﻿ 基于BEC故障模型下的极化码SC译码算法研究
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 应用科技  2017, Vol. 44 Issue (6): 32-35  DOI: 10.11991/yykj.201608012 0

### 引用本文

LIU Tong, MENG Xiangyu. Research on polar codes SC decoding algorithm based on BEC fault model[J]. Applied Science and Technology, 2017, 44(6), 32-35. DOI: 10.11991/yykj.201608012.

### 文章历史

Research on polar codes SC decoding algorithm based on BEC fault model
LIU Tong, MENG Xiangyu
College of Information and Communication, Harbin Engineering University, Harbin150001, China
Abstract: In order to reduce the decoding errors caused by hardware factors about polar codes successive cancellation (SC) in the process of actual hardware implementation, a simple fault model based on the binary erasure channel (BEC) was built. In this model, it is impossible to achieve complete and reliable communications at any rate. For this fault module, a non-uniform error protection scheme was proposed, and the simulation shows that the scheme can be ignored in hardware cost, which significantly improves decoding performance of successive cancellation decoding by the fault binary erasure model.
Key words: polar codes    successive cancellation    decoding    hardware    binary erasure channel    non-uniform    error protection    fault model

1 极化码 1.1 极化码的构造

W表示一个二进制输入离散无记忆信道，输入 $u \in \left\{ {0,1} \right\}$ ，输出 $y \in \gamma$ ，信道转移概率表示为 $W\left( {y|u} \right)$ 。极化码的构造首先是N个独立的二进制离散信道W联合生成信道WN(其中N=2nn为大于0的整数)，然后再进行信道分裂，把WN分裂回N个二进制输入信道 $W_N^{\left( i \right)}$ (其中 $1 \leqslant i \leqslant N$ )，经过联合分裂操作后，子信道与原来信道相比，在总的信道容量不变情况下，子信道的信道容量会出现变化：一部分子信道的对称信道容量增大，另一部分子信道的对称信道容量减小。当N趋近于无穷时，一部分信道的容量将会趋近于1(通过该部分子信道传输的信息比特一定会被正确接收，即为无噪信道)；而其余信道将会趋近于0(这种信道不适合传递信息，即纯噪信道)，这一现象称作信道极化现象。在信道极化的基础之上，选择一部分信道容量趋近于1的信道去传输信息比特ui $i = 1,2, \cdots ,N$ ，而剩下的信道上用以传送收双方已知的冻结比特[10]

1.2 极化码连续删除译码算法

SC译码算法对于通信系统来说至关重要，直接影响着整个系统的通信质量。SC译码算法是Arikan最早提出来的译码算法，是极化码最为经典和最适合极化码特性的译码方法。

 $L(y_1^N,{u^\prime }_1^{i - 1}\left| {{u_i}} \right.) \buildrel \Delta \over = \displaystyle\frac{{W_N^{(i)}(y_1^N,{u^\prime }_1^{i - 1}\left| {{u_i}} \right. = 0)}}{{W_N^{(i)}(y_1^N,{u^\prime }_1^{i - 1}\left| {{u_i} = 1} \right.)}}$

 {u_i}^\prime = \left\{ \begin{aligned}\,\,& 0, \quad {\rm{ }}L(y_1^N,{u^\prime }_1^{i - 1}\left| {{u_i}} \right.) > 0\;\;{\rm{and}}\;i \in A\\\,\,&1, \quad {\rm{ }}L(y_1^N,{u^\prime }_1^{i - 1}\left| {{u_i}} \right.) > 0\;\;\;{\rm{and}}\;i \in A\\\,\,&{u_i}, \quad \quad \quad i \in {A^C}\end{aligned} \right.

 ${\rm{LLRs = }}\log \frac{{W\left( {{y_i}|{x_i} = 0} \right)}}{{W\left( {{y_i}|{x_i} = 1} \right)}}$

 $m = {m_1} + {( - 1)^{{u_s}^\prime }}{m_2}$

 $m = 2{\tanh ^{ - 1}}(\tanh ({m_1}/2)\tanh ({m_2}/2))$

2 故障SC译码 2.1 信息传递

 ${T^ + }(\varepsilon ) \buildrel \Delta \over = {\varepsilon ^2}$ (1)

 ${T^ - }(\varepsilon ) \buildrel \Delta \over = 1 - {(1 - \varepsilon )^2} = 2\varepsilon - {\varepsilon ^2}$ (2)

 ${\varepsilon _{j + 1}} = \left\{ \begin{array}{l}{T^ + }({\varepsilon _j})\;\;{\rm{w}}{\rm{.p}}{\rm{.}}\;\;1/2,\\{T^ - }({\varepsilon _j})\;\;{\rm{w}}{\rm{.p}}{\rm{.}}\;\;1/2.\end{array} \right.$

2.2 故障模型与信息传递

 $T_\delta ^ + (\varepsilon ){\rm{ = }}{\varepsilon ^2} + (1 - {\varepsilon ^2})\delta$ (3)

 $T_\delta ^ - (\varepsilon ){\rm{ = }}2\varepsilon - {\varepsilon ^2} + (1 - 2\varepsilon + {\varepsilon ^2})\delta$ (4)

 ${\varepsilon _{j + 1}} = \left\{ \begin{array}{l}T_\delta ^ + ({\varepsilon _j})\;\;{\rm{w}}{\rm{.p}}{\rm{.}}\;\;{\rm{1/2}}\\T_\delta ^ - ({\varepsilon _j})\;\;{\rm{w}}{\rm{.p}}{\rm{.}}\;\;{\rm{1/2}}{\rm{}}\end{array} \right.$

 $\begin{split}& {\rm E}({\varepsilon _{j + 1}}|{\varepsilon _j}){\rm{ }} = \frac{1}{2}(T_\delta ^ + ({\varepsilon _j}) + T_\delta ^ - ({\varepsilon _j})) = \\& \frac{1}{2}((1 - \varepsilon _j^2)\delta + 2{\varepsilon _j} + (1 - 2{\varepsilon _j} + \varepsilon _j^2)\delta ) = \\& {\varepsilon _j} + (1 - {\varepsilon _j})\delta \geqslant {\varepsilon _j}{\rm{ }}\end{split}$ (5)

3 非均匀错误保护方案

SC译码过程可以看成是基于深度优先的一种树搜索过程，每一层代表一位的译码估值，每个节点通过上层节点发送的信息计算当前节点的似然信息并发送到下层节点。硬件中SC译码器译码深度为n，包含n+1层译码节点，因此一个译码器共包含的译码节点为

 ${N_{PE}} = \sum\limits_{j = 0}^n {{2^{n - j}}} = {2^{n + 1}} - 1 = 2N - 1$

np表示被保护层的序号，nu表示未被保护的层号。我们假设np层经过保护处理后δ=0，Np表示被保护的节点总数，有

 {N_P} = \left\{ {\begin{aligned}& {\displaystyle\sum\nolimits_{j = 0}^{{n_p} - 1} {{2^j} = {2^{{n_p}}} - 1} ,\;\;\;{n_p}}{>0}\\& {0,\;\;\;{n_p} = 0}& {}\end{aligned}} \right.

 {\varepsilon _{j + 1}} = \left\{ \begin{aligned}& T_\delta ^ + ({\varepsilon _j}),\;\;{\rm{w}}{\rm{.p}}{\rm{.}}\;\;1/2,\;\;\;\;\;{\rm{ }}j{\rm{ }} = {\rm{ }}0,1, \cdots ,{n_u} - 1\\& T_\delta ^ - ({\varepsilon _j}),\;\;{\rm{w}}{\rm{.p}}{\rm{.}}\;\;1/2,\\& {T^ + }({\varepsilon _j}),\;\;{\rm{w}}{\rm{.p}}{\rm{.}}\;\;1/2,\;\;\;\;\;j{\rm{ }} = {\rm{ }}{n_u},\;\;{n_u} - 1, \cdots ,n\\& {T^ - }({\varepsilon _j}),\;\;{\rm{w}}{\rm{.p}}{\rm{.}}\;\;1/2,\end{aligned} \right.
4 仿真结果

N分别为1 024、2 048、4 096；信道擦除概率0.5；故障模型下加性擦除概率δ=10−6情况下，SC译码器的帧擦除率如图1所示。

 图 1 故障模型与无故障模型极化码的帧擦除率

 图 2 码率损失 $\Delta R(\delta ,\varepsilon ,{n_u})$

 图 3 SC译码在故障模型下的误帧率曲线

 图 4 故障模型与无故障模型误帧率
5 结论

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