An asymmetric ZCZ sequence set with inter-subset uncorrelated property and flexible ZCZ length

Longye Wang; Gaoyuan Zhang; Hong Wen; Xiaoli Zeng

doi:10.3934/amc.2018032

Article Contents

2018, Volume 12, Issue 3: 541-552. Doi: 10.3934/amc.2018032

This issue Previous Article On the linear complexities of two classes of quaternary sequences of even length with optimal autocorrelation Next Article A first step towards the skew duadic codes

An asymmetric ZCZ sequence set with inter-subset uncorrelated property and flexible ZCZ length

1.
The National Key Laboratory of Science and Technology on Communications, University of Electronic Science and Technology of China, Chengdu, Sichuan 611731, China
2.
School of Electronic and Information Engineering, Henan University of Science and Technology, Luoyang, Henan 471022, China
3.
School of Information Science and Technology, Tibet University, Lhasa, Tibet 850000, China

^* Xiaoli Zeng is the corresponding author
^* Xiaoli Zeng is the corresponding author

^† Longye Wang is also affiliated with School of Engineering and Technology, Tibet University, Lhasa, Tibet 850000, China

Received: June 30, 2017

Revised: December 31, 2017

Published: July 2018

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Abstract

In this paper, we propose a novel method for constructing new uncorrelated asymmetric zero correlation zone (UA-ZCZ) sequence sets by interleaving perfect sequences. As a type of ZCZ sequence set, an A-ZCZ sequence set consists of multiple sequence subsets. Different subsets are correlated in conventional A-ZCZ sequence set but uncorrelated in our scheme. In other words, the cross-correlation function (CCF) between two arbitrary sequences which belong to different subsets has quite a large zero cross-correlation zone (ZCCZ). Analytical results demonstrate that the UA-ZCZ sequence set proposed herein is optimal with respect to the upper bound of ZCZ sequence set. Specifically, our scheme enables the flexible selection of ZCZ length, which makes it extremely valuable for designing spreading sequences for quasi-synchronous code-division multiple-access (QS-CDMA) systems.

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Mathematics Subject Classification: Primary: 94A05, 94B60.

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Figure 1. Periodic correlation properties of U-ZCZ sequence set ${B}$.

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Figure 2. Comparison of Different Families of A-ZCZ Sequence Sets

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Table 1. The A-ZCZ Sequence Set $\mathcal{C} = \{{C}^{(0)}, \, {C}^{(1)}, \, {C}^{(2)}, \, {C}^{(3)}\}$

${C}^{(0)}$	$\mathit{\boldsymbol{c}}^{(0)}_0$	1, 0, 0, 1, 0, 0, -1, 0, 1, -1, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, -1, 0, 1, -1, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, -1, 0, 1, -1, 1, 1, 0, 1, 1, 0, 0, 1, 0, 0, -1, 0, 1, -1, 1, 1, 0, 1
	$\mathit{\boldsymbol{c}}^{(0)}_1$	1, 0, 0, -1, 0, 0, -1, 0, 1, 1, 1, -1, 0, -1, 1, 0, 0, -1, 0, 0, -1, 0, 1, 1, 1, -1, 0, -1, 1, 0, 0, -1, 0, 0, -1, 0, 1, 1, 1, -1, 0, -1, 1, 0, 0, -1, 0, 0, -1, 0, 1, 1, 1, -1, 0, -1
	$\mathit{\boldsymbol{c}}^{(0)}_2$	1, 0, 1, -1, 0, 1, 1, 1, 0, 0, 0, 1, -1, 0, 1, 0, 1, -1, 0, 1, 1, 1, 0, 0, 0, 1, -1, 0, 1, 0, 1, -1, 0, 1, 1, 1, 0, 0, 0, 1, -1, 0, 1, 0, 1, -1, 0, 1, 1, 1, 0, 0, 0, 1, -1, 0
	$\mathit{\boldsymbol{c}}^{(0)}_3$	1, 0, 1, 1, 0, -1, 1, -1, 0, 0, 0, -1, -1, 0, 1, 0, 1, 1, 0, -1, 1, -1, 0, 0, 0, -1, -1, 0, 1, 0, 1, 1, 0, -1, 1, -1, 0, 0, 0, -1, -1, 0, 1, 0, 1, 1, 0, -1, 1, -1, 0, 0, 0, -1, -1, 0
${C}^{(1)}$	$\mathit{\boldsymbol{c}}^{(1)}_0$	1, 0, 0, -i, 0, 0, i, 0, 1, 1, i, {-i}, 0, 1, -i, 0, 0, -1, 0, 0, 1, i, -i, -i, 1, -1, i, -i, -1, i, i, i, i, i, -i, i, -1, -1, -i, i, i, -1, i, i, i, 1, i, i, -1, i, i, i, -1, 1, i, i
	$\mathit{\boldsymbol{c}}^{(1)}_1$	1, 0, 0, i, 0, 0, i, 0, 1, -1, i, i, 0, -1, -i, 0, 0, 1, 0, 0, 1, 0, -i, i, 1, 1, 0, i, -1, 0, 0, -i, 0, 0, -i, 0, -1, 1, -i, -i, 0, 1, i, 0, 0, -1, 0, 0, -1, 0, i, -i, -1, -1, 0, -i
	$\mathit{\boldsymbol{c}}^{(1)}_2$	1, 0, i, i, 0, 1, -i, i, 0, 0, 0, -i, 1, 0, -i, 0, 1, 1, 0, -i, -1, 1, 0, 0, 0, -1, -i, 0, -1, 0, -i, -i, 0, -1, i, -i, 0, 0, 0, i, -1, 0, i, 0, -1, -1, 0, i, 1, -1, 0, 0, 0, 1, i, 0
	$\mathit{\boldsymbol{c}}^{(1)}_3$	1, 0, i, -i, 0, -1, -i, -i, 0, 0, 0, i, 1, 0, -i, 0, 1, -1, 0, i, -1, -1, 0, 0, 0, 1, -i, 0, -1, 0, -i, i, 0, 1, i, i, 0, 0, 0, -i, -1, 0, i, 0, -1, 1, 0, -i, 1, 1, 0, 0, 0, -1, i, 0
${C}^{(2)}$	$\mathit{\boldsymbol{c}}^{(2)}_0$	1, 0, 0, i, 0, 0, -i, 0, 1, 1, -i, i, 0, 1, i, 0, 0, -1, 0, 0, 1, 0, i, i, 1, -1, 0, i, -1, 0, 0, -i, 0, 0, i, 0, -1, -1, i, -i, 0, -1, -i, 0, 0, 1, 0, 0, -1, 0, -i, -i, -1, 1, 0, -i
	$\mathit{\boldsymbol{c}}^{(2)}_1$	1, 0, 0, -i, 0, 0, -i, 0, 1, -1, -i, -i, 0, -1, i, 0, 0, 1, 0, 0, 1, 0, i, -i, 1, 1, 0, -i, -1, 0, 0, i, 0, 0, i, 0, -1, 1, i, i, 0, 1, -i, 0, 0, -1, 0, 0, -1, 0, -i, i, -1, -1, 0, i
	$\mathit{\boldsymbol{c}}^{(2)}_2$	1, 0, -i, -i, 0, 1, i, -i, 0, 0, 0, i, 1, 0, i, 0, 1, 1, 0, i, -1, 1, 0, 0, 0, -1, i, 0, -1, 0, i, i, 0, -1, -i, i, 0, 0, 0, -i, -1, 0, -i, 0, -1, -1, 0, -i, 1, -1, 0, 0, 0, 1, -i, 0
	$\mathit{\boldsymbol{c}}^{(2)}_3$	1, 0, -i, i, 0, -1, i, i, 0, 0, 0, -i, 1, 0, i, 0, 1, -1, 0, -i, -1, -1, 0, 0, 0, 1, i, 0, -1, 0, i, -i, 0, 1, -i, -i, 0, 0, 0, i, -1, 0, -i, 0, -1, 1, 0, i, 1, 1, 0, 0, 0, -1, -i, 0
${C}^{(3)}$	$\mathit{\boldsymbol{c}}^{(3)}_0$	1, 0, 0, -1, 0, 0, 1, 0, 1, -1, -1, -1, 0, 1, -1, 0, 0, 1, 0, 0, -1, 0, -1, 1, 1, 1, 0, -1, 1, 0, 0, -1, 0, 0, 1, 0, 1, -1, -1, -1, 0, 1, -1, 0, 0, 1, 0, 0, -1, 0, -1, 1, 1, 1, 0, -1
	$\mathit{\boldsymbol{c}}^{(3)}_1$	1, 0, 0, 1, 0, 0, 1, 0, 1, 1, -1, 1, 0, -1, -1, 0, 0, -1, 0, 0, -1, 0, -1, -1, 1, -1, 0, 1, 1, 0, 0, 1, 0, 0, 1, 0, 1, 1, -1, 1, 0, -1, -1, 0, 0, -1, 0, 0, -1, 0, -1, -1, 1, -1, 0, 1
	$\mathit{\boldsymbol{c}}^{(3)}_2$	1, 0, -1, 1, 0, 1, -1, -1, 0, 0, 0, -1, -1, 0, -1, 0, 1, -1, 0, -1, 1, 1, 0, 0, 0, 1, 1, 0, 1, 0, -1, 1, 0, 1, -1, -1, 0, 0, 0, -1, -1, 0, -1, 0, 1, -1, 0, -1, 1, 1, 0, 0, 0, 1, 1, 0
	$\mathit{\boldsymbol{c}}^{(3)}_3$	1, 0, -1, -1, 0, -1, -1, 1, 0, 0, 0, 1, -1, 0, -1, 0, 1, 1, 0, 1, 1, -1, 0, 0, 0, -1, 1, 0, 1, 0, -1, -1, 0, -1, -1, 1, 0, 0, 0, 1, -1, 0, -1, 0, 1, 1, 0, 1, 1, -1, 0, 0, 0, -1, 1, 0

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Table 2. Periodic correlation properties of UA-ZCZ sequence set $\mathcal{C}$.

Constructions	Parameters	Uncorrelated or not	Flexible ZCZ or not
Theorem^$\ast$1 in [12]	$\mathcal{Z}_A(LP, [L, N], [M-1, 2M-1])$	No	No
Theorem^†2 in [12]	$\mathcal{Z}_A(TL, \, [T, N], \, [M, TL])$	Yes	No
Theorem^‡2 in [16]	$\mathcal{Z}_A(TLP, \, [L, T], \, [P, TLP])$ or $\mathcal{Z}_A(TLP, \, [L, T], \, [P-1, TLP])$	Yes	No
Theorem^$\sharp$3.4	$\mathcal{Z}_A(2TP, \, [2M, T], \, [Z, 2TP])$	Yes	Yes
^$\ast$ $L$ is the order of orthogonal matrix $O_L$, $P$ is length of perfect sequence, and $L=KM$, $N=\lfloor\frac{T}{M}\rfloor>1$, $K>1$, $M>1$.
^† $T$ is the order of DFT matrix $H_T$, $L$ is the order of orthogonal matrix $O_L$, and $L=KM$, $N=\lfloor\frac{T}{M}\rfloor>1$, $K>1$, $M>1$.
^‡ $T$ is the order of DFT matrix $H_T$, $L$ is the order of orthogonal matrix $O_L$, $P$ is length of perfect sequence, and $\gcd{(T, P)}=1$, $\gcd{(L, P)}=1$ (or $L\|P$ or $P\|L$).
^$\sharp$ $T$ is the order of DFT matrix $H_T$, $P$ is length of perfect sequence, and $Z\leq2$, $M=\lfloor\frac{P-2}{Z}\rfloor$ or $M=\lfloor\frac{P-1}{Z}\rfloor$.

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