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A Novel Design of Spatially Coupled LDPC Codes for Sliding Window Decoding

Min Zhu1, David G. M. Mitchell2, Michael Lentmaier3, and Daniel J. Costello, Jr.4

1State Key Lab. of ISN, Xidian University, Xi’an, China 2Klipsch School of Electrical and Computer Engineering, New Mexico State University, Las Cruces, NM, USA

• 3Dept. of Electrical and Information Technology, Lund University, Lund, Sweden
• 4Dept. 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

1

Introduction

2

Sliding Window Decoding (SWD) of Spatially Coupled LDPC (SC-LDPC) Codes

3

Analysis of Decoder Error Propagation in SWD

4

Check Node Doped SC-LDPC Codes

5

Numerical Results

6

Summary

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 ≥ 6η, 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 (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 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

M

Protograph

• c

v

n n

• “Lifting factor” M

1 protograph node = M Tanner graph 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 nvM = 2M 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 = 2M (w + 1).

time

• 2M

symbols

w → 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 nvM = 2M 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 = 2M (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 nvM = 2M 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 = 2M (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

1

If an erroneously decoded block contains a high number of incorrect LLRs with large magnitudes, this could trigger additional block errors, resulting in an error propagation effect, i. e., a continuous sequence of erroneously decoded blocks.

2

We assume that in any given frame of length L, the decoder operates in one of two states: a random error state Sre - block errors occur independently with probability p an error propagation state Sep - block errors occur with probability 1.

3

At each time unit t = 1, 2, 3, . . . , L, the decoder transitions from state Sre to state Sep independently with probability q (typically, q ≪ p), and once in state Sep, the decoder remains there for the rest of the frame.

re

S

ep

S 1 q

• q

1 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 = αL 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 Sep at time t = τ (and thus stays in state Sep until time t = Λ) is Pτ (Sep, t = [τ : Λ]) = q(1 − q)τ−1, τ = 1, 2, . . . , Λ, (1) where the notation t = [t1 : t2] denotes the set of time units from t1 to t2. Similarly, the probability that the decoder stays in state Sre throughout the entire frame is P (Sre, t = [1 : Λ]) = 1 −

Λ

• τ=1

Pτ (Sep, t = [τ : Λ]) = (1 − q)Λ . (2)

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 Sep at time t = τ, we can express the average BLER as PBL (τ = 1) = 1, (3a) PBL (τ) = [p · (τ − 2) + 0 · 1 + 1 · (Λ − τ + 1)]/Λ = [p · (τ − 2) + Λ − τ + 1]/Λ, τ = 2, . . . , Λ, (3b) where we note that state Sep must be preceded by at least one correctly decoded block. Finally, the overall average BLER is PBL =

Λ

• τ=1

PBL (τ) · Pτ (Sep, t = [τ : Λ]) + p · P (Sre, t = [1 : Λ]) =

Λ

• τ=1

PBL (τ) · q(1 − q)τ−1 + p · (1 − q)Λ . (4)

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 PBL = 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 PBL = 1.4982 · 10−2. For α = 10, i.e., we simulate Ω = N /10 frames, each of length Λ = 10L = 1000, we obtain PBL = 5.7939 · 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 PBL = 1.4982 · 10−2. For α = 10, i.e., we simulate Ω = N /10 frames, each of length Λ = 10L = 1000, we obtain PBL = 5.7939 · 10−2. For α = 100, i.e., we simulate Ω = N /100 frames, each of length Λ = 100L = 10, 000, we obtain PBL = 3.74244 · 10−1. From the example, we clearly see that, under error propagation conditions, the simulated BLER depends on the frame length Λ = αL, becoming worse as α increases.

Min Zhu (Xidian University) A Novel Design of SC-LDPC Codes for SWD 2020 ISIT 12 / 26

Analysis of Decoder Error Propagation

Example 2: The BLER performance

W = 18, M = 2000, L = 250, N = 20, 000, frame length Λ = αL, number of frames Ω = N /α, and frame length parameter α =1, 2, and 4, where we note that, in each case, the total number of simulated blocks is B = LN = ΛΩ = 5 × 106 and the total number of simulated code symbols is nvMB = 2 × 1010.

0.75 0.8 0.85 0.9 10-4 10-3 10-2 10-1 100

Figure 2: SWD BLER performance of a (3,6)-regular SC-LDPC code for three different frame lengths but the same total number of simulated blocks.

Min Zhu (Xidian University) A Novel Design of SC-LDPC Codes for SWD 2020 ISIT 13 / 26

Motivation for CN Doped Codes

Observation 1: The irregularities of the boundaries of a (J, K)-regular spatially coupled chain improve the performance of SC-LDPC codes (due to wave propagation) compared to the underlying (J, K)-regular LDPC block code. Observation 2: When the code is terminated, any error propagation will recover by itself, beginning about W /2 blocks from the end of the frame, due to the reduced degree CNs. Error Performance (20000 frames, L = 250, M = 2000, w = 2, W = 18) Index of error frame Number of error bits in a frame Start and end error blocks Original BER = 5.979 × 10−6 Frame 2673 6 Blocks 144-147 (3,6)-regular BLER = 8.22 × 10−5 Frame 5924 6 Blocks 144-147 SC-LDPC FER=4/20000 Frame 18545 53800 Blocks 60-238 code = 2 × 10−4 Frame 19646 65697 Blocks 15-238

Min Zhu (Xidian University) A Novel Design of SC-LDPC Codes for SWD 2020 ISIT 14 / 26

Construction of CN Doped Codes

Idea

Occasionally insert additional CNs into the protograph of a regular SC-LDPC code. This additional irregularity, referred to as check node (CN) doping, emulates termination but still allows continuous encoding and decoding. a

aThe basic concept of doping was first introduced by ten Brink in [4].

• 2M

symbols

time

• 1

t

2

t

The edges of the red VNs at time τj, representing the j th doping point, are connected to the CNs at times τj + j , τj + j + 1, and τj + j + 2.

[4] S. ten Brink,“Code doping for triggering iterative decoding convergence,”in Proc. IEEE Int. Symp. Information Theory, Washington, DC, USA, June 24-29, 2001, pp. 235. Min Zhu (Xidian University) A Novel Design of SC-LDPC Codes for SWD 2020 ISIT 15 / 26

Construction of CN Doped Codes

Idea

Occasionally insert additional CNs into the protograph of a regular SC-LDPC code. This additional irregularity, referred to as check node (CN) doping, emulates termination but still allows continuous encoding and decoding. a

aThe basic concept of doping was first introduced by ten Brink in [4].

• 2M

symbols

time

• 1

t

2

t

The edges of the red VNs at time τj, representing the j th doping point, are connected to the CNs at times τj + j , τj + j + 1, and τj + j + 2. The VNs between doping points (colored black) are coupled in the same way as the preceding red VN pair.

[4] S. ten Brink,“Code doping for triggering iterative decoding convergence,”in Proc. IEEE Int. Symp. Information Theory, Washington, DC, USA, June 24-29, 2001, pp. 235. Min Zhu (Xidian University) A Novel Design of SC-LDPC Codes for SWD 2020 ISIT 15 / 26

Construction of CN Doped Codes

The construction from the base matrix view:

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 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

t

• 2

t

• The construction has the effect of inserting an additional CN at each doping point,

resulting in three degree 4 CNs (colored green).

Min Zhu (Xidian University) A Novel Design of SC-LDPC Codes for SWD 2020 ISIT 16 / 26

Construction of CN Doped Codes

The construction from the base matrix view:

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 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

t

• 2

t

• The construction has the effect of inserting an additional CN at each doping point,

resulting in three degree 4 CNs (colored green). This creates stronger “local” codes, thus facilitating the ability of a SWD to truncate error propagation, at the cost of a small rate loss.

Min Zhu (Xidian University) A Novel Design of SC-LDPC Codes for SWD 2020 ISIT 16 / 26

Decoding of CN Doped Codes

• 2M symbols
• Target symbols

window size W window shifting

For a window of size W , the block of 2M symbols at the earliest time (leftmost position in the window) are the target symbols.

Min Zhu (Xidian University) A Novel Design of SC-LDPC Codes for SWD 2020 ISIT 17 / 26

Decoding of CN Doped Codes

• 2M symbols
• Target symbols

window size W window shifting

For a window of size W , the block of 2M symbols at the earliest time (leftmost position in the window) are the target symbols. Normally, after a block of target symbols is decoded, the window (VNs and CNs) shifts by one time unit.

Min Zhu (Xidian University) A Novel Design of SC-LDPC Codes for SWD 2020 ISIT 18 / 26

Decoding of CN Doped Codes

• 2M symbols
• Target symbols

window size W window size W window shifting

For a window of size W , the block of 2M symbols at the earliest time (leftmost position in the window) are the target symbols. Normally, after a block of target symbols is decoded, the window (VNs and CNs) shifts by one time unit. When a doping point (red VN pair) becomes the target block, the window shifts by

• ne VN time unit to include one new block of VNs and by two CN time units to

include two new blocks of CNs, after which normal window shifting resumes.

Min Zhu (Xidian University) A Novel Design of SC-LDPC Codes for SWD 2020 ISIT 19 / 26

Decoding of CN Doped Codes

The decoding from the base matrix view:

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 1

Shift by two CNs doping point VNs

Min Zhu (Xidian University) A Novel Design of SC-LDPC Codes for SWD 2020 ISIT 20 / 26

CN Doping Analysis

Notation and Assumption

Let λj , j ∈ [0, 1, . . . , l − 1] be the number of blocks in each sub-frame of the graph (between doping points), where l represents the number of sub-frames in a frame. If error propagation occurs in the ith sub-frame, the decoder state Sep will be truncated at the end of that sub-frame. Since there are λj blocks, j ∈ [0, 1, . . . , l − 1], in the j th sub-frame, we can write the average BLER of this sub-frame as PBL,λj =

λj

• τ=1

PBL,λj (τ) · q(1 − q)τ−1 + p(1 − q)λj , (5) where PBL,λj (τ) is the average BLER given that the j th sub-frame enters state Sep at time t = τ. Now using (3), we obtain PBL,λj (τ = 1) = 1, (6a) PBL,λj (τ) = [p · (τ − 2) + λj − τ + 1]/λj, τ = 2, . . . , λj . (6b)

Min Zhu (Xidian University) A Novel Design of SC-LDPC Codes for SWD 2020 ISIT 21 / 26

CN Doping Analysis

Finally, we can write the overall average BLER as PBL,doped =

l−1

• j=0

PBL,λj · λj Λ . (7) To determine the best doping points for a fixed number of sub-frames l, (7) can be treated as an optimization problem with respect to the λj parameters. Here we consider

• nly the important special case that the doped CNs are spaced uniformly in the coupling

chain (that is λj = λ, j ∈ [0, 1, . . . , l − 1]), in which case (7) can be written as PBL,doped = PBL,λj|λj =λ, j = 0, 1, . . . , l − 1, =

λ

• τ=1

PBL,λ (τ) · q(1 − q)τ−1 + p(1 − q)λ. (8)

Min Zhu (Xidian University) A Novel Design of SC-LDPC Codes for SWD 2020 ISIT 22 / 26

CN Doping Analysis

Example 3: Assume q = 10−4, p = 10−2, and Λ = 1000

For the original (3,6)-regular SC-LDPC codes, we obtain PBL = 5.7939 · 10−2 (same as case 2 in Example 1). If one doped CN is spaced in the middle of the chain, i.e., λ0 = λ1 = λ = 500, then according to (8) we obtain PBL,doped = 3.4391 · 10−2. If three doped CNs are spaced uniformly, i.e., λ0 = λ1 = λ2 = λ3 = λ = 250, then according to (8) we obtain PBL,doped = 2.2321 · 10−2.

Min Zhu (Xidian University) A Novel Design of SC-LDPC Codes for SWD 2020 ISIT 23 / 26

Numerical Results

50 100 150 200 250 10-4 10-3 10-2 10-1

Figure 3: Bit error rate distribution per block.

Min Zhu (Xidian University) A Novel Design of SC-LDPC Codes for SWD 2020 ISIT 24 / 26

Numerical Results

50 100 150 200 250 10-4 10-3 10-2 10-1

Figure 3: Bit error rate distribution per block. (3,6)-regular SC-LDPC code, M = 2000, L = 250, W = 18. We see that CN doping effectively truncates the error propagation and that adding more doped CNs (a slight additional rate loss) truncates the error propagation earlier.

Min Zhu (Xidian University) A Novel Design of SC-LDPC Codes for SWD 2020 ISIT 24 / 26

Numerical Results

0.8 0.82 0.84 0.86 0.88 0.9 10-7 10-6 10-5 10-4 10-3 10-2 10-1

Figure 4: Performance comparison of doped and undoped (3,6)-regular SC-LDPC codes with L = 500, W = 18, and W = 15.

Min Zhu (Xidian University) A Novel Design of SC-LDPC Codes for SWD 2020 ISIT 25 / 26

Numerical Results

0.8 0.82 0.84 0.86 0.88 0.9 10-7 10-6 10-5 10-4 10-3 10-2 10-1

Figure 4: Performance comparison of doped and undoped (3,6)-regular SC-LDPC codes with L = 500, W = 18, and W = 15. We note that the CN doped code gains up to two orders of magnitude in BER and more than

• ne order of magnitude in BLER

compared to the undoped code at SNR operating points of interest. Also, we see that the doped code with W = 15 provides almost an

• rder of magnitude coding gain

plus a 16% reduction in decoding latency compared to the undoped code with W = 18. At larger SNRs, where error propagation is no longer a problem, undoped codes are preferred, since decoding is a little simpler and there is no rate loss.

Min Zhu (Xidian University) A Novel Design of SC-LDPC Codes for SWD 2020 ISIT 25 / 26

Summary

We investigated the error propagation problem associated with SWD of SC-LDPC codes, which can cause significant performance degradation for streaming or large frame length applications, and we noted the difficulty of using conventional computer simulation methods to assess the severity of the problem.

Min Zhu (Xidian University) A Novel Design of SC-LDPC Codes for SWD 2020 ISIT 26 / 26

Summary

We investigated the error propagation problem associated with SWD of SC-LDPC codes, which can cause significant performance degradation for streaming or large frame length applications, and we noted the difficulty of using conventional computer simulation methods to assess the severity of the problem. We presented the concept of check node doped (J,K)-regular SC-LDPC codes to limit the effects of error propagation in low latency SWD.

Min Zhu (Xidian University) A Novel Design of SC-LDPC Codes for SWD 2020 ISIT 26 / 26

Summary

We investigated the error propagation problem associated with SWD of SC-LDPC codes, which can cause significant performance degradation for streaming or large frame length applications, and we noted the difficulty of using conventional computer simulation methods to assess the severity of the problem. We presented the concept of check node doped (J,K)-regular SC-LDPC codes to limit the effects of error propagation in low latency SWD. The novel code design takes advantage of a structured irregularity in the code graph introduced by occasionally adding (doping) check nodes that allow the SWD to recover from error propagation.

Min Zhu (Xidian University) A Novel Design of SC-LDPC Codes for SWD 2020 ISIT 26 / 26

Summary

We investigated the error propagation problem associated with SWD of SC-LDPC codes, which can cause significant performance degradation for streaming or large frame length applications, and we noted the difficulty of using conventional computer simulation methods to assess the severity of the problem. We presented the concept of check node doped (J,K)-regular SC-LDPC codes to limit the effects of error propagation in low latency SWD. The novel code design takes advantage of a structured irregularity in the code graph introduced by occasionally adding (doping) check nodes that allow the SWD to recover from error propagation. Simulation results were included to illustrate the beneficial effects of the new code design.

Min Zhu (Xidian University) A Novel Design of SC-LDPC Codes for SWD 2020 ISIT 26 / 26