# Improving Polar-Coded SCMA System by Information Coupling and Parity Check

^{*}

## Abstract

**:**

## 1. Introduction

- The uplink information-coupled PC-SCMA system is proposed. At the transmitter, channel coding, SCMA mapping, and transmission are all performed based on CB. When the entire TB is received, the receiver performs iterative detection and decoding.
- The PCCA-polar code in system form is designed. Then, every two consecutive systematic PCCA-polar code blocks are connected by information coupling technology to form a new type of PIC polar code. For windowed decoder, the list algorithm based on parity check can provide more accurate extrinsic messages for the coupling CBs. In addition, the windowed decoding algorithm is improved for the requirement of iterative system.
- An extrinsic messages construction algorithm of coupled polar decoder is proposed. This algorithm enables the coupled polar decoder to exchange extrinsic messages with the MPA detector, and achieving iterative detection and decoding.
- A joint iterative detection and SCL decoding algorithm (JIDS) based on variable list size is designed. During the iteration, those CBs who failed CRC verification improved the performance in the next iteration by increasing the list size, while others maintained a smaller list size. In addition, when all CBs are correctly decoded, the iteration is stopped immediately.
- The extrinsic information transfer (EXIT) idea is used to optimize the weight factor of extrinsic messages construction algorithm and coupling ratio of PIC PCCA-polar code.

## 2. System Model

**F**. As shown in Figure 2, a SCMA factor graph with six variable nodes (VNs) and four function nodes (FNs) is illustrated, where VNs and FNs represent the users and resources, respectively. The corresponding mapping matrix is denoted as

## 3. Design of Partially Information-Coupled Polar Code based on Parity Check

#### 3.1. Systematic PCCA-Polar Code

**Theorem**

**1.**

**Proof.**

**Theorem**

**2.**

**Proof.**

Algorithm 1 Get parity check indices ${\mathcal{P}}_{c}$ |

Input: ${\mathcal{A}}^{\prime}$ and ${n}_{pc}$. |

Output: ${\mathcal{P}}_{c}$. |

1: Initialize: ${\mathcal{P}}_{c}=\xd8$. |

2: Generate submatrix ${G}_{{\mathcal{A}}^{\prime}{\mathcal{A}}^{\prime}}$ and calculate the row’s weight ${W}_{t}=\left\{{W}_{t}^{1},\dots ,{W}_{t}^{\left|{\mathcal{A}}^{\prime}\right|}\right\}$ of ${G}_{{\mathcal{A}}^{\prime}{\mathcal{A}}^{\prime}}$. |

3: ${w}_{t}=1$. |

4: while $\left|{\mathcal{P}}_{c}\right|<{n}_{pc}$ do |

5: $\mathcal{S}=\xd8$. |

6: for $i=1\to \left|{\mathcal{A}}^{\prime}\right|$ do |

7: if ${W}_{t}^{i}={w}_{t}$ then |

8: Get ${1}^{i}$ of ${g}_{i}$. |

9: ${\mathcal{S}}^{\prime}=\left|{1}^{i}\cap \mathcal{S}\right|$. |

10: if $\left|{1}^{i}\right|-\left|{\mathcal{S}}^{\prime}\right|=1$ then |

11: update the set $\mathcal{S}=\mathcal{S}\cup \left\{{{\mathcal{A}}^{\prime}}_{i}\right\}$. |

12: end if |

13: end if |

14: end for |

15: if ${n}_{pc}-\left|{\mathcal{P}}_{c}\cup \mathcal{S}\right|<0$ then |

16: Select ${n}_{pc}-\left|{\mathcal{P}}_{c}\right|$ elements with the lowest reliability of $\mathcal{S}$ as set ${\mathcal{S}}^{\u2033}$. |

17: update the set ${\mathcal{P}}_{c}={\mathcal{P}}_{c}\cup {\mathcal{S}}^{\u2033}$. |

18: else |

19: update the set ${\mathcal{P}}_{c}={\mathcal{P}}_{c}\cup \mathcal{S}$. |

20: end if |

21: ${w}_{t}={w}_{t}+1$. |

22: end while |

Algorithm 2 Systematic PCCA-polar encoding |

Input: $b$, ${N}_{m}$, ${\mathcal{A}}^{\prime}$, ${n}_{pc}$ and rate-matching pattern $\mathcal{E}$, where $\left|\mathcal{E}\right|={N}_{e}$. |

Output: ${c}^{\prime}$. |

1: Initialize: Set a p-length cyclic shift register $y\left[0\right],\dots ,y\left[p-1\right]$ to 0, and $a=0$, where $\left|a\right|={N}_{m}$. |

2: Generate the generator matrix $G$. |

3: Get the information and CRC indices $\mathcal{A}=\left\{{\mathcal{A}}^{\prime}\backslash {\mathcal{P}}_{c}\right\}$, and generate the submatrix ${G}_{\mathcal{A}\mathcal{A}}$. |

4: Calculate ${b}^{\prime}=b{G}_{\mathcal{A}\mathcal{A}}$. |

5: for $i=1\to {N}_{m}$do //PC bits generation [14] |

6: Cyclic left shift the register. |

7: If $i\in \mathcal{A}$: set $y\left[0\right]={a}_{i}\oplus y\left[0\right]$. |

8: If $i\in {\mathcal{P}}_{c}$: set ${a}_{i}=y\left[0\right]$. |

9: end for |

10: Perform the basic polar coding to obtain mother code by $x=aG$. |

11: Get systematic PCCA-polar codes ${c}^{\prime}$ by rate-matching pattern $\mathcal{E}$. |

#### 3.2. Partially Information-Coupled PCCA-Polar Code

## 4. Joint Iterative Detection and SCL Decoding Receiver

- ${M}_{{g}_{i}\to {q}_{j}}$: The messages passing from the i-th function node ${g}_{i}$ to the j-th variable node ${q}_{j}$.
- ${M}_{{q}_{j}\to {g}_{i}}$: The messages passing from the j-th variable node ${q}_{j}$ to the i-th function node ${g}_{i}$.
- ${\mathcal{Q}}_{j}$: The set of function nodes that connect to variable node ${q}_{j}$.
- ${\mathcal{G}}_{i}$: The set of variable nodes that connect to function node ${g}_{i}$.
- $\left\{{\mathcal{Q}}_{j}\backslash i\right\}$: Excluding function node ${g}_{i}$ from the set of ${\mathcal{Q}}_{j}$.
- $\left\{{\mathcal{G}}_{i}\backslash j\right\}$: Excluding variable node ${q}_{j}$ from the set of ${\mathcal{G}}_{i}$.
- $P\left({s}_{v}^{l}\right)$: The prior information of the l-th SCMA codeword for user v.

#### 4.1. Update Function Nodes of MPA Detector

#### 4.2. Updated a Priori Information of MPA Detector

#### 4.3. Update Variable Nodes of MPA Decoder

Algorithm 3 Joint iterative detection and SCL decoding |

Input: Signal $\left\{{y}_{1},\dots ,{y}_{l},\dots ,{y}_{L}\right\}$ from the channel, maximum number of iterations ${I}_{\mathrm{max}}$, windowed size ${L}_{w}$, list size ${L}_{\mathrm{min}}$ and ${L}_{\mathrm{max}}$. |

Output: The decoding decision $\widehat{U}=\left\{{\widehat{u}}_{1},{\widehat{u}}_{2},\cdots ,{\widehat{u}}_{V}\right\}$. |

1: Initialize: ${M}_{{q}_{j}\to {g}_{i}}=1/M$ and ${L}_{v,k}={L}_{\mathrm{min}}$, $1\le v\le V$, $1\le k\le {L}_{c}$. |

2: for iter_num = $1\to {I}_{\mathrm{max}}$ do |

3: for $l=1\to L$ do //Update the function nods |

4: for $v=1\to V$ do |

5: Calculate the extrinsic messages ${L}_{e,\mathrm{MPA}}^{v}$ by (4) to (6). |

6: end for |

7: end for |

8: for $v=1\to V$ do //Update the Priori Information |

9: De-interleave the ${L}_{e,\mathrm{MPA}}^{v}$ to ${z}_{v}$, and input to the coupled polar decoder. |

10: Run the decoder, calculate the extrinsic messages ${L}_{e}^{v}$ by HC algorithm. |

11: for $k=1\to {L}_{c}$ do |

12: if CRC check fails then |

13: Modify the list size by (22). |

14: end if |

15: end for |

16: Interleave the ${L}_{e}^{v}$ to ${L}_{a}^{v}$. |

17: end for |

18: for $l=1\to L$ do |

19: for $v=1\to V$ do |

20: Transform the priori information ${L}_{a}^{v}$ into the probability domain by (20). |

21: Update the information of variable nodes by (21). |

22: end for |

23: end for |

24: If all users decode correctly, exit the iteration. |

25: end for |

## 5. Optimization of Partially Information-Coupled Polar Code Based on EXIT Idea

#### 5.1. Transfer Characteristics of Base Polar Decoder

#### 5.2. Optimization of Coupling Ratio $\delta $ for PIC PCCA-Polar Code

## 6. Performance Evaluation

#### 6.1. Block Error Rate Performance Comparison

^{−3}as shown in Figure 13a. In the early stage of each CB decoding, parity-check bits are used to correct errors and save the correct decoding path as far as possible. Therefore, PIC PCCA-polar code also achieved an additional 0.22 dB coding gain compared to PIC CA-polar code over SCMA system. Similarly, for the code rate-1/3, PIC PCCA-PC-SCMA can also obtain corresponding gain as shown in Figure 13b. However, at the same time, the performance of uncoupled CA-PC-SCMA based on SCAN decoding is the worst among other systems. Since error blocks cannot provide perfect extrinsic messages to adjacent blocks in a TB, adjacent CB can only be decoded at a higher code rate. In particular, in the lower ${E}_{b}/{N}_{0}$ range, a wrong CB often causes continuous CB errors in a same TB, in other words, these errors are aggregated. Thus, with SCL decoding (for CB), the CBER performance of PIC PC-SCMA systems is worse than that of uncoupled PC-SCMA system. Similarly, we can obtain the same results from Figure 14.

^{−3}and TBER = 10

^{−2}.

#### 6.2. Complexity Analysis

## 7. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

**Proof**

**of**

**Theorem**

**1.**

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**Figure 4.**The effect of ${\mathcal{P}}_{c}$ on transformation ${x}_{{\mathcal{A}}^{\prime}}={a}_{{\mathcal{A}}^{\prime}}{G}_{{\mathcal{A}}^{\prime}{\mathcal{A}}^{\prime}}$.

**Figure 9.**Performance of systematic polar code with SCAN decoder. (

**a**) Transfer characteristics; (

**b**) code block error rate (CBER) of PC-SCMA over additive white Gaussian noise (AWGN) channel.

**Figure 10.**Transfer characteristics of systematic polar code with successive cancellation list (SCL) decoder.

**Figure 11.**Transfer characteristics of PIC PCCA-polar code with different $\delta $. (

**a**) K = 100, R = 1/2; (

**b**) K = 64, R = 1/3.

**Figure 15.**TB error rate (TBER) of SCMA over AWGN channel. (

**a**) K = 100, R = 1/2; (

**b**) K = 64, R = 1/3.

Abbreviation | Full Name | Abbreviation | Full Name |
---|---|---|---|

5G | Fifth generation | AWGN | Additive white Gaussian noise |

BC | Bayes construction | BLER | Block error rate |

BMC | Binary-input memoryless channel | BP | Belief propagation |

CA | CRC-aided | CB | Code block |

CBER | Code block error rate | CRC | cyclic redundancy check |

DCA | Distributed CRC-aided | eMBB | enhanced Mobile BroadBand |

EXIT | Extrinsic information transfer | FN | Function node |

HC | Hybrid construction | IDD | Iterative detection and decoding |

JIDD | Joint iterative detection and decoding | JIDS | Joint iterative detection and SCL decoding |

JSC | Joint successive cancellation | LDPC | Low-density parity-check |

LDPC-SCMA | LDPC coded sparse code multiple access | LDS | Low-density signatures |

LLR | Log-likelihood rate ratio | log-BP | log-Belief propagation |

mMTC | massive Machine Type Communications | MPA | Message passing algorithm |

MUD | Multi-user detection | MUSA | Multi-user shared access |

NOMA | Non-orthogonal multiple access | NR | New radio |

OMA | Orthogonal multiple access | PCCA | Joint parity check and CRC aided |

PC-SCMA | Polar-coded sparse code multiple access | PDMA | Pattern division multiple access |

PE | Processing element | PIC | Partially information-coupled |

QoS | Quality of service | SC | Successive cancellation |

SCAN | Soft cancellation | SCL | Successive cancellation list |

SCMA | Sparse code multiple access | SISO | Soft-input-soft-output |

TB | Transport block | TBER | TB error rate |

TC | Transfer characteristic | URLLC | Ultra-Reliable and Low Latency Communications |

VN | Variable node |

Parameters | Polar | LDPC | ||||
---|---|---|---|---|---|---|

PIC CA | CA (SCL) | CA (SCAN) [17] | ||||

AWGN | K = 100 R = 1/2 | CBER | 0.22 | 0.5 | 1 | 2 |

TBER | 0.22 | 0.52 | 1.03 | 2.02 | ||

K = 64 R = 1/3 | CBER | 0.22 | 0.7 | 2.09 | 1.49 | |

TBER | 0.25 | 0.75 | 2.12 | 1.52 | ||

Rayleigh | K = 100 R = 1/2 | CBER | 0.23 | 0.5 | 1.91 | 2.57 |

TBER | 0.25 | 0.52 | 1.93 | 2.59 | ||

K = 64 R = 1/3 | CBER | 0.24 | 0.92 | 2.84 | 2 | |

TBER | 0.27 | 0.95 | 2.88 | 2.04 |

**Table 3.**The average number of iterations $\overline{I}$ and the average list size $\overline{L}$ over AWGN channel.

${\mathit{E}}_{\mathit{b}}/{\mathit{N}}_{0}$ (dB) | PIC PCCA | PIC CA | CA (SCL) | CA (SCAN) [17] | |
---|---|---|---|---|---|

2.5 | $\overline{I}$ | 5 | 5 | 5 | 5 |

$\overline{L}$ | 17.83 | 18.27 | 17.78 | 1 | |

3 | $\overline{I}$ | 5 | 5 | 5 | 5 |

$\overline{L}$ | 16.2 | 17.89 | 14.94 | 1 | |

3.5 | $\overline{I}$ | 4.79 | 5 | 4.96 | 5 |

$\overline{L}$ | 11.74 | 14.42 | 11.35 | 1 | |

4 | $\overline{I}$ | 3.85 | 4.39 | 4.42 | 5 |

$\overline{L}$ | 8.08 | 10.1 | 8.45 | 1 | |

4.5 | $\overline{I}$ | 2.91 | 3.34 | 3.55 | 4.95 |

$\overline{L}$ | 6.24 | 7.17 | 6.8 | 1 |

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Wu, X.; Wang, Y. Improving Polar-Coded SCMA System by Information Coupling and Parity Check. *Sensors* **2020**, *20*, 6740.
https://doi.org/10.3390/s20236740

**AMA Style**

Wu X, Wang Y. Improving Polar-Coded SCMA System by Information Coupling and Parity Check. *Sensors*. 2020; 20(23):6740.
https://doi.org/10.3390/s20236740

**Chicago/Turabian Style**

Wu, Xi, and Yafeng Wang. 2020. "Improving Polar-Coded SCMA System by Information Coupling and Parity Check" *Sensors* 20, no. 23: 6740.
https://doi.org/10.3390/s20236740