A Novel Design of Spatially Coupled LDPC Codes for Sliding Window - PowerPoint PPT Presentation

A Novel Design of Spatially Coupled LDPC Codes for Sliding Window Decoding Min Zhu 1 , David G. M. Mitchell 2 , Michael Lentmaier 3 , and Daniel J. Costello, Jr. 4 1 State Key Lab. of ISN, Xidian University, Xi’an, China 2 Klipsch School of Electrical and Computer Engineering, New Mexico State University, Las Cruces, NM, USA 3 Dept. of Electrical and Information Technology, Lund University, Lund, Sweden 4 Dept. of Electrical Engineering, University of Notre Dame, Notre Dame, IN, USA ISIT2020, Los Angeles, California, USA, June 21-26 Min Zhu (Xidian University) A Novel Design of SC-LDPC Codes for SWD 2020 ISIT 1 / 26

Outline Introduction 1 Sliding Window Decoding (SWD) of Spatially Coupled LDPC (SC-LDPC) Codes 2 Analysis of Decoder Error Propagation in SWD 3 Check Node Doped SC-LDPC Codes 4 Numerical Results 5 Summary 6 Min Zhu (Xidian University) A Novel Design of SC-LDPC Codes for SWD 2020 ISIT 2 / 26

Introduction Streaming applications, such as interactive audio/video conferencing, mobile gaming, cloud computing, and so on, need reliable and low-latency error control coding strategies suitable for continuous transmission. Convolutional codes lend themselves naturally to a streaming environment: – low latency operation can be achieved using Viterbi decoding and sequential decoding, but coding gains are limited. Based on their capacity-approaching performance, spatially coupled low-density parity-check (SC-LDPC) codes with sliding window decoding (SWD) are desirable for streaming or large frame length applications. For SWD of SC-LDPC codes, good performance can typically be maintained as long as the window size W ≥ 6 η , where η represents the decoding constraint length [1]. [1] K. Huang, D. G. M. Mitchell, L. Wei, X. Ma, and D. J. Costello,“Performance comparison of LDPC block and spatially coupled codes over GF (q),” IEEE Transactions on Communications, vol. 63, no. 3, pp. 592-604, Mar. 2015. Min Zhu (Xidian University) A Novel Design of SC-LDPC Codes for SWD 2020 ISIT 3 / 26

Introduction However, for smaller values of W (reduced decoding latency), infrequent but severe decoder error propagation sometimes occurs. Error Propagation: following a decoding error, the decoding of the subsequent symbols is affected, which in turn causes a continuous string of errors. Related works: → Klaiber et al. proposed adapting the number of decoder iterations and shifting the window position to combat the error propagation [2]. → Zhu et al. proposed error propagation mitigation techniques for braided convolutional codes with SWD [3]. Here we introduce a novel design of SC-LDPC codes that reduces the effects of error propagation in SWD for large frame length or streaming applications. [2] K. Klaiber, S. Cammerer, L. Schmalen, and S. t. Brink, “Avoiding burstlike error patterns in windowed decoding of spatially coupled LDPC codes,”in Proc. IEEE 10th Int. Symp. on Turbo Codes & Iterative Inf. Processing (ISTC), Hong Kong, China, 2018, pp. 1-5. [3] M. Zhu, D. G. M. Mitchell, M. Lentmaier, D. J. Costello, Jr., and B. Bai, “Combating error propagation in window decoding of braided convolutional codes,”in Proc. IEEE Int. Symp. Information Theory (ISIT), Vail, CO, USA, June 17-22, 2018, pp. 1380-1384. Min Zhu (Xidian University) A Novel Design of SC-LDPC Codes for SWD 2020 ISIT 4 / 26

� � � � � � � � � � � � � LDPC Codes Based on Protographs Parity-check matrix Protograph (3,4)-regular 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 n n 1 1 1 1 c v 1 1 1 1 1 1 1 1 1 1 1 1 “Lifting factor” M 1 1 1 1 1 protograph node = M Tanner graph M nodes 1 protograph edge = M Tanner graph edges Min Zhu (Xidian University) A Novel Design of SC-LDPC Codes for SWD 2020 ISIT 5 / 26

��� Convolutional Protographs Consider the transmission of independent (3,6)-regular blocks over time, where each block contains n v M = 2 M code symbols. Spatial coupling: blocks are connected by spreading edges to their nearest w neighbors (introducing memory into the encoding process), where w is the coupling width and the decoding constraint length is η = 2 M ( w + 1) . w 2 M time symbols → The edge spreading introduces a structured irregularity at the end of the graph, which triggers decoding wave propagation , resulting in threshold saturation. Min Zhu (Xidian University) A Novel Design of SC-LDPC Codes for SWD 2020 ISIT 6 / 26

��� Convolutional Protographs Consider the transmission of independent (3,6)-regular blocks over time, where each block contains n v M = 2 M code symbols. Spatial coupling: blocks are connected by spreading edges to their nearest w neighbors (introducing memory into the encoding process), where w is the coupling width and the decoding constraint length is η = 2 M ( w + 1) . target symbols W → Decoding of a target block is jointly carried out over a window of size W blocks. Min Zhu (Xidian University) A Novel Design of SC-LDPC Codes for SWD 2020 ISIT 7 / 26

��� Convolutional Protographs Consider the transmission of independent (3,6)-regular blocks over time, where each block contains n v M = 2 M code symbols. Spatial coupling: blocks are connected by spreading edges to their nearest w neighbors (introducing memory into the encoding process), where w is the coupling width and the decoding constraint length is η = 2 M ( w + 1) . W → Then the window shifts by one block to decode the next target block. Min Zhu (Xidian University) A Novel Design of SC-LDPC Codes for SWD 2020 ISIT 8 / 26

Analysis of Decoder Error Propagation If an erroneously decoded block contains a high number of incorrect LLRs with 1 large magnitudes, this could trigger additional block errors, resulting in an error propagation effect, i. e., a continuous sequence of erroneously decoded blocks. We assume that in any given frame of length L , the decoder operates in one of two 2 states: a random error state S re - block errors occur independently with probability p an error propagation state S ep - block errors occur with probability 1. At each time unit t = 1 , 2 , 3 , . . . , L , the decoder transitions from state S re to state 3 S ep independently with probability q (typically, q ≪ p ), and once in state S ep , the decoder remains there for the rest of the frame. - 1 q 1 S S re ep q Figure 1: The state diagram describing the operation of an SWD subject to decoder error propagation. Min Zhu (Xidian University) A Novel Design of SC-LDPC Codes for SWD 2020 ISIT 9 / 26

Analysis of Decoder Error Propagation Simulation Experiment Consider the simulation of a large number N of frames, each of length L , such that ∆ the total number of simulated blocks is B = LN . Now let Λ = α L and Ω = N /α , so that B = LN = ΛΩ , where the frame length parameter α ≥ 1 corresponds to simulating a smaller number Ω ≤ N of frames of length Λ ≥ L , resulting in the same total number of simulated blocks B . The probability that the decoder first enters state S ep at time t = τ (and thus stays in state S ep until time t = Λ ) is P τ ( S ep , t = [ τ : Λ]) = q (1 − q ) τ − 1 , τ = 1 , 2 , . . . , Λ , (1) where the notation t = [ t 1 : t 2 ] denotes the set of time units from t 1 to t 2 . Similarly, the probability that the decoder stays in state S re throughout the entire frame is Λ � P ( S re , t = [1 : Λ]) = 1 − P τ ( S ep , t = [ τ : Λ]) (2) τ =1 = (1 − q ) Λ . Min Zhu (Xidian University) A Novel Design of SC-LDPC Codes for SWD 2020 ISIT 10 / 26

Analysis of Decoder Error Propagation Now, given that a frame enters state S ep at time t = τ , we can express the average BLER as P BL ( τ = 1) = 1 , (3a) P BL ( τ ) = [ p · ( τ − 2) + 0 · 1 + 1 · (Λ − τ + 1)]/Λ (3b) = [ p · ( τ − 2) + Λ − τ + 1]/Λ , τ = 2 , . . . , Λ , where we note that state S ep must be preceded by at least one correctly decoded block. Finally, the overall average BLER is Λ � P BL = P BL ( τ ) · P τ ( S ep , t = [ τ : Λ]) τ =1 + p · P ( S re , t = [1 : Λ]) (4) Λ P BL ( τ ) · q (1 − q ) τ − 1 + p · (1 − q ) Λ . � = τ =1 Min Zhu (Xidian University) A Novel Design of SC-LDPC Codes for SWD 2020 ISIT 11 / 26

Analysis of Decoder Error Propagation Example 1: Assume q = 10 − 4 , p = 10 − 2 , and L = 100 For α = 1 , i.e., we simulate Ω = N frames, each of length Λ = L = 100 . Using (3) and (4) we obtain P BL = 1 . 4982 · 10 − 2 . Min Zhu (Xidian University) A Novel Design of SC-LDPC Codes for SWD 2020 ISIT 12 / 26

Analysis of Decoder Error Propagation Example 1: Assume q = 10 − 4 , p = 10 − 2 , and L = 100 For α = 1 , i.e., we simulate Ω = N frames, each of length Λ = L = 100 . Using (3) and (4) we obtain P BL = 1 . 4982 · 10 − 2 . For α = 10 , i.e., we simulate Ω = N / 10 frames, each of length Λ = 10 L = 1000 , we obtain P BL = 5 . 7939 · 10 − 2 . Min Zhu (Xidian University) A Novel Design of SC-LDPC Codes for SWD 2020 ISIT 12 / 26