IEEE Transaction on Communications (November 2004)

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NOVEMBER 2004

VOLUME 52

NUMBER 11

IECMBT

(ISSN 0090-6778)

TRANSACTIONS LETTERS

Coding An Algorithm for Detecting Unreliable Code Sequence Segments and Its Applications . . . . . . . . . . J. Freudenberger and B. Stender Decoding of Low-Density Parity-Check Codes Over Finite-State Binary Markov Channels . . . . . . . . . . . . . . . . . . .J. Garcia-Frias Digital Communications Accurate Computation of the Performance of -ary Orthogonal Signaling on a Discrete Memoryless Channel. . . . . . . . J. Hamkins High-Rate Recursive Convolutional Codes for Concatenated Channel Codes . . . . . F. Daneshgaran, M. Laddomada, and M. Mondin Fading/Equalization A New Base-Station Receiver for Increasing Diversity Order in a CDMA Cellular System . . . W. Choi, C. Yi, J. Y. Kim, and D. I. Kim Power Control and Diversity in Feedback Communications Over a Fading Channel. . . . . . . . . . . . . . . I. Saarinen and A. Mämmelä Spread Spectrum Large Set of CI Spreading Codes for High-Capacity MC-CDMA . . . . . . . . . . . . . . B. Natarajan, Z. Wu, C. R. Nassar, and S. Shattil Synchronization On the Miller–Chang Lower Bound for NDA Carrier Phase Estimation . . . . . . . . . . G. N. Tavares, L. M. Tavares, and M. S. Piedade

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TRANSACTIONS PAPERS

Coding Performance Analysis and Design Criteria for Finite-Alphabet Source-Channel Codes. . . . . . . . . . . . . A. Hedayat and A. Nosratinia Near-Capacity Coding in Multicarrier Modulation Systems . . . . . . . . . . . . . . . . . M. Ardakani, T. Esmailian, and F. R. Kschischang Performance of Coded OQPSK and MIL-STD SOQPSK With Iterative Decoding . . . . . . . . . . . . . . . . . . . . . L. Li and M. K. Simon Design and Performance of Turbo Gallager Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G. Colavolpe Reduced Complexity MAP-Based Iterative Multiuser Detection for Coded Multicarrier CDMA Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. Li, K. B. Letaief, and Z. Cao Digital Communications A Novel Trellis-Shaping Design With Both Peak and Average Power Reduction for OFDM Systems . . . . . . . . . . . . . . . . H. Ochiai Reduced-Delay Protection of DSL Systems Against Nonstationary Disturbances . . . . . D. Toumpakaris, J. M. Cioffi, and D. Gardan Fading/Equalization Training Sequence Optimization in MIMO Systems With Colored Interference. . . . . . . . . . . . . . . . . . . . . . T. F. Wong and B. Park Distribution Functions of Selection Combiner Output in Equally Correlated Rayleigh, Rician, and Nakagami- Fading Channels. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Y. Chen and C. Tellambura Networks Delay-Limited Throughput of Ad Hoc Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . E. Perevalov and R. S. Blum Optical Communication Performance Analysis and Tradeoffs for Dual-Pulse PPM on Optical Communication Channels With Direct Detection. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . M. K. Simon and V. A. Vilnrotter

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(Contents Continued on Back Cover)

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(Contents Continued from Front Cover) Personal Communication Systems Accurate Simulation of Multiple Cross-Correlated Rician Fading Channels . . . . . . . . . . . . . . . . .K. E. Baddour and N. C. Beaulieu Signal Processing OFDM Systems in the Presence of Phase Noise: Consequences and Solutions . . . . . . . . . . . . . . . . . . . . . . . S. Wu and Y. Bar-Ness Synchronization Pilot-Assisted Maximum-Likelihood Frequency-Offset Estimation for OFDM Systems . . . . . . . . . . . . . . . . . . J. H. Yu and Y. T. Su Transmission Systems A Class of Nonlinear Signal-Processing Schemes for Bandwidth-Efficient OFDM Transmission With Low Envelope Fluctuation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R. Dinis and A. Gusmão Transmission/Reception Pre-DFT Processing Using Eigenanalysis for Coded OFDM With Multiple Receive Antennas . . . . . . . . D. Huang and K. B. Letaief

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2009 2019

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

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

2031

PAPERS SCHEDULED TO BE PUBLISHED IN THE NEXT ISSUE DECEMBER 2004

Transactions Letters On Soft-Input Soft-Output Decoding Using “Box and Match” Techniques . . . . P. A. Martin, A. Valembois, M. P. C. Fossorier, and D. P. Taylor An Efficient Algorithm to Compute the Euclidean Distance Spectrum of a General Intersymbol Interference Channel and Its Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . J. Li, K. R. Narayanan, and C. N. Georghiades Reduced-Complexity Error State Diagrams in TCM and ISI Channel Performance Evaluation . . . . . . . . . . . . . . . . . W. E. Ryan and Z. Tang A Modified Blahut Algorithm for Decoding Reed-Solomon Codes Beyond Half the Minimum Distance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . S. Egorov, G. Markarian, and K. Pickavance Lattice-Reduction-Aided Broadcast Precoding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C. Windpassinger, R. F. H. Fischer, and J. B. Huber Bit-Interleaved Turbo-Equalization Over Static Frequency-Selective Channels: Constellation Mapping Impact . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .A. Dejonghe and L. Vandendorpe Sharing of ARQ Slots in Gilbert-Elliot Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . C.-H. Hou, J.-F. Chang, and D.-Y. Chen Doppler-Channel Blind Identification for Non-Circular Transmissions in Multiple-Access Systems . . . . . . . . . . . A. Napolitano and M. Tanda An Alternative Expression for the Symbol Error Probability of MPSK in the Presence of I-Q Unbalance . . . . . . . . . . . . .S. Park and D. Yoon Matched Filter Bound for Binary Signaling Over Dispersive Fading Channels With Receive Diversity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H. Hadinejad-Mahram, D. Dahlhaus, and D. Blömker Transactions Papers Threshold Values and Convergence Properties of Majority-Based Algorithms for Decoding Regular Low-Density Parity-Check Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P. Zarrinkhat and A. H. Banihashemi Graph-Based Message-Passing Schedules for Decoding LDPC Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . .H. Xiao and A. H. Banihashemi A More Accurate One-Dimensional Analysis and Design of Irregular LDPC Codes . . . . . . . . . . . . . . . . M. Ardakani and F. R. Kschischang Subspace Algorithms for Error Localization With Quantized DFT Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . G. Rath and C. Guillemot On Finite-State Vector Quantization for Noisy Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . P. Yahampath and M. Pawlak Parametric Construction of Nyquist-I Pulses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .N. C. Beaulieu and M. O. Damen SER and Outage of Threshold-Based Hybrid Selection/Maximal-Ratio Combining Over Generalized Fading Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .X. Zhang and N. C. Beaulieu A NS Frequency Domain Approach for Continuous Time Design of CAP/ICOM Waveform . . . . . . . . . . . . . . . . . . X. Tang and I. L.-J. Thng OCDMA Coded Free Space Optical Links for Wireless Optical Mesh Networks . . . . . . . . . . . . . . . . . . . . . . . B. Hamzeh and M. Kavehrad Performance Modeling of Optical Burst Switching With Fiber Delay Lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . X. Lu and B. L. Mark Bandwidth Efficient WDM Channel Allocation for Four-Wave Mixing Effect Minimization . . . . . . . . . V. L. L. Thing, P. Shum, and M. K. Rao Characterizing Outage Rates for Space-Time Communication Over Wideband Channels . . . . . . . . . . . . . . . . . . G. Barriac and U. Madhow Parallel Interference Cancellation for Uplink Multirate Overlay CDMA Channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . H. Yan and S. Roy Abstracts of Forthcoming Manuscripts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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B. JABBARI, Wireless Multiple Access George Mason Univ., Fairfax, VA S. JAFAR, Wireless Commun. Theory & CDMA Univ. of 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

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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 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.2004.839885

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IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 11, NOVEMBER 2004

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Transactions Letters________________________________________________________________ An Algorithm for Detecting Unreliable Code Sequence Segments and Its Applications Jürgen Freudenberger and Boris Stender

Abstract—Let the Viterbi algorithm be applied for maximum-likelihood decoding of a terminated convolutional code using a trellis. We propose an additional procedure that permits a receiver to locate unreliable segments within an estimated code sequence. This reliability output may be used, for example, to request retransmissions, in systems with error concealment, or in channel-coding systems with unequal error protection. Index Terms—Convolutional codes, list decoding, reliability output, repeat request, Viterbi decoding.

I. INTRODUCTION ECODING with erasures is a form of decoding where the decoder rejects received data if a reliable decision is not apparent. Therefore, the reliability of the estimate is tested and the decoded message is only accepted if the decision is sufficiently reliable, whereas an erasure is declared if a reliable decision is not evident. There are several decoding schemes with erasures which have been applied to coded automatic repeat request (ARQ). Those schemes may be classified according to their erasure-decision rules [1], [2]. An important class of erasure-decision rules are likelihood-ratio tests where the ratio of likelihood (metric) values are tested [2]–[5]. Frequently, the erasure decision is based on explicit error detection. For this purpose, usually a high-rate cyclic code is used exclusively for error detection. This concept was generalized to errorlocating codes by Wolf and Elspas [6]. With error-locating codes, a block of received digits is regarded as subdivided into mutually exclusive subblocks. Errors occurring within particular subblocks are detected at the receiver. In addition, the receiver is able to determine which particular subblocks contain errors. Similarly, it is desirable to generalize the concept of likelihood-ratio decision. In addition to the likelihood-ratio test, the receiver should be able to determine which parts of a codeword (or which segments of a convolutional code sequence) are unreliable.

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Paper approved by R. D. Wesel, the Editor for Coding and Communication Theory of the IEEE Communications Society. Manuscript received November 13, 2003; revised May 7, 2004. This work was supported by the DFG (Deutsche Forschungsgemeinschaft) under Grant Bo 867/9-1. This paper was presented in part at the International Symposium on Information Theory, Lausanne, Switzerland, June 2002. The authors are with the Department of Telecommunications and Applied Information Theory, University of Ulm, D-89081 Ulm, Germany (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TCOMM.2004.836556

The contribution of this letter is a modified Viterbi decoder for terminated convolutional codes. The new algorithm does not only output a single estimate, but marks code sequence segments as unreliable if an erroneous decision for these segments is likely. We do not consider symbol-by-symbol reliabilities, but possible error events for Viterbi decoding, i.e., paths in the decoder trellis which diverge from the most likely code sequence and later on re-merge. In addition to this erasure-declaring decoding algorithm, we present a soft-output version. Here the decoder puts out not an erasure or estimate alone, but an estimate with a reliability indicator. This reliability indicator is the threshold for which the corresponding code segment would have been erased, had an erasure option been used. The paper is structured as follows. In Section II, we introduce notations, and in Section III, we describe the two versions of our reliability output decoding algorithm in detail. In the subsequent section, we consider their application in repeat-request schemes. First, we compare the performance of the reliability output algorithm with that of Yamamoto and Itoh’s scheme [4]. Second, we introduce an adaptive-feedback error-correction scheme. Similar to standard hybrid ARQ, the new approach is packet oriented. However, we propose to transmit incremental redundancy only for particular unreliable code sequence segments which can be determined using the new algorithm. Finally, we give some concluding remarks in Section V. II. NOTATIONS Traditional Viterbi decoding is explained by means of a trellis diagram. A trellis is a directed graph. The set of nodes of the is decomposed into a union of disjoint graph subsets that are called levels of the trellis. A node of the level may be connected with a of the level by one or several branches. node to a node Each branch is directed from a node of level of the next level . We assume that the end levels have only one node, namely, and . binary terminated In the following, we consider rate convolutional codes [7] used for communication over a memoryless channel. Assume a terminated code sequence consists of -tuples , i.e., . The encoder of the convolutional code is forced to return dummy -tuples to the to its initial state by appending denotes the memory of the information sequence, where convolutional code. The corresponding information sequence is , with . For

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Fig. 1.

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 11, NOVEMBER 2004

Diverging paths labeling segments as unreliable.

the decoder trellis of a convolutional code, a node corresponds to a possible encoder state. Each branch of the is labeled by a -tuple . There is decoder trellis in a one-to-one correspondence between each codeword the code and a path in the trellis, such that . We denote code sequence segments and path segments by and , merged if we respectively. We call the segments . Let be the received sequence. have A Viterbi decoder selects the code sequence which maximizes the conditional probability , or equivalently, which minimizes the Viterbi metric (1) with

(2) (3)

In order to evaluate (1) efficiently, we assign additional node and partial path metrics and , respectively. is the metric of the local survivor entering node from level . The complexity of the Viterbi algorithm (VA) for a , where terminated convolutional code is denotes the overall constraint length of the code, which is the minimum number of memory elements required to realize the corresponding convolutional encoder. For a detailed discussion of the trellis representation of convolutional codes or Viterbi decoding, see, for example, [8] and [9] or the textbooks [7] and [10].

III. RELIABILITY OUTPUT ALGORITHM An error event with Viterbi decoding occurs whenever the estimated path diverges from the correct path. In many practical situations, it is useful to decide whether such an error event is likely or not. For this purpose, Yamamoto and Itoh [4] be a given introduced the following reliability test. Let threshold. Consider the first and second most likely paths , such that . The estimated is considered reliable if , path otherwise unreliable. The underlying idea is that the most unreliable decision during the VA occurred at the node , where the best and the second-best paths merged. However, for moderate or high signal-to-noise ratios (SNRs), an erroneously estimated path will remerge with the correct path after a few transitions. It is therefore reasonable to declare only parts of the . estimated sequence as unreliable if We therefore consider the following criterion.

be a given threshold. ConDefinition 1: Let most likely paths sider the ordered list of the such that with and for all paths For levels where all paths in are merged, i.e., , we define the decoder output as reliable, otherwise as unreliable. Example 1: Fig. 1 shows the conceptual situation with this reliability criterion, where the figure illustrates only paths . The straight line indicates the most likely code sequence. Furthermore, there are three unreliable segments indicated, where for each segment, there are several candidate paths with metric denote the differences less or equal . The values metric differences of merging paths. Note that this reliability criterion is similar to the one Schaub and Modestino [11] introduced for a symbol-by-symbol erasure-declaring VA. It is also related to the various list-type generalizations of the VA which have appeared in the literature [8], [12]–[15]. That is, a straightforward method for labeling the decoder output would be as follows. First, determine the ordered initially as reliable. Consider all paths in the list and mark ordered list starting with and ending with . Trace the corresponding paths from node to node and mark the as unreliable if a path is not merged with decoder output at time . The complexity of this method increases linearly with increasing list size. However, it is not necessary to generate the list of paths explicitly in order to mark the decoder output according to Definition 1. The following lemma provides some insight concerning this issue. Compare any path from with the path . The two paths may diverge and re-merge many times, but the is end states of the paths are the same. We say that a path only once. a one-loop path if it diverges from the path Furthermore, let the symbol denote path concatenation, i.e., with and . Lemma 1: In order to mark the decoder output as reliable or unreliable according to Definition 1, it is sufficient to consider all one-loop paths in the ordered list . as reliable. By Proof: Assume that we initially mark from node to node , we change back-tracing a path the label of the decoder output from reliable to unreliable if the path is not merged with at time . Let a path , have two or more loops , with respect to the path . Regarding , the code-sequence

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Fig. 2. Example of a labeled trellis (ROA I).

segments corresponding to should be marked unreliable. We show that the following one-loop paths , all have a metric less than or equal to , from which we conclude that . In fact, , , has the least metric among all paths. For the path because , we have , and we obtain , , and so on. From follows . Thus, by tracing back the , we obtain the same labeling as with one-loop paths . Similarly, by tracing back all one-loop paths in , we obtain the desired labeling, according to Definition 1. It follows from Lemma 1 that we may use an ordered list of all one-loop paths contained in instead of the complete list . A recursive algorithm to construct the list of the most likely one-loop paths is given in [13]. However, the following nonrecursive algorithm determines these unreliable positions without explicitly generating a list. Algorithm 1 Reliability Output Algorithm (ROA) I: Step 1) Step 2)

Step 3)

Step 4)

Assign metric zero to the initial node and . set Each node of level is processed as follows (forward pass). Calculate the partial path metrics correthat enter the node sponding to the branches by , where is the . node metric of this branch’s predecessor node Moreover, assign the node metric to the node , where denotes the branch with the smallest partial path metric among all branches merging in node . Store a list with all partial path metrics. , then continue with the next level. Increment If by one and go to Step 2). Otherwise, label the terminating node with a threshold value , all others with threshold value 0, and go to Step 4). Each node of level with threshold greater than 0 is processed as follows (backward pass). For each element in the metric list, determine the difference be-

tween the list entry and the node metric. Trace back all branches with metric difference less than or equal to the node threshold. Label the corresponding nodes with new thresholds, which are the at level differences of the node threshold at level and the metric differences of the particular branches. If dif, find the ferent paths merge in a node at level maximum of all according thresholds at level and assign it to the node at level . between Step 5) If there exists a branch different from which connects nodes labeled with level and thresholds not equal to zero, mark the estimate as unreliable, otherwise, mark it reliable. , then continue with the next level. Decrement Step 6) If by one and go to Step 4). Theorem 1: Algorithm 1 labels the maximum-likelihood estimate according to Definition 1. A proof of Theorem 1 is given in the Appendix. Note that our reliability criterion is the same as that of Yamamoto and Itoh [4] if we declare the complete code sequence as unreliable once an unreliable segment has occurred. In particular, the bounds derived in [4] on the first event-error probability and on the event of the first erasure are also valid for our scheme. These bounds depend on the path enumerator function of the code and on the metric threshold. For low channel-error rates, these bounds can be used to determine the appropriate threshold values. However, for poor channel conditions, these union bounds diverge. In this case, threshold values have to be determined by computer simulations. Example 2: Fig. 2 depicts a labeled trellis for the rate convolutional code with memory and generator (encoder in conmatrix troller canonical form). Four information symbols followed by two dummy zeros are encoded. We consider transmission over the binary symmetric channel with a crossover probability . Therefore, we have branch metric values and . Let the received sequence be . The list under each node comprises the metrics corresponding to incoming branches. The backward pass of in Fig. 2). the ROA I starts at the terminating node (node Above each node which is reprocessed during the backward pass, the node threshold is depicted. Additionally, all branches

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Fig. 3. Example of a labeled trellis (ROA II).

are marked which will be traced back. In this example, the termi. The node nating node is initialized with threshold metric for node is 5.8, and the differences of the incoming path metrics to the node metric are 0.0 and 6.9. Consequently, only the local survivor is traced back. Note that as a result, all are labeled nodes corresponding to the global survivor path with threshold , because is always greater than or equal to any other threshold. At node , the differences are 0.0 and 2.3. As these values are smaller than 2.5, both incoming branches are traced back. Moreover, node is labeled with the new threshold . Now, two paths are traced back until they merge , we in node . If different paths merge in a node at level assign the maximum of all thresholds, e.g., node is marked with threshold 2.5. Finally, we notice that the estimated code segment corresponding to the path from node to node is marked unreliable, because there are nodes which do not correthat are labeled with thresholds spond to the survivor path greater than 0. In the remaining part of this section, we consider a modification of Algorithm 1 such that the new algorithm (ROA II) deterof reliability values. Steps 1–3 mines a vector of ROA II are the same as with ROA I, except that we do not assume a predetermined threshold . Therefore, in Step 3, we with value 0 and continue label only the terminating node of level is processed as folwith Step 4, where each node lows. For each element in the metric list, determine the difference between the list entry and the node metric. Trace back all branches and label the corresponding nodes at level with the new values. The new values are the node values at level plus the metric differences of the particular branch. If different paths merge in a node at level , find the minimum of all these values. The nodes labeled with value 0 correspond to the . The reliability value is the smallest value, survivor path except the value zero from the survivor, which can be found at a or . The labeling of the decoder output label from nodes can be performed based on these reliability values. Thus, Step 5 in Algorithm 1 can be omitted. In order to obtain an erasure labeling of the decoder output, we declare all positions , where the reliability value is less than or equal to a chosen threshold , as unreliable. This second version of the reliability output algorithm requires more calculations, because each node is reprocessed during the backward pass. However, the complexity

is now independent of the received sequence and is still of the same order as the VA. Example 3: Fig. 3 depicts a labeled trellis for the ROA II, where we used the same settings as in Example 2. Above each node, the node’s reliability value is shown. With ROA II, all branches in the trellis are traced back. The terminating node is initialized with value 0.0. The differences of the incoming branch metrics to the terminating node’s metric are 0.0 and 6.9. Both incoming branches are traced back, and the nodes at are labeled with these values. As 6.9 is the level smallest value that can be found at a node connected by the last branches, which is greater than 0, this value is given to the output. If we continue the backtracing, we have to accumulate all the metric differences until some paths merge. In this case, we continue with the minimum value. The final reliability . Finally, marking the output is estimated code segment corresponding to reliability values less as unreliable, we obtain the same labeling as in than Example 2. IV. APPLICATIONS AND SIMULATION RESULTS The above-presented algorithm is applicable for repeatrequest strategies. In this case, the complete received code sequence should be stored, and the receiver may simply request retransmission of code sequence segments that are marked unreliable. The marked symbols are then substituted by (or combined with) the newly received symbols and the decoding procedure restarts. The transmission terminates when finally no unreliable symbols are determined. The performance of such repeat-request schemes may be measured in terms of error probabilities, e.g., the bit-error probability, after the final decoding step. Furthermore, the effective transmission rate is of great importance, which is defined as the ratio of the number of decoded data symbols to the number of transmitted code symbols. In the following, we investigate such a repeat-request scheme by means of computer simulations. We compare our scheme with the strategy proposed by Yamamoto and Itoh [4]. Yamamoto and Itoh’s algorithm (YIA) is also based on Viterbi decoding, has the same order of decoding complexity, and uses the same likelihood test. However, YIA does not determine the exact lengths of unreliable segments. With Yamamoto and

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Fig. 5. ARQ scheme with ROA II.

Fig. 4. Simulation results for a memory m = 3 optimum free distance convolutional code ( (D) = (1 + D + D + D ; 1 + D + D ), M = 32). AWGN channel with SNRs E =N 2; 1; 0; 1 dB . The values beside each simulation point are the respective metric thresholds. For the ROA algorithm, the code sequences are terminated after 1000 information bits, while with YIA, the sequences are not terminated.

G

2 f0 0

g

code tuples is Itoh’s scheme, a retransmission of the latest is a requested if the survivor is declared unreliable, where preselected constant. This results in a rate loss, as can be seen from the following simulation results. For these simulations, we considered transmission over a binary-input additive white Gaussian noise (BIAWGN) channel, an ideal return channel, and maximum ratio combining of retransmitted symbols. Fig. 4 presents the corresponding results.1 Note that both schemes achieve the same BERs for a given metric threshold and value. This is due to the fact that both schemes are based on the same likelihood test. However, with the scheme based on ROA I, the effective transmission rate, necessary to achieve a certain BER, is significantly improved for all considered SNRs. In the following, we describe a different system where only a few bits are retransmitted with each repeat request. Fig. 5 shows the flow chart for a transmission of one data block with this ARQ scheme. For the initial transmission, the data block is first encoded with a convolutional encoder. Then the code sequence is punctured to get an initial code rate. After transmission over the BIAWGN channel, all code bits of a block are maximum ratio combined (MRC). Then the aggregated code sequence is decoded using the ROA II. Afterwards, error detection should be applied. The error-detection part is pictured by the term test in Fig. 5. If an error is detected, a list with unreliable segments must be sent back. The interception point labeled indicates the 1For YIA, we used M = 32 as in the original paper [4]. Small gains can be obtained by using optimized values of M for different thresholds and SNR values. However, the differences are moderate, compared with a fixed value of M .

noiseless feedback link. If actual feedback links with noise are treated, then the list to be transmitted over the return channel has to be protected by forward-error correction. If a retransmission is requested, then the transmitter calculates the code bit positions to be sent next. This is done simultaneously at the receiver, as this information is needed in the MRC unit. The scheme used for simulation can be described as follows. First, calculate the number of code bits that are to be transmitted. For each unreliable segment, we choose a value which is the sum of a small constant and a fraction of the segment length . As should be an integer, we calculate . Second, the particular code bit positions have to be found. For this purpose, the receiver registers the number of transmissions for each code bit, and selects positions which increment the redundancy. Example 4: For simulations, we assume perfect error detecwith a memory tion. We encode each data block terminated convolutional code (133 145 175). Then we perform an equidistant puncturing, such that every third code ). bit is transmitted (giving an initial code rate The parameters and for retransmission are chosen to be 2 and 1/24, respectively. The increment is fixed to be 1/10. In Fig. 6, we see the throughput reached with this scheme as well as the capacity curve for the BIAWGN channel. The asterisk at position ( 1.62; 0.331) indicates the SNR where a BER of is obtained with the terminated rate mother code. The achieved performance gain of about 2.9 dB is a result of the feedback scheme. V. CONCLUSIONS We have devised a new algorithm that permits a receiver to locate unreliable segments within an estimated convolutional code sequence. The reliability criterion is based on the list of the most likely codewords which satisfy a likelihood-ratio test. In principle, the same reliability output decoding could be performed using one of the various list-type generalizations of the VA. However, the complexity of list-output algorithms, in general, depends on the list size . It may increase linearly with , where the list size itself depends on the received sequence

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derive an equivalent expression in terms of the set . Consider the node . A path passing through can be represented as concatenation of a head and a tail path: . If , then at least the path with minimum is an element of . metric among all paths passing through This path has also minimum head and minimum tail path metric. Thus, we have (4) where the minimum is taken over all head and tail paths, ending or starting in node . With the help of the list of all branch metrics that were stored in Step 2, we can calculate the accumulated metric of any tail path Fig. 6. BIAWGN capacity and throughput of ROA-ARQ.

and increases with increasing channel error rate. The worst-case complexity of the new algorithm is independent of the list size, and is of the same complexity order as the VA. The reliability output may be used to request retransmissions. The performance of such a repeat-request scheme has been compared with that of Yamamoto and Itoh [4] and is shown to provide a significant improvement in throughput. In addition, an adaptive-feedback error-correction scheme has been presented. The corresponding simulation results indicate that near-optimum performance for the BIAWGN channel is attainable with moderate complexity and for very short block sizes. This is true for a wide range of SNRs. However, while standard hybrid ARQ schemes require only a single bit of information per packet over the feedback path, we have to send much more information in order to obtain this level of performance.

where we have used the abbreviation , i.e., denotes the difference between the list entry corresponding to the particular branch and the node metric at node . Substituting this into (4), we get

Notice that only the sum depends on the particular tail and path. In particular, we have . Therefore, canceling these terms we obtain

APPENDIX PROOF OF THEOREM 1 Proof: Steps 1–3 of the ROA I are essentially the forward pass of a VA. With the ROA I, we additionally store a list of all partial path metrics for each node in the trellis. Thus, following the branches with the smallest partial path metric in the list, starting in the terminating node , and ending in the initial node , we can find the best path . In the following proof, . we consider the additional labeling of the output sequence According to Definition 1, we have to show that our algorithm reliable if all paths in the list labels the decoder output are merged. That is, during the backward pass (Steps 4–6) the at level reliable, iff all algorithm should mark the estimate paths are merged at this level. Otherwise, has to be labeled unreliable. Instead of directly considering paths through the trellis, we consider certain trellis nodes. We therefore define there exists a path

passing through

That is, is the set of all nodes in the trellis representation which are connected by any path . For convenience, we . We will now resume

Let denote the node threshold at node with . Note that the backward pass of the ROA I (Steps 4–6) is essentially a VA which calculates the node thresholds for each node . To see this, consider Step 4, where the node threshold is calculated as . Branches which need not to be considered, because stem from a state the corresponding metric differences would result in a threshold value less than zero. Thus, by finite induction, we have . Finally, we observe that . It follows from Step 4 that we trace back all branches to nodes for which this inequality is fulfilled. Therefore, all nodes are found. Moreover, iff there exist more than one state at time or more than one state at time , then there is a branch corresponding to a path . Consequently, has to be marked unreliable. Remarks: The above proof is only valid for trellis representations which have no parallel branches from some state to . In particular, it does not hold for partial unit some state

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memory codes. However, the validity of the algorithm can also be shown for this case. Moreover, we would like to mention that in the proof corresponds to the the expression labeling of ROA-II. In order to obtain an erasure labeling of the decoder output of ROA-II, we declare all positions , where the reliability value is less than or equal to a chosen threshold as unreliable. Arguments similar to the proof of Theorem 1 show that the resulting labeling satisfies Definition 1. REFERENCES [1] T. Hashimoto and M. Taguchi, “Performance of explicit error detection and threshold decision in decoding with erasures,” IEEE Trans. Inform. Theory, vol. 43, pp. 1650–1655, Sept. 1997. [2] T. Hashimoto, “Composite scheme LR + Th for decoding with erasures and its effective equivalence to Forney’s rule,” IEEE Trans. Inform. Theory, vol. 45, pp. 78–93, Jan. 1999. [3] G. D. Forney, Jr., “Exponential error bounds for erasure, list, and decision feedback schemes,” IEEE Trans. Inform. Theory, vol. IT-14, pp. 206–220, Mar. 1968. [4] H. Yamamoto and K. Itoh, “Viterbi decoding algorithm for convolutional codes with repeat request,” IEEE Trans. Inform. Theory, vol. IT-26, pp. 540–547, Sept. 1980.

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[5] T. Hashimoto, “On the error exponent of convolutionally coded ARQ,” IEEE Trans. Inform. Theory, vol. 40, pp. 567–575, Mar. 1994. [6] J. K. Wolf and B. Elspas, “Error-locating codes—a new concept in error control,” IEEE Trans. Inform. Theory, vol. IT-9, pp. 113–117, Apr. 1963. [7] R. Johannesson and K. S. Zigangirov, Fundamentals of Convolutional Coding. Piscataway, NJ: IEEE Press, 1999. [8] G. D. Forney, Jr., “Convolutional codes {II}: Maximum likelihood decoding,” Inform. Control, vol. 25, pp. 222–266, July 1974. [9] , “The Viterbi algorithm,” Proc. IEEE, vol. 61, pp. 268–278, Mar. 1973. [10] M. Bossert, Channel Coding for Telecommunications. New York: Wiley, 1999. [11] T. Schaub and J. W. Modestino, “An erasure declaring Viterbi decoder and its applications to concatenated coding systems,” in Proc. IEEE Int. Conf. Communications, 1986, pp. 1612–1616. [12] T. Hashimoto, “A list-type reduced-constraint generalization of the Viterbi algorithm,” IEEE Trans. Inform. Theory, vol. IT-33, pp. 866–876, Nov. 1987. [13] V. V. Zyablov, V. G. Potapov, and V. R. Sidorenko, “Maximum-likelihood list decoding using trellises,” Problemy Peredachi Informatsii, vol. 29, pp. 3–10, 1993. [14] N. Seshadri and C. E. W. Sundberg, “List Viterbi decoding algorithm with applications,” IEEE Trans. Commun., vol. 42, pp. 313–323, Feb. 1994. [15] C. Nill and C. E. W. Sundberg, “List and soft symbol output Viterbi algorithms: Extensions and comparisions,” IEEE Trans. Commun., vol. 43, pp. 277–287, Feb. 1995.

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Decoding of Low-Density Parity-Check Codes Over Finite-State Binary Markov Channels Javier Garcia-Frias, Member, IEEE

Abstract—We propose a modified algorithm for decoding of lowdensity parity-check codes over finite-state binary Markov channels. The proposed approach clearly outperforms systems in which the channel statistics are not exploited in the decoding, even when the channel parameters are not known a priori at the decoder. Index Terms—Baum–Welch algorithm, finite-state binary Markov channels, Gilbert–Elliot channels, low-density paritycheck (LDPC) codes.

I. INTRODUCTION ANY practical digital communications channels exhibit statistical dependencies among errors. The error pattern of the discrete channel (modulator-real channel-demodulator) can be modeled using finite-state binary Markov channels [hidden Markov models (HMMs)]. Such channels , the are characterized by a set of states , , matrix of transition probabilities among states ( with the probability of transition from state to state , , , ), and the list giving i.e., the bit-error probability (BEP) to associate with each state , with the probability of getting the output ( in state , i.e., , , ). It is intuitive that the presence of memory in Markov chan, relative to the capacity of nels leads to increased capacity, memoryless channels, , with the same stationary BEP [1], [2]. In practice, many communications systems make use of a channel interleaver to distribute the errors so that codes designed for a memoryless channel can be used. While the application of interleaving does not change the capacity of the channel, the achievable performance of a decoder which assumes that the channel is memoryless is limited by . Exploiting the higher capacity of Markov channels, in practice, has proven to be challenging. In general [3], even if the parameters of the Markov channel are known to the receiver, maximum-likelihood strategies are too complex to implement. Both [1] and [2] use decision-feedback decoders which perform recursive state estimations that are used in the decoding process. However, since the state estimate at a given time is obtained based on hard-decision estimates of the previous transmitted symbols, these schemes are vulnerable to error propagation, and they can not be reliably

M

Paper approved by E. Ayanoglu, the Editor for Communication Theory and Coding Application of the IEEE Communications Society. Manuscript received August 14, 2002; revised January 29, 2004. This work was supported in part by the National Science Foundation CAREER/PECASE Award CCR-0093215. This paper was presented in part at the International Symposium on Information Theory, Washington, DC, June 2001. The author is with the Department of Electrical and Computer Engineering, University of Delaware, Newark, DE 19716 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TCOMM.2004.836568

Fig. 1. Bipartite graph representing an LDPC code over a finite-state binary Markov channel. fc g, fe g, and fs g represent the check, the error pattern, and the HMM nodes. The figure shows different types of messages that are propagated in the decoding process through the graph.

used when the quality of the channel degrades. In [4] and [5], we proposed modifications for the decoding of turbo codes that exploit the structure of the Markov channel and allow commu. However, those methods nications at rates greater than are either more complex or achieve worse performance than the ones proposed in this letter. In this letter, we propose the modification of the message-passing decoding algorithm to exploit the structure of finite-state binary Markov channels in low-density parity-check (LDPC) codes [6]. Decoding modifications to exploit the structure of these channels in LDPC codes were proposed first in [7] and [8]. Additional work for discrete and continued-valued Markov channels was presented in [9]. The work in [7] is based on producing hard estimates of the error pattern. The method proposed here (partially presented in [8]) considers soft estimates, which leads to performance improvements over the scheme in [7]. Furthermore, it clearly outperforms standard LDPC decoding in which the channel statistics are not exploited in the decoding, and allows communications at rates greater and close to . than II. DECODER MODIFICATIONS FOR FINITE-STATE BINARY MARKOV CHANNELS The possibility of incorporating Markov channels in the decoding of codes defined over general graph schemes is suggested in [10] (see also [11] for a more general framework), although no results are presented there. As shown in Fig. 1, the basic idea is to “link” the error-pattern nodes of the LDPC code with the HMM. The whole structure can then be considered as a Bayesian network, and decoding can be performed by proceeding with the message-passing algorithm (belief propagation [12]). The introduction of the HMM produces an increase in the number of cycles in the network. However, as we will see in Section IV, these additional cycles do not seem to strongly affect the performance of the decoding algorithm. Notice that

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due to the structure of LDPC codes, the use of a channel interleaver is unnecessary. Following the notation in [13], we denote the set of indexes of the error-pattern nodes that participate in check node as . represents the set of indexes of the check nodes that participate in error-pattern . The set with element node , being excluded is denoted by . As shown in Fig. 1, without counting the message passing among the HMM nodes, we can distinguish four different types of messages propagated through the network: 1) , , the message propagated from check node to error-pattern node ; 2) , , the message propagated from error-pattern node to HMM node ; 3) , , the message propto error-pattern node ; 4) , agated from HMM node , the message propagated from error-pattern node to check node . The steps in which the proposed algorithm is different from the standard one are detailed below. 1) Message passing from the error-pattern nodes to the HMM nodes , calculate For any , , where is a normalization factor such that . 2) Message passing from the HMM nodes to the error-pattern nodes In order to derive the equations, we consider the trellis for a finite-state binary Markov channel. The starting and ending states associated with a particular edge are represented by and , respectively, and . the error pattern corresponding to is denoted by The trellis has two parallel branches between states (one , and the associated with the error pattern ). Each one of the other with the error pattern branches in the trellis will have an associated a priori , which is obtained from the parameters probability of the HMM as . The resulting equations that implement the belief-propagation algorithm over the HMM (forward/backward recursions, with the difference that the “observation sequence” is now a random vector defined , , )1 are by the probabilities given by (1)

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value of is not used in (3). However, in order to , make decisions, we also need to calculate , where is a normalization factor such that . Message passing from the the error-pattern nodes to the check nodes The messages propagated from the error-pattern nodes to the check nodes will combine the messages proceeding from the HMM with the messages originated in the check nodes. Specifically, for every pair existing in the code graph, we calculate , , where is a normalization factor such that . III. JOINT CHANNEL ESTIMATION AND DECODING

Although in the previous section we have assumed that the parameters of the binary Markov channel are known a priori by the decoder, it is also possible to successfully perform decoding when the parameters describing the Markov channel are unknown. The idea is to apply a Baum–Welch-like re-estimation procedure [14] each time that the messages are propagated through the HMM (step 2 in Section II). In this way, each iteration in LDPC decoding also results in an iteration on the parameter estimation, and no training sequence is needed. As opposed to other approaches (e.g., [7]), parameter estimation is performed from scratch for each block, using only the information contained in the received block (i.e., no information from previous blocks is kept). This guarantees the robustness of the proposed approach if the channel parameters change from block to block. In order to derive the estimation algorithm, we will assume and that the channel is characterized by an HMM we can, therefore, calculate the value of for each branch in the corresponding trellis. After any given iteration, the probability of going through branch in the trellis section of the HMM can be calculated as (4) Since branch is determined by the initial state , final , and its associated error pattern , we can express state , the probability of the branch as the joint probability of , and ; i.e., . Therefore, by using the same approach as in the Baum–Welch algorithm [14], the joint probability for the HMM representing the channel can be expressed as

(2) (5) (3) where is a normalization factor such that . Notice that, as in any belief-propagation scheme, the 1The work in [7] uses a hard estimate of the error pattern, which is made available by step 1 (e.g., fg g 2 f0; 1g). This constraint leads to some performance degradation.

where . The product of this probability and the number of trellis transitions represents, as in the Baum–Welch algorithm, the expected number of transitions from state to state generating the error pattern . From this joint probability, it is easy to calculate the expected number of transitions from state to state , the expected number of times error pattern

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is generated in state number of times in state

, and the expected

(6)

(7) (8) The resulting equations to calculate the re-estimated model , which will be used in the next iteration of the LDPC decoder, are given by

Fig. 2. Performance for a rate-1/2 (3,6) regular LDPC code over the Gilbert–Elliot channel with parameters P = 0:5, g = 0:1092, b = 0:0308, and a variable P resulting in stationary BEP . The curve for the case requiring estimation overlaps with the curve obtained when the channel parameters are perfectly known at the decoder site. For this channel, the  giving a capacity C = 1=2 is  = 0:137. For a memoryless channel, capacity C = 1=2 occurs for  = 0:11.

(9)

(10)

,

with ,

,

,

, and

. IV. SIMULATION RESULTS Although the decoder modifications to incorporate channel statistics can be applied to general finite-state Markov channels with states, for simplicity, we consider the special case of . A Gilbert–Elliot channel conGilbert–Elliot channels , and a “bad” state tains a “good” state in which the BEP is with BEP . The transition probability of going from the good state to the bad state is denoted by , and represents the probability of going from the bad state to the good state. In order to assess the performance of the proposed method, we consider the two Gilbert–Elliot channels studied in [4]. For and the first channel, the transition probabilities are . The transition probabilities of the second channel are chosen so that the ratio is the (i.e., longer same as in the channel above, but with error burst lengths than for the first channel). In both cases, the BEP in the bad state is fixed to . The performance of the system is studied as a function of the value of , the BEP in the good state. Notice that since all the other parameters are fixed, there is a one-to-one correspondence between and the stationary BEP, , which is the

Fig. 3. Performance for a rate-1/2 (3,6) regular LDPC code over the Gilbert–Elliot channel with parameters P = 0:5, g = 0:0156, b = 0:0044, and a variable P resulting in stationary BEP . The curve for the case requiring estimation overlaps with the curve obtained when the channel parameters are perfectly known at the decoder site. For this channel, the  giving a capacity = 1=2 C = 1=2 is  = 0:154. For a memoryless channel, capacity C occurs for  = 0:11.

parameter used in Figs. 2 and 3.2 We use a rate-1/2 (3,6) regular coded bits. For rate-1/2 codes, LDPC code with the BEP corresponding to the capacity of a binary symmetric . Therefore, if the memory of the channel (BSC) is channel is ignored, it is impossible to send reliable information through any of these channels when the stationary BEP is higher than 0.11. and the proFig. 2 shows, for the first channel posed decoding method, the decoded bit-error rate (BER) as a function of the stationary BEP . After simulating 2000 blocks, no errors are found for values of . The average number is slightly less than 18. Notice that of iterations for 2For the two Gilbert–Elliot channels considered in this section, the lowest value of  is achieved when P = 0, and corresponds to  = 0:11.

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convergence is achieved at values of higher than the memoryless limit and close to the theoretical limit for this channel [1]). For comparison (which corresponds to a value purposes, the performance when the HMM is not considered in decoding is also shown in the figure. In this case, the residual (which corresponds to a value of BER for and is, therefore, the smaller value of that can be obtained and keeping the rest of the channel paramby decreasing . The same LDPC code applied eters fixed) is BER for values of BEP over a BSC achieves a BER around slightly less than 0.085, which gives an idea of the improvement obtained with the approach presented in this letter. Although more decoding iterations are required, we obtain the same performance when the parameters of the HMM are not known a priori, and are jointly estimated with the decoding process (the curve for the case requiring estimation overlaps with the curve obtained when the channel parameters are perfectly known at the decoder site). In this case, we assume in our simulations that the lack of information holds for all blocks. No pilot symbols are used, and the initial HMM is defined by the parameters , , , and . As we can see in Fig. 3, for the second channel convergence is also achieved at values of higher than the memoryless limit. The theoretical limit for this channel corresponds . After simulating 2000 blocks, no errors are to a value . In contrast with Fig. 2, in Fig. 3, found for values of the degradation in performance as increases is smoother. This is related to the higher variance in the stationary BEP for dif. For the case ferent realizations of the channel with in which the channel parameters are not known a priori, no loss in performance is observed when the method proposed in Section III is applied, and the initial model is the same as in Fig. 2. Again, the curve for the case requiring estimation overlaps with the curve obtained when the channel parameters are perfectly known at the decoder site. Notice that the residual block (frame)-error rate (FER) can be easily obtained from Figs. 2 and 3 by applying the formula , where is the average number of errors BER = FER is the block length. In for the blocks in error and (greater the range of interest of Fig. 2, ), , which values of result in a FER results in a FER curve shifted around one order of magnitude with respect to the BER curve presented in Fig. 2. Similarly, for in Fig. 3 (greater values of result in a FER ), , which again leads to a FER curve shifted around one order of magnitude with respect to the BER in Fig. 3. It is interesting to compare the performance of LDPC and turbo codes when they are used over finite-state binary Markov channels. The best schemes in turbo codes make use of supertrellises jointly describing the Markov channel and each of the constituent encoders [4]. In order to obtain the best possible performance, the encoder structure (i.e., the number of constituent encoders and the puncturing pattern) has to be adjusted to the channel parameters. Simplified approaches that do not need to adjust the encoder to the channel and consider the Markov channel as another constituent decoder block (more in

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the spirit of this letter) were proposed in [5]. The results obtained in this letter for LDPC codes outperform the simplified methods for turbo codes described in [5] when those turbo schemes are applied to the channels described here, and are very close to results obtained with turbo schemes using supertrellis approaches (which have a much higher complexity in the decoder). A very important advantage of LDPC codes is that a fixed encoder can achieve good performance for different Markov channels.

V. CONCLUSION We have introduced a modified message-passing algorithm for the decoding of LDPC codes over finite-state binary Markov channels. This approach clearly outperforms systems in which the channel statistics are not exploited in the decoder, and allows reliable communications at rates which are above the capacity of a memoryless channel with the same stationary BEP as the Markov channel. This holds even in the case in which the channel parameters are not known a priori in the decoder, since the parameters can be estimated jointly with the decoding process. The proposed approach can also be applied to irregular codes, which is expected to lead to further performance improvements.

REFERENCES [1] M. Mushkin and I. Bar-David, “Capacity and coding for the Gilbert–Elliott channels,” IEEE Trans. Inform. Theory, vol. 35, pp. 1277–1290, Nov. 1989. [2] A. J. Goldsmith and P. P. Varaiya, “Capacity, mutual information, and coding for finite-state Markov channels,” IEEE Trans. Inform. Theory, vol. 42, pp. 868–886, May 1996. [3] A. Lapidoth and P. Narayan, “Reliable communication under channel uncertainty,” IEEE Trans. Inform. Theory, vol. 44, pp. 2148–2177, Oct. 1998. [4] J. Garcia-Frias and J. D. Villasenor, “Turbo decoding of Gilbert–Elliot channels,” IEEE Trans. Commun., vol. 50, pp. 357–363, Mar. 2002. [5] J. Garcia-Frias and J. Villasenor, “Low-complexity turbo decoding for binary Markov channels,” in Proc. Vehicular Technology Conf., vol. 2, May 2000, pp. 840–843. [6] R. G. Gallager, “Low-density parity-check codes,” IEEE Trans. Inform. Theory, vol. IT-8, pp. 21–28, Jan. 1962. [7] T. Wadayama, “An iterative decoding algorithm of low density parity check codes for hidden Markov noise channels,” in Int. Symp. Information Theory, Applications, Nov. 2000, [CD-ROM]. [8] J. Garcia-Frias, “Decoding of low-density parity-check codes over finitestate binary Markov channels,” in Proc. Int. Symp. Information Theory, June 2001, p. 72. [9] E. A. Ratzer. Low-density parity-check codes on Markov channels. presented at 2nd IMA Conf. Mathematics in Communications. [CD-ROM] [10] N. Wiberg, “Codes and decoding on general graphs,” Ph.D. dissertation, Dept. Elect. Eng., Univ. Linkoping, Linkoping, Sweden, 1996. [11] A. P. Worthen and W. E. Stark, “Unified design of iterative receivers using factor graphs,” IEEE Trans. Inform. Theory, vol. 47, pp. 843–849, Feb. 2001. [12] J. Pearl, Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. San Mateo, CA: Morgan Kaufmann, 1988. [13] D. J. C. MacKay, “Good error-correcting codes based on very sparse matrices,” IEEE Trans. Inform. Theory, vol. 45, pp. 399–431, Mar. 1999. [14] L. R. Rabiner, “A tutorial on hidden Markov models and selected applications on speech recognition,” Proc. IEEE, vol. 77, pp. 257–285, Feb. 1989.

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Accurate Computation of the Performance of -ary Orthogonal Signaling on a Discrete Memoryless Channel Jon Hamkins, Senior Member, IEEE

Abstract—A formula for the error rate of maximum-likelihood detection of -ary orthogonal signaling on a discrete memoryless channel is manipulated into a form that avoids numerical imprecision when it is used to calculate low error rates. Index Terms—Optical modulation, optical signal detection, pulse-position modulation (PPM), signal detection.

and are now probability mass functions, and where is a cumulative distribution funcwhere tion. (We use the notational convention that if , there is no contribution to the sum.) This may be written more simply as (2)

I. A FORMULA FOR COMPUTING LOW SYMBOL-ERROR RATES HIS letter considers the error probability when mutually orthogonal signals are transmitted with equal likelihood correlators at and equal power, and received by a bank of the receiver. The analysis requires that the channel be memoryless and have the property that the maximum-likelihood symbol decision is the result of identifying the highest correlator output. We are motivated by the desire to calculate the performance of -ary pulse-position modulation (PPM) on a Poisson channel, which is a good model for some free-space optical communications links [3]. We present an easily computed formula that works at low bit-error rates (BERs) that some applications require. When the channel has continuous-valued outputs, the probability of incorrectly deciding which of the signals was sent is well known (see, e.g., [1] and [2] for the additive white Gaussian noise (AWGN) channel) to be , where and are the conditional probability density functions for a correlator output for the transmitted signal or one of the other signals, respectively. The remainder of the letter considers a discrete-output channel. The probability of symbol error for -ary orthogonal signaling on the Poisson channel is derived in [3] and [4], and the straightforward generalization of that result to a discrete memoryless channel whose outputs take values from the nonnegative integers is

T

Unfortunately, a direct numerical evaluation of either (1) or is small, because it involves differences (2) is difficult when that can be many orders of magnitude smaller than either term. This is problematic when numbers are stored with finite precision, such as with the IEEE 754 floating point standard [5]—a typical program would incorrectly evaluate as zero, for example. Thus, it is helpful to derive a formula for the symbol-error rate (SER) that does not involve the type of difference present in (1) and (2). This would provide an alternative to the union bound or other upper bound [6] which is typically used when is small. Using , we may rewrite (2) as (3) nearly equals , or equivalently, is very small, the th term is difficult to calculate in a numerically precise way. We let and rewrite the th term as When

(1) (4) Paper approved by A. Abu-Dayya, the Editor for Modulation and Diversity of the IEEE Communications Society. Manuscript received May 24, 2004. This work was supported by the Interplanetary Network Directorate Technology Program and performed at the Jet Propulsion Laboratory, California Institute of Technology, Pasadena, under Contract with the National Aeronautics and Space Administration. The author is with the Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TCOMM.2004.836561

(5) where in (4) and (5) we used the Taylor series . (5) becomes accurate as , since and . This leads to the main result of the letter, which we now state.

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The probability of symbol error is given by

(6) and may be freely chosen to minimize the total computational error due to numerical imprecision in the first summation, and due to the Taylor-series remainder error in the second summa, the second summation is simply a tion. Note that when union bound. Fig. 1. SER of 64-PPM on a Poisson channel, with n = 1, as computed using (1), (2), (3), and (6).

II. APPLICATION TO THE POISSON CHANNEL

(7)

imately . Using (6), the error rate could be accurately , which is the computed for for error rates down to limit of representable floating point numbers in the IEEE 754 double precision format.

(8)

REFERENCES

In the case of PPM on a Poisson channel

represents the average number of background counts where represents the average number of detected in a slot, and signal counts detected in a signal slot. Fig. 1 shows the SER as and . Using (1) or (2), a function of , when the error-rate computation became inaccurate whenever the true error rate was below 0.01. This is because the square-bracket is evalterm in (1) evaluated to zero (e.g., uated as zero) for significant terms of the sum. Using (3), the computation becomes inaccurate for error rates below approx-

[1] W. C. Lindsey and M. K. Simon, Telecommunication Systems Engineering. Toronto, ON, Canada: Dover, 1973. [2] R. Gallager, Information Theory and Reliable Communication. New York: Wiley, 1968. [3] R. M. Gagliardi and S. Karp, Optical Communications. New York: Wiley, 1976. [4] C.-C. Chen, “Figure of merit for direct detection optical channels,” TDA Progress Report, vol. 42, pp. 136–151, May 1992. [5] J. P. Hayes, Computer Architecture and Organization. New York: McGraw-Hill, 1988. [6] L. W. Hughes, “A simple upper bound on the error probability for orthogonal signals in white noise,” IEEE Trans. Commun., vol. 40, p. 670, Apr. 1992.

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High-Rate Recursive Convolutional Codes for Concatenated Channel Codes Fred Daneshgaran, Member, IEEE, Massimiliano Laddomada, Member, IEEE, and Marina Mondin, Member, IEEE

Abstract—This letter presents the results of the search for optimum punctured recursive convolutional codes (RCCs) of rate , for , suitable for concatenated channel codes whose constituent encoders are recursive, systematic convolutional codes. The mother codes that are punctured are rate-1/2 RCCs proposed for use in parallel and/or serial concatenation schemes. Extensive tables of systematic and nonsystematic puncturing patterns, optimized relative to various objective functions suitable for concatenated channel codes, are presented for several mother codes.

+1

= 2 ... 8

TABLE I RATE-1/2 SYSTEMATIC RECURSIVE CONSTITUENT ENCODERS USED IN THE DESIGN OF HIGH-RATE PUNCTURED ENCODERS

Index Terms—Convolutional codes, parallel concatenated convolutional codes (PCCCs), punctured, recursive convolutional codes (RCCs), serial concatenated convolutional codes (SCCCs), turbo codes, universal mobile telecommunications systems (UMTS) code.

I. INTRODUCTION OR applications requiring high spectral efficiency, there is often a need for high-rate codes that satisfy the system requirements in terms of the required bit-error rate (BER) or frame-error rate (FER) at a target signal-to-noise ratio (SNR). To this end, high-rate punctured convolutional codes (CCs) or a suitable concatenation of such codes are among the most commonly used for forward error correction (FEC). Puncturing, introduced in [1], is the most widely used technique to obtain high-rate CCs, since the trellis complexity of the overall code is the same as the lower rate mother code whose output is punctured. It is known that for soft-decision Viterbi decoding, the BER of a convolutional code of rate with binary phaseshift keying (BPSK) or quaternary phase-shift keying (QPSK) modulation in additive white Gaussian noise (AWGN), can be well upper bounded by the following expression:

F

(1) in which is the minimum nonzero Hamming distance of is the cumulative Hamming weight associated with the CC, all the paths that diverge from the correct path in the trellis of the code, and re-emerge with it later and are at Hamming distance from the correct path, and finally is the Gaussian integral

function, defined as . Note that (1) is valid for any linear code, provided that the summation is upper-limited to the block size of the block code. A classic approach to the design of good punctured codes consists of finding the puncturing pattern (PP) that yields a code whose distance spectrum has the property of having the max. A better approach is to obtain imum minimum distance the distance spectra of the punctured codes and to select the one which minimizes the BER upper bound based on the first few terms of the distance spectra. In this letter, the emphasis is on the use of punctured CC in serially concatenated convolutional codes (SCCCs) [2] and parallel concatenated convolutional codes (PCCCs) [3]. Because of the inherent difficulty in finding good PPs for concatenated channel codes, usually it is preferable to search for the optimal punctured constituent encoders of the concatenated codes satisfying some specific requirements. Several authors have already considered the problem of obtaining good PPs for PCCCs [4]–[7] and SCCCs [8]–[11], while many others have addressed the code-search problem for optimum punctured nonrecursive convolutional codes. There is ample literature in this area [12]–[16]. This letter presents the results of our , recursive exhaustive search for good punctured, rateconvolutional codes (RCCs) to be used in the construction of PCCCs and SCCCs. II. CODE-SEARCH TECHNIQUE

Paper approved by E. Ayanoglu, the Editor for Communication Theory and Coding Application of the IEEE Communications Society. Manuscript received August 11, 2003; revised April 1, 2004. This work was supported in part by Euroconcepts S.r.l. (http://www.euroconcepts.it), and in part by MURST (Ministero dell’Universitá per la Ricerca Scientifica Tecnologica), Italy. F. Daneshgaran is with the ECE Department, California State University at Los Angeles, Los Angeles, CA 90032 USA (e-mail: [email protected]). M. Laddomada and M. Mondin are with the Dipartimento di Elettronica, Politecnico di Torino, 10129 Torino, Italy (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TCOMM.2004.836590

Mother codes selected for puncturing in this letter are the best recursive rate-1/2 CCs with 4, 8, 16, and 32 states proposed in the literature for the construction of both PCCCs and SCCCs. Matrix generators of the considered codes are shown in the extensive PP tables presented in the letter. The first few terms of the distance spectra of the rate-1/2 recursive CCs chosen for puncturing are shown in Tables I and II. The tables also list the effective distance , that is the minimum Hamming weight of

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TABLE II OPTIMIZED RATE-1/2 SYSTEMATIC RECURSIVE CONSTITUENT ENCODERS FOR CONCATENATED CHANNEL CODES

the codewords generated by weight-2 input patterns, and the distance generated by weight-3 input patterns of the considered codes. To the best of our knowledge, all the mother CCs we have used are among the best RCCs obtained by using primitive feedback polynomials for the code generators [3], [17]. For clarity of presentation, in the following, we shall distinguish between the design of mother encoders and PPs for PCCCs and SCCCs. A. Design of High-Rate Constituent Encoders for PCCCs In this section, the focus is on the design of good high-rate constituent encoders for PCCCs. In addition to the mother codes presented in Table I, we have conducted a search for good constituent recursive rate-1/2 convolutional encoders to be used in PCCCs. Note that the design of good mother encoders and PPs for PCCCs follows the same general rules. In connection with the PCCCs, it is known that the constituent encoders must be recursive and systematic in order for the interleaver to yield a gain [3]. Furthermore, in PCCCs, the dominant patterns yielding the lowest terms of the distance spectra are due to input patterns with weight-2, especially for large interleaver sizes. In fact, it is known that the performance of the PCCC with large interleavers [17] for moderate-to-high SNR can be expressed as (2) is the code rate of the PCCC, and is the interleaver where length. For this reason, a good criteria for obtaining both good mother encoders and PPs in a PCCC consists of maximizing the effective distance . We note that the maximum effective distance achievable with a recursive systematic rate-1/2 encoder with generator macan be obtained from [17], trix . In particular, equality is achieved when the

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is primitive and under two addenominator polynomial and that ditional conditions which require that for . The search for good mother encoders having highest has been conducted by considering the . We above conditions on the polynomials associated with have used these codes as mother codes by following the generally accepted rule that “good mother codes” lead to “good punctured codes.” Indeed, practical documented results show are derived from mother that PPs with maximum possible encoders having maximum . In connection with the use of punctured encoders in a PCCC, the possible puncturing strategies can be different. Due to the complexity of finding good PPs for the overall PCCC codewords [6], a viable solution is to use punctured constituent encoders. For example, in reference to the general scheme of a PCCC, a recursive systematic encoder can be used as the rateupper encoder of the PCCC, whereas the lower encoder can be punctured so that the systematic bits and some of its parity bits are completely eliminated, in order to achieve the desired rate for the overall PCCC. Considerations above motivated us to design both systematic mother encoders and PPs by using, as objective function, the maximization of the effective distance . In a second phase, among the encoders yielding the same (if several), we chose the one requiring the minimum SNR for achieving the target , and then the one with maximum . As a cost BER function for optimization in connection with the minimization of SNR, we have used the inverse of the BER upper bound, as expressed in (1), using the first few terms of the distance spectra of the codes. In the following, we shall identify this design criteria for PPs as criterion . The results of the search for good mother encoders are shown in Table II labeled with the acronym for constituent encoders with memory equal to 2, 3, 4, and 5 (the column heading shows the number of states). The second column shows the generator matrices of the optimal encoders, the third column lists the code distance spectra up to the fifth term (each triplet represents the of all Hamming weight of the codewords , the multiplicity leading to codewords the input patterns with overall weight with weight and the input weight ), and the last column shows the effective distance and the distance generated by (the entry is used to weight-3 sequences denoted signify the fact that there are no weight-3 input patterns leading to low-distance codewords). B. Design of High-Rate Constituent Encoders for SCCCs In this section, the emphasis is on the design of good high-rate constituent encoders for SCCCs. For serial concatenation of CCs, the optimization criteria adopted is somewhat different, depending on whether the punctured codes have to be used as outer or as inner codes. As discussed in [2], the asymptotic BER of an SCCC for very large interleaver sizes is given by (3)

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, and

(4) and do for odd values of . In both equations, the terms not depend on the interleaver length , is the free distance of the outer code, is the effective distance of the inner code, is the rate of the SCCC, and is the minimum weight of the inner code codewords generated by weight-3 input sequences. Equations (3) and (4) can be used to deduce some useful design criteria for constituent encoders of SCCCs. First of all, we note that the inner encoder in an SCCC must be a recursive convolutional encoder, no matter if it is systematic or not, while the outer encoder should have maximum free distance. It is not necessary for the outer encoder to be either recursive or systematic, as is evident from the asymptotic interleaving gain given by for odd values of , and for even values of . In particular, compatible with the desired rate of the SCCC, it is better to choose outer encoders with odd values of . In summary, for the outer encoder in an SCCC, classical design methodologies for design of optimal CCs [12]–[16] are adequate. In the moderate-to-high SNRs where an interleaver gain is observed, the outer encoder should simply possess a good distance spectrum, i.e., highest minimum distance and low weight of the associated error-sequence combination not just for the minimum-distance term, but also for other low-distance terms of the distance spectrum. Let us focus on the design criteria for inner encoders in an SCCC. Equations (3) and (4) suggest that for outer encoders with even values of , a suitable criteria to design good inner encoders is to maximize their effective distance . In the case where the outer encoder has an odd value of , it is also better to choose inner encoders with the greatest possible (recall that is the smallest weight of the inner encoder codeword generated by weight-3 input sequences). In this case, an inner encoder with the feedback polynomial containing can be chosen, thus avoiding altogether termithe factor nating error patterns, and hence, low-weight codewords due to input sequences of weight-3. Once the outer and inner encoders are chosen in accordance to the previous criteria, the interleaver design should focus on optimizing the matching between the outer and the inner encoders in such a way that for each Hamof the outer codewords, the Hamming weight ming weight of the inner codewords are maximized (here, is the minimum Hamming weight of the inner codewords generated by inner input patterns with the same weight of the outer codewords). This maximization is then applied to increasing weights of the outer codewords. In addition to the mother codes presented in Table I, we have conducted a search for good constituent recursive rate-1/2 convolutional outer encoders in SCCCs, by using as an objective [16]. function the minimization of SNR at a target BER In order to resolve potential ties, in a second phase between all the encoders yielding similar performance, we choose the encoder with the best and, subsequently, the best . The results of this search are shown in Table II labeled with the acronym SNR for constituent encoders with memory equal to 2, 3, 4, and 5.

The results in Table II obtained for the 32-state recursive encoders are slightly different. During our search, we found an encoder satisfying both objective functions mentioned above codes). For this (listed in the last line of Table II for reason, in the upper line of the same row, we show the best enwhile simultaneously satisfying the coder having maximum minimum SNR requirement and achieving maximum . This encoder can be useful, for instance, as the inner encoder of an SCCC when the outer encoder is punctured so that its minimum distance is equal to three. In this case, because of the absence of inner encoder codewords generated by weight-3 input patterns, the overall SCCC yields better performance. Following the general guidelines above, let us outline the design criteria adopted in this letter for obtaining good punctured encoders for SCCCs. As far as the inner encoder of the SCCC is concerned, the design rule consists of searching for the PP for a leading to the maximum possible effective distance given rate. Between all PPs having the same maximum , we choose the one yielding the minimum SNR requirements , and finally, between all for achieving the target BER the PPs yielding the same SNR requirement, we chose the one with maximum . We shall identify this design criteria as criterion . In connection with the design of punctured encoders to be used as outer codes in an SCCC, we conducted a search for the best PPs yielding the minimum SNR requirements for . This target is such that the achieving the target BER optimization acts to maximize the minimum distance of the punctured encoder and minimize the overall weight associated with the input pattern yielding the minimum distance, followed by maximization of the successive low-distance terms and minimization of their corresponding input weights for the first four minimum-distance terms used in the formulation of the cost function. Between various PPs satisfying the requirement with the same SNR, we chose the one having first of all the maximum , and then the maximum . We shall identify this design criteria as criterion . III. CODE-SEARCH RESULTS The results of our search for the best PPs are presented in Tables III–VIII. In particular, for any encoder with memory having states (the column heading shows the number of states) in any given column, Tables III–V show the best PPs resulting in punctured encoders of rateunder criteria , , and . For any given memory size and code rate, Tables VI–VIII show the best PPs under criteria , , and , respectively. Since there was no single punctured encoder of a given memory outperforming all other encoders with the same memory over all code rates examined, we developed three global cost metrics as , for any given follows. For codes obtained using criterion code rate, we evaluated the loss in SNR between the performance of each encoder and the one achieving the desired target with the minimum SNR. Then, we summed the SNR losses of the punctured encoders over all the code rates. This was repeated for other memory sizes independently. These PPs are shown in and , for any given Table VIII. In relation to the criteria

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TABLE III BEST PPS FOR VARIOUS CODE RATES UNDER CRITERION C

TABLE VI OPTIMAL PUNCTURED ENCODERS OVER ALL CODE RATES UNDER CRITERION C

TABLE IV BEST PPS FOR VARIOUS CODE RATES UNDER CRITERION C

TABLE VII OPTIMAL PUNCTURED ENCODERS OVER ALL CODE RATES UNDER CRITERION C

TABLE V BEST PPS FOR VARIOUS CODE RATES UNDER CRITERION C

TABLE VIII OPTIMAL PUNCTURED ENCODERS OVER ALL CODE RATES UNDER CRITERION C

number of states of the considered mother encoders, we selected over the largest possible the encoders having the maximum set of code rates. Then, between these encoders, we chose the

one yielding the minimum SNR losses evaluated as above for criterion . The results are shown in Tables VI and VII.

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The tables are organized as follows. For any given memory size shown in the column with the respective number of states and any rate shown in a given row, we show the polynomial generators of the mother encoder yielding the best PP in the identifying the minimum first line, the triplet distance , the number of nearest neighbors yielding the min, and the total weight of these input patterns imum distance as the second line, and the effective distance and the of the punctured codes as the third line. weight-3 distance does not exist in the distance spectra Where not specified, of the punctured codes. As an example of how to read the table entries, consider the rate-2/3 puncturing pattern PP-13 related shown in Table III to the mother encoder yielding the best PPs under criterion . This PP, represented in octal form, leads to a code whose minimum distance three is due to one input pattern with input weight three. The effective distance of the code is four, and the weight-3 distance is three. As noted above, the PPs are represented in octal form. A given PP should be read from right to left by collecting pairs of systematic-parity bits. As an example, the PP in Table III, which yields a code with rate-2/3 for the 4-state code, should be in(the terpreted as follows: subscript denotes the base of the numbers). In this case, the PP leaves the encoder systematic and deletes the first parity bit associated with every two input bits.

IV. CONCLUSIONS In this letter, we have presented extensive optimized PP tables for recursive CCs to be used in the design of parallel and serially concatenated CCs. The optimization was conducted using three different objective functions, each one of which is suited to a certain application in connection with the design of PCCCs and SCCCs. We have further conducted exhaustive searches for mother encoders of rate-1/2 to be used for puncturing using two different selection criteria. These encoders, and several other encoders reported in the literature, were then used as mother encoders to which puncturing is applied.

ACKNOWLEDGMENT The authors wish to thank the editor and the anonymous reviewers for many useful suggestions that have improved the quality of this letter. REFERENCES [1] J. B. Cain, G. Clark, and J. M. Geist, “Punctured convolutional codes of rate (n 1=n) and simplified maximum-likelihood decoding,” IEEE Trans. Commun., vol. COM-25, pp. 97–100, Jan. 1979. [2] S. Benedetto, D. Divsalar, G. Montorsi, and F. Pollara, “Serial concatenation of interleaved codes: Performance analysis, design, and iterative decoding,” IEEE Trans. Inform. Theory, vol. 44, pp. 909–926, May 1998. [3] S. Benedetto and G. Montorsi, “Design of parallel concatenated convolutional codes,” IEEE Trans. Commun., vol. 44, pp. 591–600, May 1996. [4] F. Babich, G. Montorsi, and F. Vatta, “Design of rate-compatible punctured turbo (RCPT) codes,” in Proc. IEEE Int. Conf. Communications, vol. 3, Apr. 2002, pp. 1701–1705. , “Rate-compatible punctured serial concatenated convolutional [5] codes,” in Proc. IEEE Globecom, vol. 4, Dec. 2003, pp. 2062–2066. [6] M. A. Kousa and A. H. Mugaibel, “Puncturing effects on turbo codes,” IEE Proc. Commun., vol. 149, no. 3, pp. 132–138, June 2002. [7] O. F. Acikel and W. E. Ryan, “Punctured turbo codes for BPSK/QPSK channels,” IEEE Trans. Commun., vol. 47, pp. 1315–1323, Sept. 1999. [8] S. S. Pietrobon, “Super codes: A flexible multi-rate coding system,” in Proc. Int. Symp. Turbo Codes, Related Topics, Brest, France, Sept. 2000, pp. 141–148. , “On punctured serially concatenated turbo codes,” in Proc. 35th [9] Asilomar Conf. Signals, Systems, Computers, vol. 1, 2001, pp. 265–269. [10] F. Daneshgaran, M. Laddomada, and M. Mondin, “An extensive search for good punctured rate-k=k + 1 recursive convolutional codes for serially concatenated convolutional codes,” IEEE Trans. Inform. Theory, vol. 50, pp. 208–217, Jan. 2004. [11] O. F. Acikel and W. E. Ryan, “Punctured high-rate SCCCs for BPSK/QPSK channels,” in Proc. IEEE Int. Conf. Communications, vol. 1, 2000, pp. 434–439. [12] K. J. Hole, “Punctured convolutional codes for the 1-D partial-response channel,” IEEE Trans. Inform. Theory, vol. 37, pp. 808–817, May 1991. [13] Y. Yasuda, K. Kashiki, and Y. Hirata, “High-rate punctured convolutional codes for soft-decision Viterbi decoding,” IEEE Trans. Commun., vol. COM-32, pp. 315–319, Mar. 1984. [14] G. Begin, D. Haccoun, and C. Paquin, “Further results on high-rate punctured convolutional codes for Viterbi and sequential decoding,” IEEE Trans. Commun., vol. 38, pp. 1922–1928, Nov. 1990. [15] K. J. Hole, “New short constraint length rate (N 1=N ) punctured convolutional codes for soft-decision Viterbi decoding,” IEEE Trans. Inform. Theory, vol. 34, pp. 1079–1081, Sept. 1988. [16] P. J. Lee, “Constructions of rate-(n 1=n) punctured convolutional codes with minimum required SNR criterion,” IEEE Trans. Commun., vol. 36, pp. 1171–1174, Oct. 1988. [17] D. Divsalar and F. Pollara, “On the design of turbo codes,”, JPL TDA Progress Rep. 42-123, 1995.

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A New Base Station Receiver for Increasing Diversity Order in a CDMA Cellular System Wan Choi, Chaehag Yi, Jin Young Kim, and Dong In Kim, Senior Member, IEEE

Abstract—A new base station receiver is proposed and analyzed for a code-division multiple-access (CDMA) cellular system. The proposed receiver can achieve remarkable diversity gain by increasing diversity order with reasonable cost and complexity. From the numerical results, it is confirmed that the proposed receiver structure can be a practical solution for enhancing reverse-link capacity and improving performance in CDMA cellular system operations. The result in the letter can find its applications to legacy IS-95/cdma2000 1x base stations with simple modifications. Index Terms—Code-division multiple access (CDMA), diversity, multipath fading, receiver complexity.

I. INTRODUCTION EVERAL diversity techniques have been studied and found practical use in many communication systems in order to combat multipath fading and improve performance. Among the techniques, spatial diversity has been commonly used at cellular base station (BS) receivers due to its simplicity in implementation [1]. Although it is well known that higher order diversity can improve the receiver performance, the second-order spatial diversity with two receiving antennas at a BS has been most popular, because the higher order spatial diversity at a BS requires additional cost associated with strict zoning requirements. The antenna elements for the spatial diversity need to be separated at least ten times wavelength in order to obtain signals that fade independently [1]. The possibility of using polarization diversity has been studied with a motivation that the polarization diversity using a dual-polarized antenna can reduce cost and space for installation, compared with the traditional spatial diversity. The previous investigations on polarization diversity have revealed that the polarization diversity is able to achieve comparable performance to the spatial diversity [2]–[5]. Exploring the polarization as an additional source of diversity has been considered in [6] and [7]. Particularly, [7] investigated the benefit of increased diversity order using spatially separated polarized antennas at the BS receiver of a code-division multipleaccess (CDMA) cellular system. The paper proved that the four-

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Paper approved by V. A. Aalo, the Editor for Diversity and Fading Channel Theory of the IEEE Communications Society. Manuscript received December 30, 2002; revised June 9, 2003 and February 16, 2004. W. Choi was with KT Freetel, Inc., Seoul, Korea. He is now with the Department of Electrical Engineering, The University of Texas at Austin, Austin, TX 78712 USA (e-mail: [email protected]). C. Yi is with Solid Technologies, Inc., Seoul 138-803, Korea (e-mail: [email protected]). J. Kim is with Kwangwoon University, Seoul 139-701, Korea (e-mail: [email protected]). D. Kim is with the School of Engineering Science, Simon Fraser University, Burnaby, BC V5A 1S6, Canada (e-mail: [email protected]). Digital Object Identifier 10.1109/TCOMM.2004.836595

branch diversity receiver combining spatial and polarization diversity could considerably improve receiver performance and enhance reverse-link capacity. However, implementation of the four-branch diversity receiver using spatially separated cross-polarized antennas requires high cost and considerable modification, compared with the conventional dual-diversity receiver. The four-branch diversity receiver also needs additional hardware units and space for increased diversity branches. In this letter, we propose a new BS receiver to increase diversity order with low cost and complexity, and analyze the performance of the proposed receiver. The bit-error probability (BEP) and the outage probability are derived and compared for each type of receivers. The letter is organized as follows. In Section II, the proposed new BS receiver is shown and described. In Section III, outage probability and BEP are derived for the proposed receiver. In Section IV, some numerical results are presented. The conclusions are drawn in Section V. II. PROPOSED RECEIVER STRUCTURE In sectorized cellular CDMA systems, a typical BS supports three sectors, and the structure for the four-branch diversity receiver using spatially separated dual-polarized antennas in [7] is shown in Fig. 1(a). In each sector, the polarized antennas form four diversity branches and the signals from each branch are down-converted and digitized in the radio frequency (RF) and the intermediate frequency (IF) circuits. Then, they are transfered to the baseband RAKE receiver for matched filtering and combining. The four-branch diversity receiver requires additonal hardware units for the increased diversity branches, compared with the conventional dual-diversity receiver, because the signals from antenna elements have the same pseudonoise (PN) code offsets and suffer independent fading. In cellular CDMA systems, though the transmitted signal from a mobile station (MS) is spread by a user-specific PN code, the received signals from the MS have different PN offsets due to the multipath components. The RAKE receiver in the baseband modem resolves the multipath signals and combines them [8]. Even if the signals with the different PN offsets are combined, the RAKE receiver can resolve the combined signals based on the PN code offsets. With this motivation, we propose a new BS receiver that can increase diversity order with low cost and complexity. The structure of the proposed receiver is shown in Fig. 1(b). The RF signal from one antenna element of a cross-polarized antenna is intentionally delayed by the amount of a predetermined value and added by the RF signal from the other antenna element. In the proposed receiver, branches are for normal branches, and the remaining

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Fig. 1. BS receiver structures. (a) Four-branch diversity receiver. (b) Proposed diversity receiver.

branches are for the delay branches among the total -diversity branches [ for Fig. 1(b)]. Then, the combined signal in

the RF level is input to the RF and IF units for down-conversion and digitization, and transferred to the baseband RAKE receiver.

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All of the signals from diversity branches can be resolved in the RAKE receiver, because the sufficient delay can make the fading statistics appear independent even for the same user, and assure the resolvability as well. The predetermined delay should be larger than the maximum excess path delay for the meaningful multipath signals from the normal branches, in order to avoid overlapping with the intentionally delayed signals from the delay branches. The intentional RF delay module can be deployed with either active components or passive components. The proposed receiver can be easily applied to the legacy BSs with some modification in the antenna part, and does not need additional hardware units and space, compared with the fourbranch diversity receiver in Fig. 1(a). The additional hardware units and the space for increasing the number of branches cost too much burden in implementing the BS.

III. PERFORMANCE ANALYSIS In this section, the outage probability and BEP are derived for each type of receiver by modifying the results of [9] and [10]. Although the synchronous assumption is not valid in commercial CDMA systems, a synchronous reverse link is assumed in the performance analysis because it allows a complete comparative analysis of the diversity techniques. In addition, the following assumptions are made for simplicity of analysis: 1) all the signals from antenna elements are independent; 2) power control and channel parameters estimated for maximum ratio combining (MRC) are perfect; 3) the multipath fading channel is modeled as single-path Rayleigh fading; and 4) the crosscorrelation factor between different user-specific PN codes is . In the synchronous reverse link, the user can be effectively distinguished by appropriately cyclic shifted sequences of a PN sequence, and the crosscorrelation factor between user-specific codes can be equivalent to the autocorrelation factor. A. Conventional

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Thus, the weighting vector for the MF output given in (1) be. Then, the output signal-to-interference-plus-noise comes ratio of the MRC system (SINR_o) is given by (2)

where the superscript denotes complex conjugate transpose, and , , and . The SINR_o in (2) has a similar form to that of [9] and [10]. From the results of [9] and [10], the probability density function (pdf) of the SINR_o can be easily obtained as (3)

and are the pdfs of where respectively, and are given by

and

,

(4) (5) The outage probability is the probability that the received signal level falls below a specified level , at which the desired service quality cannot be satisfied. From (3), the outage probais obtained by [10] bility

-Branch Diversity Receiver

For the -branch diversity receiver, matched-filter (MF) output for a user 0 can be given in a vector form by (6) (1) where each element of represents the MF output corresponding to each diversity branch. is a transmitted data sequence of user , is an -dimensional vector which represents filtered outputs is the number of inof the attenuated signals for the user , terfering users, and is an -dimensional Gaussian zero. Each component of mean vector with covariance matrix is assumed to follow a complex Gaussian distribution vector with mean power of . For a perfect power-controlled system, , . In an MRC system, the signal of each branch is weighted by a corresponding complex-valued channel gain before combining.

where and denotes explicitly the dependence of on . For the MRC, the BEP can be obtained by GA for multipleaccess interference (MAI). In (1), the second term represents the MAI and can be approximated as a Gaussian random variable from the result of [11]. Then, the MAI plus additive white Gaussian noise (AWGN) is also a Gaussian random variable, and the BEP can be given by [12]

where (SINR_br).

and

(7) is an average SINR per branch

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-Branch Diversity Receiver

For the proposed receiver, the RAKE receiver resolves the signals of each diversity branch from the coupled signal. The MF output for a user 0 can be divided into two parts, which are the resolved MF outputs for the signals from normal and delay branches, respectively, and given by (8) where and are respectively, be given by

-dimensional vectors, and can,

diversity branches are independent, (9) and (10) can be effectively approximated to (12) (13) From (12) and (13), the MF output for a user 0 can be given in an -dimensional vector, and given by (14) where

(9)

(10) of represents the MF output corwhere each element responding to normal and delay diversity branches, respectively. In the equations, the denotes the intentional delay, denotes a sampling time, and denotes the floor function. and are -dimensional vectors which represent filtered outputs of the attenuated signal of user from the normal and delay branches, respectively. The and are -dimensional Gaussian vectors . Each component of with zero mean and covariance matrix vector and is assumed to follow a complex Gaussian . The relationship distribution with mean power of between the intentional delay and the crosscorrelation factor is given by (11) where and denote the length of user-specific PN codes and the chip duration, respectively. The crosscorrelation factor is , and thus, intentional delay should be minimized for larger than but smaller than the shifted value for user distinction. The proposed scheme may insert sufficient delay to make the fading statistics appear independent even for the same user, and assure the resolvability as well. In the matched filtering for user 0’s signals from the normal branches, user 0’s signals from the delay branches act as another MAI source given in the fourth term in (9). Because the self-interference in the second term in (9) is statistically equivalent to the crosscorrelated interference between different users, the crosscorrelation appears stationary, and the signals from the

The resolved MF output in (14) is similar to that of the conventional diversity receiver, except that the MAI increases by a . For the MRC, the weighting vector in (14) factor of , and the SINR_o is given by (2) with replaced becomes . Therefore, the outage probability is obtained by by in (6). In (14), the second term represents the MAI and can be approximated as a Gaussian random variable. When the average , SINR_br of the conventional receiver is the average SINR_br for the proposed receiver is given by

(15) Then the BEP of the proposed receiver can be obtained as in (7). For the case where the MAI becomes dominant, for the proposed the average SINR_br becomes receiver, and the BEP of the proposed receiver can be obtained as in (7). C. On the Validity of Gaussian Approximation Though we use the Gaussian approximation (GA) in deriving the BEP, the GA can be inaccurate when the transmitted date sequence is random, unless the number of interfering users is large. In the situations where the GA is not appropriate, a more in-depth analysis considering conditional BEP on the interference terms must be applied. But it may be very difficult to obtain a closed-form result by this approach. The method to improve the inaccuracy of GA in this situation remains our further study. However, considering the fact that the purpose of this letter is to investigate the feasibility of the proposed receiver, the derived BEP with the GA can be useful in comparative analysis by itself. For the conventional diversity receiver, the BEP with the GA can be regarded as only an optimistic loose bound because of the inaccuracy of the GA. But for the proposed receiver, the inaccuracy of the GA is obviously improved, since the proposed

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Fig. 2.

BEP versus average SINR_br.

receiver causes the effect of increasing the number of interfering . users IV. NUMERICAL RESULTS Some numerical results for the BEP and outage probability are presented and compared for the conventional and the proposed diversity receivers. To show the feasibility of the latter, we are more concerned with the comparison between conventional two-branch diversity and proposed diversity receivers. Fig. 2 shows the BEP versus average SINR_br in a Rayleigh fading channel environment in the general case, including the situation where the MAI is not dominant compared with , the four-branch receiver has about AWGN. For a BEP of 1-dB gain over the proposed four-branch receiver, while the proposed four-branch receiver has about 5-dB gain over the conventional two-branch receiver. Though the four-branch receiver shows better performance than any other receivers considered, it requires high cost and considerable complexity, compared with the conventional two-branch receiver. Meanwhile, the proposed four-branch receiver can achieve a reasonable BEP gain over the conventional two-branch receiver with low cost and complexity. The performance gain becomes larger as the average SINR_br increases. This gain can directly result in capacity and coverage increases in the mobile cellular system operations. Fig. 3 shows the outage probability versus SNR as another performance measure. Five interfering users, 5-dB outage threshold SINR, and 128-length user-specific codes are assumed for all the types of receivers for , performance comparison. For an outage probability of the gap of the required SNR between the proposed four-branch receiver and the conventional two-branch receiver is about 15 dB. That means the received power level of the conventional two-branch receiver should be 15 dB larger than that of the proposed four-branch receiver at the same level of AWGN to . Thus, transmit achieve the same outage probability of power of the MS in the proposed four-branch receiver can be 15 dB less than that of the conventional two-branch receiver.

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Fig. 3.

Outage probability versus SNR.

The gap may depend on average outage threshold SINR_o , number of interfering users, and target outage probability. The saved power can extend the lifetime of the battery in the MS and reduce the interfering power to other users. Although MAI is not always dominant compared with AWGN in the synchronous reverse link, MAI becomes dominant, and the average SINR_o converges on a certain level as the SNR increases. In the predetermined average outage threshold SINR_o condition, the converged SINR_o also makes the outage probability converge on a certain probability. The converged value of outage probability will depend on the value of the predetermined average outage threshold SINR_o. From the figure, the outage probability versus SINR can be easily derived, using the relationship between the average SINR_o and SNR, , where . From the BEP and outage performance shown in Figs. 2 and 3, the proposed four-branch diversity receiver can achieve remarkable gains over the conventional two-branch diversity receiver with low cost and complexity, and the gains can directly be linked to a capacity enhancement. V. CONCLUSION In this letter, the new BS receiver has been proposed and analyzed in order to increase diversity order in the CDMA cellular system. The BEP and outage probability are evaluated and compared for the conventional and the proposed diversity receivers. From the numerical results, it has been demonstrated that the proposed receiver can achieve remarkable diversity gain with reasonable cost and complexity. The proposed receiver is expected to provide a practical solution for enhancing reverselink capacity and improving performance in the CDMA cellular system operations. The proposed receiver can be applied to the legacy IS-95/cdma2000 1x BSs with simple modifications. ACKNOWLEDGMENT The authors would like to thank R. Padovani of Qualcomm Inc. for his helpful discussion and encouraging appreciation of

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our work. They also appreciate the comments of the editor and reviewers.

REFERENCES [1] W. C. Jakes, Microwave Mobile Communications. New York: Wiley, 1974. [2] W. C. Y. Lee and Y. S. Yeh, “Polarization diversity system for mobile radio,” IEEE Trans. Commun., vol. COM-26, pp. 912–923, Oct. 1972. [3] R. G. Vaughan, “Polarization diversity in mobile communications,” IEEE Trans. Veh. Technol., vol. 39, pp. 177–186, Aug. 1990. [4] A. M. D. Turkmani, A. A. Arowojolu, P. A. Jefford, and C. J. Kellen, “An experimental evaluation of the performance of two-branch space and polarization diversity scheme at 1800 MHz,” IEEE Trans. Veh. Technol., vol. 44, pp. 318–326, May 1995. [5] B. Lindmark and M. Nilsson, “On the available diversity gain from different dual-polarized antennas,” IEEE J. Select. Areas Commun., vol. 19, pp. 287–294, Feb. 2001.

[6] G. I. Siquiera et al., “Combined use of space and polarization diversity on mobile cellular network,” in Proc. IEEE GLOBECOM, Rio de Janeiro, Brazil, Dec. 1999, pp. 863–867. [7] L. Aydin, E. Esteves, and R. Padovani, “Reverse link capacity and coverage improvement for CDMA cellular systems using polarization and spatial diversity,” in Proc. IEEE Int. Conf. Communications, New York, NY, Apr. 2002, pp. 1887–1892. [8] G. L. Turin, “Introduction to spread spectrum antimultipath techniques and their application to urban digital radio,” Proc. IEEE, vol. 68, pp. 328–353, Mar. 1980. [9] J. Cui and A. U. H. Sheikh, “Outage probability of cellular radio systems using maximal ratio combining in the presence of multiple interferers,” IEEE Trans. Commun., vol. 47, pp. 1783–1787, Aug. 1999. [10] V. A. Aalo and C. Chayawan, “Outage probability of cellular radio systems using maximal ratio combining in Rayleigh fading channel with multiple interferers,” IEE Electron. Lett., vol. 36, pp. 1314–1315, July 2000. [11] J. S. Lehnert and M. B. Pursley, “Error probabilities for binary direct sequence spread-spectrum communication with random signature sequences,” IEEE Trans. Commun., vol. COM-35, pp. 87–98, Jan. 1987. [12] J. G. Proakis, Digital Communications, 4th ed. New York: McGrawHill, 2001.

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Power Control and Diversity in Feedback Communications Over a Fading Channel I. Saarinen and A. Mämmelä, Senior Member, IEEE

Abstract—Pilot-symbol-assisted modulation system using feedback minimum mean-square error (MMSE) power control in subject to an unavoidable feedback delay, and in conjunction with diversity, is considered over a slow Rayleigh fading channel. Feedback MMSE power control is defined as a power-control function, with feedback MMSE predictions of the current channel fading gains as input that minimizes the system-error probability. The use of feedback requires causality, and an MMSE predictor has to be employed for the purpose of power control. Previously, in the literature, the predictor was used also in detection. The pilot-symbol system with MMSE power control is shown to achieve a clear performance improvement by employing a smoother, instead of the predictor, in detection. Furthermore, the performance loss caused by a feedback delay of 10%–20% from the channel coherence time appeared to be minor with reasonable bit-error rate levels. Finally, additional performance improvement using low-order diversity was shown to be considerable. Index Terms—Channel estimation, closed-loop method, delay, feedback communications, power control.

I. INTRODUCTION AYES evaluated the optimal power control for the coherent antipodal and noncoherent binary orthogonal system in a fading multipath channel, through the ideal knowledge of the channel state fed back from the receiver [1]. Transmission power was controlled, so that the average error probability was minimized subject to the constraint of the average transmission power. Srinivasan proposed in [2] a pilot-symbol-assisted modulation system over a random time-invariant channel (Doppler frequency is zero), using noiseless and delayless feedback. The frame size was two, or, in addition to a known pilot symbol, only one antipodal data symbol was included in a time-multiplexed form in each transmitted frame. The channel-state values were estimated by a one-tap minimum mean-square error (MMSE) estimator using pilot symbols. The optimal transmission power was evaluated as a function of the estimated channel state. In [3], the analysis was extended to the case where the number of data symbols in a frame is arbitrary. Also, the channel was a slow Rayleigh fading channel, and the MMSE predictor, which optimally predicts the fading gain, was applied as an estimator

H

Paper approved by T. F. Wong, the Editor for Wideband and Multiple-Access Wireless Systems of the IEEE Communications Society. Manuscript received March 21, 2003; revised December 3, 2003 and May 4, 2004. This work was supported in part by the Nokia Foundation. This paper was presented in part at the IEEE International Symposium on Personal, Indoor and Mobile Radio Communications, September 18-21, 2000. I. Saarinen was with the University of Oulu, FIN-90014 Oulu, Finland. He is now with VTT Electronics, FIN-90571 Oulu, Finland (e-mail: [email protected]). A. Mämmelä is with VTT Electronics, FIN-90571 Oulu, Finland (e-mail: [email protected]). Digital Object Identifier 10.1109/TCOMM.2004.836588

for both the purpose of feedback MMSE power control and detection. The feedback was still noiseless and delayless. Cavers analyzed in [4] the bit-error probability (BEP) of the pilot-symbol system for different modulation methods when an MMSE smoother was used as a channel estimator. In [5], he determined the optimum variation of data rate with known channel-state values, and the effect of a feedback delay on the system performance was also considered. In this letter, the analysis presented in [3] is extended so that the smoother is employed in detection, and the additional effects of a feedback delay and diversity on the system performance are evaluated. Recently, systems using a feedback link have been proposed, where some combination of transmission quantities, like power, bit rate, constellation size, and code rate, is adjusted according to the feedback information regarding the channel state, see, e.g., [6]–[9]. In [8], it was shown that it is possible to attain the optimal performance with power control only, without using bit-rate control. Furthermore, in our case, the analysis of the more complex system model, e.g., including varying bit rate, would have been mathematically intractable. The rest of the letter is organized as follows. The system description and the performance of the pilot-symbol system with diversity and without feedback MMSE power control is presented in Section II. The performance of the pilot-symbol system using feedback MMSE power control with and without a feedback delay and diversity is evaluated in Section III, and at the end of the letter, some conclusions are drawn. II. SYSTEM DESCRIPTION The system model for the pilot-symbol system with feedback is introduced in Fig. 1. A discrete-time model is used, as explained in [10]. We assume that receiver antenna diversity is available at a base station. That is, there are antennas with enough physical separation so that fading gains are statistically independent and identically distributed (i.i.d.) random variables. The frame size is , and the pilot symbol is transantipodal mitted after every data symbols, i.e., pilot symbols are transmitted at time instants , represents the set of integers). In the case of no power control, the transmitted symbols are ,

.

(1)

When power control is not employed, the energies of the and , respectransmitted pilot and data symbol, tively, are assumed to be equal [2]. The received complex is a length- row vector signal at time , where and contain i.i.d. fading gains and noise components, respectively. The fading

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Fig. 1. Pilot-symbol-assisted modulation system with feedback of estimated fading channel gains.

gain , , is a zero-mean complex Gaussian random variable with a variance for all representing the wide-sense stationary (WSS) fre, quency-nonselective slow Rayleigh fading channel, and , is zero-mean complex WSS additive white Gaussian noise (AWGN) with a variance . In this letter, Jakes’ spectrum is used as the Doppler power spectrum, and the autocorrelation function of the fading gain is expressed [11, p. 26], where is the by is the norzeroth-order Bessel function of the first kind, malized Doppler frequency (divided by the symbol rate), and is the discrete delay. In Fig. 1, terms and denote the fading gain estimates derived by predictors and smoothers, respectively, and is the norm1 of the vector. We have used optimal MMSE estimators for estimating fading gains, and each diversity channel has its own estimator. In our system model, maximum a posteriori (MAP) and MMSE estimators are actually identical. The optimal estimator for estimating the full channel strength (a scalar) is not known. The outputs of the MMSE predictors at time are [12]

(2) where

and , , are coefficients of the predictors. Term , represents the set of natural numbers, indicates the amount of a feedback delay in power control as means the situamultiples of a frame size . Note that tion with no feedback delay. The error signals of the predictors , and the optimal coefficients are defined as are determined from the linear equations 2

( when ) where is the Kronecker delta function.3 The coand the parameter are periodic in time , or efficients [13]. We assume in this letter that the autocorrelation functions are known, and they need not to be estimated. The MMSEs of the predictors become (4) which are also periodic in time, or [13]. In the case of the smoothers , the error signals are defined as , and the summation index in (2) and (3) and the in (3) have values . term Note that in detection there is no feedback delay, and in the case in of the smoother, the delay parameter has to be [see, e.g., (2)]. The MMSEs of the smoothers yield

(5) , , are cowhere efficients of the smoothers. is a lengthThe received signal at time vector, and it can be rewritten as (“ ” is assumed to be transmitted). When conditioned on the predicted gain estimate , it is noted from the preceding equation that the channel is conditionally Rician (see . The smoother es[14]), with -factor (time index is timates omitted for simplicity) are used in detection, and the decision . In this equation, and are variable is a pair of correlated complex Gaussian random variables with nonzero mean values, and for channels considered, the pairs are mutually statistically independent. The conditional [14], [15] error probability becomes

(3)

=[

]

=

=

1Squared norm of the vector x x x is defined as kxk xx jx j jx j . Superscript H denotes complex conjugate and transpose. 2Labeling bxc means the largest integer which is smaller than or equal to x.

+

(6) 3

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and [15] (13) where the average received signal-to-noise ratio (SNR) including the effect of the pilot symbol is , and the term

(14) When

, the average error probability becomes (15)

(7) respectively, where is the Marcum -function and is the th-order modified Bessel function of the first kind. The parameters are given by (8) and

is defined in (6) and where square probability density function with given by [16]

Correspondingly, in the case of ability is

is the central chidegrees of freedom,

elsewhere. (16) , the average error prob-

(9) where is the conditional mean of (when conditioned on ) and , ( means denotes the conditional mean of and “for all”). Similarly, , . The correlation coeffi, where cient is defined as , . The ratios of squared norms of means to variances and the correlation coefficient finally yield (10) (11)

(12)

(17) where

is defined in (7). III. FEEDBACK MMSE POWER CONTROL

When power control is employed, . The average error probability is still given in (15) and (17), respectively, but has been introduced to (10), (12), , as shown in (18)–(20) at and (13), and we get the bottom of the page, where the average transmitted SNR [3] is given by . In MMSE power control, an optimal, nonnegative function has to be determined, which minimizes the average error probability (15) and (17), respectively, when the average energy ratio . The energy ratios with different SNR is values have been evaluated numerically by using the barrier method (a special case of the penalty method), as explained in [3].

(18) (19)

(20)

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Fig. 2. Performance of the pilot-symbol-assisted modulation system with and without MMSE power control (f = 0:001, L = 1).

Fig. 3. Performance of the pilot-symbol-assisted modulation system with and without MMSE power control (f = 0:01, L = 1).

The frame size has been , the number of predictor and , and normalization smoother coefficients 21 is assumed, as in [3]. As was shown in [3], the frame size is an optimal value when employing the predictor in detection dB, , and . That is, in the above with case, the BEP of the pilot-symbol system was minimized. The BEP was very unsensitive as a function of SNR, i.e., error-probability difference was negligible, whether the true optimal frame size value with each SNR or the value was used. In the dB, ), an case of the smoother in detection ( optimal frame size is 31 when and , 30 when and , 15 when and , and 8 and . The relative error-probability difwhen instead of the above-preferences, when using the value sented optimal values, were found to be only 3.0% ( , ), 5.1% ( , ), 7.9% ( , ), , ), respectively. The perand less than 0.1% ( formance of the pilot symbol system using the smoother in detection with and without MMSE power control and a feedback , is plotted in Fig. 2 as a delay, and without diversity function of SNR when the normalized Doppler frequency is . In the figures, the abbreviation PC is used for power control. The above-mentioned curves in the case of no power control use (10)–(13) and (15), and correspondingly, (11), (15), and (18)–(20) when power control is employed. Similar curves . Limited plotting range is are shown in Fig. 3 for due to very excessive numerical calculation times. In Fig. 2, the performance curves of the pilot-symbol system, using the predictor in detection [3] with and without MMSE power control , are illustrated as a comparison. Also, in both figures, the performances of the coherent antipodal system in a Rayleigh fading channel with and without optimal (channel-state values are known) [1] power control are plotted. We can notice in Fig. 2 that in the case of power control and , a clear performance improvement no feedback delay can be achieved by using the smoother, instead of the predictor, in detection. The use of a smoother introduces a time delay of half of a smoother length in the detection process. We will

now illustrate the operating range of the pilot-symbol system employing power control and smoother, with regard to the has values 1, feedback delay. The delay parameter , and 1, 3 with , 5, 15, 25 with respectively. Since the length of a feedback delay is symbols, the delay is approximately half of the coherence time of the channel (defined as symbols) and , likewise, and when . Similarly, the delay duration corresponds about , when and , and 20% 10% from from , when and . We can observe , the from Fig. 2 that with Doppler frequency feedback delay of a duration of one frame size has a negligible effect on the system performance. Also, the performance loss is small, slightly over 0.5 dB, when the (see Figs. 2 and 3). The delay is 10%–20% from loss increases with lower bit-error rate (BER) values [6]. When the delay approaches half of the coherence time, the performance degradation is found to be remarkable. Cavers showed in [5], for the noncoherent orthogonal system with the optimal BER minimizing bit-rate variation scheme, that is some decibels, the performance loss with BER level and over 20 dB when the delay is 1% and 10% from the coherence time, respectively. Thus, the optimal variable-rate scheme is very sensitive to the delay effect. Note, however, that the performance of this variable-rate system, even with delay of 10% from the coherence time, was still substantially better than with the constant-rate system. Also, it was stated in [7], for the spectral-efficiency-maximizing variable-power variable-rate -ary quadrature amplitude modulation ( -QAM) scheme with known channel-state values, that in order to remain at the (i.e., to cause only negligible performance BER target loss), the delay should not exceed 2% from the coherence time. Note that in [7], the receiver had accurate channel estimates and the transmitter used a delayed version of the true channel value as an estimate, i.e., there was only lag error, not the loss due to noise, involved. When we have , 2% from the coherence time is 10 symbols. Thus, our result with

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back delay, was considered over a slow Rayleigh fading channel. When using power control, a clear performance improvement was achieved when the smoother, instead of the predictor, is used in detection. The effect of a feedback delay of 10%–20% from the channel coherence time on the system performance was noticed to be small with reasonable BER levels. Substantial performance improvement for the pilot-symbol system with MMSE power control was gained by using, additionally, loworder diversity. ACKNOWLEDGMENT The authors would like to thank Prof. J. K. Cavers of Simon Fraser University, Canada, for indicating how to incorporate a smoother in detection and, due to that, to include the effects of a feedback delay and diversity in the analysis. Fig. 4. Performance of the pilot-symbol-assisted modulation system with and without MMSE power control and diversity (f = 0:001, L = 2 and L = 3).

and , meaning a delay of nine symbols, supports the rule presented in [7] (note that we have here slightly higher BER values than ). The bit-error performances of the pilot-symbol system with and without MMSE power control and with diversity are plotted in Fig. 4 as a function of the SNR . The diversity order is or three , and other parameters have two , , and . The performances values of the pilot-symbol system with and without MMSE power control ( , see Fig. 3), the coherent antipodal system using optimal [1] power control with and without diversity over a fading channel, and the coherent antipodal system over an AWGN channel, the performance upper limit, are shown as comparison curves. We can notice that even with diversity order of only two, considerable performance gain is achieved. , the performance In the case of diversity order of two loss (at BER ) of the pilot-symbol system with MMSE power control and diversity, and the coherent antipodal system with optimal power control and diversity, compared with the coherent antipodal system over an AWGN channel, is about 5.5 and 2.8 dB, respectively. The corresponding figures for diare 3.3 and 1.7 dB, respectively. versity order of three The performance of the pilot-symbol system can be improved further with the cost of complexity by increasing the diversity order, and by incorporating also powerful channel coding, e.g., turbo coding. IV. CONCLUSIONS Performance of the pilot-symbol-assisted modulation system, using the MMSE smoother in detection in conjuction with diversity and feedback MMSE power control, in subject to a feed-

REFERENCES [1] J. F. Hayes, “Adaptive feedback communications,” IEEE Trans. Commun., vol. COM-16, pp. 29–34, Feb. 1968. [2] R. Srinivasan, “Feedback communications over fading channels,” IEEE Trans. Commun., vol. COM-29, pp. 50–57, Jan. 1981. [3] I. Saarinen, A. Mämmelä, P. Järvensivu, and K. Ruotsalainen, “Power control in feedback communications over a fading channel,” IEEE Trans. Veh. Technol., vol. 50, pp. 1231–1239, Sept. 2001. [4] J. K. Cavers, “An analysis of pilot symbol assisted modulation for Rayleigh fading channels,” IEEE Trans. Veh. Technol., vol. 40, pp. 686–693, June 1991. , “Variable-rate transmission for Rayleigh fading channels,” IEEE [5] Trans. Commun., vol. COM-24, pp. 15–22, Jan. 1972. [6] D. L. Goeckel, “Adaptive coding for time-varying channels using outdated fading estimates,” IEEE Trans. Commun., vol. 47, pp. 844–855, June 1999. [7] A. J. Goldsmith and S.-G. Chua, “Variable-rate variable-power MQAM for fading channels,” IEEE Trans. Commun., vol. 45, pp. 1218–1230, Aug. 1997. [8] G. Caire, G. Taricco, and E. Biglieri, “Optimum power control over fading channels,” IEEE Trans. Inform. Theory, vol. 45, pp. 1468–1489, July 1999. [9] Q. Liu, S. Zhou, and G. B. Giannakis. Jointly adaptive modulation and packet retransmission over block fading channels with robustness to feedback latency. presented at Conf. Information Sciences, Systems. [CD-ROM] [10] P. Y. Kam, “Optimal detection of digital data over the nonselective Rayleigh fading channel with diversity reception,” IEEE Trans. Commun., vol. 39, pp. 214–219, Feb. 1991. [11] W. C. Jakes, Jr., Microwave Mobile Communications. New York: Wiley, 1974. [12] P. Y. Kam and C. H. Teh, “Reception of PSK signals over fading channels via quadrature amplitude estimation,” IEEE Trans. Commun., vol. COM-31, pp. 1024–1027, Aug. 1983. [13] A. Mämmelä and V.-P. Kaasila, “Smoothing and interpolation in a pilot symbol-assisted diversity system,” Int. J. Wireless Inform. Networks, vol. 4, no. 3, pp. 205–214, 1997. [14] J. K. Cavers, “Multiuser transmitter diversity through adaptive downlink beamforming,” in Proc. Wireless Communications, Networking Conf., New Orleans, LA, Sept. 21–24, 1999, pp. 251–255. [15] J. G. Proakis, “Probabilities of error for adaptive reception of M -phase signals,” IEEE Trans. Commun. Technol., vol. COM-16, pp. 71–81, Feb. 1968. , Digital Communications, 2nd ed. New York: McGraw-Hill, [16] 1989.

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Large Set of CI Spreading Codes for High-Capacity MC-CDMA Balasubramaniam Natarajan, Member, IEEE, Zhiqiang Wu, Member, IEEE, Carl R. Nassar, Senior Member, IEEE, and Steve Shattil

Abstract—In this letter, a large set of spreading codes that doubles capacity in multicarrier code-division multiple-access (MC-CDMA) systems without any cost in bandwidth and with negligible cost in performance is introduced. This large set is comprised of: 1) complex spreading codes instead of conventional real-valued spreading codes and 2) two sets, each made up of orthogonal complex spreading codes, with minimum cross correlation between sets. Simulations performed over Rayleigh fading channels demonstrate 100% gains in terms of MC-CDMA capacity with negligible loss in performance. Index Terms—Carrier interferometry (CI), complex spreading sequences, multicarrier code-division multiple access (MCCDMA), pseudo-orthogonality.

I. INTRODUCTION ULTICARRIER code-division multiple-access (MC-CDMA) [1] has emerged as a powerful alternative to conventional direct-sequence CDMA (DS-CDMA) [2]. In MC-CDMA, each user’s data bit is transmitted simultaneously over narrowband subcarriers, with each subcarrier typically encoded with a or (in the case of binary spreading sequences). Multiple users are assigned unique spreading codes to guarantee their separability at the carriers, orthogonal users receiver. Specifically, with can be supported via Hadamard–Walsh codes. If more than users are to be supported, pseudo-orthogonal codes must be employed during system design. This results in: 1) degradation in bit-error rate (BER) performance even when less than users are present and 2) high multiuser interference due to cross correlation between pseudo-orthogonal codes. In our earlier work [3], [4], we overcame many of the above limitations of MC-CDMA by introducing carrier interferometry (CI) codes. Referred to as CI/MC-CDMA, the usual binary spreading codes are replaced by complex spreading codes. Complex spreading codes have been proposed for DS-CDMA in [5] and [6]. Recently, a complex family of sequences called the S(2) family has been adopted in the UTRA/FDD standard as scrambling codes to improve DS-CDMA system performance.

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Paper approved by G. E. Corazza, the Editor for Spread Spectrum of the IEEE Communications Society. Manuscript received January 18, 2002; revised October 6, 2003, and June 24, 2004. This work was supported in part by the National Science Foundation under Grant ECS 9988665 “Ultra-Wideband Wireless Communication from Emerging Multiple Access Technology.” B. Natarajan is with the Department of Electrical and Computer Engineering, Kansas State University, Manhattan, KS 66506-5204 USA (e-mail: [email protected]). Z. Wu is with the Department of Electrical and Computer Engineering, West Virginia Tech University, Montgomery, WV 25136 USA (e-mail: [email protected]). C. R. Nassar was with the Department of Electrical and Computer Engineering, Colorado State University, Fort Collins, CO 80523-1373 USA. S. Shattil is with CIAN Systems, Boulder, CO 80303 USA. Digital Object Identifier 10.1109/TCOMM.2004.836564

In [7], multiple orthogonal quadriphase code sets were proposed to increase the CDMA capacity. However, the use of complex codes in MC-CDMA has not been explored completely, and the ones proposed in [8] have strict length restrictions and/or are capacity-limited. In CI/MC-CDMA, user ’s CI code corresponds to the lin. We early increasing phase offsets demonstrated [3], [4] that, given carriers, CI/MC-CDMA supports orthogonal users by appropriate choice of s, and, if capacity is to be increased further, an additional pseudo-orthogonal users can be added. While our research demonstrated that CI/MC-CDMA: 1) offers orthogonal performance below users and 2) easily outperforms MC-CDMA for more than users, performance still degrades severely as the number of [4]. users increases beyond In this letter, we introduce a large set of spreading codes corresponding to two sets of orthogonal CI codes. Each set is carefully selected to ensure minimum cross correlation between the two sets. When the number of users in the system is less than , one orthogonal set of spreading codes is used. When the number of users in the system exceeds , the second orthogonal set of spreading codes is introduced. Performance of users, indicating the system degrades negligibly from to that these two sets of orthogonal codes are particularly well suited for use in high-capacity MC-CDMA. We demonstrate that CI/MC-CDMA employing the large set of CI codes significantly outperform other large code sets such as extended Gold codes as well as the quadriphase sequences introduced in [7]. Section II briefly reviews the CI/MC-CDMA transmitter structure and introduces the large set of CI spreading codes. Section III presents receiver structure. Section IV provides the channel model and performance results; and the conclusions follow in Section V. II. LARGE SET OF CI CODES AND TRANSMITTER STRUCTURE In CI/MC-CDMA, user ’s transmitted signal is much like that of traditional MC-CDMA and can be represented as (1) refers to user ’s BPSK modulated bit, refers to user ’s spreading code, is a rectangular pulse of height 1 for the bit duration , and is the carrier frequency. Now, in CI/MC-CDMA, as in traditional MC-CDMA, the spreading waveform has the form where

(2)

0090-6778/04$20.00 © 2004 IEEE

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where to ensure orthogonality between subcaror , whereas in CI/MC-CDMA riers. In MC-CDMA, . That is, in MC-CDMA, the spreading sequence can or values, whereas in be treated as a sequence made up of CI/MC-CDMA the spreading sequence applied to the carriers corresponds to the complex set . for user ’s spreading waveform A careful selection of and for user ’s spreading waveform leads to orthogonality in time between users’ codes. Specifically, for a and , the real part of the CI spreading waveforms’ given correlation corresponds to (3)

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Of course, between the orthogonal set of codes constructed [represented in (6)] and the orthogonal set of with codes constructed with arbitrary [represented in (7)], there exists a nonzero cross correlation. We seek to simultaneously support both sets of orthogonal users. To do this in an optimal (characterizing the fashion, we now determine the value of orthogonal codes) that minimizes cross correlatwo sets of tion between the sets—we call the resulting codes the large set of CI codes. Let refer to the cross correlation between the th user in orthogonal code group 1 (constructed with ) and the th user in orthogonal code group 2 (constructed with ). Also, let (8)

(4) . There exist equally where . These spaced zeros at zeros indicate that a CI/MC-CDMA system can simultaneously support orthogonal users by selecting

represent the root mean square (rms) cross correlation that exists between users in groups 1 and 2. We seek to find the two sets of ) minimizing . Now, it is easily shown codes (i.e., the that

(5) These ’s correspond to the small set of CI codes. With all orthogonal users on the system, and assuming synchronous transmission, the total transmitted signal considering all users is

(9) It is also easy to show that (10)

(6) . where Now, if we introduce a fixed phase offset to all users’ by for all spreading sequences, i.e., replace values, the set of users remains orthogonal to one another. That is, the cross correlation between the spreading codes remains zero, as is evident from (4), where depends only on the differ. Hence, as an alternative ence to supporting orthogonal users with spreading sequences of , we can the form support orthogonal users with the spreading sequences of . In this the new form case, with orthogonal users in the system, the total transmit signal becomes

(7)

That is, the total cross correlation between the th user in orthogonal group 2 and all users in group 1 is identical to the cross th user in orthogonal group 2 and all correlation between the users in group 1. Using (10), we rewrite (8) as (11) and, using (9), this becomes

(12) minimizes this rms As illustrated in Appendix A, correlation of (12). orthogonal users using CI Hence, if we have one set of users, codes, and we want to increase system capacity to we can introduce a second set of orthogonal users also using CI codes. To best do that, in a minimum interference sense, introduce a second set of CI with codes phase-offset by with respect to the first set of CI codes. That is, for users on the system, numbered 0 to , the large set of CI

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codes is used where the th spreading sequence has the general and form

(13) The total transmitted signal, with a full

users, is then

where

A hard decision device is then employed to create a final decision . IV. PERFORMANCE RESULTS

(14) is selected as shown in (13). where It is important to note that (13) provides a large set of CI codes that is optimal for downlink transmission (since the zero-lag cross correlation is the metric used to determine the extended CI code set). Optimal complex spreading code design for the uplink is significantly more complex as cross correlation for all possible lags needs to be considered. However, by using evolutionary algorithms such as genetic algorithms, the authors have shown that it is possible to search the phase space (both continuous as well as discretized) to generate complex spreading code sets that are optimized for the uplink [14]. III. RECEIVER STRUCTURE At the receiver side, assuming a frequency-selective Rayleigh fading channel, the received signal is characterized by (as is typical of MC-CDMA)

(15) where is the number of active users in the system, is the is the phase offset of the th subgain of the th subcarrier, represents additive white carrier due to the channel, and Gaussian noise (AWGN). Downlink (or synchronous uplink) communication and exact phase synchronization is assumed. As is typical in MC-CDMA receivers for user , the received signal is first projected onto the orthonormal basis of the transmitted signal (i.e., decomposed into its carrier components), then despread, outputting where (16)

Here, index represents the carrier number and is a Gaussian . Minimum random variable with mean 0 and variance mean-square error combining is employed to combine the ’s, as this has been shown to demonstrate the performances close to that of maximum-likelihood (ML) detection. That is, in the case of MC-CDMA with the large set of CI cpreading codes, a via final decision statistic is created from (17)

A. Channel Model The multipath fading channel models used to assess the performance of the high-capacity CI/MC-CDMA system are the Hilly Terrain (HT) channel and Typical Urban (TU) channel taken from the COST-207 GSM standard [9]. This channel model is defined as a transversal filter with time-varying coefficients whose average power is determined by the multipath power delay profile (PDP) given in [9]. The frequency-domain characterization of the channels can be derived from the multipath power delay profile by using the approximate relationship in [10]. This leads to the following coherence bandwidth result: kHz; for TU, kHz. for HT, Assuming a typical system bandwidth (one consistent with the usual GSM data rate) and a typical number of carriers , this value satisfies total bandwidth

(18)

This indicates that the channels are frequency selective over the entire bandwidth, but not over each carrier [11]. Specifically, each carrier undergoes a flat fade, with the correlation between the th subcarrier fade and the th subcarrier fade characterized by [12] (19) where indicates the frequency separation between the th and the th subcarriers. The generation of fades with correlation has been discussed in [13]. B. Results Simulations are performed assuming carriers and the channel model of Section IV-A. Benchmark results are generated using the following systems: 1) a traditional MC-CDMA Hadamard–Walsh codes and 2) a system receiver using traditional MC-CDMA system using pseudo-orthogonal codes where the first 33 users use one set of Gold codes and next 32 users use a second set of Gold codes (for a total capacity of users). In order to compare CI/MC-CDMA performance with MC-CDMA employing complex spreading sequences, we also consider an MC-CDMA system using two orthogonal quadriphase code sets proposed in [7]. We employ the primitive basic [7] and, by irreducible polynomial adding 1 and onto the set of eight cyclically distinct sequences from the family [7], two orthogonal code sets are generated. Furthermore, by applying the Hadamard transform on these two code sets twice, two orthogonal code sets with length 32 were generated.

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Fig. 1.

Simulation result 1: HT channel, SNR

=10 dB.

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benefits when compared to the use of traditional MC-CDMA systems with pseudo-orthogonal codes for increased capacity. When compared to HW codes: without any loss of performance, capacity increases of 50% can be supported (Figs. 1 and 2), and there are negligible performance losses even when capacity increases by 100%. MC-CDMA systems with the extended Gold code set and the quadriphase code sets exhibit identical performance. Both of these systems show a sharp degradation in performance when the capacity exceeds the number supported by the corresponding first code set. That is, in the case of Gold codes and quadriphase codes, the performance degrades significantly when the capacity goes above 33 users and 32 users, respectively. This jump in BER is easily explained by observing that the second set of gold codes and quadriphase codes exhibit significant correlation with the first set. The jump in BER is less dramatic when the number of users exceeds 48, since the amount (variance) of multiple-access interference due to loss in orthogonality (between code sets 1 and 2) is close to its maximum. With the large CI code set, the BER degrades gracefully, as the second set is optimized in a way that the correlation, and therefore, the interference, are kept to a minimum relative to the first set. V. CONCLUSION In this letter, we introduced two groups of orthogonal complex spreading codes (CI codes) with minimum cross correlation between them. By employing this large set of CI codes, we demonstrated 100% increase in system capacity with no extra expense in bandwidth. Increases in capacity are achieved with only negligible performance loss relative to traditional MC-CDMA systems, with HW codes. APPENDIX A To minimize the rms correlation between the two orthogonal sets of CI codes, we impose

Fig. 2.

Simulation result 2: TU channel, SNR

=15 dB.

Fig. 1 presents bit-error probability versus the number of users for SNR 10 dB in the HT channel, and Fig. 2 presents these results in the TU channel at SNR 15 dB. The curve marked with asterisks represents the performance of traditional MC-CDMA with Hadamard–Walsh codes; the curves marked with circles represents the performance of CI/MC-CDMA; the curves marked with dots represents the performance of traditional MC-CDMA with Gold codes (supporting up to 65 users), and the curves marked with diamonds represents the performance of MC-CDMA with two orthogonal sets of quadriphase sequences (supporting up to 64 users). In CI/MC-CDMA and MC-CDMA with quadriphase sequences, the first 32 users are supported using the first orthogonal set of spreading codes, and the next 32 users are supported using the second orthogonal set of spreading codes. These curves demonstrate that the use of this novel spreading scheme to increase capacity offers tremendous performance

(20) Now

(21) where

(22)

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Now

are maxima and which are minima, we determine which calculate the second-order partial derivative at and determine

(24) Hence,

corresponds to maxima and provides minima. Selecting , we choose as our minima. REFERENCES

(23) Hence, when and, therefore, from (13),

, we have . Seeking to

[1] S. Hara and R. Prasad, “Overview of multicarrier CDMA,” IEEE Commun. Mag., pp. 126–131, Dec. 1997. [2] A. J. Viterbi, CDMA: Principles of Spread Spectrum Communication. Reading, MA: Addison-Wesley, 1995. [3] C. R. Nassar, B. Natarajan, and S. Shattil, “Introduction of carrier interference to spread spectrum multiple access,” in Proc. IEEE Emerging Technologies Symp., sec. II, Richardson, TX, Apr. 1999, pp. 11–15. [4] B. Natarajan, C. R. Nassar, S. Shattil, Z. Wu, and M. Michelini, “Highperformance MC-CDMA via carrier interferometry codes,” IEEE Trans. Veh. Technol., vol. 50, pp. 1344–1353, Nov. 2001. [5] I. Opperman and B. S. Vucetic, “Complex spreading sequences with a wide range of correlation properties,” IEEE Trans. Commun., vol. 45, pp. 365–375, Mar. 1997. [6] G. W. Wornell, “Spread-signature CDMA: Efficient multi-user communication in the presence of fading,” IEEE Trans. Inform. Theory, vol. 41, pp. 1418–1438, Sept. 1995. [7] B. M. Popovic, N. Suehiro, and P. Z. Fan, “Orthogonal sets of quadriphase sequences with good correlation properties,” IEEE Trans. Inform. Theory, vol. 48, pp. 956–959, Apr. 2002. [8] B. M. Popovic, “Spreading sequences for multicarrier CDMA systems,” IEEE Trans. Commun., vol. 47, pp. 918–926, June 1999. [9] “Digital Land Mobile Radio Communications,” Commission of the European Community, Brussels, Belgium, Final Report of the COST-Project 207, 1989. COST-207. [10] T. S. Rappaport, Wireless Communications—Principles and Practice. Upper Saddle River, NJ: Prentice-Hall, 1996. [11] J. Proakis, Digital Communications. New York: McGraw-Hill, 1995. [12] W. Xu and L. B. Milstein, “Performance of multicarrier DS CDMA systems in the presence of correlated fading,” in Proc. IEEE 47th Vehicular Technology Conf., Phoenix, AZ, May 4–7, 1997, pp. 2050–2054. [13] B. Natarajan, C. R. Nassar, and V. Chandrasekhar, “Generation of correlated Rayleigh fading envelops for spread spectrum applications,” IEEE Commun. Lett., vol. 4, pp. 9–11, Jan. 2000. [14] E. Buehler, B. Natarajan, and S. Das, “Multiobjective genetic algorithm based complex spreading code sets with a wide range of correlation properties,” in Proc. IEEE 15th Int. Conf. Wireless Communications, vol. 2, Calgary, AB, Canada, 2003, pp. 548–552.

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On the Miller–Chang Lower Bound for NDA Carrier Phase Estimation Gonçalo N. Tavares, Luis M. Tavares, and Moisés S. Piedade

Abstract—In this letter, we derive a new Miller–Chang lower bound (MCB) for the variance of unbiased non-data-aided (NDA) samples of carrier phase estimates obtained from a block of a linearly modulated pulse-amplitude modulation or quadratureamplitude modulation information signal, transmitted through an additive white Gaussian noise channel. This bound is tighter than the corresponding Cramér–Rao lower bound (CRB) for data-aided or continuous-wave carrier phase estimation (CW-CRB), particularly when is small. For some given and sufficiently high signal-to-noise ratios, the MCB is tighter than the corresponding true NDA CRB. The main limitation of this new MCB is that its application is restricted to carrier phase estimators which are unbiased for all possible values of the received symbol sequence. Index Terms—Carrier phase estimation, Cramér–Rao lower bound (CRB), Miller–Chang lower bound (MCB), synchronization.

I. INTRODUCTION ARRIER phase synchronization is a fundamental task in many communication receivers. Practical synchronizer performance is usually assessed by comparing the variance of actual estimates with some adequate theoretical lower limit. Due mainly to its simplicity and accuracy at high signal-to-noise ratios (SNRs), the Cramér–Rao lower bound (CRB) is one of the most popular and widely used bounds. Although computation of the CRB is straightforward when the observation depends only on a single parameter, it becomes quite involved and difficult when other unwanted or nuisance parameters are present [1], [2]. The Miller–Chang lower bound (MCB) is a Cramér–Rao-like bound which results from a different way to treat and eliminate the dependence of the CRB on these parameters [3]. In this letter, we address the computation of the MCB for the estimation of the carrier phase, assuming that a linearly modulated information signal is transmitted over an additive white Gaussian noise (AWGN) channel. In Section II, the MCB is derived for general -ary constellations pulse-amplitude modulation and is then particularized for quadrature-amplitude modulation ( -PAM) and square -QAM). The numerical results presented in Section III ( assess the tightness of the new bound by comparing it with other commonly used CRBs. The simulated performance of the feedforward non-data-aided (NDA) power-law carrier phase

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Paper approved by R. Reggiannini, the Editor for Synchronization and Wireless Applications of the IEEE Communications Society. Manuscript received April 24, 2002; revised January 25, 2004, and May 5, 2004. G. N. Tavares and M. S. Piedade are with the Department of Electrical and Computer Engineering, Instituto Superior Técnico (IST) and with Instituto de Engenharia de Sistemas e Computadores (INESC), 1000-029 Lisbon, Portugal (e-mail: [email protected]; [email protected]). L. M. Tavares is with the Department of Engineering, Escola Superior de Tecnologia e Gestão (ESTIG), 9800-050 Beja, Portugal, and also with INESC, 1000-029 Lisbon, Portugal (e-mail: [email protected]). Digital Object Identifier 10.1109/TCOMM.2004.836592

estimator, operating from 4-PAM signals, is also compared with the new MCB. Finally, some concluding remarks are drawn in Section IV. II. MCB DERIVATION Assuming that the effect of all angular synchronization parameters other than the carrier phase has been properly corrected, the matched filter output, sampled at one sample per symbol with perfect symbol synchronization, is given by (1) where is the carrier phase (assumed to be fixed during the -symbol observation interval), and is a sequence of independent identically distributed (i.i.d.) equiprobable symbols from an arbitrary -ary constellation with zero-mean symbols and with ; we assume that does not contain the null symbol . The sequence consists of i.i.d. samples of a zero-mean complex Gaussian noise process with variance and with indepen. dent real and imaginary parts, each with variance The symbol SNR is where is the symbol energy and is the one-sided power spectral density of the noise. With this signal model, the joint probability density function (pdf) of the observation vector , conditioned is on and on the symbol sequence

(2) The CRB is a fundamental lower bound on the error variance of any unbiased estimate. If denotes an unbiased estimate of the CRB with [4] true carrier phase , then CRB

(3)

If the symbols are known [data-aided (DA) estimation], the pdf (2) may be used as in (3). In this case, we obtain a conditional CRB which depends on the symbol sequence; this bound is denoted as CRB . On the other hand, when the symbols are unknown (NDA estimation), is viewed as a nuisance vector and the required pdf in (3) is determined by averaging (2) with respect to the a priori distribution of , i.e., . Generally, this averaging operation is very difficult to perform analytically and is usually accomplished using numerical methods [2], [5]. The resulting bound will be referred to as true NDA CRB and denoted as CRB . True NDA CRBs for carrier-phase and frequency-offset estimation have been derived in [5] for BPSK and QPSK and in [6] for general symmetric QAM constellations. Miller and Chang [3]

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proposed a different general way to treat the nuisance parameters. For the problem at hand, it consists of first determining considering as a vector of known parameters CRB and then averaging this conditional bound with respect to . This eliminates the dependence on and produces the NDA lower bound MCB

CRB

CRB

(4) where denotes the set of all possible -symbol sequences with symbols from . We note that this bound may only be applied to estimates which are unbiased for each , i.e., (locally and every possible value of unbiased estimates [3], [7]). This contrasts with the broader estimation context in which the conventional CRB may be applied: estimates need only be unbiased over the ensemble , i.e., . Using the signal model (1) and the of CRB definition (3) with the pdf (2), it is easily shown that CRB (5) From (4), the MCB MCB

is then (6)

The following remarks are of interest. for all and both 1) With -PSK, the conditional bound in (5) and the MCB in (6) become CRB

2)

(7)

which is the true CRB for DA or CW carrier phase estimation. In view of Jensen’s inequality [1], [2], we have the following relationship:

CRB does not depend on and CRB tends to unity (from above) as [2], [5]. From these properties and the previous remark, we conclude that, for any given and for any constellation other than -PSK, there is a limit SNR, say , such that MCB CRB for , i.e., the MCB is tighter than the true CRB. in (6) requires computing the individual 4) The MCB contribution from each possible sequence; this is impractical for the usual values of and . It should be pointed out that the number of terms in (6) can be reduced by noting that different sequences may originate the same contribution. In fact, if the constellation is rotationally symmetric, the . number of sequences may be reduced to Also, if we observe that the number of -collections1 -ary which can be formed with symbols from an [9, p. 49], it follows alphabet is that the minimum number of different terms in (6) is ; nevertheless, it must be noted that this solution requires some way of generating the -collections and this will ultimately add to the computational complexity. For example, with square 16-QAM (a rotationally symmetric constellation, ) and using symbols in the estimation process, it is sequences necessary to consider in (6), a con(10-collections) to compute the MCB siderable effort. In the following, we present a different method to determine this bound for general constellations, which is computationally more efficient. First, we define the discrete random variable (r.v.) (8) which takes distinct positive values (the actual value of is immaterial in this context). The MCB in (6) may be written as MCB

3)

with equality iif is the same for every sequence in [8]. Since this will only be the case with -PSK, it follows from (6) and (7) that for all other constellations MCB CRB . Moreover, the ratio MCB CRB depends on but not on (this is a consequence of the initial assumption of known symbols in the MCB derivation). This means that, regardless of the SNR and for any given , the MCB is always tighter than the CRB . Estimating the carrier phase in the presence of unknown data symbols (nuisance parameters) is always more difficult than when symbols are known, so CRB CRB for all . A formal proof of this statement may be obtained using the results in [1, Appendix A]. Also, the ratio

where is defined as

(9) . The characteristic function (cf) of

(10) over the positive imaginary semi-axis yields Integrating (note that for all , )

(11)

N

N

1An -collection is any set of elements, not necessarily distinct, chosen by any order from some other set of elements.

M

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Because data symbols are considered i.i.d. and equiprobable, , the cf. of is the product of the individual cf of each r.v. i.e., (12) Using (11) and (12) and making the change of variable the bound (9) may finally be written as MCB

,

(13)

In order for the integral in (13) to converge, it is required that . Since , this condition is fulfilled making , which is always possible (recall that, by assumption, the symbol does not belong to the symbol alphabet). In Sections II-A and II-B, the general bound (13) is individeven and for square -QAM. ualized for -PAM with -PAM Signals With

A. For

-PAM

with

Even even,

the

symbols

are

Fig. 1. Ratio between the MCB() and the CRB() modulation formats.

for several

The bound (13) is now MCB

and ( -rotationally symmetric constellation). The bound in (13) becomes MCB

(14) When , the modulation is BPSK and it may easily be checked that (14) reduces to (7). In the case of 4-PAM, there are only two terms in the sum over in (14). Using the binomial series to expand the term raised to , the required integration may be expressed in closed form; the result is MCB even odd (15) simple which only requires computing a sum with terms. The reason for writing explicitly the dependence of both (14) and (15) on will become clear shortly. B. Square For square to consider the

-QAM Signals -QAM constellations, , and it suffices first quadrant symbols, which are

(16) For 16-QAM, one may use (15) to compute , which only requires a sum with terms. III. RESULTS CRB is plotted in Fig. 1 as a funcThe ratio MCB -QAM and 4-PAM. As can be seen, tion of , for square is higher than the CRB , especially when the MCB a small number of samples is used in the estimation process. , the MCB is about 16% For example, with 4-PAM and higher than the CRB. However, it should be noted that, for high values of , the improvement of the MCB over the CW-CRB is small (with all modulation formats in this figure, the MCB is ). As inless than 5% tighter than the CW-CRB for approaches the CRB from above. creases, the MCB This is expected because with increasing , regardless of the modulation format, and, as proved earlier, MCB CRB for all . Within the -QAM modulation class, estimation difficulty increases with the number of elements in the constellation. The plots indicate that, for a given , the MCB predicts that locally unbiased

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Fig. 2. Ratio between the true NDA CRB() and the MCB() for several modulation formats with N = 8.

carrier phase estimation is more difficult with -PAM than with -QAM regardless of the SNR; however, this is only true in the range of SNR for which the MCB is tighter than the true NDA CRB, for all modulations being considered. In the sequel, we compare the MCB with the true NDA CRB for the estimation of with the signal model (1) using an arbitrary (real or complex) constellation. Following the derivation presented in [5] for BPSK and QPSK, one finds that the true is NDA CRB CRB

CRB

(17)

given by (7) and given by (18), shown with CRB at the bottom of the page, where is a zero-mean complex Gaussian random variable with independent real and imaginary components each with variance (19) The bound (17) is also found in [6] for several QAM constellations. The ratio between the CRB and the MCB is plotted in Fig. 2 as a function of the SNR for 4-PAM, square symbols. 16-QAM, and square 64-QAM with increases within the -QAM class, carrier phase esAs the timation is progressively more difficult. As bounds for 16-QAM and 64-QAM converge; this behavior is due to the fact that, in strong noise conditions, the CRB is determined by the shape of the constellation and not by its cardinality

Fig. 3. Limit SNR, (E =N ) number of symbols.

, for 4-PAM and 16-QAM as a function of the

[6]. For sufficiently high SNR, the MCB is slightly tighter than the true CRB. However, for the range of low-to-medium SNR, is a tighter bound. the MCB is loose and the CRB As stated before in Section II, for any given number , such of observations, there is a limit value that the MCB is tighter than the true NDA CRB whenever (this is visible in Fig. 2). This limit is plotted in Fig. 3 for square 16-QAM and 4-PAM signals. As expected, increases with , meaning that the is more useful when estimation is carried out from a MCB small number of observations. Finally, to compare the performance of actual phase estimates with the new MCB , we selected the NDA feedforward power-law (PL) estimator from [10] which is the extension for -rotationally symmetric signal constellations of the well-known Viterbi and Viterbi estimator, originally proposed in [11] for operation with -PSK signals. With the PL estimator, NDA feedforward carrier phase estimates are computed as (20) A note about the local unbiasedness property is required. To the best of the authors’ knowledge, there is no systematic analytical procedure to determine whether or not an arbitrary estimator is locally unbiased, and this may limit the practical application of the MCB. Using numerical simulation, we have

(18)

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istic nature. Enlightening discussions on this issue may be found in [12] and [13]. IV. CONCLUSION In this letter, we have derived a new NDA MCB on the variance of unbiased carrier phase estimates. The bound is tighter than the conventional CW-CRB for all values of and number of observation samples , especially for small values and any modulation format other than of . For any given -PSK, it was shown that there is a limit SNR, , such that the MCB is tighter than the corresponding true NDA CRB whenever . However, for low-tomedium SNR values, the true CRB is a tighter bound. Another practical limitation of the new MCB is that it can only be applied to the class of estimators which are known to be unbiased irrespective of the actual transmitted symbol sequence (locally unbiased estimators). REFERENCES Fig. 4. Comparison of the carrier phase PL estimator performance with the corresponding MCB() and other CRBs. Operation is from 4-PAM signal packets with = 4 symbols.

N

found that the PL estimator (20) provides locally unbiased estimates with -PSK ( , ), and -PAM , ) signals; this will happen even for ( provided the SNR is sufficiently high. However, if small the operation is from square -QAM signals, then estimates given by (20) are not locally unbiased (for small ), not ; with this modulation format, the even when minimum distance estimator (MDE) proposed in [6] provides if the locally unbiased estimates (again even with small SNR is appropriately high). The simulated variance of the carrier phase estimates (20) with 4-PAM signals is plotted in Fig. 4, together with the relsimulation consisted of processing evant bounds. Each random data sequences, rotated by a random uniform phase , kept fixed over the duration of the offset symbols). It is seen that the data sequence (equal to greater than MCB is tighter than the true NDA CRB for 1.4 dB (cf. Fig. 3 with ). As increases, approaches the new MCB (rather than the CW or the true NDA CRB) and attains it for 15 dB. As decreases, approaches , which is the variance of uniformly distributed carrier phase estimates in the sup. In this circumstance, all bounds diverge. This port behavior is a consequence of these bounds being local, i.e., it is assumed in their derivation that the carrier phase is of determin-

[1] A. N. D’Andrea, U. Mengali, and R. Reggiannini, “The modified Cramer-Rao bound and its application to synchronization problems,” IEEE Trans. Commun., vol. 42, pp. 1391–1399, Feb./Mar./Apr. 1994. [2] M. Moeneclaey, “On the true and the modified Cramer-Rao bounds for the estimation of a scalar parameter in the presence of nuisance parameters,” IEEE Trans. Commun., vol. 46, pp. 1536–1544, Nov. 1998. [3] R. W. Miller and C. B. Chang, “A modified Cramér-Rao bound and its applications,” IEEE Trans. Inform. Theory, vol. IT-24, pp. 398–400, May 1978. [4] H. L. Van Trees, Detection, Estimation, and Modulation Theory: PART I- Detection, Estimation, and Linear Modulation Theory. New York: Wiley, 1968. [5] W. G. Cowley, “Phase and frequency estimation for PSK packets: Bounds and algorithms,” IEEE Trans. Commun., vol. 44, pp. 26–28, Jan. 1996. [6] F. Rice, B. Cowley, B. Moran, and M. Rice, “Cramér–Rao lower bounds for QAM phase and frequency estimation,” IEEE Trans. Commun., vol. 49, pp. 1582–1591, Sept. 2001. [7] F. Gini and R. Reggiannini, “On the use of Cramér–Rao-like bounds in the presence of random nuissance parameters,” IEEE Trans. Commun., vol. 48, pp. 2120–2126, Dec. 2000. [8] E. W. Weisstein, The CRC Concise Encyclopedia of Mathematics. London, U.K./Boca Raton, FL: Chapman & Hall/CRC, 1999. [9] V. K. Balakrishnan, Introductory Discrete Mathematics. Englewood Cliffs, NJ: Prentice-Hall, 1991. [10] M. Moeneclaey and G. de Jonghe, “ML-oriented NDA carrier synchronization for general rotationally symmetric signal constellations,” IEEE Trans. Commun., vol. 42, pp. 2531–2533, Aug. 1994. [11] A. J. Viterbi and A. M. Viterbi, “Nonlinear estimation of PSK-modulated carrier phase with application to burst digital transmission,” IEEE Trans. Inform. Theory, vol. IT-29, pp. 543–551, July 1983. [12] R. Reggiannini, “A fundamental lower bound to the performance of phase estimators over Rician-fading channels,” IEEE Trans. Commun., vol. 45, pp. 775–778, July 1997. [13] K. L. Bell, Y. Steinberg, Y. Epharin, and H. L. Van Trees, “Extended Ziv-Zakai lower bound for vector parameter estimation,” IEEE Trans. Inform. Theory, vol. 43, pp. 624–637, Mar. 1997.

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Transactions Papers Performance Analysis and Design Criteria for Finite-Alphabet Source-Channel Codes Ahmadreza Hedayat, Student Member, IEEE, and Aria Nosratinia, Senior Member, IEEE

Abstract—Efficient compression of finite-alphabet sources requires variable-length codes (VLCs). However, in the presence of noisy channels, error propagation in the decoding of VLCs severely degrades performance. To address this problem, redundant entropy codes and iterative source-channel decoding have been suggested, but to date, neither performance bounds nor design criteria for the composite system have been available. We calculate performance bounds for the source-channel system by generalizing techniques originally developed for serial concatenated convolutional codes. Using this analysis, we demonstrate the role of a recursive structure for the inner code and the distance properties of the outer code. We use density evolution to study the convergence of our decoders. Finally, we pose the question: Under a fixed rate and complexity constraint, when should we use source-channel decoding (as opposed to separable decoding)? We offer answers in several specific cases. For our analysis and design rules, we use union bounds that are technically valid only above the cutoff rate, but interestingly, the codes designed with union-bound criteria perform well even in low signal-to-noise ratio regions, as shown by our simulations as well as previous works on concatenated codes. Index Terms—Concatenated coding, iterative decoding, joint source-channel coding, variable-length codes (VLCs).

I. INTRODUCTION N THIS PAPER, we consider the problem of the transmission of discrete, finite-alphabet sources over a noisy channel. Since efficient entropy codes are often variable-length codes (VLCs), a conventional channel decoding followed by a typical symbol-by-symbol entropy decoding will result in error propagation, thus a single uncorrected channel error may result in a long sequence of data errors. This difficulty has led to a search for error-resilient entropy codes. A prominent example is the reversible (R)VLC [1], used in the video-coding standard H.263 and its descendants. RVLC consists of a class of codes that have

I

Paper approved by F. Alajaji, the Editor for Source and Source-Channel Coding of the IEEE Communications Society. Manuscript received August 2, 2003; revised February 29, 2004, and June 4, 2004. This work was supported in part by the National Science Foundation under Grant CCR-9985171. The work of A. Hedayat was also supported in part by the Texas Telecommunications Engineering Consortium (TxTEC). This paper was presented in part at the Allerton Conference on Communications, Control, and Computing, Monticello, IL, 2002 and in part at the IEEE International Conference on Communications, Anchorage, AK, 2003. The authors are with the Multimedia Communications Laboratory, University of Texas at Dallas, Richardson, TX 75083-0688 USA (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TCOMM.2004.836562

Fig. 1.

System block diagram.

not only a prefix property, but also a suffix property, thus they can be decoded from both directions. A more comprehensive attempt at introducing error resilience into variable-length entropy codes was made by Buttigieg [2], [3], who studied the general class of entropy codes with errorcorrection ability, and introduced various sequence-decoding algorithms. Subbalakshmi and Vaisey also provided a trellis for describing VLCs and introduced an optimal maximum a posteriori (MAP) probability decoder for variable-length encoded sources over a binary symmetric channel (BSC) [4], [5]. Error-resilient codes mentioned above are not strong enough to handle the error rates generated by most communication channels, thus a separate layer of channel coding is usually necessary (see Fig. 1). In such a concatenated system, iterative decoding methods, originally introduced for channel codes [6], [7], provide another opportunity for improved source-channel coding. To the best of our knowledge, the first attempt at iterative decoding of source and channel codes is due to Bauer and Hagenauer [8], [9], who proposed an iterative (turbo) decoding scheme between a channel code and the residual redundancy of an RVLC.1 They reported a significant coding gain compared with a system with an equivalent transmission rate. Guyader et al. [11] proposed various algorithms in the framework of Bayesian networks for the iterative decoding of the chain in Fig. 1 with a Markov nonbinary source. Lakovic and Villasenor [12] studied the performance of VLCs followed by turbo codes, and suggested combining the trellises of the VLC and the upper convolutional code of the turbo code for more coding gain. Despite many interesting and useful results, including those mentioned above, to date, neither a comprehensive analysis nor design criteria have been available for iteratively decoded source-channel coding systems. In this paper, we analyze this concatenated system, study design criteria for the constituent codes, and present comparisons of various tradeoffs in the design of such codes, supported by extensive simulations. 1For an example of iterative source-channel decoding of fixed-length codes, see [10].

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To start, we generalize the source-channel structure by assigning the error-correcting entropy codes of Buttigieg as the outer code of the source-channel concatenated system. We study, via simulations, the performance of this generalized system. The central contribution of this paper, however, is an analysis of the performance of the concatenated source-channel codes. We employ the techniques originally developed for serially concatenated convolutional codes (SCCCs) [7], with the critical difference that our outer codes (and hence, our overall codes) are nonlinear, thus the techniques of [7] need to be appropriately extended. Our analysis is general with respect to the choice of outer VLCs and inner channel codes. The outer code can be an RVLC similar to that in [8], [9], or [12], it can be a VLC with higher redundancy, similar to the codes introduced in [2], or a VLC with minimum redundancy, such as Huffman codes. The analysis clarifies the roles of the inner and outer codes in the overall performance, allowing us to make statements about the free distance of the outer code and the desirability of a recursive structure for the inner code. To the best of our knowledge, these or similar results have not been previously reported in the literature on source-channel coding. Our analysis and design rules are based on union bounds, which are useful in the medium-to-high signal-to-noise ratios (SNRs), in particular above the cutoff rate. However, it has been observed and reported [7] that concatenated codes designed with union-bound criteria perform well even in low-SNR regions. We have observed the same phenomenon for our codes. The analysis and simulations presented in this paper enable us to make several observations with practical implications. For example, the method of Bauer and Hagenauer [9] achieved significant gain compared with systems with a similar rate. We found, however, that it is possible to improve on the scheme of [9], while maintaining the same overall rate and complexity, by using separable source decoding and an iteratively decoded SCCC. Thus, in this case, investing computational resources into the channel decoder alone gives better returns in terms of system performance. This suggests that whenever the entropy code has small free distance (such as the RVLC used in [9]), one may be better off spending the computational budget mostly on the inner code and not on iterative decoding between source and channel codes. We also found that even with outer codes having larger free distance, iterative source-channel decoding may yield only a slight advantage compared with a separable baseline system of equivalent rate and complexity. These findings are expressed in more detail in the following. II. VLCS WITH ERROR-CORRECTING CAPABILITY Buttigieg [2] introduced a class of entropy codes with error-correction ability under the name of variable-length error-correcting codes (VLECCs). These codes have entropy coding property, in the sense that low-probability symbols have longer codewords. On the other hand, these codes also have error-correction capability, arising from a careful assignment of codewords to symbols such that a minimum Hamming distance is maintained between all codeword pairs. Obviously, maintaining a minimum distance introduces redundancy into the code, such that its average length will be bounded away from the entropy of the source.

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Consider a -ary source with elements denoted by and a . The minimum and VLC whose codewords are denoted by maximum length of ’s are denoted by and , re. To perform maxspectively, and the average length by imum-likelihood decoding, we need to consider a sequence of codewords. We now define such composite codewords. Asis ensume the source sequence tropy-encoded to the bit sequence

Because the codewords are variable length, the length of the output sequence , denoted by , is variable. This leads to difficulty in analysis, therefore, we partition the overall code into subcodes such that each partition consists only of codewords of length . The free distance of , denoted , is defined as the minimum value of the minimum Hamming distances of the individual binary codes . Note that are, in general, nonlinear codes. Buttigieg [2] calculates the upper bound for the error-event probability of a VLC in the same manner as convolutional codes, by introducing the average number of converging paths on an appropriate trellis at a given Hamming distance. Unfortunately, this approach is not appropriate for our purposes since, unlike Buttigieg, we intend to use the VLCs in concatenation with another code. Instead, we use the codeword-enumeration technique [7]. Considering that , the length of the bit-sequence , is a random variable that takes value in , the upper bounds for the codeword-error probability and symbol-error probability are

(1)

(2) is the pairwise error probability (PEP), which has where value in the additive white Gaussian noise (AWGN) channel, and are multiplicities. Specifically, is the number of codeword pairs in with Hamming distance . Eventually, we are interested in the distance between symbol strings corresponding to codeword pairs with Hamming distance . The average contribution of two codewords of Hamming distance to the Levenshtein distance is denoted . Note that and are normalized by the size of the respective codebooks . The exchange of summations in (1) and (2) allow us to think of the terms inside parentheses as equivalent and for the entire code, without the need to consider individual code partitions separately. It is, in fact, more convenient to calculate instead of ; see, for example, [2] and [8].

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We use a uniform interleaver, which maps a codeword of permutations with equal probaweight into all distinct bility.2 The outer VLC has free distance , therefore

(4)

Fig. 2. Tree and bit-level trellis for C = f00; 11; 10; 010; 011g.

The computation of is based on the Levenshtein distance between the two symbol sequences, . The Levenshtein distance is defined as the minimum number of insertions, deletions, or substitutions to transform one symbol sequence into another [2], [13]. The Levenshtein distance is widely used as an error measure for VLCs, justified partly in light of the self-synchronization property of VLCs [2], and partly because of the lack of other more meaningful and useful distance measures for VLCs.

where and are the associated multiplicities for , the length- subcode of the outer code, and are the multiplicities of the inner code. Summing the contributions over all of the possible ’s gives the associated coefficients and . This derivation was facilitated by two facts. First, for two codewords and , we have , where is the interleaving function. Second, since the inner code is a linear code, we may speak equivalently of codeword weights or codeword pair distances. In particular, for the inner , will code, two sequences with distance result in two codewords with coded distance . It is due to this “invariance” of the interleaver and the linearity of the inner code that our analysis remains tractable. We can now calculate the FER and symbol-error rate (SER) of the concatenated scheme as follows:

III. TRELLIS REPRESENTATION OF VLCS In this paper, we employ the bit-level trellis proposed by Balakirsky [14] and later used by Murad and Fuja [15], as well as Bauer and Hagenauer [8]. This trellis is obtained simply by assigning the states of the trellis to the nodes of the VLC tree. The root node and all terminal nodes are assumed to represent the same state, since they all show the start of a new sequence of bits. Other nodes, the so-called internal nodes, are assigned one by one to the other states of the trellis. The number of states of the trellis is equal to the number of internal nodes of the tree plus one. As an example, Fig. 2 shows the trellis corresponding . to a Huffman code

IV. SERIAL CONCATENATION OF VLC AND CHANNEL CODES In the following, functions and variables related to the inner code will be distinguished by the superscript , and those related to the outer code with superscript . Assume that the inner code is a convolutional code with rate . The input–output weight-enumerating function (IOWEF) of the equivalent block code of the inner convolutional code, with the input length , is [7]

(3)

where represents the number of codewords with weight generated by information words of weight , and and are dummy variables.

(5)

(6) where is the free distance of the concatenated code. Similar to (1) and (2), the above results may be presented in terms of equivalent and . This alternative form is omitted here for the sake of brevity. One may also obtain bounds similar to (5) and (6) for the average interleaver size . We note that the above union bounds can be used with different choices of inner code, for example, a convolutional code, as in [8], or a turbo code, as in [12]. The asymptotic performance of the bounds above can be studied by looking at the behavior of coefficients and . We mainly present the analysis for ; similar developments are possible for . Following [16], the multiplicities can be modeled as a polynomial function of interleaver size, i.e., 2This was first proposed by Benedetto and Montorsi, and then used in [7] to analyze concatenated codes.

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is the multiplicity of the pair of codewords at diswhere tance consisting of a concatenation of exactly simple error events, with no trivial error event in between. A trivial error event is a section of the bit trellis of the two codewords that are identical, and furthermore, it starts and ends in the root state. Equation (9) illustrates the expansion of simple codepaths with error events, to compound codepaths that include trivial error events. One may obtain the coefficients of the concatenated code by substituting (8) and (9) into (4) to yield

Fig. 3. Top: Concatenation of n error events with no gap in between, used in calculating the inner code multiplicity. Bottom: A pair of codewords showing concatenation of m error events with no trivial error event in between, used for calculating the VLC multiplicity.

. We are particularly interested in the exponent of the highest order term in this polynomial, which is indicative of the asymptotic improvement of the multiplicity (and hence, code performance) with increasing interleaver length. To eliminate the dependency on , define (7) Thus, is the dominant coefficient of , where we here emphasize the dependence on , the Hamming distance. The dominant multiplicity exponent is referred to as interleaver gain in the literature [7]. The performance of the code, except in a very high SNR regime, is dependent on the multiplicities of the code, and therefore depends on . Whenever , the dominant multiplicity gets smaller with increasing interleaver size, therefore, we will be motivated to design codes with . We define as the Hamming distance of codeword pairs having the dominant multiplicity (the maximizer in the expression above). We now wish to calculate and . The inner code multiplicity can be expressed as (8) where are the multiplicities for codewords with input–output weights having exactly consecutive error events with no gap in between, as shown in Fig. 3 (top trellis). Equation (8) derives the overall multiplicities by inserting zero runs before some of the error events, such that the overall number of trellis sections is [16]. For the outer VLC, a similar expression can be derived with certain modifications. Because of the nonlinearity of VLCs, weight enumeration has to be carried out through all pairs of codewords, as is shown in Fig. 3 (bottom trellis). All error events for the VLC must initiate and terminate in a “root state,” which is the state where the bit sequence for a source symbol begins or terminates (see Fig. 2). In Fig. 3, we show the root state as the top node. A pair of codewords of a VLC are illustrated in Fig. 3, using the trellis of Fig. 2. Following a similar argument as in [16], we obtain (9)

(10) where the approximation of (10) into (7), we obtain

is used. Substituting

(11) where the bound for the outer code, , reflects the possibility of the concatenation of error events, all with minimum distance , while the bound for the inner code, , shows the maximum number of concatenated error events with minimum uncoded weight . For block codes and nonrecursive convolutional codes, , which results in a positive value for , thus the concatenated code will not have any interleaving gain. For recursive convolutional codes , since no finite error event with exists.3 Evaluating the maximum of the right-hand side of (11) for the recursive convolutional code results in (12) offering interleaving gain for FER whenever is greater than two.4 To summarize, there are two important factors in the performance of source-channel concatenated codes: the free distance of the outer code and the recursive structure for the inner code. This was demonstrated by an extension of the techniques of [7] in order to accommodate the nonlinear codes of interest in source-channel coding. The design issues of the inner code are similar to the case of ordinary SCCCs, which have been well developed in the literature [7], [16]. Our analysis is based on the concept of union bound, which diverges at low values (in particular, at SNR values below those corresponding to the channel cutoff rate [7]). There is no solid theoretical ground for using union-bound analysis below cutoff rate. However, as has been noted in the literature [7], design criteria based on these bounds perform surprisingly well, even at SNR values where the bounds do not converge. Our simulations also support that conclusion. 3In other words, in block codes and nonrecursive convolutional codes, a single “1” leads to a finite error event, while in recursive convolutional codes, at least two “1”s are needed in the data sequence for a finite error event. 4Similarly, calculations for B indicate that the interleaving gain for SER P is  ^ = ^ 1.

0

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V. ITERATIVE VLC AND CHANNEL DECODING In iterative decoding, each decoder, in turn, processes the available information about the desired signal, typically loglikelihood ratios, thus modifying and hopefully improving in each iteration the pool of available information on the received signal. The additional information is called extrinsic information [6], [17]. Extrinsic information represents the new information obtained in each half-iteration by applying the constraint of a constituent decoder. An efficient way to calculate extrinsic information is via the soft-input soft-output (SISO) algorithm [18]. In the following, we discuss the structure of the SISO module for a channel code as well as a VLC. A. SISO Channel Decoder A soft-output algorithm for channel decoding was introduced in [19]. A slightly different version of this algorithm, called the SISO module, was introduced in [18]. We give a system-level description of this block below. The SISO module for the convolutional code, shown in Fig. 4, works on the channel code trellis. It accepts two probability streams and as inputs. The former is about the coded sequence , and the latter is about the information sequence . Applying the constraints provided by the channel code, additional information (extrinsic information) is obtained and , which, in turn, is for both sequences, passed to the other decoder. Each decoder repeats this process by using the extrinsic information that was fed back as its new input. B. Bit-Level SISO VLC Decoder Many efficient channel-decoding algorithms are trellis-based. In particular, the Viterbi algorithm (VA) and SISO algorithms [18], [19] are all trellis-based. By building a trellis for a VLC, one may employ these algorithms in the decoding of VLCs. The trellis-based algorithms for the VLC are simpler than those for the inner code for two reasons. First, for the VLC trellis, only one node (root node) has multiple incoming branches, thus the compare–select operation of the VA and selection of surviving path is done only for the root node. At other nodes, only the metric is calculated. Second, for VLC we do bit-level detection, and there is no reference to the input symbols except connections of the trellis, which simplifies the SISO module. Based on the trellis representation of a VLC introduced in Section III, we derive a SISO algorithm for VLCs. Following the notation of [18], the extrinsic information is calculated as follows. At time , the output probability distribution is evaluated as (13) where represents a branch of the trellis, , and are, respectively, the branch value, the starting state, and the ending state of the branch , and the constant is a normalizing factor to ensure . The quantities

Fig. 4.

Iterative VLC and convolutional decoding.

and are calculated through forward and backward recursions, respectively, as follows:

with initial values for the root state (since the trellis always starts and ends at the root state), and for all other states. In order to exclude the input from the output probability and obtain the information so-called extrinsic information, both sides of (13) are divided by i.e., . (input probability) and (exTherefore, trinsic information) together form the a posteriori probability (APP) of the input sequence. In practice, the additive (logarithmic) version of a SISO algorithm is employed to avoid multiplications and prevent numerical problems. C. Iterative Decoding and Density Evolution An iterative decoder is shown in Fig. 4, using the SISO blocks are the interalready introduced. Blocks denoted and leaver and deinterleaver, respectively. In each iteration, only the extrinsic information generated by and , are exchanged each SISO block, between the soft-output decoders. After the final iteration, the , is decoded log-scale soft-sequence, at symbol level by the Viterbi decoder over the same bit-level trellis. The convergence of the iterative decoder can be illustrated by the density-evolution method [20], or alternatively, by extrinsic information transfer (EXIT) charts [21]. We use the density-evolution method for our analysis. We have verified the corresponding Gaussian approximation assumption in our setting. Density evolution treats the decoder as a nonlinear dynamical feedback system where the nonlinear input–output functions are calculated empirically. For further information on density evolution, we refer the reader to [20]. VI. EXPERIMENTAL RESULTS Table I shows the 5-ary source used in our experiments and is a Huffman code, various codes designed for this source. is an RVLC for this source reported in [8], and the codes , and were designed by us with successively higher redundancies. The redundancy of the VLCs can be quantified by

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TABLE I VARIABLE-LENGTH CODES USED IN SECTION VI

comparing their average length with that of the Huffman code. The Levenshtein distance [13], [2] is used in reporting SERs. It is noteworthy that despite the differences, the trellises of the different codes have roughly the same order of complexity. This arises due to the sparseness of the VLC trellises, which becomes more pronounced when the code has redundancy. For a more meaningful comparison of complexity, one may construct more compact trellises with minimal number of states. has a sparse trellis with 13 states and only 17 For example, single-bit branches, which is equivalent to a four-state compact trellis with eight branches, the same as the equivalent trellis of code . Thus, the complexities are comparable. Our inner convolutional codes are four-state codes with rates or taken from [7]. The ratecodes are the recurand nonrecursive with the following generator sive polynomials: and . The ratecodes, and , with the generator polynomials as shown below are considered

In our experiments, a packet of symbols is entropy encoded, interleaved, channel encoded, and transmitted using binary phase-shift keying (BPSK) modulation over an AWGN channel. Our first experiment is designed to test the accuracy of our analysis. Fig. 5 shows union bounds and simulation results for . There are several factors to the concatenated code consider when reading this plot. First, union bounds work in the regions, and the cutoff rate for this code is assohigh dB. Second, union bounds are calciated with culated for the optimal (ML) decoder, while the simulations, by necessity, use iterative decoding. Finally, calculation of the multiplicities for a nonlinear, VLC is a lengthy and time-consuming process, thus we present “truncated bounds” calculated with the first ten terms of the multiplicities of the outer code that were available in [8]. The decoding experiment was performed with ten iterations, with packet lengths of 20 and 200. The bounds are in agreement with simulations.

Fig. 5. SER of

C + CC K = 20 and 200 symbols. ;

As mentioned in Section IV, higher redundancy in the outer VLC leads to higher interleaver gain. To demonstrate this effect, and , each concatenated with the we simulated the VLCs . To maintain (approximately) the same overall inner code rate, in the experiment with , the inner code is punctured to rate, hence the notation . The overall rates of the two experiments are 0.445 for , and 0.471 for , where the equivalent “code rate” of a VLC is calculated by dividing the average length of the code by the average length of the Huffman code. The SERs and FERs are symbols. For small values of shown in Fig. 6 for , the code outperforms due to the more powerful inner code. However, the latter starts outperforming the former for dB. The sharper drop in error rate of the latter code is noteworthy.5 The corresponding density evolutions of the iterative decoders are shown in Fig. 7. We use the approximate Gaussian density evolution of [20]. We show density evolutions at dB (the staircase plot). The corresponding iterative decoding thresholds are demonstrated in Fig. 7 as well. with free For further comparisons, we used the code , concatenated with the inner code , a distance raterecursive convolutional code. The concatenated code has an overall rate of 0.404. The SER and FER of the this code are shown in Fig. 6. The code outperforms in the entire range of after the second iteration of the decoder. As mentioned previously, Bauer and Hagenauer [8] demonstrated coding gain via iterative source-channel decoding. However, in their case, the baseline system did not have the advantage of iterative decoding. We pose a slightly different question. Assuming we have a fixed computational and rate budget, we would like to compare the source-channel iterative decoder with a separable decoder whose channel decoder is iterative. For experimental verification of this and similar questions, we introduce three SCCCs: and , both with overall rate, and with overall rate. 5The BPSK-constrained Shannon limit for these codes is approximately 0 dB.

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Performance of

C + CC C + CC ;

8 9), and C + CC . = 2000 symbols.

(punctured to rate-

C +CC

=

C +CC

Fig. 7. Approximate Gaussian density evolution of and (punctured to rate- = ), K symbols. An instance of the convergence : dB is shown. of the decoders at E =N

89

= 2000 =15

K

. The two systems have the same overall rate nated with and decoder complexity. The simulation results are not shown in a separate figure, but one can compare the results of in Fig. 6 with the results of in Fig. 8 (both for nine iterations). The comparison indicates that the case of separable source-channel decoding is superior to joint source-channel deof , coding. We believe this is largely due to the small which is the RVLC used by Bauer and Hagenauer [8]. Therefore, it seems that separable decoding (with an iterative channel decoder) can be superior to iterative source-channel decoding when the outer code has small free distance. Then one may ask, how does a joint source-channel decoder compare with a separable decoder if we increase the free distance of the outer source code? We designed several experiments to address this question. In a comparison of the separable code with the joint source-channel decoding of , we found that especially at higher , the joint source-channel decoding works much better, while at intermediate , the two methods perform the same. We conducted two more experiments, whose results are shown in , is compared against Fig. 8. The separable code (both with rates ), where outperforms slightly. In the same Fig. 8, is compared against (both with rate ), and they perform roughly similarly. Thus, the simulations did not point to a clear and universal advantage for either the joint or separable approach. In some cases, where the outer entropy code has low redundancy, the separable case is clearly better, while in other cases, either the joint or the separable solution might be superior. The design choices must be made on a case-by-case basis. VII. CONCLUSION

Fig. 8. Comparison of concatenated redundant VLC and convolutional codes versus concatenated Huffman code and SCCCs, K symbols.

= 2000

We compare the iterative source-channel decoding of with a system consisting of the Huffman code concate-

We obtain union bounds for the SERs and FERs of concatenated source-channel codes for finite-alphabet sources. We generalize the previous notions of outer entropy code by inserting an unrestricted redundant VLC; thus, our analysis is general with respect to the choice of the outer code, including nonredundant (Huffman) codes, RVLC codes, and the so-called VLECCs. We

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use techniques originally developed for the SCCCs and adapt them so that they can be used with the nonlinear outer codes that are of interest in source-channel coding. By evaluating the union bounds of the concatenated scheme, we further studied the role of the constituent codes, and illustrated through simulations the relevance of the suggested design rules. REFERENCES [1] Y. Takishima, M. Wada, and H. Murakami, “Reversible variable length codes,” IEEE Trans. Commun., vol. 43, pp. 158–162, Feb.-Apr. 1995. [2] V. Buttigieg, “Variable-length error-correcting codes,” Ph.D. dissertation, Dept. Elec. Eng., Univ. Manchester, Manchester, U.K., 1995. [3] V. Buttigieg and P. G. Farrell, “Variable-length error-correcting codes,” Proc. IEE Commun., vol. 147, no. 4, pp. 211–215, Aug. 2000. [4] K. P. Subbalakshmi and J. Vaisey, “On the joint source-channel decoding of variable-length encoded sources: The BSC case,” IEEE Trans. Commun., vol. 49, pp. 2052–2055, Dec. 2001. , “Joint source-channel decoding of entropy coded Markov sources [5] over binary symmetric channels,” in Proc. IEEE Int. Conf. Communications, 1999, pp. 446–450. [6] C. Berrou and A. Glavieux, “Near-optimum error correcting coding and decoding: Turbo codes,” IEEE Trans. Commun., vol. 44, pp. 1261–1271, Oct. 1996. [7] S. Benedetto, D. Divsalar, G. Montorsi, and F. Pollara, “Serial concatenation of interleaved codes: Performance analysis, design, and iterative decoding,” IEEE Trans. Inform. Theory, vol. 44, pp. 909–926, May 1998. [8] R. Bauer and J. Hagenauer, “On variable length codes for iterative source-channel decoding,” in Proc. Data Compression Conf., Apr. 2001, pp. 273–282. , “Iterative source-channel decoding using reversible vari[9] able-length codes,” in Proc. Data Compression Conf., Apr. 2000, pp. 93–102. [10] N. Görtz, “On the iterative approximation of optimal joint sourcechannel decoding,” J. Select. Areas Commun., vol. 19, pp. 1662–1670, Sept. 2001. [11] A. Guyader, E. Fabre, C. Guillemot, and M. Robert, “Joint source-channel turbo decoding of entropy-coded sources,” J. Select. Areas Commun., vol. 19, pp. 1680–1696, Sept. 2001. [12] K. Lakovic and J. Villasenor, “Combining variable-length codes and turbo codes,” in Proc. IEEE Vehicular Technology Conf., Spring 2002, pp. 1719–1723. [13] T. Okuda, E. Tanaka, and T. Kasai, “A method for correction of grabled words based on the Levenshtein metric,” IEEE Trans. Comput., vol. C-25, pp. 172–176, Feb. 1976. [14] V. B. Balakirsky, “Joint source-channel coding with variable length codes,” in Proc. IEEE Int. Symp. Information Theory, June 1997, p. 419. [15] A. Murad and T. Fuja, “Robust transmission of variable-length encoded sources,” in Proc. IEEE Wireless Communications, Networking Conf., vol. 2, Sept. 1999, pp. 968–972.

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[16] D. Divsalar and R. J. McEliece, “On the design of generalized coding systems with interleavers,”, Jet Propulsion Lab. Prog. Rep., Aug. 1998. [17] J. Hagenauer, E. Offer, and L. Papke, “Iterative decoding of binary block and convolutional codes,” IEEE Trans. Inform. Theory, vol. 42, pp. 429–445, Mar. 1996. [18] 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. [19] L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal decoding of linear codes for minimizing symbol error rate,” IEEE Trans. Inform. Theory, vol. IT-20, pp. 284–287, Mar. 1974. [20] D. Divsalar, S. Dolinar, and F. Pollara, “Iterative turbo decoder analysis based on density evolution,” IEEE J. Select. Areas Commun., vol. 19, pp. 891–907, May 2001. [21] S. ten Brink, “Convergence behavior of iteratively decoded parallel concatenated codes,” IEEE Trans. Commun., vol. 49, pp. 1727–1737, Oct. 2001.

Ahmadreza Hedayat (S’00) received the B.S.E.E. and M.S.E.E. degrees from the University of Tehran, Tehran, Iran, in 1994 and 1997, respectively, and the Ph.D. degree in electrical engineering from the University of Texas at Dallas, Richardson, in 2004. From 1995 to 1999, he was with Pars Telephone Kar and Informatic Services Corporation, Tehran, Iran. His current research interests include MIMO signaling and techniques, channel coding, and source-channel schemes.

Aria Nosratinia (M’97–SM’04) received the B.S. degree in electrical engineering from the University of Tehran, Tehran, Iran, in 1988, the M.S. degree in electrical engineering from the University of Windsor, Windsor, ON, Canada, in 1991, and the Ph.D. degree in electrical and computer engineering from the University of Illinois at Urbana-Champaign in 1996. From 1995 to 1996, he was with Princeton University, Princeton, NJ. From 1996 to 1999, he was a Visiting Professor and Faculty Fellow at Rice University, Houston, TX. Since 1999, he has been with the University of Texas at Dallas, Richardson, where he is currently an Associate Professor of Electrical Engineering. His research interests are in the broad area of communication and information theory, particularly coding and signal processing for the communication of multimedia signals. Dr. Nosratinia is currently an Associate Editor for the IEEE TRANSACTIONS ON IMAGE PROCESSING. He was the recipient of the National Science Foundation Career award in 2000 and has twice received chapter awards for outstanding service to the IEEE Signal Processing Society.

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Near-Capacity Coding in Multicarrier Modulation Systems Masoud Ardakani, Student Member, IEEE, Tooraj Esmailian, and Frank R. Kschischang, Senior Member, IEEE

Abstract—We apply irregular low-density parity-check (LDPC) codes to the design of multilevel coded quadrature amplitude modulation (QAM) schemes for application in discrete multitone systems in frequency-selective channels. A combined Gray/Ungerboeck scheme is used to label each QAM constellation. The Gray-labeled bits are protected using an irregular LDPC code with iterative soft-decision decoding, while other bits are protected using a high-rate Reed–Solomon code with hard-decision decoding (or are left uncoded). The rate of the LDPC code is selected by analyzing the capacity of the channel seen by the Gray-labeled bits and is made adaptive by selective concatenation with an inner repetition code. Using a practical bit-loading algorithm, we apply this coding scheme to an ensemble of frequency-selective channels with Gaussian noise. Over a large number of channel realizations, this coding scheme provides an average effective coding gain of more than 7.5 dB at a bit-error rate of 10 7 and a block length of approximately 105 b. This represents a gap of approximately 2.3 dB from the Shannon limit of the additive white Gaussian noise channel, which could be closed to within 0.8–1.2 dB using constellation shaping. Index Terms—Discrete multitone systems, low-density parity-check (LDPC) codes, multilevel-coded modulation, frequency-selective channels.

I. INTRODUCTION N DISCRETE multitone (DMT) systems, there are many sub-channels with different signal-to-noise ratios (SNRs). Designing a coding system that meets the requirements of these channels has drawn much attention [1]–[3]. In standard asymmetric digital subscriber lines, a trellis-coded-modulation scheme in concatenation with a Reed–Solomon code is used [4], providing approximately 5-dB coding gain at a symbol-error rate (SER) of 10 . The two principal classes of codes for the high-SNR regime are lattice codes and trellis codes. The SNR gap between uncoded baseline performance of quadrature-amplitude modulation (QAM) and Shannon limit is 9 dB at an SER of 10 . At this SER, the coding gain of the Leech lattice in dimension 24 is less than 4 dB [5]. With a 512-state trellis code, an effective

I

Paper approved by W. E. Ryan, the Editor for Modulation, Coding, and Equalization of the IEEE Communications Society. Manuscript received June 19, 2003; revised March 3, 2004 and May 26, 2004. M. Ardakani and F. R. Kschischang are with the Edward S. Rogers Sr. Department of Electrical and Computer Engineering, University of Toronto, Toronto, ON M5S 3G4, Canada (e-mail: [email protected]; [email protected]). T. Esmailian was with the Edward S. Rogers Sr. Department of Electrical and Computer Engineering, University of Toronto, Toronto, ON M5S 3G4, Canada. He is now with the Research and Development Group, Edgewater Computer Systems, Inc., Kanata, ON K2K 3G6, Canada (e-mail: [email protected]). Digital Object Identifier 10.1109/TCOMM.2004.836560

coding gain of 5.5 dB can be achieved, but it seems that approaching a coding gain close to 6 dB with trellis codes is very difficult [5]. All of the coding gains that we refer to are only due to coding. Using proper shaping techniques, a shaping gain of up to 1.53 dB can be added independent of the coding gain. However, even with a 512-state trellis code and achieving the ultimate shaping gain, there is a gap of 2 dB from the Shannon limit. The goal of this paper is to provide near-capacity coding techniques for the high-SNR regime and more generally for DMT systems. At low SNR, the problem of near-capacity coding has been studied extensively, and it has been shown [6]–[9] that turbo codes and low-density parity-check (LDPC) codes can approach the capacity of many channels with practical complexity. For high-SNR channels, however, multilevel modulation is required. The main problem of using multilevel symbols in these codes is that large alphabet sizes create prohibitively large decoding complexity. To overcome this problem, one can use multilevel coding, which allows one to apply binary codes to multilevel modulation schemes. However, in DMT systems, dealing with subchannels whose SNR is different and use different constellation size is a challenge. In [10], a regular high-rate LDPC code is used for error correction in a digital subscriber line (DSL) transmission system. The maximum reported coding gain at a bit-error rate (BER) of 10 is 6.2 dB, which, compared to the maximum possible coding gain (8.3 dB for the additive white Gaussian noise (AWGN) channel), shows a gap of more than 2 dB from the Shannon limit. The goal of [10] is to provide a coding system for DSL transmission systems with practical complexity and suitable structure for hardware implementation, so highly efficient irregular codes, which are more difficult to implement, were not considered. The idea of using LDPC codes together with coded modulation is used in [11] as well. In [2], a turbo code has been used for ADSL and a coding gain of 6 dB at an (equivalent to approximately 6.8 dB at an SER SER of ) is reported. This amounts to a 1.5-dB gap from the of Shannon limit for the AWGN channel. In this paper, we address the problem of coding for DMT systems and propose a coding scheme based on a combination of irregular LDPC codes and multilevel coding, which provides an average coding gain of more than 7.5 dB at a message error rate of 10 . This is equivalent to a gap of approximately 0.8 dB from the Shannon limit for the AWGN channel. The decoding complexity in our system is comparable with a 512-state trellis code. The main difference between our approach and other work on use of turbo codes/LDPC codes for DMT systems is that our

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main goal is to maximize the coding gain for a practical decoding complexity. For this purpose, motivated by [12] and [13], after labeling symbols with binary sequences, we model an effective channel—which will be referred to as a bit-channel—for each bit level of the label. Then, for different constellation sizes, we compute the capacity of the bit-channels that each label bit sees. This allows us to choose an efficient error-correcting code for each level bit whose code rate accounts for the capacity of the associated bit-channel. Another difference is that we avoid coding all of the label bits with the same code. Although it has been shown that bitinterleaved coded modulation together with a Gray-labeling can perform very close to multilevel coding with set partitioning labeling [14], and hence the performance loss in this strategy is not an issue, we use the powerful LDPC coding only for those label bits that in fact need this high-complexity coding scheme. As a result, we propose a quasi-multilevel-coding system. We use a single LDPC code for the least significant bits of the label, but higher level bits are coded with less complex codes. Another benefit of this strategy is that we avoid high-rate LDPC codes. High-rate irregular LDPC codes need a very high check degree in order to be able to perform near capacity. Highdegree check nodes introduce a large number of short cycles in the Tanner graph of the code, which adversely affects the performance in the case of finite-length codes. Moreover, instead of choosing from a set of codes with different rates at each channel condition, we employ a fixed-rate LDPC code that is selectively concatenated with a repetition code to provide a flexible overall coding rate for the system. This is necessary, since for some realizations of the channel the capacity of the bit-channels assigned to the LDPC code can be less than the rate of the LDPC code. The repetition code is used only for bit-channels whose capacity is very low. In terms of hardware implementation, compared to the approach in [10], which uses highly structured LDPC codes in a multilevel coding system, our system is more complex where can be an because we use constellations of size even or odd integer, and our LDPC code is irregular. Although our decoding complexity is similar, the encoding complexity in our system is higher because we have not used codes with algebraic construction. All of the above differences have allowed us to approach capacity much more closely. Moreover, our system works for a very wide range of channel conditions because it has a flexible code rate. This paper is organized as follows. In Sections II and III, we briefly review some aspects of multilevel coding and LDPC codes. In Section IV, our channel model is briefly discussed. In Section V, we explain the structure of our proposed system. In Section VI, we present the specifications of our system for an ensemble of frequency-selective channels encountered in powerline channels [15]. Section VII shows some simulation results, and, finally, we conclude the paper in Section VIII. II. MULTILEVEL CODING The idea of multilevel coding was introduced in 1977 by Imai and Hirakawa [16]. Like Ungerboeck’s idea for trellis-coded

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modulation [17], the idea of multilevel coding is based on the concept of set partitioning. Assume that each point of a constellation of points is labeled . The idea of with a binary address of the multilevel coding is to protect each address bit constellation point by an individual binary code at level [12]. Since the mapping between addresses and constellation points is one-to-one, if and represent the random variables corresponding to the transmitted and received signals, respecis equal to the tively, then the mutual information mutual information between address vector and the received as the vector of random signal. Considering variables corresponding to the address bits, we have

(1) over Hence, transmission of address vector the physical channel can be separated into parallel transmission of individual digits. Notice that each address bit sees an effective channel (bit-channel) whose capacity can be different from the other bit-channels due to the labeling scheme. At the decoder, according to (1), the mutual information is used in order to decode . Then the conditional is used to decode . That is mutual information to say, the capacity of the second bit-channel has to be defined conditioned on the first address bit. In general, for the th to . bit-channel, the capacity is defined conditioned on If set partitioning labeling, e.g., an Ungerboeck labeling, is used for label bits, then conditioned on , bit sees a higher capacity bit-channel. Hence, a higher rate code can be used for it. At the next stage, conditioned on both and , bit sees an even higher capacity bit-channel, and so on. Therefore, label bits must be protected by different rate codes. It is known that, if each code achieves the capacity of its bit-channel, the total capacity of the channel is achieved [12]. Fig. 1 shows the net capacity of 4-QAM and 8-QAM signaling together with the capacity of each bit-channel as a function of SNR assuming that an Ungerboeck labeling is used. We refer the reader to [18] and [19] for a complete study of these systems and a more detailed structure of the encoder and decoder for such systems. III. LDPC CODES In the case of binary modulation, LDPC codes are capable of approaching the capacity of many different types of channels with a practical decoding complexity [8], [9], [20]. For the binary erasure channel, LDPC codes can, in fact, achieve the capacity under message-passing decoding [21]. Decoding LDPC codes is based on iterative message-passing algorithms. There are many different message-passing algorithms, among which the sum–product algorithm [22] appears to be the most accurate. The main idea in all message-passing algorithms is that each message on each edge carries a belief about the value of the adjacent variable node. This can be,

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Fig. 1. Net capacity and capacities associated with bit-channels (bit-channel 0: I (Y ; b ); bit-channel 1: I (Y ; b 4-QAM/8-QAM signaling when an Ungerboeck labeling is used.

for example, a hard decision binary value, a soft probability , or the log-likelihood ratio (LLR) of measure such as . In each iteration, all of the variprobabilities able nodes send their messages to all adjacent check nodes and then all of the check nodes send their updated messages back. The update rules depend on the decoding algorithm. In [22], different decoding algorithms are discussed. Depending on their structure, LDPC codes are said to be “regular” or “irregular.” In regular LDPC codes, all variable nodes have a fixed degree and all check nodes have another fixed degree. In the irregular case, variable nodes and check nodes of different degree are used. It has been shown that irregular codes can have a better performance than regular ones [8], [23]. An ensemble of irregular codes can be defined by their variable and check degree distributions. A degree distribution at the variable/check side defines the portion of edges incident to variable/check nodes with different degrees. For instance, if the variable degree distribution of a code is , it means that 20% of edges are connected to degree-2 variable nodes, 50% of edges are connected to degree-5 variable nodes and 30% of them are connected to degree-9 variable nodes. Similarly, the check nodes have their own degree distribution. As the block length of a randomly chosen code approaches infinity, the performance of the code depends only on its degree distributions [8]. Hence, the design of an ensemble of irregular LDPC codes is equivalent to finding good variable and check degree distributions. The main goal of design is to find the maximum-rate ensemble that guarantees some required convergence behavior for a given channel condition, in the limit of long block length. In practice, when a finite-length code is used, the performance of the code depends on the block length as well: the larger the block length, the closer the performance of the code to the predicted asymptotic behavior. For instance, with a block length of

jb

); bit-channel 2:

I (Y

;b

jb ;b

)) for

10 , there is a gap of about 0.2 dB to the predicted performance, and with a block length of 10 this gap is reduced to 0.05 dB [9]. IV. CHANNEL MODEL Although the approach of this paper in terms of providing a near-capacity coding system for DMT systems is quite general, we consider in-building power-line channels as an example, and our focus in the design will be this class of channels. A detailed study of characteristics of in-building power-line channels has been conducted in [15], and a stochastic ensemble of test channels described. The parameters of the test channels are based on limitations placed on wiring configurations by the National Electric Code, by the size and type of building in which the power lines are located, by the expected load impedances, and by experimental measurement of background and impulsive noise. We use the channel model provided in [15] to generate our sample test channels. We also use the distribution of channel SNR provided in [15] for our system design. To show the success of our design, we present results for different size buildings and different realizations of the channel. Fig. 2 shows the magnitude of the frequency response of three representative test channels. For more details about the parameters of these channels, see [15]. The application to power-line channels is intended to be representative only, and we expect the results of this paper to apply to broad classes of frequency-selective Gaussian channels. V. SYSTEM STRUCTURE Equation (1) is one of many different ways that one can partition the total capacity to the capacity of single bits. As discussed before, this suggests using an Ungerboeck labeling together

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Fig. 2. Magnitude of the frequency response of three test channels. Fig. 3.

with separate codes for each level of the label bits. Constellation labeling and decoding stages strongly depend on the way that one partitions the total capacity. Since we wish to perform near the capacity of the channel, we consider using LDPC codes for different levels of the label bits. However, we avoid using a single code for each address bit because decoding higher level bits is not possible without finishing the decoding for lower level bits. As a result, to avoid a long decoding delay, the length of the employed LDPC codes should be relatively small which hinders the performance of the code. An alternative solution for our system is bit-interleaved coded modulation, i.e., to use a single code for all label bits without partitioning the total capacity to sum of bit-channel capacities. Bit-interleaved coded modulation, together with a Gray labeling, can approach the channel capacity very closely [14], but unlike multilevel coding cannot achieve it. The capacity of Gray-coded channels for various constellation sizes and number of coded bits is analyzed in [25]. Assigning all label bits to one LDPC code has some disadvantages. For instance, it requires a high-rate LDPC code which performs close to capacity only with high-degree check nodes. High-degree check nodes introduce short cycles in the factor graph of the code and hinder the performance. Another problem with bit-interleaved coded modulation is that in terms of decoding complexity it may not be an efficient solution. Notice that the capacity of higher bit-channels under set partitioning labeling can be very close to one (typically more than 0.95 b/symbol). Hence, in some bits, no coding is required and other bits can be protected effectively by a low complexity code such as a Reed–Solomon code. Considering these facts, we propose using a single code for only the two least significant bits. This way, we can double the length of the code for the same delay and hence have a better performance. Notice that the LDPC decoding complexity scales linearly with the length of the code. Hence, the complexity per bit of information is independent of the code length. Encoding and decoding these two bits together is equivalent to bit-interleaved code modulation for these two bits. Hence, a Gray labeling must be used in order to approach the

Labeling of a 16-QAM constellation in our system.

capacity very closely [14]. In other words, these two address bits will have equal minimum Euclidean distance and almost equal error probability, which allows use of a single LDPC code for them. We use Ungerboeck labeling for higher level bits to further increase the capacity for even higher address bits. This provides us with some bit-channels whose raw error rate is very low and can be coded with low-complexity high-rate codes and some bit-channels whose raw error rate is better than the target error rate of the system and hence do not need any coding. This will further reduce the overall decoding complexity of the system. In this paper, only and need protection and higher address bits are not coded. Fig. 3 shows the labeling used for a 16-QAM constellation in our system. Notice that for and an Ungerboeck labeling is used. While one of our main reasons for bundling and together was to permit a longer blocklength for the LDPC code, we avoid including in this bundle. This is because many subchannels do not use constellation sizes more than 4-QAM. Thus, including in the LDPC code increases the complexity of the system, but they do not contribute much in the length of the LDPC code to provide a better performance. Based on the above discussion, we rewrite (1) as

(2) For and puted as

, the average bit-channel capacity can be com-

(3)

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Fig. 4. Block diagram of the encoder of the proposed system.

Fig. 5. Block diagram of the decoder of the proposed system.

where is the number of points in the constellation, is the th constellation point, is the th point on the subconstellation with fixed and , and represents samples of a complex Gaussian noise with variance . Since the rate of the irregular LDPC code that we use is fixed and very close to the bit-channel capacity, but the characteristics of the channel vary with time and frequency, we use an inner repetition code to take care of cases that the capacity of the bit-channels assigned to the LDPC code is less than the rate of the LDPC code. The repetition code is used only for bit-channels whose capacity is very low. In other words, our code (LDPC code in concatenation with the repetition code) has an adaptive rate to guarantee convergence over all channel realizations. Another advantage of the repetition code is that it allows use of low-capacity bit-channels. This in turn allows us to switch to higher constellation sizes at lower SNRs compared to previous work. As a result, the system can perform even closer to the capacity. Figs. 4 and 5 show the block diagram of the proposed system at the transmitter and the receiver, respectively. At the transmitter, the constellation mapper uses two bits from the output of LDPC encoder as the lowest address bits. For higher address bits it uses the output of other encoders plus the uncoded bits. In the case of 2-PAM, only one bit from the LDPC encoder is used. Using these address bits, a point from the constellation is chosen for each subchannel of the DMT system. After assigning data to all subchannels, the inverse fast Fourier transform (IFFT) is used to map this complex vector to a DMT symbol. At the receiver, the DMT symbol is converted to a complex vector using the FFT.

Our LDPC decoder first computes the LLR values for each subchannel. The LLR value for is computed as LLR (4) represents a point of in which is the received signal, transmitted constellation, is the subchannel gain, is the Gaussian noise variance, and represents a subconstellation of with the address bit equal to zero. Similarly, the LLR value for can be computed. Once all LLR values are ready, they will be decoded in the LDPC decoder. These decoded values are used to decode higher level address bits like any other multistage decoder. VI. SYSTEM SPECIFICATIONS In this system, we use a rate-0.6 irregular LDPC code. The rate of the LDPC code is chosen according to the expected value over all realizations of the channel. Fig. 6 depicts , of the capacity of the bit-channel that sees and the distribution of subchannels as a function of SNR. The system delay is mainly due to the following: • buffer-fills in the LDPC encoder/decoder, which can be computed as shown in the equation at the bottom of the next page; (LDPC code length) • decoding delay: (decoder throughput)

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Fig. 6.

C

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, the bit-channel capacity for b and the PDF of subchannels as a function of SNR.

Compared to above-mentioned delays, encoding delay of the LDPC code, transmission delay, and other delays due to higher level codes are negligible. Hence, the total delay can be approx. imated as In our system, the length of the LDPC code is 100 000 and its rate is 0.6. Assuming a typical throughput of 100 Mb/s for the decoder, a minimum bit rate of 1 Mb/s and assuming that of bits are being passed to the LDPC code, the overall delay is approximated as 181 ms. Hence, the actual delay is not more than 190 ms. About 95% of this delay is due to buffer-fills in the LDPC encoder/decoder. For higher bit rates, say 10 Mb/s, this delay is much shorter (approximately 19 ms) and the overall delay does not exceed 30 ms. Here we assume that the channel has the same ratio of coded to uncoded bits and the higher bit rate is due to

a higher channel bandwidth. A longer delay is expected, if the higher bit rate is due to (or partially due to) a higher SNR for the channel. sees a channel whose From Fig. 6, it can be seen that capacity is very close to one, which means that a high-rate code can be used. The expected value of this capacity is about 0.986 b/symbol and for it is more than 0.999 b/symbol. The capacity for higher levels is even greater. There are many different types of codes which can be used for these highcapacity channels. Depending on the decoding complexity that one can afford, the rate of these codes can be close to the capacity. One simple solution is to use Reed–Solomon codes. A (255, 209) Reed–Solomon code, with a rate more than 0.819, for and a can guarantee a frame error rate less than (255, 241) Reed–Solomon code, with a rate more than 0.945

LDPC code length LDPC code rate system bit rate portion of bits assigned to the LDPC code

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can guarantee the same frame error rate for . Hence, the gap from capacity in these bit-channels is small. Also notice that, due to their low SNR, many subchannels do not have and , so some rate loss in these bit-channels does not dramatically impact the overall performance of the system. The bit-loading algorithm, i.e., the algorithm that chooses the constellation size at a given SNR, affects the capacity of bit-channels. The bit-loading algorithm chooses the constellation size in order to approach the total capacity as closely as possible. The zigzag form of bit-channel capacities in Fig. 6 is due to the bit-loading algorithm, which switches to larger constellations as the SNR increases. At each switch, the capacity of all address bits drops, however the net capacity increases. This can be understood from Fig. 1. The capacity of “bit-channel 0” and “bit-channel 1” in the case of 4-QAM is always greater than that of 8-QAM. However, in the case of 8-QAM, there is an extra bit-channel with a very high capacity which makes the net capacity of 8-QAM greater than the net capacity of 4-QAM. As a result, the bit-loading algorithm, on the one hand, wishes to switch to higher constellation sizes to approach capacity more closely and, on the other hand, wishes to avoid higher constellation sizes as they may lead to bit-channels with a very poor capacity. Higher constellation sizes may also increase complexity. Hence, the bit-loading algorithm must trade off between complexity and performance. Pseudocode for the bit-loading algorithm that we have used is as follows.

Bit-Loading Algorithm 1. Input: SNR, Threshold 2. set Constellation MaximumConstellationSize 3. while C(Constellation, SNR)-C(Constellation/2, SNR) Threshold set Constellation Constellation/2 4. Output: Constellation

Here the function returns the capacity of a QAM constellation of the given size at the given channel SNR. In our setting, the maximum constellation size is 256-QAM, the threshold is 0.25 b/symbol, and we allow all constellation sizes of the form down to 2-PAM. As a result, our bit-loading algorithm assigns, at each SNR, the minimum number of bits to each subchannel in a way so that the constellation capacity is less than 0.25 b away from the capacity of the constellation one level higher. In this way, we make sure that we are quite close to the channel capacity with the minimum complexity in terms of constellation size. varies approximately between Fig. 6 also shows that the 0.5 and 0.9 b/symbol for different SNRs. Its average value over all realizations of the channel is about 0.68 b/symbol. Hence, a rate 0.6 LDPC code is used. In any given channel realization, each subchannel can be viewed as a random sample from the given distribution of SNRs in Fig. 6. Since the number of subchannels is relatively for one realization is close large, the average value of to the expected value on all realizations. However, with a

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small probability, some realizations of channel can result in less than 0.6 b/symbol. This is possible an average since each subchannel, independent of the other subchannels, can have a capacity less than 0.6 b/symbol. To handle these cases with the same fixed-rate LDPC code, we use an inner code for some subchannels with very low repetition ratecapacity. For a given realization of the channel, we first check the av. If it is less than 0.66, we decide to employ erage of the repetition inner code because the LDPC code may not conis less than a verge on its own. In every subchannel, if threshold value (typically 0.5 b/symbol), we repeat those bits. From the LDPC code point of view, this is equivalent to sending a bit over a cleaner channel with a capacity almost doubled. Making the simplifying assumption that the capacity in those subchannels is actually doubled, we compute the average over all subchannels again, and if it exceeds 0.7 we reduce the threshold value ; if it is less than 0.66, we increase the threshold value ; otherwise we proceed. This repetition algorithm makes the average capacity over all subchannels to be between 0.66–0.7 b/symbol. In this way, we maintain a reasonably small gap from capacity for rate-0.6 LDPC code. The degree distribution of the LDPC code that we have used in these simulations is . The code is regular at the check side with check nodes of degree 10. The code is designed under a Gaussian assumption on the messages and using extrinsic information transfer (EXIT) charts based on message error rate [24]. We note that we cannot make the length of the LDPC code an integer multiple of the length of the DMT symbols, because for many channels that change with time (for example, DSL or power-line channels), the length of the DMT symbols vary. To overcome this problem, we simply put as many DMT symbols as possible in the LDPC structure and fill the remaining portion of the LDPC code with zeros. This has no significant effect on the system performance, because the typical size of a DMT symbol (a few hundred bits), compared to the length of the code (100 000 bits), is very small. One may use the fact that the code is zero-padded in order to perform a more efficient decoding, but the improvement is minor.

A. System Complexity Since the encoding/decoding complexity of the rather long LDPC code dominates the overall complexity of the system, it will be our focus on this section. The encoding complexity, when the code has some structure, is relatively small in comparison with the decoding complexity as the decoder needs a large number of iterations (about 100 in our case). For example, if the degree two variable nodes are chained to form a structure similar to RA codes [26], the number of operations for encoding will be approximately equal to the number of edges (400 000 in our case), while the decoder needs more than this many operations per iterations. So the main focus of this complexity analysis will be on the decoding complexity.

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Following the notation of [22], the update rules of the sum–product algorithm at a check node and a neighboring variable node are as follows:

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TABLE I SIMULATION RESULTS ON SIX POWER-LINE CHANNELS

(5) (6) In (5), is the set of all of the neighbors of except ; similarly, in (6), is the set of all of the neighbors represents a message from a variable of except . Here represents a message from a node to the check node check node to the variable node and is the channel message to the variable node . From the above equations, it can be shown that at a variable operations are enough to compute all node of degree the output messages. Notice that one can add all input messages in operations and then, for every outgoing message, one subtraction is required. Thus, another operations should be done. As a result, the number of operations at the variable or equivalently , where is the side of the code is number of edges in the graph. Similarly, at a check node of degree , a total of operations is needed. Hence, at the check side of the code the number of operations is , where is the number of check nodes. Notice that (5) can be written as (7) Hence, having a lookup table or a circuit to map a message to and back, the situation is similar to a variable operations per iteration is node. As a result, a total of required at the decoder. Considering a maximum of 100 iterations, the number of edges (400 000) and the number of check nodes (40 000), the overall complexity of decoding is 152 Mega operations for 60 000 information bits. That is, 2534 operations per bit. Considering complexities involved with other parts of the system, we estimate an overall complexity of less than 2700 operations per bit. In our proposed system, we have four coded bits whose . The complexity of our system is comoverall rate is about parable with the complexity of Viterbi decoding on a 512-state code. Such a trellis has 512 trellis with an underlying ratestate and eight branches leaving each state. At each stage of the trellis, the decoding involves one computation and addition per branch and seven comparisons per state of the trellis. That is 7680 operations per three information bits or 2560 operations per bit. It has to be mentioned that the hardware complexity of the LDPC decoder is higher than that of the Viterbi decoder as the LDPC decoder needs more memory. However, such comparisons are highly implementation-dependant.

Table I shows the results of our simulations on six samples of power-line channels. In this table, is the average of (3) over all subchannels. is the overall code rate for each specific channel realization. When no repetition is required, the overall rate is 0.6 b/symbol, but, when the average channel capacity is smaller than 0.66 b/symbol, the repetition code is active and the overall rate of the code is less than 0.6 b/symbol. represents the gap from the capacity of In this table, binary modulation. In other words, assuming a binary input is the gap from AWGN channel when the capacity is the Shannon limit considering a rate code is used. As another measure of performance, we define excess redundancy (ER). Excess redundancy in each subchannel is the difference between the capacity of that subchannel and the actual information bits which are sent in it. ER is the average excess redundancy on all subchannels which can be computed as ER

SNR

(8)

where is the number of subchannels, is the number of bits transmitted in the th subchannel, SNR is the SNR of the th is the code rate used for the th bit of th subchannel, and subchannel. ER is a good measure of performance for a coding scheme, because it directly measures the gap from the capacity in bits/symbol. In many other papers in the coded-modulation literature, coding gain is represented as a measure of performance. In our case, however, since subchannels with different SNRs are used, there is no standard way to compute the coding gain. subchannels However, considering a DMT channel with whose capacities are given by to , and comparing it with a second DMT system whose subchannels all have the same for the two systems to capacity , we have have equal capacity. Now, assume SNR for subchannel on the first system and SNR for all of the subchannels of the second DMT system. Using SNR , we obtain SNR

SNR

When SNR values are sufficiently large, this can be simplified to VII. SIMULATION RESULTS In our simulations, we have assumed that synchronization and channel knowledge at the transmitter and receiver sides are perfect.

SNR

SNR

(9)

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or, in other words, SNR is the geometric mean of the SNR ’s. Now, comparing an uncoded system with one using coding, if in deciwe denote the effective SNR after coding with SNR bels and if each subchannel has a coding gain of in deciso bels, then the required SNR for th subchannel is SNR we have SNR

SNR

so our average coding gain in decibels is . This is in decibels which in linear form would be the geometric average of the subchannels coding gain. To find the coding gain in each subchannel, we first compute a gap from the Shannon limit as SNR

(10)

The coding gain on this subchannel at a BER of is , because the uncoded SNR for QAM modulation at this BER is 9.8 dB. The effective coding gain is then . This effective coding gain is also reported in Table I. It worth mentioning that at high SNRs there is a gap of about 1.53 dB between modulation capacity and the capacity of the Gaussian channel [5]. This can be reduced using shaping techniques [27]–[29]. The gap at lower SNRs is smaller. In a DMT system, which has subchannels with different SNRs, this gap is less than 1.53 dB on average. In Table I, the first three channels are three power-line test channels derived from buildings with different sizes. The next three channels are random channels from medium-size buildings. The channel frequency response of the first three test channels of the table is shown in Fig. 2. VIII. CONCLUSION In this paper, we have proposed a high-performance error-correction system suitable for DMT systems. In our system, we label QAM constellation points using a combined Gray–Ungerboeck labeling scheme. Low-order bits are coded with an irregular LDPC code, which is selectively concatenated with a repetition code to provide an adaptive rate. The rate of the LDPC code is carefully chosen based on the capacity of the corresponding bit channels. Higher order bits are coded with Reed–Solomon codes or are left uncoded. We provide a practical bit-loading algorithm that maintains a balance between constellation complexity and coding efficiency. The decoding complexity of the system is comparable to that of a traditional trellis-coded modulation scheme with 512 states. Using our scheme, performance is achieved equivalent to a gap of approximately 2.3 dB from the Shannon limit without constellation shaping, on a class of channels encountered in power-line communication systems. This gap could be closed to about 1 dB with appropriate shaping methods. Our approach applies not only to power-line channels, but to general frequency-selective channels; hence, we believe that a similar

system can be considered for high SNR wireless channels or DSL channels.

REFERENCES [1] T. N. Zogakis, J. T. Aslanis, and J. M. Cioffi, “Analysis of a concatenated coding scheme for a discrete multi-tone modulation,” in Proc. IEEE System Military Communications Conf., 1994, pp. 433–437. [2] L. Zhang and A. Yongacoglu, “Turbo coding in ADSL DMT systems,” in Proc. IEEE Int. Communications Conf., vol. 1, 2001, pp. 151–155. [3] Z. Cai, K. R. Subramanian, and L. Zhang, “DMT scheme with multidimensional turbo trellis code,” Electron. Lett., pp. 334–335, Feb. 2000. [4] Asymmetric Digital Subscriber Line (ADSL) Metallic Interface, 1995. ANSI Standard T1E1.4/95-007R2. [5] G. D. Forney and G. Ungerboeck, “Modulation and coding for linear Gaussian channels,” IEEE Trans. Inform. Theory, vol. 44, pp. 2384–2415, Oct. 1998. [6] C. Berrou, A. Glavieux, and P. Thitimajshima, “Near Shannon limit error-correcting coding and decoding: Turbo-codes,” in Proc. IEEE Int. Communications Conf., 1993, pp. 1064–1070. [7] D. Divsalar and F. Pollara, “On the design of turbo codes,” Jet Propulsion Lab., Pasadana, CA, TDA Progr. Rep., 1995. [8] T. J. Richardson and R. L. Urbanke, “The capacity of low-density paritycheck codes under message passing decoding,” IEEE Trans. Inform. Theory, vol. 47, pp. 599–618, Feb. 2001. [9] T. J. Richardson, A. Shokrollahi, and R. L. Urbanke, “Design of capacity-approaching irregular low-density parity-check codes,” IEEE Trans. Inform. Theory, vol. 47, pp. 619–637, Feb. 2001. [10] E. Eleftheriou and S. Ölçer, “Low-density parity-check codes for digital subscriber lines,” in Proc. IEEE Int. Communications Conf., 2002, pp. 1752–1757. [11] T. Cooklev, M. Tzannes, and A. Friedman, “Low-density parity-check coded modulation for ADSL,”, ITU-Telecommun. Standardization Sector, Temp. Doc. BI-081, Oct. 2000. [12] U. Wachsmann, R. F. H. Fischer, and J. B. Huber, “Multilevel codes: Theoretical concepts and practical design rules,” IEEE Trans. Inform. Theory, vol. 45, pp. 1361–1391, July 1999. [13] G. D. Forney, M. D. Trott, and S.-Y. Chung, “Sphere-bound-achieving coset codes and multilevel coset codes,” IEEE Trans. Inform. Theory, vol. 46, pp. 820–850, May 2000. [14] G. Caire, G. Taricco, and E. Biglieri, “Bit-interleaved coded modulation,” IEEE Trans. Inform. Theory, vol. 44, pp. 927–946, May 1998. [15] T. Esmailian, F. R. Kschischang, and P. G. Gulak, “In-building power lines as high speed communication channels: Channel characterization and a test channel ensemble,” Int. J. Commun. Syst., vol. 16, pp. 381–400, May 2003. [16] H. Imai and S. Hirakawa, “A new multilevel coding method using error correcting codes,” IEEE Trans. Inform. Theory, vol. IT-23, pp. 371–377, May 1977. [17] G. Ungerboeck, “Channel coding with multilevel/phase signals,” IEEE Trans. Inform. Theory, vol. IT-28, pp. 55–67, Jan. 1982. [18] G. J. Pottie and D. P. Taylor, “Multilevel codes based on partitioning,” IEEE Trans. Inform. Theory, vol. 35, pp. 87–98, Jan. 1989. [19] A. R. Calderbank, “Multilevel codes and multistage decoding,” IEEE Trans. Commun., vol. 37, pp. 222–229, Mar. 1989. [20] D. J. C. MacKay, “Good error-correcting codes based on very sparse matrices,” IEEE Trans. Inform. Theory, vol. 45, pp. 399–431, Mar. 1999. [21] A. Shokrollahi, “Capacity-achieving sequences,” in Codes, Systems, and Graphical Models, no. 123 of IMA Volumes in Mathematics and Its Applications, B. Marcus and J. Rosenthal, Eds. New York: SpringerVerlag, 2000, pp. 153–166. [22] F. R. Kschischang, B. J. Frey, and H.-A. Loeliger, “Factor graphs and the sum-product algorithm,” IEEE Trans. Inform. Theory, vol. 47, pp. 498–519, Feb. 2001. [23] M. G. Luby, M. Mitzenmacher, M. A. Shokrollahi, and D. A. Spielman, “Efficient erasure correcting codes,” IEEE Trans. Inform. Theory, vol. 47, pp. 569–584, Feb. 2001. [24] M. Ardakani and F. R. Kschischang, “Designing irregular LDPC codes using EXIT charts based on message error rate,” in Proc. IEEE Int. Symp. Inform. Theory, Lausanne, Switzerland, June 2002, p. 454. [25] E. Eleftheriou and S. Ölçer, “Low-density parity-check codes for multilevel modulation,” in Proc. IEEE Int. Symp. Inform. Theory, Lausanne, Switzerland, June 2002, p. 442.

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[26] H. Jin, A. Khandekar, and R. McEliece, “Irregular repeat-accumulate codes,” in Proc. 2nd Int. Symp. on Turbo Codes and Related Topics, Brest, France, Sept. 2000. [27] A. K. Khandani and P. Kabal, “Shaping multidimensional signal space—Part I: Optimum shaping, shell mapping,” IEEE Trans. Inform. Theory, vol. 39, pp. 1799–1808, Nov. 1993. [28] F. R. Kschischang and S. Pasupathy, “Optimal nonuniform signaling for Gaussian channels,” IEEE Trans. Inform. Theory, vol. 39, pp. 913–929, May 1993. , “Optimal shaping properties of the truncated polydisc,” IEEE [29] Trans. Inform. Theory, vol. 40, pp. 892–903, May 1994.

Masoud Ardakani (S’01) received the B.Sc. degree in electrical engineering from Isfahan University of Technology, Isfahan, Iran, in 1994, and the M.Sc. degree in electrical engineering from University of Tehran, Tehran, Iran, in 1997. He is currently working toward the Ph.D. degree at the University of Toronto, Toronto, ON, Canada. From 1997 to 1999, he was with the Electrical and Computer Engineering Research Center, Isfahan, Iran. His research interests are in the general area of digital communications, error-control coding, especially codes defined on graphs associated with iterative decoding, and MIMO systems. Mr. Ardakani was the recipient of a number of scholarships and awards including the Edward S. Rogers Sr. Scholarship while at the University of Toronto.

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Tooraj Esmailian received the B.Sc. degree from Isfahan University of Technology, Isfahan, Iran, in 1992, the M.Sc. degree from Sharif University of Technology, Tehran, Iran, in 1995, and the Ph.D. degree from the University of Toronto, Toronto, ON, Canada, in 2002, all in electrical engineering. While at the University of Toronto, he was supported by a grant from Motorola and received an international student award from the University of Toronto in 1998, a University of Toronto open fellowship in 1999, and Ontario graduate scholarships in 2000 and 2001. From 2002 to 2003, he was with the research and development group of Cogency Semiconductor, Inc., Kanata, ON, Canada, working on Home-Plug-based power line products. Since February 2004, he has been with the Research and Development Group, Edgewater Computer Systems, Inc., Kanata, ON, Canada. His research interests are in the general area of digital communications, especially multicarrier modulation, error control coding, and channel modeling.

Frank R. Kschischang (S’83–M’91–SM’00) received the B.A.Sc. degree (with honors) from the University of British Columbia, Vancouver, BC, Canada, in 1985 and the M.A.Sc. and Ph.D. degrees from the University of Toronto, Toronto, ON, Canada, in 1988 and 1991, respectively, all in electrical engineering. He is a Professor of Electrical and Computer Engineering and Canada Research Chair in Communication Algorithms at the University of Toronto, where he has been a faculty member since 1991. During 1997–1998, he spent a sabbatical year as a Visiting Scientist at the Massachusetts Institute of Technology (MIT), Cambridge. His research interests are focused on the area of coding techniques, primarily on soft-decision decoding algorithms, trellis structure of codes, codes defined on graphs, and iterative decoders. He has taught graduate courses in coding theory, information theory, and data transmission. Dr. Kschischang was the recipient of the Ontario Premier’s Research Excellence Award. From October 1997 to October 2000, he served as an Associate Editor for Coding Theory for the IEEE TRANSACTIONS ON INFORMATION THEORY. He also served as technical program co-chair for the 2004 IEEE International Symposium on Information Theory held in Chicago, IL.

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Performance of Coded OQPSK and MIL-STD SOQPSK With Iterative Decoding Lifang Li and Marvin K. Simon, Fellow, IEEE

Abstract—We show that military standard (MIL-STD) shapedoffset quadrature phase-shift keying (SOQPSK), a highly bandwidth-efficient constant-envelope modulation, can be represented in the form of a cross-correlated trellis-coded quadrature modulation. Similarly, we show that offset QPSK (OQPSK) can be decomposed into a “degraded” trellis encoder and a memoryless mapper. Based on the representations of OQPSK and MIL-STD SOQPSK as trellis-coded modulations (TCMs), we investigate the potential coding gains achievable from the application of simple outer codes to form a concatenated coding structure with iterative decoding. For MIL-STD SOQPSK, we describe the optimum receiver corresponding to its TCM form and then propose a simplified receiver. The bit-error rate (BER) performances of both receivers for uncoded and coded MIL-STD SOQPSK are simulated and compared with that of OQPSK and Feher-patented QPSK (FQPSK). The asymptotic BER performance of MIL-STD SOQPSK is also analyzed and compared with that of OQPSK and FQPSK. Simulation results show that, compared with their uncoded systems, there are significant coding gains for both OQPSK and MIL-STD SOQPSK, obtained by applying iterative decoding to either the parallel concatenated coding scheme or the serial one, even when very simple outer codes are used. Index Terms—Bit-error rate (BER), iterative decoding, offset quadrature phase-shift keying (OQPSK), trellis-coded modulation (TCM).

I. INTRODUCTION FFSET quadrature phase-shift keying (OQPSK) is a constant-envelope modulation that has no 180 phase shifts, and therefore has a much higher spectral containment than nonoffset QPSK when transmitted over bandlimited nonlinear channels. To further bandlimit an OQPSK signal, shaped OQPSK (SOQPSK) was introduced by Dapper and Hill [1] in the early 1980s, and its initial version was referred to as MIL-STD SOQPSK after it was adopted as part of a military standard. The frequency-shaping pulse for MIL-STD SOQPSK in its continuous phase modulation (CPM) representation is rectangular, and it lasts one bit interval [2]. Later on, more spectrally efficient versions of SOQPSK were developed by Hill [3], [4], and these variants are comparable to or even better than Feher-patented QPSK (FQPSK) [5] with regard

O

Paper approved by W. E. Ryan, the Editor for Modulation, Coding, and Equalization of the IEEE Communications Society. Manuscript received October 29, 2003; revised May 12, 2004. This work was performed at the Jet Propulsion Laboratory, California Institute of Technology, under a contract with the National Aeronautics and Space Administration. The authors are with the Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109-8099 USA (e-mail: marvin.k.simon@ jpl.nasa.gov). Digital Object Identifier 10.1109/TCOMM.2004.836557

to spectral and/or power efficiency [3], [4], [6], [7].1 These variants replace the linear variation of the phase over the single bit interval inherent to MIL-STD SOQPSK and analogous to that of minimum-shift keying (MSK), with other variations that extend over several bit intervals, and as such, are analogous to partial-response types of CPM systems such as Gaussian MSK (GMSK). Our focus in this paper is on MIL-STD SOQPSK, and thus, the more recent work of Hill and others will not be further elaborated on.2 Cross-correlated trellis-coded quadrature modulation (XTCQM) was introduced by Simon and Yan [8] as a generic modulation scheme, containing both memory and cross-correlation between the inphase (I) and quadrature (Q) channels. One specific embodiment of XTCQM is FQPSK [9]–[13]. In this paper, we show how MIL-STD SOQPSK can also be represented in the form of an XTCQM. Analogous to FQPSK, the representation of MIL-STD SOQPSK in the form of XTCQM allows identification of an optimum receiver for it, and allows for its inherent memory to be used in the iterative decoding of its coded and interleaved systems.3 We describe such an optimum receiver and investigate the potential improvement in power efficiency obtained from exploring the inherent memory of MIL-STD SOQPSK in a coded system with iterative decoding. Furthermore, we introduce a similar representation for OQPSK to that of XTCQM for MIL-STD SOQPSK. Based on this representation of OQPSK and the XTCQM representation of MIL-STD SOQPSK, we present a symbol-by-symbol mapping for both OQPSK and MIL-STD SOQPSK that is performed directly on the input I and Q data sequences in every symbol (2-bit) interval. This direct symbol-by-symbol mapping results in a clear interpretation for MIL-STD SOQPSK as being composed of a cross-correlated trellis encoder and a memoryless mapper and, for OQPSK, a “degraded” trellis encoder and a memoryless mapper. Such decomposition of OQPSK makes it possible to apply iterative decoding to coded OQPSK, where the degraded trellis code of OQPSK, after being remapped to its recursive version, can be viewed as an inner code of a concatenated coding structure. The performance of coded OQPSK with iterative decoding provides a lower bound to that of coded MIL-STD SOQPSK and FQPSK. In an effort to reduce receiver complexity while maintaining reasonable performance, we also propose a simplified receiver for MIL-STD SOQPSK. This simplified receiver only requires 1Note

that MIL-STD SOQPSK and its variants are unlicensed technologies.

2Suffice it to say that the ideas presented in this paper can be extended to those

more bandwidth-efficient SOQPSK variants. 3Note that iterative decoding of coded CPM waveforms in general can be found in [14]–[16] and the references therein.

0090-6778/04$20.00 © 2004 IEEE

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half of the matched filters used in the optimum receiver, and it has the same complexity as the simplified receiver for FQPSK [17]. We then simulate the bit-error rate (BER) performance of the optimum receiver and the simplified receiver for uncoded and coded MIL-STD SOQPSK, and compare it with that of OQPSK and FQPSK. In the coded case, we investigate the serially concatenated system with two very simple codes of different rates as outer codes and the parallel concatenated (turbo-like) system without any outer codes. Simulation results show that, compared with the uncoded case, there are significant coding gains for both parallel and serially concatenated systems, even with these simple codes. The remainder of the paper is organized as follows. In Section II, we show the decompositions of OQPSK and MIL-STD SOQPSK by describing a time-invariant symbol-interval trellis representation for them obtained from their CPM forms, and give equivalent transmitter implementations for them. Based on their symbol-interval trellis representations, in Section III, we present the explicit trellis-coded modulation (TCM) forms for OQPSK and MIL-STD SOQPSK. The optimum receiver and the simplified receiver for MIL-STD SOQPSK are described in Section IV. In Section V, we give the analytical asymptotic BER performance as well as the simulated BER results of uncoded MIL-STD SOQPSK with both receivers, and compare them with those of OQPSK and FQPSK. In Section VI, we describe the coded OQPSK and MIL-STD SOQPSK systems with iterative decoding and present the simulated BER performance of them, followed by our conclusions in Section VII. II. DECOMPOSITIONS OF OQPSK AND MIL-STD SOQPSK In order to show the time-invariant symbol-interval trellis representations of OQPSK and MIL-STD SOQPSK, as well as the corresponding equivalent transmitter implementations, we first give a brief review of the CPM representations of OQPSK and MIL-STD SOQPSK, and describe an eight-state bit-interval trellis diagram of OQPSK based on its CPM representation. A. CPM Representations of OQPSK and MIL-STD SOQPSK It is known that a conventional OQPSK signal can be represented as a full-response CPM signal in the form of [18]

where and denote the energy and duration of a bit, respectively, is the carrier frequency, and is an arbitrary phase constant that, without loss of generality, can be set to zero. In is the phase-modulation process that can be addition, expressed as (1) where, for unshaped OQPSK, the modulation index , the normalized phase pulse is simply a step function in each bit period (equivalently, the frequency pulse is a delta function, i.e., ), and the th element of the effective ternary ( 1, 0, 1) data sequence , , is related to the true input

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binary ( 1) data sequence by [18] (2) Detailed explanations on how (2) can be obtained through an eight-state (3-bit-state) trellis diagram of OQPSK (which will be described in the next subsection) are given in [19]. Note that the CPM representation of OQPSK can be implemented with the cascade of a precoder satisfying (2) and a conventional CPM modulator, such as in [18, Fig. 2-7]. To improve bandwidth efficiency, SOQPSK introduces pulse shaping into the above CPM representation. In particular, MIL-STD SOQPSK uses a rectangular frequency pulse in each ]. Equivalently, the phase pulse bit interval [i.e., varies linearly with time over each bit interval, and as such, is still characterized by a full-response precoded CPM as that of OQPSK. B. Eight-State Bit-Interval Trellis Diagram of OQPSK We now describe an eight-state bit-interval trellis diagram of OQPSK based on its CPM representation, which can be used to obtain (2) as well as to provide a means for demodulation of OQPSK using a Viterbi algorithm (VA). The 3-bit trellis state variable for describing the CPM representation of OQPSK is defined as follows. The first bit corresponds to whether the coming input bit, i.e., , corresponds to an even interval (I) or odd interval (Q). In particular, we choose a “1” if the incoming bit will be assigned to the I channel (even bit interval), and a “0” if the incoming bit will be assigned to the Q channel (odd bit interval). Note that successive input bits will be alternately assigned to even (I) and odd (Q) intervals. Therefore, states starting with a “0” can only transition to states starting with a “1,” and vice versa. The second and third bits of a trellis state correspond to the current phase state, which is represented by the current I and Q bits, respectively. Specifically, assuming a conventional Gray code mapping, the phase states , , , and are assigned the bit mappings (in the form of “IQ”) 00, 10, 11, 01, respectively. Note that at most one bit (I or Q) of the phase state can change during each state transition, i.e., the phase change is constrained to be rad or 0 in each bit interval. For example, the phase state 00 can only transition to phase states 10 and 01 , , depending on whether the or remain in phase state 00 incoming bit is an I bit or a Q bit, and whether its value is 1 rad, rad, or 0. The corresponding phase changes are and 0, respectively, which implies that the corresponding ’s are 1, 1, and 0, respectively. Fig. 1 is the eight-state trellis diagram illustrating the transitions from state to state in accordance with the above. The branches of the trellis are labeled with the value of that results in the transition due to the corresponding phase-state change. The input bits of the upper and lower branches emerging from each trellis state are “0” and “1,” respectively. It can be easily verified that the value of on each branch is in accordance with (2). A detailed example is given in [19]. As shown in Fig. 1, in each bit (half-symbol) interval, the eight-state for an trellis diagram of OQPSK gives the corresponding

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, denoted by , 0, 1, waveforms that can result for , 0, 1, and , 2, 3. In addition, the expressions for 0, 1, 2, 3, in each symbol interval are

Similarly, for MIL-STD SOQPSK, there are eight possible waveforms that can result for , denoted by , , and eight possible waveforms that can result for , denoted by , . The expressions , 0, 1, 2, 3, and , 0, 1, 2, 3, are for

Fig. 1. Eight-state bit-interval trellis diagram of OQPSK.

input bit associated with the transition between two states. Based on this trellis diagram, in Section II-C, we consider transitions between the four phase states corresponding to a pair of input bits in each symbol interval. C. Time-Invariant Symbol-Interval Trellis Representations of OQPSK and MIL-STD SOQPSK Without loss of generality, assume that in each symbol interval, the first bit of the input pair is always an I bit. Then, given the eight-state trellis diagram of OQPSK in Fig. 1, we can easily obtain the trellis between the four phase states , , , and , which is illustrated in Fig. 2. Note that we have drawn the trellis in expanded form with each transition interval (now 2 bits in duration) showing the transitions leaving from one of the four phase states. In Fig. 2, each branch is now labeled with a pair of output values, i.e., , . The corresponding pair of input bits is the same as the pair of bits representing the terminating phase state. Given the pair of outputs ( , ) for each transition, there is a pair of waveforms ( , ) associated with it, which represents the pair of symbols synchronously transmitted on the I and Q chanis the initial phase of each transition indicated by nels. Here the starting phase state, and in each symbol interval

and, in addition, , , , 0, 1, 2, 3. and for OQPSK Graphical illustrations of each and MIL-STD SOQPSK are given in [19] and [20]. Note that, for each phase-state transition in Fig. 2, the corresponding waveform pair ( , ) for MIL-STD SOQPSK is indicated above the associated branch, and the waveform pair , ) for OQPSK is indicated below the branch. ( D. Equivalent Transmitter Implementations of OQPSK and MIL-STD SOQPSK

for OQPSK. For MIL-STD SOQPSK, we have

Given the 16 possible combinations of output pair and initial phase as shown in Fig. 2, it is easily verified that for OQPSK, there are only two possible waveforms that can , denoted by , 0, 1, and only four possible result for

Based upon the foregoing trellis representations of OQPSK and MIL-STD SOQPSK, as well as the labeling of their I and Q waveforms described in Section II-C, for both OQPSK and MIL-STD SOQPSK, we can express the indexes of the specific waveforms transmitted for and in each symbol (2-bit) interval in terms of the two values in this interval and the phase state at the beginning of the interval (which itself depends on the previous values of ). Specifically, corresponding and in the symbol interval to ( even) and phase state at the start of this interval, we have

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Fig. 2. Expanded (branch leaving each state) time-invariant symbol-interval phase trellis for OQPSK and MIL-STD SOQPSK.

and , where the binary-coded decimal (BCD) representations of and are if if if if for OQPSK, with “ ” denoting the XOR operation for binary (0, 1) data. For MIL-STD SOQPSK, we have

th symbol interval is simply “ ” and it will be” in the th symbol interval, with the input come “ data and . Therefore, given the indexes of the cor, ], as indicated responding output waveform pair [ on each branch in the trellis representations of OQPSK and MIL-STD SOQPSK in Fig. 2, it is straightforward to express the indexes in terms of the input data. In particular, assume that , ] in the th symbol inthe output waveform pair is [ terval, and define the indexes and by

if if Then, for MIL-STD SOQPSK, we have if if The BCD representations of and for OQPSK and MIL-STD SOQPSK can be easily verified from Fig. 2. Therefore, the OQPSK and MIL-STD SOQPSK transmitters can be equivalently implemented with the cascade of a precoder satisfying (2), a signal mapper for choosing I and Q waveforms based on the above considerations, and a quadrature modulator. Block diagrams of these equivalent transmitters are given in [19] and [20]. III. INTERPRETATION OF OQPSK AND MIL-STD SOQPSK AS TCM In Section II, we expressed for OQPSK and MIL-STD SOQPSK the indexes of the specific waveforms transmitted for and in terms of the two values in each symbol interval and the starting phase at the beginning of each interval. In this section, we will show for both OQPSK and MIL-STD SOQPSK that the indexes of the transmitted waveforms for and can be directly expressed in terms of the I and Q channel input binary (0, 1) data. Specifically, in each symbol interval , we denote the I and Q input binary data as and , respectively. Note that the phase state in the

and for OQPSK

Graphical illustration of the implementations of MIL-STD SOQPSK and OQPSK based on the above mappings is given in Fig. 3. From Fig. 3(a), we see that MIL-STD SOQPSK can be clearly decomposed into a four-state trellis encoder and a memoryless signal mapper. This inherent four-state trellis encoder of MIL-STD SOQPSK has two binary (0, 1) inputs and and two waveform outputs , , where the trellis state is defined by the 2-bit sequence and . The trellis of this four-state encoder is exactly the one illus, ] indicated above each branch. trated in Fig. 2 with [ Since both the I and Q channel output waveform indexes depend on the cross-channel input data in addition to their own channel-input data, it is obvious that MIL-STD SOQPSK is a form of XTCQM. Similarly, from Fig. 3(b), we see that OQPSK can be interpreted as being composed of a “degraded” four-state

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ciency, a simplified receiver can be formed by grouping sets of waveforms together based on their similarities. In particular, the are divided into eight waveforms of the I channel output four groups, and so are the eight waveforms of the Q channel . For , the th ( 0, 1, 2, 3) group consists of output and . For , when 0, 2, the waveforms th group consists of waveforms and , and when 1, 3, of waveforms and . By defining and ( 0, 1, 2, 3) as the average of the waveforms in each and , respectively, we have group for

. Fig. 3. Alternative implementations of MIL-STD SOQPSK and OQPSK baseband signals in each symbol interval. (a) MIL-STD SOQPSK. (b) OQPSK.

trellis encoder and a memoryless signal mapper.4 The trellis of this degraded trellis encoder is exactly the one depicted in Fig. 2 , ] indicated below each branch. For OQPSK, with [ it is obvious from Fig. 3(b) that the I and Q channel-output waveform indexes depend only on their own channel-input data. Therefore, the signal mapping for OQPK is not cross-correlated, and independent I and Q channel detection is possible for OQPSK. Note that the decompositions of OQPSK and MIL-STD SOQPSK into a (degraded) trellis encoder and a memoryless mapper is important since, as will be shown in Section VI, it allows iterative decoding of the outer codes and these inherent trellis codes in their corresponding coded systems. IV. RECEIVER STRUCTURES FOR MIL-STD SOQPSK In accordance with the foregoing representation of MIL-STD SOQPSK as a TCM with four states, in this section, we present the corresponding optimum receiver structure and propose a simplified receiver structure for it. A. Optimum Receiver The optimum receiver employing a VA for maximum-likelihood sequence detection is illustrated in Fig. 4. It consists of a bank of eight matched filters (four in each of the I and Q channels) followed by a four-state trellis decoder. Note that although members of the I and Q signaling sets and (, ) do not all have equal energy, energy biases in the matched filters are not necessary when the matched-filter outputs are used in a VA. This is because the sum of the energies from allowable pairs of I and Q signals is constant, which has been explicitly verified in [20], and which comes from the fact that MIL-STD SOQPSK is a constant-envelope modulation. B. Simplified Receiver In a desire to reduce the complexity of the optimum receiver in Fig. 4 with the hope of not sacrificing significant power effi4What is meant by “degraded” trellis encoder is a degenerate form of such an encoder having no memory.

and , 0, Note that since 1, 2, 3, we have , , 0, 1. and ( 0, 1) are illustrated in The waveforms for and ( 0, 1) are [20], and the waveforms for and , but with opposite of the same shape as those of signs. Now we replace the waveform assignments of the group and by their corresponding average members for and both become ; and waveform, i.e., both become , and so on. Then, because of the relation between the I and Q coded bits and the BCD signal mapping in Fig. 3(a), the cross-correlation between the I and Q channel would disappear. This is because what distinguishes the two waveforms in each group for is the least-significant bit , and it is the middle bit for . If no distinction needs to be made in each group, we can simply drop the bits and , and just use the remaining two bits , and , in each channel to specify the transmitted waveform pair ( , ), , 0, 1, 2, 3. That is

By inspecting Fig. 3(a), we see that this is equivalent to the I channel signal being chosen based only on the I encoder outputs, and the Q channel signal being chosen based only on the Q encoder outputs. Thus, the cross-correlation of the encoder outputs in choosing the I and Q waveforms disappears, and the trellis structure of the modulation decouples into two independent (I and Q) two-state trellises. The simplified Viterbi receiver corresponding to the two-state trellises is illustrated in Fig. 5. In this simplified receiver, the I and Q decisions are separately generated by individual two-state VAs using the energy-biased correlations derived from the I and Q demodulated signals, respectively. Note that, since the energy and ( 0, 2) as well as per symbol is different for and ( 0, 2), the energy biases must be set in for the matched-filter outputs, as shown in Fig. 5. In this figure, and denote the energy per symbol for and ( 0, 1), respectively. It is easily calculated that and that . Of course, the two VAs for the I and Q channels can be combined into a single four-state VA if desired. Compared with the optimum Viterbi receiver, the simplified one reduces the number of correlators by half. This simplified receiver for MIL-STD SOQPSK is very

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Fig. 4. Optimum receiver structure for MIL-STD SOQPSK.

similar to the one for FQPSK described in [17, Fig. 5] in both structure and complexity. In addition, it is also similar in concept to the average matched filter (AMF) receivers for continuousphase frequency-shift keying (CPFSK) [21] and CPM schemes [2].

, ], respectively. The output waveform pair assoand [ ciated with each zero-input pair for initial phase state is [ , ]. Therefore, the minimum squared Euclidean distance between the shortest-length path and the all-zero sequence path is

V. PERFORMANCE COMPARISON OF UNCODED OQPSK, MIL-STD SOQPSK, AND FQPSK In this section, we first analyze the asymptotic BER performance of MIL-STD SOQPSK and then compare the simulated BER performance of uncoded MIL-STD SOQPSK with both the optimum receiver and the simplified receiver to that of OQPSK and FQPSK.

The average signal from

energy per symbol

is obtained

A. Asymptotic BER Performance of MIL-STD SOQPSK In this section, we obtain an expression for the minimum Euclidean distance associated with the symbol-by-symbol trellis representation of MIL-STD SOQPSK shown in Fig. 2. and the Supposing that the initial phase state is all-zero sequence is transmitted, from Fig. 2, we see that there is a path of length two that starts and ends at the same phase state but differs from the all-zero sequence path. This shortest-length error-event path passes through the phase state first and then back to the phase state . The two corresponding output waveform pairs are [ , ]

Therefore, the normalized minimum squared Euclidean distance is (3) Examination of other length-two error-event paths relative to transmitted sequences other than the all-zero sequence reveals that the smallest value of normalized squared Euclidean distance

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Fig. 5. Simplified Viterbi receiver for MIL-STD SOQPSK.

is still given by (3).5 Furthermore, error-event paths longer than length two correspond to larger values of squared Euclidean distance. Thus, the normalized minimum squared Euclidean distance for MIL-STD SOQPSK is given by (3). An alternative method for obtaining the result in (3) is by using the bit-by-bit CPM representation of MIL-STD SOQPSK, the details of which are given in [20]. When compared to OQPSK, which has the same normalized minimum squared Euclidean distance as binary (B)PSK, i.e., , there is a loss of 0.638 dB for MIL-STD SOQPSK. In addition, when compared with FQPSK, which is more spectrally efficient (see [4, Figs. 6 and 9]) and for which it was shown , there is an asymptotic gain of in [9] and [10] that 0.441 dB for MIL-STD SOQPSK. B. Simulation Results We have simulated the optimum receiver structure shown in Fig. 4 for uncoded MIL-STD SOQPSK, as well as the simplified receiver structure shown in Fig. 5. The numerical results are illustrated in Fig. 6. Also shown in Fig. 6 are the simulated 5In fact, we believe that SOQPSK, in general, has the uniform error property, since the Euclidean distance between two effective data sequences  and  in its CPM representation only depends on their difference sequence   [6].

= 0

BER performance of FQPSK with the optimum receiver and with a simplified receiver, which are taken from [9], [10], and [17]. Furthermore, in Fig. 6, we have also given the simulated BER performance of uncoded OQPSK, which provides a lower bound for both MIL-STD SOQPSK and FQPSK at high values. To simulate the performance of OQPSK with Viterbi decoding, the bit-interval eight-state trellis illustrated in Fig. 1 is used. In addition, we have also simulated the case where the symbol-interval trellis representation of OQPSK illustrated in Fig. 2 is used. The simulation results using these two different trellis representations of OQPSK with a Viterbi decoder are the same, and they also match the theoretical BER of BPSK and QPSK. , MIL-STD SOQPSK From Fig. 6, we see that at BER with the optimum receiver is about 0.31 dB worse than OQPSK, but is about 0.46 dB better than FQPSK with optimal reception. The simplified MIL-STD SOQPSK receiver has a performance , that is very close to the optimum receiver: at BER loss is only about 0.12 dB. For FQPSK, the perforthe mance gap between the simplified receiver and the optimum one is bigger: the loss at BER is roughly 0.27 dB. The smaller performance gap between the simplified receiver and the optimum receiver for MIL-STD SOQPSK may be due to the fact that the simplified MIL-STD SOQPSK receiver only reduces the number of matched filters in the optimum receiver

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Fig. 7. Recursive version of the I and Q encoders for MIL-STD SOQPSK and OQPSK. (a) MIL-STD SOQPSK. (b) OQPSK.

Fig. 6. BER performance comparison of uncoded systems.

by a factor of two, while the simplified FQPSK receiver reduces it by a factor of four. VI. CODED OQPSK AND MIL-STD SOQPSK WITH ITERATIVE DECODING Since both OQPSK and MIL-STD SOQPSK have inherent trellis codes, as shown in Fig. 3, these trellis codes can be viewed as the inner code of a concatenated code in coded OQPSK and MIL-STD SOQPSK systems. As was true for the FQPSK applications [11]–[13], in order to realize coding gains from the concatenation of the outer code and the inherent inner code of OQPSK or MIL-STD SOQPSK, the I and Q inner encoders of the equivalent transmitters in Fig. 3 must be replaced by their recursive equivalents. Therefore, before presenting the serially and parallel concatenated coding structures for coded OQPSK and MIL-STD SOQPSK systems and introducing the simplified iterative decoding process for MIL-STD SOQPSK, we first describe the recursive I and Q encoders for both OQPSK and MIL-STD SOQPSK. A. Recursive I and Q Encoders of OQPSK and MIL-STD SOQPSK For OQPSK and MIL-STD SOQPSK, given the original trellis of each nonrecursive I or Q encoder, the remapped recursive encoder must have a trellis for which the output bits corresponding to each transition between states remain unchanged. The only changes allowed are the input bit(s) associated with each transition. This is to guarantee that the allowable OQPSK or MIL-STD SOQPSK encoder output sequences remain unchanged so that the remapping does not change the envelope and spectral characteristics of the modulated signals [11]. Under this consideration, it can be easily

shown that only one recursive version is allowed for each of the encoders in Fig. 3, and the recursive equivalents of these encoders are illustrated in Fig. 7. Note that if we replace the original encoders of OQPSK and MIL-STD SOQPSK, shown in Fig. 3, with their recursive equivalents, shown in Fig. 7, simulation results (not illustrated) show that the BERs of the uncoded OQPSK and MIL-STD SOQPSK are now twice those values, of the original systems shown in Fig. 6 at high similar to the phenomena observed for turbo codes and serially concatenated codes [22]. However, for the coded cases, the recursive versions of the encoders provide significant coding gains, which will be demonstrated through a few examples in the following subsections. B. Serial Concatenation We first consider the serially concatenated coded OQPSK and MIL-STD SOQPSK systems, similar to the coded FQPSK system illustrated in [13, Fig. 5(b)]. Specifically, the input data are first encoded by an outer encoder, interleaved, and then applied to the I and Q channels of the equivalent baseband transmitter for OQPSK or the equivalent baseband transmitter for MIL-STD SOQPSK shown in Fig. 3, where the I and Q inner encoders are replaced by their recursive counterparts in Fig. 7. After transmission over the additive White Gaussian noise (AWGN) channel, for MIL-STD SOQPSK, the received signals of the I and Q channels are passed through a bank of eight matched filters to generate a total of 16 correlator outputs, as shown in Fig. 4. For OQPSK, the I channel received signal is passed through one matched filter, and the Q channel received signal is passed through two matched filters to generate a total of six correlator outputs. These correlator outputs are then used by a four-state soft-input soft-output (SISO) [23] iterative decoder as branch metrics. For OQPSK, since there is no correlation between the I channel and the Q channel, two separate two-state iterative decoders can be used instead of a combined four-state decoder. These two decoding schemes have the same BER performance, which is verified by our simulations. For MIL-STD SOQPSK, the four-state joint I and Q channel iterative decoder must be used, since there exist correlations between these two channels. Note that in addition to the SISO module for decoding the inner code provided by OQPSK or MIL-STD SOQPSK, there is also a SISO module for decoding the outer code. To reduce the complexity of the inner and outer SISOs, we simulate the max-log versions of them, which are equivalent to modified soft-output VAs (SOVAs) [24]. The simulation results will be given in Section VI-E.

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C. Parallel Concatenation We now consider a parallel concatenated coding scheme of the turbo-coding type similar to the coded FQPSK system illustrated in [13, Fig. 5(c)]. In this turbo-coding scheme, there is no explicit outer code, but rather the input bits and their interleaved versions are applied to the inherent I and Q channel encoders of OQPSK or MIL-STD SOQPSK directly. Therefore, there is no corresponding outer SISO decoder at the receiver. Instead, in each iteration, the output extrinsic information of the I and Q input bits from the inner SISO decoder(s) are fed back as reliabilities of the opposite (I to Q and Q to I) bits after appropriate interleaving and deinterleaving. At the end of the last iteration, the extrinsic information for the I input bits is combined with the deinterleaved extrinsic information for the Q input bits to produce a decision on the input information bits. D. Simplified Iterative Decoding for MIL-STD SOQPSK For both serial and parallel concatenations of coded MIL-STD SOQPSK, it is obvious that the inner SISO decoder for decoding the inherent code of MIL-STD SOQPSK can be replaced with two separate and simplified SISO decoders, using the trellis diagrams of the simplified I and Q encoders shown in Fig. 5. Now a joint SISO decoder for decoding the I and Q input bits of the inherent encoder in MIL-STD SOQPSK is not necessary, due to the decoupling of the I and Q channel transmissions. Consequently, the number of matched filters required at the receiver can be reduced by half. It will be shown in the next subsection that although there is a noticeable performance loss due to the simplification, this reduced-complexity scheme has a performance comparable to that of coded FQPSK with iterative decoding using the full-blown matched-filter configuration [10], and it is superior to coded FQPSK with iterative decoding using a simplified receiver [11]–[13], [17].6 E. Simulation Results In this section, we present simulation results for serially and parallel concatenated coded OQPSK and MIL-STD SOQPSK with iterative decoding. Also presented are simulation results of the simplified iterative decoding for MIL-STD SOQPSK, as described in the previous subsection. In addition, for comparison, we have simulated in each case the performance of coded FQPSK with iterative decoding using full-blown matched filters [10], as well as using a simplified receiver structure [11]–[13], [17]. In our simulations, for both parallel and serially concatenated cases, no termination bits are added anywhere and the decoder state metrics are uniformly initialized in each data block. For serial concatenation, we have investigated two different outer codes. The first one is a recursive rate-1/2 optimum four-state convolutional code with minimum free distance . The generator polynomials and of its nonrecursive equivalent have an octal representation of 5 and 7, respectively. The second outer code considered is a higher rate (i.e., 3/4) recursive code, obtained by treating the the first one as a rate-3/6 code, and then 6Note, however, that MIL-STD SOQPSK is less bandwidth-efficient than FQPSK.

Fig. 8. BER performance of coded systems with serial and parallel concatenations.

puncturing two of the output bits. The minimum free distance . Compared to the rate-1/2 code, this of this code is code is more bandwidth-efficient,7 but we will see that it is less power-efficient. In our simulations, the number of iterations is , and the interleaver block size is 2048 bits (1024 information bits) for the rate-1/2 outer code. For the rate-3/4 1364 outer code, the interleaver block size is chosen to be bits (1364 (3/4) = 1023 information bits). Note that since is small, for better performance, we have scaled the extrinsic information from the inner SISO(s) and the outer SISO by a factor of 0.75 for the rate-1/2 outer code [25]. For the rate-3/4 outer code, we have scaled the extrinsic information from the inner SISO(s) to the outer SISO by a factor of 0.7, while leaving the extrinsic information from the outer SISO to the inner SISO(s) unchanged. For parallel concatenation,8 the number of iterations , and the interleaver block size is 2048 bits (1024 is information bits). In this case, no scaling factor is applied to the extrinsic information from the inner SISO decoder. Fig. 8 shows the BER performance of the three parallel concatenated coding systems as well as that of the three serially concatenated coding systems with both the rate-1/2 outer code and the rate-3/4 outer code. From Fig. 8 we see that, for serial concatenation with the rate-1/2 outer code, coded OQPSK has the best performance, as expected, and its performance is -differvery similar to that of serially concatenated coded ential QPSK (DQPSK) with the same rate-1/2 outer code [26]. The performance of coded MIL-STD SOQPSK is only slightly worse than that of coded OQPSK, and it is noticeably better than , the required gap that of coded FQPSK. At BER between coded OQPSK and coded MIL-STD SOQPSK is less 7The rate-1/2 code gives an overall rate of 1 bit/symbol for the serially concatenated system, while the rate-3/4 code gives an overall rate of 1.5 bits/symbol. 8The overall rate of the parallel concatenated coding scheme is 1 bit/symbol.

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LI AND SIMON: PERFORMANCE OF CODED OQPSK AND MIL-STD SOQPSK WITH ITERATIVE DECODING

than 0.02 dB, while it is about 0.09 dB between coded OQPSK and coded FQPSK. The simplified iterative decoding of coded MIL-STD SOQPSK is about 0.1 dB worse than the nonsimpli, and it is very close to that of fied decoding at BER coded FQPSK with iterative decoding using full-blown matched filters. The simplified decoding of coded FQPSK, however, is about 0.15 dB worse than the nonsimplified decoding at BER . Compared with the uncoded cases shown in Fig. 6, , the coding gains for OQPSK, MIL-STD SOat BER QPSK, and FQPSK are 7.05, 7.35, and 7.78 dB, respectively, and the coding gains for the simplified receivers of MIL-STD SOQPSK and FQPSK are 7.43 and 7.88 dB, respectively. Therefore, of the three modulation schemes, the more bandwidth-efficient a scheme is, the greater the coding gain. For serial concatenation with the rate-3/4 outer code, the relations of the five BER curves are very similar to those for the rate-1/2 outer code, except that now the performance of the simplified decoding of coded MIL-STD SOQPSK is almost indistinguishable from that of coded FQPSK with nonsimplified decoding. In addition, unlike the rate-1/2 outer-code case, the increases, BERs are not decreasing so dramatically as ranges. This is because the interleaver especially at high size is smaller in the rate-3/4 outer-code case, and the rate-3/4 code has a smaller free distance than the rate-1/2 code [22]. Still, with this rate-3/4 outer code, there are significant coding gains , for when compared with the uncoded case. At BER OQPSK, MIL-STD SOQPSK, and FQPSK, they are 5.77, 6.03, and 6.41 dB, respectively, and for the simplified decoding of MIL-STD SOQPSK and FQPSK, they are 6.10 and 6.45 dB, respectively. Finally, we observe from Fig. 8 that, with the simple parallel concatenated coding scheme, the relations of the five BER curves are very similar to those of their serial concatenation counterparts, except that now the BERs are decreasing much increases. In fact, the five BER curves start slower as to show leveling off even before the BER reaches , which is similar to the “error floor” scenario of parallel concatenated convolutional codes [22]. For this simple parallel concatenated coding scheme, when compared with the uncoded systems, are 4.76, 5.03, and 5.42 dB the coding gains at BER for OQPSK, MIL-STD SOQPSK, and FQPSK, respectively, and they are 5.08 and 5.46 dB for the simplified decoding of MIL-STD SOQPSK and FQPSK, respectively. Although these coding gains are not as big as those obtained with serial concatenation of the rate-1/2 outer code, or even the higher rate (i.e., 3/4) outer code, the iterative decoding complexity of the parallel concatenated coding scheme is much lower than its serial concatenation counterparts due to the lack of necessity of an outer SISO decoder. Therefore, tradeoffs must be made between receiver complexity, coding gain, and bandwidth efficiency when designing coded OQPSK, MIL-STD SOQPSK, and FQPSK systems. VII. CONCLUSION We have shown how both OQPSK and MIL-STD SOQPSK can be decomposed into a (degraded) trellis encoder and a memoryless mapper. When concatenated with an outer code, coded

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OQPSK with iterative decoding provides a lower bound to the performance of coded MIL-STD SOQPSK and FQPSK. For MIL-STD SOQPSK, we have analyzed its asymptotic BER performance, presented the optimum receiver structure, and proposed a simplified receiver. The simplified receiver maintains good performance with reduced complexity. Simulation results show that the performance of coded MIL-STD SOQPSK comes very close to that of coded OQPSK, and is noticeably better than that of coded FQPSK, though MIL-STD SOQPSK is less bandwidth-efficient than FQPSK. When compared with their uncoded systems, there are significant coding gains for both coded OQPSK and MIL-STD SOQPSK applying iterative decoding to either the parallel concatenated coding scheme or the serial one, even when very simple outer codes are used. ACKNOWLEDGMENT The authors would like to thank Dr. D. Divsalar and D. Lee for many helpful discussions. REFERENCES [1] M. J. Dapper and T. J. Hill, “SBPSK: A robust bandwidth-efficient modulation for hard-limited channels,” in Proc. IEEE Military Commun. Conf., Los Angeles, CA, Oct. 1984. [2] J. B. Anderson, Digital Phase Modulation. New York: Plenum, 1986. [3] T. J. Hill, “An enhanced, constant envelope, interoperable shaped offset QPSK (SOQPSK) waveform for improved spectral efficiency,” in Proc. Int. Telemetering Conf., San Diego, CA, Oct. 2000, pp. 127–136. [4] , “A nonproprietary, constant envelope, variant of shaped offset QPSK (SOQPSK) for improved spectral containment and detection efficiency,” in Proc. IEEE Military Commun. Conf., vol. 1, Los Angeles, CA, Oct. 2000, pp. 347–352. [5] P. S. K. Leung and K. Feher, “F-QPSK—A superior modulation technique for mobile and personal communications,” IEEE Trans. Broadcast., vol. 39, pp. 288–294, June 1993. [6] M. Geoghegan, “Implementation and performance results for trellis detection of SOQPSK,” in Proc. Int. Telemetering Conf., Las Vegas, NV, Oct. 2001, pp. 531–540. , “Bandwidth and power efficiency trade-offs of SOQPSK,” in Proc. [7] Int. Telemetering Conf., San Diego, CA, Oct. 2002, pp. 780–789. [8] M. K. Simon and T.-Y. Yan, “Cross-Correlated Trellis-Coded Quadrature Modulation,” U.S. patent filed Oct. 1999. [9] M. K. Simon and T.-Y. Yan. (1999, May) Performance evaluation and interpretation of unfiltered Feher-patented quadrature phase-shift keying (FQPSK). Telecommunications and Mission Operations Prog. Rep. [Online]. Available: http://tmo.jpl.nasa.gov/tmo/progress_report/42-137/137C.pdf , “Unfiltering Feher-patented quadrature phase-shift keying [10] (FQPSK): Another interpretation and further enhancements: Parts 1, 2,” Appl. Microwave Wireless Mag., vol. 12, p. 76–96/100–105, Feb./Mar. 2000. [11] M. K. Simon and D. Divsalar. (2001, May) A reduced-complexity, highly power-/bandwidth-efficient coded Feher-patented quadrature phase-shift keying system with iterative decoding. Telecommunications and Mission Operations Prog. Rep. [Online]. Available: http://tmo.jpl.nasa.gov/tmo/progress_report/42-145/145A.pdf , (2001, Aug.) Further results on a reduced-complexity, highly [12] power-/bandwidth-efficient coded Feher-patented quadrature phaseshift keying system with iterative decoding. Interplanetary Network Prog. Rep. [Online]. Available: http://tmo.jpl.nasa.gov/tmo/progress_report/42-146/146I.pdf [13] , “A reduced complexity highly power/bandwidth efficient coded FQPSK system with iterative decoding,” in Proc. IEEE Int. Conf. Communications, vol. 7, Helsinki, Finland, June 2001, pp. 2204–2210. [14] P. Moqvist and T. M. Aulin, “Serially concatenated continuous phase modulation with iterative decoding,” IEEE Trans. Commun., vol. 49, pp. 1901–1915, Nov. 2001. [15] K. R. Narayanan and G. L. Stüber, “Performance of trellis-coded CPM with iterative demodulation and decoding,” IEEE Trans. Commun., vol. 49, pp. 676–687, Apr. 2001.

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[16] M. R. Shane and R. D. Wesel, “Reduced complexity iterative demodulation and decoding of serial concatenated continuous phase modulation,” in Proc. IEEE Int. Conf. Commununications, vol. 3, Apr. 2002, pp. 1672–1676. [17] D. Lee, M. K. Simon, and T.-Y. Yan, “Enhanced performance of FQPSK-B receiver based on trellis-coded Viterbi demodulation,” in Proc. Int. Telemetering Conf., San Diego, CA, Oct. 2000, pp. 631–640. [18] M. K. Simon, Bandwidth-Efficient Digital Modulation With Application to Deep-Space Communication. New York: Wiley, 2003. [19] M. K. Simon and L. Li. (2003, Aug.) A cross-correlated trellis-coded quadrature modulation representation of MIL-STD shaped offset quadrature phase-shift keying. Interplanetary Network Prog. Rep. [Online]. Available: http://ipnpr.jpl.nasa.gov/progress_report/42-154/154J.pdf [20] L. Li and M. K. Simon. (2004, Feb.) Performance of coded offset quadrature phase-shift keying (OQPSK) and MIL-STD shaped OQPSK (SOQPSK) with iterative decoding. Interplanetary Network Prog. Rep. [Online]. Available: http://ipnpr.jpl.nasa.gov/progress_report/42-156/156A.pdf [21] W. Osborne and M. B. Luntz, “Coherent and noncoherent detection of CPFSK,” IEEE Trans. Commun., vol. COM-22, pp. 1023–1034, Aug. 1974. [22] S. Benedetto, D. Divsalar, G. Montorsi, and F. Pollara, “Serial concatenation of interleaved codes: Performance analysis, design, and iterative decoding,” IEEE Trans. Inform. Theory, vol. 44, pp. 909–926, May 1998. , “A soft-input soft-output APP module for iterative decoding of [23] concatenated codes,” IEEE Commun. Lett., vol. 1, pp. 22–24, Jan. 1997. [24] M. P. C. Fossorier, F. Burkert, S. Lin, and J. Hagenauer, “On the equivalence between SOVA and max-log-MAP decodings,” IEEE Commun. Lett., vol. 2, pp. 137–139, May 1998. [25] D. Kim, T. Kwon, J. R. Choi, and J. J. Kong, “A modified two-step SOVA-based turbo decoder with a fixed scaling factor,” in Proc. IEEE Int. Symp. Circuits and Systems, vol. 4, Geneva, Switzerland, May 2000, pp. 37–40. [26] K. R. Narayanan and G. L. Stüber, “A serial concatenation approach to iterative demodulation and decoding,” IEEE Trans. Commun., vol. 47, pp. 956–961, July 1999.

Lifang Li received the B.S. and M.S. degrees from Tsinghua University, Beijing, China, in 1993 and 1996, respectively, and the Ph.D. degree from the California Institute of Technology, Pasadena, in 2000, all in electrical engineering. From February 2000 to April 2000, she worked as a postdoctoral scholar in the Advanced Communications Systems Concepts Group, Jet Propulsion Laboratory (JPL), Pasadena, CA. In May 2000, she joined the Exeter Group, Inc., Los Angeles, CA. Since November 2002, she has been a Member of the Technical Staff with the Information Processing Group, JPL.

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Marvin K. Simon (S’60–M’66–SM’75–F’78) is currently a Principal Scientist at the Jet Propulsion Laboratory (JPL), California Institute of Technology (Caltech), Pasadena, where for the last 36 years he has performed research as applied to the design of NASA’s deep-space and near-earth missions, and which has resulted in the issuance of nine U.S. patents, 25 NASA Tech Briefs, and four NASA Space Act awards. He is known as an internationally acclaimed authority on the subject of digital communications with particular emphasis in the disciplines of modulation and demodulation, synchronization techniques for space, satellite, and radio communications, trellis-coded modulation, spread spectrum and multiple access communications, and communication over fading channels. He has also held a joint appointment with the Electrical Engineering Department at Caltech. He has published over 195 papers on the above subjects and is coauthor of 10 textbooks, including Telecommunication Systems Engineering (Englewood Cliffs, NJ: Prentice-Hall, 1973, and New York: Dover Press, 1991), Phase-Locked Loops and Their Application (New York: IEEE Press, 1978), Spread Spectrum Communications, Vols. I, II, and III (Computer Science Press, 1984 and New York: McGraw-Hill, 1994), An Introduction to Trellis Coded Modulation with Applications (MacMillan, 1991), Digital Communication Techniques: Vol. I (Englewood Cliffs, NJ: Prentice-Hall, 1994) and Digital Communication Over Fading Channels: A Unified Approach to Performance Analysis (New York: Wiley, 2000, 2nd ed. available Nov. 2004), Probability Distributions Involving Gaussian Random Variables—A Handbook for Engineers and Scientists (Norwell, MA: Kluwer, 2002) and Bandwidth-Efficient Digital Modulation with Application to Deep-Space Communication (New York: Wiley, 2003). His work has also appeared in the textbook Deep Space Telecommunication Systems Engineering (New York: Plenum, 1984), and he is coauthor of a chapter entitled “Spread Spectrum Communications” in the Mobile Communications Handbook (Boca Raton, FL: CRC Press, 1995), Communications Handbook (Boca Raton, FL: CRC Press, 1997), and the Electrical Engineering Handbook (Boca Raton, FL: CRC Press, 1997). His work has also appeared in the textbook Deep Space Telecommunication Systems Engineering (New York: Plenum, 1984). Dr. Simon is the corecipient of the 1986 Prize Paper Award in Communications of the IEEE Vehicular Technology Society and the 1999 Prize Paper Award of the IEEE Vehicular Technology Conference (VTC’99-Fall), Amsterdam, The Netherlands. He is a Fellow of the IAE, and winner of a NASA Exceptional Service Medal, a NASA Exception Engineering Achievement Medal, the IEEE Edwin H. Armstrong Achievement Award, and most recently, the IEEE Millennium Medal.

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Design and Performance of Turbo Gallager Codes Giulio Colavolpe

Abstract—The most powerful channel-coding schemes, namely, those based on turbo codes and low-density parity-check (LDPC) Gallager codes, have in common the principle of iterative decoding. However, the relative coding structures and decoding algorithms are substantially different. This paper shows that recently proposed novel coding structures bridge the gap between these two schemes. In fact, with properly chosen component convolutional codes, a turbo code can be successfully decoded by means of the decoding algorithm used for LDPC codes, i.e., the belief-propagation algorithm working on the code Tanner graph. These new turbo codes are here nicknamed “turbo Gallager codes.” Besides being interesting from a conceptual viewpoint, these schemes are important on the practical side because they can be decoded in a fully parallel manner. In addition to the encoding complexity advantage of turbo codes, the low decoding complexity allows the design of very efficient channel-coding schemes. Index Terms—Belief propagation (BP), iterative decoding, lowdensity parity-check (LDPC) codes, turbo codes.

I. INTRODUCTION N RECENT years, great attention has been devoted to powerful coding schemes which achieve near-Shannon limit performance with affordable decoding complexity. In 1993, turbo codes were first introduced [1], [2]. A turbo encoder is the parallel concatenation of two simple constituent encoders interconnected through an interleaver. The corresponding decoding process is based on an iterative algorithm in which each component decoder takes advantage of the extrinsic information produced by the other decoder at the previous step. This iterative decoding process usually employs soft-output component decoders based on the algorithm by Bahl et al. (BCJR) [3] or its simplified logarithmic versions [4]. After the invention of turbo codes, this decoding technique was extended to serially concatenated convolutional codes (SCCCs). SCCCs are based on a serial concatenation, through an interleaver, of an outer code and an inner recursive code [5]. The extraordinary success of turbo codes has stimulated the rediscovery of another class of codes exhibiting similar performance and characteristics [6]. These codes, called low-density parity-check (LDPC) codes, were first introduced by Gallager [7] in their original regular version. In terms of performance, regular LDPC Gallager codes are only slightly inferior to parallel and serial concatenated convolutional codes. However, in

I

Paper approved by R. D. Wesel, the Editor for Coding and Communication Theory of the IEEE Communications Society. Manuscript received October 11, 2002; revised January 2, 2004, and April 9, 2004. This paper was presented in part at the Third International Symposium on Turbo Codes and Related Topics, Brest, France, September 2003. The author is with the Dipartimento di Ingegneria dell’Informazione, Università di Parma, I-43100 Parma, Italy (e-mail: [email protected]). Digital Object Identifier 10.1109/TCOMM.2004.836566

their irregular version, they exhibit an impressive performance outperforming the best known turbo codes [8]–[10]. LDPC codes are linear codes specified by a sparse -regular LDPC code, as origparity-check matrix. A inally defined by Gallager, is a binary linear code such that every code bit participates in exactly parity-check equations, code bits. In and every check equation involves exactly has other words, the corresponding parity-check matrix ones in each column and ones in each row. As originally suggested by Tanner [11], LDPC codes are well represented by bipartite graphs in which one set of nodes, the variable nodes, represents the elements of a codeword, and the other set of nodes, the check nodes, corresponds to the set of parity-check constraints which define the code. In the following, we will denote these bipartite graphs as Tanner graphs. Regular LDPC codes are such that all nodes of the same type have an equal number of edges. On the contrary, for irregular LDPC codes, the node degree of each node in each set is not equal, but chosen according to an optimized distribution [8]–[10]. As already mentioned, some irregular LDPC codes have better performance with respect to turbo codes for equal codeword length. In addition, they have other advantages over turbo codes. In fact, belief propagation (BP), the iterative decoding algorithm for LDPC codes which works on the code Tanner graph, can be fully parallelized and potentially implemented at high speed. Moreover, in [12], low-complexity decoders that closely approximate BP in performance have been designed for these codes, extending the original work in [7]. In the following, all these decoders, i.e., BP and the low-complexity decoders described in [12], will be referred to as message-passing decoders. On the negative side, one major drawback of LDPC codes is represented by their high encoding complexity [13]. In this paper, we will describe and analyze two classes of coding schemes, originally proposed in [14], which can be easily decoded at high speed by means of message-passing decoders. The first class is represented by single convolutional codes with some specific algebraic properties which ensure that message-passing decoders can operate successfully on the code Tanner graph. In this case, a message-passing schedule, specifically tailored for this class of codes, will be described. Since message-passing decoding algorithms are very simple and characterized by a decoding complexity which does not directly depend on the code constraint length, they can be used to decode convolutional codes with a large constraint length, and hence, potentially characterized by a large free distance.1 1The decoding complexity of a BCJR or a Viterbi algorithm grows exponentially with the code constraint length, therefore, these algorithms cannot be practically used for codes with a very large free distance.

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The second class of codes is based on code concatenation. The resulting coding schemes and the corresponding decoding algorithms combine the advantages of turbo codes and LDPC codes. In fact, the coding structures are very simply based on the parallel or serial concatenations of simple convolutional codes through interleavers, in which the component convolutional codes are constrained to have the above-mentioned specific algebraic properties which ensure that message-passing decoders can work successfully on the Tanner graph of the overall code. In this way, combining the coding structure of parallel or serial concatenated codes and the decoding algorithm for LDPC codes, a new architecture results, which can be effectively implemented with simple coding and decoding operations. In terms of decoding complexity, the proposed schemes represent a valid alternative to classical turbo codes and give a solution to the encoding problem of LDPC codes [13]. The proposed codes are different from low-density convolutional codes proposed in [15] which are time-varying, periodical, binary convolutional codes.

II. BACKGROUND: TANNER GRAPHS AND BELIEF PROPAGATION Message-passing decoding algorithms for LDPC codes are based on an iterative exchange of messages along the edges of a code Tanner graph [7], [11], [16]. This graph, which can be of the drawn by direct inspection of a parity-check matrix code, is composed of two classes of nodes: variable nodes corresponding to the code bits, and check nodes corresponding to the code constraints. Check nodes are connected by edges to variable nodes on which they depend. A cycle is a closed path in the graph, and its length is defined as the corresponding number of path edges. The length of the shortest cycle is the girth of the graph. We now review the BP decoding algorithm. Let us denote by a suitable message propagating on an edge of the code Tanner graph. This message represents the log-likelihood ratio related to the code bit corresponding to the variable node from which the considered edge originates. At the th iteration, we denote by a message sent from a variable node to a check a message in the opposite direction. A node, and by variable node of degree receives and processes the messages , , and sends back to its th neighboring check node the message [7], [12] (1)

where is the initial message received by the considered variable node as a function of the channel output corresponding to the considered code bit. When , the variable node simply propagates its initial received message . It may be observed that the message in (1) does not depend on the message previously received on the same edge, i.e., only extrinsic information is exchanged.

receives and processes the mesA check node of degree sages , , and sends back to its th neighboring variable node the message [7], [12]

(2)

A simplified min-sum version of the updating rule (2) is described in [17]. The decoding algorithm proceeds iteratively until the code parity-check constraints are all verified or a maximum number of iterations is reached. Although this decoding algorithm is provably optimal for bipartite graphs without cycles [12], in practice, it is necessary to only avoid cycles of length up to four to attain good performance [10]. Other message-passing algorithms can be devised based on different types of messages, possibly taking on values in some finite message alphabet, and different updating rules [12]. All these algorithms can be applied to decode the codes described in Sections III–V. A message-passing schedule in a factor graph is the specification of the order in which messages are updated. In general, the so-called flooding schedule is adopted to decode LDPC codes [18]. In this case, in each iteration, all variable nodes, and subsequently, all check nodes, pass new messages to their neighbors. As can be easily understood, this schedule is well suited for a fully parallel implementation of the decoder. III. SINGLE CONVOLUTIONAL CODE In this section, we describe a class of convolutional codes which can be decoded by means of the described message-passing algorithms. We consider a single convolutional code truncated to a block code of a given codeword length. Tailbiting can be used to convert the convolutional code to a block code with no rate loss [19]. As mentioned in the previous section, a message-passing decoding algorithm operating on a code Tanner graph is an effective technique if the graph girth is at least six. In the following, we analyze the conditions that a convolutional code must satisfy in order to have a Tanner graph without cycles of length four.2 The girth of a code Tanner graph can be easily determined by analyzing the corresponding code parity-check matrix. In fact, a bipartite graph has girth of at least six if and only if (iff) the corresponding parity-check matrix has no rectangles, i.e., four one’s in two separate rows which define the corners of a rectangle. This property can be easily verified by observing that if a rectangle is present, the row and column indexes of its corners determine the variable and check nodes connected by a cycle of length four. The condition for the absence of cycles of length four can be directly translated into a condition on the code parity-check equations. For illustration purposes, ratesystematic 2We emphasize that in this paper, the term “Tanner graph” is used to denote a graph directly derived from the code parity-check matrix and without the explicit representation of hidden (state) variable nodes present in the so-called Wibergtype graphs [17].

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COLAVOLPE: DESIGN AND PERFORMANCE OF TURBO GALLAGER CODES

codes are considered, but the conditions being described can be convolutional codes. easily generalized to raterecursive systematic convolutional (RSC) A ratecode is described by a polynomial generator matrix of the form (3) where and

is a identity matrix, is a unit-delay operator, is a column vector of elements of the form (4)

where , , and denote the feedforward and feedback polynomials, respectively. These polynomials may be expressed in the form3

(5) where , , and denote suitable integers which specify the considered code. RSC code by using the We may also describe a ratecorresponding parity-check equation at discrete time . This description has an immediate visual impact on the code Tanner graph. The encoder for the code described by (3) receives, at , and discrete time , the information bits , a parity produces, assuming without loss of generality bit given by the following parity-check equation:

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six, a necessary and sufficient condition is that the index differ, ences are all distinct. and In particular, for a rate-1/2 code, the following proposition holds. Proposition 2: For a systematic rate-1/2 code: and , the bipartite 1) if graph has no cycles; and , denoting by 2) if and , the graph has girth four if , six if and or , and otherwise the graph has girth eight; or , the graph can have cycles of at 3) if , most length six iff all differences are distinct. For a general rateconvolutional code, possibly punctured, similar conditions may be derived. As an example, for a systematic convolutional code of rate, in the absence of puncturing, the following proposition holds. Proposition 3: Considering a systematic convolutional code as the interlacing of the parity sequences of of raterate-1/2 subcodes operating on the same information sequence, the condition for the absence of cycles of length four is that the above-mentioned index differences computed on all the rate-1/2 subcodes must be different. The proofs of these propositions are omitted for the reasons explained in the following Remark 1. Example 1: Let us consider a rate-1/2 systematic recursive code with generator matrix (7)

(6) where is the total number of transmitted parity bits and it is understood that, if the lower limit of a sum is greater than the , upper limit, no terms are summed. Therefore, denote the number of information and parity bits in and each parity check, respectively. In order to simplify the noand in the following vectation, we collect integers tors: , , and . The corresponding Tanner graph can be easily built. In fact, variable nodes for each discrete-time instant , we have and corresponding to the information bits the parity bit , and a check node representing (6). This check node will be connected to some variable nodes, according to (6). For a given number of decoding iterations, the complexity of a message-passing algorithm working on this graph depends on the number of edges in the graph, and therefore, on , and is independent of the code-constraint length. Proposition 1: For a systematic rateconvolutional code, in order to ensure that the bipartite graph has girth at least 3In the following, we employ the usual symbols addition.

+ and

to denote modulo-2

In this case, and , and the code is defined by the following two vectors: and . For this code, the described conditions are not satisfied, because . Hence, the Tanner graph has girth four, as can be easily verified. Example 2: Let us now consider a rate-1/2 systematic recursive code with generator matrix (8) , , , This code is characterized by . According to Proposition 2, the corresponding and Tanner graph has girth eight. Remark 1: Nonrecursive systematic codes are a special case of the codes described in this section. In this case, the conditions in Propositions 1 and 3 for the absence of cycles of length four (or, equivalently, for the absence of rectangles in the parity-check matrix) define the so-called convolutional self-orthogonal codes (CSOCs) [20]–[22]. These codes form a class of nonrecursive systematic codes adopted for satellite communications in the 1960s and 1970s since they admit simple decoding schemes, namely, the so-called hard-input majority-logic decoding and soft-input threshold decoding [20], [22]. The conditions in Propositions 1 and 3 were recognized as necessary for the applicability of majority logic and threshold decoding [20],

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[21]. However, this is not surprising, since it can be shown that threshold decoding coincides with the previously described BP algorithm when a single iteration is performed. Similarly, it can be shown that majority-logic decoding is related with the first iteration of the so-called Gallager B algorithm described in [7] and [12]. The codes described in this section may be viewed as a generalization of CSOCs to recursive codes. Proofs of Propositions 1–3 can be obtained as a straightforward extension of the proofs in [21] for conventional CSOCs, and are therefore omitted. Remark 2: The code description is not unique, nor is the corresponding code Tanner graph. As a consequence, even if a given generator matrix does not comply with the conditions for the absence of cycles of length four, it could be possible to find an equivalent code description that does. For instance, this is the case of the code in Example 1. In fact, multiplying numer, we obtain the ator and denominator by the common factor equivalent generator matrix given in (8), whose corresponding Tanner graph has girth eight. This code and its equivalent descriptions will be reconsidered in the numerical results. Schedule: When used to decode convolutional codes, message-passing algorithms can also adopt a schedule different from that previously described and usually adopted for LDPC codes. This schedule, referred to as the wave schedule, is made possible by the fact that the parity-check matrix of a convolutional code is block triangular, and by the relevant structure of the code Tanner graph. In fact, at each iteration, after variable have been processed, and check nodes up to discrete-time the edges connecting “future” check nodes may be immediately updated without waiting for the successive iteration. In practice, each check node is “activated” when it can benefit from the updating, at the same iteration, of the messages coming from previous check and variable nodes. This schedule is pictorially exemplified in Fig. 1, with reference to the Tanner graph of a rate-1/2 systematic convolutional code with parity-check . In the figure, three successive equation steps of this schedule are shown under the letters (a), (b), and (c). Although this new schedule allows a speed up of the convergence process, it is not compatible with a fully parallel implementation of the decoder. For high-speed communication systems, the flooding schedule could be preferred. IV. CONCATENATED SCHEMES In this section, we discuss the concatenation of the convolutional codes previously described. The resulting coding schemes are classical turbo codes or serially concatenated codes, but may be decoded as LDPC codes. For this reason, these new concatenated schemes are nicknamed “turbo Gallager codes” (TGCs). More generally, code networks can be designed by concatenating a few convolutional component codes in mixed parallel and serial configurations [5], [23]. TGC are based on the following two key concepts: 1) at the encoder, the use of recursive CSOCs in code networks as component codes; 2) at the decoder, the use of a message-passing algorithm which works on the Tanner graph of the overall code by adopting the flooding or the wave schedule.

Fig. 1.

Wave schedule.

Fig. 2. Overall Tanner graph for a turbo code.

Let us consider the overall Tanner graph of a turbo code. In Fig. 2, we show, as an example, the graph for a rate-1/3 code. The component codes are rate-1/2 systematic codes with , , and, for simplicity, the code termination is ignored. In the upper part of the graph, the information bits and the parity bits of the first component code are connected with the corresponding check nodes. The information bits are also connected, in a permuted order, with the lower check nodes . related to the second component code having parity bits The decoding process can be performed by means of a message-passing decoder working on the Tanner graph of the overall code. In this case, the variable nodes of both codes, and subsequently, the check nodes of both codes, are activated simultaneously. Alternatively, a “turbo-like” decoder such as that described in [2] could be used, in which each soft-output decoder is based on the message-passing algorithm working on the Tanner graph of the corresponding component code, and both decoders

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iteratively exchange soft information.4 For a given performance, the convergence speed of a message-passing decoder working on the Tanner graph of the overall code with the flooding or wave schedule was observed to be greater than that of the above-mentioned turbo-like decoder. In fact, each component decoder of this turbo-like scheme must perform some inner iterations in order to pass reliable information to the other decoder. Then, additional iterations on the overall scheme are necessary. Note that even if the graphs of each component code have girth six, the overall Tanner graph may have girth four, depending on the interleaver used. As an example, consider a turbo code composed of rate-1/2 convolutional codes. In this case, a sufficient condition to avoid the appearance of cycles of length four in a parallel concatenation is that information symbols which differ for less than have a distance, after the interleaver, greater than or equal to . This may be obtained by the use of an -random interleaver with [24]. Turbo codes based on CSOCs as component codes were previously proposed in [25], where, however, nonrecursive CSOCs were adopted, and the decoding scheme was “turbo-like” using threshold decoders as soft-output component decoders. The performance of this scheme has a twofold limitation: 1) the use of nonrecursive codes prevents the possibility of an interleaver gain; and 2) the suboptimal threshold decoders further limit the overall performance. This limitation may be overcome by the proposed schemes based on recursive codes and a more efficient decoding. As a design criterion to select the component codes as well as the type of interleaver, puncturing, and concatenation (parallel, serial, or mixed), the method of density evolution can be used. Density evolution is a tool for jointly analyzing the code and the message-passing decoding algorithm, evaluating the relevant performance when averaged over the ensemble of codes with a common degree distribution of variable and check nodes [10], [12]. This method, which assumes the absence of cycles, analyzes the evolution of the probability density functions of the messages propagating in the graph during decoding, with the aim of determining the critical channel parameter (the so-called threshold) which separates the region where reliable transmission is possible from that where it is not [12]. Notice that for TGCs, the condition of absence of cycles for increasing codeword length is not satisfied. Hence, no rigorous claim on the achievable iterative threshold can be made. Nevertheless, density evolution can provide useful information in the code design. The degree distribution of variable and check nodes, necessary for the application of the density evolution method, can be and defined concisely described by two polynomials in [9] and [10]. For TGCs, these two polynomials depend on the type of concatenation, interleaver, puncturing (if any), and component codes. Example 3: Let us consider a rate-1/2 turbo code obtained from two identical rate-1/2 RSC codes by means of puncturing. Let us assume that the interleaver is odd–odd, i.e., bits in odd 4This

turbo-like scheme corresponds to a particular message-passing schedule in the overall graph. In fact, in this case, messages are exchanged in the upper or in the lower part of the graph alternatively.

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Fig. 3. BER of the convolutional code in Examples 1 and 2, by using the BCJR and the BP algorithm.

(even) position remain in odd (even) position after the permutation,5 and that the odd parity bits of the first encoder and the even parity bits of the second encoder are punctured. When a bit is punctured, the corresponding variable node in the graph disappears, along with the neighboring check nodes and the edges connected to them. As a consequence, by means of simple considerations on the overall Tanner graph, it can be shown that the corresponding degree polynomials are

(9) By properly choosing the integers and , simple irregular , a regular code codes can be obtained. However, if , a (3,6) regular LDPC results. As an example, if code is obtained. V. NUMERICAL RESULTS Computer simulations are used to assess the performance of the proposed codes, whether concatenated or not, in terms of bit-error rate (BER) versus , being the received signal the two-sided noise power energy per information bit, and spectral density. For TGCs, density evolution is used to select the component codes. To this purpose, discretized density evolution [10] is considered with 512-level quantization in the range from 32 to 32.6 Finally, a comparison between the flooding and the wave schedules in terms of convergence speed is reported. Single Convolutional Code: The application of the BP algorithm to a single convolutional code is first analyzed, and in Fig. 3, the code in Examples 1 and 2 is considered (see also Remark 2). In this figure, the performance of the optimal BCJR algorithm is also given as a benchmark. For the BP algorithm, computer simulations were performed by using early detection of convergence, i.e., at each iteration the decoder determines if 5Examples of such an interleaver are described in [2] and [26]. A dithered relative prime (DRP) interleaver [27] can be also odd–odd for particular choices of the dither functions. 6We assume that the transmitted code symbols are 1.

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TABLE I THRESHOLDS FOR A RATE-1/2 TURBO CODE WITH EQUAL COMPONENT CODES, AN ODD–ODD INTERLEAVER AND AN ODD–EVEN PUNCTURING

checks are all satisfied, stopping decoding when they are. When the BP algorithm is applied to the Tanner graph with girth four derived from the generator matrix in (7), the performance does not approach the optimal one, even for a maximum allowed number of 50 iterations. On the contrary, on a Tanner graph with girth greater than four, such as the graph which derives from the generator matrix in (8) for the same code, the BP algorithm has a performance which is very close to the optimal one in two to three iterations (the relevant curve in Fig. 3 refers to ten maximum allowed iterations). In general, when the code Tanner graph has girth six, we observed in our simulations that the performance of the BCJR algorithm is always reached. On the contrary, when the code has girth four, the BER never converges to the optimal performance. Concatenated Schemes: Code Design Based on Density Evolution: As a tool to perform the code design, the density evolution technique was used. The threshold of the signal-to-noise ratio (SNR), i.e., the value above which the expected fraction of incorrect messages approaches zero, assuming absence of cycles, is computed. As already mentioned, these threshold values can be used to select the component codes as well as the type of interleaver, puncturing, and code concatenation. As an example, let us consider the case of turbo codes, such as those described in Example 3. In this case, the threshold values are shown in Table I for different values of , along with the corresponding optimal values of and . It can be noted that an optimal value of exists and corresponds to the couple (or equivalently, ). Hence, component codes with these values of and represent the better choice in this case. The corresponding threshold value is 0.797 dB, whereas the Shannon limit for binary inputs and rate-1/2 is 0.187 dB. Lower values of the threshold (0.584 dB) may be obtained by using different component codes, i.e., an asymmetric turbo code, in order to make the overall code more irregular. As a side result, we observed a significant increase of the threshold values when the component codes are nonrecursive, according to well-known results for turbo codes. As a final comment, we summarize the results we obtained by applying the threshold analysis to serial concatenated schemes. We found that lower threshold values are obtained for a recursive inner code according to the results in [5]. As a further de-

Fig. 4. BER of the considered TGCs and comparison with regular LDPC codes for different codeword lengths L.

gree of freedom which can allow lower threshold values, code networks obtained by concatenating component codes in mixed parallel and serial configurations may be considered. Concatenated Schemes: BER Analysis: In Example 3, we observed that, with a proper choice of component codes, interleaver type, and puncturing, it is possible to obtain a TGC which is also a (3,6)-regular LDPC code. In Fig. 4, we compare the performance of these (3,6)-regular TGCs with that of classical (3,6)-regular LDPC codes [6], [28] for different codeword lengths . For the proposed schemes, the component codes have , , and we use tailbiting in both component codes, as described in [29] for recursive codes, in order to avoid a rate loss due to code termination. Note that no code search was performed—the component codes are only chosen following Proposition 3 and simple intuitive considerations. Specifically, as both component codes are punctured, inmust be all even (or all odd) in order to prevent all of tegers the check nodes from disappearing. In addition, integers are chosen small in order to reduce the probability of appearance of cycles of length four due to the interleaver, whereas larger values of the integers are chosen in order to allow the propagation of a message to a wide region of the graph in a few iterations. 496, For TGCs, different codeword lengths, namely, 2624, and 8000, which correspond to interleaver lengths of 248, 1312, and 4000, respectively, are considered.7 We emphasize the simplicity in obtaining codes with different codeword lengths. The component codes remain the same and only the interleaver size changes. Computer simulations were performed by using early detection of convergence. The maximum allowed number of iterations is 400 for TGCs, as well as for (3,6)-regular LDPC 504, 2340, and 8000, reported in Fig. 4 as a codes of length comparison [28]. We observe that the considered TGCs perform slightly better than classical (3,6)-regular LDPC codes, despite their simpler construction and encoding. In Fig. 5, we compare the performance of TGCs with that of the original rate-1/2 turbo code by Berrou and Glavieux (B&G) 7DRP

interleavers with M = 8 are used [27].

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Fig. 5. BER of the considered TGCs and comparison with the original turbo code by B&G [2].

[2]. Unlike the code considered in Examples 1 and 2 and Remark 2, it can be shown that it is not possible to rearrange the generator matrix of the component code of the B&G turbo code in order to obtain a bipartite graph with girth greater than four. For this reason, we use different component codes for the considered TGCs. A symmetric TGC, such as that described in Example 3, is considered with component codes having and , since these values correspond to the lowest threshold. This code is denoted TGC1 in Fig. 5. The identical component codes have and . We also consider an asymmetric turbo code, denoted as TGC2 in Fig. 5, which corresponds to a threshold of 0.584 dB. In this case, one compoand , whereas the nent code has and . For second one has these three turbo codes (B&G, TGC1, and TGC2), DRP interor 16 000 bits are leavers with lengths 1024 used. For TGC1 and TGC2, the maximum number of iterations is 200, for interleavers of length 1024, or 800, for interleavers of length 16 000. However, as shown below, the mean number of iterations really necessary is significantly lower. From Fig. 5, it can be observed that, despite the lower level of decoding complexity, TGC1 exhibits a performance degradation of less than 0.3 dB at a BER of . This performance loss reduces to about 0.15 dB by using TGC2. A serial code concatenation is considered in Fig. 6, where the performance of an overall code obtained by concatenating and (hence, a systematic outer code with this code in nonrecursive), and a systematic recursive inner code and , is shown for different with interleaver lengths. The overall rate of 1/3 is obtained by properly puncturing the inner code. Hence, the codeword length is 1.5 times the interleaver length. A density evolution analysis showed that, corresponding to the degree distribution of this code, the threshold is 0.697 dB. From the figure, it can be observed that by increasing the interleaver length, the performance tends to this limiting value. Schedule: Finally, in Fig. 7, we compare the wave and flooding schedules in terms of mean value and standard devi-

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Fig. 6. BER of the considered TGCs in the case of serial concatenation.

Fig. 7. Mean and standard deviation of the number of iterations for the wave and flooding schedules.

ation of the number of iterations until convergence versus the SNR. The considered TGC code is TGC1 from Fig. 2 with an interleaver length of 1024 bits. We observe that the wave schedule reduces the number of iterations. However, as already mentioned, this schedule is not compatible with a fully parallel decoder implementation—it represents a viable solution for serial implementations. VI. CONCLUSION In this paper, we showed that code networks, designed by concatenating convolutional codes in mixed serial and/or parallel configuration through interleavers, may be effectively decoded by means of the decoding algorithm used for LDPC Gallager codes and working on the code Tanner graph, provided that the component codes are chosen according to proper conditions which guarantee that the graph has girth at least and , these conditions six. For RSC codes of rate are explicitly given, and their analogy with the conditions which define CSOCs were drawn. Hence, the proposed code

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networks combine the simple coding operations of turbo and serial concatenated codes with the fully parallel decoding implementation of LDPC codes. These codes were compared, in terms of BER performance, with classical regular LDPC codes and the original B&G turbo code showing that, despite the lower overall complexity, very powerful schemes may be obtained. ACKNOWLEDGMENT The author acknowledges the contribution of Dr. M. Bertinelli for his help in developing part of the computer programs, and that of M. Franceschini for the density evolution results.

REFERENCES [1] C. Berrou, A. Glavieux, and P. Thitimajshima, “Near-Shannon limit error-correcting coding and decoding: Turbo codes,” in Proc. Int. Conf. Communications, Geneva, Switzerland, 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, pp. 1261–1271, Oct. 1996. [3] L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal decoding of linear codes for minimizing symbol error rate,” IEEE Trans. Inform. Theory, vol. IT-20, pp. 284–287, Mar. 1974. [4] P. Roberston, E. Villebrun, and P. Hoeher, “Optimal and sub-optimal maximum a posteriori algorithms suitable for turbo decoding,” Eur. Trans. Telecommun., vol. 8, no. 2, pp. 119–125, Mar./Apr. 1997. [5] S. Benedetto, D. Divsalar, G. Montorsi, and F. Pollara, “Serial concatenation of interleaved codes: Performance analysis, design, and iterative decoding,” IEEE Trans. Inform. Theory, vol. 44, pp. 909–926, May 1998. [6] D. J. C. MacKay, “Good error correcting codes based on very sparse matrices,” IEEE Trans. Inform. Theory, vol. 45, pp. 399–431, Feb. 1999. [7] R. G. Gallager, Low-Density Parity-Check Codes. Cambridge, MA: MIT Press, 1963. [8] M. Luby, M. Mitzenmacher, A. Shokrollahi, D. Spielman, and V. Stemann, “Practical loss-resilient codes,” IEEE Trans. Inform. Theory, vol. 47, pp. 569–584, Feb. 2001. [9] T. J. Richardson, M. A. Shokrollahi, and R. L. Urbanke, “Design of capacity-approaching irregular low-density parity-check codes,” IEEE Trans. Inform. Theory, vol. 47, pp. 619–637, Feb. 2001. [10] S.-Y. Chung, G. D. Forney, T. J. Richardson, and R. L. Urbanke, “On the design of low-density parity-check codes within 0.0045 dB of the Shannon limit,” IEEE Commun. Lett., vol. 5, pp. 58–60, Feb. 2001. [11] R. M. Tanner, “A recursive approach to low complexity codes,” IEEE Trans. Inform. Theory, vol. IT-27, pp. 533–547, Sept. 1981. [12] T. J. Richardson and R. L. Urbanke, “The capacity of low-density parity-check codes under message-passing decoding,” IEEE Trans. Inform. Theory, vol. 47, pp. 599–618, Feb. 2001. , “Efficient encoding of low-density parity-check codes,” IEEE [13] Trans. Inform. Theory, vol. 47, pp. 638–656, Feb. 2001.

[14] G. Colavolpe, “Decoding of Concatenated Self-Orthogonal Convolutional Codes on Graphs,” Italian Patent MI2002A001438, June 28, 2002, Int. Patent Applicat. PCT/EP03/06337, Int. Patent WO 2004/004134, June 16, 2003. [15] A. J. Felström and K. S. Zigangirov, “Time-varying periodic convolutional codes with low-density parity-check matrix,” IEEE Trans. Inform. Theory, vol. 45, pp. 2181–2191, Sept. 1999. [16] N. Wiberg, “Codes and decoding on general graphs,” Ph.D. dissertation, Linköping Univ., Linköping, Sweden, 1996. [17] F. R. Kschischang, B. J. Frey, and H.-A. Loeliger, “Factor graphs and the sum-product algorithm,” IEEE Trans. Inform. Theory, vol. 47, pp. 498–519, Feb. 2001. [18] F. R. Kschischang and B. J. Frey, “Iterative decoding of compound codes by probability propagation in graphical models,” IEEE J. Select. Areas Commun., vol. 16, pp. 219–231, Feb. 1998. [19] H. H. Ma and J. K. Wolf, “On tail biting convolutional codes,” IEEE Trans. Commun., vol. COM-34, pp. 104–111, Feb. 1986. [20] J. L. Massey, Threshold Decoding. Cambridge, MA: MIT Press, 1963. [21] J. P. Robinson and A. J. Bernstein, “A class of binary recurrent codes with limited error propagation,” IEEE Trans. Inform. Theory, vol. IT-13, pp. 106–113, Jan. 1967. [22] G. C. Clark, Jr and J. B. Cain, Error-Correction Coding for Digital Communications, 3rd ed. New York: Plenum, 1988. [23] S. Benedetto, D. Divsalar, G. Montorsi, and F. Pollara, “Soft-input softoutput modules for the construction and distributed iterative decoding of code networks,” Eur. Trans. Telecommun., vol. 9, no. 2, pp. 155–172, Mar./Apr. 1998. [24] D. Divsalar and F. Pollara, “Turbo codes for PCS applications,” in Proc. IEEE Int. Conf. Communications, Seattle, WA, June 1995, pp. 54–59. [25] S. Riedel and Y. V. Svirid, “Iterative (“turbo”) decoding of threshold decodable codes,” Eur. Trans. Telecommun., vol. 6, no. 5, pp. 527–534, Sept./Oct. 1995. [26] A. S. Barbulescu and S. S. Pietrobon, “Interleaver design for turbo codes,” Electron. Lett., vol. 30, pp. 2107–2108, Dec. 1994. [27] S. Crozier and P. Guinand, “High-performance low-memory interleaver banks for turbo codes,” in Proc. IEEE Vehicular Technology Conf., Atlantic City, NJ, Oct. 2001, pp. 2394–2398. [28] D. J. C. MacKay. Regular LDPC Online Database. [Online]. Available: http://www.inference.phy.cam.ac.uk/mackay/ [29] P. Ståhl, J. B. Anderson, and R. Johannesson, “A note on tailbiting codes and their feedback encoders,” IEEE Trans. Inform. Theory, vol. 42, pp. 529–534, Feb. 2002.

Giulio Colavolpe was born in Cosenza, Italy, in 1969. He received the Dr. Ing. degree (cum laude) in telecommunication engineering from the University of Pisa, Pisa, Italy, in 1994 and the Ph.D. degree in information technology from the University of Parma, Parma, Italy, in 1998. Since 1997, he has been with the University of Parma, where he is now an Associate Professor of Telecommunications. In 2000, he was a Visiting Scientist with the Insitut Eurécom, Valbonne, France. His main research interests include digital transmission theory, channel coding, and signal processing. His reesearch activity has led to more than 60 scientific publications in leading international journals and conference proceedings and several industrial patents.

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Reduced-Complexity MAP-Based Iterative Multiuser Detection for Coded Multicarrier CDMA Systems Junqiang Li, Khaled Ben Letaief, Fellow, IEEE, and Zhigang Cao, Senior Member, IEEE

Abstract—In recent years, combining multiuser detection (MUD) and channel decoding has received considerable attention. The maximum a posteriori (MAP) criterion-based iterative multiuser detector greatly improves the system performance and can approach the performance of single-user coded systems. However, its complexity increases exponentially with the number of users and can become prohibitive for systems with a medium-to-large number of users. In this paper, a reduced complexity MAP-based iterative MUD based on the use of a soft sensitive bits algorithm is proposed for coded multicarrier code-division multiple-access systems. It is shown that it can greatly reduce the computational complexity with a minimal penalty in performance compared to the conventional optimal scheme. Index Terms—Coded multicarrier code-division multiple access (MC-CDMA), iterative multiuser detection (MUD), maximum a posteriori (MAP) detection, turbo processing.

I. INTRODUCTION ODE-DIVISION multiple access (CDMA) is a well-known scheme for providing multirate applications and is one of the main air interfaces in the third-generation cellular systems. However, conventional CDMA is not practical when the data transmission rate is very high, for obvious reasons [1], [2]. Multicarrier CDMA (MC-CDMA) systems can efficiently deal with this problem by combining multicarrier modulation and CDMA [3]–[5]. As a result, MC-CDMA has been recently receiving considerable attention. In MC-CDMA systems, however, the multiple-access interference (MAI) problem in the uplink is always encountered, even if orthogonal spread codes are adopted by all users [6]. The MAI problem is particularly severe if there are some users with a spreading code that is highly correlated with the desired user’s spreading code and/or in near–far scenarios. A significant amount of research has addressed various multiuser detection (MUD) methods for MAI suppression in CDMA systems and synchronous MC-CDMA systems. Various

C

Paper approved by R. Kohno, the Editor for Spread Spectrum Theory and Applications of the IEEE Communications Society. Manuscript received May 11, 2001; revised January 22, 2002. This work was supported in part by the Hong Kong Research Grant Council. J. Li and Z. Cao are with the State Key Laboratory on Microwave and Digital Communications, Department of Electronic Engineering, Tsinghua University, Beijing 100084, China, (e-mail: [email protected]; [email protected]). K. Ben Letaief is with the Center for Wireless Information Technology, Department of Electrical and Electronic Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong (e-mail: [email protected]). Digital Object Identifier 10.1109/TCOMM.2004.836565

MUD algorithms can also be employed in quasi-synchronous MC-CDMA systems, where a cyclic extension is added to each frame in the time domain, with the length of the guard interval being larger than the sum of the maximum time delay of the multipath fading channels and the maximum difference in the synchronous time among all users [7], [8]. Most of the previous work on MUD, however, focused on uncoded MC-CDMA systems. In practice, most communication systems employ forward error control (FEC) coding to improve the transmission performance. In particular, joint MUD and FEC decoding can be used to further improve the bit-error rate (BER) performance while mitigating the detrimental effects due to MAI. Several efforts have been undertaken to derive and analyze optimal joint MUD and FEC decoding. Unfortunately, the computational complexity associated with such an optimal approach is usually prohibitively high. For example, it is shown in [9] that the optimal decoding scheme for an asynchronous convolutionally coded CDMA system combines the trellises of both the asynchronous multiuser detector and the convolutional code. As a result, the computational complexity will increase exponentially with the number of users and the number of states in each user’s encoder. This has motivated the study of a number of low-complexity suboptimal solutions that are more attractive for use in practical applications. In this paper, we propose a low-complexity iterative MUD approach for MC-CDMA systems, which can effectively alleviate the harmful effects of MAI. The proposed method is based upon the use of iterative or “turbo” processing techniques, which have already been successfully applied to many digital communication systems and, in particular, to the MUD case. For instance, a suboptimal iterative multiuser receiver structure, consisting of optimal maximum a posteriori (MAP)-based iterative MUD along with parallel single-user soft-input/soft-output (SISO) channel decoders, has been derived for coded CDMA systems in [9] and [10]. It was shown that such an iterative receiver could achieve near-signal-user performance. Unfortunately, its computational complexity is still very high and increases exponentially with the number of users. More recently, studies in the field of soft iterative MUD were directed at reducing the complexity of the optimal MAP-MUD at the cost of a small degradation in system performance. In [12], [13], interference cancellation based soft iterative MUD methods combined with MMSE filtering were proposed. Although the complexity of these low-complexity methods increases linearly as a function of the number of users, the computational burden for each user is still high, especially in fast fading channels. Furthermore, it was shown that a large number of

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Fig. 1. Low-complexity iterative MAP-based MUD for coded MC-CDMA systems.

iterations are needed to approach the performance of the optimal MAP-based scheme. To deal with this complexity issue while achieving good system performance, this paper presents a novel low-complexity iterative MAP-based MUD based on the use of a soft sensitive bits (SBs) algorithm. It is demonstrated that we can significantly reduce the complexity of the optimal MAP-based MUD while achieving performance that can approach that of the optimal MAP-based scheme. This paper is organized as follows. In Section II, the coded MC-CDMA system model is described. The proposed low-complexity iterative MAP-based MUD algorithm is presented in Section III. Numerical results are then presented in Section IV. Finally, we present our conclusions in Section V.

[14]. In this system, the information bits of the th user can be encoded using a convolutional code.1 Interleaving is emto avoid burst errors ployed for the coded bits sequence due to occasional deep fades. The coded bit, , of the th user at the th time interval is then spread by a PN sequence and transmitted using MC-CDMA, where the total number of subcarriers is equal to the length of the signature sequence . Throughout this paper, we shall assume flat fading at each subcarrier. In particular, the channel responses of the subcarriers are assumed to be independent. At the receiver, the signal of each user is de-spread and maximum ratio combined (MRC) in the frequency domain. The received signal at the th time interval is given by IFFT

II. SYSTEM MODEL Consider a synchronous coded MC-CDMA system, which consists of users as shown in Fig. 1(a). We also assume perfect frame synchronization, which can be achieved using various timing synchronization techniques such as that given in

(1)

where is an matrix with the th column of denoting a signal vector of the th user that includes the channel 1However,

note that a Turbo code can also be used.

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response and spreading codes in all carriers, and is the transusers. Likewise, denotes mitted coded bits vector of the the average power of all users. Specifically (3)

Finally

where is the channel frequency response at the th subcaris the received power of the th rier of the th user, and user at the th time interval. The matched filter (MF) outputs , which includes the de-spreading and MRC operations, can be written as FFT IFFT

The first term in (3) is known as the extrinsic information, which is derived from the MAP-based MUD and is denoted by . In order to calculate the extrinsic information of the th user , the a priori information of all coded bits should be known (the subscript and shall denote the extrinsic information and the a priori information, respectively). The calculations of the extrinsic information in MAP-MUD are conducted in an iterative fashion given the a priori information of all coded bits. As in [10], at a given iteration, the a priori information is achieved according to the channel decoder of the th user in the previous iteration (i.e., ). By considering (2), the conditional probability distribution of with a -dimensional Gaussian probability density function can be given by

(2) is the cross-correlation matrix as . By where FFT , then is another complex Gaussian letting , where white noise random process with variance is the variance of the Gaussian noise . As a result, it follows that is a colored complex Gaussian random process . From with zero mean and variance (2), it is clear that the signal model of a MC-CDMA system is similar to a direct-sequence (DS)-CDMA system. Hence, our proposed algorithm can also be effective in conventional DS-CDMA systems. III. PROPOSED SOFT ITERATIVE MUD ALGORITHM In this section, we propose a novel low-complexity MAPbased multiuser detector based on the use of a soft SBs algorithm. We begin by briefly describing the optimal MAP-based MUD scheme.

(4) is a constant and does not depend on . According to where (3), the conditional probability distribution of in terms of the coded bit of the th user is required to compute . Since (5) and since it is assumed that the coded bits of different users are independent, it follows that the conditional likelihood probability is given by

A. Optimal MAP-Based Multiuser Detector The basic structure of the soft iterative multiuser detector is illustrated in Fig. 1(b). As in the case of a conventional serial Turbo code, the detector consists of two main parts: a MAP-based MUD structure and parallel single-user MAPbased decoders. It is shown that iterations between the two parts separated by de-interleavers ( ) and interleavers ( ) are performed. In this case, two extrinsic information, 2 of the and th user, from the MAP-based MUD and single-user MAP-based decoders, respectively, are exchanged during the iterations. The MAP-based multiuser detector gives the a posteriori log-likelihood ratio (LLR) of a transmitted “ ” or a transmitted “ ” for the code bit of the th user at the th time interval. The LLR is given by

2The subscripts 1 and 2 denote the MAP-based MUD and MAP-based channel decoder, respectively.

(6) Because no prior information is available in the first iteration, it is assumed that the coded bits are equally likely. In the following iterations, the a priori information of MAP-MUD is obtained from the extrinsic information delivered by the th user’s channel decoder in a previous iteration as (7) where is given in Section III-C. From (3), it is evident that is not influenced by the a priori the extrinsic information information . After de-interleaving, is fed into the th user’s decoder for further processing in the MAP-based channel

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decoding block. Finally, the channel decoder computes the a posteriori LLR of the information bits during the last iteration. This is then used to make a decision regarding the decoded bits. It is clear from (6) that the computational complexity of the optimal MAP-based multiuser detector is exponential in terms 3 compuof the number of users. In particular, note that tations are required for the calculation of . B. Proposed Low-Complexity MAP-Based Multiuser Detector It is well known that the MAP detector will have a better performance than the ML-based one when the a priori information is available. However, the optimal MAP-based multiuser detector is impractical when the number of users is high (e.g., ). The same high computational complexity problem is also true in the case of the conventional MLD. In our previous work [15], [16], an effective SBs algorithm was proposed, which can greatly reduce the computational complexity of the conventional maximum-likelihood detector (MLD). The basic idea behind this algorithm is as follows. By obtaining initial estimates of all the user bits using any detection algorithm (e.g., the conventional correlator receiver), we identify some specific bits, which we refer to as “sensitive bits.” These bits correspond to those bits that are most likely to be in errors. Assuming that all the other bits are correctly detected in the initial estimation (with the exception of the SBs), we can then use the MLD algorithm to correct these SBs in a much smaller coset that corresponds to these bits. In this paper, an iterative MAP-based SB algorithm is considered for reducing the complexity of the MAP detector in MC-CDMA systems. In this scheme, we begin by identifying the SBs as described above, where a low-complexity detection algorithm can be employed. Since the SBs algorithm could classify which coded bits are most likely to be correctly estimated or not, by doing so, we have some knowledge or “coarse” a priori information. With such a priori information, we can operate a posteriori detection in a much smaller coset, which corresponds to the identified SBs. We denote a likelihood metric as (8) (9) where

is the transmitted multiuser coded bits vector estimate.

According to the SB algorithm, a new updated bits vector can be obtained from

In other words, the larger

is the higher the likelihood

that its corresponding “updated” bit is wrong. In general, the number of incorrect bits in the initial estimated coded bits vector is not too large even if we use the conventional detector. As example, if the coded bit-error rate (BER) is , then only one bit out of one hundred is incorrectly decoded on average. Hence, in general, the number of SBs is not expected to be large after the initial estimation stage. In this paper, these SBs are identified as follows. The initial estimated coded bits of all the users are obtained by using the conventional single-user MF receiver. Let the total number of SBs equal to , the SBs algorithm based on (8)–(10) is then used to identify SBs. Among all possible metric values, we identify the largest

ones.

The SBs are then defined as those bits corresponding to those identified coded bits vectors. Once the SBs are identified, bits to be correct. we assume that the other remaining In order to proceed with the iterative-based MAP algorithm, we begin by assuming that the prior probabilities, which correspond to the SBs, are equally likely in the first iteration (i.e., ). On the other hand, we shall assume that the prior probabilities for the nonsensitive bits to be 1 because these bits are assumed to be correct in the initial estimation stage. The MAP-based iterative algorithm can then be employed in a smaller coset corresponding to the coded bits vectors that represent the SBs. Thus, by modifying the optimal MAP criterion (6), the calculation of the conditional likelihood probability can be simplified to

(11) where only or (if the th coded bit is an SB) “important” coded bits vectors are considered. The other vectors are referred to as unimportant vectors and their contributions to (6) are not taken into consideration as such contributions are considered to be nonsignificant compared to that of the important vectors. As described above, we initialize the prior probability in the first iteration as follows:

by “reversing” the polarity of one and

(i.e., or ). Furthermore, only one bit in we have demonstrated in [16] that, if there is one or more than

if

is an SB

one error bits in

if

is not an SB

error bit in

, then by simply reversing the polarity of one

, it follows that (10)

3By O(2

2 .

), we mean that the computational complexity is proportional to

(12)

where is the estimate of the transmitted bit . In the next iterations, with the improved a priori information feedback from the bank of single-user decoders, improved hard-decision based estimates of the transmitted coded bits are obtained as with

(13)

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Updated SBs can then be obtained with the improved estimate for the following iterations. In contrast to the first iteration, the prior probabilities, , used for MAP-based MUD in the next iterations are given by

(14)

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(MF) receiver. By using the SBs algorithm in [15] and [16], SBs are identified. Based on the SBs information, the extrinsic information delivered by a MAP-based MUD is

where

C. Single-User Decoder The a posteriori probabilities (APP) of the coded bits and the information bits at the end of the last iteration is obtained by the single-user decoders following the MAP-based MUD by using the MAP algorithm [12]. By using a rate convolutional code, we have coded bits for every uncoded information bit . Throughout this paper, the channel bits are denoted by

. We also have

(15) is the received signal sequence of the th user, is where the state at time with the state index range being over all of the possible states. Note that (14) can be realized using the MAP algorithm or a simplified log-MAP algorithm as in [17] and [18]. After achieving the extrinsic information from the MAP-based MUD, the transition branch metric of the th decoder is obtained as

(16) Thus, the a posteriori LLR of each coded bit of the th user is given by

(17)

Finally, the extrinsic information delivered by the single-user decoder is obtained as (18) Such information is fed back to the MAP-based MUD scheme and an improved a priori information can be obtained according to (13). We conclude this section by briefly summarizing the proposed scheme. Let denote the maximum number of SBs and denote the maximum number of iterations. Then, the proposed low-complexity soft iterative MAP-based MUD can be described as follows. Initialization: Obtain the initial estimates of the entire user coded bits using the conventional single-user matched filter

and the initial prior probability is given by (11). to : Iterations: For from the Iteration 1: With the extrinsic information MAP-based MUD, the extrinsic information of the th user’s decoder is obtained with (17). Then, is fed back to the MAP-based MUD. If , the a posteriori LLR of the information bits are computed. These are then used for the decoding of the received bits. The algorithm is then stopped. Iteration 2: With the improved a priori information fed back from the bank of the single user decoder (i.e., ), better estimates of all the users coded bits are obtained. As in Iteration ( ), the extrinsic information is calculated with updated SBs, and the prior probabilities are used for the MAP-based MUD as described in (11). Iteration 3: Go to Iteration 1.

IV. NUMERICAL RESULTS In this section, we present some simulation results and comparisons that demonstrate the potential of our proposed algorithm over an AWGN channel and frequency-selective fading channels. Throughout this section, it is assumed that all users employ the same rate-1/2 convolutional code with constraint length 5 and generators (23,35) in octal notation. The information bits block size for each user is 128 and a random interleaver is used for all the simulations results. All users have equal power ). We assume that the receiver has perfect knowl(i.e., and the spreading codes of all edge of the noise variance users. Finally, the signal-to-noise ratio (SNR) is defined as the ratio of the information bit power to noise power, and the fol) will mean SBs and iterations. lowing notation ( We begin by considering Fig. 2 where we list the system performance as a function of SNR for an MC-CDMA system with users and a PN sequence of length . (Recall that the total number of subcarriers is equal to .) Specifically, this figure lists the average bit error probability when we vary the number of SBs and the number of iterations against SNR. Note that, when , the proposed system reduces to the

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Fig. 2. Performance of the proposed soft iterative MAP-based MUD in AWGN channels ( = 5, = 7).

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Fig. 4. Performance of the proposed soft iterative MAP-based MUD in frequency-selective fading channels ( = 5, = 7).

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of the single-user case, within only 0.15 dB from the single user performance at a BER , for instance. Also the computational complexity of the MAP-based MUD is greatly reduced to . from We conclude this section by presenting Fig. 4, where we list the performance of the proposed method in frequency-selective fading channels. It is shown that the proposed method can work effectively and achieve almost the same performance as the conventional optimal MAP-based MUD. Compared with the simulation results in the AWGN channel, it is shown that the performance of the proposed method can approach that of the single-user system in frequency-selective fading channels with a smaller number of SBs and a smaller number of iterations (e.g., and ). V. CONCLUSION Fig. 3. Performance of the proposed soft iterative MAP-based MUD in AWGN channels ( = 10, = 15).

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conventional receiver with no feedback information used to improve the system performance. It is also noted that, when the , then the proposed method becomes number of SBs equivalent to the optimal MAP-based MUD. A close observation of Fig. 3 indicates that the proposed method performance can approach that of the optimal MAP-based MUD with significant reduction in system complexity. Indeed, note that the proposed approach reduces the computational complexity of the to . In parMAP-based MUD from and , the performance of the ticular, note that when proposed method is very close to that of the single user system, and the complexity is reduced from to In Fig. 3, it is demonstrated that the proposed method is effective even when the number of SBs is much smaller than that of the number of users. In particular, note that the performance of and , can approach that the proposed method, when

In this paper, an iterative MUD based on the use of a soft SBs algorithm for coded MC-CDMA systems was presented. The proposed scheme is featured as an effective MAP-based MUD scheme that can greatly reduce the computational complexity with a minimal penalty in performance compared with the conventional optimal scheme. Numerical results have demonstrated that the performance of our proposed algorithm can approach that of the optimal MAP-based MUD even when the number of SBs is much smaller than the total number of users. REFERENCES [1] S. Ohmori, Y. Yamao, and N. Nakajima, “The future generations of mobile communications based on broadband access technologies,” IEEE Commun. Mag., vol. 38, pp. 134–142, Dec. 2000. [2] N. Morinaga, M. Nakagawa, and R. Kohno, “New concepts and technologies for achieving highly reliable and high-capacity multimedia wireless communications systems,” IEEE Commun. Mag., vol. 35, pp. 34–40, Jan. 1997. [3] X. Gui and T. S. Ng, “Performance of asynchronous orthogonal multicarrier CDMA system in frequency selective fading channel,” IEEE Trans. Commun., vol. 47, pp. 1084–1092, July 1999. [4] S. Hara and R. Prasad, “Overview of multicarrier CDMA,” IEEE Commun. Mag., vol. 35, pp. 126–133, Dec. 1997.

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[5] N. Yee, J. P. Linnartz, and G. Fettweis, “Multi-carrier CDMA in indoor wireless radio,” in Proc. PIMRC’93, Yokohama, Japan, Dec. 1993, pp. D1.3.1–5. [6] S. Verdú, “Minimum probability of error for asynchronous Gaussian multiple-access channels,” IEEE Trans. Inform. Theory, vol. IT-32, pp. 85–96, Jan. 1986. [7] R. Esmailzadeh and M. Nakagawa, “Quasisynchronous time division duplex CDMA,” in Proc. IEEE GLOBECOM, vol. 3, San Francisco, CA, 1994, pp. 1637–41. [8] S. Tsumura and S. Hara, “Design and performance of quasisynchronous multi-carrier CDMA system,” in Proc. IEEE VTC Fall, vol. 2, Atlantic City, NJ, Oct. 2001, pp. 843–847. [9] T. R. Giallorenzi and S. G. Wilson, “Multiuser ML sequence estimator for convolutional coded asynchronous DS-CDMA systems,” IEEE Trans. Commun., vol. 44, pp. 997–1008, Aug. 1996. [10] M. C. Reed, C. B. Schelegel, P. D. Alesander, and J. A. Asenstorfer, “Iterative multiuser detection for DS-CDMA with FEC: Near-signaluser performance,” IEEE Trans. Commun., vol. 46, pp. 1693–1699, Dec. 1998. [11] M. Moher, “An iterative multiuser decoder for near-capacity communications,” IEEE Trans. Commun., vol. 46, pp. 870–880, July 1998. [12] X. D. Wang and H. V. Poor, “Iterative (Turbo) soft interference cancellation and decodig for coded CDMA,” IEEE Trans. Commun., vol. 47, pp. 1046–1061, July 1999. [13] H. E. Gamal and E. Geraniotis, “Iterative multiuser detection for coded CDMA signals in AWGN and fading channels,” IEEE J. Select. Areas Commun., vol. 18, pp. 30–41, Jan. 2000. [14] S. Nahm and W. Sung, “A synchronization scheme for multi-carrier CDMA systems,” in Proc. IEEE ICC 98, vol. 3, Atlanta, GA, 1998, pp. 1330–1334. [15] J. Li, K. B. Letaief, R. S. Cheng, and Z. Cao, “Multi-stage low complexity maximum likelihood detection for OFDM/SDMA wireless LANs,” in Proc. IEEE ICC , Helsinki, Finland, June 2001. [16] J. Li, K. B. Letaief, and Z. Cao, “A reduced-complexity maximum-likelihood method for multiuser detection,” IEEE Trans. Commun., vol. 52, pp. 289–295, Feb. 2004. [17] L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal decoding of linear codes for minimizing symbol error rate,” IEEE Trans. Inform. Theory, vol. IT-20, pp. 284–287, Mar. 1974. [18] B. Vucetic and J. Yuan, Turbo Codes: Principles and Applications. Norwell, MA: Kluwer, 2000.

Junqiang Li received the B.S.E degree in modern physical electronics and the M.S. and Ph.D. degrees in communication and information systems from Tsinghua University, Beijing, China, in 1997 and 2002, respectively. From 2000 to 2001, he was a Research Assistant with the Center of Wireless Information Technology (CWIT), Hong Kong University of Science and Technology (HKUST), Kowloon, Hong Kong. From 2002 to 2003, he was a Senior Researcher with the I-Networking Laboratory, Samsung Advanced Institute of Technology (SAIT), Seoul, Korea. He is currently with Broadcom Corporation, Matawan, NJ. His current research interests include broadband wireless and mobile communications, multicarrier modulation, CDMA technologies, space–time processing for wireless systems, turbo processing, and multiuser detection.

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Khaled Ben Letaief (S’85–M’86–SM’97–F’03) received the B.S. degree (with distinction) and M.S. and Ph.D. degrees from Purdue University, West Lafayette, IN, in 1984, 1986, and 1990, respectively, all in electrical engineering. Since January 1985 and as a Graduate Instructor in the School of Electrical Engineering at Purdue University, he has taught courses in communications and electronics. From 1990 to 1993, he was a faculty member with the University of Melbourne, Melbourne, Australia. Since 1993, he has been with the Hong Kong University of Science & Technology, Kowloon, where he is currently a Professor and Head of the Electrical and Electronic Engineering Department. He is also the Director of the Hong Kong Telecom Institute of Information Technology as well as the Director of the Center for Wireless Information Technology. His current research interests include wireless and mobile networks, broad-band wireless access, space–time processing for wireless systems, wide-band OFDM, and CDMA systems. Dr. Letaief served as consultants for different organizations and is currently the founding Editor-in-Chief of the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS. He has served on the editorial board of other journals including the IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS Wireless Series (as Editor-in-Chief). He served as the Technical Program Chair of the 1998 IEEE Globecom Mini-Conference on Communications Theory, held in Sydney, Australia. He was also the Co-Chair of the 2001 IEEE ICC Communications Theory Symposium, held in Helsinki, Finland. He is the Co-Chair of the 2004 IEEE Wireless Communications, Networks, and Systems Symposium, held in Dallas, TX, and is the Vice-Chair of the International Conference on Wireless and Mobile Computing, Networking, and Communications, WiMob’05, to be held in Montreal, QC, Canada. He is currently serving as the Chair of the IEEE Communications Society Technical Committee on Personal Communications. In addition to his active research activities, he has also been a dedicated teacher committed to excellence in teaching and scholarship. He was the recipient of the Mangoon Teaching Award from Purdue University in 1990, the Teaching Excellence Appreciation Award by the School of Engineering at HKUST (four times), and the Michael G. Gale Medal for Distinguished Teaching (the highest university-wide teaching award and only one recipient/year is honored for his/her contributions). He is a Distinguished Lecturer of the IEEE Communications Society.

Zhigang Cao (M’85–SM’85) graduated from the Department of Radio Electronics at Tsinghua University, Beijing, China, in 1962. He was a Visiting Scholar with Stanford University, Stanford, CA, from 1984 to 1986. Since 1962, he has been with Tsinghua University, where he is now a Professor with the Electronic Engineering Department, Deputy Director of the State Key Laboratory on Microwave and Digital Communications, member of Academic Committee of Tsinghua University, and Vice Chair of the Distance Learning Expert Committee of the Education Ministry of China. He is the author and coauthor of six books and has published more than 200 papers on Communications and Signal Processing, and he holds three patents. His current research interests include mobile communications, satellite communications, modulation/demodulation, coding/decoding, and digital signal processing. Prof. Cao is a fellow of Chinese Institute of Communications (CIC), a senior member of CIE, and a member of IEICE, and the New York Academy of Sciences. He is a member of the board of directors of Beijing Telecommunication Institute, chairman of the Information/Signal Processing Committee of Beijing Telecommunication Institute, and a member of the board of directors of China Satellite Communication Broadcasting & Television Association and serves as a vice chairman of Academic Committee of CIC. He also serves an Editor of the Chinese Journal of Electronics, China Telecommunications Construction, China SatCom and Satellite & Network. For his research achievements, he was the recipient of seven awards from the National Science Congress, the Ministry of Electronic Industry, and the State Education Commission, among others.

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A Novel Trellis-Shaping Design With Both Peak and Average Power Reduction for OFDM Systems Hideki Ochiai, Member, IEEE

Abstract—A new trellis shaping design is proposed for reducing the peak-to-average power ratio of the bandlimited orthogonal frequency-division multiplexing (OFDM) signals. The approach is based on recursive minimization of the autocorrelation sidelobes of an OFDM data sequence. A novel metric in conjunction with the Viterbi algorithm is devised. The performance of the trellis shaping depends on signal mapping strategy, and the two types of mapping, referred to as Type-I and Type-II, are proposed. The Type-I mapping has no capability of reducing the average power, but it can achieve a significant reduction of the peak-to-average power ratio. On the other hand, the Type-II mapping is designed to achieve both peak and average power reduction. The bit error probability of the system over an AWGN channel is evaluated based on the simulations, which confirms the effectiveness of the proposed scheme. Index Terms—Normalized instantaneous power, orthogonal frequency-division multiplexing (OFDM), peak-to-average power ratio, trellis shaping.

I. INTRODUCTION UE TO AN increasing demand on multimedia communications, future generation communications systems should support a very high data rate transmission with high reliability. However, the wide-band nature of the signal renders a wireless communication channel frequency-selective and thus reliable communications difficult. The orthogonal frequency-division multiplexing (OFDM) technique can cope with this difficulty with reasonable complexity and thus is expected to continuously serve as a promising modulation technique for future broad-band communications systems [1]. Nevertheless, the bandwidth is a costly resource, and achieving a higher information data rate without resorting to further expansion of bandwidth necessitates the use of bandwidth-efficient modulation techniques such as a high level quadrature amplitude modulation (QAM). Therefore, the combination of a higher order QAM with the OFDM signaling bears particular importance.

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Paper approved by Y. Li, the Editor for Wireless Communication Theory of the IEEE Communications Society. Manuscript received April 15, 2002; revised May 15, , 2003. This work was supported in part by the Support Center for Advanced Telecommunications Technology Research, in part by the Telecommunications Advancement Foundation, and in part by MEXT under a Grant-in-Aid for Young Scientists (B) 15760255. This paper was presented in part at the IEEE International Symposium on Wireless Personal Multimedia Communications (WPMC’02), Honolulu, HI, October 2002, and in part at the IEEE International Symposium on Information Theory (ISIT’03), Yokohama, Japan, June 2003. The author is with the Division of Electrical and Computer Engineering, Graduate School of Engineering, Yokohama National University, Yokohama, 240-8501, Japan (e-mail: [email protected]). Digital Object Identifier 10.1109/TCOMM.2004.836593

One major problem associated with the OFDM signaling technique is its signal waveform with extremely large dynamic range [2]. In many wireless applications where out-of-band radiation of signal power is restricted, nonlinear amplification of the OFDM signal is prohibitive. However, in order to linearly amplify the bandlimited signal with large dynamic range, a power amplifier (PA) must be operated with a large input back-off (IBO). This results in a severe penalty in terms of the PA efficiency. Thus, even though the OFDM signal is known to posses nearly rectangular power spectra, it is also well known that the power efficiency is prohibitively low. Therefore, a dynamic range reduction of the OFDM signal plays an important role for its application to both power and bandlimited communications systems. In order to alleviate this power efficiency problem, a growing number of techniques have been proposed in the recent literature. The simplest and most effective approach appears to be a straightforward clipping and filtering [3]. As shown in [4], if the information bit rate per subcarrier is low (e.g., QPSK signaling), the use of powerful channel coding may significantly alleviate the degradation caused by clipping distortion. However, as also recognized in [4], if the data rate per subcarrier needs to be increased with a larger constellation size, substantial performance degradation is no longer avoidable even with the help of powerful channel coding. This fact motivates us to seek another peak power reduction technique, and a trellis shaping approach [5] may be one of the possible candidates. The general purpose of constellation shaping is to reduce the average power of the multilevel signals, and the original work of Forney [5] focuses on an efficient average power reduction of the QAM signals based on the Viterbi algorithm. The application of the trellis shaping to dynamic range reduction of a bandlimited single-carrier signal was first proposed in [6] and further developed in [7]. Recently, Henkel and Wagner [8] have shown that the trellis shaping is also applicable to the peak power reduction of the OFDM signals, and heuristic metrics based on either time or frequency domain are proposed. However, the time-domain metric in [8] appears to be computationally demanding for the bandlimited OFDM signals, whereas generalization of the simple frequency-domain approach in [8] to the OFDM system with larger constellation size or a large number of subcarriers seems difficult. In this paper, we propose a new trellis shaping system design based on the control of autocorrelation sidelobes of an OFDM data sequence. Since the autocorrelation of a data sequence (in the frequency domain) and its power spectrum (in the time domain) form a Fourier transform pair, minimizing the autocorrelation sidelobes of a data sequence may help flatten the resulting

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bandlimited OFDM signals. A novel metric in conjunction with the Viterbi algorithm is devised for this purpose. The proposed system expands a constellation size of each subcarrier similar to the tone injection method in [9]. However, while the approach in [9] may typically increase the average power associated with constellation expansion, some of the proposed signal mapping in this paper may also help reduce the average power. The paper is organized as follows. In Section II, the OFDM system and its associated figures of merit are defined. Section III provides a brief review of the trellis shaping system and introduces the two types of signal constellation mapping considered throughout the paper. The proposed algorithm with a novel metric is given in Section IV, and Section V evaluates the dynamic range reduction capability and the resulting bit-error rate (BER) performance of the proposed system by computer simulations. Finally, concluding remarks are given in Section VI.

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’s are independent and identically distributed (i.i.d.) If the is chosen with equal probability and each complex point in (which is the case for the system with i.i.d. information bits without any constellation shaping), the average power can be a priori. For example, the avdetermined as a function of erage power of the square -ary QAM is given by (5) A constellation shaping is generally designed such that for a given fixed MSED . A gain due to the average power reduction achieved by shaping can thus be defined as (6) In this paper, we consider the following two figures of merit that describe statistical characteristics of signal dynamic range.

II. OFDM SYSTEM AND FIGURES OF MERIT We consider an expressed as

A. Normalized Instantaneous Power

-subcarrier baseband OFDM signal

(1) is the symbol period and is a windowing where function that controls the spectral spread due to the abrupt transition of consecutive OFDM symbols. For simplicity, the rectangular window is considered. The th complex baseband is given by OFDM symbol (centered at zero frequency)

The first-order statistics of the instantaneous power may be insightful for characterizing the dynamic range of the baseband signal and is easy to evaluate by simulation [3]. The normalized instantaneous power of the OFDM signal with trellis shaping is defined as (7) In particular, we are interested in the complementary cumulative distribution of this random process, which is the probability that a certain signal process exceeds a given threshold, i.e.,

(2) is the complex modulated symbol (after trellis where shaping) on the th subcarrier of the th OFDM symbol and is the OFDM symbol period without guard interval. In this paper, the effect of the guard interval is not considered since it does not change a peak power property of the OFDM signal. Also, each subcarrier is assumed to be modulated by an -ary QAM, i.e., , where denotes a set of complex-dimensional constellation points, which is normally chosen as a subset of the two dimensional lattice. With ergodicity of the signal in mind, the average power can be expressed as (3) where denotes the mean of a random variable . Now, let denote the minimum squared Euclidean distance (MSED) in , i.e., (4)

(8) It can be easily shown that if is a complex stationary Gaussian process, the complementary cumulative distribution is given by (9) Apparently, the distribution of the instantaneous power does not depend on the number of subcarriers as long as the signal is well approximated by a Gaussian process. B. Peak-to-Average Power Ratio The peak-to-average power ratio (PAR)1 is another figure of merit that describes the dynamic range of the OFDM symbols. The PAR can be used to estimate how much back-off is needed 1The peak-to-average power ratio is often abbreviated as PAPR in the OFDM scenario, whereas PAR has been consistently used in the literature of coded modulation and constellation shaping [10]. This paper follows the latter convention for simplicity. It should be noted, however, that these earlier studies define the PAR as a deterministic ratio, typically for the two-dimensional single-carrier constellation without pulse-shaping filtering, whereas the definition of PAR in this paper is that of bandlimited signals, which is a more practical measure in many communications systems.

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Fig. 1. Trellis shaping system: (a) transmitter and (b) receiver.

in order to avoid nonlinear amplification or clipping at the transmitter. Assuming the PAR as a random variable, the conventional definition of the PAR for the th OFDM symbol may be expressed as

can be modulo-2 added to the Any valid code word in output sequence without changing the original data sequence at the receiver; provided the correct sequence is received, where denotes a modulo-2 addition operation, the data sequence can be retrieved by the syndrome former as

(10) (12) With this definition, the complementary cumulative distribution of the PAR can be used to estimate the clipping probability of the OFDM symbols. Unlike the normalized instantaneous power, the distribution of the PAR depends on the number of subcarriers , and theoretical approximation formulas based on Gaussian approximation of the bandlimited OFDM signal can be found in [11].

for any valid code word in . since denote a mapping function which maps a binary Now, let data sequence to the corresponding signal constellation, i.e., (13) The constellation of the th subcarrier is then given by2

III. TRELLIS SHAPING The principle of the trellis shaping can be found in [5]–[8]. We begin with a brief review of its basic procedure and describe the signal mapping strategy, which is one of the key design criteria in the proposed system. A basic block diagram of the trellis shaping system is dedenote a convolutional code with rate scribed in Fig. 1. Let , which is referred to as a shaping code [5], and let be the corresponding generator matrix. Let and denote the parity check matrix and its left in) matrix for this code, respectively. verse (i.e., Let be a binary information data sequence to be transmitted by each -subcarrier OFDM symbol. At the transmitter with trellis shaping, information data bits are first divided into the two sets of sequences, and , where the former is used to choose the most significant bits (MSB) of the mapping constellation labeling, and the latter chooses its least significant bits (LSB). In -bit sequence is first encoded choosing the MSB, an by the inverse syndrome former to generate an -bit sequence , i.e., (11)

(14) corresponds to the th shaping symbol, is the where th output symbol of the shaping decoder, and deis the th data symbol that detertermines the MSB. Also, mines the LSB and thus is chosen a priori independent of the shaping symbol. In the original trellis shaping, is chosen such that the -complex-dimensional signal constellation chosen leads to minimization of the average by the sequence power [5]. In this paper, the optimization criterion lies in minimization of the autocorrelation sidelobes described in the next section. A. Constellation Mapping for Sign-Bit Shaping We first focus on the sign-bit shaping described in [5], since it not only serves as a lucid example but also offers significant peak and average power reduction capabilities. For the sign-bit 2In the rest of the paper, the OFDM symbol index l will be dropped from the subcarrier symbol A for brevity.

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Fig. 2.

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Constellation mapping for the sign-bit shaping (16-QAM): (a) MSB, (b) LSB: Type-I, and (c) LSB: Type-II.

shaping, the rate-1/2 convolutional code is used. The data seto yield a binary sequence as quence is encoded by

Suppose that the following code word above sequence:

is added to the

In this paper, we consider the bit mappings depicted in Fig. 2 for the square 16-QAM example with sign-bit shaping. In this are both two-bit symbols and thus the MSB will case, and choose a quadrant in Fig. 2(a). For the LSB, we evaluate the two types of mapping shown in Fig. 2(b) and (c). The mapping shown in Fig. 2(b) is referred to as Type-I in this paper and is designed such that the four points in the equivalent class (i.e., a set of the points with the same LSB labeling) form symmetry with respect to both of the axes. Apparently, this constellation mapping does not have the average power reduction capability, since

the energy of all the four points in the equivalent class chosen by the MSB is identical. However, the absence of average power reduction also suggests that the shaping decoder can work exclusively for the PAR reduction. In fact, as will be shown in the simulation results later, this mapping strategy offers significant reduction in the PAR. The other LSB mapping, shown in Fig. 2(c), is referred to as Type-II and is designed such that all the quadrants are identical. Unlike the Type-I mapping, the shaping decoder can now choose a signal point with different energy for a given LSB. This mapping thus offers the average power reduction capability. In both cases, the constellation within each quadrant is designed to employ Gray mapping such that the BER of an uncoded system is minimized.

B. Constellation Mapping for Multidimensional Shaping In order to enhance the bandwidth efficiency, the dimensionality of shaping should be increased as in [8]. In the case of -dimensional shaping (or, equivalently, -complex-dimensubcarriers will form one shaping symbol. sional shaping), Therefore, the code word consists of symbols, each

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Fig. 3. Constellation mapping for multidimensional shaping (16-QAM): (a) MSB, (b) LSB: Type-I, (c) LSB: Type-II.

symbol composed of bits that determine the MSB of secutive subcarriers. That is, is given by

con-

reduction capability similar to that of the sign-bit shaping. In the Type-II mapping shown in Fig. 3(c), the LSB mapping is designed such that the constellation of the diagonally opposite quadrant is identical. Again, in both cases, Gray mapping is used for the constellation within each half plane. C. Redundancy Ratio

Similar to the case of the sign-bit shaping, the mapping used for the multidimensional case is defined in Fig. 3 for the 16-QAM example, where each MSB will choose the half plane according to Fig. 3(a). In the Type-I mapping of the LSB shown in Fig. 3(b), the two points in the equivalent class form symmetry with respect to the origin. Again, this constellation mapping does not have the average power reduction capability but is expected to offer a good PAR

Similar to coded modulation, the trellis shaping requires redundancy and thus increases the constellation size for a given information rate. In other words, it reduces the information rate compared to the same -ary QAM system without shaping. For the -complex-dimensional trellis shaping with total transmission bits per complex dimension, one bit (out of bits) is reserved for shaping and this bit is encoded by rate shaping code defined by . Consequently, the overall information rate of the -QAM OFDM system with -complex-dimensional shaping is given by (15)

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where we define a redundancy ratio of this system3 as . Likewise, in the sign-bit shaping, since two bits are used for shaping with rate-1/2 convolutional code, one may write

B. Metric Design for the Viterbi Algorithm Instead of minimizing the absolute value of the sidelobes as in (19), we consider minimizing the square of them as (20)

(16) with . The redundancy ratio can thus be made small by increasing the constellation size or dimensionality. IV. METRIC DESIGN FOR TRELLIS SHAPING FOR THE OFDM SIGNAL An important issue when applying the trellis shaping principle to the OFDM system is the design of metric calculation used in the (shaping) Viterbi decoder. In this section, we derive a novel metric that can be used for reducing the autocorrelation sidelobes of an OFDM data sequence in conjunction with the Viterbi algorithm.

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The minimization process (20) can be transformed into a recursive process as follows. Let denote the number of subcarriers controlled by each shaping symbol . At the th stage (where here denotes a shaping symbol index and thus increases as ), the shaping symbol is chosen according to (21) is a set of the th output symbols of the shaping code where , is defined as (22)

A. Autocorrelation Representation of the OFDM Signal From (2), it may be straightforward to show that

(17) is the aperiodic autocorrelation function of the comwhere plex data sequence , defined for as

and is the aperiodic autocorrelation function of the complex data sequence of length , i.e., . Note that the index corresponds to the total number of subcarriers processed up to the th stage and is given by . The recursive process thus terminates at . In the following subsections, we derive an additive rein conjunction with the cursive structure of the metric Viterbi algorithm.

(18) C. Metric for Sign-Bit Shaping The first term in (17) is the direct current (dc) component, while the second term represents the fluctuation of the OFDM signal. Therefore, choosing a code word such that the sidelobes of the autocorrelation function are minimized, i.e., finding (19) may help reduce the dynamic range of the bandlimited OFDM signals.4 Note that finding optimum requires all the search over . , the Since the cardinality of the shaping code is brute force code search is too complex to implement in practice even for a moderately large value of . Thus, we shall develop a metric design such that the Viterbi algorithm can be applied for the search of . 3In the conventional constellation shaping scenario, the redundancy required for shaping is characterized by the shaping constellation expansion ratio (CER) [10]. The CER is defined as a ratio of the increased number of points (due to shaping) to the minimum number of the necessary points required for the same information rate without shaping, each per two dimensions or per subcarrier. . Thus, the CER and the redundancy ratio is related by CER 4In fact, Golay complementary sequences [12] are known to have very low autocorrelation sidelobes, and, due to this property, the OFDM signal modulated by this set of sequences results in a waveform with little fluctuation or very low PAR (see, e.g., [13] and references therein).

r

=M

and the following recurFor the sign-bit shaping case, sive relationship can be easily derived:

(23) where (24) All possible entries for can be computed and tabulated beforehand for computational complexity reduction. Substituting (23) into (22), can be expressed in the following recursive form:

(25) The third term in the right-hand side of (25) is a function of the power of subcarrier symbols, which remains constant for the Type-I mapping regardless of the chosen path. Therefore, it can be omitted for mapping strategies without average powerreduction capability, such as the Type-I mapping. Otherwise, may be computed and tabulated all possible entries for beforehand in order to alleviate computational overhead.

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After the minimum path for each state is chosen by the Viterbi algorithm, the autocorrelation can be revised based on (23), and are those attributed to the chosen path. where This value can be used for the recursive calculation of the next in (25). Note that, unlike the conventional Viterbi metric decoder for convolutional codes, the calculation of (25) requires that each state should store and refer to all the past subcarrier symbols associated with the chosen path. Thus, the complexity of metric calculation required for the proposed algorithm is considerably higher than that of a typical Viterbi decoder.

where (32) -complex-dimensional case, the update of the For the in (28) is expressed as metric

D. Metric for Multidimensional Shaping

(33)

Likewise, an extension to the multidimensional shaping is straightforward. In this case, , and we obtain the recursive relationship of the autocorrelation from (23) as

(26) where

The total number of complex multiplication can then be estias mated for

(34) Let us define the complexity reduction ratio achieved by the trellis window truncation as (35)

(27) Consequently, the metric for each state can be expressed as

(28) . Again, the third term of (28) can be for omitted if one uses the Type-I mapping. E. Complexity Reduction by Trellis Window Truncation Computational complexity of the proposed algorithm is dominated by the metric calculation of (25) or (28). For an implementation based on the digital signal processor (DSP), the number of complex multiplications in the metric of (25) or (28), which increases linearly with the trellis section index , is the denote the number of states major part of the complexity. Let in the Viterbi decoder for . For the -complex-dimensional case, the total number of complex multiplications can be estimated as (29) Therefore, for a large value of , the complexity is still prohibitive. In this case, instead of working with (20), we shall consider minimizing the truncated version of the autocorrelation sidelobes, i.e., (30) where can be seen as a window size of the trellis and should . Then, (21) is rewritten as be chosen as (31)

From (29) and (34),

can be approximated as (36)

is a normalized window size. From this exwhere . pression, the reduction ratio of 25% is achieved with , reaches 56.25%, in which case the Moreover, with complexity can be reduced by more than half. The price for this significant complexity reduction is its loss in the dynamic range reduction capability. This tradeoff will be examined in subsequent computer simulations. V. SIMULATION RESULTS We study the performance of the proposed trellis shaping in terms of dynamic range and average power reduction capabilities by computer simulations. The 4-state and 64-state convolutional codes with maximum free distance are used for shaping code . However, the choice of convolutional codes may not have significant impact on resulting shaping performance, as noted in [8]. As figures of merit, we evaluate the complementary cumulative distributions of the normalized instantaneous power and PAR, as well as the gain due to average power reduction defined in Section II. In the simulation, each subcarrier is modulated by the square 256-QAM constellation with either the Type-I or Type-II mapping. Although all of the simulation results presented are based on 256-QAM, very similar results have been obtained with other constellations such as the square 16-QAM. The bandlimited OFDM symbol is generated with eight times oversampling for the PAR evaluation. Note that, while a sufficient oversampling factor is required to characterize the distribution of the PAR of bandlimited signals, it does not have much influence on the distribution of the normalized instantaneous power with a sufficiently long observation interval.

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Fig. 4. Complementary cumulative distribution of the 256-QAM-modulated OFDM signal with and without the proposed 4-state sign-bit trellis shaping. (a) Normalized instantaneous power. (b) Peak-to-average power ratio. Note that the instantaneous power is given in a linear scale, whereas the peak-to-average power ratio is given in a logarithmic scale.

A. Dynamic Range-Reduction Capability of Sign-Bit Shaping We begin with evaluation of the dynamic range reduction capability of the simple 4-state sign-bit shaping. The redundancy for 256-QAM. Fig. 4(a) and (b) shows ratio is the complementary cumulative distributions of the normalized instantaneous power and PAR, respectively. In these figures, the theoretical distributions based on the Gaussian assumption, i.e., (9) for the instantaneous power and the exact form derived in [11] for the PAR, are plotted as a reference for the cases without shaping. As observed, the proposed shaping can significantly mitigate the occurrence of high peak power, and the Type-I mapping offers better performance than the Type-II mapping. It is interesting to note that very little difference is observed for 64- and 256-subcarrier OFDM signals in the case of the normalized instantaneous power. It is thus conjectured that regardless of the number of subcarriers, the proposed trellis shaping offers a similar reduction capability in terms of the signal dynamic range.

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Fig. 5. Complementary cumulative distribution of the 256-QAM-modulated 64-subcarrier OFDM signal using the proposed sign-bit shaping with different number of states. (a) Normalized instantaneous power. (b) Peak-to-average power ratio.

We now evaluate the performance improvement by increasing the number of states in the trellis shaping decoder. The results are shown in Fig. 5 with 64-subcarrier OFDM system. It is observed that, by increasing the number of states, and thus by augmenting the complexity of the shaping Viterbi decoder, further reduction of the dynamic range can be achieved. B. Dynamic Range Reduction Capability With Multidimensional Shaping The use of multidimensional trellis shaping will reduce the redundancy ratio for a given constellation size as well as complexity of the shaping Viterbi decoder defined in (29) for a given . The consequence of this cost reduction is its relative performance loss in terms of the dynamic range reduction capability. Comparisons of the two distributions with respect to dimensionality of 1 (sign-bit), 2, and 8 are shown in Fig. 6. 1/8, Note that the corresponding redundancy ratios are 1/16, and 1/64, respectively. As observed from the figure, an increase of dimensionality causes a loss in the dynamic range reduction performance.

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Fig. 7. Gain in average power reduction achieved by the trellis shaping with the Type-II mapping for different number of subcarriers, states, and dimensionality of the 256-QAM-modulated OFDM signals.

Fig. 6. Complementary cumulative distribution of the 256-QAM-modulated 64- and 256-subcarrier OFDM signals using the proposed 4-state trellis shaping with the Type-I mapping and different order of dimensionality. (a) Normalized instantaneous power. (b) Peak-to-average power ratio.

C. Average Power Reduction Capability of the Type-II Mapping Although the Type-II mapping shows an inferior PAR reduction performance compared to that of the Type-I mapping, its benefit lies in the additional capability of average power reduction for a given MSED. The average power reduction achieved by the Type-II mapping is shown in Fig. 7 with a various number of subcarriers. It is observed that the sign-bit shaping can achieve a gain of approximately 3 dB in terms of average power reduction, and the gain decreases as the dimensionality increases. Also, increasing the number of states in the Viterbi decoder helps augment this gain. D. Effect of Complexity Reduction via Trellis Window Truncation We examine the performance of the trellis shaping with computational reduction by trellis window truncation discussed in Section IV-E. Fig. 8(a) shows the distribution of the normalized instantaneous power of 64- and 256-subcarrier OFDM systems with the Type-I mapping and various normalized window size

Fig. 8. Complementary cumulative distribution of the 256-QAM-modulated OFDM signal using the proposed 4-state trellis shaping with reduced complexity via trellis window truncation and the Type-I mapping. (a) Normalized instantaneous power of 64- and 256-subcarrier OFDM signals. (b) Peak-to-average power ratio of 256-subcarrier OFDM signal.

. It is interesting to observe the similarity in the two distributions with and 256 for a given . From this result, it is

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Fig. 9. Gain in average power reduction achieved by the 4-state trellis shaping with the Type-II mapping for 256-QAM-modulated 64- and 256-subcarrier OFDM signals with different window size and dimensionality.

conjectured that the distribution of the normalized instantaneous power is well characterized by the normalized window size , rather than the number of subcarriers or actual window size themselves. The distribution of the PAR is shown in Fig. 8(b) for , where it can be seen that the PAR reduction capability is affected only gradually by a decrease of . A similar tendency has been observed for the cases with the Type-II mapping and a different number of subcarriers. These results suggest that most reduction can be achieved by the range of between 1/2 and 1/4, and further increments may result in a minor improvement of the dynamic range reduction capability. Fig. 9 shows the relation between the gain due to average power reduction and the normalized window size for the 4-state trellis shaping with the Type-II mapping and different dimensionality. From this figure, it is observed that most reduction gain is already achieved with very small , and the gain is less sensitive to an increase of the window size. Considering the performance–complexity tradeoff, the choice of around 1/2 to 1/4 may thus be reasonable in practice. E. BER Evaluation Over AWGN Channel The BER performance of the proposed system remains to be evaluated. In the following simulation, the additive white Gaussian noise (AWGN) channel without frequency-selecis tive fading is considered for simplicity. By definition, is the the transmit energy per information bit, whereas one-sided power spectral density of an AWGN process. Let and denote the required transmit symbol energy in order to achieve a given MSED with and without shaping, with respectively. Then, noticing that given by (15), one may write

Fig. 10.BERs of the uncoded trellis shaped 256-QAM-modulated 64-subcarrier OFDM system with the Type-I or Type-II mapping and different dimensionality. (a) Ideal AWGN channel. (b) After envelope limiter with IBO = 5 (7 dB). The BERs of 128-QAM (crosses) and 256-QAM (squares) signal without shaping are also plotted.

(37) where the second term of (37), which is always negative for , corresponds to the loss due to constellation redundancy introduced by shaping, and the third term corresponds to the gain due to average power reduction by shaping with the Type-II mapping. The simulation results of a 64-subcarrier OFDM system over an AWGN channel are shown in Fig. 10(a), with various dimensionality of the 4-state trellis shaping. For comparison, the BER of the cross 128-QAM signal without shaping [14] (which achieves the same information rate as the sign-bit 256-QAM) is shown, along with that of the 256-QAM signal without shaping. It can be seen that the Type-I mapping exhibits only the loss associated with reduction of an information rate, i.e., , since there is essentially no average power reduction (i.e., ). On the other hand, the shaping with the Type-II mapping achieves a lower BER than that of the 256-QAM, and the performance of the sign-bit shaping is even

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comparable to that of the cross 128-QAM without shaping for high SNR.5 F. BER Evaluation With Envelope Limiter The benefit of the proposed shaping techniques becomes apparent when their performances are compared over channels with peak-to-average power limitation. In the following simulation, we consider the envelope limiter as a simple example of nonlinear devices, where the output complex OFDM signal is given by otherwise.

(38)

The IBO above corresponds to that of the envelope limiter, and is the normalized instantaneous power defined by (7). The resulting BERs of the same systems in Fig. 10(a) after envelope dB are compared in Fig. 10(b). Note limiting with IBO that the SNR in this figure is defined as the average signal power before envelope limiter to the noise ratio. It can be observed that, due to the nonlinearity of the channel, the performance of the systems without shaping exhibits severe error floor, whereas those with shaping can significantly reduce the resulting error floor. It is also observed that, with this particular value of IBO, the performance of shaping with the Type-I mapping can eventually exceed those with the Type-II mapping for high SNR. In particular, the Type-I with sign-bit shaping exhibits error floor , which may be low enough for uncoded perforbelow mance in many applications. This error-floor effect can be inferred from the corresponding statistical behavior of the PAR dB , the clipping in Fig. 6(b). With the threshold PAR probability is lowest for the Type-I sign-bit shaping, whereas the clipping probability for those without shaping is close to 1. VI. CONCLUSION In this paper, we have proposed a new trellis shaping design based on the recursive autocorrelation sidelobe minimization of an OFDM data sequence. It has been shown that the performance depends on signal constellation mapping, and the Type-II mapping introduced in the paper can reduce both dynamic range and average power, while the Type-I mapping reduces dynamic range only, but with better reduction capability than that of the Type-II mapping. Throughout the paper, the OFDM system is assumed to be uncoded, but in a practical system some form of error control coding is employed in order to reduce the required SNR with improved reliability. In fact, the proposed shaping system can be combined naturally with the multilevel coded modulation [15], [16]. The extension to a coded OFDM system will be further investigated in the subsequent work. 5This may be best explained by means of shaping gain [5]. For the sign-bit shaping, the shaping gain of complex-dimensional constellation signal is given (1=2)3, which turns out to be about 1 (0 dB) from Fig. 7. This means by that the proposed Type-II sign-bit shaping achieves almost the same bit error performance as the 128-QAM without shaping for high SNR.



ACKNOWLEDGMENT The author wishes to thank the reviewers for the constructive comments that helped to improve the presentation of this paper. REFERENCES [1] J. Chuang and N. Sollenberger, “Beyond 3G: wideband wireless data access based on OFDM and dynamic packet assignment,” IEEE Commun. Mag., vol. 38, pp. 78–87, July 2000. [2] H. Sari, G. Karam, and I. Jeanclaude, “Transmission techniques for digital terrestrial TV broadcasting,” IEEE Commun. Mag., vol. 33, pp. 100–109, Feb. 1995. [3] X. Li and L. J. Cimini Jr, “Effects of clipping and filtering on the performance of OFDM,” in Proc. IEEE Vehicular Technology Conf. (VTC’97), Phoenix, AZ, May 1997, pp. 1634–1638. [4] H. Ochiai and H. Imai, “Performance analysis of deliberately clipped OFDM signals,” IEEE Trans. Commun., vol. 50, pp. 89–101, Jan. 2002. [5] G. D. Forney Jr., “Trellis shaping,” IEEE Trans. Inform. Theory, vol. 38, pp. 281–300, Mar. 1992. [6] I. S. Morrison, “Trellis shaping applied to reducing the envelope fluctuations of MQAM and band-limited MPSK,” in Proc. Int. Conf. Digital Satellite Commun. (ICDSC’92), May 1992, pp. 143–149. [7] M. Litzenburger and W. Rupprecht, “Combined trellis shaping and coding to control the envelope of a bandlimited PSK-signal,” in Proc. IEEE Int. Conf. Communications (ICC’94), New Orleans, LA, May 1994, pp. 630–634. [8] W. Henkel and B. Wagner, “Another application for trellis shaping: PAR reduction for DMT (OFDM),” IEEE Trans. Commun., vol. 48, pp. 1471–1476, Sept. 2000. [9] J. Tellado and J. Cioffi, “Peak power reduction for multicarrier transmission,” in Proc. IEEE Communication Theory Mini Conf., GLOBECOM ’98, Sydney, Australia, Nov. 1998, pp. 219–224. [10] G. D. Forney Jr. and L. F. Wei, “Multidimensional constellations—Part I: introduction, figures of merit, and generalized cross constellations,” IEEE J. Select. Areas Commun., vol. 7, pp. 877–892, Aug. 1989. [11] H. Ochiai and H. Imai, “On the distribution of the peak-to-average power ratio in OFDM signals,” IEEE Trans. Commun., vol. 49, pp. 282–289, Feb. 2001. [12] M. J. E. Golay, “Complementary series,” IRE Trans. Inform. Theory, vol. IT-7, pp. 82–87, Apr. 1961. [13] K. G. Paterson, “Generalized Reed-Muller codes and power control in OFDM modulation,” IEEE Trans. Inform. Theory, vol. 46, pp. 104–120, Jan. 2000. [14] J. G. Proakis, Digital Communications, 3rd ed. New York: McGrawHill, 1995. [15] H. Imai and S. Hirakawa, “A new multilevel coding method using error correcting codes,” IEEE Trans. Inform. Theory, vol. IT-23, pp. 371–377, May 1977. [16] U. Wachsmann, R. Fischer, and J. Huber, “Multilevel codes: Theoretical concepts and practical design rules,” IEEE Trans. Inform. Theory, vol. 45, pp. 1361–1391, July 1999.

Hideki Ochiai (S’97–M’01) Hideki Ochiai received the B.E. degree in communication engineering from Osaka University, Osaka, Japan, in 1996, and the M.E. and Ph.D. degrees in information and communication engineering from The University of Tokyo, Tokyo, Japan, in 1998 and 2001, respectively. From 1994 to 1995, he was with the Department of Electrical Engineering, University of California, Los Angeles, under the scholarship of the Ministry of Education, Science, and Culture. From 2001 to 2003, he was with the Department of Information and Communication Engineering, The University of Electro-Communications, Tokyo, Japan. Since April 2003, he has been with the Division of Electrical and Computer Engineering, Yokohama National University, Yokohama, Japan, where he is currently an Assistant Professor. From 2003 to 2004, he was a Visiting Scientist at the Division of Engineering and Applied Sciences, Harvard University, Cambridge, MA. Dr. Ochiai was a recipient of a Student Paper Award from the Telecommunications Advancement Foundation in 1999 and the Ericsson Young Scientist Award in 2000.

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Reduced-Delay Protection of DSL Systems Against Nonstationary Disturbances Dimitris Toumpakaris, Member, IEEE, John M. Cioffi, Fellow, IEEE, and Daniel Gardan

Abstract—In addition to being designed to successfully cope with stationary noise, crosstalk, and intersymbol interference, digital subscriber loop (DSL) systems need to be shielded from nonstationary disturbances, such as impulse noise and RF interference. Currently, deployed DSL systems achieve protection against nonstationary interference using a combination of Reed–Solomon (RS) codes and interleaving. However, interleaving results in delay. Long delays are undesirable in high-rate systems that support interactive applications. In this study, it is shown that the interleaving delay of DSL systems can be significantly reduced by performing erasure decoding of the RS codewords at the receiver. Three different techniques for determining the erasures are proposed. Use of the techniques results in a reduction of the interleaving delay that is required to mitigate worst-case impulse noise by up to a factor of 2, which is verified by simulation. Moreover, the techniques do not require any changes at the transmitter and therefore guarantee compatibility with currently deployed systems. Index Terms—Electromagnetic interference, impulse noise, interleaved coding, subscriber loops.

I. INTRODUCTION N RECENT years, digital subscriber loop (DSL) transmission systems that utilize the existing copper infrastructure of the telephone network between the provider’s line terminal and the customer’s network terminal, have been experiencing growing popularity. As an example, asymmetric DSL (ADSL) [1] is a rapidly expanding service, delivering data rates of up to 6 Mb/s to residential and business customers, whereas very-high bit-rate DSL (VDSL) [2], [3] will provide substantially higher rates, possibly up to 100 Mb/s. In order to attain high rates using the telephone wires, DSL systems rely on sophisticated designs that compensate for intersymbol interference (ISI), stationary noise, the crosstalk induced by other twisted pairs in the same bundle, and the nonstationary disturbances that appear on the copper lines. To combat ISI, DSL systems employ equalizers or, as is the case for ADSL, discrete multitone (DMT) transmission. The levels of stationary noise and crosstalk determine

I

Paper approved by E. Ayanoglu, the Editor for Communication Theory and Coding Application for the IEEE Communications Society. Manuscript received April 15, 2003; revised February 9, 2004. This work was supported in part by France Télécom R&D. This paper was presented in part at the IEEE International Conference on Communications, Anchorage, AK, May 2003, and at the IEEE Global Communications Conference (Globecom), San Francisco, CA, December 2003. D. Toumpakaris is with Marvell Semiconductor, Inc., Sunnyvale, CA 94089 USA (e-mail: [email protected], [email protected]). J. M. Cioffi is with the Department of Electrical Engineering, Stanford University, Stanford, CA 94305-9515 USA (e-mail: [email protected]). D. Gardan is with France Télécom R&D, 22 307 Lannion, France (e-mail: [email protected]). Digital Object Identifier 10.1109/TCOMM.2004.836589

the maximum rate that can be transmitted through a given subscriber loop. Recent work in this area has shown that the performance of DSL systems can be further improved by coordinating the connections that utilize the same bundle [4]–[7]. In this case, the conservative signal-to-noise ratio (SNR) margins and power spectral density (PSD) masks that are used in current systems to protect neighboring services can be relaxed, leading to higher data rates. Moreover, DSL systems are shielded from nonstationary interference using a combination of Reed–Solomon (RS) codes and interleaving. Nonstationary interference in DSL typically consists of impulse noise and RF interference (or igress) (RFI). Impulse noise is the term commonly used to describe noise bursts of nonstationary nature and relatively high energy that are caused by electromagnetic interference due to physical phenomena, electrical switches, motors, and home appliances. Lately, there has been an increased interest in modeling impulse noise by many researchers and operators. Despite the ongoing effort, because of the highly nonstationary nature and the variety of the noise bursts observed on the twisted pairs, a universally accepted model is yet to be agreed upon [8]–[12]. RFI usually refers to nonstationary interference constrained to a relatively narrow part of the spectrum and is due to similar sources as impulse noise, as well as to HAM radio interference. Although most DSL systems operate in frequencies below the standardized amateur bands, and VDSL is designed so that both ingress and egress are avoided, RFI can still appear in the used frequency band due to nonlinearities of the RF circuits, imperfect filtering, and signal leakage due to the fast Fourier tranforms (FFTs) at the receiver of DMT-based DSL systems. As SNR margins are lowered to allow for exploiting the full potential of DSL systems, protection against nonstationary disturbances becomes more crucial. Although the combination of RS codes and interleaving can provide any level of protection against impulse noise, this comes at the expense of interleaving delay that is highly undesirable for some applications requiring fast interaction, such as teleconferencing or video games. Reducing the interleaving delay would facilitate higher throughputs at higher layers. Regarding RFI, it can also lead to large delay requirements, as will be explained below. Improving the correction capability of RS codes can lead to delay reduction without compromising the immunity of DSL systems to impulse noise when RFI is also present. In this study, it is shown that the interleaving delay can be significantly reduced by performing erasure decoding of the RS codewords as long as the delay remains below the shortest interarrival time between consecutive impulses. The reduction is achieved without any rate penalty. Hence, the main focus of this paper is on developing reliable

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Fig. 1. System model.

erasure techniques that locate the potentially erroneous bytes of each received RS codeword and erase them before decoding. The proposed techniques utilize the existing structure already in place in DSL systems and do not affect processing at the transmitter, therefore ensuring compatibility with currently deployed systems. It should be noted that a byte erasure technique that utilizes unused tones in DMT systems is proposed in [13]. The techniques in this study follow a different approach compared to that of [13] in order to mark the potentially erroneous bytes. Section II describes the nonstationary interference mitigation scheme that is used in typical DSL systems. Section III focuses on interleaving in the context of DSL systems in order to facilitate the discussion of the proposed erasure techniques that are presented in Section IV. In Section V, the achieved delay reduction is verified by simulating a DMT-VDSL system. The contributions of the study are summarized in Section VI. II. SYSTEM MODEL The proposed block diagram of the nonstationary noise protection system is shown in Fig. 1. The only difference with current DSL systems, where the soft information link from the demodulator to the RS decoder does not exist, is that the receiver structure is augmented in order to allow use of the techniques of Section IV. To simplify the diagram, and due to the fact that the effect of RFI depends on the modulation scheme [DMT or single-carrier modulation (SCM)], only the effect of impulse noise on the received data is depicted. A worst-case noise burst of very high energy will destroy most of the transmitted symbols of an SCM system while it lasts. Moreover, in DMT systems, even after the FFT, most tones will be affected due to the large amplitude of the time-domain samples of the burst. This is verified by actual impulse noise data. Hence, worst-case impulse noise creates bursts of erroneous bytes in the data stream. Since the interleaved buffer of DSL systems needs to be shielded from worst-case noise, this will be the focus of this study, and, in the development of the impulse noise mitigation techniques of Section IV, no distinction will be made between DMT and SCM DSL systems.

Although the effect of worst-case impulse noise is similar for DMT and SCM systems, this does not hold for RFI. For SCM systems, the time-domain samples of the RFI signal will be superimposed on the received symbols, creating errors at the time instants where the amplitude of the RFI signal is large. In the case of DMT systems, the FFT of the RFI signal will be superimposed on the symbols in each tone. Due to FFT leakage, an RFI signal will affect more than one tone in general, creating a short burst in the frequency domain which translates to a burst in the data stream. This study does not get into the details of the error generation due to RFI. More details on the effect of RFI on DMT systems can be found in [14]–[16]. The user data to be transmitted over the subscriber loop are grouped in blocks of bytes and sent to the input of an RS encoder that adds parity bytes, thus increasing the block size to . RS codes are block codes with alphabets . Current DSL systems use byte-oriented RS codes in that operate on , restricting the maximum block size to bytes. At the receiver, the decoder is able to reconRS codeword provided that no struct the transmitted bytes are in error. When the exact location of more than the erroneous bytes is known, the receiver can perform erasure decoding and can reconstruct the transmitted codeword as long as the number of erased bytes does not exceed . More generally, if the number of parity bytes is equal to , the decoder can reconstruct the original codeword provided that , where is the number of errors whose location is unknown, and is the number of erasures. Assuming correct erasures, if is , the decoder will either declare a failure, such that in which case the received bytes are left intact, or will miscorrect to a wrong codeword. The larger the portion of allocated to the detection of errors of unknown location is, the smaller the probability of miscorrection becomes. For more details on how RS codes work, the reader is referred to error-correction coding texts, such as [17]. It should be emphasized that the assumption of correct erasures is very critical, and care should be taken in order for the set of erased bytes to always contain at least the erroneous ones. All of the techniques of Section IV guarantee that this condition is satisfied. The encoded bytes are then interleaved in order to randomize the error bursts caused by impulse noise, effectively creating a code, where is the interleaver depth, i.e., the separation between two consecutive bytes after interleaving. This way, the correction capability of the system increases by a factor of . However, the interleaving delay is proportional to , which means that there is a tradeoff between the error-correction capability of the system and the delay. Nevertheless, if erasure decoding is used, the error-correction capability of the RS codes doubles for a given . Thus, protection can be improved without the need to increase the delay or reduce the user rate or, equivalently, for a given level of protection, the interleaving delay can be decreased. Section III examines interleaving more closely in order to facilitate the understanding of the erasure techniques proposed in this paper. Although, in general, interleaving does not contribute to protection against RFI, as will be explained in more detail in Section III, when erasure decoding is used the correction capability of the RS code doubles, and protection against impulse noise in the presence of RFI can be achieved with lower

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delays. Finally, it should be noted that the full interleaving gain can be achieved when the interarrival times between impulses are longer than the interleaving delay. In this study, it is assumed that this condition is satisfied. The interleaved stream is then sent to the DSL system for transmission. The details on how the stream will be transmitted depend on the specific standard that is used. Before transmission, the data may be further encoded using an inner code. For example, ADSL and DMT-VDSL systems include the option of using Wei’s four-dimensional (4-D), 16-state trellis code [18]. At the receiver side, the data are demodulated (and, if an inner code is used, decoded) and sent to a deinterleaver that has the same parameters as the interleaver at the transmitter. The bytes at the output of the deinterleaver form the original RS blocks. However, some RS codewords may contain bytes in error. The RS decoder attempts to reconstruct the original bytes either relying on the parity alone or also assisted by the erasure information.

Fig. 2.

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Generalized triangular interleaver.

III. INTERLEAVING Fig. 1 shows a data-stream error burst caused by impulse noise. Note that, in general, interleaving does not help combat RFI. For the case of DMT systems, and due to its long duration, RFI may be present in most DMT symbols of the data stream. Therefore, even when interleaving is used, there will be no long RFI-free part of the data stream where the short data bursts can be spread out. In the case of SCM systems, since errors will most probably be caused at the bytes where the RFI signal reaches its peak values, the errors will already be spread across the data stream. Hence, in most cases, interleaving does not improve protection from RFI, and therefore the interleaving gain only applies to impulse noise. However, when RFI is present, the interleaving delay that is required in order to mitigate impulse noise increases, as will be shown in this section. Note that, although the duration of RFI is significantly larger than that of impulse noise, RFI is nonstationary since the frequencies where it appears change, usually every few seconds. A. Triangular Interleaver The byte interleaver proposed for VDSL systems [19] is the so-called generalized triangular convolutional interleaver. Its block diagram is shown in Fig. 2. Before interleaving, the data are grouped in blocks of bytes, where , is an integer, and is the size of the RS codewords. Each byte of each block of bytes is sent to a buffer with different delay. More specifically, the th byte of each block is delayed by bytes. is a parameter that can be changed according to the desired level of impulse protec, each RS codeword is tion and depends on . When divided in blocks of bytes. The first bytes of each block will not be delayed. Another bytes are delayed by , and so on. The largest delay, i.e., the interleaving delay, is experienced ) and is by the th byte of each block (bytes . Typical values that are used in DSL equal to systems are for , which results in eight blocks of size per RS codeword, and for

Fig. 3. Mapping of corrupted bytes to a user stream when using a generalized triangular interleaver.

, which yields four blocks of size . The operation of the generalized triangular interleaver is illus, , and to simplify trated in Fig. 3, where the diagram. In the remainder of this paper, the generalized triangular interleaver is used as an example to demonstrate the improvement achieved by the erasure methods. B. Calculation of the Required Interleaving Delay Consider a system subject to impulse noise that can corrupt up to consecutive bytes of the interleaved data. Suppose that , the correction capability of the RS code (equal to or depending on whether erasures are used) satisfies and that the th byte of an RS block gets mapped to a location in the interleaved stream that is corrupted by impulse noise. The . As explained above, delay of this byte is in each RS codeword, bytes experience the same delay. In the worst case, all remaining bytes that are delayed by by the interleaver will fall on the corrupted area of the interleaved stream. Suppose now that the next group of bytes in the same RS codeword whose delay is also falls in the corrupted area of the interleaved stream. Then, up to bytes of the received RS codeword may be in error. By induction, if the error-correction capability of the RS code is equal to bytes, for the errors to be correctable, no more than such “groups” of bytes should get mapped to the corrupted area. Hence, the difference in the delay of bytes that same-delay groups apart should be larger are more than than the worst-case length of the corrupted area. Therefore, (1) To evaluate the effect of RFI on the interleaving delay, it is assumed that up to bytes of each RS codeword can be in error

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due to RFI. Note that the value of depends on . Suppose, as an example, that ten evenly spread erroneous bytes are con, the RS codeword tained in a block of 200 bytes. If contains ten errors, and . On the other hand, if the , then 200 bytes belong to two RS codewords of size . In the following, the notation will be used in order to emphasize this fact. In addition to , the function depends on the modulation parameters of the system and can be obtained using models or by simulation. Most of the time, the system will only be subject to RFI. To should satisfy ensure error-free transmission, or else the RS decoder may not be able to reconstruct the orig, inally transmitted data. When erasures are used, else . When a noise burst appears, the erroneous bytes will be deinterleaved into RS codewords that may also contain erroneous bytes due to RFI. The remaining correction capability of the code that can be used to mitigate impulse noise . Hence, the expression for becomes is

When RFI does not appear on the lines or is successfully dealt with using RFI cancellation, (1) is a sufficient condition for error-free transmission. On the other hand, when RFI protection relies solely on the RS code, the sufficient condition becomes (2) , . To avoid this, the value Note that, if of should be reduced by changing the block size . In practice, the following conditions should be satisfied: (3) Fig. 4.

Although interleaving does not, in general, improve immunity against RFI, has to be increased in order for a system to cope with both RFI and impulse noise compared to the case where only impulse noise affects transmission. If the amount of RFI-induced errors leads to an unacceptable interleaving delay, of the RS block should be reduced in order to rethe size or should be increased. Alternatively, it may be duce possible to use some RFI cancellation scheme that corrects or reduces the number of erroneous bytes before they are sent to the RS decoder. IV. ERASURE TECHNIQUES This section presents three different erasure techniques. The first one is targeted to systems that do not use an inner code, whereas the second assumes a coded system. Neither technique can handle the case of RFI. The third technique can be employed when both impulse noise and RFI affect transmission at the expense of increased complexity. As was previously stated,

Double-pass technique.

all techniques assume that the interarrival time between consecutive impulses is longer than the interleaving delay. A. Double-Pass Technique The double-pass technique [20], which is described in detail in Fig. 4, can be used in systems that do not employ an inner code as long as they are not subject to RFI or if they mitigate the effect of RFI using some cancellation method. Although it can be applied to coded systems without any changes, when access to the decoder metrics is available it may be preferable to use the more robust, and also cheaper in terms of additional calculations, technique of Section IV-B. The double-pass technique relies on the observation that, provided that some conditions are met, the RS codewords in the deinterleaved stream that contain the largest number of possibly erroneous bytes will always be preceded by RS codewords that contain fewer bytes that were

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Fig. 5. Steps of the double-pass technique.

affected by a noise burst. This is described by the following lemma, which is proven in the Appendix. Lemma 1: Consider a block of consecutive bytes in the interleaved stream, all of which are in error due to a noise burst. , and same-delay byte “groups” (of size ) of an RS If codeword in the deinterleaved stream contain bytes mapped to the corrupted block in the interleaved stream, the following RS codeword in the deinterleaved stream cannot contain more than such groups. Moreover, in each RS codeword, the samedelay groups containing erroneous bytes will be consecutive. are larger For most VDSL deployment rates, the values of than . Consider, for example, a VDSL system with a transmission rate equal to 20 Mb/s. Suppose that a noise burst corrupts data corresponding to 500 s. This is the maximum impulse noise length specified in the standards [3]. Then, in the worst b bytes will case, and be in error. When a (240, 224) RS code is used with , from (1), . Similarly, it can be shown when a (144, 128) code is used in a typical VDSL that system. Therefore, when Lemma 1 holds, before encountering RS codewords with the maximum number of bytes in error, there will be an indication of the occurrence of a noise burst because of errors in previous RS codewords. If the number of errors , those errors can be corrected, is less than or equal to and therefore the locations of the bytes in error can be determined. Using the interleaving function, one can then determine the byte blocks of the interleaved stream to which those bytes belong, and the entire blocks that contain erroneous bytes can be marked as corrupted. In a DMT system, the byte blocks in the interleaved buffer will be blocks containing all bytes coming from the same DMT symbol, whereas in an SCM system the blocks would be groups of bytes of fixed size. As long as the RS decoder does not declare failure, it does not use any erasure information. However, when it fails to decode an RS codeword, the table of the corrupted DSL blocks is used and all the bytes that come from those blocks are erased. The technique satisfies (1) and that the maximum works well provided that number of corrupted DSL blocks is correctly estimated off-line when choosing the interleaver parameters. A tricky situation arises when some bytes of a codeword that is decoded using erasures belong to byte blocks of the interleaved stream that have not been marked yet as corrupted. Under

the assumption of consecutive corrupted bytes, those bytes can be erased without the risk of overerasing, as long as the total number of marked consecutive DSL blocks is less than or equal , so that satisfies to , where (bytes per DSL block) (1). A conceptual diagram of the double-pass technique is presented in Fig. 5. The technique is robust to random errors due to background noise and crosstalk, since such events will only result in one byte error and they will not divert the RS decoder to erasure mode. A problem can only occur when a byte that is in error due to a random error falls inside an RS codeword that also contains erroneous bytes due to impulse noise. In this case, it is possible for the technique to enter the erasure mode and fail to decode correctly a certain number of RS blocks. However, the probability that a random error coincides with a severe noise burst is very low for typical target bit-error rates (BERs). Another common practice is to erase fewer than bytes in order to be able to locate random errors with the remaining parity bytes. Finally, it should be noted that the double-pass technique relies on the assumption that all of the bytes of the corrupted byte block are in error. If this is not the case (as, for example, when a mild impulse appears), there may be some very unfortunate situations where only the RS codewords that contain many same-delay “groups” mapped to the corrupted area contain erroneous bytes. However, the probability of such a situation where there is no indication is exceeded is very low, of trouble before the threshold of as is also verified by simulation, and does not affect the overall error probability of the system. Although in some cases an RS block has to be decoded twice, the double-pass technique does not have a major impact on the overall delay of the system, since the time required for the decoding of RS codes is small compared to the delay due to interleaving. Even if the decoding delay is an issue, a pipelining structure could be used that passes each received RS block through two consecutive decoders. The first one always decodes without using erasure information. If decoding succeeds, the second decoder does not use erasure information either, whereas if decoding at the first RS decoder fails, the technique passes the erasure information to the second RS decoder in order to assist decoding. Finally, the double-pass technique can also be used in systems employing the more general convolutional interleaver, and the conditions that need to hold can be found in a similar manner.

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B. Inner Code-Based Technique As shown in Fig. 1, a DSL system may be using an inner code in order to achieve higher rates or lower the transmitted energy. The improvement in the performance of the system caused by the insertion of an inner code is usually quantified by the coding gain. However, it should be emphasized that typical inner codes do not improve the robustness of the systems against high-energy and bursty disturbances, such as impulse noise. Nevertheless, by monitoring the output metrics of the decoder, it is possible to determine if a high-energy disturbance has appeared on the line. Consider an inner code decoder that evaluates a reliability metric in order to decode the received data symbols. Typical metrics include the cost of the surviving sequences when maximum-likelihood sequence detection (MLSD) is implemented using the Viterbi algorithm or the a posteriori probabilities (APPs) of the received symbols when MAP detection is performed, using, for example, the “forward–backward” algorithm [21]. In the absence of noise outliers, there will be a sequence with significantly lower metric compared to the others in the case of MLSD or, when MAP is used, one APP whose value is very close to 1 for each received symbol. Occasionally, a stationary noise outlier may be causing the MLSD path metrics to converge or the APP values to get closer. However, in the MLSD case, the overall metrics of the surviving sequences will still be small, and when MAP is used the symbols with uncertain values will be limited. In this case, the designer can rely on the inner or outer code to correct the error, since the system is designed to achieve a certain BER assuming stationary noise. Even though there may be very rare occasions that neither the inner code nor the outer RS code will be able prevent errors appearing in the output, the overall BER will not exceed the design value. When a noise burst appears, it will typically cause more than one symbol error on the interleaved stream. Moreover, some of the received symbols will typically be very far from the originally transmitted ones due to the relatively high energy of impulse noise compared to the stationary interference. Therefore, the costs of all of the surviving paths will increase substantially or, in the MAP case, there will be many symbols whose APPs will be such that the decoder will not be able to decide with high reliability. Hence, when the decoder notices such effects, it is almost certain that a high-energy disturbance has appeared and that most probably it cannot be corrected by the inner code. Thus, erasures can be provided to the RS decoder by monitoring the output metrics of the inner code decoder [22]. When the Viterbi path metrics are small or for all received symbols (except possibly for very few in a block) there is an APP with a value very close to 1, the data block under consideration has not been corrupted by impulse noise, and no bytes are erased. However, when the path metrics of the Viterbi decoder exceed some threshold value or there are many received symbols with no APP above some threshold (close to 1) in a block, the entire block is marked as corrupted and the corresponding bytes are erased. The erasure information is then sent to the RS decoder, using the deinterleaver mapping function.

Fig. 6.

Square-distance method.

Note that this method implicitly assumes that the data are encoded in blocks. This is always the case for DMT-based systems who transmit data in DMT symbols of 250 s each. ADSL and DMT-VDSL systems use Wei’s code [18] that encodes the bytes across the tones of each DMT symbol. The code is terminated at the end of each DMT symbol, so the erasure decision can be made each time a DMT symbol is received and decoded. In order to use the method in SCM systems, the designer needs to define an appropriate window for which the path metrics or the APPs are calculated. The inner code-based technique has the obvious advantage of not requiring any additional calculations other than the ones performed by the inner code decoder, provided that it has access to the decoder metrics. However, the technique only works for systems using an inner code and cannot be employed when high-energy RFI appears on the line, since the method offers no way of determining the length of the disturbance and cannot distinguish between impulse noise and the shorter duration but more frequently occurring RFI. C. Square-Distance Technique The square-distance technique achieves more reliable erasure calculation compared to the techniques of Sections IV-A and IV-B. It does so by examining each received symbol before the demodulation (or, equivalently, by comparing the received and demodulated symbols). The use of distances between received and demodulated symbols to find erasures was first proposed in [25] for DMT-based DSL systems. In this paper, it is shown that use of the technique can lead to reduction of the interleaving delay by a factor of 2 for worst-case impulse noise regardless of the modulation scheme used by the DSL system. Moreover, use of the technique is extended to the case where both impulse noise and RFI affect transmission [26]. Consider the received symbol in Fig. 6 after the equalizer. The receiver knows the constellation to which the symbol belongs, but not the actual symbol. In an SCM system, the constellation is the same for all transmitted symbols, whereas for DMT it depends on the tone. To illustrate the concept of the technique, a 16-QAM constellation is depicted. Suppose that was transmitted. Then, with very high probability, the received symbol will be very close to the transmitted one. Hence, if the around received symbol is inside one of the circles of radius the constellation points, with very high probability no error has occurred, since the probability that the noise vector added to

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the transmitted symbol brings it to another circle of radius is very small for the low BERs for which DSL systems are decircles of signed and the high SNR margins that are used (the Fig. 6 are exaggerated to allow better illustration). Occasionally, a stationary noise outlier may send a symbol outside the circle. However, this will happen very infrequently. The designer can either rely on the inner code to correct the error if the received symbol is still close enough to a constellation point or erase the bytes that the symbol affects. When impulse noise appears, a significant number of received symbols will be outside the circles. This is a clear indication that a high-energy disturbance has occurred. The proposed scheme can also handle the case of RFI, since, by examining each symbol, the granularity is refined to the symbol (and therefore the byte) level. Hence, in contrast to the previous techniques, the square-distance technique can distinguish between impulse noise that appears infrequently and affects several consecutive bytes and RFI that appears regularly, but usually affects smaller areas. As explained above, in order for the square distances to be obtained, the received symbol needs to be compared to the constellation where it belongs. Ideally, the square distance can be obtained by the demodulator (or the decoder when an inner code is used). A demodulator (or decoder) will need to calculate the square distances in order to demodulate (decode) so it can make the smallest square distance for each symbol available to the technique. If this is not possible, the square distances can be obtained by comparing the demodulated and received symbols. In that case, any distance metric can be used and not necessarily the square distances. D. Comparison of the Techniques As was also mentioned above, the square-distance technique is the most reliable and most useful, since it can provide erasures even when DSL systems are also affected by RFI. However, it is more complex and is more easily implemented when it can get soft information for every symbol from the demodulator or decoder of the inner code. If this is not possible, an additional stage that computes the distance metrics is required. The inner code-based technique also requires soft information. Nevertheless, the required amount of information is smaller when MLSD is used and can simply be the metric of the ML sequence. However, it only applies to systems using an inner code and cannot help combat RFI. Finally, the double-pass technique has the advantage of being applicable to any system and of not requiring any cooperation with the demodulator or the decoder, but is also vulnerable to RFI, and occasionally requires decoding a RS block twice, an operation that some designers may find undesirable. Table I compares the three techniques that were presented in this section.

V. SIMULATION RESULTS The remainder of this paper considers the case of DMT-VDSL, although, as has been mentioned in the previous sections, the proposed techniques can be applied to SCM systems as well.

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TABLE I COMPARISON OF THE 3-BYTE ERASURE TECHNIQUES

A. Interleaver Parameters for the DMT-VDSL Case In DMT-VDSL the duration of each DMT symbol is equal to 250 s. A noise burst of 500 s, which is the maximum duration specified in the standards [2], [3], will affect up to three DMT symbols depending on the alignment of the FFT and the temporal occurrence of the noise impulse. In general, the receiver’s FFT spreads the impulse so that some tones of the DMT symbol will not be affected. However, some of the impulses measured in actual local loops have very high energies that can corrupt most bytes of each DMT symbol. Moreover, it has been reported that in some cases the duration of these impulses can exceed 500 s [23]. Therefore, the simulations focus on the design of a system that can mitigate the effect of impulse noise that can corrupt up to four consecutive DMT symbols in the interleaved buffer. bytes. Then, in Suppose that each DMT symbol carries consecutive bytes (belonging to four the worst case, up to consecutive DMT symbols) will be corrupted by impulse noise. From (3), has to satisfy

Then, the interleaving delay is bytes. To convert the delay to seconds, recall that it takes 250 s to transmit bytes. Therefore, s

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TABLE II DMT-VDSL PARAMETERS USED FOR THE SIMULATIONS

With parameters given in Table II, the top rows of Table III enumerate the minimum delay that is required in order to protect a DMT-VDSL system that uses (240, 224), (144, 128), and (64, 48) RS codes from the worst-case impulses. The assumptions will be given in Section V-B. It used for the calculation of is clear that, since the standards impose a 20-ms limit on the end-to-end delay of systems subject to impulse noise, use of (240, 224) codes is only possible when erasures are used. As can also be seen, when RFI is present and protection relies solely on the RS code, the required interleaving delay can increase conand can vary slightly siderably. Note that the values of depending on the transmission rate. However, the variation is small and is due to the boundary effects because of the grouping of bytes into DMT symbols. This is also expected from intuition. An impulse of fixed duration will affect many bytes in a high-rate system and a few bytes in a low-rate system. However, the high-rate system will also be transmitting faster, so bytes interleaved further apart will be transmitted after the same (more or less) time delay compared to the lower rate system whose interleaved bytes are closer to each other but need more time to be transmitted. B. Simulation Setup and Obtained Delays Simulated transmission on a DMT-VDSL system with different noise impulses is used to verify the improvement in error correction when using the erasure techniques. A data buffer is constructed and then encoded using an RS code with (240, 224), (144, 128), or (64, 48). Then, the data are transmitted in the downstream direction using a DMT-VDSL system with parameters shown in Table II. Note that the system uses Wei’s 4-D Trellis code [18]. The transmission is impaired by a deterministic noise burst that is superimposed on the received modulated data. The reason for using deterministic bursts is because no widely acceptable model is available,

and the impulses measured by France Télécom are not described well by the frequently used Cook impulse model. The received data are then decoded using the Viterbi algorithm. It is assumed that each technique has access to the required information (square distances and surviving sequence metric). Three representative impulses provided by France Télécom are used. The first two represent the worst-case scenario for which the system is designed. They are of long duration and high energy and corrupt a large number of bytes of four consecutive DMT symbols. The third impulse is shorter (115 s) and, for simulation purposes, is superimposed onto two DMT symbols. The length of the loop and the waveforms of the impulses are not given in order to protect company proprietary information. To compare the three techniques, it is first assumed that there is no RFI corrupting any of the tones. Random data are encoded into RS codewords. For a given value of , the bytes are interleaved, transmitted, and corrupted by stationary noise, and each of the deterministic impulses are deinterleaved and sent to is increased until no errors the RS decoder of the receiver. appear at the output of the RS decoder. Table III compares the interleaving delay that is required in order to protect the system from each of the three impulses. It also gives the corresponding values of , , and for the target rate of Table II. Note some deviations from the theoretical values, since, in general, not all bytes of the affected DMT symbols are in error. However, in most cases, it is not possible to know in advance if some bytes will never be in error, unless a very accurate model of impulse noise is available. Therefore, in order to guarantee error-free opobtained by (3) and given eration, the worst-case values of in the top rows of Table III should be used. As expected, when erasure decoding is used, the interleaving delay is significantly reduced without the need to compromise the level of impulse noise protection or decrease the length of the DSL loop in order to make up for extra parity. The reduction is larger (approaching a factor of 2) for the worst-case noise bursts and more redundant RS codes and smaller for milder impulses, since all three techniques are conservative in marking possibly erroneous bytes to avoid missing the erroneous ones. However, systems that comply with the standards (that require complete impulse noise immunity for the interleaved buffer) are designed for worst-case bursts, so the resulting interleaving delay is reduced by a factor of approximately 2 when erasures are used. Note that all three techniques result in the same delay. This was expected for the double-pass and the inner code-based technique that mark entire DMT symbols even if one carrier symbol is actually in error. The reason why the delay is also the same for the square-distance technique is because a very conwas used for the square distance in servative threshold the simulations, in order to ensure that all possibly erroneous bytes are erased. This leads to many erased bytes in each DMT symbol. For shorter loops and milder impulses, the square-distance technique can lead to a slight reduction of the delay. However, if the goal is to shield a system against all possible noise should be used bursts, then the most conservative value of by the interleaver even if the square-distance technique is employed. It should be pointed out that, for the case of impulse 3 that only lasts for two DMT symbols , the maximum number of possibly corrupted DMT symbols (defined in Fig. 4) should

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TABLE III MINIMUM DELAY FOR IMPULSE NOISE AND RFI PROTECTION

be changed to two for the double-pass technique to work with 30, 14, and 9 for the (240, 224), (144, 128), and (64, 48) code, respectively. If were equal to 4, the condition would not be satisfied. However, if are chosen, the impulse-noise mitigation the above values of scheme will only be able to protect the DSL system from noise bursts corrupting up to two DMT symbols. and the delay Finally, the effect of RFI on the values of is verified using the same test impulses. The square-distance technique is used to determine the erasures in systems subject to both impulse noise and RFI. As shown in Section III-B, the . For the resulting interleaving delay depends strongly on simulations, it is assumed that up to seven tones are affected by RFI in each DMT symbol that are located in two spectrum areas is obtained of four and three corrupted tones, respectively. by simulation and is shown in Table III. While it is true that the assumption on RFI is somewhat arbitrary, considering RFI in

more detail would distract from the purpose of this paper, which is to demonstrate the erasure techniques and the effect of RFI on the interleaving delay. In reality, RFI may be affecting fewer or more tones depending on the location of the loop, the type of the twisted pairs that is used, and whether any RFI cancellation is smaller when no erasures are used, method is used. since in practice some of the erased bytes may not actually be in is obtained by simulation, it can vary slightly error. Since for different interleaver parameters. Note that, as mentioned in Section III-B, a fine-tuning of the values of and is needed in order to obtain the minimum interleaving delay. Moreover, it is interesting to observe the large penalty in terms of interleaving delay incurred due to the errors that are induced by RFI. Clearly, in many cases, the delay exceeds the allowable limit of 20 ms and, to avoid compromising the nonstationary noise protection of the system, either more redundant RS codes should be used or an RFI canceling scheme should be employed. Similar to the

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Fig. 7.

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Interleaving delay as a function of the RS code.

case where there is no RFI, the theoretical values of the delay are larger, since, in practice, some of the bytes in the corrupted areas may not be in error. However, in many cases, the interleaving delay still remains above 20 ms. The resulting delay for the three impulses and the tradeoff between the code redundancy and the interleaving delay is illustrated in Fig. 7. It is easy to discern the reduction achieved using erasures and the adverse effect that RFI has on the interleaving delay.

do not contain any corrupted bytes. If corrupted bytes and belong to groups and , respectively, where denotes addition modulo- . Since precedes , expeand riences smaller interleaving delay compared to . Let denote the position of the bytes in the deinterleaved stream and their position in the interleaved stream and is the interleaver mapping function): ( • If , then obviously, . Since both and are corrupted, at least all bytes between and are corrupted (recall that it is assumed that all bytes in the corrupted area are in error). This means that all bytes between and that belong to the groups between and and whose interleaver delays are smaller than and that of (and therefore fall between when interleaved) will be corrupted. , suppose that and have the largest possible • If separation within the RS codeword. That means that is located in the last block of size in the RS codeword, whereas is in the first -byte block of the RS codeword. . Hence Therefore,

Comparing the positions of stream, we have

and

in the interleaved

VI. CONCLUSION In this paper, it was shown that the interleaving delay of DSL systems can be significantly reduced using erasure techniques before the RS decoder of the receiver. Three techniques were proposed that can lead to a reduction of the delay by a factor close to 2 when the goal is to protect the DSL system from worst-case impulse noise. Moreover, the square-distance technique can also be employed with systems that are also subject to RFI in addition to impulse noise. As was shown, RS-code-based RFI protection can lead to a significant increase of the interleaving delay. Although the square-distance technique can allow operation of systems even when RFI is present, the required interleaver depth may still be too large. One way to lower the interleaving delay is by using more redundant RS codes. However, this leads to the reduction of the data rates that are available to the users or, equivalently, to shorter maximum reach lengths for the subscriber loop. Efficient RFI-cancellation techniques that attempt to mitigate the effect of RFI falling within the transmission band based on the ideas contained in [14] and [16] is a topic of current research.

which holds for any when . Obviously, if and were closer within the RS codeword, the inequality and would still hold, since the difference between would increase. Therefore, when , all bytes of group will be mapped after all bytes of group in the interleaved stream. Thus, similar to the case where , all of the bytes between and will be corrupted. Consider , the first byte of the group that is the group after , i.e., the group whose bytes get delayed by . Then, ( is still the last byte of group ). Hence

Comparing

with

, we have

APPENDIX PROOF OF LEMMA 1 It is first proven that corrupted bytes should belong to consecutive same-delay groups. Suppose that and that in a given RS codeword the groups containing corrupted bytes are not consecutive. Consider two groups and that contain at least one corrupted byte each and are separated by groups that

since . Since belongs to a group before , it will fall within the corrupted area, so it will also be corrupted. Hence, contains corrupted bytes as well. The same can be group induced for all other groups up to since their bytes belonging to the first -byte block of the RS codeword will be mapped

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Fig. 8. Interleaver mapping for

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m > k.

, but before , and will therefore be in the after corrupted area as well. guarantees that Having established that the condition the groups containing corrupted bytes are consecutive, it can be shown that their number can only increase by 1 between neighboring RS codewords. First, observe that the first bytes to be , since corrupted belong to group it is the one whose bytes get mapped further away in the interleaved stream (assuming that the noise burst does not appear immediately after the system has been turned on). So, the groups , etc. Suppose will be corrupted in decreasing order: , contains groups containing corrupted that RS codeword to . It can be shown by contradicbytes, i.e., groups can contain no more than such tion that codeword contains groups with groups. Assume that codeword corrupted bytes, whereas codeword contains such groups. . After passing Consider the bytes belonging to group . through the interleaver, they will be delayed by in RS Therefore, if is the last byte of the group , whose position in the deinterleaved stream is codeword as shown in Fig. 8, it will in get mapped at the interleaved stream. Since at least one of the bytes of group in symbol is corrupted and, as was shown will be mapped furabove, all bytes from groups , of the ther down the interleaved stream when interleaved stream is corrupted. Consider now , the last byte in symbol which gets mapped from of group to . Comparing and , we have

since . Therefore, will be mapped after in the interleaved stream. Moreover, it will be mapped before any of the of codeword . Since there is at least bytes of group one corrupted byte in codeword belonging to group , will be inside the corrupted area. However, this contradicts the assumption that codeword contains only groups with corrupted bytes. Hence, it cannot happen that the number of corrupted groups increases by more than 1 between neighboring RS codewords.

ACKNOWLEDGMENT The authors would like to acknowledge Dr. W. Yu for many helpful discussions and insights and Dr. F. Gauthier for the acquisition and formatting of the impulse noise data used in the simulations. REFERENCES [1] Network to Customer Installation Interfaces—Asymmetric Digital Subscriber Line (ADSL) Metallic Interface, ANSI T1.413-1998. [2] VDSL Technical Specification Part 1. Functional Requirements and Common Specification, T1E1.4/2000-009R3. [3] Transmission and Multiplexing (TM); Access Transmission Systems on Metallic Access Cables; Very High Speed Digital Subscriber Lines (VDSL); Part 1: Functional Requirements, ETSI TS 101 270-1 v1.2.1 (1999-10). [4] G. Ginis and J. M. Cioffi, “Vectored transmission for digital subscriber line systems,” IEEE J. Select. Areas Commun. , vol. 20, pp. 1085–1104, June 2002. [5] W. Yu, G. Ginis, and J. M. Cioffi, “An adaptive multiuser power control algorithm for VDSL,” IEEE J. Select. Areas Commun., vol. 20, pp. 1105–1115, June 2002. [6] M. L. Honig, K. Steiglitz, and B. Gopinath, “Multichannel signal processing for data communications in the presence of crosstalk,” IEEE Trans. Commun., vol. 38, pp. 551–558, Apr. 1990. [7] M. L. Honig, P. Crespo, and K. Steiglitz, “Suppression of near- and far-end crosstalk by linear pre- and post-Filtering,” IEEE J. Select. Areas Commun., vol. 10, pp. 614–629, Apr. 1992. [8] I. Mann, S. McLaughlin, and W. Henkel, “Impulse generation with appropriate amplitude, length, inter-arrival, and spectral characteristics,” IEEE J. Select. Areas Commun., vol. 20, pp. 1–8, May 2002. [9] R. Kirkby, D. B. Levey, and S. McLaughlin, “Statistics of impulse noise,”, Edinburgh, U.K., ETSI UK TM6, TD18, TD19, TD20, TD21, 1999. [10] I. Mann, S. McLaughlin, and D. B. Levey, “A new statistics for impulse noise measurement,”, Amsterdam, The Netherlands, ETSI UK TM6, TD 55, 1999. [11] B. Rolland, D. Bardouil, F. Clérot, D. Collobert, D. Gardan, and J. Le Roux, “Classification du Bruit Impulsif par Méthode D’apprentissage non Supervisé,” France Télécom, Internal Rep. RP/FTR&D/7133, 2000. [12] D. Collobert et al., “Proposal for impulse noise classification,” Sophia Antipolis, France, ETSI TM6, TD12, 2002. [13] F. Sjöberg, “The Zipper Duplex Method in Very High-Speed Digital Subscriber Lines,” Doctoral, Lulea Univ. of Technology, 2000. [14] J. A. C. Bingham, ADSL, VDSL, and Multicarrier Modulation. New York: Wiley, 2000. [15] Digital RFI Cancellation With SDMT, T1E1.4/96-083. [16] F. Sjöberg, R. Nilsson, N. Grip, P. O. Börjesson, S. K. Wilson, and P. Ördling, “Digital RFI suppression in DMT-based VDSL systems,” in Proc. Int. Conf. Telecommunications, vol. 2, Chalkidiki, Greece, June 1998, pp. 189–193. [17] S. Wicker, Error Control Systems for Digital Communication and Storage. Englewood Cliffs, NJ: Prentice-Hall, 1995. [18] L. F. Wei, “Trellis-coded modulation with multidimensional constellations,” IEEE Trans. Inform. Theory, vol. IT-33, pp. 483–501, July 1987.

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[19] “Very-high bit-rate digital subscriber lines (VDSL) metallic interface, Part 3: Technical specification of a multi-carrier modulation transceiver,”, Rep. T1E1.4/2000-013R4. [20] D. Toumpakaris, W. Yu, J. M. Cioffi, D. Gardan, and M. Ouzzif, “A simple byte-erasure method for improved impulse immunity in DSL,” in Proc. IEEE Int. Conf. Communications, vol. 4, Anchorage, AK, May 2003, pp. 2426–2430. [21] L. R. Bahl, J. Cocke, F. Jelinek, and J. Raviv, “Optimal decoding of linear codes for minimizing symbol error rate,” IEEE Trans. Inform. Theory, vol. IT-20, pp. 284–287, Mar. 1974. [22] D. Toumpakaris, W. Yu, J. M. Cioffi, D. Gardan, and M. Ouzzif, “A byte-erasure method for improved impulse immunity in DSL systems using soft information from an inner code,” in Proc. IEEE Int. Conf. Communications, vol. 4, Anchorage, AK, May 2003, pp. 2431–2435. [23] D. Toumpakaris, “The effect of impulse noise on the performance of VDSL systems, Internal report,” France Télécom, 2001. [24] T. N. Zogakis, J. T. Aslanis Jr, and J. M. Cioffi, “Analysis of a concatenated coding scheme for a discrete multitone modulation system,” in IEEE MILCOM Conf. Rec., vol. 2, 1994, pp. 433–437. [25] T. N. Zogakis, P. S. Chow, J. T. Aslanis Jr, and J. M. Cioffi, “Impulse noise mitigation strategies for multicarrier modulation,” in Proc. IEEE Int. Conf. Communications, vol. 2, Geneva, Switzerland, May 1993, pp. 23–26. [26] D. Toumpakaris, J. M. Cioffi, D. Gardan, and M. Ouzzif, “A square distance-based byte-erasure method for reduced-delay protection of DSL systems from mon-stationary interference,” in Proc. IEEE Globecom, vol. 4, San Francisco, CA, Dec. 2003, pp. 2114–2119.

Dimitris Toumpakaris (S’97–M’04) received the Diploma degree in electrical and computer engineering from the National Technical University of Athens, Athens, Greece, in 1997, and the M.S. and Ph.D. degrees in electrical engineering from Stanford University, Stanford, CA, in 1999 and 2003, respectively. He was a Research Assistant with Stanford University from 1997 to 2003. Since September 2003, he has been with Marvell Semiconductor, Inc., Sunnyvale, CA. His current interests include signal processing and baseband system design for OFDM communications systems.

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John M. Cioffi (S’77–M’78–SM’90–F’96) received the B.S. degree from the University of Illinois in 1978 and the Ph.D. degree from Stanford University, Stanford, CA, in 1984, both in electrical engineering. He was with Bell Laboratories from 1978 to 1984 and IBM Research from 1984 to 1986. He has been a Professor of electrical engineering with Stanford University since 1986. He is currently on sabbatical leave and consults for SBC Network System Engineering on dynamic spectrum management through his consulting firm (Adaptive Spectrum and Signal Analysis, ASSIA). He founded Amati Communications Corporation in 1991 (purchased by Texas Instruments in 1997) and was Officer/Director from 1991 to 1997. He currently is on the Board of Directors of Marvell, Teknovus, ASSIA, Teranetics, and ClariPhy. He is on the advisory boards of Halisos Networks, Ikanos, and Portview Ventures. His specific interests are in the area of high-performance digital transmission. He has published over 250 papers and holds over 40 patents. Prof. Cioffi is a member of the National Academy of Engineering. He was the recipient of the Hitachi America Professorship in Electrical Engineering at Stanford (2002), the IEEE Kobayashi Medal (2001), the IEEE Millennium Medal (2000), the IEE J.J. Tomson Medal (2000), the 1999 University of Illinois Outstanding Alumnus Award, the 1991 IEEE Communications Magazine Best Paper Award, the 1995 ANSI T1 Outstanding Achievement Award, and the National Science Foundation Presidential Investigator Award (1987–1992).

Daniel Gardan received the Diploma in physics from the National Institute of Applied Sciences (INSA), Rennes, France, in 1975. In 1976, he joined the National Center for Telecommunications Studies (CNET, now France Télécom), Lannion, France. A considerable part of his work was on the reliability, maintainability, and availability of transmission systems. Since 1981, he has been focusing on techno-economic studies of access networks and, in particular, of optical fiber networks. He has contributed to several European projects together with various European telephone operators. Since 1995, his work is primarily targeted on the characterization of copper networks in the context of the deployment of xDSL transmission systems.

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Training Sequence Optimization in MIMO Systems With Colored Interference Tan F. Wong, Senior Member, IEEE, and Beomjin Park

Abstract—In this paper, we address the problems of channel estimation and optimal training sequence design for multiple-input multiple-output systems over flat fading channels in the presence of colored interference. In practice, knowledge of the unknown channel is often obtained by sending known training symbols to the receiver. During the training period, we obtain the best linear unbiased estimates of the channel parameters based on the received training block. We determine the optimal training sequence set that minimizes the mean square error of the channel estimator under a total transmit power constraint. In order to obtain the advantage of the optimal training sequence design, long-term statistics of the interference correlation are needed at the transmitter. Hence, this information needs to be estimated at the receiver and fed back to the transmitter. Obviously it is desirable that only a minimal amount of information needs to be fed back from the receiver to gain the advantage in reducing the estimation error of the short-term channel fading parameters. We develop such a feedback strategy in this paper. Index Terms—Best linear unbiased estimator (BLUE), feedback design, multiple-input multiple-output (MIMO) channel estimation, training sequences.

I. INTRODUCTION ECENTLY, wireless communication systems using multiple antennas, usually referred to as multiple-input multiple-output (MIMO) systems, have drawn considerable attention, because MIMO systems promise higher capacity [1], [2] than single-antenna systems over fading channels. Different space–time coding techniques [3]–[6] have been proposed to practically achieve the capacity advantages of MIMO systems. To be able to achieve the coding advantage, it is required, for many space–time coding schemes, to obtain accurate channel information at the receiver. In practice, it is common that the unknown channel parameters are estimated by sending known training symbols to the receiver. This training-based channel estimation approach at the receiver is suitable for quasi-static or slowly varying fading channels. Much work has been done to design training sequences for channel estimation. There have been two major approaches to designing optimal training sequences for both single-antenna systems [7]–[12] and multiple-antenna systems [13]–[18]. One

R

Paper approved by C. Tellambura, the Editor for Modulation and Signal Design of the IEEE Communications Society. Manuscript received May 22, 2003; revised March 19, 2004, and May 31, 2004. This work was supported in part by the Office of Naval Research under Grant N000140210554 and in part by the National Science Foundation under Grant ANI-0020287. This paper was presented in part at the 2004 IEEE GlobeCom Conference. The authors are with the Wireless Information Networking Group, University of Florida, Gainesville, FL 32611-6130 USA (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TCOMM.2004.836558

approach is to find training sequences that minimize the channel estimation error [7]–[11], [14], [16] and the other approach is to maximize a lower bound of the channel capacity [15]. A more recent paper [19] presents optimal training sequence designs under these two approaches for correlated MIMO channels. Most of these works assume the presence of white noise. Not much consideration has been given to the case of colored interference, except in [20], where we address the training sequence optimization problem in the presence of a single interferer. In this paper, we extend our result in [20] to the case in which the colored interference is composed of thermal noise and interference signals transmitted by multiple interferers. We employ the best linear unbiased (BLUE) channel estimator to estimate the channel matrix during the training period. The mean squared error (MSE) of the BLUE channel estimator is used as a performance metric for selecting the training sequence set. We show that the interference covariance matrix decomposes into a Kronecker product of temporal and spatial correlation matrices and that only the temporal correlation needs to be considered in obtaining the optimal training sequence set. The memory of the colored interference induces a nontrivial eigenstructure of the temporal correlation matrix in that some subspaces are less contaminated by the interference. This motivates the problem of judiciously allocating training power to these subspaces. Based on this observation, we determine the optimal training sequence set that minimizes the MSE under a total transmit power constraint. We note that the optimization problem treated in [19] turns out to be similar to the one treated in this paper. In order to obtain the advantage of the optimal training sequence design, we develop an information feedback scheme that requires a minimal amount of information to be fed back from the receiver to approximately obtain the optimal training sequence set at the transmitter. Numerical results show that we can reduce the MSE of the BLUE channel estimator significantly by using the optimal training sequence set instead of a usual orthogonal training sequence set. We can also achieve comparable estimation performance with the approximate optimal training sequence set obtained by the proposed feedback scheme. The rest of this paper is organized as follows. In Section II, we describe the MIMO system model and the BLUE channel estimator for the channel matrix based on the received training sequence block. In Section III, the training sequence optimization problem is considered and its solution is given. We also develop the feedback scheme to approximately obtain the optimal training sequence set in Section III. Numerical examples for the

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cases of autoregressive jammers and cochannel interferers are provided in Section IV, and conclusions are drawn in Section V.

We note that the interference-plus-noise vector is (4)

II. BLUE CHANNEL ESTIMATOR We consider a transmitter–receiver pair with transmit receive antennas over a frequency flat fading antennas and channel in the presence of colored interference and white thermal noise. We assume that the interference is composed of interferers. The th interferer has signals transmitted by transmit antennas. We assume that the transmission from the transmitter to the receiver is packetized. Each packet contains a training frame that is composed of a set of known training sequences, each of which is sent out by a transmit antenna. In matrix notation, the observed training symbols at the receiver for a packet are given by

Let be the symbol transmitted by the th antenna of the th interferer at time and be the time correlation between the symbols at time instants and from the th antenna of the th interferer. Then it is not hard to noise correlation matrix is given by show that the (5) where .. .

(1)

where is the transmitted training symbol matrix that is known to the receiver, is the number of training symbols is the interference signal per transmit antenna, and matrix from the th interferer. We assume that symbols in are zero-mean, complex random variables, correlated across both space and time. In addition, the interference processes are assumed to be wide-sense stationary. We assume that the is larger than . The number of training symbols matrix and matrix are the channel matrices from the transmitter and the th interferer to the receiver, respectively. and are independent, We assume that the elements in identically distributed (i.i.d.) zero-mean, circular-symmetric, complex Gaussian random variables with variance and , respectively. In addition, is an additive white Gaussian noise are assumed to be (AWGN) matrix and the elements in independent, zero-mean, circular-symmetric, complex Gaussian . Finally, , , random variables with variance , and are all independent of one another. Let , , and , where is the vector obtained by stacking the columns of on top of each other [21]. Taking transpose and then vectorizing on both sides of (1), we have

..

.. .

.

(6) We note from (5) and (6) that the space correlations between interference symbols from different antennas play no role in the correlation matrix . This is due to the i.i.d. assumption , for we made on the elements of the channel matrices . This turns out to be a crucial property of the interference model, as illustrated below. The Kronecker product form of in (5) leads to the following simplification of the BLUE for the channel vector : (7) where (8) Writing (7) back into matrix form, we have (9) Moreover, the MSE of the BLUE for

is given by

(2) denote the Kronecker product and where and identity matrix, respectively. In (2), is the channel vector with and is the interferzero mean and covariance matrix ence-plus-noise vector with zero mean and covariance matrix . From the above, we note that is independent of . During the training period, the BLUE [22] of the channel vector based on the received training block can be obtained as

(3)

(10) We assume that the channel matrices and for are short-term statistics that may change from packet to packet. On the other hand, the interference correlation matrix varies at a rate that is much slower than that of the channel matrices. As a result, it is possible for the receiver to estimate using a number of previous packets and feed back relevant information to the transmitter, which can then make use of this information to select the optimal training sequence set for the estimation of during the current packet.

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WONG AND PARK: TRAINING SEQUENCE OPTIMIZATION IN MIMO SYSTEMS WITH COLORED INTERFERENCE

III. TRAINING SEQUENCE OPTIMIZATION We note that the MSE of the BLUE channel estimator depends on the choice of the training symbol matrix (training sequence set) . Hence, it is natural to ask whether there is an optimal set of training sequences that gives the best estimation performance. Moreover, it is conceivable that the optimal training sequence set will depend on the characteristic of the interference. Hence, in order to obtain the advantage of employing the optimal training sequence set, information about the interference has to be measured at the receiver and fed back to the transmitter so that it can construct the optimal sequence set. The obvious questions are what information about the interference we should feed back to the transmitter and whether this feedback design is practical or not. We study these questions in this section.

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This relaxed optimization problem can be solved by the standard Karush–Kuhn–Tucker condition technique [25] since the cost function and the constraint are both convex. Indeed, let

(15) above is well defined and that the inequality We note that in (15) holds for , while it does that defines not hold for . With this definition, it is not hard to show that the optimal solution is given by

for A. Optimal Training Sequence Set To have a meaningful formulation of the sequence optimization problem, we need to limit the maximum total transmit power of the transmit antenna array to . The following discussion provides a constructive method to obtain the optimal training sequence set under this restriction. Our goal is to minimize the MSE of the BLUE channel estimator by selecting the optimal training sequence set with the total energy constraint . Therefore, we can express the training sequence set optimization problem as follows:

for

. (16) We note that this solution has the standard water-filling [26] interpretation. If we can construct a matrix such that the eigenvalues of are exactly the solution of the above relaxed optimiza, then this tion problem and that choice of will be a solution of the original sequence optimization problem in (12). It is easy to see that this can be done and the resulting optimal training sequence set is given by

subject to (17)

(11) . We can rewrite the optimization problem in Let (11) in the following form: subject to

where

is an arbitrary unitary matrix and is the matrix whose columns are the eigenvectors of cor. With this opresponding to the smallest eigenvalues of timal choice of training sequence set, the minimum estimation error achieved is given by

(12) be the nonnegative eigenvalues of Further, let arranged in a descending order and be the posiarranged in an ascending order. To protive eigenvalues of ceed, we need to make use of the following result, whose proof can be found, for example, in [23, pp. 249]. Lemma 1: Suppose that and are two Hermitian matrices. Arrange the eigenvalues of in a deof in an scending order and the eigenvalues . ascending order. Then Applying this lemma to the constraint in (12), we can bound (13) Now, consider the following relaxed optimization problem:

(14)

MSE

(18)

A physical interpretation of this solution is that the optimal training sequence set put its power to where the effect of the interference is the smallest, hence the estimation error can be minimized. We note that the optimal training sequence set is an orthogonal set if is chosen to be an identity matrix. However, the optimal training sequence set, in general, is not necessarily orthogonal. For instance, it is possible to obtain a choice of which spreads power evenly across the transmit antennas with the use of nonorthogonal sequences. To do so, we need to construct a unitary to make the diagonal elements of (19) the same. This is shown to be possible in [24], and such a can be constructed using a simple iterative procedure. We note that not only does the choice of optimal sequence set minimize the estimation error, but this choice also simplifies the

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implementation complexity of the BLUE for . It is easy to see that the BLUE channel estimator in (9) reduces to (20)

and denote the ( , )th element the training matrix as defined in (1) for the of the observed training matrix th packet, respectively. Then it is not hard to see that, for and ,

where for for

(21)

(25)

.

Thus, the complexity of the BLUE with the optimal sequence . set reduces to

where (26)

B. Feedback Design In order to obtain the advantage of the optimal training sequence design, long-term statistics of the interference correlation need to be estimated at the receiver and fed back to the transmitter. We note from Section III-A that the optimal training and the sequence set depends on the channel gain variance eigenstructure of the matrix in (8). As a result, only these two long-term statistics need to be estimated at the receiver. Obviously it is desirable that only a minimal amount of information is needed to be fed back from the receiver to gain the advantage in reducing the estimation error of the short-term channel fading matrix. In this section, we develop such a feedback scheme based on the fact that a suitable Toeplitz matrix can be approximated by a circulant matrix. Since the interferer signals are wide-sense stationary, takes the form of a Toeplitz matrix. Indeed, consider a sequence of complex numbers such that and the elements of at the th row and th column is given by . The sequence is obtained by sampling the autocorrelation function of the interference at the symbol rate. In addition, if the is absolutely summable, then it is shown in sequence [27] that the Toeplitz matrix can be approximated by the circulant matrix (22)

In addition, let be the projection matrix onto the subspace perpendicular to the one spanned by the rows of the training matrix for the th packet. Denote the by and the ( , )th ( , )th element of element of by . Then it can be shown that, for , we have

(27) where (28) We note that (25) for together with (27) provides us equations to solve for the unknowns , (both real-valued), and for (all complexvalued). Thus, estimates of and for can be obtained by solving this set of linear equations with the expectation terms replaced by their usual estimates as follows:

where is the FFT matrix, i.e., the ( , )th element of is , and is an diagonal matrix with as its diagonal elements. A reasonable way [27] to obtain , for , is (23) , it can be shown that With this choice of approaches as approaches infinity [27]. Moreover, if we arrange in an ascending order, we have (24) for

, and

.. .

.. .

is the th smallest eigenvalue among

the set . Now we turn to the estimation of and . As mentioned before, they are both long-term statistics and hence should be estimated based on the observed training frames of the previous packets, where is smaller than the number of packets during which the long-term statistics remain the denote the ( , )th element of same. Toward this end, let

(29)

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In the above, the biased estimators , for , have been employed to approximate the corresponding expectations on the right-hand side of (25). We note that the use of these biased estimators is similar to the use of biased autocorrelation function estimators in the Yule–Walker method of estimating the power spectral density of an AR process [28]. In summary, we can use the solution of (29) to estimate the autocorrelation function of the interference at the receiver and ’s by (23). Then the estimated value obtain estimates of the ’s, and the corresponding indices are of , the smallest fed back to the transmitter. At the transmitter, we can replace by the columns of that are indexed by the feedback indices to construct the optimal training sequence set for the current packet. We note that the estimates of the ’s obtained by solving (29) do not guarantee the resulting estimates of the ’s and to be positive, although this is almost always the case when , , and are sufficiently large. When the estimate of is negative, we heuristically use the absolute value of the estimate instead. In addition, we do not use those ’s with negative estimates in finding the minimum values of as described before.

C. Asymptotic Estimation Performance Gain It is illustrative as well as practical to develop a simple measure that can tell us how much advantage we can obtain by employing the optimal training sequence set over other choices of training sequences. For instance, if the receiver determines that there is not much to gain by using the optimal training sequences, it can inform the transmitter to keep on using the current ones. To this end, we employ equal-power orthogonal training sequences as our baseline for comparison, since these training sequences are commonly [13]–[16] suggested when the noise is white. , when First, we want to obtain the worst-case MSE, MSE equal-power orthogonal training sequences are employed and the total transmit power is . It is not too hard to see that

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and are the minimum and maxwhere imum eigenvalues of a Hermitian matrix, respectively, and are the eigenvalues of arranged in an ascending order. From (10), we can bound the worst-case MSE as MSE

(31)

On the other hand, from (18), when the optimal training sequence set is employed, the minimum MSE can be bounded by MSE

(32)

Combining (31) and (32), we can bound the ratio between the minimum MSE and the worst-case MSE by MSE MSE (33) This MSE ratio gives the maximum possible relative reduction in the estimation error that we can obtain by using the optimal sequence set under a specific set of interferers. Here we obtain a simpler performance metric by considering the asymptotic value of this MSE ratio when is very large. To do so, we employ the following results regarding the extremal eigenvalues of the sequence of Toeplitz matrices in [29, Ch. 5]. Suppose that the sampled autocorrelation se, of the wide-sense stationary interference quence, process is absolutely summable. Let

be the discrete-time Fourier transform of , we have

. Then, for

(34) From (15) and (34), we see that . Applying this and (34) to (33), the asymptotic maximum MSE reduction ratio is given as follows: MSE MSE

(35)

IV. NUMERICAL EXAMPLES

(30)

In this section, we consider two examples to illustrate the potential advantage of employing the optimal training sequence set. The first example considers the case when the interference signals are described by first-order autoregressive (AR) random processes. The second example considers the case in which the interference is caused by cochannel interferers whose signal structures are exactly the same as that of the desired signal. We assume that the desired user has two transmit antennas and three

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receive antennas. In each example, we evaluate the MSEs of the BLUE channel estimator when the following three different training sequence sets are employed: 1) the Hadamard sequence set, i.e., the first two rows of a Hadamard matrix are used as the training sequences; 2) the optimal training sequence set described in Section III-A; and 3) the approximate optimal training sequence set described in Section III-B. A. AR(1) Jammers We assume that there are two jammers in the system. Both jammers have one transmit antenna. The interference signals from the jammers are modeled by two first-order AR processes , , respectively. For instance, with AR parameters the AR model of the first jammer is given by (36) is a white Gaussian random process with zero mean where . The AR parameter can be interpreted as the and variance intensity of correlation among the symbols of the jammer. It is easy to verify that, for this case, we have

Fig. 1. Comparison of MSEs obtained by using different training sequence sets. Two AR(1) jammers with  = 0:3 and  = 0:5.

(37) Hence, the asymptotic maximum MSE reduction ratio is given by

(38)

where is the transmit power of the th jammer. The MSEs of the BLUE channel estimator with the three different training sequence sets are shown and compared in Figs. 1 and 2 for the two different combinations of and . In each case, we consider different lengths ( , 32, 64, 128, 256, 512, and 1024) of the training sequences and different dB and received signal-to-interference ratios ( 20 dB for and 2). The received signal-to-noise ratio (SNR) is set to 10 dB. The technique described in Section III-B is employed to obtain the approximate optimal training sequences. We have assumed that the received training signals from ten previous packets are employed to estimate the jammer information at the receiver. The training sequences used in the previous ten packets are the Hadamard sequences described above. From these figures, we observe that there exist only minimal differences between the MSEs of using the optimal training sequence set and approximate optimal training sequence set. Obviously, this is a desirable result, because it indicates that we can obtain comparable performance to the optimal training sequence set by estimating the jammer information at the receiver and feeding back only a small amount of information to the transmitter. In general, we see that the optimal training sequence set significantly outperforms the Hadamard sequence set in all the cases considered. The advantage of using the optimal

Fig. 2. Comparison of MSEs obtained by using different training sequence sets. Two AR(1) jammers with  = 0:7 and  = 0:9.

training sequence set increases as the correlation parameters ’s increase. The asymptotic maximum MSE reduction ratios for the cases considered above are shown in Table I. For comparison, the MSE reduction ratios obtained by using the optimal sequenceset against the Hadamard sequence set for are also included in Table I. We can deduce from the table that the Hadamard sequence set is rather inefficient. In addition, much more reduction in MSE can be obtained using the optimal sequence set when both of the ’s are close to 1. B. Cochannel Interferers In this example, we assume that the interference is caused by two cochannel interferers whose signal format is similar to that of the desired user. More precisely, let us assume that the transmitted signal at the th transmit antenna of the th interferer is given by (39)

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TABLE I COMPARISON OF ASYMPTOTIC MAXIMUM MSE REDUCTION RATIO AND MSE RATIO BETWEEN USING OPTIMAL AND HADAMARD SEQUENCES IN THE CASE OF AR JAMMERS

where is the sequence of data symbols, which are assumed to be i.i.d. binary random variables with zero mean and unit is variance, from the th antenna of the th interferer, the symbol waveform, is the symbol interval, and is the symbol timing difference between the th interferer and the desired signal. Without loss of generality, we can assume that . We also assume that . With the model described above, the elements of the interferin (1) are samples at the matched filter ence signal matrix output at the receiver at time . Specifically, the ( , )th element of is given by

Fig. 3. Comparison of MSEs obtained by using different training sequence sets. Two cochannel interferers with rectangular waveforms and delays  = 0:3T ,  = 0:5T .

1) Rectangular Symbol Waveform: In this case

otherwise

and otherwise.

From (43), we have

(40) (44)

where Hence, the asymptotic maximum MSE reduction ratio is (41)

(45) is the autocorrelation of the symbol waveform. Thus, it is easy to see that the sampled autocorrelation sequence

(42) and its discrete-time Fourier transform is given by

(43) where

is the transmit power from the th interferer, , and is the Fourier transform of the symbol waveform . To illustrate how the use of the optimal training sequence set can benefit the channel estimation process, let us consider the following two common symbol waveforms:

From (45), the use of the optimal training sequence provides no gain when the cochannel interferers are symbol-synchronous to . On the other hand, the desired user signal, i.e., when , the asymptotic maximum MSE reduction ratio attains its smallest possible value. This means that we can almost completely eliminate the effect of the interferers by using the set of long optimal training sequences. As before, we compare the MSEs of the BLUE channel estimator with the three different training sequence sets in Fig. 3 and . by considering the case in which The other parameters are chosen as in the AR jammer example before. Again, from Fig. 3, we observe that there exist only minimal differences between the MSEs of using the optimal training sequence set and approximate optimal training sequence set, and that the optimal training sequence set significantly outperforms the Hadamard sequence set in all the cases considered. The asymptotic maximum MSE reduction ratios for the cases considered above are shown in Table II. For comparison, the MSE reduction ratios obtained by using the optimal sequence set against the Hadamard sequence set for are also included in Table II. We can deduce from the table that the Hadamard sequence set is rather inefficient.

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TABLE II COMPARISON OF ASYMPTOTIC MAXIMUM MSE REDUCTION RATIO AND MSE RATIO BETWEEN USING OPTIMAL AND HADAMARD SEQUENCES IN THE CASE OF COCHANNEL INTERFERERS

2) ISI-Free Symbol Waveform With Raised Cosine Spectrum [30]: In this case, we have

where

is the roll-off factor. Since for all and is positive, it can be deduced from (43) that . To find , because of symmetry of

consider the interval (43), we have

, it is enough to

calculus

can almost completely eliminate the effect of the interferers by using the set of long optimal training sequences. As before, we compare the MSEs of the BLUE channel estimator with the three different training sequence sets in Fig. 4 by and . The considering the case in which roll-off factor of the ISI-free waveform is chosen to be . The other parameters are chosen as in the AR jammer example before. The conclusions from Fig. 4 are similar to those for the rectangular waveform.

. Over this interval, by V. CONCLUSION

(46) Simple

Fig. 4. Comparison of MSEs obtained by using different training sequence sets. Two cochannel interferers with ISI-free waveform waveforms and delays  = 0:3T ,  = 0:5T .

reveals

that

. totic maximum MSE reduction ratio is

Thus,

the

asymp-

(47) From (47), the use of the optimal training sequence provides no gain when the co-channel interferers are symbol-synchronous to the desired user signal, i.e., . On the other hand, when , the asymptotic maximum MSE reduction ratio attains its smallest possible value. This means that we

We have solved the problem of optimal training sequence design for MIMO systems over flat fading channels in the presence of colored interfering signals. In order to obtain the advantage of the optimal training sequence design, we have also developed an information feedback scheme that requires a minimal amount of information from the receiver to approximately construct the optimal training sequence set. Numerical results show that the MSE of the channel estimator at the transmitter can be significantly reduced by using the optimized training sequence set over the Hadamard training sequence set that is often used in the case of white noise. In addition, we observe that comparable estimation performance can be achieved by using the approximate optimal training sequence set obtained by the proposed feedback scheme. REFERENCES [1] 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. [2] I. E. Telatar, “Capacity of multi-antenna Gaussian channels,” Eur. Trans. Telecommun., vol. 10, pp. 585–595, Nov. 1999. [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. Inform. Theory, vol. 44, pp. 744–765, Mar. 1998. [4] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block coding for wireless communications: performance results,” IEEE J. Select. Areas Commun., vol. 17, pp. 452–460, Mar. 1999.

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[5] B. M. Hochwald and T. L. Marzetta, “Unitary space-time modulation for multiple-antenna communication in Rayleigh flat fading,” IEEE Trans. Inform. Theory, vol. 46, pp. 543–564, Mar. 2000. [6] B. Hassibi and B. M. Hochwald, “High-rate codes that are linear in space and time,” IEEE Trans. Inform. Theory, vol. 48, pp. 1804–1824, July 2002. [7] S. N. Crozier, D. D. Falconer, and S. A. Mahmoud, “Least sum of squared error (LSSE) channel estimation,” Proc. Inst. Elect. Eng., pt. F, vol. 138, pp. 371–378, Aug 1991. [8] G. Caire and U. Mitra, “Training sequence design for adaptive equalization of multi-user systems,” in Proc. 32nd Asilomar Conf. Signals, Systems and Computers, vol. 2, Nov. 1998, pp. 1479–1483. [9] W. Chen and U. Mitra, “Training sequence optimization: comparison and an alternative criterion,” IEEE Trans. Commun., vol. 48, pp. 1987–1991, Dec. 2000. [10] C. Tellambura, M. G. Parker, Y. J. Guo, S. J. Shepherd, and S. K. Barton, “Optimal sequences for channel estimation using discrete Fourier transform technique,” IEEE Trans. Commun., vol. 47, pp. 230–238, Feb. 1999. [11] C. Tellambura, Y. J. Guo, and S. K. Barton, “Channel estimation using aperiodic binary sequences,” IEEE Commun. Lett., vol. 2, pp. 140–142, May 1998. [12] H. Vikalo, B. Hassibi, B. Hochwald, and T. Kailath, “Optimal training for frequency-selective fading channels,” in Proc. Int. Conf. Acoustics, Speech, Signal Processing, vol. 4, Salt Lake City, UT, May 7–11, 2001, pp. 2105–2108. [13] A. F. Naguib, V. Tarokh, N. Seshadri, and A. R. Calderbank, “Space-time coding modem for high data rate wireless communications,” IEEE J. Select. Areas Commun., vol. 16, pp. 1459–1478, Oct. 1998. [14] T. L. Marzetta, “BLAST training: estimating channel characteristics for high capacity space-time wireless,” in Proc. 37th Annual Allerton Conf. Communication, Control, and Computing, Monticello, IL, Sept. 22–24, 1999. [15] B. Hassibi and B. M. Hochwald, “How much training is needed in multiple-antenna wireless links?,” IEEE Trans. Inform. Theory, vol. 49, pp. 951–962, Apr. 2003. [16] C. Fragouli, N. Al-Dhahir, and W. Turin, “Training-based channel estimation for multiple-antenna broadband transmissions,” IEEE Trans. Wireless Commun., vol. 2, pp. 384–391, Mar. 2003. [17] Y. Song and S. D. Blostein, “Data detection in MIMO systems with co-channel interference,” in Proc. IEEE 56th Vehicular Technology Conf., vol. 1, Fall 2002, pp. 3–7. [18] X. Ma, L. Yang, and G. B. Giannakis, “Optimal training for MIMO frequency-selective fading channels,” IEEE Trans. Wireless Commun., to be published. [19] J. H. Kotecha and A. M. Sayeed, “Transmit signal design for optimal estimation of correlated MIMO channels,” IEEE Trans. Signal Processing, vol. 52, no. 2, pp. 546–557, Feb. 2004. [20] B. Park and T. F. Wong, “Training sequence optimization in MIMO systems with colored noise,” in Proc. IEEE MILCOM, vol. 1, Boston, MA, Oct. 2003, pp. 135–140. [21] A. Graham, Kronecker Products and Matrix Calculus With Applications. New York: Halsted, 1981. [22] S. M. Kay, Fundamental of Statistical Signal Processing: Estimation Theory. Englewood Cliffs, NJ: Prentice-Hall, 1993. [23] A. W. Marshall and I. Olkin, Inequalities: Theory of Majorization and Its Applications. New York: Academic, 1968.

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[24] T. F. Wong and T. M. Lok, “Transmitter adaptation in multicode DS-CDMA systems,” IEEE J. Select. Areas Commun., vol. 19, pp. 69–82, Jan. 2001. [25] E. K. P. Chong and S. H. Zak, An Introduction to Optimization. New York: Wiley, 1996. [26] T. M. Cover and J. A. Thomas, Elements of Information Theory. New York: Wiley, 1991. [27] R. M. Gray. (2002, Aug.) Toeplitz and Circulant Matrices: A Review [Online]. Available: http://ee.stanford.edu/~gray/toeplitz.pdf [28] S. M. Kay, Modern Spectral Estimation: Theory & Application. Englewood Cliffs, NJ: Prentice-Hall, 1988. [29] U. Grenander and G. Szego, Toeplitz Forms and Their Applications, 2nd ed. New York: Chelsea, 1984. [30] J. G. Proakis, Digital Communications, 4th ed. New York: McGrawHill, 2001.

Tan F. Wong (S’96–M’98–SM’03) received the B.Sc. degree (with first-class honors) in electronic engineering from the Chinese University of Hong Kong in 1991, and the M.S.E.E. and Ph. D. degrees in electrical engineering from Purdue University, West Lafayette, IN, in 1992 and 1997, respectively. He was a Research Engineer with the Department of Electronics, Macquarie University, Sydney, Australia, where he was involved with the high-speed wireless networks project. He also served as a Postdoctoral Research Associate with the School of Electrical and Computer Engineering, Purdue University. Since August 1998, he has been with the University of Florida, Gainesville, where he is currently an Associate Professor of electrical and computer engineering. Prof. Wong serves as the Editor for Wideband and Multiple Access Wireless Systems for the IEEE TRANSACTIONS ON COMMUNICATIONS and as an Editor for the IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY.

Beomjin Park received the B.S. degree in electronic engineering from Hankuk Aviation University, Koyang City, Korea, in 1995, the M.S. degree in electrical engineering from the University of Southern California, Los Angeles, in 1999, and the Ph.D. degree in electrical and computer engineering from the University of Florida, Gainesville, in 2004, in the area of wireless communications. From 1995 to 1996, he was with Samsung Electronics Company, Ltd., Korea, in the Division of Semiconductors as a Product Engineer. He is currently with Samsung Electronics Company, Ltd., Korea, in the Division of Telecommunication Network as a Research Engineer. His research interests include space–time processing and channel estimation in multiple-input multiple-output systems.

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Distribution Functions of Selection Combiner Output in Equally Correlated Rayleigh, Rician, and Nakagami- Fading Channels

m

Yunxia Chen, Student Member, IEEE, and Chintha Tellambura, Senior Member, IEEE

Abstract—We develop a novel approach to derive the cumulative distribution functions (cdfs) of the selection-combining (SC) output signal-to-noise ratio (SNR) in equally correlated Rayleigh, Ricean, and Nakagami- fading channels. We show that a set of equally correlated channel gains can be transformed into a set of conditionally independent channel gains. Single-fold integral expressions are, therefore, derived for the cdfs of the SC output SNR. Infinite series representations of the output cdfs are also provided. New expressions are applied to analyze the average error rate, the outage probability, and the output statistics of SC. Numerical and simulation results that illustrate the effect of fading correlation on the performance of -branch SC in equally correlated fading channels are provided. Index Terms—Cumulative distribution function (cdf), diversity, equally correlated fading channel, outage probability, selection combining (SC).

I. INTRODUCTION ORRELATED fading among diversity branches can significantly degrade the performance of spatial diversity systems, such as maximal ratio combining (MRC), equal gain combining (EGC), and selection combining (SC). In practice, independent fading across diversity branches can rarely be achieved due to the insufficient separation between the antennas. Thus, quantifying the resultant degradation of the performance of diversity systems is a long standing problem of importance.

C

A. Equally Correlated Model In this paper, we study the distribution of the SC output signal-to-noise ratio (SNR) in equally correlated fading channels. The equally correlated model may be valid for a set of closely placed antennas [1] and be used as a worst-case Paper approved by V. A. Aalo, the Editor for Diversity and Fading Channel Theory of the IEEE Communications Society. Manuscript received July 31, 2003; revised March 19, 2004, and May 17, 2004. This work was supported in part by the Natural Sciences and Engineering Research Council and in part by iCORE. This paper was presented in part at Wireless 2003, the 15th Annual International Conference on Wireless Communications, Calgary, AB, Canada, July 7–9, 2003. Y. Chen was with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB T6G 2V4, Canada. She is now with the Department of Electrical and Computer Engineering, University of California at Davis, Davis, CA 95616 USA (e-mail: [email protected]). C. Tellambura is with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB T6G 2V4, Canada (e-mail: [email protected]; [email protected]). Digital Object Identifier 10.1109/TCOMM.2004.836596

benchmark or as a rough approximation by replacing every in the correlation matrix with the average value of . However, this model (unlike the exponentially correlated model) may have limited usefulness in representing a scenario with equally placed antennas. Experimental measurements have shown that for three equally placed antennas (a triangular), the fading correlation among them with does not follow the equally correlated model [2]. B. Relation to the Previous Papers While comprehensive theoretical results for the bit-error rate (BER) and the outage performance of MRC systems in various correlated fading channels are available, by comparison, such performance analysis for -branch EGC and SC is not available. This is due to the lack of explicit expressions for the joint prob, ability density function (pdf) of the branch SNRs for when the branch signals are correlated. The SC performance has therefore been comprehensively treated for various independent fading models and modulation methods. For correlated or fading, almost all results deal with two branches (see [3]–[11]). three branches Accordingly, in the extensive list of papers dealing with SC, we have been able to find only a few papers that address -branch SC in correlated fading. Exceptionally, Ugweje and Aalo [12] derive the pdf of the SC output SNR in correlated Nakagami fading as a multiple series of generalized Laguerre polynomials using [13]. However, the convergence properties of this series appear to be poor and the complexity of . Zhang and Lu [14] this approach increases rapidly for have derived a general approach for -branch SC in correlated fading, which requires -dimensional integration. For large , this method is also fairly complicated. Mallik and Win [15] analyze the generalized SC (GSC) in equally correlated Nakagami- fading channels using its output characteristic function (chf). All of these methods utilize the joint chf of the branch SNRs and the complexity increases as increases. Following Miller [16], Mallik [17] derives the joint pdf of the multivariate Rayleigh distribution. However, the pdf expression requires -fold integration. Karagiannidis et al. derive the joint pdf of the exponentially correlated Nakagami- distribution [18] and apply this result with the Green’s matrix to approximate multivariate Nakagami- distribution [19]. Although such an approximation is accurate for exponentially and linearly correlated models, it may not be good for the equally correlated model.

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C. Our Contributions

A. Correlated Rayleigh Envelopes

We thus develop a new approach to derive the cdfs of the -branch SC output SNR in equally correlated fading channels and apply the new results to analyze the SC performance. The novel insight of this paper is that a set of independent Gaussian random variables (RVs) can be linearly combined to form a set of equally correlated complex Gaussian RV’s. Using this insight, we translate the problem of the SC output cdf in equally correlated fading to the problem of the SC output cdf in a conditionally independent fading environment. This reformulation allows us to extend known results for independent fading to analyze the -branch SC performance in correlated fading. It should be emphasized that this approach can be used to analyze not only SC, but also more general diversity combining schemes. Further, the new representations developed for equally correlated channel gains may be useful in other applications such as cochannel interference modeling and multiple-antenna systems. This paper is organized as follows. Section II develops new representations for equally correlated Rayleigh, Rician, and Nakagami- channel gains. Section III derives the cdfs of the SC output in three types of equally correlated fading channels. Section IV derives infinite series representations for the output cdfs. Section V uses our new expressions to evaluate the average error rates of various modulation schemes, the outage probability, and the output statistics of SC. Section VI presents some numerical and simulation results, and Section VII concludes the paper.

Rayleigh envelopes are frequently used to model the amplitudes of received signals in urban and suburban areas [20]. We represent the Rayleigh envelopes using a set of zero-mean complex Gaussian RVs given by (2) , where

, , and , , are independent. That is, , , and for any , . The validity of (2) for positive only may at first seem a significant limitation. However, the entire and (the lower limit follows range for is between from the positive-definiteness constraint on the covariance ma. trix). For large , we may therefore ignore However, a more complicated representation than (2) can be developed to handle negative values. For brevity, we omit the details and do not pursue it further. Since , is a set of Rayleigh envelopes with mean-square value . The cross-correand equals lation coefficient between any . This specifies the correlation (also known as power correlation) between two complex Gaussian samples. However, it is required to relate this to the envelope correlation (i.e., the correlation between two Rayleigh samples). The cross-correlation coefficient between and can be expressed in terms the envelopes of power cross-correlation coefficient as [24, eq. (1.5-26)] for

,

II. REPRESENTATION OF EQUALLY CORRELATED CHANNEL GAINS Rayleigh, Ricean, and Nakagami- distributions are widely used to model the amplitude fluctuations of received signals from different multipath fading channels [20]–[22]. We develop new representations of equally correlated channel gains for these fading models, which can be used to evaluate the performance of various diversity combiners. The following notation will be used throughout the paper. We denote the average, the complex conjugate, and the abso, , and , respectively. We write lute value of as to denote that is Gaussian distributed with denote that mean and variance . We let is complex Gaussian distributed with mean and co. variance degrees of The noncentral chi-square distribution with freedom and noncentrality parameter will be denoted by . The central chi-square distribution with degrees . The th order Marcum of freedom is denoted as -function is defined as [23]

(3)

where denotes the complete elliptic integral of the second kind with modulus . For a given , solving (3) yields . Several solution methods are discussed in [25] and [26], and we also provide a new, more general solution in (10). Thus, using (2) and (3), we can readily represent a set of equally correlated Rayleigh envelopes with a specified value of envelope correlation. Next, we introduce a “trick” that enables performance analand to be fixed. Then, ysis. We consider . Consequently, the branch are independent. powers Performance analysis can now be carried out in two steps. First, conditional performance results are obtained for the set of conditionally independent branch powers, and the conditional re. The second step is to average the sults are functions of . conditional results over the distribution of B. Correlated Ricean Envelopes

(1) is the th-order modified Bessel function of the where to denote . The first kind. For brevity, we write Kronecker delta will be defined as and for .

Ricean distribution is usually used where line-of-sight (LOS) propagation exists. We can also represent the Ricean fading envelopes by a set of complex Gaussian RVs

(4)

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for , where , , , is the nonzero LOS component. are independent, and , is Ricean distributed Since with the Ricean factor and the mean-square . The power correlation between value and is equal to . Thus, (4) can be used to represent the equally correlated Ricean fading envelopes. C. Correlated Nakagami-

(10)

Envelopes

The Nakagami- distribution is a versatile statistical distribution which can accurately model a variety of fading environments. It has greater flexibility in matching some empirical data than the Rayleigh, Lognormal, or Ricean distribution. It also includes the Rayleigh and the one-sided Gaussian distributions as special cases. Moreover, the Nakagami- can closely approximate the Ricean distribution and the Hoyt distribution is an integer, a Nak[27]. When the fading severity index agami- envelope is the square root of the sum of squares of independent Rayleigh variates. Hence, we can represent the zero-mean Nakagami- fading envelopes using a set of complex Gaussian RVs (5) for

and , where , and are independent. That is, for any , , , and . and The cross-correlation coefficient between

where obtain .

. Then, using (8), we can immediately

III. DISTRIBUTION FUNCTIONS OF THE SC OUTPUT In this section, we derive the expressions for the cdfs of the SC output SNR in equally correlated Rayleigh, Rician, and Nakagami- fading channels. We assume that the received signals at different branches are identically distributed and equally correlated with each other. The noise components at different branches are assumed to be independent of the signal components and uncorrelated with each other. A. Rayleigh Fading Channels

and , , and

In SC, the branch with the largest instantaneous SNR is selected as the output as follows:

is

where is the number of diversity branches. The instantaneous , , SNR of the th branch is is the energy of the transmitted signals and is the where noise power spectral density per branch. Note that, when the , i.e., all of the branches experience the fading correlation same fading, the -branch SC reduces to the single-branch SC whose performance is well known. In our derivation, we will not consider this case. and are fixed, and Recall that, when are independent, whose cdf is given by [30, eq. (2-1-124)]

and . (6) Let

for a given , we may use a polynomial approximation for . Expressing the hypergeometric function in the form of Gauss series [28, eq. (15.1.1)] and using the reversion of power series [29, p. 138], we obtain the approximation for as

denote the summation of the absolute square of (7)

From (6), we can see that, for any fixed , , , are independent. Thus, is the sum of squares of independent Rayleigh envelopes with cross-correlation coefficient [27] (8)

(11)

Evaluating the cdf of SC output for fixed obtain the conditional cdf

(12) , we

Therefore, is a set of equally correlated Nakagamifading envelopes with mean-square value . The relationship between the power correlation and the envelope correlation is [27] (13) (9) where and is the hypergeometric function [28, eq. (15.1.1)]. Note that for , (9) reduces to (3). Since there is no closed-form solution,

where is the average , where , branch SNR. Notice that and its pdf is given by a special case of [30, eq. (2-1-110)] (14)

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Averaging the conditional cdf (13) over the distribution of [see (14)], we obtain the cdf for the output of -branch SC in equally correlated Rayleigh fading as

There is no closed-form solution to this integral. However, in , the output pdf can be written in terms of the the case of Marcum -function

(15)

(18) This special case is already known [7, eq. (11)]. Using the new expression for output pdf (17), we can also derive the moment generating function (mgf) of the output SNR. The mgf is defined as the statistical average

Notice that Matlab provides the function NCX2CDF to compute the cdf of the noncentral chi-square distribution, which is in the form of the Marcum -function. Therefore, we can readily evaluate (15) numerically using Matlab. To the best of our knowledge, this is a novel result. It reduces the -dimensional integration [14, eq. (9)] necessary for the cdf of correlated SC output to a single-fold integral, enabling the analysis for -branch SC in equally correlated Rayleigh fading. This new expression (15) for the output cdf reduces to the previous results for two special cases. : Case 1) Independent Rayleigh fading channel and Using the relation , the output cdf (15) simplifies to , which is equivalent to the well-known result [3, eq. (10-4-12)]. SC in equally correlated Case 2) Dual-branch Rayleigh fading channels: Using integration by parts, we can show that the output cdf can now be written as

(19) In the special case of dual-branch SC, the mgf can be expressed in a closed-form [7] equation as shown in (20), at the bottom of the page. B. Ricean Fading Channels Using the representations of the equally correlated Ricean channel gains (4) and following the same approach, we can also derive the output cdf of SC in equally correlated Ricean channels. , Fix , are independent, whose cdf is given by [30, eq. (2-1-124)]. Also notice , where that , and its pdf is given by [30, eq. (2-1-118)]. Hence, the output cdf can be obtained as

(21) (16) This result is equivalent to the well-known expression [3, eq. (10-10-8)]. Differentiating (15) yields the output pdf

(17)

where is the Ricean factor. To the best of our knowledge, (21) is also a new result. We are not aware of any study dealing with -branch SC in correlated Ricean fading. C. Nakagami-

Fading Channels

To derive the output cdf of SC in equally correlated fading channels, we use the channel gain Nakagamirepresentations (5) and (7). Now we fix and , . The branch power is independent, whose cdf is given by [30, eq. (2-1-124)]. Noticing that

(20)

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, where , and its pdf may be found as [30, eq. (2-1-110)], we can readily obtain the output cdf

TABLE I NUMBER OF TERMS REQUIRED IN (26) TO ACHIEVE

= 1) SIGNIFICANT FIVE-FIGURE ACCURACY (

(22) This novel result reduces to several well-known special cases. The cdf of SC in independent Nakagami- fading channels can in (22) to yield be obtained by setting

where

(23) which is equivalent to the result [31]. As expected, when , this new expression reduces to (15) for the output cdf of SC in , we equally correlated Rayleigh fading channels. For have also checked (22) numerically against an infinite series for the cdf derived from [32, eq. (3)]. Both of the methods give exactly the same numerical values. Differentiating (22) yields the output pdf of SC in equally correlated Nakagami- fading channels. It can be shown that, for dual-branch SC, the output pdf reduces to

(24)

(27) is the incomplete gamma function [28, eq. where (6.5.4)], which is available in standard numerical softwares, such as Maple V and Matlab. Compared with the integral expression (15) for the SC output cdf, the infinite series representation (26) may be more useful for computation. The number of terms required in (26) to achieve a target accuracy depends on several factors, including the fading , and the divercorrelation , the normalized branch SNR sity order as well. Table I lists the results for a significant five-figure accuracy over a range of values of , , and with . As the fading correlation , the the average branch SNR branch SNR or the diversity order increases, more terms are required in (26) to achieve the target accuracy. Following the same approach as above, we obtain the infinite series representation for output cdfs in Ricean and Nakagamifading channels, respectively, as

This result is equivalent to that of [7]. These special cases reaffirm the rightness of (22). IV. INFINITE SERIES REPRESENTATION In this section, we provide infinite series representations for computation of the cdfs of SC output SNR in equally correlated Rayleigh, Ricean, and Nakagami- fading channels. We start our derivation from the equally correlated Rayleigh fading case. , Noting the property of the Marcum -function we may rewrite the output cdf (15) as

(25) Using an infinite series [28, eq. (9.6.10)] for and interchanging the order of integration and summation, we obtain an infinite series representation of the output cdf after some algebra as (26)

(28) (29) where

(30) V. PERFORMANCE ANALYSIS Using the output cdfs (15), (21), and (22), we can readily evaluate the average error rate, the outage probability, and the

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output statistics of -branch SC in equally correlated fading channels. We assume that channel fading is nonselective and changing slowly enough so that the channel parameters remain constant for the duration of the signaling interval.

readily adapted to handle all integral expressions derived in this section. : The CEP of noncoherent BPSK 2) (NCBFSK) and differential BPSK (DBPSK) can be expressed in the exponential form [30]

A. Average Error Rate The average BER or symbol-error rate (SER) is one of the most commonly used performance criterion of digital communication systems. Several methods can be used to obtain the error rates. Conventionally, the average error rate is obtained by inte, over the grating the conditional error probability (CEP) [see (17)] to yield pdf of the SC output SNR

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(36) for DBPSK and for where NCBFSK. Following the same procedure as the above, we obtain the BER as

(31) (37) Using (19), the mgf approach [33], [34] can be readily employed for evaluating the error-rate performance of SC. We can also . express the average error probability in terms of the cdf of This can readily be done using the integration-by-parts method as follows:

The CEP for noncoherent -ary frequency-shift keying (MFSK) can be expressed as a sum of exponential forms [30, eq. (5-4-46)]. Thus, we can readily write down the SER for MFSK using (37) to yield

(32) where denotes the negative derivative of the CEP. Next, we will present the BER or SER for various modulation schemes with SC in equally correlated Rayleigh fading channels according to their CEP forms. : Binary phase-shift keying (BPSK) 1) is used for reverse link in CDMA2000 due to its high power efficiency. The CEP for coherent binary frequency-shift keying (BFSK) and -ary pulse amplitude modulation (PAM) is in the form of [30] (33) for BPSK, for coherent where for -ary BFSK, is the area under the tail of the Gaussian pdf PAM, and and defined as [30, eq. (2-1-97)]. Substituting (33) into (32), we obtain the following expression for the average BER:

(34) Notice that there is a removable singularity at That is the integrand approaches 0 as , i.e.,

(38) where is the number of bits per symbol. : Since it is both easy to 3) implement and fairly resistant to noise, quadrature phase-shift keying (QPSK) has been adopted in various third-generation (3G) standards, such as European Telecommunications Standards Institute (ETSI), Europe, and Association of Radio Industries and Business (ARIB), Japan. Quadrature-amplitude modulation (QAM) is also an attractive modulation scheme due to its spectral efficiency. Both 16-QAM and 64-QAM have been adopted in the IEEE 802.11a standard. The CEP for QPSK, QAM, minimum-shift keying (MSK) and coherent detected DPSK is in the following form [30]: (39) where

for QPSK or MSK, for coherent detected DPSK and

for QAM. The average SER is obtained as

in (34).

(35) When a numerical quadrature technique such as Gaussian quadrature is used, this singularity can be readily handled. Common mathematical software provides both Gaussian quadrature techniques and also the straight forward techniques such as Newton–Cotes formulas. All such techniques can be

(40) 4) : The performance analysis of -shifted DQPSK) has received differentially encoded QPSK (

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considerable attention, owing to its adoption in the second generation of North American and Japanese digital cellular standards, such as the North American IS-54 and Japanese Personal Digital Cellular (PDC) (see [35] and its references). DQPSK and noncoherent correlated BFSK The CEP for can be written as [30], [36] (41) where

for

DQPSK and

for correlated NCBFSK, where is the correlation coefficient between the binary signals. Substituting (41) into (31), we obtain the SER as

Fig. 1.

CDF of the SC output SNR in equally correlated Rayleigh fading;  =

0:5. Circles denote simulation points.

(42)

Using the above expressions, we can readily derive other useful measures, such as the central moments, the skewness, the kurtosis, and the Karl Pearson’s coefficient of variation (amount of fading) [37]. For brevity, we do not develop such results here.

B. Outage Probability Outage probability is another standard performance measure of digital communication systems. It is defined as the probability that the output instantaneous SNR falls below a certain given . Hence, evaluating the output cdfs (15), (21), and threshold (22) at , we immediately obtain the outage probability for -branch SC in equally correlated fading channels as follows: (43)

C. Output Quality Indicators The mean output SNR of a diversity combiner is often used as a comparative performance measure. We only consider Rayleigh fading for brevity. Since we have derived the output pdf [(17)] and output mgf [(19)], the moments of the output SNR can be determined. The mean output SNR is obtained as

(44) More generally, we obtain higher order moments as

(45)

VI. NUMERICAL RESULTS Several numerical (represented by lines) and simulation (represented by circles and plus signs) results are given to illustrate the effect of correlation on the performance of SC in several fading channels. In all of the figures, is the correlation among underlying Gaussian RVs, is the diversity order, and is the normalized branch SNR. Note that semi-analytical simulation results are provided as an independent check of our analytical results. We use the Cholesky decomposition approach [38] to generate the equally correlated complex Gaussian variables and transform them to the Rayleigh, Ricean, and Nakagami- envelopes. of Fig. 1 shows the effect of on the output cdf repreSC in equally correlated Rayleigh. The case of sents a situation with no diversity. As expected, diversity gain can still be achieved even with correlated fading. The maximum additional diversity gain is achieved with dual-branch diversity. With increasing diversity order , additional diversity gain diminishes, as is the case for independent fading. on the output cdf of Figs. 2–4 show the impact of four-branch SC in equally correlated Rayleigh, Ricean, and Nakagami- fading channels, respectively. The case of represents independent fading. The case of represents a single-branch case. Observe that the maximum diversity gain will not be achieved when correlated fading exists. The diversity gain decreases as increases. However, the diversity gain is still available even with high correlation. We can also see that, in the low correlation case, where is small, the performance of SC is comparable to that in the independent case. However, in heavily correlated fading channels, where

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Fig. 2. Impact of fading correlation on the cdf of the SC output SNR in equally correlated Rayleigh fading; L = 4. Circles denote simulation points.

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Fig. 5. Average BER of BPSK with four-branch SC in 2 different equally correlated Rayleigh fading channels;  f0; 0:3; 0:5; 0:6; 0:7; 0:8; 0:9; 0:95; 1g. Circles denote simulation points.

tends to 1, a minute increase of will cause severe degradation of the SC performance. Fig. 5 shows the effect of on the BER of BPSK with SC in equally correlated Rayleigh fading channels. The correlation among the available signals results in significant loss in performance. As an example, the BER of BPSK with SC increases to when increases from 0.3 to 0.9 from at an average branch SNR of 15 dB. VII. CONCLUSION

Fig. 3. Impact of fading correlation on the cdf of the SC output SNR in equally correlated Ricean fading; L = 4.

We have derived new representations for equally correlated Rayleigh, Ricean, and Nakagami- fading gains. We showed that the cdfs of the SC output SNR can be represented as singlefold integral and derived infinite series representations. Consequently, unlike the other two existing methods [12], [14], any number of diversity branches can be handled as a single-fold integral (for the outage probability) or two-dimensional integral (for the error rate and the mean output SNR). The complexity of our approach does not increase with the number of diversity branches. These representations therefore resolve the long-standing open problem of SC performance in equally correlated fading channels. Consequently, we have derived SER expressions and outage and output statistics. Numerical results show that diversity benefits still exist in correlated fading channels, although the maximum diversity gain will not be achieved when fading is correlated. The representations developed in this paper can also be used to analyze EGC and other generalized SC schemes. These applications are currently being investigated. Finally, using bounds of the Marcum -function, we may develop performance bounds for -branch SC in equally correlated fading channels. ACKNOWLEDGMENT

Fig. 4. Impact of fading correlation on the cdf of the SC output SNR in equally correlated Nakagami-m fading; L = 4.

The authors wish to thank the anonymous reviewers for their detailed comments.

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REFERENCES [1] V. A. Aalo, “Performance of maximal-ratio diversity systems in a correlated Nakagami-fading environment,” IEEE Trans. Commun., vol. 43, pp. 2360–2369, Aug. 1995. [2] Q. T. Zhang, “Maximal-ratio combining over Nakagami fading channels with an arbitrary branch covariance matrix,” IEEE Trans. Veh. Technol., vol. 48, pp. 1141–1150, July 1999. [3] M. Schwartz, W. R. Bennett, and S. Stein, Communication Systems and Techniques. New York: McGraw-Hill, 1966. [4] Y.-C. Ko, M.-S. Alouini, and M. K. Simon, “Average SNR of dual selection combining over correlated Nakagami-m fading channels,” IEEE Commun. Lett., vol. 4, pp. 12–14, Jan. 2000. [5] M. K. Simon and M.-S. Alouini, “A unified performance analysis of digital communication with dual selective combining diversity over correlated Rayleigh and Nakagami-m fading channels,” IEEE Trans. Commun., vol. 47, pp. 33–44, Jan. 1999. [6] C. Tellamura, A. Annamalai, and V. K. Bhargava, “Contour integral representation for generalized Marcum Q-function and its application to unified analysis of dual-branch selection diversity over correlated Nakagami-m fading channels,” in Proc. IEEE Vehicular Technology Conf., vol. 2, May 2000, pp. 1031–1034. [7] C. Tellambura, A. Annamalai, and V. K. Bhargava, “Closed-form and inifinite series solutions for the MGF of a dual-diversity selection combiner output in bivariate Nakagami fading,” IEEE Trans. Commun., vol. 51, pp. 539–542, Apr. 2003. [8] L. Fang, G. Bi, and A. C. Kot, “Performance of antenna diversity reception with correlated Rayleigh fading signals,” in Proc. IEEE Int. Conf. Communications, vol. 3, June 1999, pp. 1593–1597. [9] Y. Wan and J. C. Chen, “Fading distribution of diversity techniques with correlated channels,” in Proc. IEEE Int. Symp. Personal, Indoor, and Mobile Communications PIMRC’95, vol. 3, Sept. 1995, pp. 1202–1206. [10] G. K. Karagiannidis, “Performance analysis of SIR-based dual selection diversity over correlated Nakagami-m fading channels,” IEEE Trans. Veh. Technol., vol. 52, pp. 1207–1216, Sept. 2003. [11] G. K. Karagiannidis, D. A. Zogas, and S. A. Kotsopoulos, “Performance analysis of triple selection diversity over exponentially correlated Nakagami-m fading channels,” IEEE Trans. Commun., vol. 51, pp. 1245–1248, Aug. 2003. [12] O. C. Ugweje and V. A. Aalo, “Performance of selection diversity system in correlated Nakagami fading,” in Proc. IEEE Vehicular Technology Conf., vol. 3, New York, 1997, pp. 1488–1492. [13] A. S. Krishnamoorthy and M. Parthasarathy, “A multivariate gammatype distribution,” Ann. Math. Statist., vol. 22, pp. 549–557, 1951. [14] Q. T. Zhang and H. G. Lu, “A general analytical approach to multi-branch selection combining over various spatially correlated fading channels,” IEEE Trans. Commun., vol. 50, pp. 1066–1073, July 2002. [15] R. K. Mallik and M. Z. Win, “Analysis of hybrid selection/maximal-ratio combining in correlated Nakagami fading,” IEEE Trans. Commun., vol. 50, pp. 1372–1383, Aug. 2002. [16] K. S. Miller, Complex Scochastic Processes. Reading, MA: AddisonWesley, 1974. [17] R. K. Mallik, “On the multivariate Rayleigh and exponential distributions,” IEEE Trans. Inform. Theory, vol. 49, pp. 1499–1515, June 2003. [18] G. K. Karagiannidis, D. A. Zogas, and S. A. Kotsopoulos, “On the multivariate Nakagami-m distribution with exponential correlation,” IEEE Trans. Commun., vol. 51, pp. 1240–1244, Aug. 2003. [19] G. K. Karagiannidis, D. A. Zogas, and S. A. Kotsopoulos, “An efficient approach to multivariate Nakagami-m distribution using Green’s matrix approximation,” IEEE Trans. Wireless Commun., vol. 2, pp. 883–889, Sept. 2003. [20] W. R. Young, “Comparison of mobile radio transmission at 150, 450, 900, and 3700 MHz,” Bell Syst. Tech. J., vol. 31, pp. 1068–1085, 1952. [21] H. W. Nylund, “Characteristics of small-area signal fading on mobile circuits in the 150 MHz band,” IEEE Trans. Veh. Technol., vol. VT-17, pp. 24–30, Oct. 1968. [22] Y. Okumura, E. Ohmori, T. Kawano, and K. Fukuda, “Field strength and its variability in VHF and UHF land mobile radio services,” Rev. Electron. Commun. Lab., vol. 16, pp. 825–873, Sept./Oct. 1968. [23] A. H. Nuttall, “Some integrals involving the Q function,” Naval Underwater Systems Center, New London Lab., CT, Tech. Rep. 4297, May 1974. [24] W. C. Jakes, Microwave Mobile Communications. New York: IEEE Press, 1994.

[25] R. B. Ertel and J. H. Reed, “Generation of two equal power correlated Rayleigh fading envelopes,” IEEE Commun. Lett., vol. 2, pp. 276–278, Oct. 1998. [26] N. C. Beaulieu, “Generation of correlated Rayleigh fading envelopes,” IEEE Commun. Lett., vol. 3, pp. 172–174, June 1999. [27] M. Nakagami, “The m-distribution, a general formula of intensity distribution of rapid fading,” in Statistical Methods in Radio Wave Propagation, W. G. Hoffman, Ed. Oxford, U.K.: Pergamon, 1960. [28] M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions With Formulas, Graphs, and Mathematical Tables. New York: Dover, 1972. [29] M. R. Spiegel and J. Liu, Mathematical Handbook of Formulas and Tables, 2nd ed. New York: McGraw-Hill, 1999. [30] J. G. Proakis, Digital Communications, 3rd ed. New York: McGrawHill, 1995. [31] A. Annamalai and C. Tellambura, “Error rates for Nakagami-m fading multichannel reception of binary and M -ary signals,” IEEE Trans. Commun., vol. 49, pp. 58–68, Jan. 2001. [32] C. C. Tan and N. C. Beaulieu, “Infinite series representations of the bivariate Rayleigh and Nakagami-m distributions,” IEEE Trans. Commun., vol. 45, no. 10, pp. 1159–1161, Oct. 1997. [33] C. Tellambura, A. J. Mueller, and V. K. Bhargava, “Analysis of M -ary phase-shift keying with diversity reception for land-mobile satellite channels,” IEEE Trans. Veh. Technol., vol. 46, pp. 910–922, Nov. 1997. [34] M. K. Simon and M.-S. Alouini, “A unified approach to the performance analysis of digital communication over generalized fading channels,” Proc. IEEE, vol. 86, pp. 1860–1877, Sept. 1998. [35] C. Tellambura and V. K. Bhargava, “Unified error analysis of DQPSK in fading channels,” Electron. Lett., vol. 30, no. 25, pp. 2110–2111, Dec. 1994. [36] L. E. Miller and J. S. Lee, “BER expressions for differentially detected =4 DQPSK modulation,” IEEE Trans. Commun., vol. 46, pp. 71–81, Jan. 1998. [37] A. Stuart and K. Ord, Kendall’s Advanced Theory of Statistics, 6th ed. New York: Oxford Univ. Press, 1994, vol. 1. [38] M. C. Jeruchim, P. Balaban, and K. S. Shanmugan, Simulation of Communication Systems: Modeling, Methodology and Techniques, 2nd ed. New York: Kluwer/Plenum, 2000.

Yunxia Chen received the B.Eng. degree in information engineering from Shanghai Jiaotong University, Shanghai, China, in 1998, and the M.Sc. degree from the University of Alberta, Edmonton, AB, Canada in 2004. She is currently working toward the Ph.D. degree in the Department of Electrical and Computer Engineering, University of California, Davis. Her research interests are in diversity techniques, MIMO systems, and wireless sensor networks.

Chintha Tellambura received the B.Sc. degree (with first-class honors) from the University of Moratuwa, Moratuwa, Sri Lanka, in 1986, the M.Sc. degree in electronics from the University of London, London, U.K., in 1988, and the Ph.D. degree in electrical engineering from the University of Victoria, Victoria, BC, Canada, in 1993. He was a Postdoctoral Research Fellow with the University of Victoria (1993–1994) and the University of Bradford (1995–1996). He was with Monash University, Melbourne, Australia, from 1997 to 2002. Presently, he is an Associate Professor with the Department of Electrical and Computer Engineering, University of Alberta. His research interests include coding, communication theory, modulation, equalization, and wireless communications. Prof. Tellambura is an Associate Editor for both the IEEE TRANSACTIONS ON COMMUNICATIONS and the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS. He is a Co-Chair of the Communication Theory Symposium in Globecom’05 to be held in St. Louis, MO.

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Delay-Limited Throughput of Ad Hoc Networks Eugene Perevalov, Member, IEEE, and Rick S. Blum, Senior Member, IEEE

Abstract—The delay-limited throughput of an ad hoc wireless network confined to a finite region is investigated. An approximate expression for the achievable throughput as a function of the maximum allowable delay is obtained. It is found that: 1) for moderate values of the delay , the throughput that can be achieved by taking advantage of the motion increases as 2 3 and 2) for a fixed value of , the dependence of the achievable throughput on the number 1 3 . A transmission and relaying strategy ensuring of nodes is continuous information flow is constructed. It is shown that there exists a critical value of the delay such that: 1) for values of the delay below the critical delay, the throughput does not benefit appreciably from the motion and 2) the dependence of the critical delay on the number of nodes is a very slowly increasing function ( 1 14 ). Finally, asymptotic optimality of the proposed strategy in a certain class is shown. Index Terms—Ad hoc networks, delay-limited throughput.

I. INTRODUCTION D HOC wireless networks [1], represent a promising new technology in communications that is currently receiving significant attention. Since the early research efforts, including the DARPA research program on packet radio networks, [2], [3], significant progress has been made. However, some important fundamental problems remain unsolved. Here we focus on a theoretical investigation of the capacity of ad hoc networks for delay-constrained data and movable terminals. More precisely, we are interested in the relationship between the end-to-end delay and the throughput. This study is motivated by the results of Gupta and Kumar [4] and Grossglauser and Tse [5] (see also [7] for a different approach). For an ad hoc network of size , it was shown in [4] that the capacity per node decreases with , thus making large networks impractical. It was demonstrated in [5] that, if the nodes’ mobility is taken advantage of, the effect of decreasing capacity can be overcome. The price one has to pay for such a dramatic increase in capacity is an end-to-end delay no smaller than the time scale characterizing the nodes’ motion. In this paper, we make an attempt to quantify the relationship between the maximum allowable delay and the throughput. First, we find the delay-limited throughput, as a function of the maximum allowable delay , within the class of “one-relay” strategies in the spirit of [5]. We find an analytical expression for

A

Paper approved by E. Ayanoglu, the Editor for Communication Theory and Coding Application of the IEEE Communications Society. Manuscript received August 14, 2002; revised August 6, 2003, and January 8, 2004. This work was supported in part by the Air Force Research Laboratory under Agreement F49620-03-1-0214 and in part by the National Science Foundation under Grant CCR-0112501. E. Perevalov is with the Department of Industrial and Systems Engineering, Lehigh University, Bethlehem, PA 18015 USA (e-mail: [email protected]). R. S. Blum is with the Department of Electrical and Computer Engineering, Lehigh University, Bethlehem, PA 18015 USA (e-mail: [email protected]). Digital Object Identifier 10.1109/TCOMM.2004.836587

the throughput and use it to obtain a simple approximate result valid for moderate values of the maximum delay. We then impose an additional requirement that the information flow from every source to every destination be continuous and construct a transmission and relaying strategy that satisfies this requirement. Our approach is based on the combination of the diversity routing idea of Grossglauser and Tse [5] and the multipath routing methodology of Tsirigos and Haas [9] that relies on the diversity coding approach from [8]. Namely, just as in [5], a source node transmits to its current nearest neighbor at each time slot allocated for transmission. The difference is that, in our approach, we do not send a packet to its destination via one relay node. Instead, after adding redundant information, we split the resulting “enlarged” packet into many blocks and send the blocks to the destination via different relay nodes. As a result, in order to achieve the desired level of service (measured as the probability of correct reconstruction of the message by the destination node within time from the moment of the message origination), one needs to employ a certain redundancy level which in turn directly affects the maximum throughput. We calculate the required redundancy level approximately. We find that there exists a critical value of the delay such that, for delays below the critical value, the gain in the achievable throughput due to the use of the motion is negligible. In other words, for such delays, the result of [4] applies: the per-node . For delays throughput of the network scales roughly as just above the critical value, the throughput benefits from the . It is intermotion as in [5] and increases approximately as esting to note that the value of the critical delay increases only ) with the number of nodes , which is a very slowly (as welcome feature. A. Model and Previous Results The model we adopt is similar to those used in [4] and [5]. The network consists of nodes located on a sphere of area . All nodes are mobile, and we assume that the motion of any node is described by the same stationary ergodic random process such that at each time there is no preferred direction. The trajectories of all nodes are assumed to be independent and identically distributed (i.i.d.). Moreover, we assume that the motion of the nodes is completely “memoryless,” i.e., the increments in a node’s position over nonoverlapping time intervals are independent. Note that this assumption does not hold for mobility models with differentiable trajectories which are realistic for ad hoc networks. Nevertheless, the assumption can be justified by observing that the typical time scale relevant for the ensuing dis, and all we need to guarantee is cussion is of the order of is such that that the typical change of direction time scale and the direction after the change is independent

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from that before the change. That is, we assume that the nodes move in straight lines with constant speed in between “memoryless” direction changes which are separated by time periods . negligible compared to We assume that every node has an infinite amount of data for and that the source–destination association its destination does not change with time. In the transmission model we use (which is the same as in b/s to node at time [5]), a node is capable of transmitting if (1)

as well. Generally speaking, (2) is a consequence of the propare i.i.d. random variables such that erty that, if decays slower than as , then the their cdf largest of them is of the same order as the sum. (In this context, ’s are the received powers from the transmitting nodes.) More specifically, the main technical complication in the proof of [5, eq. (2)] was due to the fact that that the distribution of the received power from the sender and the distribution of the interference depends on the location of the receiver because of the edge effects of the disk. This complication is not present in the case of a sphere which makes the proof of (2) for this case only simpler. B. Main Results and Discussion

where is the transmitting power of node , is the channel gain from node to node , is the background noise power, is the processing gain of the system, and is the signal-to-interference ratio (SIR) requirement for successful communication. The channel gain is assumed to depend only on the shortest disbetween the respective nodes as in tance

where is a parameter greater than 2. At any time a scheduler decides which nodes transmit bits and the corresponding power levels. The objective is to ensure a high average throughput for every source–destination pair. Let the number of bits that the destination us denote by receives in time slot . We say that an average throughput of is feasible if for every source–destination pair

Gupta and Kumar [4] demonstrated that, if a node can bits per second over a common wireless channel, transmit there exist constants and such that

As discussed above, the throughput of [5] can be achieved in the assumption that the end-to-end delay is unimportant. It is essentially an asymptotic value. However, we can roughly estimate the delay necessary to get close to this throughput. Let us denote by the average length of time during which the identity of the nearest neighbor remains unchanged. It is easy to see that, for large values of ,

where is a numerical constant (see (42) and the discussion following it). Then, in order to obtain the throughput of the order of , one needs to wait

time units, where is another numerical constant. Introducing dimensionless “natural units” for the delay as

and combining the above formulas, we obtain

or, equivalently,

is feasible is feasible i.e., the throughput per source–destination pair roughly goes to . zero as Grossglauser and Tse [5] constructed a scheduling policy according to which, in any transmission time slot , (where is the sender density parameter to be determined) nodes nodes are designated as senders and the remaining as potential receivers. Each sender node then transmits packets to to its nearest neighbor using unit transmit power. Among the sender–receiver pairs, the policy retains those for which the interference generated by the other senders is sufficiently low so that a successful transmission is possible. If is the number of such pairs, then, as was shown in [5],

for the delay in “natural units” necessary to achieve throughput of the order . In this paper, we are mostly interested in calculating throughput for the range of delays such that or, in asymptotic notation,

Our first main result is the following. Main Result 1: A delay-limited throughput that can be achieved for the motion model described above within the class of “one-relay” strategies for the values of the delay such that has the form

(2) Indeed, in [5], this statement was proven for nodes on a disk of unit area. It is straightforward to see that (2) holds for a sphere

where is the numerical coefficient that we compute explicitly [see (38)].

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Further, for the same range of delays, if we impose the additional requirement that the information flow from the source to the corresponding destination be continuous (each packet reaches the destination with high probability ), we obtain the following main result for the throughput achievable within the same class of strategies. Main Result 2: A delay-limited throughput that can be achieved within the class of “one-relay” strategies for the values of the delay such that with the additional constraint of information flow continuity has the form

for the delays in the excess of the critical delay be calculated as

which can

where is the same numerical coefficient as in Main Result 1, and is the numerical coefficient that can be calculated up to a factor characterizing the specific motion model. Remark: Comparing the above expressions for and and taking the dependence of into account, we can see that, for large values of , the ratio is close to 1. In other words, one gets the continuity of the information flow from source to destination “for free.” C. Structure of the Paper In Section II, we calculate the throughput achievable within the class of “one-relay” strategies under the constraint that the end-to-end delay is not to exceed . For this purpose, we develop expressions for the probability that two nodes come within range of direct transmission and the probability that the resulting transmissions are successful. We then determine the optimal transmission range. In Section III, we consider the additional requirement that the information flow from every source to every destination be continuous. We construct a different transmission and relaying strategy that has this property and find an approximate expression for its delay-limited throughput. We show that the price that one ends up paying for the continuity of the information flow is the existence of the critical delay such that the throughput is small for the values of delay below critical. We show also that, in the limit of a very large number of nodes in the system (and sufficient delay), the throughput achieved under the continuity condition is equal to that achieved with this condition relaxed. Conclusions are given in Section IV. II. ACHIEVABLE DELAY-LIMITED THROUGHPUT Let us denote by the per-node throughput of the network achieved by the one-relay node approach [5] in the abbe the unsence of end-to-end delay constraints. Also let conditional probability that two randomly chosen nodes come within a transmission range in time units and let be the corresponding capture probability at the second stage (from relay to destination). We now find an achievable throughput in the presence of an end-to-end delay constraint.

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Theorem 1: In the class of relaying strategies where each packet goes through at most one relay node, an achievable of an ad hoc network under the constraint that throughput the end-to-end delay not exceed can be calculated as (3) for sufficiently large . Proof: Since the relative contribution of direct source-todestination transmissions to the total throughput is negligible, we will ignore it similar to [5]. Consider the first-stage transmission (origin to relay). In the absence of end-to-end delay constraints, the number of pairs for which a successful transmissions takes place at the first stage is on average equal to . If the end-to-end delay constraint of is imposed, among the relay nodes that successfully received the packets from the of them will be successful. corresponding origins, Therefore, if in the absence of end-to-end delay constraints , then, if a the resulting throughput achieved is equal to uniform delay constraint of is imposed, the corresponding throughput becomes . A. Probability of Two Nodes Coming Within Range of Direct Transmission We would like to find an approximate expression for the unconditional probability that two nodes come within range of direct transmission within time . More precisely, we wish to find the unconditional probability that a given node comes within distance from another node in an interval of time of length . Let us introduce the following notation. is the unconditional probability that two nodes come • within a distance of in a time period of • is the probability that two nodes come within a distance of in a time period provided they were separated by a distance greater than at time 0 Theorem 2: These two quantities are related by (4) be the event that two nodes come within Proof: Let range of in time , and let be the event that two random points on the sphere are within distance from each other at time 0. Then the probability and can be found as Noting that

1, and , we obtain the statement of the theorem. In order to find the probability , we determine the probability of the opposite event: . Let us denote by the probability of two nodes not coming within range provided that at time 0 the two nodes in question are located at points and , respectively. Then the probability can be defined as the average of over all (uniformly distributed on the sphere) starting points and , which we denote as

(5) It can be shown that 1Strictly

speaking, this is an approximate relation valid for r=R

 1.

(6)

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where is the probability that the two nodes will not come within range in time provided that at time 0 the first node is at point and the second is at the north pole. The simplification of (5) to (6) follows from the uniform distribution of on the sphere. To see this, note that, for any realization of , since is distributed uniformly on the sphere, the relative position of and (which is obtained by rotating the sphere such that is at the north pole) also has a uniform distribution on the sphere. Therefore, the relative position of and is distributed uniformly for any distribution of (including uniform). . Denote by Now fix a point in time such that the probability that the two nodes have not come into the range within time units, and they were at the relative position at time . Then, under the assumption of independence, we can write

, the solution of Note that, in the absence of the term for some constant . (12) would be Let us write as (13) is a new function to be determined. where Theorem 3: The function is equal to the solution of the following ordinary differential equation: (14) where (15) (16)

(7) The conditional probability

is related to

Now, taking expectations of both sides in (7) over all , we obtain

via

and

Proof: If we set

then, from (13), (17)

(8) stands for the expectation with respect to the diswhere tribution of the position of the first node at time , provided the nodes did not come within range between 0 and . Now let be the probability density function (pdf) characterizing the above probability distribution, which depends on . We will write it as

Substituting (13) into (12), we obtain: (18) Next, we let only terms linear in

in (18), for infinitesimal , obtain

, and, keeping

(9) is the area of the sphere less the circle where or radius around the north pole. Remark: In the following, we will neglect the quantity compared to 1. Thus, we will not acknowledge any difference between and the area of the sphere . , coincides with , and its probability Note that, for distribution is uniform on the sphere: , which means that . Substituting (9) into the expression for the expectation over all possible starting points, we arrive at

Using (17), simplifying the above expression and dividing by yields

The expression is a function of only, and we denote it by . Thus, we arrive at the following ordinary differential equation for :

(10) where

with

Theorem 4: The function pressed as

from (6) and

defined above can be ex(19)

(11) accounts for the difference of the initial distribution in (10) from the uniform. Now, substituting (10) into (8), we obtain (12)

where is a point at a distance from the north pole. Proof: Recalling the definition of , we can write (20)

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Now notice that, since the partial derivative of in the above , it can be nonzero only for points at expression is taken at a distance from the north pole. On the other hand, from (15), we have

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Now we can compute in the following way. Let be an infinitesimal time interval. Then the increase in probability during this interval can be calculated as

which yields, using (25), the following expression for

which implies that

:

(21) where is the standard polar coordinate, and is the Dirac’s delta function.2 Substituting (21) into (20), we obtain the statement of the theorem. Substituting (19) into (14) and solving the latter with the ini, we obtain tial condition (22) Thus, for

, we obtain

Introducing a dimensionless parameter rewrite the above expression as

, we can (26)

Now we obtain an approximate expression for the probability valid for small values of . Theorem 6: To the first order of the parameter , the probability is (27)

(23) The latter expression involves the function whose exact form depends on the particular model of random motion of the nodes. Thus, if we are willing to characterize , (23) provides the required expression for . To avoid this and to promote simple expressions, we take an alternative approach and restrict attention to to find approximate expressions. The expression for to be obtained shortly will show how these assumptions allow us to obtain results for of interest. Before we perform a partial characterization of the function and establish an approximate expression for the probability , we determine the form of the parameter as a function of the problem’s input data. Theorem 5: The parameter has the following form: (24)

Proof: To prove the theorem, we use the expression (23) and partial characterization of the function . First note that, since for any point , the Taylor expansion of the function at the point reads (28) where, in particular, is a coefficient whose dimension is ( stands for the dimension of time). In other words, we can write , where is a dimensionless coefficient. First, we show that is a numerical coefficient of order 1. From the definition of , we have

and, using arguments similar to those employed in the proof of the previous theorem, we can show that

Proof: In order to find the value of , we note that or equivalently To evaluate from this definition, we must calculate the number of nodes that enter a circle of radius during a differential time interval assuming uniformly distributed nodes over a sphere of radius which are moving at speed . First let us calculate an average relative speed of two nodes each moving with speed in random directions on a plane. If is the angle between vectors, and , then the relative speed is equal to

Averaging over all (uniformly distributed) angles , we obtain for the average relative speed (25) 2Dirac’s

delta function  (x) can be defined by  (x) = 0 for x 6= 0 and  (x)dx = 1=2)

 (x)dx = 1 (and

where is a numerical coefficient that has to be of order 1. Now, using (23) and (28), and keeping only the terms linear in , we obtain the first-order expression for as follows:

and, therefore, with the same precision

B. Probability of Capture A typical information block on its way from the source node to the destination performs two hops: from the source to a relay node, and from the relay to the destination. We describe these two stages in turn below. For simplicity, we assume, following

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[5], that in odd time slots the first stage is affected and in even time slots the second is affected. 1) Source to Relay: The source-to-relay transmission is effected to the nearest neighbor as described in [5]. There, it was shown that the capture probability approaches a finite number for a very large number of nodes . Let us denote this number by . In this paper, we do not go into the details of the transmission policy, postponing it to future treatments. 2) Relay to Destination: In the second stage of our transmission policy, the relay will transmit to the destination once it is at the distance from the destination. It is clear that choosing a larger value of increases the probability of coming into the range within limited time and, at the same time, decreases the probability of capture. So one can hope to be able to choose the optimal value of given the parameters of the system. The probability of coming in the range was studied in the previous section. Here, we approximately compute the probability of capture. It is convenient to introduce the dimensionless parameter which measures the delay in “natural” units, counting how many times a node could traverse the spherical region if it moved in a straight line. Theorem 7: In the above scheme, the probability of successful transmission from relay to destination can approximately be calculated as

(29) where and . Proof: First, note that, if any other transmitter is at a distance no more than from our destination, then capture is impossible. Let us denote by the event that none of the other transmitting relays are within distance or less from our destination. The probability of can be computed as (30)

and standard deviation equal to , where and are the mean and standard deviation of the interfering power of one other node, respectively, given that this node is outside the circle of radius . If the transmitting power of each node is equal to 1, we can as calculate

where and are standard spherical coordinates. Taking into , we can simplify the account the fact that, for large , above expression by: 1) setting to ; 2) setting to ; and 3) neglecting compared to . Then we obtain

Evaluating the integral and neglecting terms of the order of 1 , we arrive at compared to

In order to calculate

, we express it as

and employ the same approximations as above. The result reads

So, given that is true, the probability that the transmission is successful can be computed as4 (32) where denotes the standard normal cdf. Finally, combining (31) and (32), we obtain an approximate (valid for large ) equation for the probability of capture as follows:

where is the number of simultaneously transmitting relays which is on average equal to where is the sender density in the first stage. Using the fact that (as a consequence of large ), we can write an approximate expression for (30) as follows:

(33) Using the definition of and remembering that, in the first order in , , we can rewrite (33) as

(31) If is true, i.e., none of the other transmitting nodes are within distance from our destination, the capture may still be impossible due to the total interference power from all of the other transmitting nodes. Since the number of such nodes is large (proportional to ), we can approximate the distribution of the interfering power by the normal one3 with the mean equal to 3It is known that, for  > 2, most of the total interfering power is due to a few nodes close to the point in question. The statement made here does not contradict this fact as it pertains to the a priori distribution of the interfering power as opposed to a concrete realization where the phenomenon of a few close nodes contributing the most power manifests itself.

C. Delay-Limited Throughput Thus far, we have not chosen the transmission range . Recall that the quantities and depend on (or its dimensionless version ) as shown in (26) and (29), respectively. Thus, we can choose the value of such that the throughput is maximized for any fixed value of the delay . 4We

have set N = 0 and L = 1 in (1), for simplicity.

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Substituting (4), (27), (26), and (29) into (3), and neglecting compared to terms of order 1, we obtain the small terms which can an approximate expression for the throughput be symbolically expressed as

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TABLE I

g() AND F () FOR SOME VALUES OF 

(34) We see that one can maximize the throughput by choosing the value of so that is maximized. Thus, we need to maximize the function

with respect to

. Introducing the new parameter , the above expression can be rewritten as

(35) Fig. 1. Ratio y = C =C as a function of dimensionless delay w as given in (38). Here we use n = 10 000,  = 4, = 2,  = 0:4, and  = 0:14.

Differentiating (35) with respect to , we obtain

on the delay The dependence of the throughput in Fig. 1. Several observations are now in order. (36) where is a new dimensionless variable. Now, let be the root of (36). Then the optimal value of becomes (37) The root of (36) for any fixed can be easily found numerifor the optimal transmission cally. The capture probability range (37) depends on only and is equal to

The quantities and for some values of greater than and 2 are shown in Table I. Note that, for large values of , approach their asymptotic values equal to and , respectively. Substituting (37) into (34), we obtain for the delay-limited throughput

or, using the dimensionless delay in “natural” units (38)

is shown

, so sig• The dependence of the throughput on is as nificant throughput is reached relatively quickly. • As can be seen from (37), the optimal transmission range , i.e., for large values of , it can be decreases as significantly larger than the typical internode distance that scales as . It is also interesting to note that the op. timal transmission range decreases with as • For a fixed value of , the throughput decreases as as a function of the number of nodes . , and, for optimal transmission range, • Note that , where is a numerical constant of order 1. So we can see that, for large values of , one can have fairly large and , which is the regime for which our approximations are valid.

III. THROUGHPUT WITH CONTINUOUS INFORMATION FLOW The throughput we have found in the previous section has the property that, for a given packet, the probability of its successful delivery with delay less than is typically substantially less than 1. Thus, only a relatively small fraction of all packets sent by a source to its destination reach the latter in time . On the other hand, it is natural to ask what throughput can be achieved under the condition that the resulting information flow from source to destination is continuous (modulo delay ), i.e., what throughput can be achieved if we demand that the probability of each packet reaching the destination within time be close to 1. In this section, we construct a transmission and relaying strategy that satisfies such continuity requirement and calculate the corresponding achievable throughput.

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We now describe our relaying strategy. The goal is to get close to the throughput found in the previous section. We achieve it by using a different approach from that in [5]; specifically, we spread the packet traffic between many nodes. Namely, after adding redundant information, we split the resulting packet into blocks and send the latter via different routes (relay nodes). We extra bits employ the coding scheme used in [9] in which are added to the packet of information bits as overhead, thus resulting in bits that are treated as one new networklayer packet. The additional bits are calculated as a function bits so that the original bits can be correctly of the original reconstructed from any subset of the bits of size no less than . The quantity (39) is the overhead factor. Our strategy consists of splitting the resulting -bit packet into equal-size blocks and sending them via different (consecutive) relay nodes. Note that a source can hand off as many blocks as it can to the same nearest neighbor as long as they come from different packets. So, the available transmission time slots for every source are fully utilized. Similarly, the blocks are communicated from the relays to the destination.

. Obviously, is the average length of time during which the identity of a nearest neighbor of a node remains unchanged. In the limit of very large , one can see that, in the order of magnitude, is equal to the time it takes a node to travel an average ) so that distance between the nodes (proportional to (42) where is some constant depending on the details of the motion model. Indeed, when is very large, the average distance between the nodes is substantially smaller than the average distance a node travels along a straight line , and decreasing the average distance between the nodes by a factor of results in decreasing by the same factor. On the other hand, increasing the speed by the factor of results in the increase of by the factor of , which leads to (42). With the above notation, we have . be equal to We demand that the success probability so that

(43)

A. Approximate Calculation of the Throughput The key question we have to answer is, given the maximum allowable delay , how do we select the overhead ratio in order to achieve the required probability (close to 1) that a correct message is received in the required time? If we are able to find the minimum sufficient overhead ratio , then obviously the throughput such a strategy can achieve will be equal to (40) In this paper, we will limit ourselves to the case where, for each -bit packet, exactly one block is sent to the destination via each relay node. Let us fix the desired level of service , the average probability that a packet will reach the destination within time . Our task is to determine the minimum overhead ratio such that the desired level of service can be achieved. Using the results of [9], we can write an approximate expression for the probability of successful reconstruction of a packet at the destination within time as

(41)

where is the number of blocks the packet is split into, is the probability that the th block reaches the destination within time , and is the employed overhead ratio. is the time that the th block has to reach the destination. So , and for . Let us denote —the difference be, respectween the time moments in which blocks and tively, were sent. We will denote the average value of by

The left-hand side of (43) seems to be hard to evaluate, so we simplify it by first noting that and, hence, if we demand that

(44)

then the resulting level of service will be no less than denote, for convenience,

. Let us

Then (44) reads (45) Equation (45) can be solved for

numerically as follows:

Table II shows the numerical values of for some fixed levels of service . We can now find the value of by solving the following equation: (46) The above equation states that we want to reach the desired level of service for at least one (optimal) value of the number of . The maximization over will blocks which we denote lead to the smallest value of , as illustrated in Fig. 2.

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TABLE II (Q) FOR SOME LEVELS OF SERVICE Q

Fig. 3.

“Saw” function (m).

by its maximum value of 1/2 and finding the value of maximizing the resulting expression:

we obtain

Fig. 2. g (m; z ) [as given by (47)] as a function of m for different values of the required level of service z and Q = 0:99. We can see that for z < z ( = 2:19) is not achieved for any m, while for z > z it is achieved or exceeded for a whole range of values of m.

We can now substitute the approximate expression (27) for into (46). Strictly speaking, since the time periods during which the identity of the nearest neighbor of a node is unchanged are random variables, we would have to take the averages over the corresponding distributions for calculating the quantities for all values of the index . However, since, for large , the standard deviation of is much smaller than its mean (see Theorem 8), we can get a good approximation by replacing the random variables , by their means. In this way, we obtain

On the other hand, if we drop the term altogether, we obtain

(49) from (48)

(50) Comparing (49) and (50), we see that the difference can be ne. Inglected provided deed, using (26), (37), and (42), we obtain (51) and therefore

Also, as we will see below [cf. (57)] (47) and therefore We can rewrite (47) as follows:

Thus (48) is the “saw” function depicted in Fig. 3. Let where us demonstrate that we can find the maximizer of (48) by neglecting the “saw” function part provided the number of nodes is large. First, note that, since the number of blocks has to be large by design, the exact shape of the function does not play a substantial role since , and, therefore, the error in determining will not exceed , which is small compared to for all reasonable values of . Next, let us convince ourselves that the into account maximizers of (48) with and without taking are close to each other for large values of . Indeed, replacing

This ratio approaches zero for large values of and therefore comcan be neglected in (49) in that limit. Finally, dropping pared to for the same reason, we obtain the approximate expression for (52) and its maximizer with respect to (53)

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Theorem 8: Assume that the quantities dependent and the ratio

are mutually inis bounded. Then

where

is another numerical constant of order 1. Since the ratio is bounded, and , we conclude that

If we relax the independence assumption above, then Now, substituting (53) back into (52), we obtain

Proof: First, we obtain a lower bound on

(56)

.

Finally, solving (46) for using (56), we arrive at the following expression for : where we have used (53). , the variance of . Next, we find an upper bound on Since , , are mutually independent, we obtain (54) Therefore, we can write

(57)

Thus, the throughput under the constraint that the end-to-end delay not exceed with probability no less than is approximately equal to

(58) Finally, since the ratio tain that

is bounded, we ob-

and, using (42) and the fact that the delays we are interested in are of the order or more [cf. (63)], we arrive at the first statement of the theorem. If one relaxes the assumption that the quantities are mutually independent, then, instead of (54) above, we can claim that (55)

where is a numerical constant of order 1. Indeed, in case the intervals between nearest-neighbor changes of a node are not independent, the corresponding “memory” can only last for approximately as long as the average node travels along a straight line. In other words, in the worst case, the number of positive entries in the covariance matrix of the vector does not exceed , which leads to (55). Then

where the function is implicitly described by (45). Analyzing (56), we can see that, for a given level of service , there exists a critical delay such that: , the transmission strategy discussed above is • for unable to achieve the required level of service. • for , the throughput increases with , as illustrated in Fig. 4. Then we can see that, if is chosen too large, the quantity on the right-hand side of (57) becomes negative for small , which implies there was no solution to (46) for small . This leads to the first bullet above. However as we increase , it is clear that the quantity in (57) becomes positive, as per the second bullet above. The change occurs at . Thus, we can use these same ideas to estimate the value of from (58) by equating to 0 as (59) Expanding (58) to the first order in , we see that function of delay behaves approximately as if if

as a

(60)

provided . Switching to the “natural” units for the delay, and substituting the explicit for the elements of (60), we obtain or

if if (61)

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Fig. 4. Ratio y = C =C as a function of delay w . We use c = 1,  = 2:19, and the rest of the parameters as in Fig. 1.

Fig. 5. Ratio R = C =C as a function of number of nodes n. We use  = 0:05 and the rest of the parameters as in Figs. 1 and 4.

It is interesting to compare (60) with the corresponding ap. Namely proximate expression for the throughput

So, if we fix and increase the number of nodes , we see approaches 1. Thus, we have proved the that the ratio following theorem. Theorem 9: There exists a transmission and relaying strategy satisfying the continuity of information flow requirement that asymptotically achieves the unrestricted delay-limited of an ad hoc network. throughput

(62) . can be obtained from (60) by setting Substituting (37), (26), and (29) into (59), and solving the resulting equation for , we obtain

IV. CONCLUSION (63) or, in the “natural” units (64) We can now summarize our observations as follows. , the dependence of the • For a fixed value of above throughput on is as , just as in the previous section. • Beyond , the dependence of the throughput on for a fixed value of is as . • Comparing (60) with (62), we see that the “price” that one ends up paying for the continuity of the packet stream is precisely the existence of the critical delay and the corresponding drop in the throughput by the factor for values of the delay above the critical value. • The increase in the critical delay with the number of , which is a denodes turns out to be very slow sirable feature (see Fig. 5). Let us now set the delay equal to where is some . Then small number

For the ratio

, we obtain

In this paper, we have conducted a preliminary exploration of the problem of the influence of the end-to-end delay on the throughput of a wireless ad hoc network confined to a certain area. Aiming at general results, we have made a number of simplifying assumptions. Thus, we adopted a “totally random” model of motion ignoring both the details of the corresponding distributions and possible patterns in the nodes’ motion. We also limited ourselves to the “one-relay” class of strategies in the spirit of [5]. Confining the analysis to the above class of strategies, we found an expression for the achievable delay-limited throughput that involves the unconditional probability of two nodes coming within a certain range within the maximum delay time and the corresponding capture probability. We then obtained a general expression for the required probability which depends on the transmission range from the relay to the destination. Next, we found an approximate expression for the probability of success of the corresponding transmission as a function of the transmission range. Finally, we have obtained an approximate expression for the delay-limited throughput as a function of the maximum allowable delay . We found that, for moderate values of , the achievable throughput increases as and, for a fixed value of , the throughput goes down as with the number of nodes . We then proceeded to construct a relaying strategy that would asymptotically achieve the above throughput while satisfying the continuity of the information flow requirement. We used the diversity coding approach in combination with the “secondary” diversity routing of [5] in order to asymptotically achieve the delay-limited throughput for this class of strategies in the presence of information flow-continuity requirement. We made use of a number of approximations that allowed us

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to end up with closed-form expressions. We found that, for moderate delays, the dependence of the optimal throughput on the delay is characterized by the “critical delay,” below which our relaying and transmission strategy does not lead to any appreciable throughput. For the values of the delay larger than the critical value, the achievable throughput grows approximately as . The critical delay has a very slow dependence on the number of nodes , so that it practically is independent of . It is interesting to note that the existence of that minimum delay is precisely the price one has to pay for the continuity of the information flow. Finally, we have shown that this transmission and relaying strategy is asymptotically optimal in the sense that the ratio of its achieved throughput and the unrestricted throughput approaches 1 (albeit rather slowly) as the number of nodes grows. REFERENCES [1] A. Alwan et al., “Adaptive mobile multimedia networks,” IEEE Pers. Commun. Mag., vol. 2, pp. 34–51, Mar. 1996. [2] A. Ephremides, J. E. Wieselthier, and D. Baker, “A design concept for reliable mobile radio networks with frequency hopping signaling,” Proc. IEEE, vol. 75, pp. 56–73, Jan. 1987. [3] J. Jubin and J. D. Tornow, “The DARPA packet radio network protocols,” Proc. IEEE, vol. 75, pp. 21–32, Jan. 1987. [4] P. Gupta and P. R. Kumar, “The capacity of wireless networks,” IEEE Trans. Inform. Theory, vol. 46, pp. 388–404, Mar. 2000. [5] M. Grossglauser and D. Tse, “Mobility increases the capacity of ad hoc wireless networks,” in Proc. INFOCOM, vol. 3, 2001, pp. 1360–1369. [6] P. Gupta and P. R. Kumar, “Toward an information theory of large networks: An achievable rate region,” in Proc. IEEE Int. Symp. Information Theory, 2001, p. 159. [7] S. Toumpis and A. Goldsmith, “Capacity regions for wireless ad hoc networks,” IEEE Trans. Wireless Commun., vol. 2, pp. 736–748, July 2003. [8] E. Ayanoglu et al., “Diversity coding for transparent self-healing and fault-tolerant communications networks,” IEEE Trans. Commun., vol. 41, pp. 1677–1686, Nov. 1993.

[9] A. Tsirigos and Z. Haas, “Multipath routing in the presence of frequent topological changes,” IEEE Commun. Mag., vol. 39, pp. 132–138, Nov. 2001.

Eugene Perevalov (M’01) received the M.S. degree in physics from the Moscow Engineering Physics Institute, Moscow, Russia, in 1993 and the Ph.D. degree in mathematical physics from the University of Texas at Austin in 1998. From 1998 to 1999 and from 1999 to 2001, he was a Postdoctoral Research Associate with Harvard University, Cambridge, MA, and the Massachusetts Institute of Technology, Cambridge, respectively. Since 2001, he has been with the Industrial and Systems Engineering Department, Lehigh University, Bethlehem, PA, where he is currently an Assistant Professor. His research interests include wireless networking and related topics in communications as well as optimization and financial engineering.

Rick S. Blum (S’83–M’84–SM’94) received the B.S. degree from the Pennsylvania State University, State College, in 1984 and the M.S. and Ph.D. degree from the University of Pennsylvania, Philadelphia, in 1987 and 1991, respectively, all in electrical engineering. From 1984 to 1991, he was a Member of Technical Staff with General Electric Aerospace, Valley Forge, PA, and he graduated from GE’s Advanced Course in Engineering. Since 1991, he has been with the Electrical and Computer Engineering Department, Lehigh University, Bethlehem, PA, where he is currently a Professor and holds the Robert W. Wieseman Chair in Electrical Engineering. His research interests include signal detection and estimation and related topics in the areas of signal processing and communications. He holds a patent for a parallel signal and image processor architecture. Dr. Blum is a member of Eta Kappa Nu and Sigma Xi. He was the recipient of an Office of Naval Research Young Investigator Award in 1997 and a National Science Foundation Research Initiation Award in 1992. He is currently an Associate Editor for the IEEE TRANSACTIONS ON SIGNAL PROCESSING and for IEEE COMMUNICATIONS LETTERS. He was a member of the Signal Processing for Communications Technical Committee of the IEEE Signal Processing Society.

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1969

Performance Analysis and Tradeoffs for Dual-Pulse PPM on Optical Communication Channels With Direct Detection Marvin K. Simon, Fellow, IEEE, and Victor A. Vilnrotter, Senior Member, IEEE

Abstract—The performance and tradeoffs of multipulse position modulation (MPPM) are investigated and compared with that of the traditional (single) pulse position modulation (PPM) scheme typically employed on the optical direct-detection channel. While the primary motivation for the consideration of the problem is to provide performance improvement for deep-space optical communications where narrow high-peak-power transmitted pulses offer significant advantages in terms of detection probabilities and background suppression capabilities at the receiver, the results obtained are sufficiently generic as to apply to other applications. Index Terms—Direct detection, multipulse position modulation (MPPM), optical communications, photon counting.

I. INTRODUCTION ULSE position modulation (PPM) is an accepted technique for transmitting information over the optical direct-detection channel [1], [2]. At the transmitter, the encoder maps blocks of consecutive binary symbols, or bits, into a single PPM channel symbol by placing a single laser pulse into one of time slots. The PPM symbols are orthogonal, since there is no overlap between pulses in any pair of symbols. After establishing slot and symbol synchronization, the receiver detects the uncoded PPM symbols by determining which of the slots contains the laser pulse and performs the inverse mapping operation to recover the bit-stream. Each correctly decoded PPM symbol conveys bits of information; however, the receiver must operate with much greater bandwidth than the actual data rate to affect the decoding operation. If each bit is seconds in seconds to transmit: this means duration, then bits takes that the receiver must process PPM time slots in seconds to avoid overflow. The processing rate of the system (both transtimes as mitter and receiver) must therefore be a factor of great as the transmitted bit rate, implying a required bandwidth expansion by a corresponding amount. For large , this bandwidth expansion can be severe, ultimately limiting the information throughput of the system due to limitations on the sampling rate.

P

Paper approved by K. Kitayama, the Editor for Optical Communication of the IEEE Communications Society. Manuscript received November 29, 2003; revised April 13, 2004. This work was performed at the Jet Propulsion Laboratory, California Institute of Technology under a contract with the National Aeronautics and Space Administration. The authors are with the Jet Propulsion Laboratory, California Institute of Technology, Pasadena, CA 91109-8099 USA (e-mail: victor.vilnrotter@ jpl.nasa.gov). Digital Object Identifier 10.1109/TCOMM.2004.836554

A natural extension of single-pulse PPM is the use of two or more pulses to convey information in each channel symbol [3]. This can be accomplished by placing more than one pulse slots, generating a much larger in all possible ways among is large. However, number of usable channel symbols when not all of the channel symbols are orthogonal, and the set of symbols thus generated are not necessarily powers of two, thus complicating the encoding operation. Nevertheless, multipulse PPM1 has desirable properties, namely the potential for significantly reducing bandwidth requirements at fixed average power as compared to the “best” single-pulse PPM strategy or increasing information throughput at a given bandwidth without incurring significant performance penalties. In Section II, we shall examine the information throughput and bandwidth requirement properties of multipulse PPM and compare it to both conventional single-pulse PPM and the “best” single-pulse PPM strategy that maximizes throughput with an average power constraint. The maximum-likelihood (ML) strategy for optimally decoding multipulse PPM will be derived in Section III for direct detected optical signals, in the presence of multimode background radiation. Exact performance of two-pulse PPM will be determined in Section IV, both for the erasure channel (no background radiation) and for the general case with arbitrary background, and performance comparisons and numerical results will be presented in Section V. II. INFORMATION THROUGHPUT AND BANDWIDTH REQUIREMENTS slots, as with conWith a single pulse placed in one of ventional PPM, the number of bits per PPM symbol is . The number of orthogonal symbols with “single-pulse PPM” is . It is assumed that each laser pulse contains an average of photons, where denotes the duration of the PPM time slot and denotes the signal intensity in photons/second (assumed to be constant over the time slot).2 The symbol time , and associated with a single PPM transmission is . thus, the average signal power of such a transmission is Suppose that, instead of using a single PPM pulse, we employed two pulses and consider the number of possible “dualpulse PPM” symbols that can be generated in this manner. The ; however, not number of such symbols is clearly 1For the specific case of two pulses per channel symbol, we shall refer to this modulation scheme as either two-pulse PPM or dual-pulse PPM. 2The bandwidth of a particular modulation scheme is directly proportional to the inverse of the time-slot duration  . Thus, any PPM schemes having the same slot duration will occupy the same bandwidth.

0090-6778/04$20.00 © 2004 IEEE

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all “dual-pulse PPM” symbols are orthogonal. In general, each other sym“two-pulse” PPM symbol is orthogonal to columns bols. With any two slots occupied, that leaves among which we can arrange two pulses in distinct ways, none of which have any overlap with the original symbol. The number of symbols not orthogonal with a given two-pulse PPM . symbol is therefore For conventional single-pulse PPM, the number of orthogonal symbols is equal to the number of slots ; thus, the information bits/symbol. As stated above, the number throughput is of symbols for two-pulse PPM is . With

slots, the two-pulse PPM set contains

the latter is constrained to have no more than one pulse in any -slot symbols. It is clear that -pulse PPM always of the improves upon single-pulse (conventional or compound) PPM in terms of information throughput, the improvement becoming significant as the number of pulses per symbol is increased. Another way to compare these schemes is in terms of bandwidth required for the same information throughput. Denoting the total number of slots for -pulse, compoundsingle-pulse, and conventional single-pulse PPM by , , and , respectively, then, analogous to the development in [3], equating the total number of symbols for the three schemes, their information throughputs are equal when

(1) bits of information per symbol. The extension to “ -pulse PPM” is straightforward. The number of symbols generated by pulses arranged in all possible patterns among slots is

(5) Solving for

in terms of

, for example, yields (6)

(2) Hence, the average number of bits per symbol is

(3) Thus, for the case , the information throughput has over single-pulse PPM been increased by nearly a factor of with the same number of slots . However, if each pulse contains the same average photon count as a single PPM pulse, then , has effectively been the average signal power, i.e., increased by a factor of in order to achieve this gain. Perhaps then, it might be more appropriate to compare each multipulse PPM symbol not just to a single PPM symbol with the same average photon count as one of the multipulse PPM symbols, but consecurather to a constellation of symbols composed of tive single-pulse PPM symbols, each of symbol duration and dimensionality . This modulation scheme, which then has the same average energy per unit time or, equivalently, the same average power as the previously described -pulse PPM, shall be referred to as compound- single-pulse -PPM.3 By dividing the slots into groups, each occupying slots and assigning a conventional single-pulse PPM constellation to each group as was done in [3], the number of comwith a correpound- single-pulse PPM symbols is sponding throughput (4) , , it is clear bits per symbol. Since that -pulse PPM contains more symbols, and hence, conveys more information than compound- single-pulse PPM, since 3We point out that an equivalent comparison to compound-K M -PPM is a comparison to conventional single-pulse M=K -PPM having symbol rate K=T . Specifically, both schemes have the identical bit rates, i.e., (1=T ) log (M=K ) for the former and (K=T ) log (M=K ) for the latter, and, furthermore, since the K symbols in compound-K PPM are independently chosen, optimal detection should yield identical bit-error rates (BERs). We will discuss this further later on in the paper.

The factor involving on the right-hand side (RHS) of (6) is equal to when and approaches the number as approaches infinity. Since for a fixed symbol time the , the above indicates that, in slot time is given by order to convey the same amount of information, compoundsingle-pulse PPM requires significantly greater bandwidth than -pulse PPM. With limitations on digital sampling rates imposed by hardware considerations, this property of -pulse PPM could become a significant factor for implementing high-data-rate telemetry. It should be emphasized that the number of multipulse PPM signals is not an exact power of two, hence, some thought must be given to the mapping of information bits to multipulse PPM symbols. For example, it is easily shown that the possible slots number of dual-pulse PPM symbols among is , which is symbols short of being an exact power of two. For large , this difference is negligible when compared to the total number of dual-pulse PPM symbols, however, it does imply that not every pattern of information bits can be represented. A possible solution is to single-pulse PPM symbols; however, this results append in an asymmetric signal set that is difficult to implement and decode. An alternative to this scheme is based on the observa, , hence tion that, for information bits can always be encoded by using a subset of the available dual-pulse PPM symbols. However, this strategy is large, discards nearly half of the available symbols when reducing throughput by nearly an entire bit. An improved strategy is to create compound symbols by cascading enough multipulse PPM symbols to ensure that the total number just exceeds a power of two and then encoding suitably larger blocks of information bits into this compound signal , there are 28 dual-pulse PPM set. For example, with bits. A cascade of two symbols, representing dual-pulse PPM symbols yields 9.6 bits, while a cascade of four represents 19.2 bits; therefore, 19 bits can be encoded into four dual-pulse PPM symbols, while making use of nearly the entire compound symbol set.

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III. MAXIMUM-LIKELIHOOD DECISION METRIC The ML metric for deciding optimally between an arbitrary number of intensity-modulated optical symbols has been previously derived [1]. Here we summarize the specific results pertinent to the problem under investigation in this paper. We assume that Poisson statistics apply to both the detected signal and the detected background fields and to their sum when appropriate. This simplifying assumption is valid whenever a large number of space–time modes are observed, and is generally true for signal-plus-background radiation under nominal operating conditions. The decision is based on a vector of time-disjoint count observables covering the symbol duration. Here we assume that distinct and disjoint counting intervals are sufficient to characterize each received symbol and base the decision . We on the -component count-vector denote the th hypothesis, corresponding to the th distinct intensity-modulated symbol, by . The probability of each count component is Poisson distributed with average value ( denotes the background intensity) when only when both background radiation is observed, and signal and background are present. Here the subscript refers to the hypothesis, whereas the subscript identifies the observation interval within the count vector. Since, conditioned on the intensity (hence, the hypothesis), Poisson counts from disjoint intervals are independent, the joint probability for the count vector, conditioned on a given hypothesis, can be expressed as the product of the individual probabilities corresponding to each count (7) The ML decision rule is to compute the conditional probability for each hypothesis given the observed count vector and select that hypothesis that yields the greatest value (i.e., that is most likely). Since the logarithm is a monotone increasing function of its argument, a simpler computation results if we equivalently base the decision on the log-likelihood function (LLF), which for the th hypothesis is denoted by and is given by

(8) Note that the second term on the RHS of (8) does not depend on the hypothesis, and hence, can be ignored in so far as the decision is concerned. The third term on the RHS of (8) is the total energy of the signal and background for the th symbol, denoted by , and contributes to the decision only if the total symbol energy depends on the hypothesis, i.e., for the case of equal-energy symbols, it too can be ignored. Furthermore, dividing the argument of the logarithm in the first term of (8) by is the background energy and recognizing that independent of the hypothesis, the LLF for the general case can be rewritten as (9)

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is the energy associated with the th hypothesis. For where equal-energy symbols, and with equal average signal power per laser pulse, the th LLF simply sums up the counts for the signal slots associated with the th symbol and selects the symbol corresponding to the greatest value. Note that, for the case of compound symbols generated by cascading multipulse symbols together, there are now slots per compound symbol, and the LLFs must collect counts over every possible pattern of slots. For the equal-energy case, the LLF for compound symbols can be written as

(10) This is the LLF to be maximized over the range of pulse patterns for -compound multipulse PPM symbols, expressed in terms of the partial sums over each component multipulse PPM symbol. However, since maximizing the compound LLF is equivalent to maximizing the partial sum over each multipulse PPM symbol, it follows that this maximization can be implemented sequentially over each multipulse PPM symbol. Therefore, maximizing the LLF for compound PPM symbols is equivalent to sequential symbol detection. Note that, if some of the compound symbols were left out in order to construct a signal set that can accommodate an integral number of information bits, the LLF as written in (10) is no longer valid, since not every possible pattern of dual-pulse PPM symbols are included in the abridged signal set—ML detection then requires the comparison of every compound symbol with every other compound symbol in the codebook. Next, we apply these results to the case of two-pulse PPM and determine its performance first with negligible background and then in the presence of background radiation.

IV. PERFORMANCE The decision rule derived above is defined in terms of log-likelihood metrics consisting of the sum of logarithmically weighted counts obtained from each “signal slot” of the various hypotheses. The decision rule is to select that hypothesis corresponding to the greatest LLF, given the vector of observables. For the special case of equal-energy signal pulses, the logarithmic weights for all likelihood functions can be ignored, and, as such, the LLFs consist of the sum of counts from all slots available for possible patterns of two slots, among the each symbol.

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The symbol-error probability (SEP) performance of conventional single-pulse PPM is well known and is repeated here for reference in the form derived in [4], namely

second slot and vice versa or 3) receive no counts in either signal slot, hence no counts over the entire observation vector. The probabilities of these events are through through through

(11) We now proceed to derive an expression for the equivalent performance characterization of dual-pulse PPM. Our goal is to obtain an exact expression for the average SEP performance of this modulation scheme (in contrast to an upper bound on such performance obtained previously in [5]), thereby allowing for more accurate numerical results to be obtained for small values of . With no loss in generality, we shall assume that the first hypoth, corresponding to the transmission esis is true, denoted by slots. of laser pulses in the first and second of the 1) Negligible Background Radiation: Background radiation is ever present, although narrowband optical filters that pass only the modulated signal fields can reduce it to negligible values, particularly at night. However, for daytime operation of free-space terrestrial optical links, or when a bright planetary source is in the receiver’s field of view in deep-space applications, background radiation often cannot be eliminated entirely. In these cases, which comprise a large class of realistic scenarios, significant amounts of background photons enter the receiver along with the signal, degrading receiver performance. A proper accounting of the effects of background radiation is therefore essential to a thorough understanding of the performance of the multipulse PPM signals. We begin by first considering the performance of dual-pulse PPM for the noiseless case, so as to introduce the key underlying concepts that will have to be generalized when we relax this constraint. Because of symmetry, we assign equal a priori probabilities to all hypotheses, so that the probability of the th two-pulse . The following are pattern occurring is the events that arise in the analysis of the error-probability performance of dual-pulse PPM. Under the condition of negligible background radiation, photon counts can only occur in the signal slots, but detection of no photon counts in either or both of the signal slots is also possible due to the nonzero probability of getting zero counts with the assumed Poisson statistics. The probability of getting zero counts in the slots that do not contain the signal is one, since we assumed zero-average background energy. Therefore, the probability of getting: 1) a count of one or more in the first ) and two slots (taken to be the signal slots for hypothesis zero everywhere else is

(13) (14)

When 2) occurs, there is no clear decision strategy, because we do not have enough information to distinguish between those patterns that have a signal pulse in the first (or second) slot; in this case, optimality is not compromised by choosing randomly among the remaining possibilities. Since the number of , patterns with a pulse in the first (or second) slot is a random choice among these possibilities yields a correct de. Similarly, when event 3) cision with probability occurs, a random choice among all possibilities yields a cor. Considering rect decision with probability all possibilities, the probability of a correct decision in the absence of background radiation is

(15) Note that for , we obtain , meaning that we always choose correctly; however, since with dual-pulse PPM there can only be one hypothesis in this case, there is no transfer of information. Finally, since the result in (15) is independent of the hypothesis chosen, the average SEP is given by (16) For conventional single-pulse PPM, the corresponding relationship to (16) is (17) , we observe that, in the limit of Thus, for a given value of and large , single-pulse PPM will outperform duallarge pulse PPM by as much as a factor of two in SEP. For compound-2 single-pulse PPM, a correct symbol decision requires that both pulses from the two consecutive single-pulse PPM constellations are correctly detected. Thus, the probability of error for this modulation scheme is

(18) through

(12)

When this event occurs, the transmitted hypothesis is identified correctly with probability one. However, it is also possible to: 2) receive a nonzero count in the first slot but no counts in the

That is, the probability of an error in the compound symbol is equal to the probability of error in the first pulse (with the second pulse correct or not), plus the probability that the second pulse is in error (with the first pulse correct or not), and, since this

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sum contains the simultaneous (squared) error term twice, we need to subtract out one of the squared error terms in order to obtain the exact error probability. Here again, for a given value , we observe that, in the limit of large , compound-2 of single-pulse PPM will approach the performance of dual-pulse PPM. 2) Nonnegligible Background Radiation: To begin the derivation, we point out that, if any of the noise-only slots contains a number of counts greater than the smaller of the number of counts in the two signal slots, then the signal slot containing this smaller number of counts will not be included as part of the slot-pair decision, i.e., a decision error will , occur. Mathematically speaking, if, for any , then such an event cannot contribute to the probability of correct detection. We now spell out the events that contribute to this probability when background radiation is included. Event 1: One or more photons are detected in slots 1 and ) and all other slots have fewer detected 2( , photons than in either slot 1 or 2 Here, a correct decision will be made with certainty, and thus the contribution of this event to the probability of a correct decision corresponds to the probability of occurrence of the event itself, namely

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(Note that could also be denoted by since and are equal. This will be convenient later on ways when combining events.) This event can occur in (corresponding to the number of possible combinations of the noise-only slots that contain detected photons). Furthermore, a correct decision is no longer guaranteed since there now are equally likely ways for a pair of slots to have detected photons, only one of which produces the true correct decision. Thus, the contribution to the probability of correct decision stemming from this event is

(21) Event 4: One or more photons are detected in slot 1 , and all other slots have zero detected photons. Furthermore, a correct decision is no longer guaranteed since there now are equally likely ways for a slot (other than the first) to have zero detected photons, only one of which produces the true correct decision. Thus, the contribution to the probability of a correct decision stemming from this event is

(19) Event 2: An unequal number (but at least one) of pho; any other tons are detected in slots 1 and 2 slots have detected photons, and the remaining slots have less than detected phoways (corresponding to the tons. This event can occur in number of possible combinations of the noise-only slots that detected photons). Furthermore, a correct decision contain equally likely is no longer guaranteed since there now are ways for a slot (other than the signal slot not corresponding to ) to have detected photons, only one of which produces the true correct decision. Thus, the contribution to the probability of a correct decision stemming from this event is

(22) Event 5: This is the counterpart to Event 4—one or more photons are detected in slot 2 , and all other slots have zero detected photons. By symmetry, the probability of correct decision stemming from this event is identical to . Event 6: All slots have zero detected counts. Here a possible slot pairs is random decision among made, resulting in a contribution to the probability of correct decision of

(23)

(20) Event 3: An equal number (but at least one) of photons ; any other are detected in slots 1 and 2 slots have detected photons, and the slots have less than detected photons. remaining

Examination of the probability of Event 1 as given in (19) reveals that it can be included in Events 2 and 3 by allowing the summation on in (20) and (21) to run from zero to infinity instead of one to infinity. Then the total probability of correct detection is obtained from the sum of (20) and (21), (22) (modified as above), and (23). Some simplification of the various terms that contribute to this summation is possible. Before proceeding with this, however, we first check that the end result agrees with the result previously obtained for the special case of no background noise. For the special case , each of the contributing probabilities is evaluated as follows.

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For Event 1, only the Thus, from (19), we have

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term in the third sum survives.

and , it is possible to Due to the symmetry of (28) in reduce the infinite sum on one of these indices to a finite sum. can be replaced In particular, the double sum by , leading to the simplification

(24) The probabilities corresponding to Events 2 and 3 are equal to zero. For the equiprobable Events 4 and 5, we have (25) Finally, for Event 6, we obtain (26) Thus, summing up (24)–(26) gives (27) which agrees with the previously obtained result in (19). Including (19) as the term in (20) and (21) as well as reordering the summations, then summing (20)–(23) and subtracting from 1, one finally obtains the desired expression for average SEP [6]

(31) The performance of dual-pulse PPM and its comparison with that of the conventional and compound single-pulse PPM approaches under average power, peak power, and bandwidth constraints is the subject of Section V. V. NUMERICAL RESULTS

(28) where (29) With some abuse of the notation, it is possible to write (28) in yet a more compact form that incorporates the second and third terms in the summations on and . In particular, after some manipulation, it can be shown that

(30) where it is understood that, if equals zero and thus , in which case

, then the summation on , unless .

The advantages of multipulse PPM over the best single-pulse strategy in terms of information throughput have been described in Section II. However, throughput is not the only criterion for evaluating and comparing modulation formats: performance must also be taken into account. Since single-pulse PPM is a completely orthogonal modulation while dual or multipulse PPM is only partially orthogonal, performance degradation for the multipulse modulation formats is expected since not all of the energy is available for differentiating between nonorthogonal symbols. For example, in the case of two-pulse PPM, where the maximum overlap between nonorthogonal pulses is equal to the energy of one pulse, only half of the energy is used to differentiate between the correct pulse and competing nonorthogonal pulses. On the other hand, multipulse PPM signals contain more channel symbols than the corresponding repeated single-pulse PPM sets, and they therefore enjoy a bandwidth advantage when equal throughput is the design criteria. In view of the above discussion, performance should be compared both at equal bandwidth and at equal information throughput. Finally, since increased bandwidth (narrower slot time) necessitates greater peak signal power to maintain constant average pulse energy, average-power and peak-power constraints should also be applied when comparing single-pulse and multipulse PPM signals. The performance of single-pulse (conventional or compound) PPM can be compared with multipulse PPM symbols in several ways, namely, under a bandwidth constraint or under an

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information-throughput constraint: each of these can further be divided into average-power or peak-power constrained signals. For the purpose of numerical comparison, we consider only the . To facilitate the notation associated with the varcase of ious comparisons, consistent with what was done previously in and to this paper, we shall assign the superscripts the parameters associated with dual-pulse, compound-2 single pulse, and conventional PPM modulations, respectively. A. Peak Power Constraint The first set of comparisons to be made assumes that we impose a peak-power constraint on the signal, i.e., . The peak-power constraint is particularly important in deep-space optical communications applications, where narrow high-peak-power transmitted pulses offer significant advantages in terms of detection probabilities and background suppression capabilities at the receiver. Therefore, operation with a peak-power constraint below the damage threshold is required, and, hence, modulation formats capable of delivering high information throughput under a peak-power constraint are highly desirable in deep-space applications. 1) Case of Equal Bandwidths: Requiring that all three schemes have equal bandwidths is equivalent to requiring that they have identical slot durations, i.e., or, since the symbol time is fixed, the equivalent relation . Combining this with the peak-power constraint of above is tantamount to setting . Also, since the background radiation intensity is fixed at , then we also have . A pictorial representation of the , modulations for this case is illustrated in Fig. 1 for which depicts conventional single-pulse PPM, compound-2 single pulse PPM formed from two single-pulse PPM symbols , and a typical two-pulse PPM each of dimension symbol, all chosen from constellations using a total of 16 equal duration slots. The SEP of the three schemes is plotted (peak power normalized by the in Fig. 2 versus 16 slot time or, equivalently, the energy per pulse) for 0, 0.1, and 0.5. Solid curves refer to dual-pulse and PPM, dotted curves to conventional single-pulse PPM, and the dashed curves refer to the corresponding compound symbol formed from the pair of single-pulse PPM symbols of half the dimension. The expressions used to arrive at these plots and (11), (18), and (31) are given by (16)–(18) for for . We observe that, consistent with the previous discussion in Section IV, dual-pulse and compound-2 pulse PPMs both perform worse than conventional single-pulse PPM, dual-pulse PPM being approximately a factor of two worse in SEP. Again the reader is reminded of the fact that, with an only equal to 16, the equal bandwidth constraint, even for information throughput of dual-pulse PPM is still somewhat higher than that of conventional or even compound-2 PPM. In particular, the number of bits per symbol for the three cases , , and , are which represents a valuable tradeoff against the relatively small penalty in symbol probability performance.

Fig. 1. Illustration of three different PPM modulation schemes with equal bandwidths and peak power levels; = 16.

M

Fig. 2. Average SEP versus normalized peak power for conventional single-, compound single-, and dual-pulse PPM with normalized background power as a parameter; equal bandwidths.

We observe here that, while conventional single-pulse PPM and compound-2 PPM both result in an integral number of bits per symbol, dual-pulse PPM does not. However, as suggested previously, several dual-pulse PPM symbols can be concatenated to form compound symbols carrying close to (but slightly more than) an integral number of information bits. For example, a concatenation of two symbols yields 13.81 bits, four symbols 27.63 bits, and 8 symbols carry 55.26 bits, of information. Therefore, blocks of 55 information bits can be efficiently mapped into eight consecutive 16-dimensional dual-pulse PPM symbols, with only an insignificant number of compound symbols remaining unused.

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It is interesting to observe that, if all of the compound symbols were used, then optimum decoding would correspond to making individual decisions on each two-pulse PPM symbol, as shown in (10), hence both the decoding strategy and the symbol-error performance remain exactly the same as for the individual dual-pulse PPM symbols derived above. When a small number of compound symbols remain unused, this interpretation is no longer strictly correct, and a slight improvement in performance could then be obtained by computing the LLF for each compound symbol, at the expense of much more complicated processing due to the much greater number of LLFs that have to be computed and compared. If each compound symbol was detected according to the ML rule, this would require the computation of the LLF for every possible compound symbol and selecting the compound symbol with the greatest metric, resulting in somewhat improved performance over symbol-bysymbol detection. In this case, the results obtained here for dualpulse PPM symbols should be viewed as an upper bound on the optimum compound symbol performance; however, the bound becomes arbitrarily tight as the number of unused compound symbols decreases. The analysis required for determining the performance of the optimum compound-PPM symbols when not every symbol is employed is beyond the scope of this paper. An alternative way of characterizing the performance for this case, which also takes into account the difference in information throughputs of the three schemes, is to consider a plot of bit-error probability (BEP) versus peak power (normalized by where, as before, dethe slot time) per bit, i.e., notes the bits/symbol for each scheme, as given in Section II. The conversion of the axis (ordinate) of Fig. 2 from symbolto bit-error probability is, in principle, dependent on the modulation scheme. Since conventional single-pulse PPM is truly an orthogonal modulation scheme, then one can apply the wellknown relation between symbol- and bit-error probability for such a scheme, namely [6, ch. 4, eq. (4.96)] (32) Although compound-2 single-pulse PPM and dual-pulse PPM are not truly orthogonal modulation schemes, for large , the number of symbols that are not orthogonal to a given member of the signal constellation becomes quite small relative to the number that are indeed orthogonal. Thus, to a first-order approximation, one can apply the same relation to these two modulation forms, accepting as well the fact that the number of bits per symbol may not always be integral. Thus, for compound-2 single-pulse PPM, we have

(33)

Fig. 3. Average BEP versus normalized peak power per bit for conventional single-, compound single-, and dual-pulse PPM with normalized background power as a parameter; equal bandwidths.

For the case at hand, , and for thus these conversions become: for conventional single-pulse PPM, compound-2 single-pulse PPM, and for dual-pulse PPM, all of which are close to . The resulting plot is illustrated in Fig. 3. Here we see that dual-pulse PPM is the best performer of the three, particularly when compared to conventional single-pulse -PPM. Note that the BEP of compound-2 -PPM is identical to that -PPM, as explained in footof conventional single-pulse note 3, and thus the exact expression for this probability could replaced by combined with be obtained from (11) with (32). 2) Case of Equal Information Throughputs per Symbol: Suppose now that we compare the three schemes based on equal information throughputs as characterized by (5) but still maintaining the same peak-power constraint. Then, if the symbol time is fixed, by necessity the bandwidths of the three schemes will be different. In particular, relative to the slot , we have width of single-pulse conventional PPM

(35) or, equivalently, in terms of the normalized average signal energy of the single-pulse PPM scheme,

(36) That is, the probability distributions characterizing the signal photon counts of the three different modulations schemes now have different Poisson parameters. Similarly, because of the unequal slot widths, we also have

whereas for dual-pulse PPM we have

(34)

(37) A comparison of the SEP performances of the three different cases under a peak-power constraint and equal information throughputs is illustrated in Fig. 4 for dual-pulse PPM with

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SIMON AND VILNROTTER: DUAL-PULSE PPM ON OPTICAL COMMUNICATION CHANNELS

Fig. 4. Average SEP versus normalized peak power for conventional single-, compound single-, and dual-pulse PPM with normalized background power as a parameter; equal information throughputs.

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Fig. 5. Average SEP versus average energy per symbol for dual-pulse PPM with normalized background power as a parameter; equal bandwidths.

which, from the relation in (5), results in for compound-2 single-pulse PPM and for conventional single-pulse PPM. The curves in the plot depict (the energy per pulse average SEP versus of conventional single-pulse PPM) with as a parameter. From the results in this figure, we see the dramatic improvement in error-probability performance achieved by dual-pulse PPM, as compared with the other two schemes over and above the bandwidth advantage described by (35). B. Average-Power Constraint For the second set of comparisons, we impose an average-power constraint on the signal. Unlike the peak-power constraint examined above, an average-power constraint is usually not imposed artificially to avoid damage, but rather is a natural consequence of the limited power resources available on the spacecraft. Therefore, the average-power constraint is a fundamental limit related to the conservation of energy and imposes a bound on the number of photons the transmitter can deliver per unit time. As before, modulation formats capable of maintaining high data throughput at the receiver under an average-power constraint, and at the required fidelity, are highly desirable. 1) Case of Equal Bandwidths: If we require that all three schemes have equal bandwidths, i.e., equal slot durations, then, because compound-2 single-pulse and dual-pulse PPM schemes each have two slots filled per symbol, whereas the signals in conventional single-pulse PPM occupy only a single slot per symbol. In this case we have or, equivalently, . Thus, for an averagepower constraint and equal transmission bandwidths, conventional single-pulse PPM requires twice the peak power of the other two schemes. Fig. 5 shows a plot analogous to Fig. 2 for the average-power-constraint case where the abscissa now corresponds to the average energy per symbol . Since the Poisson distribution depends only on the average photon count parameter , we observe, as expected, that now conventional single-pulse PPM has a greater advantage than before, at the expense of a doubled peak power.

Fig. 6. Average BEP versus normalized average power per bit for conventional single-, compound single-, and dual-pulse PPM with normalized background power as a parameter; equal bandwidths.

Once again, to take into account the difference in information throughputs of the three schemes, we proceed analogous to Fig. 3 and consider a plot of BEP versus average energy per bit . The relations between bit- and symbol-error probabilities are still given by (32)–(34), but now, as was the case in Fig. 5, the effective for the Poisson distribution of each of the two-pulse schemes is half that of the conventional single-pulse PPM scheme. The results are illustrated in Fig. 6. Here we see that dual-pulse PPM once again outperforms compound-2 single-pulse PPM but is still inferior to conventional single-pulse PPM. We reiterate the fact that the BEP of compound-2 -PPM is identical to that of conventional singlepulse -PPM as explained in footnote 3. 2) Case of Equal Information Throughputs per Symbol: Next, we compare the three schemes based on equal information throughputs as characterized by (5) but still maintaining the average power constraint. In addition, in view of the peak power inequality between single- and dual-pulse schemes, instead of (36), we now have (38) Since the background radiation level is not affected by the nature (peak or average) of the power constraint imposed on the signal, the relation in (37) still applies here. Fig. 7 is the analogous plot

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Fig. 7. Average SEP versus average energy per symbol for conventional single-, compound single-, and dual-pulse PPM with normalized background power as a parameter; equal information throughputs.

to Fig. 4 for the average power constraint case. Here we see that, despite the bandwidth advantage of the two-pulse PPM schemes compared to that of conventional single-pulse PPM, the loss of 3 dB in the average signal photon count for the former compared to the latter, as predicted by (38), dominates the differences in their relative performance. To take into account the differences in the bandwidth of the three schemes, we modify the comparison to allow for equal , which is average symbol energy per slot-time duration, tantamount to assuming . Equivalently, the relationship among the average photon counts results in a combination of (36) and (38), namely

(39) Once again, the average photon count for the background noise in the three cases satisfies (37). Based on the above, we consider a plot of average SEP versus average symbol energy per slot-time duration where for convenience we fix the slot time of the conventional single-pulse PPM case, . Such a comparison is illustrated in Fig. 8 where is plotted versus for a typical value of . Here we see the superior performance of dual-pulse PPM over the other two schemes brought about by the bandwidth saving. As before, one could alternatively characterize the performance in terms of a plot of BEP versus average energy (photons) per bit. Since, in the equal throughput case, the number of bits/symbol is the same for all three modulation schemes, e.g., , then the axis (abscissa) would merely scale by this factor. The conversion of the axis (ordinate) from symbol- to bit-error probability would follow the relations in (32)–(34). For , as is the case in Fig. 8, these conversions become: for conventional single-pulse PPM, for compound-2 single-pulse PPM, and for dual-pulse PPM all of which are very close to . Thus, in conclusion, a plot of bit error probability versus average energy per slot time per bit would resemble Fig. 8 where the axis would be scaled

Fig. 8. Average SEP versus average symbol energy per slot time for conventional single-, compound single-, and dual-pulse PPM with normalized background power as a parameter; equal information throughputs.

(divided) by a factor of a factor of 1/2.

and the axis by approximately

VI. CONCLUSION Higher order PPM signaling has been examined and evaluated in terms of information throughput, bandwidth requirements, and error performance, under both peak and average power constraints. It was demonstrated that higher order PPM, where multiple pulses are used per channel symbol, is a viable solution to bandwidth- and power-constrained optical communications, enabling higher data rates at a given bandwidth, without sacrificing BER performance. ML decision strategies were developed and exact performance expressions for two-pulse PPM were derived. Two different forms of multipulse PPM signals were examined, one that is a direct extension of conventional single-pulse PPM, and which could therefore be substituted directly into existing single-pulse systems to enhance performance without the need for extensive redesign, and a more general form that employs a new detection strategy that is somewhat more complicated, but not prohibitively so. Both of these multipulse signaling formats, when operating with equal throughput/symbol and equal symbol rates, were shown to significantly outperform conventional single-pulse PPM when operating under a peak power constraint. REFERENCES [1] R. M. Gagliardi and S. Karp, Optical Communications. New York: Wiley, 1976. [2] J. R. Pierce, “Optical channels: practical limits with photon counting,” IEEE Trans. Commun., vol. COM-26, pp. 1819–1821, Dec. 1978. [3] H. Sugiyama and K. Nosu, “MPPM: a method of improving the bandutilization efficiency in optical PPM,” J. Lightwave Technol., vol. 7, pp. 465–472, Mar. 1989. [4] V. A. Vilnrotter and M. Srinivasan, “Adaptive detector arrays for optical communications receivers,” IEEE Trans. Commun., vol. 50, pp. 1091–1097, July 2002. [5] C. Georghiades, “Modulation and coding for throughput-efficient optical systems,” IEEE Trans. Inform. Theory, vol. 40, pp. 1313–1326, Sept. 1994. [6] J. Hamkins and B. Moision, “Multi-pulse PPM on memoryless channels,” in Proc. Int. Symp. Information Theory, Chicago, IL, June–July 2004, p. 336. [7] M. K. Simon, S. M. Hinedi, and W. C. Lindsey, Digital Communication Techniques: Signal Design and Detection. Upper Saddle River, NJ: Prentice-Hall, 1995.

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Marvin K. Simon (S’60–M’66–SM’75–F’78) is currently a Principal Scientist at the Jet Propulsion Laboratory (JPL), California Institute of Technology (Caltech), Pasadena, where for the last 36 years he has performed research as applied to the design of NASA’s deep-space and near-earth missions, and which has resulted in the issuance of nine U.S. patents, 25 NASA Tech Briefs, and four NASA Space Act awards. He is known as an internationally acclaimed authority on the subject of digital communications with particular emphasis in the disciplines of modulation and demodulation, synchronization techniques for space, satellite, and radio communications, trellis-coded modulation, spread spectrum and multiple access communications, and communication over fading channels. He has also held a joint appointment with the Electrical Engineering Department at Caltech. He has published over 195 papers on the above subjects and is coauthor of 10 textbooks, including Telecommunication Systems Engineering (Englewood Cliffs, NJ: Prentice-Hall, 1973, and New York: Dover Press, 1991), Phase-Locked Loops and Their Application (New York: IEEE Press, 1978), Spread Spectrum Communications, Vols. I, II, and III (Computer Science Press, 1984 and New York: McGraw-Hill, 1994), An Introduction to Trellis Coded Modulation with Applications (MacMillan, 1991), Digital Communication Techniques: Vol. I (Englewood Cliffs, NJ: Prentice-Hall, 1994) and Digital Communication Over Fading Channels: A Unified Approach to Performance Analysis (New York: Wiley, 2000, 2nd ed. available Nov. 2004), Probability Distributions Involving Gaussian Random Variables—A Handbook for Engineers and Scientists (Norwell, MA: Kluwer, 2002) and Bandwidth-Efficient Digital Modulation with Application to Deep-Space Communication (New York: Wiley, 2003). His work has also appeared in the textbook Deep Space Telecommunication Systems Engineering (New York: Plenum, 1984), and he is coauthor of a chapter entitled “Spread Spectrum Communications” in the Mobile Communications Handbook (Boca Raton, FL: CRC Press, 1995), Communications Handbook (Boca Raton, FL: CRC Press, 1997), and the Electrical Engineering Handbook (Boca Raton, FL: CRC Press, 1997). His work has also appeared in the textbook Deep Space Telecommunication Systems Engineering (New York: Plenum, 1984). Dr. Simon is the corecipient of the 1986 Prize Paper Award in Communications of the IEEE Vehicular Technology Society and the 1999 Prize Paper Award of the IEEE Vehicular Technology Conference (VTC’99-Fall), Amsterdam, The Netherlands. He is a Fellow of the IAE, and winner of a NASA Exceptional Service Medal, a NASA Exception Engineering Achievement Medal, the IEEE Edwin H. Armstrong Achievement Award, and most recently, the IEEE Millennium Medal.

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Victor A. Vilnrotter (M’79–SM’02) received the Ph.D. degree in electrical engineering and communications theory from the University of Southern California, Los Angeles, in 1978. He joined the Jet Propulsion Laboratory (JPL), California Institute of Technology, Pasadena, in 1979, where he is a Senior Engineer with the Digital Signal Processing Research Group. He has conducted research on various topics in deep-space communications, including real-time electronic compensation for gravity- and wind-induced deformation of large antennas with focal-plane arrays, adaptive algorithms for optimum combining and tracking of spacecraft with large arrays, improved optical communications through atmospheric turbulence, and the application of quantum communications theory to deep-space optical communications. He has written or coauthored over 100 papers in conferences, JPL publications, and refereed journals. Dr. Vilnrotter was the recipient of numerous NASA awards for technical innovations.

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Accurate Simulation of Multiple Cross-Correlated Rician Fading Channels Kareem E. Baddour, Student Member, IEEE, and Norman C. Beaulieu, Fellow, IEEE

Abstract—The computer generation of multiple cross-correlated Rician fading channels is investigated. We prove that the output sequences of existing multichannel fading simulators are restricted to have cross-correlation statistics that have the same functional form as the component autocorrelation functions. To overcome this limitation, vector autoregressive stochastic models are proposed for the generation of multiple Rician fading processes with specified realizable autocorrelation and cross-correlation statistics. This capability is desirable, for example, to permit realistic performance assessments of space–time modem designs by enabling the simulation of space–time-selective wireless channel models. The utility of the simulation approach is demonstrated by the accurate synthesis of some bandlimited multichannel Rayleigh and Rician processes. Index Terms—Autoregressive (AR) processes, band-limited stochastic processes, cross-correlation, fading channels, Rician channels, simulation.

I. INTRODUCTION OMPUTER simulation of cross-correlated fading processes has become an important research topic due to the increased interest in using antenna arrays, both at the transmitter and at the receiver, to improve cellular radio communications. Simulators which can accurately capture the characteristics of correlated diversity channels are needed to enable realistic performance assessments of multiple-antenna systems. The simulation of narrow-band vector fading channels, in particular, requires the generation of cross-correlated Rayleigh and Rician sequences. Typically, the sequences must have specified autocorrelation and cross-correlation statistics. Since the desired fading coefficients are complex Gaussian variates, they can be generated in principle by factorization of the desired correlation matrix, followed by linear transformation of sequences of uncorrelated variates [1, pp. 254–256]. While this direct approach is valid in principle, it cannot be applied to the correlation matrices of interest in wireless communications problems. The

C

Paper approved by R. A. Valenzuela, the Editor for Transmission Systems of the IEEE Communications Society. Manuscript received August 3, 2003; revised March 16, 2004. This work was supported in part by a postgraduate scholarship from the Natural Sciences and Engineering Research Council of Canada (NSERC), in part by an Industry Canada Fessenden Postgraduate Scholarship, in part by an Ontario Graduate Scholarship in Science and Technology, and in part by the Alberta Informatics Circle of Research Excellence (iCORE). This paper was presented in part at the 2002 IEEE International Conference on Communications, New York, NY, May 2002. K. E. Baddour is with the Department of Electrical and Computer Engineering, Queen’s University, Kingston, ON K7L 3N6 (e-mail: baddourk@ ee.queensu.ca). N. C. Beaulieu is with the Department of Electrical and Computer Engineering, University of Alberta, Edmonton, AB T6G 2V4 (e-mail: [email protected]). Digital Object Identifier 10.1109/TCOMM.2004.836559

reason is that the matrices involved are very nearly singular and the factorization cannot be done in a straightforward manner. The expensive computational requirements of the direct method also makes it impractical to implement. Special techniques must, therefore, be employed as evidenced by the large body of work on practical simulator designs [2]–[13]. Caution, however, must be exercised in using more efficient, simplified simulator models. References [11]–[14] discuss limitations and shortcomings of some simplified simulators. In particular, an advantage of the autoregressive (AR) approach advocated in this paper is that the simulator output is wide-sense stationary (WSS), autocorrelation-ergodic, and possesses a Rayleigh or Rician distribution. Recently, several authors have published efficient methods for generating two [3], [4] or any number [5]–[9] of cross-correlated Rayleigh fading envelopes. There is a crucial difference between their approach and the one presented in this paper. In these approaches, independent fading processes with desired autocorrelations are first generated and then multiplied by a coloring matrix. We prove in this paper, that the generated sequences in these cases are restricted to have cross-correlation functions (CCFs) that have the same time-dependencies as the autocorrelation functions (ACFs). Meanwhile, a realizable CCF need not necessarily have this form. In general, the cross-correlation behavior of a vector fading channel is a function of the mean direction of arrival of the received signal, the angle spread, and the antenna array geometry. For some space–time-selective fading models [15], the temporal correlation and spatial correlation statistics are separable and have the same functional form. In these cases, application of the methods in [5]–[9] is highly motivated. However, for the synthesis of more general models with general joint space–time cross-correlation statistics (e.g., [16]–[18]), a more comprehensive variate generation method is required. In this paper, a multichannel generalization of the AR variate generation method [11] is proposed to simulate vector Rayleigh and Rician fading processes that possess specified ACFs and CCFs. As well as being useful for simulating general space–time-selective fading models, this capability is also desirable, for example, to simulate correlated tap delay line models of frequency-selective fading channels [19]. The technique can also be used to synthesize a more general class of complex Gaussian fading channels with correlated quadrature components, which is useful for simulating nonisotropic fading models. The remainder of this paper is organized as follows. Section II defines the problem in more detail. Section III describes the vector AR method of generating multiple Rayleigh processes

0090-6778/04$20.00 © 2004 IEEE

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with specified second-order statistics. Our treatment briefly discusses the numerical problems that must be overcome for the accurate simulation of band-limited fading processes, which are of interest in wireless applications. We also describe an extension which allows the AR fading generator to control quadrature component correlations. In Section IV, the utility of the vector AR approach is demonstrated by the accurate synthesis of several band-limited vector fading processes. Section V concludes the paper. II. VECTOR FADING SIMULATION The problem at hand is to generate a vector of complex Gaussian fading processes

stationary

1981

crossIn principle, exact generation of time samples of correlated Gaussian sequences can be accomplished by a linear uncorrelated (zero cross-correlation transformation of and zero autocorrelation) samples. Factorization of an correlation matrix, which can be accomplished in the genoperations using Cholesky reduction, is eral case in required to obtain the coloring matrix. This expensive computational burden makes implementation of the method impractical. Efficient methods of generating cross-correlated Rayleigh fading envelopes have recently been published [5]–[9]. In these approaches, independent Gaussian sequences are first colored in time and then between sequences to achieve a complexity savings. While any ACF can be specified, typically the fading sequences are modeled to have the normalized real autocorrelation (3)

for successive time instants with a specified covariance matrix

.. .

.. .

..

.

.. .

for

, where denotes expectation and indicates Hermitian transpose. In this paper, we focus on the generation of vector Rayleigh processes for simplicity of exposition. In this case, the in-phase and quadrature components of each Gaussian process must have zero mean to represent Rayleigh fading. The extension required for the generation of multiple Rician fading processes is straightforward. Rician channels can be generated as (1) where is the zero-mean complex Gaussian process representing the scattering/diffuse Rayleigh component, denotes the specular or line-of-sight (LOS) component, and represents the ratio of the LOS power to the scattered power (Rice factor) for the th channel. The specular compois typically modeled as a deterministic complex nent exponential [20] (2) is the maximum Doppler frequency normalized by where the sampling rate, and and represent the angle of arrival and the initial phase of the LOS component, respectively. An alternative Rician model suggested in [12] uses a uniformly distributed random initial LOS phase, in which case the specular component becomes a stochastic sinusoid. For notational simrepresents a Rayleigh fading process plicity, in the sequel, unless otherwise noted. An example in Section IV will illustrate the extension to Rician simulation.

is the zeroth-order Bessel function of the first kind where [21] and is the maximum Doppler frequency normalized by the sampling rate. This ACF, which corresponds to the wellknown U-shaped band-limited power spectral density [2], results from the assumption that the propagation environment is two-dimensional (2-D) with isotropic scattering. Of great importance, we have discovered that while the cross-correlation coefficients (zeroth lag of the CCFs) can be set arbitrarily by the methods of [5]–[9], the CCFs generated by these methods are restricted to have the same shape as the ACFs. For example, if the specified autocorrelations are given by (3), then the generated sequences can only have the CCFs (4) where is the cross-correlation coefficient between processes and . We give a proof of this, perhaps, unexpected fact in Appendix A. However, a realizable CCF in this case need not necessarily have this form and, further, does not have this form in some practical cases (see [16]–[18]). Thus, the methods of [5]–[9] have limited applicability. In the next section, a vector AR modeling approach is proposed to accurately synthesize any stationary multichannel Rayleigh fading process. III. AR GENERATION OF VECTOR RAYLEIGH PROCESSES In this section, we describe the vector AR methodology for generating multichannel Rayleigh processes. For simplicity, we focus first on the synthesis of vector Rayleigh processes with uncorrelated quadrature components. However, the modification required to synthesize unconstrained cross-channel quadrature components is not difficult and is summarized later for completeness. A. Uncorrelated Quadrature Components By adopting a vector AR model, the correlated complex Gaussian processes are assumed to evolve according to the th-order autoregression (5)

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where

.. .

.. .

..

.. .

.

are matrices containing the multichannel AR model coefficients. The driving noise process, , is a complex white vector Gaussian sequence with zero mean and covariance matrix . That is, is uncorrelated in time but not necessarily across sequences. For convenience, the observation matrix

is defined. The model covariance matrix, which is defined as , is a block matrix of dimension . To correspond to a realizable multichannel WSS random must be positive semidefinite and have a block process, Toeplitz structure [22]. The relationship between the desired and the vector AR model parameters is given by the multichannel Yule–Walker equations, which are [22]

.. .

.. .

..

.

.. .

.. .

.. .

(6)

where the submatrices in are the Toeplitz ma. Note that trices , but the are not, in general, Hermitian. The system of equations in (6) can be solved efficiently for using the Levinson–Wiggins–Robinson (LWR) algorithm [22], which exploits the block Toeplitz structure of . coefficient matrices have been determined, the Once the covariance matrix of the driving noise vector process can be computed using [22] (7) or obtained directly from the LWR algorithm. After obtaining , the realization of the driving noise process in (5) can be accomplished by first computing the factorization . The Cholesky decomposition can be used here. The driving , where is process is then generated by the product an vector of independent zero-mean complex Gaussian variates with unit variance.

We note that the application of vector AR time series to fading channel modeling was briefly considered in [23] to generate cross-correlated fading processes and in [24] and [25] to model the fading channel dynamics for the purposes of Kalman-filter-based channel estimation. In these papers, simple first- and second-order vector AR processes were proposed. For example, in [23], vector AR(2) time series were used and a simple exponential (nonband-limited) function was adopted to model the ACF and CCF for each generated sequence and between sequences, respectively. While simple AR(1) or AR(2) models are adequate to model exponential correlations [26], higher order models are required to accurately synthesize band-limited processes [11]. However, the multichannel Yule–Walker equations in the band-limited spectrum cases are for all but very plagued by a numerically ill-conditioned small AR orders. This phenomenon arises due to the highly deterministic nature of band-limited processes (see [11] for more details). As for the single-channel case, a spectral bias must be added as explained in [11] for the vector AR simulation technique to work. That is, the numerical problems that arise in these cases can be resolved by approximating any band-limited processes to be generated with nondeterministic processes by adding a very small positive bias to the zeroth lag of their is ignored, corresponding ACFs. If the ill-conditioning in the LWR algorithm typically produces a meaningless solution that is not realizable or with either a covariance matrix a multichannel infinite-impulse response (IIR) filter that is unstable. To obtain guidelines for the selection of , it proves useful to observe the numerical problems due to the ill-conditioning . If the eigenvalues of are computed (e.g., using of MATLAB), several very small negative eigenvalues will typically be found even if the underlying system is a positive semidefinite system corresponding to a realizable multichannel WSS band-limited process. A suitable bias has been found to , for be one which results in a well-conditioned matrix which the computed eigenvalues are all positive and with the smallest eigenvalue approximately equal to , and where the matrix denotes the identity matrix. The minimum bias which satisfies these conditions typically varies depending and the chosen model parameters. on the size of The above technique is suitable for ill-conditioned positive semidefinite matrices and has been found to result in biases that are insignificant when compared to the output variance [10]. We caution, however, that the method is not applicable for nonmatrices. For example, realizable, nonpositive semidefinite if empirical measurements are made to determine the simulation is not positive semidefinite, a parameters and the resulting large bias may have to be added using the proposed approach to obtain a stable multichannel filter. For such cases, which are identifiable by an with computed negative eigenvalues that are significant, we recommend the use of techniques which apmatrix with the nearest realizable proximate the desired positive semidefinite matrix.1 A recent conference paper [27] 1Although classification between an ill-conditioned positive semidefinite matrix and a nonpositive semidefinite matrix based on the sign of computed eigenvalues appears to be an imprecise task, our experience suggests that an insignificant negative eigenvalue is one that is more than three orders of magnitude smaller than the largest eigenvalue [10].

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addresses this issue and proposes a stable approximation technique, which involves finding the closest positive semidefinite matrix in a Frobenius sense. Additional details can be found in [27]. B. Correlated Quadrature Components In this section, we briefly describe an extension which allows the vector AR generator to handle correlations between the quadrature (real and imaginary) components on each channel and between channels. This capability enables the simulation of a more general class of multichannel fading models, including those with nonisotropic scattering (e.g., see [16]). The technique involves the specification of temporal, cross-channel, and quadrature component correlations directly within the multichannel Yule–Walker equations. Following [28], this can be accomplished as follows. First, consider the real, concatenated vector of in-phase and quadrature fading coefficients

Fig. 1. Envelope sample sequences of three cross-correlated Rayleigh fading channels.

covariance matrix of the driving noise vector The process can then be computed using

where

and

(9)

denote the real and imaginary components of correlation matrix for is The

The submatrices in

are the

, respectively.

correlation matrices

which contain the desired quadrature correlations. The real, concatenated quadrature form of the multichannel Yule–Walker equations can, thus, be expressed as

.. .

.. .

..

.

.. .

.. .

.. . In this case, we must solve for the matrices , where

(8)

AR coefficient

A Cholesky decomposition of provides the real matrix , which serves the same role as in the previous secis able to control not only the tion. In this case, however, cross-channel correlations, but also the quadrature correlations when multiplied by a vector of uncorrelated Gaussian variates. IV. SIMULATION EXAMPLES In this section, we show how to use the vector AR method to generate multiple accurately correlated, band-limited Rayleigh processes. For simplicity, we first consider the generation of Rayleigh channels with identical ACFs and CCFs given in (3) and (4) with The cross-correlation coefficients , , and cannot be arbitrarily assigned, but must correspond to a positive semidefinite correlation matrix (see [29] for characterization of allowable values for this case). Realizable , , choices are the real-valued coefficients [29]. A vector AR(80) model was chosen to and synthesize the processes and the spectral bias was added to , to well condition the correlation matrix. The output Rayleigh envelopes are shown in Fig. 1. The corresponding phase sequences are presented in Fig. 2, demonstrating that the generated deep amplitude fades have corresponding “hits” in the phase. This matches real world behavior, which is of interest to some applications such as fading channel equalizer design [30]. The empirical autocorrelations output samples, are and cross-correlations, computed over plotted in Fig. 3, and these are found to provide very accurate approximations to the desired statistics.

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Fig. 2. Phase sample sequences corresponding to the cross-correlated sequences of Fig. 1.

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Fig. 4. Empirical autocorrelations corresponding to the 2 synthesized using a four-channel AR(40) model.

2 2 MIMO example,

In the sequel, we will use the multiple-input multiple-output (MIMO) space–time correlation model of [16], which includes many known correlated fading models as special cases. Consider the 2 2 fading channel scenario involving transmission by a dual-base-station-antenna array and reception by a mobile user with a dual-antenna array. Under the assumption of 2-D isotropic scattering at the receiver antennas, the ACF for each channel is identically given by (3). Following the MIMO correlation model in [16, eq. (12)], the sampled CCFs can be expressed in terms of the array configurations and Doppler spread as

(10a)

Fig. 3. Empirical ACFs and CCFs corresponding to the sequences in Fig. 1.

We note that the preceding simple example can be synthesized more efficiently by first generating each channel separately and then coloring the sequences using a simple linear transformation as in [5]. In general, the accuracy is dependent on which technique is chosen to obtain independent temporally correlated fading sequences prior to introducing the cross-correlations. The accuracy of our new approach, in this special case, is not better than the inverse discrete Fourier transform (IDFT) advocated in [5], but is arguably better than sum-ofsinusoids approaches. We refer the reader to [11] for a related discussion on the relative accuracies of the different techniques. However, such efficient methods have limited applicability in that they can only produce CCFs which have the same time-dependency as the ACFs. As a demonstration of the generality of the AR approach, we consider the generation of four Rayleigh channels with CCF shapes that are different than the ACFs. Although characterizing the class of realizable CCFs appears to be very difficult, several joint space–time cross-correlation models have been recently proposed in the literature [16]–[18].

(10b)

(10c) where , , , denotes the distance between the receiver antennas, denotes the separation between base antennas, specifies the beamwidth at the base, and , , and are angles which specify the orientations of the base station and mobile arrays and the direction of user motion, respectively (see [16] for geometrical details). A four-channel was used to synAR(40) model with spectral bias thesize an accurate approximation to this vector process for the , , , , scenario with (parallel arrays), and . The empirical autocorrelations and cross-correlations, computed over variates, are plotted in Figs. 4 and 5, respectively. These are found to provide a good match to the desired model statistics, thus demonstrating the AR simulator’s ability to synthesize multiple Rayleigh processes with arbitrary CCF shapes.

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Fig. 5. Empirical CCFs corresponding to the four-channel AR(40) MIMO synthesis example.

Fig. 6. Empirical autocorrelations of the corresponding to the Rician synthesis example.

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quadrature

components

Next, we consider a dual diversity scenario with one transmit antenna and two receive antennas and with both nonisotropic scattering and Rician fading at the mobile receiver. Following the space–time fading model in [16, eq. (12)], which assumes a parametric Von Mises distribution function for the scattering angle of arrival (AOA) at each antenna [31], the sampled ACF and CCF are given as the sum of a diffuse/scattering component and a LOS component (11a) (11b) with

(12a) Fig. 7. Empirical cross-correlations of the quadrature components corresponding to the Rician synthesis example.

(12b) (12c) (12d) where controls the width of the scattering AOA, represents the mean direction of the scattering AOA, and where, for simplicity, we have assumed that the Rice factors of both channels are identical and denoted by . For more details regarding this model, including an empirical justification of the Von Mises AOA distribution, the reader may refer to [16] and [31]. The model corresponds to isotropic scattering scenarios when , in which case and are real-valued. For such cases, a two-channel vector AR model can be directly used as described in Section III to synthesize a multichannel zero-mean complex Gaussian process which approximates the desired statistics in (12a) and (12b). For general nonisotropic , and are scattering scenarios with complex-valued and the concatenated vector AR methodology of Section IV must be used in order that both the correlations

between the in-phase and quadrature scattering components of each fading channel and between channels can be correctly specified. For Rician channels, either a deterministic [16] or a random [12] complex exponential representing the LOS component is added to each complex Gaussian AR random process to complete the fading simulation. In either case, the LOS component contributes (12c) and (12d) to the output’s second-order statistics. As a synthesis example, a vector AR(40) model with spectral was used to simulate the dual-diversity model in bias , , (12a) and (12b) for the scenario with , , , and . Following [16], a deterministic Rician component as given by (2) . The empirical was added to both channels with sample autocorrelations and cross-correlations of the quadraoutput samples, are plotted ture components, averaged over in Figs. 6 and 7, respectively. The close match to the desired statistics demonstrates the simulator’s ability to synthesize vector

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Rician processes with arbitrary realizable correlation functions. As further demonstration of the capabilities of the simulator, we point the reader to [17], where the authors made use of our AR design as reported in [13] to investigate the outage capacity of a MIMO channel with receiver motion and nonisotropic scattering at both ends of the radio link.

The cross correlation between the fading sequences is then commonly introduced by means of a coloring matrix

V. CONCLUSION

with the multichannel simulator outputs at each time instant given by

The computer generation of multiple cross-correlated complex Gaussian fading channels was investigated in this paper. It was proved that the output sequences of previous multichannel fading simulators are restricted to have CCFs that have the same shape as the ACFs. To resolve this shortcoming, vector AR stochastic models were proposed as a general simulation methodology. The numerical difficulties faced by this approach were resolved by approximating the deterministic band-limited Doppler processes with regular processes. The method was demonstrated to be capable of accurately generating multiple band-limited Rayleigh and Rician WSS processes with specified realizable cross-correlation statistics. Such a capability cannot be provided by previous methods and is desirable, for example, to simulate space–time-selective Rician fading channel models.

.. .

.. .

..

.

.. .

(A2) where

See [3]–[9] for details regarding particular designs of the coloring matrix . If we consider the CCF of the output sequences, we find using (A1) and some algebra that

(A3) APPENDIX A FORM OF THE CCF OF THE OUTPUT OF MULTICHANNEL RAYLEIGH FADING SIMULATORS WHICH USE A COLORING MATRIX In this Appendix, we prove that the output sequences produced by previous multichannel fading simulators are restricted to have CCFs that have the same functional form as the ACFs. Let

be uncorrelated time sequences of complex Gaussian fading signal amplitude samples. That is, one has that the correlation matrix

(A1) where is the identity matrix and is the zero matrix. The sequences can have any specified autocorrelation , . For wireless simulations, the sequences typically have the same autocorrelation function . For example, the autocorrelation function is often given by , , which assumes a 2-D isotropic scattering environment. Rayleigh sequences with this ACF or a close approximation can be generated using various approaches such as the IDFT, sum of sinusoids, or white noise filtering methods [11].

. For the typical scenario when all chanwhere , nels have the same autocorrelation, the CCF simplifies to (A4) where the constants are , , and . Thus, the coloring matrix approach can only be used to generate multiple fading channels for which the CCFs have the same temporal dependencies as the component ACFs. REFERENCES [1] A. Leon-Garcia, Probability and Random Processes for Electrical Engineering. Reading, MA: Addison-Wesley, 1990. [2] W. C. Jakes, Ed., Microwave Mobile Communications. New York: Wiley, 1974. [3] R. B. Ertel and J. H. Reed, “Generation of two equal power correlated Rayleigh fading envelopes,” IEEE Commun. Lett., vol. 2, pp. 276–278, Oct. 1998. [4] N. C. Beaulieu, “Generation of correlated Rayleigh fading envelopes,” IEEE Commun. Lett., vol. 3, pp. 172–174, June 1999. [5] N. C. Beaulieu and M. Merani, “Generation of multiple Rayleigh fading sequences with specified cross-correlations,” Eur. Trans. Telecommun., vol. 15, pp. 471–476, Sept.-Oct. 2004. [6] B. Natarajan, C. Nassar, and V. Chandrasekhar, “Generation of correlated Rayleigh fading envelopes for spread spectrum applications,” IEEE Commun. Lett., vol. 4, pp. 9–11, Jan. 2000. [7] A. Hansson and T. Aulin, “Generation of correlated Rayleigh fading processes for the simulation of space-time-selective radio channels,” in Proc. European Wireless Conf., Munich, Germany, Oct. 1999, pp. 269–272. [8] S. Kim, J. Yoo, and H. Park, “A spatially and temporally correlated fading model for array antenna applications,” IEEE Trans. Veh. Technol., vol. 48, pp. 1899–1905, Nov. 1999.

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[9] J. Han, J. Yook, and H. Park, “A deterministic channel simulation model for spatially correlated Rayleigh fading,” IEEE Commun. Lett., vol. 6, pp. 58–60, Feb. 2002. [10] K. E. Baddour, “Simulation, estimation and prediction of flat fading wireless channels,” Ph.D. dissertation (in progress), Queen’s Univ., Kingston, ON, Canada. [11] K. E. Baddour and N. C. Beaulieu, “Autoregressive modeling for fading channel simulation,” IEEE Trans. Wireless Commun., to be published. [12] C. Xiao, Y. R. Zheng, and N. C. Beaulieu, “Statistical simulation models for Rayleigh and Rician fading,” in Proc. IEEE Int. Conf. Communication, Anchorage, AK, May 2003, pp. 3524–2529. [13] K. E. Baddour and N. C. Beaulieu, “Accurate simulation of multiple cross-correlated fading channels,” in Proc. IEEE Int. Conf. Communication, New York, May 2002, pp. 267–271. [14] M. F. Pop and N. C. Beaulieu, “Limitations of sum-of-sinusoids fading channel simulators,” IEEE Trans. Commun., vol. 49, pp. 699–708, Apr. 2001. [15] G. Raleigh, S. Diggavi, A. Naguib, and A. Paulraj, “Characterization of fast fading vector channels for multi-antenna communication systems,” in Proc. Asilomar Conf. Signals, Systems, and Computers, 1995, pp. 853–857. [16] A. Abdi and M. Kaveh, “A space-time correlation model for multielement antenna systems in mobile fading channels,” IEEE J. Select. Areas Commun., vol. 20, pp. 550–560, Apr. 2002. [17] G. Byers and F. Takawira, “Spatially and temporally correlated MIMO channels: modeling and capacity analysis,” IEEE Trans. Veh. Technol., vol. 53, pp. 634–643, May 2004. [18] T. A. Chen, M. P. Fitz, M. D. Zoltowski, and J. H. Grimm, “A spacetime model for frequency nonselective Rayleigh fading channels with applications to space-time modems,” IEEE J. Select. Areas Commun., vol. 18, pp. 1175–1190, July 2000. [19] S. Fechtel, “A novel approach to modeling and efficient simulation of frequency-selective fading radio channels,” IEEE J. Select. Areas Commun., vol. 11, pp. 422–431, Apr. 1993. [20] G. Stuber, Principles of Mobile Communication. Norwell, MA: Kluwer, 1996. [21] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions. New York: Dover, 1965. [22] S. M. Kay, Modern Spectral Estimation. Englewood Cliffs, NJ: Prentice-Hall, 1988. [23] S. Gordon and J. Ritcey, “Generating correlated Nakagami fading channels,” in Proc. Asilomar Conf. on Signals, Systems, and Computers, 1997, pp. 684–688. [24] M. Tsatsanis and Z. Xu, “Pilot symbol assisted modulation in frequency selective fading wireless channels,” IEEE Trans. Signal Processing, vol. 48, pp. 2353–2365, Aug. 2000. [25] C. Komninakis, C. Fragouli, A. Sayed, and R. Wesel, “Multi-input multi-output fading channel tracking and equalization using Kalman estimation,” IEEE Trans. Signal Processing, vol. 50, pp. 1065–1076, May 2002. [26] W. Wei, Time Series Analysis: Univariate and Multivariate Methods. New York: Addison-Wesley, 1990. [27] S. Sorooshyari and D. G. Daut, “Generation of correlated Rayleigh fading envelopes for accurate performance analysis of diversity systems,” in Proc. 14th IEEE Int. Symp. Personal, Indoor and Mobile Radio Communications, Beijing, China, Sept. 2003, pp. 1800–1804. [28] J. Michels, P. Varshney, and D. Weiner, “Synthesis of correlated multichannel random processes,” IEEE Trans. Signal Processing, vol. 42, pp. 367–375, Feb. 1994. [29] T. Hattori and K. Hirade, “Generation method of mutually correlated multipath fading waves,” Electron. Commun. Japan, vol. 59-B, pp. 69–76, 1976. [30] M. Fattouche and H. Zaghloul, “Equalization of =4 offset DQPSK transmitted over fast fading channels,” in Proc. 1992 IEEE Int. Conf. Communication (ICC ’92), Chicago, IL, June 1992, pp. 296–298.

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[31] A. Abdi, H. Barger, and M. Kaveh, “A parametric model for the distribution of the angle of arrival and the associated correlation function and power spectrum at the mobile station,” IEEE Trans. Veh. Technol., vol. 51, pp. 425–434, May 2002.

Kareem E. Baddour (S’93) received the B.Eng. degree in electrical engineering from Memorial University of Newfoundland, St. John’s, NL, Canada, in 1996, and the M.Sc.(Eng.) degree from Queen’s University, Kingston, ON, Canada, in 1998, where he is currently working toward the Ph.D. degree in the Department of Electrical and Computer Engineering. He is also affiliated with the iCORE Wireless Communications Laboratory at the University of Alberta, Edmonton, AB, Canada. Previously, he worked for Northern Telecom, Newbridge Networks, Microtel Pacific Research, as well as for the Canadian Coast Guard’s Telecommunications and Electronics Branch. His current research interests lie in the general area of wireless communications, with a focus on fading channel simulation, estimation, prediction, and equalization. Mr. Baddour was the recipient of numerous scholarships, including two Natural Sciences and Engineering Research Council of Canada (NSERC) Postgraduate Scholarships, two Industry Canada Fessenden Postgraduate Scholarships, and an Ontario Graduate Scholarship in Science and Technology. He was the recipient of the Memorial University Medal for Electrical Engineering in 1996.

Norman C. Beaulieu (S’82–M’86–SM’89–F’99) received the B.A.Sc. (honors), M.A.Sc., and Ph.D. degrees in electrical engineering from the University of British Columbia, Vancouver, BC, Canada, in 1980, 1983, and 1986, respectively. He was a Queen’s National Scholar Assistant Professor with the Department of Electrical Engineering, Queen’s University, Kingston, ON, Canada, from September 1986 to June 1988, an Associate Professor from July 1988 to June 1993, and a Professor from July 1993 to August 2000. In September 2000, he became the iCORE Research Chair in Broadband Wireless Communications at the University of Alberta, Edmonton, AB, Canada, and in January 2001, the Canada Research Chair in Broadband Wireless Communications. His current research interests include broadband digital communications systems, fading channel modeling and simulation, interference prediction and cancellation, decision-feedback equalization, and space–time coding. Dr. Beaulieu is a Member of the IEEE Communication Theory Committee and served as its Representative to the Technical Program Committee of the 1991 International Conference on Communications and as Co-Representative to the Technical Program Committee of the 1993 International Conference on Communications and the 1996 International Conference on Communications. He was General Chair of the Sixth Communication Theory Mini-Conference in association with GLOBECOM 97 and Co-Chair of the Canadian Workshop on Information Theory 1999. He has been an Editor for Wireless Communication Theory of the IEEE TRANSACTIONS ON COMMUNICATIONS since January 1992, and was Editor-in-Chief from January 2000 to December 2003. He served as an Associate Editor for Wireless Communication Theory of the IEEE Communications Letters from November 1996 to August 2003. He has also served on the Editorial Board of the Proceedings of the IEEE since November 2000. He received the Natural Science and Engineering Research Council of Canada (NSERC) E. W. R. Steacie Memorial Fellowship in 1999. He was awarded the University of British Columbia Special University Prize in Applied Science in 1980 as the highest standing graduate in the faculty of Applied Science. He is a Fellow of The Royal Society of Canada.

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OFDM Systems in the Presence of Phase Noise: Consequences and Solutions Songping Wu, Student Member, IEEE, and Yeheskel Bar-Ness, Fellow, IEEE

Abstract—In this paper, we provide an exact analysis of orthogonal frequency-division multiplexing (OFDM) performance in the presence of phase noise. Unlike most methods which assume small phase noise, we examine the general case for any phase noise levels. After deriving a closed-form expression for the signal-to-noise-plus-interference ratio (SINR), we exhibit the effects of phase noise by precisely expressing the OFDM system performance as a function of its critical parameters. This helps in understanding the meaning of small phase noise and how it reflects on the proper parameters selection of a specific OFDM system. In order to combat phase noise, we also provide in this paper a general phase-noise suppression scheme, which, by analytical and numerical results, proves to be quite effective in practice. Index Terms—Orthogonal frequency-division multiplexing (OFDM), phase noise.

I. INTRODUCTION AVING high spectral efficiency and being quite robust against intersymbol interference (ISI) caused by time-dispersive channels, orthogonal frequency-division multiplexing (OFDM) has been widely adopted and implemented in wire and wireless communications, such as digital terrestrial TV broadcasting (dTTB), digital subscriber line (DSL), European highperformance local area networks (HIPERLANs), and the IEEE 802.11a standard for wireless local area networks (WLANs) [1]–[4]. It is also very well suited to future high-data-rate wireless multimedia communications. Implementing multicarrier modulation with a discrete Fourier transform (DFT), OFDM is quite effective in eliminating channel multipath fading effects while having high-frequency efficiency. However, it is quite sensitive to frequency offset and phase noise. Caused by the frequency deviation between the transmitter and the receiver, or by Doppler shift, frequency offset has been thoroughly analyzed, and many methods have been proposed for its estimation and correction [5]–[8]. Unlike constant frequency offset, phase noise is a random process

H

Paper approved by Editor Y. Li, the Editor for Wireless Communication Theory of the IEEE Communications Society. Manuscript received January 22, 2003; revised October 22, 2003. This work was supported by National Science Foundation under Grant CCR-0085846. S. Wu was with the Center for Communications and Signal Processing Research (CCSPR), the Electronic and Computer Engineering Department, New Jersey Institute of Technology, Newark, NJ 07102 USA. He is now with Marvell Semiconductor Inc., Sunnyvale, CA 94087 USA. Y. Bar-Ness is with the Center for Communications and Signal Processing Research (CCSPR), the Electronic and Computer Engineering Department, New Jersey Institute of Technology, Newark, NJ 07102 USA (e-mail: barness@[email protected]). Digital Object Identifier 10.1109/TCOMM.2004.836441

caused by the fluctuation of the receiver and transmitter oscillators. In this paper, we will specifically concentrate on the random phase-noise problem. Phase noise causes leakage of DFT, which subsequently destroys the orthogonalities among subcarrier signals. There are two effects of phase noise on OFDM subcarriers: common phase error (CPE) and intercarrier interference (ICI). CPE causes subcarrier phase rotation, which does not change within an OFDM symbol period, while ICI introduces interferences to any subcarrier of a certain symbol from all the other subcarriers of that symbol, and therefore exhibits noise-like characteristics. In fact, phase noise in OFDM systems has been analyzed in many papers [9]–[15]. The original work done in [9] has successfully derived the expression of OFDM system degradation in a closed form which is valid for small phase noise and a large number of subcarriers. In [10] and [11], the characterization of phase noise and its effects on OFDM have been carefully studied, and the bit-error rate (BER) performance has been further analyzed. A different approach in [12] has provided the upper and lower bounds of the signal-to-interference-plus-noise ratio (SINR), revealing the dependence of OFDM system performance on some critical parameters. Nevertheless, the analytical results in [10]–[12] have assumed that phase-noise variance is much less than unity, and is only suitable for additive white Gaussian noise (AWGN) channels, even though some simulation results have been provided over multipath fading channels. Later in [13], the robustness of modulation methods against phase noise has been studied in a general OFDM system, while the effect of the phase-noise-bandwidth-to-subcarrier-spacing ratio on system performance has been analyzed in [14] and further extended in [15] to introduce the effect of the number of subcarriers on system performance, both with and without phase-noise correction. The approaches in [13]–[15], however, do not provide, even for AWGN channels, a closed-form analytical result that shows the exact quantitative relations between various parameters and OFDM performance; therefore, computer simulation is quite necessary. We derive in this paper the exact SINR expression in a closed form for all possible phase-noise levels and different numbers of OFDM subcarriers. SINR is expressed as a function of various critical system parameters, and provides a quantitative understanding of how system behavior changes with a certain parameter, which also enables one to determine under which condition phase noise can be treated as small, leading to adequate solutions, and what are proper parameter settings for a specific OFDM system to help mitigate phase noise. While high phase-noise levels are catastrophic, as it makes the ICI dominant over the transmitted signal [15], small phase noise

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can be corrected by estimating and compensating for CPE only (e.g., in [14], [16], and references therein) or by considering both CPE and ICI to minimize the mean square error (MSE) and get a better result [17]. However, the performance of the proposed algorithm in [17] specifically deals with the IEEE 802.11a standard, which provides few pilots such that decision feedback is needed to maintain adequate CPE estimation performance. Moreover, the estimation of ICI-plus-noise energy is affected by the front-end analog filter design at the receiver. The work in [17] can be further extended to a more general case in such a way that the optimal performance is guaranteed in the sense of the minimum mean-square error (MMSE), by estimating CPE without using decision feedback, as well as ICI-plus-noise energy that is independent of filter design. Such an approach can satisfy different system requirements and exhibit a great flexibility. The paper is organized as follows. In Section II, the phasenoise process is reviewed and a discrete phase-noise model is presented, together with a multipath channel model. Then the OFDM system model is given in the presence of phase noise and multipath fading. In Section III, the exact performance analysis of the phase-noise effects is provided without restrictions on phase-noise level and number of subcarriers; and by introducing SINR in a closed form, system performance is precisely described as a function of various parameters in a multipath fading environment. Based on the discussion given in Section III, a general phase-noise suppression (GPNS) scheme is described in Section IV, and analytical results are provided for its performance. In Section V, numerical results are given to illustrate the effectiveness of the proposed GPNS scheme, and the paper is concluded in Section VI.

II. SYSTEM MODEL In this section, we introduce the models of phase noise, channel, and a general OFDM system with phase noise. For simplicity, our discussion focus on baseband systems which can readily be extended to bandpass without any difficulties. , , and denote the DFT length, the cyclic prefix length, and the OFDM symbol period, respectively. Mutual independence is assumed among different kinds of signals and responses, e.g., transmitted signals, channel response, phase noise, and AWGN noise. A. Phase-Noise Model Phase noise , generated at both transmitter and receiver oscillators, can be described as a continuous Brownian motion , where denotes process with zero mean and variance the phase-noise linewidth, i.e., frequency spacing between 3-dB points of its Lorentzian power spectral density function [9], [13], [18]. Such noise has independent Gaussian increments [18], which, from a spectral point of view, can be presented as a finite-power Wiener process [10], [14], [15]. To better characterize phase noise, Demir et al. [19] developed a unifying theory using a nonlinear method which proves to be more accurate for its description. With such a method [19, Remark 7.1],

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is shown to become, asymptotically with time, phase noise a Gaussian random process having a constant mean, a variance increasing linearly with time, and the correlation function that . Fursatisfies thermore, as indicated in [20], the aforementioned discussion suggests a discrete Markov process which, in an OFDM system, illustrates the phase noise on the th sample of the th symbol as (1) ’s denote mutually independent Gaussian random where . In variables with zero mean and variance for particular, (1) reduces to ( , when ). B. Channel Model We assume that the transmitted OFDM signals propagate through a multipath fading channel which is modeled in the delayed impulses time domain by (2) where and are a Dirac delta function and zero-mean complex Gaussian random variables, respectively. For a slow-varying channel whose response does not vary within an OFDM symbol, the parameter can be omitted, is uniformly and as shown in [21]. Delay spread independently distributed within . For the th symbol of an -subcarrier OFDM system, the corresponding channel gains in the frequency domain are expressed by with autocorrelation function , as given in [21]. C. OFDM System Model In OFDM systems, inverse DFT (IDFT) uses the th symbol, subcarrier signals in the frewhich includes quency domain, to produce the corresponding modulated signals in the time domain. A cyclic prefix of length is then added to the beginning of the output stream of IDFT to produce the baseband signal (3) where ranges from to . Due to multipath fading, AWGN noise, and phase noise, the received signal can be written as (4) where and respectively, while mean and variance

denote the circular convolution and IDFT, indicates the AWGN noise with zero . After DFT, (4) yields (5)

where and

is defined as is the DFT response of

, . Since DFT does not

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change the noise energy, is still a Gaussian random variis given by able with zero mean and variance . (6) Note that the effects of phase noise, CPE, and ICI are repand , respectively. In the case of resented by perfect phase synchronization, phase noise does not exist, and and therefore (6) and (5) reduce to , respectively. III. PERFORMANCE ANALYSIS A. Exact SINR Expression Many approaches to OFDM performance analysis in the literature are based on small phase-noise levels and large numbers of subcarriers. Although this assumption may be true in practice, a thorough and exact evaluation without such restrictions will not only extend those results, but also provide an insight into phase-noise effects on the actual systems by introducing it as a function of critical system parameters. In particular, with the exact analysis, it is possible to determine an acceptable level of phase noise for a certain OFDM system and how it can be designed to avoid severe degradation in the presence of such noise. Therefore, in our paper, we assume a general OFDM system where phase-noise variance is not necessarily much less than unity, and the number of subcarriers is not necessarily very large. Since SINR is a very important indicator of system performance, its exact expression will be found and used to analyze system performance. From (5), SINR can be given by (7) where we have used the condition given in Section II-B. We also assume that the OFDM subcarrier signals are mutually independent random variables with zero mean and variance . From the definition in (5) and mutual independence of different kinds of signals, we have (8) Note that for a slow-fading channel of Section II-B, given and , one can realize from (7), together with (8), that the SINR expression has the same form as what we have obtained previously [12] for AWGN channels, if transmitted signals are independent and channel energy on each subcarrier is is given by (A7) unity. From Appendix A, the energy of as

(9) where equals , with denoting the transmission data rate. Equation (9) shows the dependence on phase-noise linewidth , number of of the energy of

Fig. 1. Effect of phase-noise linewidth on SINR performance for different numbers ofsubcarriers, with SNR = 20 dB and R = 10 samples/s.

subcarriers , and transmission data rate into (8) yields

. Substituting (9)

(10) From the definition of is shown that the sum of in (10) is real, hence

, we have

, with which it

(11) Substituting (9) and (11) into (7) yields the exact SINR expression (12) denotes the signal-to-noise ratio (SNR) per where subcarrier. Both (9) and (12) indicate that, in the presence of phase noise, several parameters affect OFDM system performance, resulting in severe performance degradation which is unacceptable in practice. In the case of perfect phase synchrobecomes a Dirac delta function, and the SINR nization, . expression of (12) reduces to SNR, i.e., B. Phase-Noise Effects In the presence of phase noise, (12) indicates that SINR is a function of various system parameters , , , , and their corresponding ratios. These relations are depicted, respectively, in Figs. 1–5. 1) It is well known that OFDM system performance with imperfect oscillators is strongly dependent on phasenoise linewidth [15]. As shown in Fig. 1, the larger is, the worse the SINR is, degrading as a logarithmically linear function of phase-noise linewidth for

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Fig. 2.

Effect of number ofsubcarriers N on SINR performance for different

=R ratios, with SNR = 20 dB.

Fig. 3. SINR as a function of R= for different SNR levels, with N = 256.

2)

3)

Hz. Hence, as we stated earlier, the best way to eliminate the detrimental effects of phase noise is to improve oscillator accuracy, and thus decrease phase-noise linewidth. It is quite straightforward to see that a larger number of subcarriers leads to worse system performance due to the shorter subcarrier spacing distance, hence, more sensitive to phase noise. In particular, Fig. 2 suggests ratio is of the order of or less, that, when doubling always causes approximately 3-dB loss of SINR for all values. This implies that, when is below a certain level, SINR becomes inversely proportional to the number of subcarriers . Higher transmission data rate results in better system performance. In fact, the transmission-datais of interest. rate-to-phase-noise-linewidth ratio Fig. 3 shows that the SINR has a limiting low value , regardless of the SNR value. This for small

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Fig. 4. SINR as a function of phase-noise-linewidth-to-subcarrier-spacing ratio N=R with different SNR settings.

Fig. 5. SINR degradation as a function of SNR with different N =R settings.

comes directly from (12) by letting zero, namely

approach (13)

Hence, for very large , this limit can be approximated as , which suggests that, for large ), a high SNR does not help phase noise (thus, low the performance. Therefore, a well-designed system ratio in order to achieve must have a reasonable adequate performance. In particular, (12) shows that increases to infinity, SINR becomes SNR when i.e., (14) This makes sense by noticing that, when goes to infinity, the system is equivalent to a normal one without phase noise.

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From (12), the SINR is strongly dependent on . As shown in Fig. 4, for high data rate , when is very small in comparison to the subcarrier spacing , i.e., is of the order of or less, phase noise becomes negligible, and it is equivalent to the nonphase-noise case. When is between and , phase noise is small, and thus, effective schemes are available for its correction. Higher SNR leads to better performance in the presence of phase noise. But systems with high SNR are more sensitive to phase noise, as shown in Fig. 5. For , SINR degrahigh phase-noise levels with dation exceeds the value of SNR itself. This implies that the ICI overwhelms the desired signals. In particular, all phase-noise correction schemes are based on CPE estimation, which, as indicated in [15], does not improve or even makes the performance worse with high phase-noise levels. We see from Fig. 5 that, when , though system loss does not exceed the value of the SNR, it is pretty high, e.g., the when the SINR degrades 20 dB for SNR equals 30 dB. In this case, phase-noise correction can be applied, but the performance requirement may not be guaranteed.

for small with further yields

and

, (9)

(15) For large subcarrier number

, (15) becomes (16)

With the difference of the phase-noise linewidth definition in [9] (one-sided linewidth) and in Section II-A of this paper (two-sided linewidth), the result of (16) is exactly the same as that derived in [9]. It is known that the Fourier transform does is the not change signal energy. Since (6) shows that divided by a factor of , then IDFT output of we have , which yields . This equation, together with (8) and (16), leads to (17) Substituting (16) and (17) into (7), we end up with (18)

C. Parameter Design As was discussed earlier, phase noise proves to be a problem for OFDM systems whose performance depends on a few system parameters. Thus, the derived SINR expression provides a good guide for how to choose these parameters to keep sufficient performance in the presence of phase noise. Using the derived expression follows several steps. Step 1) Get the knowledge of some fixed system parameters, e.g., transmission data rate, number of subcarriers, or SNR level. Step 2) Specify the requirement for the system, such as the tolerable SINR level. Step 3) Choose the remaining parameters to meet the requirement, according to the SINR expression in (12). We take the IEEE 802.11a standard as an example, which , and the transmisspecifies the number of subcarriers sample/s. With the SNR ranging besion data rate tween 0–25 dB, if we require that the SINR degradation cannot be larger than 3 dB, especially for high SNR levels, we obtain , in terms of (12). In other words, the phase-noise linewidth must be less than 296 Hz. On the other hand, if the SNR level is between 0–20 dB, and other conditions remain the same, the phase-noise linewidth is required to be less than 2900 Hz. D. Small Phase-Noise Approximation Equation (12) can be further simplified in the case of small much phase noise, which requires less than unity. Using the approximation

IV. PHASE-NOISE SUPPRESSION A. GPNS Scheme Due to its severe effects on OFDM system performance, phase-noise correction should be considered in practice. As shown in the previous section, we cannot predict the effectiveness of any correction schemes for large phase noise. In fact, system performance with large phase noise may not improve or, in many cases, become even worse after phase-noise correction. Nevertheless, it was shown in the literature [10], [16], [17] that correction can be successful for small phase-noise levels (phase-noise variance is much less than unity or, more is between and , as we discussed specifically, earlier). Random phase noise itself proves to be quite difficult to deal with directly, whereas the results of phase noise, CPE and ICI, as indicated by (5), may be compensated. In particular, due to the invariance of CPE within an OFDM symbol, several methods have been developed in the literature [15], [16] for its mitigation using pilot signals, but none addressed ICI suppression. In [17], both CPE and ICI have been considered for phase-noise correction in the OFDM-based IEEE 802.11a standard, by assuming unknown noise (both AWGN and phase noise) statistics. Unlike the systems used in [15] and [16], the IEEE 802.11a standard has within a data symbol only four pilot signals designated for channel- and frequency-offset correction [4], which makes it difficult to provide a useful CPE estimation result, unless additional information, such as decision feedback, is provided for estimation purposes [17]. Using the MMSE equalization technique [17], we could simultaneously suppress both CPE and

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ICI caused by phase noise, which results in better performance, accordingly.1 In fact, the proposed approach in [17] can be further extended to a general case, where an optimal receiver performance is achieved by minimizing the MSE, namely

1993

and denoting the set and number of guard-band subwith carriers, respectively. B. Normalized MMSE (NMMSE) The estimate of the desired signal can be written as

(19) and Given is shown to be

, the optimal

, referring to (5),

(23)

(20)

yields the

Substituting (20) into minimum conditional MSE (MMSE) Since OFDM channel estimation has been considered in many papers, e.g., [21] and [23], channel frequency response are assumed known in this paper. However, for the complete solution of (20), the CPE term has to be estimated. Minimizing the cost function leads to the estimate (21) indicates the set of pilot signals. where The ICI-plus-noise energy in (20) can be precalculated using (8) and (9), if we have a prior knowledge of both AWGN and phase-noise statistics. In this case, the optimal can be used for data equalization. However, in the case of unknown noise statistics, it is required to estimate the ICI plus AWGN noise energy from received signals. As derived in (17), for small phase noise, ICI energy is constant within a symbol, which implies that we possibly can estimate the ICI-plus-noise energy using pilot subcarriers. If these specific pilot subcarriers carry at the transmitter null data, we can readily see from (5) that, after DFT at the receiver, they will carry ICI-plus-noise only, which suggests their possible usage for ICI-plus-noise energy estimation. The estimation method can, therefore, be expressed by (22) where and denote the set and number of these pilot subcarriers, respectively, used for estimation. Note that transmitting pilot signals for ICI-plus-noise energy estimation is not always feasible, due to limited bandwidth, e.g., IEEE 802.11a standard. In such a case, the subcarriers in the guard band (excluding cyclic prefix) can then be used to estimate ICI-plus-noise energy with guaranteed accuracy, even with analog filtering at the front end of the receiver [17]. However, this is suitable for sufficiently high SNR, as the estimation accuracy is affected by the noise colored by the analog filter. Equation (22) can still be used, 1Since the submission of this paper, and motivated by the fact that the neighboring subcarrier signals are correlated, we proposed in [22] a simultaneous CPE and ICI correction scheme, which, by exploiting such correlation, results in some performance gain in comparison with the proposed GPNS, but requires extra complexity.

(24) which

leads

to

the

normalized

MMSE

(NMMSE) . With the approximation of by its expectation , the conditions on and can be removed, which gives rise to the average unconditional NMMSE (25) where we used the independency between and . The numerator in (25) gives the energy of ICI plus AWGN noise, while the denominator indicates the entire energy of the received signal after DFT. Note that for the GPNS scheme, the assumpresult in (16) and (17), tion of small phase noise and large yielding

(26) Equation (26) suggests that, in case of small phase noise, NMMSE can be expressed as a simple function of the phasenoise-linewidth-to-subcarrier-spacing ratio and SNR . or increasing It further shows that either decreasing will reduce the residual MMSE, and subsequently enhance receiver performance. This conclusion coincides with what we have found in Section III-B. Fig. 6 illustrates how NMMSE is related to and . With smaller , which indicates better frequency and phase synchronization, SNR has a domiis of the order of nant effect on NMMSE. In case or less, NMMSE decreases almost linearly with SNR, and the effects of phase noise are negligible. Larger , however, results in more errors that overwhelm the positive effect of high SNR over NMMSE, as indicated by the inevitable error floor in Fig. 6, even after phase-noise correction.

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Fig. 6. Normalized MMSE for a 1024-subcarrier system with different N=R values.

Fig. 7. Estimated and actual values of the amplitude and angle of CPE I (0), versus phase-noise variance as well as the ICI-plus-noise energy  2 T , in a 1024-subcarrier OFDM system.

V. NUMERICAL RESULTS In order to examine the effectiveness of the proposed GPNS scheme, we resort to computer simulation, where we assume 16-quadrature amplitude modulation (QAM) which is more sensitive to phase noise than -ary phase-shift keying ( -PSK) schemes with the same transmitting power. We set subcarriers per OFDM symbol with a cyclic prefix of length and 64 guard-band subcarriers. Guard-band subcarriers are filtered, as is customarily done in practice. We use 32 pilots for CPE estimation, as well as 32 null pilots for ICI-plusnoise energy estimation. Pilots for both CPE and ICI are assumed available and evenly interpolated into 1024 subcarriers. The proposed scheme is evaluated with a frequency-selective Rayleigh fading channel by Monte Carlo trials. The statistics of AWGN and phase noise is not available in computer simulations. Simulation results are shown in Figs. 7–9, where the proposed GPNS scheme is compared with the scheme using CPE estimation only, e.g., as proposed in [16]. For reference purposes, the results based on theoretical noise statistics calculation are added to these figures. and ICI-plus-noise Fig. 7 illustrates how the CPE changes with phase-noise levels. The actual energy values of these parameters are also given to facilitate comparison with the estimates. It can be seen from this figure are very that the amplitude and angle of the estimated close to their actual values within a wide range of phase-noise levels. This suggests the effectiveness of the estimation method is given by (21). On the other hand, the estimate of quite accurate, compared with its actual value for small phase noise, while the estimation error increases significantly when . With such phase-noise variance exceeds the order of phase-noise levels, the rapid decreasing amplitude of and the fast increasing make phase-noise correction ineffective. Fig. 8 shows the catastrophic effects of phase noise with variance 0.0193 on symbol-error rate (SER) performance. The

Fig. 8. SER performance of a 1024-subcarrier OFDM system in the presence of phase noise over Rayleigh fading channels with 2 T = 0:0193 rad .

CPE estimation and correction helps improve the system performance, but exhibits an error floor at high SNRs, as it does not consider ICI. The proposed GPNS scheme outperforms the CPE estimation scheme by considering ICI and minimizing the overall errors. The theoretical curve indicated in this figure is with a prior based on the theoretical calculation of knowledge of AWGN and phase-noise statistics. We notice that results with the GPNS scheme are completely comparable with the theoretical calculation (both curves are almost identical). This suggests the effectiveness and capability of the proposed scheme in a general OFDM system. Fig. 9 demonstrates the performance of the proposed scheme with different phase-noise levels. It always performs better than the CPE estimation and correction scheme. Moreover, the GPNS scheme has approximately the same performance as the . no-phase-noise case with phase-noise variance less than Since such a requirement for small phase noise can practically

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In addition, from (1), we rewrite the discrete phase-noise model as

(A2) where

Fig. 9. SER performance versus phase-noise variance 2 T 1024-subcarrier OFDM system, where SNR = 30 dB.

is defined by and by . Substituting (A2) into (A1) yields

in a

be met in many cases, the proposed GPNS scheme is thus suitable for practical applications. VI. CONCLUSIONS In this paper, we have analyzed the effects of phase noise on the OFDM system performance and provided the corresponding solution for its mitigation. In order to perform the analysis, we have derived an exact closed-form expression for the SINR, with which, system behavior can clearly be judged for any phasenoise levels. In the presence of phase noise, the quantitative relations of critical parameters to system performance, such as phase-noise linewidth, number of subcarriers, transmission data rate, and SNR, have been clearly presented by means of mathematical functions. The derived SINR expression could indicate the condition under which phase noise is considered small and how to design proper parameters for any OFDM systems. With the understanding that phase noise could be corrected for a small level, a general suppression scheme, termed GPNS, has been proposed in this paper, dealing with a general OFDM system impaired by small phase noise. Since the proposed GPNS scheme has been shown to be quite feasible under various circumstances, this exhibits its wide applications in OFDM-based systems. This scheme is presented analytically by the minimization for residual MSE and provides an optimal performance. Numerical results are used to show the effectiveness of the proposed scheme. APPENDIX A ENERGY OF

(A3) is defined as , with where denoting the sign operation. We notice that of (A2) does , and is independent of not affect symbol index . Since are mutually independent Gaussian random variables with zero mean and covariance , then are also mutually independent Gaussian random variables with the same mean and variance ; and the sum of such Gaussian random as , is Gaussian variables, namely, with the characteristic function given by [24]

(A4) From (A4), fore

gives the term

in (A3), there-

(A5) as a function of . Since , with indicating the transmission data rate, then we have . Substituting into (A5), after several algebraic manipulations, yields Now we have

The derivation of the energy of is quite crucial for the exact SINR expression. From (6), the energy of can be written as (A1)

(A6)

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where indicates the real part of a complex variable. It can be shown that . Therefore, (A6) gives

(A7) REFERENCES [1] J. Bingham, “Multicarrier modulation for data transmission: An idea for whose time has come,” IEEE Commun. Mag., vol. 28, pp. 5–14, May 1990. [2] H. Sari, G. Karam, and I. Jeanclaude, “Transmission techniques for digital terrestrial TV broadcasting,” IEEE Commun. Mag., vol. 33, pp. 100–109, Feb. 1995. [3] Broadband Radio Access Networks (BRAN); HIPERLAN Type 2; Physical (PHY) Layer, ETSI TS 101 475 V1.3.1 (2001-12), http://www.etsi.org, Dec. 2001. [4] Supplement to IEEE Standard for Information Technology—Telecommunications and Information Exchange Between Systems—Local and Metropolitan Area Networks—Specific Requirements. Part II: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications: High-Speed Physical Layer in the 5 GHz Band, IEEE Std 802.11a-1999, http://www.ieee.org, Dec. 1999. [5] P. H. Moose, “A technique for orthogonal frequency division multiplexing frequency offset correction,” IEEE Trans. Commun., vol. 42, pp. 2908–2914, Oct. 1994. [6] H. Sari, G. Karam, and I. Jeanclaude, “Channel equalization and carrier synchronization in OFDM systems,” in Proc. Tirrenia Int. Workshop Digital Communications, Tirrenia, Italy, Sept. 1993. [7] M. A. Visser and Y. Bar-Ness, “OFDM frequency offset correction using an adaptive decorrelator,” in Proc. Personal, Indoor, Mobile Radio Conf., Boston, MA, Sept. 1998, pp. 816–820. [8] M. A. Visser, P. Zong, and Y. Bar-Ness, “A novel method for blind frequency offset correction in OFDM systems,” in Proc. CISS, Princeton, NJ, Mar. 1998, pp. 483–488. [9] T. Pollet, M. Bladel, and M. Moeneclaey, “BER sensitivity of OFDM systems to carrier frequency offset and Wiener phase noise,” IEEE Trans. Commun., vol. 43, pp. 191–193, Feb. 1995. [10] M. S. El-Tanany, Y. Wu, and L. Hazy, “Analytical modeling and simulation of phase noise interference in OFDM-based digital television terrestrial broadcasting systems,” IEEE Trans. Broadcast., vol. 47, pp. 20–31, Mar. 2001. [11] J. Scott, “The effects of phase noise in COFDM,” EBU Tech. Rev., Summer 1998. [12] S. Wu and Y. Bar-Ness, “Performance analysis on the effect of phase noise in OFDM systems,” in Proc. Int. Symp. Spread-Spectrum Techniques, Applications, Prague, Czech Republic, Sept. 2002, pp. 133–138. [13] L. Tomba, “On the effect of Wiener phase noise in OFDM systems,” IEEE Trans. Commun., vol. 46, pp. 580–583, May 1998. [14] A. G. Armada and M. Calvo, “Phase noise and sub-carrier spacing effects on the performance of an OFDM communication system,” IEEE Commun. Lett., vol. 2, pp. 11–13, Jan. 1998. [15] A. G. Armada, “Understanding the effects of phase noise in orthogonal frequency division multiplexing (OFDM),” IEEE Trans. Broadcast., vol. 47, pp. 153–159, June 2001. [16] P. Robertson and S. Kaiser, “Analysis of the effects of phase noise in orthogonal frequency division multiplexing (OFDM) systems,” in Proc. Int. Conf. Communications, Seattle, WA, 1995, pp. 1652–1657. [17] S. Wu and Y. Bar-Ness, “A phase noise suppression algorithm for OFDM based WLANs,” IEEE Commun. Lett., vol. 6, pp. 535–537, Dec. 2002. [18] G. J. Foschini and G. Vannucci, “Characterizing filtered light waves corrupted by phase noise,” IEEE Trans. Inform. Theory, vol. 34, pp. 1437–1448, Nov. 1988. [19] A. Demir, A. Mehrotra, and J. Roychowdhury, “Phase noise in oscillators: A unifying theory and numerical methods for characterization,” IEEE Trans. Circuits Syst. I, vol. 47, pp. 655–674, May 2000. [20] K. Nikitopoulos and A. Polydoros, “Compensation schemes for phase noise and residual frequency offset in OFDM systems,” in Proc. GLOBECOM, San Antonio, TX, Nov. 2001, pp. 331–333.

[21] O. Edfors, M. Sandell, J. V. D. Beek, S. K. Wilson, and P. O. Borjesson, “OFDM channel estimation by singular value decomposition,” IEEE Trans. Commun., vol. 46, pp. 931–939, July 1998. [22] S. Wu and Y. Bar-Ness, “A new phase noise mitigation method in OFDM systems with simultaneous CPE and ICI correction,” in Proc. 4th Int. Workshop Multi-Carrier Spread Spectrum, Oberpfaffenhofen, Germany, Sept. 2003. [23] Y. Li, L. J. Cimini, Jr., and N. R. Sollenberger, “Robust channel estimation for OFDM systems with rapid dispersive fading channels,” IEEE Trans. Commun., vol. 46, pp. 902–915, July 1998. [24] J. G. Proakis, Digital Communications, 3rd ed. New York: McGrawHill, 1995.

Songping Wu (S’04) received the B.S. and M.S. degrees from the Department of Electronic Engineering, Tsinghua University, Beijing, China, in 1996 and 1998, respectively, and the Ph.D. degree from the Department of Electrical and Computer Engineering, New Jersey Institute of Technology (NJIT), Newark, in 2004. From 1996–1998, he was a Research Assistant in the Department of Electronic Engineering, Tsinghua University. From 1998–1999, he was a Wireless Engineer with Nokia, Beijing, China. From 1999–2001, he was a Wireless Engineer with Ericsson, Beijing, China. From 2001 to 2004, he was a Research Assistant at the Center for Communications and Signal Processing Research (CCSPR) at the Department of Electrical and Computer Engineering, NJIT, Newark, NJ. Currently, he is with Marvell Semiconductor Inc., Sunnyvale, CA. His research interests include OFDM, CDMA, MIMO, phase noise, frequency offset, and channel estimation.

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, Brown University, Providence, RI. Currently, he is a Distinguished Professor of Electrical and Computer Engineering and Foundation Chair of Center for Communication and Signal Processing Research (CCSPR) at the New Jersey Institute of Technology (NJIT), Newark. He worked for Rafael Armament Development Authority, Israel, in the field of communications and control; and for the Nuclear Medicine Department, Elscint Ltd., Haifa, 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, where he held the position of 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, 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 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 Wireless Personal Communications, an international journal. 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 (ISSSTA 2000). He is a recipient of the Kaplan Prize (1973), which is awarded annually by the government of Israel to the ten best technical contributors.

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1997

Pilot-Assisted Maximum-Likelihood Frequency-Offset Estimation for OFDM Systems Jiun H. Yu and Yu T. Su, Member, IEEE

Abstract—For orthogonal frequency-division multiplexing (OFDM) signals that suffer from frequency-selective fading, we derive the maximum-likelihood (ML) pilot-assisted carrier frequency offset (CFO) estimate and show that most proposals based on repetitive pilot symbols did not use the complete set of sufficient statistics. We convert the problem of obtaining the ML solution from searching exhaustively over the entire uncertainty range to that of solving a spectrum polynomial, thereby greatly reducing the computational load. By properly truncating the polynomial, we obtain a closed-form expression for the corresponding zeros so that the root-searching procedure is greatly simplified. The complexity of locating the desired root is further reduced at almost no expense of performance degradation by an alternate algorithm that uses the fact that the solution is related to the root of a special factor of the polynomial. This alternate method is very attractive for its simplicity and excellent performance that, even at low signal-to-noise ratios (SNRs), is very close to the corresponding Cramér–Rao lower bound. A detailed analysis of the mean-squared error performance is presented and the analysis is validated by simulations. Index Terms—Frequency estimation, orthogonal frequencydivision multiplexing (OFDM).

I. INTRODUCTION RTHOGONAL frequency-division multiplexing (OFDM) is an effective antifading modulation scheme for broad-band wireless communications. It has been adopted by several standardization groups for various applications; see [1] and the references therein. A shortcoming of OFDM systems is the sensitivity to the carrier frequency offset (CFO). The presence of a CFO causes reduction of amplitude of the desired subcarrier and induces intercarrier interference (ICI) because the desired subcarrier is no long sampled at the zero-crossings of its adjacent carriers’ spectrum. Due to the inherent characteristics of OFDM signals, the tolerable frequency offset range is very limited [1].

O

Paper approved by G. M. Vitetta, the Editor for Equalization and Fading Channels of the IEEE Communications Society. Manuscript received July 7, 2003; revised December 14, 2003 and May 7, 2004. This work was supported by the National Science Council of Taiwan under Grant NSC 91-2219-E-009. J. H. Yu was with the Department of Communication Engineering, National Chiao Tung University, Hsinchu 30056, Taiwan. He is now with the Research and Development Center, Realtek Semiconductor Corporation, Hsinchu 30056, Taiwan (e-mail: [email protected]). Y. T. Su is with the Department of Communication Engineering, National Chiao Tung University, Hsinchu 30056, Taiwan (e-mail: [email protected]). Digital Object Identifier 10.1109/TCOMM.2004.836555

There have been many CFO estimation schemes for OFDM signals [2]–[15]. These schemes can be conveniently categorized into blind and pilot-assisted schemes. Pilot-assisted schemes use well-designed pilot symbols to estimate CFO and, because these schemes are capable of achieving rapid and reliable frequency synchronization, are often used by packet-oriented systems. Moose [2] proposed a correlation-based technique that uses two consecutive identical pilot symbols to estimate CFO. Although Moose’s algorithm is a maximum-likelihood (ML) estimate, its maximum frequency subcarrier spacing. Following acquisition range is only Moose’s proposal, subsequent techniques use multiple identical pilot symbols with a smaller symbol period to increase the estimation range of CFO. be the frequency offset and Let be the normalized CFO with respect to the subcarrier spacing , where is the OFDM symbol period, is the integer part of , while is the fractional part. Schmidl and Cox (SC) [5] used two identical half-period symbols to estimate the fractional part of the CFO and a second full-period symbol that has a special correlation relation with the first pilot symbol to estimate . Lim [11] also proposed a similar method and exploited only two identical half-period symbols to estimate both and . Morelli and Mengali (MM) [6] increased the acquisisubcarrier spacing by dividing a symbol into tion range to repetitive parts, as shown in Fig. 1(a). Their algorithm has been proved to be better than the SC estimate for yielding a smaller minimum mean-square error (MMSE). In addition, their algorithm needs only one symbol period for computing the CFO. The MM algorithm was further improved by the two-stage method of Minn, Tarasak, and Bhargava (MTB) [10]. Song [7] exploited the same pilot symbol structure and suggested a multistage correlation method to acquire CFO, but performance results were the same as those of Schmidl’s. A simplified version of Song’s estimation method that requires only two correlation steps was proposed by Patel [8]. The performance of these methods, except for the MM and MTB algorithms, depends on the correlation of two half-period identical blocks. MM and MTB used differential phases of the correlations between different pairs of adjacent fractional-period blocks to form an improved CFO estimate. The blind schemes, on the other hand, exploit the structural and statistical properties of the transmitted OFDM signals such as cyclic prefix [3], virtual subcarrier [4], or constant-modulus [15]. Since no training symbols are required, blind methods are

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Fig. 1. (a) Short pilot symbol sequence of the IEEE 802.11a. (b) Symbol arrangement and definitions of the proposed ML estimate.

attractive for saving bandwidth and having higher throughput and are more suitable for circuit-switched transmissions. In this paper, we present ML CFO estimates that use an of identical fractional-period OFDM arbitrary number blocks. An efficient algorithm is provided to solve the associated highly nonlinear ML equation. Instead of searching within the candidate CFO range exhaustively, we only need . Two methods to to solve a polynomial of degree further reduce the complexity of extracting the desired root are presented. Numerical results indicate that even at low SNR the performance of the proposed methods still approach the corresponding Cramér–Rao lower bound. The rest of this paper is organized as follows. Section II describes our signal model and defines related parameters. We derive the corresponding optimal frequency-offset estimate and its simplified version in Section III and Appendix I. A detailed performance analysis is presented in Section IV and Appendix II. We discuss the simulation results in Section V and, finally, in Section VI, we summarize our main results.

ceded by a cyclic prefix longer than the maximum channel delay spread to form an “extended” symbol so that ISI can be eliminated at the receiving end by simply discarding the prefix part. One can also have a short OFDM symbol whose duration is a fraction of by placing data symbols only at the subcarriers , i.e., data sequence is transmitted at the frequencies while zeros are inserted at the rewhich are multiples of maining frequencies [6]. be the th sample of the th (time-domain) short Let pilot symbol and assume that the preamble part of a transmitted identical short pilot symbols with a package consists of total preamble duration of seconds. We thus have for and the relation . Shown in Fig. 1(a) is the IEEE 802.11a , , and to form a standard that uses training sequence of ten identical short symbols. Consider a frequency-selective channel with a maximum delay spread shorter than a short symbol duration. Assuming that the combined frequency response of the prefilters is flat , where is the signal within the range is the maximum frequency offset, the bandwidth and received baseband waveform is matched-filtered and sampled samples/s. After discarding the first symbol, at a rate of the remaining received pilot symbols can be represented as (1) , and , where for are uncorrelated circularly symmetric Gaussian random vari. ables (rvs) with zero mean and variance is the channel output corresponding to the transmitted . Due to the assumption that the channel pilot symbol delay spread is shorter than the length of one short symbol and the channel impulse response remains the same during the preare periodic. Note amble period, the remaining samples that the above signal model implies that the maximum CFO one can recover is subcarrier spacings. Define the two vectors (2) and (3)

II. SIGNAL MODEL AND PARAMETERS data symbols Parallel transmission of a block of drawn from a quadrature-amplitude modulation (QAM) or phase-shift keying (PSK) constellation is efficiently im-point inverse discrete Fourier transform plemented by an (IDFT). The transformed block of (time-domain) samples

where denotes the matrix transpose. Then, as shown in Fig. 1(b), we have

(4) where

, and . The received samples can thus be

expressed compactly as forms a long OFDM symbol, where the equally spaced data-bearing subcarriers are mutually orthogonal over a symbol interval of seconds, where is the IDFT output sample interval. Oftentimes, an OFDM symbol is pre-

(5) where

, , and . Hence, given the received sample

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vectors , we have to estimate through the deterministic . For notational simplicity, we shall drop the arvector in the subsequent discussion. gument in The above signal model (5) assumes that perfect symbol timing has been established prior to frequency synchronization. However, it is still valid even if timing error does exist provided that the selected received pilot symbols are within the range of the preamble and the number of identical short pilot symbols is larger than (excluding the first discarded received short symbol). Therefore, the beginning position received pilot symbols are selected to form the where the signal is very flexible. More specifically, even if we do not have the symbol timing, we still can use the above signal model selected short symbols are located within provided that the the legitimate interval that spans from the start of the second received short symbol to the last sample of the last transmitted short pilot symbol. Thus, the CFO estimators derived from (5) are expected to be insensitive to timing error. III. ML ESTIMATE OF CFO Since the noise is temporally white Gaussian, is a multivariate Gaussian distributed random vector with covariance ma, where is the identity matrix. The joint ML estimates trix of and , treating as a deterministic unknown vector, are obtained by minimizing the joint probability density function

1999

, it is a Hermitian matrix such that , where denotes the complex conjugate. The desired CFO estimate is then given by sample vectors

(9) It can be proved that the above ML solution is the same as [9, eq. (9)] whose computing load, however, is much heavier. Although (9) gives a compact representation of the ML CFO estimate, it requires an exhaustive search over the entire uncertainty range. The resulting complexity may make its implementation infeasible. We observe, however, that has a special structure that can be of use to reduce the complexity of searching the desired CFO solution of (9). Invoking an approach similar to that used by the and define the MUSIC algorithm [16], we set parametric vector (10) can be expressed as

so that the log-likelihood a polynomial of order

as follows: (11)

(6) The corresponding log-likelihood function, after dropping constant and unrelated terms, is given by (7) For a given , setting , where denotes complex gradient operation with respect to , we obtain the conditional ML estimate , where and denotes the Hermitian into operation. Substituting the least-square solution (7), we obtain

where

, for , and . To highlight the usefulness of this important observation, we restate it in the form of the following proposition. Proposition 1: The log-likelihood function for a candidate CFO is given by (12) Some remarks about this proposition are in order. Remarks: R1. is the summation of diagonal entries of and is also equivalent to the aperiodic autocorrelation value of the waveform at time difference seconds, i.e., . R2. It can be shown that, in the absence of noise (13)

(8)

, and When noise is present, is the same as its noisethe mean value of less value except for ; more specifically, , where is the Kronecker delta function. Evaluating (11) at the unit circle , we obtain the discrete-time Fourier transform of the sequence , where

.

denotes the trace of a matrix [18], , and . Note the th entry of the matrix , , is correlation value of th and th received symbols, . As is (time-averaged) autocorrelation matrix of the received

where that the i.e., the

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which has an envelope similar to whose maximum value is at the correct “modified” frequency . R3. Due to the Hermitian nature of , is a con. The jugate symmetric sequence of length guarantees that its Fourier symmetric property of is real and nonnegative. This transform also follows from the semi-positive definiteness of the quadratic form . Because and the log-likelihood function constitute a Fourier transas the form pair, we will henceforth refer to log-likelihood spectrum or spectrum, for short, and the polynomial defined by (11) the spectrum polynomial. constitutes a set R4. of sufficient statistic for estimating . Almost all previous correlation-based algorithms use only a subset , the Moose’s algorithm [2] uses of , e.g., For , the GM algorithm [7] uses . The MM and MTB algorithms use only the phases of elements of . Furthermore, the former achieves its best performance when it uses only half of the phases [10]. It is expected that an algorithm that uses the sufficient statistic would outperform those that use only a part of the sufficient statistic. R5. Computing the desired CFO estimate through (11) is equivalent to searching for the peak of the candidate . Hence, the spectrum can be comspectrum puted using a discrete Fourier transform (DFT), but the resolution of the CFO estimate depends on the size of the DFT. Padding more zeros in the sequence results in higher resolution at the expense of inducing higher computation complexity. As the spectrum is a real smooth function of , taking with respect to and setting a derivative of , we obtain (14) where is a polynomial of order . As mentioned before, in a noiseless environment, , , is a scaled version of the the Fourier transform of , and all roots of function are on the unit circle. The presence of noise and multipath will modify the Fourier transform and move some roots away from the unit circle so that the solutions of become a proper subset of those of . In , where that case, we have is a complex constant, , , and , . Although has a fixed number of roots, the distribution of these roots among or, equivalently, the degrees of and depend upon SNR for the existence of nonunit-amplitude roots due to the , which, in turn, merging of neighboring sidelobes of results from large noise perturbation. We can either restrict our search to those roots that are on the unit circle or normalize those nonunit-amplitude roots. Several reasons convince us that both approaches will most likely give the same estimate. First, the desired root is associated with the

Fig. 2. Normalized log-likelihood spectrum and the associated root distribution where the spectrum is normalized by the noiseless mainlobe peak value; CFO = 1:2 subcarrier spacings.

peak of and the mainlobe-peak-to-sidelobe-peak ratio is greater than 25 dB. As the relative height difference between any two neighboring sidelobe peaks is far smaller than that between the mainlobe and its two neighboring sidelobes, it is much more likely for the merging of neighboring sidelobes than that of the mainlobe and one of its neighboring sidelobes and, even if the latter merge occurs, the associated peak will most probably be the peak of the spectrum. In other words, the desired root is likely to stay at the unit circle with a very high probability. Second, our simulation has shown that, even in the presence of still strong noise and severe multipath, the resulting and the bears a close resemblance to a scaled version of roots on the unit circle are within small neighborhoods of their is a linear transnoiseless locations; see Fig. 2. Finally, and, as (B8) indicates (see Appendix II), form of can be decomposed into two deterministic terms and two zeromean complex perturbation terms that are uncorrelated with the deterministic part. For simplicity, we shall use the first approach, i.e., the desired estimate is to be obtained by (15) where (16) Note that we have converted the exhaustive search problem of (9) to a root-finding problem, reducing the candidate solution number from infinity to at most . A. A ML CFO Estimation Algorithm We summarize the procedure leading to (16) as follows. 1) Collect received symbols and construct the sample correlation matrix . 2) Calculate the coefficients of based on . 3) Find the nonzero unit-magnitude roots of (14). 4) Obtain the CFO estimate from (15) and (16).

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We will refer to the above procedure as Algorithm . It can , the resulting estimate is equivalent be shown that, when to the Moose estimate [2]. The above algorithm needs to locate all of the roots of . We can reduce the order of by truncating the , i.e., using a lesser number of length of the sequence autocorrelation values. When we use the truncated version, , to carry out Algorithm , the resulting algorithm is referred to as Algorithm . The complexity of is much less than that of Algorithm since Algorithm the associated has closed-form solutions. We do expect some performance degradation as less autocorrelation values are used. In the following subsection, we present a method to further reduce the complexity of solving (14) in step 3) with little or no performance degradation. B. A Simplified CFO Estimate We notice that the solutions of (14) are the nonzero roots of the polynomial (17) satisfy On the other hand, (14) implies that the roots of the equation , where is the imaginary part of . This observation indicates that the nonzero roots (the root of is undesired) of . When is an are a subset of the roots of arbitrary polynomial, its roots are not necessarily a subset of defined by (17). However, in those of the corresponding and do divide our case, as shown in Appendix I, both when there is no noise in the received signal vector or the ensemble average is used to construct and , i.e., the roots of are indeed a subset of . Moreover, we can show (see Appendix I) that those of the following proposition holds. Proposition 2: In the absence of noise, the polynomial de, can be decomposed into fined by (17),

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have a closed-form expression for its roots. We will show later via simulations and analysis (see Appendix II) that (18) is a valid approximation that incurs only negligible performance loss even when the system is operating at an SNR as low as 0 dB. The above discussion suggests the following simplified CFO estimate algorithm. 1) Follow 1) of Algorithm . based on two correlation 2) Compute the coefficients and . values for the unit-magnitude roots of , 3) Solve . 4) Find the estimate from (15) and (16). The above algorithm will be referred to as Algorithm . Note [see (13)] indicates that that the definition of and suggests that steps 3) and 4) of Algorithm can be replaced , where is the by using the estimate principal value of the argument of . The disadvantage of this estimate is that it cannot be applied to the situation when the CFO is such that . On the other hand, (13) also implies that , the CFO can be estimated by . Amongst these candidate estimates, is the only one that does not violate the no-phase-ambiguity requirement and, in fact, is the same as the generalized Moose (GM) algorithm that uses averaged autocorrelation values [7]. IV. PERFORMANCE ANALYSIS Note that the CFO estimate derived from Algorithm is , identical to (9). This estimate is unbiased, for as , with , , and is the channel response at the subcarrier , hence . In Appendix II, we show that the variance of the ML estimate is given by (20)

(18) where the desired CFO estimate is one of the roots of defined by (19) . where When noise is present, the above equality becomes an approximation only. Nevertheless, the desired CFO estimate can still be derived immediately from taking the th root of . Fig. 2 highlights the locations of the normalized roots of , and for , i.e., various local extremes of . The global maximum that colocated with a root of corresponds to the desired CFO estimate while the remaining locate at a local minimum (null) of the spectrum. roots of On the other hand, the roots of are at the local sidelobe peaks of the spectrum. It is clear that the union of the roots of and is the set of the roots of . Hence, the complexity of extracting the roots is significantly reduced, for we , which happens to only have to solve the equation

where . Following the analysis that leads to [16, eq. (4.1)] and upon substituting the parameter, , we obtain the corresponding Cramer–Rao bound (CRB) CRB

(21)

Equation (20) indicates that is a decreasing function of . Therefore, replacing one long pilot symbol with several identical short symbols and using the proposed ML method would yield a performance superior to that resulting from using the correlation-based method that correlates two , i.e., Moose algorithm), identical half-period symbols ( even though both use the same number of data samples. For Algorithm , the associated variance can be approximated by (22), shown at the bottom of the next page (see Appendix II). Equation (22) reveals that, even at low SNRs, Algorithm still gives a satisfactory performance. For Algorithm , we expect its performance to be between those of

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Algorithms and , since the former uses all of the available autocorrelation values while one of the two autocorrelation , which involves a smaller values the latter uses is number of time-correlation samples than and is, therefore, less reliable. The simulation results shown in Section V do confirm this conjecture. It is also clear that, as long as the true CFO does not incur ), then the th roots of are phase ambiguity (i.e., , . Although given by in a noiseless environment (assuming, without loss of generality, the desired root is ), the variance analysis presented in Apyields a smaller variance. As a pendix II shows that result, Algorithm yields a performance superior to that of the generalized Moose (GM) estimate obtained by averaging over two consecutive overlapped median symbols [see Fig. 1(a)] times. Later simulation results indeed reveal the superiority of Algorithm . Furthermore, it also suggests that step 4) can be replaced by picking up the th root whose principal argument is closest to .

V. SIMULATION RESULTS AND DISCUSSION Numerical examples are provided in this section to examine the behavior of the proposed CFO estimation technique. As shown in Fig. 1(a), eight short training symbols which are the same as those used in the IEEE 802.11a preamble are used in our simulation. Results reported in Figs. 3–6 assume a static frequency-selective fading channel with ten paths whose complex amplitudes are independent identically distributed (i.i.d.) complex Gaussian random variables. CFO is normalized by subcarrier spacing and the mean values and mean-squared errors independent (MSE) of various estimates are computed by trials. For comparison purposes, the corresponding CRBs and the behaviors of four other estimates are provided as well. These estimates are the MTB estimate (which outperforms the MM estimate [10]), the Moose estimate [2], the GM estimate, and the Patel–Song [7], [8] (PS) estimate. The last estimate uses the GM estimate as the initial estimate and selects the final estimate , from the family of candidate estimates where is the Moose estimate and is the set of integer that is the multiples of twice the acquisition range of closest to the initial estimate. The GM estimate is a special case of our proposals, corresponding to one that uses the truncated . version of the spectrum polynomial, The GM estimate has a frequency acquisition range larger than that of the Moose estimate but renders a less accurate estimation when CFO is within the latter’s acquisition range. The PS estimate is designed to retain the advantages of both algorithms,

Fig. 3. (a) Averaged estimation values for various CFO estimates. (b) MSE performance of CFO estimates; K = 8, and true CFO = 0:48 subcarrier spacings.

i.e., having an acquisition range the same as that of the GM estimate while achieving the performance of the Moose estimate. The coefficients used by the MTB estimate to optimally combine the differential phases of the elements in the set depend 20 dB on the operating SNR. In our simulation, a design is assumed. The MM estimate is not compared for it best percorrelations are used [10]. formance is achieved when only For the purpose of fair comparison, all algorithms whose performance is shown in the same figure use the same number of samples. Fig. 3(a) and (b) depicts the mean and MSE of various CFO estimates as a function of SNR. Since the maximum CFO , the that can be corrected by the Moose estimate is

(22)

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Fig. 4. MSE performance of CFO estimates; subcarrier spacings.

K

= 8, and true CFO = 1:8

corresponding performance degrades rapidly at low SNR when the true CFO approaches the maximum correctable value. For and the MTB and GM estimates, Algorithms , , and , hence they however, the maximum correctable CFO is give a much better MSE performance with Algorithm yielding a performance almost the same as that of the CRB. Both and (corresponding to ) outperform Algorithms ) and GM estimates. the Moose (corresponding to For 802.11a systems, using eight short training symbols not only allows a larger CFO offset range but also yields better performance. At CFO , as shown in Fig. 3, the Moose estimate is worse than the GM estimate at low SNRs. is simpler but its performance is a little worse Algorithm than those of Algorithms and , though it is still better than the GM estimate at low-to-medium SNRs. The performance of the MTB estimate is almost the same as that of Algorithm A except at low SNR. . AlgoFig. 4 plots the MSE performance for CFO rithms and and the MTB estimate still maintain superior performance in this case. For the MTB estimate, however, the effects of SNR mismatch and initial CFO estimation error become apparent at low SNRs. Fig. 5 plots the MSE performance when CFO using four short symbols. The Moose esshort timate uses two identical symbols: each lasts for symbol durations so that the maximum offset range it can correct . Fig. 6 compares the MSE performance of our becomes algorithms and the GM and PS estimates when CFO . Figs. 5 and 6 clearly indicate that the PS (or Moose) estimate is not necessarily better than the GM estimate but that the proposed methods still provide the best performance although the improvement is less impressive. This confirms our analysis in Section IV where it is shown that the performance of Algorithm improves as increases. Finally, we examine the performance in a time-varying channel composed of ten uncorrelated fading paths generated by the modified Jakes’ model [17]. The maximum path delay is equal to ten data sample intervals, assuming a sampling rate of 20 MHz. The performance of our CFO estimates is independent of the path numbers so long as

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Fig. 5. MSE performance of CFO estimates; K = 4, and true CFO = 0:94 subcarrier spacings.

Fig. 6. MSE performance of CFO estimates; subcarrier spacings.

K

= 4, and true CFO = 1:6

the channel’s maximum delay spread is shorter than the length of the cyclic prefix. Assuming a mobile speed of 100 km/h, corresponding to a Doppler frequency of approximately 463 Hz when the carrier frequency is 5 GHz, we plot the corresponding MSE performance in Figs. 7 and 8. As our derivations assume a quasi-static channel that remain unchanged during the preamble period, the estimation performance is degraded due to the fact that the received signal model (4) is no longer valid. In sumand the MTB estimate render the mary, Algorithms and best performance, followed by Algorithm , and then the other correlation-based algorithms. When is small, Algorithms , , and yield almost the same MSE performance. The proof posed methods can be used when an arbitrary number identical pilot symbols are available. As for the computational complexity issue, we first noticed . Different that the basic requirement is the computing of methods call for the computation of different subsets of the sufficient statistic . The detailed computational complexity

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the initial phase value for searching for the roots. An elementary root-finding algorithm like Newton’s method can then be in just a applied to find the first unit-magnitude root of few iterations, as the desired root has a phase very close to that . of VI. CONCLUSION

Fig. 7. MSE performance of CFO estimates in a time-varying frequency-selective Rayleigh fading channel; K = 8, and true CFO = 1:8 subcarrier spacings.

The optimal ML CFO estimate that uses several identical pilot symbols has been derived. By transforming the loglikelihood function into a spectrum polynomial, we reduce the ML estimate’s complexity from that of an exhaustive search over a continuum to that of solving a polynomial. Besides the ML estimate (Algorithm ), we propose two simplified and ) and show that some of the versions (Algorithms previous correlation-based algorithms are special cases of our proposals. The MSE performance of the proposed algorithms are analyzed in detail, and our analysis does match the simulation results. Both the analysis and simulations indicate that the performance of Algorithm is very close to the corresponding and require much less computational CRB. Algorithms load but they do not incur a noticeable loss of performance. Numerical results also demonstrate that, in a time-varying Rayleigh fading channel, the performance of the proposed methods suffers from minor performance degradation only. APPENDIX I DERIVATION OF (18) AND THE ROOT DISTRIBUTION OF Dividing by , we obtain the quotient polynomial defined by (19) and the remainder polynomial

Fig. 8. MSE performance of CFO estimates in a time-varying frequency-selective Rayleigh fading channel; K = 4, and true CFO = 1:6 subcarrier spacings.

(A1) , . In the absence of noise, we have with . Invoking the definition and substituting the resulting alter, , and native expressions , for , we for . Therefore, and find that the polynomial are indeed a factor of . Similarly, it is is easy to see that, if the ensemble average and , we still have the factorization used to construct (18). Obviously, the roots of the quotient polynomial are the set , as one of its members. Next we which has the desired root prove that the other roots of are located at the sidelobes where

and storage requirement for the CFO estimation algorithms short pilot are listed in Table I, assuming that there are symbols available with each symbol of -sample duration. Clearly, Algorithm and the MTB estimate require the highest complexity while Algorithm is the simplest among the three proposed algorithms. The MTB algorithm needs to perform down-converting (multiplying exponentials) that require table loop-ups. The storage requirement for building the table is not included in Table I. The computing effort for finding the roots needs one additional real is also not included. Algorithm division only. Algorithm has to solve a polynomial of order four that renders a closed-form expression while Algorithm has to solve a polynomial of order , and the associated complexity is relatively high. A method to significantly reduce as this complexity can be found if one uses the phase of

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TABLE I COMPLEXITY COMPARISON OF VARIOUS CFO ESTIMATION ALGORITHMS

of the log-likelihood spectrum . Assuming

. Recall that is even, we obtain

, and the discrepancy matrix consists of three components (B2) where

(A2) It is obvious that if , for . Hence, when , for , which are the remaining roots of . The case is odd can be similarly proved. Note that the roots of are the extreme values of the spectrum of . Since the are located at either the global maximum or the roots of local minima of , the roots of should sit at the local maxima, i.e., sidelobe peaks of the spectrum, as shown in Fig. 2. The above discussion assumes a noiseless environment. When noise is present, we can easily show that the factorization still holds in the mean sense. Furthermore, the analysis of presented in Appendix II convinces us that factorization of (18) will remain valid with a probability close to one unless SNR is very small (say, 0 dB).

, . The first component, , is independent of SNR and the CFO estimation algorithm. Using the , we exdefinition press the second component as the sum of two matrices , where the entries of the second matrix are given by while the first matrix is a diagonal matrix. The third component, , represents the cross correlation of the received samples and affect the and noise. We will show that only mean-squared performance of our CFO estimates. are i.i.d. We first note that the assumption that zero-mean Gaussian rvs with variance leads immediately to for , and then the equations

APPENDIX II MSE PERFORMANCE ANALYSIS

(B3)

A. Decomposition of the Correlation Coefficients The performance of the proposed algorithms depends on whose coefficients , as shown in R2 the behavior of of the main text, are functions of the autocorrelation matrix . We thus begin our analysis by examining

for

(B4)

(B1)

where discrepancy matrix between erage, , i.e., It can be shown that

. Let be the and its ensemble av. , where

, and for . for Furthermore, it can be shown that all of the elements in the upper or lower triangular part of are zero mean uncorrelated rvs and . with identical variance Therefore, the rvs (B5)

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have zero mean and variances for . Moreover, , , and is a sequence of uncorrelated rvs. Next, let us examine the statistical properties of . Recall that , whose entry is given by

is a zero-mean rv. where Thus, the MSE of the estimate can be obtained as soon as the second moment of is known. Invoking the alternate expression and the facts that and are uncoris white but not , we obtain related and

(B6) . It follows immediately that , for is assumed to be uncorrelated with , for all . Equation (B6) thus gives , for . Similarly, we can show that the entries of the cross-correlation matrix are uncorrelated unless they belong to the same column or . row in the upper (or lower) triangular part of , for , In other words, , for , but and , for , and , for . The new zero-mean rvs

(B11)

with

(B7)

have variances for and the Hermitian property but they are not independent. as Now we can express the sequence

(B8) Note that

,

, and

are uncorrelated.

B. Performance of Algorithm As Algorithm calls for locating the point (or ) that maximizes (or ), it follows that , where . Assuming small estimation errors , we obtain , where and . The fact that implies , and thus the estimation error can be approximated by

(B9) The statistical properties of and derived in the previous section convince us that, with high probability,

and tions due to simplify (B9) to

where is the sum of the cross-correlation value of corre’s, i.e., lated elements

(B12) As mentioned in Appendix II-A, and for , after some algebra, we can show that the first term on the right-hand side of (B11) is equal to

(B13) remains to be determined. For convenience, The value of we define the auxiliary rv

(B14) for , , and . If , the associated rv is zero with probability 1. It follows that entries are uncorrelated of the upper triangular matrix unless they belong to the same row or column. In other words, for and for , but for and for . By using (B7), (B12), and (B14), we can prove that is equivalent to the sum of cross correlations of the entries of , i.e.,

, if 0 dB. Ignoring the perturbain the denominator, we can further

(B10)

(B15)

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(B22)

where

denotes the “exclusive or” operator. As , and the recursive relation

,

is simply the corresponding variance of , denoted by , given by (B23)

(B16) holds for

Numerical results show that the difference between and is about 1 dB within the range of interest. , and the Equation (B21) implies gives polar coordinate representation

, we obtain (B24)

(B17) This equation, along with (B10), (B11), and (B13), yields

where and are the amplitude and phase of spectively. When , i.e., and we have

, re,

(B18) (B25) The relation

then leads to (B19)

C. Performance of Algorithm Algorithm calls for solving evaluation of the corresponding roots . Since

define intermediate rvs

or, equivalently, the , , , and , it is convenient to and (whose variances are given by and , respec-

tively) so that we can write (B20) where and . Note that and are correlated but their correlation is a decreasing function of and their variances are and , respectively. After some algebra, we obtain (B21) . It can be shown that , the variance of , is given by (B22), shown at the top of the page. However, if we assume that and are uncorrelated, where

where we have used the approximation , for . Hence, the variance of can be obtained from . Without loss of generin ality, we assume that the phase of the desired root of (19) is , then the CFO estimate becomes , and, as a result, by substituting , we obtain the variance of the estimate (22) as (B26)

REFERENCES [1] J. Terry and J. Heiskala, OFDM Wireless LANs: A Theoretical and Practical Guide. Indianapolis, IN: Sams, 2001. [2] P. H. Moose, “A technique for orthogonal frequency division multiplexing frequency offset correction,” IEEE Trans. Commun., vol. 42, pp. 2908–2914, Oct. 1994. [3] F. Daffara and O. Adami, “A novel carrier recovery technique for orthogonal multicarrier systems,” Eur. Trans. Telecommun., vol. 7, pp. 323–334, July/Aug. 1996. [4] H. Liu and U. Tureli, “An high-efficiency carrier estimator for OFDM communation,” IEEE Commun. Lett., vol. 2, pp. 104–106, Apr. 1998. [5] T. M. Schmidl and D. C. Cox, “Robust frequency and timing synchronization for OFDM,” IEEE Trans. Commun., vol. 45, pp. 1613–1621, Dec. 1997. [6] M. Morelli and U. Mengali, “An improved frequency offset estimator for OFDM applications,” IEEE Commun. Lett., vol. 3, pp. 75–77, Mar. 1999. [7] H.-K. Song, Y.-H. You, J.-H. Paik, and Y.-S. Cho, “Frequency-offset synchronization and channel estimation for OFDM-based transmission,” IEEE Commun. Lett., vol. 4, pp. 95–97, Mar. 2000. [8] S. Patel, L. S. Cimini, and B. McNair, “Comparison of frequency offset estimation techniques for burst OFDM,” in Proc. 55th IEEE Vehicular Technology Conf., Birmingham, Al, May 2002, pp. 772–776.

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[9] J. Li, G. Liu, and G. B. Giannakis, “Carrier frequency offset estimation for OFDM based WLANs,” IEEE Signal Processing Lett., vol. 8, pp. 80–82, Mar. 2001. [10] H. Minn, P. Tarasak, and V. K. Bhargava, “OFDM frequency offset estimation based on BLUE principle,” in Proc. IEEE Vehicular Technology Conf., Vancouver, BC, Canada, Sept. 2002, pp. 1230–1234. [11] Y. S. Lim and J. H. Lee, “An efficient carrier frequency offset estimation scheme for an OFDM system,” in Proc. 52nd IEEE Vehicular Technology Conf., Sept. 2000, pp. 2453–2457. [12] A. J. Coulson, “Maximum likelihood synchronization for OFDM using a pilot symbol: part 1: algorithms, part 2: analysis,” IEEE J. Select. Areas Commun., vol. 19, pp. 2486–2503, Dec. 2001. [13] T. Keller, L. Piazzo, P. Mandarini, and L. Hanzo, “Orthogonal frequency division multiplex synchronization techniques for frequency-selective fading channels,” IEEE J. Select. Areas Commun., vol. 19, pp. 999–1008, June 2001. [14] L. Piazzo, T. Keller, A. Falaschi, and P. Mandarini, “Time and frequency synchronization in DQPSK-OFDM based high speed wireless local area networks,” Eur. Trans. Telecommun., vol. 13, no. 3, pp. 279–284, May/June 2002. [15] M. Ghogho and A. Swami, “Blind frequency-offset estimator for OFDM systems transmitting constant-modulus symbols,” IEEE Commun. Lett., vol. 6, pp. 343–345, Aug. 2002. [16] P. Stoica and A. Nehorai, “MUSIC, maximum likelihood, and Cramér-Rao bound,” IEEE Trans. Acoust., Speech, Signal Processing, vol. 37, pp. 720–741, May 1989. [17] Y. Li and Y. L. Guan, “Modified Jakes’ model for simulating multiple uncorrelated fading waveforms,” in Proc. 51st IEEE Vehicular Technology Conf., Tokyo, Japan, May 2000, pp. 1819–1822. [18] G. H. Golub and C. F. VanLoan, Matrix Computations, 3rd ed. Baltimore, MD: Johns Hopkins Univ. Press, 1996.

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Jiun H. Yu was born in Nantou, Taiwan, on March 16, 1979. He received the B.S. and M.S. degrees in communication engineering from National Chiao Tung University, Hsinchu, Taiwan, in 2001, and 2003, respectively. He is currently a System Design Engineer with the Research and Development Center, Realtek Semiconductor Corporation, Hsinchu. His research interests include digital transmission systems, detection and estimation theory, and communication signal processing.

Yu T. Su (S’81–M’83) received the Ph.D. degree in electrical engineering from the University of Southern California, Los Angeles, in 1983. From 1983 to 1989, he was with LinCom Corporation, Los Angeles, where he was a Corporate Scientist, where his work involved the design of various measurement and digital satellite communication systems. Since September 1989, he has been with National Chiao Tung University, Hsinchu, Taiwan, where he was head of the Communication Engineering Department between 2001 and 2003. He is also affiliated with the Microelectronics and Information Systems Research Center of the same university and served as a Deputy Director from 1997 to 2000. His main research interests include communication theory and statistical signal processing.

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2009

A Class of Nonlinear Signal-Processing Schemes for Bandwidth-Efficient OFDM Transmission With Low Envelope Fluctuation Rui Dinis, Member, IEEE, and António Gusmão

Abstract—This paper presents a wide class of digital signal-processing schemes for orthogonal frequency-division multiplexing (OFDM) transmission which combine a nonlinear operation in the time domain and a linear filtering operation in the frequency domain. The ultimate goal of these schemes is to reduce the envelope fluctuation of ordinary OFDM, while keeping its high spectral efficiency and allowing a low-cost, power-efficient implementation. An appropriate statistical model concerning the transmitted frequency-domain blocks is developed, which is derived from well-established results on Gaussian stochastic processes distorted by memoryless nonlinearities. This model can be employed for performance evaluation by analytical means, with highly accurate results whenever the corresponding conventional OFDM signals exhibit quasi-Gaussian characteristics. Cases where the signal-processing scheme is repeatedly used, in an iterative way, are treated through an extension of the proposed statistical modeling. A set of numerical results is presented and discussed so as to show the practical interest of both the proposed schemes and the analytical methods for evaluation of their performance. For the sake of comparisons, this paper includes numerical results concerning the partial transmit sequence technique, which is an alternative peak-to-mean envelope power ratio-reducing technique of higher complexity, often recommended due to its distortionless nature. The superior performance/complexity tradeoffs through the proposed class of nonlinear signal-processing schemes is emphasized. Index Terms—Envelope fluctuations, Gaussian stochastic processes, nonlinear signal processing, orthogonal frequency-division multiplexing (OFDM) transmission.

I. INTRODUCTION WELL-KNOWN, major drawback of conventional orthogonal frequency-division multiplexing (OFDM) transmission schemes [1] is their strong envelope fluctuation and high peak-to-mean envelope power ratio (PMEPR), leading to power amplification difficulties. In order to avoid the out-of-band radiation levels which are inherent to nonlinear distortion, power amplifiers for OFDM transmission are required to have linear characteristics and/or a significant input backoff has to be adopted. Therefore, a reduced power efficiency is the price to pay for a high bandwidth efficiency. In recent years, several methods have been proposed to reduce these amplification difficulties by means of digital signal

A

Paper approved by N. Al-Dhahir, the Editor for Space–Time, OFDM, and Equalization of the IEEE Communications Society. Manuscript received May 11, 2003; revised September 2, 2003 and March 6, 2004. The authors are with CAPS, Instituto Superior Técnico, 1049-001 Lisboa, Portugal (e-mail: [email protected]). Digital Object Identifier 10.1109/TCOMM.2004.836567

processing. Some of these methods operate in the frequency domain, using a reasonable amount of redundancy to avoid high amplitude peaks when the number of subcarriers is low [2]. However, as this number increases, the code rate of the required PMEPR-reducing frequency-domain codes becomes lower and lower. The so-called partial transmit sequence (PTS) techniques [3]–[5], which also operate in the frequency domain, can achieve a strongly reduced PMEPR for a large number of subcarriers while using a very small redundancy. The main drawback of these techniques is their very high computational complexity for large OFDM blocks, mainly due to the optimization procedures which are required on a block-by-block basis. Some other signal-processing methods rely on a time-domain operation, such as a digital clipping of the high amplitude peaks, for PMEPR-reduction purposes. Following the pioneering contributions in this direction (namely, in [6] and [7]), it has been recognized that the clipping operation should be performed on oversampled OFDM bursts and followed by a filtering procedure, so as to reduce the out-of-band radiation levels while alleviating the peak regrowth problem. A class of low-complexity signal-processing schemes for reduced PMEPR, spectrally efficient OFDM transmission was proposed in [8] by the authors of this paper. Such schemes combine a nonlinear operation in the time domain (possibly according to a clipping characteristic) with a linear, filtering operation in the frequency domain. This frequency-domain filtering, besides not requiring an increased guard time to avoid intersymbol interference (ISI), can be very selective, e.g., completely removing the out-of-band radiation effects of the preceding nonlinear operation. More recently, we analyzed in [9] a similar class of signal-processing schemes for the same goals. The difference between [8] and [9] lies in the type of nonlinear operation with each class: a nonlinearity operating on the complex OFDM samples in [8], and a nonlinearity which separately operates on their real (I) and imaginary (Q) parts in [9]. This paper considers the former class only, which provides better performance, and studies the cases where a given scheme (nonlinear time-domain operation followed by frequency-domain filtering) is repeatedly used, in an iterative way. The several operations along the signal-processing chain are described in Section II. An appropriate statistical modeling is then presented, in detail, in Section III; this modeling approach is derived from well-established results on memoryless nonlinearities with Gaussian inputs (e.g., [10]). Section IV is concerned with the performance evaluation of the proposed

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Fig. 1.

Block diagram for the proposed class of signal-processing schemes.

schemes by analytical means: we derive the power spectral density (PSD) of the transmitted signals, as well as the bit-error rate (BER) performance, conditioned on a given channel realization, taking advantage of the statistical modeling previously developed. Section V presents performance results for both the basic (one iteration) signal-processing schemes and, as a natural extension, the multi-iteration schemes; for the sake of comparisons, some results concerning a PTS technique are included. Finally, Section VI presents the conclusions of this paper.

II. A WIDE CLASS OF NONLINEAR SIGNAL-PROCESSING SCHEMES With the basic signal-processing schemes considered here for OFDM transmission (see Fig. 1), first proposed in [8], each block of time-domain samples is generated as follows. • An augmented frequency-domain block , where is a power of two, is formed as folzeros to the original frequency-dolows, by adding main block , directly related to data: for , for , for . and • The inverse discrete Fourier transform (IDFT) of this frequency-domain block is computed, leading to the time, with domain block . • Each sample is submitted to a nonlinear operation ac(with a selected cording to function of the envelope ), leading to the . This nonlinear modified block operation can be regarded as a deliberate signal distortion by a bandpass memoryless nonlinearity, with underlying amplitude modulation (AM)/AM and AM/phase modulaand tion (PM) characteristics given by , respectively. • A discrete Fourier transform (DFT) brings the nonlinearly modified block back to the frequency domain, where a shaping operation is performed by a multiplier bank with , , so as to selected coefficients obtain the block , with (some of coefficients are equal to zero in the out-of-band these region). • The final frequency-domain block results from by removing the zeros. • This final frequency-domain block is then used for generating the reduced-PMEPR OFDM burst, as usual: IDFT,

cyclic extension and time-domain windowing, reconstruction filtering. For the sake of simplicity, we will assume in , which means that the following that . zeros to each input block prior to comAppending puting the required IDFT is a well-known OFDM implementation technique, which is equivalent to oversampling, by a factor , the ideal OFDM burst. The subsequent nonlinear operation is crucial for reducing the envelope fluctuations, whereas the frequency-domain filtering using the can reduce the resulting specset tral spreading (of course, with some regrowth of the envelope subcarriers with zero fluctuations). The removal of amplitude reduces the computational effort and corresponds to and a careful a decimation in the time domain. For a given , the nonlinear characteristic (for a selection of set can ensure small envelope given input level) and the fluctuations while maintaining low out-of-band radiation and in-band self-interference levels. When the nonlinear operation is chosen to be an ideal envelope clipping, with clipping level

(1) It should be mentioned that this wide class of signal-processing schemes (following our original proposal in [8]) includes, as specific cases, schemes proposed by other authors in and the meantime. This is the case in [11] ( envelope clipping). This is also the case in [12] and [13], where , with the same clipping is adopted when assuming for the in-band subcarriers, and out-of-band. It should also be mentioned that, for an ideal envelope clipping, the proposed signal-processing scheme can be shown (see [14]) to be equivalent to the peak cancellation method reported in [15, Ch. 6], according to the ideas of [16]. A more sophisticated technique, allowing improved PMEPRreducing results, could be simply developed on the basis of the signal-processing approach described above. Such a technique consists of repeatedly using, in an iterative way, the signal-proto in Fig. 1. (Of cessing chain which leads from for and , course, where each superscript concerns a given iteration.) The technique proposed in [17] corresponds to the particular case where the nonlinear operation is an envelope clipping, and the frefor the quency-domain filtering is characterized by out-of-band subcarriers. III. MODELING OF THE TRANSMITTED BLOCKS ALONG THE SIGNAL-PROCESSING CHAIN This section shows how to obtain an appropriate statistical characterization for the modified block of frequency-domain

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samples in Fig. 1, , that replaces of conventional OFDM the block schemes. For this purpose, we need to statistically characterize the blocks along the signal-processing chain, as shown in Sections III-A–C. Section III-D briefly addresses the modeling process for the iterative technique reported at the end of Section II.

2011

It can be shown [18] that

(6)

A. Time-Domain Block at the Input to the Nonlinearity In this paper, it is assumed that and ( for , and 0, otherwise), with ( denotes ensemble average). Therefore, it and can be easily demonstrated that

denotes the total power associated where the coefficient . This with the intermodulation product (IMP) of order coefficient can be obtained as follows [19]: (7) where

(2) , , with . The generalization for the case where the power is not uniformly distributed across all subcarriers is straightforward. When the number of subcarriers is high , the time-domain coefficients can be approximately regarded as samples of a zeromean complex Gaussian process with autocorrelation given by (2). B. Time-Domain Block at the Output of the Nonlinearity In the following, we take advantage of the quasi-Gaussian nature of the samples for obtaining the statistical characterization of the time-domain block at the output of the nonlinearity. In fact, it is well known that the output of a memoryless nonlinear device with a Gaussian input can be written as the sum of two uncorrelated components [10]: a useful one, proportional to the input, and a self-interference one. As a consequence, the nonlinearly modified samples can be decomposed into uncorrelated useful and self-interference components, as follows: (3) where

and the gain of the useful component is

with (see [19]) ( denotes a generalized Laguerre polynomial of order [20]). Since , it can be easily recognized that and

(8) The total self-interference power is . This method for statistical characterization of the transmitted blocks is quite appropriate whenever the power series in (6) can be reasonably truncated while ensuring an accurate computation. However, for strongly nonlinear conditions, the required number of terms becomes very high. In such cases, one can simplify the computation as explained below. When , for and 0, otherwise, which means that the frequency-domain distribution of the power associated to a given IMP is almost constant . As a consequence, the contribution of the for th IMP to the output autocorrelation can be approximated by , leading to

(9)

(4) The average power of the useful component is , and the average power of the self-interference component is given by , where denotes the average power of the signal at the nonlinearity output, given by

(5)

, with . This means that, besides computing , we just have to calculate the terms inherent to the first IMPs. C. Final Frequency-Domain and Time-Domain Blocks Having in mind (3) and the signal-processing chain in Fig. 1, the frequency-domain block can obviously be decomposed into useful and self-interference components (10)

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where . Clearly,

is the DFT of and

otherwise.

(11)

, where denotes the DFT of the block . Similarly, it can be shown that for and , where de[given by notes the DFT of DFT (6)], with . Therefore, for , and . By employing the statistical characterization of the frequency-domain block to be transmitted, one can calculate a signal-to-interference ratio (SIR) for each subcarrier, given by (12) . Without oversampling, (2) leads to From (6), for and for ; therefore, , which is independent of , when . It should be noted that for (i.e., when , leading to ), is a depends on . function of , since ) It can be shown that when is high enough (say, to validate a Gaussian approximation for the samples at the nonlinearity input, our modeling approach is quite accurate. It typcan also be shown that, under this high assumption, ically is quasi-Gaussian for any , provided that the nonlinear distortion effort is significant for many time-domain samples in each OFDM block. This explains the similarity of the analytical and the simulated BER results (see Section V).

as a generalization of (10). As with the first iteration, our can exhibit simulations indicate that quasi-Gaussian characteristics, provided that is high enough; moreover, , for , which means that , for . The values of and can be obtained by simulation in the following way: (14) (15) Therefore, we can use (14) and (15) to calculate, for each subcarrier

(16)

IV. PERFORMANCE EVALUATION A. Spectral Characterization of the Transmitted Signals The complex envelope of the transmitted OFDM signal can , where corbe written as responds to the th OFDM block and is the block interval. The OFDM samples for each block, taken at a rate [i.e., with an oversampling factor ], are given by , where (with denoting an appropriate window shape). The block is the (for the sake of notational IDFT of simplicity, we ignore the dependency with ). When assuming that the time-domain samples and the frequency-domain samples (i.e., the DFT pairs) are periodic with period , the complex envelope of a given OFDM burst can be written as

D. Modeling Issues for the Iterative Technique Let us consider the repeated use of the signal-processing chain leading from to in Fig. 1. After the first iteration, the Gaussian approximation for the samples at the input to the nonlinearity ceases to be reasonable; therefore, the computational method described in Sections III-B and C for modeling of the transmitted blocks has to be appropriately modified. In fact, our extensive simulation work shows that the th component of the frequency-domain block, for the th iteration, can still be decomposed as a sum of two uncorrelated components (13) with depending on for a given . Therefore, the frequency-domain samples at the output of the nonlinearity can still be decomposed into a useful component and a self-interference component, but that useful component is no longer proportional to the input after the first iteration (this means ). Of that a decomposition similar to (3) is not valid for course, in (13), and this equation can be regarded

(17) where denotes the impulse response of the low-pass resequence is periodic, but the construction filter (the sequence is not, of course). For the raised-cosine-shaped windows considered in [15] (see Fig. 2), we can write (18) where and

is a rectangular pulse with duration (19)

( for and zero, otherwise), with denoting the guard interval duration, used to cope with time-dispersive channel effects. The window duration is .

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DINIS AND GUSMÃO: A CLASS OF NONLINEAR SIGNAL PROCESSING SCHEMES FOR BANDWIDTH-EFFICIENT OFDM

depends on , and, consequently, on concerned with the reconstruction filter.

2013

; and

is

B. Peak-to-Mean Envelope Power Ratio (PMEPR)

Fig. 2. Window shape.

Of course,

corresponds to since , i.e., the conventional rectangular windowing; by , we have an additional filtering effect that allows using a reduced out-of-band radiation. This means that the generation of the OFDM signals considered here involves three filtering effects. There is an early filtering effect in the frequency-domain , associated with the coblock , ; another filtering effect is efficients concerned with the reconstruction filter, with impulse response ; and, finally, from (18) and (19), there is inherently an additional filtering effect on each subchannel, which is associated with . By assuming that the different OFDM bursts are uncorrelated and have the same statistical properties, resulting from and , the complex envelope of the transmitted OFDM signals has a PSD given by (20) denotes the Fourier transform of the OFDM burst where [see (17)–(19)]. It can be written as (21) with (22) since

(23) denotes Fourier transform). Therefore, by using (21) in ( (20), we get

(24) since it can be assumed that for . The three filtering effects associated with the generation of OFDMdetype signals as proposed here can be found in (24). pends on the frequency-domain filtering through ;

, each When the number of subcarriers is high can be regarded a sample of a zero-mean complex Gaussian process with autocorrelation given by (2). In this case, the envelope has a quasi-Rayleigh distribution, which means a strong envelope fluctuation. Since the highest envelope values have a very small probability, it is reasonable to define the PMEPR in a statistical way, as follows: (25) is the envelope value that is exceeded with probawhere bility . For a Rayleigh-distributed envelope

(26) and

. A reasonable value for is , which corresponds to PMEPR dB, regardless of ). the value of (as long as Due to the nonlinearity of the proposed signal-processing schemes, the transmitted signals are no longer of the Gaussian type. Thanks to this nonlinearity, the resulting PMEPR can be made much lower than the PMEPR of the corresponding conventional OFDM signals. C. BER Issues It should be emphasized that the proposed signal-processing schemes only involve modifications at the transmitter side, being compatible with conventional OFDM receivers. The received signal is submitted to the receive filter, and , leading then sampled at a rate to blocks after removal of the guard period. A square-root raised-cosine (SRRC) reis assumed (as well as for ceive filter characteristic ; it is also asthe transmit filter), with rolloff sumed that in-band. Next, a DFT DFT operation leads to , where is concerned with the th subchannel. It is well known that, thanks to the cyclic prefix, (when the guard interval is longer than the overall channel impulse response), where and denote the noise component and the overall channel frequency response (CFR), respectively, at the th subchannel. It can also be shown that these frequency-domain noise samples, uncorrelated and Gaussian, are zero mean, and the variance of their I , when denotes and Q parts is given by the PSD of the white Gaussian noise at the receiver input. As described by (13), the frequency-domain samples can be decomposed into two uncorrelated components, even when the repeated use of the basic signal-processing chain iterations, is assumed: a useful compo(see Fig. 1), with nent, , and a self-interference component .

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This means that, besides the Gaussian channel-noise component, each received frequency-domain block can also be decomposed into uncorrelated useful and self-interference compo. Moreover, nents, since for high , each zero-mean term is quasi-Gaussian with regards to (see Section III-D), adding to the Gaussian . noise term For an ideal, coherent receiver with perfect synchronization and channel estimation, the BER at the th in-band subchannel or ) can be ( expressed as (27) and the overall BER is then the average of the BERs associated with the in-band subchannels. In the equation above, and are functions of the adopted constellation (for -ary quadrature amplitude modulation ( -QAM) constellations and and Gray mapping, ), denotes the well-known Gaussian error function, and SNR denotes an equivalent signal-to-noise ratio. This ratio is given by (28) with and the other expectations computed as explained in Section III (analytically for , ). by simulation for Asymptotically, when the channel-noise effects become negligible, (27) takes the form , where , which means that the nonlinear distortion should lead to an irreducible BER, only, for any given constellation. When depending on and , as reported in Section III, is , with denoting the overall independent of and given by useful power and denoting the overall self-interference power. This allows very simple BER computations, since we do not need to separately evaluate the powers of the different IMPs, , required for the computation of . For an ideal Gaussian channel, in-band; therefore, (28) can be written as (29) where (

is the degradation factor due to the guard interval if ) (30)

(independent of carriers) and

when

is constant for the

in-band sub-

(31)

Fig. 3. FOBP when N = N , s = = 2:0, and T =T = 0:2, for T = 0 (dashed line) or T =T = 0:025 (dash-dotted line). For the sake of comparison, we include the FOBP when G = 1 for any k and T =T = 0 (dotted line) or 0.025 (solid line), respectively.

Clearly, the degradation factor is associated with the useless power spent in the transmitted self-interference; the is due to the fact that the received degradation factor quasi-Gaussian self-interference is added to the Gaussian channel noise on each subcarrier. V. PERFORMANCE RESULTS In the following, we present a set of performance results concerning the proposed class of nonlinear signal-processing schemes. We consider an OFDM modulation with subcarriers and an -QAM constellation, with a Gray mapping rule, on each subcarrier (this value of is high enough to allow the Gaussian approximation of the OFDM signals). The set of multiplying coefficients has a trapezoidal shape, with for the data subcarriers (in-band region), dropping linearly to 0 along the first out-of-band subcarriers at both sides of the in-band nonzero subcarriers. The nonlinear region, which means operation is chosen to be an ideal envelope clipping [see (1)]. Both transmit and receive analog filter characteristics have an SRRC shape with and a one-sided bandwidth . The transmitter employs a power amplifier which is quasi-linear within the range of variations of the input envelope, and the coherent detection operates under perfect synchronization and channel estimation. Let us consider the basic, single-iteration signal-processing schemes. Fig. 3 is concerned with the bandwidth-ef, ficiency issues when using these schemes, with a clipping level , and (similar results were observed for other practical values of ). A well-known PSD-related function was adopted: the so-called fractional out-of-band power (FOBP), defined as , for a symmetrical PSD. Clearly, this clipping can lead to high out-of-band radiation levels. However, by using a frequency-domain filfor the data subcarriers and 0 for the tering with remaining ones (i.e., ), we can reduce these

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=1 =1 =2 =15 ( )

(2) = () =4

Fig. 4. PMEPR in the following cases: M and G for any k ; and G for any k ;M and N N o ; M and N N ;M and N : N ;M and N : N .

M

= +1 =1 =4 = (5) = 1 5 (1)

=2

(+)

levels to those of conventional OFDM. Further reduction is achieved if this filtering is combined with the windowing procedure reported in Section IV-A, in this example, with (such windowing, if for any , practically should not provide any improvement in the FOBP). , we can still have out-of-band radiFor other values of ation levels similar to those of conventional OFDM schemes, is low enough. provided that Fig. 4 shows the PMEPR, defined according to (25) with , when , or, for the sake of for comparisons, with no frequency-domain filtering ( . This figure shows any ), for different values of that the PMEPR increases with and is slightly higher for (i.e., when the lowest out-of-band radiation levels are intended). As expected, the nonlinear operation allows an overall reduction of several decibels on the PMEPR, but the subsequent frequency-domain filtering leads to some PMEPR , the PMEPR for is alregrowth. When ; with higher ready close to the ideal PMEPR for values of , a PMEPR close to the one for could only be observed for . Let us consider now the corresponding BER performances on an ideal additive white Gaussian noise (AWGN) channel ), still with an envelope clipping as the non(allowing decreases linear operation. As expected, the required when is increased, and the BER performances with are already close to the corresponding performances with . However, increased values of imply a higher PMEPR, i.e., an increased power amplification backoff. Fig. 5 shows the required to have BER dB dB PMEPR dB . This peak is appropriate for an overall power-efficiency comparison between the different transmission alternatives, since it combines the detection-efficiency issue (through the required ) and the requirements on power amplification backoff (through the PMEPR). From Fig. 5, we can see that the opare 2.0 for 4, 2.6 for 16, and timum values of

2015

= 10 , with = 1 for any

Fig. 5. Required E =N for BER clipping, for the following cases: G : N (solid line). (dashed line); N

=15

M k

=2

and an envelope (dotted line); N N

=

Fig. 6. Envelope distribution for iterations 1–4 (solid lines), together with the envelope distribution for conventional OFDM (dashed line).

64. Moreover, these optimum values of are 3.2 for almost independent of the frequency-domain filtering effort. Let us consider now the iterative procedure analyzed in Section III-D, where we repeatedly use the signal-processing chain to in Fig. 1, so as to provide which leads from an additional reduction of the envelope fluctutions while preserving a compact spectrum. We assume iterations, , and an envelope clipping with , while for . Fig. 6 the in-band subcarriers and 0 for the remaining shows the impact of the number of iterations on the distribution of the signal envelope. From this figure, it is clear that we can strongly reduce the envelope fluctuations by using the iterative technique. The maximum envelope can be already close to with just three or four iterations. This corresponds to a PMEPR of about 5.8, 4.9, 4.7, and 4.5 dB for iterations 1, 2, 3, and 4, respectively. Since out-of-band, and taking into account the PSD of the transmitted signals [see (24)], it is clear that these PMEPR reductions are still achieved with out-of-band radiation levels similar to those of conventional OFDM.

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Fig. 7.

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Evolution of

SIR

,l

= 1 to 4, for an envelope clipping with s

= =

Fig. 8. BER for iterations 1–4 (solid lines), together with BER for conventional OFDM (dashed line) (the circles denote simulated values).

As shown in [21], we must take into account that there is an increase of the self-interference levels in the in-band region, , when inas well as decreased values of the gain factor levels. This is illuscreases, leading to a reduction in the trated in Fig. 7, and Fig. 8 shows the corresponding BER performances, when an ideal AWGN channel is assumed. These results, obtained by using (27) and (29)–(31), are shown to be closely matched to the results obtained through Monte Carlo simulations, which indicates the high accuracy of the proposed statistical modeling. On the other hand, the detection efficiency decreases when the number of iterations is higher, but this degradation is not too high. With four iterations, leading to a PMEPR of 4.5 dB (instead of a PMEPR of 5.8 dB for just one iteration), the corresponding degradation is only about 1.9 dB for , 1.2 dB for BER , and 0.6 dB for BER BER . This means that for BER , the additional degradation on the detection efficiency when we increase the number of iterations is lower than the corresponding PMEPR reduction , (see also Fig. 9, where the BER is plotted against as defined regarding Fig. 5, instead of ). Moreover, the

Fig. 9. BER for iterations 1 (solid line), 2 (dashed line), 3 (dotted line), and 4 (dash-dotted line).

= = =4

Fig. 10. PMEPR distribution with the PTS technique, when N 64 (solid line) or 256 (dashed line), and, for the sake of comparison, with a conventional OFDM scheme (dotted line): o for V and W , for V W , for V and W . and

()

=8

() =2

=4

= 2 (1)

PMEPR reduction itself is very useful, since it allows a simpler and more power-efficient amplifier implementation. An alternative method for reducing the PMEPR of OFDM signals, while keeping a high spectral efficiency, is to employ PTS techniques [3]–[5], which have been described as distortionless methods, where the frequency-domain OFDM subblocks that are phase roblock is decomposed into tated and combined to generate a minimum-PMEPR OFDM signal. In this paper, it is assumed that the first subblock is not phase rotated; for each of the remaining blocks, possible phase rotations, selected from the set there are (a continous partition is assumed). With these PTS techniques, we obtain the same BER performance and about the same spectral efficiency as with the corresponding conventional OFDM schemes. In Fig. 10, we show the envelope distribution of the samples of the transmitted OFDM signal, when the PTS technique is emis assumed ployed and an -QAM constellation with

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on each subcarrier (similar results could be obtained for other values of ). The reduction on the PMEPR is as follows. For , the the PMEPR values are approximately 6.8 dB with and , 6.4 dB with , and 5.8 dB with and ; for , these PMEPR values increase to 7.5, 7.2, and 6.9 dB, respectively. For the signal-processing schemes considered in this paper, with an envelope-clipping op, we obtain a PMEPR of 5.7 dB eration according to iteration (corresponding to a degradation of 0.2 dB for and 0.4 dB at BER ), and a PMEPR of at BER 4.5 dB for iterations (corresponding to a degradation of 0.6 dB at BER and 1.2 dB at BER ). This means that the PMEPR values are worse with the PTS technique than , unless the with the basic signal-processing scheme number of subcarriers is very low; even when we take into account the BER performance degradations due to the nonlinear distortion, the proposed schemes are shown to provide better re, the sults, especially for high . When used iteratively signal-processing schemes studied here exhibit an improved advantage. With very large constellations, however, the optimum values of (and the corresponding PMEPR values) are a bit higher, therefore decreasing the difference between the two techniques. Moreover, as an additional advantage, the signal-processing schemes presented in this paper do not require any modification in the OFDM receiver, and the implementation complexity at the transmitter side is much lower than with the PTS techniques; essentially, three fast Fourier transform (FFT) operations (for a single iteration) per OFDM transmitted block, versus FFTs plus the signal processing for the optimization of the phase rotations. The PTS techniques also require the transmission of an extra overhead for each block carrying the information on the selected set of phase rotations.

VI. CONCLUSIONS AND COMPLEMENTARY REMARKS In this paper, we presented and evaluated a wide class of digital signal-processing schemes for OFDM transmission which combine a nonlinear operation in the time domain, and a linear filtering operation in the frequency domain. The ultimate goal of these schemes is to reduce substantially the envelope fluctuation of ordinary OFDM, while keeping its high spectral efficiency and allowing a low-cost, power-efficient implementation. An appropriate statistical model concerning the transmitted frequency-domain blocks has been developed so as to provide analytical support for a computationally efficient evaluation of power-bandwidth tradeoffs. A detailed evaluation of OFDM transmission techniques which employ the signal-processing schemes considered here was carried out, involving computations of power spectra, BER performances, and achieved PMEPR values. Such evaluation has taken advantage of our statistical characterization of the transmitted blocks and included other implementation issues (such as the impact of the time-windowing procedures). A set of performance results was presented and discussed, showing that the proposed basic schemes can provide a significant PMEPR reduction while keeping a high spectral efficiency. The iterative technique

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based on the basic schemes was shown to allow a further PMEPR reduction, also maintaining the spectral efficiency of conventional OFDM, with only a moderately increased implementation complexity. When compared with the distortionless PTS techniques, also capable of reducing the PMEPR of OFDM signals, these signalprocessing techniques involving deliberate nonlinear distortion were shown to offer improved performance/complexity tradeoffs, for small constellations and a high number of subcarriers. REFERENCES [1] L. Cimini, Jr., “Analysis and simulation of a digital mobile channel using orthogonal frequency-division multiplexing,” IEEE Trans. Commun., vol. COM-33, pp. 665–675, July 1985. [2] A. Jones and T. Wilkinson, “Combined coding for error control and increased robustness to system nonlinearities in OFDM,” in Proc. IEEE Vehicular Technology Conf., vol. 2, Atlanta, GA, May 1996, pp. 904–908. [3] S. Müller and J. Huber, “OFDM with reduced peak-to-average power ratio by optimum combination of partial transmit sequences,” Electron. Lett., vol. 33, no. 5, pp. 368–369, Feb. 1997. [4] S. Müller, R. Bäuml, R. Fischer, and J. Huber, “OFDM with reduced peak-to-average power ratio by multiple signal representation,” Ann. Telecommun., vol. 52, pp. 58–67, Feb. 1997. [5] L. Cimini, Jr. and N. Sollenberger, “Peak-to-average power reduction of an OFDM signal using partial transmit sequences,” IEEE Commun. Let., vol. 4, pp. 86–88, Mar. 2000. [6] R. O’Neill and L. Lopes, “Envelope variations and spectral splatter in clipped multicarrier signals,” in Proc. IEEE Personal, Indoor, Mobile Radio Conf., vol. 1, Sept. 1995, pp. 71–75. [7] X. Li and L. J. Cimini, Jr., “Effects of clipping and filtering on the performance of OFDM,” IEEE Commun. Lett., vol. 2, pp. 131–133, May 1998. [8] R. Dinis and A. Gusmão, “A class of signal processing algorithms for good power/bandwidth tradeoffs with OFDM transmission,” in Proc. IEEE Int. Symp. Information Theory, June 2000, pp. 216–216. [9] , “A new class of signal processing schemes for bandwidth-efficient OFDM transmission with low envelope fluctuation,” in Proc. IEEE Vehicular Technology Conf., vol. 1, May 2001, pp. 658–662. [10] H. Rowe, “Memoryless nonlinearities with Gaussian input: Elementary results,” Bell Syst. Tech. J., vol. 61, pp. 1519–1525, Sept. 1982. [11] H. Ochiai and H. Imai, “Performance of deliberate clipping with adaptive symbol selection for strictly bandlimited OFDM systems,” IEEE J. Select. Areas Commun., vol. 18, pp. 2270–2277, Nov. 2000. [12] J. Armstrong, “New OFDM peak-to-average power reduction scheme,” in Proc. IEEE Vehicular Technology Conf., vol. 1, May 2001, pp. 756–760. [13] H. Ochiai and H. Imai, “Performance analysis of deliberately clipped OFDM signals,” IEEE Trans. Commun., vol. 50, pp. 89–101, Jan. 2002. [14] R. Dinis and A. Gusmão, “Signal processing schemes for power/bandwidth efficient OFDM transmission with conventional or LINC transmitter structures,” in Proc. IEEE Int. Conf. Communications, vol. 4, June 2001, pp. 1021–1027. [15] R. Nee and R. Prasad, OFDM for Wireless Multimedia Communications. Norwood, MA: Artech House, 2000. [16] T. May and H. Rohling, “Reducing the peak-to-average power ratio in OFDM radio transmission systems,” in Proc. IEEE Vehicular Technology Conf., vol. 3, May 1998, pp. 2474–2478. [17] J. Armstrong, “Peak-to-average power reduction for OFDM by repeated clipping and frequency-domain filtering,” IEE Electron. Lett., vol. 38, no. 5, pp. 246–247, Feb. 2002. [18] R. Dinis and A. Gusmão, “On the performance evaluation of OFDM transmission using clipping techniques,” in Proc. IEEE Vehicular Technology Conf., vol. 5, Sept. 1999, pp. 2923–2928. [19] G. Stette, “Calculation of intermodulation from a single carrier amplitude characteristic,” IEEE Trans. Commun., vol. COM-22, pp. 319–323, Mar. 1974. [20] M. Abramowitz and I. Stegun, Handbook of Mathematical Functions. New York: Dover, 1972. [21] R. Dinis and A. Gusmão, “Performance evaluation of an iterative PMEPR-reducing technique for OFDM transmission,” in Proc. IEEE GLOBECOM, vol. 1, Dec. 2003, pp. 20–24.

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Rui Dinis (S’96–M’00) received the Ph.D. degree from the Instituto Superior Técnico (IST), Technical University of Lisbon, Lisbon, Portugal, in 2001. Since 2001, he has been a Professor with IST. He has also been a member of the Digital Communications Group of CAPS/IST since 1992. During 2003, he was a visiting Professor at Carleton University, Ottawa, ON, Canada. He has been involved in several research projects in the broadband wireless communications area. His main research interests include modulation, equalization, and channel coding.

António Gusmão received the Ph.D. degree from the Instituto Superior Técnico (IST), Technical University of Lisbon, Lisbon, Portugal, in 1987. Since 1987, he has been a Professor with IST, with both teaching and research activities in the digital communications area, on which he has published several tens of journal and conference papers. He has also been a member of the Digital Communications Group of CAPS/IST, and the leader of that group since 1998. Since 1992, he has been involved in European projects concerning the development of broadband wireless communication systems. His main research interests concern modeling and simulation of communication systems, modulation and coded modulation issues, and signal processing for broadband wireless communications.

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2019

Pre-DFT Processing Using Eigenanalysis for Coded OFDM With Multiple Receive Antennas Defeng (David) Huang, Student Member, IEEE, and Khaled Ben Letaief, Fellow, IEEE

Abstract—In broadband wireless communications, coded orthogonal frequency-division multiplexing (OFDM) can be used with multiple receive antennas to achieve both frequency diversity and space diversity. In this scenario, the optimal approach is subcarrier-based space combining. However, such an approach is quite complex, because multiple discrete Fourier transform (DFT) blocks, each per receive antenna, are used. In this paper, we propose a pre-DFT processing scheme based upon eigenanalysis. In the proposed scheme, the received signals are weighted and combined both before and after the DFT processing. As a result, the required number of DFT blocks can be significantly reduced. With perfect weighting coefficients, the margin of the performance improvement decreases along with the increase of the number of DFT blocks, thus enabling effective performance and complexity tradeoff. To achieve a maximum average pairwise codeword distance, it will be shown that the maximum number of DFT blocks required is equal to the minimum of the number of receive antennas and the number of distinct paths in the channel. When the number of distinct paths is larger than the number of receive antennas and with a smaller number of DFT blocks, extensive simulation results will also show that near-optimal performance can still be achieved for most channels. Finally, in an OFDM system with differential modulation, we use a signal covariance matrix to obtain the weighting coefficients before the DFT processing. In this case, simulation results will demonstrate that the performance of the proposed scheme can be better than subcarrier-based space combining, but with much lower complexity. Index Terms—Broadband wireless communications, diversity, eigen-analysis, multiple receive antennas, orthogonal frequency-division multiplexing (OFDM).

I. INTRODUCTION ULTIPLE receive antennas can be employed with orthogonal frequency-division multiplexing (OFDM) to improve system performance, where space diversity is achieved using subcarrier-based space combining [1]. However, in subcarrier-based space combining, it is required that multiple discrete Fourier transform (DFT) processing, each

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Paper approved by N. Al-Dhahir, the Editor for Space–Time, OFDM, and Equalization of the IEEE Communications Society. Manuscript received October 9, 2003; revised January 27, 2004 and March 21, 2004. This work was supported in part by the Hong Kong Telecom Institute of Information Technology. This paper was presented in part at the IEEE Wireless Communications and Networking Conference, New Orleans, LA, March 2003. D. Huang was with the Center for Wireless Information Technology, Electrical and Electronic Engineering Department, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong. He is now with the Department of Electronic Engineering, Tsinghua University, Beijing, China (e-mail: [email protected]). K. Ben Letaief is with the Center for Wireless Information Technology, Electrical and Electronic Engineering Department, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong (e-mail: [email protected]). Digital Object Identifier 10.1109/TCOMM.2004.836594

per receive antenna, be used. As a result, such systems are quite complicated, because the complexity of DFT is a major concern for system implementation [2], [3]. Recently, some schemes [4]–[7] have been proposed to reduce the number of DFT blocks required. In [6], and later extended by [7] using the principle of orthogonal designs [8], the number of DFT blocks is reduced to a half with 3 dB performance degradation. In [4], the received time-domain OFDM symbols from each antenna are first weighted and then combined before the DFT processing. By doing so, the number of DFT blocks required is reduced to one. In [5], OFDM with multiple transmit and multiple receive antennas (MIMO) is investigated, and a reduced-complexity algorithm is proposed to reduce the number of DFT blocks required to one. Motivated by the work in [4] and based upon eigenanalysis, we propose a receive space-diversity scheme to effectively trade off system performance and complexity. The scheme in [4] can be regarded as a special case of our proposed scheme. However, we will show that the good performance in [4] is only applicable when the number of distinct paths in the channel is very limited.1 When the number of distinct paths is large, it will be shown that more DFT blocks are needed. In our proposed scheme, the received signals are weighted and combined both before and after the DFT processing, and the margin of the performance improvement decreases along with the increase of the number of DFT blocks. As a result, system complexity and performance can be effectively traded off. When the weighting coefficients are obtained assuming perfect channel information, we will show that the maximum number of DFT blocks required is the minimum of the number of receive antennas and the number of distinct paths in the channel. Such an achievement is obtained without performance loss, compared with subcarrier-based space combining. When the number of distinct paths is larger than the number of receive antennas, the number of DFT blocks required is not necessarily equal to the number of receive antennas. For example, extensive simulation results will show that good performance can be achieved by using only two DFT blocks for four receive antennas and eight-ray channels. In this paper, we also simulate an OFDM system with differential modulation, where the weighting coefficients before the DFT processing are obtained using the signal covariance matrix [4]. In this case, when the number of distinct paths is less than the number of receive antennas, the proposed scheme can achieve better performance than the subcarrier-based space combining scheme, but with lower complexity. 1In [4], it was shown that system performance is good with a two-ray equalgain Rayleigh fading channel, but poor with a large number of distinct paths (e.g., a 12-ray exponentially decaying Rayleigh fading channel).

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Fig. 1. Transmitter of the COFDM system.

The rest of this paper is organized as follows. The coded OFDM (COFDM) system model and the optimal receiver are given in Section II. In Section III, the proposed scheme based upon eigenanalysis is presented. Simulation results are listed in Section IV, and our conclusion is given in Section V. Throughout this paper, the following notations will be used. Vectors and matrices shall be denoted by bold letters and underlined bold letters, respectively. Furthermore, lowercase letters will denote any entities in the time domain such as the signals, codewords, and channels. Such entities in the frequency domain will be represented by uppercase letters. II. SYSTEM MODEL AND OPTIMAL RECEIVER

Assuming that the delay of the th tap is equivalent to samples, the CIR in its sampled equivalent low-pass form can then be written as follows: (4) matrix with “1” in its entries, and “0” in other entries. We also CIRs are aligned (i.e., is constant for assume that all the all ). This is a reasonable assumption because the distinct impulses of the CIRs are most often caused by large scatters and the distances between the receive antennas are relatively small in size, compared with those between the dominant scatters [10]. Accordingly, we have where

is an

A. System Model We consider an OFDM system with one transmit antenna and receive antennas. The structure of the transmitter is as shown in Fig. 1. For simplicity, it is assumed that the coding and interleaving are across one whole OFDM symbol (it can be easily extended to the case across multiple OFDM symbols as long as the channel does not vary). The interleaver is assumed to be an ideal random interleaver. The coded bits are interleaved and then mapped to the signal constellation. The codeword is then given by

(5) where is an identity matrix and in the superscript stands for conjugate transpose. Given the CIRs, the channel frequency response for the th vector: receive antenna can be denoted by the following (6) where

, for , . For convenience, we use the following notations:

(1)

(7)

in the superscript denotes the transpose operation, where is the number of subcarriers in an OFDM symbol, and is the coded information symbol at the th subcarrier. For the convenience of mathematical development, the codeword is also represented by a diagonal matrix as follows:

(8)

(2)

(9)

where denotes a diagonal matrix with the main diagonal elements given by . The channels, each per receive antenna, can be modeled by delayed transmission lines with distinct taps [9]. It is assumed that the channels do not vary within one OFDM symbol, but vary from symbol to symbol.2 During the transmission of one OFDM symbol, the attenuation coefficients of the taps can be represented by the following vector: (3)

Then, we have (10) In the frequency domain (i.e., after the guard-interval removal and at the output of the DFT processing), the received signals at the th antenna can be denoted by the following vector: (11) where

is an

complex Gaussian noise matrix with , and is the noise variance per dimension. For convenience, we rewrite (11) as follows:

where represents the gain at the th tap of the channel impulse response (CIR) corresponding to the th receive antenna. 2When the coding and interleaving is across multiple OFDM symbols (which consist of one frame), it is assumed that the channels do not vary within the frame.

(12) where

.

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Fig. 2. Pre-DFT processing scheme based upon eigenanalysis with R DFT blocks.

B. Optimal Receiver At the output of the receive antennas, after sampling, guard-interval removal, and DFT processing, the received signals can be regarded as an -length codeword. An optimal decoder, based upon the maximum-likelihood sequence detection (MLSD) criterion [11], can then be employed to detect the codeword. In this case, the pairwise error probability of deciding erroneously in favor of a coded sequence instead of the transmitted coded sequence , conditioned on the channel , is given by

(13) where is the average energy of the coded information symbol per receive antenna, and the pairwise codeword distance is given by

(14) is the sum of the main diagonal elements of where and . The complexity of the optimal MLSD receiver is proportional to the length of the received code sequence. Subcarrier-based maximum ratio combining (MRC) [11] can be used to reduce -length received code sequence to an -length sethe quence. In this case, after the DFT processing, the received

signals are first combined in the space domain for each subcarrier. After that, the original codeword is restored according to the MLSD criterion. The following proposition shows that the subcarrier-based MRC scheme and the optimal MLSD receiver are equivalent. Proposition 1: In an OFDM system with multiple receive antennas employing subcarrier-based MRC, the pairwise codeword distance is given by

Proof: See Appendix A. In this paper, due to the optimality3 of the subcarrier-based MRC scheme, we will take it as our baseline system. III. PRE-DFT PROCESSING USING EIGENANALYSIS In the proposed scheme (as shown in Fig. 2) and before the DFT processing, the received signals from the output of the receive antennas are weighted by normalized orthogonal weighting vectors, and then equally combined to form branches. Each branch is then processed by a DFT. After the DFT processing and for each subcarrier, the signals are combined in the space domain according to the principle of MRC. In the following, we analyze the proposed scheme using the pairwise codeword distance. 3The frame-error rate (FER) is minimized using the MLSD criterion. As a result, the subcarrier-based space combining can also be regarded optimal in the sense of minimum FER.

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At the output of the DFT processing, the signal at the th branch is given by

(15) where is an normalized vector and is the weighting coefficient for the th receive antenna at the th branch. In (15), the variance of the noise term remains unchanged, since is a normalized vector. The pairwise codeword distance between codewords and contributed by the th branch is then given by (16) It can be seen from (16) that the optimal weighting coefficients are related to the specific codeword pair. To make the weighting coefficients and the codeword pair independent, we average (16) over all codewords. As a result, we get (17) where the overbar stands for the average over the codeword pair ensemble. When a random bit interleaver is used, we have the following assumption: (18) where is a constant and is related to the coding and modulation scheme. The average pairwise codeword distance contributed by the th branch is then given by

In the proposed scheme, by increasing , more eigenvalues of are collected, and the average pairwise codeword distance approaches (21). Furthermore, since we arrange the eigenvalue in a decreasing order, the marginal improvement of the average pairwise codeword distance decreases along with the increase of . To achieve the maximum average pairwise codeword distance given by (21), and considering the number of DFT blocks required, we have the following proposition. Proposition 2: Let denote the rank of the matrix . The proposed scheme can then achieve the optimal average pairwise codeword distance given by (21) by using DFT blocks. Proof: See Appendix B. and the rank of is equal to Note that that of , it follows that . As a result, to achieve the maximum average pairwise codeword distance, the maximum number of DFT blocks required is at most , according to Proposition 2. The performance of the proposed scheme is related to the parameter , and the probability density function (pdf) of the ordered eigenvalues of , as well. However, the pdf of the ordered eigenvalues of is difficult to obtain analytically. In the next section, we will study some pdfs of through simulations. In the following, we investigate a way to obtain the eigenvectors of without the explicit knowledge of the CIRs. This is especially important for differential modulation, where the CIRs are not supposed to be explicitly known. For coherent modulation, the complexity of channel estimation can also be reduced, since only the equivalent channel for each branch is required to be estimated, rather than one estimation per receive antenna. As shown in [4], the weighting coefficients before the DFT processing can also be achieved using the signal covariance matrix, which is defined by

(19) For convenience, we define . Since the noise terms among the branches are independent, it follows that the average pairwise codeword distance after the DFT processing and subcarrier-based MRC combining is the sum of those contributed by each branch. It is given by

(20) where . We can arrange the order of to make . Since , are normalized and orthogonal, it can be easily seen that , are the eigenvalues of in a decreasing order, and are the corresponding eigenvectors. It can also be seen that the scheme in [4] is only a special case of our proposed scheme when and only the maximum eigenvalue is used. Now, let us compare the proposed scheme with the optimal receiver in Section II using the average pairwise codeword distance as a performance measure. From (14) and (18), the maximum average pairwise codeword distance achieved by the optimal receiver is given by (21)

(22) where

in the superscript denotes conjugate operation, is the received signal vector, and is the number of training samples used to calculate the signal covariance matrix. As long as the number of subcarriers is large, the transmitted signals can be regarded as white. As a result, when is large, the signal covariance matrix approaches the following matrix [4]: (23) From (23), it can be seen that the eigenvectors of the signal covariance matrix are exactly the same as those of when the noise term is ignored. Note that the contribution of each branch to the average pairwise codeword distance is proportional to the corresponding eigenvalue of . From (23), it can be seen that the estimation error due to the noise has much more impact on the branches with low eigenvalues than those with large eigenvalues. Furthermore, the rank of the signal covariance matrix is equal to the number of receive antennas with probability 1. When and using the signal covariance matrix to obtain the

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weighting coefficients, it can be expected that the system performance should be better by using branches than by using branches, since the extra branches only introduce noise. In the following, we consider the complexity of the proposed pre-DFT processing scheme, which consists of the eigenanalysis, the pre-DFT combining, DFT processing, and the post-DFT MRC combining. For the eigenanalysis, it is only required to be calculated once, as long as the channels do not vary. Furthermore, the number of receive antennas is much smaller, compared with the number of subcarriers in an OFDM symbol. As a result, the complexity of the eigenanalysis is ignored, since it is trivial compared with that of pre-DFT combining, DFT processing, and post-DFT MRC combining. In the following, the complexity is measured by the number of complex multiplications for simplicity.4 The pre-DFT processing component of the proposed scheme can be either implemented using analog amplifiers before the analog/digital (A/D) converters, or using digital multipliers after the A/D converters. When using analog amplifiers, the ratio of the number of multiplications used between the proposed scheme and the subcarrier-based MRC scheme is given by (24) When using digital multipliers, the ratio of the number of multiplications used between the proposed scheme and the subcarrier-based space combining scheme is as follows:

, , For example, when multiplications is reduced by 21%. When close to .

(25) , the number of , (25) is

IV. SIMULATION RESULTS We simulate an OFDM system with four receive antennas over quasi-static channels (the CIRs are kept constant during one OFDM frame). In Section IV-A, we show the pdf of for several channel models. In the ordered eigenvalues of Sections IV-B and C, the performance of an OFDM system with coherent and differential modulation is analyzed by simulations. A. PDF of The pdf of the ordered eigenvalues of depends on the channel models used. A common practice is to assume a channel that is uncorrelated for different antennas and distinct paths [1]. This assumption is reasonable, as long as the receive antennas are far away and in an environment rich in scatters. For example, at the mobile stations in an outdoor urban area, the antenna branches are required to be more than half a wavelength apart. random matrix with indeIn this case, we model by an pendent circularly symmetric Gaussian random variables in its entries. In our simulations, is set to eight, and 20 000 channel realizations are used. For the equal-gain uncorrelated Rayleigh 4DFT can be implemented using the fast Fourier transform (FFT), which consists of multiplications, summations, and interconnections between them. For the sake of easy comparison, we only consider the number of multiplications required. In practice, the summations and the interconnections of an FFT are not trivial, especially for ASIC implementation.

Fig. 3. PDF of the ordered eigenvalues of h h (L = 8). (a) Equal-gain uncorrelated Rayleigh fading channel. (b) Exponentially decaying uncorrelated Rayleigh fading channel.

fading channel,5 it can be seen from Fig. 3(a) that the eigenare mainly distributed between 12 and 5 dB. values of For the exponentially decaying uncorrelated Rayleigh fading channel with the root mean square (rms) delay spread set to one sample, they are mainly distributed between 22 and 8 dB [Fig. 3(b)]. On average, the fourth eigenvalue is about 10 dB away from the first eigenvalue in Fig. 3(a), and it is about 20 dB in Fig. 3(b). As a result, it can be expected that the fourth DFT blocks will contribute very little to system performance, especially for the exponentially decaying Rayleigh fading channel. This will be further justified by the simulation results presented in Section IV-B. Measurement campaigns have also shown that the channel can be correlated in the space domain [12]. In this case, the entries of are correlated, and the columns of may be linearly dependent with nonzero probability. As a result, can be less than with nonzero probability, and a lower number 5In the following, for all equal-gain channels, the time delay between two adjacent paths is set to one sample.

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Fig. 4. PDF of the ordered eigenvalues of correlated channel.

h h (L

= 8) in a spatially

of required DFT blocks are expected. In this part, and for simplicity, we use the same model as that employed in [13] and [14], where the channel is assumed to be correlated in space but uncorrelated between distinct paths. Assume that at the th path, the actual angle of arrival is given by

where is the average angle of arrival, and is a Gaussian random variable with the variance being proportional to the angular spread. Then, the correlation matrix of the receive signals at the th path, denoted by , is given by (26) is the path gain where is the relative antenna spacing, and that relates to the power delay profile. For small angular spreads, is given by the correlation function (27) The th row of the matrix erated as follows:

, denoted by

Fig. 5. FER perfomance with coherent modulation and equal-gain uncorrelated Rayleigh fading channel. (a) L = 2. (b) L = 8.

, can then be genB. Performance of the Proposed OFDM System With Coherent Modulation (28)

where , and is an uncorrelated vector with independent and identically distributed (i.i.d.) random variables with zero mean, unit variance, circularly symmetric complex Gaussian distribution. In our simulations, as in [13], we assume a total angle spread of 90 and a uniform , and . We also set angle spacing with and assume an equal-gain power delay profile. The pdf of the ordered eigenvalues for this channel is shown in Fig. 4. Compared with Fig. 3, it can be seen that the first and the second eigenvalues are significantly larger in this case, with the fourth eigenvalue being almost negligible. As a result, the correlated channel is the best one for the proposed scheme, in terms of tradeoff between the number of required DFT blocks and system performance.

In this part, we simulate an OFDM system with quaternary phase-shift keying (QPSK) modulation and 64 subcarriers. The length of the guard interval is 12 samples, which is equivalent to 3/19 of the whole OFDM symbol. For the coding scheme, we use a convolutional code with code rate-1/2 and constraint length 7. We also group 10 OFDM symbols into one frame. The interleaver used is a random interleaver across the whole OFDM frame. Furthermore, we assume that perfect channel information is available. As a result, the weighting coefficients are perfect, both before and after the DFT processing. The FER performance is shown in Figs. 5–7. For the two-ray equal-gain is uncorrelated Rayleigh fading channels, the rank of equal to two. As a result, using our proposed method, two DFT blocks are required to achieve the optimal performance. This can be justified by Fig. 5(a), where it is shown that by using

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equal-gain spatially correlated channel, the FER performance of the OFDM system is shown in Figs. 6 and 7, respectively. In and . As a result, the rank of this case, we also set is equal to four, and the number of DFT blocks required is four to achieve optimal performance. However, it can be seen from Figs. 6 and 7 that the performance achieved by using four or three DFT blocks is almost the same. Furthermore, the performance gap between using two, three, or four DFT blocks is within 1 dB, while it is quite large between using one and two DFT blocks. As a result, we can conclude that using two DFT blocks is appropriate for an OFDM system with four receive antennas. C. Performance of the Proposed OFDM System With Differential Modulation

Fig. 6. FER performance with coherent modulation and exponentially decaying uncorrelated Rayleigh fading channel (L = 8).

Fig. 7. FER performance with coherent modulation and spatially correlated equal-gain channel (L = 8).

two DFT blocks, the performance of the proposed scheme is as good as that of the conventional subcarrier-based space combining. Over an eight-ray equal-gain uncorrelated Rayleigh is equal to four, and four fading channel, the rank of DFT blocks are required. It can also be seen from Fig. 5(b) that the margin of the performance improvement decreases as the number of DFT blocks increases. By comparing the curves over the two-ray channels with the eight-ray ones, it can be seen that the performance difference is quite limited when two DFT blocks are employed. For example, when FER , the performance gap is only about 0.7 dB. For the subcarrier-based space combining scheme, the performance gap is about 2.5 dB. As a result, it can be concluded that our proposed scheme is robust over various channels when an appropriate number of DFT blocks is used. For the exponentially decaying uncorrelated Rayleigh fading channel (with the rms delay spread set to be one sample) and the

In this part, we simulate an OFDM system with differential QPSK modulation, where the channel information cannot be assumed to be known. In this case, we group 11 OFDM symbols into a frame and take the first OFDM symbol as the reference symbol. For other parameters, they are the same as those in Section IV-B. In the simulations, the weighting coefficients before the DFT processing are calculated by using the signal covariance matrix as defined in (22). Furthermore, as shown in [4] and [5], the performance is good enough by setting to be 16. From Sections IV-A and B of this section, it can be seen that the equal-gain uncorrelated Rayleigh fading channel is the worst one for our proposed scheme, in terms of tradeoff between the number of DFT blocks required and system performance. In this part, we show the simulation results of differential QPSK modulation over such a channel. For a two-ray equal-gain uncorrelated Rayleigh fading channel, it can be seen from Fig. 8(a) that the proposed scheme with two DFT blocks has the best FER performance (it is even better than the subcarrier-based space combining scheme). This can be justified by the analysis in Section III [see (23)]. Over an equal-gain eight-ray uncorrelated Rayleigh fading channel, it can be observed from Fig. 8(b) that the performance gap is huge between the case of employing one DFT block and that of employing two DFT blocks. Note that when one DFT block is used, the proposed scheme is exactly the same as that in [4], which implies that the scheme in [4] is not appropriate for such a channel. Finally, Fig. 9 is used to present the bit-error rate (BER) performance of the proposed OFDM system with four receive antennas over an eight-ray equal-gain Rayleigh fading channel. Compared with Fig. 8(b), it can be seen that the BER performance and the FER performance have the same trend. As a result, the same conclusion can be made for the BER curves.

V. CONCLUSION In this paper, we presented a receive space-diversity scheme based upon eigenanalysis for an OFDM system with multiple receive antennas. The proposed scheme can be used to effectively trade off system performance and complexity. For example, extensive simulation results showed that for an OFDM system with four receive antennas, near-optimal performance

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Fig. 9. BER performance with differential modulation and equal-gain uncorrelated Rayleigh fading channel (L = 8).

where , matrices , we have

, and , and

are the transpose of the th rows of the , respectively. Then, for the th subcarrier, (30)

After the MRC, for the th subcarrier, the combined signal is given by

(31) Fig. 8. FER performance with differential modulation and equal-gain uncorrelated Rayleigh fading channel. (a) L = 2. (b) L = 8.

can be achieved by using two DFT blocks. For coherent modulation and with the same performance as the conventional subcarrier-based space combining, the complexity of the proposed scheme can be much lower. For differential modulation and using the signal covariance matrix to obtain the weighting coefficients before the DFT processing, the performance of the proposed scheme can be even better than that of the subcarrier-based space combining scheme. As a result, the proposed scheme can be used to reduce system complexity and achieve good performance as well. APPENDIX A PROOF OF PROPOSITION 1 We can rewrite (12) row by row as follows:

where denotes the norm of tance between codeword and

. The pairwise codeword disis then given by

(32)

APPENDIX B PROOF OF PROPOSITION 2 corSuppose that the normalized eigenvectors of , responding to the nonzero eigenvalues are given by . In Fig. 2, by using DFT blocks, the signal before the MRC combining can be written as follows: (33)

(29)

where and . By noting the similarity between (12) and (33), we can calculate the average pair-

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HUANG AND LETAIEF: PRE-DFT PROCESSING USING EIGENANALYSIS FOR CODED OFDM WITH MULTIPLE RECEIVE ANTENNAS

wise codeword distance by replacing follows:

in (14) with

, as

(34)

REFERENCES [1] K. K. Wong, R. S. K. Cheng, K. B. Letaief, and R. D. Murch, “Adaptive antennas at the mobile and base stations in an OFDM/TDMA system,” IEEE Trans. Commun., vol. 49, pp. 195–206, Jan. 2001. [2] E. Grass, K. Tittelbach-Helmrich, U. Jagdhold, A. Troya, G. Lippert, O. Kruger, J. Lehmann, K. Maharatna, K. F. Dombrowski, N. Fiebig, R. Kraemer, and P. Mahonen, “On the single-chip implementation of a Hiperlan/2 and IEEE 802.11a capable modem,” IEEE Pers. Commun. Mag., vol. 8, pp. 48–57, Dec. 2001. [3] M. Speth, S. Fechtel, G. Fock, and H. Meyr, “Optimum receiver design for OFDM-based broadband transmission—Part II: A case study,” IEEE Trans. Commun., vol. 49, pp. 571–578, Apr. 2001. [4] M. Okada and S. Komaki, “Pre-DFT combining space diversity assisted COFDM,” IEEE Trans. Veh. Technol., vol. 50, pp. 487–496, Mar. 2001. [5] D. Huang and K. B. Letaief, “Symbol-based space diversity for coded OFDM systems,” IEEE Trans. Wireless Commun., vol. 3, pp. 117–127, Jan. 2004. [6] S. B. Slimane, “A low complexity antenna diversity receiver for OFDM based systems,” in Proc. IEEE Int. Conf. Communications, June 2001, pp. 1147–1151. [7] D. Huang, K. B. Letaief, and J. Lu, “A receive space diversity architecture for OFDM systems using orthogonal designs,” IEEE Trans. Wireless Commun., vol. 3, pp. 992–1002, May 2004. [8] V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space–time block codes from orthogonal designs,” IEEE Trans. Inform. Theory, vol. 45, pp. 1456–1467, July 1999. [9] P. Hoeher, “A statistical discrete-time model for the WSSUS multipath channel,” IEEE Trans. Veh. Technol., vol. 41, pp. 461–468, Nov. 1992. [10] H. Hashemi, “The indoor radio propagation channel,” Proc. IEEE, vol. 81, pp. 943–967, July 1993. [11] J. G. Proakis, Digital Communications, 4th ed. New York: McGrawHill, 2001. [12] C.-C. Chong, C. M. Tan, D. I. Laurenson, S. Mclaughlin, M. A. Beach, and A. R. Nix, “A new statistical wideband spatio–temporal channel models for 5-GHz band WLAN systems,” IEEE J. Select. Areas Commun., vol. 21, pp. 139–150, Feb. 2003. [13] H. Bölcskei, D. Gesbert, and A. J. Paulraj, “On the capacity of OFDMbased spatial multiplexing systems,” IEEE Trans. Commun., vol. 50, pp. 225–234, Feb. 2002. [14] H. Bölcskei, M. Borgmann, and A. J. Paulraj, “Impact of the propagation environment on the performance of space-frequency coded MIMOOFDM,” IEEE J. Select. Areas Commun., vol. 21, pp. 427–439, Apr. 2003.

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Defeng (David) Huang (M’01–S’02) received the B.S.E.E. and M.S.E.E. degrees in electronic engineering from Tsinghua University, Beijing, China, in 1996 and 1999, respectively, and the Ph.D. degree from the Department of Electrical and Electronic Engineering, Hong Kong University of Science and Technology (HKUST), Kowloon, in 2004. From 1998 to 2001, he was an Assistant Teacher and later a Lecturer with Tsinghua University. Currently, he is with the Department of Electronic Engineering, Tsinghua University. His research interests include wireless communications, OFDM, multiple-access protocol, space–time processing, channel estimation, and digital implementation of communication systems. Dr. Huang received the Hong Kong Telecom Institute of Information Technology Postgraduate Excellence Scholarship in 2004.

Khaled Ben Letaief (S’85–M’86–SM’97–F’03) received the B.S. degree (with distinction) and M.S. and Ph.D. degrees from Purdue University, West Lafayette, IN, in 1984, 1986, and 1990, respectively, all in electrical engineering. Since January 1985 and as a Graduate Instructor in the School of Electrical Engineering at Purdue University, he has taught courses in communications and electronics. From 1990 to 1993, he was a faculty member with the University of Melbourne, Melbourne, Australia. Since 1993, he has been with the Hong Kong University of Science & Technology, Kowloon, where he is currently a Professor and Head of the Electrical and Electronic Engineering Department. He is also the Director of the Hong Kong Telecom Institute of Information Technology as well as the Director of the Center for Wireless Information Technology. His current research interests include wireless and mobile networks, broad-band wireless access, space–time processing for wireless systems, wide-band OFDM, and CDMA systems. Dr. Letaief served as consultants for different organizations and is currently the founding Editor-in-Chief of the IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS. He has served on the editorial board of other journals including the IEEE JOURNAL ON SELECTED AREAS IN COMMUNICATIONS Wireless Series (as Editor-in-Chief). He served as the Technical Program Chair of the 1998 IEEE Globecom Mini-Conference on Communications Theory, held in Sydney, Australia. He was also the Co-Chair of the 2001 IEEE ICC Communications Theory Symposium, held in Helsinki, Finland. He is the Co-Chair of the 2004 IEEE Wireless Communications, Networks, and Systems Symposium, held in Dallas, TX, and is the Vice-Chair of the International Conference on Wireless and Mobile Computing, Networking, and Communications, WiMob’05, to be held in Montreal, QC, Canada. He is currently serving as the Chair of the IEEE Communications Society Technical Committee on Personal Communications. In addition to his active research activities, he has also been a dedicated teacher committed to excellence in teaching and scholarship. He was the recipient of the Mangoon Teaching Award from Purdue University in 1990, the Teaching Excellence Appreciation Award by the School of Engineering at HKUST (four times), and the Michael G. Gale Medal for Distinguished Teaching (the highest university-wide teaching award and only one recipient/year is honored for his/her contributions). He is a Distinguished Lecturer of the IEEE Communications Society.

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2028

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 11, NOVEMBER 2004

Abstracts of Forthcoming Manuscripts Characterizing Outage Rates for Space–Time Communication Over Wideband Channels G. Barriac and U. Madhow Abstract—We provide a compact characterization of outage rates for a wideband communication system whose parameters are chosen to model an outdoor cellular downlink. The base station transmitter is equipped with an antenna array, while the mobile receiver has a single antenna. Our analysis quantifies the effects of frequency and spatial diversity for measurement-based channel models available in the literature. Design prescriptions based on our framework would apply, for example, to fourth-generation cellular systems using orthogonal frequency-division multiplexing. Our information-theoretic computations yield the following findings. 1) Complex models typically employed in simulations can be replaced by simple, bandwidth-dependent, tap-delay-line models without loss of accuracy. 2) The spectral efficiency (i.e., the achievable rate, divided by the bandwidth) is well approximated as a Gaussian random variable, so that it is only necessary to specify its mean and variance in order to compute the outage rates. We provide analytical formulas for the means and variance as a function of the space–time channel model, and verify that the resulting outage rates match closely with simulation. 3) For a wide class of outdoor channels, the mean spectral efficiency depends only on the spatial diversity, while the variance depends on the spatial and frequency diversity via a product. Our definitions of frequency and spatial diversity have physically motivated interpretations, and do not rely on high signal-to-noise ratio asymptotics, as in prior work. The authors are with the Department of Electrical and Computer Engineering, University of California, Santa Barbara, Santa Barbara, CA 93106 USA. Digital Object Identifier 10.1109/TCOMM.2004.836601

Error Analysis of the Simulated Impulse Response on Indoor Wireless Optical Channels Using a Monte Carlo-Based Ray-Tracing Algorithm Oswaldo Gonzalez Hernandez, Silvestre Rodriguez, Rafael Perez-Jimenez, Alejandro Ayala, and Beatriz R. Mendoza Abstract—This paper describes a method to determine the error in a Monte Carlo-based ray-tracing algorithm used to compute the impulse response on indoor wireless optical channels. The algorithm, which accounts for multiple reflections of any order on irregularly shaped furnished rooms with diffuse and specular reflectors, allows for their analysis. Equations that estimate algorithm-produced error are given. We also report several simulation results concerning the error estimation which verify the reliability of the equations. The authors are with Departmento de Fisica Fundamental y Experimental, Electronica y Sistemas, Universidad de La Laguna, Tenerife, Canary Islands, Spain. Digital Object Identifier 10.1109/TCOMM.2004.836598

Performance Bounds on Chip-Matched-Filter Receivers for Bandlimited DS/SSMA Communications Yeon K. Jeong, Joon Ho Cho, and James S. Lehnert Abstract—Performance bounds on chip-matched-filter (CMF) receivers for bandlimited direct-sequence spread-spectrum multiple-access systems with aperiodic random spreading sequences are obtained. First, the op-

timum transmit–receive chip waveform pairs that maximize the conditional signal-to-interference ratio are derived. This leads to performance bounds on CMF receivers when the conditional Gaussian approximation for cyclostationary multiple-access interference (MAI) is exploited. The bounds are used to examine the dependence of the MAI suppression capability of the CMF receivers on the excess bandwidth of the system and the delay profile of multiple-access users. The system employing the flat spectrum chip waveform pair is shown to have near optimum average bit-error rate performance among the fixed CMF (FCMF) receiver systems. Numerical results are provided for an adaptive CMF receiver and for FCMF receivers employing several different fixed chip waveforms. Y. K. Jeong and J. S. Lehnert are with the School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907 USA. J. H. Cho is with the Department of Electrical and Computer Engineering, University of Massachusetts, Amherst, MA 01003 USA. Digital Object Identifier 10.1109/TCOMM.2004.836600

A Unified Analysis for Coded DS-CDMA With Equal-Gain Chip Combining in the Downlink of OFDM Systems Anders Persson, Tony Ottosson, and Erik G. Strom Abstract—This paper presents a novel unified analysis for the bit-error rate in the downlink of convolutionally coded and single-user detected MC-CDMA, MCDS-CDMA, TFL-CDMA, and OFDMA. Performance is analyzed under the assumption of Hadamard spreading codes, equal-gain chip combining, and a highly correlated frequency-selective Rayleigh-fading channel. Closed-form expressions are presented for the cumulative distribution function, probability density function, and moment generating function for the signal-to-noise-plus-interference ratio after despreading. The presented results assume error-free channel estimates, a perfectly synchronized receiver, and are found to agree reasonably well with simulation results. The authors are with Chalmers University of Technology, Department of Signals and Systems, SE-412 96 Goteborg, Sweden. Digital Object Identifier 10.1109/TCOMM.2004.836599

Convergence of a Maximum-Likelihood Parameter-Estimation Algorithm for DS/SS Systems in TimeVarying Channels With Strong Interference Shiauhe Tsai, James S. Lehnert, and Mark R. Bell Abstract—An unbiased, maximum-likelihood (ML), channel parameter-estimation algorithm for direct-sequence spread-spectrum systems with strong interference is discussed in this paper. The algorithm includes correcting terms to the extended Kalman filter (EKF) based on the gradient of the negative log-likelihood function of the output of a conventional matched filter. By an asymptotic analysis, the algorithm is shown to determine the actual parameters. A complete implementation of the algorithm is given, and its transient behavior is examined by computer simulations. Results show that ML algorithm, albeit optimal in the sense of unbiased parameter estimation, is less robust than the modified EKF described in our first reference. The authors are with Purdue University, West Lafayette, IN 47907-1285 USA. Digital Object Identifier 10.1109/TCOMM.2004.836597

0090-6778/04$20.00 © 2004 IEEE

TLFeBOOK

2028

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 11, NOVEMBER 2004

Abstracts of Forthcoming Manuscripts Characterizing Outage Rates for Space–Time Communication Over Wideband Channels G. Barriac and U. Madhow Abstract—We provide a compact characterization of outage rates for a wideband communication system whose parameters are chosen to model an outdoor cellular downlink. The base station transmitter is equipped with an antenna array, while the mobile receiver has a single antenna. Our analysis quantifies the effects of frequency and spatial diversity for measurement-based channel models available in the literature. Design prescriptions based on our framework would apply, for example, to fourth-generation cellular systems using orthogonal frequency-division multiplexing. Our information-theoretic computations yield the following findings. 1) Complex models typically employed in simulations can be replaced by simple, bandwidth-dependent, tap-delay-line models without loss of accuracy. 2) The spectral efficiency (i.e., the achievable rate, divided by the bandwidth) is well approximated as a Gaussian random variable, so that it is only necessary to specify its mean and variance in order to compute the outage rates. We provide analytical formulas for the means and variance as a function of the space–time channel model, and verify that the resulting outage rates match closely with simulation. 3) For a wide class of outdoor channels, the mean spectral efficiency depends only on the spatial diversity, while the variance depends on the spatial and frequency diversity via a product. Our definitions of frequency and spatial diversity have physically motivated interpretations, and do not rely on high signal-to-noise ratio asymptotics, as in prior work. The authors are with the Department of Electrical and Computer Engineering, University of California, Santa Barbara, Santa Barbara, CA 93106 USA. Digital Object Identifier 10.1109/TCOMM.2004.836601

Error Analysis of the Simulated Impulse Response on Indoor Wireless Optical Channels Using a Monte Carlo-Based Ray-Tracing Algorithm Oswaldo Gonzalez Hernandez, Silvestre Rodriguez, Rafael Perez-Jimenez, Alejandro Ayala, and Beatriz R. Mendoza Abstract—This paper describes a method to determine the error in a Monte Carlo-based ray-tracing algorithm used to compute the impulse response on indoor wireless optical channels. The algorithm, which accounts for multiple reflections of any order on irregularly shaped furnished rooms with diffuse and specular reflectors, allows for their analysis. Equations that estimate algorithm-produced error are given. We also report several simulation results concerning the error estimation which verify the reliability of the equations. The authors are with Departmento de Fisica Fundamental y Experimental, Electronica y Sistemas, Universidad de La Laguna, Tenerife, Canary Islands, Spain. Digital Object Identifier 10.1109/TCOMM.2004.836598

Performance Bounds on Chip-Matched-Filter Receivers for Bandlimited DS/SSMA Communications Yeon K. Jeong, Joon Ho Cho, and James S. Lehnert Abstract—Performance bounds on chip-matched-filter (CMF) receivers for bandlimited direct-sequence spread-spectrum multiple-access systems with aperiodic random spreading sequences are obtained. First, the op-

timum transmit–receive chip waveform pairs that maximize the conditional signal-to-interference ratio are derived. This leads to performance bounds on CMF receivers when the conditional Gaussian approximation for cyclostationary multiple-access interference (MAI) is exploited. The bounds are used to examine the dependence of the MAI suppression capability of the CMF receivers on the excess bandwidth of the system and the delay profile of multiple-access users. The system employing the flat spectrum chip waveform pair is shown to have near optimum average bit-error rate performance among the fixed CMF (FCMF) receiver systems. Numerical results are provided for an adaptive CMF receiver and for FCMF receivers employing several different fixed chip waveforms. Y. K. Jeong and J. S. Lehnert are with the School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907 USA. J. H. Cho is with the Department of Electrical and Computer Engineering, University of Massachusetts, Amherst, MA 01003 USA. Digital Object Identifier 10.1109/TCOMM.2004.836600

A Unified Analysis for Coded DS-CDMA With Equal-Gain Chip Combining in the Downlink of OFDM Systems Anders Persson, Tony Ottosson, and Erik G. Strom Abstract—This paper presents a novel unified analysis for the bit-error rate in the downlink of convolutionally coded and single-user detected MC-CDMA, MCDS-CDMA, TFL-CDMA, and OFDMA. Performance is analyzed under the assumption of Hadamard spreading codes, equal-gain chip combining, and a highly correlated frequency-selective Rayleigh-fading channel. Closed-form expressions are presented for the cumulative distribution function, probability density function, and moment generating function for the signal-to-noise-plus-interference ratio after despreading. The presented results assume error-free channel estimates, a perfectly synchronized receiver, and are found to agree reasonably well with simulation results. The authors are with Chalmers University of Technology, Department of Signals and Systems, SE-412 96 Goteborg, Sweden. Digital Object Identifier 10.1109/TCOMM.2004.836599

Convergence of a Maximum-Likelihood Parameter-Estimation Algorithm for DS/SS Systems in TimeVarying Channels With Strong Interference Shiauhe Tsai, James S. Lehnert, and Mark R. Bell Abstract—An unbiased, maximum-likelihood (ML), channel parameter-estimation algorithm for direct-sequence spread-spectrum systems with strong interference is discussed in this paper. The algorithm includes correcting terms to the extended Kalman filter (EKF) based on the gradient of the negative log-likelihood function of the output of a conventional matched filter. By an asymptotic analysis, the algorithm is shown to determine the actual parameters. A complete implementation of the algorithm is given, and its transient behavior is examined by computer simulations. Results show that ML algorithm, albeit optimal in the sense of unbiased parameter estimation, is less robust than the modified EKF described in our first reference. The authors are with Purdue University, West Lafayette, IN 47907-1285 USA. Digital Object Identifier 10.1109/TCOMM.2004.836597

0090-6778/04$20.00 © 2004 IEEE

TLFeBOOK

2028

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 11, NOVEMBER 2004

Abstracts of Forthcoming Manuscripts Characterizing Outage Rates for Space–Time Communication Over Wideband Channels G. Barriac and U. Madhow Abstract—We provide a compact characterization of outage rates for a wideband communication system whose parameters are chosen to model an outdoor cellular downlink. The base station transmitter is equipped with an antenna array, while the mobile receiver has a single antenna. Our analysis quantifies the effects of frequency and spatial diversity for measurement-based channel models available in the literature. Design prescriptions based on our framework would apply, for example, to fourth-generation cellular systems using orthogonal frequency-division multiplexing. Our information-theoretic computations yield the following findings. 1) Complex models typically employed in simulations can be replaced by simple, bandwidth-dependent, tap-delay-line models without loss of accuracy. 2) The spectral efficiency (i.e., the achievable rate, divided by the bandwidth) is well approximated as a Gaussian random variable, so that it is only necessary to specify its mean and variance in order to compute the outage rates. We provide analytical formulas for the means and variance as a function of the space–time channel model, and verify that the resulting outage rates match closely with simulation. 3) For a wide class of outdoor channels, the mean spectral efficiency depends only on the spatial diversity, while the variance depends on the spatial and frequency diversity via a product. Our definitions of frequency and spatial diversity have physically motivated interpretations, and do not rely on high signal-to-noise ratio asymptotics, as in prior work. The authors are with the Department of Electrical and Computer Engineering, University of California, Santa Barbara, Santa Barbara, CA 93106 USA. Digital Object Identifier 10.1109/TCOMM.2004.836601

Error Analysis of the Simulated Impulse Response on Indoor Wireless Optical Channels Using a Monte Carlo-Based Ray-Tracing Algorithm Oswaldo Gonzalez Hernandez, Silvestre Rodriguez, Rafael Perez-Jimenez, Alejandro Ayala, and Beatriz R. Mendoza Abstract—This paper describes a method to determine the error in a Monte Carlo-based ray-tracing algorithm used to compute the impulse response on indoor wireless optical channels. The algorithm, which accounts for multiple reflections of any order on irregularly shaped furnished rooms with diffuse and specular reflectors, allows for their analysis. Equations that estimate algorithm-produced error are given. We also report several simulation results concerning the error estimation which verify the reliability of the equations. The authors are with Departmento de Fisica Fundamental y Experimental, Electronica y Sistemas, Universidad de La Laguna, Tenerife, Canary Islands, Spain. Digital Object Identifier 10.1109/TCOMM.2004.836598

Performance Bounds on Chip-Matched-Filter Receivers for Bandlimited DS/SSMA Communications Yeon K. Jeong, Joon Ho Cho, and James S. Lehnert Abstract—Performance bounds on chip-matched-filter (CMF) receivers for bandlimited direct-sequence spread-spectrum multiple-access systems with aperiodic random spreading sequences are obtained. First, the op-

timum transmit–receive chip waveform pairs that maximize the conditional signal-to-interference ratio are derived. This leads to performance bounds on CMF receivers when the conditional Gaussian approximation for cyclostationary multiple-access interference (MAI) is exploited. The bounds are used to examine the dependence of the MAI suppression capability of the CMF receivers on the excess bandwidth of the system and the delay profile of multiple-access users. The system employing the flat spectrum chip waveform pair is shown to have near optimum average bit-error rate performance among the fixed CMF (FCMF) receiver systems. Numerical results are provided for an adaptive CMF receiver and for FCMF receivers employing several different fixed chip waveforms. Y. K. Jeong and J. S. Lehnert are with the School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907 USA. J. H. Cho is with the Department of Electrical and Computer Engineering, University of Massachusetts, Amherst, MA 01003 USA. Digital Object Identifier 10.1109/TCOMM.2004.836600

A Unified Analysis for Coded DS-CDMA With Equal-Gain Chip Combining in the Downlink of OFDM Systems Anders Persson, Tony Ottosson, and Erik G. Strom Abstract—This paper presents a novel unified analysis for the bit-error rate in the downlink of convolutionally coded and single-user detected MC-CDMA, MCDS-CDMA, TFL-CDMA, and OFDMA. Performance is analyzed under the assumption of Hadamard spreading codes, equal-gain chip combining, and a highly correlated frequency-selective Rayleigh-fading channel. Closed-form expressions are presented for the cumulative distribution function, probability density function, and moment generating function for the signal-to-noise-plus-interference ratio after despreading. The presented results assume error-free channel estimates, a perfectly synchronized receiver, and are found to agree reasonably well with simulation results. The authors are with Chalmers University of Technology, Department of Signals and Systems, SE-412 96 Goteborg, Sweden. Digital Object Identifier 10.1109/TCOMM.2004.836599

Convergence of a Maximum-Likelihood Parameter-Estimation Algorithm for DS/SS Systems in TimeVarying Channels With Strong Interference Shiauhe Tsai, James S. Lehnert, and Mark R. Bell Abstract—An unbiased, maximum-likelihood (ML), channel parameter-estimation algorithm for direct-sequence spread-spectrum systems with strong interference is discussed in this paper. The algorithm includes correcting terms to the extended Kalman filter (EKF) based on the gradient of the negative log-likelihood function of the output of a conventional matched filter. By an asymptotic analysis, the algorithm is shown to determine the actual parameters. A complete implementation of the algorithm is given, and its transient behavior is examined by computer simulations. Results show that ML algorithm, albeit optimal in the sense of unbiased parameter estimation, is less robust than the modified EKF described in our first reference. The authors are with Purdue University, West Lafayette, IN 47907-1285 USA. Digital Object Identifier 10.1109/TCOMM.2004.836597

0090-6778/04$20.00 © 2004 IEEE

TLFeBOOK

2028

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 11, NOVEMBER 2004

Abstracts of Forthcoming Manuscripts Characterizing Outage Rates for Space–Time Communication Over Wideband Channels G. Barriac and U. Madhow Abstract—We provide a compact characterization of outage rates for a wideband communication system whose parameters are chosen to model an outdoor cellular downlink. The base station transmitter is equipped with an antenna array, while the mobile receiver has a single antenna. Our analysis quantifies the effects of frequency and spatial diversity for measurement-based channel models available in the literature. Design prescriptions based on our framework would apply, for example, to fourth-generation cellular systems using orthogonal frequency-division multiplexing. Our information-theoretic computations yield the following findings. 1) Complex models typically employed in simulations can be replaced by simple, bandwidth-dependent, tap-delay-line models without loss of accuracy. 2) The spectral efficiency (i.e., the achievable rate, divided by the bandwidth) is well approximated as a Gaussian random variable, so that it is only necessary to specify its mean and variance in order to compute the outage rates. We provide analytical formulas for the means and variance as a function of the space–time channel model, and verify that the resulting outage rates match closely with simulation. 3) For a wide class of outdoor channels, the mean spectral efficiency depends only on the spatial diversity, while the variance depends on the spatial and frequency diversity via a product. Our definitions of frequency and spatial diversity have physically motivated interpretations, and do not rely on high signal-to-noise ratio asymptotics, as in prior work. The authors are with the Department of Electrical and Computer Engineering, University of California, Santa Barbara, Santa Barbara, CA 93106 USA. Digital Object Identifier 10.1109/TCOMM.2004.836601

Error Analysis of the Simulated Impulse Response on Indoor Wireless Optical Channels Using a Monte Carlo-Based Ray-Tracing Algorithm Oswaldo Gonzalez Hernandez, Silvestre Rodriguez, Rafael Perez-Jimenez, Alejandro Ayala, and Beatriz R. Mendoza Abstract—This paper describes a method to determine the error in a Monte Carlo-based ray-tracing algorithm used to compute the impulse response on indoor wireless optical channels. The algorithm, which accounts for multiple reflections of any order on irregularly shaped furnished rooms with diffuse and specular reflectors, allows for their analysis. Equations that estimate algorithm-produced error are given. We also report several simulation results concerning the error estimation which verify the reliability of the equations. The authors are with Departmento de Fisica Fundamental y Experimental, Electronica y Sistemas, Universidad de La Laguna, Tenerife, Canary Islands, Spain. Digital Object Identifier 10.1109/TCOMM.2004.836598

Performance Bounds on Chip-Matched-Filter Receivers for Bandlimited DS/SSMA Communications Yeon K. Jeong, Joon Ho Cho, and James S. Lehnert Abstract—Performance bounds on chip-matched-filter (CMF) receivers for bandlimited direct-sequence spread-spectrum multiple-access systems with aperiodic random spreading sequences are obtained. First, the op-

timum transmit–receive chip waveform pairs that maximize the conditional signal-to-interference ratio are derived. This leads to performance bounds on CMF receivers when the conditional Gaussian approximation for cyclostationary multiple-access interference (MAI) is exploited. The bounds are used to examine the dependence of the MAI suppression capability of the CMF receivers on the excess bandwidth of the system and the delay profile of multiple-access users. The system employing the flat spectrum chip waveform pair is shown to have near optimum average bit-error rate performance among the fixed CMF (FCMF) receiver systems. Numerical results are provided for an adaptive CMF receiver and for FCMF receivers employing several different fixed chip waveforms. Y. K. Jeong and J. S. Lehnert are with the School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907 USA. J. H. Cho is with the Department of Electrical and Computer Engineering, University of Massachusetts, Amherst, MA 01003 USA. Digital Object Identifier 10.1109/TCOMM.2004.836600

A Unified Analysis for Coded DS-CDMA With Equal-Gain Chip Combining in the Downlink of OFDM Systems Anders Persson, Tony Ottosson, and Erik G. Strom Abstract—This paper presents a novel unified analysis for the bit-error rate in the downlink of convolutionally coded and single-user detected MC-CDMA, MCDS-CDMA, TFL-CDMA, and OFDMA. Performance is analyzed under the assumption of Hadamard spreading codes, equal-gain chip combining, and a highly correlated frequency-selective Rayleigh-fading channel. Closed-form expressions are presented for the cumulative distribution function, probability density function, and moment generating function for the signal-to-noise-plus-interference ratio after despreading. The presented results assume error-free channel estimates, a perfectly synchronized receiver, and are found to agree reasonably well with simulation results. The authors are with Chalmers University of Technology, Department of Signals and Systems, SE-412 96 Goteborg, Sweden. Digital Object Identifier 10.1109/TCOMM.2004.836599

Convergence of a Maximum-Likelihood Parameter-Estimation Algorithm for DS/SS Systems in TimeVarying Channels With Strong Interference Shiauhe Tsai, James S. Lehnert, and Mark R. Bell Abstract—An unbiased, maximum-likelihood (ML), channel parameter-estimation algorithm for direct-sequence spread-spectrum systems with strong interference is discussed in this paper. The algorithm includes correcting terms to the extended Kalman filter (EKF) based on the gradient of the negative log-likelihood function of the output of a conventional matched filter. By an asymptotic analysis, the algorithm is shown to determine the actual parameters. A complete implementation of the algorithm is given, and its transient behavior is examined by computer simulations. Results show that ML algorithm, albeit optimal in the sense of unbiased parameter estimation, is less robust than the modified EKF described in our first reference. The authors are with Purdue University, West Lafayette, IN 47907-1285 USA. Digital Object Identifier 10.1109/TCOMM.2004.836597

0090-6778/04$20.00 © 2004 IEEE

TLFeBOOK

2028

IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 52, NO. 11, NOVEMBER 2004

Abstracts of Forthcoming Manuscripts Characterizing Outage Rates for Space–Time Communication Over Wideband Channels G. Barriac and U. Madhow Abstract—We provide a compact characterization of outage rates for a wideband communication system whose parameters are chosen to model an outdoor cellular downlink. The base station transmitter is equipped with an antenna array, while the mobile receiver has a single antenna. Our analysis quantifies the effects of frequency and spatial diversity for measurement-based channel models available in the literature. Design prescriptions based on our framework would apply, for example, to fourth-generation cellular systems using orthogonal frequency-division multiplexing. Our information-theoretic computations yield the following findings. 1) Complex models typically employed in simulations can be replaced by simple, bandwidth-dependent, tap-delay-line models without loss of accuracy. 2) The spectral efficiency (i.e., the achievable rate, divided by the bandwidth) is well approximated as a Gaussian random variable, so that it is only necessary to specify its mean and variance in order to compute the outage rates. We provide analytical formulas for the means and variance as a function of the space–time channel model, and verify that the resulting outage rates match closely with simulation. 3) For a wide class of outdoor channels, the mean spectral efficiency depends only on the spatial diversity, while the variance depends on the spatial and frequency diversity via a product. Our definitions of frequency and spatial diversity have physically motivated interpretations, and do not rely on high signal-to-noise ratio asymptotics, as in prior work. The authors are with the Department of Electrical and Computer Engineering, University of California, Santa Barbara, Santa Barbara, CA 93106 USA. Digital Object Identifier 10.1109/TCOMM.2004.836601

Error Analysis of the Simulated Impulse Response on Indoor Wireless Optical Channels Using a Monte Carlo-Based Ray-Tracing Algorithm Oswaldo Gonzalez Hernandez, Silvestre Rodriguez, Rafael Perez-Jimenez, Alejandro Ayala, and Beatriz R. Mendoza Abstract—This paper describes a method to determine the error in a Monte Carlo-based ray-tracing algorithm used to compute the impulse response on indoor wireless optical channels. The algorithm, which accounts for multiple reflections of any order on irregularly shaped furnished rooms with diffuse and specular reflectors, allows for their analysis. Equations that estimate algorithm-produced error are given. We also report several simulation results concerning the error estimation which verify the reliability of the equations. The authors are with Departmento de Fisica Fundamental y Experimental, Electronica y Sistemas, Universidad de La Laguna, Tenerife, Canary Islands, Spain. Digital Object Identifier 10.1109/TCOMM.2004.836598

Performance Bounds on Chip-Matched-Filter Receivers for Bandlimited DS/SSMA Communications Yeon K. Jeong, Joon Ho Cho, and James S. Lehnert Abstract—Performance bounds on chip-matched-filter (CMF) receivers for bandlimited direct-sequence spread-spectrum multiple-access systems with aperiodic random spreading sequences are obtained. First, the op-

timum transmit–receive chip waveform pairs that maximize the conditional signal-to-interference ratio are derived. This leads to performance bounds on CMF receivers when the conditional Gaussian approximation for cyclostationary multiple-access interference (MAI) is exploited. The bounds are used to examine the dependence of the MAI suppression capability of the CMF receivers on the excess bandwidth of the system and the delay profile of multiple-access users. The system employing the flat spectrum chip waveform pair is shown to have near optimum average bit-error rate performance among the fixed CMF (FCMF) receiver systems. Numerical results are provided for an adaptive CMF receiver and for FCMF receivers employing several different fixed chip waveforms. Y. K. Jeong and J. S. Lehnert are with the School of Electrical and Computer Engineering, Purdue University, West Lafayette, IN 47907 USA. J. H. Cho is with the Department of Electrical and Computer Engineering, University of Massachusetts, Amherst, MA 01003 USA. Digital Object Identifier 10.1109/TCOMM.2004.836600

A Unified Analysis for Coded DS-CDMA With Equal-Gain Chip Combining in the Downlink of OFDM Systems Anders Persson, Tony Ottosson, and Erik G. Strom Abstract—This paper presents a novel unified analysis for the bit-error rate in the downlink of convolutionally coded and single-user detected MC-CDMA, MCDS-CDMA, TFL-CDMA, and OFDMA. Performance is analyzed under the assumption of Hadamard spreading codes, equal-gain chip combining, and a highly correlated frequency-selective Rayleigh-fading channel. Closed-form expressions are presented for the cumulative distribution function, probability density function, and moment generating function for the signal-to-noise-plus-interference ratio after despreading. The presented results assume error-free channel estimates, a perfectly synchronized receiver, and are found to agree reasonably well with simulation results. The authors are with Chalmers University of Technology, Department of Signals and Systems, SE-412 96 Goteborg, Sweden. Digital Object Identifier 10.1109/TCOMM.2004.836599

Convergence of a Maximum-Likelihood Parameter-Estimation Algorithm for DS/SS Systems in TimeVarying Channels With Strong Interference Shiauhe Tsai, James S. Lehnert, and Mark R. Bell Abstract—An unbiased, maximum-likelihood (ML), channel parameter-estimation algorithm for direct-sequence spread-spectrum systems with strong interference is discussed in this paper. The algorithm includes correcting terms to the extended Kalman filter (EKF) based on the gradient of the negative log-likelihood function of the output of a conventional matched filter. By an asymptotic analysis, the algorithm is shown to determine the actual parameters. A complete implementation of the algorithm is given, and its transient behavior is examined by computer simulations. Results show that ML algorithm, albeit optimal in the sense of unbiased parameter estimation, is less robust than the modified EKF described in our first reference. The authors are with Purdue University, West Lafayette, IN 47907-1285 USA. Digital Object Identifier 10.1109/TCOMM.2004.836597

0090-6778/04$20.00 © 2004 IEEE

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Digital Object Identifier 10.1109/TCOMM.2004.839890

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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-

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Digital Object Identifier 10.1109/TCOMM.2004.839887

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IEEE Communications Society 2004 Board of Governors President C. A. SILLER, JR. Cetacean Networks, Inc.

Officers Past President C. DESMOND, World Class—Telecommunications

Treasurer H. BLANK

Secretary/Executive Director J. M. HOWELL, IEEE ComSoc

VP-Technical Activities H. FREEMAN, Booz Allen Hamilton Inc.

VP-Society Relations N. CHEUNG, Telcordia Technologies

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—LA Region R. VEIGA

Director—On-Line Content J. HONG POSTECH

Director—Sister Society Relations N. OHTA Sony Corp.

Director—Meetings/Conferences S. GOYAL

Director—Education W. H. TRANTER Virginia Tech. Director—NA Region R. SHAPIRO

Members at Large V. BHARGAVA (’04) Univ. of Victoria

H. BRADLOW (’04) Telstra

C-L. I (’04) G000CCL/ITRI

J. PETERSON (’04) Lucent Technologies—Bell Labs

H. BLANK (’05)

J. LIEBEHERR (’05) Univ. of Virginia

J. LOCICERO (’05) Illinois Inst. of Technology

C.-S. LI (’05) IBM

T. S. ATKINSON (’06) ICSI Consulting Services, Inc.

S. MOYER (’06) Telcordia Technol.

N. OHTA (’06) Sony Corp.

H. STUETTGEN (’06) NEC Europe Ltd.

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

Committee Chairs Emerging Technologies Fellow Evaluation L. CIMINI R. CALDERBANK Princeton Univ. Univ. Delaware

Management D. ZUCKERMAN Strategy C. K. TOH TRW Systems 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.

Industry Initiatives G. JAKOBSEN Altusys Corp.

Magazines Board Director of Magazines A. JAJSZCZYK AGH Univ. Technol.

Strategic Planning M. EJIRI Fujitsu Ltd.

Communications Magazine R. GLITHO, EIC Ericsson Res. Canada

Network Magazine C. BISDIKIAN, EIC IBM T. J. Watson Res. Ctr. Wireless Communications Magazine M. ZORZI, EIC Univ. of Ferrara

Nomination & Elections J. R. B. DEMARCA Cetuc-Puc/Rio Technical Activities Council Vice President H. FREEMAN Booz Allen Hamilton Inc. Secretary F. BAUER Nokia

Global Communications Newsletter Communications Quality & Reliability J. GARCIA HARO, Editor K. RAUSCHER Polytechnic Univ. Lucent Technologies Interactive Magazines Communications Software Y.-C. CHANG, Editor A. MISHRA LG Electronics Virginia Tech. IEEE Communications Communications Switching Surveys & Tutorials & Routing M. REISSLEIN, EIC H. STEUTTGEN Arizona State Univ. NEC Europe Ltd. ComSoc e-News Communications Systems N. FONSECA, Editor Integration & Modeling State Univ. of Campinas M. DEVETSIKIOTIS NC State Univ. IEEE Press M. SHAFI, ComSoc Liaison Communication Theory Telecom New Zealand Ltd. S. BENEDETTO Multimedia Magazine C. W. CHEN, ComSoc Liaison Univ. of Missouri–Columbia Internet Computing Magazine G. S. KUO, ComSoc Liaison TRW Systems Optical Communications Supplement, Area Editors C. QIAO, S. KARTALOPOULOS SUNY Buffalo Pervasive Computing M. NAGHSHINEH, ComSocLiaison IBM T. J. Watson Res. Ctr.

Politecnico di Torino

Strategic Planning C. DESMOND World Class—Telecommun.

Internet J. TOUCH USC/ISI C. KALMANEK AT&T Bell Labs—Research Multimedia Communications H. YU Panasonic Network Operations & Management C. RAD AT&T Optical Networking B. MUKHERJEE Univ. of California Personal Communications K. B. LETAIEF Hong Kong Univ. of Sci. & Technol. Radio Communications M. CHIANI Univ. of Bologna Satellite & Space Communications A. JAMALIPOUR Univ. of Sydney

Computer Communications J. LIEBEHERR Univ. of Virginia

Signal Processing & Communications Electronics R. SMITH Northrup Grumman Space

Enterprise Networking P. RAY Univ. of New South Wales

Signal Processing for Storage G. SILVUS Seagate Technology

High-Speed Networking (formerly Gigabit Networking) J. EVANS Univ. of Kansas

Tactical Communications K. C. YOUNG, JR. Telcordia Technologies

Information Infrastructure M. ULEMA Manhattan College

Transmission Access & Optical Systems J. M. H. ELMIRGHANI Univ. of Wales Swansea

Digital Object Identifier 10.1109/TCOMM.2004.839886

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