# Design and Analysis of Joint Group Shuffled Scheduling Decoding Algorithm for Double LDPC Codes System

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Related Work and Motivation of Shuffled Scheduling Decoding

#### 1.2. Related Work on D-LDPC Codes Systems

#### 1.3. Main Contribution

- (1)
- From a global viewpoint, the D-LDPC codes structure is considered as a whole, and a joint shuffled scheduling decoding strategy is introduced to the D-LDPC codes system.
- (2)
- A grouping method for the joint shuffled scheduling decoding strategy, which relies on the types or the length of the VNs, is introduced.
- (3)
- A novel EXIT algorithm to calculate the channel and source decoding thresholds for the general D-LDPC coding structure with the JGSSD algorithm is proposed.
- (4)
- A comparison between the SSSD algorithm and the JGSSD algorithm is conducted, including decoding performance, decoding complexity and decoding latency.

#### 1.4. Paper Organization

## 2. Preliminaries of D-LDPC Codes Systems

#### 2.1. The D-LDPC Coding Structure

#### 2.2. Transmission System Model

- (1)
- Update all the C2V messages for each of the ${M}_{c}$ CNs in the channel part;
- (2)
- Update all the V2C messages for each of the ${N}_{s}$ VNs in the source part;
- (3)
- Update all the C2V messages for each of the ${M}_{s}$ CNs in the source part;
- (4)
- Update all the V2C messages for each of the ${N}_{c}$ VNs in the channel part;
- (5)
- The source part and channel part exchange decoding information through ${\mathit{H}}_{l1}$ and ${\mathit{H}}_{l2}$ (i.e., the dashed blue and red lines in Figure 1);
- (6)
- Estimate the codeword $\widehat{\mathit{u}}$ based on the posterior LLRs at the VNs;
- (7)
- Repeat Steps (1) to (6), unless (i) the estimated codeword $\widehat{\mathit{s}}$ and $\widehat{\mathit{d}}$ satisfy $\widehat{\mathit{s}}{\left({\mathit{H}}_{s}\right)}^{T}=0$ and $\widehat{\mathit{d}}{\left({\mathit{H}}_{c}\right)}^{T}=0$ (ii) the maximum iteration number K is reached.

## 3. Joint Group Shuffled Scheduling Decoding Algorithm

- ${z}_{n}^{s}$ represents the LLR of the n-th bit of original source $\mathit{s}$.
- ${z}_{n}^{d}$ represents the LLR of the n-th bit of codeword $\mathit{d}$.
- ${\epsilon}_{mn}^{k}$ represents the LLR from the m-th CN to the n-th VN at k-th iteration.
- ${\varphi}_{mn}^{k}$ represents the LLR from the n-th VN to the m-th CN at k-th iteration.
- ${\Phi}_{n}^{k}$ represents the LLRs of the n-th bit at k-th iteration.

**Initialization:**The initial LLR of VNs can be calculated by

**Step 1:**Update C2V messages, for $1\le n\le {N}_{s}+{N}_{c}$ and $1\le m\le {M}_{s}+{M}_{c}$,

**Step 2:**Update V2C messages, for $1\le n\le {N}_{s}+{N}_{c}$ and $1\le m\le {M}_{s}+{M}_{c}$,

**Step 3:**Stopping condition: if $\widehat{\mathit{u}}\xb7{\mathit{H}}_{J}=0$ or $k={K}_{max}$, the iteration will stop; otherwise, set $k=k+1$ and go to Step 1, where ${K}_{max}$ is the maximum iteration of decoding.

## 4. Analysis of the D-LDPC Codes System with JGSSD Algorithm

#### 4.1. Joint Shuffled Extrinsic Information Algorithm

- ${I}_{ij,\left(k\right)}^{Ev}$: the extrinsic MI from j-th VN to i-th CN at k-th iteration;
- ${I}_{ij,\left(k\right)}^{Ec}$: the extrinsic MI from i-th CN to j-th VN at k-th iteration;
- ${I}_{ij,\left(k\right)}^{Av}$: the a priori MI from j-th VN to i-th CN at k-th iteration;
- ${I}_{ij,\left(k\right)}^{Ac}$: the a priori MI from i-th CN to j-th VN at k-th iteration;
- ${I}_{j,\left(k\right)}^{APP}$: the MI between a posteriori LLR evaluated by j-th VN and the corresponding source bit ${s}_{j}$ at k-th iteration.

**Step 1: The MI update from VNs to CNs**

**Step 2: The MI update from CNs to VNs**

**Step 3: The APP-LLR MI evaluation**

**Remarks**: If the maximum iteration is set to a large value, like the conventional EXIT algorithm, then the JSEXIT algorithm cannot reflect its advantage of shuffled scheduling, as with the larger iteration number in shuffled scheduling decoding, which has similar performance to that of the standard BP algorithm. We set the maximum iteration to 20 here for this reason. Therefore, the decoding threshold has a gap compared with that of the conventional EXIT algorithm, but it can provide comparable results.

#### 4.2. Decoding Threshold Calculation

## 5. Simulation and Comparison

#### 5.1. BER Performance

#### 5.2. Decoding Complexity and Latency

## 6. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Abbreviations

AWGN | additive white Gaussian noise |

APP | a posteriori probability |

BER | bit error rate |

BP | belief propagation |

CNs | check nodes |

C2V | check-to-variable |

D-LDPC | double low-density parity-check |

EXIT | extrinsic information transfer |

IBP | iterative belief propagation |

IoT | Internet of things |

JBP | joint belief propagation |

JEXIT | joint extrinsic information transfer |

JPEXIT | joint protograph EXIT |

JGSSD | joint group shuffled scheduling decoding |

JSCC | joint source-channel coding |

JSEXIT | joint shuffled EXIT |

LLR | log-likelihood-ratio |

MI | mutual information |

PEG | progressive edge growth |

SPEXIT | source protograph EXIT |

SSSD | separated shuffled scheduling decoding |

SWD | sliding window decoding |

V2C | variable-to-check |

VNs | variable nodes |

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**Figure 2.**BER performance of ${\mathit{B}}_{J1}$ with ${G}^{H}=1,2,8,400,3200,6400$ at source statistic $\eta =0.04$.

**Figure 3.**BER performance of ${\mathit{B}}_{J1}$ with ${G}^{H}=1,8,6400$ at source statistic $\eta =0.07$.

**Figure 4.**BER performance of ${\mathit{B}}_{J2}$ with ${G}^{H}=1,2,8,400,3200,6400$ at source statistic $\eta =0.07$.

**Figure 5.**BER performance of ${\mathit{B}}_{J2}$ with ${G}^{H}=1,8,6400$ at source statistic $\eta =0.11$.

**Figure 6.**Average iteration number ${K}_{avg}$ of ${\mathit{B}}_{J1}$ with ${G}^{H}=1,2,8,400,3200,6400$ at source statistic $\eta =0.04$.

IBP (e.g., [2]) | SSSD [7,8] | JGSSD | |
---|---|---|---|

Decoding order | Parallel | Serial | Parallel and Sequential |

EXIT algorithm | ∖ | only for algorithm SSSD with the case ${\mathit{B}}_{l2}=0$ | both for the SSSD and JGSSD with the case ${\mathit{B}}_{l2}=0$ and ${\mathit{B}}_{l2}\ne 0$ |

Complexity | High | Low | Adaptive |

Latency | Low | High | Adaptive |

Adaptive judgement | No | No | Yes |

**Table 2.**Source decoding thresholds ${\eta}_{th}$ and channel decoding thresholds for ${({E}_{s}/{N}_{0})}_{th}$ at $\eta =0.04$ and $0.07$ for ${\mathit{B}}_{J1}$ with different grouping strategies.

${\mathit{B}}_{\mathit{J}1}$ | ${\mathit{\eta}}_{\mathit{th}}$ | ${({\mathit{E}}_{\mathit{s}}/{\mathit{N}}_{0})}_{\mathit{th}}$ | |
---|---|---|---|

0.04 | 0.07 | ||

${G}^{B}=1$ | 0.072 | $-3.13$ dB | $-1.39$ dB |

${G}^{B}=2$ | 0.073 | $-3.27$ dB | $-1.41$ dB |

${G}^{B}=4$ | 0.077 | $-3.46$ dB | $-2.00$ dB |

${G}^{B}=8$ | 0.078 | $-3.54$ dB | $-2.19$ dB |

${G}^{B}=16$ | 0.079 | $-3.58$ dB | $-2.25$ dB |

**Table 3.**Source decoding thresholds ${\eta}_{th}$ and channel decoding thresholds for ${({E}_{s}/{N}_{0})}_{th}$ at $\eta =0.07$ and $0.11$ for ${\mathit{B}}_{J2}$ with different grouping strategies.

${\mathit{B}}_{\mathit{J}2}$ | ${\mathit{\eta}}_{\mathit{th}}$ | ${({\mathit{E}}_{\mathit{s}}/{\mathit{N}}_{0})}_{\mathit{th}}$ | |
---|---|---|---|

0.07 | 0.11 | ||

${G}^{B}=1$ | 0.118 | $-1.67$ dB | $0.65$ dB |

${G}^{B}=2$ | 0.119 | $-1.77$ dB | $0.55$ dB |

${G}^{B}=4$ | 0.125 | $-1.98$ dB | $-0.27$ dB |

${G}^{B}=8$ | 0.128 | $-2.07$ dB | $-0.50$ dB |

${G}^{B}=16$ | 0.129 | $-2.10$ dB | $-0.62$ dB |

**Table 4.**${E}_{b}/{N}_{0}$ Gain at BER = $1\times {10}^{-6}$ and error floor levels for different grouping strategies.

Grouping Strategy | ${\mathit{E}}_{\mathit{b}}/{\mathit{N}}_{0}$ Gain at BER = $1\times {10}^{-6}$ | Error Floor Level | ||
---|---|---|---|---|

$\mathit{\eta}=\mathit{0}.\mathit{04},{\mathit{B}}_{\mathit{J}\mathit{1}}$ | $\mathit{\eta}=\mathit{0}.\mathit{07},{\mathit{B}}_{\mathit{J}\mathit{2}}$ | $\mathit{\eta}=\mathit{0}.\mathit{07},{\mathit{B}}_{\mathit{J}\mathit{1}}$ | $\mathit{\eta}=\mathit{0}.\mathit{11},{\mathit{B}}_{\mathit{J}\mathit{2}}$ | |

${G}^{H}=1$ | 0 | 0 | $7\times {10}^{-4}$ | $2\times {10}^{-4}$ |

${G}^{H}=2$ | 0.05 dB | 0.01 dB | - | - |

${G}^{H}=8$ | 0.08 dB | 0.12 dB | $5\times {10}^{-4}$ | $1\times {10}^{-4}$ |

${G}^{H}=400$ | 0.18 dB | 0.18 dB | - | - |

${G}^{H}=3200$ | 0.19 dB | 0.22 dB | - | - |

${G}^{H}=6400$ | 0.19 dB | 0.22 dB | $3\times {10}^{-4}$ | $9\times {10}^{-5}$ |

**Table 5.**The decoding complexity (average iteration number ${K}_{avg}$) and the decoding latency (average decoding time ${T}_{avg}$) for different grouping strategies of ${\mathit{B}}_{J1}$ at $\eta =0.04$ and ${E}_{s}/{N}_{0}=-2.4$ dB.

Grouping Strategy | ${\mathit{K}}_{\mathit{avg}}$ | ${\mathit{T}}_{\mathit{avg}}$ |
---|---|---|

${G}^{H}=1$ | $9.9$ | $9.9{t}_{uni}$ |

${G}^{H}=2$ | $9.7$ | $19.4{t}_{uni}$ |

${G}^{H}=8$ | $6.5$ | $52{t}_{uni}$ |

${G}^{H}=400$ | $5.9$ | $2360{t}_{uni}$ |

${G}^{H}=3200$ | $5.1$ | $\mathrm{16,320}{t}_{uni}$ |

${G}^{H}=6400$ | $5.1$ | $\mathrm{32,640}{t}_{uni}$ |

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## Share and Cite

**MDPI and ACS Style**

Chen, Q.; Ren, Y.; Zhou, L.; Chen, C.; Liu, S.
Design and Analysis of Joint Group Shuffled Scheduling Decoding Algorithm for Double LDPC Codes System. *Entropy* **2023**, *25*, 357.
https://doi.org/10.3390/e25020357

**AMA Style**

Chen Q, Ren Y, Zhou L, Chen C, Liu S.
Design and Analysis of Joint Group Shuffled Scheduling Decoding Algorithm for Double LDPC Codes System. *Entropy*. 2023; 25(2):357.
https://doi.org/10.3390/e25020357

**Chicago/Turabian Style**

Chen, Qiwang, Yanzhao Ren, Lin Zhou, Chen Chen, and Sanya Liu.
2023. "Design and Analysis of Joint Group Shuffled Scheduling Decoding Algorithm for Double LDPC Codes System" *Entropy* 25, no. 2: 357.
https://doi.org/10.3390/e25020357