面向不等差错保护的低误码平台LT编码算法

doi:10.11959/j.issn.1000-436x.2022123

通信学报 ›› 2022, Vol. 43 ›› Issue (6): 85-97.doi: 10.11959/j.issn.1000-436x.2022123

面向不等差错保护的低误码平台LT编码算法

宋鑫¹, 倪淑燕², 张喆³, 廖育荣², 雷拓峰¹

¹ 航天工程大学研究生院，北京 101416
² 航天工程大学电子与光学工程系，北京 101416
³ 北京遥感信息研究所，北京 100192

修回日期:2022-05-09 出版日期:2022-06-01 发布日期:2022-06-01
作者简介:宋鑫（1995- ），男，山西临汾人，航天工程大学博士生，主要研究方向为信道编码、信道均衡、稀疏码分多址技术等
倪淑燕（1981- ），女，河北清河人，航天工程大学副教授，主要研究方向为阵列信号处理、正交频分复用技术等
张喆（1993- ），男，河南南阳人，北京遥感信息研究所助理研究员，主要研究方向为卫星信号处理、OFDM 技术、信道编码等
廖育荣（1972- ），男，四川德阳人，航天工程大学研究员，主要研究方向为目标探测识别理论与技术、航天测控技术等
雷拓峰（1998- ），男，陕西西安人，航天工程大学博士生，主要研究方向为正交频分复用技术、稀疏码分多址技术、信道编码等
基金资助:
国家自然科学基金资助项目(61805283);国家高技术研究发展计划基金资助项目(7026085)

Low error floor LT coding algorithm for unequal error protection

Xin SONG¹, Shuyan NI², Zhe ZHANG³, Yurong LIAO², Tuofeng LEI¹

¹ Department of Graduate Management, Space Engineering University, Beijing 101416, China
² Department of Electronic and Optical Engineering, Space Engineering University, Beijing 101416, China
³ Beijing Institute of Remote Sensing Information, Beijing 100192, China

Revised:2022-05-09 Online:2022-06-01 Published:2022-06-01
Supported by:
The National Natural Science Foundation of China(61805283);The National High Technology Research and Development Program of China(7026085)

摘要/Abstract

摘要：

目的：无速率LT码旨在为大规模数据分发和可靠广播提供一种理想的传输协议。无速率LT码具有三个优良的特点：（1）链路自适应；（2）码率值可无隙切换；（3）反馈方式较为简单。在无线数据传输中有一类重要的应用场景，即不等差错保护（UEP）数据传输。作为第一种可实现的无速率码，无速率LT码可以便捷地与UEP算法联合设计以实现自适应数据传输。但是，传统的UEP-LT码算法在加性白高斯噪声（AWGN）信道中存在的高误码平台、收敛速度慢等问题。为此，本文设计了一种改进的系统UEP-LT编码方案。

方法：本文考虑面向AWGN信道设计独立的系统UEP-LT码，给出了与该方案相匹配的校验度分布设计方法，并提出了一种具有分段特点的编码方案。该方案中设计了与消息节点一一相连的系统节点，用以提供来自信道的非零对数似然比（LLR）信息。之后是固定长度的校验节点，即固定段，该段校验节点只连接到重要比特（MIB），其目的是使MIB最接近译码成功状态。最后是无速率编码段，该段中的校验节点会选取MIB或者次要比特（LIB）作为邻居节点，并且连接至MIB和LIB的校验节点占比可以灵活调整，从而使MIB始终优先于LIB被成功译码。本文还提出了与上述编码方案相适配的度分布设计模型。该设计模型旨在为MIB提供足够宽的外信息译码通道，为LIB提供一个开放且不太窄的译码通道。在只传输固定段时，待设计的度分布应使得MIB最接近于成功译码状态。在开始传输无速率段时，待设计的度分布应确保MIB尽快地被正确恢复，且在LIB处于临界译码状态时，MIB的译码通道足够宽。在上述约束条件下，所设计的校验度分布可以为MIB提供优于LIB的收敛性能。

结果：以码长K等于6000时为例进行仿真，结果显示：（1）信噪比较低时，本文方案的MIB具有最低的误码平台。例如，当码率倒数R^-1=2.05时，与参考方案相比，本文方案MIB的误比特率（BER）降低了近一个数量级，最低可达10^-7量级。此外，本文方案还具有最优的收敛性能，即能够以较小的编码开销进入BER瀑布区。以10^-7作为BER衡量标准，则本文方案可节省的开销至少为码长K的10%，这体现了本文方案的优势。（2）信噪比较高时，无论是MIB还是LIB，本文方案的BER性能是最优的。对于MIB而言，本文方案的BER始终低于参考方案一个数量级以上。若考虑以10^-6为BER标准，本文方案可节省的开销约为码长K的7%；不过，此时的信噪比较高，若进一步考虑以10^-7为BER标准，则本文方案可节省的开销约为码长K的15%。这意味着当信噪比增加时，本文方案的性能提升幅度要高于参考方案。

结论：本文设计了一种系统不等差错保护LT编码方案，并构建了适用于该方案的校验度分布设计模型，以解决传统UEP-LT算法在AWGN信道中存在的高误码平台的问题。该模型的主要思想是设计固定编码段、无速率编码段以及系统节点段，并将它们按顺序传输，优势在于可为消息节点尽可能早地提供非零的LLR信息，并使MIB和LIB获得不同的且可灵活调整的平均节点度数值。此外，基于外信息传递（EXIT）图法为固定段和无速率段设计了校验度分布，使MIB获得最优的保护性能，同时尽可能地提高LIB的收敛性能。在后续工作中，可以考虑设计能够更加逼近信道容量的校验度分布模型，从而在给定BER标准时实现进一步提升方案编码效率的效果。

关键词: 信道编码, 无速率码, 不等差错保护, 收敛性, 误比特率

Abstract:

Objectives:Rateless LT code is designed to provide an ideal transport protocol for large-scale data distribution and reliable broadcasting.The rateless LT code has three excellent characteristics,namely link adaptation,the code rate can be switched seamlessly, and the feedback method is relatively simple. There is an important application scenario in wireless data transmission, namely unequal error protection (UEP) data transmission. As the first achievable rateless code,the rateless LT code can be conveniently used in conjunction with the UEP algorithm to realize adaptive data transmission.However,the conventional UEP-LT code algorithm has problems such as high error floor and poor convergence performance in the additive white Gaussian noise(AWGN)channel.Therefore,an improved systematic UEP-LT coding scheme is designed in this paper.

Methods: This paper considers designing independent systematic UEP-LT codes for AWGN channels.A design method of check distribution matching this scheme is given,and a coding scheme characterized by segmentation is proposed.In this scheme,systematic nodes connected with information nodes one by one are designed to provide non-zero log-likelihood ratio (LLR) information from the channel.After that, there is a fixed number of check nodes,that is,the fixed segment.The check node of this segment is only connected to the important bit(MIB),and its purpose is to make the MIB the closest to the successful decoding state.The final part of the coding scheme is the rateless coding segment. The check nodes in this segment will select MIB or least important bit (LIB) as neighbor nodes,and the proportion of check nodes connected to MIB and LIB can be flexibly adjusted,so that the MIB is always successfully decoded before the LIB.This paper also proposes a degree distribution design model adapted to the above coding scheme.This design model aims to provide a sufficiently wide extrinsic information decoding tunnel for the MIB,and an open and not too narrow decoding tunnel for the LIB.When only the fixed segment is transmitted, the degree distribution to be designed should bring the MIB closest to the successful decoding state.When starting to transmit the rateless segment,the degree distribution to be designed should ensure that the MIB is recovered correctly as soon as possible,and the decoding tunnel of the MIB is wide enough when the LIB is in a critical decoding state. Under the above constraints, the designed check degree distribution can provide the MIB with better convergence performance than the LIB.

Results:Taking the code length K equal to 6000 as an example to simulate,and the results are as follows.(i)When the signal-to-noise ratio(SNR) is low,the MIB in this scheme has the lowest error floor.For example, when the reciprocal code rate ^-1=2.05, compared with the reference scheme, the bit error rate (BER) of the MIB proposed in this paper is reduced by nearly an order of magnitude,and the lowest value can reach the order of 10^-7. In addition,the scheme in this paper also has the optimal convergence performance,that is,it can enter the BER waterfall region with a small coding overhead. Taking 10^-6as the BER standard, the overhead saved by the proposed scheme is at least 10% of the code length K,which reflects the advantages of the proposed scheme.(ii) When the SNR is high,whether it is the MIB or the LIB,the BER performance of the proposed scheme is optimal. For the MIB,the BER of the proposed scheme is always more than an order of magnitude lower than the reference scheme. If 10^-6is considered as the BER standard, the overhead saved by this scheme is about 7% of the code length K.However,if 10^-7is further considered as the BER standard,the overhead saved by this scheme is about 15% of the code length K.This means that when the SNR increases,the performance improvement of the proposed scheme is higher than that of the reference scheme.

Conclusions: In this paper, a systematic unequal error protection LT coding scheme is designed, and a check degree distribution design model suitable for this scheme is constructed to solve the problem of high error floor existing in the conventional UEP-LT algorithm in the AWGN channel.The main idea of this scheme is to design fixed coding segments,rateless coding segments and systematic node segments and transmit them in sequence.The advantage of this scheme is that it can provide non-zero LLR information as early as possible for information nodes, and make the MIB and the LIB obtain different and flexibly adjustable average degrees.In addition,based on the extrinsic information transfer(EXIT)chart,the check degree distribution is designed for the fixed segment and the rateless segment,so that the MIB can obtain the optimal protection performance,and the convergence performance of the LIB can be improved as much as possible. In the follow-up work, it can be considered to design a check degree distribution model that can more closely approximate the channel capacity, so as to further improve the coding efficiency of the scheme when the BER standard is given.

Key words: channel coding, rateless code, unequal error protection, convergence, bit error rate

中图分类号:

TN911.22

宋鑫, 倪淑燕, 张喆, 廖育荣, 雷拓峰. 面向不等差错保护的低误码平台LT编码算法[J]. 通信学报, 2022, 43(6): 85-97.

Xin SONG, Shuyan NI, Zhe ZHANG, Yurong LIAO, Tuofeng LEI. Low error floor LT coding algorithm for unequal error protection[J]. Journal on Communications, 2022, 43(6): 85-97.

图/表 13

图1

图2

表1

不同信噪比下的信道容量值"

信噪比/dB	$σ_{n}$	$C_{σ_{n}}$
-2.82	0.978 3	0.500 0
-1.53	0.843 3	0.600 5
-0.27	0.729 4	0.700 4
1.07	0.625 2	0.799 4
2.73	0.516 4	0.900 0

表1

图3

图4

图5

表2

校验度分布的设计结果"

期望参数	UEP参数	固定段和无速率段的校验度分布	度分布模型目标函数值
$\begin{array}{l} σ_{n} = 0.843 3 ， \\ α_{G} = 7.3 ， \\ α_{F} = 13.5 \end{array}$	Γ_M =0.25,Γ_L=0.75	$\begin{array}{l} Ω_{G} (x) = 0.238 8 x^{7} + 0.707 2 x^{8} + 0.054 0 x^{66} \\ Ω_{F} (x) = 0.850 8 x^{7} + 0.035 4 x^{14} + 0.019 2 x^{15} + 0.094 6 x^{66} \end{array}$	η=0.020 19
	Γ _M =0.3,Γ_L=0.7	$\begin{array}{l} Ω_{G} (x) = 0.238 8 x^{7} + 0.707 2 x^{8} + 0.054 0 x^{66} \\ Ω_{F} (x) = 0.010 1 x^{6} + 0.846 1 x^{7} + 0.032 0 x^{13} + 0.028 9 x^{14} + 0.082 9 x^{66} \end{array}$	η=0.021 43
	Γ_M =0.35,Γ_L=0.65	$\begin{array}{l} Ω_{G} (x) = 0.238 8 x^{7} + 0.707 2 x^{8} + 0.054 0 x^{66} \\ Ω_{F} (x) = 0.854 2 x^{7} + 0.026 0 x^{8} + 0.049 1 x^{14} + 0.070 7 x^{66} \end{array}$	η=0.022 47
$\begin{array}{l} σ_{n} = 0.729 4 ， \\ α_{G} = 5.3, \\ α_{F} = 12.5 \end{array}$	Γ_M =0.25,Γ_L=0.75	$\begin{array}{l} Ω_{G} (x) = 0.736 8 x^{10} + 0.225 7 x^{11} + 0.037 5 x^{66} \\ Ω_{F} (x) = 0.376 7 x^{9} + 0.485 1 x^{10} + 0.138 2 x^{66} \end{array}$	η=0.027 17
	Γ _M =0.3,Γ_L=0.7	$\begin{array}{l} Ω_{G} (x) = 0.736 8 x^{10} + 0.225 7 x^{11} + 0.037 5 x^{66} \\ Ω_{F} (x) = 0.034 9 x^{8} + 0.837 6 x^{9} + 0.127 5 x^{66} \end{array}$	η=0.028 71
	Γ_M =0.35,Γ_L=0.65	$\begin{array}{l} Ω_{G} (x) = 0.736 8 x^{10} + 0.225 7 x^{11} + 0.037 5 x^{66} \\ Ω_{F} (x) = 0.708 2 x^{9} + 0.182 2 x^{10} + 0.109 6 x^{66} \end{array}$	η=0.030 85
$\begin{array}{l} σ_{n} = 0.625 2 ， \\ α_{G} = 4.3, \\ α_{F} = 11.5 \end{array}$	Γ_M =0.25,Γ_L=0.75	$\begin{array}{l} Ω_{G} (x) = 0.704 1 x^{16} + 0.277 4 x^{17} + 0.018 5 x^{66} \\ Ω_{F} (x) = 0.213 2 x^{12} + 0.566 4 x^{13} + 0.220 4 x^{66} \end{array}$	η=0.034 87
	Γ _M =0.3,Γ_L=0.7	$\begin{array}{l} Ω_{G} (x) = 0.704 1 x^{16} + 0.277 4 x^{17} + 0.018 5 x^{66} \\ Ω_{F} (x) = 0.537 4 x^{13} + 0.270 2 x^{14} + 0.192 4 x^{66} \end{array}$	η=0.036 17
	Γ_M =0.35,Γ_L=0.65	$\begin{array}{l} Ω_{G} (x) = 0.704 1 x^{16} + 0.277 4 x^{17} + 0.018 5 x^{66} \\ Ω_{F} (x) = 0.194 8 x^{12} + 0.629 6 x^{13} + 0.175 7 x^{66} \end{array}$	η=0.039 90

表2

图6

图7

图8

图9

图10

图11

参考文献 39

[1]	BYERS J W , LUBY M , MITZENMACHER M . A digital fountain approach to asynchronous reliable multicast[J]. IEEE Journal on Selected Areas in Communications, 2002,20(8): 1528-1540.
[2]	HUANG J X , FEI Z S , CAO C Z ,et al. Reliable broadcast based on online fountain codes[J]. IEEE Communications Letters, 2021,25(2): 369-373.
[3]	HE H , SU J , CHEN Y R ,et al. Reliable cross-technology communication with physical-layer acknowledgement[J]. IEEE Transactions on Communications, 2020,68(8): 5175-5187.
[4]	CAI P X , ZHANG Y , WU Y C ,et al. Feedback strategies for online fountain codes with limited feedback[J]. IEEE Communications Letters, 2020,24(9): 1870-1874.
[5]	WEN J L , SHIRVANIMOGHADDAM M , ABBAS R ,et al. Analysis and design of analog fountain codes for short packet communications in IoT[J]. IEEE Transactions on Vehicular Technology, 2021,70(12): 12662-12674.
[6]	HU Y M , LIU R K , KAUSHIK A ,et al. Performance analysis of NOMA multicast systems based on rateless codes with delay constraints[J]. IEEE Transactions on Wireless Communications, 2021,20(8): 5003-5017.
[7]	ZHANG K , JIAO J , HUANG Z X ,et al. Finite block-length analog fountain codes for ultra-reliable low latency communications[J]. IEEE Transactions on Communications, 2020,68(3): 1391-1404.
[8]	JAIN S , BOSE R . Rateless-code-based secure cooperative transmission scheme for industrial IoT[J]. IEEE Internet of Things Journal, 2020,7(7): 6550-6565.
[9]	LUO S X , MA T , SHAN W ,et al. Efficient multisource data delivery in edge cloud with rateless parallel push[J]. IEEE Internet of Things Journal, 2020,7(10): 10495-10510.
[10]	SCHULZ P , TRABL A , BARRETO A N ,et al. Efficient and reliable wireless communications via multi-connectivity using rateless codes in single- and multi-user scenarios[J]. IEEE Transactions on Wireless Communications, 2021,20(9): 5714-5729.
[11]	CHEN H , ZHANG X , XU Y L ,et al. Efficient mobile video streaming via context-aware RaptorQ-based unequal error protection[J]. IEEE Transactions on Multimedia, 2020,22(2): 459-473.
[12]	RAHNAVARD N , VELLAMBI B N , FEKRI F . Rateless codes with unequal error protection property[J]. IEEE Transactions on Information Theory, 2007,53(4): 1521-1532.
[13]	LUBY M , . LT codes[C]// Proceedings of 43rd Annual IEEE Symposium on Foundations of Computer Science. Piscataway:IEEE Press, 2002: 271-280.
[14]	WU S , GUAN Q Y , MIAO Z . A new class of LT-based UEP rateless codes for satellite image data transmission[J]. AEU - International Journal of Electronics and Communications, 2020,122: 153256.
[15]	CAI P X , ZHANG Y , PAN C Y ,et al. Online fountain codes with unequal recovery time[J]. IEEE Communications Letters, 2019,23(7): 1136-1140.
[16]	AHMAD S , HAMZAOUI R , AL-AKAIDI M M , . Unequal error protection using fountain codes with applications to video communication[J]. IEEE Transactions on Multimedia, 2011,13(1): 92-101.
[17]	SORENSEN J H , POPOVSKI P , OSTERGAARD J . UEP LT codes with intermediate feedback[J]. IEEE Communications Letters, 2013,17(8): 1636-1639.
[18]	DENG K Y , YUAN L , WAN Y ,et al. Expanding window fountain codes with intermediate feedback over BIAWGN channels[J]. IET Communications, 2018,12(8): 914-921.
[19]	SHOKROLLAHI A . Raptor codes[J]. IEEE Transactions on Information Theory, 2006,52(6): 2551-2567.
[20]	ZHANG Y F , HOU W , LI Y ,et al. Robust rateless spatially coupled repeat-accumulate-repeat multi-user codes on IDMA systems[J]. IEEE Transactions on Vehicular Technology, 2021,70(11): 11523-11537.
[21]	LI A M , WU S H , JIAO J ,et al. Spinal codes over fading channel:error probability analysis and encoding structure improvement[J]. IEEE Transactions on Wireless Communications, 2021,20(12): 8288-8300.
[22]	XU X , WU S H , DONG D ,et al. High performance short polar codes:a concatenation scheme using spinal codes as the outer code[J]. IEEE Access, 2018,6: 70644-70654.
[23]	HUSSAIN I , XIAO M , RASMUSSEN L K . Error floor analysis of LT codes over the additive white Gaussian noise channel[C]// Proceedings of 2011 IEEE Global Telecommunications Conference. Piscataway:IEEE Press, 2011: 1-5.
[24]	HUSSAIN I , XIAO M , RASMUSSEN L K . Design of LT codes with equal and unequal erasure protection over binary erasure channels[J]. IEEE Communications Letters, 2013,17(2): 261-264.
[25]	CHEN C M , CHEN Y P . Connection choice codes[J]. IEICE Transactions on Communications, 2014,97(7): 1350-1357.
[26]	CHANG L J , WANG C H , ZAO J K . An error-floor reduction technique for short-length LT codes[C]// Proceedings of 2014 International Symposium on Information Theory and its Applications. Piscataway:IEEE Press, 2014: 279-283.
[27]	SONG X , CHENG N P , LIAO Y R ,et al. Design and analysis of LT codes with a reverse coding framework[J]. IEEE Access, 2021,9: 116552-116563.
[28]	BRINK S T . Convergence behavior of iteratively decoded parallel concatenated codes[J]. IEEE Transactions on Communications, 2001,49(10): 1727-1737.
[29]	ETESAMI O , SHOKROLLAHI A . Raptor codes on binary memoryless symmetric channels[J]. IEEE Transactions on Information Theory, 2006,52(5): 2033-2051.
[30]	JAYASOORIYA S , SHIRVANIMOGHADDAM M , JOHNSON S J . A design of reconfigurable raptor codes for wide SNR ranges using a multi-edge framework[J]. IEEE Communications Letters, 2018,22(8): 1532-1535.
[31]	KHAREL A , CAO L . On the performance of LT codes over BIAWGN channel[C]// Proceedings of 2016 International Conference on Computer,Information and Telecommunication Systems (CITS). Piscataway:IEEE Press, 2016: 1-5.
[32]	ZHANG M X , KIM S . Efficient encoding scheme for LT codes with soft iterative decoding[J]. IET Communications, 2018,12(13): 1624-1629.
[33]	KHAREL A , CAO L . Improved fountain codes for BI-AWGN channels[C]// Proceedings of 2017 IEEE Wireless Communications and Networking Conference. Piscataway:IEEE Press, 2017: 1-6.
[34]	YUAN L , LI J , WAN Y . Design of expanding window fountain codes with unequal power allocation over BIAWGN channels[J]. IET Communications, 2016,10(14): 1786-1794.
[35]	SONG X , CHENG N P , NI S Y ,et al. An improved LT encoding algorithm based on reselecting check nodes[C]// Proceedings of 2021 13th International Conference on Communication Software and Networks (ICCSN). Piscataway:IEEE Press, 2021: 75-79.
[36]	BRINK S T , KRAMER G , ASHIKHMIN A . Design of low-density parity-check codes for modulation and detection[J]. IEEE Transactions on Communications, 2004,52(4): 670-678.
[37]	XU S K , XU D Z . Optimization design and asymptotic analysis of systematic luby transform codes over BIAWGN channels[J]. IEEE Transactions on Communications, 2016,64(8): 3160-3168.
[38]	ZHANG W Z , HRANILOVIC S , SHI C . Soft-switching hybrid FSO/RF links using short-length raptor codes:design and implementation[J]. IEEE Journal on Selected Areas in Communications, 2009,27(9): 1698-1708.
[39]	冯钟葵, 葛小青, 张洪群 ,等. 遥感数据接收与处理技术[M]. 北京: 北京航空航天大学出版社, 2015.
	FENG Z K , GE X Q , ZHANG H Q ,et al. Remote sensing data receiving and processing technology[M]. Beijing: Beihang University Press, 2015.

面向不等差错保护的低误码平台LT编码算法

Low error floor LT coding algorithm for unequal error protection

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[7]	徐志江，凌晓，王亢，孟利民. 二进制差分随机过程键控的结构与性能分析[J]. 通信学报, 2014, 35(12): 21-189.
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[11]	柴争义,刘芳. 基于免疫克隆选择优化的认知无线网络频谱分配[J]. 通信学报, 2010, 31(11): 92-100.
[12]	刘星成,刘庆. 基于部分外信息的分组Turbo码译码算法研究[J]. 通信学报, 2008, 29(3A): 20-24.
[13]	刘军清,李天昊. 基于纠错算术码的信源信道自适应联合编解码系统[J]. 通信学报, 2007, 28(9): 112-118.
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[15]	林志远,魏平. 一种新的UWB通信脉冲设计[J]. 通信学报, 2006, 27(7): 122-126.