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

PRACTICAL DECODERS FOR STANDARD AND

NON-STANDARD CHANNELS

MO HUISI, ELISA

NATIONAL UNIVERSITY OF SINGAPORE

2009

PRACTICAL DECODERS FOR STANDARD AND

NON-STANDARD CHANNELS

MO HUISI, ELISA

(B.Eng.(Hons.), NUS )

A THESIS SUBMITTEDFOR THE DEGREE OF DOCTOR OF PHILOSOPHY

DEPARTMENT OF ELECTRICAL AND COMPUTER ENGINEERING

NATIONAL UNIVERSITY OF SINGAPORE

2009

The four-year pursuit of a doctorate degree has not been easy, but it is definitely anenriching journey I had the opportunity to work closely with Dr Marc Armandand Prof Kam Pooi Yuen Dr Armand, well-versed in algebra and codingtheory, has always provided directions to guide my research His comments andencouragements also served as my motivation to persevere, especially in timeswhen I was stuck with a problem for weeks (or even months) Combined withProf Kam’s knowledge in communications, our joint discussions has often led toimportant insights and interesting research problems I would also like to thankthem for their patience and time spent to help craft and edit (and re-edit ) mymanuscripts until they are in their best form for submission Of course, this thesiswould not have been possible without their advice and guidance.

I would like to express my gratitude towards DSO National Laboratories forthe financial support offered through my course of study, and also towards Dr NgBoon Chong, my mentor from DSO, for his advice and support rendered I amlooking forward to applying my research completed, knowledge accumulated andthinking skills cultivated when I return to DSO in 2010

Research can be a lonely process A special thanks goes to my lunch buddywho has always listened to my grumbles, encouraged me and sometimes studied

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 unwaveringunderstanding, support and love from Mummy, Papa and my Significant Other.Thank you!

Acknowledgments ii

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

1.3 Contributions of the Thesis 20

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 27 2.1 Construction of Mixed-Alphabet Codes 28

2.1.1 Simulation Studies 28

2.2 Addition of Redundant Check Nodes to Tanner Graphs 30

2.2.1 Simulation Results and Discussion 33

2.3 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

3.2 Latin Squares 40

3.2.1 Definition and Application to Galois Fields 40

3.2.2 Extended Application to Multiplicative Groups over Integer Residue Rings 41

3.3 Structured LDPC Codes overZ2a 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

4 Iterative Decoding using Binary Differential PSK 56

4.1 System Model 584.2 Metric Derivation 604.2.1 The optimal TSOI-PN-LLR and its approximations 604.2.2 The G-PN-LLR 674.2.3 Case with no phase noise 684.3 Simulation Study 704.3.1 Performance of LDPC/turbo codes with different metrics

and no phase noise 714.3.2 Effects of SNR estimation error on performance of metrics 754.3.3 Effects of phase noise on performance of metrics 774.3.4 Effects of phase noise estimation error on performance of

metrics 794.4 Conclusion 79

5.1 System Model 835.2 Metric Derivation 855.3 Simulation Study 895.3.1 Performance of Binary LDPC Codes using Different Metrics 895.3.2 Performance of Mixed Alphabet LDPC Codes using TSOI-

SA-LLR 915.3.3 Effects of SNR Mis-estimation 925.4 Conclusion 93

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

7.6 Conclusion 128

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 135

8.2.2 Performance of LDPC codes with noisy, unknown carrier

phase 1378.2.3 Effects of SNR estimation error 1388.3 Conclusion 139

9.1 Conclusion 1419.2 Proposals for Future Research 144

Low-density parity-check (LDPC) codes are known for their near Shannon-limitperformance Since non-binary LDPC codes are generally capable of outperform-ing binary LDPC codes, much interest were involved in the construction of goodnon-binary codes However, decoding complexity increases with the alphabet size.Following recent work on mixed-alphabet codes, we design near-regular LDPCcodes where the information symbols and majority of the parity-check symbolsare defined over an integer residue ring, while the remaining parity-check symbolsare defined over another integer residue ring of a larger size Further, it has beenshown that performance of the iterative decoder improves when redundant checknodes are added to the Tanner graph This motivates our research on structuredLDPC codes over integer residue rings, where the corresponding Tanner graphswith constant variable and check node degrees contain redundant check nodes.The original decoding algorithm proposed by Gallager is designed fortransmission over the additive white Gaussian noise channel Since then,performance of LDPC codes transmitted using modulation other than the binaryphase shift keying (BPSK) over other types of channels was investigated However,the decoding algorithm, the computation of the log-likelihood ratios (LLRs) inparticular, is either executed with assumptions on the channel or altered based

on unnecessary approximations The calculation of the LLR is revisited and theoptimal 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 aswell as mixed-alphabet LDPC codes over the noncoherent channel are examined

We analyse finite-length binary and mixed-alphabet LDPC codes under BDPSKand QDPSK, and explain the difference in error performance under these twotransmissions using the notion of pseudocodewords Further, we derive the LLRfor pilot-symbol-assisted BPSK transmission which yields better performance thanBDPSK transmission but requires higher bandwidth Extension to higher ordermodulations and non-binary codes shall be left for possible future research

1.1 Tanner graph for Example 1.1 41.2 Tanner graph for Example 1.2 101.3 Tanner graph corresponding to a length 21 projective geometryLDPC code 131.4 System model 26

2.1 BER performance of Z4 codes extended with N2 parity-checksymbols defined over Z16 for N2 = 0, 10, 20 292.2 BER performance of the mixed-alphabet codes with and withoutredundant check nodes in their Tanner graph representations for

N2 = 0 332.3 BER performance of the mixed-alphabet codes with and withoutredundant check nodes in their Tanner graph representations for

N2 = 10 342.4 BER performance of the mixed-alphabet codes with and withoutredundant check nodes in their Tanner graph representations for

N2 = 20 343.1 Portion of parity check matrix constructed in each step 46

3.2 Tree constructed for a = 2, s = 2 after (a) steps 1-3 and (b) step 4

(the final structure) 473.3 Performance of structured and random LDPC codes over Z4 withQPSK signaling over the AWGN channel 533.4 Performance of structured and random LDPC codes transmittedusing matched signals over the AWGN channel 54

4.1 System model 594.2 BER performances of (2640,1320) and (1008,504) LDPC codes overnoncoherent AWGN channel without phase noise 724.3 BER performances of (3072,1024) SCCC and (3072,1024) PCCCover noncoherent AWGN channel without phase noise 724.4 BER performances of codes over coherent and noncoherent AWGNchannels 744.5 BER performances of (1008,504) LDPC code over noncoherentAWGN channel without phase noise using TSOI-LLR, TSOI-SA-LLR and G-LLR, subjected to SNR estimation error 764.6 BER performances of (3072,1024) SCCC over noncoherent AWGNchannel without phase noise using TSOI-LLR, TSOI-SA-LLR andG-LLR, subjected to SNR estimation error 764.7 BER performances of (1008,504) LDPC code over noncoherent

AWGN channel with phase noise where σ2 = 10−2 784.8 BER performances of (1008,504) LDPC code over noncoherent

AWGN channel with phase noise where σ2 = 4× 10 −2 . 78

4.9 BER performances of (1008,504) LDPC codes over noncoherentAWGN channel using TSOI-PN-LLR, TSOI-PN-SA-LLR and G-PN-LLR, subjected to phase noise variance estimation error 80

5.1 System model 835.2 BER performances of LDPC codes over noncoherent AWGN channels 905.3 BER performances of LDPC codes over coherent and noncoherentAWGN channels 905.4 BER performances of mixed alphabet LDPC codes over noncoher-ent AWGN channels 925.5 BER performances of (1008,504) LDPC code with QDPSK trans-mission over noncoherent AWGN channels using TSOI-LLR, sub-jected to SNR misestimation 945.6 BER performances of mixed alphabet LDPC code with QDPSKtransmission over noncoherent AWGN channels using TSOI-SA-LLR, subjected to SNR misestimation 94

6.1 BER performance of a (1008,504) binary LDPC code on thenoncoherent AWGN channel with optimal and suboptimal LLRs

of each code bit fed to the BP decoder for which the maximumnumber of iterations was set to 50 1086.2 BER performance of (8,4) and (8,3) code on the noncoherentAWGN channel with the G-LLR of each code bit fed to the BPdecoder for which the maximum number of iterations was set to 50 1096.3 Weight distribution of pseudocodewords arising from the M -covers

of the Tanner graph corresponding to H(8,4) for M = 1, 2, 3 110

6.4 Weight distribution of pseudocodewords arising from the M -covers

of the Tanner graph corresponding to H(8,3) for M = 1, 2, 3 1117.1 System model 1147.2 Frame model 1147.3 BER performances of (256,128) LDPC code over noncoherentAWGN channel using PSAM-LLR with varying number of pilotsymbols 1237.4 BER performances of (1008,504) LDPC code over noncoherentAWGN channel using PSAM-LLR with varying number of pilotsymbols 1247.5 BER performances of (1008,504) LDPC code over noncoherentAWGN channel with various metrics 1257.6 BER performances of (1008,504) LDPC code over noncoherentAWGN channel with various block lengths, subjected to phase noise

where σ2 = 10−6 1267.7 BER performances of (1008,504) LDPC code over noncoherentAWGN channel subjected to SNR mis-estimation 128

8.1 BER performances of (256,128) LDPC codes over noncoherentAWGN channel using PSAM-R-LLR 1368.2 BER performances of (1008,504) LDPC codes over noncoherentAWGN channel using PSAM-R-LLR 1368.3 BER performances of (1008,504) LDPC code over noncoherentAWGN channel with phase noise 138

8.4 BER performances of (256,128) LDPC code over noncoherentAWGN channel subjected to SNR mis-estimation 1408.5 BER performances of (1008,504) LDPC code over noncoherentAWGN channel subjected to SNR mis-estimation 140

List of Abbreviations

BDPSK binary differential phase shift keyingQDPSK quadrature differential phase shift keying

TSOI-PN-LLR Two-Symbol-Observation-Interval Phase-Noise LLRTSOI-PN-A-LLR TSOI-PN-Approximate-LLR

TSOI-PN-SA-LLR TSOI-PN-Simplified-Approximate-LLR

List of Symbols and Notations

x denotes scalar variable

x denotes vector

X denotes matrix

xi ith row of matrix X

x i ith element of vector x

x(i) value x assumes at time instant i

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

ρ row weight of a parity-check matrix or degree of a check node in a

Tanner graph

m message vector

s signal vector sent

r signal vector received

E b energy per message bit

E s energy per symbol

N0 power spectral density of noise

n noise vector

λ( ·) log-likelihood ratio

T tanner graph

Chapter 1

Introduction

Low-density parity-check (LDPC) codes are a class of linear error-correcting blockcodes introduced in [36] Contrary to other linear codes, e.g., convolutional codesand Reed Solomon codes, LDPC codes are constructed and represented by sparseparity-check matrices As its name suggest, in a sparse parity-check matrix, theratio 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 decodingalgorithm relative to the availability of technology during that time Tanner’sgeneralization and graphical representation of LDPC codes [112] aside, there waslittle 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 ofonly 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 mathematicalpreliminaries and give a descriptive overview of the discovery and development ofLDPC codes Readers are referred to Appendix A for the algorithms pertaining

to the construction and decoding of LDPC codes Further in this chapter, we

describe the current challenges and research interests that motivate the researchproblems undertaken in the thesis A summary of the main contributions of thisthesis 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 h 1· · · hM]T is such

that for each c ∈ C, ch T

i = 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 {h i } 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 notcontain an equal number of non-zero elements in each column and row Thevariable node and check node degree distribution polynomials are denoted by

γ(x) and ρ(x), respectively In the polynomial

ρ i denotes the fraction of check nodes with degree i, and d c denotes the maximumcheck 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 vertexset can be partitioned into two disjointed subsets such that every edge connects anode in one subset to a node in the other subset, and no two vertices are connectedwithin each subset In a Tanner graph, the two disjoint subsets of nodes are thecheck 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 h ji in H is nonzero, there exists an edge

of weight h ji that connects check node f j and variable node v i

Example 1.1 Consider a (7, 3) linear block code with the following parity-check

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, acycle of length six is shown in bold in Fig 1.1 The smallest possible girth of any

f3 f2

f1 f0

Figure 1.1: Tanner graph for Example 1.1

Tanner graph is four To decode an LDPC code iteratively, it is important thatthe Tanner graph does not contain short cycles, especially cycles of length four.The original method of construction [36] yields regular LDPC codes rep-resented by a parity-check matrix that is a concatenation of submatrices, suchthat the corresponding Tanner graph does not contain short cycles In MacKay’sconstruction, the main objectives are to generate random/semi-random sparseparity-check matrices and to avoid short cycles in their corresponding Tannergraphs [75] Due to the lack of structure, MacKay codes do not allow low-complexity encoding The generator matrix in systematic form is obtained byperforming Gaussian elimination Although the parity check matrix is sparse, theresultant 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) Toovercome this, an efficient encoding technique was proposed in [105] which requiressome 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

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 LDPCcode depends on the variable and check node degree distributions of its Tannergraph The optimization of these distributions is found by density evolution, theevolution of the probability density functions of the messages passed between thevariable and check nodes in a belief propagation decoder However, the optimizeddistributions only provide a good code when the block length approaches infinity.The distributions applied to short or medium length codes give rise to high error-rate floor Similar to MacKay codes, efficient encoding for irregular LDPC codescan be performed using the algorithms proposed in [105]

Since LDPC codes are generally not structured, they cannot be decodedalgebraically Iterative algorithms were hence devised to perform decoding Whenapplied to a Tanner graph, these algorithms are simple and easy to implement.They execute maximum-likelihood (ML) decoding in each iteration, but aresuboptimal on the whole due to the presence of cycles in the Tanner graph Thebit-flipping decoding of LDPC codes is a very simple iterative hard decision basedalgorithm introduced with the code itself [36]

On the other hand, the probabilistic decoding algorithm performs softdecision 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 Tannergraph, it can be perceived as repeatedly passing messages along the edges, fromthe variable nodes to the check nodes and back, while updating the information

contained in the nodes Similar to the bit flipping algorithm, the probabilisticdecoding algorithms performs ML decoding in each iteration, but is suboptimal

on the whole due to presence of cycles in the Tanner graph The sum-productalgorithm (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 domain Information about each code bit is represented in the form of a log-likelihood ratio (LLR) of the probability that the code bit assumes the value ‘0’

log-to that of the value ‘1’ Some multiplication operations are reduced log-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 multiplicationoperations to comparisons These algorithms are generally developed assumingthat 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 N O , N A , N M , and N X denote the numbers of sum-product or sum operators, signed adders, registers and connections respectively Theimplementation complexity for decoding is estimated by [40]

N X = N (¯ γ2+ ¯γ ¯ ρ + 2), (1.4)

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 C O and the average number of

additions C A per coded symbol as [40]

1.2 Current Research and Challenges

Following the rediscovery of LDPC codes and the excellent error performance

of binary LDPC codes over the AWGN channel, LDPC codes over finite fieldswere constructed and applied to the binary symmetric channel (BSC) and thebinary AWGN channel [21] Significant improvement over the performance ofthe binary codes reported motivated recent works on the analysis and design ofnon-binary LDPC codes on binary and nonbinary channels In [6], nonbinarycodes under ML decoding were shown to provide reliable communication atrates very close to the capacity of any discrete memoryless channel Analysis ofiterative decoding were performed using extrinsic information transfer charts [7],Gaussian approximation [65] and density evolution [101] Unlike previous worksthat designed LDPC codes over finite fields, [29, 109] designed LDPC codes overrings These codes can be mapped to matched nonbinary signal constellations toimprove bandwidth efficiency In particular, when mapped to PSK signal sets, thecodes become geometrically uniform signal space codes Empirical results in [109]

showed that LDPC codes over rings provide coding gain over coded modulationbased on binary LDPC codes.

Despite the superior error correcting performance of nonbinary codes overbinary codes, computational efficiency remains a challenge due to its exponentialincrease with the alphabet size Very recently, a new class of nonbinary LDPCcodes, LDPC codes over mixed alphabets, has been introduced Studies on thesecodes are motivated by the potential improvement in error performance from usinglarger 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 −K2

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 manyalphabets, were constructed as single-error-correcting perfect codes applied toproblems in distribution of resources such as speech coding The parity-checkmatrix of a mixed-covering code consists of sub-matrices, each defined over asingle alphabet As the sub-matrices are arranged in a disjoint manner in theparity-check matrix, its corresponding Tanner graph is simply a collection ofdisjoint subgraphs, each corresponding to one sub-matrix defined over a singlealphabet Thus, performing iterative decoding on the Tanner graph is equivalent

to performing iteratively decoding on each individual subgraph, which seeminglydoes not provide any coding gain

A mixed-alphabet LDPC code may be represented by a sparse parity-checkmatrix with rows and columns defined over different alphabets Correspondingly,its Tanner graph has variable nodes and check nodes defined over more than one

For practical applications, one would like to keep the number of nodes definedover the larger alphabets small to maintain a relatively low decoding complexity.Since the message passing algorithm, in its most general form, does not have anyrestriction on the alphabet on which each node is defined, iterative decoding caneasily be modified to operate on the Tanner graph with nodes defined over mixedalphabets.

In [85], LDPC codes over two finite Galois fields were introduced Thesecodes were shown to perform better than their single alphabet counterparts of thesame rate and equivalent binary length as the number of code symbols definedover the larger alphabet increases In [9], LDPC codes constructed with multipleGalois fields were introduced The codes were optimized according to the profile

of the channel and applied to different frequency selective channels On the otherhand, [107] proposed irregular LDPC codes defined over groups of different ordersand optimized the distribution of the degrees and groups of both the variable and

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 codesoutperform single alphabet LDPC codes of the same length and rate with a slightincrease 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 allowablecan be further increased when working with nonbinary codes A simple andefficient method of constructing Tanner graphs with large girths by progressivelyestablishing edges between code nodes and check nodes was proposed in [46] Theedge 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-algebraicmethod of constructing graphs with large girth, simulation results show thatLDPC codes from progressive edge-growth construction significantly outperformrandomly constructed ones A similar construction method was presented in [120].However, the memory space required for storing the parity-check matrix stillposes a problem in hardware implementation if very long LDPC codes are used.Furthermore, although the parity-check matrix of an LDPC code is sparse, itscorresponding 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 thelines 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

have low density, do not contain cycles of length four and thus fall into the class ofLDPC codes [61,62] The parity-check matrices can be expressed in cyclic or quasi-cyclic forms which require little storage space during implementation Efficientencoding can be performed using shift-register circuits The cyclic finite geometrycodes tend to have relatively large minimum distance while the quasi-cyclic codestend to have small minimum distance The connectivity of the correspondingTanner graph, though is deterministic, appear random at the decoder Fig 1.3,for example, shows a Tanner graph corresponding to a length-21 LDPC codeconstructed using PG Thus, these codes perform well under iterative decoding.Although the densities of the parity-check matrices are low, the row and columnweights increase with block length and are typically larger than that of randomlygenerated LDPC codes Therefore, short codes are favored Not only are thesecodes favored over randomly constructed LDPC codes due to reduction in storagespace of the large parity check matrices and ease in performance analysis, theycould also achieve relatively similar performance compare to random LDPC codes.Similar to FG and PG codes, LDPC codes may be designed by combinatorialapproaches, exploiting well developed topics in Mathematics LDPC codes weredesigned 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 thatunital designs exist with incidence matrices which are rank deficient, thus givingrise to the high-rate LDPC codes with large number of parity-check equations

Figure 1.3: Tanner graph corresponding to a length 21 projective geometry LDPCcode.

Such codes are well structured and have low-complexity implementation LDPCcodes constructed using combinatorial design share common characteristics; theircorresponding 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 tothe ML decoders However, unlike the ML decoder, the iterative decoder doesnot give the optimum bit-error-rate (BER) performance In the limit as codelength goes to infinity, analysis of LDPC codes can be performed using densityevolution 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

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 goodfinite length LDPC codes Further, it is also not known if the LDPC code designedhas an error floor, or where the error floor exists

Analyzing the performance of finite length LDPC codes is of researchinterest currently The suboptimality of iterative decoding has been attributed

to the emergence of pseudocodewords arising from the Tanner graph due tosuboptimal computation in an iterative manner Finite-length analysis ofthis iterative decoding behavior of LDPC codes was first proposed in [121].Pseudocodewords that arises from computation trees were introduced and theconcept was subsequently extended in [34] These pseudocodewords were used tomodel 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 somenecessary and sufficient conditions for the decoder to converge were provided.However, since computation trees grow exponentially with each iteration, thetracking of all pseudocodewords that may arise from a computation tree is virtuallyimpossible after a few iterations Similar works in [33] and [58] explained thebehavior of iterative decoders using the lifts of the base Tanner graph Thecommon 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 informationfrom code (check) nodes directly connected to it Thus, the iterative decoding

algorithm cannot differentiate if it operates on a Tanner graph or a finitecover of the graph Further, a codeword that arises from a finite cover ofthe Tanner graph, after normalization, does not necessarily reduces to a validcodeword from the Tanner graph In short, a codeword is a pseudocodewordbut a pseudocodeword may not be a codeword These pseudocodewords can berepresented very elegantly with the notion of fundamental cones and polytopes[30, 119] It is shown in [119] that adding redundant check nodes to the Tannergraph representation of a code improves its performance under iterative decoding.Although such pseudocodewords can only perfectly characterize the iterativedecoding behavior of the graph cover decoder (GCD) and linear programmingdecoder (LPD), they also provide substantial insights to the behavior of min-sum decoding Pseudocodewords of a Tanner graph play an analogous role indetermining convergence of an iterative decoder as codewords do for a ML decoder.The error performance of a ML decoder can be computed analytically using thedistance distribution of the codewords in the code Similarly, an iterative decoder’sperformance maybe characterized by the pseudocodeword distance For linearcodes, distance reduces to weight with respect to the all-zero codeword Thus,

in the context of iterative decoding, a minimum weight pseudocodeword [33] ismore fundamental than a minimum weight codeword In [55], lower bounds onthe minimum pseudocodeword weight for the BSC and AWGN channels werepresented

1.2.4 Transmission over Nonstandard Channels

LDPC codes have been shown to achieve reliable transmission at SNR extremelyclose to the Shannon limit on the AWGN channel [104] Despite the promising

error control capability of LDPC codes on coherent channels, knowledge of theexact 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 Themain drawbacks of phase-locked loops circuits, used to approximately implementcoherent detection, are false-locks, phase slips, loses due to severe fading, Dopplershifts, phase noise, and oscillator frequency instabilities Inaccuracy in phaseestimation degrades the performance of the iterative decoder

When knowledge of carrier phase is not available, a simple solution is toapply 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 improvenoncoherent detection Multiple-symbol differential detection [22–25, 59, 60, 63,122] is block-based ML detection of information symbols given the correspondingblock of received signals On the other hand, noncoherent sequence detection[1, 15, 16, 77, 78, 100] based on Viterbi algorithm approximates the optimal MLsequence 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 iterativedecoding is required Recent research focus is thus on the development of softdecision noncoherent decoding

Iterative decoding requires the evaluation of the LLR of the two possiblevalues of each code bit, based on the received signals pertaining to that bit TheLLR is fed into the decoder as soft information input For noncoherent channels

where LDPC codes are modulated using binary differential phase-shift keying(BDPSK), the detection metric was recently derived in [113] However, theirresults are not the optimal metrics for two reasons First, the authors assumedthat 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 twosignals ˜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 ofthe bit Much information is lost in using the PDF of Re[˜r(k)˜ r(k −1) ∗], compared

to using the joint PDF of ˜r kand ˜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 theLLR encompasses the PDF of the unknown carrier phase Thus, explicit carrierphase estimation is not required However, the performance is a few decibels worsethan 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 theoreticalanalysis of this code based on a cut-off rate bound was proposed in [94] (onlyfor noniterative decoding) Turbo processing is performed, where reliabilityinformation is iteratively fed into and updated in the inner (modulation) decoderand the outer (convolutional/LDPC) decoder Channel estimation can beexplicitly carried out in the inner decoder [42] uses linear prediction and per-

survivor processing to estimate the channel response of the frequency-flat fadingchannel, resulting in an exponential expansion of the decoding trellis [44]updated the channel estimates of a noncoherent AWGN channel model using softinformation from the outer decoder and assumed three different phase models:constant but unknown phase offset, Gaussian random walk and constant frequencyoffset Alternatively, channel estimation can be incorporated in the inner decoderusing a modified trellis-based decoding algorithm [79] An iterative algorithmspecific for noisy-phase channels was proposed in [35] In [93], the capacity ofnoncoherent channels has been proven to be very similar to that corresponding tocoherent channels In [96] and [11], receivers for the block-constant phase modeland discrete random-walk phase model are developed by using a discrete phaseapproach More recently, [106] introduced a joint carrier phase estimator andturbo 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 ineach 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 yieldsgood error performance but is computationally costly

When phase dynamics are slow enough, pilot symbols can be multiplexed intothe transmitted signal sequence Pilot symbols, containing no information fromthe transmitter, are periodically inserted into the transmitted signal sequence totrack the carrier phase A branch metric modified for turbo-coded systems usingpilot symbols is presented in [64] However, this metric requires knowledge ofthe carrier phase In [48] and [64], estimation was only performed prior to the

first iteration of turbo decoding Since turbo decoding is an iterative process,performance can be improved by re-estimating the channel after each decoderiteration [114] Iterative estimation and decoding was proposed for convolutionalcodes in [37], for BPSK modulated turbo codes in [115], and for quadratureamplitude modulation (QAM) modulated turbo codes in [124] More recentapproaches 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 arerefined with the aid of the tentative decisions fed back from the decoder Similarly,joint carrier synchronization and decoding algorithms that recursively performcarrier phase and/or frequency synchronization based on updated soft decisionsfrom the decoder were proposed in [4, 90, 99] These receivers are computationallyintensive and require explicit estimation Furthermore, the channel estimatesobtained 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 isassumed in each iteration that the channel estimates obtained are exact, andmuch 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 graphsand expectation-maximization algorithm, though not in focus here, shall be brieflysummarized The use of factor graphs, including code constraints and channelparameter statistics, is introduced in a broad context in [123] and modified tocater for noncoherent decoding under certain phase statistics subsequently Using