最优高斯整数周期组间互补序列集的构造

doi:10.12263/DZXB.20210600

电子学报 ›› 2022, Vol. 50 ›› Issue (8): 1866-1874.DOI: 10.12263/DZXB.20210600

最优高斯整数周期组间互补序列集的构造

刘凯¹^,², 倪佳¹^,², 焦佳旺¹^,², 李玉博¹^,²

^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

收稿日期:2021-05-11 修回日期:2021-08-16 出版日期:2022-08-25
- 作者简介:
- 刘凯女，1977年5月出生，黑龙江齐齐哈尔人.现为燕山大学信息科学与工程学院副教授、硕士生导师.研究方向为无线通信编码理论.E-mail: liukai@ysu.edu.cn
  倪佳女，1995年12月出生，河北秦皇岛人.现为燕山大学信息科学与工程学院硕士研究生.研究方向为扩频序列设计.E-mail: 2606494794@qq.com
  焦佳旺男，1994年2月出生，河北廊坊人.现为燕山大学信息科学与工程学院硕士研究生.研究方向为扩频序列设计.E-mail: 623233828@qq.com
  李玉博男，1985年8月出生，河北衡水人.现为燕山大学信息科学与工程学院副教授、硕士生导师.研究方向为无线通信中的序列设计、编码理论.E-mail: liyubo6316@ysu.edu.cn
- 基金资助:
- 河北省自然科学基金 (F2020203043); 河北省高等学校科学技术研究项目 (ZD2020179)

Construction of Optimal Gaussian Integer Periodic Inter-Group Complementary Sequence Sets

LIU Kai¹^,², NI Jia¹^,², JIAO Jia-wang¹^,², LI Yu-bo¹^,²

^1.School of Information Science and Engineering，Yanshan University，Qinhuangdao，Hebei 066004，China
^2.Hebei Province Key Laboratory of Information Transmission and Signal Processing，Qinhuangdao，Hebei 066004，China

Received:2021-05-11 Revised:2021-08-16 Online:2022-08-25 Published:2022-09-08

摘要/Abstract

摘要：

本文在高斯整数集上，基于完备高斯整数序列和正交矩阵，构造了一类最优周期组间互补序列集，可实现任意长度的零相关区内互补码数量和互补码集数量的灵活选择.通过设计2阶和3阶核正交矩阵，将其对角拼接与完备高斯整数过滤，得到一类任意阶正交矩阵.利用该正交矩阵，结合任意长完备高斯整数序列，可构造零相关区与完备序列等长的周期组间互补序列集，其参数达到理论界.构造结果与已有文献比较，参数可在无条件限制下实现最优.组间互补序列集应用于多载波系统可消减邻区间的通信串扰并提升用户容量.

关键词: 组间互补序列集, 正交矩阵, 完备高斯整数序列, 理论界

Abstract:

Based on perfect Gaussian integer sequences and orthogonal matrices, a class of optimal periodic inter-group complementary(IGC) sequence set is constructed on the Gaussian integer set, which can realize the flexible number of complementary codes and complementary code sets within any zero-correlation zone length. By designing 2-order and 3-order kernel orthogonal matrices, splicing them on the diagonal way and filtering with perfect Gaussian integer sequences, a class of orthogonal matrices of arbitrary order is obtained. By using the orthogonal matrices and perfect Gaussian integer sequences of arbitrary length, the periodic inter-group complementary sequence set can be constructed, in which the length of zero correlation zone equals to that of perfect sequence and the set parameters reach the theoretical bound. Compared with the existing literature, the construction results show that the parameters can be optimized without any restriction. IGC sequence sets can be applied to the multi-carrier code division multiple access communication system to reduce adjacent cell interference and increase user capacity.

Key words: inter-group complementary sequence sets, orthogonal matrices, perfect Gaussian integer sequence, theoretical bound

中图分类号:

TN92

刘凯, 倪佳, 焦佳旺, 李玉博. 最优高斯整数周期组间互补序列集的构造[J]. 电子学报, 2022, 50(8): 1866-1874.

LIU Kai, NI Jia, JIAO Jia-wang, LI Yu-bo. Construction of Optimal Gaussian Integer Periodic Inter-Group Complementary Sequence Sets[J]. Acta Electronica Sinica, 2022, 50(8): 1866-1874.

图/表 5

图1 周期IGC序列集构造方法示意图

图2 集合S周期相关函数图

图3 S00与Syx的周期互相关函数图

表1 IGC序列集参数比较

文献	构造基础	集合参数	是否最优	周期性
[5]	完备互补码 $P C (M, L)$ 、正交矩阵 $H N × N$	$(M, N, L) I G C M L N$	是	非周期
[6]	两个正交矩阵 $H N × N$	$(N / Z, N, Z) I G C N N$	是	非周期
[7]	二阶广义布尔函数	$(2 k + 1, 2 p, 2 m - p) I G C 2 k + 1 2 m$	是	非周期
[8]	完备互补码 $P C (G, P, L)$ 、移位序列集 $E M × 2$	$(G, 2 M, Z) I G C P 2 L$	$2 G M = 2 q P + P 2 r / Z$ , L=qZ+r, 其中q, r为非负整数,当Z为偶数且 $1 < r < Z / 2$ 、Z为奇数且 $0 < r < Z / 2$ 时,达到最优	周期
[9]	Z互补码 $(M, Z) A C S P N$ 、完备周期互相关码 $P P C C (G, L)$	$(G, M, Z) I G C P L (N + Z - 1)$	否	周期
[10]	完备互补码 $P C (M, P, T q k)$ 、正交矩阵 $H N × N$	$(M, N T, k o r k - 1) I G C P T N q k$	q=1时,达到最优	周期
[11]	非相关序列集 $H T 2 × T 1$ 、 $(M, Z) Z P C S P L$ 序列集	$(T 2, M, Z) I G C P T 1 L$	$T 2 M = P T 1 L / Z$ 时,达到最优	周期
[12]	周期互补序列集 $(M, P, L) P C S$ 、正交矩阵 $H N × N$	$(M, N, L + 1 / N ⋅ N - 1) I G C P N L$	$r < L / N + 1$ , $M = P$ 时,达到最优	周期
定理2	Z长完备高斯整数序列、正交矩阵 $H 1 M × M$ 和 $H 2 K × K$	$(K, M, Z) I G C K M Z$	是	周期

表1 IGC序列集参数比较

文献	构造基础	集合参数	是否最优	周期性
[5]	完备互补码 $P C (M, L)$ 、正交矩阵 $H N × N$	$(M, N, L) I G C M L N$	是	非周期
[6]	两个正交矩阵 $H N × N$	$(N / Z, N, Z) I G C N N$	是	非周期
[7]	二阶广义布尔函数	$(2 k + 1, 2 p, 2 m - p) I G C 2 k + 1 2 m$	是	非周期
[8]	完备互补码 $P C (G, P, L)$ 、移位序列集 $E M × 2$	$(G, 2 M, Z) I G C P 2 L$	$2 G M = 2 q P + P 2 r / Z$ , L=qZ+r, 其中q, r为非负整数,当Z为偶数且 $1 < r < Z / 2$ 、Z为奇数且 $0 < r < Z / 2$ 时,达到最优	周期
[9]	Z互补码 $(M, Z) A C S P N$ 、完备周期互相关码 $P P C C (G, L)$	$(G, M, Z) I G C P L (N + Z - 1)$	否	周期
[10]	完备互补码 $P C (M, P, T q k)$ 、正交矩阵 $H N × N$	$(M, N T, k o r k - 1) I G C P T N q k$	q=1时,达到最优	周期
[11]	非相关序列集 $H T 2 × T 1$ 、 $(M, Z) Z P C S P L$ 序列集	$(T 2, M, Z) I G C P T 1 L$	$T 2 M = P T 1 L / Z$ 时,达到最优	周期
[12]	周期互补序列集 $(M, P, L) P C S$ 、正交矩阵 $H N × N$	$(M, N, L + 1 / N ⋅ N - 1) I G C P N L$	$r < L / N + 1$ , $M = P$ 时,达到最优	周期
定理2	Z长完备高斯整数序列、正交矩阵 $H 1 M × M$ 和 $H 2 K × K$	$(K, M, Z) I G C K M Z$	是	周期

表2 周期IGC序列集2≤Z≤10，2≤M≤50参数举例

ZCZ	互补序列数M	K
2	3,5,7,9,11,13,15,17,19,21,23,25,27, 29,31,33,35,37,39,41,43,45,47,49	任意≥2的正整数
3	2,4,5,7,8,10,11,13,14,16,17,19,20,22,23,25,26,28, 29,31,32,34,35,37,38,40,41,43,44,46,47,48,49,50
4	3,5,7,9,11,13,15,17,19,21,23,25,27, 29,31,33,35,37,39,41,43,45,47,49
5	2,3,4,6,7,8,9,11,12,13,14,16,17,18,19,21,22,23,24,26,27, 28,29,31,32,33,34,36,37,38,39,41,42,43,44,46,47,48,49
6	5,7,11,13,17,19,23,25,29,31,35,37,41,43,47,49
7	2,3,4,5,6,8,9,10,11,12,13,15,16,17,18,19, 20,22,23,24,25,26,27,29,30,31,32,33,34, 36,37,38,39,40,41,43,44,45,46,47,48,50
8	3,5,7,9,11,13,15,17,19,21,23,25,27, 29,31,33,35,37,39,41,43,45,47,49
9	2,4,5,7,8,10,11,13,14,16,17,19,20,22,23,25,26, 28,29,31,32,34,35,37,38,40,41,43,44,47,48,49,50
10	3,7,9,11,13,17,19,21,23,27,29,31,33,37,39,41,43,47,49

参考文献 19

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最优高斯整数周期组间互补序列集的构造

Construction of Optimal Gaussian Integer Periodic Inter-Group Complementary Sequence Sets

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