基于Zadoff-Chu矩阵的最优码本构造方法

doi:10.11959/j.issn.1000-436x.2020021

通信学报 ›› 2020, Vol. 41 ›› Issue (3): 112-119.doi: 10.11959/j.issn.1000-436x.2020021

基于Zadoff-Chu矩阵的最优码本构造方法

李玉博^1,²,刘胜毅^1,²,张景景^1,²,贾冬艳³

¹ 燕山大学信息科学与工程学院，河北秦皇岛 066004
² 河北省信息传输与信号处理重点实验室，河北秦皇岛 066004
³ 河北科技师范学院数学与信息科技学院，河北秦皇岛 066004

修回日期:2019-12-07 出版日期:2020-03-25 发布日期:2020-03-31
作者简介:李玉博（1985– ），男，河北衡水人，博士，燕山大学副教授，主要研究方向为编码理论、序列设计、信息处理等|刘胜毅（1994– ），男，河北沧州人，燕山大学硕士生，主要研究方向为编码理论、压缩感知|张景景（1995– ），女，河北石家庄人，燕山大学硕士生，主要研究方向为编码理论、压缩感知|贾冬艳（1983– ），女，河北衡水人，博士，河北科技师范学院讲师，主要研究方向为推荐系统、智能信息处理
基金资助:
国家自然科学基金资助项目(61501395)

Construction method of optimal codebook based on Zadoff-Chu matrix

Yubo LI^1,²,Shengyi LIU^1,²,Jingjing ZHANG^1,²,Dongyan JIA³

¹ School of Information Science ＆Engineering,Yanshan University,Qinhuangdao 066004,China
² Hebei Key Laboratory of Information Transmission and Signal Processing,Qinhuangdao 066004,China
³ School of Mathematics and Information Science ＆Technology,Hebei Normal University of Science ＆Technology,Qinhuangdao 066004,China

Revised:2019-12-07 Online:2020-03-25 Published:2020-03-31
Supported by:
The National Natural Science Foundation of China(61501395)

摘要/Abstract

摘要：

具有低相关性质的码本在同步码分多址系统（CDMA）、量子信息理论以及压缩感知领域都有重要应用。为扩展码本数量，放宽了变换矩阵的限制条件。基于Zadoff-Chu矩阵，利用差集、几乎差集以及有限域特征和构造了新的码本，得到的码本依照Welch界限或Levenstein界是最优或几乎最优的。通过实验仿真发现，基于该类码本构造的确定性测量矩阵在压缩感知中具有良好的性能。

关键词: 码本, 差集, 几乎差集, Welch界, Levenstein界

Abstract:

Codebooks with low-coherence have wide utilization in code division multiple access (CDMA) communications,quantum information theory,compressed sensing and so on.In order to expand the number of codebooks,the restrictions on the transformation matrix were relaxed.Based on the Zadoff-Chu matrix,new codebooks were constructed using the difference set,almost difference set,and finite field character sum.The proposed codebooks were optimal or near optimal according to the Welch bound or Levenstein bound.Through experimental simulation,it is found that the deterministic measurement matrices constructed using these codebooks also have good performance in the process of compressed sensing.

Key words: codebook, difference set, almost difference set, Welch bound, Levenstein bound

中图分类号:

TN911.2

李玉博,刘胜毅,张景景,贾冬艳. 基于Zadoff-Chu矩阵的最优码本构造方法[J]. 通信学报, 2020, 41(3): 112-119.

Yubo LI,Shengyi LIU,Jingjing ZHANG,Dongyan JIA. Construction method of optimal codebook based on Zadoff-Chu matrix[J]. Journal on Communications, 2020, 41(3): 112-119.

图/表 2

表1

几类最优码本的构造方法"

方法	码本参数	最大相关幅度值	最优性	矩阵Φ	集合D
文献[4,18]	(p,K)	$I_{\max} (C_{Φ}) = \sqrt{\frac{p - K}{(p - 1) K}}$	Welch界最优	IDFT矩阵	差集
文献[5]	$(p, \frac{p - 1}{2})$	$I_{\max} (C_{Φ}) = \frac{1 + \sqrt{p}}{p - 1}$	Welch界渐进最优	Hadamard矩阵	几乎差集
文献[19]	$(q - 1, \frac{q}{p^{2}} - 1)$	$I_{\max} (C_{Φ}) = \frac{(p^{2} - 1) \sqrt{q}}{\sqrt{q} - p^{2} }$	Welch界几乎最优	IDFT矩阵	二元序列支撑集
文献[20]	$(q, \frac{p - 1}{2 p} (q + \sqrt{q}) + 1)$	$I_{\max} (C_{Φ}) = \frac{1}{K} \frac{(p + 1) \sqrt{q}}{2 p}$	Welch界几乎最优	Hadamard矩阵	二元序列支撑集
定理2	(p,K)	$I_{\max} (C_{Φ}) = \sqrt{\frac{p - K}{(p - 1) K}}$	Welch界最优	Zadoff-Chu矩阵	差集
定理3	$(p, \frac{p - 1}{2})$	$I_{\max} (C_{Φ}) = \frac{1 + \sqrt{p}}{p - 1}$	Welch界渐进最优	Zadoff-Chu矩阵	几乎差集
定理4	(q ²+q-1,q)	$I_{\max} (C_{Φ} \cup ξ_{K}) = \frac{1}{\sqrt{q}}$	Levenstein界渐进最优	Zadoff-Chu矩阵	Eisenstein sum定义的集合

表1

图1

参考文献 22

[1]	DING C S , GOLIN M,KL?VE T . Meeting the Welch and Karystinos-Pados bounds on DS-CDMA binary signature sets[J]. Design Codes and Cryptography, 2003,30(1): 73-84.
[2]	冯克勤, 金玲飞 . 量子信息理论中的几个数学问题[J]. 中国科学:数学, 2017,47(11): 1387-1408.
	FENG K Q , JIN L F . Several mathematical problems in quantum information theory[J]. Scientia Sinica (Mathematics), 2017,47(11): 1387-1408.
[3]	LI S X , GE G . Deterministic sensing matrices arising from near orthogonal systems[J]. IEEE Transactions on Information Theory, 2014,64(4): 2291-2302.
[4]	XIA P F , ZHOU S L , GIANNAKIS G B . Achieving the Welch bound with difference sets[J]. IEEE Transactions on Information Theory, 2005,51(5): 1900-1907.
[5]	DING C S . Complex codebooks from combinatorial designs[J]. IEEE Transactions on Information Theory, 2006,52(9): 4229-4235.
[6]	DING C S , FENG T . Codebooks from almost difference sets[J]. Design Codes and Cryptography, 2008(46): 113-126.
[7]	张爱仙, 冯克勤 . 一类近似最佳码本的构造[J]. 中国科学:信息科学, 2015,45(12): 1632-1639.
	ZHANG A X , FENG K Q . Construction of a class of codebooks nearly meeting the Welch bound[J]. Scientia Sinica (Informations), 2015,45(12): 1632-1639.
[8]	LI C J , YUE Q , HUANG Y W . Two families of nearly optimal codebooks[J]. Design Codes and Cryptography, 2015,75(1): 43-57.
[9]	ZHANG A X , FENG K Q . Construction of cyclotomic codebooks nearly meeting the Welch bound[J]. Design Codes and Cryptography, 2012,63(2): 209-224.
[10]	张爱仙, 何春燕, 吉喆 . 几类近似达到 Welch 界码本的构造[J]. 纯粹数学与应用数学, 2018,34(3): 323-330.
	ZHANG A X , HE C Y , JI Z . Constructions of some classes of codebooks nearly meeting the Welch bound[J]. Pure and Applied Mathematics, 2018,34(3): 323-330.
[11]	ZHOU Z C , DING C S , LI N . New families of codebooks achieving the Levenstein bound[J]. IEEE Transactions on Information Theory, 2014,60(11): 2507-2511.
[12]	QU L J . A new approach to constructing quadratic pseudo-planar functions over F₂n[J]. IEEE Transactions on Information Theory, 2016,62(11): 6644-6658.
[13]	HENG Z L , DING C S , YUE Q . New constructions of asymptotically optimal codebooks with multiplicative characters[J]. IEEE Transactions on Information Theory, 2017,63(10): 6179-6187.
[14]	HENG Z L . Nearly optimal codebooks based on generalized Jacobi sums[J]. Discrete Applied Mathematics, 2018(250): 227-240.
[15]	LUO G J , CAO X W . Two constructions of asymptotically optimal codebooks via the hyper Eisenstein sum[J]. IEEE Transactions on Information Theory, 2018,64(10): 6498-6505.
[16]	LUO G J , CAO X W . Two constructions of asymptotically optimal codebooks[J]. Cryptography and Communications, 2018,11(4): 825-838.
[17]	XIANG C , DING C S , MESNAGER S . Optimal codebooks from binary codes meeting the Levenstein bound[J]. IEEE Transactions on Information Theory, 2015,61(12): 6526-6535.
[18]	YU N Y . A construction of codebooks associated with binary sequences[J]. IEEE Transactions on Information Theory, 2012,58(8): 5522-5533.
[19]	CAO X W , CHOU W S , ZHANG X Y . More constructions of near optimal codebooks associated with binary sequences[J]. Advances in Mathematics of Communications, 2017,11(1): 187-202.
[20]	HONG S , PARK H , NO J . Near optimal partial Hadamard codebook construction using binary sequences obtained from quadratic residue mapping[J]. IEEE Transactions on Information Theory, 2014,60(6): 3698-3705.
[21]	WANG X , ZHANG J , GE G . Deterministic convolutional compressed sensing matrices[J]. Finite Fields and Their Applications, 2016(42): 102-117.
[22]	BERNDT B , EVANS R , WILLIAMS K . Gauss and Jacobi sums[M]. New York: WileyPress, 1990.

基于Zadoff-Chu矩阵的最优码本构造方法

Construction method of optimal codebook based on Zadoff-Chu matrix

在线阅读

PDF下载

可视化

摘要/Abstract

引用本文

使用本文

图/表 2

参考文献 22

相关文章 10

Metrics

推荐阅读 0

[1]	张伟, 王宇, 陈新怡, 王延文, 景庆阳, 雷为民. 基于图像块码本模型的监控视频背景参考帧生成方法[J]. 通信学报, 2023, 44(1): 129-141.
[2]	雷拓峰, 倪淑燕, 程乃平, 宋鑫. 瑞利衰落信道下SCMA码本设计[J]. 通信学报, 2022, 43(4): 107-113.
[3]	赵耿,马艳艳,马英杰. 基于混沌序列的SCMA码本设计及系统性能分析[J]. 通信学报, 2019, 40(2): 137-144.
[4]	刘剑飞,何利平,陶颖,刘迪,曾祥烨,王蒙军. 大规模MIMO系统中低复杂度的码本搜索方法[J]. 通信学报, 2019, 40(1): 79-86.
[5]	邵小桃,郭鸣坤,杨维. 基于资源块星座图的稀疏码多址接入码本设计[J]. 通信学报, 2018, 39(9): 1-9.
[6]	居美艳，葛欣，李岳衡，谭国平. 空时相关MIMO信道下的空时联合Huffman有限反馈预编码[J]. 通信学报, 2013, 34(11): 20-183.
[7]	居美艳,葛欣,李岳衡,谭国平. 空时相关MIMO信道下的空时联合Huffman有限反馈预编码[J]. 通信学报, 2013, 34(11): 179-183.
[8]	何世文,黄永明,杨绿溪. 基于信道相关统计及有限反馈的多基站协作波束成形[J]. 通信学报, 2011, 32(9): 61-67.
[9]	贾彦国,许成谦. 差集偶与周期互补二元序列偶的研究[J]. 通信学报, 2007, 28(8): 123-127.
[10]	许成谦,郑娟,；李刚. 广义相对差集偶与二元序列偶的研究[J]. 通信学报, 2005, 26(8): 44-50.