Analysis and Design of Staircase Codes for High Bit-Rate Fibre-Optic Communication. Lei Zhang

Transcription

1 Analysis and Design of Staircase Codes for High Bit-Rate Fibre-Optic Communication by Lei Zhang A thesis submitted in conformity with the requirements for the degree of Doctor of Philosophy Graduate Department of Electrical and Computer Engineering University of Toronto c Copyright 2017 by Lei Zhang

2 Abstract Analysis and Design of Staircase Codes for High Bit-Rate Fibre-Optic Communication Lei Zhang Doctor of Philosophy Graduate Department of Electrical and Computer Engineering University of Toronto 2017 Low-complexity, iterative hard-decision decoded staircase codes for hard-decision optical transport-networks (OTNs) are designed, with overheads (OH) between 6.25% and 33.3%. Extensive software simulations are performed for all code designs. Net coding gain gaps to hard-decision capacity are found to range from 0.45 db at 6.25% OH to 1.38 db at 33.3% OH. All code designs with 25.0% OH achieved net coding gains within 1.0 db of hard-decision channel capacity. The spatially-coupled split-component (SCSC) ensemble is defined, generalizing the structures of staircase and braided block codes. An analysis of SCSC ensembles is given for the binary erasure channel, using the differential equation method of random graph theory and the potential function analysis of spatially-coupled systems. The erasure channel analysis is used to approximate the binary symmetric channel performance. Simulation results show that the analysis gives an accurate prediction of staircase code performance when the size of a staircase block is large (more than 10 6 bits). Generalizations such as mixture ensembles containing different component codes and SCSC ensembles decoded by beyond bounded-distance component code decoders are also analyzed. A concatenated coding scheme for soft-decision OTN is proposed, consisting of an inner low-density generator-matrix (LDGM) code and an outer staircase code. The LDGM code is designed while taking into account its decoding complexity, measured as the product of the maximum number of iterations and the number of edges in the ii

3 code graph. The Pareto frontiers between decoding complexity and net coding gain of the concatenated coding scheme are evaluated for several OTN overheads. Simulations of concatenated coding scheme examples at 20% overhead show the same net coding gains as the best existing soft-decision OTN coding schemes, with up to 68% reduction in decoding complexity. iii

4 Acknowledgements First and foremost, I am deeply indebted to my advisor, Professor Frank Kschischang, for his immense knowledge on a vast array of topics, always insightful suggestions, and infinite patience. His professionalism and dedication to his students, in both research and teaching, have been a sustained source of inspiration for me. I will not soon forget the long discussion or revision sessions in his office. It has truly been a privilege for me to work with Professor Kschischang for the past seven years. I would like to thank my external examiner, Professor Pfister, and the other members of my defense committee: Professor Draper, Professor Frey, and Professor Sousa, for taking valuable time out of their busy schedules to appraise this work and critically evaluate its contributions. Thank you for your insightful comments which have greatly improved the content and presentation of this work. I have had the fortunate opportunity during my graduate studies to visit Bell Labs in Stuttgart, Germany and collaborate with world leading researchers on channel coding for fibre-optic communications. I would like to thank Dr. Laurent Schmalen for his support and guidance. The collaboration was made possible by a Deutscher Akademischer Austauschdienst (DAAD) Rise Professional scholarship. I am also very lucky to have been among an amazing group of fellow graduate students. I would like to thank Benjamin Smith and Chen Feng, two senior PhDs who showed me what hard work and dedication really means in graduate school as I was starting out. I would also like to thank Christopher Blake, whose enthusiasm and ability for research is breathtaking and inspiring. Finally, I would like to thank Chunpo Pan, Siddarth Hari, and Siyu Liu, for helping me through the last several years with their knowledge, humour, and friendship. We did it! To my parents, I dedicate this thesis to you as a token of my appreciation for your love and support. Finally, to Cecilia, this thesis is as much yours as it is mine. It only exists because iv

5 of your unwavering love, support, and encouragement. You believed this was possible, always. Thank you. v

6 Contents 1 Introduction Overview of the Thesis Notation Background System and Channel Models Coding Gain and Channel Capacity Rate and Overhead Net Coding Gain Capacity Prior Work HD FEC Schemes for OTN SD FEC Schemes for OTN Technical Background Fundamentals of Error-Correcting Codes BCH Codes Concatenated Codes Product Codes Staircase Codes The Differential Equation Method vi

7 3 Staircase Codes with 6% to 33% Overhead Code search Simulation Results Conclusion Spatially-Coupled Split-Component Codes Introduction Related Work Ensemble Definition Related ensembles BEC Decoding Analysis Preliminaries Analysis Outline Single-interleaver ensemble Coupled ensemble Potential threshold Weight-pulling threshold Mixture ensembles BSC Decoding Analysis BSC Performance Beyond Bounded-Distance Decoding Conclusion Concatenated LDGM-Staircase Codes Introduction Low-Density Generator-Matrix Codes Concatenated LDGM-Staircase Code Structure vii

8 5.3.2 Error Floor Concatenated Code Design Pareto Optimal LDGM Design Concatenated Code Optimization Results Conclusion Conclusion and Future Work Conclusion Future Work A Appendix 110 A.1 Decoupling interleavers A.2 Proof of Lemma A.3 Potential function analysis Bibliography 116 viii

9 List of Tables 2.1 Hard-decision decodable codes for OTN from ITU-G Binary BCH codes with m {8, 9, 10} and t {2, 3, 4} Staircase code designs with t = 4 component codes. NCG calculated from extrapolated p. Gap is NCG gap to BSC capacity Staircase code designs with t = 5 component codes. NCG calculated from extrapolated p. Gap is NCG gap to BSC capacity Staircase codes with t c = 4 BCH component codes Variational Distance V (P s, Q n/s ) LDGM optimization input parameters % overhead concatenated code designs for BP decoder ix

10 List of Figures 2.1 Block diagram of communication system and channel model Net coding gains at the Shannon limit vs. E s /N 0 (in db) for the BSC (HD) and the BIAWGNC (SD) Net coding gain at the Shannon limit vs. overhead for the BSC (HD) and the BIAWGNC (SD) Block diagram of a concatenated coding scheme with S = Structure of a product code Staircase code block structure. Information bits (white) and parity bits (shaded) are shown. Bits in block B0 T are fixed to all-zeros Sample paths of normalized degree distributions 1 m Y (i,m) mx (thin lines) and expected normalized degree distributions z i (x) (thick lines) for the MinDeg process. For illustration, only i [5] are plotted. 10 sample paths are shown for each i Simulated staircase code bit-error rate and their extrapolations to Arranged by decreasing code rate from left to right as listed in Table 3.1 (circles) and 3.2 (squares). Note the different x-axis scales used in each of the shaded and unshaded regions. Solid lines indicate extrapolation based on high confidence performance data. Dashed lines indicate extrapolation based on low confidence performance data x

11 3.2 Simulated NCG of staircase code designs. BSC capacity corresponds to the HD Capacity NCG shown in Fig Filled and open symbols corresponding to staircase codes are explained in Sec Ref 1 refers to [1]. Ref 2 refers to [2] Illustration of 2 consecutive spatial-locations of a spatially-coupled splitcomponent ensemble with coupling-width w = 2. The number of nodes and edges shown is for illustration only and does not satisfy any edge-count constraints Illustration of a E(C, v, M, 3) spatially-coupled split-component ensemble. Elements of the ensemble completely enclosed by the dotted line represent the single-interleaver ensemble at spatial-location 0 as defined in Sec Multiplicative gaps to capacity of analytical (labelled Threshold) and simulated (labelled Simulation) thresholds of E(C, 2, M, 2) spatially-coupled split-component ensembles transmitted over the BEC. Analytical thresholds were obtained by using potential function analysis. Simulated thresholds were obtained by sampling code graphs and simulating transmission and decoding of over a BEC. A simulated threshold is defined as the maximum input erasure probability that results in an output erasure probability of Analytical thresholds (in terms of the multiplicative gap to capacity) of E(T, 2, M, w) mixture ensemble with a set of mixture ratios. T = {(510, 492, 5), (126, 112, 5)} Multiplicative gaps to capacity of analytical (labeled Threshold) and simulated (labeled Syndrome decoding) thresholds of E(C, 2, M, 2) spatiallycoupled split-component ensembles over the BSC xi

12 4.6 Multiplicative gaps to capacity of analytical thresholds of E(C, 2, M, 2) ensembles under bounded-distance and beyond bounded-distance decoding. Thresholds for beyond bounded-distance decoding were obtained by setting decoding profiles (β) to the values given by the lower and upperbounds given in Theorem 5. Different line styles distinguish thresholds due to different decoders Bipartite representation of an LDGM ensemble. Node degrees are denoted by d v and d c, which are different for different nodes in general. The rectangle labelled Π represents the set of all possible edge permutations BERs at fixed points of rate 1/2 LDGM and LDPC ensembles. Insets show the probability of error EXIT charts at SNRs below (left) and above (right) the LDPC threshold Pareto frontiers of concatenated code designs with 15%, 20%, and 25% overhead, under BP decoding Simulated output BER of sample LDGM codes under BP decoding. The BER curves, ordered from right to left, correspond to codes sampled from LDGM ensembles given in the rows of Table 5.4, ordered from top to bottom NCG and η comparison of soft-decision FEC schemes xii

13 List of Publications Works containing portions of this thesis L. M. Zhang and F. R. Kschischang, Staircase Codes With 6% to 33% Overhead, J. Lightw. Technol., vol. 32, no. 10, pp , May L. M. Zhang, D. Truhachev, and F. R. Kschischang, Spatially-coupled Split-component Codes with Bounded-distance Component Decoding, Proc. IEEE Int. Symp. Information Theory (ISIT), June 2015, Hong Kong, China. L. M. Zhang, D. Truhachev, and F. R. Kschischang, Spatially-coupled Split-component Codes with Iterative Algebraic Decoding, submitted to IEEE Trans. Inf. Theory, Oct 2015, Available: L. M. Zhang and F. R. Kschischang, Concatenated LDGM-Staircase Codes for Longhaul Optical-fibre Networks, Biennial Symp. Communications (BSOC), Jun 2016, Kelowna, Canada. xiii

14 Chapter 1 Introduction Long-haul optical transport networks (OTNs), such as the system standardized in ITU-T Recommendation G.709 [3], carry a majority of the inter-continental internet protocol (IP) traffic. The high bandwidth available in the fibre-optic medium allows the fastest networks to achieve a bit rate of 100 gigabits-per-second (Gbps) per wavelength, in a wavelength-division multiplexed system. However, given the growth of IP traffic, OTNs with bit-rates of 400 Gbps or 1 terabit-per-second (Tbps) per channel are expected to be used within the next 5 to 10 years [4]. This thesis focuses on the design of forward error-correction (FEC) schemes for OTNs with bit-rates of 100 Gbps and beyond. The unique combination of high bit-rate, low latency, and a stringent reliability constraint makes FEC design for OTN a unique and interesting problem for coding theory research. Good reviews of historical and recent developments can be found in [5 7] (see also Sec. 2.3). In [8], a G.709-compatible FEC scheme referred to as a staircase code was described. It was shown to have better efficiency (in terms of the net coding gain, see Sec ) than all of the OTN FEC schemes presented in ITU-T Recommendation G [9] (also known as enhanced or super FEC). Motivated by the potential of this particular code and the continuously evolving requirements of OTN systems, we aim to analyze, design, 1

15 Chapter 1. Introduction 2 and apply codes developed from the original staircase code, with better performance, predictability, and flexibility. In particular, we answer the following questions, among others: Q1. Can we design staircase codes of different rates for the hard-decision channel? Q2. Can we generalize staircase codes while preserving their essential properties? Q3. Can we analytically predict the performance of staircase codes? Q4. What are the non-trivial limits to staircase code performance? Q5. How can staircase codes be used for the soft-decision channel? 1.1 Overview of the Thesis In Chp. 2, we provide a brief review of the development of FEC schemes for long-haul OTN systems, which also serves as a survey of prior literature related to the analysis and design of such codes. The necessary technical background is also given, both to specify the technical scope of the thesis and to introduce the relevant concepts and technical results used in later chapters. In Chp. 3, we answer Q1 by designing staircase codes for a hard-decision output channel over a range of code rates. An efficient search procedure is used for code design. Simulation results of coded transmission over the binary symmetric channel are given. Net coding gains are obtained for all codes based on extrapolations of simulated data. In Chp. 4, we answer Q2 Q4 by analyzing the class of spatially-coupled split-component (SCSC) codes. This class of codes is a generalization of the staircase code structure described in [8]. It preserves the properties of the staircase code most important to high bit-rate OTN: a low data-flow, hard-decision decoder and a very low output bit-error probability. We prove that SCSC codes (hence staircase codes) are an instance of a spatially-coupled system [10, 11] and make connections to the current understanding of

16 Chapter 1. Introduction 3 such systems. We compare analytically predicted performance to simulated performance over the binary erasure and binary symmetric channels and observe a close match between prediction and simulation. We derive a non-trivial upper-bound on the performance of SCSC codes, called the weight-pulling threshold, which allows for quick performance estimates during code design. We also study further generalizations such as mixture ensembles and decoding under beyond bounded-distance decoders, which can provide performance improvements at a cost of higher complexity. In Chp. 5, we answer Q5 by applying staircase codes to a soft-decision output channel in a concatenation of an inner low-density generator-matrix (LDGM) code and an outer staircase code. The LDGM decoding complexity, in terms of the number of edges and iterations, is taken into account during code design. Using the staircase codes designed in Chp. 3 as outer codes, we present concatenated schemes for several important OTN code rates. The schemes are shown to achieve the performance of best-known codes while requiring lower decoding complexity. 1.2 Notation In this section, we introduce the general notation used throughout this thesis. More notation will be introduced in each chapter, with their scope limited to only the chapter in which they appear. We denote the set of natural numbers including 0 by N, the set of real numbers by R, and the set of complex numbers by C. We denote the finite field containing q elements by F q. For any positive integer a, we denote the set {0, 1,..., a 1} by [a]. For any integer i, we denote the set {i, i + 1,..., i + a 1} by i + [a] or [a] + i. For example, we can convert zero-based indexing of n elements i [n] to one-based indexing by 1 + [n]. We use lower-case bold font, for example x, to denote a finite or countably infinite sequence of values. An element in the sequence is denoted by lower-case regular font

17 Chapter 1. Introduction 4 with the subscript i, for example x i. Vectors are denoted by lower-case bold font since they are finite-length sequences. The default vector is a row vector. Upper-case bold font is used to denote matrices, for example X. The n n identity matrix is denoted by I n. Similarly, the n n all-zeros matrix is denoted by 0 n. Vector and matrix transpose is denoted by ( ) T. Given a finite or countably infinite set of elements S, we denote the n-fold Cartesian product of S by S n {(x 0, x 1,..., x n 1 ) : x i S for all i [n]}. We define the indicator function I[P ] to indicate the truth of predicate P, taking value 1 when P is true, and taking value 0 when P is false. We use the summary notation v(i 0,..., i w 1 ) i k to denote the sum over index vectors (i 0,..., i w 1 ) of length w with the kth component fixed, i.e., v(i 0,..., i w 1 ). i 0 i k+1 i w 1 i k 1 We denote the probability of an event E by Pr(E) and the conditional probability of an event E given the event F has occurred by Pr(E F ). The probability mass (density) function of a discrete (continuous) random variable X is denoted by p X (x). The conditional probability mass (density) function of X conditioned on the random variable Y is denoted by p X Y (x y). For a discrete or continuous random variable X, we denote the expectation of a function g of X by E X [g(x)]. The conditional expectation of a function g of X conditioned on the realization y of the random variable Y is denoted by E X Y [g(x) y]. We denote a binomial probability mass function with parameters (n, p) for n 0, p [0, 1] as Bi[n, p](i) = ( ) n p i (1 p) n i I[i [n + 1]]. i

18 Chapter 1. Introduction 5 Similarly, we denote a Poisson probability mass function with mean α > 0 as Po[α](i) = e α αi I[i N]. i! To indicate that X is a Gaussian random variable with mean µ, variance σ 2, and probability density function p X (x) = ( ) 1 exp (x µ)2 2πσ 2 2σ 2 we write X N (µ, σ 2 ). We use the standard definition of the error function erf(x) 2 x e t2 dt π 0 and the complementary error function erfc(x) 1 erf(x). The inverse of the complementary error function is denoted by erfc 1 (x). The related Q-function is defined as Q(x) 1 2 erfc ( x 2 ), and its inverse is denoted by Q 1 (x). Note that Q 1 (x) = 2 erfc 1 (2x). (1.1)

19 Chapter 2 Background 2.1 System and Channel Models In this thesis, we will focus only on the binary symmetric channel (BSC) and the binaryinput additive white Gaussian noise channel (BIAWGNC), in order to make direct comparisons with prior works in fibre-optic FEC design. The BSC is defined by the input alphabet X = {0, 1}, output alphabet Y = {0, 1}, and channel transition probabilities P Y X (y x) = (1 p) I[x=y] p I[x y] for x X, y Y. Parameter p [ 0, 1 2) is the channel output bit-error probability (BER). Since the channel properties are fully described by p, we denote a BSC with parameter p by BSC(p). The BIAWGNC is defined by the input alphabet X = { 1, +1}, output alphabet Y = R and the channel transition probability densities p Y X (y x) = ( ) 1 exp (y x)2 for x X, y Y. 2πσ 2 2σ 2 Figure 2.1 shows a block diagram of the communication system under study. The 6

20 Chapter 2. Background 7 n Source m Encoder c Mapper x + y Slicer z Decoder ˆm Sink BIAWGNC/SD BSC/HD Figure 2.1: Block diagram of communication system and channel model. blocks with solid borders are necessary while those with dashed borders are optional and will be added or removed under different system settings. When optional blocks are excluded, any mappings within those blocks are assumed to be replaced by the identity map. The goal of the communication system is to reliably transmit a sequence of symbols taking values in F 2, referred to as bits, from the information source to the information sink. In the simplest system setup, a stream of source symbols m is mapped to the transmitted symbols x and corrupted by additive white Gaussian noise with distribution n i N (0, σ 2 ). Under Quadrature Phase-Shift Keying (QPSK) modulation, every pair of source bits: (m i, m i+1 ) for i even, is mapped to a transmitted symbol x i { E s,qpsk /2(±1, ±1)}. We assume that the mapping is Gray, i.e., nearest neighbours (in Euclidean distance) in the set of transmitted symbols only differ by 1 bit under the inverse map. The noise added to each transmitted symbol is 2-dimensional, zero-mean, with covariance matrix σ 2 I 2. We note that QPSK transmission with AWGN is widely used to model current fibre-optic communication systems. When E s,qpsk = 2, the above system is equivalent to a pair of Binary Phase-Shift Keying (BPSK) modulated systems with normalized average energy per transmitted symbol E s,bpsk = 1 (or simply E s = 1), transmitted under AWGN. Specifically, they are equal to the system which maps every source bit m i to a transmitted symbol x i { 1, 1} using

21 Chapter 2. Background 8 f : F 2 { 1, 1}, f(m) ( 1) I[m=1] and each received symbol is given by y i = x i + n i with n i N (0, σ 2 ). We refer to this system as uncoded soft-decision transmission, since no coding is used to transmit the source symbols m and the channel output symbols belong to R. Note that this system is also equivalent to a BIAWGNC. In Fig.2.1, the blocks involved in this system are surrounded by the dotted grey rectangle labelled BIAWGNC/SD. We can optionally apply a slicer, also referred to as a hard-decision, at the output of uncoded soft-decision transmission, defined as z = I[y < 0] (2.1) The overall system between m and z form the uncoded hard-decision transmission. This system is equivalent to a BSC(p) with p = Q(1/σ). (2.2) In Fig.2.1, the blocks involved in this system are surrounded by the dotted grey rectangle labelled BSC/HD. 2.2 Coding Gain and Channel Capacity The goal of the communication system is to transmit a sequence of N binary symbols over either the SD or HD channel while satisfying a reliability requirement in terms of the system output BER, defined as P e E[I[x i z i ]].

22 Chapter 2. Background 9 Standard long-haul OTN systems usually have a required output BER of P e [3, 9, 12]. Define the signal-to-noise ratio (SNR) with respect to the energy per transmitted symbol as snr E s /σ 2 = 1/σ 2. From (2.2), the SNR required to achieve P e < under uncoded SD transmission is given by snr [Q 1 (10 15 )] 2. We will also reference the SNR in decibels (db), defined and denoted by snr db 10 log 10 (snr) Rate and Overhead When a channel code, also referred to and used interchangeably in this thesis as a FEC code, FEC scheme, error-correcting code, or simply code, is used in addition to uncoded SD or HD transmission, the SNR required to achieve the same reliability constraint can be reduced. A channel code maps each block m of K source bits to a sequence c of N coded bits, c i F 2, called a codeword. The coded bits are input into uncoded transmission, as shown in Fig The mapping of information bits to coded bits is called encoding and is achieved by an encoder. At the output of uncoded SD or HD transmission, the received sequence of symbols y or z is input into a decoder, as shown in Fig. 2.1, which outputs a decoded sequence ˆm consisting of K decoded bits. The overall system with channel coding is referred to as coded SD or HD transmission. The two commonly used measures of channel code redundancy are the code rate (or rate) and overhead. The rate of a code is defined as R K N. (2.3)

23 Chapter 2. Background 10 It is the fraction of source bits transmitted by each channel use in a coded system. The overhead (OH) is defined as OH N K K = N K 1 = 1 1. (2.4) R The OH is most often presented as a percentage OH 100%. It is the additional redundancy (in bits) required to transmit each source bit in a coded system Net Coding Gain In order to make comparisons between the transmit energy required by the uncoded and coded systems, we account for the redundancy added by the channel code by defining the energy per information (source) bit E b E s /R = 1/R. Furthermore, we define the information bit SNR by snr b E b σ 2 = snr R = 1 Rσ 2. Operationally, snr b is the energy required to reliably communicate one information bit from the source, regardless of whether a channel code is used. Note that for the uncoded system R = 1, hence snr b = snr. For coded HD transmission, assume that we have a channel code with rate R that can achieve a decoder output BER of at a channel output (decoder input) BER of p. We define the net coding gain (NCG), in db, of the coded system by ( ) uncoded NCG HD (p snrb ) 10 log 10 coded snr b ( ) [Q = 10 log 10 [Q 1 (10 15 )] 2 1 (p )] 2 10 log 10 = 20 log 10 Q 1 (10 15 ) 20 log 10 Q 1 (p ) + 10 log 10 R. R Using (1.1) we obtain the commonly used expression for HD NCG in fibre-optic commu-

24 Chapter 2. Background 11 nications [9] NCG HD (p ) = 20 log 10 erfc 1 ( ) 20 log 10 erfc 1 (2p ) + 10 log 10 R. (2.5) For coded SD transmission, assume that we have a channel code with rate R that can achieve a decoder output BER of at snr. The SD NCG, in db, is defined as ( ) uncoded snrb NCG SD (snr) 10 log 10 coded snr b = 20 log 10 Q 1 (10 15 ) 10 log 10 (snr) + 10 log 10 R. Using (1.1) we obtain the expression for SD NCG NCG SD (snr) = 20 log 10 erfc 1 ( ) 10 log 10 (snr) + 10 log 10 2R. (2.6) Finally, we define an alternative measure of SNR, E s /N 0 10 log 10 (snr/2). The SD NCG with respect to E s /N 0 is given by NCG SD (E s /N 0 ) = 20 log 10 erfc 1 ( ) E s /N log 10 R. (2.7) In all NCG calculations, the required output BER (e.g., ) may be replaced by any other output BER requirement, e.g., or We refer to the NCG calculated at a given output BER requirement p o as the NCG at an output BER of p o Capacity Given any channel, the channel capacity or Shannon limit, C, is the theoretical upperbound on the code rate such that for all R < C, there exists a channel code satisfying the reliability requirement P e < ɛ for all ɛ > 0, as N. The capacity of a BSC(p) is

25 Chapter 2. Background 12 given by C HD 1 H(p) bits/channel use (2.8) where H(p) p log 2 p (1 p) log 2 (1 p). The capacity of an BIAWGNC can be obtained by estimating the mutual information expression, by Monte-Carlo sampling or numerical integration, between the random variables X { 1, +1} and Y R (see Sec. 2.1) C SD 1 E Y [ EX Y [log 2 p X Y (x y) y] ]. (2.9) In Fig. 2.2, we plot the NCG at the Shannon limit for the BSC (HD) and the BI- AWGNC (SD) at different values of E s /N 0. For the BSC, given E s /N 0 the BER at the output of the slicer is calculated as p and the rate at the Shannon limit is calculated by evaluating (2.8) at p. The NCG is then given by (2.5). For the BIAWGNC, given E s /N 0 the rate at the Shannon limit is calculated by evaluating (2.9) and the NCG is given by (2.7). In Fig. 2.3, we plot the NCG at the Shannon limit for the BSC (HD) and the BI- AWGNC (SD) at different overheads (or equivalently, code rates). For the BSC, the plot is given by the parametric curve (NCG HD (E s /N 0 ), C HD (p ) 1 1), where p is the BER at the output of the slicer for an SD channel with SNR E s /N 0. For the BIAWGNC, the plot is given by the parametric curve ( NCG SD (E s /N 0 ), C 1 SD 1), where C SD is the SD capacity of a BIAWGNC with SNR E s /N Prior Work In this section, we overview the most important prior work in the design and analysis of FEC schemes for fibre-optic communication systems. The discussion largely follows the historical developments in the field. It is brief on technical details since a thorough

26 Chapter 2. Background SD HD 14 Net Coding Gain (NCG) [db] E s /N 0 [db] Figure 2.2: Net coding gains at the Shannon limit vs. E s /N 0 (in db) for the BSC (HD) and the BIAWGNC (SD) Net Coding Gain (NCG) [db] SD HD Overhead [%] Figure 2.3: Net coding gain at the Shannon limit vs. overhead for the BSC (HD) and the BIAWGNC (SD).

27 Chapter 2. Background 14 overview of the technical background relevant to this thesis will be given in Sec We focus first on FEC schemes developed for coded hard-decision transmission, consisting of classical error-correction codes with efficient, algebraic decoding algorithms. We then review FEC schemes developed for coded soft-decision transmission, characterized by structures defined by sparse-graphs and decoded by message-passing algorithms HD FEC Schemes for OTN We use the notation (N, K) to denote a linear code with length N and dimension K. The first standardized (in ITU Recommendation G.975 [12]) error-correcting code for OTN was the (255, 239) Reed-Solomon (RS) code defined over 8-bit symbols, capable of correcting 8 symbol errors per coded block. Its NCG was 5.6 db at an output BER of Often referred to as the first generation of FEC for OTN [6, 13], it was chosen mainly for its efficient, algebraic encoding and decoding algorithms. The design and analysis of Reed-Solomon codes is a well-studied topic in coding theory and we refer the reader to the textbooks [14, 15] for more detail. The (255, 238) Reed-Solomon code was also designed to satisfy a framing structure defined by the ITU G.709 standard [3]. In this thesis, we shall refer to a frame defined by the G.709 standard as an OTN frame. Each OTN frame contains 16 interleaved coded blocks for a total of = bit symbols or bits. Any subsequent OTN FEC must also adhere to the framing standard and contain the same number of bits per OTN frame. This requirement largely accounts for the choice of parameters in many proposed coding schemes following the first Reed-Solomon code. In ITU Recommendation G [9], a number of enhanced FEC schemes were proposed. Often referred to as the second generation FEC schemes for long-haul OTN, they have the same code rate as the (255, 239) Reed-Solomon code. All except the I.6 code were based on the combination of several smaller algebraic codes into a more powerful code.

28 Chapter 2. Background 15 The I.3 code is a concatenation of inner (2040, 1930) Bose-Chaudhuri-Hocquenghem (BCH) codes 1 with outer (3860, 3824) BCH codes 2. In order to satisfy the structure of an OTN frame, during encoding 8 outer codes are combined (with added empty bits) to form a block of bits. After bit-interleaving, the empty bits are filled by encoding 16 blocks of the inner code. The I.3 code has an NCG of 8.99 db at an output BER of or 3.39 db higher than the (255, 239) RS code. The additional coding gain is obtained at the cost of greater decoding latency, due to concatenation and interleaving. The I.9 code consists of interleaved copies of the (1020, 988) doubly-extended BCH code 3. The BCH codes are divided into two categories: column codes and diagonal codes. A given number of BCH codes are arranged in columns and encodes over the information bits. The resulting coded bits, arranged in a rectangular array, are encoded by the diagonal codes along the diagonals of the array. The NCG of this code is 8.5 db at an output BER of With a 2.9 db gain over the (255, 239) RS code, the I.9 code has the best performance of all second generation FEC schemes. However, this is achieved with 10 iterations of decoding between the column and diagonal decoders. Table 2.1 gives a summary of the hard-decision decodable codes given in the Appendices of Recommendation G All of the codes have an overall block-length of bits and an overhead of 6.67%. All NCGs were calculated at an output BER of except for the I.9 code, which was based on an output BER of The I.3, I.4, and I.9 codes are related to a class of codes called product codes [14,16]. The performance of product codes in the asymptotic regime was studied in [17]. The analysis and design of product codes and related structures for fibre-optic communications was given in [18]. Iterative decoding of interleaved block codes for the BSC was analyzed in [19]. 1 For BCH codes, we denote the field extension degree by m, the unique decoding radius by t, and the number of shortened bits by s (see also Sec ). The inner BCH code has parameters: m = 11, t = 10, s = 7. 2 m = 12, t = 3, s = m = 10, t = 3, s = 7

29 Chapter 2. Background 16 Table 2.1: Hard-decision decodable codes for OTN from ITU-G Appendix Construction Outer Code Inner Code NCG (db) I.3 Concatenated BCH (3860, 3824) BCH (2040,1930) 8.99 I.4 Concatenated RS (1023, 1007) BCH (2047, 1952) 8.67 I.5 Concatenated RS (1901, 1855) Hamming Product Code 8.50 I.7 Single Code RS (2720, 2550) 8.00 I.9 Concatenated BCH (1020, 988) BCH (1020, 988) 8.67 The concept of combining short block codes into a larger code that can be decoded iteratively was also proposed by Tanner [20]. Instead of using a rectangular array, a sparse graph was used to combine the short block codes. This idea includes the class of lowdensity parity-check (LDPC) codes [14,21, 22], in which the block code is a single paritycheck code. The class of generalized LDPC (GLDPC) codes based on more powerful block codes was introduced in [23]. The performance of GLDPC codes based on BCH and RS codes over the BSC was studied in [24]. In [8], the authors introduced the class of staircase codes as an alternative to the second generation FECs for OTN. It was shown that a staircase code with rate R = 239/255 can achieve an NCG of 9.41 db at an output BER of This is an improvement of 0.42 db over the I.9 code. Staircase codes are decoded by iterative algebraic decoding of short block codes, with complexity similar to the iteratively-decoded second generation FECs. The structure of staircase codes is similar to the class of braided block codes (BBCs), originally designed for soft-decision channels and iterative message-passing decoding [25, 26]. A modified BBC was proposed for high bit-rate OTN in [27] SD FEC Schemes for OTN As shown in Figs. 2.2 and 2.3, significant coding gains can be obtained by using the soft-decision outputs from the BIAWGNC directly. The difference in capacity is more significant at lower code rates. Lower rate channel codes designed for coded soft-decision

30 Chapter 2. Background 17 transmission over high bit-rate OTN have been referred to as third generation FECs [6, 13]. This is a topic of current research and no codes have been standardized yet. Third generation FECs can be broadly categorized into: soft-decision decoding of product codes, LDPC codes, and spatially-coupled LDPC codes. The redundancy of these codes is typically higher, with particular focus on 12%, 20%, and 25% OH. A product code with soft-decision channel outputs may be decoded by an iterative soft-decision message-passing decoder. Such a code is referred to as a block turbo code [28] or turbo product code (TPC) [29,30]. TPCs not only take in soft-decision channel outputs, but also decode according to the extrinsic message-passing (EMP) principle [22,28]. They can achieve an NCG of 10.3 db at an output BER of [30]. LDPC codes are known to be universal codes, i.e., code designs with good performance for one particular channel are likely also good for another channel [22]. This property allows LDPC codes to be developed for both the BSC/HD and BIAWGNC/SD channels. A 20% OH LDPC code designed for OTN with a NCG of 11.3 db at an output BER of was given in [31]. In order to minimize complexity and latency, LDPC codes designed for OTN usually adopt a quasi-cyclic (QC) structure. An example of a 20% OH QC-LDPC code for OTN FEC was given in [32], with a NCG of 11.3 db at an output BER of Concatenation of LDPC and QC-LDPC codes with an outer algebraic block code can help achieve the stringent reliability constraint of long-haul OTN standards. Examples of such coding schemes have been proposed for OTN [33 35]. The most recent development in code design for soft-decision channels are spatiallycoupled (SC) LDPC codes [10, 11]. SC-LDPC codes promise capacity-approaching performance while using a relatively low complexity message-passing decoder. However, the structural complexity of SC-LDPC codes is much higher than standard LDPC codes. Several works have studied the use of SC-LDPC codes for OTN [36 38], with a best reported NCG of 12 db at an output BER of and 25.5% OH.

31 Chapter 2. Background Technical Background In this section, we provide the necessary technical background for the rest of the thesis. The material covered in Secs is standard, and can be found in any textbook on coding theory, e.g., [14, 15, 39]. Those familiar with coding theory may wish to skip directly to Sec , beginning at the introduction to staircase codes Fundamentals of Error-Correcting Codes An (N, K) binary linear code C is a K-dimensional sub-space of the vector space F N 2. The elements of C are called codewords. A code C is given as the row space of a K N matrix G, known as a generator matrix, i.e., C = {ug : u F K 2 }. A generator matrix is systematic if it has the form G = [I K P]. A binary linear code C is also given as the null space of an (N K) N matrix H known as the parity-check matrix, i.e., C = {v F K 2 : vh T = 0}. Given a systematic generator matrix, the corresponding parity-check matrix is given by H = [ P T I N K ]. The Hamming weight of a word x F N 2 is defined by w H (x) I[x i 0]. i [N] Given x, y F N 2, the Hamming distance between them is defined as d H (x, y) w H (x y).

32 Chapter 2. Background 19 The minimum distance of C is defined as d min d H(x, y). x,y C, x y The unique decoding radius of a code with minimum distance d is given as d 1 t. 2 At most one codeword lies at distance t or less from any vector in F N 2. Thus, if a code C with minimum distance d is used for coded transmission over a binary additive noise channel (such as the BSC) given by y = x + e with x C, e, y F N 2, then there exists a decoder that can correct all error patterns e with w H (e) t BCH Codes The binary BCH codes are an important class of binary, linear, block codes used in this thesis [14,15]. Given field extension degree m 3 and unique decoding radius t, primitive BCH codes have parameters given by N = 2 m 1, K N mt, d 2t + 1. We refer to a BCH code with unique decoding radius t as a t-error-correcting BCH code. Table 2.2 shows a short list of 2 to 4-error-correcting binary BCH codes. Note that K = N mt for all of the codes listed in Table 2.2. In fact, this is true for all BCH codes with m 6 and t 4 (See [14, Table 6.1]). Since we will be using BCH codes with parameters similar to those given in Table 2.2, we will assume that K = N mt in the rest of this thesis. Since N mt is a lower-bound on K, our assumption is pessimistic

33 Chapter 2. Background 20 Table 2.2: Binary BCH codes with m {8, 9, 10} and t {2, 3, 4}. m N K t in the code rate. The rate of a BCH code can be lowered by shortening, i.e., setting to zero and not transmitting s [K] information bits during encoding. The rate of a shortened BCH code is given by R s = K s N s. Decoding of binary BCH codes is usually achieved by using a syndrome decoder. Assume that an (N, K) BCH code C with a parity-check matrix H is used to transmit codewords x C over a BSC with output y = x + e for e F N 2. The decoder first calculates a syndrome yh T from y. An error-locator polynomial (ELP) is calculated from the syndrome. Error locations, as determined by the decoder, are obtained by finding the roots of the ELP. If w H (e) t, then the error locations determined by the decoder correspond to the actual error locations and decoding succeeds. The ELP can be calculated from the syndrome by direct computation, which can be efficiently implemented using look-up tables [8, Appendix A]. However, the size of the look-up tables increases quickly with t and this method is impractical for t 5. For larger t, the Berlekemp-Massey algorithm [14,15] can be used to calculate the ELP, with higher complexity. Roots of the ELP can also be efficiently found by direct solution for t 4 [40]. For larger t, an exhaustive search algorithm called the Chien search [14,15] can be used, with

34 Chapter 2. Background 21 higher latency. If w H (e) > t and there exists a codeword z C such that d H (z, y) t, then the syndrome decoder outputs the mis-corrected codeword z instead of the transmitted codeword x. In other words, the error locations determined by the decoder are incorrect and the decoder introduces additional errors into the received word. If w H (e) > t and there does not exist such a codeword, then a decoding failure is declared and the syndrome decoder outputs y. A decoding failure does not introduce any additional errors into the received word. Such a syndrome decoder is an instance of a bounded-distance decoder (BDD), defined as a decoding function D : F N 2 F N 2 such that z, d H (z, y) t for some z C D(y) = y, otherwise. Decoding of shortened BCH codes can be achieved by filling the shortened bitpositions with zeros and then applying the syndrome decoder for the non-shortened code Concatenated Codes Consider two error-correcting codes C o (n o, k o ) and C i (n i, k i ), defined over the same field. We denote code parameters by lower-case n and k when the code is an element or component of a larger code. The respective generator matrices are denoted by G o and G i. We define the concatenation of these codes by the following encoding procedure. First, encode an information word u o of length k o by c o = u o G o. Let π be a permutation over n o elements. The purpose of the permutation is to minimize any correlation between biterrors in the inner decoded words. Apply π to c o to obtain the permuted outer codeword ĉ o π(c o ). Assume that k i divides n o by the choice of code parameters. Let S = n o /k i. Partition the permuted outer codeword into S sub-blocks u i,j for j [S], each containing

35 Chapter 2. Background 22 C i Enc Channel C i Dec C o Enc π C i Enc Channel C i Dec π 1 C o Dec C i Enc Channel C i Dec Figure 2.4: Block diagram of a concatenated coding scheme with S = 3. k i bits, according to ĉ o = (u i,0 u i,1... u i,s 1 ). Each u i,j is encoded by the inner code using c i,j = u i,j G i. The final codeword is given by (c i,0 c i,1... c i,s 1 ). The rate of a concatenated code is given by R = k o Sn i = k ik o n i n o. Decoding of a concatenated code follows the exact reverse of the encoding process. A received word is partitioned into S blocks of n i bits. Each block is decoded by the inner code decoder. The S decoded words are combined into one large word consisting of Sk i bits. The inverse permutation of π is applied and the result is decoded by the outer decoder. Figure 2.4 illustrates the encoding and decoding of a concatenated code with S = Product Codes Consider again two error correcting codes C r (n r, k r ) and C c (n c, k c ) defined over the same field, with minimum distances d r, d c and canonical systematic generator matrices G r, G c, respectively. A codeword of the product code [14,16] is defined by the n c n r rectangular

36 Chapter 2. Background 23 A B k c C D n c k c k r n r k r Figure 2.5: Structure of a product code. array, as shown in Fig At the beginning of encoding, block A in Fig. 2.5 is filled with information bits. For each row of A, parity bits of C r are calculated using G r and used to populate each row of block B. For each column of the concatenated block [A B], parity bits of C c are calculated using G c and used to populate blocks C and D. The encoding process results in the same product code codeword if the columns were encoded first, followed by rows. The rate of the product code is given by R = k rk c n r n c. The minimum distance of the product code is given by d = d r d c [14, 16]. As a result of the row/column encoding process, each bit in the codeword is protected by one row block code and a column block code. Decoding of a received word proceeds by decoding each row in the array by using the decoder for C r, followed by decoding each column om the array by using the decoder for C c. Each complete pass through the array by the row and column decoders constitutes an iteration of product code decoding. The decoding process continues until a maximum number of iterations has been reached or if

37 Chapter 2. Background 24 n c M 1 P 1 M B 0 B 1 B 2 B 3 r c B 4 B 5. Figure 2.6: Staircase code block structure. Information bits (white) and parity bits (shaded) are shown. Bits in block B0 T are fixed to all-zeros. a codeword is found. Note that for a row/column permutation (also known as a matrix transpose) π and S = n r, a product code is a special case of a concatenated code with C o equivalent to n c copies of C r and C i = C c Staircase Codes A staircase code is a generalization of the product code, built from a binary linear block code C(n c, k c ), referred to as a component code. We denote the component code rate by R c k c /n c, its systematic generator matrix by G c, and its unique decoding radius by t c. We also denote the number of parity bits in a component node by r c n c k c. Let M n c /2. The staircase code structure consists of a semi-infinite chain of blocks B i, i N, as shown in Fig Each staircase block contains M 2 bits. Encoding begins by filling block B 0 with all-zeros. For i 1, first fill the M (M r c ) matrix M i with information bits. The matrix [Bi 1 T M i ] is formed and each row of the

38 Chapter 2. Background 25 matrix is encoded by the systematic generator matrix G c. The parity-bits from the encoding are stored in the M r c matrix P i. Finally, the staircase block B i is formed by the concatenation of matrices B i = [M i P i ]. The rate of a staircase code is given by R = 2R c 1. (2.10) Note that to have a positive staircase code rate, the component code must satisfy R c > 1/2. Decoding is performed by an iterative sliding window decoder containing L received blocks. Consider Fig. 2.6 now to be L = 6 received staircase blocks in the decoding window. At each iteration, for each i [L 1] the decoder forms the matrix [Bi T B i+1 ] and decodes each row of the matrix by using the component code decoder. Decoding continues until the maximum number of iterations, I, has been reached or no more corrections are made by any component decoder. The window then slides by shifting out the decoded block B 0 and shifting in the newly received block B L. The process continues indefinitely. The original staircase code [8] used a binary BCH code with t c = 3 as the component code. Decoding was achieved by sliding window decoding with syndrome decoding of component codes. Since t c = 3 is small, efficient table look-up methods were used to calculate the ELP and direct computation was used to find its roots. As a result, the sliding window decoder required at most t c M(L 1)I M(M r c ) = t ci(l 1) M r c (2.11) binary updates per information bit. Since not all component codes need to be updated at each iteration, the actual decoding data-flow is much lower than this upper-bound.

39 Chapter 2. Background The Differential Equation Method We now introduce a result from random graph theory with applications in the analysis of certain error-correcting codes defined by sparse graphs. The differential equation method (DEM) was developed by Wormald [41 43] to analyze random graph processes. It was introduced into coding theory for the analysis of random ensembles of irregular LDPC codes transmitted over the binary erasure channel [44, 45]. It has also been used to analyze the finite block-length performance of random ensembles of irregular LDPC codes [46] and spatially-coupled LDPC codes [47], and to analyze the performance of certain message-passing algorithms [48, 49]. In this section, we use the analysis of phase 0 of the min-degree graph process (Min- Deg), adapted from [50, Sec. 2.2], as a motivating example for the DEM and to help illustrate the technical details involved in its application. The MinDeg process was first described in [43] as a non-trivial random graph process for the same purpose. We use the standard notation K m to denote the complete graph over m vertices. Assign each vertex a unique label from [m]. At time n = 0, the MinDeg process is initialized to the empty graph Q (m) 0 = K m. For n > 0, the following random edgeaddition procedure is applied: 1. Choose an isolated vertex u in Q (m) n 1, uniformly at random from all isolated vertices. 2. Choose any vertex v u, uniformly at random. 3. Add the edge {uv} to the graph Q (m) n 1 and denote the new graph by Q (m) n. Phase 0 of MinDeg terminates when no isolated vertices remain. The time at which it terminates is called the stopping time of the process and denoted by n s. In this example, we set Q (m) n = Q (m) n s for all n > n s. At the end of phase i, the MinDeg process continues on to phase i + 1, where i + 1 is the minimum vertex degree in the graph at the end of phase i. In phase i+1, a vertex u of degree i + 1 is chosen at each time step and a vertex v is chosen so that no parallel edges