参数灵活的二维格雷-零相关区阵列集构造方法

王美悦; 刘涛; 陈晓玉; 李玉博

doi:10.11999/JEIT251360

参数灵活的二维格雷-零相关区阵列集构造方法

doi: 10.11999/JEIT251360 cstr: 32379.14.JEIT251360

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.
燕山大学深圳研究院深圳 518063

基金项目: 国家自然科学基金(62501520, 62471427)，河北省中央引导地方科技发展资金项目(246Z0403G)，河北省自然科学基金(F2025203055)

详细信息

作者简介:
王美悦：女，博士生，研究方向为序列设计与编码理论

刘涛：女，讲师，研究方向为序列设计与编码理论

陈晓玉：女，副教授，研究方向为为信号设计与无线通信技术

李玉博：男，教授，研究方向为为序列设计与编码、码域非正交多址技术、通信与雷达信号设计

通讯作者:
李玉博　liyubo6316@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: 2025-12-24
- 修回日期: 2026-01-31
- 录用日期: 2026-02-05
- 网络出版日期: 2026-02-15

Construction Methods of Two-Dimensional Golay-Zero Correlation Zone Array Sets with Flexible Parameters

1.
School of Information Science and Engineering, Yanshan University, Qinhuangdao 066004, China
2.
Hebei Province Key Laboratory of Information Transmission and Signal Processing, Qinhuangdao 066004, China
3.
Shenzhen Research Institute of Yanshan University, Shenzhen 518063, China

Funds: The National Natural Science Foundation of China (62501520, 62471427), The Central Guiding Local Science and Technology Development Fund Project of Hebei Province (246Z0403G), The Natural Science Foundation of Hebei Province (F2025203055)

摘要

摘要: 二维(2D)格雷-零相关区(Golay-ZCZ)阵列集在多输入多输出(MIMO)全向传输系统中具有潜在的应用前景，例如用于预编码矩阵、相控阵天线和声源阵列。然而，针对其构造研究的现有文献仍较为有限。该文分别基于2D多变量函数和级联法提出了3种2D Golay-ZCZ阵列集的构造方法，构造的阵列集具有灵活阵列尺寸和大的零相关区。与已有文献相比，本文所构造的阵列集在阵列尺寸和零相关区宽度方面不再局限于2的幂次形式，因此能够获得现有文献中没有的新参数，阵列集参数更加灵活，同时可将已有结果作为特例包含在内。
- 二维格雷-零相关区阵列集 /
- 二维多变量函数 /
- 零相关区 /
- 全向传输
Abstract: Objective Sequences with good correlation properties are widely used in wireless communications, cryptography, and radar systems. However, a sequence set cannot simultaneously achieve ideal autocorrelation and ideal cross-correlation. This limitation has led to the study of two signal classes with ideal correlation properties: Zero Correlation Zone (ZCZ) sequences and Golay Complementary Sets (GCS). A Golay-ZCZ sequence set combines the advantages of both. Its constituent sequences exhibit ideal periodic autocorrelation and cross-correlation within the ZCZ, and the sums of their aperiodic autocorrelations are zero at all nonzero shifts. Therefore, a Golay-ZCZ set is both a ZCZ set and a GCS. It can thus be used in the applications of both sequence classes. An array set is a two-dimensional extension of a sequence set. Although Golay-ZCZ sequence sets have been widely studied and constructed, research on Two-Dimensional (2D) Golay-ZCZ array sets remains limited. This study proposes three constructions of 2D Golay-ZCZ array sets based on 2D multivariable functions and the concatenation operator. These array sets can be used as precoding matrices for massive Multiple Input Multiple Output (MIMO) omnidirectional transmission. Methods Three construction methods for 2D Golay-ZCZ array sets are proposed, including one direct construction and two indirect constructions. The resulting parameters have not been reported in existing studies. In the first construction, a 2D Golay-ZCZ array set is generated using 2D multivariable functions, with parameters expressed as prime powers. This direct function-based approach enables efficient synthesis of the target arrays. The second and third constructions generate 2D Golay-ZCZ array sets through horizontal and vertical concatenation of Two-Dimensional Complete Complementary Codes (2D CCC), respectively. In these indirect constructions, the parameters are not restricted to prime powers. This property broadens the applicability of the methods and increases parameter flexibility. Results and Discussions The first construction generates a 2D Golay-ZCZ array set with array size $ p_{1}^{{m}_{1}}\times p_{2}^{{m}_{2}} $ and ZCZ size $ ({p}_{1}-1)p_{1}^{{\pi }_{1}(2)-1}\times ({p}_{2}-1)p_{2}^{{\sigma }_{1}(2)-1} $ through a direct function-based method, where $ {p}_{1} $ and $ {p}_{2} $ are prime numbers. For clarity, the magnitudes of the 2D periodic cross-correlation function of the constructed array set are illustrated in Example 1 (Fig. 1). The second construction generates a ZCZ array set with array size $ {L}_{1}\times {N}^{2}{L}_{2} $ and ZCZ size $ ({L}_{1}-1)\times (N-1){L}_{2} $ based on the horizontal concatenation of $ (N,N,{L}_{1},{L}_{2}) $ 2D CCC. The third construction generates a ZCZ array set with array size $ {N}^{2}{L}_{1}\times {L}_{2} $ and ZCZ size $ (N-1){L}_{1}\times ({L}_{2}-1) $ based on the vertical concatenation of $ (N,N,{L}_{1},{L}_{2}) $ 2D CCC. An illustrative example of Construction 2 is provided, and the corresponding correlation magnitudes are shown in (Figs. 2 and 3). As summarized in (Table 1), the construction methods proposed in this paper generate parameter sets that have not been reported in the existing literature. The constructed array sets provide considerable flexibility in array dimensions and ZCZ sizes. This flexibility is valuable for the design of precoding matrices in MIMO omnidirectional transmission systems. In practical implementations, the dimension of a precoding matrix is typically determined by the number of transmit antennas, whereas the ZCZ size must match the maximum multipath delay spread of the channel. Owing to this parameter flexibility, the proposed 2D Golay-ZCZ array sets support adaptive selection under different antenna configurations and channel conditions. Conclusions Three construction methods for 2D Golay-ZCZ array sets are proposed. These methods generate array sets with flexible array sizes and large ZCZ widths. The first construction is based on a 2D multivariable function and can include previous results as special cases without using kernels. The second and third constructions rely on the concatenation operator and provide greater parameter flexibility. The proposed 2D Golay-ZCZ arrays have potential applications in MIMO omnidirectional transmission. The parameter-flexible array sets can be selected according to different antenna configurations and channel conditions. This property suppresses multi-antenna interference within the zero-correlation zone and maintains uniform transmitted energy.
- 2D Golay-Zero Correlation Zone (Golay-ZCZ) array set /
- 2D multivariable function /
- Zero Correlation Zone (ZCZ) /
- Omnidirectional transmission

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图 1 例1中阵列集的二维周期相关函数幅值

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图 2 例2中阵列集${\mathcal{A}}$的二维非周期自相关函数幅值

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图 3 例2中阵列集的二维周期相关函数幅值

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表 1 二维Golay-ZCZ阵列集参数比较

方法	集合大小	阵列尺寸	零相关区宽度	基于	相位$ q $
文献[26]定理1	2	$ (m,4n) $	$ (m-1,n) $	GCAP	同GCAP的相位
文献[27]定理1	$ {2}^{k} $	$ ({2}^{n},{2}^{m}) $	$ ({2}^{n}-1,{2}^{{{\pi }_{1}}(2)-1}) $	2D GBF	$ q\geq 2 $且为偶数
文献[27]定理2	$ {4}^{k} $	$ ({2}^{n},{2}^{m}) $	$ ({2}^{{{\sigma }_{1}}(2)-1},{2}^{{{\pi }_{1}}(2)-1}) $	2D GBF	$ q\geq 2 $且为偶数
本文定理1	$ p_{1}^{{k}_{1}}p_{2}^{{k}_{2}} $	$ (p_{1}^{{m}_{1}},p_{2}^{{m}_{2}}) $	$ \left(({p}_{1}-1)p_{1}^{{\pi }_{1}(2)-1},({p}_{2}-1)p_{2}^{{\sigma }_{1}(2)-1}\right) $	2D MVF	$ \text{lcm}({p}_{1},{p}_{2})\left\| q\right. $，$ {p}_{1},{p}_{2} $为素数
本文定理2	$ N $	$ ({L}_{1},{N}^{2}{L}_{2}) $	$ \left(\begin{array}{c}({L}_{1}-1),(N-1){L}_{2}\end{array}\right) $	2D CCC和DFT矩阵	$ \text{lcm}(Q,N)\left\| q\right. $，$ Q $为2D CCC相位， $ N $为DFT矩阵阶数
本文定理3	$ N $	$ ({N}^{2}{L}_{1},{L}_{2}) $	$ \left((N-1){L}_{1},({L}_{2}-1)\right) $	2D CCC和DFT矩阵

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