a given value of the PLL–bandwidth . The techniques of this paper symbol–interval product are also useful when the receiver employs open-loop phase estimation based on a pilot tone. Previous publications on trellis-coded modulation (TCM) for additive white Gaussian noise (AWGN)/Tikhonov channels (e.g., [3] and [4]) used squared Euclidean distance (SED) as the Viterbi algorithm (VA) decoding metric. Here, we use the optimal maximum-likelihood (ML) metric. VA simulations show that the ML metric gives gains of 2–5 dB over SED when (i.e., for moderate-to-large phase errors). A previous paper on parallel concatenated TCM (PCTCM) for the AWGN/Tikhonov channel [5] employed the ML decoding metric, but used SED as the code design criterion, rather than the pairwise error probability (PEP) considered here. Section V

T

Paper approved by I. Lee, the Editor for Wireless Communication Theory of the IEEE Communications Society. Manuscript received October 29, 2003; revised December 1, 2004. This work was supported by the National Science Foundation under Grant CCR-0098357. This paper was presented in part at the IEEE Symposium on Advances in Wireless Communications, Victoria, BC, Canada, September 2002, and in part at the 37th Conference on Information Sciences and Systems, Johns Hopkins University, Baltimore, MD, March 2003. The authors are with the School of Electrical Engineering and Computer Science, Washington State University, Pullman, WA 99164-2752 USA (e-mail: [email protected]; [email protected]; [email protected]). Digital Object Identifier 10.1109/TCOMM.2005.857158

contains examples demonstrating that PEP is a better code-design criterion than SED for the AWGN/Tikhonov channel, . especially at higher values of Previously proposed PCTCM encoder architectures can be classified into two main types. The bit-interleaved architecture of [6] uses a separate interleaver for each input bit shared between thetwoconstituentencoders.Thesymbol-interleavedapproaches of [7] and [8] employ symbol interleavers, which permute the symbol sequence, but preserve the mapping of input bits to symbols. The authors of [8] obtain about a 0.4-dB gain over [6] in the waterfall region of the bit-error rate (BER) performance curve; however, they point out that the gain in performance at lower SNRs given by symbol interleaving comes at the cost of a higher error floor. In this paper, we employ the the bit-interleaved architecture of [6] and propose a Gaussian approximation technique to improve performance in the waterfall region. It is important to note the difference between this paper, where we optimize turbo TCM codes by taking into account the phase-noise statistics of an (noniterative) estimate of the channel phase, and related work (e.g., [9]–[11]), where phase detection and error-correction decoding are combined in an iterative “turbo” scheme employing turbo TCM codes optimized for the AWGN channel without phase noise. Joint detection/decoding generally performs better, but requires more computational complexity, than the phase-robust codes studied in this paper. In Section V, a performance and complexity comparison (with [9]) indicates that the codes proposed in this paper require about 4 dB more SNR than joint detection decoding, but need only about two-thirds the computational complexity. Our codes are “plug-in compatible” with installed receivers similar to that described in Section II of this paper and, therefore, offer significant performance improvement for such receivers without the need to integrate phase estimation with iterative decoding. An outline of this paper is as follows. Section II describes the assumed system model. Section III presents two methods for computing the ML-metric-based PEP and a fast algorithm to compute the maximum PEP of a TCM code. Section IV discusses a Gaussian approximation method to predict convergence thresholds for PCTCM. Section V describes examples of TCM and PCTCM codes designed using the PEP computation and the Gaussian approximation. Section VI draws conclusions. Appendix I explains the details of PEP computation using the ML metric. Appendix II presents the PEP computation using the SED metric. II. SYSTEM MODEL Fig. 1 shows a coherent receiver with PLL. The transmitter sends a carrier pilot tone , where is the av-

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Fig. 1. Block diagram of correlator receiver with pilot-tone PLL.

erage transmitted symbol power, and is the power fraction allotted to the pilot tone; it is assumed throughout this paper that . Received pilot tone has channel phase distortion ; it is assumed that the channel amplitude gain . The PLL uses to form an estimate of channel phase . The PLL output is mixed with the data-bearing signal , yielding a baseband signal with phase-estimation error . After passing through a correlator receiver and deinterleaver, the discrete-time complex baseband model is

(1) where and are the transmitted and received symbols in the th symbol interval, respectively, and is a complex zero-mean Gaussian random variable (r.v.) with independent real and imaginary parts, each having variance . For a first-order PLL with zero frequency error, the phase-esin (1) has the Tikhonov probability density timation error function (PDF) [12] (2) , where is the with loop SNR is the one-sided PLL bandwidth, is the symbol interval, is the zeroth-order modaverage energy per symbol, and ified Bessel function of the first kind. The r.v.s , , and are independent, and the use of ideal interleaving/deinterleaving makes the channel memoryless. The model assumes that the carrier pilot tone is not interleaved, so the PLL tracks the timevarying channel phase with an effective observation interval of symbols. There is a tradeoff in choosing the value of . Reducing reduces the effect of noise on the PLL should be less than or equal to the tracking; however, coherence time of the channel phase process , so that the PLL responds quickly enough to track . In this paper, we assume that has been chosen to satisfy the tradeoff conditions, and we simulate only the discrete-time channel model (1).

When the phase error is considered to be modulo , the above Tikhonov model accounts for cycle slips in the first-order PLL [12]. For higher order PLLs, the Tikhonov model is only accurate at loop SNRs high enough to make cycle slips rare. The effect of cycle slips in higher order PLLs is not considered in this paper, but may be considered in future publications. The Tikhonov model has been widely used in past publications on phase-robust modulation (e.g., [3], [4], [12], and [13]). The Tikhonov model also applies to pilot-tone-aided systems with open-loop phase tracking, as the Tikhonov PDF closely approximates the phase-error PDF of simple one-shot estimators (see, e.g., [14, pp. 266–269]). The conditional channel PDF is [13]

(3) where (4) The communication channel defined by (3) is parameterized by the SNR , as well as by the value of ; by contrast, the standard AWGN channel is parameterized only by . Taking the logarithm of (3) leads to the optimal ML metric employed in this paper as follows: (5) III. PEP COMPUTATION A. Definitions and Background and denote two In a coded modulation scheme, let length- complex symbol sequences. The PEP is the probability that is chosen by the receiver when is actually sent, when the receiver chooses only between and . From [3], the union bound on the probability of an error event can be expressed in terms of the PEP (6)

ZHU et al.: DESIGN OF TURBO-CODED MODULATION FOR THE AWGN CHANNEL WITH TIKHONOV PHASE ERROR

At high SNR, the maximum PEP codeword pairs dominate the TCM performance. A code which has a relatively small maximum PEP will have relatively good performance. Based on (5), we have

Let terms of as

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and . Solving for and in and , and using (9), gives the joint PDF of and

(10) where (7) where is

,

Then the PEP can be computed exactly by doing a double integral [19]

is given by (4), and

(8) For the AWGN channel, it is typical [6], [8], [15] to search , the minimum for turbo-TCM constituent codes with large SED between path pairs differing by input Hamming weight two, or by two input symbols for symbol-interleaved systems. (However, we note that other publications, e.g.,[16]–[18], have achieved even better high-SNR BER performance by optimizing several low-weight lines in the turbo code’s distance spectrum.) For PCTCM on the AWGN/Tikhonov channel, this paper selects constituent codes so as to minimize the maximum PEP ; PEP denotes the PEP [based on ML metric (5)] of error events in which the information bits of the correct and incorrect paths differ by Hamming distance two. The maximum PEP events dominate the high-SNR performance of bit-interleaved PCTCM . A justification of this fact as the interleaver length follows by repeating the “approximate analytical explanation” given by Benedetto et al. in [15, p. 425], but with PEP replacing , which is the minimum error-event weight among all error events with minimum information weight . (For the recursive convolutional codes employed in this paper, .) Because the uniform error property cannot be assumed for TCM, we compute the PEP over all pairs of remerge paths whose information-bit sequences differ in two places.

is one, the PEP is

Using (4) and (8) to find and in terms of and , and using the channel PDF (3), we find the joint PDF of and (9) where

(11) appears in [12, p. 207] for

C. PEP Computation for Sequences With The method of exact computation is impractical for , because it requires difficult -dimensional numerical integrals. Instead, when , we use the central limit theorem (CLT) with correction terms to approximate the PEP. Let , , and . Let , where and denote the mean and variance of . Since the ’s are independent (due to the symbol-wise independence of and ), the cumulative districan be written as a series [20] bution function (CDF) of (12) , and where can be expressed in terms of the conditional moments of and . Using (12), the PEP Chebyshev–Hermite polynomials is computed as (13) The conditional moments of numerically using (3), to yield

B. PEP Computation for When the symbol sequence length

An alternate computation for . the special case

needed in (12) are computed

We map the region of integration from to the rectangle by letting , , and . Comparison of the simulated and computed PEPs shows that three or four correction terms are sufficient to attain a reasonable approximation of the true PEP [19]. D. A Fast Computational Algorithm for the Maximum PEP

For values of

where no solution exists for is set to zero.

,

One way to find the maximum PEP of a code is by directly enumerating the correct paths and the remerge paths with length less than or equal to . The time cost of this method increases

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Fig. 2. Example of the remerge property.

exponentially as increases. Thus, we would like to find a fast computational algorithm to compute the maximum PEP. Mulligan and Wilson [3], [21] have derived an efficient algobetween rithm for the computation of the minimum SED two trellis paths. The Mulligan–Wilson (M-W) algorithm up, whose elements are the minimum dates a matrix SED between all pairs of paths starting from any initial state, and reaching states and at discrete time . The diagonal elare the minimum distances among all remerge ements paths (i.e., error events) that start from any state and end at state in stages. The algorithm terminates at the stage when all possible are greater than the minimum value among the , which is . The SED metric between length- symbol sequences and has the additive property . The M-W algorithm exploits the additive property in two ways. First, the set of possible minimum at SEDs between trellis paths passing through states time is easily updated by adding the SEDs for links between and to the minimum predecessor state pairs of the predecessor states at the th distances , which is simply the stage. Second, the minimum SED minimum element from , is guaranteed to be the global minimum SED once the termination condition is reached, since path pairs with greater than minimum SED at time will , remain so, if they are extended into the future from states and can therefore be rejected at the th stage. We denote this last property as the remerge property, because it allows a (minimum or maximum) metric selection between two pairs of remerge paths as soon as they merge into one pair at states , instead of at their common endpoint, which may be far in the future. To illustrate the remerge property, consider two and , as shown in Fig. 2. Starting pairs of paths from some time , and merge together, and and merge together. Let , , , and denote the segments of paths , , , and from time 0 to . If implies that , then we say that the metric has the remerge property. The ML-metric-based PEP , when considered as a metric, is not additive, but experiments strongly suggest that it has the remerge property. We therefore modify the M-W algorithm to update a matrix , whose elements are

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the maximum PEP between all pairs of paths starting from any initial trellis state, and reaching states and at time . From (13), the PEP at time cannot be computed by simple addition with the ’s. However, the components of (13), namely, , , and , can be computed at time by up) and dating their running sums (stored along with each renormalizing, since the terms of are independent. The series expansion terms in (12) can also be additively updated th stage; specifically, we from information stored in the of every in the path. From need the th-order moments the PEP correction terms [20] and [19], we see that the terms , , , , , and are additive. In addition, only these terms are associated with the moof , we store the ments. Thus, for each element . Then, the terms corresponding value of used to compute the PEP at time can be obtained from and the moments . The modified algorithm is as follows. Step 1) For each state , find the predecessor states from which a transition to is possible and store them in a table. Initialize by setting for all and , where . Step 2) Find . For each pair of states , using the exact PEP computation method for , find , and store the corresponding . Step 3) Update for . For each pair of states , using the table of Step 1), find the predecessors of : , and the predecessors of : . Compute the maximum PEP among all paths passing through at time using the moments stored for whenever is not equal to MAX. (If , then no pair of paths can pass through the states and at .) When , represents the PEP betime tween two paths remerging on the state at time . So if , and , then will be updated by . Step 4) Let for any such that . If there exists at least one pair of such that and , then set and go back to Step 3). Otherwise, stop the iteration and set the maximum PEP to be . IV. PREDICTING CONVERGENCE THRESHOLDS USING THE GAUSSIAN APPROXIMATION Although the AWGN/Tikhonov channel variables are not Gaussian, we have observed experimentally that the extrinsic information from constituent maximum a posteriori (MAP) decoders for the AWGN/Tikhonov channel can be well approximated as Gaussian. Thus, the decoder convergence threshold can be predicted using a modified version of the Gaussian approximation technique in [22]. The convergence threshold is , under which the iterative decoder will the minimum

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Fig. 3.

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Decoder structure for Gaussian approximation.

converge with zero error probability. The extrinsic information , where output SNR is measured as SNR is the error probability of the decoder extrinsic information outputs, and . By feeding a sequence of independent Gaussian r.v.s with a fixed SNR into the extrinsic information inputs of the decoders, a point on the extrinsic information SNR input/output curve is obtained. When the turbo decoders are symmetric, the converunder which there is gence threshold is the minimum no intersection between the SNR input/output curve and the SNR . The Gaussian approximation techline SNR nique of [22] is based on the assumption that all intrinsic and extrinsic information variables are Gaussian and statistically independent. A problem arises when applying the above Gaussian approximation method to PCTCM. Consider, e.g., a bit-interleaved and two-encoder PCTCM where two binary inputs are applied to each encoder, and the encoder output bits are mapped into constellation symbols. (Although the following discussion assumes two input bits, generalization to more than two input bits is straightforward.) Due to the nonlinear mapping of encoder output bits to constellation symbols, the associated decoder extrinsic information r.v.s and are correlated, so that at each decoder iteration, a specific ratio exists between SNR and SNR . Since is the systematic output of decoder SNR at the output of decoder 1; for 1, initially SNR SNR for the same reason. After many decoder 2, SNR . However, assuming iterations, the ratio SNR SNR that SNR SNR at the first iteration leads to an incorrect convergence-threshold prediction. Thus, the correct ratio SNR SNR must be simulated during threshold prediction. By concatenating the two decoders as in Fig. 3, we get the correct extrinsic information SNR input/output relations for and . At each , the initial and are set to zero. At decoder 1’s output, we measure the SNRs of and . These extrinsic information r.v.s are approximated by independent Gaussian r.v.s (with SNRs equal to the measured SNRs) injected into decoder 2. After one iteration, the SNRs of and are measured at decoder 2’s output. Then, we use the ratio SNR SNR as the ratio SNR SNR for the next point on the SNR input/output curve. The and are Gaussian r.v.s with mean = sign SNR and variance SNR , where . Under this scheme, the correct ratio between extrinsic information SNRs for and is maintained, so that the correct extrinsic SNR input/output relation is obtained. The conunder which neither vergence threshold is the minimum nor that for intersects the the SNR input/output curve for line SNR SNR .

TABLE I PREDICTED AND SIMULATED E =N CONVERGENCE THRESHOLDS (QUADRATURE AMPLITUDE MODULATION (QAM) RADIAL MAPPING, B T = 0:1, SIMULATION USES TWO 16 384-B INTERLEAVERS)

Fig. 4. Eight-star QAM radial and nonradial signal set mappings, for coded communication at 1 b/symbol/Hz. The nonradial mapping is shown in parenthesis. In both mappings, the information bit is the left-most bit.

The above convergence-threshold prediction is much faster than an actual turbo-decoding simulation. In an actual simulation, much time is needed to achieve an accurate estimate of the at which falls below some threshold (typically lowest or ). However, convergence prediction requires only one iteration for each value of the extrinsic information SNR. At each computation of the output extrinsic information SNR, the intersection condition is checked. If intersection occurs, the is increased. The compucomputation is stopped, and for which intersection does not tation stops at the first occur; this is the predicted convergence threshold. The predicted convergence threshold and the actual simulation results are listed in Table I, for several different codes in the search space discussed in Section V-A. The simulation converat which . gence threshold is the minimum The simulation used two 16 384-b -random interleavers, one for each input bit, as described in Section V-C. From the table, it is seen that the predicted results track the simulation results reasonably well. The simulated results are about 0.2-dB higher than the predicted threshold, due to use of finite-length interleavers in the simulation. The Gaussian approximation technique described in this section is useful for any turbo-decoding scheme where the MAP decoders have multiple extrinsic inputs which can be modeled as correlated Gaussian r.v.s.

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Fig. 5.

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Generalized form of constituent encoder 1.

V. DESIGN EXAMPLES FOR TCM AND TURBO TCM A. Turbo Encoder and Search Space The examples presented in this section employ turbo codes with two inputs and use the bit-interleaved architecture of [6], but with a factor of four constellation expansion. In [23] and [24], the capacity-achieving input PDF for the average-power-constrained AWGN/Tikhonov channel is shown to be discrete in amplitude and uniform in phase (DAUP), i.e., it consists of concentric rings about the origin. Numerical experiments in [23] and [24] show that the DAUP input -ary capacity is closely approached by using multiring phase-shift-keying (PSK) constellations, when the number of , where is the code rate in constellation points b/symbol/Hz. Our 1-b/symbol/Hz PCTCM codes use the 8-ary constellations and bit-mapping schemes discussed in [25]: 8-PSK with natural mapping and 8-star QAM with radial and nonradial mappings, as shown in Fig. 4. In all mappings, the information bit is the most significant bit. For each constellation, : and we consider two values of (corresponding to large and moderate phase errors). We consider the 1-b/symbol/Hz, rate-2/6 bit-interleaved turbo TCM code [26], where the constituent codes are 16-state, rate-2/3 recursive systematic convolutional codes with one punctured input bit; the structure of the constituent encoder is shown in Fig. 5. The constituent code has two inputs, one systematic output bit, and two parity bits. Bits U1 and U2 are connected to the lower encoder by separate bit interleavers; in this paper, the interleavers are of length 16 384 b each. For the upper encoder, the systematic bit is U1, and U2 is punctured. For the lower encoder, the systematic bit is U2, and U1 is punctured. The state-space equations [over GF(2)] for the constituent encoder are [8]

where is the 1 4 state vector at time , , is the 1 2 input vector, and is the 1 3 output (i.e., codeword) vector. The matrices are

We use binary numbers (in decimal notation) to represent is represented by the 3-b matrices , , , and , e.g., . Since all variables are over GF(2), there number are possible codes. We apply a number of constraints to narrow the search space [26]; in particular, we require noncatastrophic codes without parallel transitions. The avoidance of parallel transitions is an intuitive requirement inspired by well-known TCM design rules for the Rayleigh fading channel (e.g., [3, pp. 411–412] and [27]), which state that the time-diversity benefit of coding is obtained by maximizing the length of the shortest error-event path. in (1) can be considered a fading gain, Indeed, the factor and time-diversity techniques are known to be beneficial for the AWGN/Tikhonov channel [3, pp. 458–459]. After applying the constraints, the total number of codes is 2 752 512. B. Predicting TCM Performance In order to assess the ability of the PEP to predict turbo-TCM performance, we first examine the PEP’s performance as a code-selection metric for individual (nonconcatenated) TCM codes. For each TCM code in the class described in Section V-A, we find the maximum PEP; then, among all codes, we find the code which has the minimum maximum PEP. Table II shows the computed maximum PEP for six codes and SNR dB, and when when and SNR dB, for the radially mapped 8-QAM shown in ) is also given Fig. 4. The minimum pairwise SED (i.e., for comparison.

ZHU et al.: DESIGN OF TURBO-CODED MODULATION FOR THE AWGN CHANNEL WITH TIKHONOV PHASE ERROR

TABLE II MAXIMUM PEP OF SELECTED CODES

Fig. 6. TCM simulation results for B

T

= 0:01 and B

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TABLE III CODE SEARCH RESULTS

T

= 0 :1 .

Fig. 6 presents TCM simulation results for the individual constituent TCM codes listed in Table II; because the codes are not concatenated, no interleavers are used (or needed) to produce Fig. 6. The decoders use the VA with ML decoding metric (5). , ranking the codes by maximum or by When minimum/maximum PEP predicts two good and four bad codes, , which is consistent with the simulation. When path the AWGN dominates the phase noise, so that the is identical to the maximum PEP path. When ,a cannot explain the simulation results, while the ranking by PEP ranking gives a good prediction of code performance. We conclude that the PEP is a good code-selection metric for the AWGN/Tikhonov channel. We also believe the corrected-CLT PEP computation can be applied to other channels of interest. C. Design of Parallel Concatenated Turbo TCM To find good constituent codes for turbo TCM, we use the PEP to find codes with low error floors and then use the Gaussian approximation to find codes with low convergence thresholds. At medium SNR, the maximum PEP of weight two paths, PEP , is the dominant error-floor predictor [15]. At high SNR, the maximum PEP among all unconstrained paths, PEP , predicts the error floor. The smaller the maximum PEP, the lower the error floor. Since there are many more unconstrained paths than weight-two paths, it takes longer to compute PEP than PEP . So, we design the code search procedure as follows.

Step 1) Find the PEP for each code in the search space given in Section V-A. Rank the codes by their PEP , and choose those codes whose PEP is less than a suitable threshold. Those codes having the smallest PEP are predicted to have the lowest error floors at medium SNR. Step 2) For each code chosen by Step 1), use the Gaussianapproximation method to compute the convergence threshold. Choose the codes with smallest convergence threshold. Step 3) Find the PEP for each code chosen by Step 2). Those codes having the smallest PEP are predicted to have the lowest error floors at high SNR. value, we apply the code For each constellation and search over the search space given by Section V-A, and choose one code with low error floor and low convergence threshold as the constituent code for the turbo-TCM simulation. The top four codes for each case are listed in Table III. Simulated codes are shown in bold typeface. The PEP computation for each comwas conducted at an SNR suffibination of mapping and ciently high to make the PEP “small” (i.e., about or less) [26]. For all simulations presented in this section, a data block length of 16 384 b is used; this is also the length of each of the two interleavers. The two interleavers are designed as in [28], and . The decoder uses a modwith parameters ified MAP algorithm based on the ML decoding metric (5). All simulations are run with eight iterations. and are The simulation results for shown in Figs. 7 and 8. We see that these best codes have no . error floor down to a BER of The constellation constrained capacities for 8-QAM and 8-PSK, computed in [25], are shown as dotted and solid vertical lines, respectively, in the figures; these capacities depend on the value of , and hence, are different in the two figures. When

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D. Performance and Complexity Comparison With Iterative Joint Estimation and Decoding

Fig. 7. Parallel concatenated, bit-interleaved, turbo-TCM simulation results for the AWGN/Tikhonov channel with B T = 0:1. Two 16 384-b S -random interleavers (S-40 and S-32) were used in all simulations shown.

Fig. 8. Parallel concatenated, bit-interleaved, turbo-TCM simulation results for the AWGN/Tikhonov channel with B T = 0:01. Two 16 384-b S -random interleavers (S-40 and S-32) were used in all simulations shown.

, the BER curves for the best mappings (QAM radial and PSK natural) are both about 0.6 dB away from . The QAM constellation constrained capacity at radial mapping has a 1.3-dB gain over the nonradial mapping. This is because the radial mapping maps the information bit to the two radii of the constellation; when the channel phase error is large, errors are more likely to occur in favor of other points on the same radius as the correct symbol. , the BER curves for all mapping schemes When are also about 0.6 dB away from the constellation constrained capacity. Since the phase error is small, the two QAM mappings have almost the same performance. For the QAM radial mapping, the code performance when (incorrect) AWGN statistics are used in the MAP decoder is also given. There is a 2-dB penalty compared with using the true AWGN/Tikhonov channel PDF.

This subsection compares the foregoing proposed codes with the iterative channel estimation and decoding system proposed by Komninakis and Wesel (K&W) [9]. This is done by estimating the performance in [9] if radial-mapped 8-QAM had been used, based on the actual performance (relative to capacity) with 4-PSK. In [9], simulations are presented for 1-b/codedsymbol 4-PSK pilot-symbol-aided modulation (PSAM) [29] on , the finite-state Markov channel (FSMC) , where and are the transmitted and reis ceived symbols, and is complex AWGN. Phase process a finite-state Markov model designed to capture the phase dynamics of the standard Clarke fading channel [30]. The FSMC is very similar to our channel (1); in fact, (1) is what would be expected if the receiver of Fig. 1 were running on an analog channel matched to the FSMC. The FSMC is parameterized , the by the Doppler frequency symbol interval product number of symbols over which the phase process is approximately constant. PSAM inserts known pilot symbols every data symbols; the receiver uses the pilot symbols to estimate and , K&W achieve the channel gain. When a BER of at an about 6.3 dB greater than , which is the 4-PSK constrained FSMC capacity upper bound computed by assuming perfect channel state information for the coded symbols. is the phase process coherence time, the appropriate As for a PLL-based system on the setting for FSMC is . However, for fair comparison with a PSAM, our code’s performance at should be compared to K&W’s performance at , since PSAM must operate with half of the symbol interval of the PLL to maintain the same data throughput rate. For the 16-state FSMC and 8-QAM modulation, at with dB. We assumed that the system in [9] would perfor 8-QAM, as it did for 4-PSK, form within 6.3 dB of giving a required of 8.3 dB. From Fig. 7 for , the 8-QAM radial-mapping code proposed in this paper needs , i.e., about 4.1 dB more than 12.4 dB at a BER of that of K&W. K&W employ two Q-single input–single outputs (“Q-SISOs”) for phase estimation, and two “C-SISOs” for decoding; both Q-SISOs and both C-SISOs are updated in each detection/decoding iteration. For 8-QAM, the Q-SISOs would need at least 16 states to avoid significant quantization error; the C-SISOs in [9] use 8 states. By comparison, our system updates two 16-state C-SISOs per iteration. Since a Q-SISO’s perstate complexity is roughly the same as a C-SISOs, an 8-QAM K&W system with 8-state C-SISOs would require about 50% more computational complexity, per iteration, than the 8-QAM PCTCM proposed here, and would require twice the computational complexity if 16-state C-SISOs were used. The estimate that the proposed codes would perform about 4 dB worse than the system in [9] must be taken in context. Our system assumes perfect frequency synchronization and a firstorder PLL, whereas, in practice, higher order PLLs are typically channel employed. Use of higher order PLLs on the is not possible, as the low value of PLL SNR leads to loss of

ZHU et al.: DESIGN OF TURBO-CODED MODULATION FOR THE AWGN CHANNEL WITH TIKHONOV PHASE ERROR

phase lock. However, a filtered “one-shot” ML phase estimation from a pilot tone, together with our proposed PCTCM codes, should give similar results to those reported here, and may be an attractive alternative to joint detection/decoding in applications requiring lower computational complexity.

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for

B. Computing

To compute the PEP of (11), it is easier to compute the range of when is fixed than the range of when is fixed. Also, since is nonnegative and integrable, the integration order can be changed. Then we can find the minimum and maximum value of for each and do the double integral numerically as

VI. CONCLUSION This paper has demonstrated that significant coding gain can be realized by taking into account the effect of phase-estimation errors when designing turbo TCM with coherent signaling. New techniques for computing the PEP of the AWGN channel with Tikhonov-distributed phase error using the optimal ML metric have been described and used to find PCTCM codes with low error floors. A modified Gaussian approximation technique has been developed for design of PCTCM codes with low convergence thresholds. The PEP computation and Gaussian approximation techniques have been combined into a code-search algorithm. Simulation results show that the algorithm finds codes with both low error floors and low convergence thresholds, and that phase-robust bit-mapping schemes give significant gains when the phase-estimator observation interval is small. Generalizations of the code-search techniques in this paper to other phase-estimation schemes and other channel models, such as PSAM on fading channels, are currently under investigation.

if (15) where (16) (17) The values and are found from the nonzero con. From (14), we have dition of , and (10) gives (18)

APPENDIX I PEP COMPUTATION USING ML METRIC FOR

Then, (16) and (17) follow from (18) and the fact that .

A. Nonzero Condition of For values of where no solution exists for , is set to 0. Now we will look at the nonzero con. We rewrite as dition of

Let , , and We see that, if , , and the SNR are known, the locus of is a circle with origin and radius . Similarly, can be rewritten in the form

if if

. and

Let , , and . If , , and the SNR are known, the locus of and is a circle with origin and radius . , we need to find the interTo find the solutions for section points of these two circles. Let denote the distance beand ; then, when tween the two origins or , the two circles do not intersect, and there is no solution for and . Thus, the nonzero condition of is (14)

APPENDIX II PEP COMPUTATION USING SED METRIC For the AWGN channel with moderate Tikhonov phase error, e.g., , the VA simulation based on the SED metric has a 2-dB loss compared to that of the ML metric. Nonetheless, it may be desirable to use the SED decoding metric due to its lower computational complexity or to ensure backward compatibility with a preexisting decoder. The PEP based on the SED metric can be considered as the code design criterion on such a channel. In [4], the CLT approximation was used to derive the SED-based PEP for M-PSK constellations, with an upper bound . The following derivation gives an given for the case exact expression when and a CLT approximation for ; both cases are valid for arbitrary constellations. A. Exact PEP Computation for the Length-One Sequence Let vector

be

where is the unit vector Conditioned on , the PEP is

.

(19)

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and

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is a Gaussian r.v. with mean

and variance

, where

In (27) and (28), is a Gaussian r.v. with zero mean and variand is an r.v. given by ance (29)

(20) (21) Then

where

,

, and . If the error sequence is long enough, the r.v. can be approximated by a Gaussian r.v. according to the CLT [4] with mean and variance

(22) and

(30) (23)

B. Approximate PEP Computation for LengthUsing the CLT Conditioned on where

Sequences

, the PEP is and

is the

(31)

Gaussian r.v. [4]

Since is approximated by a Gaussian r.v., the PEP for the length- sequence is simplified as (24)

(25) Rearranging the inequality

(32)

,

We note that so that

where and (31).

is the Gaussian PDF with

and

given in (30)

REFERENCES

by using (25) gives

(26) For simplicity, we rewrite the probability equation as

(27) where (28)

[1] C. Berrou, A. Glavieux, and P. Thitimajshima, “Near-Shannon-limit error-correcting coding: Turbo codes,” in Proc. IEEE Int. Conf. Commun., May 1993, pp. 1064–1070. [2] C. Berrou and A. Glavieux, “Near-optimum error-correcting coding and decoding: Turbo codes,” IEEE Trans. Commun., vol. 44, no. 10, pp. 1261–1271, Oct. 1996. [3] E. Biglieri, D. Divsalar, P. J. McLane, and M. K. Simon, Introduction to Trellis-Coded Modulation with Applications. New York: Macmillan, 1991. [4] B. Vucetic and J. Du, “The effects of phase noise on trellis-coded modulation over Gaussian and fading channels,” IEEE Trans. Commun., vol. 43, no. 2–4, pp. 252–260, Feb.–Apr. 1995. [5] A. Risley, B. Belzer, and Y. Zhu, “Turbo trellis coded modulation on partially coherent fading channels,” in Proc. IEEE Int. Symp. Inf. Theory, Jun. 2000, p. 222. [6] S. Benedetto, D. Divsalar, G. Montorsi, and F. Pollara, “Parallel concatenated trellis coded modulation,” in Proc. IEEE Int. Conf. Commun., Jun. 1996, pp. 974–978. [7] P. Robertson and T. Wörz, “Bandwidth-efficient turbo trellis-coded modulation using punctured component codes,” IEEE J. Sel. Areas Commun., vol. 16, no. 2, pp. 206–218, Feb. 1998. [8] C. Fragouli and R. D. Wesel, “Turbo-encoder design for symbol-interleaved parallel concatenated trellis-coded modulation,” IEEE Trans. Commun., vol. 49, no. 3, pp. 425–435, Mar. 2001.

ZHU et al.: DESIGN OF TURBO-CODED MODULATION FOR THE AWGN CHANNEL WITH TIKHONOV PHASE ERROR

[9] C. Komninakis and R. D. Wesel, “Joint iterative channel estimation and decoding in flat correlated Rayleigh fading,” IEEE J. Sel. Areas Commun., vol. 19, no. 9, pp. 1706–1717, Sep. 2001. [10] M. C. Valenti and B. D. Woerner, “Iterative channel estimation and decoding of pilot symbol assisted turbo codes over flat-fading channels,” IEEE J. Sel. Areas Commun., vol. 19, no. 9, pp. 1697–1705, Sep. 2001. [11] A. Anastasopoulos and K. M. Chugg, “Adaptive iterative detection for phase tracking in turbo coded systems,” IEEE Trans. Commun., vol. 49, no. 12, pp. 2135–2144, Dec. 2001. [12] A. J. Viterbi, Principles of Coherent Communication. New York: McGraw-Hill, 1966. [13] G. J. Foschini, R. D. Gitlin, and S. B. Weinstein, “On the selection of a two-dimensional signal constellation in the presence of phase jitter and Gaussian noise,” Bell Syst. Tech. J., vol. 52, pp. 927–965, Jul./Aug. 1973. [14] J. G. Proakis, Digital Communications, 4th ed. Boston, MA: McGrawHill, 2001. [15] S. Benedetto and G. Montorsi, “Unveiling turbo codes: Some results on parallel concatenated coding schemes,” IEEE Trans. Inf. Theory, vol. 42, no. 2, pp. 409–428, Mar. 1996. [16] J. Yuan, B. Vucetic, and W. Feng, “Combined turbo codes and interleaver design,” IEEE Trans. Commun., vol. 47, no. 4, pp. 484–487, Apr. 1999. [17] D. Rowitch and L. Milstein, “On the performance of hybrid FEC/ARQ systems using rate compatible punctured turbo (RCPT) codes,” IEEE Trans. Commun., vol. 48, no. 6, pp. 948–959, Jun. 2000. [18] D. Tujkovic, “Constituent code optimization for space–time turbo coded modulation over fast fading channels,” in Proc. IEEE Int. Symp. Inf. Theory, Yokohama, Japan, Jun.-Jul. 2003, p. 408. [19] Y. Zhu and B. J. Belzer, “New results for the pairwise error probability on the AWGN channel with Tikhonov phase, with applications to coded modulation,” in Proc. IEEE Symp. Adv. Wireless Commun., Victoria, BC, Canada, Sep. 2002, pp. 93–94. [20] V. V. Petrov, Sums of Independent Random Variables. New York: Springer-Verlag, 1975. [21] M. M. Mulligan and S. G. Wilson, “An improved algorithm for evaluating trellis phase codes,” IEEE Trans. Inf. Theory, vol. IT-30, no. 6, pp. 846–851, Nov. 1984. [22] H. El Gamal and A. R. Hammons Jr., “Analyzing the turbo decoder using the Gaussian approximation,” IEEE Trans. Inf. Theory, vol. 47, no. 2, pp. 671–686, Feb. 2001. [23] P. Hou, “Capacity and coding/modulation study for partially coherent additive white Gaussian noise channels,” Ph.D. dissertation, School Elect. Eng. Computer Sci., Washington State Univ., Pullman, WA, 2002. [24] P. Hou, B. J. Belzer, and T. R. Fischer, “On the capacity of the partially coherent additive white Gaussian noise channel,” in Proc. IEEE Int. Symp. Inf. Theory, Yokohama, Japan, Jun.-Jul. 2003, p. 372. [25] A. Risley, “Turbo TCM for quasi-coherent fading channels,” Master’s thesis, School Elect. Eng. Computer Sci., Washington State Univ., Pullman, WA, 1999. [26] Y. Zhu, L. Ni, and B. J. Belzer, “Design of turbo coded modulation for the AWGN channel with Tikhonov phase error,” in Proc. 37th Conf. Inf. Sci. Syst., Baltimore, MD, Mar. 2003, Paper 27, pp. 1–6, [CD-ROM]. [27] D. Divsalar and F. Pollara. Turbo trellis coded modulation with iterative decoding for mobile satellite channels. presented at Int. Mobile Satellite Conf. [Online]. Available: http://www331.jpl.nasa.gov/public/tcodesbib.html

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, “On the design of turbo codes,” Telecommunications and Data Acquisition, Prog. Rep. 42–123, 1995. [29] J. K. Cavers, “An analysis of pilot symbol assisted modulation for Rayleigh fading channels,” IEEE Trans. Veh. Technol., vol. 40, no. 6, pp. 686–693, Nov. 1991. [30] R. H. Clarke, “A statistical theory of mobile radio reception,” Bell Syst. Tech. J., pp. 957–1000, Jul. 1968.

Ying Zhu received the B.S. degree in communication engineering from Xidian University, Xi’an, China, in 1992, and the M.S. degree in electrical engineering from Washington State University, Pullman, in 2002, where she is currently working toward the Ph.D. degree in electrical engineering. Her research interests include digital data transmission, channel coding, signal processing for communications, and mobile communication networks.

Li Ni received the B.S. and M.S. degrees in electrical engineering from Zhejiang University, Hangzhou, China, in 1993 and 1996, respectively. She is currently working toward the Ph.D. degree in electrical engineering at Washington State University, Pullman. Her current research interests include coded modulation design, error-correction codes, and nonlinear shaping codes.

Benjamin J. Belzer (S’93–M’96) received the B.A. degree in physics from the University of California, San Diego, in 1982, and the Ph.D. degree in electrical engineering from the University of California, Los Angeles, in 1996. From 1981 to 1991, he was a Software Engineer with Beckman Instruments, Hughes Aircraft, Northrop Corporation, and Source Scientific in southern California, and Develco, Inc., in northern California. Since 1996, he has been on the faculty of the School of Electrical Engineering and Computer Science, Washington State University, Pullman, where he is currently an Associate Professor. His research interests include coded modulation for wireless communications, iterative detection for two-dimensional intersymbol-interference channels, combined source and channel coding, and image and video communication systems.

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Adaptive Opportunistic Fair Scheduling Over Multiuser Spatial Channels Chuxiang Li and Xiaodong Wang, Senior Member, IEEE

Abstract—We consider the problem of opportunistic fair scheduling (OFS) of multiple users in downlink time-division multiple-access (TDMA) systems employing multiple transmit antennas and beamforming. OFS is an important technique in wireless networks to achieve fair bandwidth usage among users, which is performed on a per-frame basis at the media access control layer. Multiple-transmit-antenna beamforming provides TDMA systems with the capability of supporting multiple concurrent transmissions, i.e., multiple spatial channels at the physical layer. Given a particular subset of users and their channel conditions, the optimal beamforming scheme can be calculated. The multiuser opportunistic scheduling problem then refers to the selection of the optimal subset of users for transmission at each time instant to maximize the total throughput of the system subject to a certain fairness constraint on each individual user’s throughput. We propose discrete stochastic approximation algorithms to adaptively select a better subset of users. We also consider scenarios of time-varying channels for which the scheduling algorithm can track the time-varying optimal user subset. We present simulation results to demonstrate the performance of the proposed scheduling algorithms in terms of both throughput and fairness, their fast convergence, and the excellent tracking capability in time-varying environments. Index Terms—Beamforming, discrete stochastic approximation, multiantenna system, opportunistic fair scheduling (OFS), tracking.

I. INTRODUCTION PPORTUNISTIC fair scheduling (OFS) is an important technique in wireless networks [8], [11]–[13], [18]. It aims at balancing two conflicting goals: fairness and efficient resource use. On the one hand, the wireless resource normally should be allocated to users with good channel conditions to achieve high use of the limited system resource [15]. On the other hand, the efficiency-based strategy will inevitably cause fairness problems [4]. Although it is impractical to ensure fairness for all users over short time scales, lagging flows can catch up those forwarding ones as the channels vary over long time scales. Thus, in general, a tradeoff between the resource efficiency and the users’ fairness is needed. Several approaches have been proposed in the literature. In [8], a revenue-based rate controller for time-division multiple-access (TDMA)

O

Paper approved by A. Lozano, the Editor for Wireless Communication of the IEEE Communications Society. Manuscript received October 6, 2003; revised August 22, 2004 and April 10, 2005. This work was supported in part by the National Science Foundation under Grant CCR-0207550, and in part by the Office of Naval Research under Grant N00014-03-1-0039. The authors are with the Department of Electrical Engineering, Columbia University, New York, NY 10027 USA (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TCOMM.2005.857163

systems is proposed, where the revenue for each user is defined as the product of the rate and a fairness weight, so that the revenue-based decision depends on both the throughput and the fairness constraint. In [11] and [12], a utility-based time slot scheduler for TDMA systems is developed, where an additive fairness weight is introduced into the utility of each user, and both the channel conditions and the fairness constraints are considered in the utility-based policy. In [13], a general methodology is treated, where a similar revenue as that in [8] and a similar fairness updating procedure as that in [11], [12] are adopted, and a multiuser scheduler is proposed for CDMA systems. Note that, instead of incorporating the physical-layer constraints and implementation details, simplified and abstracted models for the physical layer are employed in existing works on transmission scheduling. In [11] and [12], the physical-layer condition for each user is captured by a single value of channel-dependent performance, and, in [13], a per-bit power consumption for guaranteeing a certain signal-to-interference-plus-noise ratio (SINR) is used to describe the physical-layer condition. Multiple antennas provide TDMA systems with the capability of supporting multiple concurrent transmissions. In a cellular system, multiple antennas can be employed at the base station, whereas at the mobile unit, a single antenna is typically employed, due to the size and power limits. In this paper, we consider multiuser transmission scheduling in downlink systems employing multiple transmit antennas and beamforming. For multiantenna systems, transmit antenna beamforming is an effective technique to control the SINR by adjusting the radiation patterns (beams) of the users. In particular, the joint downlink multiuser beamforming and power control is a coupled optimization problem, where one user’s beam may affect the crosstalk experienced by others. This problem has been extensively analyzed in the recent literature. In [19] and [20], the optimal power allocation under fixed beamformers is treated. In [5] and [6], an iterative joint power and beamforming approach is proposed, and its convergence is analyzed in [7] and [17]. Also note that only single-user schedulers have been considered for TDMA systems in the current literature. In [18], a single-user opportunistic scheduler combined with a random beamforming is proposed for the downlink multiple-input single-output (MISO) system. Such a single-user scheduling strategy was motivated from an information-theoretic result on the single-transmit-antenna downlink model [18]. It is also pointed out in [18] that multiple transmit antennas provide the potential for multiuser scheduling, which is an interesting and open problem for the downlink MISO broadcasting system. Also, a multiuser scheduling strategy is discussed in [18]

0090-6778/$20.00 © 2005 IEEE

LI AND WANG: ADAPTIVE OPPORTUNISTIC FAIR SCHEDULING OVER MULTIUSER SPATIAL CHANNELS

from the information-theoretic point of view. In this paper, we treat the multiuser scheduling problem for the downlink MISO system employing multiuser beamforming scheme from a practical system point of view. We can also draw an analogy between the multiuser schedulers considered in this paper and the opportunistic multiuser scheduling framework proposed in [13] for code-division multiple-access (CDMA) systems. As pointed out in [17], many results found in the context of synchronous CDMA systems can be transformed to the multiuser beamforming scenario. The opportunistic multiuser scheduling with a beamforming approach treated in this paper is analogous to that for CDMA systems. Given a particular user subset and their channel conditions, the optimal beamforming scheme can be calculated. The multiuser opportunistic scheduling problem then refers to the selection of the optimal subset of users for transmission at each time slot to maximize the total throughput of the system subject to a certain fairness constraint on each individual user’s throughput. Straightforward implementation of the user subset selection by simple exhaustive enumeration suffers from several problems in practice. One is the high computational complexity. In particular, denote and as the number of users requesting services and the number of users simultaneously supported by the system, respectively. Also denote as the maximum . Then, number of simultaneously active users, i.e., the number of all possible user subsets is given by , is large and is which is tremendous because normally relatively small. Another problem is that the exact channel knowledge is not available in practice. In addition, when the physical channel is time-varying, the algorithm should be able to track the time-varying optimal user subset. In this paper, using the advanced stochastic optimization techniques developed in the recent operation research literature [1], [2], [9], we propose discrete stochastic approximation algorithms to adaptively select the optimal user subset. Note that such a discrete stochastic approximation technique has been recently applied to solve several other problems in wireless communications [3], [10]. Moreover, the algorithm can also be applied for the scenario of time-varying channels, where the time-varying optimal user subset can be effectively tracked. Specifically, a fixed step size is adopted for the adaptive tracking process, which acts as a forgetting factor. The remainder of this paper is organized as follows. Section II describes the opportunistic downlink MISO system, including the basic OFS architecture and the multiuser beamforming scheme. Section III formulates the user subset selection problem as a discrete stochastic optimization problem and then proposes the discrete stochastic approximation algorithm for solving it. Simulation results are given in Section IV, and Section V contains the conclusions. II. SYSTEM DESCRIPTIONS A. Opportunistic Fair Scheduling A generalized architecture for the OFS in downlink wireless systems is proposed in [13] and is shown in Fig. 1. The objective is to simultaneously achieve two goals: maximizing the total system throughput and ensuring fairness among all users. It is

Fig. 1.

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Generalized architecture of OFS.

seen that the system consists of a controller and a scheduler. The scheduler intends to choose the users with high-quality channels whenever possible to maximize the total system throughput. On the other hand, the controller aims at guaranteeing the fairness among all users; otherwise, those users with low-quality channels will not get the opportunity to transmit data. Note that the architecture is a centralized one and both the scheduling and the control are implemented at the base station. users in the system. At Suppose that there are in total each scheduling interval, a certain number of users are selected as the active ones. It is seen from Fig. 1 that, at the scheduling interval corresponding to the th time slot, the inputs to the scheduler are the data flows of all users, the channel conditions, and the control parameters given by the controller . The scheduler makes the decision for on the rates of all users the current scheduling interval. We have , . Those satisfying are chosen as active users. The objective of the scheduler is to maximize the weighted instantaneous system throughput [8], [13], i.e., (1) subject to certain physical constraints, e.g., the total power constraint. On the other hand, the inputs to the controller at the control interval corresponding to the th time slot are the throughput priorities of all users and the rate decioutput by (1). The deterministic fairness constraint is sion given by [13] (2) Note that, in practical systems, can normally be calculated as the average throughput over a finite-length window [18]. The controller adaptively updates the fairness weights of

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all users according to and , where the least meansquare-type algorithm is employed [11]–[13]. In particular, denote a vector as

given by [5], [6]

where

Denote and as the beamformers and the powers of all users, respectively. Also denote as the total power constraint at the base station and , , as the minimum SINR requests of all users. One design strategy for the downlink multiuser beamforming system is to find the beamformers and the powers for all active users so as to maximize the minimum ratio between the achievable SINR and the SINR requirement [7], [17]. That is

denotes the rate decision [i.e., solution to (1)] based . Then, the deterministic fairness constraint on the given (2) is equivalent to and, thus, the controller needs to find the solution for this equation, i.e., the . Note that contains expected values, which zeros of , a stochastic cannot be directly observed. To find the zeros approximation algorithm is used [13]. In particular, define the as unbiased noisy observation of

Then, at each control interval, the weight vector as [13]

is updated (3)

is the step size and normally chosen to conwhere . In this way, the fact that verge to zero, e.g., guarantees that will converge to the zeros of with probability one. Remark: Note that the scheduling interval and the control interval are at different time scales. It is seen from Fig. 1 that the scheduling interval is much smaller than the control interval, i.e., there are many scheduling updates [given by (1)] between each two consecutive control updates [given by (3)]. The scheduling process is an iterative one (cf. Section III), and the weight vector is kept fixed during this process so that the algorithm can converge to the optimum.

. The received SINR at the th mobile is then SINR

(5)

SINR

subject to

(6) , , and SINR , as the solution to (6). Following [7] and [17], the algorithm for solving (6) involves two steps. The first step is to given , and the second step is to compute compute and given and . Step 1) Compute . An iterative algorithm is proposed in [6] and [17] to obtain the optimal in (6). Denote the optimal beamformers as beamformers under a given power set . Then, is given by the dominant generalized eigenvector of the matrix pair Denote

On the other hand, given and a set of beamformers , it is shown in [17] that the optimal powers can be computed as follows. Define , , , and

B. Multiuser Downlink Beamforming In this paper, we consider the downlink multiantenna system employing transmit antenna beamforming. It is worth noting that, although we focus our concern in a single-cell scenario, the results can be easily extended to multicell scenarios [17]. We assume that there are transmit antennas at the base station and a single receive antenna at each mobile. Suppose that there are users simultaneously active at each time slot. In a downlink multiuser beamforming system, the received signal by the th user during one symbol interval is given by (4) where is the channel response of the th user, , , and are the beamformer, the transmit power, and the transmitted symbol, respectively, of the th user, and is the additive noise at the th mobile. Define

.. .

where

.. .

..

.

.. .

. Let and be the maximal eigenvalue and the corresponding eigenvector of . is given by Then, the optimal power vector . The optimal beamformers in (6) can now be iteratively computed as follows. Starting with , calculate . Then, compute and the corresponding maximum eigenvalue, denoted . Repeat this procedure, i.e., compute as

LI AND WANG: ADAPTIVE OPPORTUNISTIC FAIR SCHEDULING OVER MULTIUSER SPATIAL CHANNELS

,

, and . The iteration stops if for some preset value . The optimal beamformer set is then given by . and . Given and , it is Step 2) Compute shown in [19] and [20] that the optimal power and the optimal achievable SINR ratio in (6) can be computed as follows. Let and be the maximal eigenvalue and the corresponding eigenvector of ,

record

where

,

, and

are defined as before. Then and . Furthermore, we have SINR , . III. ADAPTIVE USER SUBSET SELECTION In this paper, we assume that the channel , , keeps invariant within each scheduling interval. For the scenario keeps invariant for different scheduling of fixed channels, intervals; for the scenario of time-varying channels, the autore, gressive (AR) model is adopted to describe the dynamic of which will be detailed later. In what follows, we first state the optimal user subset selection problems under two scheduling strategies and formulate them under the framework of discrete stochastic optimization. We then propose solutions based on discrete stochastic approximation algorithms for both static and time-varying channels, which are detailed in Sections III-B and III-C, respectively. A. Optimal User Subset Selection for Multiuser Scheduling At each scheduling interval, the multiuser opportunistic scheduler in (1) selects the optimal subset of users for transmission to maximize the weighted instantaneous system as the number throughput. Denote as a user subset and of users in (e.g., selecting the first, second, and fourth users corresponds to and ). Let be the set of as the set of all possible candidate user subsets. Denote the channels for the users in at the interval . Then, for a given , is a function user subset , the rate of a user in , , which will be specified below. Denote the objective of function in (1) as (7) , Then, we can rewrite (1) as where denotes the best user subset at the interval . Note that we assume that and remain fixed within each interval . For notational simplicity, hereafter we drop the time slot index . Then we can rewrite the optimization problem (1) . as Note that, in practice, is estimated and therefore noisy. Denote as a sequence of the noisy estimates of . For each estimate , we can compute the optimal beamformers and power vector as discussed in Section II-B and the corresponding noisy estimate of the

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at the th iteration, denoted as objective function . Then, we obtain the sequence of the noisy observafor the fixed . If each tions is an unbiased estimate of , the discrete optimization problem can then be reformulated as the following discrete stochastic optimization problem: (8) We next consider two multiuser schedulers with different forms . of Scheduler I: Fixed-Size User Subset: Given a particular user of all users in , the optimal subset and the channels , , ) can be calculated as beamforming scheme ( as the corresponding discussed in Section II-B. Denote for user subset and channel under the optimal beamforming. To guarantee the minimum SINR constraints, the bal. anced SINR ratio of the user subset should satisfy , . Then the SINRs of the users are also Let balanced, i.e., SINR , . Then the achievable rate for the th user is given by SINR for each . In the first scheduling scheme, at each time slot, users simultaneously receive data where is typically chosen as , where is the number of transmit antennas. Suppose that users in total in the system. Then, the size of the there are whole solution space is given by . The objective function is then given by (9) Note that there exists no SINR threshold requirement in Scheduler I, i.e., is no longer a necessary constraint. Scheduler I treats the candidate user subset with fixed size and without SINR threshold requirement. Such a scheduler is defined from the practical concerns as follows. Note that, if certain SINR threshold is required, then only the user subset satisfying can be viewed as the possible candidate in ; if no SINR threshold exists, any user subset is the possible candidate. Scheduler I is suitable for applications which have no strict rate requirement for individual users (i.e., no hard SINR threshold), e.g., image communications. Also note that, to achieve all of the degrees of freedom of the channel, the size of the candi. Thus, date user subset should be variable, i.e., , is trementhe size of the solution space, dous. The fixed-size constraint on the candidate user subset can significantly reduce the implementation complexity, since the . Therefore, size of the solution space is reduced to Scheduler I can be viewed as a suboptimal multiuser scheduler to approximately achieve all degrees of freedom of the downlink MISO system. Scheduler II: Variable-Size User Subset With Minimum PerUser Rate Requirement: In this case, the number of users in the subset is not fixed but only upper bounded by . Thus, the total number of possible subsets is . Here, a feasibility condition is imposed on a subset, namely, each user in the chosen subset should meet its minimum SINR requirement, i.e., , . From [5], [6], and [16], this SINR implies that . The objective function of the scheduler

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can be detailed as follows. Given two possible subsets, and , both satisfying the minimum SINR requirement, we would prefer the subset with more users; if they both have the same number of users, we would prefer the subset with the higher total rate. Denote

and for Schedulers I and II, estimates are respectively. Among all computational costs, however, only the estimates corresponding to the optimal user subset are eventually useful. Moreover, if the channels are time-varying, such a scheme cannot track the time-varying optimal subset. Another approach for solving the discrete stochastic optimization problem (8) is to exploit an iterative procedure which resembles a stochastic approximation algorithm. In the algorithm, a sequence of estimates of the solution is generated, and each new estimate is obtained using the previous one and a good direction toward the global optimizer [1], [2], [9]. One favorite property of such an algorithm is its high computational efficiency in the sense that most of the computational cost is spent close to the optimizer of the objective function. We now present the stochastic approximation algorithm for user subset selection based on [1], which generates a random walk over the different user subsets where each new subset is obtained from the old one by moving in a good direction toward the global optimizer. Note that in this section, we focus on the scenario of static channels, and the scenario of time-varying channels will be addressed in Section III-C. unit vectors to denote all possible user We use the subsets. That is, , where denotes a vector, with a one at the th position and zeros elsewhere. At vector each iteration, the algorithm updates the , which represents the state occupation probabilities with the elements satisfying and . Denote as the user subset visited at the th iteration. For notational simplicity, we map a sequence of user subsets to the sequence , whose element is a unit vector if , . The satisfying discrete stochastic approximation algorithm for solving the optimum user subset selection problem (8) within each time slot is summarized as follows. Note that the quantities with the superscript ( ) denote those at the th iteration.

(10) Then, the objective function through the difference sets and as follows:

for Scheduler II is defined between the two sub-

(11) The first two terms in (11) correspond to the case that both of the two user subsets satisfy the minimum SINR requirements, where the first one implies that the candidate with higher sum throughput is selected when the two candidates have the same size, and the second one implies that the scheduler prefers to select the user subset with a larger size. The latter two items in (11) correspond to the cases of only one user subset satisfying the minimum SINR requirement. Scheduler II, which treats the candidate user subset with variable size and certain SINR threshold requirement, also arises from some practical concerns. Note that a certain SINR threshold is required in several applications, e.g., voice transmission in cellular systems. In such scenarios, the sum throughput can be roughly denoted by the number of users accommodated by the system. B. Discrete Stochastic Approximation Algorithm Several algorithms can be exploited to solve the discrete stochastic optimization problem (8). One method is the exhaustive search in the whole solution space, i.e., enumerate all possible solutions and compute an empirical average. That is, for each , compute , and then the . optimum user subset is obtained by Since for any fixed , is an independent identically distributed (i.i.d.) sequence of random variables, by the strong law of large numbers, almost surely as . Using the , finite number of user subsets in implies that, as then

Although such an exhaustive search can in principle find the optimum solution, it is inefficient in the sense that most computational costs are useless. In particular, for each , estimates of the channels and the corresponding objective function values are needed, and thus the total numbers of required

Algorithm 1 [Adaptive user subset selection] ; randomly select the initial user 1) Initialization: and ; set the probability subset vector by and for all . , estimate the cor2) Sampling and evaluation: Given ; solve the optimal beamforming responding problem to obtain ; calculate based on (9) or (11); uniformly choose another candidate , compute the corresponding . , then set 3) Acceptance: If ; otherwise, set . 4) Adaptive filter for updating state occupation probabilities: Update the state occupation probabilities by , where is a decreasing step size. 5) Updating the estimate of the optimizer at the current iter, then set ation: If ; otherwise, set . 6) , and go to step 2).

LI AND WANG: ADAPTIVE OPPORTUNISTIC FAIR SCHEDULING OVER MULTIUSER SPATIAL CHANNELS

The sequence generated in steps 2) and 3) is a Markov chain on the state space and, in general, is not expected to converge. In step de4), notes the empirical state occupation probability of the at the th iteraMarkov chain tion. In step 5), the optimization problem is equivalent to . Hence, the algorithm essentially chooses the state most frequently visited by the Markov chain . The sequence contains the estimates of the optimal user subset at different iterations. As discussed below, under certain conditions, almost as surely as or, equivalently, the Markov chain generated by the algorithm spends more time in the optimal user subset than in any other subset. Note on Convergence: A sufficient condition for Algorithm 1 to converge to the global optimizer of the objective function is as follows [1]. Denote as the optimal subset and and as two nonoptimal subsets. Let , , and be the independent observations corresponding to , , and , respectively. If the following two conditions are satisfied: (12) (13) generated by Algothen the sequence rithm 1 converges almost surely to the global optimizer , in visits the sense that the Markov chain more frequently than any other state, i.e., for all as . Condition (12) states that it is more probable for the Markov chain to move to from a state that is not the optimizer than in the reverse direction. Also, (13) states that if the Markov chain is not in the state corresponding to the optimizer, it will more probably move to the optimizer than to any other state. It remains an open problem to analytically verify the convergence of the stochastic approximation algorithm for user subset selection considered in this paper, though simulation results in Section IV demonstrate the excellent performance of the alis obtained gorithm. Note that the objective function through a sophisticated computational from the channel procedure involving the optimal beamformer calculation discussed in Section II-B (to obtain ) and the calculation of (9) (for Scheduler I) or (11) (for Scheduler II). Hence, even can be assumed to though the noisy channel estimate have a Gaussian distribution, it seems impossible to obtain the analytically. Nevertheless, to gain exact distribution of some insight on the convergence behavior of Algorithm 1, we carried out extensive numerical study to verify conditions (12) and (13). Specifically, for Scheduler I, defined by (9), we found is biased and approximately has a Gaussian distrithat , where bution, i.e., , and there are no closed-form expressions for the mean and the variance. Fig. 2 shows the histogram of for Scheduler I, and the corresponding Gaussian fit by using the empirical mean and variance. The simulation conditions are as follows: the number of simultaneous transmissions ; the number of transmit antennas is ; the total is

Fig.

2.

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(m; ) and the Gaussian fit T = 4, jj = 4,  = 0:05, = 0:05, Scheduler I, 8(H ) = 13:64, and f(m; )g = 13:58. Histogram

of

N ( f(m; )g; varf(m; )g):

 = 1, 

transmit power constraint at the base station is ; and the noise level is . It is seen from Fig. 2 that the districan be well approximated as Gaussian. The bution of is also shown. Since is biased, to exact value guarantee the solution given by Algorithm 1 coinciding with the original optimal solution , the following condition should be satisfied: If (14) Analytical proof of (14) remains open, although numerical results indicate that it seems to hold. Assuming that the above Gaussian assumption holds, then the independent observations of the objective function satisfy , , and . Then, condition (12) is equivalent to , which in turn can be written as (15) , i.e., Now (14) implies that . Thus, (12) is satisfied, since both sides of (15) have the same variance. Furthermore, condition (13) is equivalent to , which can be rewritten as . In addition, after normalizing, this is equivalent to . Although no analytical results are available on the mean and the variance of the estimate of the objective function, numerical results indicate that it seems to hold. C. Adaptive Algorithm for Time-Varying Channels Thus far, we have assumed that the user channels are static and, therefore, for a fixed , the optimal user subset is timeinvariant. Next, we consider the time-varying channels case,

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where the user subset selection problem has time-varying optimal solutions. Consequently, the user subset selection algorithm should be capable of tracking the optimal user subset when the channel variation is not so fast. Under the static channel condition, in Algorithm 1, a decreasing step size is employed, . With such an approach, the method gradually becomes more and more conservative as the number of iterations increases, and thus it will not move away from a promising point unless there appears to be strong evidence that the move can result in an improvement. On the other hand, in the time-varying channel case, we need such a step size that moving away from a state is permitted when the optimum user subset changes [10]. Hence, the step of adaptive filter for updating state occupation probabilities in Algorithm 1 is replaced by (16) . The fixed step where is a fixed step size satisfying size in (16) introduces an exponential forgetting factor of the past occupation probabilities and allows us to track the slowly time-varying optimum user subset.

Fig. 3. Total rate of the chosen user subsets versus the iteration number. T =

4, N = 8, and jj = 4, for Scheduler I fixed-channel case.

IV. SIMULATION RESULTS The simulation conditions are as follows. The number of transmit antennas at the base station is ; the total number in total; the total power of users in the system is constraint at the base station is ; and the noise level is . For Scheduler I, the size of the user subset is for any (and therefore, the total number ), and the minimum SINR of user subsets is requests are set as , . For Scheduler II, the maximum number of users for all possible user subset is for any (and therefore, the total three, i.e., number of user subsets is ); and the minimum SINR requests are , . Moreover, in a realistic communications scenario, for each in step 3) of Algorithm 1, the noisy estimates iteration and are needed to compute the noisy oband , respectively. Note that servations several algorithms can be employed to obtain the channel estimate in multiantenna systems [14]. In the simulations, is generated according to , where the contains i.i.d. zero-mean Gaussian random error term variables , where . A. Optimal User Subset Selection in Static Channels We first show the effectiveness of the discrete stochastic approximation algorithm for optimal user subset selection. The fairness weights are all set as 1. Hence, in this case, Scheduler I maximizes the instantaneous throughput, whereas Scheduler II maximizes the instantaneous throughput and the number of simultaneous transmissions while guaranteeing the minimum rate requirement of each user. The channel responses for all users are randomly generated and fixed for all simulation runs. Fig. 3 shows the total instantaneous rate of the chosen subset versus the iteration number for Scheduler I. The curve obtained in one simulation run and the curve averaged over 100 runs are both

Fig. 4. Total rate of the chosen user subsets versus the iteration number. T = 3, for Scheduler II fixed-channel case.

4, N = 8, and jj



shown in Fig. 3 together with the throughput of the optimal user subset obtained by an exhaustive search. Fig. 4 shows the similar results for Scheduler II. It is seen from both Figs. 3 and 4 that the discrete stochastic approximation algorithm can effectively find the optimum subset. Moreover, it is seen that the algorithm can quickly lock on a user subset with performance close to the optimal performance. B. Time-Varying Optimal User Subset Tracking Next, we demonstrate the tracking performance of the discrete stochastic approximation algorithm in time-varying channel scenarios. Suppose the channel responses for all users keep fixed within symbols. The first-order AR model over is adopted to describe the dynamic of the channel as , where is for the th user, contains i.i.d. elements , and the parameters and satisfy . We set the values of

LI AND WANG: ADAPTIVE OPPORTUNISTIC FAIR SCHEDULING OVER MULTIUSER SPATIAL CHANNELS

Fig. 5.

Total rate of the chosen user subsets versus the iteration number. T =

Fig. 6.

Total rate of the chosen user subsets versus the iteration number. T =

4, N = 8, and jj = 4, for Scheduler I time-varying channel case.

4, N = 8, and jj  3, for Scheduler II time-varying channel case.

the parameters as and . The fixed step size . The fairness weights in Algorithm 1 is chosen as are all set as 1. Fig. 5 shows the total rate of the chosen user subset versus the iteration number for Scheduler I over a single simulation run. Comparing the instantaneous total throughput of the optimal user subset with that of the chosen user subset, it is seen that the algorithm can track the time-varying optimum user subset closely. Similar results for Scheduler II are shown in Fig. 6, from which the same conclusion can be drawn. C. Fairness Guarantee Finally, we show the system performance in terms of the fairness among all users. Here, we assume that there are in total five users in the system ( ). For Scheduler I, the throughput requests of all users are all equal, i.e., the normalized rate re, . For Schedquests of the users are uler II, the normalized throughput requests of all users are set

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Fig. 7. Normalized throughput of all users versus the time slot number. T = 4, N = 5,  = 3, Scheduler I, fixed channel case, and  =  =  =  =  = 1=5.

jj

Fig. 8. Normalized throughput of all users versus the time slot number. T = 4, N = 5,  3, Scheduler II, fixed channel case,  =  = 1=8, and  =  =  = 1 =4 .

jj

and . Moreover, the as , and the size of size of the user subset for Scheduler I is user subset for Scheduler II is upper bounded by . First, we consider the fixed channel case. Within each time is kept fixed; the optimum user subset is selected, and slot , is updated for the next time slot. Note that the channel then responses for all user subsets over all time slots are fixed. There are 100 iterations within each time slot. Fig. 7 shows the normalized throughput of all users versus the time slot number for Scheduler I. Fig. 8 shows a similar result for Scheduler II. It is seen that, at the beginning, the gap between the achieved throughput of the users and their own throughput requests is relatively large, because only the users with better channel conditions are chosen for transmissions. As time elapses, the fairness weights of those users with better channel conditions are reduced, and thus the users with worse channel conditions but

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V. CONCLUSION

Fig. 9. Normalized throughput of all users versus the time slot number. T = 4, N = 5,  = 3, Scheduler I, time-varying channel case, and  =  =  =  =  = 1=5.

jj

We have developed a framework for downlink TDMA systems employing multiple transmit antennas and beamforming to achieve efficient resource use over multiuser channels while guaranteeing the long-term fairness. Multiple-transmit-antenna beamforming is employed at the base station to support multiple user-concurrent transmissions. Given the user subset and their channel conditions, the optimal downlink multiuser beamforming scheme can be calculated. The multiuser opportunistic scheduling problem then refers to the optimum user subset selection for each time slot to maximize the instantaneous system throughput subject to certain fairness constraints. We have developed the discrete stochastic approximation algorithm to achieve efficient and effective optimal user subset selection. The algorithm is able to track the time-varying optimal user subset when the channels vary. The algorithm is iterative in nature and we have presented two scheduling schemes, with or without minimum rate constraints on individual users. We present simulation results to demonstrate that the algorithms can effectively find the optimal user subset with good convergence performances, and adaptively track the time-varying optimum solutions in the nonstationary environments. The system can also achieve fairness among all users over large time scales. REFERENCES

Fig. 10. Normalized throughput of all users versus the time slot number. T = 4, N = 5, jj  3, Scheduler II, time-varying channel case,  =  = 1=8, and  =  =  = 1=4.

higher fairness weights obtain more opportunities to be selected for transmissions. In this way, the lagging users will catch up with those forwarding ones. This phenomena can be clearly seen from Figs. 7 and 8. Finally, we consider the time-varying channel case. The same AR channel model as that in Figs. 5 and 6 is used. The values are all the same as those in Figs. 5 and 6. of , , , and Note that there are 400 iterations within each time slot, during is kept fixed. Fig. 9 shows which the fairness weight vector the normalized rates of all users versus the time slots for Scheduler I. Fig. 10 shows similar results for Scheduler II. It is seen that, although the rate requirements for all users are not satisfied within a short time scale, the long-term fairness among all users can be well guaranteed.

[1] S. Andradóttir, “A global search method for discrete stochastic optimization,” SIAM J. Control Optim., vol. 6, no. 6, pp. 513–530, May 1996. , “Accelerating the convergence of random search methods for dis[2] crete stochastic optimization,” ACM Trans. Model. Comput. Simul., vol. 9, no. 4, pp. 349–380, Oct. 1999. [3] C. R. N. Athaudage and V. Krishnamurthy, “A low complexity timing and frequency synchronization algorithm for OFDM systems,” in Proc. IEEE Global Telecommun. Conf., Taipei, Taiwan, R.O.C., Nov. 2002, pp. 244–248. [4] V. Bharghavan, S. Lu, and T. Nandagopal, “Fair scheduling in wireless networks: Issues and approaches,” IEEE Pers. Commun., vol. 6, no. 1, pp. 44–53, Feb. 1999. [5] H. Boche and M. Schubert, “SIR balancing for multiuser downlink beamforming—A convergence analysis,” in Proc. IEEE Int. Commun. Conf., New York, NY, Apr. 2002, pp. 841–845. , “Solution of the SINR downlink beamforming problem,” in Proc. [6] Conf. Inform. Sci. Syst., Princeton, NJ, Mar. 2002, [CD-ROM]. , “Optimal multi-user interference balancing using transmit beam[7] forming,” Wireless Pers. Commun., vol. 26, no. 4, pp. 305–324, Sep. 2003. [8] S. Borst and P. Whiting, “Dynamic rate control algorithms for HDR throughput optimization,” in Proc. IEEE INFOCOM, Anchorage, AK, Apr. 2001, pp. 976–985. [9] W. B. Gong, Y. C. Ho, and W. Zhai, “Stochastic comparison algorithm for discrete optimization with estimation,” SIAM J. Control Optim., vol. 10, no. 2, pp. 384–404, Feb. 1999. [10] V. Krishnamurthy and X. Wang, “Spreading code adaptation in multipath fading channels for CDMA systems,” in Proc. 36th Asilomar Conf. Signals, Syst., Computers, Pacific Grove, CA, Nov. 2002, pp. 843–848. [11] X. Liu, E. K. P. Chong, and N. B. Shroff, “Opportunistic transmission scheduling with resource-sharing constraints in wireless networks,” IEEE J. Sel. Areas Commun., vol. 19, no. 10, pp. 2053–2064, Oct. 2001. , “Transmission scheduling for efficient wireless utilization,” in [12] Proc. IEEE INFOCOM, Anchorage, AK, Apr. 2001, pp. 776–785. [13] Y. Liu and E. W. Knightly, “Opportunistic fair scheduling over multiple wireless channels,” in Proc. IEEE INFOCOM, San Francisco, CA, Mar. 2003, pp. 1106–1115. [14] T. L. Marzetta, “BLAST training: Estimating channel characteristics for high capacity space-time wireless,” in Proc. 37th Annu. Allerton Conf. Commun., Control, Computing, Monticello, IL, Sep. 1999, pp. 958–966.

LI AND WANG: ADAPTIVE OPPORTUNISTIC FAIR SCHEDULING OVER MULTIUSER SPATIAL CHANNELS

[15] S. Nanda, K. Balachandran, and S. Kumar, “Adaptation techniques in wireless packet data services,” IEEE Commun. Mag., vol. 38, no. 1, pp. 54–64, Jan. 2000. [16] M. Schubert and H. Boche, “Solvability of coupled downlink beamforming problems,” in Proc. IEEE Global Telecommun. Conf., San Antonio, TX, Nov. 2001, pp. 614–618. [17] , “Solution of the multiuser downlink beamforming problem with individual SINR constraints,” IEEE Trans. Veh. Technol., vol. 53, no. 1, pp. 18–28, Jan. 2004. [18] P. Viswanath, D. N. C. Tse, and R. Laroia, “Opportunistic beamforming using dumb antennas,” IEEE Trans. Inf. Theory, vol. 48, no. 3, pp. 1277–1294, Jun. 2002. [19] W. Yang and G. Xu, “Optimal downlink power assignment for smart antenna systems,” in Proc. IEEE Int. Conf. Acoust., Speech, Signal Process., Seattle, WA, May 1998, pp. 3337–3340. , “The optimal power assignment for smart antenna downlink [20] weighting vector design,” in Proc. IEEE Veh. Tech. Conf., Ottawa, ON, Canada, May 1998, pp. 485–488.

Chuxiang Li received the B.S. and M.S. degrees in electronic engineering from Tsinghua University, Beijing, China, in 1999 and 2002, respectively. He is currently working toward the Ph.D. degree in electrical engineering at Columbia University, New York, NY. From January to June 2002, he was a Visiting Student with Microsoft Research Asia (MSRA), Beijing, China. From February to June 2005, he was a Research Assistant with NEC Laboratories America, Princeton, NJ. His research interests fall in the general areas of wireless communications, statistical signal processing, wireless resource allocation and scheduling, coding techniques, and error-resilient techniques for wireless video.

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Xiaodong Wang (S’98–M’98–SM’04) received the B.S. degree in electrical engineering and applied mathematics (with highest honors) from Shanghai Jim Tong University, Shanghai, China, in 1992, the M.S. degree in electrical and computer engineering from Purdue University, West Lafayette, IN, in 1995, and the Ph.D. degree in electrical engineering from Princeton University, Princeton, NJ, in 1998. From July 1998 to December 2001, he was an Assistant Professor with the Department of Electrical Engineering, Texas A&M University. In January 2002, he joined the Department of Electrical Engineering, Columbia University, New York, NY, as an Assistant Professor. His research interests fall in the general areas of computing, signal processing, and communications. Among his publications is a recent book entitled Wireless Communication Systems: Advanced Techniques for Signal Reception (Englewood Cliffs, NJ: Prentice-Hall PTR, 2003, with H. V. Poor). His current research interests include wireless communications, statistical signal processing, and bioinformatics. Dr. Wang received the 1999 National Science Foundation CAREER Award and the 2001 IEEE Communications Society and Information Theory Society Joint Paper Award. He currently serves as an Associate Editor for the IEEE TRANSACTIONS ON COMMUNICATIONS, the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, the IEEE TRANSACTIONS ON SIGNAL PROCESSING, and the IEEE TRANSACTIONS ON INFORMATION THEORY.

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Systolic Array Implementation of a Real-Time Symbol-Optimum Multiuser Detection Algorithm Chung-Chin Lu, Member, IEEE, Jau-Yuan Hsu, and Chih-Chung Cheng, Student Member, IEEE

Abstract—This paper presents the systolic array implementation of a real-time symbol-optimum multiuser detection (MUD) algorithm for a direct-sequence code-division multiple-access system by truncating the backward recursions in the generalized forward/backward schedule. Simulation results show that the real-time algorithm provides negligible performance loss compared to the original symbol-optimum detection algorithm. The systolic array implementation is derived in this paper through the factor graph language of the real-time algorithm in order to exploit the suitability of the algorithm for parallel signal processing. Index Terms—Code-division multiple access (CDMA), factor graphs, multiuser detection (MUD), parallel signal processing, systolic array.

I. INTRODUCTION ULTIUSER detection (MUD) for direct-sequence codedivision multiple-access (DS-CDMA) systems has received considerable attention over the past decade. The symboloptimum MUD based on the backward–forward dynamic programming is described in [1]. The maximum-likelihood (ML) multiuser sequence detection based on the Viterbi algorithm is proposed in [2]. Several linear multiuser detectors are studied in [3]–[5]. Multiuser detectors that employ interference cancellation methods are studied in [6]–[9]. A rich list of references can be found in [10], in which other kinds of multiuser detectors are also discussed. Among these multiuser detectors, the symbol-optimum detector achieves the minimum symbol error rate (SER) [1]. Since this detector requires the processing of the whole sequence of received bits to make an optimum decision of any specific transmitted bit, it appears to be prohibitively complex for a real-time implementation of this symbol-optimum detection. In this paper, we present a modified algorithm called the truncated symbol-optimum MUD which lends itself to a real-time systolic array implementation. Such a modification is based on the observation that the detection of a given symbol is more affected by its neighboring symbols than the distant ones. The

M

Paper approved by C. Schlegel, the Editor for Coding Theory and Techniques of the IEEE Communications Society. Manuscript received March 28, 2003; revised February 3, 2005. This work was supported by the National Science Council, Taiwan, R.O.C., under Contract NSC89-2213-E-007-060. This paper was presented in part at the 2003 International Symposium on Communications, Taoyuan, Taiwan, R.O.C., December 2003. C.-C. Lu and C.-C. Cheng are with the Department of Electrical Engineering, National Tsing Hua University, Hsinchu 30013, Taiwan, R.O.C. (e-mail: [email protected]; [email protected]). J.-Y. Hsu is with Acer Communications and Multimedia, Inc., Taipei 114, Taiwan, and also with the BenQ Corporation, Taipei 114, Taiwan, R.O.C. (e-mail: [email protected]). Digital Object Identifier 10.1109/TCOMM.2005.857161

truncated algorithm is derived by applying the backward–forward dynamic programming within a sliding window and makes it possible to trade off the performance with the latency and the computation load. Simulation results show that this real-time algorithm performs almost the same as the original symbol-optimum detection. A real-time maximum a posteriori (MAP) decoding algorithm applied to convolutional codes has been studied before in [11]. This algorithm is based on one-way recursion and is inappropriate for parallel processing. To exploit its applicability for parallel processing, the truncated algorithm is described by the factor graph language [12] in this paper. Such a view from factor graphs will reveal the natural connection between the truncated algorithm and the systolic array implementation. The remainder of this paper is organized as follows. In Section II, we brief the basic asynchronous CDMA model and the symbol-optimum MUD. The factor graph language is introduced in Section III. In Section IV, we present the truncated symbol-optimum MUD, together with the simulation results. The systolic array implementation of the truncated symbol-optimum MUD is presented in Section V. A conclusion is given in Section VI.

II. SYMBOL-OPTIMUM DETECTION IN ASYNCHRONOUS CDMA SYSTEMS A. System and Channel Model Consider that each of asynchronous users transmits data with a preassigned from the antipodal binary alphabet signature waveform over a common additive white Gaussian noise (AWGN) channel. The received waveform will be a noisy sum of all users’ signals

(1) where , , , , and represent the received bit energy, the th transmitted bit, the signature waveform, the time delay, and the carrier phase of the th user, respectively. Also, is the bit interval, is the length of the data stream of each is the carrier frequency, and is a bandpass white user, Gaussian noise with two-sided power spectral density . We assume that all signature waveforms are normalized to have unit energy and zero outside the interval , and , , ,

0090-6778/$20.00 © 2005 IEEE

LU et al.: SYSTOLIC ARRAY IMPLEMENTATION OF A REAL-TIME SYMBOL-OPTIMUM MULTIUSER DETECTION ALGORITHM

, , , , and for each are either known or estimated by some mechanism to facilitate the MUD. The interference between users are quantified by the cross correlations of , we denote the signature waveforms. If

for

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and

(2) (3) Though all users transmit their own data streams, it is convenient to take these data streams of length as a single data . Hence, we define an -vector with stream of length components for , where and represent the modulo- remainder and quotient . We also write the of , respectively. Note that , outputs of the matched filters matched to the signature waveforms, in an -vector with components .

for , with . The backward recursion operates from the last stage to the first stage as

(8) for tion by

, with for any . After both recursions are completed, the objective funccan be evaluated for and

B. Symbol-Optimum MUD For the th bit , MUD is to select a value in posteriori probability

, the symbol-optimum which maximizes the a

(4) As studied in [10], when , the a posteriori probability in (4) is proportional to the function (5), shown at the bottom of the page, where is the th state vector with for and

(6) with the th component of the -dimensional state . vector and efficiently, the backward–forward dyTo calculate namic programming such as the Bahl–Cocke–Jelinek–Raviv (BCJR) algorithm [13] can be applied to a trellis with state vector , input , and branch metric . There are two independent iterations in the backward–forward dynamic programming, namely, the forward recursion and the backward recursion. The forward recursion operates from the first stage to the last stage as

(7)

(9) Although the symbol-optimum MUD achieves the minimum SER, the decoding delay due to the forward and the backward is recursions will be intolerable if a long sequence length under consideration. We will present a real-time version of this algorithm through the factor graph language in the remainder of this paper. III. VIEW BY FACTOR GRAPHS A. Factor Graphs and the Sum–Product Algorithm The notations introduced here follow those used in [12]. Let be a collection of variables indexed by a finite set . For each , the variable takes on values from is a subset of , then the alphabet set . If is the subset of variables indexed by . A particular is called assignment of a value to each of the variables of a configuration of the variables. Clearly, configurations of the , variables are elements of the Cartesian product called the configuration space. For a subset , the set of is denoted by all subconfigurations with respect to . be a real-valued function with the elements Let as arguments. Suppose, for some collection of subsets of of , that the function factors as (10) where, for each , is a real-valued func. is called a local function. tion with arguments in The factor graph is a bipartite graph that expresses the strucin (10) ture of the factorization of the global function

(5)

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[12]. A factor graph has a variable node for each variable , , and a (nondirected) a factor node for each local function connecting variable node to factor node edge if and only if (iff) is an argument of . For a node , denotes the neighbors of . be a real-valued function with arguments in , Let , we define the marginal where is a subset of . For each function as

In a GFB schedule, exactly one updated message will pass in either direction on every edge of the tree, and an updated mes[according to the sum–product update rules sage in (12) or in (13)] is scheduled to be sent through the directed at time , i.e., , only if all neighbors of edge but have passed an updated message to before time . Thus, only leaf nodes of the tree can send an updated message at the very first time(s). Specifically, at a leaf variable node and at a leaf factor node , the update rules in (12) and (13) become

(11) The marginal function is a function of only. An is algorithm for computing the marginal functions based on the notion of message passing on the factor graph and is called the sum–product algorithm, as self-explained shortly. denote the message sent from the variable node Let to the factor node , and let denote the message sent from the factor node to the variable node . The message computations performed by the sum–product algorithm is subject to the following sum–product update rules: • variable node to factor node : (12) • factor node

(15) (16) All subsequent message passings among the nonleaf nodes will be scheduled by the all-but-one rule, and a GFB schedule will terminate when all of the leaf nodes receive an updated message from their neighbors. It is shown in [12] that, after a GFB schedule is completed on a is just cycle-free factor graph, the marginal function the product of the messages received at variable node , as given in (14). A GFB schedule can be said to be efficient since only one message passes in either direction on every edge of the tree. C. Symbol-Optimum MUD

to variable node :

Note that the messages sent through the edge in either directions are functions of the variable associated with that edge. In many applications, since variables are discrete (and the alphabet sets are finite), the messages are usually in vector form. To initiate the message passing, we will set all messages to be a unit message, i.e., a message representing the unit function, before the sum–product algorithm starts. When the factor graph is cycle-free, i.e., a tree, the sum–product algorithm is guaranteed to give the marginal funcas tions

The objective functions for the symbol-optimum MUD are , , in (5) for . These objective functions can be represented as the marginal functions of a global function as defined below. For a simple detailed example made to illustrate how to represent an objective function as a marginal function of a global function, please refer to [12]. , Let the variables considered be input bits , and matched-filter outputs state vectors , denoted as , , and , respectively, in vector , form for brevity. Note that the ’s take on values in ’s take on values in for the and values in for , and the ’s always take on the values of the matched-filter outputs (i.e., variables with given value). Then, the global function is defined as

(14)

(17)

after a suitable message passing schedule is completed [12]. A particularly useful schedule on a cycle-free factor graph will be discussed in the next subsection.

where the local functions are given in (18) at the bottom of the are next page. It is now clear that the objective functions , i.e., just marginal functions of the global function

(13)

B. Generalized Forward–Backward Schedules Assume that there is a global discrete-time clock synchronizing the message passing. A message-passing schedule for the sum–product algorithm on the factor graph is a sequence , where is a set of edges, now considered to be directed, over which messages pass at time . We will restrict ourselves to the so-called generalized forward–backward (GFB) schedules on cycle-free factor graphs [12].

(19) As shown in Fig. 1, the factor graph associated with (17) is , and , as a tree with , , leaf nodes and , , and , as nonleaf nodes. Hence, the sum–product algorithm is guaranteed to give the marginal function (19) after a GFB schedule is completed on the factor graph. The message passing in a fragment of the tree is also shown in Fig. 1. A GFB schedule

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Fig. 1.

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b X y).

Factor graph representation for the global function g (

;

;

for the sum–product algorithm on this tree is specified as follows:

From (20) and (21), we have

(22)

(23) Now, at the last time clock for all sages passing on edge

As shown in (15), the updated messages passing on edges in at the first time clock are unit messages, i.e., and , for all . , , the updated messages At time passing on edges in are (20)

, the updated mesis

(24) of which is just the marginal functions the global function for by the sum–product algorithm and by (19), is equal to the object function in (9). Note that it is not necessary to update the for all . message on edges Also, by letting and , (22) and (23) are equivalent to (7) and (8), respectively, by (18) and (11).

(21) where and to be the unit message 1. Next, at time the updated message passing on edges in

, for are

are defined ,

IV. TRUNCATED SYMBOL-OPTIMUM MULTIUSER DETECTION As shown in the previous two sections, for the optimum detection of the th bit for any , the backward recursion in the sum–product algorithm can be triggered only when the last matched-filter output is observed. Thus, the decoding delay of a long data sequence will be long enough to prohibit a real-time implementation. However, the decoding delay problem can be alleviated by conjecturing that the observation of will have negligible effect on the detection of whenever . This suggests a suboptimum detection of by performing the sum–product algorithm on a segment of the factor graph, centered at the variable node . The size of

if otherwise

(18)

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the segment is called the window size and is assumed to be an . odd number Now, the localized global function associated with the th window is

(25) where , , . The object funcand tion of the sum–product algorithm for the th window is . Thus, once the th matched-filter output is observed, the sum–product algorithm for the th window to detect the th bit can be activated. is observed consecutively As the matched-filter output as time goes on, a sliding window on the factor graph will be created to activate consecutive sum–product algorithms. On the other hand, the message passing on the factor graph for the forward recursions of the consecutively generated localized sum–product algorithms are synchronized. Thus, all of the “local” forward recursions can be merged into a single “global” forward recursion and, as a good side effect, all of the previously , can be observed matched-filter outputs , starting from used to detect the th bit with the truncated global function

(26) In addition, the “local” backward recursion of the sum–product algorithm for the th window can be terminated when it meets with the forward recursion at the center of the th window since the object function for the detection of the th bit is . Since the “local” backward recursion for the detection of starts once additional matched-filter outputs the th bit are observed and terminates when it meets with the single “global” forward recursion at the center of the th window, the decoding delay is limited and upper bounded by , no matter how long the data sequence is. Thus, it is possible to real-time implement this truncated symbol-optimum MUD. In the next section, a real-time parallel processing for the multiple backward recursions with a systolic array architecture will be proposed. Without a rigid proof, we conduct a simulation study of the performance of the truncated symbol-optimum MUD. Two asynchronous DS-CDMA multiuser systems are considered. The first is a two-user system with a medium cross correlation , (as studied in [6]) and a high cross corre, , respectively. As shown in Fig. 2, lation under the medium cross-correlation condition, the performance loss on the averaged bit-error rate (BER) can be neglected as compared with the original symbol-optimum MUD (dashed lines) when a window size of five is used, no matter how power is assigned to either user. Indeed, using a window size of three is sufficient, and an obvious improvement is observed when the window size is increased from three to five. Next, under the , , heavy cross-correlation condition, i.e., it can be seen in Fig. 3 that a larger window size up to 11 is needed to combat the heavy multiple-access interference

Fig. 2. Average BERs of the truncated symbol-optimum detection in a two-user system with heavy cross correlation ~ = 0:4, ~ = 0:2 in different window sizes. The original symbol-optimum detection is shown by dashed lines for all cases.

(MAI) in this case. However, a window size of seven is good enough to neglect the performance loss as compared with the original symbol-optimum MUD. In Fig. 4, individual BERs of user 0 and user 1 are compared with the single-user bounds , (dashed curves) under different rela. It tive power assignments for the fixed window size can be seen that the performance of the truncated detection is very close to the single-user bound. Also, as shown in Fig. 4, larger gaps exist between the individual BER of each user and the single-user bound due to the heavy MAI. However, similar to the medium MAI case, the weaker user has a BER close to the single-user bound when the power difference of the two users is enlarged. Second, we consider a four-user system with signature waveforms derived from Gold sequences of length 7. We assume that . Then, the cross correlations ’s are (27) We also let correlations

such that the effective cross in (6) have the largest mag-

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Fig. 4. BER of each user of the truncated symbol-optimum detection in a two-user system with medium cross correlation ~ = 0:4, ~ = 0:2, and window size W = 5 (denoted by #1), and heavy cross correlation ~ = 0:5, ~ = 0:4 and window size W = 11 (denoted by #2). Single-user bounds are shown by dashed lines.

section will be derived in this section, first in an intuitive way and second in a systematic way. Fig. 3. Average BERs of the truncated symbol-optimum detection in a two-user system with heavy cross correlation ~ = 0:5, ~ = 0:4, in different window sizes. The original symbol-optimum detection is shown by dashed lines for all cases.

nitudes for worst case consideration. It can be seen from Fig. 5 that, for a window size of 15, the averaged BER is almost the same as that of the original symbol-optimum MUD (dashed lines), no matter how power is assigned in the four-user system. Also, a window size of 11 is sufficient to ignore the performance loss as compared with the optimum detector. When the power assignment is more even, a more obvious performance improvement is observed when window size is increased from 3 to 11. The individual BER of each user is shown in Fig. 6 with a fixed window size , where the BERs of the weaker users (user 2 and user 3) tend to meet the single-user bound as the SNR difference is enlarged. Our simulation results have demonstrated that the truncated symbol-optimum MUD algorithm with a small window size (5–15) performs almost the same as the original symbol-optimum MUD. Note that the length of data sequences in our simulation is of order . V. SYSTOLIC ARRAY IMPLEMENTATION A systolic array implementation of the sum–product algorithm on the truncated factor graph discussed in the previous

A. Intuitive Derivation There are two things to be observed. First, since each varihas exactly two neighbors, it simply passes the able node message received from one neighbor to the other without any , a buffer can be assigned. Second, change. Hence, to each leaf nodes and only pass unit messages out. Furthermore, variable nodes ’s always take on the values of the matchedfilter outputs, and the messages received at the variable nodes ’s are just the object functions for the detection. Thus, the variable nodes ’s will be grouped into the input device and the variable nodes ’s into the output device. Now, the main operations of the sum–product algorithm will be in factor nodes. As illustrated in Fig. 7 with window ( ), at time , only factor node size is involved in the single global forward recursion, the th local backward recursion activated at time , and the summary operation for the conclusion th bit , i.e, . Also, only factor nodes are involved in the simultaneously active th th local backward recursions activated , respectively. Thus, only at times factor nodes are active at time in the sum–product algorithm. We now use processing element (PE) cells to implement the operations active factor nodes, respectively. The PE cell on the corresponding to the active factor node dealing with the single

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Fig. 6.

BER of each user of the truncated symbol-optimum detection with

W = 15 in a four-user system. Single-user bounds are shown by dashed lines.

Fig. 5. Average BERs of the truncated symbol-optimum detection in a four-user system in different window sizes with the original symbol-optimum detection shown by dashed lines.

global forward recursion and the summary operation will be labeled as PE , and the PE cells corresponding to the active factor nodes dealing with local backward recursions are labeled PE . These PE cells are interconnected as PE to form a systolic array, together with an input device, an output device, and a control logic, as shown in Fig. 8. There are three data flows passing in the systolic array. The passing first is the forward message flow within the PE cell. The second is the backward message flow passing through all backward recursion PE cells to PE , and finally to the PE cell. The third from PE is the local function flow , generated in the input device with given as the th matched-filter output and passing through all backward recursion PE cells PE from the right end of the systolic array to the PE cell at the left end. As mentioned before, the forward and backward and can be repremessages sented as vectors of dimension . Also, the local function can be represented as a vector of dimension , since must be equal to and is given and fixed.

As described above, there are two kinds of PE cells in the systolic array. One is the “FS” type (where F stands for forward recursion and S stands for summary operation), and the other is the “B” type (B stands for backward recursion). A B-type PE cell, as shown in Fig. 8, should perform the following functions. by per1) Update the output backward message forming a backward recursion backward as in (8) or (23), with the input local function and the . input backward message 2) Defer the local function flow to make it fall one step behind the backward message flow. 3) Latch the output backward message , the output local function , and the deferred local function . An FS-type PE cell, as shown in Fig. 8, should perform the following functions. by performing 1) Update the output forward message a forward recursion forward as in (7) or (22), with the input local function and the input forward message . 2) Perform the summary operation to produce the marginal function summary according to the input local function , the input forward message , and the input backward message by (9) or (24). and the output 3) Latch the output marginal function forward message . The operation of the systolic array is described as follows. The first is an initialization phase which starts with the input of

LU et al.: SYSTOLIC ARRAY IMPLEMENTATION OF A REAL-TIME SYMBOL-OPTIMUM MULTIUSER DETECTION ALGORITHM

b

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W

Fig. 7. Time diagram of the backward recursions, the forward recursion, and the summary operations for ’s for = 7. Here, the numbers in the hexagons mean the time instants. The thin boxes mean the factor nodes, the circles mean the buffers for variable nodes x , and the thick boxes mean potential PE cells.

Fig. 8.

Block diagram of the systolic array architecture with the PE cell definitions shown in dashed-line boxes.

from the input device in the first local function data and ends with the arrival of the first local function data at the . At the beginning of , the control cell PE in unit sets the loop switch (SW ) in Fig. 8 to connect to the unit , it switches WS to conmessage and at the end of to of PE to form a loop. The control unit sets nect the output switch (SW ) in Fig. 8 to be opened at the beginning of . It then switches SW to connect of PE to the . Next, the systolic array output device at the end of enters a steady-state phase, which starts with the output of the to the output defirst objective function vice in , and ends with the input of the last local function in . During this phase, object functions data for are output to the output device consecutively. Finally, the array enters an ending phase from to . During this phase, the input device inputs dummy local function data, and the object functions for are output to the output device. A scenario example is shown in Fig. 9 with , where ( ) is used to denote the forward (backward) message as the result of the global forward (local backward) recursion which is activated at time , and has processed the local function data from to . Note that and are the initial values of the forward and backward , the evaluated recursions, and hence, are unit messages. At local function data is input to the first B-type PE cell of the

, the evaluated local function data is systolic array. At is deferred in the cell. input to the first B-type PE cell, while , the evaluated local function data and are input At to the first and the second B-type PE cells, respectively, while is deferred in the first cell. At , the first backward meswith initial value appears at the input to the first sage flows B-type PE cell, while the evaluated local function data into the systolic array. At , the first backward message has processed the data and appears, together with , at the input to the second B-type PE cell, while the second with initial value appears, together backward message with , at the input to the first B-type PE cell. At , the has processed one additional first backward message and appears, together with , at the input to the third data B-type PE cell, and the second backward message has processed the data and appears, together with , at the input to the second B-type PE cell, while the third backward meswith initial value appears, together with , at sage , the single forthe input to the first B-type PE cell. At ward message with initial value appears at the left input to the single FS-type PE cell, while the first backward has processed one more additional data message and reaches, together with , to the right input to the FS-type PE cell, the second and the third backward messages and has processed data and , respectively, and appear at the input to the third and the B-type PE cells, and a appears at the input to the first new backward message

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Fig. 10. Composite data-dependency graph of the recurrence equations (28)–(31) in the 2-D space–time domain, where h = 3. Transformed graphs by matrices (33) and (34) are also shown, respectively.

Fig. 9. Scenario example of

W = 7.

and are initialization data. Data satwhere each isfy summary To map the computation into the 2-D space–time domain, we need to fully index the above three equations as backward (28) backward (29)

B-type PE cell. At

, the summarized first objective function is output from the FS-type PE cell, while and the second backward message the forward message have processed the data and , respectively, and appear, together with the data , at the input to the FS-type PE cell. From now on, the systolic array enters a steady-state phase.

forward (30) summary where

B. Systematic Derivation We now derive the systolic array implementation in a formal systematic way [14]. , we let be the reFor each sult of a local backward recursion that has processed the local to , where function data from . Also, for , we let be the result of the global forward recursion that has processed the local function data from to and let be the object function for . Data and satisfy the following recurrent relations: backward forward

,

, and

(31) are pipelined as (32)

The composite data-dependency graph of the recurrence equations (28)–(31) in the 2-D space–time domain is shown in the first part of Fig. 10. To find an appropriate array topology, we have to find a time axis along which the composite data-depenseems to be a good dency graph is projected. Direction positive time direction, because it will produce a finite number cells). However, with such an arrangement, of cells ( will flows along the positive time direction, while will flow along the negative time direction. This is impossible. Thus, we transform the graph with the matrix (33) The modified graph is shown in the middle part of Fig. 10. Now, the direction is the positive time direction. and

LU et al.: SYSTOLIC ARRAY IMPLEMENTATION OF A REAL-TIME SYMBOL-OPTIMUM MULTIUSER DETECTION ALGORITHM

will flow along the positive time direction, but the flow of is perpendicular to the time direction which implies that appear simultaneously. Hence, some modification is still needed. A second linear transformation (34) is then selected to form a suitable version of the data-dependency graph, as shown in the final part of Fig. 10. By projecting , a systolic array consisting of the graph along direction two types of PE cells, as shown in Fig. 8, is derived. In the Appendix, we show that the derived systolic array implementation is optimum in some sense. C. Tradeoff Between Real-Time Implementation and Complexity The computation load of the symbol-optimum detection and the truncated symbol-optimum detection is mainly contributed by three parts: the metric data computation, the forward and backward iteration, and the summary operation for ’s. Now, , , and be the computation load for calculating , let or , and , respectively. To detect an -bit-long frame, the computation load of the conventional symbol-optimum detection is

algorithms with reduced complexity but yet maintaining negligible performance loss. APPENDIX LINEAR TRANSFORMATION OF THE COMPOSITE DATA-DEPENDENCY GRAPH We consider a linear transformation

where – are arbitrary integers. In order to produce a finite number of PE cells, we have to choose in the original graph as the positive time direction in the transformed graph. Thus, in (39) and we require that becomes direction of

for convenience. The (40)

The direction of

becomes (41)

(35) If

, detecting one bit takes

Also, the new direction of

is

(36) On the other hand, the computation load of the truncated symbol-optimum detection is (37) Thus, if

, detecting one bit takes (38)

Hence, the truncated symbol-optimum detection takes additional computations to detect one bit. This is the cost we pay for real-time implementation. However, we should note that, though a larger window-size increases the computation load and the latency, the throughput is unchanged. VI. CONCLUSION We have presented the systolic array implementation of a real-time symbol-optimum MUD algorithm by first truncating the backward recursion in the original symbol-optimum detection algorithm and then describing the truncated algorithm with the factor graph language. Simulation results have demonstrated that the proposed real-time truncated algorithm has negligible performance loss as compared to the original algorithm when the window size of the truncated algorithm is properly selected. However, since the complexity per bit of the original symbol-op[10], our real-time symbol-optimum timum detection is detection algorithm still suffers from such a high complexity. For further research, it is plausible to develop real-time MUD

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(42) which is the same as the positive time direction. We require that and , so that all the data flow along the positive as a unit time in the transformed time direction. Taking will result in a lower efficiency of graph, an increase of the systolic array (i.e., fewer PE cells work simultaneously). and can be interpreted as the time it took and to and simply move one step in the systolic array. Larger delay and longer. determines the spaces between the cells in the array, which will not change the result in effect. Actually, the previously chosen linear transformation (43) is equivalent to choosing , , and . With the recurrence equations we have written in (28)–(31), it seems to be an optimum choice in some sense, because the result under the constraint smallest (44) Rewriting the recurrence equations might be the only way to derive a better systolic algorithm. ACKNOWLEDGMENT The authors would like to thank the National Center for High-Performance Computing (NCHC), Taiwan, R.O.C., for supporting the computing resource for their simulation. They

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would also like to thank the Associate Editor and the anonymous referees for their valuable comments and suggestions that helped improve the presentation of this paper.

Chung-Chin Lu (S’86–M’88) was born in Taiwan, R.O.C. He received the B.S. degree from National Taiwan University, Taipei, Taiwan, R.O.C., in 1981, and the Ph.D. degree from the University of Southern California, Los Angeles, in 1987, both in electrical engineering. Since 1987, he has been with the Department of Electrical Engineering, National Tsing Hua University, Hsinchu, Taiwan, R.O.C., where he is currently a Professor and Department Chair. His current research interests include coding theory, digital communications, bioinformatics, and quantum communications.

REFERENCES [1] S. Verdú, “Optimum multiuser signal detection,” Ph.D. dissertation, Univ. Illinois, Urbana-Champaign, IL, 1984. , “Minimum probability of error for asynchronous Gaussian mul[2] tiple access channels,” IEEE Trans. Inf. Theory, vol. IT-32, no. 1, pp. 85–96, Jan. 1986. [3] R. Lupas and S. Verdú, “Linear multiuser detectors for synchronous code-division multiple-access channels,” IEEE Trans. Inf. Theory, vol. 35, no. 1, pp. 123–136, Jan. 1989. , “Near–far resistance of multiuser detectors in asynchronous chan[4] nels,” IEEE Trans. Commun., vol. 38, no. 4, pp. 496–508, Apr. 1990. [5] Z. Xie, R. Short, and C. Rushforth, “A family of suboptimum detectors for coherent multiuser communications,” IEEE J. Sel. Areas Commun., vol. 8, no. 5, pp. 683–690, May 1990. [6] A. Duel-Hallen, “A family of decision-feedback detectors for asynchronous code-division multiple-access channels,” IEEE Trans. Commun., vol. 43, no. 4, pp. 421–434, Apr. 1995. [7] T. Giallorenzi and S. Wilson, “Suboptimum multiuser receivers for convolutionally coded asynchronous DS-CDMA systems,” IEEE Trans. Commun., vol. 44, no. 9, pp. 1183–1196, Sep. 1996. [8] M. Varanasi and B. Aazhang, “Multistage detection in asynchronous code-division multiple-access communications,” IEEE Trans. Commun., vol. 38, no. 4, pp. 509–519, Apr. 1990. [9] L. Wei and C. Schlegel, “Synchronous DS-CDMA system with improved decorrelating decision-feedback multiuser detection,” IEEE Trans. Veh. Technol., vol. 43, no. 3, pp. 767–772, Aug. 1994. [10] S. Verdú, Multiuser Detection. Cambridge, U.K.: Cambridge Univ. Press, 1998. [11] L. N. Lee, “Real-time minimal bit-error probability decoding of convolutional codes,” IEEE Trans. Commun., vol. COM-22, no. 2, pp. 146–151, Feb. 1974. [12] F. R. Kschischang, B. J. Frey, and H.-A. Loeliger, “Factor graphs and the sum-product algorithm,” IEEE Trans. Inf. Theory, vol. 47, no. 2, pp. 498–519, Feb. 2001. [13] L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal decoding of linear codes for minimizing symbol-error rate,” IEEE Trans. Inf. Theory, vol. IT-20, no. 3, pp. 284–287, Mar. 1974. [14] G. M. Megson, An Introduction to Systolic Algorithm Design. Oxford, U.K.: Clarendon, 1992.

Jau-Yuan Hsu was born in Taiwan, R.O.C. He received the B.S. and M.S. degrees in electrical engineering from National Tsing Hua University, Hsinchu, Taiwan, R.O.C., in 1998 and 2000, respectively. He is currently a Research and Development Engineer with Acer Communications and Multimedia, Inc., Taipei, Taiwan, R.O.C. His research interest is in cellular and mobile communications.

Chih-Chung Cheng (S’00) was born in Taiwan, R.O.C. He received the B.S. and M.S. degrees in electrical engineering in 1996 and 1998, respectively, from National Tsing Hua University, Hsinchu, Taiwan, R.O.C., where he is currently working toward the Ph.D. degree. His research interests are in coding theory, wireless communication networks, and mobile communications.

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An Overlay Architecture for Managing Lightpaths in Optically Routed Networks Vinay A. Vaishampayan, Senior Member, IEEE, and Mark D. Feuer, Senior Member, IEEE

Abstract—Existing solutions for optically routed networks (ORNs) lack certain key management functions available to electronically routed networks. In particular, ORNs have extremely limited capabilities to trace the path of an optical signal through the network. In this paper, we present a method for embedding path identification and other management information into the transport stream in such a way that the management information can be read by a low-bandwidth, low-cost receiver, without having to terminate or decode the full-rate payload stream. We outline a method for embedding such management information, using a digital coding process at the transmitter, and two distinct digital decoding processes for receiving the management and payload data streams, respectively. Feasibility of the method is demonstrated by computing the bit-error performance of example codes under realistic operating conditions, including multiple management streams in multiwavelength systems. Index Terms—Code-division multiple access (CDMA), constant weight codes, optical fiber communications, optically routed networks (ORNs), path-trace information, reconfigurable optical add/drop multiplexers (ROADMs), spectral null codes, spread-spectrum communications.

I. EMBEDDING INFORMATION IN OPTICAL STREAMS ONVENTIONAL networks for optical communication transmit information from one node to another as optical signals, but require full conversion of all data from optical form to electrical form at every node. In a ring or mesh network, the data may have to traverse many nodes as it passes from a source node to a destination node. Thus, the data will be converted from optical to electrical form and back many times before reaching its destination. Equipment associated with these optical–electrical–optical (OEO) conversions makes up the bulk of the capital cost of a conventional optical communication network. In addition, the electrical routing equipment used in the OEO nodes is specific to a particular modulation format and data rate, so that an upgrade to increase the capacity of a particular channel will require replacement of OEO equipment all along the route, from source to destination. This implies a substantial expense in both capital budget and operating budget

C

Paper approved by R. Hui, the Editor for Optical Transmission and Switching of the IEEE Communications Society. Manuscript received October 26, 2004; revised April 19, 2005. This paper was presented in part at the European Conference on Optical Communication, Stockholm, Sweden, September 2004, in part at the Brazilian Telecommunictions Symposium, Belem, Brazil, September 2004, and in part at the Optical Fiber Communications Conference, Anaheim, CA, March 2005. V. A. Vaishampayan is with AT&T Labs-Research, Florham Park, NJ 07932 USA (e-mail: [email protected]). M. D. Feuer is with AT&T Labs-Research, Middletown, NJ 07748 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TCOMM.2005.857129

associated with channel upgrades, and often results in delays until a complete new build can be finished. Problems with OEO conversions are particularly pronounced in networks which use wavelength division multiplexing (WDM) technology. WDM allows a single fiber to carry many distinct data channels by encoding each data stream onto its own optical wavelength, and then combining the wavelengths for transport through the fiber. Multiple fiber spans can be concatenated by inserting optical amplifiers between them. Up to 80 wavelengths can be handled in a single amplifier, so the cost per data channel can be greatly reduced through the use of WDM. However, existing WDM networks still require separation of individual wavelengths whenever OEO conversion is needed, and each OEO converter handles only one wavelength, so the cost of OEO routing nodes scales very unfavorably as traffic demands grow. Optical communication networks with optical–optical (OO) nodes, based on all-optical routing systems which do not require conversion of signals to electronic format, have been developed to greatly reduce the initial capital cost of networks, while providing a flexible method for capacity upgrades of channels. The OO nodes traversed by the optical signal as it passes from source to destination are transparent to modulation format and data rate, so an upgrade can be achieved by changing the equipment at the source and destination nodes only. Thus, optically routed networks (ORNs) are expected to yield substantial savings in both capital expense and operating expense associated with channel upgrades. The ORNs being developed today are WDM-capable: that is, the reconfigurable optical add/drop multiplexers (ROADMs) and photonic cross connects (PXCs) at the network nodes can control signal routing on a wavelength-bywavelength basis. The route followed by a wavelength from source node to destination node through the ORN is called a lightpath. Fig. 1 shows an example of an ORN in which four separate data signals are carried on the same wavelength. The solid lines show fiber routes, and the four gray lines (solid, dotted, dashed, and dot–dashed) represent the four distinct lightpaths. Despite these advantages, the problem with existing solutions for ORNs is that they lack several key management functions provided by the OEO nodes in conventional networks. In particular, existing solutions for ORNs provide only very limited capability to trace a signal path through the network. Presence of a particular wavelength at a node is easily checked by optical means, but these methods cannot distinguish between different lightpaths which use the same wavelength. If mechanical failure or operator error causes incorrect routing of an optical signal, so that data is delivered to the wrong destination, the network may be unable to identify the cause and location of the fault.

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Fig. 1. In an optically routed WDM network, multiple lightpaths may be carried on the same wavelength. Then a path ID technique is needed to assure correct routing or to locate routing faults.

This suggests the need for a method for embedding a unique optical path identifier (path ID) into each optical signal in the network, so that each lightpath can be traced throughout the ORN. The method must allow use of a very inexpensive detector mechanism, so that path ID detectors can be ubiquitous, to permit path tracing and fault localization to very precise locations within the network. A path ID detector at each OO node would be highly desirable. Since the OO nodes are fairly complex assemblies, it might be even better to have multiple detectors within each node, to identify which subassembly within a node is at fault. More generally, one can replace the fixed path ID word by a time-varying data stream that carries network management information. Such an in-band supervisory channel (ISC) might provide distribution of network routing tables, a voice wire for communication between offices, or element manager communications needed to monitor the health of components and subsystems throughout the network domain. The most widely studied method of implementing such an ISC has been overmodulation with one or more subcarrier tones [1]–[3]. For photonic networks, continuous wave (CW) subcarriers could provide an optical path ID, by assigning a unique tone frequency to each lightpath present in the network. The ubiquitous detectors for such a path ID signal could be relatively inexpensive photoreceivers that do not have a tunable optical filter or optical wavelength selector. However, there are several problems with the overmodulation-based ISC, primarily associated with the interference between payload and side data. 1) The modulation depth of the path ID tone must be kept small to minimize degradation of the data payload. This limits the sensitivity of the path ID function. In [2], the pilot-tone modulation index was set to 3% in order to limit the payload SNR penalty to 0.5 dB. 2) The power spectrum of the optical payload signal also produces interference to the subcarrier channel. To achieve reasonable side data error rates, the subcarrier frequency must be chosen either above [2] or below [1], the main spectral peak of the payload signal. Adopting the former choice, Hamazumi et al. [2] find that the subcarrier frequency must be larger than 4 GHz when the payload is at 10 Gb/s, leading to expensive side-data receivers with relatively noisy front ends.

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 53, NO. 10, OCTOBER 2005

3) Pilot-tone frequencies below the spectral peak of the payload are possible, but for OC-48 SONET signals, they must be below about 50 kHz, severely limiting the available side-data rate. For slower SONET signals, such as OC-12 (622 Mb/s) or OC-3 (155 Mb/s), the low-frequency pilot tones appear quite impractical. 4) Low pilot-tone frequencies also lead to cross-gain modulation in the optical amplifiers in the network [3], causing unintended “ghost” tones to appear on other wavelengths passing through the same optical amplifier. To minimize the crosstalk in optical amplifiers, the path ID signals should operate at frequencies above 0.3 MHz. 5) In WDM systems with a wavelength-insensitive receiver, the payload-to-side-channel crosstalk will be multiplied, and crosstalk will also appear among the various side channels. In particular, Chung et al. [3] have shown that crosstalk due to stimulated Raman scattering (SRS) can limit the use of pilot tones in WDM systems with high wavelength count and high launch power. 6) Finally, the overmodulation method may also require an extra optical modulator, which adds cost to the system. The digital method described here addresses all of the above issues related to the tone overmodulation method, while maintaining the virtue of ubiquitous monitoring by low-cost, wavelength-insensitive photoreceivers. Any additional optical modulator is eliminated, while no new optical components are added. The method permits the path ID signal to operate at higher, but still moderate, data rates where the cross-gain modulation is insignificant. Path IDs are distinguished in the receiver by a digital decoding process which does not demand narrowbandpass filtering, and the new method can be used in conjunction with lower payload data rates. In addition, the new method may also provide enough bandwidth for per-channel signaling, which can be used to transmit more detailed management information from one node to another within the network. It is important to note that no OEO conversion is required at the path ID monitoring locations. Rather, the path ID receivers operate from low-power optical tap couplers which introduce negligible degradation to the propagating signals. Since it involves the modulation of the time-average optical power at rates slower than the payload rate, the new method is susceptible to the kind of SRS-induced crosstalk discussed in [3]. However, the integrated encoding and digital reception processes can eliminate both payload-to-side-channel crosstalk and side-to-payload channel penalties, enabling chip rates (see below) as high as 100 Mchips/s or more, with appropriately chosen code parameters. As noted in [3], such higher frequencies will substantially relieve SRS crosstalk. In addition, the use of code-division multiple-access (CDMA) techniques in the receiver provides enhanced rejection of any residual crosstalk, ensuring error-free reception of the side-channel data. The coding process presented here may also be seen as a form of asymmetric multiplexing, in which digital coding combines a high-rate data stream and a low-rate data stream into a single data stream. The special feature of this method is that the cost of decoding the lower rate stream is smaller than that of decoding the higher rate stream, since the low-rate data stream can be extracted by an inexpensive receiver with low bandwidth. The

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Fig. 2. Encoder and decoders for the system with embedded side data. Note that the chip rate is much smaller than the coded payload rate.

inherent fairness of the decoding costs might make such techniques more generally applicable, as well. The remainder of the paper is organized as follows. The system design is presented in Section II, followed by noise and interference analysis in Section III, performance results in Section IV, and ending with a summary and conclusions in Section V. In this paper, we refer to “side information” or “side data” as general terms that include the path ID as well as other supervisory information to be sent via the overlay channel constructed here. This work is an extended version of that presented in [4]. II. OVERLAY COMMUNICATIONS SYSTEM Some of the interference issues that limit the performance of the tone-modulation overlay system can be mitigated through the use of spectral shaping codes. The idea is to use digital coding techniques to shape the data spectrum, so as to make room for the side channel to be used for carrying path-trace data. Examples are Manchester coding and coded marked inversion (CMI) coding, which insert a spectral null at dc, thus enabling use of an ac-coupled channel. Spectral approaches such as the above presume that the sidechannel data will be mixed with the payload data in the optical domain. Here, we consider the possibility of combining data in the digital domain. The result is that there is no interference between the path ID and payload signals, and the need for an extra optical modulator is removed. Additionally, a wavelengthselective filter is not required for side-data demodulation. The lack of a wavelength-selective filter in the side-data demodulator leads to interference between various side-channel

signals. That is, the path ID signal at any given node will include components encoded on many wavelengths, generated at many different source nodes, and the path ID receiver must be capable of distinguishing these components. Our method addresses multiple-user interference (MUI) issues by using CDMA. Note that this is electrical CDMA and not optical CDMA. Also, it is worth noting that the shared medium problem is somewhat unusual, due to the fact that the bandwidth of the spread-spectrum signal, though much higher than the side-channel data rate, is several orders of magnitude smaller than the payload rate. The low-cost side-data receivers can be deployed in a ubiquitous way throughout the network, while the (more expensive) payload data receiver is deployed only at the destination node. At any given point in the network, the maximum number of users will be limited to , the number of wavelengths used by the optical equipment, though the total number of lightpaths could be much larger. We assign a unique signature to each wavelength during the side-data coding process. Thus, we design the CDMA system for users. A. Encoding Method and Data Frame Structure The encoding system that we have devised integrates a directsequence CDMA (DS-CDMA) technique with a complementary constant weight code (CCWC) and is illustrated in Fig. 2. information bits select a codeword from a CCWC of length bits. The code consists of codewords, half of constant weight ( ) and the other half of weight , obtained by complementing each codeword in the first group. The information bits always select a codeword of weight . Groups of codewords are output in serial order and XOR’ed with

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Fig. 3.

Data block structure for the overlay system.

Fig. 4. SNR.

Representative example of an ORN. Signals entering the network from different points may traverse different numbers of amplifiers, affecting their final

a spreading sequence which has been modulated by the side information. This XOR’ed bit stream is used to generate an on–off keying (OOK) signal at a specific wavelength and transmitted over the optical fiber. The following bit streams are associated with each lightpath or user , each at a different time scale: the raw payload bit stream with a bit duration ; the coded paywith a bit duration ; the signature bit load bit stream , a periodically repeated signature sequence of stream chips, with chip duration ; and the side-data stream , with a bit duration . (The subscript identifying the lightpath will be added when necessary.) Streams of different rates are combined using the XOR function to yield a stream at the higher . of the two rates, e.g., we obtain Here, denotes the value obtained by rounding down to the for . In nearest integer. We use the special notation other words, the signature stream is modulated by the side-data bit stream, and the resulting stream modulates the coded payload bit stream to generate . The structure of the encoded data frame is shown in Fig. 3. bits long, such blocks Each block of the CCWC code is constitute a single chip of the spreading sequence, and chips

of the spreading sequence constitute a single period of the spreading code, hereafter referred to as a frame, of equivalent duration as a side information bit. Thus, each chip of the s. Since spreading sequence has a duration of side information bits are transmitted using nonoverlapping but contiguous blocks of the spreading sequence, the frame s. duration is given by III. NOISE AND INTERFERENCE ANALYSIS The baseband models describe electrical current waveforms sent to the optical transmitter and received from the analog photoreceiver. The optical fiber network and intervening components, including electrical amplifiers and filters in the optical receiver, make up the channels. Two channel models are needed: one for the payload transmission and one for the side-data transmission. We have used Gaussian approximations consistent with those used in [5]. The optical network example to be modeled is shown in Fig. 4. Optical wavelength multiplexing occurs both within local terminal units and at the PXC and ROADM nodes

VAISHAMPAYAN AND FEUER: AN OVERLAY ARCHITECTURE FOR MANAGING LIGHTPATHS IN ORNs

throughout the network. Optical amplifiers are provided after each fiber span and each lossy network node to restore the optical signal power of each wavelength to its original launch level. Each amplifier adds spontaneous emission light which will contribute noise when the signal is detected at the receiver. In optically amplified networks, this amplified spontaneous emission (ASE) noise is typically the primary factor limiting the network reach, so we will include only ASE noise and receiver noise in our present model. In the network of Fig. 4, each wavelength is used in only a single lightpath, so there is no wavelength reuse, but that simplification should not materially affect the results given below. Various lightpaths traverse differing numbers of fiber spans and amplifiers, leading to wide variations in the spontaneous emission noise present at different wavelengths. Since the sidedata photodiode PD-S has no wavelength-selective filter, it will receive the full complement of wavelength signals and ASE noise, leading to both larger noise terms and multiple-channel interference. The payload photodiode, on the other hand, will receive only a single wavelength signal and the ASE noise within that one wavelength slot. Information is transmitted on the optical fiber via rectangular s signaling pulses of effective duration

elsewhere.

(1)

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is the optical bandwidth, assumed to be equal to the where channel spacing of 50 GHz. The payload demodulator averages the received current over and then applies a single-bit binary a payload bit interval decision process. The output current of the averager at time is denoted by . It has expected value (5) where ance

is the autocorrelation function of the noise, and vari-

(6)

where the terms represent two-sided electrical current noise power spectral densities from signal-spontaneous beat noise, spontaneous–spontaneous beat noise, and circuit noise. The ciris calculated from the circuit noise coefficient cuit noise (in units of A Hz) obtained from a specification sheet . These three noise sources are via the formula the most significant for the ORNs considered here. Following [5, (4.8) and (4.9)], we use the formulas

Assuming perfect extinction ratio and equal probability of ones and zeros, the average optical launch power is equal to . First, consider the payload channel. Since the payload receiver follows the wavelength demultiplexing unit (WDU), there is no MUI, and noise-inducing ASE power is limited to a single spectrum slice surrounding wavelength . The optical signal power is restored to its nominal launch value by the final associated optical amplifier, but then is reduced by the loss with the tap coupler and WDU (see Fig. 4). Then, the optical amplitude corresponding to the modulated signal of user incident on the photodetector is , where

which take into account both fundamental polarization modes. The circuit noise coefficient value is indicated in Section IV. We is larger than the have assumed that the optical bandwidth electrical bandwidth ( ) of the payload receiver. The two optical terms can be expressed in terms of the optical SNR , conventionally defined as optical signal power divided by the ASE power contained within 0.1 nm (12.5 GHz) of optical bandwidth, by setting

(2)

(9)

The optical power reaching the photodetector is converted to electrical current with responsivity factor , and receiver noise is added. The electric current generated by the photodetector is given by (3) where represents the spontaneous emission noise from the is assumed to be a zero-mean stationary optical amplifier. Gaussian random process with autocorrelation function and power spectral density

otherwise

(4)

(7) (8)

A single-symbol binary decision yields the conventional formula for the bit-error probability (BEP) [5] snr

(10)

where (11)

snr and Marcum’s

function is given by (12)

The side-data channel requires a more complex model, including the sum of signal-spontaneous noise contributions from

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all users plus a large spontaneous–spontaneous contribution integrated over the entire wavelength range. Clearly, the noise level at the side-data receiver will be highest when all user wavelengths are present, and every user is operating at the worst optical SNR (OSNR) condition tolerated by the payload receiver. If the side data can operate with an acceptable error rate under this condition, it will operate acceptably under all conditions. The side-data detector must also accommodate MUI. The multiple users may be synchronous or asynchronous; due to the practical issues with synchronizing a large mesh network, we are most interested in asynchronous operation. However, for the purpose of the analysis presented here, we will assume that a limited form of synchronism exists, i.e., streams are synchronized at the chip level. The side-data demodulator first averages the received signal over a chip interval , removes the bias, and then projects the resulting discrete-time signal on a suitably chosen “despreading” sequence. The current generated by the side-channel photodetector is given by

, and is lightpath . This is a bipolar signal, taking values through . The effective related to (two per lightpath) spreading sequences due to lightpath are denoted and and are given in terms of the spreading sequence for lightpath , , and , the delay of lightpath relative to lightpath 1 by

(13) The chip-level filter output is given by

By using the fact that , it follows that the discrete-time chip-level signal is now given by

(20) where . Observe that the chip-level representation is independent1 of the payload bit sequence . The detector projects the received vector onto a suitably chosen vector . The choice of the vector depends on the decoder. In the decorrelating decoder, whose analysis we present here (21) where

is the projection of on the subspace spanned by . The decision variable given by (22)

bias (14) where, in terms of

, we have (15) (16)

bias

(17)

is compared with zero in order to decide the value of (this is justified by the analysis that follows). In the following analysis, we derive the statistics of the terms and . Then, following [6]–[8], we will consider random delays and randomly chosen signature sequences to arrive at an asymptotic characterization of the signal-to-interference ratio (SIR). This asymptotic approach developed in the cited references allows for a particularly clean and elegant expression for the SIR. From (21), it follows that the signal term in (22) is given by (23) The variance of

is given by (24)

(18)

where (25) (26)

(19) Let

bias, and let . Let denote the spreading sequence allocated to

1In a nonideal case, where the filter defining the chip integrator is not a rectangular pulse, or when there is phase offset between transmitter and receiver chip-level clocks, the samples of the spreading waveform will exhibit an information sequence-dependent variation or cross modulation.

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and the value of the circuit noise is obtained from the circuit by . The circuit noise conoise coefficient efficient is given in Section IV. Thus we obtain the SIR sir , given by sir

(27)

and the associated error probability is sir Asymptotics for which the ratio is driven to infinity lead to

(28) is kept constant while

(29) where the convergence is in probability. We thus use the limiting form of the SIR in our calculations, which is given by sir

(30)

IV. PERFORMANCE AND TRADEOFFS Suppose we wish to support an 80-wavelength network with 100 nodes and must detect the path ID word in 1 ms. Each wavelength would receive a unique CDMA signature, and a 7-bit path ID word would suffice to identify a source node. The data rate required per wavelength would be 7 kb/s, and the total data rate needed would be 560 kb/s. To accomodate additional management functions, the per-wavelength rate could be increased to 16 kb/s. Then, the path ID function could still be accomplished within 1 ms, and a dedicated ISC with data rate of about 9 kb/s would be available to each lightpath, while the total bandwidth used by the path ID/ISC band would increase to 1.3 Mb/s. Performance of the side-data coding method is assessed by comparing its error rate with that of the high-rate payload data, under conditions of varying OSNR . For general acceptance, we use the performance measure which has been used in [2], namely, an average of one error in the path ID in ten years. Two approaches are followed for generating an alarm based on the number of bytes that must be in error. In method a, an alarm is sounded if a single byte (which is a path ID) is in error compared with a stored value, and, in method b, three consecutive bytes must be in error to raise an alarm. The probability of this and for method a and happening is approximately method b, respectively. Assuming a side-data rate of less than 200 kb/s, we may conservatively require that for method a and for method b. Achievable performance is determined using realistic parameters for various optical components in the system. For the pay, the responsivity is load receiver, the optical loss is A/W, and the circuit noise coefficient is 1 nA/ Hz appropriate for a high-bandwidth InGaAs avalanche photodiode. , the reFor the side-data receiver, the optical loss is sponsivity is A/W, and the circuit noise coefficient

Fig. 5. Error rates as a function of OSNR in an example ORN. Error probability for payload data, for side data considering only noise, and for side data considering both noise and MUI. Channel model parameters are listed in text. We denote the coded payload bit rate and the side bit rate by r := 10 =R b/s and r := r =N BC b/s, respectively.

is 0.2 nA/ Hz, appropriate for a moderate-bandwidth InGaAs GHz. The PINFET receiver. The optical bandwidth is raw payload rate is 10.0 Gb/s, , and W. The resulting bit-error rates (BERs) plotted in Fig. 5 show particular parameter settings which achieve the required BERs for methods a and b at an OSNR of 14 dB. In order to validate the approximation used for the MUI, we simulated 500 random sets of signature sequences and delays and measured the SIR obtained by a decorrelating receiver. were measured at The mean, median, and variance of and 0.9597, 0.9596, and 0.0044, respectively, for . The closeness to the theoretically predicted value of 0.96 is to be noted. For and , we obtain mean, median, and standard deviations of 0.8312, 0.8313, and 0.0173, respectively, as compared with the expected value of 0.8348. In addition to noise, optical fiber communication channels are subject to a variety of impairments, including chromatic dispersion, polarization-mode dispersion, and fiber nonlinearities. Since the side-data chip rate is much less than the payload rate,

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the side data should be quite insensitive to the residual dispersion of real systems. The effect of fiber nonlinearity is more complex, and will be addressed in future work. The following investigations are of a more theoretical nature, intended to study the tradeoffs between the various system parameters. In particular, one of the tradeoffs demonstrates the efficiency of the method. The performance of the overlay communication system is characterized by two SNRs and two rates (31) (32) snr

(33)

sir

(34)

where the and subscripts denote payload and side-channel, and are suitable constants. In the above, respectively, and the payload SNR snr is based on the uncoded stream. A constant weight code and the CCWC has a minimum Hamming distance of 2. Thus, the BEP is smaller for the coded stream, though and since it does not scale with any of the code parameters , it will not play a role in parameter selection. The most important observation is regarding the tradeoff and sir , obtained by varying the parameter . between Through Stirling’s approximation for , we are able to express this tradeoff by sir

(35)

where

is the inverse of the binary entropy function , on the domain , . For fixed , and we vary (which changes , , and hence, sir ) and plot . This tradeoff is plotted the resulting values of sir against in Fig. 6(a). The steep rise in the SIR experienced by the side channel shows how valuable this tradeoff is. We assume that , , , , snr , and sir are given. In above a certain value, is constrained from order to achieve below. We thus have three equations (the one for snr does not involve these parameters) and four unknowns, , , , and . We now show that is optimal. To see this, observe that if we lower , we must increase the product in order to satisfy allows us to choose a relathe constraint on . Increasing to achieve the payload rate constraint, tively smaller value of and this results in a larger value for sir , because increases. Increasing reduces , also increasing sir . Thus, is optimal. Nevertheless, it is worth considering the penalty incurred by choosing to be large. To understand this, note that the complexity of a constant weight code grows with its block length. Since this code must operate at the payload rate, a designer may wish to place an upper limit on the block length of the scales inversely code. However, the noise standard deviation with the product . Thus, if we have a specific noise variance target, is constrained from below, and it may be necessary to use a value of which is greater than 1. The parameter that appears in the expression for sir is twice

Fig. 6. (a) Tradeoff between sir and R for different values of N with B = 1, C = 512, U = 80, and channel model parameters as listed in the text. (b) Tradeoff between distance and rate using two methods.

the minimum distance between constant weight symbols. We investigate the nature of the tradeoff between rate and distance . blocks of a constant weight code of block length of weight or less and their complements are used. We set and derive expressions for the coding efficiency, i.e., the total number of payload and side-information bits per transmitted bit as a function of the minimum distance achieved for the side-information bit stream. In order to achieve distance in bits, we require a distance of per -bit block. Let be the number of binary . We thus transmit words of length bits with weight bits every -bit block, and (36) in a There are two extremal ways of achieving a distance block of bits. For convenience, we will assume that , . The first method sets and . Thus, we must enumerate a very long blocks of bits. The second method sets , , and ,

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There is much work still to be done. A thorough analysis is required of the effect of chip-level asynchronism on the performance, as well as a more detailed performance analysis that quantifies the effect of nonidealities such as dispersion and SRS. Another important and interesting subject will be the design and construction of constant weight codes that also have a strong error-correction capability. REFERENCES

Fig. 7. Illustrating the behavior of sir = 0:95, B = 1.

with C for constant N C

R

 62 000.

thus accruing unit distance per -bit block. In both cases, we compute the total rate (i.e., the payload plus the side-information rate) as a function of the distance achieved. For method 1, we obtain (37) For method 2, we obtain

[1] R. M. Brooks, A. Sharland, S. Whitt, R. J. Higginson, and C. J. Lilly, “An optical fiber supervisory sub-system employing an entirely optical telemetry path,” in Proc. Eur. Conf. Opt. Commun., Sep. 1982, pp. 400–405. [2] Y. Hamazumi and M. Koga, “Transmission capacity of optical path overhead transfer scheme using pilot tone for optical path network,” J. Lightw. Technol., vol. 15, no. 12, pp. 2197–2205, Dec. 1997. [3] H. S. Chung, S. K. Shin, K. J. Park, H. G. Woo, and Y. C. Chung, “Effects of stimulated Raman scattering on pilot-tone-based WDM supervisory technique,” IEEE Photon. Technol. Lett., vol. 12, no. 6, pp. 731–733, Jun. 2000. [4] M. D. Feuer and V. A. Vaishampayan, “In-band management channel for lightpaths in photonic networks,” in Proc. Eur. Conf. Opt. Commun., Sep. 2004, Paper Tu3.6, [CD-ROM]. [5] R. Ramaswami and K. N. Sivarajan, Optical Networks: A Practical Prerspective. San Francisco, CA: Morgan Kaufmann, 1998. [6] S. Verdu and S. Shamai (Shitz), “Spectral efficiency of CDMA with random spreading,” IEEE Trans. Inf. Theory, vol. 45, no. 2, pp. 622–640, Mar. 1999. [7] D. N. C. Tse and S. V. Hanly, “Linear multiuser receivers: Effective interference, effective bandwidth and user capacity,” IEEE Trans. Inf. Theory, vol. 45, no. 2, pp. 641–657, Mar. 1999. [8] Kiran and D. N. C. Tse, “Effective interference and effective bandwidth of linear multiuser receivers in asynchronous systems,” IEEE Trans. Inf. Theory, no. 4, pp. 1426–1447, Jul. 2000.

(38) These rates are plotted in Fig. 6. The penalty for choosing to be large is seen to be large. We also examine, numerically, the nature of the tradeoff be. Once is fixed, so is the product tween and with . Subject to this constraint, we plot the behavior of sir as is changed ( is chosen to satisfy the constraint for each ). It is interesting to note that the error probability for the side channel exhibits a (broad) minimum with respect to . This behavior is exhibited in Fig. 7.

Vinay A. Vaishampayan (M’85–SM’03) received the B.Tech degree from the Indian Institute of Technology, Delhi, India, in 1981, and the M.S. and Ph.D. degrees from the University of Maryland, College Park, in 1986 and 1989, respectively, all in electrical engineering. From 1989 to 1996, he was with the Electrical Engineering Department, Texas A&M University, College Station. Since 1996, he has been with AT&T Labs-Research, Florham Park, NJ. His interests are in the mathematical aspects of communications and signal processing.

V. SUMMARY AND FUTURE WORK We have designed a new digital coding method for embedding management and supervisory information into lightpaths in an optically routed WDM network. The code design explicitly provides for two different types of receivers with distinct functions. The method overcomes many of the drawbacks associated with subcarrier-based overmodulation systems, and it incorporates a multiple-access strategy for sharing of low-cost wavelength-insensitive receivers. Analysis of the BEP shows that useful side data rates can be achieved with acceptable coding overheads under the demanding conditions of a large-scale photonic network. The new technique may offer a practical and desirable all-digital alternative to conventional systems.

Mark D. Feuer (M’87–SM’04) received the B.A. degree from Harvard University, Cambridge, MA, and the Ph.D. degree from Yale University, New Haven, CT, both in physics. He is a member of the Optical Systems Research Department, AT&T Labs, Middletown, NJ. His current work focuses on enabling technologies for dynamic photonic networks, comprising topics ranging from optical device physics through network design. Prior to rejoining AT&T Labs in 2002, he was with JDS Uniphase, AT&T Labs, and Bell Labs, researching optical subsystems, high-speed measurements, and the physics and fabrication of compound semiconductor electronics. Dr. Feuer is a member of the American Physical Society. He will serve as chair of the Optical Fiber Communications Conference OFC2007.

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Multiple-Subcarrier Optical Communication Systems With Subcarrier Signal-Point Sequence Shota Teramoto and Tomoaki Ohtsuki, Senior Member, IEEE

Abstract—We propose a multiple-subcarrier (MS) optical communication system with subcarrier signal-point sequence (SSPS). We use the SSPSs having a large minimum value and large Euclidean distances, so that the required dc bias is minimized and the error-rate performance is improved. Note that in the proposed system, the signal points having the larger minimum value are selected, while the signal points having a lower peak-to-mean-envelope-power ratio (PMEPR) are selected in orthogonal frequencydivision multiplexing (OFDM) systems. Therefore, the SSPSs good for OFDM with phase shifting by rad are not necessarily effective for MS optical communication systems. The main contributions of our paper are: 1) we derive transmit sequences having a large minimum value and large Euclidean distances by using 8-phase-shift keying and (8+1)-amplitude phase-shift keying; and 2) since designing optimal sequences would be prohibitively complex, we introduce a reasonable procedure for suboptimal sequence design, obtaining good results. We show that the normalized power requirements and normalized bandwidth requirements of the MS systems with SSPS are smaller than those of the conventional MS systems. Index Terms—dc bias, (8+1)-amplitude phase-shift keying (APSK), multiple-subcarrier modulation (MSM), subcarrier signal-point sequence (SSPS).

I. INTRODUCTION PTICAL wireless communication systems have attracted much attention for high-speed indoor wireless communications. Compared with radio systems, infrared radiation (IR) offers many advantages, such as a higher bit rate, an enormous unregulated bandwidth, and no interference between channels operating in the adjacent rooms. In optical wireless communications, optical wireless systems using intensity modulation with direct detection (IM/DD) are most popular because of their simplicity.Multiple-subcarrier(MS)opticalcommunicationsystems using IM/DD (IM/DD MS) are also attractive, because the use of several narrow-band subcarriers promises to minimize intersymbolinterference(ISI)onmultipathchannelsandbecausemultiple-subcarrier modulation (MSM) can provide immunity to fluorescent-light noise near dc. [1]. The main fault of IM/DD MS systems is their poor average optical power efficiency. An optical intensity must be nonnegative, and thus a dc bias must be added to an MS electrical signal to modulate it onto an intensity of an optical carrier. As the number of subcarriers increases, the minimum value of the MS electrical signal decreases (becomes more

O

Paper approved by K. Kitayama, the Editor for Optical Communication of the IEEE Communications Society. Manuscript received March 15, 2004. This paper was presented in part at the IEEE Global Telecommunications Conference, Taipei, Taiwan, R.O.C., November 2002. S. Teramoto is with the Department of Electrical Engineering, Tokyo University of Science, Chiba, 278-8510 Japan. T. Ohtsuki is with the Department of Information and Computer Science, Keio University, Yokohama, 223-8522 Japan. Digital Object Identifier 10.1109/TCOMM.2005.857152

negative), and, thus, the required dc bias increases. The average optical power efficiency depends on the bias signal. Therefore, it is important to minimize the bias signal. In IM/DD MS systems, information bits are mapped onto the intensity of an optical carrier. Some block codes have been proposed to reduce the dc bias and to improve the power efficiency of IM/DD MS systems [1]. In this paper, we propose an IM/DD MS optical communication system with subcarrier signal-point sequence (SSPS) to improve the power efficiency of IM/DD MS systems. The SSPS is a set of sequences that consists of a signal point of signal points is each subcarrier. An SSPS consisting of selected according to input data. The proposed system uses the SSPSs having a large minimum value and large Euclidean distance, so that the required dc bias is minimized and the error-rate performance is improved. Note that in the proposed systems, the signal points having the larger minimum value are selected, while the signal points having a lower peak-to-mean-envelope-power ratio (PMEPR) are selected in orthogonal frequency-division multiplexing (OFDM) systems. Therefore, the SSPSs good for OFDM in [2] with phase shifting by rad are not necessarily effective for MS optical communication systems. We derive the SSPS that is suitable for MS optical communication systems. We analyze the power and bandwidth requirements of the proposed system. The main contributions of our paper are as follows: 1) we derive transmit sequences having a large minimum value and large Euclidean distances by using 8-ary phase-shift keying (8-PSK) and (8+1)-ary amplitude phase-shift keying [(8+1)-APSK] and 2) since designing optimal sequences would be prohibitively complex, we introduce a reasonable procedure for suboptimal sequence design, obtaining good results. We show that the normalized power requirements and the normalized bandwidth requirements of the MS systems with SSPS (MS-SSPS) are smaller than those of the MS systems with the minimum-power block coding (MS-Min-Power), respectively. We also show that the MS-SSPS with (8+1)-APSK MS-SSPS whose constellation consists of 8-PSK signal points plus zero-signal point [3], [4] can achieve better normalized power requirement and normalized bandwidth requirement than the MS-SSPS with 8-PSK [MS-SSPS (8-PSK)]. II. MULTIPLE-SUBCARRIER OPTICAL COMMUNICATION SYSTEMS A. IM/DD Optical Channel Model In the IM/DD optical channel with the impulse response the received photocurrent is given by [5]

0090-6778/$20.00 © 2005 IEEE

, (1)

TERAMOTO AND OHTSUKI: MULTIPLE-SUBCARRIER OPTICAL COMMUNICATION SYSTEMS WITH SSPS

where represents the photodetector responsivity, repreis the receiver sents the transmitted optical intensity, and thermal noise and the intense ambient shot light noise. The channel is modeled by a linear system having the impulse and the frequency response . The noise response can be modeled as Gaussian, independent of , and . We white with two-sided power spectral density (PSD) assume that the multipath distortion is negligible, as in [1] and [6]. Wireless infrared links are subject to intense ambient light that gives rise to a high-rate, signal-independent shot noise, which can be modeled as white and Gaussian [5]. Note that must be nonnegative. Note also that the electrical input is limited because of power the average amplitude of consumption and eye safety considerations. Scaling the current for convenience, of the photodiode at the receiver by has the same unit as . The average optical power is given ; . Note that in electrical by the mean value of channels, the average power is the mean value of . B. MS Optical Communication Systems An MS system is a system that multiplexes subcarriers with different frequencies in the electrical domain and modulates an optical subcarrier by the multiplexed signal. The transmitter uses a set of subcarrier frequencies. During each symbol interval of duration , it transmits a vector of information bits. A block coder maps a vector of information bits to a corresponding vector of symbol amplitudes, and it is modulated onto subcarriers. The MS electrical signal is formed by summing the modulated subcarriers. We assume the rectangular transmit pulse of width and unit amplitude. We also assume that each subcarrier is mutually orthogonal. The average optical power is , where is a nonnegative scale given by [1] factor and is a bias signal. C. Block Code In the MS systems, the block coder maps the vector of information bits to the corresponding vector of symbol amplitudes. The vector of symbol amplitudes is modulated onto the subcarriers. Some block codes have been proposed for MS systems to reduce the bias signal and improve the average power efficiency [1]. The normal block code is generally used in the MS system, subcarriers are used for transmission of information whose bits. Information bits can be mapped independently to the corresponding vector of symbol amplitudes. At the receiver, each vector of the detected symbol amplitudes can be demapped independently to information bits. Therefore, the number of transfor binary phase-shift keying (BPSK). mitted bits is In the reserved-subcarrier block code [1], subcarriers are used to enlarge the minimum value of the MS electrical signal, thereby reducing the average optical power, where the amplitudes of the reserved subcarriers are 1 and the phases of the reserved subcarriers are 0 or . The number of transmitted bits in the reserved-subcarrier block code is for BPSK. For each choice of and , there exists an optimal set of reserved subcarriers, though this set is often not unique. In the minimum-power block code [1], information bits are mapped to the set of all of the subcarriers. Thus, the information

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bits are not mapped onto each subcarrier independently. It was shown in [1] that the average optical power of the minimumpower block code is smaller than that of the reserved-subcarrier block code. III. MULTIPLE-SUBCARRIER OPTICAL COMMUNICATION SYSTEMS WITH SUBCARRIER SIGNALPOINT SEQUENCE (MS-SSPS) We propose the MS optical communication system with SSPS to reduce the optical power requirement. The block diagram of the proposed MS system with quaternary phase-shift keying (QPSK) is illustrated in Fig. 1. The transmitter uses a set subcarriers. During each symbol interval of duration , of information bits . it transmits a vector of The SSPS coder first determines the phases of SSPS with reference to the look-up table (LUT), where the suboptimum phases of SSPSs selected beforehand are written in the LUT. Then, the SSPS block coder maps a set of information bits to a set of corresponding vectors of symbol amplitudes as follows: (2) is a mapping function of the proposed system. Exwhere amples of the proposed mapping function are shown in Tables I and II. Note that, in the conventional MS systems (except for the system with the minimum-power block code), the block to the corcoder independently maps information bits responding vectors of symbol amplitudes as follows: (3) where is a mapping function of the conventional system. In this paper, we assume the rectangular transmit pulse of width and unit amplitude. We also assume that each subcarrier is mutually orthogonal. The bias signal is chosen so , where is a nonnegative scale that , factor. We set the subcarrier frequencies to where . We refer to as the position of the subcarrier with . With choosing , , and as shown above, , and the average optical power is given by [1]. In evaluating the average transmit power requirement, we thus consider the minimum value of the MS electrical signal described with the amplitude over the symbol interval (4) is converted to the At the receiver, the MS optical signal MS electrical signal by photodetector (PD), and each subcarrier component is obtained using a matched filter. The receiver detects the transmitted symbols based on the maximum-likelihood detection (MLD). According to the Euclidean distance between the received SSPS and the locally generated SSPS, the detector selects the SSPSs having the minimum value of Euclidean distance as the transmitted symbol. The Euclidean distance is given by (5)

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Fig. 1. Block diagram of the proposed system. TABLE II

TABLE I

N = 4, K = 4, 8-PSK, TIME-VARYING BIAS)

SSPS (

where is the total number of subcarriers, is the th prepared is the th received SSPS for the SSPS for the th subcarrier, th subcarrier, and is mapped onto with reference to the LUT. The receiver detects the transmitted symbols based on the MLD, and the vector of the detected symbol amplitudes is obtained. The SSPS decoder maps the vector of the detected symbol amplitudes to a vector of the detected information bits .

SSPS (

N = 4, K = 4, (8+1)-APSK, TIME-VARYING BIAS)

The SSPS is a set of sequences that consists of a signal point signal points is of each subcarrier. The SSPS consisting of selected according to input data. As we explained above, the proposed MS system uses the SSPSs having the large minimum value of the MS electrical signal and the large minimum Euclidean distance between each pair of SSPSs. Thus, the required dc bias is minimized, and the error-rate performance is

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IV. PERFORMANCE ANALYSIS In the proposed system, the receiver detects the transmitted SSPSs while comparing the received sequence with all of the possible sequences, and selects the sequence having the smallest Euclidean distance from the received sequence. The received SSPS has a noise component. The probability that the MLD when , , is transmitted detects the incorrect sequence is derived as

Fig. 2. (8+1)-APSK signal constellations.

improved. The SSPSs used in the proposed system are selected as follows. The bias signal corresponding to each sequence sequences is calculated for all of the possible sequences: for -PSK and sequences for -APSK. Fig. 2 shows (8+1)-APSK signal constellations. The (8+1)-APSK signal constellation consists of 8-PSK signal points plus a zero-signal point. As shown in [3] and [4], the modulation that has the signal constellation in Fig. 2 is generally referred to as (8+1)-APSK. Note that the available SSPSs are limited by the acceptable bias value and the minimum Euclidean distance. The computation for selecting the optimum SSPSs is prohibitive because of a large number of candidate sets. Therefore, we use the suboptimal selection method shown below. First, we choose the set of the SSPSs having the large minimum value of the MS electrical signal (6) where represents the set of the SSPSs, represents the selected set of the SSPSs, represents the specific threshold of is defined minimum value of MS electrical signal, and as (7) Second, we choose the set of the SSPSs having the large minimum Euclidean distance between each pair of SSPSs among (8) where represents the selected set of the SSPSs, represents the specific threshold of minimum Euclidean distance between is defined as each pair of SSPSs, and (9) Tables I and II show sets of SSPSs and the bias values for four subcarriers ( ) with time-varying bias when four bits per sequence can be transmitted ( ), where each signal point of the sequences in Tables I and II corresponds to a signal point of 8-PSK and (8+1)-APSK signal constellations, respectively. In these tables, the bias value of the MS-SSPS with (8+1)-APSK is smaller than that of the MS-SSPS with 8-PSK in most sequences. This is because, in the MS-SSPS with (8+1)-APSK, some subcarriers corresponding to the zero-signal point in (8+1)-APSK are not transmitted. In addition, the error probability can be improved, because we can choose sequences from more candidates in the MS-SSPS with (8+1)-APSK.

(10)

, is the transmitted signal where energy per symbol, and the noise is white Gaussian noise with . Assuming that all symbols are equally two-sided PSD likely, we get a union bound of symbol-error probability by averaging (10) over all the possible sequences, that is (11)

V. NUMERICAL RESULTS In this section, we evaluate the bandwidth and power requirements of the proposed systems (MS-SSPS (8-PSK), MS-SSPS [(8+1)-APSK], and the MS system with the minimum-power block coding (MS-Min-Power), comparing them with those of on–off keying (OOK). For each scheme, represents the information bit rate, represents the total electrical bandwidth represents the probability of inrequired at the receiver, and formation bit error. We set the required bit-error probability to . We get the power requirement by using the amplitude scale factor of subcarrier . For comparison, the required power is normalized with that of OOK at the same bit rate . We first consider a reference system using OOK with . rectangular pulses of duration and the symbol rate of We have , , and

(12) For the MS-SSPS, we use (11), and for the MS-Min-Power, we [1]. In each scheme, the informause tion bit rate is given by , where is the number of information bits transmitted by an MS symbol. The electrical . We assume bandwidth requirement is given by in all of the systems. We also assume in the MS-Min-Power, where is the number of reserved subcarriers. We use two bias signals for all of the systems: the fixed bias and the time-varying bias [1]. The fixed bias always adds the fixed bias to all the symbols, while the time-varying bias adds the corresponding bias to each symbol, that is, the symbol-dependent bias.

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Fig. 3. Normalized power requirement versus the number of subcarriers for the systems with BPSK. (a) Fixed bias. (b) Time-varying bias.

A. Normalized Power Requirement Versus the Total Number of Bits Fig. 3(a) and (b) shows the normalized power requirement versus the total number of bits for the systems with BPSK, with fixed bias and time-varying bias. In Fig. 3(a), the normalized power requirements of the MS-SSPS are smaller than those of the MS-Normal and the MS-Min-Power with . For instance, at 8 b, the normalized power requirement of the MS-SSPS (8-PSK) is 4.5 dB smaller than that of the MS-Normal, and 1.9 dB smaller than that of the MS-Min-Power, respectively. Similar trends can be seen in Fig. 3(b); the normalized power requirements of the MS-SSPS are smaller than those . This of the MS-Normal and the MS-Min-Power for is because the required dc bias of the proposed system is smaller than those of the conventional MS systems. In Fig. 3(a), the normalized power requirement of the MS-SSPS [(8+1)-APSK] is smaller than that of the MS-SSPS (8-PSK) for . For instance, at 8 b, the normalized power requirement of the MS-SSPS [(8+1)-APSK] is 1.6 dB smaller than that of the MS-SSPS (8-PSK). Similar trends can be seen in Fig. 3(b); the normalized power requirement of the MS-SSPS [(8+1)-APSK] . is smaller than that of the MS-SSPS (8-PSK) for

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 53, NO. 10, OCTOBER 2005

Fig. 4. Normalized power requirement versus normalized bandwidth requirement for the systems with BPSK. (a) Fixed bias. (b) Time-varying bias.

This is because in the MS-SSPS with (8+1)-APSK, some subcarriers are not transmitted, and thus, the negative peak value is large. Comparing the performances of the MS-SSPS with both biases, the performance improvement of the MS-SSPS over the other systems is large for the fixed bias. Note that the normalized power requirements of all of the MS systems with the time-varying bias are smaller than those of the systems with the fixed bias, respectively. B. Normalized Power Requirement Versus Normalized Bandwidth Requirement Fig. 4(a) and (b) shows the normalized power requirement versus the normalized bandwidth requirement for the systems with BPSK, with fixed bias and time-varying bias. In Fig. 4(a), the MS-SSPS can reduce the normalized power requirement at the same normalized bandwidth requirement as the other systems. For instance, at the normalized bandwidth requirement of 1.25, the normalized power requirement of the MS-SSPS (8-PSK) is 4.0 dB smaller than that of the MS-Normal, and 3.0 dB smaller than that of the MS-Min-Power, respectively. This is because the required dc bias of the proposed system is smaller than those of the conventional MS systems. When the normalized bandwidth requirement is smaller than 1.25, the

TERAMOTO AND OHTSUKI: MULTIPLE-SUBCARRIER OPTICAL COMMUNICATION SYSTEMS WITH SSPS

normalized power requirement of the MS-SSPS [(8+1)-APSK] is smaller than that of the MS-SSPS (8-PSK). For instance, at the normalized bandwidth requirement of 1.2, the normalized power requirement of the MS-SSPS [(8+1)-APSK] is 0.6 dB smaller than that of the MS-SSPS (8-PSK). Similar trends can be seen in Fig. 4(b). The normalized power requirement of the MS-SSPS is smaller than that of the other systems. When the normalized bandwidth requirement is smaller than 1.25, the normalized power requirement of the MS-SSPS [(8+1)-APSK] is smaller than that of the MS-SSPS (8-PSK). Comparing the performances of the MS-SSPS with both biases, the normalized power requirement is small for the time-varying bias. In Figs. 3 and 4, the proposed system needs more power than the OOK system in many cases. However, note that at high band), the width efficiency (when the number of subcarriers is proposed system needs less optical power. In addition, similar to the MSM in [1], the proposed system has the following advantages: 1) the use of several narrowband subcarriers promises to minimize ISI on multipath channels; and 2) MSM can provide immunity to fluorescent-light noise near dc. Therefore, the proposed system is attractive. VI. CONCLUSION We have proposed an MS optical communication system with SSPS to reduce the power requirement. In the proposed MS system, an SSPS is a set of sequences that consists of a signal point of each subcarrier, and the received signal is detected with MLD. In the proposed system, the signal-point sequence having the large minimum value and the large Euclidean distances are used, so that the required dc bias is minimized and the errorrate performance is improved. We showed that the normalized power requirements of the MS systems with SSPS (MS-SSPS) are smaller than that of the MS system with the normal block coding (MS-Normal), and that of the MS system with the minimum-power block coding (MS-Min-Power), respectively. We also showed that the MS-SSPS can reduce the normalized power requirement at the same normalized bandwidth requirement as the other systems. Moreover, we showed that the normalized power requirement and the normalized bandwidth requirement of the MS-SSPS with (8+1)-APSK are smaller than those of the MS-SSPS with 8-PSK, respectively. Comparing the performances of the MS-SSPS with the fixed bias and the time-varying bias, the normalized power requirement is small for the timevarying bias.

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REFERENCES [1] R. You and J. M. Kahn, “Average power reduction techniques for multiple-subcarrier intensity-modulated optical signals,” IEEE Trans. Commun., vol. 49, no. 12, pp. 2164–2171, Dec. 2001. [2] M. Harada, T. Yamazato, M. Katayama, and A. Ogawa, “A study on reducing the nonlinear distortion in multicarrier systems,” IEICE Trans. Fundam., vol. E83-A, no. 10, pp. 1992–1995, Oct. 2000. [3] A. D. S. Jayalath and C. Tellambura, “Interleaved PC-OFDM to reduce the peak-to-average power ratio of OFDM signal,” in Proc. Int. Symp. DSP Commun. Syst., Manly-Sydney, Australia, Jan. 2002, pp. 224–228. [4] P. K. Frenger and N. A. B. Svensson, “Parallel combinatory OFDM signaling,” IEEE Trans. Commun., vol. 47, no. 4, pp. 558–567, Apr. 1999. [5] J. R. Barry, Wireless Infrared Communications. Boston, MA: Kluwer, 1994. [6] J. M. Kahn and J. R. Barry, “Wireless infrared communications,” Proc. IEEE, vol. 85, no. 2, pp. 265–298, Feb. 1997. [7] J. A. C. Bingham, “Multicarrier modulation for data transmission: An idea whose time has come,” IEEE Commun. Mag., vol. 28, no. 5, pp. 5–14, May 1990. [8] T. E. Darcie, “Subcarrier multiplexing for lightwave networks and video distribution systems,” IEEE J. Sel. Areas Commun., vol. 8, no. 7, pp. 1240–1248, Sep. 1990. [9] J. B. Carruthers and J. M. Kahn, “Multiple-subcarrier modulation for nondirected wireless infrared communication,” IEEE J. Sel. Areas Commun., vol. 14, no. 3, pp. 538–546, Apr. 1996.

Shota Teramoto received the B.E. and M.E. degrees in electrical engineering from Tokyo University of Science, Chiba, Japan, in 2002 and 2004, respectively. His research interests were in optical communication systems.

Tomoaki Ohtsuki (M’90–SM’01) received the B.E., M.E., and Ph.D. degrees in electrical engineering from Keio University, Yokohama, Japan, in 1990, 1992, and 1994, respectively. From 1994 to 1995, he was a Post Doctoral Fellow and a Visiting Researcher in electrical engineering with Keio University. From 1993 to 1995, he was a Special Researcher of Fellowships with the Japan Society for the Promotion of Science for Japanese Junior Scientists. From 1995 to 2005, he was with Tokyo University of Science, Chiba, Japan. From 1998 to 1999, he was with the Department of Electrical Engineering and Computer Sciences, University of California, Berkeley. He is now an Associate Professor with Keio University, where he is engaged in research on wireless communications, optical communications, signal processing, and information theory. Dr. Ohtsuki is a member of the IEICE and the Symposium on Information Theory and Its Applications (SITA). He was a recipient of the 1997 Inoue Research Award for Young Scientist, the 1997 Hiroshi Ando Memorial Young Engineering Award, the Ericsson Young Scientist Award in 2000, the 2002 Funai Information and Science Award for Young Scientists, and the IEEE First AsiaPacific Young Researcher Award in 2001.

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On the Tradeoff Between Two Types of Processing Gains Eran Fishler and H. Vincent Poor, Fellow, IEEE

Abstract—One of the features characterizing almost every multiple-access (MA) communication system is the processing gain. Through the use of spreading sequences, the processing gain of random code-division multiple-access (RCDMA) systems, or any other code-division multiple-access (CDMA) systems, is devoted to both bandwidth expansion and orthogonalization of the signals transmitted by different users. Another type of MA system is impulse radio (IR). IR systems promise to deliver high data rates over ultra-wideband channels with low-complexity transmitters and receivers. In many aspects, IR systems are similar to time-division MA systems, and the processing gain of IR systems represents the ratio between the actual transmission time and the total time between two consecutive transmissions (on-plus-off-to-on ratio). While CDMA systems, which constantly excite the channel, rely on spreading sequences to orthogonalize the signals transmitted by different users, IR systems transmit a series of short pulses, and the orthogonalization between the signals transmitted by different users is achieved by the fact that most of the pulses do not collide with each other at the receiver. In this paper, a general class of MA communication systems that use both types of processing gain is presented, and both IR and RCDMA systems are demonstrated to be two special cases of this more general class of systems. The bit-error rate of several receivers as a function of the ratio between the two types of processing gain is analyzed and compared, under the constraint that the total processing gain of the system is large and fixed. It is demonstrated that in non-intersymbol interference (ISI) channels, there is no tradeoff between the two types of processing gain. However, in ISI channels, a tradeoff between the two types of processing gain exists. In addition, the suboptimality of RCDMA systems in frequency-selective channels is established. Index Terms—Code-division multiple access (CDMA), impulse radio (IR), multiple access (MA), random code-division multiple access (RCDMA), time-division multiple access (TDMA).

I. INTRODUCTION A. Motivation ULTIPLE-access (MA) communication systems are in widespread use. It is enough to mention that almost every cellular phone system and wireless local area network is

M

Paper approved by M. Z. Win, the Editor for Equalization and Diversity of the IEEE Communications Society. Manuscript received November 26, 2002; revised April 14, 2004. This work was supported in part by the U.S. Army Research Laboratory under Contract DAAD 19-01-2-0011, and in part by the New Jersey Center for Wireless Telecommunications. This paper was presented in part at the 40th Annual Allerton Conference on Communication and Control, Allerton Park, IL, October 2002. E. Fishler was with the Department of Electrical Engineering, Princeton University, Princeton, NJ 08544 USA. He is now with the Stern School of Business, New York University, New York, NY 10010 USA (e-mail: [email protected]). H. V. Poor is with the Department of Electrical Engineering, Princeton University, Princeton, NJ 08544 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TCOMM.2005.855001

an MA system. Many approaches for implementing MA systems exist; for example, direct-sequence code-division multiple access (DS-CDMA), frequency hopping, and time division, to name a few [8]. Recently, impulse radio (IR) systems have been suggested as a simple way of implementing MA systems [10]. IR systems promise to deliver high data rates in MA channels with low-complexity transmitters and receivers. Currently, IR systems are being considered for use in many applications, and mainly as the preferred solution for communication systems transmitting over ultra-wideband (UWB) channels [2], [3], [16]–[19]. The two most popular ways for implementing MA systems are CDMA and time-division multiple access (TDMA). These two types of systems are based on two different, and even “orthogonal,” ideas. Consider a CDMA system and a TDMA system assigned with equal bandwidth and supporting identical users. In the CDMA system, each user’s transmitted signal’s bandwidth is expanded using a distinct spreading sequence, and all the users transmit simultaneously over the same channel. The purposes of the spreading sequences are to spread the transmitted energy over the assigned bandwidth and to make different users’ transmitted signals as close to orthogonal as possible. Alternatively, in a TDMA system, each user transmits for only a small fraction of the time, but at a high data rate (and hence, the need for large bandwidth). By preventing simultaneous transmissions from two different users, collisions between the signals transmitted by different users are avoided, and hence, the required rate from each user is achieved. One type of CDMA system is the long-code CDMA system, also known as the random CDMA (RCDMA) system [11], [13]. In RCDMA systems, each user uses a random spreading sequence for expanding the bandwidth of the transmitted signal. Several MA communication systems are based on long-code CDMA, with the IS-95 mobile phone system being the most famous one [8]. As mentioned previously, IR systems have been suggested as a new approach to implementing MA communication systems. IR systems transmit a series of very short pulses, typically on the order of a fraction of a nanosecond in duration. Each user transmits each pulse at a time slot randomly chosen, and each pulse is repeated several times. The receiver, using appropriate signal-processing algorithms, recovers the transmitted bits [15]. IR systems can be regarded as random TDMA systems, where each user transmits for a very short time at a time slot randomly chosen. It is interesting to note that RCDMA systems and IR systems represent two extremes of a wide range of MA systems. In the first, the processing gain is devoted to increasing the signal bandwidth and making the different users’ transmitted

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signals as close to orthogonal as possible, while in the second, the processing gain is mostly devoted to reducing the transmission time, which, in turn, reduces the probability of several users transmitting simultaneously. Although both IR and RCDMA systems have been analyzed in the past, the literature still lacks a study that examines the tradeoff between the types of processing gain represented by these two systems. Moreover, systems that use both types of processing gain have not been suggested and analyzed. This paper offers such a study by examining the performance of IR systems and the performance tradeoff between the two types of processing gain as a function of the system parameters for a fixed signaling environment. The processing gain of IR systems, denoted hereafter by , is . The pulse rate represents the first type of processing gain, that is, the number of times each pulse is repeated (either uncoded or coded). Alternatively, , which is the ratio between the average total time between two consecutive transmissions and the actual transmission time, is the second type of processing gain. Assume that the total processing gain is fixed. , the ratio between the two types By changing the pulse rate of processing gain is changed, as well. The effect of this change on the system’s bit-error rate (BER) is the main interest of this paper. Thus, throughout the rest of the paper, we will compare systems that have equal total processing gain, but which divide this total processing gain between the two types of processing gain differently.

is the information symbol transframe. Thus, th frame. represents white, mitted during the per Hertz. Gaussian noise with power spectrum Two types of IR systems are considered in this paper. In the , while in the second, the ’s are binary first type, random variables, independent for , taking each of the values 1 with probability 1/2. The first type of system was the first to be proposed in the literature for transmission over UWB channels [7], while a different variant of the second was recently proposed in [9]. In the following, the two types of systems are referred to as uncoded and coded systems, respectively. Note that a coded IR systems can model an RCDMA system by and letting be the processing gain of the taking RCDMA system. the following sequence: Denote by

B. Signal Model

Although (3) describes a continuous-time TH-IR signal, discrete-time-equivalent models for CDMA systems can be used for describing the more general TH-IR systems. Assume that the received signal is passed through a linear filter , and sampled at the pulse rate. Denote by matched to the collection of all the samples corresponding to the th frame. can be described by the following linear model:

Consider the case of downlink channel of a -user time-hopping impulse radio (TH-IR) synchronous cellular-type system transmitting over a frequency-flat channel. Note that generalizing the results reported herein to more complex network configurations, e.g., uplink channels or asynchronous systems, can be easily carried out at the expense of complicating most of the already complicated computations, while the main results will essentially remain the same. The received signal of, say, the th user (out of total users in the system), in a binary phase-shift keyed (BPSK) random TH-IR system can be described by the following continuous time model:

(1) is the average pulse repetition time, is the transwhere mitted unit-energy pulse, also referred to (in the UWB literature) is the transmitted energy per bit for as the monocycle, and user . In order to allow the channel to be exploited by many users and to avoid catastrophic collisions, a long pseudorandom , such that is an integer taking one of the values sequence , is assigned to each user. Each sequence, in usually referred to as the TH sequence, provides an additional seconds to the th pulse of the th user. In time shift of order to avoid interpulse interference (IPI), it is usually required that , so that overlaps between pulses originating from the same user are avoided. In typical IR systems, each data symbol is transmitted over a set of multiple monocycles called denotes the number of pulses that correspond a frame. Here, to one information symbol, i.e., the number of monocycles per

otherwise.

(2)

can be regarded as a pulse (or chip) rate The sequence whenever spreading sequence where takes the value a pulse is transmitted, and zero otherwise. Assuming, without , the received signal loss of generality (wolog), that can be described by the following model: (3)

(4) where is the matrix whose columns are the spreading sequences used for spreading the th information symbol of all the users, that is, ; is a diagonal matrix with the users’ amplitudes on its diagonal, that is, ; is the vector containing the th transmitted symbols of all the users; is the additive noise. Note that a coded and IR system can model an RCDMA system by taking and letting be the processing gain of the RCDMA system. Also note that as decreases, the transmitted signal becomes more and more impulsive, and hence, the name impulse radio system [6], [7], [9]. It is well known that is a sufficient statistic for detecting the transmitted symbols. It is also well known that is also a sufficient statistic for detecting the transmitted symbols. It is easily seen that can be described by the following model: (5) where with

is the unnormalized cross-correlation matrix, on its main diagonal and

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as the off-diagonal elements; and where is a zero-mean, Gaussian random vector with correlation matrix . Discrete-time-equivalent models for frequency-selective channels are more complex. In order to have a tractable model that allows for analysis, we assume that a guard time equal to the length of the channel impulse response (CIR) that exists at the end of each symbol. This assumption is usually made in order to simplify the analysis [4]. As such, the following model for is used: (6) is a lower triangular Toeplitz matrix whose first where . column equals C. Organization of the Paper The rest of the paper is organized as follows. In Section II, IR systems transmitting over frequency-flat channels are analyzed, and in Section III, IR systems transmitting over frequency-selective channels are analyzed. In Section IV, some conclusions and concluding remarks are given. For convenience, numerical examples are presented at the end of each section. II. TRANSMISSION OVER FLAT-FADING CHANNELS In this section, we focus on systems transmitting over flatfading channels. Both coded and uncoded systems are analyzed, and are shown to behave differently as a function of the pulse rate. A. Coded System In this subsection, coded-user systems are analyzed. The following simple lemma will be very useful in the analysis of both the matched filter (MF) detector and the optimal multiuser detector. Lemma 1: Denote by , the vector containing cross-correlations between any two spreading sequences. Assume that and that . Then is asymptotically normally distributed with zero mean . and correlation matrix Proof of Lemma 1: See Appendix A. Consider the model for the received signal (5) scaled by

processing gain and not on the ratio between the two types of processing gains. Simulation results, which are not reported here due to space limitations, show that this result holds for . systems with processing gains as low as In the following, we refer to a system satisfying as a low-pulse-rate system, while a system such that is is referred to as a high-pulse-rate system. on the order of RCDMA is an example of a high-pulse-rate system, since in . It should be noted that as the pulse rate this system, decreases, the energy per transmitted pulse increases. This general behavior characterizes the main hardware complexity tradeoff between high-pulse-rate and low-pulse-rate systems. We demonstrate this hardware complexity by examining the MF detector. The MF detector for detecting the th symbol of the first user is . Denote the symbol rate by . In order to implement the MF detector, the system sampling rate can be as low , while the transmitted energy per pulse is . These as two terms represent a hardware complexity tradeoff between high- and low-pulse-rate systems. While the sampling rate of low-pulse-rate systems can be lower than the sampling rate used by high-pulse-rate systems, the receiver dynamic range of lowpulse-rate systems must be higher than the receiver dynamic range of high-pulse-rate systems. The increase in the receiver dynamic range is due to the increase in the signal peak-to-average power ratio exhibited by low-pulse-rate systems. B. Uncoded System In a way similar to the proof of Lemma 1, it is easy to verify that in uncoded-user systems, is asymptotically normally disand covariance matrix tributed with mean . 1) Two-User Systems: The MF Detector: Consider a two-uncoded-user system. For large , the BER of the MF detector can be approximated as follows:

(7) According to Lemma 1, asymptotically (for large ) the vector of elements of is distributed as a normal random vector with zero mean and correlation matrix . Thus, for systems with large total processing gain, the distribution of , which is a sufficient statistic for detecting the transmitted symbols, is essentially independent of the pulse rate, and depends solely on . Consequently, the BER of any multiuser detection (MUD) algorithm which is based on is essentially independent of the pulse rate. In particular, the performances of the optimal, MF, minimum mean-square error (MMSE), and zero-forcing (ZF) multiuser detectors are independent of the pulse rate, and depend solely on the total

(8) is due to [12], and we used the identity for . It can be easily seen that asymptotically (as ), the BER of the system depends on the ratio between the two types of processing gain, and hence, there is a tradeoff between the two types of processing gains. The main question and minimizes that arises is, “What ratio between the system BER?”. In Appendix B, the following result is and are less than , and that proven. Given that , the BER of (8) is a monotonically where

FISHLER AND POOR: ON THE TRADEOFF BETWEEN TWO TYPES OF PROCESSING GAINS

increasing function of the pulse rate. The above sufficient conditions mean that the transmitted energy per chip is lower than the background noise level, and that the energies transmitted by the two users do not differ by a factor larger than the square of the processing gain. These conditions are almost always met in practical systems. Simulation results we conducted confirm that unless one of the users is much stronger than the other, low-pulse-rate systems are preferable to high-pulse-rate systems. The superiority of low-pulse-rate systems can be intuitively deduced from (8) quite easily. Let us assume that . It can be easily seen that under this condition, the approximate BER of the MF detector is the average of the -function over a simple random variable that can take one of two possible values. Due to our assumptions, on one hand, the average of these two values is approximately a constant independent of the pulse rate, and on the other, as the pulse rate increases, the distance between these two values increases, as well. Since the -function is a convex function, Jensen’s inequality implies that the BER of the MF detector is a monotonically increasing function of the pulse rate. Optimal Detector: In uncoded systems, it is quite clear from the asymptotic distribution of the correlation between the two users’ spreading sequences that the distribution of any sufficient statistic for detecting the transmitted symbols depends on the pulse rate. Thus, it is of interest to study the BER of the optimal MUD as a function of the pulse rate. the probability of error of optimal MUD Denote by given that the correlation between the two users’ spreading sequences is . It is well known that no general closed-form expression for exists. Nevertheless, upper bounds for exist, and it is easily seen that the one reported in [12] is a monotonically increasing function of . As indicated by a large is number of simulation studies, it is widely believed that a monotonically increasing function of , as well. Since in our system, is a random variable, the overall BER is given by avwith respect to the distribution of . Denote by eraging this overall BER of optimal MUD, given that the pulse rate equals , i.e., . . Note that since the system is an unAssume that with probability one. In Appendix C, it coded one, is proven that for all and large , . In the statistical literature, this kind of relation between two random variables is usually termed first sto. It is chastic dominance, and it is denoted as and is a monotonically inwell known that if creasing function, then [14]. Thus, , and the upper bound for the probasince bility of error given is a monotonically increasing function of , the average upper bound, which is also an upper bound for the average BER, is a monotonically increasing function , as well. By using the conjecture that the probability of of error is a monotonically increasing function of , and by using , we also conjecture that the BER of the optimal MUD is a monotonically increasing function of pulse rate. 2) Multiple-User Systems: In this section, the BER of uncoded-user systems with arbitrary number of users using the MF detector or the optimal detector is examined. It is demonstrated

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Fig. 1. Probability of error as a function of the pulse rate. Two equal-power users transmitting over a frequency-flat channel.

that the general behavior observed for two-uncoded-user systems carries over to the case of a large number of users. The MF detector for detecting the transmitted symbol, of, say, the first user, is , where is the th element of the first row . Assuming that is large, and all the users transmit of with equal power, by invoking the central limit theorem is asymptoti(CLT) cally normally distributed with zero mean and variance . As a result, the BER of the MF detector can be approximated by (9) It is clear that the approximate system BER is a monotonically increasing function of the pulse rate, as is the case for the twouser uncoded system. It is also clear that when the users transmit at different powers, a similar conclusion can be reached. Analyzing the performance of the optimal multiuser detector is quite difficult, due to the lack of closed-form expressions, or even simple upper bounds, for the system BER as a function of the users’ gains and correlation matrix. Nevertheless, we conjecture that similar to the case of a two-uncoded-user system, the system BER is a monotonically increasing function of the pulse rate. This conjecture is based on the observation that assuming , then . That is, when the pulse rate decreases, the off-diagonal elements of tend to be smaller, and hence, the spreading sequences of the various users tend to be less correlated. C. Numerical Example In this subsection, we present a numerical example that confirms the results reported thus far. We consider a system with . Figs. 1 and 2 depict the BER of both processing gain the MF detector and the optimal MUD as a function of the pulse rate. In Fig. 1, we assume two equal-power users transmitting at

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is an indicator function taking the value 1 if the where th pulse transmitted by the first user collides via the second path with the th pulse transmitted by the first user, and zero otherwise. is a function taking the value 1 if the th pulse transmitted from the second user collides with the th pulse transif the th pulse transmitted from the first user, the value mitted from the second user and arriving via the second path collides with the th pulse transmitted from the first user, and is an indicator function taking the value 1 if zero otherwise. th pulse transmitted by the second user collides via the the second path with the th pulse transmitted by the first user, and zero otherwise. The MF detector is shown in (11) at the bottom of the page. In Appendix D, it is shown that the multiple-access interference (MAI)

Fig. 2. Probability of error as a function of the pulse rate. Two nonequal-power users transmitting over a frequency-flat channel.

a signal-to-noise ratio (SNR) of 6 dB, while in Fig. 2, we assume that the first user transmits at an SNR of 5 dB, and the second user at an SNR of 8 dB. The theoretical expressions for the performance of the MF detector are depicted as well [6]. It is evident from the graph that the BERs of both the MF detector and the optimal multiuser detector in the coded system are unaffected by the pulse rate. Also, the BERs of both the MF detector and the optimal multiuser detector in the uncoded system degrade considerably as the pulse rate increases. This is in accordance with the analysis conducted in this section. Moreover, we can see that the empirical and the theoretical curves agree well. III. TRANSMISSION OVER FREQUENCY-SELECTIVE CHANNELS A. Analysis In this section, coded-user systems transmitting over frequency-selective channels are analyzed. The analysis will be carried out in two stages. In the first stage, it is assumed that only two paths arrive at the receiver, that is, the CIR is , and that . In the second step, we will consider more general channels. Denote by the sample at the output of the MF at the time instant corresponding to the arrival time of the th pulse from the user of interest, say, the first user. The following model for can be easily deduced from (6):

(10)

is asymptotically (as ) normally distributed with . zero mean and variance Thus, for systems with large processing gain, the distribution of the MF test statistic is approximately , and the BER can be approximated by (12) It can be easily seen from Appendix D that the MAI is the sum of two independent terms. The first is the MAI created by the , and second user, the second is the self-interference processes created by the first . Note that the user upon itself, self-interference created by the first user is due to pulses arriving via the second path colliding with different pulses arriving via the first path; this represents IPI. We now turn to the computation of the BER of the MF when more than two paths arrive at the receiver. The MAI created ) normally by the second user is asymptotically (as . distributed with zero mean and variance This MAI can be modeled as the sum of two independent, zero-mean Gaussian random variables, with variances and , respectively. The first (second) random variable represents the MAI caused by pulses originating from the second user and arriving at the receiver via the first (second) path. Hence, asymptotically, the MAI resulting from pulses arriving through different paths are independent. Using the same method used in Appendix D, this can be generalized to channels with more than two paths, or with two paths such that . Thus, when the CIR is arbitrary, the MAI created by the second user is asymptotically normally distributed, with zero

(11)

FISHLER AND POOR: ON THE TRADEOFF BETWEEN TWO TYPES OF PROCESSING GAINS

mean and variance , which is the sum of the interference created by the different paths. If the number of users in the system is larger than two, it is easy to verify that the MAI processes due to different users are independent. Hence, the total MAI due to various users is ) normally distributed, with zero asymptotically (as . mean and variance The self-interference created by the first user due to pulses arriving via the second path is asymptotically normally distributed, with zero mean and variance . It is , the self-interference easy to verify (Appendix D) that if caused by the first user is asymptotically normally distributed, . It should be noted that with zero mean and variance , the average power of the self-interference is the when , , average power of the self-interference when multiplied by the probability that a transmitted pulse will arrive via the second path at times when the next transmitted pulse will arrive via the first path. If more than two paths exist, it can be seen that the self-interference terms created by any two paths are asymptotically independent. Thus, similar to the MAI created by the second user, the total self-interference created by the first user is the sum of the individual interferences. The conclusion follows readily from this discussion: in arbitrary channels, the self-interference created by the first user is asymptotically normally distributed, with zero mean and . variance Combining the asymptotic distribution of the MAI and the self-interference results in the following simple approximate expression for the BER of the MF detector:

(13) Note that (12) is a special case of (13). It is very easy to see that the BER of the MF detector is in-function fluenced by the pulse rate. The argument of the appearing in the expression for the BER, (13), is a monotoni, or equivalently, monotoncally nonincreasing function of ically nondecreasing function of the pulse rate , and hence, the BER is a monotonically nondecreasing function of the pulse rate. Moreover, the effect of the pulse rate on the BER is due to collisions between the transmitted pulses and pulses received via the multipath from the user of interest. Thus, as the pulse rate increases, the probability of such collisions increases as well, and hence, the BER increases. On the other hand, the interference caused by the other users is independent of the pulse rate. The result just obtained raises a question as to whether the pulse rate has any effect on the performance of the (jointly or individually) optimal multiuser detector. The answer to this question is simple: a numerical example, discussed in the next subsection, demonstrates that the pulse rate does affect the BER of the system. This answer raises an even more interesting question, as to whether the pulse rate affects the BER in the same way, regardless of the scenario. The answer to this question

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Fig. 3. BER of the MF detector as a function of SNR, for an RCDMA system and a low-pulse-rate system.

is much more complicated, and the lack of a closed-form expression for the BER of the optimal multiuser detector prohibits a definitive answer. In the numerical example just mentioned, the BER of the optimal MUD is a monotonically increasing function of the pulse rate. We conjecture that performance improvement occurs with the decrease of the pulse rate. Although we do not have mathematical proof for this claim, we do have some evidence supporting it. By invoking the CLT, it could be argued that the mean of the correlation matrix, , is independent of the pulse rate, and that a decrease in the pulse rate uniformly “concentrates” the distribution of the correlation matrix around its mean; that is, with some abuse of notation, for . Combining this with the convexity of the probability of error as a function of provides evidence supporting the conjecture. Numerous simulations we have conducted supports this conjecture, as well. B. Numerical Example In this subsection, we present a numerical example that confirms the results of Section III-A. We consider a coded system . For simplicity, the channel was with processing gain taken to be . Fig. 3 depicts the BER of the MF detector as a function of the SNR. We consider a three-user system, one user of which has SNR 3 dB higher than the SNR of the other two. The curves , and a shown correspond to an RCDMA system, system with pulse rate equal to 32, i.e., . The theoretical expressions for the BER are depicted, as well. As can be seen from the figure, the BER of the low-pulse-rate system is lower than the BER of the RCDMA system, and a gain of more than 0.5 dB can be achieved by using the low-pulse-rate system. It is evident that the performance gap between the low-pulse-rate and high-pulse-rate systems increases as the SNR increases. Recall that the self-interference noise level increases as the pulse rate increases [see (13)]. Therefore, as long as the additive noise level and the MAI level are high compared with the self-interference level, then the difference in BER of high- and low-pulse-rate systems is

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Fig. 4.

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BER of the optimal MUD as a function of the number of users. Fig. 5.

negligible. However, when the SNR increases and the self-interference noise becomes dominant, then the differences between low- and high-pulse-rate systems become evident. In Fig. 4, the optimal detector’s BER is depicted as a function of the number of users. We consider an RCDMA system and a low-pulse-rate system transmitting 16 pulses per symbol, with equal-power users. We examined two SNRs, 4 and 6 dB. As can be seen from Fig. 4, the BER increases as the number of users increases, and the low-pulse-rate system outperforms the RCDMA system for any number of users. In addition, it can be seen from the figure that for fixed SNR and BER, the lowpulse-rate system can support two additional users, compared with the RCDMA system. In Fig. 5, we examine the optimal detector’s BER as a function of the SNR and the pulse rate. We consider six equal-power users, and an RCDMA system and two low-pulse-rate systems transmitting 32 and 8 pulses per symbol. We can see from the figure that the RCDMA system requires an additional 0.3 dB to achieve the same BER as the low-pulserate system. The performance improvement due to the use of the low-pulse-rate system is not large. However, this performance improvement demonstrates the validity of our theoretical results, as well as the advantages of using low-pulse-rate systems. IV. SUMMARY AND CONCLUDING REMARKS In this paper, the tradeoff between two types of processing gain, namely, spreading and time division, has been analyzed under the assumption that the total processing gain is fixed and large. These two types of processing gain are interchangeable, and the analysis reveals that in some cases, one should favor the second type of processing gain over the first. Specifically, it has been argued that when coded systems transmitting over non-ISI channels are used, the two types of processing gains are reciprocal. That is, the BER of many MUDs is independent of the ratio between the two types of processing gain, as long as the total processing gain is fixed. Nevertheless, the system complexity varies as the ratio between the two types of processing gain is changed. In systems that devote some of their processing gain to reducing the transmission time, the sampling

BER of the optimal MUD as a function of the SNR.

rate can be decreased at the expense of large dynamic-range requirements, when compared with high-pulse-rate systems. In the context of UWB systems, this result is very important. Under today’s regulations, the bandwidth of UWB systems could be up to 7 GHz. It is obvious that RCDMA systems that use the whole bandwidth will have to sample the received signal at a rate of at least 7 GHz. On the other hand, some of the processing gain canbe devoted to reducing the transmissiontime,and thus, alower sampling rate could be used. Moreover, MUD algorithms specifically designed for low-pulse-rate systems have very low complexity, compared with their high-pulse-rate counterparts [5]. In frequency-selective channels, it has been shown that there is a tradeoff between the two types of processing gain, and this tradeoff is in favor of reducing the pulse rate, that is, reducing the total transmission time. Although the decrease in the BER due to the use of low-pulse-rate systems can be low, the system complexity will be much lower than that of high-pulse-rate systems. It can be seen from (13), for the approximate BER of the MF detector, that the effect of the pulse rate on the total noise level is only via the signal transmitted from the user of interest. If the number of users is large, or all the users transmit with equal power, the part of the noise level depending on the pulse rate is negligible, compared with the part that is independent of the pulse level. In the equal-power-users case, it is easy to verify that the part depending on the pulse rate is always smaller than the part independent of the pulse rate. Therefore, in these cases, the advantage of the low-pulse-rate systems is negligible, compared with high-pulse-rate systems. APPENDIX I PROOF OF LEMMA 1 Recall that

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In order to prove the lemma, we first show that for , the random vectors and are independent and identically distributed, with zero mean and . It is easy to see from covariance matrix equal to the definition of that for every , is independent of for . Thus, for , the random variables are jointly independent of , and so the random vectors and are independent, as well. We now turn to examine the random variable , where . This sum will be equal to zero if the th pulse of the th and th users are transmitted at different time slots. The probability of this event is . If the th pulse of the th and th users are transmitted at the same time slots, then the probability that both of them will transmit a pulse with equal phase is one-half. Combining is these observations, it is easy to see that , a ternary random variable equaling zero with probability . By and one or minus one, each with probability using the same technique, one can verify that the expectation of equals zero. Now, , where is a zero-mean random vector with covariance matrix . Invoking the CLT on this sum proves the lemma; i.e.,

In order for the BER to be a monotonically decreasing func, the derivative of the BER should be negative for tion of every . This is equivalent to the following condition:

(14)

In this appendix, the correlation between the spreading sequences of two uncoded users is examined. In particular, it is for proven that asymptotically in , , or equivalently, that for , where denotes first stochastic domination. In order to prove that , we have to prove that for a fixed , the probability of the event is a monotonically increasing function of . Recall that the correlation between the spreading sequences, given , , , and is asymptotically normally distributed with mean . Thus, the probability of the event variance is, asymptotically, . After some manipulation, and using the relation , the asymptotic probability of error is thus given by

APPENDIX II ANALYSIS OF THE MF RECEIVER IN UNCODED SYSTEM Define two functions

,

as follows:

(15) The BER (8) can be easily expressed with the aid of , , and it is given by . In what follows, we find sufficient conditions such that the BER is a monotonically decreasing function of , which is equivalent to proving that the BER is a monotonically increasing function of the pulse rate. In order to prove that the BER is a monotonically decreasing function of , we , which, first take the derivative of the BER with respect to after some manipulation, can be seen to be given by

(16) Assume that , . Substituting (15) into (16), and by using our assumption, the following sufficient condition is deduced:

(17) Upper bounding the right-hand side (RHS) using the bound results in the following sufficient condition:

(18) Bounding the denominator of the left-hand side from above by , and the denominator of the RHS from below by , re. sults in the following sufficient condition: APPENDIX III PROPERTIES OF THE CORRELATION COEFFICIENT IN AN UNCODED SYSTEM

(19) Assume that . In order to prove that (19) is a monotonically increasing function of , it suffices to prove that the argument of the -function in (19) is a monotonically increasing function of . Differentiating the argument of the -function , and omitting some positive scaling in (19) with respect to factors, results in the following:

(20)

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It is easy to see that for

, (20) is positive, and hence, is a monotonically increasing . Thus, is a monotonically infunction of creasing function of , as well. Note that since the system is , so the case has an uncoded system, no interest. . It is easy to verify that Assume that for , and . Combining for this result with the monotonicity of concludes the proof.

, respectively. Although is a white random sequence, it is not an independent one. Nevertheless, it is a 1-dependent random sequence, and hence, it is a -mixing random sequence for which the conditions in [1] hold. Thus, a CLT can be invoked, implying the asymptotic normality of the total interference

APPENDIX IV PROPERTIES OF THE INTERFERENCE It is easy to verify from the definition of that the , , and are binary random random variables with probability variables taking each of the values 1/2. Let us derive the distribution of the random variable . The th pulse transmitted by the first user will arrive via the second path at times when the th pulse transmitted by the first user might be received if and only if , and the probability of this event is . Given that the probability that (the probability of that arrival a collision will occur is time of the th pulse via the main path is equal to the arrival th pulse via the second path). Thus, the time of the probability of the event is . Combining the distribution of and the distribution of results in the following distribution for : can take on the values , 0, with probabilities , , and , respectively. It is easy to show in a similar way is equal to the that the marginal distribution of marginal distribution of . Similar arguments can lead to the following marginal distri: takes on the values 1, , 0, , bution of with probabilities , , , , and , respectively. It is , , are easy to verify that zero-mean white, mutually uncorrelated random sequences. For , and let and be two distinct time example, take indexes. The mean of is

(21) where for the last equality, we used the fact the is independent of all the other random variables, and we assumed that (if , take instead). The total interference is given by

Since the three random processes , , and are zero mean and mutually uncorrelated, the mean and the variance of are zero and

(22)

REFERENCES [1] P. Bilingsly, Probability and Measure, 2nd ed. New York: Wiley, 1986. [2] D. Cassioli, M. Z. Win, and A. F. Molisch, “The ultra-wide bandwidth indoor channel: From statistical model to simulations,” IEEE J. Sel. Areas Commun., vol. 20, no. 9, pp. 1247–1257, Dec. 2002. [3] R. J. Cramer, R. A. Scholtz, and M. Z. Win, “An evaluation of the ultrawideband propagation channel,” IEEE Trans. Antennas Propag., vol. 50, no. 5, pp. 561–570, May 2002. [4] J. Evans and D. N. C. Tse, “Large system performance of linear multiuser receivers in multipath fading channels,” IEEE Trans. Inf. Theory, vol. 46, no. 9, pp. 2059–2078, Sep. 2000. [5] E. Fishler and H. V. Poor, “Low-complexity multi-user detection in time hopping impulse radio system,” IEEE Trans. Signal Process., vol. 52, no. 9, pp. 2561–2571, Sep. 2004. , “On the tradeoff between two types of processing gain,” in Proc. [6] 40th Annu. Allerton Conf. Commun., Control, Comput., Allerton Park, IL, Oct. 2002, [CD-ROM]. [7] C. J. Le Martret and G. B. Giannakis, “All-digital PAM impulse radio for multiple-access through frequency-selective multipath,” in Proc. IEEE Global Telecommun. Conf., vol. 1, San Francisco, CA, Nov. 2000, pp. 77–81. [8] T. S. Rappaport, Wireless Communications: Principles and Practice, 2nd ed. Upper Saddle River, NJ: Prentice-Hall, 2001. [9] B. Sadler and A. Swami, “On the performance of UWB and DS-spread spectrum communication systems,” in Proc. IEEE Conf. Ultra Wideband Syst. Technol., Baltimore, MD, May 2002, pp. 289–292. [10] R. A. Scholtz, “Multiple access with time-hopping impulse modulation,” in Proc. IEEE Military Commun. Conf., vol. 2, Boston, MA, Oct. 1993, pp. 447–450. [11] D. Tse and S. Hanly, “Linear multiuser receivers: Effective interference, effective bandwidth and user capacity,” IEEE Trans. Inf. Theory, vol. 45, no. 2, pp. 641–657, Mar. 1999. [12] S. Verdú, Multiuser Detection. Cambridge, U.K.: Cambridge Univ. Press, 1998. [13] S. Verdú and S. Shamai (Shitz), “Spectral efficiency of CDMA with random spreading,” IEEE Trans. Inf. Theory, vol. 45, no. 2, pp. 622–640, Mar. 1999. [14] G. A. Whitmore and M. C. Findlay, Stochastic Dominance. Lexington, MA: Lexington Books, 1982. [15] M. Z. Win and R. A. Scholtz, “Impulse radio: How it works,” IEEE Commun. Lett., vol. 2, no. 2, pp. 36–38, Feb. 1998. , “On energy capture of ultra-wide bandwidth signals in dense mul[16] tipath environments,” IEEE Commun. Lett., vol. 2, no. 9, pp. 245–247, Sep. 1998. [17] , “Ultra-wide bandwidth time-hopping spread-spectrum impulse radio for wireless multiple access communications,” IEEE Trans. Commun., vol. 48, no. 4, pp. 679–689, Apr. 2000. [18] , “Characterization of ultra-wide bandwidth wireless indoor communications channel: A communication theoretic view,” IEEE J. Sel. Areas Commun., vol. 20, no. 9, pp. 1613–1627, Dec. 2002. [19] , “On the robustness of ultra-wide bandwidth signals in dense multipath environments,” IEEE Commun. Lett., vol. 2, no. 2, pp. 51–53, Feb. 1998.

FISHLER AND POOR: ON THE TRADEOFF BETWEEN TWO TYPES OF PROCESSING GAINS

Eran Fishler was born in Tel Aviv, Israel, in 1972. He received the B.Sc. in mathematics, and the M.Sc. and Ph.D. degrees in electrical engineering, in 1995, 1997, and 2001, respectively, all from Tel Aviv University, Tel Aviv, Israel. He is currently working toward the MBA degree at the Stern School of Business, New York University, New York, NY. From 1993 to 1995, he was with the Israeli Navy as a Research Engineer. From 1996 to 2000, he was with Tadiran Electronic System, Holon, Israel. During 2002 and parts of 2003, he was a Research Fellow at Princeton University, Princeton, NJ, where he worked on various topics related to UWB communication systems. His research interests include statistical signal processing and communication theory.

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H. Vincent Poor (S’72–M’77–SM’82–F’77) received the Ph.D. degree in electrical engineering and computer science from Princeton University, Princeton, NJ, in 1977. From 1977 until 1990, he was on the faculty of the University of Illinois at Urbana-Champaign. Since 1990, he has been on the faculty at Princeton University, where he is the George Van Ness Lothrop Professor in Engineering. His research interests are in the areas of statistical signal processing and its applications in wireless networks and related fields. Among his publications in these areas are the recent books Wireless Communication Systems: Advanced Techniques for Signal Reception (Prentice-Hall: Upper Saddle River, NJ, 2004) and Wireless Networks: Multiuser Detection in Cross-Layer Design (Springer: New York, 2005). Dr. Poor is a member of the U.S. National Academy of Engineering and the American Academy of Arts and Sciences, and is a Fellow of the Institute of Mathematical Statistics, the Optical Society of America, and other organizations. In 1990, he served as President of the IEEE Information Theory Society, and in 1991–1992, he was a member of the IEEE Board of Directors. He is currently serving as the Editor-in-Chief of the IEEE TRANSACTIONS ON INFORMATION THEORY. Recent recognition of his work includes the Joint Paper Award of the IEEE Communications and Information Theory Societies (2001), the NSF Director’s Award for Distinguished Teaching Scholars (2002), a Guggenheim Fellowship (2002–2003), and the IEEE Education Medal (2005).

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An Iterative Extension of BLAST Decoding Algorithm for Layered Space–Time Signals Ke Liu, Student Member, IEEE, and Akbar M. Sayeed, Senior Member, IEEE

Abstract—We propose an iterative extension of the Bell Laboratory Layered Space–Time (BLAST) algorithm and its variant, vertical BLAST (VBLAST). A characteristic feature of the BLASTtype algorithm is that symbol decisions with low reliability are fed back to decode other symbols. Both performance analysis based on Gaussian approximation of residual interference, and simulation results demonstrate that error propagation due to unreliable decision feedback can severely limit system performance. The extended algorithm exploits inherent signal diversity in BLAST to mitigate residual interference, thus overcoming the performance bottleneck due to error propagation. It yields an impressive performance gain over BLAST. In particular, the extension of BLAST with zero-forcing interference nulling admits a simple QR implementation and exhibits excellent performance with low complexity. Index Terms—Bell Laboratory Layered Space–Time (BLAST) algorithm, layered space–time processing, multiantenna systems, vertical BLAST (VBLAST).

I. INTRODUCTION UGMENTING temporal and frequency dimensions, the spatial dimension afforded by antenna arrays holds great promise for improving wireless system performance. Information-theoretic studies in [1] and [2] lay out a foundation for deploying multiple antennas at both the transmitter and the receiver. In these works, multiple-antenna systems are shown to increase wireless channel capacity. In particular, channel capacity grows at least linearly with the number of transmit antennas, provided that the number of receive antennas is greater than or equal to the number of transmit antennas. The multiantenna channel can be described as a multiple-input multiple-output (MIMO) system, where the number of degrees of freedom (DOFs) is given by the product of the number of transmit and receive antennas. However, the relatively high dimensional nature of MIMO systems poses a nontrivial complexity problem on practical system design. Code design criteria and constructions for quasi-static slow and fast fading channels have been treated extensively in the seminal paper [3]. The general coding approach is most

A

Paper approved by R. A. Valenzuela, the Editor for Transmission Systems of the IEEE Communications Society. Manuscript received December 16, 2002; revised December 20, 2004 and January 14, 2005. This work was supported in part by the National Science Foundation under Grant CCR-9875805 and in part by the Office of Naval Research under Grant N00014-01-1-0825. This paper was presented in part at the 39th Annual Allerton Conference, Monticello, IL, October 2001. K. Liu is with the Department of Electrical and Computer Engineering, The Ohio State University, Columbus, OH 43210 USA (e-mail: [email protected]). A. M. Sayeed is with the Department of Electrical and Computer Engineering, University of Wisconsin-Madison, Madison, WI 53705 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TCOMM.2005.857154

suitable to realize spatial diversity gain for a small number of transmit antennas. A few handcrafted space–time codes for two transmit antennas were demonstrated to operate very close to channel outage capacity. However, the prohibitive decoding complexity limits the general coding approach when there are a large number of transmit antennas in the system. Reduced-complexity approaches that build on [3] have also been proposed (see, e.g., [4]). Signal processing techniques can be used to harness channel capacity with affordable complexity. Foschini et al. devised a multilayer structure, called Bell Laboratories Layered Space–Time (BLAST) [5], which decodes signals from a particular transmit antenna by using linear filtering and decision feedback to suppress interference from other transmit antennas. Linear filters in BLAST can be designed using minimum mean-square error (MMSE) or zero-forcing (ZF) criterion. Vertical BLAST (VBLAST) [6] represents an improvement over the original BLAST by dynamically optimizing the order of symbol decoding to reduce the effect of erroneous decision feedback. We will call BLAST, including its variants, the baseline algorithm. Different variants of the baseline algorithm generally exhibit different performance. BLAST with ZF filtering (ZF-BLAST) ranks the worst among all of the baseline algorithms in terms of performance. However, an efficient QR implementation makes it an attractive low-complexity solution to space–time decoding. On the contrary, VBLAST with MMSE filtering (MMSE-VBLAST) entails the most computational cost, but is capable of achieving excellent performance. In an important contribution [7], Hassibi has derived an efficient square-root algorithm which reduces the complexity of MMSE-VBLAST to roughly the cubic order of the number of transmit antennas. The algorithm is more involved than the simple ZF-BLAST, and hence, a nonnegligible implementation complexity may be expected. In this paper, we study the performance of the baseline algorithm based on a Gaussian approximation of residual interference. The analysis and simulation identify the error propagation due to unreliable decision feedback as a bottleneck in limiting system performance. We observe that the quality of symbol decision gradually improves as the baseline algorithm proceeds—the last decoded symbols enjoy the highest reliability. However, potential benefits of having more reliable decisions at the last decoded symbols are largely unexploited in the baseline algorithm. Therefore, we propose an iterative extension, the extended algorithm, which subtracts signals due to these symbols from the received signal, and hence, reduces interference toward other symbols. The extended algorithm successively refines decisions for every transmitted symbol using a baseline algorithm. The iterative BLAST extension proposed in

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LIU AND SAYEED: ITERATIVE EXTENSION OF BLAST DECODING ALGORITHM

this paper belongs to a larger family of iterative decision-feedback methods. The exact analysis on such decision-feedback systems seems very difficult. However, improving the quality of feedback decisions is key to performance enhancement. Among all of the variants of the extended algorithm, we focus on the extension of ZF-BLAST (EXT-ZF-BLAST) because of its relatively small complexity and clear interpretation of algorithm behavior. It is shown via simulations that the extended algorithm is able to significantly reduce the effects of error propagation. Moreover, the extended algorithm is also very efficient, in that few iterations are needed to achieve an impressive gain. In our investigation of several extended algorithms, we found that simple EXT-ZF-BLAST can easily outperform much more complicated dynamically ordering algorithms such as ZF-VBLAST. However, the study confirms the superior performance of MMSE-VBLAST; it yields a substantial gain over ZF-BLAST, and its iterative extension only produces a marginal improvement. The contribution of this paper is in studying the effects and performance gains associated with the extended algorithm. From a complexity point of view, simple EXT-ZF-BLAST based on QR implementation performs reasonably well, and holds the merit of low complexity. If the complexity of MMSE-VBLAST can be afforded, the Hassibi algorithm for MMSE-VBLAST should be applied [7]. The outline of the paper is as follows. Section II briefly reviews various baseline algorithms. The extended algorithm is formalized in Section III. The ZF-BLAST algorithm and its extension are presented from a particular implementation, QR decomposition, which concisely represents the layered space–time structure in BLAST and simplifies signal processing and decoding. Performance of the baseline or extended algorithm can be analyzed by approximating residual interference as Gaussian variables. We use ZF-BLAST and its extension to illustrate the proposed analysis technique. Simulation results are provided in Section IV to demonstrate the strength of the extended algorithm and to investigate performance of different algorithm variants. Finally, we make some concluding remarks in Section V. The following notations are used throughout this paper. and denote the Hermitian (complex) transpose Let and transpose, respectively. A -dimensional (complex) real and correlation Gaussian random distribution with mean is denoted by . Let be matrix independent, identically distributed (i.i.d.) random variables. Then, is a chi-squared random variable with DOFs, denoted by . The expectation is denoted by , and “orthogonal to” by . The identity matrix is denoted by . II. BASELINE PROCESSING FOR LAYERED SPACE–TIME SIGNALS Consider a narrowband multiantenna system with transmit antennas and receive antennas, denoted by an system. At a given discrete time instant, the -dimensional received signal and the -dimensional transmitted signal are related by (1)

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is assumed to be where the noise vector in (1) uncorrelated in time as well. The channel matrix represents couplings between different pairs of transmit and receive antennas. In rich scattering environments, elements of random the channel matrix can be modeled as i.i.d. variables. We adopt a quasi-static approximation of the fading channel, that is, the channel remains unchanged during a codiscrete time) and changes independently herence period ( from one period to another. In this paper, the channel state information is available at the receiver, but not known to the transmitter. is used The same signal constellation with average power at every transmit antenna. Since the noise has unit variance in the channel model, the signal-to-noise ratio (SNR) is then given by SNR

(2)

Note that the total transmitted power is fixed independent of the number of transmit antennas. A layered space–time system modulates incoming bit streams onto symbols and transmits them across all of the transmit antennas. For simplicity, we assume that symbols are uncoded, that is, no error-control code is used in the conversion from information bits into transmitted symbols. Essential ideas in this paper would remain unchanged in the case of coded systems. The uncoded layered space–time signal can be regarded as a special example of space–time codes in [3], which gave an upper bound of the pairwise error probability of decoding while is transmitted (3) where

for are eigenvalues of . Let rank rank . The number of nonzero eigenvalues is in (3). Therefore, if every codeword difference matrix has rank not less than , the corresponding order of diversity. space–time code can guarantee an Since a codeword in layered space–time systems is simply the uncoded transmitted signal vector, it is easy to see that the codeword difference between two distinct codewords is only of rank 1. Thus, one can conclude from (3) that (uncoded) layered space–time signals attain a diversity of , which is the number of receive antennas. This implies that uncoded multiantenna systems could deliver good performance while operating at a high rate, if a large number of receive antennas are available. However, such performance would require the use of maximum-likelihood (ML) decoding at the receiver. As increases, ML decoding quickly becomes impractical. Hence, alternative decoding algorithms of low complexity, although suboptimum, are sought instead. If transmit antennas are treated as “users,” the channel (1) resembles that of multiple-access channels, which suggests that a vast amount of multiuser detection literature [8], [9] can be carried over to the decoding of layered space–time signals. BLAST and its variant VBLAST are those signal-processing algorithms that draw insights from multiuser detection techniques.

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A. BLAST

the decoding algorithm chooses the “best” symbol at each decoding step. VBLAST [6] ordered symbols according to their mean square error (MSE) after interference nulling. It chooses the one with minimal MSE to perform actual decoding and interference cancellation. In order to determine the order among symbols, one needs to determine the nulling vector for every symbol. Fortunately, all denote of the nulling vectors can be computed at once. Let the nulling vector for the th symbol. The interference nulling for each symbol is performed as

The BLAST [5] algorithm is centered around the notion of interference nulling and interference cancellation. Suppose the , the symbol transmitted at the th processing starts from transmit antenna. Rewrite (1) as (4) where is the th column of for . Hence, the is in the direction of , and the signal due to the symbol lies within a linear subspace spanned by interference toward . Interference nulling tries to reduce the amount of interference toward by applying a linear filter , the nulling vector, to the received signal. Typically, the linear filter is derived under the ZF or MMSE criterion, which gives rise to ZF-BLAST or MMSE-BLAST, respectively. In ZF-BLAST, interference from other transmit antennas is completely removed by projecting the received signal into a direction orthogonal and denote the to the interference subspace. Let by removing its th column and the matrix obtained from corresponding linear subspace. Therefore, ZF-BLAST requires . It can be solved by (5) Although the ZF interference nulling completely removes interference, it weakens signal as well especially when and form a large angle so that the projection of onto has small power. As an improvement, MMSE-BLAST chooses the linear . Note that . The filter to minimize solution is given by (6) After interference nulling, the decision of , denoted by , . Then, the algorithm reconstructs is generated from the signal due to and subtracts it from the received signal . In other words, the interference from toward the rest of the undecoded symbols is canceled. If the above interference cancellation is perfect, the system has one less transmit antenna

(7) where the superscript denotes the reduced order of the problem. Applying the same interference nulling and cancellation procedures to the order-reduced problem, that is, an system, BLAST successively decodes symbols transmitted at every transmit antenna. B. VBLAST Close examination of BLAST reveals that decision feedback comes from “weaker” symbols that experience more interferis interfered ence from other symbols. For example, symbol symbols. However, it is the first one by all of the rest of to be decoded in BLAST, and its decision is used in the subsequent interference cancellation. The unreliable decision results in nonnegligible error propagation, which can cost a severe performance penalty. Thus, an improvement can be that

(8) where the th column of is . The filter matrix MMSE nulling is given by [6], [7]

for ZF or

(9) The MSE of interference nulling for every symbol is given by ZF Nulling MMSE Nulling. (10) VBLAST decodes the “strongest” symbol whose MSE is smallest among all of the symbols, thus improving the quality of decision feedback. After the symbol has been decoded, its interference toward other symbols is subtracted. Ordering and interference nulling/cancellation operations repeat until all of the symbols have been decoded. As we will demonstrate via simulations, dynamic ordering in VBLAST can effectively mitigate error propagation due to unreliable decision feedback. In this paper, BLAST-type algorithms will be termed baseline algorithms, as they are not iterative. We also use the names such as ZF-BLAST and MMSE-VBLAST to categorize different baseline variants. III. EXTENDED BLAST DECODING As discussed above, error propagation due to imperfect decision feedback in the baseline algorithm represents a bottleneck in system performance. Although dynamic ordering, as in the VBLAST, is an effective method to overcome this bottleneck, there exist other remedies that yield low-complexity solutions. Roughly speaking, the last symbol to be decoded in the baseline algorithm would experience the least amount of interference from other symbols, because interference from the other symbols has been canceled. Thus, its decisions are relatively more reliable. However, this observation is not used in the baseline algorithm. So, a simple way to improve system performance is to subtract from the received signal the signal of the last decoded symbol after an initial application of the baseline algorithm. This forms a feedback flow in the reverse order, which we term loopback. Assuming good loopback cancellation, loopback operation can effectively remove the contribution of one transmit antenna, which helps to improve the next iteration of the baseline algorithm, since the number of interfering sources is reduced by one. Loopback operation defines the distinct characteristics of the extension of the BLAST-type algorithm, which we call the extended algorithm. Moreover, the algorithm can be configured to loopback more symbols, and thus operates in an

LIU AND SAYEED: ITERATIVE EXTENSION OF BLAST DECODING ALGORITHM

iterative fashion. We formalize the proposed extension in the following algorithm. Algorithm 1 (Extended Algorithm): Let denote the number of loopback iterations. Fix a baseline algorithm to be used in space–time decoding. Maintain a set whose elements are the symbols used in loopback operations. system using the baseStep 1) Decode the original line algorithm. Add the last decoded symbol into . Step 2) Subtract the signal due to the symbols in from previous iterations of the baseline algorithm. Step 3) Apply the baseline algorithm to generate decisions for symbols that are not in . Step 4) Update symbol decision for every element in by subtracting interference from all the other symbols. Add the last decoded symbol in Step 3) into . times. Step 5) Repeat Steps 2)–4) by Note that corresponds to the baseline algorithm, and different variants of the baseline algorithm give rise to different variants of the extended algorithm. For example, EXT-ZF-BLAST stands for the extended algorithm with ZF-BLAST as its baseline algorithm. A. EXT-ZF-BLAST Among many variants of the extended algorithm, we primarily focus on EXT-ZF-BLAST because of its relatively low implementation complexity through QR decomposition [7], [10], [11]. We may assume that the number of transmit antennas is not larger than the number of receive antennas, that is, . , with Essentially the same idea applies for the case of a small modification given in [4]. The QR decomposition [12] of the channel matrix is (11) where is an triangular matrix

unitary matrix and the has the following form: .. .. .

.

.. .

upper

(12)

It can be shown [1], [4] (13) The upper triangle structure of simplifies the ZF-BLAST algorithm considerably. All of the interference-nulling operations in BLAST can be done in one step prior to decoding by multi. More specifically plying by (14) . Since is an upper trianwhere gular matrix, symbols with larger indexes avoid interference from symbols with small indexes, which exactly reflects the pattern of interference nulling in BLAST. We summarize the implementation in the following algorithm.

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Algorithm 2 (QR Implementation of ZF-BLAST): The processing proceeds as follows. Step 1) Perform the unitary transformation in (14) Step 2) Decode symbol of the th transmit antenna ac. Denote the corcording to . responding decision by Step 3) The algorithm successively decodes symbols from to the first transmit antenna. Given ( ), subtract interference generated by as

(15) is then generated from . Decision Algorithm 2 concisely represents the key BLAST-type signalprocessing techniques. Assuming perfect interference cancellanoninterfering one-dimensional tion, the algorithm creates (1-D) subchannels, corresponding to every transmit antenna, as described in [5]. The th subchannel can be expressed as (16) where , that is, the th subchannel has diversity order of . Besides the layered space–time processing, codes can be designed to improve system performance (see, e.g., [13]). However, as we elaborated before, the decision from the “weakest” subchannel, that is, the one with lowest diversity, is used to create a subchannel with higher diversity. Therefore, the performance of those subchannels with higher diversity may be compromised, due to error propagation. The extended algorithm EXT-ZF-BLAST can be used to overcome the performance bottleneck while maintaining low complexity of QR implementation. We use a (6, 6) system as an example to explain the operation of EXT-ZF-BLAST. Algorithm 2 is applied to generate decisions of the first subchannel, which is the strongest one. The signal emitted by the first transmit antenna can be estimated from the decisions. We subtract it from the received signal, which now can be treated as being generated from a (5, 6) system. If we apply Algorithm 2 for this system, the diversity order of all subchannels, ranging from 2 to 6, is increased by one level, compared with the original system. Improved subchannels produce better decisions, which, in turn, help the decoding of the first subchannel. This completes the first loopback operation, after which, the diversity order of all subchannels (starting from 1 to 6) is 6, 6, 5, 4, 3, and 2, respectively. We see that the second subchannel has full diversity order besides the first subchannel. We can start the next loopback operation by canceling signals from both the first and the second subchannels, thus effectively forming a (4, 6) , system. Define the depth of loopback operation to be that is, subchannel 1 to are used successively in the loopback cancellation. Since the processing structure evolves after each loopback operation, the final structure has the propsubchannels have full diversity erty that the first order, while the rest have five down to the level of

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diversity. The formal EXT-ZF-BLAST is given in the following system. algorithm for an , we define Algorithm 3 (EXT-ZF-BLAST): For

Recall that the second term in (15) is the interference term due to imperfect decision feedback. In order to quantify the effect of interference, we approximate it as a Gaussian random variable, for is , that is, denoted by . Suppose . Given equal probability of and , it is easy to verify that

(17) where

is the th column of . Also, let . Denote the loopback depth by . The algorithm proceeds as follows. Step 1) Decode each symbol using Algorithm 2 for the origsystem. Denote the initial decision of inal the symbol by . Step 2) For to , do the following: a) Subtract signals due to symbols from to as

(18) b) Apply Algorithm 2 to the system expressed in (18) to generate decisions . c) From to 1, cancel interference from other symbols as (19) The updated decision

is generated from

.

B. Approximate Performance Analysis The subchannels created by the layered space–time processing from Algorithm 2 are coupled through decision feedback. Since subchannels with higher indexes have less diversity, information bits transmitted through them need more protection. If the corresponding decisions suffer considerable degradation in quality, the higher diversity gain afforded by subchannels with lower indexes cannot be realized, due to nonnegligible interference. We could analytically study the performance of the baseline algorithm and its extension by modeling residual interference in every step of the algorithm as Gaussian variables, and evaluating the performance based on this approximation. In this section, we use ZF-BLAST ant its extension to illustrate the method. For simplicity, we assume that binary phase-shift keying (BPSK) is used. A th-order diversity channel can be described as

(23) (24) Therefore, the mean and variance of

are given by (25) (26)

where we have used the fact that , and it is independent . of Now we are ready to give an algorithm that calculates of all subchannels under Gaussian approximation for interference. Algorithm 4 (Performance Approximation for SNR in (2). ZF-BLAST): Given SNR, Step 1) Calculate of th subchannel by (22) with diverand . sity order successively where . Step 2) Calculate is calculated by For given , interference power (26). Then is obtained using (22) with and . is obtained by averaging across all Step 3) The overall the subchannels, that is, SNR . Similarly, we can analytically study the performance of EXT-ZF-BLAST by approximating the residual interference after decision feedback as Gaussian random variables. Hence, we have the following algorithm parallel to Algorithm 4. Algorithm 5 (EXT-ZF-BLAST): Let be a vector of ’s of all subchannels at each step in . Denote Algorithm 3 with loopback parameter . Given SNR, SNR by (2). Step 1) Calculate using Algorithm 4 for the original system using ZF-BLAST. Step 2) For to , do the following. )By Gaussian approximation of residual interference, in (18) is distributed as with (27)

(20) where and . Let denote signal . The is the average power, that is, SNR per diversity branch. The probability of error for BPSK modulation given channel realization is (21) is the tail integration of where Gaussian function. The average , , is the expectation of in (21) with respect to random variable , and is given in [14] as (22) where

.

Apply Algorithm 4 for the (18) to generate . From to 1, update and sity order in view of (19).

system in using (22) with diver-

IV. NUMERICAL RESULTS Fig. 1 shows simulation performance of the baseline space–time processing (ZF-BLAST) for a (6, 6) uncoded system. is chosen to be 10 000 time instants. For each static channel coherence period, is generated randomly by sampling independent variables for all its entries. Information bits are modulated using BPSK onto six transmit

LIU AND SAYEED: ITERATIVE EXTENSION OF BLAST DECODING ALGORITHM

Fig. 1. Baseline space–time processing performance for a (6, 6) BPSK k 6 denotes probability of bit error for kth uncoded system. p where 1 subchannel. Simulated performance, plotted as the solid line, is obtained by averaging over 10 000 channel static periods with T = 10 000. Performance from approximate analysis is also plotted in dashed lines.

 

antennas at every time instant. Algorithm 2 is applied at the is averaged over 10 000 channel coherence pereceiver. riods, and thus system performance is estimated over time instants. We plot overall and for each subchannel from simulations in Fig. 1. We also plot the corresponding analytical performance curves from Algorithm 4. It is seen that our approximate analysis deviates from simulation data at higher SNR. However, they are very close for overall . Furthermore, we observe from the figure that subchannels with higher indexes tend to have worse performance, since those subchannels have less diversity. The Gaussian approximation is motivated by the argument of the central limit theorem, as the interference consists of a sum of individual error signals. However, the approximation is not accurate for small values of antenna dimension. Fig. 2 plots the actual histogram of the interference, collected from BLAST simulations. It is seen that the interference has much higher concentration around the center than a corresponding Gaussian distribution with the same mean and variance. This may explain the discrepancy between the approximate analysis and simulation data. However, as clearly demonstrated in Fig. 1, the Gaussian approximation is able to capture the essential behavior of decoding algorithm, thus serving as a useful tool to study the qualitative algorithm characteristics. Imperfect decision feedback in the baseline algorithm compromises the performance of the subchannels with large diverperformance of each subchannel assity order. The ideal suming perfect interference cancellation is computed using (22), performance. In and serves as a lower bound on the actual for several subFig. 3, we plot both the ideal and the actual channels in the (6, 6) system used in the above simulation. Comparing ideal performance with actual simulation data, we can see that imperfect decision feedback severely compromises the performance of subchannels with large diversity order. Moreover, the overall system performance is limited by the worst

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Fig. 2. Histogram of the interference signal corresponding to the first subchannel of a (6, 6) BLAST simulation. The horizontal scale is with respect to the data variance.

Fig. 3. Ideal subchannel performance assuming perfect decision feedback, the lower bound for P is compared with actual simulated performance for the simulation in Fig. 1.

subchannel, as evident from the figure. Therefore, the worst subchannel in layered space–time structures is the bottleneck that limits system performance. Also, the significant gap between ideal and actual performance of subchannels with high diversity indicates significant room for improvement. Fig. 4 demonstrates performance improvement with loopback cancellation in the extended space–time algorithm (EXT-ZFBLAST) for a (6, 6) system. As before, bits are sent using BPSK at each transmit antenna. We vary the loopback depth from 1 to 5. The performance of the baseline algorithm (ZFBLAST) is also included as a comparison. As is evident from the figure, a large performance gain is achieved by using loopback cancellation. For example, the system with full loopback at 4 dB of the BLAST system at 9 dB, resulting in already achieves a 5-dB saving. We note that performance gain increases as SNR increases, which projects more power savings at higher SNR.

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Fig. 4. Simulated (6, 6) BPSK uncoded system performance using . Performance of the baseline EXT-ZF-BLAST with different I algorithm is also included for comparison. Dramatic improvement in performance compared with the baseline algorithm is evident.

Fig. 5. Performance comparison between variants of baseline (extended) algorithm on a simulated (4, 4) 4-QAM system. Extended algorithm is seen to improve upon its corresponding baseline algorithm. EXT-ZF-BLAST outperforms ZF-VBLAST, which suggests that the ordering overhead could be avoided in this case. It is also evident from the figure that MMSE-BLAST ranks the best in various algorithms, and its extension only brings marginal performance gain.

Moreover, a few loopback cancellations are sufficient. Fig. 4 shows that the algorithm is able to achieve large performance improvement with only one or two levels of loopback. Different layered space–time decoding algorithms entail different complexity. MMSE filtering is more expensive than the ZF filtering, because of the QR implementation used by the latter. Dynamic order in the algorithm adds additional complexity. We are interested in comparing the performance of different variants of the baseline (extended) algorithms. As demonstrated by the simulations and analysis above, ZF-BLAST has the minimal complexity, but its performance is limited. Since EXT-ZF-BLAST maintains the low-complexity advantage as its baseline algorithm, we single out EXT-ZF-BLAST in our simulation study. Fig. 5 plots the performance of ZF-BLAST, ZF-VBLAST, EXT-ZF-BLAST, MMSE-VBLAST, and EXT-VBLAST on a simulated (4, 4) system with 4-ary quadrature amplitude modulation (4-QAM). It is seen that one loopback operation in EXT-ZF-BLAST is sufficient to bring significant gain over its baseline algorithm ZF-BLAST. EXT-ZF-BLAST outperforming ZF-VBLAST suggests that the cost of dynamic ordering with ZF filtering is unnecessary. The performance enhancement due to loopback is critically linked to the relative improvement in terms of the quality of feedback decisions. In an MMSE-VBLAST system, each subchannel decoding is dynamically optimized to best exploit channel diversity. Thus, the loopback can only bring in a marginal gain on decision quality, which is demonstrated in the figure, showing that MMSE-VBLAST exhibits exceptional performance and its iterative extension seems not worthy of the extra cost. However, the implementation of MMSE-VBLAST is much more complicated, compared with that of ZF-BLAST and EXT-ZF-BLAST. Therefore, EXT-ZF-BLAST could well serve as a low-complexity option for layered space–time processing. As we increase the number of antennas in the system, the performance gain of EXT-ZF-BLAST gradually increases.

Fig. 6. Performance comparison between variants of baseline (extended) algorithm on a simulated (6, 6) 4-QAM system. Extended algorithm outperforms ZF-VBLAST.

We plot the performance of EXT-ZF-BLAST on a simulated (6, 6) system using 4-QAM modulation in Fig. 6. Comparing the extended Fig. 5 with Fig. 6, one can see at algorithm achieves a 6-dB gain in a (6, 6) system, while only a 4-dB gain in a (4, 4) system. For the same (6, 6) system as in Fig. 6, we show the performance of MMSE-VBLAST in Fig. 7, which again demonstrates that an extended algorithm for MMSE-VBLAST could not offer much benefit. V. CONCLUSION We have presented an iterative extension of the baseline BLAST-type algorithm. The extended algorithm is aimed at mitigating error propagation due to imperfect decision feedback

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[9] M. K. Varanasi, “Decision feedback multiuser detection: A systematic approach,” IEEE Trans. Inf. Theory, vol. 45, no. 1, pp. 219–240, Jan. 1999. [10] T. Marzetta, “BLAST training: Estimating channel characteristics for high-capacity space-time wireless,” in Proc. 37th Annu. Allerton Conf., Monticello, IL, Sep. 1999, pp. 958–966. [11] K. Liu and A. M. Sayeed, “Improved layered space-time processing in multiple antenna systems,” in Proc. 39th Annu. Allerton Conf., Monticello, IL, Oct. 2001. [12] R. A. Horn and C. R. Johnson, Matrix Analysis. New York: Cambridge Univ. Press, 1988. [13] D. S. Shiu and J. M. Kahn, “Scalable layered space–time codes for wireless communications performance and analyis and design criteria,” in Proc. WCNC, New Orleans, LA, Sep. 1999, pp. 159–163. [14] J. G. Proakis, Digital Communications, 3rd ed. New York: McGrawHill, 1995.

Fig. 7. Performance of EXT-MMSE-VBLAST on a simulated (6, 6) 4-QAM system. The extended algorithm does not seem to significantly improve over the baseline algorithm.

in BLAST. Performance and cost tradeoffs play an important role in choosing the appropriate algorithm for a particular application. Among all of the variants of the baseline (extended) algorithm, EXT-ZF-BLAST seems a good candidate for low-complexity applications, yet is capable of achieving good performance. However, if the complexity is not a primary concern, MMSE-VBLAST gives the best performance. REFERENCES [1] G. J. Foschini and M. J. Gans, “On limits of wireless communications in a fading environment when using multiple antennas,” Wireless Pers. Commun., vol. 6, no. 3, pp. 311–335, Mar. 1998. [2] I. E. Telatar, “Capacity of Multi-Antenna Gaussian Channels,” AT&T Bell Labs., Tech. Rep., 1995. [3] V. Tarokh, N. Seshadri, and A. R. Calderbank, “Space–time codes for high data rate wireless communication: Performance criterion and code construction,” IEEE Trans. Inf. Theory, vol. 44, no. 2, pp. 744–765, Mar. 1998. [4] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Combined array processing and space-time coding,” IEEE Trans. Inf. Theory, vol. 45, no. 3, pp. 1121–1128, May 1999. [5] G. J. Foschini, “Layered space-time architecture for wireless communication in a fading environment when using multi-element antennas,” Bell Labs Tech. J., vol. 1, no. 2, pp. 41–59, 1996. [6] P. W. Wolniansky, G. J. Foschini, G. D. Golden, and R. A. Valenzuela, “V-BLAST: An architecture for realizing very high data rates over the rich-scattering wireless channel,” in Proc. ISSSE, Pisa, Italy, Sep. 1998, pp. 295–300. [7] B. Hassibi, “An efficient square-root algorithm for BLAST,” in Proc. ICASSP, Istanbul, Turkey, Jun. 2000, pp. 11737–11740. [8] S. Verdú, Multiuser Detection. New York: Cambridge Univ. Press, 1998.

Ke Liu (S’99) received the B.S. degree in electrical engineering from Tsinghua University, Beijing, China, in 1999, and the M.S. degree in electrical engineering, the M.A. degree in mathematics, and the Ph.D. degree in electrical engineering from the University of Wisconsin, Madison, in 2001, 2002, and 2004, respectively. From 2000 to 2004, he was a Research Assistant with the Wireless Communications Laboratory, University of Wisconsin, Madison. Since October 2004, he has been with The Ohio State University, Columbus, where he currently serves as a Postdoctoral Researcher with the Information Processing Systems Laboratory. His research interests include information theory, wireless communications, wireless ad hoc networks, and statistical signal processing. Dr. Liu was the recipient of the Graduate School Fellowship from 1999 to 2000 and the Harold A. Peterson Graduate Student Paper Award for his work on space–time coding for multiantenna systems. He has served as a reviewer for the IEEE Communications, Signal Processing, and Information Theory Societies.

Akbar M. Sayeed (S’89–M’97–SM’02) received the B.S. degree from the University of Wisconsin, Madison in 1991, and the M.S. and Ph.D. degrees from the University of Illinois at Urbana-Champaign, Urbana, in 1993 and 1996, respectively, all in electrical and computer engineering. While with the University of Illinois, he was a Research Assistant with the Coordinated Science Laboratory and was also the Schlumberger Fellow in signal processing from 1992 to 1995. During 1996–1997, he was a Postdoctoral Fellow with Rice University, Houston, TX. Since August 1997, he has been with the University of Wisconsin, Madison, where he is currently an Assistant Professor of Electrical and Computer Engineering. His research interests are in wireless communications, sensor networks, statistical signal processing, wavelets, and time-frequency analysis. Dr. Sayeed was the recipient of the National Science Foundation CAREER Award in 1999 and the Office of Naval Research Young Investigator Award in 2001. He served as an Associate Editor for the IEEE SIGNAL PROCESSING LETTERS from 1999 to 2002.

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Blind Carrier Frequency Tracking for Filterbank Multicarrier Wireless Communications Vincenzo Lottici, Marco Luise, Senior Member, IEEE, Cosimo Saccomando, and Filippo Spalla

Abstract—Filterbank multicarrier modulation (FBMCM) is an attractive technology for high-speed twisted-pair transmission and for broadband wireless communications, as well. In wireline applications, signal transmission takes place at baseband, so the issue of carrier acquisition and tracking for coherent demodulation does not apply. In wireless communications, on the contrary, carrierfrequency recovery reveals the Achille’s heel of multicarrier modulation, so that robust signal processing algorithms are needed in this respect. In this paper, we derive a nondata-aided carrier-frequency offset recovery method for wireless FBMCM modems. In particular, we illustrate how to derive a low-complexity closed-loop tracker starting from a maximum-likelihood approach. We then show that the proposed simplifications do not entail large performance losses. In this respect, we derive the standard performance metrics of a closed-loop tracker (S-curve, root mean square estimation error, acquisition time) both on the additive white Gaussian noise channel and on a typical static frequency-selective wireless channel. We also demonstrate by simulation good robustness of the frequency tracker with respect to FBMCM symbol-timing errors. Index Terms—Filterbank, frequency recovery, multicarrier (MC) transmission, synchronization.

I. INTRODUCTION ULTICARRIER (MC) modulation has received considerable interest in the last decades in the field of high-bit-rate transmission, over both wired and wireless frequency-selective communication channels [1]–[3]. The most prominent examples of MC techniques are orthogonal frequency division multiplexing (OFDM) and discrete multitone (DMT). The former was adopted in the European standard for terrestrial digital audio broadcasting (DAB) [4] and terrestrial digital video broadcasting (DVB-T) [5], and is also embedded into the popular Wi-Fi standard for wireless LAN’s 802.11a/g [6]. DMT is the workhorse for high-rate communications over the copper-wire access network, and represents the basis for digital subscriber line (DSL) services [7]. Recently, the concept of MC transmission has been generalized with the introduction of filterbank multicarrier modulation (FBMCM) in the area of high-speed wired access networks [8], [9]. In FBMCM-based systems, the data symbols are transmitted over different subchannels after suitable pulse shaping. In particular,

M

Paper approved by S. K. Wilson, the Editor for Multicarrier Modulation of the IEEE Communications Society. Manuscript received March 12, 2004; revised July 30, 2004 and March 1, 2005. This paper was presented in part at the IEEE International Conference on Communications, Paris, France, June 2004. V. Lottici, M. Luise, and C. Saccomando are with the University of Pisa, Department of Information Engineering, I-56122 Pisa, Italy (e-mail: [email protected]; [email protected]; [email protected]). F. Spalla is with STMicroelectronics, AST Division, Agrate Brianza (MI), Italy (e-mail: [email protected]). Digital Object Identifier 10.1109/TCOMM.2005.855003

the pulse shape in FBMCM is significantly longer than the subchannel symbol period, so, unlike OFDM or DMT, the pulse waveforms for different symbols overlap in time. The effect of pulse shaping is that the spectra of the data signals on the different subcarriers are bandlimited, while preserving orthogonality amongst subcarriers in spite of the time overlap. This, in a sense, is a dual arrangement with respect to conventional OFDM, wherein pulses are non-overlapping in time, while spectra do overlap in frequency. The almost null frequency overlap of subchannels brings forth a number of advantages [9]: 1) reduction of sensitivity to near-end crosstalk (NEXT), echo signals, and narrowband interferers; 2) more flexibility to allocate groups of subchannels to different services or even service providers over the same cable (as is the case in the context of unbundling of the local loop); 3) frequency-domain equalization can be applied without the adoption of the cyclic extension, and without the consequent spectral efficiency loss. This is why FBMCM modulation was included in the interim very-high-speed DSL (VDSL) standard [9], and is being considered for wireless LAN and broadband wireless access network applications [10]. In addition, FBMCM together with OFDM has also recently become part of the standard for the terrestrial return channel of DVB (DVB-RCT) [11]. As far as signal detection is concerned, the major drawback of cyclic-extensionless FBMCM is caused by the introduction of pulse shaping; correct channel equalization and data demodulation require prior symbol-timing recovery [12], as opposed to conventional cyclic-extension-based OFDM, which does not. The increasing interest in FBMCM is also demonstrated by a number of relevant contributions that have recently appeared in the literature on the topics of channel equalization and signal synchronization [8], [12]–[16]. The performance of an optimal postdetection combiner that yields maximum signal-to-interference-plus-noise ratio (SINR) is evaluated for a version of FBMCM referred to as discrete wavelet multitone (DWMT) modulation [8]. The sensitivity of FBMCM to timing errors is discussed in [12] and [13], whereas in [14], fractionally spaced linear and decision-feedback equalizers for FBMCM are designed and analyzed. The issue of timing recovery is addressed in [15], leading to a data-aided or decision-directed early–late (EL) timing-error detector (TED), as well as in [16], presenting a simpler TED based on symbol-rate signal samples. With respect to synchronization, very little effort has been devoted so far (to the best of the authors’ knowledge) to the issue of carrier-frequency offset (CFO) estimation in FBMCM-based systems. This is mainly due to the fact that most studies were geared toward baseband communications. Previous contributions on the subject were primarily focused on OFDM formats.

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Fig. 1.

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FBMCM transmission system.

Indeed, a maximum-likelihood (ML)-based approach is proposed in [17], where the authors end up with a blind estimator. Others either exploit training sequences [18], [19], or are based on the redundancy of the cyclic prefix [20], [21]. All of them are valuable references, but cannot be directly applied to FBMCM, due to the different structure of the data signal. In this paper, we derive a carrier-frequency tracking method for FBMCM based on an ML approach. We also show how to derive simplified algorithms from the truly-ML solution to get an efficient, nondata-aided (NDA) algorithm with modest computational complexity. Although such algorithms are derived under the hypothesis of frequency-flat additive white Gaussian noise (AWGN) channel and ideal timing synchronization, they turn out to be remarkably robust against multipath transmission and timing offset, as well. The relevant performance analysis is carried out with a suitable mixture of analytical results (S-curve) and simulation [acquisition time, root-mean-square estimation error (RMSEE)]. In the next section, we describe the FBMCM system model, and, in particular, develop the architecture of the ML receiver. In Section III, we derive the ML-based carrier-frequency estimator and resort to some simplifications to reduce the computational complexity. The open-loop and closed-loop performance of the synchronizers are illustrated by the results reported in Section IV, and the paper is eventually concluded by some final remarks in Section V. II. FILTER-BANK MULTICARRIER MODULATION A. FBMCM Signal Model In an FBMCM-based system, the available frequency band is subdivided into a set of contiguous subbands through which the data symbols are transmitted after suitable pulse shaping. Specifically, the source -ary quadrature amplitude modulation ( –QAM) symbols (at the rate ) are grouped into blocks of size , the symbol being the index of the symbol within each block ( ) (and is also the subcarrier index after MC modulation), while denotes the generic block index. Each subcarrier is spectrally shaped

with a conventional square-root raised cosine (SRRC) Nyquist has roll-off factor and filter whose impulse response signaling interval . The center frequency for the th subcarto prevent spectral overlapping between the rier is subchannels. To simplify the subsequent derivations, we also be an integer. The require that the quantity time-continuous transmitted FBMCM signal is then

(1) A reasonable implementation of the FBMCM modulator is based on digital processing. Adhering to this approach, we adopt the sampling frequency , and use a polyphase decomposition for the time index of the th sampling , . The digitized instant FBMCM signal is thus

(2) . As in [9], where, following a standard notation, the signal in (2) can be generated via a conventional OFDM modulator followed by a polyphase filterbank, as shown in Fig. 1. The th -symbol block undergoes an inverse discrete Fourier transform (IDFT), and then each element of the parallel IDFT output is processed at the rate by a different filter (the filterbank) whose impulse response is obtained by the polyphase decomposition of the prototype SRRC filter

(3) Notice that the number of filters is (where is the number of signal samples to be computed in every block of source symbols), while the number of the IDFT outputs is (the number of subcarriers). As a result, the filterbank in Fig. 1 also contains a

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suitable signal memory that feeds the diverse filters of the bank with appropriately permuted versions of the IDFT outputs.

The problem of maximizing such quantity with respect to equivalent to maximizing

B. Derivation of the FBMCM ML Data Demodulator In this section, our aim is to derive the (optimum) ML architecture of the data demodulator for FBMCM. We will follow an elementary approach that provides more insight than the method based on the filterbank theory presented in [9]. We assume an AWGN channel, and that signal synchronization, i.e., carrier frequency/phase and symbol timing, is ideal. Under such hypotheses, the analog baseband received signal plus WGN can be used anti-alias filtered, and digitized at the same rate in the digital modulator, thereby obtaining a stream of digital samples as in (2), corrupted by complex-valued time-discrete . AWGN with variance the data seLet us denote by quence to be detected, given by the concatenation of data with , for a blocks source symbols. Cortotal observation length of respondingly, the signal samples relevant to the same interval can be arranged in the vector with . The log-likelihood function for (neglecting irrelevant factors) can be written as

is

(11) As is typical with ML detection, (11) shows that the ML receiver whose Euclidean distance searches for that data sequence from the sequence of signal samples given by (8) is minimum. This means that in the case of uncoded transmission, one into a (multilevel, complex) threshold has simply to feed detector. Now the question is, how can one compute the sufficient statistics from the received signal ? The simplest and most efficient way to do this is obtained by changing the time , with index in (8), i.e., letting . This leads to

(12) where (13)

(4) where

is the “local” trial signal replica, defined as (5)

can be interpreted as the response of the receiver polyphase filterbank (14) to the input signal

. Hence, (12) becomes

and indicates a trial value for . Therefore, the ML estimate for the data sequence can be found as (6) Using (5), we can cast the first term at the right-hand side (RHS) of (4) in the following form:

(15) where (16)

(7) where

(8) After some algebra, we obtain (9) Therefore, using (7) and (9) in (4), we find

(10)

. is the discrete Fourier transform (DFT) of Summing up, we obtain the efficient digital receiver architecture shown in Fig. 1 that appears to implement the dual functions of the transmitter. Once the received signal samples within a data block are match-filtered on each subcarrier by the receiver is Fourier polyphase filterbank (14), the parallel outputs transformed into the sequence . Finally, each DFT output is phase-shifted by the factor to yield the sufficient statistics depending on and , namely, the FBMCM symbol and the subcarrier indexes, respectively. As regards the final phase shift that depends both on the time index and on the subcarrier index , we can observe that the FBMCM demodulator will certainly encompass a frequency-domain equalizer (as shown in Fig. 1) to compensate for the channel frequency response [10], [12]. Therefore, the phase-shift factors in (15) will be directly compensated for by the function of channel equalization.

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III. CARRIER-FREQUENCY ESTIMATION A. Likelihood Function Assuming ideal timing recovery, the received signal on the AWGN channel is

(17) where is the frequency offset normalized to the sampling rate , is the phase offset, and represents complex-valued . The joint likelihood AWGN with zero mean and variance function (LF) for the transmitted symbol sequence and the two unknown parameters and is (up to irrelevant multiplicative factors) [22]

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is the modified Bessel function of the first kind and and order zero. Derivation of a blind estimator of the frequency offset calls with respect to to obtain for further averaging of . Since this opthe final (data-independent) marginal LF eration is very difficult to do analytically, we resort to the following approximations. Assuming a low signal-to-noise-ratio in (22) is small with high prob(SNR) regime, where the can be replaced by a power ability, the Bessel function . series truncated up to the quadratic term, i.e., Under the assumption of independent, zero-mean data symbols, and discarding irrelevant factors, it can be shown that (24) thus obtaining the blind carrier-frequency estimator

(18)

(25)

where B. Frequency-Error Detector (FED) (19) is the “local” trial signal replica, with and as trial values for the sync parameters. Replacing (19) into (18) yields

The ML frequency estimator (25) requires a grid search, which is not computationally efficient, and accordingly, a different solution has to be pursued. An alternative approach can be based on a recursive solution of (25) via the stochastic gradient algorithm (26)

(20) with

where the error signal (with the assumption that by

) is given

(27) (21) Let us now consider the phase offset as a nuisance parameter, and average (20) with respect to . We get the following marginal LF that is a function of and only

and is the step-size parameter. Elaborating on (21) we have (28), shown at the bottom of the page, or equivalently, (29), shown at the bottom of the page. Thus, using (21) and (29) and skipping irrelevant terms, we obtain

(22) where (23)

(30)

(28)

(29)

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FC-FED recovery loop.

where

. Indeed, using (33) and (34) in (35), the error signal can be rearranged as (31)

(32) Therefore, letting

(36) since

(33) (34)

(37) Therefore, dropping the factor as

the final result is

, the error signal can be written

(38) (35)

C. Efficient Realization of the FED The error signal given by (35) used in the recursion (26) is evaluated at the FBMCM symbol rate of at the end of each data block. First, the two signals and , as seen from (31) and (32), are obtained from the after preliminary frequency received signal precompensation by the (current) “trial” value , and then by processing the compensated received signal with the two polyphase filter units and , having as prototypes the matched filter and the frequency-derivative matched , respectively. The error signal (35) is finally evalufilter ated by cross-correlating the DFTs of the two sequences and above. Such processing is clearly nontrivial, in particular, for the need of an additional DFT unit to compute (the DFT output is needed for data detection, see Fig. 1). Fortunately, a simple application of Parseval’s theorem shows that cross-correlating the two DFTs and as in (35) is equivalent to directly correlating the two frequency-precorrected time signals and

thus achieving a considerable complexity reduction, since the extra DFT is no longer required. The resulting frequency recovery loop, which we refer to as full-complexity FED (FCFED), is outlined in Fig. 2. D. Low-Complexity FED The complexity of the blind FC-FED loop in Fig. 2 is still considerable due to the presence of the additional frequencyderivative polyphase filterbank . Therefore, we will develop in the following a low-complexity approximation of the FC-FED through replacing the additional polyphase filterbank with a simpler filtering scheme that provides an approximation of . To this aim, we decompose the signal given by (31) into the sum of the following three terms:

(39a)

(39b) (39c)

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Fig. 3. LC-FED recovery loop.

so that (40) The key idea in obtaining a low-complexity FED, as originally proposed in [23] in the context of single-carrier transmission, is letting

(41) Now, replacing (41) in (32) and considering (39), we find the desired approximation (42) that can be computed with no additional filterbank, provided that the main output of the filterbank is conveniently reorganized according to (39). Substituting (42) and (40) into (38), we obtain

S-curve is evaluated both analytically and by simulation in the AWGN channel and on a typical static wireless multipath channel, both with ideal timing recovery and in the presence of a static timing offset. In the second part, we concentrate our attention on closed-loop tracking, that is to say on the issues of loop acquisition time and steady-state RMSEE. In the evaluation of those parameters, evaluation by theoretical analysis is quite tedious, so that we resort to simulation. In all of the charts illustrated in the following, the frequency error is normalized to the subcarrier spacing. Our numerical results refer specifically to the DVB-RCT con, which text [11]. The number of the subcarriers is corresponds to the size of the (I)DFT. The SRRC prototype filter of the transmit and receive polyphase filters has roll-off . The modulation format is 4-QAM, the frefactor kHz, and quency spacing between subcarriers is therefore, the length of the FBMCM signaling interval is 1.25 ms. The DVB-T multipath channel is static with propagation paths, with complex gains and delays given in [11] (see also Appendix A for further details on channel modeling). A. Open Loop

(43) A further simplification that leads to the low-complexity FED (LC-FED) can be motivated as follows. Indeed, we found out by simulation that the second RHS term in (43) is negligible with respect to the first one, so that the error signal can be finally simplified as (44) The resulting low-complexity frequency-recovery loop is sketched in Fig. 3. IV. PERFORMANCE RESULTS This section is dedicated to the performance analysis of both the FC- FED and the LC-FED loops with signal errors given by (38) and (44), respectively. The first part deals with the open-loop analysis of the FEDs that we derived above. The

As is known, the S-curve is the expectation of the error signal for a given value of the frequency error . As illustrated in Appendix A, the S-curve for the FC-FED in the multipath channel is S

(45) is the frequency errornormalized to the samwhere pling rate , is the Fourier transform of the root-raised , while is cosine Nyquist pulse, the frequency response of the multipath channel at the th subcarrier. From (45), we see that each subcarrier contributes to the in overall S-curve with a common term given by the factor (45), weighted by a specific frequency-flat factor . Hence, regardless of the basic assumption of the AWGN channel we formulated above, the proposed FED exhibits a considerable robustness with respect to the multipath channel as well, as confirmed by the simulation results shown later on.

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Fig. 4. S-curves for the FC-FED given by theory (solid lines) and simulation (marks) for AWGN and multipath DVB-T channels with roll-off factors  0.25, 0.5.

=

The analytical (solid line) and simulated (marks) S-curves for the FC-FED in the AWGN and multipath channels (DVBT/DVB-RCT benchmark channel [11]) are shown in Fig. 4 assuming ideal timing recovery. Some remarks on these results are in order. 1) The S-curves are periodic with unit period (the subcarrier spacing), and exhibit an acquisition range of roughly two-thirdsof the subcarrier spacing. Therefore, as with most blind frequency-recovery algorithms for MC modulation, the loop has an intrinsic ambiguity of one subcarrier that has to be solved by other means. 2) The major pitfall of the FC-FED is that its S-curve is characterized by the presence of a “dead zone” centered around (characterized by , see Fig. 4) that may cause hang-ups of the loop during initial acquisition, and so may lead to long acquisition times. This is a direct consequence of the shape , or, more specifically, of its roll-off of the prototype filter factor . Indeed, choosing a different value of the roll-off factor, , it is apparent that the effect of a wider roll-off namely, interval, equal to , is to reduce the “dead zone” to the points , as illustrated in Fig. 4. The effect of nonideal timing recovery is illustrated in Fig. 5. To be specific, we consider as timing offset the values , where is the length of the FBMCM signaling interval, and the DVB-T multipath channel. Our results indicate that the larger is the timing offsets, the weaker is the FC-FED sensitivity to frequency errors. However, for timing offsets restricted to 20%–30% of the symbol interval, the degradation experienced by the FED, as acquisition range and detector gain, is definitely tolerable. The S-curves for the FC- and the LC-FEDs are compared in Fig. 6, wherein we show both simulation and theoretical results (that exhibit perfect agreement). The low-complexity algorithm exhibits a lower intrinsic discrimination gain (the slope of the curve), and this is motivated by the suboptiat mality of the approach that led to its derivation, as compared with the truly ML derivation of the FC-FED. On the other hand, , the LC-FED does not have any “dead zone” around and so it is preferable to its FC counterpart. As remarked above for the FC-FED, the reason can be found again in the approximation we followed for the LC-FED. That can be thought to be equivalent to taking a different shape of the prototype filter

Fig. 5. S-curves for the FC-FED with various timing offsets for the DVB-T multipath channel.

Fig. 6. S-curves for the FC-FED and the LC-FED given by theory (solid lines) and simulation (marks) with ideal timing for the DVB-T multipath channel.

for the additional polyphase filterbank . Indeed, the prototype (note that is assumed to be real and even) in the FC-FED is substituted in the LC-FED with , whose frequency response is not confined as the in the roll-off region case for , but spans the entire frequency axis. Further results (not reported here) show that the LC-FED is quite insensitive to timing errors in the range of 20%–30% of the symbol interval, as already observed in Fig. 5 for the FC-FED. B. Closed Loop During closed-loop tracking, the frequency estimate is updated according to (26), where the error signal is given either by (38) or by (44). The step size is related to the normalized noise-equivalent bandwidth of the (first-order) loop by [22], where is the slope of the S-curve at . The bandwidth is normalized to the updating rate of (26), i.e., the data block rate. The inverse of the noise bandwidth also gives a rough indication about the time that the loop takes to acquire lock, starting from an incorrect initial estimate of the frequency offset. To investigate both

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Fig. 7. Frequency acquisitions of the FC-FED loop with various timing offsets for the DVB-T multipath channel.

the acquisition and the steady-state behavior of the loop, we resorted to simulations. Sample acquisitions of the FC-FED loop are shown in Fig. 7 for and dB on the DVB-(RC)T multipath channel. It is apparent that the frequency error eventually settles to zero, independent of the value of the timing error ( ), even though the acquisitions with timing errors are longer than with ideal timing recovery, due to the smaller . The figures to follow illustrate the frequency value of RMSEE (normalized to the subcarrier spacing) as a function of for the the energy per bit to noise spectral density ratio DVB-T multipath channel and several system configurations. Fig. 8 shows the RMSEE with ideal timing recovery for two and normalized loop bandwidths, namely, , together with the relevant modified Cramér–Rao bound (MCRB) [24] computed in the same conditions. We observe that the RMSEE exhibits a floor as the SNR increases. This is just the effect of the loop self-noise, and is typical of blind, NDA synchronization schemes: the error signal has a considerable AWGN-independent noise component caused by data modulation and arising by the interaction of the two signal terms in (38) or (44). The considerable distance of the RMSEE curves in Fig. 8 from their respective MCRBs is typical of blind closedloop frequency recovery, as well [22]. Fig. 9 illustrates the performance degradation caused by nonideal timing recovery. In full agreement with the results illustrated above for both the S-curve and the acquisition transients, the RMSEE increases in the presence of a static timing error. Finally, a comparison of the RMSEE of the FC-FED and the LC-FED discussed above is presented in Fig. 10. It makes clear that the LC-FED loop exhibits a worse performance than the FC-FED, i.e., the former is affected by a larger RMSEE floor component at high SNR. However, the performance degradation can be considered as acceptable, and therefore, we conclude that the LC-FED loop represents a reasonable tradeoff between performance and implementation complexity. V. SUMMARY AND CONCLUSIONS After presenting the basics of FBMCM and the relevant optimum data demodulation for wireless communications,

Fig. 8. Frequency RMSEEs of the FC-FED loop with ideal timing recovery for the DVB-T multipath channel with B 0:1; 0:01.

=

Fig. 9. Frequency RMSEEs of the FC-FED loop with various timing offsets for the DVB-T multipath channel and with B = 0:1.

this paper showed how to derive blind, NDA, closed-loop frequency-recovery schemes based on the ML criterion for such signaling formats. Due to their intrinsic estimation ambiguity equal to one subcarrier spacing, the proposed algorithms are suitable to apply for carrier-frequency tracking. The performance in terms of the S-curve and RMSEE was investigated, taking also into account the effect of the multipath channel and imperfect timing recovery. Our results indicate that the proposed frequency-recovery schemes: 1) perform remarkably well on a typical DVB-T

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and are the complex gains and the delays where (normalized to the sampling interval ) of the paths, respectively, and using (17), (31) and (32) can be written as shown in (A.2)–(A.3) at the bottom of the page, where is the frequency . Hence, substituting error normalized to the sampling rate (A.2) and (A.3) into (38), and computing the expectation under the assumption of independent and identically distributed data symbols, the S-curve turns out to be as shown in (A.4) at the . Now, let us conbottom of the page, where centrate on the first series contained in the square brackets. Resorting to a frequency-domain representation and making use of Poisson’s formulas, this can be written as

(A.5) Fig. 10. Frequency RMSEE of FC-FED and LC-FED loops with ideal timing recovery over the DVB-T multipath channel and with B 0:1; 0:01.

=

multipath channel; 2) exhibit an unambiguous acquisition range around plus or minus one-third of the subcarrier spacing for the full-complexity FED, and up to plus or minus half the subcarrier spacing for the low-complexity version; 3) can operate with slight performance degradation in the presence of a timing inaccuracy up to 20%–30% of the FBMCM symbol interval; 4) have satisfactory RMSEE performance, both in the full- and the low-complexity versions, although orders of magnitude worse than the relevant MCRB, as is typical for NDA closed-loop frequency synchronizers. APPENDIX

where is the Fourier transform of the Nyquist SRRC pulse . Likewise, the second term of (A.4) enclosed in the square brackets can be expressed as the derivative, with respect to , . Thereof the conjugate of (A.5) divided by the constant fore, substituting into (A.4), after some algebra, the S-curve of the FC-FED in the multipath channel can be put (except for a multiplicative factor independent of ) in the form shown in (A.6) at the top of the next page. In order to simplify (A.6), is we now make the following remarks: 1) the support of ; 2) the indexes and can assume the values ; 3) , i.e., the maximum delay spread of the wireless transmission channel is much shorter than the FBMCM symbol interval. Owing to 2) and 3), we can assume , and therefore, (A.6) can be rearranged as that

In this appendix we give some details about the evaluation of the S-curve of the FC-FED for the multipath channel. Assuming that the impulse response of the static multipath channel is (A.1)

(A.7)

(A.2)

(A.3)

(A.4)

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(A.6)

where is the multipath channel frequency response. Further, taking into account remarks that 1) and 2), we can assume over the support of . Finally, (A.7) can be written as

(A.8) REFERENCES [1] B. R. Saltzberg, “Performance of an efficient parallel data transmission system,” IEEE Trans. Commun. Technol., vol. COM-15, no. 6, pp. 805–811, Dec. 1967. [2] S. B. Weistein and P. M. Ebert, “Data transmission by frequency-division multiplexing using the discrete Fourier transform,” IEEE Trans. Commun. Technol., vol. COM-19, no. 5, pp. 628–634, Oct. 1971. [3] J. A. C. Bingham, “Multicarrier modulation for data transmission: An idea whose time has come,” IEEE Commun. Mag., pp. 5–14, May 1990. [4] B. Le Floch, H. Lassalle, and D. Castelain, “Digital sound broadcasting to mobile receivers,” IEEE Trans. Consum. Electron., vol. 35, no. 3, pp. 493–503, Aug. 1989. [5] H. Sari, G. Karam, and I. Jeanclaude, “Transmission techniques for digital terrestrial TV broadcasting,” IEEE Commun. Mag., pp. 100–109, Feb. 1995. [6] R. van Nee et al., “New high-rate wireless LAN standards,” IEEE Commun. Mag., pp. 82–88, Dec. 1999. [7] J. S. Chow, J. C. Tu, and J. M. Cioffi, “A discrete multitone transceiver system for HDSL applications,” IEEE J. Sel. Areas Commun., vol. 9, no. 6, pp. 895–908, Aug. 1991. [8] S. D. Sandberg and M. A. Tzannes, “Overlapped discrete multitone modulation for high-speed copper wire communications,” IEEE J. Sel. Areas Commun., vol. 13, no. 9, pp. 1571–1585, Dec. 1995. [9] G. Cherubini, E. Eleftheriou, and S. Olcer, “Filtered multitone modulation for high-speed digital subscriber lines,” IEEE J. Sel. Areas Commun., vol. 20, no. 5, pp. 1016–1028, Jun. 2002. [10] M. Renfors, H. Xing, A. Viholainen, and J. Rinne, “On channel equalization in filter bank based multicarrier wireless access systems,” in Proc. Veh. Technol. Conf., Sep. 1999, pp. 228–232. [11] European Telecommunications Standard Institute (ETSI), EN 301 958 DVB-RCT Standard, 2001. [12] K. M. Wong, J. Wu, and T. N. Davidson, “Wavelet packet division multiplexing and wavelet packet design under timing error effects,” IEEE Trans. Signal Process., vol. 45, no. 12, pp. 2877–2890, Dec. 1997. [13] J. Louveaux, L. Vandendorpe, L. Cuvelier, and T. Pollet, “Bit-rate sensitivity of filter-bank-based VDSL transmission to timing errors,” IEEE Trans. Commun., vol. 49, no. 2, pp. 375–384, Feb. 2001. [14] L. Vandendorpe, L. Cuvelier, F. Deryck, J. Louveaux, and O. van de Wiel, “Fractionally spaced linear and decision feedback detectors for transmultiplexers,” IEEE Trans. Signal Process., vol. 46, no. 4, pp. 996–1011, Apr. 1998. [15] J. Louveaux, L. Vandendorpe, L. Cuvelier, and T. Pollet, “An early–late timing recovery scheme for filter-bank-based multicarrier transmission,” IEEE Trans. Commun., vol. 48, no. 10, pp. 1746–1754, Oct. 2000. [16] J. Louveaux, L. Cuvelier, L. Vandendorpe, and T. Pollet, “Baud rate timing recovery scheme for filter-bank-based multicarrier transmission,” IEEE Trans. Commun., vol. 51, no. 4, pp. 652–663, Apr. 2003. [17] F. Daffara and A. Chouly, “Maximum-likelihood frequency detectors for orthogonal multicarrier systems,” in Proc. IEEE Int. Conf. Commun., 1993, pp. 766–771.

[18] P. H. Moose, “A technique for OFDM frequency offset correction,” IEEE Trans. Commun., vol. 42, no. 10, pp. 2908–2914, Oct. 1994. [19] T. M. Schmidl and D. C. Cox, “Robust frequency and timing synchronization for OFDM,” IEEE Trans. Commun., vol. 45, no. 12, pp. 1613–1621, Dec. 1997. [20] F. Daffara and O. Adami, “A novel carrier recovery technique for orthogonal multicarrier system,” Euro. Trans. Telecommun., vol. 7, no. 4, pp. 323–334, Jul.-Aug. 1996. [21] J. J. Van de Beek, M. Sandell, and P. O. Borjesson, “ML estimation of time and frequency offset in OFDM systems,” IEEE Trans. Signal Process., vol. 45, no. 7, pp. 1800–1805, Jul. 1997. [22] U. Mengali and A. N. D’Andrea, Synchronization Techniques for Digital Receivers. New York: Plenum, 1997. [23] G. Karam, F. Daffara, and H. Sari, “Simplified versions of the maximumlikelihood frequency detector,” in Proc. IEEE GLOBECOM, Dec. 1992, pp. 345–349. [24] A. N. D’Andrea, U. Mengali, and R. Reggiannini, “The modified Cramer–Rao bound and its applications to synchronization problems,” IEEE Trans. Commun., vol. 42, no. 2–4, pp. 1391–1399, Feb.-Apr. 1994.

Vincenzo Lottici received the Dr. Ing. degree (cum laude) in electronic engineering from the University of Pisa, Pisa, Italy, in 1985. From 1987 to 1993, he worked in the design and development of sonar digital signal processing systems. Since 1993, he has been with the Department of Information Engineering, University of Pisa, where he is currently a Research Fellow and Assistant Professor in Telecommunications. His research interests include the area of wireless multicarrier and UWB systems, with particular emphasis on synchronization and channel-estimation techniques. Dr. Lottici received the “Best Thesis SIP Award” from the University of Pisa in 1986.

Marco Luise (M’88–SM’97) was born in Livorno, Italy, in 1960. He received the M.D. and Ph.D. degrees in electronic engineering from the University of Pisa, Pisa, Italy. He is currently a Full Professor of Telecommunications at the University of Pisa. In the past, he was a Research Fellow of the European Space Agency (ESA) at the European Space Research and Technology Centre (ESTEC), Noordwijk, The Netherlands, and a Research Scientist of CNR, the Italian National Research Council, at the Centro Studio Metodi Dispositivi Radiotrasmissioni (CSMDR), Pisa. He co-chaired four editions of the Tyrrhenian International Workshop on Digital Communications, and in 1998 was the General Chairman of the URSI Symposium ISSSE’98. He has been the Technical Co-Chairman of the 7th International Workshop on Digital Signal Processing Techniques for Space Communications and of the Conference European Wireless 2002. He has served as Editor for Communications Theory of the European Transactions on Telecommunications. He has authored more than 100 publications on international journals and contributions to major international conferences, and holds a few international patents. His main research interests lie in the broad area of wireless communications, with particular emphasis on CDMA systems and satellite communications. Dr. Luise served as Editor for Synchronization of the IEEE TRANSACTIONS ON COMMUNICATIONS.

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Cosimo Saccomando was born in Cacciano, Italy, in 1975. He received the telecommunications engineering degree in 2001 from the University of Pisa, Pisa, Italy, where he is currently working toward the Ph.D. degree. His research interests are in the field of synchronization and equalization of multicarrier communications.

Filippo Spalla was born in 1974. He received the M.S. degree in electrical engineering in 1999 from the Universitá di Pavia, Pavia, Italy. Since 2000, he has been with STMicroelectronics (Italy) working on baseband signal processing, synchronization, and channel estimation for several wireless telecommunication systems (DVB-T, DVB-RCT, WLAN), based on the OFDM technique. From 2002 to 2004, he was WorkPackage Leader of the MEDEA+ European project with major European R&D departments, where he coordinated the development of an OFDM-A technology-based baseband prototype for interactive digital terrestrial TV user terminals. During 2004, he spent a period at the University of California, Los Angeles, focusing on channel estimation techniques for MIMO OFDM systems, and contributing to the definition of a 4 4 on-air MIMO prototype. Currently, he is the STM representative in the IEEE 802.11 TGn standardization process, where he is contributing mainly to the MIMO preamble definition, packet detection, synchronization, and MIMO channel estimation aspects.

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Abstracts of Forthcoming Manuscripts Nonsystematic Turbo Codes Adrish Banerjee, Francesca Vatta, Bartolo Scanavino, and Daniel J. Costello, Jr. Abstract—In this paper, we introduce the concept of nonsystematic turbo codes and compare them with classical systematic turbo codes. Nonsystematic turbo codes can achieve lower error floors than systematic turbo codes because of their superior effective free-distance properties. Moreover, they can achieve comparable performance in the waterfall region if the nonsystematic constituent encoder has a lower weight feedforward in12 3 verse. A uniform inteleaver analysis is used to show that rate turbo codes using nonsystematic constituent encoders have larger effective free distance than when systematic constituent encoders are used. Also, mutual information-based transfer characteristics and EXIT charts are = 1 3 turbo codes with nonsystematic conused to show that rate stituent encoders having low-weight feedforward inverses achieve convergence thresholds comparable to those achieved with systematic constituent encoders. Catastrophic encoders, which do not possess a feedforward inverse, are shown to be capable of achieving low convergence thresholds by doping the code with a small fraction of systematic bits. Finally, we give tables of good nonsystematic turbo codes and present simulation results comparing the performance of systematic and nonsystematic turbo codes. A. Banerjee is with the Department of Electrical Engineering, Indian Institute of Technology, Kanpur 208026, India. F. Vatta is with DEEI, Universita di Trieste, I Trieste, Italy. B. Scanavino is with CERCOM, Politecnico di Torino, 10129 Torino, Italy. D. J. Costello, Jr. is with the Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN 46556 USA. Digital Object Identifier 10.1109/TCOMM.2005.857226

The Effect of Narrowband Interference on Wideband Wireless Communication Systems Andrea Giorgetti, Marco Chiani, and Moe Z. Win Abstract—This paper evaluates the performance of wideband communication systems in the presence of narrowband interference. In particular, we derive closed bit-error probability expressions for spread-spectrum systems by approximating narrowband interferers as independent asynchronous tone interferers. The scenarios considered include additive white Gaussian noise channels, flat-fading channels, and frequency-selective multipath fading channels. For multipath fading channels, we develop a new analytical framework based on perturbation theory to analyze the performance of a Rake receiver in Nakagami- channels. Simulation results for narrowband interference such as GSM and Bluetooth are in good agreement with our analytical results, showing the approach developed is useful for investigating the coexistence of ultrawide bandwidth systems with existing wireless systems. A. Giorgetti and M. Chiani are with IEIITBO/CNR, DEIS, University of Bologna, 40136 Bologna, Italy. M. Z. Win is with the Laboratory for Information and Decision Systems (LIDS), Massachusetts Institute of Technology, Cambridge, MA 02139 USA. Digital Object Identifier 10.1109/TCOMM.2005.857224

Accurate Evaluation of Bit-Error Rates of Optical Communication Systems Using the Gram–Charlier Series Moshe Nazarathy Abstract—The probability densities and cumulative distribution functions of decision statistics of optical communications systems are expanded

as a Gram–Charlier series, leading to arbitrarily accurate systematic evaluation of bit-error rates and optimal decision thresholds of optical communication systems. The method displays negligible computational complexity, and is applicable whenever the moment or cumulant generating function of the decision statistics are analytically available. We applied the technique to a birth-and-death Markoffian model of a direct-detection receiver with optical preamplifier in a two-level amplitude-shift keying system. The modal expansion series rapidly converged, whereas the alternative saddlepoint approximation method predicted a bit-error rate which deviated by 7% from the Gram–Charlier result. The author is with the Faculty of Electrical Engineering, Technion, Israel Institute of Technology, Haifa, Israel. Digital Object Identifier 10.1109/TCOMM.2005.857227

Generalized Stokes-Parameters Shift Keying: A New Perspective on Optimal Detection Over Electrical and Optical Incoherent Channels Moshe Nazarathy and Erez Simony Abstract—We consider block transmission over the incoherent channel, whereby several symbols either launched consecutively in time, or in parallel over several diversity paths, are equally affected by common isotropic phase and additively corrupted by circularly Gaussian additive noise. Our novel approach toward transmission over such a vector incoherent channel (VIC) model consists of a coherency matrix and generalized Stokes parameters descriptions of optimal detection inspired by classical optical polarization theory, but further extended to the dual context of generic electrical, as well as optical, communication. The four classical real-valued parameters introduced into optics by Stokes in 1853 to characterize the state of polarization of an optical beam, recently generalized Stokes extended in a quantum-mechanical context to parameters, underlie our new alternative novel description of maximum a posteriori/maximum-likelihood (MAP/ML) optimal detection over the VIC channel. The modulation formats of polarization-shift keying (POLSK), differential phase-shift keying, and amplitude-shift keying, as well as recent and novel combinations thereof, will all be seen to be equivalent to a new conceptual form of modulation called generalized Stokes parameters shift keying. Several forms of the optimal MAP/ML receiver for vector multienergy transmission are developed, generalizing the classical scalar incoherent receiver and the known POLSK receivers. The authors are with the Technion, Israel Institute of Technology, Haifa, Israel. Digital Object Identifier 10.1109/TCOMM.2005.857223

Simulation of Rayleigh-Faded Mobile-to-Mobile Communication Channels Chirag S. Patel, Gordon L. Stuber, and Thomas G. Pratt Abstract—Mobile-to-mobile channels find increasing applications in futuristic intelligent transport systems, ad hoc mobile wireless networks, and relay-based cellular networks. Their statistical properties are quite different from typical cellular radio channels, thereby requiring new methods for their simulation. This paper proposed a “double-ring” model to simulate the mobile-to-mobile local scattering environment, and develops sum-of-sinusoids-based models for simulating such channels. The proposed models produce waveforms having desired statistical properties with good accuracy, and also remove some drawbacks of an existing model derived by using the discrete line spectrum-simulation method. C. S. Patel and G. L. Stuber are with the School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0250 USA. T. G. Pratt is with the Georgia Tech Research Institute, Atlanta, GA 303320250 USA. Digital Object Identifier 10.1109/TCOMM.2005.857225

0090-6778/$20.00 © 2005 IEEE

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Abstracts of Forthcoming Manuscripts Nonsystematic Turbo Codes Adrish Banerjee, Francesca Vatta, Bartolo Scanavino, and Daniel J. Costello, Jr. Abstract—In this paper, we introduce the concept of nonsystematic turbo codes and compare them with classical systematic turbo codes. Nonsystematic turbo codes can achieve lower error floors than systematic turbo codes because of their superior effective free-distance properties. Moreover, they can achieve comparable performance in the waterfall region if the nonsystematic constituent encoder has a lower weight feedforward in12 3 verse. A uniform inteleaver analysis is used to show that rate turbo codes using nonsystematic constituent encoders have larger effective free distance than when systematic constituent encoders are used. Also, mutual information-based transfer characteristics and EXIT charts are = 1 3 turbo codes with nonsystematic conused to show that rate stituent encoders having low-weight feedforward inverses achieve convergence thresholds comparable to those achieved with systematic constituent encoders. Catastrophic encoders, which do not possess a feedforward inverse, are shown to be capable of achieving low convergence thresholds by doping the code with a small fraction of systematic bits. Finally, we give tables of good nonsystematic turbo codes and present simulation results comparing the performance of systematic and nonsystematic turbo codes. A. Banerjee is with the Department of Electrical Engineering, Indian Institute of Technology, Kanpur 208026, India. F. Vatta is with DEEI, Universita di Trieste, I Trieste, Italy. B. Scanavino is with CERCOM, Politecnico di Torino, 10129 Torino, Italy. D. J. Costello, Jr. is with the Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN 46556 USA. Digital Object Identifier 10.1109/TCOMM.2005.857226

The Effect of Narrowband Interference on Wideband Wireless Communication Systems Andrea Giorgetti, Marco Chiani, and Moe Z. Win Abstract—This paper evaluates the performance of wideband communication systems in the presence of narrowband interference. In particular, we derive closed bit-error probability expressions for spread-spectrum systems by approximating narrowband interferers as independent asynchronous tone interferers. The scenarios considered include additive white Gaussian noise channels, flat-fading channels, and frequency-selective multipath fading channels. For multipath fading channels, we develop a new analytical framework based on perturbation theory to analyze the performance of a Rake receiver in Nakagami- channels. Simulation results for narrowband interference such as GSM and Bluetooth are in good agreement with our analytical results, showing the approach developed is useful for investigating the coexistence of ultrawide bandwidth systems with existing wireless systems. A. Giorgetti and M. Chiani are with IEIITBO/CNR, DEIS, University of Bologna, 40136 Bologna, Italy. M. Z. Win is with the Laboratory for Information and Decision Systems (LIDS), Massachusetts Institute of Technology, Cambridge, MA 02139 USA. Digital Object Identifier 10.1109/TCOMM.2005.857224

Accurate Evaluation of Bit-Error Rates of Optical Communication Systems Using the Gram–Charlier Series Moshe Nazarathy Abstract—The probability densities and cumulative distribution functions of decision statistics of optical communications systems are expanded

as a Gram–Charlier series, leading to arbitrarily accurate systematic evaluation of bit-error rates and optimal decision thresholds of optical communication systems. The method displays negligible computational complexity, and is applicable whenever the moment or cumulant generating function of the decision statistics are analytically available. We applied the technique to a birth-and-death Markoffian model of a direct-detection receiver with optical preamplifier in a two-level amplitude-shift keying system. The modal expansion series rapidly converged, whereas the alternative saddlepoint approximation method predicted a bit-error rate which deviated by 7% from the Gram–Charlier result. The author is with the Faculty of Electrical Engineering, Technion, Israel Institute of Technology, Haifa, Israel. Digital Object Identifier 10.1109/TCOMM.2005.857227

Generalized Stokes-Parameters Shift Keying: A New Perspective on Optimal Detection Over Electrical and Optical Incoherent Channels Moshe Nazarathy and Erez Simony Abstract—We consider block transmission over the incoherent channel, whereby several symbols either launched consecutively in time, or in parallel over several diversity paths, are equally affected by common isotropic phase and additively corrupted by circularly Gaussian additive noise. Our novel approach toward transmission over such a vector incoherent channel (VIC) model consists of a coherency matrix and generalized Stokes parameters descriptions of optimal detection inspired by classical optical polarization theory, but further extended to the dual context of generic electrical, as well as optical, communication. The four classical real-valued parameters introduced into optics by Stokes in 1853 to characterize the state of polarization of an optical beam, recently generalized Stokes extended in a quantum-mechanical context to parameters, underlie our new alternative novel description of maximum a posteriori/maximum-likelihood (MAP/ML) optimal detection over the VIC channel. The modulation formats of polarization-shift keying (POLSK), differential phase-shift keying, and amplitude-shift keying, as well as recent and novel combinations thereof, will all be seen to be equivalent to a new conceptual form of modulation called generalized Stokes parameters shift keying. Several forms of the optimal MAP/ML receiver for vector multienergy transmission are developed, generalizing the classical scalar incoherent receiver and the known POLSK receivers. The authors are with the Technion, Israel Institute of Technology, Haifa, Israel. Digital Object Identifier 10.1109/TCOMM.2005.857223

Simulation of Rayleigh-Faded Mobile-to-Mobile Communication Channels Chirag S. Patel, Gordon L. Stuber, and Thomas G. Pratt Abstract—Mobile-to-mobile channels find increasing applications in futuristic intelligent transport systems, ad hoc mobile wireless networks, and relay-based cellular networks. Their statistical properties are quite different from typical cellular radio channels, thereby requiring new methods for their simulation. This paper proposed a “double-ring” model to simulate the mobile-to-mobile local scattering environment, and develops sum-of-sinusoids-based models for simulating such channels. The proposed models produce waveforms having desired statistical properties with good accuracy, and also remove some drawbacks of an existing model derived by using the discrete line spectrum-simulation method. C. S. Patel and G. L. Stuber are with the School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0250 USA. T. G. Pratt is with the Georgia Tech Research Institute, Atlanta, GA 303320250 USA. Digital Object Identifier 10.1109/TCOMM.2005.857225

0090-6778/$20.00 © 2005 IEEE

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Abstracts of Forthcoming Manuscripts Nonsystematic Turbo Codes Adrish Banerjee, Francesca Vatta, Bartolo Scanavino, and Daniel J. Costello, Jr. Abstract—In this paper, we introduce the concept of nonsystematic turbo codes and compare them with classical systematic turbo codes. Nonsystematic turbo codes can achieve lower error floors than systematic turbo codes because of their superior effective free-distance properties. Moreover, they can achieve comparable performance in the waterfall region if the nonsystematic constituent encoder has a lower weight feedforward in12 3 verse. A uniform inteleaver analysis is used to show that rate turbo codes using nonsystematic constituent encoders have larger effective free distance than when systematic constituent encoders are used. Also, mutual information-based transfer characteristics and EXIT charts are = 1 3 turbo codes with nonsystematic conused to show that rate stituent encoders having low-weight feedforward inverses achieve convergence thresholds comparable to those achieved with systematic constituent encoders. Catastrophic encoders, which do not possess a feedforward inverse, are shown to be capable of achieving low convergence thresholds by doping the code with a small fraction of systematic bits. Finally, we give tables of good nonsystematic turbo codes and present simulation results comparing the performance of systematic and nonsystematic turbo codes. A. Banerjee is with the Department of Electrical Engineering, Indian Institute of Technology, Kanpur 208026, India. F. Vatta is with DEEI, Universita di Trieste, I Trieste, Italy. B. Scanavino is with CERCOM, Politecnico di Torino, 10129 Torino, Italy. D. J. Costello, Jr. is with the Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN 46556 USA. Digital Object Identifier 10.1109/TCOMM.2005.857226

The Effect of Narrowband Interference on Wideband Wireless Communication Systems Andrea Giorgetti, Marco Chiani, and Moe Z. Win Abstract—This paper evaluates the performance of wideband communication systems in the presence of narrowband interference. In particular, we derive closed bit-error probability expressions for spread-spectrum systems by approximating narrowband interferers as independent asynchronous tone interferers. The scenarios considered include additive white Gaussian noise channels, flat-fading channels, and frequency-selective multipath fading channels. For multipath fading channels, we develop a new analytical framework based on perturbation theory to analyze the performance of a Rake receiver in Nakagami- channels. Simulation results for narrowband interference such as GSM and Bluetooth are in good agreement with our analytical results, showing the approach developed is useful for investigating the coexistence of ultrawide bandwidth systems with existing wireless systems. A. Giorgetti and M. Chiani are with IEIITBO/CNR, DEIS, University of Bologna, 40136 Bologna, Italy. M. Z. Win is with the Laboratory for Information and Decision Systems (LIDS), Massachusetts Institute of Technology, Cambridge, MA 02139 USA. Digital Object Identifier 10.1109/TCOMM.2005.857224

Accurate Evaluation of Bit-Error Rates of Optical Communication Systems Using the Gram–Charlier Series Moshe Nazarathy Abstract—The probability densities and cumulative distribution functions of decision statistics of optical communications systems are expanded

as a Gram–Charlier series, leading to arbitrarily accurate systematic evaluation of bit-error rates and optimal decision thresholds of optical communication systems. The method displays negligible computational complexity, and is applicable whenever the moment or cumulant generating function of the decision statistics are analytically available. We applied the technique to a birth-and-death Markoffian model of a direct-detection receiver with optical preamplifier in a two-level amplitude-shift keying system. The modal expansion series rapidly converged, whereas the alternative saddlepoint approximation method predicted a bit-error rate which deviated by 7% from the Gram–Charlier result. The author is with the Faculty of Electrical Engineering, Technion, Israel Institute of Technology, Haifa, Israel. Digital Object Identifier 10.1109/TCOMM.2005.857227

Generalized Stokes-Parameters Shift Keying: A New Perspective on Optimal Detection Over Electrical and Optical Incoherent Channels Moshe Nazarathy and Erez Simony Abstract—We consider block transmission over the incoherent channel, whereby several symbols either launched consecutively in time, or in parallel over several diversity paths, are equally affected by common isotropic phase and additively corrupted by circularly Gaussian additive noise. Our novel approach toward transmission over such a vector incoherent channel (VIC) model consists of a coherency matrix and generalized Stokes parameters descriptions of optimal detection inspired by classical optical polarization theory, but further extended to the dual context of generic electrical, as well as optical, communication. The four classical real-valued parameters introduced into optics by Stokes in 1853 to characterize the state of polarization of an optical beam, recently generalized Stokes extended in a quantum-mechanical context to parameters, underlie our new alternative novel description of maximum a posteriori/maximum-likelihood (MAP/ML) optimal detection over the VIC channel. The modulation formats of polarization-shift keying (POLSK), differential phase-shift keying, and amplitude-shift keying, as well as recent and novel combinations thereof, will all be seen to be equivalent to a new conceptual form of modulation called generalized Stokes parameters shift keying. Several forms of the optimal MAP/ML receiver for vector multienergy transmission are developed, generalizing the classical scalar incoherent receiver and the known POLSK receivers. The authors are with the Technion, Israel Institute of Technology, Haifa, Israel. Digital Object Identifier 10.1109/TCOMM.2005.857223

Simulation of Rayleigh-Faded Mobile-to-Mobile Communication Channels Chirag S. Patel, Gordon L. Stuber, and Thomas G. Pratt Abstract—Mobile-to-mobile channels find increasing applications in futuristic intelligent transport systems, ad hoc mobile wireless networks, and relay-based cellular networks. Their statistical properties are quite different from typical cellular radio channels, thereby requiring new methods for their simulation. This paper proposed a “double-ring” model to simulate the mobile-to-mobile local scattering environment, and develops sum-of-sinusoids-based models for simulating such channels. The proposed models produce waveforms having desired statistical properties with good accuracy, and also remove some drawbacks of an existing model derived by using the discrete line spectrum-simulation method. C. S. Patel and G. L. Stuber are with the School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0250 USA. T. G. Pratt is with the Georgia Tech Research Institute, Atlanta, GA 303320250 USA. Digital Object Identifier 10.1109/TCOMM.2005.857225

0090-6778/$20.00 © 2005 IEEE

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 53, NO. 10, OCTOBER 2005

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Abstracts of Forthcoming Manuscripts Nonsystematic Turbo Codes Adrish Banerjee, Francesca Vatta, Bartolo Scanavino, and Daniel J. Costello, Jr. Abstract—In this paper, we introduce the concept of nonsystematic turbo codes and compare them with classical systematic turbo codes. Nonsystematic turbo codes can achieve lower error floors than systematic turbo codes because of their superior effective free-distance properties. Moreover, they can achieve comparable performance in the waterfall region if the nonsystematic constituent encoder has a lower weight feedforward in12 3 verse. A uniform inteleaver analysis is used to show that rate turbo codes using nonsystematic constituent encoders have larger effective free distance than when systematic constituent encoders are used. Also, mutual information-based transfer characteristics and EXIT charts are = 1 3 turbo codes with nonsystematic conused to show that rate stituent encoders having low-weight feedforward inverses achieve convergence thresholds comparable to those achieved with systematic constituent encoders. Catastrophic encoders, which do not possess a feedforward inverse, are shown to be capable of achieving low convergence thresholds by doping the code with a small fraction of systematic bits. Finally, we give tables of good nonsystematic turbo codes and present simulation results comparing the performance of systematic and nonsystematic turbo codes. A. Banerjee is with the Department of Electrical Engineering, Indian Institute of Technology, Kanpur 208026, India. F. Vatta is with DEEI, Universita di Trieste, I Trieste, Italy. B. Scanavino is with CERCOM, Politecnico di Torino, 10129 Torino, Italy. D. J. Costello, Jr. is with the Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN 46556 USA. Digital Object Identifier 10.1109/TCOMM.2005.857226

The Effect of Narrowband Interference on Wideband Wireless Communication Systems Andrea Giorgetti, Marco Chiani, and Moe Z. Win Abstract—This paper evaluates the performance of wideband communication systems in the presence of narrowband interference. In particular, we derive closed bit-error probability expressions for spread-spectrum systems by approximating narrowband interferers as independent asynchronous tone interferers. The scenarios considered include additive white Gaussian noise channels, flat-fading channels, and frequency-selective multipath fading channels. For multipath fading channels, we develop a new analytical framework based on perturbation theory to analyze the performance of a Rake receiver in Nakagami- channels. Simulation results for narrowband interference such as GSM and Bluetooth are in good agreement with our analytical results, showing the approach developed is useful for investigating the coexistence of ultrawide bandwidth systems with existing wireless systems. A. Giorgetti and M. Chiani are with IEIITBO/CNR, DEIS, University of Bologna, 40136 Bologna, Italy. M. Z. Win is with the Laboratory for Information and Decision Systems (LIDS), Massachusetts Institute of Technology, Cambridge, MA 02139 USA. Digital Object Identifier 10.1109/TCOMM.2005.857224

Accurate Evaluation of Bit-Error Rates of Optical Communication Systems Using the Gram–Charlier Series Moshe Nazarathy Abstract—The probability densities and cumulative distribution functions of decision statistics of optical communications systems are expanded

as a Gram–Charlier series, leading to arbitrarily accurate systematic evaluation of bit-error rates and optimal decision thresholds of optical communication systems. The method displays negligible computational complexity, and is applicable whenever the moment or cumulant generating function of the decision statistics are analytically available. We applied the technique to a birth-and-death Markoffian model of a direct-detection receiver with optical preamplifier in a two-level amplitude-shift keying system. The modal expansion series rapidly converged, whereas the alternative saddlepoint approximation method predicted a bit-error rate which deviated by 7% from the Gram–Charlier result. The author is with the Faculty of Electrical Engineering, Technion, Israel Institute of Technology, Haifa, Israel. Digital Object Identifier 10.1109/TCOMM.2005.857227

Generalized Stokes-Parameters Shift Keying: A New Perspective on Optimal Detection Over Electrical and Optical Incoherent Channels Moshe Nazarathy and Erez Simony Abstract—We consider block transmission over the incoherent channel, whereby several symbols either launched consecutively in time, or in parallel over several diversity paths, are equally affected by common isotropic phase and additively corrupted by circularly Gaussian additive noise. Our novel approach toward transmission over such a vector incoherent channel (VIC) model consists of a coherency matrix and generalized Stokes parameters descriptions of optimal detection inspired by classical optical polarization theory, but further extended to the dual context of generic electrical, as well as optical, communication. The four classical real-valued parameters introduced into optics by Stokes in 1853 to characterize the state of polarization of an optical beam, recently generalized Stokes extended in a quantum-mechanical context to parameters, underlie our new alternative novel description of maximum a posteriori/maximum-likelihood (MAP/ML) optimal detection over the VIC channel. The modulation formats of polarization-shift keying (POLSK), differential phase-shift keying, and amplitude-shift keying, as well as recent and novel combinations thereof, will all be seen to be equivalent to a new conceptual form of modulation called generalized Stokes parameters shift keying. Several forms of the optimal MAP/ML receiver for vector multienergy transmission are developed, generalizing the classical scalar incoherent receiver and the known POLSK receivers. The authors are with the Technion, Israel Institute of Technology, Haifa, Israel. Digital Object Identifier 10.1109/TCOMM.2005.857223

Simulation of Rayleigh-Faded Mobile-to-Mobile Communication Channels Chirag S. Patel, Gordon L. Stuber, and Thomas G. Pratt Abstract—Mobile-to-mobile channels find increasing applications in futuristic intelligent transport systems, ad hoc mobile wireless networks, and relay-based cellular networks. Their statistical properties are quite different from typical cellular radio channels, thereby requiring new methods for their simulation. This paper proposed a “double-ring” model to simulate the mobile-to-mobile local scattering environment, and develops sum-of-sinusoids-based models for simulating such channels. The proposed models produce waveforms having desired statistical properties with good accuracy, and also remove some drawbacks of an existing model derived by using the discrete line spectrum-simulation method. C. S. Patel and G. L. Stuber are with the School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0250 USA. T. G. Pratt is with the Georgia Tech Research Institute, Atlanta, GA 303320250 USA. Digital Object Identifier 10.1109/TCOMM.2005.857225

0090-6778/$20.00 © 2005 IEEE

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 53, NO. 10, OCTOBER 2005

1773

Abstracts of Forthcoming Manuscripts Nonsystematic Turbo Codes Adrish Banerjee, Francesca Vatta, Bartolo Scanavino, and Daniel J. Costello, Jr. Abstract—In this paper, we introduce the concept of nonsystematic turbo codes and compare them with classical systematic turbo codes. Nonsystematic turbo codes can achieve lower error floors than systematic turbo codes because of their superior effective free-distance properties. Moreover, they can achieve comparable performance in the waterfall region if the nonsystematic constituent encoder has a lower weight feedforward in12 3 verse. A uniform inteleaver analysis is used to show that rate turbo codes using nonsystematic constituent encoders have larger effective free distance than when systematic constituent encoders are used. Also, mutual information-based transfer characteristics and EXIT charts are = 1 3 turbo codes with nonsystematic conused to show that rate stituent encoders having low-weight feedforward inverses achieve convergence thresholds comparable to those achieved with systematic constituent encoders. Catastrophic encoders, which do not possess a feedforward inverse, are shown to be capable of achieving low convergence thresholds by doping the code with a small fraction of systematic bits. Finally, we give tables of good nonsystematic turbo codes and present simulation results comparing the performance of systematic and nonsystematic turbo codes. A. Banerjee is with the Department of Electrical Engineering, Indian Institute of Technology, Kanpur 208026, India. F. Vatta is with DEEI, Universita di Trieste, I Trieste, Italy. B. Scanavino is with CERCOM, Politecnico di Torino, 10129 Torino, Italy. D. J. Costello, Jr. is with the Department of Electrical Engineering, University of Notre Dame, Notre Dame, IN 46556 USA. Digital Object Identifier 10.1109/TCOMM.2005.857226

The Effect of Narrowband Interference on Wideband Wireless Communication Systems Andrea Giorgetti, Marco Chiani, and Moe Z. Win Abstract—This paper evaluates the performance of wideband communication systems in the presence of narrowband interference. In particular, we derive closed bit-error probability expressions for spread-spectrum systems by approximating narrowband interferers as independent asynchronous tone interferers. The scenarios considered include additive white Gaussian noise channels, flat-fading channels, and frequency-selective multipath fading channels. For multipath fading channels, we develop a new analytical framework based on perturbation theory to analyze the performance of a Rake receiver in Nakagami- channels. Simulation results for narrowband interference such as GSM and Bluetooth are in good agreement with our analytical results, showing the approach developed is useful for investigating the coexistence of ultrawide bandwidth systems with existing wireless systems. A. Giorgetti and M. Chiani are with IEIITBO/CNR, DEIS, University of Bologna, 40136 Bologna, Italy. M. Z. Win is with the Laboratory for Information and Decision Systems (LIDS), Massachusetts Institute of Technology, Cambridge, MA 02139 USA. Digital Object Identifier 10.1109/TCOMM.2005.857224

Accurate Evaluation of Bit-Error Rates of Optical Communication Systems Using the Gram–Charlier Series Moshe Nazarathy Abstract—The probability densities and cumulative distribution functions of decision statistics of optical communications systems are expanded

as a Gram–Charlier series, leading to arbitrarily accurate systematic evaluation of bit-error rates and optimal decision thresholds of optical communication systems. The method displays negligible computational complexity, and is applicable whenever the moment or cumulant generating function of the decision statistics are analytically available. We applied the technique to a birth-and-death Markoffian model of a direct-detection receiver with optical preamplifier in a two-level amplitude-shift keying system. The modal expansion series rapidly converged, whereas the alternative saddlepoint approximation method predicted a bit-error rate which deviated by 7% from the Gram–Charlier result. The author is with the Faculty of Electrical Engineering, Technion, Israel Institute of Technology, Haifa, Israel. Digital Object Identifier 10.1109/TCOMM.2005.857227

Generalized Stokes-Parameters Shift Keying: A New Perspective on Optimal Detection Over Electrical and Optical Incoherent Channels Moshe Nazarathy and Erez Simony Abstract—We consider block transmission over the incoherent channel, whereby several symbols either launched consecutively in time, or in parallel over several diversity paths, are equally affected by common isotropic phase and additively corrupted by circularly Gaussian additive noise. Our novel approach toward transmission over such a vector incoherent channel (VIC) model consists of a coherency matrix and generalized Stokes parameters descriptions of optimal detection inspired by classical optical polarization theory, but further extended to the dual context of generic electrical, as well as optical, communication. The four classical real-valued parameters introduced into optics by Stokes in 1853 to characterize the state of polarization of an optical beam, recently generalized Stokes extended in a quantum-mechanical context to parameters, underlie our new alternative novel description of maximum a posteriori/maximum-likelihood (MAP/ML) optimal detection over the VIC channel. The modulation formats of polarization-shift keying (POLSK), differential phase-shift keying, and amplitude-shift keying, as well as recent and novel combinations thereof, will all be seen to be equivalent to a new conceptual form of modulation called generalized Stokes parameters shift keying. Several forms of the optimal MAP/ML receiver for vector multienergy transmission are developed, generalizing the classical scalar incoherent receiver and the known POLSK receivers. The authors are with the Technion, Israel Institute of Technology, Haifa, Israel. Digital Object Identifier 10.1109/TCOMM.2005.857223

Simulation of Rayleigh-Faded Mobile-to-Mobile Communication Channels Chirag S. Patel, Gordon L. Stuber, and Thomas G. Pratt Abstract—Mobile-to-mobile channels find increasing applications in futuristic intelligent transport systems, ad hoc mobile wireless networks, and relay-based cellular networks. Their statistical properties are quite different from typical cellular radio channels, thereby requiring new methods for their simulation. This paper proposed a “double-ring” model to simulate the mobile-to-mobile local scattering environment, and develops sum-of-sinusoids-based models for simulating such channels. The proposed models produce waveforms having desired statistical properties with good accuracy, and also remove some drawbacks of an existing model derived by using the discrete line spectrum-simulation method. C. S. Patel and G. L. Stuber are with the School of Electrical and Computer Engineering, Georgia Institute of Technology, Atlanta, GA 30332-0250 USA. T. G. Pratt is with the Georgia Tech Research Institute, Atlanta, GA 303320250 USA. Digital Object Identifier 10.1109/TCOMM.2005.857225

0090-6778/$20.00 © 2005 IEEE

Digital Object Identifier 10.1109/TCOMM.2005.859282

Digital Object Identifier 10.1109/TCOMM.2005.859283

INFORMATION FOR AUTHORS THE IEEE TRANSACTIONS ON COMMUNICATIONS invites the submission of technical manuscripts on topics within the scope of the IEEE Communications Society, which includes all areas indicated on the inside front cover and those shown under the Technical Committees listing. Manuscripts reporting on original theoretical and/or experimental work and tutorial expositions of permanent reference value are welcome. In general, material which has been previously copyrighted, published, or accepted for publication will not be considered for publication in this TRANSACTIONS. Exceptions to this rule include items that have limited distribution, have appeared in Abstract form only, or have appeared only in conference proceedings; notice of such prior publication or concurrent submission elsewhere must be given at the time of submission to this TRANSACTIONS. A manuscript identical to, or largely based on, a conference paper must be so identified. All papers are reviewed by competent referees and are considered on the basis of their significance, novelty, and usefulness to the TRANSACTIONS readership. Contributions may be in one of two forms: Papers and Letters. Transactions Papers must be concisely written and be no longer than 20 double-spaced pages (12-point font, approximately 26 lines per page with 6.5-in. line length) excluding figures. Double-sided printing is encouraged. Figures must be used sparingly and repetitive patterns or results omitted. Transactions Papers must have no more than a total of ten figures and tables. Submitted manuscripts significantly exceeding these guidelines will be returned to the authors for revision before being reviewed. It is essential that each manuscript be accompanied by a 75- to 200-word abstract clearly outlining the scope and contributions of the paper and a list of up to five keywords should be included on the manuscript. The list of IEEE keywords is available by going to http://www.ieee.org/organizations/pubs/ani_prod/keywrd98.txt. Transactions Letters, limited to seven double-spaced pages and four figures, are intended to articulate new results and are to be published in the most expeditious manner. This category includes comments on published papers, corrections, and open problems, as well as new high-quality technical contributions primarily representing enhancements of a previous paper. It is the intention to provide a very rapid review process as well as publication of Transactions Letters. A Letter must be accompanied by an Abstract of 50 words or less. All papers should be written in English. Introductory discussion should be kept at a minimum and material published elsewhere should be referenced rather than reproduced or paraphrased. Authors should strive for maximum clarity of expression, bearing in mind that the purpose of publication is the dissemination of technical knowledge and that an excessively complex or poorly written presentation can only obscure the significance of the work described. Care should be taken in the organization of the material such that the contributions of the work and a logical, consistent progression of thought are evident. It is strongly suggested that material which is not essential to the continuity of the text (e.g., proofs, derivations, or calculations) be placed in Appendixes. Either hard copy or electronic submissions that conform to the respective guidelines following will be accepted for review: 1) Hard Copy: Six copies each of the manuscript (including the abstract and illustrations, but not the original illustrations themselves) should be submitted to the Editor-in-Chief, whose ad-

Digital Object Identifier 10.1109/TCOMM.2005.859281

dress is given on the inside front cover. Papers submitted to any other Editor of this TRANSACTIONS will be returned to the author. Double-sided copies of the initial manuscript are preferred, although an accepted manuscript must be submitted with a singlesided copy when in final form. A separate signed letter should indicate the preferred mailing address (including postal code), telephone and fax numbers, and email address (when available) for correspondence. Supplementary material, such as biographies and original figures, will be requested when needed for publication. 2) Electronic Copy: Electronic submissions should be made using the IEEE Transactions on Communications Web-based editorial processing system at mc.manuscriptcentral.com/tcom. Submission instructions, including how to create an author account on the site, are available on line. Submissions in PostScript (PS) or Portable Document Format (PDF) are accepted. When printed, the manuscript must satisfy all the page and figure count guidelines noted above. In addition, the author will be asked to provide (i) the name, email, and contact information of all the authors; (ii) the manuscript category; and (iii) the title and a text-only version of the abstract. All authors must also enclose, or mail separately in the case of electronic submission, a signed IEEE Copyright Form on submitting a paper and should have company clearance before submission. Submitted papers are assumed to contain no proprietary material unprotected by patent or patent application; responsibility for technical content and for protection of proprietary material rests solely with the author(s) and their organizations and is not the responsibility of the IEEE or its Editorial Staff. The format of papers submitted to the IEEE TRANSACTIONS ON COMMUNICATIONS should follow IEEE editorial and typographical standards, as described in “Information for IEEE Authors,” available on request from the IEEE Transactions/Journals Department, 445 Hoes Lane, Piscataway, NJ 08855-1331 USA or by email to [email protected]. After a manuscript has been accepted for publication, the author’s company or institution will be requested to pay a page charge of $110.00 per printed TRANSACTIONS page, for the first seven pages of a paper, to cover part of the cost of the publication. Payment of page charges for this IEEE TRANSACTIONS (in keeping with journals of other professional societies) is not a prerequisite for publication. The authors will receive 100 free reprints if the page charge is honored. All submissions after January 1, 1994 that are accepted for publication are subject to a mandatory page charge of $220.00 for each Transactions page exceeding seven printed pages. Please Note: Authors must provide to the Publications Editor an electronic version of their paper after acceptance (PostScript files cannot be accepted). Make sure that as you make changes to the paper version, you incorporate these changes into your disk version. Try to adhere to the accepted style of this TRANSACTIONS as much as possible. A list of various types of electronic media that the IEEE accepts can currently be obtained from the IEEE Transactions/Journals Department, 445 Hoes Lane, Piscataway, NJ 08855-1331 USA or from the Publications Editor of this TRANSACTIONS. Transactions Abstracts are abstracts of Transactions Papers accepted for publication in later issues. Transactions Abstracts are selected for publication from the accepted manuscripts.

IEEE Communications Society 2005 Board of Governors President C. A. SILLER, JR., Enginnovation, LLC

Officers President-Elect N. CHEUNG, Telcordia Technologies

Treasurer H. BLANK

Secretary/Executive Director J. M. HOWELL, IEEE ComSoc

VP-Technical Activities H. FREEMAN, Booz Allen Hamilton Inc.

VP-Society Relations R. SARACCO, Telecom Italia Lab

VP-Membership Services D. ZUCKERMAN

VP-Membership Development A. GELMAN, Panasonic Technologies

Chief Information Officer M. KAROL Avaya Inc. Director—Membership Programs Development P. PERRA Director—Marketing S. WEINSTEIN

Director—Journals D. P. TAYLOR Univ. of Canterbury Director—Asia/Pacific Region I. SASASE Keio Univ. Director—Related Societies J. LOCICERO Illinois Inst. of Technol.

CIO and Directors Director—Magazines A. JAJSZCZYK AGH Univ. of Technol. Director—EAME Region M. JAGODIC

Director—Meetings/Conferences S. GOYAL St. Petersburg College Director—LA Region R. VEIGA

Director—On-Line Content J. HONG POSTECH

Director—Sister Society Relations Director—Standards N. OHTA R. BOUTABA Sony Corp. Univ. of Waterloo

Director—Education W. H. TRANTER Virginia Tech. Director—NA Region R. SHAPIRO

Members at Large H. BLANK (’05)

J. LIEBEHERR (’05) Univ. of Virginia

J. LOCICERO (’05) Illinois Inst. of Technology

C.-S. LI (’05) IBM T. J. Watson Res. Ctr.

T. S. ATKINSON (’06) ICSI Consulting Services, Inc.

S. MOYER (’06) Telcordia Technol.

N. OHTA (’06) Sony Corp.

H. STUETTGEN (’06) NEC Europe Ltd.

R. BLAKE (’07) BT Labs.

L. CIMINI (’07) Univ. of Delaware

L. GREENSTEIN (’07) WINLAB, Rutgers Univ.

J. GIBSON (’07) Southern Methodist Univ.

Awards M. KINCAID MKS & Assoc.

Distinguished Lecturers Selection F. BAUER Nokia

Meetings & Conferences Board Director S. GOYAL St. Petersburg College TAC Liaison M. ULEMA Manhattan College Standing Member C. DESMOND World Class—Telecommun. Standing Member M. KINCAID MKS & Associates Standing Member J. LOCICERO Illinois Inst. of Technol. Standing Member C-L. I G000CCL/ITRI Standing Member S. MARCUS Wireless S. DIXIT Optical N. CHEUNG Telcordia Technologies Internet H. STUETTGEN NEC Europe Ltd. Network Management R. BOUTABA Univ. of Waterloo Industry Initiatives G. JAKOBSON Altusys Corp. Industry Initiatives R. SMITH Northrop Grumman Space Tech.

Committee Chairs Fellow Evaluation Emerging Technologies R. CALDERBANK L. CIMINI Univ. Delaware Princeton Univ.

Strategic Planning M. EJIRI Fujitsu Ltd. Journals Board Director of Journals D. P. TAYLOR Univ. of Canterbury Transactions on Communications E. AYANOGLU, EIC Univ. of California, Irvine Communications Letters E. BIGLIERI, EIC Politecnico di Torino Journal on Selected Areas in Communications N. F. MAXEMCHUK, EIC Columbia Univ. Transactions on Wireless Communications K. B. LETAIEF, EIC Hong Kong Univ. of Sci. & Technol. IEEE/ACM Transactions on Networking E. ZEGURA, EIC Georgia Inst. of Technol.

Network Magazine C. BISDIKIAN, EIC IBM T. J. Watson Res. Ctr. Wireless Communications Magazine M. ZORZI, EIC Univ. of Ferrara Global Communications Newsletter N. OACA, Editor

Interactive Magazines Y.-C. CHANG, Editor LG Electronics IEEE Communications Surveys & Tutorials M. REISSLEIN, EIC Arizona State Univ. ComSoc e-News N. FONSECA, Editor State Univ. of Campinas IEEE Press M. SHAFI, ComSoc Liaison Telecom New Zealand Ltd. Multimedia Magazine C. W. CHEN, ComSoc Liaison Univ. of Missouri–Columbia Internet Computing Magazine G. S. KUO, ComSoc Liaison Nat. Chengchi Univ.

Magazines Board Director of Magazines A. JAJSZCZYK AGH Univ. Technol.

Optical Communications Supplement, Area Editors C. QIAO, S. KARTALOPOULOS SUNY Buffalo

Communications Magazine R. GLITHO, EIC Ericsson Res. Canada

Pervasive Computing M. NAGHSHINEH, ComSocLiaison IBM T. J. Watson Res. Ctr.

Digital Object Identifier 10.1109/TCOMM.2005.859280

Nomination & Elections C. DESMOND World Class—Telecommun. Technical Activities Council Vice President H. FREEMAN Booz Allen Hamilton Inc. Secretary F. BAUER Nokia

Strategic Planning N. CHEUNG Telcordia Technol. Internet J. TOUCH USC/ISI C. KALMANEK AT&T Bell Labs—Research Multimedia Communications G. PAU UCLA

TAC Liaison to M&C K. SOHRABY Univ. of Arkansas

Network Operations & Management C. RAD AT&T

Communications Quality & Reliability K. MASE Niigata Univ.

Optical Networking B. MUKHERJEE Univ. of California

Communications Software A. MISHRA Virginia Tech.

Personal Communications M. GUIZANI Western Michigan Univ. & Technol.

Communications Switching & Routing H. STEUTTGEN NEC Europe Ltd.

Radio Communications M. WIN MIT

Communications Systems Integration & Modeling M. DEVETSIKIOTIS NC State Univ.

Satellite & Space Communications A. JAMALIPOUR Univ. of Sydney

Communication Theory C. GEORGHIADES Texas A&M Univ.

Signal Processing & Communications Electronics B. QIN Monash Univ.

Computer Communications J. LIEBEHERR Univ. of Virginia

Signal Processing for Storage G. SILVUS Seagate Technology

Enterprise Networking G. JAKOBSON Altusys Corp.

Tactical Communications K. C. YOUNG, JR. Telcordia Technologies

High-Speed Networking J. EVANS Univ. of Kansas

Transmission Access & Optical Systems J. M. H. ELMIRGHANI Univ. of Wales Swansea

Information Infrastructure M. ULEMA Manhattan College