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

  • * 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

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  • 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.

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

    Citation:

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

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

    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
    Theorem2 in [12] $\mathcal{Z}_A(TL, \, [T, N], \, [M, TL])$ Yes No
    Theorem2 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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