新的Ⅱ型偶长Z-互补对构造

doi:10.11959/j.issn.1000-436x.2023052

通信学报 ›› 2023, Vol. 44 ›› Issue (3): 138-144.doi: 10.11959/j.issn.1000-436x.2023052

新的Ⅱ型偶长Z-互补对构造

陈晓玉¹^,², 孙连峰¹^,², 张杰坤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

修回日期:2022-12-05 出版日期:2023-03-25 发布日期:2023-03-01
作者简介:陈晓玉（1983- ），女，内蒙古赤峰人，博士，燕山大学副教授、硕士生导师，主要研究方向为序列设计、无线通信技术
孙连峰（1999- ），男，山东济南人，燕山大学硕士生，主要研究方向为序列设计
张杰坤（1999- ），男，河南郑州人，燕山大学硕士生，主要研究方向为序列设计
李玉博（1985- ），男，河北衡水人，博士，燕山大学副教授、硕士生导师，主要研究方向为压缩感知技术、序列设计与编码理论
基金资助:
河北省自然科学基金资助项目(F2021203078);河北省自然科学基金资助项目(F2020203043);河北省高等学校科学技术研究基金资助项目(ZD2022026);河北省重点实验室基金资助项目(202250701010046)

Construction of new even-length type-II Z-complementary pair

Xiaoyu CHEN¹^,², Lianfeng SUN¹^,², Jiekun ZHANG¹^,², Yubo LI¹^,²

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

Revised:2022-12-05 Online:2023-03-25 Published:2023-03-01
Supported by:
Hebei Natural Science Foundation(F2021203078);Hebei Natural Science Foundation(F2020203043);Science and Technology Project of Hebei Education Department(ZD2022026);Hebei Key Laboratory Project(202250701010046)

摘要/Abstract

摘要：

为丰富 Z-互补对（ZCP）的参数形式，研究了新的 II 型 ZCP 的构造方法。基于迭代法，实现了 II 型 $(2^{k + 2} N, (2^{k + 2} - 1) N) - ZCP$ 的构造，当N=1时，可得Z-最优II型 $(2^{k + 2}, (2^{k + 2} - 1)) - ZCP$ 。进一步，将迭代法与插值法相结合，构造了 II 型 $(2^{k + 1} N + 2^{k + 1}, 2^{k + 1} N + 2^{k + 1} - \frac{N}{2}) - ZCP$ ；当 N=2 时，可得 Z-最优 II 型 $(3 \times 2^{k + 1}, 3 \times 2^{k + 1} - 1) - ZCP$ 。所提方法可以生成新的偶长ZCP，实现了对现有ZCP数量的扩展。

关键词: 非周期, 迭代法, Golay互补对, Z-互补对

Abstract:

To enrich the parameter forms of Z-complementary pair (ZCP), the constructions of new Type-II ZCP were investigated.First, Type-II $(2^{k + 2} N, (2^{k + 2} - 1) N) - ZCP$ was proposed by using iterative method.When N= 1, Z-optimal type-II $(2^{k + 2}, (2^{k + 2} - 1)) - ZCP$ could be obtained.Then, type-II $(2^{k + 1} N + 2^{k + 1}, 2^{k + 1} N + 2^{k + 1} - \frac{N}{2}) - ZCP$ was presented by combining the iterative method with the insertion method.When N= 2, Z-optimal type-II $(3 \times 2^{k + 1}, 3 \times 2^{k + 1} - 1) - ZCP$ could be obtained.The proposed constructions can generate new even-length ZCP, and expand the number of existing ZCP.

Key words: aperiodic, iterative method, Golay complementary pair, Z-complementary pair

中图分类号:

TN911.2

陈晓玉, 孙连峰, 张杰坤, 李玉博. 新的Ⅱ型偶长Z-互补对构造[J]. 通信学报, 2023, 44(3): 138-144.

Xiaoyu CHEN, Lianfeng SUN, Jiekun ZHANG, Yubo LI. Construction of new even-length type-II Z-complementary pair[J]. Journal on Communications, 2023, 44(3): 138-144.

图/表 5

表1

Golay核序列对"

长度	$(a, b)$
2	$(\begin{matrix} + + \\ + - \end{matrix})$
10	$(\begin{matrix} + + - + - + - - + + \\ + + - + + + + + - - \end{matrix})$
26	$(\begin{matrix} + + + + - + + - - + - + - + - - + - + + + - - + + + \\ + + + + - + + - - + - + + + + + - + - - - + + - - - \end{matrix})$

表1

表2

本文所得长度为2k+2的II型ZCP"

长度	ZCZ宽度	$ZCP (s, t)$	${\| ρ_{s} (τ) + ρ_{t} (τ) \|}_{τ = 0}^{N - 1}$
4	3	$(\begin{matrix} + + - - \\ + + + + \end{matrix})$	$(8, 4, 0_{2})$
8	7	$(\begin{matrix} + + - - + + + + \\ + + - - - - - - \end{matrix})$	$(16, 8, 0_{6})$
16	15	$(\begin{matrix} + + - - + + + + + + - - - - - - \\ + + - - + + + + - - + + + + + + \end{matrix})$	$(32, 16, 0_{14})$
32	31	$(\begin{matrix} + + - - + + + + + + - - - - - - + + - - + + + + - - + + + + + + \\ + + - - + + + + + + - - - - - - - - + + - - - - + + - - - - - - \end{matrix})$	$(64, 32, 0_{30})$

表2

表3

长度为6和24的II型ZCP比较"

构造方法	长度	ZCZ宽度	$ZCP (s, t)$	${\| ρ_{s} (τ) + ρ_{t} (τ) \|}_{τ = 0}^{N - 1}$
文献[16]	6	4	$(\begin{matrix} + - + + - + \\ - + - + - + \end{matrix})$	$(12, 8, 4, 0_{3})$
	24	22	$(\begin{matrix} + - + + - + + - + - + - + - + + - + - + - + - + \\ - + - + - + - + - - + - - + - + - + + - + + - + \end{matrix})$	$(48, 32, 16, 0_{21})$
本文定理2	6	5	$(\begin{matrix} + + + + + - \\ + + + - - + \end{matrix})$	$(12, 4, 0_{4})$
本文定理3	24	23	$(\begin{matrix} + + + + + - + + + - - + + + + + + - - - - + + - \\ + + + + + - + + + - - + - - - - - + + + + - - + \end{matrix})$	$(48, 16, 0_{22})$

表3

表4

本文构造方法与文献[15]构造方法比较"

构造方法	ZCP长度	ZCZ宽度	零相关区外的值(v)	ZCZ比率	是否Z-最优
文献[15]定理2	$\begin{array}{l} 2^{k} N + 2^{k - 1} \\ N \in ℤ^{+}, k \in {2, 3, 4 \dots} \end{array}$	2^kN+2^k-1-N	4≤v≤2^k(2N-1)	$\frac{2^{k} - 1}{2^{k}}$	当N=1时，Z-最优
文献[15]定理3	$3 N, N = 2^{α} 10^{β} 26^{γ}$	3N-1	2N	1	是
	$14 N, N = 2^{α} 10^{β} 26^{γ}$	14N-1	4N	1	是
本文定理1	$\begin{array}{l} 2^{k + 2} N \\ k \in {1, 2, 3, \dots} \end{array}$	(2^{k+ 2}-1)N	2^k × 4N	$\frac{2^{k + 2} - 1}{2^{k + 2}}$	当N=1时，Z-最优
	$N = 2^{α} 10^{β} 26^{γ}$
本文定理2	$\begin{array}{l} 2 N + 2 \\ N = 2^{α} 10^{β} 26^{γ} \end{array}$	$\frac{3 N}{2} + 2$	4	$\frac{3}{4}$	当N=2时，Z-最优
本文定理3	$\begin{array}{l} 2^{k + 1} N + 2^{k + 1} \\ k \in {1, 2, 3, \dots} \end{array}$	$2^{k + 1} N + 2^{k + 1} - \frac{N}{2}$	2^{k+ 2}	$\frac{2^{k + 2} - 1}{2^{k + 2}}$	当N=2时，Z-最优
	$N = 2^{α} 10^{β} 26^{γ}$

表4

表5

II型ZCP比较"

构造方法	长度	ZCZ宽度	零相关区外的值	${\| ρ_{e_{i}} (τ) + ρ_{f_{i}} (τ) \|}_{τ = 0}^{N - 1}, i \in {0, 1}$	PMEPR
	10	8	4≤v≤12	$(20, 12, 4, 0_{7})$	(4.4 444,3.600 0)
文献[15]定理2	18	14	4≤v≤28	$(36, 4, 20, 4, 4, 0_{13})$	(4.3 622,3.410 6)
	34	26	4≤v≤60	$(68, 20, 20, 12, 4, 4, 12, 12, 4, 0_{25})$	(4.2 141,4.446 5)
	10	8	4	$(20, 4, 4, 0_{7})$	(3.6 000,2.868 1)
本文定理2	18	14	4	$(36, 4, 4, 4, 4, 0_{13})$	(2.5 060,2.711 8)
	34	26	4	$(68, 4, 4, 4, 4, 4, 4, 4, 4, 0_{25})$	(2.9 412,2.521 6)

表5

参考文献 17

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[17]	FAN P Z , DARNELL M . Sequence design for communications applications[M]. London: Research Studies Press LTD, 1996.

新的Ⅱ型偶长Z-互补对构造

Construction of new even-length type-II Z-complementary pair

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