高斯整数零相关区序列集构造方法的研究

陈晓玉; 李冠敏; 孔德明; 李玉博

doi:10.11999/JEIT180703

高斯整数零相关区序列集构造方法的研究

doi: 10.11999/JEIT180703

陈晓玉^{1, 2},
李冠敏^{1, 2},
孔德明^3, ,,
李玉博^{1, 2}

1.
燕山大学信息科学与工程学院秦皇岛 066004
2.
河北省信息传输与信号处理重点实验室秦皇岛 066004
3.
燕山大学电气工程学院秦皇岛 066004

基金项目: 国家自然科学基金(61601399, 61501395, 61501394)，河北省自然科学基金(F2016203155)，河北省高等学校科学研究计划(QN2016120)

详细信息

作者简介:
陈晓玉：女，1983年生，副教授，研究方向为扩频序列设计

李冠敏：女，1993年生，硕士生，研究方向为扩频序列设计

孔德明：男，1983年生，副教授，研究方向为数字信号处理技术

李玉博：男，1985年生，副教授，研究方向为无限通信中的序列设计

通讯作者:
孔德明　demingkong@ysu.edu.cn

中图分类号: TN911.2
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- 被引次数: 0
出版历程
- 收稿日期: 2018-07-13
- 修回日期: 2019-01-28
- 网络出版日期: 2019-02-20
- 刊出日期: 2019-06-01

Research on the Constructions of Gaussian Integer Zero Correlation Zone Sequence Set

Xiaoyu CHEN^{1, 2},
Guanmin LI^{1, 2},
Deming KONG^{3
, ,},
Yubo LI^{1, 2}

1.
College of Information Science and Engineering, Yanshan University, Qinhuangdao 066004, China
2.
Hebei Key Laboratory of Information Transmission and Signal Processing, Qinhuangdao 066004, China
3.
College of Electrical Engineering, Yanshan University, Qinhuangdao 066004, China

Funds: The National Natural Science Foundation of China (61601399, 61501395, 61501394), The Natural Science Foundation of Hebei Province (F2016203155), The Science Research Programs of Hebei Educational Committee (QN2016120)

摘要

摘要: 该文提出两类高斯整数零相关区(ZCZ)序列集的构造方法。方法1以ZCZ序列集为基础，利用插零滤波法构造高斯整数ZCZ序列集，并给出了所构造的高斯整数ZCZ序列集度的计算方法。方法2提出了两种高斯整数正交矩阵的构造方法，进而基于正交矩阵构造最优高斯整数ZCZ序列集。该文所构造的高斯整数ZCZ序列集可以应用于准同步码多分址(QS-CDMA)、正交频分复用(OFDM)和多输入多输出(MIMO)等多种通信系统中，在抑制干扰的同时，提高系统的频谱效率。
- 信号处理 /
- 高斯整数 /
- 零相关区序列 /
- 度 /
- 正交矩阵
Abstract: Two constructions of Gaussian integer Zero Correlation Zone (ZCZ) sequence set are researched. In Construction I, the method of zero padding is implemented on the ZCZ sequence set, and then the Gaussian integer ZCZ can be obtained by the filtering operation. Furthermore, the degree of the Gaussian integer ZCZ sequence set is calculated in this paper. In Construction II, two constructions of Gaussian integer orthogonal matrix are proposed. In addition, the optimal Gaussian integer ZCZ sequence sets are constructed based on the orthogonal matrix. The two classes of Gaussian integer ZCZ sequence sets presented in this paper can be applied to many communication systems such as Quasi-Synchronous Code Division Multiple Access (QS-CDMA), Orthogonal Frequency Division Multiplexing (OFDM) and Mutiple-Input Multiple-Output (MIMO) system to suppress the interference and improve the spectrum efficiency.
- Signal processing /
- Gaussian integer /
- Zero Correlation Zone (ZCZ) sequence /
- Degree /
- Orthogonal matrix

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表 1 ${c\,^0}$ 和 ${c\,^1}$ 序列元素

$\begin{aligned}{c\,^0} =& [6 + 11j,6 + 2j, - 6 + 10j,6 + 11j, - 6 - 2j,\\& \quad 6 - 10j, - 6 - 11j, - 6 - 2j, - 6 + 10j,6 + 11j, - 6 - 2j,\\& \quad - 6 + 10j,6 + 11j,6 + 2j, - 6 + 10j, 6 + 11j,6 + 2j,\\& \quad 6 - 10j, - 6 - 11j,6 + 2j, - 6 + 10j,6 + 11j, - 6 - 2j,\\& \quad - 6 + 10j, - 6 - 11j,6 + 2j, - 6 + 10j, - 6 - 11j,6 + 2j,\\& \quad 6 - 10j,- 6 - 11j,6 + 2j,6 - 10j,6 + 11j, - 6 - 2j,\\& \quad 6 - 10j, - 6 - 11j,6 + 2j, - 6 + 10j, - 6 - 11j,- 6 - 2j,6 - 10j,\\& \quad - 6 - 11j, - 6 - 2j,{6 - 10j,6 + 11j, - 6 - 2j,6 - 10j]}\end{aligned}$

$\begin{aligned}{c\,^1} =& [ - 6 - 11j, - 6 - 2j, - 6 + 10j, - 6 - 11j, - 6 - 2j,6 - 10j,\\& \quad 6 + 11j, - 6 - 2j,6 - 10j, - 6 - 11j, - 6 - 2j,6 - 10j,6 + 11j,\\& \quad 6 + 2j,6 - 10j,6 + 11j, - 6 - 2j, - 6 + 10j, - 6 - 11j, - 6 - 2j,\\& \quad - 6 + 10j,6 + 11j,6 + 2j, - 6 + 10j,6 + 11j, - 6 - 2j, - 6 + 10j,\\& \quad {6 + 11j,6 + 2j,6 - 10j,6 + 11j,6 + 2j, - 6 + 10j, - 6 - 11j,}\\& \quad { - 6 - 2j, - 6 + 10j, - 6 - 11j,6 + 2j,6 - 10j, - 6 - 11j,6 + 2j,}\\& \quad { - 6 + 10j, - 6 - 11j,6 + 2j,6 - 10j,6 + 11j,6 + 2j,6 - 10j]}\end{aligned}$

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表 2 正交矩阵的构造实例

矩阵	构造方法	参数	矩阵实例
${{\text{H}}_2}$	基于多电平	$\begin{array}{l} a = (1 + j) \\ b = (1 - j) \\ \end{array}$	${{\text{H}}_2} = \left[ {\begin{array}{*{20}{c}} 2&{2j} \\ {2j}&2 \end{array}} \right]$
${{\text{H}}_3}$	基于DFT变换	$\begin{array}{l} {m_i} = (18,1,6) \\ {a_i} = ( - 1, - 1, - 3) \\ {b_i} = (3, - 4,3) \\ \end{array}$	${{\text{H}}_3} = \left[ {\begin{array}{*{20}{c}} {18 + 6j,18 - 6j,18} \\ {1 - 8j,1 + j,1 + 7j} \\ {6 + 6j,6 - 12j,6 + 6j} \end{array}} \right]$
${{\text{H}}_5}$	基于多电平	$\begin{array}{l} a = (1 + j, - 1 + j) \\ b = (1 - j, - 1 - j) \\ c = (2 + j) \\ \end{array}$	${{\text{H}}_5} = \left[ {\begin{array}{*{20}{c}} {1 + 5j,1 - 5j, - 4, - 4, - 4,} \\ {5 + j, - 5 + j, - 4j, - 4j, - 4j} \\ {4,4, - 1 + 5j, - 1 - 5j,4} \\ { - 4j, - 4j, - 5 + j,5 + j, - 4j} \\ { - 4 - 2j, - 4 - 2j, - 4 - 2j, - 4 - 2j,6 + 3j} \end{array}} \right]$
${{\text{H}}_6} = {{\text{H}}_2} \otimes {{\text{H}}_3}$	Kronecker积	同 ${{\text{H}}_2}$ 和 ${{\text{H}}_3}$ 的参数	${{\text{H}}_6} = \left[ \begin{array}{l} 36 + 12j,36 - 12j,36, - 12 + 36j,12 + 36j,36j; \\ 2 - 16j,2 + 2j,2 + 14j,16 + 2j, - 2 + 2j, - 14 + 2j; \\ 12 + 12j,12 - 24j,12 + 12j, - 12 + 12j,24 + 12j, - 12 + 12j; \\ - 12 + 36j,12 + 36j,36j,36 + 12j,36 - 12j,36; \\ 16 + 2j, - 2 + 2j, - 14 + 2j,2 - 16j,2 + 2j,2 + 14j; \\ - 12 + 12j,24 + 12j, - 12 + 12j,12 + 12j,12 - 24j,12 + 12j \\ \end{array} \right]$

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表 3 高斯整数ZCZ序列集参数比较

文献定理	构造基础	序列集参数	$\eta$
文献[10] 定理3	2元正交矩阵	${\rm{ZCZ}}(L,N,Z - 1),\gcd (N,Z) = 1$ ${\rm{ZCZ}}(L,N,Z - 2),\gcd (N,Z) > 1$	$\eta \le 1$
文献[11] 定理1	多元 ${\rm{ZCZ}}(L,N,K)$ 序列集	${\rm{ZCZ}}(L,N,K)$	$\eta \le 1$
文献[14] 定理1	移位序列和完备高斯整数序列	${\rm{ZCZ}}(2N,2M,Z)$	$\eta \le 1$
文献[19] 定理1	2元伪随机序列	${\rm{ZCZ}}(2N,2M,Z),N = {2^n} - 1$	$\eta \le 1$
本文定理1	${\rm{ZCZ}}(N,M,Z)$ 序列集和完备高斯整数序列	${\rm{ZCZ}}(NK,M,ZK)$	$\eta \le 1$
本文构造法2	高斯整数正交矩阵	${\rm{ZCZ}}(NK,N,K)$	$\eta = 1$

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