Design of LDPC codes and reliable practical decoders for standard and non standard channels

.. .DESIGN OF LDPC CODES AND RELIABLE PRACTICAL DECODERS FOR STANDARD AND NON -STANDARD CHANNELS MO HUISI, ELISA (B.Eng.(Hons.), NUS ) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY... performances of LDPC codes over noncoherent AWGN channels 90 5.3 BER performances of LDPC codes over coherent and noncoherent AWGN channels 5.4 BER performances of. .. improvement over the performance of the binary codes reported motivated recent works on the analysis and design of non- binary LDPC codes on binary and nonbinary channels In [6], nonbinary codes under ML

DESIGN OF LDPC CODES AND RELIABLE PRACTICAL DECODERS FOR STANDARD AND NON-STANDARD CHANNELS MO HUISI, ELISA NATIONAL UNIVERSITY OF SINGAPORE 2009 DESIGN OF LDPC CODES AND RELIABLE PRACTICAL DECODERS FOR STANDARD AND NON-STANDARD CHANNELS MO HUISI, ELISA (B.Eng.(Hons.), NUS ) A THESIS SUBMITTED FOR THE DEGREE OF DOCTOR OF PHILOSOPHY DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING NATIONAL UNIVERSITY OF SINGAPORE 2009 Acknowledgments The four-year pursuit of a doctorate degree has not been easy, but it is definitely an enriching journey. I had the opportunity to work closely with Dr. Marc Armand and Prof. Kam Pooi Yuen. Dr. Armand, well-versed in algebra and coding theory, has always provided directions to guide my research. His comments and encouragements also served as my motivation to persevere, especially in times when I was stuck with a problem for weeks (or even months). Combined with Prof. Kam’s knowledge in communications, our joint discussions has often led to important insights and interesting research problems. I would also like to thank them for their patience and time spent to help craft and edit (and re-edit..) my manuscripts until they are in their best form for submission. Of course, this thesis would not have been possible without their advice and guidance. I would like to express my gratitude towards DSO National Laboratories for the financial support offered through my course of study, and also towards Dr. Ng Boon Chong, my mentor from DSO, for his advice and support rendered. I am looking forward to applying my research completed, knowledge accumulated and thinking skills cultivated when I return to DSO in 2010. Research can be a lonely process. A special thanks goes to my lunch buddy who has always listened to my grumbles, encouraged me and sometimes studied ii Acknowledgments together with me till late. Bundled with its fair share of long hours of work and frustration, research is not always smooth-sailing. Fortunately, I am blessed with unwavering understanding, support and love from Mummy, Papa and my Significant Other. Thank you! iii Contents Acknowledgments ii Contents iv Abstract ix List of Tables xi List of Figures xii List of Abbreviations xvii List of Symbols and Notations xix 1 Introduction 1 1.1 An Overview of LDPC Codes . . . . . . . . . . . . . . . . . . . . 2 1.2 Current Research and Challenges . . . . . . . . . . . . . . . . . . 7 1.2.1 Nonbinary LDPC Codes . . . . . . . . . . . . . . . . . . . 7 1.2.2 Non-random construction of LDPC Codes . . . . . . . . . 11 1.2.3 Finite Length Analysis of LDPC Codes . . . . . . . . . . . 13 1.2.4 Transmission over Nonstandard Channels . . . . . . . . . . 15 Contributions of the Thesis . . . . . . . . . . . . . . . . . . . . . 20 1.3 iv CONTENTS 1.3.1 Code Design . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.3.2 Decoder Design . . . . . . . . . . . . . . . . . . . . . . . . 22 1.4 Organization of The Thesis . . . . . . . . . . . . . . . . . . . . . 23 1.5 Channel Model and Simulation Methodology . . . . . . . . . . . . 25 2 Construction of LDPC Codes over Mixed-Alphabets 2.1 2.2 2.3 27 Construction of Mixed-Alphabet Codes . . . . . . . . . . . . . . . 28 2.1.1 Simulation Studies . . . . . . . . . . . . . . . . . . . . . . 28 Addition of Redundant Check Nodes to Tanner Graphs . . . . . . 30 2.2.1 Simulation Results and Discussion . . . . . . . . . . . . . . 33 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 3 Construction of Structured LDPC Codes over Integer Residue Rings 36 3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.1.1 An Overview of Codes over Z2a . . . . . . . . . . . . . . . 38 3.1.2 The Matched Signal Set . . . . . . . . . . . . . . . . . . . 39 Latin Squares . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 3.2.1 Definition and Application to Galois Fields . . . . . . . . . 40 3.2.2 Extended Application to Multiplicative Groups over Integer 3.2 Residue Rings . . . . . . . . . . . . . . . . . . . . . . . . . 41 Structured LDPC Codes over Z2a . . . . . . . . . . . . . . . . . . 45 3.3.1 Construction of Graphs using Latin Squares . . . . . . . . 45 3.3.2 Properties of C(a, s) . . . . . . . . . . . . . . . . . . . . . 48 3.4 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . 50 3.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.3 v CONTENTS 4 Iterative Decoding using Binary Differential PSK 56 4.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 4.2 Metric Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.2.1 The optimal TSOI-PN-LLR and its approximations . . . . 60 4.2.2 The G-PN-LLR . . . . . . . . . . . . . . . . . . . . . . . . 67 4.2.3 Case with no phase noise . . . . . . . . . . . . . . . . . . . 68 Simulation Study . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.3 4.3.1 4.4 Performance of LDPC/turbo codes with different metrics and no phase noise . . . . . . . . . . . . . . . . . . . . . . 71 4.3.2 Effects of SNR estimation error on performance of metrics 75 4.3.3 Effects of phase noise on performance of metrics . . . . . . 77 4.3.4 Effects of phase noise estimation error on performance of metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5 Iterative Decoding using Quadrature Differential PSK 82 5.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.2 Metric Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 5.3 Simulation Study . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 5.3.1 Performance of Binary LDPC Codes using Different Metrics 89 5.3.2 Performance of Mixed Alphabet LDPC Codes using TSOISA-LLR . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 Effects of SNR Mis-estimation . . . . . . . . . . . . . . . . 92 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93 5.3.3 5.4 6 Pseudocodeword Weights under Differential PSK Transmission vi 95 CONTENTS 6.1 Pseudocodeword weights under BDPSK and QDPSK over the noncoherent AWGN channel . . . . . . . . . . . . . . . . . . . . . 96 6.1.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . 96 6.1.2 Pseudocodeword Weight under BDPSK . . . . . . . . . . . 97 6.1.3 Pseudocodeword Weight under QDPSK . . . . . . . . . . . 100 6.2 Pseudocodeword Weight Analysis of (8,4) & (8,3) Binary Codes . 106 6.3 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 7 Iterative Decoding of LDPC codes Transmitted using BPSK with PSAM 112 7.1 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 7.2 Metric Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 116 7.3 Comparison with the metric for BDPSK transmission . . . . . . . 119 7.4 Convergence of PSAM-LLR to the metric for coherent channel . . 120 7.5 Simulation Study . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 7.5.1 Performance of LDPC codes with different metrics . . . . . 123 7.5.2 Effects of phase noise on performance of metrics . . . . . . 125 7.5.3 Effects of SNR estimation error . . . . . . . . . . . . . . . 127 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 7.6 8 Iterative Decoding using PSA BPSK with Reference Phasor 130 8.1 Metric Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 8.2 Simulation Study . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 8.2.1 Performance of LDPC codes with constant, unknown carrier phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii 135 CONTENTS 8.2.2 phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Effects of SNR estimation error . . . . . . . . . . . . . . . 138 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 8.2.3 8.3 Performance of LDPC codes with noisy, unknown carrier 9 Conclusion and Proposals for Future Research 141 9.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 9.2 Proposals for Future Research . . . . . . . . . . . . . . . . . . . . 144 Bibliography 146 Publications 163 Appendix A 165 Appendix B 172 Appendix C 174 Appendix D 176 Appendix E 177 viii Abstract Low-density parity-check (LDPC) codes are known for their near Shannon-limit performance. Since non-binary LDPC codes are generally capable of outperforming binary LDPC codes, much interest were involved in the construction of good non-binary codes. However, decoding complexity increases with the alphabet size. Following recent work on mixed-alphabet codes, we design near-regular LDPC codes where the information symbols and majority of the parity-check symbols are defined over an integer residue ring, while the remaining parity-check symbols are defined over another integer residue ring of a larger size. Further, it has been shown that performance of the iterative decoder improves when redundant check nodes are added to the Tanner graph. This motivates our research on structured LDPC codes over integer residue rings, where the corresponding Tanner graphs with constant variable and check node degrees contain redundant check nodes. The original decoding algorithm proposed by Gallager is designed for transmission over the additive white Gaussian noise channel. Since then, performance of LDPC codes transmitted using modulation other than the binary phase shift keying (BPSK) over other types of channels was investigated. However, the decoding algorithm, the computation of the log-likelihood ratios (LLRs) in particular, is either executed with assumptions on the channel or altered based ix Abstract on unnecessary approximations. The calculation of the LLR is revisited and the optimal LLR for LDPC codes transmitted using binary differential PSK (BDPSK) over the noncoherent channel is derived. The computation is further generalized to the case with quadrature DPSK (QDPSK) and performances of binary as well as mixed-alphabet LDPC codes over the noncoherent channel are examined. We analyse finite-length binary and mixed-alphabet LDPC codes under BDPSK and QDPSK, and explain the difference in error performance under these two transmissions using the notion of pseudocodewords. Further, we derive the LLR for pilot-symbol-assisted BPSK transmission which yields better performance than BDPSK transmission but requires higher bandwidth. Extension to higher order modulations and non-binary codes shall be left for possible future research. x List of Tables 2.1 Number of m-cover Pseudocodewords in Example 2.1 . . . . . . . 32 3.1 Properties of C(a, s) . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.1 Average number of iterations required for convergence of a received sequence at the decoder . . . . . . . . . . . . . . . . . . . . . . . xi 74 List of Figures 1.1 Tanner graph for Example 1.1. . . . . . . . . . . . . . . . . . . . . 4 1.2 Tanner graph for Example 1.2. . . . . . . . . . . . . . . . . . . . . 10 1.3 Tanner graph corresponding to a length 21 projective geometry LDPC code. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.4 System model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 2.1 BER performance of Z4 codes extended with N2 parity-check symbols defined over Z16 for N2 = 0, 10, 20 . . . . . . . . . . . . . 2.2 29 BER performance of the mixed-alphabet codes with and without redundant check nodes in their Tanner graph representations for N2 = 0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 33 BER performance of the mixed-alphabet codes with and without redundant check nodes in their Tanner graph representations for N2 = 10. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 34 BER performance of the mixed-alphabet codes with and without redundant check nodes in their Tanner graph representations for 3.1 N2 = 20. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 Portion of parity check matrix constructed in each step . . . . . . 46 xii LIST OF FIGURES 3.2 Tree constructed for a = 2, s = 2 after (a) steps 1-3 and (b) step 4 (the final structure). . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Performance of structured and random LDPC codes over Z4 with QPSK signaling over the AWGN channel. . . . . . . . . . . . . . . 3.4 47 53 Performance of structured and random LDPC codes transmitted using matched signals over the AWGN channel. . . . . . . . . . . 54 4.1 System model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.2 BER performances of (2640,1320) and (1008,504) LDPC codes over noncoherent AWGN channel without phase noise . . . . . . . . . 4.3 BER performances of (3072,1024) SCCC and (3072,1024) PCCC over noncoherent AWGN channel without phase noise . . . . . . . 4.4 72 BER performances of codes over coherent and noncoherent AWGN channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 72 74 BER performances of (1008,504) LDPC code over noncoherent AWGN channel without phase noise using TSOI-LLR, TSOI-SALLR and G-LLR, subjected to SNR estimation error . . . . . . . 4.6 76 BER performances of (3072,1024) SCCC over noncoherent AWGN channel without phase noise using TSOI-LLR, TSOI-SA-LLR and G-LLR, subjected to SNR estimation error . . . . . . . . . . . . . 4.7 BER performances of (1008,504) LDPC code over noncoherent AWGN channel with phase noise where σ 2 = 10−2 . . . . . . . . . 4.8 76 78 BER performances of (1008,504) LDPC code over noncoherent AWGN channel with phase noise where σ 2 = 4 × 10−2 . . . . . . . xiii 78 LIST OF FIGURES 4.9 BER performances of (1008,504) LDPC codes over noncoherent AWGN channel using TSOI-PN-LLR, TSOI-PN-SA-LLR and GPN-LLR, subjected to phase noise variance estimation error . . . 80 5.1 System model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 5.2 BER performances of LDPC codes over noncoherent AWGN channels 90 5.3 BER performances of LDPC codes over coherent and noncoherent AWGN channels . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 BER performances of mixed alphabet LDPC codes over noncoherent AWGN channels . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 90 92 BER performances of (1008,504) LDPC code with QDPSK transmission over noncoherent AWGN channels using TSOI-LLR, subjected to SNR misestimation . . . . . . . . . . . . . . . . . . . . . 5.6 94 BER performances of mixed alphabet LDPC code with QDPSK transmission over noncoherent AWGN channels using TSOI-SALLR, subjected to SNR misestimation . . . . . . . . . . . . . . . 6.1 94 BER performance of a (1008,504) binary LDPC code on the noncoherent AWGN channel with optimal and suboptimal LLRs of each code bit fed to the BP decoder for which the maximum number of iterations was set to 50 . . . . . . . . . . . . . . . . . . 6.2 108 BER performance of (8,4) and (8,3) code on the noncoherent AWGN channel with the G-LLR of each code bit fed to the BP decoder for which the maximum number of iterations was set to 50 109 6.3 Weight distribution of pseudocodewords arising from the M -covers of the Tanner graph corresponding to H(8,4) for M = 1, 2, 3 . . . . xiv 110 LIST OF FIGURES 6.4 Weight distribution of pseudocodewords arising from the M -covers of the Tanner graph corresponding to H(8,3) for M = 1, 2, 3 . . . . 111 7.1 System model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 7.2 Frame model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 7.3 BER performances of (256,128) LDPC code over noncoherent AWGN channel using PSAM-LLR with varying number of pilot symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.4 123 BER performances of (1008,504) LDPC code over noncoherent AWGN channel using PSAM-LLR with varying number of pilot symbols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5 BER performances of (1008,504) LDPC code over noncoherent AWGN channel with various metrics . . . . . . . . . . . . . . . . 7.6 124 125 BER performances of (1008,504) LDPC code over noncoherent AWGN channel with various block lengths, subjected to phase noise where σ 2 = 10−6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.7 BER performances of (1008,504) LDPC code over noncoherent AWGN channel subjected to SNR mis-estimation . . . . . . . . . 8.1 136 BER performances of (1008,504) LDPC codes over noncoherent AWGN channel using PSAM-R-LLR . . . . . . . . . . . . . . . . 8.3 128 BER performances of (256,128) LDPC codes over noncoherent AWGN channel using PSAM-R-LLR . . . . . . . . . . . . . . . . 8.2 126 136 BER performances of (1008,504) LDPC code over noncoherent AWGN channel with phase noise . . . . . . . . . . . . . . . . . . xv 138 LIST OF FIGURES 8.4 BER performances of (256,128) LDPC code over noncoherent AWGN channel subjected to SNR mis-estimation . . . . . . . . . 8.5 140 BER performances of (1008,504) LDPC code over noncoherent AWGN channel subjected to SNR mis-estimation . . . . . . . . . xvi 140 List of Abbreviations AWGN additive white Gaussian noise BPSK binary phase shift keying QPSK quadrature phase shift keying BDPSK binary differential phase shift keying QDPSK quadrature differential phase shift keying BEC binary erasure channel BER bit-error-rate BIBD balanced incomplete block designs BSC binary symmetric channel FG finite geometry GCD graph cover decoder G-(PN)-LLR Gaussian (Phase-Noise) LLR LDPC low-density parity-check LLR log-likelihood ratio LPD linear programming decoder ML maximum-likelihood MOLS mutually orthogonal Latin squares PDF probability density functions xvii List of Abbreviations PG projective geometry PSAM pilot-symbol-assisted modulation PSAM-LLR PSAM LLR PSAM-A-LLR PSAM-Approximate-LLR PSAM-SA-LLR PSAM-Simplified-Approximate-LLR QAM quadrature amplitude modulation SER symbol-error-rate SNR signal-to-noise ratio SPA sum product algorithm TSOI-PN-LLR Two-Symbol-Observation-Interval Phase-Noise LLR TSOI-PN-A-LLR TSOI-PN-Approximate-LLR TSOI-PN-SA-LLR TSOI-PN-Simplified-Approximate-LLR xviii List of Symbols and Notations x denotes scalar variable x denotes vector X denotes matrix xi ith row of matrix X xi ith element of vector x x(i) value x assumes at time instant i (·)∗ conjugate operator R Galois ring Z integer ring G group C code N block length or number of variable nodes in a Tanner graph M number of rows in a parity-check matrix or number of check nodes in a Tanner graph K dimension of code H parity-check matrix γ column weight of a parity-check matrix of degree of a variable node in a Tanner graph xix List of Symbols and Notations ρ row weight of a parity-check matrix or degree of a check node in a Tanner graph R code rate m message vector c codeword s signal vector sent r signal vector received Eb energy per message bit Es energy per symbol N0 power spectral density of noise n noise vector λ(·) log-likelihood ratio T tanner graph xx Chapter 1 Introduction Low-density parity-check (LDPC) codes are a class of linear error-correcting block codes introduced in [36]. Contrary to other linear codes, e.g., convolutional codes and Reed Solomon codes, LDPC codes are constructed and represented by sparse parity-check matrices. As its name suggest, in a sparse parity-check matrix, the ratio of the number of nonzero entries to the total number of elements is small. Unfortunately, LDPC codes were ignored due to the complexity of the decoding algorithm relative to the availability of technology during that time. Tanner’s generalization and graphical representation of LDPC codes [112] aside, there was little research on LDPC codes until their rediscovery made by MacKay [75, 76]. Some long LDPC codes have been shown to achieve an error rate performance of only a few tenths of a decibel away from the Shannon’s limit [18, 75, 76, 102, 104, 125]. In this opening chapter of the thesis, we introduce the mathematical preliminaries and give a descriptive overview of the discovery and development of LDPC codes. Readers are referred to Appendix A for the algorithms pertaining to the construction and decoding of LDPC codes. Further in this chapter, we 1 1. Introduction describe the current challenges and research interests that motivate the research problems undertaken in the thesis. A summary of the main contributions of this thesis is provided, followed by a breakdown on the organization of the thesis. 1.1 An Overview of LDPC Codes LDPC codes are represented by sparse parity-check matrices. Let C denote an LDPC code. Its M × N sparse parity-check matrix H = [h0 h1 · · · hM ]T is such that for each c ∈ C, chTi = 0 for all i. The rate of C is related to the size of H by the expression R ≥ N −M , N where equality holds if the rows {hi } are linearly independent. For a regular LDPC code, its parity-check matrix contains an equal number of non-zero elements in each column and row, i.e., H contains exactly γ non-zero elements in each column and ρ = γN M non-zero elements in each row. Conversely, the parity-check matrix of an irregular LDPC code does not contain an equal number of non-zero elements in each column and row. The variable node and check node degree distribution polynomials are denoted by γ(x) and ρ(x), respectively. In the polynomial γ(x) = dv ∑ γi X i−1 , i=1 where γi denotes the fraction of variable nodes with degree i, and dv denotes the maximum variable node degree. Similarly, in polynomial ρ(x) = dc ∑ i=1 2 ρi X i−1 , 1. Introduction ρi denotes the fraction of check nodes with degree i, and dc denotes the maximum check node degree. Tanner showed that LDPC codes may be represented by a bipartite graph, also known as a Tanner graph [112]. A bipartite graph is a graph whose vertex set can be partitioned into two disjointed subsets such that every edge connects a node in one subset to a node in the other subset, and no two vertices are connected within each subset. In a Tanner graph, the two disjoint subsets of nodes are the check nodes and the variable nodes (also known as bit or symbol nodes). Each of the M check nodes represents a row in H while each of the N variable nodes represents a column in H. If the element hji in H is nonzero, there exists an edge of weight hji that connects check node fj and variable node vi . Example 1.1 Consider a (7, 3) linear block code with the following parity-check matrix.            H=         1 1 0 1 0 0 0   0 1 1 0 1 0 0     0 0 1 1 0 1 0    0 0 0 1 1 0 1    1 0 0 0 1 1 0    0 1 0 0 0 1 1    1 0 1 0 0 0 1 This is a regular LDPC code where γ = ρ = 3. Its corresponding Tanner graph is depicted in Fig. 1.1. A cycle of length l in a Tanner graph is a closed path comprising l edges. The girth of a Tanner graph is the smallest cycle length of the graph. For example, a cycle of length six is shown in bold in Fig. 1.1. The smallest possible girth of any 3 1. Introduction v0 v1 v2 v3 v4 f0 f1 f2 f3 f4 v5 v6 variable nodes check nodes f5 f6 Figure 1.1: Tanner graph for Example 1.1. Tanner graph is four. To decode an LDPC code iteratively, it is important that the Tanner graph does not contain short cycles, especially cycles of length four. The original method of construction [36] yields regular LDPC codes represented by a parity-check matrix that is a concatenation of submatrices, such that the corresponding Tanner graph does not contain short cycles. In MacKay’s construction, the main objectives are to generate random/semi-random sparse parity-check matrices and to avoid short cycles in their corresponding Tanner graphs [75]. Due to the lack of structure, MacKay codes do not allow lowcomplexity encoding. The generator matrix in systematic form is obtained by performing Gaussian elimination. Although the parity check matrix is sparse, the resultant generator matrix is usually not since the parity-check matrix is not in standard form. Thus the number of operations required for encoding O(n2 ). To overcome this, an efficient encoding technique was proposed in [105] which requires some preprocessing before encoding. A similar method was also proposed in [97], in which the parity-check matrix is constructed with a semi-random structure. For 4 1. Introduction an arbitrary regular or irregular LDPC code, encoding can be performed based on LU factorization [111]. These encoding algorithms reduce the encoding complexity to O(n). It has been shown that long random irregular LDPC codes perform very close to the Shannon limit [18,72,102,104]. The error performance of an irregular LDPC code depends on the variable and check node degree distributions of its Tanner graph. The optimization of these distributions is found by density evolution, the evolution of the probability density functions of the messages passed between the variable and check nodes in a belief propagation decoder. However, the optimized distributions only provide a good code when the block length approaches infinity. The distributions applied to short or medium length codes give rise to high errorrate floor. Similar to MacKay codes, efficient encoding for irregular LDPC codes can be performed using the algorithms proposed in [105]. Since LDPC codes are generally not structured, they cannot be decoded algebraically. Iterative algorithms were hence devised to perform decoding. When applied to a Tanner graph, these algorithms are simple and easy to implement. They execute maximum-likelihood (ML) decoding in each iteration, but are suboptimal on the whole due to the presence of cycles in the Tanner graph. The bit-flipping decoding of LDPC codes is a very simple iterative hard decision based algorithm introduced with the code itself [36]. On the other hand, the probabilistic decoding algorithm performs soft decision decoding by iteratively updating the probability of each node assuming a certain value based on the values of the nodes connected to it. On a Tanner graph, it can be perceived as repeatedly passing messages along the edges, from the variable nodes to the check nodes and back, while updating the information 5 1. Introduction contained in the nodes. Similar to the bit flipping algorithm, the probabilistic decoding algorithms performs ML decoding in each iteration, but is suboptimal on the whole due to presence of cycles in the Tanner graph. The sum-product algorithm (SPA) was introduced in [36] and was later generalized for application to nonbinary codes [75]. For binary codes, the SPA decoding algorithm can be performed in the logdomain. Information about each code bit is represented in the form of a loglikelihood ratio (LLR) of the probability that the code bit assumes the value ‘0’ to that of the value ‘1’. Some multiplication operations are reduced to additions in the log-domain, thus reducing the decoding complexity. The min-sum decoder [121] performs iterative decoding in the same steps as the log-domain SPA decoder, except for an approximation that further reduces the remaining multiplication operations to comparisons. These algorithms are generally developed assuming that the codewords are transmitted using the BPSK over the additive-white Gaussian noise (AWGN) channel, and code bit c = 0 is mapped to s = 1 and c = 1 is mapped to s = −1 in the signal constellation. The correct mapping is particularly important for the log-domain decoder. Let NO , NA , NM , and NX denote the numbers of sum-product or minsum operators, signed adders, registers and connections respectively. The implementation complexity for decoding is estimated by [40] NO = N (1 − R)¯ ρ (1.1) NA = N (¯ γ + 1) (1.2) NM = N (¯ γ + 2) (1.3) NX = N (¯ γ 2 + γ¯ ρ¯ + 2), (1.4) 6 1. Introduction where γ¯ and ρ¯ are the average variable and check node degrees respectively. The computational decoding complexity for each iteration is estimated by the average number of sum-product or min-sum operations CO and the average number of additions CA per coded symbol as [40] 1.2 1.2.1 CO = (1 − R)¯ ρ (1.5) CA = γ¯ 2 . (1.6) Current Research and Challenges Nonbinary LDPC Codes Following the rediscovery of LDPC codes and the excellent error performance of binary LDPC codes over the AWGN channel, LDPC codes over finite fields were constructed and applied to the binary symmetric channel (BSC) and the binary AWGN channel [21]. Significant improvement over the performance of the binary codes reported motivated recent works on the analysis and design of non-binary LDPC codes on binary and nonbinary channels. In [6], nonbinary codes under ML decoding were shown to provide reliable communication at rates very close to the capacity of any discrete memoryless channel. Analysis of iterative decoding were performed using extrinsic information transfer charts [7], Gaussian approximation [65] and density evolution [101]. Unlike previous works that designed LDPC codes over finite fields, [29, 109] designed LDPC codes over rings. These codes can be mapped to matched nonbinary signal constellations to improve bandwidth efficiency. In particular, when mapped to PSK signal sets, the codes become geometrically uniform signal space codes. Empirical results in [109] 7 1. Introduction showed that LDPC codes over rings provide coding gain over coded modulation based on binary LDPC codes. Despite the superior error correcting performance of nonbinary codes over binary codes, computational efficiency remains a challenge due to its exponential increase with the alphabet size. Very recently, a new class of nonbinary LDPC codes, LDPC codes over mixed alphabets, has been introduced. Studies on these codes are motivated by the potential improvement in error performance from using larger code alphabets whilst maintaining a manageable decoding complexity. The concept of deploying more than one alphabet in a code is not new. The (N, K) Chinese Remainder Theorem code [38], defined over mixed number fields, performs encoding with N relatively prime integers and can correct up to N −K 2 errors. However, since its parity-check matrix is not sparse, iterative decoding of the code is not feasible. Mixed-covering codes [41, 91], defined over many alphabets, were constructed as single-error-correcting perfect codes applied to problems in distribution of resources such as speech coding. The parity-check matrix of a mixed-covering code consists of sub-matrices, each defined over a single alphabet. As the sub-matrices are arranged in a disjoint manner in the parity-check matrix, its corresponding Tanner graph is simply a collection of disjoint subgraphs, each corresponding to one sub-matrix defined over a single alphabet. Thus, performing iterative decoding on the Tanner graph is equivalent to performing iteratively decoding on each individual subgraph, which seemingly does not provide any coding gain. A mixed-alphabet LDPC code may be represented by a sparse parity-check matrix with rows and columns defined over different alphabets. Correspondingly, its Tanner graph has variable nodes and check nodes defined over more than one 8 1. Introduction alphabet. Example 1.2 Let the following parity-check matrix   Hmixed  1 1 0 1 0 0     =  0 1 1 0 1 0    0 0 1 1 0 1 represent a mixed-alphabet code. For simplicity, we fix all nonzero inputs to be 1’s. The second and third rows, as well the the fifth and sixth columns, are defined over GF (4). The remaining rows and columns are defined over GF (2). The corresponding Tanner graph is depicted in Fig. 1.2. Each codeword that arises from the Tanner graph contains four bits and two symbols over GF (4). For practical applications, one would like to keep the number of nodes defined over the larger alphabets small to maintain a relatively low decoding complexity. Since the message passing algorithm, in its most general form, does not have any restriction on the alphabet on which each node is defined, iterative decoding can easily be modified to operate on the Tanner graph with nodes defined over mixed alphabets. In [85], LDPC codes over two finite Galois fields were introduced. These codes were shown to perform better than their single alphabet counterparts of the same rate and equivalent binary length as the number of code symbols defined over the larger alphabet increases. In [9], LDPC codes constructed with multiple Galois fields were introduced. The codes were optimized according to the profile of the channel and applied to different frequency selective channels. On the other hand, [107] proposed irregular LDPC codes defined over groups of different orders and optimized the distribution of the degrees and groups of both the variable and 9 1. Introduction GF (2) v0 v1 GF (4) v2 v3 v4 variable nodes check nodes s0 s1 GF (2) s2 GF (4) Figure 1.2: Tanner graph for Example 1.2. 10 v5 1. Introduction check nodes. The main objective is to combine the advantages of both families of codes, binary and non-binary. That is, these mixed-alphabet LDPC codes outperform single alphabet LDPC codes of the same length and rate with a slight increase in decoding complexity incurred. 1.2.2 Non-random construction of LDPC Codes In Mackay’s LDPC code construction [75], one may design a parity-check matrix to contain up to a maximum of (N − K)/2 weight-two columns without significant increase in decoding errors. The number of weight-two columns allowable can be further increased when working with nonbinary codes. A simple and efficient method of constructing Tanner graphs with large girths by progressively establishing edges between code nodes and check nodes was proposed in [46]. The edge selection procedure is such that the insertion of a new edge on the graph has as small an impact on the girth as possible. Through this general, non-algebraic method of constructing graphs with large girth, simulation results show that LDPC codes from progressive edge-growth construction significantly outperform randomly constructed ones. A similar construction method was presented in [120]. However, the memory space required for storing the parity-check matrix still poses a problem in hardware implementation if very long LDPC codes are used. Furthermore, although the parity-check matrix of an LDPC code is sparse, its corresponding generator matrix is usually not. Thus, encoding is also an issue. Codes constructed based on finite geometry (FG) and projective geometry (PG) have been introduced long ago [67]. The codes are constructed based on the lines and points of Euclidean and PG over finite fields. It was later discovered that by limiting some design parameters, the regular parity-check matrices constructed 11 1. Introduction have low density, do not contain cycles of length four and thus fall into the class of LDPC codes [61,62]. The parity-check matrices can be expressed in cyclic or quasicyclic forms which require little storage space during implementation. Efficient encoding can be performed using shift-register circuits. The cyclic finite geometry codes tend to have relatively large minimum distance while the quasi-cyclic codes tend to have small minimum distance. The connectivity of the corresponding Tanner graph, though is deterministic, appear random at the decoder. Fig. 1.3, for example, shows a Tanner graph corresponding to a length-21 LDPC code constructed using PG. Thus, these codes perform well under iterative decoding. Although the densities of the parity-check matrices are low, the row and column weights increase with block length and are typically larger than that of randomly generated LDPC codes. Therefore, short codes are favored. Not only are these codes favored over randomly constructed LDPC codes due to reduction in storage space of the large parity check matrices and ease in performance analysis, they could also achieve relatively similar performance compare to random LDPC codes. Similar to FG and PG codes, LDPC codes may be designed by combinatorial approaches, exploiting well developed topics in Mathematics. LDPC codes were designed using balanced incomplete block designs (BIBD) [2, 118]. A BIBD is defined as a collection B of equal size blocks, comprising elements drawn from a set V , such that each pair of distinct elements (x, y) of V occurs in exactly λ blocks of B. Johnson [51] constructed irregular quasi-cyclic LDPC codes derived from difference families. High-rate LDPC codes based on the incidence matrices of unital designs were constructed in [52]. This construction exploits the fact that unital designs exist with incidence matrices which are rank deficient, thus giving rise to the high-rate LDPC codes with large number of parity-check equations. 12 1. Introduction Figure 1.3: Tanner graph corresponding to a length 21 projective geometry LDPC code. Such codes are well structured and have low-complexity implementation. LDPC codes constructed using combinatorial design share common characteristics; their corresponding Tanner graphs have girth of six, and they can be designed for very high rates (R ≥ 0.8) and of relatively short length. They also perform well under iterative decoding. 1.2.3 Finite Length Analysis of LDPC Codes Iterative decoders are well-known for their computational efficiency compared to the ML decoders. However, unlike the ML decoder, the iterative decoder does not give the optimum bit-error-rate (BER) performance. In the limit as code length goes to infinity, analysis of LDPC codes can be performed using density evolution. This was first introduced in [72] for the binary erasure channel (BEC) and subsequently in [102, 103] for more general channels. This technique may be 13 1. Introduction used to design and optimize the code node and check node degree distributions of the Tanner graph corresponding to a good performing irregular LDPC code. However, since infinite length is assumed, the distributions do not guarantee good finite length LDPC codes. Further, it is also not known if the LDPC code designed has an error floor, or where the error floor exists. Analyzing the performance of finite length LDPC codes is of research interest currently. The suboptimality of iterative decoding has been attributed to the emergence of pseudocodewords arising from the Tanner graph due to suboptimal computation in an iterative manner. Finite-length analysis of this iterative decoding behavior of LDPC codes was first proposed in [121]. Pseudocodewords that arises from computation trees were introduced and the concept was subsequently extended in [34]. These pseudocodewords were used to model the behavior of min-sum decoding of LDPC codes. Both [121] and [43] examined the convergence behavior of the min-sum decoder [32] on cycle codes, a special class of LDPC codes having only degree two variable nodes, and some necessary and sufficient conditions for the decoder to converge were provided. However, since computation trees grow exponentially with each iteration, the tracking of all pseudocodewords that may arise from a computation tree is virtually impossible after a few iterations. Similar works in [33] and [58] explained the behavior of iterative decoders using the lifts of the base Tanner graph. The common underlying concept in all these works is the role of pseudocodewords in determining decoder convergence and the decoding performance. Pseudocodewords that arise from graph covers were studied in [55, 119]. In each iteration, one particular check (code) node only receives information from code (check) nodes directly connected to it. Thus, the iterative decoding 14 1. Introduction algorithm cannot differentiate if it operates on a Tanner graph or a finite cover of the graph. Further, a codeword that arises from a finite cover of the Tanner graph, after normalization, does not necessarily reduces to a valid codeword from the Tanner graph. In short, a codeword is a pseudocodeword but a pseudocodeword may not be a codeword. These pseudocodewords can be represented very elegantly with the notion of fundamental cones and polytopes [30, 119]. It is shown in [119] that adding redundant check nodes to the Tanner graph representation of a code improves its performance under iterative decoding. Although such pseudocodewords can only perfectly characterize the iterative decoding behavior of the graph cover decoder (GCD) and linear programming decoder (LPD), they also provide substantial insights to the behavior of minsum decoding. Pseudocodewords of a Tanner graph play an analogous role in determining convergence of an iterative decoder as codewords do for a ML decoder. The error performance of a ML decoder can be computed analytically using the distance distribution of the codewords in the code. Similarly, an iterative decoder’s performance maybe characterized by the pseudocodeword distance. For linear codes, distance reduces to weight with respect to the all-zero codeword. Thus, in the context of iterative decoding, a minimum weight pseudocodeword [33] is more fundamental than a minimum weight codeword. In [55], lower bounds on the minimum pseudocodeword weight for the BSC and AWGN channels were presented. 1.2.4 Transmission over Nonstandard Channels LDPC codes have been shown to achieve reliable transmission at SNR extremely close to the Shannon limit on the AWGN channel [104]. Despite the promising 15 1. Introduction error control capability of LDPC codes on coherent channels, knowledge of the exact carrier phase is usually not available in practice. Tracking the time- varying phase present in most communication channels is usually not easy, due to the low SNR environments that LDPC codes are expected to operate in. The main drawbacks of phase-locked loops circuits, used to approximately implement coherent detection, are false-locks, phase slips, loses due to severe fading, Doppler shifts, phase noise, and oscillator frequency instabilities. Inaccuracy in phase estimation degrades the performance of the iterative decoder. When knowledge of carrier phase is not available, a simple solution is to apply differential encoding which incurs no additional bandwidth. The detection of a transmitted symbol is based on two consecutive received signals r˜(k − 1) and r˜(k), where k denotes a time instant. The leading signal r˜(k − 1) serves as a reference. There are two main classes of algorithms that serve to improve noncoherent detection. Multiple-symbol differential detection [22–25, 59, 60, 63, 122] is block-based ML detection of information symbols given the corresponding block of received signals. On the other hand, noncoherent sequence detection [1, 15, 16, 77, 78, 100] based on Viterbi algorithm approximates the optimal ML sequence detection. These algorithms approach ideal coherent detection and can be used when hard decision decoding on channel codes is deployed. They are, however, not applicable on LDPC or turbo codes, or when soft and iterative decoding is required. Recent research focus is thus on the development of soft decision noncoherent decoding. Iterative decoding requires the evaluation of the LLR of the two possible values of each code bit, based on the received signals pertaining to that bit. The LLR is fed into the decoder as soft information input. For noncoherent channels 16 1. Introduction where LDPC codes are modulated using binary differential phase-shift keying (BDPSK), the detection metric was recently derived in [113]. However, their results are not the optimal metrics for two reasons. First, the authors assumed that the decision statistic is Re[˜ r(k)˜ r(k − 1)∗ ], where (·)∗ denotes the complex conjugate, and derived the LLR based on the probability density functions (PDF) of Re[˜ r(k)˜ r(k −1)∗ ] conditioned on each possible value of the transmitted code bit. Clearly, the correct metric should be the LLR based on the joint PDF of the two signals r˜(k) and r˜(k − 1), i.e., the joint PDF of the two signals conditioned on one value of the bit, divided by the same joint PDF conditioned on the other value of the bit. Much information is lost in using the PDF of Re[˜ r(k)˜ r(k − 1)∗ ], compared to using the joint PDF of r˜k and r˜(k −1). Second, following [39], the product noise term in Re[˜ r(k)˜ r(k − 1)∗ ] is assumed to have a Gaussian PDF, an approximation that further contributes to the inaccuracy of the metric. The computation of the LLR encompasses the PDF of the unknown carrier phase. Thus, explicit carrier phase estimation is not required. However, the performance is a few decibels worse than that of coherent decoding [39]. A receiver for convolutional encoded, interleaved and differentially encoded M -ary PSK, based on a modified multiple-symbol differential detection algorithm that allows iterative decoding, was proposed for the noncoherent AWGN channel [92]. An extension to turbo codes was introduced in [14, 95], and a theoretical analysis of this code based on a cut-off rate bound was proposed in [94] (only for noniterative decoding). Turbo processing is performed, where reliability information is iteratively fed into and updated in the inner (modulation) decoder and the outer (convolutional/LDPC) decoder. Channel estimation can be explicitly carried out in the inner decoder. [42] uses linear prediction and per- 17 1. Introduction survivor processing to estimate the channel response of the frequency-flat fading channel, resulting in an exponential expansion of the decoding trellis. [44] updated the channel estimates of a noncoherent AWGN channel model using soft information from the outer decoder and assumed three different phase models: constant but unknown phase offset, Gaussian random walk and constant frequency offset. Alternatively, channel estimation can be incorporated in the inner decoder using a modified trellis-based decoding algorithm [79]. An iterative algorithm specific for noisy-phase channels was proposed in [35]. In [93], the capacity of noncoherent channels has been proven to be very similar to that corresponding to coherent channels. In [96] and [11], receivers for the block-constant phase model and discrete random-walk phase model are developed by using a discrete phase approach. More recently, [106] introduced a joint carrier phase estimator and turbo decoder. The carrier phase is a continuous random variable distributed over a 2π interval. This 2π interval is quantized into equally-sized sub-intervals denoted as phase states, and the probability of the phase states are updated in each turbo decoding iteration. The accuracy of modeling the carrier phase using phase states can be improved by dividing the 2π range into finer sub-intervals at the expense of increased decoding complexity. Thus, the approach therein yields good error performance but is computationally costly. When phase dynamics are slow enough, pilot symbols can be multiplexed into the transmitted signal sequence. Pilot symbols, containing no information from the transmitter, are periodically inserted into the transmitted signal sequence to track the carrier phase. A branch metric modified for turbo-coded systems using pilot symbols is presented in [64]. However, this metric requires knowledge of the carrier phase. In [48] and [64], estimation was only performed prior to the 18 1. Introduction first iteration of turbo decoding. Since turbo decoding is an iterative process, performance can be improved by re-estimating the channel after each decoder iteration [114]. Iterative estimation and decoding was proposed for convolutional codes in [37], for BPSK modulated turbo codes in [115], and for quadrature amplitude modulation (QAM) modulated turbo codes in [124]. More recent approaches that perform iterative channel estimation and decoding can be found in [66, 86, 87, 116]. Initial channel estimates are obtained from the pilot signals using an estimation filter. After each iteration of decoding, the channel estimates are refined with the aid of the tentative decisions fed back from the decoder. Similarly, joint carrier synchronization and decoding algorithms that recursively perform carrier phase and/or frequency synchronization based on updated soft decisions from the decoder were proposed in [4,90,99]. These receivers are computationally intensive and require explicit estimation. Furthermore, the channel estimates obtained in each iteration are assumed exact, and the accuracy of these estimates is ignored. The received signal sequence corrected by the channel estimates, i.e., multiplied by the conjugate of the corresponding channel estimates, is perceived as the received sequence from a coherent BPSK transmission. Moreover, it is assumed in each iteration that the channel estimates obtained are exact, and much information concerning the estimation accuracy is discarded in the process. Such an approach is ad-hoc and therefore not optimal statistically. There are many other approaches in literature, such as the using factor graphs and expectation-maximization algorithm, though not in focus here, shall be briefly summarized. The use of factor graphs, including code constraints and channel parameter statistics, is introduced in a broad context in [123] and modified to cater for noncoherent decoding under certain phase statistics subsequently. Using 19 1. Introduction density evolution, LDPC code distributions are optimized assuming a simplified block-constant phase model quantized over the two levels 0 and π [49,50]. Constant and random-walk phase noise models with Gaussian increments are considered and approximations of the sum product algorithm are derived and evaluated [19], and the case of strong phase noise is considered in [13]. The messages in the sum product algorithm are modeled using Dirac impulses located at the estimated values for the corresponding variable nodes in [20], and different estimation methods are examined. In [89], a phase model with the unknown carrier phase is constant over a block of N symbols and independent from block to block is considered and the power allocation to the pilot symbols is optimized using density evolution. A general Bayesian approach to LDPC decoding with an explicit representation of the channel parameters into the factor graph has also been proposed and applied to different channel models [12]. In [70, 71, 88, 110, 127], the concept of soft-decision-directed estimation is introduced. The channel parameters are estimated using the expectation-maximization algorithm [70, 71, 88, 110] or an ad hoc procedure [127] and the estimation algorithm is embedded into the iterative decoding process. These approaches consider the channel phase as a deterministic unknown constant and track time-variations in phase noise by some form of sliding observation window. Thus, these approaches are suitable for the block-constant phase model since the algorithms are not designed by exploiting the statistical knowledge of the phase time-variations. 1.3 Contributions of the Thesis The overall objective of the research undertaken is to design LDPC codes and practical decoders that work well in both standard and nonstandard channels. 20 1. Introduction Standard channels refers to channel models commonly assumed in information theory research, e.g., the BEC, the BSC and the coherent AWGN channel. On the other hand, nonstandard channels refers to the interference in transmission due to obstructions in the transmission paths and physical impairments at the transmitters and receivers, e.g., the noncoherent AWGN channel and the Rayleigh/Rician (multipath) fading channel. The studies here focus on the coherent and noncoherent AWGN channels. Thus, this thesis serves to bridge the gap between the design of capacity achieving codes, and the efficient and practical implementation of these codes. The main contributions of the thesis can be broadly classified into two categories: code design and decoder design. 1.3.1 Code Design In code design, we take into consideration important aspects such as error performance, storage memory required, and ease in implementing and complexity of the encoder and decoder. Mixed-alphabet codes proposed in [9, 85, 107] were constructed over multiple fields and groups. We introduce a new class of mixed-alphabet codes defined over two integer residue rings. In particular, a message sequence over the smaller ring is encoded into a codeword sequence over the smaller ring, extended with a small number of parity-check symbols over the larger ring. Contrary to what one might expect, we show that while keeping other parameters constant, there is a limit to the error performance gain obtained by increasing the number of parity-check symbols from the larger alphabet. In fact, performance degrades when this number exceeds an optimal value. However, further performance gains can be obtained by adding redundant check nodes in the corresponding Tanner graphs. 21 1. Introduction Extending our studies on codes over rings to structured codes, we propose a class of LDPC codes defined over an integer residue ring based on Latin square construction. Since the connectivity of the corresponding Tanner graphs is deterministic, storage space for the parity-check matrices is not required. The code properties such as length, rate and minimum distance are also determined analytically. 1.3.2 Decoder Design In decoder design, we focus on the derivation of the LLR for LDPC codes transmitted over the noncoherent AWGN channel. In general, we study two transmission schemes, DPSK and pilot-symbol-assisted modulation (PSAM). For DPSK, we develop LLRs for binary and quadrature DPSK transmissions. We also propose simplified LLRs approximated from the optimal LLRs as a tradeoff between performance and implementation complexity. These LLRs can also be applied to turbo codes and other receivers that require soft information processing. We compare the empirical performance using BDPSK and QDPSK, and justify their differences in performance using the notions of pseudocodeword weights. Although DPSK transmission does not require additional bandwidth and performs well for the noncoherent AWGN channel with relatively high phase noise, its error performance is still a few decibels inferior to the case of coherent AWGN channel. For the noncoherent AWGN channel with relatively constant and unknown carrier phase, i.e., relatively low phase noise, we propose using PSAM transmission instead. We derive the optimal LLR as well as its low computational complexity approximates, and show, theoretically and through simulations, that they perform close to the case of coherent AWGN channel when the number of 22 1. Introduction pilot symbols (pilot set) used to compute the LLR is large. We also propose an alternative method of computing the LLR. Instead of using the joint observation of the pilot set, we form a reference phasor which is the sum of the pilot signals in the pilot set. The resultant LLR, though not optimal, is very much simpler to implement compared to the optimal LLR. 1.4 Organization of The Thesis In this chapter, we provided a brief overview of LDPC codes, as well as literature review of recent research interests that have motivated the research problems undertaken and led to the key contributions as summarized. The rest of the thesis is organized as follows. In Chapter 2, we construct LDPC codes over mixed integer residue rings. In particular, we study the performance of LDPC codes where a codeword defined over a smaller ring is extended by a small number of parity-check symbols defined over a larger ring. Further, we add redundant check nodes in the corresponding Tanner graphs and examine the effects on the error performance of the mixedalphabet codes. Chapter 3 extends the study of codes over integer rings to structured LDPC codes. We construct regular Tanner graphs over integer rings based on Latin squares. Connectivity of the nodes is deterministic and thus requires minimal storage memory for the parity-check matrices. Based on the structure of the Tanner graphs, we study the properties of the codes designed analytically and examine their error performance empirically. In Chapter 4, we shift our focus to decoder design and derive the LLR metric for noncoherent transmission and detection of binary LDPC codes over 23 1. Introduction the AWGN channel with unknown carrier phase using BDPSK. The derivation, which is based on the joint observation of two consecutive received signals, takes into consideration the case where carrier phase is subjected to phase noise. We also introduce approximate metrics which require less computational complexity in exchange with a slight performance loss. Finally, we compare our metrics with that in the literature by examining their error performances, iterations required for convergence and robustness to phase noise and SNR mis-estimation. In Chapter 5, we extend the derivation of the LLR metrics to the case of QDPSK. Although the derivation only assumes unknown and constant carrier phase, we similarly provide empirical studies on the performances of the metrics for the case with phase noise and SNR mis-estimation. We also provide simulation results for the performance of mixed-alphabet codes studied in Chapter 2 transmitted using QDPSK over the noncoherent AWGN channel. Chapter 6 provides an analytic perspective to explain the differences in performance for BDPSK and QDPSK transmissions. Based on the LLR metrics in Chapter 4 and Chapter 5, we derive the pseudocodeword weights corresponding to each transmission. Using a simple toy code, we plot the pseudocodeword weight spectrum under each transmission and explain their performance difference. The theoretical analysis is supported by empirical results of an LDPC code of practical code length. In Chapter 7, we deploy the PSAM BPSK transmission which is not as bandwidth efficient as DPSK transmission, but offers performance closer to that of coherent detection. Based on the joint observation of the received signal and a set of pilot signals located within a certain observation window centered at the received signal, we derive the LLR metric for the PSAM transmission. 24 1. Introduction Approximate LLR metrics with simpler computation are also proposed. Further, we examine the effects of the length of the observation window and SNR misestimation on the performances of the metrics. Chapter 8 is a short extension of the work undertaken in Chapter 7. Instead of basing the derivation of the LLR on the joint observation of the pilot signals, we form a reference phasor which is the summation of all the pilot signals in the observation window. The LLR is then derived using the product of the received signal and the conjugate of the corresponding reference phasor. Although this LLR does not perform as well as the optimal LLR in Chapter 7, its computation is less cumbersome. Effects of phase noise and SNR mis-estimation are also investigated. Finally in Chapter 9, the thesis is concluded with a summary of the work done thus far, and a discussion on some suggestions for further research. 1.5 Channel Model and Simulation Methodology In this section, we provide an overview of the channel model assumed and the simulation methodology deployed throughout the thesis. Deviations, if any, will be discussed in detail in each chapters. A message sequence m is first encoded to a codeword c before transmission PSK. The signal sequence sent is ˜s. The signal sequence is transmitted over an AWGN channel with unknown carrier phase modeled as a Gaussian random-walk. This system model is used to model optical communications [47, 128, 129], where attenuation is exceptionally low compared to electrical transmission and laser phase noise is modeled as a Gaussian random-walk process. It should be noted 25 1. Introduction m LDPC c ˜s PSK Encoder Modulator ejθ ˜ n ˆ m LDPC LLR LLR ˜r Metric Calculator Decoder Channel Figure 1.4: System model that for Chapters 2 and 3 on code design, it is assumed that the carrier phase is known at the receiver. At the receiver, the LLR of code bit c(k) is computed based on the received signal ˜r. These metrics are then passed to the belief propagation ˆ is decoder where iterative processing is performed and the estimated message m obtained as the output. For all error performance curves shown in this thesis, a maximum of 50 iterations are allowed when the belief propagation decoding algorithm is used. It should be noted, however, that each received sequence typically converges to a valid codeword within ten iterations at high SNR. Each data point is obtained by simulating the system model discussed until at least 104 erroneous message bits are collected. For example, for a data point where the BER obtained is 10−5 , 104 erroneous message bits are collected for a total of 109 message bits simulated. The 98% confidence interval is given by [0.977 × 10−5 , 1.023 × 10−5 ]. 26 Chapter 2 Construction of LDPC Codes over Mixed-Alphabets Unlike in [9, 85, 107], this chapter focuses on the construction of LDPC codes where all the information symbols and some of the redundant symbols are defined over one alphabet while the remaining redundant symbols are defined over a larger alphabet. These codes may be viewed as extended LDPC codes where the additional redundant symbols are defined over a larger alphabet. A construction for mixed-alphabet codes is proposed in Section 2.1. In Section 2.2, we show that adding a redundant row with fewer zero divisors to a parity-check matrix can more effectively constrict the fundamental polytope, compared to adding one with more zero divisors. Further, we show that cycles of length four in a Tanner graph are not bad so long one of the four corresponding edge weights is a zero divisor. Finally, we simulate and study the performance improvement of both single-alphabet and mixed-alphabet LDPC codes when redundant check nodes are added to their corresponding Tanner graphs. 27 2. Construction of LDPC Codes over Mixed-Alphabets 2.1 Construction of Mixed-Alphabet Codes (N −K)×N1 1 We start with a full-rank parity-check matrix   A1 ∈ Z2b for an LDPC code  A1 0  N2 ×(N1 +N2 ) over Z2b . Further, let Hmixed =  , d > b and  where A2 ∈ Z2d A2 0 denotes the (N1 − K) × N2 null matrix. Trivially, if N2 = 0, then Hmixed = A1 represents an LDPC code over a single alphabet. For N2 > 0, Hmixed is a paritycheck matrix for a rate Kb N1 b+N2 d mixed-alphabet code Cmixed which is a (N1 , K) Z2b code extended with N2 parity-check symbols defined over Z2d . Thus, Cmixed is a subgroup of the additive group G2Nb1 G2Nd2 where Gq is the additive group of Zq . Motivated by [82, Section IIA], we take A1 to be lower-triangular. Further, we take Hmixed to be lower-triangular as well. Thus, encoding can be performed without the need to compute a generator matrix, and encoding complexity is kept low. 2.1.1 Simulation Studies Fig. 2.1 shows the BER performance of three Z4 codes, extended with N2 paritycheck symbols defined over Z16 for N2 = 0, 10, 20. For each code, an (N1 + N2 − K) × (N1 + N2 ) sparse, binary lower triangular matrix Hbin is first generated. The non-zeros in the first N1 − K rows of Hbin are then replaced by units from Z4 , while the remaining non-zeros in the last N2 rows of Hbin by units from Z16 . For a fair comparison on performance of these three codes, we maintain the code rate and codelength of each code to be 0.5 and 1000 (bits), respectively. At a BER of 10−5 , we observe a coding gain of 0.12 dB when N2 is increased from 0 to 10. However, a further increase of N2 to 20 results in a degradation in BER performance. 28 2. Construction of LDPC Codes over Mixed-Alphabets 0 10 N =0 2 N2=10 −1 10 N2=20 −2 BER 10 −3 10 −4 10 −5 10 −6 10 1 1.5 2 2.5 3 Eb/N0 (dB) 3.5 4 4.5 5 Figure 2.1: BER performance of Z4 codes extended with N2 parity-check symbols defined over Z16 for N2 = 0, 10, 20 This phenomenon may be explained as follows. For each row hmixed,i of the parity-check matrix Hmixed , there is a corresponding N -dim convex hull conv(Cmixed,i ) of Cmixed,i where 1 Cmixed,i = { c ∈ ZN ZN2 : ⟨hmixed,i , c⟩ ≡ 0 (mod 2t ) 2b 2d where t = b if i ≤ N1 − K and t = d otherwise } Here, ⟨hmixed,i , c⟩ denotes the inner-product of hmixed,i and c. The pseudocodewords of the Tanner graph T (Hmixed ) corresponding to Hmixed is contained in the fundamental polytope P(Hmixed ) of Hmixed , which is defined as the intersection ∩N1 +N2 −K conv(Cmixed,i ) of the N 1 + N2 − K convex hulls [119]. Since the binary i=1 29 2. Construction of LDPC Codes over Mixed-Alphabets length of Cmixed is Nbin = N1 b + N2 d, the number of convex hulls is ( N1 + N2 − K = d 1− b ) N2 + Nbin − K. b With Nbin and K fixed, and d > b, increasing N2 leads to a decrease in the number of convex hulls. This results in the relaxation of the fundamental polytope. Since the number of pseudocodewords which correspond to codewords remains constant, the number of pseudocodewords that do not correspond to codewords increases. Thus the performance of the code degrades. 2.2 Addition of Redundant Check Nodes to Tanner Graphs To further improve the performance of a mixed-alphabet code, we consider the addition of redundant rows to the parity-check matrix as this has been shown to constrict the fundamental polytope [55, 119]. We shall begin our investigation with the following theorem. (n−k)×n Theorem 2.1 Let Hf ull rank ∈ Zq be a full rank matrix and a, a′ ∈ Zqn−k \ {0} such that a · Hf ull rank contains more zero divisors than a′ · Hf ull rank . Then the number of pseudocodewords of T (H′rank def ) is less than that of T (Hrank def )      Hf ull rank   Hf ull rank  where Hrank def =   and H′rank def =  . a · Hf ull rank a′ · Hf ull rank Proof The fundamental polytope of Hrank def and H′rank def are P(Hrank def ) = P(Hf ull rank ) ∩ P(a · Hf ull rank ) 30 2. Construction of LDPC Codes over Mixed-Alphabets and P(H′rank def ) = P(Hf ull rank ) ∩ P(a′ · Hf ull rank ), respectively. By definition of a and a′ , clearly, each vertex of P(a′ · Hf ull rank ) is also a vertex of P(a · Hf ull rank ), i.e., P(a′ · Hf ull rank ) ⊂ P(a · Hf ull rank ) and this proves the theorem. Example 2.1 illustrates Theorem 2.1. Example 2.1 Let  Hf ull rank   1 1 0 1 0 0     = 0 1 1 0 1 0     0 0 1 1 0 1 define a code C over Z4 . Let a = (2, 2, 2) and a′ = (1, 3, 1), we have two rankdeficient matrices    Hrank def =  Hf ull rank a · Hf ull rank 1 1       0 1 =  0 0   2 0  0 1 0 0   1 0 1 0    1 1 0 1    0 0 2 2  and   H′rank def =   1 1 0     0 1 1 =     0 0 1  1 0 0  Hf ull rank a′ · Hf ull rank 31  1 0 0   0 1 0   .  1 0 1   2 3 1 2. Construction of LDPC Codes over Mixed-Alphabets Table 2.1: Number of m-cover Pseudocodewords in Example 2.1 m 1 2 3 4 5 Hf ull rank 64 2944 98752 2488504 48650368 Hrank def H′rank def 64 64 2688 2464 85696 78528 2123320 1994296 41489408 - Table 2.1 presents the number of m-cover pseudocodewords that arise from an mcover of T (H) corresponding to each parity-check matrix up to m = 5. Observe that a redundant row with only zero divisors as its non-zero elements reduces the number of m-cover pseudocodewords. However, this number is further reduced by replacing it with another redundant row that has only one zero divisor among its non-zero elements. Thus, we propose the following method to construct a parity-check matrix with redundant rows from one without redundant rows. Beginning with a (N − K) × N parity-check matrix without redundant rows, we choose some j rows and compute all possible redundant rows that can result from a non-zero weighted summation of the j rows, for j = 2, · · · , r, where r is some fixed integer. We then choose K rows and append to the (N − K) × N parity-check matrix to obtain an N × N parity-check matrix which contains redundant rows. In view of Theorem 2.1, we only choose the rows that does not result in a Tanner graph with cycles of length four, where all four corresponding edge weights are units. Further, in order to maintain a low decoding complexity, we select the most sparse rows. 32 2. Construction of LDPC Codes over Mixed-Alphabets 0 10 without redundant check nodes with redundant check nodes −1 10 −2 10 −3 BER 10 −4 10 −5 10 −6 10 −7 10 1 1.5 2 2.5 3 Eb/N0 (dB) 3.5 4 4.5 5 Figure 2.2: BER performance of the mixed-alphabet codes with and without redundant check nodes in their Tanner graph representations for N2 = 0. 2.2.1 Simulation Results and Discussion Using the method proposed, we add K redundant check nodes to each of the Tanner graphs representing the three codes considered in Section 2.1.1 and simulate them over an AWGN channel with BPSK signaling. Fig. 2.2-2.4 show a significant improvement in BER performance for all three codes when decoding is performed on the Tanner graphs with redundant check nodes compared to those without. At BER of 10−5 , the redundant check nodes yield an additional coding gain of 0.48 dB, 0.86 dB and 0.70 dB, when N2 is equal to 0, 10 and 20, respectively. In particular, the degradation in performance as N2 is increased from 10 to 20 still persists. This shows that the increase in the number of pseudocodewords that do not correspond to codewords as N2 increases, is too sharp to be compensated by a subsequent reduction due to the redundant check nodes. It will be interesting 33 2. Construction of LDPC Codes over Mixed-Alphabets 0 10 without redundant check nodes with redundant check nodes −1 10 −2 BER 10 −3 10 −4 10 −5 10 −6 10 1 1.5 2 2.5 3 Eb/N0 (dB) 3.5 4 4.5 5 Figure 2.3: BER performance of the mixed-alphabet codes with and without redundant check nodes in their Tanner graph representations for N2 = 10. 0 10 without redundant check nodes with redundant check nodes −1 10 −2 BER 10 −3 10 −4 10 −5 10 −6 10 1 1.5 2 2.5 3 Eb/N0 (dB) 3.5 4 4.5 5 Figure 2.4: BER performance of the mixed-alphabet codes with and without redundant check nodes in their Tanner graph representations for N2 = 20. 34 2. Construction of LDPC Codes over Mixed-Alphabets to see how many redundant check nodes are needed before such a compensation can be realized, and whether decoding complexity remains manageable at that point. 2.3 Conclusion We have introduced a class of mixed-alphabet LDPC codes over integer residue rings. These codes may be viewed as LDPC codes over a ring, extended with additional parity-check symbols over a larger ring. By increasing the number of redundant check nodes, coding gain can be obtained. This is nevertheless, at the expense of increased decoding complexity, since the complexity of the BP decoder depends on the size of the parity-check matrix. We also observe shown that there is a limit to the number of additional parity-check symbols over the larger ring which can be added, while keeping the over codelength constant, before degradation in error performance sets in. We have also shown that further coding gain can be obtained by adding redundant check nodes to Tanner graphs on the error performance for both single-alphabet and mixed-alphabet codes. 35 Chapter 3 Construction of Structured LDPC Codes over Integer Residue Rings Since the symbols of a non-binary code over a finite field cannot be matched to any signal constellation, it is not possible to construct a geometrically uniform code from a non-binary, finite field code. Geometrically uniform codes have been well studied, see [31, 108] for example. More recently, geometrically uniform, non-binary LDPC codes over rings were introduced in [109]. However, the codes therein were constructed randomly. Unlike random LDPC codes, structured LDPC codes such as the FG codes [62] and the BIBD codes [118] are favored due to the reduction in storage space for the parity check matrix and the ease in performance analysis. Moreover, they achieve relatively similar performance, compared to random codes. However, structured non-binary LDPC codes that have been proposed thus far are constructed over finite fields, e.g., [21, 28], and therefore cannot be geometrically uniform. In this chapter, we design structured, geometrically uniform, non-binary 36 3. Construction of Structured LDPC Codes over Integer Residue Rings LDPC codes over integer residue rings. We shall focus our investigations on codes of short code length since short non-binary LDPC codes have been shown to outperform their binary counterparts [7, 45, 98]. Studies in [56, 58, 121] have shown that a code’s performance under iterative decoding depends on the weight distribution of the pseudocodewords that arise from the finite covers of its Tanner graph. This is analogous to the dependency of the code’s performance under ML decoding on the weight distribution of the codewords. In particular, the presence of pseudocodewords of low weights is detrimental to the code’s performance under iterative decoding, especially so if the weights are less than the minimum Hamming distance of the code. With the aim to maximize the minimum pseudocodeword weight of a code, we follow the approach in [57] and construct structured codes based on Latin squares over integer residue rings. Although codes based on Latin squares were also studied in [26, 28, 80, 117], these research were not performed using the pseudocodeword framework. Similarly, codes constructed using other combinatorial approaches, e.g., in [27,81,118] focused on the optimization of design parameters such as girth, expansion, diameter and stopping sets. For practical reasons, we only consider linear codes over Z2a . In the next section, we provide an overview of codes over Z2a and their natural mapping to a matched signal constellation, that is, the 2a -PSK constellation. Section 3.2 introduces the notion of Latin squares over finite fields, followed by our extension of Latin squares to multiplicative groups of a Galois ring. We propose a method to construct Tanner graphs using Latin squares (over a multiplicative group of a Galois ring) in Section 3.3 and show that a wide range of code rates may be obtained from our construction. Further, we analyze the codes’ properties. In particular, we present the key contribution in this chapter, that is, we show that 37 3. Construction of Structured LDPC Codes over Integer Residue Rings the minimum pseudocodeword weight of each code equals its minimum Hamming distance. Finally in Section 3.4, we present simulated error performance of our codes and show that they outperform their random counterparts of similar length and rate, when mapped to matched signal sets and transmitted over the additivewhite-Gaussian-noise (AWGN) channel. 3.1 3.1.1 Preliminaries An Overview of Codes over Z2a Let C be a Z2a -submodule of the free Z2a -module Zn2a . Its nG × n generator matrix G can be expressed in the form [10]   λ1  2 g1     2λ2 g2    G= , ..   .     λnG 2 gnG (3.1) where 0 ≤ λi ≤ a − 1 for i = 1, 2, · · · , nG and {g1 , g2 , · · · , gnG } ⊂ Zn2a is a set of linearly independent elements. The rate of C is G nG 1∑ a − λi = − r= n i=1 a n n 38 ∑nG i=1 an λi . 3. Construction of Structured LDPC Codes over Integer Residue Rings The dual code C ⊥ is generated by the nH × n parity-check matrix of C, which can be expressed in the form   µ1  2 h1     2µ2 h2    H= , ..   .     µnH 2 hnH (3.2) where 0 ≤ µi ≤ a − 1 for i = 1, 2, · · · , nH and {h1 , h2 , · · · , hnH } ⊂ Zn2a is a set of linearly independent elements. Alternatively, the rate of C can be obtained by H 1∑ a − µi nH r =1− =1− + n i=1 a n n ∑nH i=1 an µi . If G (or H) is not already in the form in (3.1) (or (3.2)), one could perform Gaussian elimination without dividing a row by a zero divisor to obtain the nG (or nH ) linearly independent rows. Remark 3.1 C is a free Z2a -submodule if λi = 0 for i = 1, 2, · · · , nG . This also implies that µi = 0 for i = 1, 2, · · · , nH . Remark 3.2 ∑nG i=1 I(λi = j) = ∑nH i=1 I(µi = a − j) for j = 1, 2, · · · , a − 1, where I(·) is the indicator function. Remark 3.3 3.1.2 ∑nG i=1 I(λi = 0) = n − nH . ∑nH i=1 I(µi = 0) = n − nG . The Matched Signal Set Consider a 2a -PSK signal set containing 2a points that are equidistant from the origin while maximally spread apart on a two-dimensional space. Projecting one dimension on the real axis and the other on the imaginary axis, a symbol x ∈ Z2a 39 3. Construction of Structured LDPC Codes over Integer Residue Rings is mapped to sx = √ Es exp(j2πx/2a ) of the signal set, where Es is the energy assigned to each symbol [109]. Observe that for any x, y ∈ Z2a , d2E (sx , sy ) = d2E (sx−y , s0 ), where d2E (sx , sy ) denotes the squared Euclidean distance between sx and sy . Thus, the 2a -PSK signal set is matched to Z2a [69]. Let cx , cy ∈ C where cx = [x1 , x2 , · · · , xn ] and cy = [y1 , y2 , · · · , yn ]. They are mapped symbol-by-symbol to [sx1 , sx2 , · · · , sxn ] and [sy1 , sy2 , · · · , syn ] respectively. The squared Euclidean distance between these two signal vectors is d2E ([sx1 , sx2 , · · · , sxn ], [sy1 , sy2 , · · · , syn ]) n ∑ = d2E (sxi , syi ) i=0 = = n ∑ d2E (sxi −yi , s0 ) i=0 d2E ([sx1 −y1 , sx2 −y2 , · · · , sxn −yn ], [s0 , s0 , · · · , s0 ]). Observe that the Hamming distance between two codewords is mapped to the Euclidean distance between their corresponding signal vectors. 3.2 3.2.1 Latin Squares Definition and Application to Galois Fields The following definition and example are taken from [68, Chapter 17]. Definition 3.4 A Latin square of order q is denoted as (R, C, S; L) where R, C 40 3. Construction of Structured LDPC Codes over Integer Residue Rings and S are sets of cardinality q and L is a mapping L(i, j) = k, where i ∈ R, j ∈ C and k ∈ S, such that given any two of i, j and k, the third is unique. A Latin square can be expressed as a q × q array, where the cell in row i and column j contains the symbol L(i, j). Two Latin squares with mapping functions L and L′ are orthogonal if (L(i, j), L′ (i, j)) is unique for each pair (i, j). Further, a complete family of q − 1 mutually orthogonal Latin squares (MOLS) exists for q = ps where p is prime. The notion of Latin squares is easily applied to Galois fields by setting R = C = S = GF(ps ) and mapping function Lβ (i, j) = i + βj for β ∈ GF(ps ) \ {0}. Example 3.1 Let R = C = S = GF(22 ) = {0, 1, α, α2 }. Setting the mapping functions as L1 (i, j) = i + j, Lα (i, j) = i + αj and Lα2 (i, j) = i + α2 j, we obtain a     2 2 1 α complete family of three MOLS M1 =  α 0 α2 α α2 0 α2 α 1 0 1  2 1 α α 0 α2 1 α2 0 0 α 1 α 0 1 Mα2 =  α α2  α 0 α α 0 1 , Mα =  α α2 0 1 α2 0 α2 1 0 α 1 α 1  and  , respectively. In addition, the mapping function L0 (i, j) =  0 0 i yields a matrix M0 =  α 1 1 1 α α α α2 α2 α2 α2 1 0 0   which is orthogonal to each Latin square in the complete family of MOLS. 3.2.2 Extended Application to Multiplicative Groups over Integer Residue Rings Extending the notion of Latin squares over integer residue rings is not trivial because a complete family of 2s − 1 MOLS cannot be obtained by simply setting R = C = S = Z2s and mapping functions Lβ (i, j) = i + βj for β ∈ Z2s \ {0}. This is illustrated in the following example. 41 3. Construction of Structured LDPC Codes over Integer Residue Rings Example 3.2 Letting R = C = S = Z22 = {0, 1, 2, 3} and mapping functions   0 be L1 (i, j) = i + j, L2 (i, j) = i + 2j and L3 (i, j) = i + 3j, M1 =  2 1  0 1 M2 =  2 3   2 0 2 0 3 1 3 0 2 0  and M3 =  1 2 1 3 1 3 3 2 1 0 3 2 1 0 3 2 1 0  3 1 2 3 2 3 0 3 0 1 0 1 2 ,  are obtained, respectively. Since the elements in each row of M2 is not unique, M2 is not a Latin square. Therefore, we do not have a complete family of three MOLS. We propose an alternative way of constructing Latin squares over integer residue rings. Let extension ring R = GR(2a , s) = Z2a [y]/⟨ϕ(y)⟩, where ϕ(y) is a degree s basic irreducible polynomial over Z2a . Embedded in R is a multiplicative group G2s −1 of units of order 2s −1. Further, we let a′ < a and define z = z mod 2a ′ where z ∈ R, and extend this notation to polynomials, n-tuples and matrices over R. However, this still may not guarantee a family of MOLS. Example 3.3 Let R = GR(22 , 2) = Z4 [y]/⟨y 2 + y + 3⟩. Embedded in R is G3 = {1, α, α2 } = {1, y + 2, 3y + 1}, generated by α = y + 2. Let R = C = G3 ∪ {0}. Mapping functions L1 (i, j) = i + j, Lα (i, j) = i + αj and Lα2 (i, j) = i + α2 j     1 y+2 3y + 1 2 y+3 3y + 2 y+3 2y 3  3 2y + 2 0 yield matrices M1 =  y + 2 1  3y + 1 0 Mα2 =  y + 2 1 3y + 1 3y + 2 1 y+2 3y + 2 2 y+3 3 y+3 2y 2y + 2 3y + 2 3 3y + 1 0 , Mα =  y+2 1 3y + 1 y+2 3y + 1 y+3 3y + 2 2 2y 3 y+3 3 2y + 2 3y + 2 1  and  , respectively. Since G3 ∪ {0} is not closed under R-addition, S ⊂ R such that |S| ̸= |R| = |C| = 2s . Thus, all three matrices are not Latin squares. To overcome this problem, we propose a slight alteration of the mapping functions. The aim is to obtain mapping functions that map i ∈ R and j ∈ C uniquely to Lβ (i, j) ∈ S and |R| = |C| = |S|. 42 3. Construction of Structured LDPC Codes over Integer Residue Rings 1 (a) 1 Definition 3.5 Lβ (i, j) = ((i) 2a−1 + (βj) 2a−1 )2 a−1 where i, j ∈ G2s −1 ∪ {0} and β ∈ G2s −1 . (a) Theorem 3.6 Lβ (i, j) ∈ G2s −1 ∪ {0}. 1 1 Proof It is apparent that (i) 2a−1 , (βj) 2a−1 ∈ G2s −1 ∪ {0}. Since G2s −1 ∪ {0} is 1 1 not closed under R-addition, (i) 2a−1 + (βj) 2a−1 = u + 2v, where u ∈ G2s −1 ∪ {0} and v ∈ R. Using binomial expansion, the mapping function can be expressed as (a) Lβ (i, j) 2a−1 = (u + 2v) = a−1 2∑ x=0 Observe that (a) Lβ (i, j) = u2 (2a−1 ) x a−1 u2 a−1 −x ( ) 2a−1 2a−1 −x u (2v)x . x (2v)x = 0 mod 2a for x = 1, 2, · · · , 2a−1 . Thus, ∈ G2s −1 ∪ {0}. Theorem 3.7 Consider the two multiplicative groups G2s −1 ⊂ GR(2a , s) = ′ Z2a [y]/⟨ϕ(y)⟩ and G′2s −1 ⊂ GR(2a , s) = Z2a′ [y]/⟨ϕ(y)⟩, where ϕ(y) is a degrees basic irreducible polynomial over Z2a . Letting i, j ∈ G2s −1 ∪ {0} and β ∈ G2s −1 , (a′ ) we have i, j ∈ G′2s −1 ∪ {0} and β ∈ G′2s −1 . Then, Lβ (i, j) = Lβ (i, j). (a) Proof Using binomial expansion, (a) Lβ (i, j) = a−1 2∑ x=0 ( ) 1 1 2a−1 a−1 ′ ((i) 2a−1 )2 −x ((βj) 2a−1 )x mod 2a . x Now, observe that (  ( ′ )  )  2a −1 , x = y · 2a−a′ where y is an integer. 2a−1 ′ y mod 2a =  x  0 , otherwise. 43 3. Construction of Structured LDPC Codes over Integer Residue Rings Thus, ′ ′ ∑2a′ −1 (2a′ −1 ) 1 1 ′ 2a−1 −y·2a−a y·2a−a 2a−1 ) 2a−1 ) ((i) ((βj) mod 2a y=0 y 1 1 ∑ a′ −1 ( a′ −1 ) a′ −1 = 2y=0 2 y ((i) 2a′ −1 )2 −y ((βj) 2a′ −1 )y (a) Lβ (i, j) = (a′ ) = Lβ (i, j). Remark 3.8 When a′ = 1, the mapping function Lβ (i, j) = i + βj coincides (1) (1) (a) with the mapping function in the finite field case. Since Lβ (i, j) = Lβ (i, j) (a) (from Theorem 3.7), Lβ (i, j) is unique for a given pair (i, j). It follows that two (a) (a) Latin squares constructed by Lβ0 (i, j) and Lβ1 (i, j), where β0 , β1 ∈ G2s −1 and β0 ̸= β1 , are orthogonal. (a) Let R = C = S = G2s −1 ∪ {0}. A complete family {(R, C, S; Lβ ) : β ∈ 1 (a) 1 G2s −1 } of MOLS is obtained by defining Lβ (i, j) = ((i) 2a−1 + (βj) 2a−1 )2 a−1 . Example 3.4 Let R = C = S = G3 ∪ {0} ⊂ GR(22 , 2) and mapping functions 1 (2) 1 1 (2) 1 1 (2) L1 (i, j) = ((i) 2 + j 2 )2 , Lα (i, j) = ((i) 2 + (αj) 2 )2 and Lα2 (i, j) = ((i) 2 +     2 2 1 α (α2 j) )2 . The resultant MOLS are M1 =  α 0 α2 α α2 0 α2 α 1 0 1 2 1  0 1 and Mα2 =  α α2 α2 1 α α 0 α2 1 0 2 0 α 1 α  α 0 α α 0 1 , Mα =  α α2 0 1 α2 0 α2 1 0 α 1 α 1  , respectively. A complete family of three MOLS is obtained. In addition, the mapping function L0 (i, j) = i yields a matrix M0 =    0 0 0 1 1 1 0 1 α α α α α2 α2 α2 α2  which is orthogonal to each Latin square in the complete family of MOLS. 44 3. Construction of Structured LDPC Codes over Integer Residue Rings Structured LDPC Codes over Z2a 3.3 3.3.1 Construction of Graphs using Latin Squares Here, the construction method proposed in [57, Section IV-A] is generalized to construct graphs for different values of a and s by altering the mapping functions according to the value of a. The graph is a tree that has three layers that enumerate from its root; the root is a variable node, the first layer has 2s + 1 check nodes, the second layer has 2s (2s + 1) variable nodes and the third layer has 22s check nodes. Thus there are 22s + 2s + 1 variable nodes and 22s + 2s + 1 check nodes. The connectivity of the nodes are executed in the following steps: 1. The variable root node is connected to each of the check nodes in the first layer. 2. Each check node in the first layer is connected to 2s consecutive variable nodes in the second layer. 3. Each of the first 2s variable nodes in the second layer is connected to 2s consecutive check nodes in the third layer. 4. For i, j, k, β ∈ G2s −1 ∪ {0}, label the remaining variable nodes in the second layer (β, i) and all check nodes in the third layer (j, k). If β = 0, variable node (0, i) is connected to check node (j, i). If β ∈ G2s −1 , variable node (a) (β, i) is connected to check node (j, Lβ (i, j)). The tree is completed once all possible combinations of (i, j, k, β) are exhausted. Let T (a, s) denote the resultant tree constructed using the complete family of MOLS derived from G2s −1 ∪ {0} ⊂ R. T (a, s) is a degree-2s + 1 regular tree. Reading the variable (check) nodes as columns (rows) of a matrix H(a, s) ∈ 45 3. Construction of Structured LDPC Codes over Integer Residue Rings step 1 2s + 1 step 2 0 step 3 1 2s step 4 22s 22s Figure 3.1: Portion of parity check matrix constructed in each step (22s +2s +1)×(22s +2s +1) Z2a in the top-bottom, left-right manner, the portion of H(a, s) constructed at each step is illustrated in Figure 3.1. Setting the edge weights to be randomly chosen units from Z2a , the null space of H(a, s) yields an LDPC code C(a, s) over Z2a . Example 3.5 Let a = 2 and s = 2, where the Latin squares are as shown in Example 3.4. Figure 3.2a illustrates the resultant tree after Steps 1-3. This can be perceived as the non-random portion of the parity-check matrix. On the other hand, the pseudo-random portion of the parity-check matrix is obtained from Step 4. The final tree is shown in Figure 3.2b. 46 3. Construction of Structured LDPC Codes over Integer Residue Rings variable node check node (a) (0, 0) (0, 1) (0, α) (0, α2 ) (1, 0) (1, 1)(1, α) (1, α2 ) (0, 0) (0, 1) (0, α) (0, α2 ) (1, 0) (1, 1) (1, α) (1, α2 ) (α, 0) (α, 1)(α, α) (α, α2 ) (α2 , 0)(α2 , 1)(α2 , α)(α2 , α2 ) (α, 0) (α, 1) (α, α) (α, α2 ) (α2 , 0) (α2 , 1) (α2 , α)(α2 , α2 ) (b) Figure 3.2: Tree constructed for a = 2, s = 2 after (a) steps 1-3 and (b) step 4 (the final structure). 47 3. Construction of Structured LDPC Codes over Integer Residue Rings 3.3.2 Properties of C(a, s) Since the construction is deterministic, properties of the codes can be analytically derived. The resultant code C(a, s) is a length n(s) = 22s + 2s + 1 regular LDPC code represented by H(a, s) (or T (a, s)). The minimum distance of C(a, s) is denoted by dmin (a, s). Following the definition given in [57], we denote by wmin (a, s), the minimum pseudocodeword weight of the pseudocodewords that arise from the Tanner graph of C(a, s) for the 2a -ary symmetric channel. Theorem 3.9 Let T (a, s) denote the graph resulting from reducing all edge ′ weights of T (a, s) by the operation mod 2a . We have T (a′ , s) = T (a, s), i.e. H(a′ , s) = H(a, s). Proof First, the connection procedure is regardless of a in steps 1-3, and (a′ ) (a) similarly for β = 0 in step 4. Since Lβ (i, j) = Lβ (i, j) (from Theorem 3.7), the (a′ ) (a) edge ((β, i), (j, Lβ (i, j))) in T (a, s) is equivalent to the edge ((β, i), (j, Lβ (i, j))) in T (a′ , s). Remark 3.10 The graphs constructed by setting a = 1 yield binary codes in [57, Section IVA]. Further, it has also been shown that these codes are the binary PG LDPC codes introduced in [62]. Thus, it is known that dmin (1, s) = 2s + 2. Before deriving dmin (a, s), we state two relationships between the codewords in C(a, s) and C(a′ , s). Corollary 3.11 (i) If c ∈ C(a, s), then c ∈ C(a′ , s). ′ (ii) If c ∈ C(a, s) can be expressed as c = 2a−a c′ where c′ ∈ Zn2a′ , then c′ ∈ C(a′ , s) and is unique. 48 3. Construction of Structured LDPC Codes over Integer Residue Rings Proof Corollary 3.11(i) is a simple consequence of Theorem 3.9 while for 3.11(ii), ′ 2a−a c′ HT (a, s) = 0 mod 2a ⇒ c′ HT (a, s) = 0 mod 2a ′ ′ ⇒ c′ HT (a′ , s) = 0 mod 2a (from Theorem 3.9) ′ The uniqueness of c′ follows from the natural group embedding, GR(2a , s) → R : ′ r → 2a−a r. Theorem 3.12 dmin (a, s) = dmin (1, s). Proof Let dc be the Hamming weight of c ∈ C(a, s) \ {0}. Case 1: c contains at least one unit. From Corollary 3.11(i), when a′ = 1, c ∈ C(1, s). Further, dc ≥ dc . If dc = dmin (1, s), dc ≥ dmin (1, s). ′ Case 2a: c can be expressed as c = 2a−a c′ where c′ contains at least one unit of Z2a′ . From Corollary 3.11(ii), c′ ∈ C(a′ , s). Further, dc = dc′ and from Case 1, dc′ ≥ dc′ . When a′ = 1, c = 2a−1 c′ and c′ ∈ C(1, s). If dc′ = dmin (1, s), dc = dmin (1, s). ′ Case 2b: c can be expressed as c = 2a−a c′ where c′ does not contain any unit of Z2a′ . Similarly, from Corollary 3.11(ii), c′ ∈ C(a′ , s). Therefore, dc = dc′ and the bounds on dc′ follow Case 2a. Thus, dmin (a, s) = dmin (1, s). It has already been shown in [57, Section IVA] that wmin (1, s) = dmin (1, s). The following theorem states the relationship between wmin (a, s) and dmin (a, s). Theorem 3.13 wmin (a, s) = dmin (a, s). 49 3. Construction of Structured LDPC Codes over Integer Residue Rings Proof Since T (1, s) = T (a, s) when a′ = 1 (from Theorem 3.9) and all edge weights in T (a, s) are units of Z2a , wmin (a, s) and wmin (1, s) share the same tree bound [57], i.e. wmin (a, s) ≥ 2s + 2, ∀a. Further, dmin (a, s) = dmin (1, s) = 2s + 2 (from Theorem 3.12). Thus, 2s + 2 ≤ wmin (a, s) ≤ dmin (a, s) = 2s + 2 ⇒ wmin (a, s) = dmin (a, s) = 2s + 2 To compute the code rate r(a, s), the parity-check matrix H(a, s) has to be reduced to the form as discussed in Section 3.1. The code rate is bounded by 22s + 2s − 3s 22s + 2s − 3s ≤ r(a, s) ≤ , a(22s + 2s + 1) 22s + 2s + 1 where the upper bound corresponds to the code rates of the binary PG-LDPC codes [62]. We observe that by setting the edge weights of T (a, s) as randomly chosen units from Z2a , r(a, s) tends to the lower bound which results in codes suitable for low-rate applications. On the other hand, by setting all edge weights to be unity, r(a, s) increases significantly. The corresponding codes can thus be used in moderate-rate applications. Table 3.1 presents the properties of C(a, s) for various values of a and s. 3.4 Simulation Results Fig. 3.3 and Fig. 3.4 show the BER and symbol-error-rate (SER) performance of our structured codes over the AWGN channel. In Fig. 3.3a, the corresponding edge weights of the codes simulated are randomly chosen units of Z4 , while those 50 3. Construction of Structured LDPC Codes over Integer Residue Rings Table 3.1: Properties of C(a, s) a 1 2 3 4 1 2 3 4 1 2 3 4 1 2 3 4 s n(s) degree of dmin (a, s) T (a, s) = wmin (a, s) 2 21 5 6 3 73 9 10 4 273 17 18 5 1057 33 34 51 r(a, s) r(a, s) (lower bound) (unity edge weights) 0.5238 0.5238 0.2619 0.4762 0.1746 0.3175 0.1309 0.2381 0.6164 0.6164 0.3082 0.5548 0.2055 0.4932 0.1541 0.3699 0.6996 0.6996 0.3498 0.6337 0.2332 0.5653 0.1749 0.4982 0.7692 0.7692 0.3846 0.7053 0.2564 0.6367 0.1923 0.5669 3. Construction of Structured LDPC Codes over Integer Residue Rings in Figs. 3.3b and 3.4 are set to unity. The codewords are transmitted using the matched signals discussed in Section 3.1.2. The received signals are decoded using the sum-product algorithm. For comparison purposes, the performance of random, near-regular LDPC codes with constant variable node degree of 3, are also shown. These codes have similar code length and rates to that of the structured codes. For each data point, 104 error bits are obtained for a maximum of 100 iterations allowed for decoding each received signal vector. Fig. 3.3a shows our structured Z4 code outperforming the random code when the codelength is small, i.e., 42 bits. On the other hand, Fig. 3.3b shows our structured code performing worse than its random counterpart when the codelength is much larger, specifically, 2114 bits. Thus, it appears that our structured codes are only better than random codes for short codelengths. For a more thorough comparison, we simulate the BER performance of random and structured codes over Z4 and Z8 , for increasing codelengths of 21, 146 and 546 bits, and 63, 219 and 819 bits, respectively. This is shown in Figs. 3.4a and 3.4b, respectively. Observe that our codes significantly outperform their random counterparts over a wide BER range for very small codelengths, i.e., less than 100 bits. On the other hand, for larger codelengths, random codes perform better in the higher BER region while our structured codes are superior at lower BER’s, specifically, 10−4 and below for codelengths close to 1000 bits and 10−6 and below for larger codelengths. This phenomenon may be attributed to the fact that the minimum distance of our codes grow linearly with the square root of their codelength. On the other hand, from [17, Theorem 26], we have that the minimum distance of a random, regular LDPC code with constant variable node degree of 3, grows linearly with its codelength with high probability. As the random codes 52 3. Construction of Structured LDPC Codes over Integer Residue Rings 0 10 Structured BER Structured SER Random BER Random SER −1 10 −2 Error Rate 10 −3 10 −4 10 −5 10 −6 10 1 2 3 4 5 Eb/N0 (dB) 6 7 8 9 (a) a=2, s=2, random edge weights 0 10 Structured BER Structured SER Random BER Random SER −1 10 Error Rate −2 10 −3 10 −4 10 1 1.5 2 2.5 3 Eb/N0 (dB) 3.5 4 4.5 (b) a=2, s=5, unity edge weights Figure 3.3: Performance of structured and random LDPC codes over Z4 with QPSK signaling over the AWGN channel. 53 3. Construction of Structured LDPC Codes over Integer Residue Rings 0 10 Structured Random −1 10 −2 BER 10 −3 10 −4 10 s=2 s=4 −5 10 s=3 −6 10 1 2 3 4 5 Eb/N0 (dB) 6 7 8 (a) a=2, unity edge weights, transmitted using QPSK signaling −1 10 Structured Random −2 10 −3 BER 10 −4 10 s=4 −5 s=2 10 s=3 −6 10 7 8 9 10 Eb/N0 (dB) 11 12 13 (b) a=3, unity edge weights, transmitted using 8-PSK signaling Figure 3.4: Performance of structured and random LDPC codes transmitted using matched signals over the AWGN channel. 54 3. Construction of Structured LDPC Codes over Integer Residue Rings considered here are near-regular, we believe that they have superior minimum distances compared to our structured codes. 3.5 Conclusion We have extended the notion of Latin squares to multiplicative groups of a Galois ring. Using the generalized mapping function, we have constructed Tanner graphs that represent a family of structured LDPC codes over Z2a that covers a wide range of code rates. Most importantly, we have shown that the minimum pseudocodeword weight of these codes are equal to their minimum Hamming distance which is desirable under iterative decoding. Finally, our simulation results show that these codes, when transmitted by matched signal sets over the AWGN channel, can significantly outperform their random counterparts of similar length and rate, at BER’s of practical interest. 55 Chapter 4 Iterative Decoding using Binary Differential PSK This chapter considers the design of the iterative decoder for LDPC/turbo codes transmitted using BDPSK on the noncoherent channel. The key step to the design of the iterative receiver is the computation of the LLR of the two possible values of each code bit, based on the received signals pertaining to that bit. The output of this computation serves as the soft information input to the iterative decoder. Since this metric calculator is executed once at the start of each decoding process, it only incurs a one-off overhead in the computational cost. This method was first applied on the noncoherent decoding of turbo codes [39]. However, their results are not optimal due to two reasons. The authors based the computation of the LLR metrics on the PDF of Re[˜ r(k)˜ r∗ (k − 1)] conditioned on each possible value of the transmitted code bit, where r˜(k) and r˜(k − 1) are two consecutive received signals at time instants k and k − 1, respectively, and (·)∗ denotes the complex conjugate. As demonstrated in [83], the correct metric should be the 56 4. Iterative Decoding using Binary Differential PSK LLR based on the joint PDF of the two signals r˜(k) and r˜(k − 1), i.e., the joint PDF of the two signals conditioned on one value of the bit, divided by the same joint PDF conditioned on the other value of the bit. Much information is lost in using the PDF of Re[˜ r(k)˜ r∗ (k − 1)], compared to using the joint PDF of r˜(k) and r˜(k − 1). Second, the assumption that the product noise term in Re[˜ r(k)˜ r∗ (k − 1)] has Gaussian PDF further degrades the performance of the LLR. Since the LLR can be applied to any receiver which requires soft information input, we study the performance of BDPSK-transmitted LDPC codes and turbo codes (referring to both parallel and serial concatenated convolutional codes) over the AWGN channel with unknown carrier phase in the presence of phase noise. The system model is presented in Section 4.1. In Section 4.2, we develop the LLR for each code bit using the joint PDFs of the two consecutive received signals, conditioned on each possible value of the code bit concerned. Specifically, we consider a phase-noisy transmission and model the unknown carrier phase as a Gaussian random walk. Hence, the metric is obtained by averaging each joint PDF, conditioned on a hypothesized value of the carrier phase, over the PDF of the unknown carrier phase. We shall call this metric the Two-SymbolObservation-Interval Phase-Noise LLR (TSOI-PN-LLR). Further, we introduce the TSOI-PN Approximate LLR (TSOI-PN-A-LLR) and the TSOI-PN SimplifiedApproximate LLR (TSOI-PN-SA-LLR), two approximations of the TSOI-PNLLR, which yield significant reductions in implementation complexity. Close examination of the TSOI-PN-LLR leads to the observation of Re[˜ r(k)˜ r∗ (k − 1)] as a reasonable approximate statistic from which to compute the LLR, followed by the derivation of the Gaussian PN-LLR (G-PN-LLR) based on this approximation. While Re[˜ r(k)˜ r∗ (k −1)] is the decision statistic used in ML hard decision detection 57 4. Iterative Decoding using Binary Differential PSK of the information, our work shows that it does not provide the full reliability information required for soft information processing. By constraining the phase noise to have zero variance, the TSOI-LLR, TSOI-A-LLR, TSOI-SA-LLR and GLLR are readily obtained. The TSOI-LLR and the TSOI-SA-LLR correspond to the metrics we derived in [83], while the G-LLR corresponds to that in [39], where both works assumed constant carrier phase. Simulation studies in Section 4.3 compare the performances of these metrics on the decoding of LDPC and turbo codes. In addition to the case of a constant carrier phase, we examine the effects of a random-walk phase that fluctuates from one bit interval to the next, on the performances of the metrics. Since all metrics require knowledge of the exact SNR and phase noise variance, the performances of the metrics in the presence of estimation error of each quantity at the decoder are also considered. Section 4.4 summarizes the key results. 4.1 System Model The overall system model is depicted in Fig. 4.1. Consider a binary message sequence m = [m(1) m(2) · · · m(K)] where m(k) takes on the value 0 or 1 with equal probabilities for k = 1, 2, · · · , K. The message is first encoded by a rate K/N encoder to a codeword c = [c(1) c(2) · · · c(N )] before each code bit c(k) is transmitted using BDPSK. The BDPSK signal sequence sent is ˜s = [˜ s(0) s˜(1) · · · s˜(N )]. Here, 1 s˜(k) = Es2 ejϕ(k) (4.1) denotes the complex baseband signal to be transmitted at time instant k, where Es is the energy of the signal and ϕ(k) ∈ {0, π}. The initial phase ϕ(0) = 0 serves 58 4. Iterative Decoding using Binary Differential PSK m LDPC c ˜s BDPSK Encoder Modulator ejθ ˜ n ˆ m LDPC L Differential Detection ˜r Metric Calculator Decoder Channel Figure 4.1: System model as a reference and does not carry any information. The information of each code bit c(k) is carried in the phase difference △ϕ(k) of two consecutive signals s˜(k) and s˜(k − 1), i.e., ϕ(k) = ϕ(k − 1) + ∆ϕ(k), where k = 1, 2, · · · , N, (4.2)    0 if c(k) = 0 ∆ϕ(k) = .   π if c(k) = 1 The signal sequence is transmitted over an AWGN channel with unknown carrier phase. The received signal ˜r = [˜ r(0) r˜(1) · · · r˜(N )] is modeled as r˜(k) = s˜(k)ejθ(k) + n ˜ (k), k = 0, 1, · · · , N. (4.3) The sequence {˜ n(k)} is an AWGN sequence with E[˜ n(k)] = 0 and E[|˜ n(k)|2 ] = N0 . Here θ(k) ∈ [−π, π) denotes the unknown carrier phase modeled as a random-walk, 59 4. Iterative Decoding using Binary Differential PSK i.e., θ(k) = θ(k − 1) + w(k) mod 2π, (4.4) where w(k) is the phase noise and the noise sequence {w(k)} is a set of iid Gaussian random variables with mean zero and variance σ 2 . Note that (4.4) has a modulo 2π operation such the resultant θ(k) is always in the range [−π, π). Assuming that θ(0) is uniformly distributed over the range [−π, π), it can easily be verified (Appendix B) that θ(k) is also uniformly distributed over the same range, for k = ˜ 1, 2, · · · , N . The sequences {˜ s(k)}, {˜ n(k)} and {θ(k)} are mutually independent of one another. At the receiver, the LLR of code bit c(k) is computed based on the received signal samples r˜(k) and r˜(k − 1), for k = 1, 2, · · · , N . These metrics are then passed to the decoder where iterative processing is performed and the estimated ˆ is obtained as the output. message m 4.2 4.2.1 Metric Derivation The optimal TSOI-PN-LLR and its approximations Since the LLR is the only input to the iterative decoder, deriving the correct metric is therefore crucial. Information concerning the code bit c(k) is contained in two consecutive received signals r˜(k) and r˜(k − 1). Thus, the metric for c(k), denoted by λ(k), is defined as ( ) p c(k) = 0|˜ r(k), r˜(k − 1) ), λ(k) = ln ( p c(k) = 1|˜ r(k), r˜(k − 1) 60 (4.5) 4. Iterative Decoding using Binary Differential PSK for k = 1, 2, · · · , N . The metric λ(k) can be perceived as a measure of reliability for the code bit c(k). Given the received signals r˜(k) and r˜(k − 1), a positive (or negative) value of λ(k) indicates a higher probability that c(k) = 0 (or c(k) = 1). The magnitude of λ(k) relates the reliability of the estimate of c(k) based only on r˜(k) and r˜(k − 1), i.e., a large value of |λ(k)| reflects a high level of certainty while a small value of |λ(k)| indicates that a decision about the code bit based on the received signals is not reliable. From (4.2), the event {c(k) = i} is equivalent to the event {∆ϕ(k) = iπ}, which is equally likely to be due to the events {ϕ(k) = iπ, ϕ(k − 1) = 0} and {ϕ(k) = (1 − i)π, ϕ(k − 1) = π}, for i = 0, 1. Thus the LLR can be expressed as λ(k) ( ) ( ) p ϕ(k) = 0, ϕ(k−1) = 0|˜ r(k), r˜(k−1) +p ϕ(k) = π, ϕ(k−1) = π|˜ r(k), r˜(k−1) ) ( ). = ln ( p ϕ(k) = π, ϕ(k−1) = 0|˜ r(k), r˜(k−1) +p ϕ(k) = 0, ϕ(k−1) = π|˜ r(k), r˜(k−1) (4.6) Using Baye’s rule, we have ( ) p ϕ(k), ϕ(k − 1)|˜ r(k), r˜(k − 1) ( ) p ϕ(k), ϕ(k − 1), r˜(k), r˜(k − 1) ( ) = p r˜(k), r˜(k − 1) ( ) ( ) ( ) p r˜(k)|˜ r(k−1), ϕ(k), ϕ(k−1) p r˜(k−1)|ϕ(k), ϕ(k−1) p ϕ(k), ϕ(k−1) ( ) = . (4.7) p r˜(k), r˜(k−1) ( ) Clearly, p ϕ(k), ϕ(k−1) takes on the value 1 4 for all four combinations of ϕ(k), ϕ(k− 1) ∈ {0, π}. Next, we show in Appendix C, that [ 1 ] ( ) ( ) 2Es2 p r˜(k−1)|ϕ(k), ϕ(k−1) = p r˜(k−1) = C(k − 1)I0 |˜ r(k − 1)| , N0 61 (4.8) 4. Iterative Decoding using Binary Differential PSK where [ ] 1 |˜ r(k)|2 + Es C(k) = exp − πN0 N0 and 1 I0 [|x|] = 2π ∫ (4.9) π exp [|x| cos θ] dθ (4.10) −π is the zeroth-order modified Bessel function of the first kind. The result in (4.8) follows because the unknown carrier phase θ(k − 1) randomizes the phase of the received signal r˜(k−1), and one cannot extract any information concerning ϕ(k−1) and ϕ(k) from the received signal r˜(k − 1). Hence we have ( ) p ϕ(k), ϕ(k−1)|˜ r(k), r˜(k−1) = ( ) ( ) p r˜(k − 1) ( ) p r˜(k)|˜ r(k−1), ϕ(k), ϕ(k−1) , 4p r˜(k), r˜(k − 1) (4.11) and (4.6) reduces to λ(k) ( ) ( ) p r˜(k)|˜ r(k−1), ϕ(k) = 0, ϕ(k−1) = 0 +p r˜(k)|˜ r(k−1), ϕ(k) = π, ϕ(k−1) = π ) ( ). = ln ( p r˜(k)|˜ r(k−1), ϕ(k) = π, ϕ(k−1) = 0 +p r˜(k)|˜ r(k−1), ϕ(k) = 0, ϕ(k−1) = π (4.12) ( ) We proceed to evaluate the likelihood function p r˜(k)|˜ r(k−1), ϕ(k), ϕ(k−1) . In the presence of θ(k), the likelihood function is obtained by taking the average ( ) of the function p r˜(k)|˜ r(k − 1), ϕ(k), ϕ(k − 1), θ(k) over the conditional PDF of θ(k), i.e., ( ) p r˜(k)|˜ r(k − 1), ϕ(k), ϕ(k − 1) ∫ π ( ) ( ) = p r˜(k)|˜ r(k − 1), ϕ(k), ϕ(k − 1), θ(k) p θ(k)|˜ r(k − 1), ϕ(k), ϕ(k − 1) dθ(k). −π (4.13) 62 4. Iterative Decoding using Binary Differential PSK Conditioned on ϕ(k) and θ(k), the only randomness in r˜(k) is due to n ˜ (k). Thus, from (4.3), we have ( ) ( ) p r˜(k)|˜ r(k−1), ϕ(k), ϕ(k−1), θ(k) = p r˜(k)|ϕ(k), θ(k) ] [ 2 1 1 1 j(ϕ(k)+θ(k)) = exp − r˜(k) − Es2 e πN0 N0 [ 1 ] 2Es2 = C(k) exp |˜ r(k)| cos(θ(k) − ∠˜ r(k) + ϕ(k)) , N0 (4.14) where C(k) is as introduced in (4.9). ( ) To evaluate p θ(k)|˜ r(k − 1), ϕ(k), ϕ(k − 1) , we first note that θ(k) is independent of ϕ(k), i.e., ( ) ( ) p θ(k)|˜ r(k − 1), ϕ(k), ϕ(k − 1) = p θ(k)|˜ r(k − 1), ϕ(k − 1) . (4.15) From (4.4), we have ( ) ( ) p θ(k)|˜ r(k − 1), ϕ(k − 1) = p θ(k − 1) + w(k)|˜ r(k − 1), ϕ(k − 1) . (4.16) Although θ(k − 1) is independent of ϕ(k − 1), however, given r˜(k − 1), θ(k − 1) and ϕ(k − 1) become conditionally dependent. In the absence of AWGN, θ(k − 1) can easily be determined if r˜(k − 1) and ϕ(k − 1) are known. When AWGN is present, the conditional PDF of θ(k − 1) given r˜(k − 1) and ϕ(k − 1) is shown in Appendix D to be a Tikhonov PDF centered at the mean value ∠˜ r(k − 1) − ϕ(k − 1), i.e., [ ( ) p θ(k−1)|˜ r(k−1), ϕ(k−1) = exp 1 2Es2 N0 ] |˜ r(k−1)| cos (θ(k−1) − ∠˜ r(k−1) + ϕ(k−1)) [ 1 ] . 2Es2 2πI0 N0 |˜ r(k−1)| (4.17) 63 4. Iterative Decoding using Binary Differential PSK ≫ 1, the above PDF may be [ 1 ]−1 2Es2 approximated as a Gaussian PDF with mean zero and variance N0 |˜ r(k−1)| . For a reasonably high SNR such that Es N0 Further, w(k) is a Gaussian random variable independent of r˜(k − 1) and ϕ(k − 1), i.e., ( ) ( ) p w(k)|˜ r(k − 1), ϕ(k − 1) = p w(k) . (4.18) From (4.17) and (4.18), θ(k − 1) and w(k) remain independent of each other given r˜(k − 1) and ϕ(k − 1). Thus, the conditional PDF of θ(k) in (4.16), given r˜(k − 1) and ϕ(k − 1), is a Gaussian PDF, where its mean and variance are the sum of the ( ) means and variances, respectively, of the PDFs p θ(k −1)|˜ r(k −1), ϕ(k −1) and ( ) p w(k) . Assuming a reasonably high SNR, we can approximate this conditional PDF as a Tikhonov PDF, that is,  ( ) p θ(k)|˜ r(k−1), ϕ(k−1) ≈ exp  1 2Es2 N0  |˜ r (k−1)| 1 2σ 2 E 2 1+ N s 0 |˜ r(k−1)| cos (θ(k) − ∠˜ r(k − 1) + ϕ(k − 1))  2πI0  1 2Es2 N0  |˜ r (k−1)| 1 2σ 2 E 2 1+ N s 0 .  |˜ r(k−1)| (4.19) Now substituting the conditional PDFs in (4.14) and (4.19) back into (4.13), we have ( ) C ′ (k) · p r˜(k)|˜ r(k − 1), ϕ(k), ϕ(k − 1) = 2π ] [ 1 ∫ π ( ) 2Es2 exp r˜(k)e−jϕ(k) + α(k−1)˜ r(k−1)e−jϕ(k−1) cos θ(k) − ψ(k) dθ(k), N 0 −π (4.20) 64 4. Iterative Decoding using Binary Differential PSK where ( ) ψ(k) = ∠ r˜(k)e−jϕ(k) + α(k − 1)˜ r(k − 1)e−jϕ(k−1) , (4.21) and C ′ (k) = [ I0 C(k) 1 2Es2 N0 ] (4.22) α(k − 1) |˜ r(k − 1)| is independent of ϕ(k) and ϕ(k − 1). Here, we have introduced a quantity ( α(k − 1) = 2 1+ 1 2 )−1 2σ Es |˜ r(k − 1)| N0 (4.23) which is dependent on σ 2 . Observe that the maximum value of α(k − 1) is one and this occurs when phase noise is absent, i.e., σ 2 = 0. To obtain (4.20), we have made use of the following property: |˜ a| cos(θ − ∠˜ a) + |˜b| cos(θ − ∠˜b) = |˜ a + ˜b| cos(θ − ∠(˜ a + ˜b)), 1 where a ˜= 2Es2 N0 r˜(k)e −jϕ(k) 1 and ˜b = 2Es2 N0 α(k−1)˜ r(k−1)e−jϕ(k−1) are easily identified in (4.14) and (4.19), respectively. Completing the integration yields the following conditional PDF p(˜ r(k)|˜ r(k − 1), ϕ(k), ϕ(k − 1)) [ 1 ] 2 2E s = C ′ (k)I0 r˜(k)e−jϕ(k) + α(k − 1)˜ r(k − 1)e−jϕ(k−1) . (4.24) N0 By substituting (4.24), with appropriate values of ϕ(k) and ϕ(k − 1), into (4.12), 65 4. Iterative Decoding using Binary Differential PSK [ we have I0 [ λ(k) = ln I0 1 2Es2 N0 1 2Es2 N0 ] |˜ r(k) + α(k − 1)˜ r(k − 1)| ]. |˜ r(k) − α(k − 1)˜ r(k − 1)| (4.25) We refer to this metric as the Two-Symbol-Observation-Interval Phase-Noise LLR (TSOI-PN-LLR). In practice, the I0 (·) and ln(·) functions can be executed using look-up tables. To further reduce computational complexity, the | · | operation may be computed by the alpha-max-plus-beta-min algorithm [73], i.e., |xr +jxi | ≈ a max(xr , xi ) + b min(xr , xi ), where a and b are predefined amplifiers. √ For high SNR, each I0 (x) term may be approximated by: I0 (x) ≈ ex / 2πx. Hence, the TSOI-PN-LLR in (4.25) easily reduces to 1 2Es2 λ(k) ≈ N0 [ ] r˜(k) + α(k − 1)˜ r(k − 1) − r˜(k) − α(k − 1)˜ r(k − 1) [ ] 1 r(k − 1) − ln r˜(k) − α(k − 1)˜ r(k − 1) . (4.26) − ln r˜(k) + α(k − 1)˜ 2 This metric shall be referred to as the TSOI-PN Approximate LLR (TSOI-PN-ALLR). This metric does not require any I0 (·) operations. Since ln |x| < |x|, an even more simplified metric is obtained by removing the ln(·) terms in the TSOI-PN-A-LLR. Thus the simplified approximate LLR is 1 2Es2 λ(k) ≈ N0 [ ] r˜(k) + α(k − 1)˜ r(k − 1) − r˜(k) − α(k − 1)˜ r(k − 1) . (4.27) This metric shall be referred to as the TSOI-PN Simplified-Approximate LLR (TSOI-PN-SA-LLR). This metric does not need any I0 (·) or ln(·) computations. 66 4. Iterative Decoding using Binary Differential PSK 4.2.2 The G-PN-LLR Using series expansion on the I0 (·) terms and expressing the square of magnitude terms as |x + y|2 = |x|2 + |y|2 + 2Re[xy ∗ ], the numerator and denominator within the ln(·) term of the TSOI-PN-LLR (4.25) are [ ] Es 2 2 ∗ 1 + 2 |˜ r(k)| + |α(k − 1)˜ r(k − 1)| + 2α(k − 1)Re[˜ r(k)˜ r (k − 1)] N0 [ ]2 Es2 2 2 ∗ + 4 |˜ r(k)| + |α(k − 1)˜ r(k − 1)| + 2α(k − 1)Re[˜ r(k)˜ r (k − 1)] + · · · 4N0 and [ ] Es 2 2 ∗ 1 + 2 |˜ r(k)| + |α(k − 1)˜ r(k − 1)| − 2α(k − 1)Re[˜ r(k)˜ r (k − 1)] N0 [ ]2 Es2 2 2 ∗ + 4 |˜ r(k)| + |α(k − 1)˜ r(k − 1)| − 2α(k − 1)Re[˜ r(k)˜ r (k − 1)] + · · · , 4N0 respectively. Comparing these two expressions, their differences lie only in the signs of the terms containing Re[˜ r(k)˜ r∗ (k − 1)] and its powers. Thus, it is reasonable to approximate the reliability information based on the statistics of Re[˜ r(k)˜ r∗ (k − 1)], which, as is well known, is the decision statistics for differential detection. The approximate LLR metric based on Re[˜ r(k)˜ r∗ (k − 1)] is ( ) p Re[˜ r(k)˜ r∗ (k − 1)]|c(k) = 0 ). λ (k) = ln ( p Re[˜ r(k)˜ r∗ (k − 1)]|c(k) = 1 ′ (4.28) From (4.2), (4.3) and (4.4), we have [ 1 Re[˜ r(k)˜ r (k − 1)] = Re (−1)c(k) Es ejw(k) + Es2 ej(ϕ(k)+θ(k)) n ˜ ∗ (k − 1) ∗ 1 2 +Es e −j(ϕ(k−1)+θ(k−1)) 67 ] n ˜ (k) + n ˜ (k)˜ n (k − 1) . (4.29) ∗ 4. Iterative Decoding using Binary Differential PSK As in [39], we assume for simplicity that Re[˜ n(k)˜ n∗ (k −1)] is a zero-mean Gaussian random variable with variance (N0 /2)2 and is independent of n ˜ (k) and n ˜ ∗ (k − 1). The PDF of Re[˜ r(k)˜ r∗ (k − 1)] is thus Gaussian with mean [ ] [ ] [ ] 1 2 ∗ c(k) jw(k) E Re[˜ r(k)˜ r (k − 1)] = Re E (−1) Es e = (−1)c(k) Es e− 2 σ and variance [ ] ( )2 N0 ∗ Var Re[˜ r(k)˜ r (k − 1)] = Es N0 + , 2 and the resultant Gaussian Phase-Noise LLR (G-PN-LLR) is r(k)˜ r∗ (k − 1)] 2Es e− 2 σ Re[˜ . λ (k) ≈ ( )2 Es N0 + N20 1 2 ′ 4.2.3 (4.30) Case with no phase noise The derivation of the LLR based on the joint PDF of received signals over two observation intervals was first introduced in [83]. The definition of the metric is the same as in (4.5) which, using Bayes’ rule, is equivalent to ( ) p r˜(k), r˜(k − 1)|∆ϕ(k) = 0 ). λ(k) = ln ( p r˜(k), r˜(k − 1)|∆ϕ(k) = π (4.31) In [83], it has been assumed that the unknown carrier phase is constant over two consecutive observation intervals, i.e., θ(k) = θ(k − 1) = θ and θ is a random variable, uniformly distributed in the interval [−π, π). The likelihood function ( ) p r˜(k), r˜(k − 1)|∆ϕ(k) = iπ is evaluated by taking the average of the function ( ) p r˜(k), r˜(k − 1)|∆ϕ(k) = iπ, θ over all possible values of the unknown phase θ, 68 4. Iterative Decoding using Binary Differential PSK i.e., ( ) ∫ ( ) ( ) p r˜(k), r˜(k − 1)|∆ϕ(k) = iπ, θ p θ dθ, (4.32) p r˜(k), r˜(k − 1)|∆ϕ(k) = iπ = θ for i = 0, 1. This has been shown in [83] to be given by ( [ ) p r˜(k), r˜(k − 1)|∆ϕ(k) = iπ = C(k)C(k − 1)I0 ] 1 2Es2 |˜ r(k) + (−1)i r˜(k − 1)| . N0 (4.33) Substituting (4.33) in (4.31) with appropriate values of i, the TSOI-LLR [83, eqn. (14)] is ] I0 |˜ r(k) + r˜(k − 1)| ]. λ(k) = ln [ 1 2Es2 I0 N0 |˜ r(k) − r˜(k − 1)| [ 1 2Es2 N0 (4.34) This is the same metric as the TSOI-PN-LLR in (4.25) when phase noise is absent, i.e., when σ 2 = 0. The above derivation has assumed constant carrier phase. Thus, our TSOI-LLR in [83] is applicable only when there are no phase fluctuations. In this chapter, we approach the problem without making such an assumption. The new derivation takes into account the distribution of the phase noise w(k). Hence, the TSOI-PN-LLR is applicable for the general case where phase noise is present. Similarly, when phase noise is absent, the TSOI-PN-A-LLR in (4.26) is reduced to the TSOI-A-LLR 1 2Es2 λ(k) ≈ N0 [ ] r˜(k) + r˜(k − 1) − r˜(k) − r˜(k − 1) [ ] 1 − ln r˜(k) + r˜(k − 1) − ln r˜(k) − r˜(k − 1) , 2 69 (4.35) 4. Iterative Decoding using Binary Differential PSK while the TSOI-PN-SA-LLR in (4.27) is reduced to the TSOI-SA-LLR 1 2Es2 λ(k) ≈ N0 [ ] r˜(k) + r˜(k − 1) − r˜(k) − r˜(k − 1) . (4.36) The latter corresponds to the approximate metric in [83, Eqn. (15)]. We note that the joint estimation approach in [83, Section IIIA] is only an approximate approach. The metric resultant from the joint estimation approach has also been shown to perform worse than the TSOI-LLR. Thus, this joint estimation approach shall not be explored any further in this chapter. When phase noise is absent, the G-PN-LLR reduces to λ′ (k) ≈ 2Es Re[˜ r(k)˜ r∗ (k − 1)] . ( )2 Es N0 + N20 (4.37) which shall be referred to as the G-LLR. This metric was first derived in [39], where constant unknown carrier phase over two consecutive transmission intervals is assumed. 4.3 Simulation Study The computer simulations follow the system model introduced in Section 4.1. The rate-half, regular (2640,1320) and (1008,504) binary LDPC codes from [74] are used. Each code is represented by a sparse parity-check matrix with constant column weight of three. The sum-product algorithm in [36] is employed for decoding LDPC codes, with up to 50 iterations. The rate-1/3 (3072,1024) serially concatenated convolutional code (SCCC) simulated employs two four-state recursive convolutional codes; the first (outer code) is rate-1/2 70 4. Iterative Decoding using Binary Differential PSK ] [ 1+D 2 generated by 1 1+D+D and the second (inner code) is rate-2/3 generated 2   1+D 2  1 0 1+D+D2  by  , joined by an interleaver of length 2048. The rate-1/3 1+D 0 1 1+D+D 2 (3072,1024) parallel concatenated convolutional code (PCCC) simulated is formed by two equal, four-state, rate-1/2, recursive convolutional constituent codes [ ] 1+D2 generated by 1 1+D+D2 and concatenated with an interleaver of length 1024. Both the PCCC and SCCC are iteratively decoded using concatenated a-posteriori probability (APP) decoders [5] for six iterations. We note that since the performances of the TSOI-A-LLR and the TSOIPN-A-LLR are, respectively, relatively similar to those of the TSOI-LLR and the TSOI-PN-LLR, only the performances of the TSOI-LLR and the TSOI-PN-LLR are shown in subsequent figures. Thus, any observations or discussions pertaining to the performance of the TSOI-LLR and the TSOI-PN-LLR are also applicable to the TSOI-A-LLR and the TSOI-PN-A-LLR, respectively. 4.3.1 Performance of LDPC/turbo codes with different metrics and no phase noise It is assumed in this sub-section that the noise variance N0 is known at the receiver and the unknown carrier phase is constant over the transmission period of the signal sequence ˜s, i.e., σ 2 = 0. Therefore, TSOI-PN-LLR is the same as TSOILLR. The bit error rate (BER) is plotted against SNR per information bit, Eb /N0 , where Eb is the energy per information bit. Thus, for the rate-R codes used, Eb = Es /R. Fig. 4.2 highlights the performances of the (2640,1320) and (1008,504) LDPC codes, over the AWGN channel using different metrics. The TSOI-LLR yields the 71 4. Iterative Decoding using Binary Differential PSK −2 10 TSOI−LLR −3 BER 10 TSOI−A−LLR G−LLR −4 10 −5 10 (1008,504) LDPC code (2640,1320) LDPC code −6 10 4.6 4.8 5 5.2 Eb/N0 (dB) 5.4 5.6 Figure 4.2: BER performances of (2640,1320) and (1008,504) LDPC codes over noncoherent AWGN channel without phase noise −2 10 (3072,1024) PCCC −3 TSOI−LLR BER 10 TSOI−SA−LLR G−LLR −4 10 (3072,1024) SCCC −5 10 4.8 4.9 5 5.1 5.2 E /N (dB) b 5.3 5.4 0 Figure 4.3: BER performances of (3072,1024) SCCC and (3072,1024) PCCC over noncoherent AWGN channel without phase noise 72 4. Iterative Decoding using Binary Differential PSK best performance, followed by the TSOI-SA-LLR, while the G-LLR performs the worst. On the other hand, the G-LLR outperforms the TSOI-SA-LLR for the (3072,1024) SCCC, and similarly for the (3072,1024) PCCC (Fig. 4.3). However, their performances converge for the latter code for Eb /N0 above 5.5 dB. Again, the TSOI-LLR achieves the best performance for these two codes. Next, we examine the effects of the metrics on the number of iterations required for convergence. Table 4.1 shows the average number of iterations needed for convergence of a received sequence at the decoder. In the event of nonconvergence, the number of iterations for that particular received sequence is 50. The TSOI-LLR results in the fastest convergence, while the G-LLR corresponds to the slowest. As SNR decreases, the average number of iterations required for convergence increases. The average number for the G-LLR increases most rapidly, and the difference in the average required number of iterations between the metrics increases. This is even more apparent for the longer code. This observation is congruent with our theoretical analysis in [126], where using the mutual information approach, the TSOI-LLR has been shown to provide more information to the decoder than the G-LLR, and, thus, the G-LLR requires more iterations for convergence, particularly when the SNR is low. Not only does the average required number of iterations data reflect the speed of processing, it is also proportional to the computational cost. While the TSOI-LLR may incur a higher initial computational overhead than the G-LLR, it leads to faster convergence and lower overall decoding cost. Our experimental results here and theoretical analysis in [126] point to the importance of starting with the most accurate value of the LLR in iterative decoding, especially for low rate codes that operate at very low SNR. 73 4. Iterative Decoding using Binary Differential PSK Table 4.1: Average number of iterations required for convergence of a received sequence at the decoder SNR (dB) 4.6 4.8 5.0 5.2 5.4 5.6 4.6 4.7 4.8 4.9 5.0 5.1 (1008,504) LDPC code (2640,1320) LDPC code Average number of iterations TSOI-LLR TSOI-SA-LLR G-LLR 23.3953 24.6526 30.7456 15.1874 16.5391 20.2983 10.6642 11.4389 12.6271 7.8696 8.2477 8.9102 6.3058 6.4951 6.9233 5.3304 5.4324 5.7610 22.0284 23.5515 30.4545 16.6563 18.9924 23.0022 13.2786 14.5738 17.3373 11.0964 11.8350 13.6074 9.6408 10.0946 11.2735 8.5840 8.8773 9.7716 (1008,504) LDPC code (2640,1320) LDPC code (3072,1024) SCCC (3072,1024) PCCC −1 10 −2 10 coherent noncoherent, TSOI−LLR −3 BER 10 −4 10 −5 10 0 1 2 3 4 Eb/No (dB) 5 6 7 8 Figure 4.4: BER performances of codes over coherent and noncoherent AWGN channels 74 4. Iterative Decoding using Binary Differential PSK These two-symbol-observation-interval metrics, just like the case of differential detection, basically amount to using one previous signal to recover the carrier phase, and are therefore expected to lose substantially (more than 2 dB) in performance, compared to the case when the carrier phase is perfectly known. This can be observed in Fig. 4.4 which compares the performances of the TSOILLR and coherent detection. Thus, it provides the motivation to devise better schemes whereby performance closer to the coherent case may be achieved. In the same figure, we also notice that LDPC codes suffer less performance degradation with noncoherent decoding, compared to the other codes simulated. In particular, the performance loss for the LDPC codes using BDPSK with TSOI-LLR over that using coherent BPSK is around 2–3 dB, while the performance loss for the SCCC and the PCCC is about 4 dB. 4.3.2 Effects of SNR estimation error on performance of metrics Calculation of all the metrics requires the estimation of the SNR at the receiver. Fig. 4.5 and Fig. 4.6 show the effect of an estimation error in SNR on the performance of the (1008,504) LDPC code and the (3072,1024) SCCC, respectively, over the AWGN channel with time-invariant unknown carrier phase, using each metric. The horizontal axis indicates the difference between the estimated and the actual SNR, measured in decibels. A positive value represents an overestimation, i.e., the actual SNR is lower than the estimated SNR. In general, our metrics perform better when there is a slight SNR underestimation. Using the TSOI-LLR, TSOI-SA-LLR and G-LLR, the optimal BER for the (1008,504) LDPC code occurs at the SNR underestimation of 1 dB, 2 75 4. Iterative Decoding using Binary Differential PSK −1 10 TSOI−LLR TSOI−SA−LLR −2 10 G−LLR BER Eb/N0=5.1 dB −3 10 Eb/N0=5.3 dB −4 10 −5 0 SNR mis−estimation (dB) 5 Figure 4.5: BER performances of (1008,504) LDPC code over noncoherent AWGN channel without phase noise using TSOI-LLR, TSOI-SA-LLR and G-LLR, subjected to SNR estimation error −1 10 Eb/N0=5.0 dB −2 BER 10 TSOI−LLR TSOI−SA−LLR Eb/N0=5.2 dB −3 10 G−LLR −4 10 −5 0 SNR mis−estimation (dB) 5 Figure 4.6: BER performances of (3072,1024) SCCC over noncoherent AWGN channel without phase noise using TSOI-LLR, TSOI-SA-LLR and G-LLR, subjected to SNR estimation error 76 4. Iterative Decoding using Binary Differential PSK dB and 0 dB, respectively, while that for the (3072,1024) SCCC occurs at the SNR underestimation of 1 dB, 1 dB and 0 dB, respectively. However, when the SNR mis-estimation is lower than the optimum, the performance degradation is significant. Thus, compared to the G-LLR, the TSOI-LLR and TSOI-SA-LLR display higher degrees of robustness against SNR estimation error. 4.3.3 Effects of phase noise on performance of metrics The performance of the metrics in the presence of phase noise is investigated. Again, it is assumed that the variance of the AWGN is known at the receiver. We simulated the (1008,504) LDPC code in the presence of phase noise and found that significant performance loss only occurs for σ 2 ≥ 10−2 radians2 . Fig. 4.7 and Fig. 4.8 depict the performance degradation due to phase noise where σ 2 = 10−2 radians2 and σ 2 = 4 × 10−2 radians2 , respectively. We first examine the performance of the code when the phase noise variance 2 is not known and the receiver assumes constant carrier phase (σest = 0) over the transmitted signal samples. In other words, the TSOI-LLR, TSOI-SA-LLR and G-LLR are used. The performance loss of the TSOI-LLR corresponding to phase noise where σ 2 = 10−2 radians2 and σ 2 = 4 × 10−2 radians2 is 0.04 dB and 0.17 dB, respectively, at a BER of 10−5 , compared to the case without phase noise. The TSOI-SA-LLR and the G-LLR also suffer relatively similar amounts of performance degradation as the TSOI-LLR in the presence of phase noise. Hence, performance gains of our metrics over the G-LLR remain constant. Next, we assume that the receiver has perfect knowledge of the phase noise 2 = σ 2 . It can be observed that the performance of the codes variance, that is, σest improves when the variance of the phase noise is considered in the computation 77 4. Iterative Decoding using Binary Differential PSK σ2=σ2est=0 2 −2 2 σ =10 , σest=0 −3 10 2 σ =σ2est=10−2 BER −4 10 TSOI−PN−LLR TSOI−PN−SA−LLR −5 10 G−LLR −6 10 5 5.1 5.2 5.3 5.4 Eb/N0 (dB) 5.5 5.6 5.7 Figure 4.7: BER performances of (1008,504) LDPC code over noncoherent AWGN channel with phase noise where σ 2 = 10−2 σ2=σ2est=0 2 −2 2 σ =4x10 , σest=0 −3 10 BER σ2=σ2est=4x10−2 −4 10 TSOI−PN−LLR TSOI−PN−SA−LLR −5 10 G−PN−LLR −6 10 5 5.1 5.2 5.3 5.4 Eb/N0 (dB) 5.5 5.6 5.7 5.8 Figure 4.8: BER performances of (1008,504) LDPC code over noncoherent AWGN channel with phase noise where σ 2 = 4 × 10−2 78 4. Iterative Decoding using Binary Differential PSK of the metrics, compared to the case where constant carrier phase is assumed. The performance gains are more significant for a larger value of σ 2 . Also, the performance gain of the TSOI-PN-LLR over the TSOI-LLR is larger, compared to that of the TSOI-PN-SA-LLR over the TSOI-SA-LLR, and that of the G-PNLLR over the G-LLR. In fact, the performance gain of the G-PN-LLR over the G-LLR is negligible. 4.3.4 Effects of phase noise estimation error on performance of metrics To gain further insights into the importance of the knowledge of σ 2 , we simulate the effect of an estimation error in phase noise variance on the performance of the (1008,504) LDPC code, as shown in Fig. 4.9. The horizontal axis indicates, in 2 log-scale, the ratio of the estimated variance to the actual variance, i.e., σest /σ 2 . 2 At an SNR value of 5.0 dB, the lowest BER occurs at σest = σ 2 . On the other 2 hand, at an SNR value of 5.2 dB, the lowest BER occurs at σest /σ 2 = 10, that is, the estimated variance of the phase noise is ten times the actual value. While 2 the BER performances degrade slightly when the ratio σest /σ 2 is less than the optimum, the degradations are severe when the ratio is greater than the optimum. 4.4 Conclusion Based on the joint PDF of two consecutive received signals conditioned on each possible code bit, we have derived the TSOI-PN-LLR for BDPSK-transmitted LDPC/turbo codes on the noncoherent channel with phase noise. We also introduced the TSOI-PN-A-LLR and the TSOI-PN-SA-LLR, two approximations 79 4. Iterative Decoding using Binary Differential PSK −1 10 2 TSOI−PN−LLR TSOI−PN−A−LLR G−PN−LLR −2 σ =4x10 , E /N =5.0 dB b 0 σ2=10−2, E /N =5.0 dB b 2 0 −2 σ =10 , Eb/N0=5.2 dB −2 BER 10 −3 10 −6 10 −4 10 −2 0 10 10 2 10 4 10 σ2est/σ2 Figure 4.9: BER performances of (1008,504) LDPC codes over noncoherent AWGN channel using TSOI-PN-LLR, TSOI-PN-SA-LLR and G-PN-LLR, subjected to phase noise variance estimation error of the TSOI-PN-LLR that incur less computational cost. Examination of our metrics led to the derivation of the G-PN-LLR as an approximate metric based on a reasonable approximate statistic from which to compute the LLR. Although TSOI metrics may incur higher initial computational overheads compared to the G-PN-LLR, they generally outperform the G-PN-LLR. They also lead to faster convergence and thus lower average overall decoding computational cost. In addition, they are more robust against SNR estimation error. Furthermore, with the variance of phase noise taken into consideration in their computations, our metrics yield even better performance gains in the presence of phase noise, when compared to the G-PN-LLR. The metrics derived offer a low-complexity solution to noncoherent soft decoding without the need for explicit carrier phase recovery. Hence these metrics have potential applications in coherent optical 80 4. Iterative Decoding using Binary Differential PSK communications using differential PSK transmission. 81 Chapter 5 Iterative Decoding using Quadrature Differential PSK Since BPSK transmission is not bandwidth efficient, we extend the approach in Chapter 4 to derive the TSOI-LLR for QDPSK transmission. We describe the system model in the next section. In Section 5.2, we derive the LLR metric, based on the joint PDFs of r˜(k − 1) and r˜(k), given each possible value of the code bit sent. Since two code bits are sent at each time instant, we obtain an LLR metric for each bit at the receiver. Here, we model the unknown carrier phase as a random variable uniformly distributed over the interval [−π, π). Hence, the TSOI-LLR is obtained by averaging each joint PDF over all possible values of the unknown carrier phase. Similarly, assuming high SNR, we introduce the TSOI-SA-LLR which yields a significant reduction in computing complexity with negligible performance loss, compared to the TSOI-LnLR. Section 5.3 compares the performances of the metrics applied to the decoding of binary LDPC codes and the mixed-alphabet LDPC codes in Chapter 2 using QDPSK transmission, 82 5. Iterative Decoding using Quadrature Differential PSK m c LDPC ˜s DQPSK Encoder Modulator ejθ ˜ n ˆ m ˜r L LDPC Metric Calculator Decoder Channel Figure 5.1: System model as well as their performances when different modulation schemes, namely BDPSK and QDPSK, are employed. In addition, we investigate the performances of the metrics when the channel is subjected to phase noise, i.e., the carrier phase is a random-walk that fluctuates from one transmission interval to the next. The performances of the metrics in the presence of SNR estimation error at the decoder are also examined. 5.1 System Model Fig. 5.1 illustrates the overall system model. Consider a binary message sequence m = [m1 m2 . . . mK ], where each bit mk takes on the value 0 or 1 with equal probabilities, for k = 1, 2, · · · , K. The message sequence is encoded by a rate K/N encoder to a codeword c = [c1,M SB c1,LSB c2,M SB c2,LSB . . . cN/2,M SB cN/2,LSB ], 83 5. Iterative Decoding using Quadrature Differential PSK where N is assumed to be an even integer. Using QDPSK transmission and denoting the signal sequence sent as s = [s(0) s(1) . . . s(N/2)], the complex 1 baseband signal sent at time instant k is s˜(k) = Es2 ejϕ(k) , for k = 0, 1, · · · , N/2, where Es and ϕ(k) are the energy and the phase of the signal, respectively. Initially, ϕ(0) acts as a reference and thus do not contain any information on the codeword. In every transmission interval, information of two code bits, ck,M SB and ck,LSB , are jointly carried in the phase difference of two consecutive signals s˜(k) and s˜(k − 1), i.e., ϕ(k) = ϕ(k − 1) + △ϕ(k), where △ϕ(k) =          (5.1) π/4 if ck,M SB = 0, ck,LSB = 0, 3π/4 if ck,M SB = 0, ck,LSB = 1,   −π/4 if ck,M SB = 1, ck,LSB = 0,       −3π/4 if c k,M SB = 1, ck,LSB = 1, (5.2) for k = 1, 2, · · · , N/2. The signal sequence is transmitted over an AWGN channel with unknown carrier phase. The received signal ˜r = [˜ r(0) r˜(1) · · · r˜(N/2)] is modeled as r˜(k) = s˜(k)ejθ + n ˜ (k), k = 0, 1, · · · , N/2, (5.3) where θ denotes the unknown carrier phase that is statistically modeled as a random variable, uniformly distributed in the interval [−π, π). The noise sequence {˜ n(k)} is composed of independent and identically distributed (iid) complex Gaussian random variables with E[˜ n(k)] = 0 and E[|˜ n(k)|2 ] = N0 . At the receiver, the metric of each code bit is computed using the metric 84 5. Iterative Decoding using Quadrature Differential PSK calculator. These values are passed to the iterative LDPC decoder where the ˆ is obtained as the output. estimated message sequence m 5.2 Metric Derivation The LDPC decoder requires the LLR of the a-posteriori probability for each code bit, i.e., the probability of each code bit conditioned on the signal sequence received, as input. Since the information concerning code bits ck,M SB and ck,LSB is contained in two consecutive received signals r˜(k) and r˜(k −1), the LLR metrics of ck,M SB and ck,LSB are given by λM SB (k) = ln p(ck,M SB = 0|˜ r(k), r˜(k − 1)) p(ck,M SB = 1|˜ r(k), r˜(k − 1)) (5.4) λLSB (k) = ln p(ck,LSB = 0|˜ r(k), r˜(k − 1)) , p(ck,LSB = 1|˜ r(k), r˜(k − 1)) (5.5) and respectively, for k = 1, 2, · · · , N/2. We assume that the a-priori probabilities of ck,M SB and ck,LSB are equally likely, that is, p(ck,M SB = 0) = p(ck,M SB = 1) = 1 2 and p(ck,LSB = 0) = p(ck,LSB = 1) = 12 . Thus, we can rewrite the metrics using Bayes’ rule as λM SB (k) = ln p(˜ r(k), r˜(k − 1)|ck,M SB = 0) p(˜ r(k), r˜(k − 1)|ck,M SB = 1) (5.6) λLSB (k) = ln p(˜ r(k), r˜(k − 1)|ck,LSB = 0) , p(˜ r(k), r˜(k − 1)|ck,LSB = 1) (5.7) and respectively, for k = 1, 2, · · · , N/2. Due to the presence of the unknown carrier phase, the metrics are evaluated as follows. We first note that for the memoryless channel described in Section 5.1, 85 5. Iterative Decoding using Quadrature Differential PSK the likelihood function p(˜ r(k), r˜(k − 1)|˜ s(k), s˜(k − 1), θ), for a fixed hypothesized value of θ is p(˜ r(k), r˜(k − 1)|˜ s(k), s˜(k − 1), θ) = p(˜ r(k)|˜ s(k), θ) · p(˜ r(k − 1)|˜ s(k − 1), θ) { 1 } [( ) ] 2Es2 = C exp Re r˜(k) + r˜(k − 1)ej△ϕ(k) e−j(ϕ(k)+θ) (5.8) N0 where [ ] 1 |˜ r(k)|2 + |˜ r(k − 1)|2 + 2Es C= exp − . (πN0 )2 N0 We proceed to evaluate the likelihood function p(˜ r(k), r˜(k − 1)|ck,M SB = 0, θ). Since the event {ck,M SB = 0} is equally likely due to {△ϕ(k) = π/4} and {△ϕ(k) = 3π/4}, using (5.8), the likelihood function is p(˜ r(k), r˜(k − 1)|ck,M SB = 0, θ) [ 1 ] { [( ] π) 1 2Es2 = C exp Re r˜(k) + r˜(k − 1)ej 4 e−j(ϕ(k)+θ) 2 N0 ] [ 1 [( ) ] } 3π 2Es2 + exp Re r˜(k) + r˜(k − 1)ej 4 e−j(ϕ(k)+θ) . (5.9) N0 Given the statistical model of the unknown carrier phase in Section 5.1, the conditional joint PDF of r˜(k) and r˜(k − 1), assuming ck,M SB = 0, is evaluated by taking the average of the function p(˜ r(k), r˜(k − 1)|ck,M SB = 0, θ) over the probability distribution of θ, i.e., ∫ p(˜ r(k), r˜(k − 1)|ck,M SB = 0) = p(˜ r(k), r˜(k − 1)|ck,M SB = 0, θ)p(θ)dθ. (5.10) θ 86 5. Iterative Decoding using Quadrature Differential PSK Substituting (5.9) in (5.10), the conditional joint PDF is p(˜ r(k), r˜(k − 1)|ck,M SB = 0) { [ 1 ] [ 1 ]} 2 π 3π 1 2Es2 2E s r˜(k) + r˜(k − 1)ej 4 + I0 = C I0 r˜(k) + r˜(k − 1)ej 4 ,(5.11) 2 N0 N0 where I0 (α|˜ x|) = 1 2π ( ) jθ exp αRe[˜ x e ] dθ is the zeroth-order modified Bessel −π ∫π function. Similarly, the joint PDFs, conditioned on ck,M SB = 1, ck,LSB = 0 and ck,LSB = 1, are p(˜ r(k), r˜(k − 1)|ck,M SB = 1) { [ 1 ] [ 1 ]} 2 π 3π 1 2Es2 2E s = C I0 r˜(k)+ r˜(k−1)e−j 4 + I0 r˜(k)+˜ r(k−1)e−j 4 , (5.12) 2 N0 N0 p(˜ r(k), r˜(k − 1)|ck,LSB = 0) { [ 1 ] [ 1 ]} 1 2Es2 2Es2 j π4 −j π4 = C I0 r˜(k) + r˜(k − 1)e + I0 r˜(k) + r˜(k − 1)e ,(5.13) 2 N0 N0 and p(˜ r(k), r˜(k − 1)|ck,LSB = 1) [ 1 { [ 1 ] ]} 2 3π 3π 2E 2Es2 1 s , (5.14) = C I0 r˜(k)+˜ r(k−1)ej 4 + I0 r˜(k)+˜ r(k−1)e−j 4 2 N0 N0 respectively. 87 5. Iterative Decoding using Quadrature Differential PSK Thus the TSOI-LLRs are obtained as λM SB (k) = [ 1 [ 1 ] ] 2Es2 2Es2 j π4 j 3π I0 N0 r˜(k) + r˜(k − 1)e + I0 N0 r˜(k) + r˜(k − 1)e 4 ] (5.15) ] [ 1 ln [ 1 2Es2 2Es2 −j 3π −j π4 I0 N0 r˜(k) + r˜(k − 1)e + I0 N0 r˜(k) + r˜(k − 1)e 4 and λLSB (k) = ] ] [ 1 [ 1 2Es2 2Es2 j π4 −j π4 + I0 N0 r˜(k) + r˜(k − 1)e I0 N0 r˜(k) + r˜(k − 1)e ] ] , (5.16) [ 1 ln [ 1 2 3π 2Es2 2E j 3π −j s I0 N0 r˜(k) + r˜(k − 1)e 4 + I0 N0 r˜(k) + r˜(k − 1)e 4 respectively. It should be noted that since ej 4 = −e−j π 3π 4 and ej = −e−j 4 , only π 3π 4 two phasors are required to perform the necessary phase rotations. Each numerator or denominator on the RHS of (5.15) and (5.16) may be expressed in the form I0 (x1 ) + I0 (x2 ), where x1 and x2 are some scalar multiples 1 of Es2 /N0 . At high SNR such that Es N0 ≫ 1, each I0 (·) term can be approximated as an exponential function. Further, as a result of the different phase rotations of the r˜(k − 1) term, one of the input terms, either x1 or x2 , will be larger than the other. Therefore, I0 (x1 ) + I0 (x2 ) can be approximated by exp(max(x1 , x2 )). Consequently, the TSOI-SA-LLRs are 1 2Es2 λM SB (k) ≈ N0 [ max △ϕ(k)= π4 , 3π 4 − r˜(k) + r˜(k − 1)ej△ϕ(k) ] max △ϕ(k)=− π4 ,− 3π 4 88 r˜(k) + r˜(k − 1)e j△ϕ(k) (5.17) 5. Iterative Decoding using Quadrature Differential PSK and 1 2Es2 λLSB (k) ≈ N0 [ max △ϕ(k)=± π4 − r˜(k) + r˜(k − 1)ej△ϕ(k) max △ϕ(k)=± 3π 4 r˜(k) + r˜(k − 1)e j△ϕ(k) ] , (5.18) respectively. Note that corresponding to each code bit, the four I0 (·), one division and one ln(·) operations required for implementing the TSOI-LLR are reduced to only two max(·) and one subtraction operations required for the TSOI-SA-LLR. Thus the overall computing complexity is significantly reduced in the TSOI-SALLRs, compared to the TSOI-LLR. 5.3 Simulation Study 5.3.1 Performance of Binary LDPC Codes using Different Metrics Rate-half, regular (256,128) and (1008,504) binary LDPC codes, each represented by a sparse, column weight 3 parity-check matrix, from [74] are simulated. The sum-product algorithm in [36] is employed to decode these codes for up to 50 iterations. It is assumed that the noise variance is known at the receiver and the unknown carrier phase θ is constant over the transmission period of the signal sequence. From Fig. 5.2, we observe negligible performance loss in using the TSOISA-LLR instead of the TSOI-LLR, although the former requires significantly less implementation complexity. The BER performances are the same for coherent BPSK and QPSK 89 5. Iterative Decoding using Quadrature Differential PSK A−TSOI−LLR AA−TSOI−LLR −2 10 −3 BER 10 −4 10 (256,128) LDPC (1008,504) LDPC −5 10 4.5 5 5.5 6 Eb/N0 (dB) 6.5 7 7.5 Figure 5.2: BER performances of LDPC codes over noncoherent AWGN channels −1 QPSK BPSK 10 (1008,504) LDPC −2 10 −3 (256,128) LDPC BER 10 A−TSOI−LLR coherent detection −4 10 −5 10 −6 10 0 1 2 3 4 Eb/N0 (dB) 5 6 7 Figure 5.3: BER performances of LDPC codes over coherent and noncoherent AWGN channels 90 5. Iterative Decoding using Quadrature Differential PSK transmission (Fig. 5.3). This is expected since one QPSK transmission amounts to two simultaneous and independent BPSK transmissions over the same period. In the same figure, we compare the performance of the TSOI-LLR using QDPSK transmission with that using BDPSK transmission [83]. We observe that while the LDPC codes sent using QDPSK outperform those using BDPSK at higher BER, they are outperformed at lower BER. Moreover, one might infer from the figure that the value of BER at which performances using the two transmission schemes crosses-over decreases as block-length increases. 5.3.2 Performance of Mixed Alphabet LDPC Codes using TSOI-SA-LLR We investigate the performance of mixed alphabet LDPC codes [82] using the TSOI-SA-LLR. These codes are essentially non-binary LDPC codes over a ring Za , extended with a small percentage of redundant symbols defined over a larger ring Zb , i.e., b > a. It is known that performance improvement may be achieved by increasing the alphabet size which a code is defined over, but at the expense of significant increase in decoding complexity. Mixed alphabet LDPC codes, on the other hand, outperform their single alphabet counterparts of the same blocklengths and code rates with only a slight increase in decoding complexity. Using the method in Chapter 2, we construct a systematic rate-half code with 450 message symbols and 444 parity-check symbols over Z4 , extended with three parity-check symbols over Z16 . Thus, the overall binary block-length is 1800 bits. Since the code is non-binary, the code symbols have to be mapped to their corresponding binary representations before transmission using QDPSK described in Section 5.2. At the receiver, the a-posteriori probabilities of the code 91 5. Iterative Decoding using Quadrature Differential PSK −2 10 BER −3 10 BDPSK QDPSK −4 10 −5 10 4 4.2 4.4 4.6 4.8 5 Eb/N0 (dB) 5.2 5.4 5.6 5.8 Figure 5.4: BER performances of mixed alphabet LDPC codes over noncoherent AWGN channels symbols are computed using the LLRs derived from the metric calculator before probabilistic iterative decoding is performed. From Fig. 5.4, these codes performs better using QDPSK transmission, compared to BDPSK. Congruent with our assertion in Section 5.3.1, we do not observe any cross-over of the BER curves corresponding to the two transmission schemes. Furthermore, the performance gain of the mixed alphabet LDPC code sent using QDPSK over that using BDPSK is more prominent, compared to the case of the binary LDPC codes in Section 5.3.1. 5.3.3 Effects of SNR Mis-estimation Calculation of all the metrics requires the estimation of the SNR at the receiver. Fig. 5.5 and Fig. 5.6 show the effects of an estimation error in SNR on 92 5. Iterative Decoding using Quadrature Differential PSK the performances of the (1008,504) LDPC code in Section 5.3.1 and the mixed alphabet code in Section 5.3.2, respectively, with QDPSK transmission over the AWGN channel, assuming time-invariant unknown carrier phase. The horizontal axis indicates the difference between the estimated and the actual SNR, measured in decibels. A positive value represents an overestimation, i.e., the estimated SNR is higher than the actual SNR. Simulation results show that for the (1008,504) LDPC code, both the TSOILLR and the TSOI-SA-LLR perform better in the presence of a slight SNR underestimation. Since the performance of the TSOI-SA-LLR remains very close to that of the TSOI-LLR, hence only the latter is presented. Similarly, for the mixed alphabet LDPC code, the TSOI-SA-LLR performs better when there is a slight SNR under-estimation. In all cases, the optimum BER occurs at the SNR underestimation of 1 dB. 5.4 Conclusion Performance of LDPC codes using QDPSK transmission has been studied. Based on the joint PDFs of two consecutive received signals, conditioned on each possible code bit, we have developed the TSOI-LLR. In addition, the TSOI-SA-LLR, an approximation of the TSOI-LLR with significant reduction in complexity, has been introduced. Simulation results have shown that both metrics offer similar performance in the noncoherent decoding of LDPC codes. Further, it has been observed that a slight SNR underestimation improves performances of both metrics. 93 5. Iterative Decoding using Quadrature Differential PSK −1 10 −2 4.6 dB BER 10 4.8 dB 5.0 dB −3 10 5.2 dB optimum SNR mis−estimation = −1 dB −4 10 −10 −5 0 SNR Mis−estimation (dB) 5 10 Figure 5.5: BER performances of (1008,504) LDPC code with QDPSK transmission over noncoherent AWGN channels using TSOI-LLR, subjected to SNR misestimation −1 10 4.2 dB −2 10 BER 4.4 dB 4.6 dB 4.8 dB −3 10 optimum SNR mis−estimation = −1 dB −4 10 −10 −5 0 SNR Mis−estimation (dB) 5 10 Figure 5.6: BER performances of mixed alphabet LDPC code with QDPSK transmission over noncoherent AWGN channels using TSOI-SA-LLR, subjected to SNR misestimation 94 Chapter 6 Pseudocodeword Weights under Differential PSK Transmission Pseudocodeword weights have been defined for the BEC, BSC and AWGN channel [33,121]. For the AWGN channel, codewords are assumed to be transmitted using BPSK over a coherent channel. However, in real-life systems such as optical fiber applications, transmission is usually noncoherent. A simple method to eradicate the phase ambiguity is to apply differential encoding before PSK transmission. Motivated by this fact, we derive the pseudocodeword weights of binary codes for the AWGN channel under BDPSK in Section 6.1. As shown in Chapter 5, QDPSK yields better BER performance compared to BDPSK in the lower SNR region, while BDPSK outperforms QDPSK in the higher SNR region. In order to compare these two transmissions, we further derive the pseudocodeword weights of binary codes for the AWGN channel under QDPSK in Section 6.1. In Section 6.2, we plot the pseudocodeword weight distribution of an (8,4) binary Hamming code and simulate the corresponding BER performance under these two transmissions. The 95 6. Pseudocodeword Weights under Differential PSK Transmission performance gain of the case with QDPSK over that with BDPSK is attributed to spectral thinning in the pseudocodeword weight distribution when QDPSK is used in place of BDPSK. 6.1 Pseudocodeword weights under BDPSK and QDPSK over the noncoherent AWGN channel 6.1.1 System Model Consider a binary message sequence m = [m1 m2 · · · mK ] where mk takes on the value 0 or 1 with equal probabilities for k = 1, 2, · · · , K. The energy allocated for each information bit is Eb . The message is first encoded by a rate R = K N LDPC encoder to a codeword c ∈ C and then transmitted using DPSK. The DPSK 1 signal sequence sent is ˜s, where s˜k = Es2 ejϕk denotes the complex baseband signal transmitted at time instant k. Here, Es is the energy of each transmitted signal. The initial phase ϕ0 = 0 serves as a reference and does not carry any information. The information of the code bits is carried in the phase difference of consecutive signals. This shall be further elaborated in the subsequent subsections. The signal sequence is transmitted over an AWGN channel with unknown carrier phase. The received signal ˜r is modeled as r˜k = s˜k ejθ + n ˜k, (6.1) where θ denotes the constant unknown carrier phase taking on a value in the 96 6. Pseudocodeword Weights under Differential PSK Transmission interval [−π, π). The sequence {˜ nk } is a complex, additive, white, Gaussian noise sequence with E[˜ nk ] = 0 and E[|˜ nk |2 ] = N0 . In addition to being identically distributed, the n ˜ k are also independent. 6.1.2 Pseudocodeword Weight under BDPSK The codeword, transmitted signal sequence and received signal sequence are denoted by c = [c1 c2 · · · cN ], ˜s = [˜ s0 s˜1 · · · s˜N ] and ˜r = [˜ r0 r˜1 · · · r˜N ] respectively. Under BDPSK, the energy of each transmitted signal is Es = REb and the information of each code bit ck is carried in the phase differences of two consecutive signals s˜k and s˜k−1 , i.e., k = 1, 2, · · · , N, ϕk = ϕk−1 + ∆ϕk , (6.2) where ∆ϕk = π · ck . At the decoder, the log-likelihood ratio (LLR) of each code bit, i.e., λk = r) k =0|˜ ln p(c , is computed as soft input into the iterative decoder. The Gaussianp(ck =1|˜ r) LLR (G-LLR) [39], λk ≈ ∗ 2Es Re[˜ rk r˜k−1 ] ( N )2 , Es N0 + 20 (6.3) was derived for BDPSK transmission over the noncoherent channel. Although the optimal LLR has been derived in Chapter 4, the G-LLR shall be deployed for the ease of analysis given its form. Suppose ˜ c = [˜ c1,1 : c˜1,2 · · · : c˜1,M , c˜2,1 : c˜2,2 · · · : c˜2,M , · · · , c˜N,1 : c˜N,2 · · · : c˜N,M ] is an M -cover codeword that arises from an M -cover T˜ of the Tanner graph T of code C, where 1 ≤ M < ∞. Then, the corresponding pseudocodeword is ∑M ˜k,m , for k = 1, 2, · · · , N . Observe wB = [w1 w2 · · · wN ], where wk = M1 m=1 c 97 6. Pseudocodeword Weights under Differential PSK Transmission that the GCD’s decision rule on the M -codeword estimate ˆ ˜ c, N M N M 1 ∑∑ 1 ∑∑ ˆ ˜ c = arg max p (ck = c˜k,m |˜r) = arg max (1 − c˜k,m ) λk , (6.4) ˜ c M ˜ c M k=1 m=1 k=1 m=1 is equivalent to deciding on the corresponding pseudocodeword estimate w ˆ B using the rule w ˆ B = arg min wB N ∑ wk λ k . k=1 Therefore, the GCD favors a pseudocodeword wB over a codeword c if ∑N k=1 wk λk , which, by substituting the G-LLR, readily reduced to SBDP SK > 0, where SBDP SK = N ∑ (6.5) ∑N k=1 ck λk > (6.6) ∗ (ck − wk )Re[˜ rk r˜k−1 ]. k=1 ∗ Based on the system model, we expand the term Re[˜ rk r˜k−1 ] as [ ∗ ] Re r˜k r˜k−1 [ ] 1 1 j△ϕk j(ϕ +θ) ∗ −j(ϕ +θ) ∗ k k−1 = Re REb e + (REb ) 2 e n ˜ k−1 + (REb ) 2 e n ˜k + n ˜kn ˜ k−1 .(6.7) The term n ˜kn ˜ ∗k−1 can be ignored since it is typically much smaller compared to the other terms. We introduce a phase-rotated noise term n ˜ ′k = ejθ n ˜ ∗k which is a zeromean complex Gaussian variable with variance N0 , for k = 1, 2, · · · , N . Further, by observing that Re(˜ z ) = Re(z˜∗ ), we obtain the following simplified expression 98 6. Pseudocodeword Weights under Differential PSK Transmission ∗ for Re[˜ rk r˜k−1 ], [ ∗ ] [ ′ ] 1 1 Re r˜k r˜k−1 = REb ej△ϕk + (REb ) 2 ejϕk Re n ˜ k−1 + (REb ) 2 ejϕk−1 Re [˜ n′k ] .(6.8) Given that c is sent, we have SBDP SK = REb N ∑ { (ck − wk )(1 − 2ck ) + (REb ) 1 2 (c1 − w1 )ejϕ1 Re [˜ n′0 ] k=1 +(cN −wN )ejϕN −1 Re [˜ n′N ]+ N −1 ∑ } ((ck −wk )ejϕk−1 +(ck+1 −wk+1 )ejϕk+1 )Re [˜ n′k ] .(6.9) k=1 SBDP SK is a Gaussian random variable with mean E(SBDP SK ) = REb N ∑ (ck − wk )(1 − 2ck ), (6.10) k=1 and variance Var(SBDP SK ) } { N N −1 ∑ ∑ (ck − wk )(ck+1 − wk+1 )(1 − 2|ck+1 − ck |) . (6.11) = REb N0 (ck − wk )2 + k=1 k=1 Thus the pairwise error probability using GCD is Pc→wB    = Q  = p(SBDP SK > 0|c sent)  ]2 [∑ N  REb k=1 (ck − wk )(1 − 2ck )  ] [∑  .(6.12) ∑ N −1 N  2 N0 k=1 (ck −wk )(ck+1 −wk+1 )(1−2|ck+1 −ck |) k=1 (ck −wk ) + 99 6. Pseudocodeword Weights under Differential PSK Transmission If an all-zero codeword is sent, the pairwise probability becomes  P0→wB   = Q  N0 (∑ REb N k=1 (∑ N k=1 wk2 + )2 wk ∑N −1 k=1 wk wk+1   )   (6.13) which yields the following definition. Definition 6.1 The pseudocodeword weight wpBDP SK of wB is (∑ wpBDP SK 6.1.3 )2 wk ). = (∑ ∑N −1 N 2 w + w w k k+1 k=1 k k=1 N k=1 (6.14) Pseudocodeword Weight under QDPSK We denote the codeword, transmitted signal sequence and received signal sequence by c = [c1,M SB c1,LSB c2,M SB c2,LSB · · · cN/2,M SB cN/2,LSB ], ˜s = [˜ s0 s˜1 · · · s˜N/2 ] and ˜r = [˜ r0 r˜1 · · · r˜N/2 ] respectively. Since two code bits are transmitted in each time interval under QDPSK, the energy of each transmitted signal is Es = 2REb . The information of each pair of code bits {ck,M SB ck,LSB } is carried in the phase difference of two consecutive signals s˜k and s˜k−1 , i.e., k = 1, 2, · · · , N/2, ϕk = ϕk−1 + ∆ϕk , (6.15) where ∆ϕk = (1 − 2ck,M SB )(1 + 2ck,LSB ) π4 . Again, the G-LLR shall be deployed although the optimal LLR metrics were derived in Chapter 5 due to the ease of analysis of the former, as demonstrated in the previous section. However, the G-LLR was only derived for BDPSK transmission in [39]. Thus, we derive the G-LLR corresponding to transmission 100 6. Pseudocodeword Weights under Differential PSK Transmission using QDPSK as follows. ∗ ∗ The decision statistics for ck,M SB and ck,LSB are Im[˜ rk r˜k−1 ] and Re[˜ rk r˜k−1 ], respectively. Based on the system model, we have ∗ r˜k r˜k−1 = 2REb ej△ϕk + (2REb ) 2 ej(ϕk +θ) n ˜ ∗k−1 + (2REb ) 2 e−j(ϕk−1 +θ) n ˜k + n ˜kn ˜ ∗k−1 . (6.16) 1 1 The term n ˜kn ˜ ∗k−1 can be ignored since it is typically much smaller compared to ∗ ∗ ] are Gaussian distributed where ] and Re[˜ rk r˜k−1 the other terms. Hence Im[˜ rk r˜k−1 ∗ Im[˜ rk r˜k−1 ]∼N and ∗ Re[˜ rk r˜k−1 ]∼N (√ (√ 2REb (1 − 2ck,M SB ), 2REb N0 + (N0 /2)2 ) ) 2REb (1 − 2ck,LSB ), 2REb N0 + (N0 /2)2 . Thus, the G-LLR corresponding to QDPSK transmission are λk,M SB ( ) ∗ p Im[˜ rk r˜k−1 ]|ck,M SB = 0 ) = ln ( ∗ p Im[˜ rk r˜k−1 ]|ck,M SB = 1 √ 2 2REb ∗ = Im[˜ rk r˜k−1 ] 2REb N0 + (N0 /2)2 (6.17) and λk,LSB ( ) ∗ p Re[˜ rk r˜k−1 ]|ck,LSB = 0 ) = ln ( ∗ p Re[˜ rk r˜k−1 ]|ck,LSB = 1 √ 2 2REb ∗ Re[˜ rk r˜k−1 ], = 2 2REb N0 + (N0 /2) respectively, for ck,M SB and ck,LSB . 101 (6.18) 6. Pseudocodeword Weights under Differential PSK Transmission Let wQ = [w1,M SB w1,LSB w2,M SB w2,LSB · · · wN/2,M SB wN/2,LSB ] be a pseudocodeword. Similar to the case of BDPSK, we have the GCD choosing wQ over c if N/2 ∑ (ck,M SB − wk,M SB )λk,M SB + k=1 N/2 ∑ (ck,LSB − wk,LSB )λk,LSB > 0, (6.19) k=1 which, by substituting the G-LLR, is easily reduced to SQDP SK > 0, (6.20) where SQDP SK = N/2 ∑ (ck,M SB − ∗ wk,M SB )Im[˜ rk r˜k−1 ] + N/2 ∑ ∗ (ck,LSB − wk,LSB )Re[˜ rk r˜k−1 ]. k=1 k=1 We introduce a phase-rotated noise term n ˜ ′k = ejθ n ˜ ∗k which is a zero-mean complex Gaussian variable with variance N0 , for k = 1, 2, · · · , N . Thus, we have ∗ r˜k r˜k−1 = 2REb ej△ϕk + (2REb ) 2 ejϕk n ˜ ′k−1 + (2REb ) 2 e−jϕk−1 n ˜ ′∗ k, 1 1 (6.21) where its real and imaginary parts are { [ ∗ ] [ ′ ] 1 2 Re r˜k r˜k−1 = 2REb cos △ϕk + (2REb ) cos ϕk Re n ˜ k−1 + cos ϕk−1 Re [˜ n′k ] } [ ′ ] − sin ϕk Im n ˜ k−1 − sin ϕk−1 Im [˜ n′k ] (6.22) 102 6. Pseudocodeword Weights under Differential PSK Transmission and { [ ∗ ] [ ′ ] 1 Im r˜k r˜k−1 = 2REb sin △ϕk + (2REb ) 2 cos ϕk Im n ˜ k−1 − cos ϕk−1 Im [˜ n′k ] } [ ′ ] ′ nk ] ,(6.23) + sin ϕk Re n ˜ k−1 − sin ϕk−1 Re [˜ respectively. We introduce two quantities, △wk,M = ck,M SB − wk,M SB and △wk,L = ck,LSB − wk,LSB . 103 6. Pseudocodeword Weights under Differential PSK Transmission [ ∗ ] [ ∗ ] Substituting Re r˜k r˜k−1 and Im r˜k r˜k−1 into the expression for SQDP SK , we have SQDP SK = N/2 ∑ (ck,M SB − ∗ wk,M SB )Im[˜ rk r˜k−1 ] k=1 = 2REb N/2 ∑ + N/2 ∑ ∗ (ck,LSB − wk,LSB )Re[˜ rk r˜k−1 ] k=1 △wk,M sin △ϕk + 2REb k=1 +(2REb ) { [ 1 2 N/2 ∑ △wk,L cos △ϕk k=1 ] △w1,M sin ϕ1 + △w1,L cos ϕ1 Re [˜ n′0 ] [ ] + △w1,M cos ϕ1 − △w1,L sin ϕ1 Im [˜ n′0 ] N/2−1 [ ∑ + △wk+1,M sin ϕk+1 − △wk,M sin ϕk−1 ] +△wk+1,L cos ϕk+1 + △wk,L cos ϕk−1 Re [˜ n′k ] k=1 N/2−1 [ ∑ + △wk+1,M cos ϕk+1 − △wk,M cos ϕk−1 ] k=1 −△wk+1,L sin ϕk+1 − △wk,L sin ϕk−1 Im [˜ n′k ] [ ] [ ] + − △wN/2,M sin ϕN/2−1 + △wN/2,L cos ϕN/2−1 Re n ˜ ′N/2 } ] [ ] + − △wN/2,M cos ϕN/2−1 − △wN/2,L sin ϕN/2−1 Im n ˜ ′N/2 .(6.24) [ SQDP SK is Gaussian distributed with mean E(SQDP SK ) = 2REb N/2 ∑ △wk,M sin △ϕk + 2REb √ ∑ △wk,L cos △ϕk k=1 k=1 = N/2 ∑ N/2 2REb [△wk,M (1 − 2ck,M SB ) + △wk,L (1 − 2ck,LSB )] (6.25) k=1 104 6. Pseudocodeword Weights under Differential PSK Transmission and variance Var(SQDP SK )   N/2 ∑ ( ) 2 2 = 2REb N0  △wk,M + △wk,L + γ , (6.26) k=1 where N/2−1 [ γ = ∑ (△wk+1,M △wk,L + △wk,M △wk+1,L ) sin(△ϕk + △ϕk+1 ) ] −(△wk+1,M △wk,M − △wk+1,L △wk,L ) cos(△ϕk + △ϕk+1 ) . k=1 Thus, the pairwise error probability using GCD is Pc→wQ    = Q  = p(SQDP SK > 0|c sent)  {∑ }2 N/2  REb k=1 [△wk,M (1 − 2ck,M SB ) + △wk,L (1 − 2ck,LSB )]  [∑ ( ]  . (6.27) ) N/2  2 2 N0 △w + △w + γ k=1 k,M k,L If an all-zero codeword is sent, the pairwise probability becomes P0→wQ       = Q        N0  REb ∑N/2 ( k=1 [∑ N/2 ]2 k=1 (wk,M SB + wk,LSB ) ) 2 2 wk,M SB + wk,LSB ∑N/2−1 + k=1 (wk+1,M SB wk,LSB +wk,M SB wk+1,LSB ) which yields Definition 2. 105      (6.28)     6. Pseudocodeword Weights under Differential PSK Transmission Definition 6.2 The pseudocodeword weight wpQDP SK of wQ is wpQDP SK [∑ ]2 (wk,M SB + wk,LSB ) .(6.29) = ∑N/2 ( ) ∑N/2−1 2 2 + w (w w + w w ) w + k+1,M SB k,LSB k,M SB k+1,LSB k=1 k,M SB k,LSB k=1 N/2 k=1 6.2 Pseudocodeword Weight Analysis of (8,4) & (8,3) Binary Codes We consider two codes to explain the performance improvements afforded by DQPSK, over DBPSK, at lower SNR values. To this end, let  H(8,4)  0 1 1   1 0 1  =  1 1 0   1 1 1    1 1 0 1 0 0 0 0 1 1 0 0 0   1 0 1 0 0   ; 1 0 0 1 0    0 0 0 0 1 H(8,3)    0 1 1   =  0 0 1    0 0 0  0 0 0 0 1 1 0 1 1 0 1   0 0 0    1 0 0  .   0 1 0   1 0 1 H(8,4) is a parity-check matrix for an extended (8,4) binary Hamming code while H(8,3) is a parity-check matrix for an (8,3) binary code obtained by extending the dual of the (7,4) Hamming code with an additional parity-check bit. Fig. 6.2 shows the BER performance of these two codes on the noncoherent AWGN channel under BP decoding with the G-LLRs as inputs. Observe that both codes perform better under DQPSK signaling at lower SNRs. Using the results of the previous section, we plot in Figs. 6.3 and 6.4, the fraction of the total number of pseudocodewords against the pseudocodeword 106 6. Pseudocodeword Weights under Differential PSK Transmission weights corresponding to the two signaling schemes. Only pseudocodewords arising from the M -covers of the Tanner graph associated with H(8,4) and H(8,3) for M = 1, 2, 3 are considered due to space constraints. Observe that Figs. 6.3 and 6.4 show a general rightward-shift in the pseudocodeword weight spectrum for both the (8,4) and (8,3) code as we move from DBPSK to DQPSK. Noting that the same code is used, the set of pseudocodewords that arises from the corresponding Tanner graph is the same for both transmissions. Thus, the difference in pseudocodeword weight spectrum is solely due to the change in transmission. Recalling that GC decoding is a very good approximation of BP decoding and noting that at lower SNRs, too many low-weight pseudocodewords is detrimental to the error performance of the GC decoder under both signaling schemes, it follows that this rightward-shift in the pseudocodeword weight spectrum explains the improved BER performance reported in Fig. 6.2. 6.3 Conclusion We have derived the pseudocodeword weights under BDPSK and QDPSK transmission over the noncoherent AWGN channel. Based on the the pseudocodeword weight distribution of an (8,4) binary Hamming code and the corresponding BER performance arising from the two signaling schemes, we have attributed the superiority in BER performance that coded QDPSK exhibits over coded BDPSK in the low SNR region to some kind of spectral thinning (in the pseudocodeword weight distribution) that occurs when we move from BDPSK to QDPSK. 107 6. Pseudocodeword Weights under Differential PSK Transmission −2 10 −3 BER 10 −4 10 −5 10 suboptimal LLR, DBPSK optimal LLR, DBPSK suboptimal LLR, DQPSK optimal LLR, DQPSK 4 4.2 4.4 4.6 4.8 5 Eb/N0 (dB) 5.2 5.4 5.6 5.8 Figure 6.1: BER performance of a (1008,504) binary LDPC code on the noncoherent AWGN channel with optimal and suboptimal LLRs of each code bit fed to the BP decoder for which the maximum number of iterations was set to 50 108 6. Pseudocodeword Weights under Differential PSK Transmission −1 (8,4) code, DBPSK (8,4) code, DQPSK (8,3) code, DBPSK (8,3) code, DQPSK 10 −2 BER 10 −3 10 −4 10 −5 10 3 4 5 6 7 8 Eb/N0 (dB) 9 10 11 Figure 6.2: BER performance of (8,4) and (8,3) code on the noncoherent AWGN channel with the G-LLR of each code bit fed to the BP decoder for which the maximum number of iterations was set to 50 109 6. Pseudocodeword Weights under Differential PSK Transmission fraction of total number of pseudocodewords 0.35 DBPSK DQPSK 0.3 0.25 0.2 0.15 0.1 0.05 0 2.5 3 3.5 pseudocodeword weight 4 4.5 (a) M=1 0.4 0.25 DBPSK DQPSK fraction of total number of pseudocodewords fraction of total number of pseudocodewords 0.45 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 2.5 3 3.5 pseudocodeword weight 4 DBPSK DQPSK 0.2 0.15 0.1 0.05 0 4.5 (b) M=2 2 2.5 3 3.5 pseudocodeword weight 4 4.5 (c) M=3 Figure 6.3: Weight distribution of pseudocodewords arising from the M -covers of the Tanner graph corresponding to H(8,4) for M = 1, 2, 3 110 6. Pseudocodeword Weights under Differential PSK Transmission fraction of total number of pseudocodewords 0.5 DBPSK DQPSK 0.45 0.4 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 2 2.5 3 3.5 pseudocodeword weight 4 4.5 (a) M=1 0.08 DBPSK DQPSK fraction of total number of pseudocodewords fraction of total number of pseudocodewords 0.35 0.3 0.25 0.2 0.15 0.1 0.05 0 2 2.5 3 3.5 4 pseudocodeword weight 4.5 0.06 0.05 0.04 0.03 0.02 0.01 0 5 (b) M=2 DBPSK DQPSK 0.07 2 2.5 3 3.5 4 pseudocodeword weight 4.5 5 (c) M=3 Figure 6.4: Weight distribution of pseudocodewords arising from the M -covers of the Tanner graph corresponding to H(8,3) for M = 1, 2, 3 111 Chapter 7 Iterative Decoding of LDPC codes Transmitted using BPSK with PSAM As a measure of reliability, the LLR for each code bit is dependent not only on the received signal sample that carries that bit, but also on the set of pilot signal samples from which information about the channel is retrieved. Hence, its computation should encompass both the received signal sample and the pilot set. This is crucial because the LLR is the sole input to the decoder and determines the latter’s performance. In this chapter, we consider the design of the receiver for LDPC codes transmitted using BPSK with PSAM over the AWGN channel with unknown carrier phase, or the so-called noncoherent channel. The key step to this design is the computation of the soft information input to the LDPC decoder, i.e., the LLR of the two possible values of each code bit. This computation only incurs a one-off overhead since it is executed once at the beginning of each decoding process. 112 7. Iterative Decoding of LDPC codes Transmitted using BPSK with PSAM The LLR is shown to depend on the accuracy of the carrier phase estimate and on the estimated phase reference. In the limit of a large number of pilot symbols, the LLR here converges to that for the case of coherent PSK. The system model, along with the notations, is introduced in Section 7.1. In Section 7.2, the LLR of each code bit is derived based on the joint PDF of the corresponding received signal and a set of received pilot signals. Due to the presence of an unknown carrier phase, modeled as a random variable uniformly distributed over the interval [−π, π), the metric is obtained by averaging the joint PDF conditioned on a hypothesized value of the carrier phase, over all possible values of the unknown carrier phase. This metric shall be referred to as the PSAMLLR. In addition, we introduce the PSAM Approximate LLR (PSAM-A-LLR) and the PSAM Simplified-Approximate LLR (PSAM-SA-LLR), two approximations of the PSAM-LLR, which yield significant reductions in implementation complexity. We compare the PSAM-LLR with the TSOI-LLR metric derived for BDPSK transmission in [83] and the metric corresponding to coherent BPSK transmission. Simulation studies in Section 7.5 verify our theoretical analysis. Since all metrics require knowledge of the exact SNR, the performance of each metric in the presence of SNR estimation error at the decoder is also considered. Section 7.6 summarizes the key results. 7.1 System Model Consider a binary message sequence m encoded by a rate R LDPC encoder to a binary codeword c. Pilot symbols are multiplexed into the codeword before each bit of the resultant sequence is transmitted using BPSK. As shown in Fig. 7.2, suppose the length-LB BPSK signal sequence sent, ˜ s = [ s˜(0) s˜(1) · · · s˜(LB −1) ], 113 7. Iterative Decoding of LDPC codes Transmitted using BPSK with PSAM m c LDPC ˜s BPSK Encoder Modulator ejθ pilot symbols ˜ n ˆ m ˜r λ LDPC Metric Calculator Decoder Noncoherent AWGN Channel Figure 7.1: System model block 0 i=0 i-th signal pilot signals i=0 block L − 1 block l i-th signal i=0 i-th signal time k 0 lB k = lB + i Figure 7.2: Frame model 114 (L − 1)B LB − 1 7. Iterative Decoding of LDPC codes Transmitted using BPSK with PSAM is partitioned into L blocks of B signal samples per block. At time instant k = lB + i, (7.1) 1 s˜(k) = Es2 eϕ(k) (7.2) is the i-th complex baseband signal in the l-th block transmitted, for i = 0, 1, · · · , B − 1 and l = 0, 1, · · · , L − 1. Here, Es and ϕ(k) ∈ {0, π} denote the energy of and the information contained in s˜(k), respectively. The first symbol in each block is a pilot symbol and thus does not carry any information, that is, the transmitted phase is fixed at ϕ(lB) = 0, for l = 0, 1, · · · , L − 1. The remaining B − 1 symbols of each block correspond to LDPC-encoded bits. The signal sequence is transmitted over an AWGN channel with unknown carrier phase. The received signal sequence ˜ r = [ r˜(0) r˜(1) · · · r˜(LB − 1) ] is modeled as r˜(k) = s˜(k)ejθ + n ˜ (k), (7.3) for k = 0, 1, · · · , LB − 1, where θ denotes the unknown carrier phase, assumed to be constant during the entire transmission and modeled as a random variable with uniform distribution in the interval [−π, π). The sequence {˜ n(k)} is a complex AWGN sequence with E[˜ n(k)] = 0 and E[|˜ n(k)|2 ] = N0 . s, θ and {˜ n(k)} are independent. It shall be noted that due to the insertion of pilot symbols, the effective energy allocated for each coded bit is in fact Es′ = B E. B−1 s Thus the ratio of the energy per information symbol to noise, which is referred to as the effective SNR, is Eb /N0 = B E /N0 . R(B−1) s At the receiver, the LLR metric of each transmitted signal, excluding the pilot signals, is first computed based on the corresponding received signal and 115 7. Iterative Decoding of LDPC codes Transmitted using BPSK with PSAM some pilot signals. These values are then passed to the iterative LDPC decoder where the estimated message m ˆ is obtained as the output. 7.2 Metric Derivation The LLR of the code bit sent at time k is computed based on the corresponding received signal sample r˜(k), and the received pilot signal samples in the W preceding blocks, current block and W succeeding blocks, i.e., the set of 2W + 1 received pilot signals {˜ r((l + w)B)}W w=−W . Thus, the metric for ϕ(k), denoted by λ(k), is defined as ( ) Prob ϕ(k) = 0|˜ r(k), {˜ r((l + w)B)}W w=−W ( ), λ(k) = ln Prob ϕ(k) = π|˜ r(k), {˜ r((l + w)B)}W w=−W (7.4) for the corresponding values of i = 1, 2, · · · , B − 1 and l = 0, 1, · · · , L − 1. The relationship between k, l and i follows (7.1) and is clearly depicted in Fig. 7.2. Since the information sent is independent of the pilot symbols, that is, ( ) Prob ϕ(k)|{˜ r((l + w)B)}W = Prob(ϕ(k)), and the a-priori probabilities of w=−W the information are equal, i.e., Prob(ϕ(k) = 0) = Prob(ϕ(k) = π) = 0.5, we can rewrite (7.4) using Bayes’ rule as ( ) Prob r˜(k)|ϕ(k) = 0, {˜ r((l + w)B)}W w=−W ( ). λ(k) = ln Prob r˜(k)|ϕ(k) = π, {˜ r((l + w)B)}W w=−W 116 (7.5) 7. Iterative Decoding of LDPC codes Transmitted using BPSK with PSAM ( ) To evaluate the likelihood function Prob r˜(k)|ϕ(k) = aπ, {˜ r((l + w)B)}W w=−W , for a = 0, 1, we first write ) ( Prob r˜(k)|ϕ(k) = aπ, {˜ r((l + w)B)}W w=−W ∫ π ( ) = Prob r˜(k)|ϕ(k) = aπ, θ, {˜ r((l + w)B)}W w=−W −π ( ) ·Prob θ|ϕ(k) = aπ, {˜ r((l + w)B)}W dθ. (7.6) w=−W Conditioned on ϕ(k) = aπ and θ, the only randomness in r˜(k) is due to n ˜ (k). Hence, we have ( ) Prob r˜(k)|ϕ(k) = aπ, θ, {˜ r((l + w)B)}W w=−W = Prob (˜ r(k)|ϕ(k) = aπ, θ) [ 1 ] 2Es2 = C(k) exp (−1)a |˜ r(k)| cos (∠˜ r(k) − θ) ,(7.7) N0 where C(k) = 1 πN0 [ exp 2 − |˜r(k)|N0+Es ] is independent of ϕ(k). Since θ is independent of ϕ(k) = aπ, we have ( ) ( ) Prob θ|ϕ(k) = aπ, {˜ r((l + w)B)}W r((l + w)B)}W w=−W = Prob θ|{˜ w=−W . (7.8) The conditional PDF of θ given {˜ r((l + w)B)}W w=−W , as shown in the Appendix E, is [ ( Prob θ|{˜ r((l + w)B)}W w=−W ) exp = 117 ] |˜ v (l)| cos (θ − ∠˜ v (l)) [ 1 ] , 2Es2 2πI0 N0 |˜ v (l)| 1 2Es2 N0 (7.9) 7. Iterative Decoding of LDPC codes Transmitted using BPSK with PSAM where I0 [|x|] = 1 2π ∫π −π exp [|x| cos θ] dθ is the zeroth-order modified Bessel function of the first kind. This conditional PDF is a Tikhonov PDF centered at the mean value ∠˜ v (l) where W ∑ v˜(l) = r˜((l + w)B) (7.10) w=−W is the reference phasor formed from the received pilot signals. Now, substituting the conditional PDFs in (7.7) and (7.9) back to (7.6), we have ( ) Prob r˜(k)|ϕ(k) = aπ, {˜ r((l + w)B)}W w=−W [ = I0 Observe that the term [ I0 C(k) 1 2Es2 N0 C(k) ] 1 2Es2 N0 ] I0 |˜ v (l)| [ ] 1 2Es2 |˜ v (l) + (−1)a r˜(k)| .(7.11) N0 is independent of a. Thus, the log-likelihood |˜ v (l)| metric, referred to as PSAM-LLR, is [ I0 [ λ(k) = ln I0 1 2Es2 N0 1 2Es2 N0 ] |˜ v (l) + r˜(k)| ]. |˜ v (l) − r˜(k)| (7.12) Although the final expression for the PSAM-LLR is only dependent on the reference phasor v˜(l), the conditional PDF of θ given the pilot signals is considered in the derivation. From [53], ∠˜ v (l) is the maximum likelihood estimate of θ given the pilot symbols and the estimation error variance is inversely proportional to the number of pilot symbols. Thus, it is clear that the entire information about the carrier phase that is retrieved from the pilot signals is considered in the derivation and contained in the PSAM-LLR. We observe that I0 (·) and ln(·) operations are required for the computation 118 7. Iterative Decoding of LDPC codes Transmitted using BPSK with PSAM of the PSAM-LLR and hence introduce two approximations which yield lower Es computational cost. For high SNR such that N ≫ 1, each I0 (x) term may be 0 √ approximated by ex / 2πx. Hence, the PSAM Approximate LLR (PSAM-A-LLR) is 1 [ ] 2Es2 λ(k) ≈ |˜ v (l) + r˜(k)| − |˜ v (l) − r˜(k)| N0 ] [ 1 v (l) + r˜(k)| − ln |˜ v (l) − r˜(k)| . (7.13) − ln |˜ 2 To further reduce computational complexity, this metric can be simplified by removing the ln(·) terms which leads to the PSAM Simplified-Approximate LLR (PSAM-SA-LLR) 1 [ ] 2Es2 λ(k) ≈ |˜ v (l) + r˜(k)| − |˜ v (l) − r˜(k)| . N0 7.3 (7.14) Comparison with the metric for BDPSK transmission This metric appears in a form similar to the metric derived for the case of BDPSK transmission over the noncoherent channel [83, eqn. 14]. We state here, for ease of reference, that the Two-Symbol-Observation-Interval LLR (TSOI-LLR) derived therein for the information sent at the k-th instant is [ I0 [ µ(k) = ln I0 1 2Es2 N0 1 2Es2 N0 ] |˜ r(k − 1) + r˜(k)| ]. |˜ r(k − 1) − r˜(k)| 119 (7.15) 7. Iterative Decoding of LDPC codes Transmitted using BPSK with PSAM Notice that the difference between (7.12) and (7.15) lies in the set of received symbols that serves as reference from which information about the carrier phase is retrieved. While only one previous received signal sample serves as the reference in the BDPSK transmission, we obtain information about the carrier phase from a set of pilot symbols in the PSAM BPSK transmission. Thus, the TSOI-LLR can be perceived as a special case of the PSAM-LLR. This follows naturally from the observation in [53] that the decision-aided receiver which detects a current signal based on a set of past detected signals as reference can be interpreted as a generalized differentially coherent detector. Since the PSAM-LLR is computed based on a set of pilot signal samples instead of only the previous signal sample, we would expect a more accurate reliability information about the carrier phase and thus better performance, compared to the TSOI-LLR. In fact, we prove in the following section that as the number of pilot symbols increases, the PSAM-LLR converges to the metric corresponding to the coherent channel. 7.4 Convergence of PSAM-LLR to the metric for coherent channel As shown in [53], as the number of pilot signals increases, the estimation error variance becomes smaller and performance of PSAM BPSK transmission over the noncoherent channel approaches that of coherent BPSK transmission. Thus, we should also expect the PSAM-LLR to converge to the metric for coherent BPSK transmission in the limit as the number of pilot symbols become large. To observe 120 7. Iterative Decoding of LDPC codes Transmitted using BPSK with PSAM this, we first express the PSAM-A-LLR as 1 1+ 2Es2 4Re[˜ r(k)e−j∠˜v(l) ] 1 λ(k) ≈ − ln N0 1 + r˜(k) + 1 − r˜(k) 2 1− v˜(l) v˜(l) r˜(k) v˜(l) r˜(k) v˜(l) . (7.16) This is done by applying the property [ ] [ ] |x + y| − |x − y| · |x + y| + |x − y| = 4Re[x∗ y] to (7.13) and normalizing the numerators and denominators of each fraction by |˜ v (l)|. Now, we note that       r˜(k) =  v˜(l)     where v˜n (l) = ∑W w=−W / / 1 1+˜ n(k) Es2 [ ] 1 2W +1 1+˜ vn (l) (2W +1)Es2 / / , if ϕ(k) = 0. , 1 −1+˜ n(k) Es2 [ ] 1 2W +1 1+˜ vn (l) (2W +1)Es2 (7.17) , if ϕ(k) = π. n ˜ ((l+w)B) is the sum of the noise terms of the pilot signals. For a large value of W , the magnitude of v˜nW(l) approaches zero. Further, since the / 1 magnitude of n ˜ (k) Es2 is typically much less than one, we make an approximation    1 , if ϕ(k) = 0. r˜(k) 2W +1 ≈  v˜(l)  − 1 , if ϕ(k) = π. 2W +1 (7.18) Following the approximation, we have 1+ r˜(k) r˜(k) + 1− ≈2 v˜(l) v˜(l) 121 (7.19) 7. Iterative Decoding of LDPC codes Transmitted using BPSK with PSAM and 1+ r˜(k) v˜(l) 1− r˜(k) v˜(l) ≈    1+ 1 W , if ϕ(k) = 0.   1− 1 W +1 , if ϕ(k) = π. (7.20) Observe that in the limit as W increases, (7.20) converges to the value 1 regardless of ϕ(k). As a result, (7.16) readily reduces to 1 4Es2 λ(k) ≈ Re[˜ r(k)e−j∠˜v(l) ], N0 (7.21) which requires each received signal sample r˜(k) to be rotated by the negative of the carrier phase estimate, i.e., −∠˜ v (l). As W becomes large, the carrier phase estimate ∠˜ v (l) converges to the actual value of θ and the estimation error variance goes to zero. Thus, the sequence {˜ r(k)e−j∠˜v(l) } can be treated as if it was sent over the coherent channel and the resultant metric corresponds to that for coherent BPSK transmission. 7.5 Simulation Study The computer simulations follow the system model introduced in Section 7.1. Rate-half, regular (256,128) and (1008,504) binary LDPC codes from [74] are used. Each code is represented by a sparse parity-check matrix with column weight of three. Based on the system model, the signal sequence sent has length BL. Since the value of L does not affect the performance of the codes, we send the signal sequences in a continuous stream which is equivalently an infinitely long transmitted signal sequence, i.e., L → ∞. Thus, for ease of implementation, the length of the LDPC codes used will not be bounded by the value of B. The sum-product algorithm in [36] is employed for decoding LDPC codes. It should 122 7. Iterative Decoding of LDPC codes Transmitted using BPSK with PSAM −1 10 noncoherent, DBPSK −2 BER 10 coherent, BPSK −3 10 noncoherent, PSA BPSK, 2W+1 = 11 −4 10 noncoherent, PSA BPSK, 2W+1 = 7 −5 10 0 1 2 3 4 Eb/N0 (dB) 5 6 7 Figure 7.3: BER performances of (256,128) LDPC code over noncoherent AWGN channel using PSAM-LLR with varying number of pilot symbols be noted that although a maximum of 50 iterations are allowed, each received sequence typically converges to a valid codeword within ten iterations at high SNR. 7.5.1 Performance of LDPC codes with different metrics It is assumed that the noise variance is known at the receiver and the unknown carrier phase θ is constant over the transmission period of the signal sequence ˜s. BPSK transmission over coherent AWGN channel, PSAM BPSK transmission using the PSAM-LLR and BDPSK transmission using the TSOI-LLR over noncoherent AWGN channel are simulated. In the case of PSAM BPSK transmission, since the block size has little effect on the performance under the assumption of constant carrier phase, it is arbitrarily set to 100. In the other 123 7. Iterative Decoding of LDPC codes Transmitted using BPSK with PSAM −1 10 −2 10 noncoherent, DBPSK BER coherent, BPSK noncoherent, PSA BPSK, 2W+1 = 15 −3 10 noncoherent, PSA BPSK, 2W+1 = 11 −4 10 noncoherent, PSA BPSK, 2W+1 = 7 −5 10 0 1 2 3 Eb/N0 (dB) 4 5 Figure 7.4: BER performances of (1008,504) LDPC code over noncoherent AWGN channel using PSAM-LLR with varying number of pilot symbols two transmissions where pilot symbols are not inserted, the effective SNR is Eb /N0 = 1 E /N0 . R s Fig. 7.3 and 7.4 show the BER performances of (256,128) and (1008,504) LDPC codes, respectively. Congruent to our theoretical analysis, the PSAM-LLR outperforms the TSOI-LLR and its performance approaches that of coherent transmission as the number of pilot symbols (2W + 1) increases. These results are obtained based purely on the PSAM-LLR. After a few decoding iterations, one can enlarge the reference set by including the tentative decisions of the code bits, assumed to be reasonably accurate. With that, performance closer to that for coherent transmission is expected. Due to space constraints, results and discussion for this strategy will be reported in future. It is observed that the longer LDPC code needs a larger set of pilot symbols to attain similar performance deviation from the coherent case, compared to the shorter code. This is because the longer code typically reaches asymptotic error 124 7. Iterative Decoding of LDPC codes Transmitted using BPSK with PSAM PSA−LLR PSA−A−LLR PSA−SA−LLR −1 10 −2 BER 10 2W+1 = 7 −3 10 2W+1 = 11 2W+1 = 15 −4 10 0 0.5 1 1.5 2 Eb/N0 (dB) 2.5 3 3.5 Figure 7.5: BER performances of (1008,504) LDPC code over noncoherent AWGN channel with various metrics performance at the lower SNR range. Thus, more pilot symbols are necessary to provide the same amount of reliability information about the carrier phase. Fig. 7.5 compares the performance of the three metrics, PSAM-LLR, PSAMA-LLR and PSAM-SA-LLR. All three metrics yield similar BER performances. Since the PSAM-SA-LLR requires the least computation complexity, it offers a suitable alternative to the PSAM-LLR. 7.5.2 Effects of phase noise on performance of metrics The performance of the metrics in the presence of phase noise is investigated. Here, we denote θk as the value of θ at time instant k. The unknown carrier phase θk is modeled as a random-walk, i.e., θk = θk−1 + δk , 125 (7.22) 7. Iterative Decoding of LDPC codes Transmitted using BPSK with PSAM −1 10 −2 BER 10 B=10000 B=3000 −3 10 B=1000 B=500 B=200 −4 10 0 0.5 1 1.5 2 Eb/N0 (dB) 2.5 3 3.5 4 Figure 7.6: BER performances of (1008,504) LDPC code over noncoherent AWGN channel with various block lengths, subjected to phase noise where σ 2 = 10−6 where δk is the phase noise. The noise sequence δk is a set of iid Gaussian random variables with mean zero and variance σ 2 . Again, it is assumed that the variance of the AWGN is known at the receiver. In Fig. 7.6, keeping the window size constant at 15, we compare the performance of the (1008,504) LDPC code when various block lengths L are deployed in the presence of phase noise where σ 2 = 10−6 . Empirical results show similar performances exhibited by the three metric, thus only the performance of the PSAM-SA-LLR is presented. Since there is one pilot symbol at the beginning of each block, the block length is inversely proportional to the percentage of pilot symbols, e.g., L = 1000 yields 0.1% pilot symbols. Consequently, the overall energy available to information sequence has to be evenly assigned to all 126 7. Iterative Decoding of LDPC codes Transmitted using BPSK with PSAM transmitted symbols. When L decreases from 10000 to 1000, since transmission time that elapse between consecutive pilot symbols are shorter, information about the varying carrier phase is better captured. Hence the BER performance improves. However, when L is further decreased to 200, the BER performance degrades although time interval between successive pilot symbols are even shorter. This can be attributed to the decrease in the average energy per transmitted symbol due to the increase in the concentration of pilot symbols. Since LDPC codes achieve asymptotic performances in tight SNR ranges, they are particularly sensitive to slight changes in the average energy per symbol. 7.5.3 Effects of SNR estimation error Calculation of all the metrics requires estimation of the SNR at the receiver. Fig. 7.7 shows the effect of an estimation error in effective SNR on the performance of the (1008,504) LDPC code over the AWGN channel with time-invariant unknown carrier phase, using each metric. Following 7.5.1, the block size is set to 100. The horizontal axis indicates the difference between the estimated and the effective SNR, measured in decibels. A positive value represents an overestimation, i.e., the effective SNR is lower than the estimated SNR. Performances of all three metrics are similar in the presence of SNR mis-estimation. In general, they perform better when there is a slight SNR underestimation of 1 dB. However, SNR underestimation beyond 2 dB results in detrimental degradation in their performances. On the other hand, although the metrics also perform worse when there is an SNR overestimation, the degradation in performances stays within an order of magnitude. We also observe that the number of pilot symbols does not contribute to any further effects on the 127 7. Iterative Decoding of LDPC codes Transmitted using BPSK with PSAM PSA−LLR PSA−A−LLR PSA−SA−LLR −1 10 −2 10 BER 2W+1 = 11, Eb/N0 = 2 dB 2W+1 = 15, Eb/N0 = 2 dB −3 10 2W+1 = 11, Eb/N0 = 3 dB −4 10 −10 −5 0 SNR Mis−estimation (dB) 5 10 Figure 7.7: BER performances of (1008,504) LDPC code over noncoherent AWGN channel subjected to SNR mis-estimation performances of the metrics subjected to SNR mis-estimation. 7.6 Conclusion Iterative decoding of low-density parity-check codes transmitted using PSAM BPSK has been studied. Based on the joint probability density function of the corresponding received signal and a set of received pilot signals, we have derived the LLR of the two values of each transmitted code bit, referred to as the PSAM-LLR. We have also introduced the PSAM-A-LLR and the PSAM-SALLR, two approximations of the PSAM-LLR, which require less computational complexity but yield similar performances to the PSAM-LLR. We observed that the PSAM-LLR is a generalized version of the metric derived for differential BPSK transmission over the noncoherent channel. Moreover, we have proven that the 128 7. Iterative Decoding of LDPC codes Transmitted using BPSK with PSAM PSAM-LLR converges to the metric corresponding to the coherent channel in the limit as the set of pilot symbols becomes large and verified this through simulation studies, and the effects of signal-to-noise ratio estimation error on the performance of these metrics are also studied. 129 Chapter 8 Iterative Decoding using PSA BPSK with Reference Phasor This chapter is an extension of the design of the receiver for LDPC codes with BPSK transmission with PSAM over the noncoherent AWGN channel in the previous chapter. Instead of using the joint PDF of the received signal and the corresponding set of received pilot signals, we derive the LLR of the two possible values of each code bit based on the pdf of the product of the corresponding received signal and conjugate of a reference phasor, the sum of the received pilot signals in an observation window centered about the received signal concerned. This LLR is referred to as the PSAM-Reference-LLR (PSAMR-LLR). We compare the PSAM-R-LLR with the existing Gaussian Metric (GM) derived for BDPSK transmission in [39] and the LLR corresponding to coherent BPSK transmission, and provide simulation studies in Section 8.2. We also study the effect of the size of the observation window on the performance of the LLR when the unknown phase is subjected to phase noise. Since all metrics require 130 8. Iterative Decoding using PSA BPSK with Reference Phasor knowledge of the exact SNR, the performance of the PSAM-R-LLR in the presence of SNR estimation error at the decoder is also considered. Section 8.3 summarizes the key results. 8.1 Metric Derivation The system model follows that discussed in Chapter 7. The LLR of the code bit sent at time k is computed based on the corresponding received signal sample r˜(k) and the set of received pilot signals in an observation window centered at the received signal concerned. The pilot set consists of received pilot signals from the W -th preceding block to the W -th succeeding block, i.e., 2W + 1 received pilot signals {˜ r((l + w)B)}W w=−W . We define the metric for ϕ(k), denoted by λ(k), as λ(k) = ln p (ϕ(k) = 0|Re[˜ r(k)˜ v ∗ (l)]) , p (ϕ(k) = π|Re[˜ r(k)˜ v ∗ (l)]) (8.1) for the corresponding values of i = 1, 2, · · · , B − 1 and l = 0, 1, · · · , L − 1, and k = lB + i. We have introduced a reference phasor v˜(l) = W ∑ r˜((l + w)B) (8.2) w=−W that is the sum of the received pilot signals. Since the a-priori probabilities of the information are equal, i.e., p(ϕ(k) = 0) = p(ϕ(k) = π) = 0.5, we can rewrite (8.1) using Bayes’ rule as λ(k) ≈ ln p (Re[˜ r(k)˜ v ∗ (l)]|ϕ(k) = 0) . p (Re[˜ r(k)˜ v ∗ (l)]|ϕ(k) = π) 131 (8.3) 8. Iterative Decoding using PSA BPSK with Reference Phasor From the system model and (8.2), we have 1 v˜(l) = (2W + 1)Es2 ejθ + n ˜ v (l), where n ˜ v (l) = W ∑ (8.4) n ˜ ((l + w)B) (8.5) w=−W is the sum of the noise terms in the pilot set. Since the noise signals {˜ n((l + w)B)}W ˜ v (l) is a zero-mean Gaussian random w=−W are independent of each other, n variable with variance (2W + 1)N0 . We proceed to evaluate the conditional pdf of Re[˜ r(k)˜ v ∗ (l)], hypothesized on a fixed value of ϕ(k). From (8.4), we have [ 1 ˜ ∗v (l) Re[˜ r(k)˜ v (l)]=Re (2W+1)Es ejϕ(k)+Es2 ej(ϕ(k)+θ) n ∗ 1 2 +(2W+1)Es e −jθ n ˜ (k) + ] . n ˜ (k)˜ n∗v (l) (8.6) 1 1 ˜ ∗v (l) and (2W + 1)Es2 e−jθ n ˜ (k) are scalar multiples of two The terms Es2 ej(ϕ(k)+θ) n phase-rotated complex Gaussian random variables that are independent of each other. Hence, their real parts are simply two independent, zero-mean Gaussian random variables with variances (2W + 1)Es N0 and (2W + 1)2 Es N0 , respectively. Following [39], we make a similar assumption here, that Re[˜ n(k)˜ n∗v (l)] is a zeromean Gaussian random variable with variance (2W +1)N02 4 and is independent of n ˜ (k) and n ˜ v (l). Conditioned on a hypothesized value of ϕ(k), the pdf of Re[˜ r(k)˜ v ∗ (l)] is thus Gaussian with mean [ ] ∗ E Re[˜ r(k)˜ v (l)] = (2W + 1)Es ejϕ(k) 132 8. Iterative Decoding using PSA BPSK with Reference Phasor and variance [ ] [ ] N02 ∗ Var Re[˜ r(k)˜ v (l)] = (2W + 1) (W + 1)Es N0 + . 4 Substituting this condition pdf with appropriate values for ϕ(k) in (8.3), the resultant LLR is 2Es Re[˜ r(k)˜ v ∗ (l)] λ(k) = (W + 1)Es N0 + N02 4 , (8.7) which shall be referred to as the PSAM Reference LLR (PSAM-R-LLR). Although the final expression for the PSAM-R-LLR is only dependent on the reference phasor v˜(l), the pdf of carrier phase θ is considered in the derivation. Observe that when W = 0, the PSAM-R-LLR becomes λ(k) = 2Es Re[˜ r(k)˜ v ∗ (l)] Es N0 + N02 4 . This is similar to the GM in [39] Λ(k) = 2Es Re[˜ r(k)˜ r∗ (k − 1)] Es N0 + N02 4 , cited here for ease of reference. Although BDPSK transmission was deployed in [39], the approach therein obtained information about each code bit using the the corresponding received signal with the previous received signal as reference. In our approach using PSAM, information is extracted from a reference set containing 2W +1 pilot symbols. As the number of pilot signals in the reference set is typically larger than one, we would expect our metric to perform better than that using BDPSK transmission, as shown in the following section. As the number of pilot signals forming the reference phasor increases, the 133 8. Iterative Decoding using PSA BPSK with Reference Phasor angle of the phasor ∠˜ v (l) becomes more accurate. From [53], ∠˜ v (l) is the maximum likelihood estimate of θ given the pilot symbols and the estimation error variance is inversely proportional to the number of pilot symbols (2W + 1). As W increases, the estimation error variance decreases and ∠˜ v (l) becomes a more reliable estimate of θ. Hence, the term r˜(k)e−j∠˜v(l) can be perceived as a signal received from a coherent BPSK transmission and its metric is used to compute the LLR. As follows, we show that the PSAM-R-LLR indeed converges to the metric corresponding to coherent BPSK transmission as W increases. Normalizing the denominator and numerator of the RHS of (8.7) by Es (2W 2 + 1), we have λ(k) ≈ 4|˜ v (l)| Re[˜ r(k)e−j∠˜v(l) ] 2W +1 N02 N0 + 2WN0+1 + 2Es (2W +1) , As W increases, we observe that 1 |˜ v (l)| → (2W + 1)Es2 , and N0 + N0 N02 + → N0 . 2W + 1 2Es (2W + 1) Thus, we have 1 4Es2 λ(k) → Re[˜ r(k)e−j∠˜v(l) ], N0 which is the metric for coherent BPSK transmission. 134 (8.8) 8. Iterative Decoding using PSA BPSK with Reference Phasor 8.2 Simulation Study Rate-half, regular (256,128) and (1008,504) binary LDPC codes from [74] are used. Since the value of L does not affect the performance of the codes, we send the signal sequences in a continuous stream which is equivalently an infinitely long transmitted signal sequence, i.e., L → ∞. Thus, for ease of implementation, the codeword length of the LDPC codes used will not be bounded by the value of B. Moreover, the ‘end-effect’ caused by the reduction in the number of pilot signals in the observation window at the beginning and end of the transmission is negligible since a relatively long bit stream is transmitted. The sum-product algorithm in [36] is employed for decoding each received sequence. It should be noted that although a maximum of 50 iterations are allowed, each received sequence typically converges to a valid codeword within ten iterations at high SNR. 8.2.1 Performance of LDPC codes with constant, unknown carrier phase It is assumed that the noise variance is known at the receiver and the unknown carrier phase θ is constant over the transmission period of the signal sequence ˜s. For PSA BPSK transmission, the block size B is arbitrarily set to 100, while for BDPSK transmission, the TSOI-LLR derived in Chapter 4 is used since it is the optimal metric and has been shown to outperform the GM. In transmissions where pilot symbols are not inserted, the effective SNR is Eb /N0 = 1 E /N0 . R s Fig. 8.1 and 8.2 show the bit-error-rate (BER) performances of (256,128) and (1008,504) LDPC codes, respectively. The PSA-LLR outperforms the TSOILLR. Moreover, its performance approaches that of coherent transmission as the 135 8. Iterative Decoding using PSA BPSK with Reference Phasor −1 10 noncoherent, DBPSK −2 10 PSAM−R−LLR 2W+1 = 7 BER −3 10 PSAM−R−LLR 2W+1 = 11 PSAM−R−LLR 2W+1 = 15 −4 10 coherent, BPSK −5 10 (256,128) LDPC code. Constant, unknown carrier phase. 0 1 2 3 4 Eb/N0 (dB) 5 6 7 Figure 8.1: BER performances of (256,128) LDPC codes over noncoherent AWGN channel using PSAM-R-LLR −1 10 noncoherent, DBPSK −2 10 PSAM−R−LLR 2W+1 = 7 −3 BER 10 PSAM−R−LLR 2W+1 = 11 −4 10 PSAM−R−LLR 2W+1 = 15 coherent, BPSK −5 10 (1008,504) LDPC code. Constant, unknown carrier phase. 0 1 2 3 Eb/N0 (dB) 4 5 6 Figure 8.2: BER performances of (1008,504) LDPC codes over noncoherent AWGN channel using PSAM-R-LLR 136 8. Iterative Decoding using PSA BPSK with Reference Phasor number of pilot symbols (2W + 1) increases. We also observe that the longer LDPC code requires a larger set of pilot symbols to attain similar performance deviation from the coherent case, compared to the shorter code. This is because the longer code typically reaches asymptotic error performance at the lower SNR range. Thus, more pilot symbols are necessary to provide the same amount of reliability information about the carrier phase. 8.2.2 Performance of LDPC codes with noisy, unknown carrier phase In the case where the unknown carrier phase is constant throughout the transmission, it is apparent that a highly reliable LLR can be obtained by considering a a very large number of pilot signals. Apart from the increased latency when the observation window is large, inserting more pilot symbols also reduces the effective SNR. In this section, we consider practical applications where phase noise is present. Let θ(k) denote the value of θ at time instant k. The unknown carrier phase θ(k) is modeled as a random-walk, i.e., θ(k) = θ(k − 1) + δ(k), (8.9) where δ(k) is the phase noise. The noise sequence δ(k) is a set of iid Gaussian random variables with mean zero and variance σ 2 . Again, it is assumed that the variance of the AWGN is known at the receiver. Fig. 8.3 shows the performance of the (1008,504) LDPC code over the noncoherent AWGN channel with phase noise where σ 2 = 1.6 × 10−5 . The number of pilot symbols in the pilot set is kept constant (W = 15) but the block size B is 137 8. Iterative Decoding using PSA BPSK with Reference Phasor −1 10 B=10000 B=3000 −2 10 BER B=1000 B=500 B=200 −3 10 −4 10 0 0.5 1 1.5 2 Eb/N0 (dB) 2.5 3 3.5 4 Figure 8.3: BER performances of (1008,504) LDPC code over noncoherent AWGN channel with phase noise varied. Decreasing the block size increases the frequency of the pilot signals. Thus the pilot signal set provides more reliability information about the unknown phase present in the received signal when phase noise is present. However, decreasing the block size also results in a decrease in the effective SNR. Hence, for block sizes below the optimal value (L = 1000 in this case), the performance loss due to the decrease in the effective SNR is greater than the performance gain due to the increased frequency of pilot signals. 8.2.3 Effects of SNR estimation error Since calculation of the PSAM-R-LLR requires estimation of the SNR at the receiver, we simulate the effect of an estimation error in the effective SNR on the performance of the (256,128) and (1008,504) LDPC codes over the AWGN 138 8. Iterative Decoding using PSA BPSK with Reference Phasor channel with time-invariant unknown carrier phase, as shown in Fig. 8.4 and 8.5 respectively. Following 8.2.1, the block size is set to 100. The horizontal axis indicates the difference between the estimated and the effective SNR, measured in decibels. A positive value represents an overestimation, i.e., the effective SNR is lower than the estimated SNR. In general, the PSAM-R-LLR performs best when there is a slight or no SNR overestimation. However, it performs worse when SNR overestimation exceed the optimum. On the other hand, SNR underestimation results in detrimental degradation in its performance. We observe that the number of pilot symbols does not contribute to any further effects on the performances of the metrics subjected to SNR mis-estimation. However, we notice that the effect of SNR misestimation is more apparent for the longer code, which suggests its performance is more susceptible to mis-estimation of the LLR. 8.3 Conclusion Iterative decoding of low-density parity-check codes using BPSK with PSAM has been studied. Using a reference phasor formed from the summation of the received pilot signals, we have derived the PSAM-R-LLR, the LLR of the two values of each transmitted code bit. We have highlighted its relationship with the existing Gaussian metric for BDPSK transmission. We have also shown that it outperforms the metric for BDPSK transmission and converges to that corresponding to the coherent channel in the limit as the set of pilot symbols becomes large, these we have shown through simulation studies. Further, we have studied the effects of phase noise and signal-to-noise ratio estimation error on the performance of the metric. 139 8. Iterative Decoding using PSA BPSK with Reference Phasor E /N = 3 dB b 0 Eb/N0 = 3.5 dB −2 10 BER 2W+1 = 7 2W+1 = 11 −3 10 2W+1 = 15 (256,108) LDPC code. Constant, unknown carrier phase. −5 0 SNR Mis−estimation (dB) 5 Figure 8.4: BER performances of (256,128) LDPC code over noncoherent AWGN channel subjected to SNR mis-estimation −1 10 Eb/N0 = 2.5 dB Eb/N0 = 2.75 dB 2W+1 = 7 −2 BER 10 2W+1 = 11 −3 10 2W+1 = 15 (1008,504) LDPC code. Constant, unknown carrier phase. −5 0 SNR Mis−estimation (dB) 5 Figure 8.5: BER performances of (1008,504) LDPC code over noncoherent AWGN channel subjected to SNR mis-estimation 140 Chapter 9 Conclusion and Proposals for Future Research 9.1 Conclusion In this thesis, we designed LDPC codes and practical decoders that work well in both standard and nonstandard channels. In code design, we highlighted recent works on mixed-alphabet codes and structured codes which considered important aspects such as error performance, storage memory required, and ease in implementing and complexity of the encoder and decoder which motivated our work on the design of LDPC codes over integer residue rings. In particular, we introduced two new classes of LDPC codes: the mixed-alphabet LDPC codes defined over two integer residue rings and the Latin-square based stuctured LDPC codes defined over an integer residue ring. In decoder design, we focused on the derivation of the LLR for LDPC codes transmitted over the noncoherent AWGN channel using DPSK and PSAM. Further, we introduced simplified alternatives 141 9. Conclusion and Proposals for Future Research which require less computational complexity in the expense of slight performance losses. Using the notion of pseudocodeword weights, we performed theoretical analysis on iterative decoding of LDPC codes transmitted using BDPSK and QDPSK, and explained their difference in error performance. As we summarise the key results obtained in this thesis, a system engineer may select the code design or LLR computation which can best fulfill the system requirements. In Chapter 2, we introduced a class of mixed-alphabet LDPC codes over integer residue rings. These codes may be viewed as LDPC codes over a ring, extended with additional parity-check symbols over a larger ring. By increasing the number of redundant check nodes, coding gain can be obtained. This is nevertheless, at the expense of increased decoding complexity, since the complexity of the BP decoder depends on the size of the parity-check matrix. We also observe shown that there is a limit to the number of additional parity-check symbols over the larger ring which can be added, while keeping the over codelength constant, before degradation in error performance sets in. We have also shown that further coding gain can be obtained by adding redundant check nodes to Tanner graphs on the error performance for both single-alphabet and mixed-alphabet codes. We extended the notion of Latin squares to multiplicative groups of a Galois ring in Chapter 3. Using the generalized mapping function, we constructed Tanner graphs that represent a family of structured LDPC codes over Z2a that covers a wide range of code rates that significantly outperformed their random counterparts of similar length and rate. Most importantly, the minimum pseudocodeword weight of these codes are equal to their minimum Hamming distance which is desirable under iterative decoding. In Chapter 4, we derived the TSOI-PN-LLR based on the joint PDF of 142 9. Conclusion and Proposals for Future Research two consecutive received signals conditioned on each possible code bit. We also introduced the TSOI-PN-A-LLR and the TSOI-PN-SA-LLR that incur less computational cost. The metrics derived offer a low-complexity solution to noncoherent soft decoding without the need for explicit carrier phase recovery. Compared to the G-PN-LLR, our metrics general perform better with and without phase noise, lead to faster convergence and are more robust against SNR estimation error. We extend the derivation of the TSOI-LLR to the case with QDPSK transmission in Chapter 5, and applied the LLR to binary and mixed-alphabet codes. In Chapter 6, We derived the pseudocodeword weights under BDPSK and QDPSK transmission over the noncoherent AWGN channel. Based on the the pseudocodeword weight distribution of an (8,4) binary Hamming code and the corresponding BER performance arising from the two signaling schemes, we explained the superiority in BER performance that coded QDPSK exhibits over coded BDPSK in the low SNR region to some kind of spectral thinning (in the pseudocodeword weight distribution) that occurs when we move from BDPSK to QDPSK. In Chapter 7, we derived the PSAM-LLR for iterative decoding of LDPC codes using PSAM BPSK transmission based on the joint PDF of the received signal and a set of received pilot signals. We also introduced the PSAM-A-LLR and the PSAM-SA-LLR which require less computational complexity but yield similar performances to the PSAM-LLR. We observed that the PSAM-LLR is a generalized version of the metric derived for differential BPSK transmission over the noncoherent channel and showed that it converges to the metric corresponding to the coherent channel in the limit as the set of pilot symbols becomes large. 143 9. Conclusion and Proposals for Future Research We also observe that although the PSAM-LLR performs better than the TSOILLR when the unknown carrier phase is constant, it is not as robust to phase noise. In Chapter 8 is a short extension of the work done in Chapter 7. Using a reference phasor which is the summation of the received pilot signals, we derived the PSAM-R-LLR. We highlighted its relationship with the existing Gaussian metric for BDPSK transmission. The PSAM-R-LLR outperformed the Gaussian metric and converges to that corresponding to the coherent channel in the limit as the set of pilot symbols becomes large. 9.2 Proposals for Future Research Extension of the work undertaken in the following directions may be considered. We have shown that for a mixed-alphabet LDPC code defined over two integer residue rings, there is a limit to which the fraction of parity-check symbols can be defined over the larger ring for optimal BER performance. It is, however, not known if this number changes for different codelengths and rates. Thus, more work can be done on the optimization of the BER performance through careful selection of the check and code nodes of the corresponding Tanner graphs which are defined over the larger ring. Our structured LDPC codes, though were shown to outperform random LDPC codes, are limited to short codelengths. Similar to PG and FG LDPC codes, the number of edges of the corresponding Tanner graphs increases significantly with the codelength. Thus, one might examine systematic elimination of some of the edges on the Tanner graphs without significantly altering the codes’ properties. In this thesis, we have limited the LLR computations to a one-off procedure at the beginning of the decoding process. Seemingly, the BER performance can 144 9. Conclusion and Proposals for Future Research be improved by including the computation of the LLR in the iterative decoding process. That is, the LLR can be re-computed after every few iterations in the LDPC decoder to provide more accurate soft information based on feedback from the LDPC decoder. When applied to PSAM transmission, this method can possibly reduce the number of pilot symbols required to attain a desired BER for a particular SNR value and lead to improved bandwidth efficiency. 145 Bibliography [1] F. 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Lightwave Technol., vol. 28, no. 11, pp. 1597–1607, Jun. 2010 162 Publications The contributions in this thesis that have been published or are pending publication are as follows. Journals 1. E. Mo and M. A. Armand, “Structured LDPC codes over integer residue rings,” EURASIP J. Wireless Commun. and Networking - Special Issue on Advances in Error Control Coding Techniques, 2008. 2. E. Mo and M. A. Armand, “Pseudocodeword weights of LDPC codes under DPSK transmission over the noncoherent channel,” submitted to IEEE Trans. Commun., 2010. Conferences 1. E. Mo and M. A. Armand, “Design and performance of LDPC codes extended with parity-check symbols from a larger alphabet,” in Proc. Int. Conf. Info., Commun. and Signal Process., Singapore, Dec. 2007. 2. E. Mo, P. Y. Kam, “Log-likelihood metrics based on two-symbol-interval observations for LDPC codes with BDPSK transmission,” in Proc. 2008 IEEE 68th Veh. Technol. Conf., Calgary, Alberta, Sep. 2008. 163 Publications 3. E. Mo, P. Y. Kam and M. A. Armand, ”LLR metrics for LDPC codes with QPSK transmission and their performances”, in Proc. Int. Sym. Info. Theory and its Appl., Auckland, New Zealand, Dec. 2008. 4. E. Mo, P. Y. Kam, “Log-likelihood ratios for LDPC codes with pilot-symbolassisted BPSK transmission over the noncoherent channel,” in Proc. IEEE Wireless Commun. and Networking Conf., Budapest, Hungary, Apr. 2009. 5. E. Mo, P. Y. Kam, “Reference phasor based log-likelihood ratios for pilotsymbol-assisted BPSK Transmission of LDPC codes over the noncoherent channel,” in Proc. IEEE 71st Veh. Technol. Conf., Taipei, Taiwan, May 2010. 164 Appendix A Gallager Codes The original method of construction [36] yields regular LDPC codes represented by a parity-check matrix in the form      H=     H1   H2   , ..  .    Hγ where H1 , H2 , · · · , Hγ are µ×µρ submatrices with column weight 1 and row weight ρ. H1 has a regular structure; the ith row contains 1’s in columns (i − 1)ρ + 1 to iρ. The other submatrices are column permutations of H1 that are chosen such that the code generated does not contain cycles of short lengths (especially four) in its Tanner graph. This is commonly performed by computer searches. Thus, H has dimension µγ × µρ and is regular with constant row and column weights ρ and γ, respectively. 165 Appendix A MacKay Codes MacKay described the following methods to construct parity-check matrices with no cycles of length four. 1. H is constructed by randomly generating columns of weight γ. The row weights are kept as uniform as possible and the maximum overlap of nonzero entries between any pair of columns is kept to one. 2. H is generated by letting up to (N − K)/2 columns have weight two and have a regular structure. The rest of the columns are created as in 1. 3. H is generated by either 1 or 2. Columns that result in short cycles of length less than a specified girth are eliminated. Bit-Flipping Decoding Algorithm A binary (demodulated) received vector is initialized to the variable nodes and decoded in the following steps: 1. For each check node, compute the parity-check sum of the variable nodes connected to it. Flag the check node if parity-check fails, i.e. the paritycheck sum is not zero. 2. Identify the variable node(s) with the most number of flagged check nodes connected to it (them). Flip these node(s), i.e., a bit 0 becomes 1 and vice versa. 3. Repeat 1 and 2 until there are no flagged check nodes or when the maximum number of iterations is reached. 166 Appendix A Probabilistic Decoding Algorithms We assume that the codewords are transmitted using the BPSK, where code bit c = 0 is mapped to s = 1 and c = 1 is mapped to s = −1 in the signal constellation. Further, the transmission is assumed to be over the additive-white Gaussian noise (AWGN) channel with signal-to-noise ratio (SNR) Eb . N0 The received channel output is r = s + n, Eb where n is a real Gaussian variable of mean zero and variance 1/(2R N ) and R is 0 the code rate. Thus, the likelihood of r is given by P (r|c) = √ 1 2πσ 2 exp(− (r − s)2 ) 2σ 2 Bayes’ rule states that P (r|c) = P (c|r)P (r) P (c) Thus P (r|c = 0) P (c = 0|r)P (c = 1) = . P (r|c = 1) P (c = 1|r)P (c = 0) Assuming that the a priori probabilities of the code bit c are equally likely, i.e., P (c = 0) = P (c = 1) = 0.5, we obtain the ratio of the a posteriori probabilities (APP) as P (r|c = 0) P (c = 0|r) = . P (c = 1|r) P (r|c = 1) Substituting the likelihood of r, Eb P (c = 0|r) = exp(4rR ). P (c = 1|r) N0 167 Appendix A Since P (c = 0|r) + P (c = 1|r) = 1, we can obtain each APP as P (c = 0|r) = 1 , Eb 1 + exp(−4rR N ) 0 1 . Eb 1 + exp(4rR N ) 0 P (c = 1|r) = The calculation of the APP is naturally extended to q-ary symbols. In some of the decoding algorithms discussed later, we will be interested in the log-likelihood ratio (LLR), L(c) = log P (c = 0|r) Eb = 4rR . P (c = 1|r) N0 The sum-product algorithm (SPA) was introduced in [36] and was later generalized for application to nonbinary codes [75]. Here, we revisit the algorithm for nonbinary codes. Notations involved in the SPA are a an element of GF(q) c set of code symbols F set of check nodes Fi set of check nodes connected to variable node vi Fi \ j set of check nodes connected to variable node vi , excluding fj V set of variable nodes Vj set of variable nodes connected to check node fj Vj \ i set of variable nodes connected to check node fi , excluding vi xaij probability that symbol i of c is a, given the information obtained via the check nodes other than check node fj a yji probability that check node fj is satisfied when symbol i of c is a and 168 Appendix A a ′ other symbols are independent with probabilities qji ′ , i ∈ Vj \ i The SPA is as follows: 1. Initialization. xaij is set to P (ci = a|r). 2. Message passing. Bottom-up. For each i, j and a, compute ∑ a yji = P (fj |c′ ) c′ :c′i =a where ∏ c′ xi′ij′ , i′ ∈Vj \i    1 , if c′ satisfies check node fj . ′ P (fj |c ) =   0 , otherwise. Top-down. For each i, j and a compute xaij = αij P (ci = a|r) ∏ yja′ i , j ′ ∈Fi \j where αij is chosen such that ∑ a xaij = 1. 3. Tentative hard decision. cˆi = arg maxa P (ci = a|r) ∏ j∈Fi a . yji If ˆ cHT = 0, a valid codeword is found and the algorithm terminates. Else, steps 2 and 3 are repeated until a valid codeword is obtained or the maximum number of iterations is reached. 169 Appendix A The log-domain SPA decoder only applies to decoding binary codes, i.e., a is limited to 0 and 1. Further, the messages are updated in the form of the log-likelihood ratios, λ(·), of the probability of a = 0 over that of a = 1, eg., x0 λ(xij ) = log( x1ij ). ij The APP’s are updated in the following steps: 1. Initialization. λ(xij ) is set to λ(ci ). 2. Message passing. Bottom-up. For each i and j, compute λ(yji ) = ∏  sign[λ(xi′ j )] · ϕ  ∑  ϕ(|λ(xi′ j )|) , i′ ∈Vj \i i′ ∈Vj \i x +1 where ϕ(x) = − log[tanh(x/2] = log( exp ). expx −1 Top-down. For each i and j, compute λ(xij ) = λ(ci ) + ∑ j ′ ∈F λ(yj ′ i ). i \j 3. Tentative  hard decision.   0 , if λ(ci ) + ∑ j∈Fi λ(yji ) > 0. cˆi =   1 , else. Similarly, if ˆ cHT = 0, a valid codeword is found and the algorithm terminates. Else, steps 2 and 3 are repeated until a valid codeword is obtained or the maximum number of iterations is reached. 170 Appendix A Notice that by using the log-domain SPA, the multiplications in the top-down message passing in Step 2 and tentative hard decision in Step 3 are reduced to additions. The min-sum decoder [121] performs iterative decoding in the same steps as the log-domain decoder, except for the approximation  ϕ ∑ i′ ∈Vj \i  ϕ(|λ(xi′ j )|) ≈ ′min |λ(xi′ j )| i ∈Vj \i in the bottom-up message passing in Step 2. Thus the multiplications involved are reduced to comparisons. 171 Appendix B Suppose θ(0) is uniformly distributed over [−π, π), i,e, pθ(0) (θ(0)) = 1 {u(θ(0) + π) − u(θ(0) − π)} , 2π (B.1) where u(x) is the unit step function. w(1) is independent of θ(0) and its pdf is denoted by pw(1) (w(1)). We first consider β(1) = θ(0) + w(1), without mod 2π operation. The pdf of β(1) is ∫ pβ(1) (β(1)) = ∞ −∞ pθ(0) (β(1) − w(1))pw(1) (w(1))dw(1). (B.2) Substituting (B.1) in (B.2), we have 1 pβ(1) (β(1)) = 2π ∫ β(1)+π pw(1) (w(1))dw(1). (B.3) β(1)−π The pdf of θ(1) = β(1) mod 2π, for −π ≤ θ(1) < π, is pθ(1) (θ(1)) = {u(θ(1) + π) − u(θ(1) − π)} ∞ ∑ k=−∞ 172 pβ(1) (β(1) + 2kπ). (B.4) Appendix B Substituting (B.3) in (B.4) yields ∫ β(1)+2kπ+π ∞ ∑ 1 pθ(1) (θ(1)) = {u(θ(1) + π) − u(θ(1) − π)} pw(1) (w(1))dw(1). 2π β(1)+2kπ−π k=−∞ (B.5) Since ∞ ∫ ∑ k=−∞ ∫ β(1)+2kπ+π pw(1) (w(1))dw(1) = β(1)+2kπ−π ∞ −∞ pw(1) (w(1))dw(1) = 1, (B.6) (B.5) reduces to pθ(1) (θ(1)) = 1 {u(θ(1) + π) − u(θ(1) − π)} . 2π (B.7) Thus, θ(1) is uniformly distributed over [−π, π), regardless of the pdf of w(1). The pdfs of θ(2), θ(3), · · · can be easily induced. 173 Appendix C In the presence of the unknown carrier phase θ(k − 1), we have ( ) p r˜(k−1)|ϕ(k), ϕ(k−1) ∫ π ( ) ( ) = p r˜(k−1)|ϕ(k), ϕ(k−1), θ(k − 1) p θ(k − 1)|ϕ(k), ϕ(k−1) dθ(k − 1). (C.1) −π Conditioned on ϕ(k − 1) and θ(k − 1), the only randomness in r˜(k −1) is due to n ˜ (k−1). Hence, we have ( ) p r˜(k−1)|ϕ(k), ϕ(k−1), θ(k−1) ] [ 2 1 1 1 2 j(ϕ(k−1)+θ(k−1)) = exp − r˜(k−1) − Es e πN0 N0 [ 1 ] 2Es2 = C(k − 1) exp |˜ r(k − 1)| cos(θ(k − 1) − ∠˜ r(k − 1) + ϕ(k − 1)) , (C.2) N0 where [ ] 1 |˜ r(k)|2 + Es C(k) = exp − . πN0 N0 174 Appendix C Furthermore, θ(k − 1) is uniformly distributed over [−π, π) and is independent of ϕ(k) and ϕ(k − 1). Thus, (C.1) becomes ( ) p r˜(k−1)|ϕ(k), ϕ(k−1) [ 1 ] ∫ π C(k−1) 2Es2 exp |˜ r(k−1)| cos(θ(k−1) − ∠˜ r(k−1) + ϕ(k−1)) dθ(k−1) = 2π N0 −π [ 1 ] 2 2Es = C(k − 1)I0 |˜ r(k − 1)| . (C.3) N0 Observe that the final expression in (C.3) is not dependent on the values of ϕ(k) and ϕ(k − 1), and is in fact the PDF of r˜(k − 1). 175 Appendix D The conditional PDF of θ(k − 1) given r˜(k − 1) and ϕ(k − 1) is p (θ(k − 1)|˜ r(k − 1), ϕ(k − 1)) p (˜ r(k − 1)|θ(k − 1), ϕ(k − 1)) p(θ(k − 1)|ϕ(k − 1)) p (˜ r(k − 1)|θ(k − 1), ϕ(k − 1)) p(θ(k − 1)|ϕ(k − 1)) dθ(k − 1) −π N = . (D.1) D = ∫π By noting that p(θ(k − 1)|ϕ(k − 1)) = p(θ(k − 1)) = 1 , 2π we have [ ] 1 2Es2 N = exp |˜ r(k − 1)| cos (θ(k − 1) − ∠˜ r(k − 1) + ϕ(k − 1)) N0 and ∫ [ π D= −π N dθ = 2πI0 176 ] 1 2Es2 |˜ r(k − 1)| . N0 Appendix E The conditional PDF of θ given {˜ r((l + w)B)}W w=−W is ( p θ|{˜ r((l + where w)B)}W w=−W ) ( ) p {˜ r((l + w)B)}W w=−W |θ p(θ) ) = ∫π ( W p {˜ r ((l + w)B)} |θ p(θ) dθ w=−W −π N = , D (E.1) [ ] 1 W 2Es2 ∑ N = exp |˜ r((l + w)B)| cos (∠˜ r((l + w)B)−θ) N0 w=−W and ∫ π D= N dθ −π are the numerator and denominator of the RHS of (E.1), respectively. We denote the summation of the received pilot symbols {˜ r((l + w)B)}W w=−W by v˜(l) = W ∑ r˜((l + w)B). w=−W The numerator easily reduces to [ ] 1 2Es2 N = exp |˜ v (l)| cos (θ − ∠˜ v (l)) N0 177 (E.2) Appendix E and the denominator is readily obtained by taking the integral of N over the range of θ, that is, [ D = 2πI0 where I0 [|x|] = 1 2π ∫π −π ] 1 2Es2 |˜ v (l)| , N0 exp [|x| cos θ] dθ is the zeroth-order modified Bessel function of the first kind. 178 [...]... larger than that of randomly generated LDPC codes Therefore, short codes are favored Not only are these codes favored over randomly constructed LDPC codes due to reduction in storage space of the large parity check matrices and ease in performance analysis, they could also achieve relatively similar performance compare to random LDPC codes Similar to FG and PG codes, LDPC codes may be designed by combinatorial... performances of LDPC codes over coherent and noncoherent AWGN channels 5.4 BER performances of mixed alphabet LDPC codes over noncoherent AWGN channels 5.5 90 92 BER performances of (1008,504) LDPC code with QDPSK transmission over noncoherent AWGN channels using TSOI-LLR, subjected to SNR misestimation 5.6 94 BER performances of. .. combine the advantages of both families of codes, binary and non- binary That is, these mixed-alphabet LDPC codes outperform single alphabet LDPC codes of the same length and rate with a slight increase in decoding complexity incurred 1.2.2 Non- random construction of LDPC Codes In Mackay’s LDPC code construction [75], one may design a parity-check matrix to contain up to a maximum of (N − K)/2 weight-two... performances of (2640,1320) and (1008,504) LDPC codes over noncoherent AWGN channel without phase noise 4.3 BER performances of (3072,1024) SCCC and (3072,1024) PCCC over noncoherent AWGN channel without phase noise 4.4 72 BER performances of codes over coherent and noncoherent AWGN channels 4.5 72 74 BER performances of (1008,504) LDPC code over noncoherent... performances of (1008,504) LDPC codes over noncoherent AWGN channel using PSAM-R-LLR 8.3 128 BER performances of (256,128) LDPC codes over noncoherent AWGN channel using PSAM-R-LLR 8.2 126 136 BER performances of (1008,504) LDPC code over noncoherent AWGN channel with phase noise xv 138 LIST OF FIGURES 8.4 BER performances of (256,128) LDPC code over noncoherent... constructed and applied to the binary symmetric channel (BSC) and the binary AWGN channel [21] Significant improvement over the performance of the binary codes reported motivated recent works on the analysis and design of non- binary LDPC codes on binary and nonbinary channels In [6], nonbinary codes under ML decoding were shown to provide reliable communication at rates very close to the capacity of any... parity-check (LDPC) codes are a class of linear error-correcting block codes introduced in [36] Contrary to other linear codes, e.g., convolutional codes and Reed Solomon codes, LDPC codes are constructed and represented by sparse parity-check matrices As its name suggest, in a sparse parity-check matrix, the ratio of the number of nonzero entries to the total number of elements is small Unfortunately, LDPC codes. .. complexity for each iteration is estimated by the average number of sum-product or min-sum operations CO and the average number of additions CA per coded symbol as [40] 1.2 1.2.1 CO = (1 − R)¯ ρ (1.5) CA = γ¯ 2 (1.6) Current Research and Challenges Nonbinary LDPC Codes Following the rediscovery of LDPC codes and the excellent error performance of binary LDPC codes over the AWGN channel, LDPC codes over... implementation LDPC codes constructed using combinatorial design share common characteristics; their corresponding Tanner graphs have girth of six, and they can be designed for very high rates (R ≥ 0.8) and of relatively short length They also perform well under iterative decoding 1.2.3 Finite Length Analysis of LDPC Codes Iterative decoders are well-known for their computational efficiency compared to the ML decoders. .. check matrix constructed in each step 46 xii LIST OF FIGURES 3.2 Tree constructed for a = 2, s = 2 after (a) steps 1-3 and (b) step 4 (the final structure) 3.3 Performance of structured and random LDPC codes over Z4 with QPSK signaling over the AWGN channel 3.4 47 53 Performance of structured and random LDPC codes transmitted using matched signals over the AWGN