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Volume 43 Issue 2
Feb.  2021
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HUAI Nan, ZENG Zhaofa, LI Jing, WANG Zhuo. Envelope-waveform Inversion Based on Multi-offset Ground Penetrating Radar Data[J]. Journal of Electronics & Information Technology, 2022, 44(4): 1212-1221. doi: 10.11999/JEIT211078
Citation: Xiaoyu CHEN, Heru SU, Xichao GAO. Construction of Optimal Zero Correlation Zone Aperiodic Complementary Sequence Sets[J]. Journal of Electronics & Information Technology, 2021, 43(2): 461-466. doi: 10.11999/JEIT190703

Construction of Optimal Zero Correlation Zone Aperiodic Complementary Sequence Sets

doi: 10.11999/JEIT190703
Funds:  The National Natural Science Foundation of China (61601399)
  • Received Date: 2019-09-10
  • Rev Recd Date: 2020-08-19
  • Available Online: 2020-12-10
  • Publish Date: 2021-02-23
  • The construction of ZCZ Aperiodic Complementary Sequence (ZACS) sets are researched based on orthogonal matrices. The proposed approach can provide optimal ZACS sets and the length of ZCZ can be chosen flexibly under the condition of Z|N. The resultant sequence sets have ideal autocorrelation properties and intra-group complementary properties. By adjusting the parameter q, different ZACS sets can be obtained. Moreover, based on the multilevel perfect sequence over integer, Gaussian integer orthogonal matrix is constructed which can be used as the initial sequence in the construction of ZACS. The sequence sets can be applied to Multi-Carrier Code Division Multiple Access (MC-CDMA) system to remove multipath interference and multiple access interference. Furthermore, it can be used as training sequence in Multiple Input Multiple Output (MIMO) channel estimation.

  • 互补序列包含多个子序列,各个子序列自相关函数之和在零位移之外均为零, 是一类具有理想相关特性的序列,广泛应用于雷达系统、信道估计、扩频通信中[1-3]。在多载波码分多址(Multi-Carrier Code Division Multiple Access, MC-CDMA)系统中可采用完全互补序列作为用户地址码。每个用户分配一个互补序列,可通过不同的序列来区分不同的用户,不同用户的地址码之间具有理想相关特性,不同的子序列通过不同的子载波调制发送,接收端将各个子载波信号相叠加以恢复用户信号。文献[1]的结果表明,在MC-CDMA系统异步上行链路信道中,完全互补序列的性能要比Gold码和m序列好很多,这是由于Gold码、m序列相关特性不理想,正交性被来自不同移动台的异步比特流所破坏。但完全互补序列数目受子序列长度的限制,于是学者们提出了零相关区(Zero Correlation Zone, ZCZ)互补序列的概念,序列数量得以扩展,可以支持更多用户。只要不同用户信号的传输时延和不同路径间的传输时延不超过零相关区长度,扩频序列正交性就得以保证,从而可以抑制传输中的多径干扰、多址干扰。

    根据相关函数的不同,零相关区互补序列可分为零相关区周期互补序列(ZCZ Periodic Complementary Sequence, ZPCS)和零相关区非周期互补序列(ZCZ Aperiodic Complementary Sequence, ZACS)。目前,ZPCS[4-7]的研究已经取得了丰富的成果。ZACS用途更多,但研究相对较少。基于有限域${\rm{GF}}(p)$${\rm{GF}}({p^n})$,文献[8]构造了一类参数达到最优的ZACS,但正交矩阵阶数$N$与零相关区$Z$的比值被限制为素数$p$${p^n}$。为减小MC-CDMA系统的峰均功率比(Peak-to-Mean Envelope Power Ratio, PMEPR),文献[2]构造了列序列PMEPR最大为2的完全互补序列,但序列数目受限。文献[9]利用Golay序列构造了具有低列序列PMEPR的ZACS,可以支持更多用户。文献[10-12]研究的ZACS分别称为非周期组间互补序列集和非周期组内互补序列集,是两类特殊的零相关区非周期互补序列集。高斯整数序列是一类实部、虚部均为整数的复数序列,包含四元和QAM序列,具有较高的传输速率和频谱利用率。基于四元ZACS,文献[13]提出了扩展ZACS序列个数和ZCZ长度的方法。文献[14]基于ZACS和格雷映射,构造了一类四元ZACS,但ZCZ长度由初始序列决定。基于正交序列集,文献[15]构造了一类8QAM+零相关区非周期互补序列集,在满足Z|N的条件下,零相关区长度可灵活设定。目前,高斯整数互补序列的研究较少,且多集中在四元、8QAM+等特殊元素上。

    本文给出了参数$\theta $的设定方法,从而基于正交序列集构造了ZACS,所得的ZACS参数达到最优,且在满足Z|N的条件下零相关区长度可以灵活设定。同时,本文基于多电平完备序列,提出了高斯整数正交序列集的构造方法,可以为ZACS的构造提供大量初始序列,进而得到高斯整数零相关区非周期互补序列集,丰富了高斯整数互补序列集的研究成果。

    定义1 设$a = (a(0),a(1), ··· ,a(L - 1))$$b = (b(0), b(1), ··· ,b(L - 1))$是两个长度为$L$的复数序列,其非周期相关函数定义为

    Ca,b(τ)={L1τt=0a(t)b(t+τ),0τL1L1+τt=0a(t+τ)b(t),1Lτ10,|τ|L
    (1)

    其中,$b^*{( \cdot )}$$b( \cdot )$的复共轭。若$a = b$,则称为非周期自相关函数,简写为${C_a}(\tau )$。若$a \ne b$,则${C_{a,b}}(\tau )$称为非周期互相关函数。

    序列$a$和序列$b$的周期相关函数和非周期相关函数具有以下关系

    Ra,b(τ)=Ca,b(τ)+Ca,b(τL)
    (2)

    $a = b$,则称为周期自相关函数,简写为${R_a}(\tau )$。若$a \ne b$,则${R_{a,b}}(\tau )$称为周期互相关函数。

    定义2 设序列集$O = \{ {o^0},{o^1}, ··· ,{o^{M - 1}}\} $含有$M$个序列,任意序列表示为${o^m} = (o_0^m,o_1^m, ··· ,o_{N - 1}^m)$,其中$0 \le m \le M - 1$。若任意两个序列${o^{{m_0}}}$${o^{{m_1}}}$$\tau = 0$时的周期相关函数满足

    Rom0,om1(0)=N1t=0om0t(om1t)={N1n=0|om0n|2,m0=m10, m0m1
    (3)

    其中,$0 \le {m_0},{m_1} \le M - 1$,则称序列集$O$为正交序列集。

    定义3 设序列集$A = \{ {A^0},{A^1}, ··· ,{A^{M - 1}}\} $包含$M$个序列,每个序列${A^m} = \{ A_0^m,A_1^m, ··· ,A_{N - 1}^m\} $含有$N$个长度为$L$子序列$A_n^m = (A_n^m(0),A_n^m(1), ··· , A_n^m(L - 1))$。如果序列非周期相关函数满足

    CAm0,Am1(τ)=N1n=0CAm0n,Am1n(τ)={N1n=0EAm0n,m0=m1,τ=00,  m0=m1,0<|τ|ZW10,  m0m1,|τ|Z1
    (4)

    其中,$0 \le {m_0},{m_1} \le M - 1$, ${E_{A_n^{{m_0}}}} = \displaystyle\sum {_{l = 0}^{L - 1}} {\left| {A_n^{{m_0}}(l)} \right|^2}$,则称序列集$A$为零相关区非周期互补序列集,表示为$(M,Z){\rm{ACS}}_N^L$,其中$M$为序列集中序列的数目,$N$为序列中子序列的个数,$L$为子序列的长度。若$M = N$$Z = L$,则序列集$A$为非周期完备互补(PC)序列集,记为${\rm{PC}}(M,L)$

    定义4 对于参数为$(M,Z){\rm{ACS}}_N^L$的零相关区非周期互补序列集,序列集参数之间的关系满足

    MNL/Z
    (5)

    当等号成立时,称序列集的参数达到理论界限,是最优的零相关区非周期互补序列集。

    步骤 1 取$N \times N$阶正交矩阵${{A}}$, ${{A}} = {\left[ {a_j^i} \right]_{N \times N}}$。取正交序列集$B = \{ {b^0},{b^1}, ··· ,{b^{T - 1}}\} $,包含$T$个序列,每个序列${b^t} = (b_0^t,b_1^t, ··· ,b_{T - 1}^t)$的长度为$T$,其中$0 \le t \le T - 1$。设正整数$Z$,令$N \ge T$, $L = N/Z$, $K = T/Z$

    步骤 2 构造含有$M = LT$个序列的序列集$S = \{ {S^0},{S^1}, ··· ,{S^{M - 1}}\} $,每个序列${S^m} = \{ S_0^m,S_1^m, ··· , S_{N - 1}^m\} $包含$N$个长度为$T$的子序列$S_n^m = (S_n^m(0), S_n^m(1), ··· ,S_n^m(T - 1))$,其中$0 \le m = lT + g \le M - 1$, $0 \le l \le L - 1$, $0 \le t,g \le T - 1$, $0 \le n \le N - 1$,子序列的具体构造如式(6)

    Smn(t)=aθnbgt
    (6)

    其中,$\theta \!=\! (tod Z)L \!+\! {\left[ {l{{( - 1)}^{tod Z}} \!+\! \left\lfloor {{t/Z}} \right\rfloor \!+\! q} \right]_{od L}}$, $0 \le q \le Q - 1$,当$L$为偶数时,正整数$Q = L/2$,当$L$为奇数时,正整数$Q = L$

    定理1 序列集$S$是零相关区非周期互补序列集,参数表示为$(LT,Z){\rm{ACS}}_N^T$

    证明 ${S^{{m_0}}},\,{S^{{m_1}}}$为序列集$S$中的任意两个序列,有${m_0} = {l_0}T + {g_0}$, ${m_1} = {l_1}T + {g_1}$。令$t = {t_0}Z + {t_1}$, $\tau = {\tau _0}Z + {\tau _1}$,其中$0 \le {t_0},{\tau _0} \le K - 1$, $0 \le {t_1}, {\tau _1} \le Z - 1$,计算序列非周期相关函数如式(7)

    CSm0,Sm1(τ)=N1n=0CSm0n,Sm1n(τ)=N1n=0T1τt=0Sm0n(t)[Sm1n(t+τ)]=N1n=0T1τt=0aθ0nbg0t[aθ1nbg1t+τ]=Raθ0,aθ1(0)T1τt=0bg0t(bg1t+τ)
    (7)

    其中,${\theta _0} = {t_1}L + {[{l_0}{( - 1)^{{t_1}}} + {t_0} + q]_{od L}}$, ${\theta _1} = ({t_1} + {\tau _1})_{od Z}L + {[{l_1}{( - 1)^{{{\left( {{t_1} + {\tau _1}} \right)}_{od Z}}}} + {t_0} + {\tau _0} + q]_{od L}}$,令${\theta _2} = {\theta _1} - {\theta _0} = \left( {{{\left( {{t_1} + {\tau _1}} \right)}_{od Z}} - {t_1}} \right)L + {\theta _3}$,其中${\theta _3} = {[{l_1}{( - 1)^{{{\left( {{t_1} + {\tau _1}} \right)}_{od Z}}}} - {l_0}{\left( { - 1} \right)^{{t_1}}} + {\tau _0}]_{od L}}$

    分以下两种情况讨论:

    情况1:${m_0} = {m_1}$, $0 < \tau \le T - 1$,即${l_0} = {l_1}$${g_0} = {g_1}$。当${\tau _1} = 0$${\tau _0} \ne 0$时, 有${\theta _2} \!=\! {\theta _3} \!= \! [{l_1}{( - 1)^{{t_1}}} \!- {l_0}{\left( { - 1} \right)^{{t_1}}} + {\tau _0}]_{od L}$,可得${\theta _2} \ne 0$,即${\theta _0} \ne {\theta _1}$,根据正交矩阵的性质,有${R_{{a^{{\theta _0}}},{a^{{\theta _1}}}}}(0) = 0$,即${C_{{S^{{m_0}}},{S^{{m_1}}}}}(\tau ) = 0$。 当${\tau _1} \ne 0$时,分两种情况进行探讨:

    (1) $0 \le {t_1} + {\tau _1} \le Z - 1$时,可得${\theta _2} = {\tau _1}L + {[{l_1}{( - 1)^{{t_1} + {\tau _1}}} - {l_0}{\left( { - 1} \right)^{{t_1}}} + {\tau _0}]_{od L}} \ne 0$,即${\theta _0} \ne {\theta _1}$,根据正交矩阵的性质,有${R_{{a^{{\theta _0}}},{a^{{\theta _1}}}}}(0) = 0$,即${C_{{S^{{m_0}}},{S^{{m_1}}}}}(\tau ) = 0$

    (2) $Z \le {t_1} + {\tau _1} \le 2Z - 2$时,${\theta _2} = \left( {{\tau _1} - Z} \right)L + {[{l_1}{( - 1)^{{t_1} + {\tau _1} - Z}} - {l_0}{\left( { - 1} \right)^{{t_1}}} + {\tau _0}]_{od L}}{\rm{ = }}cL + {\theta _3}$,其中$c$为非零整数。又$0 \le {\theta _3} \le L - 1$,故${\theta _2} \ne 0$成立,即${\theta _0} \ne {\theta _1}$。同理可得${C_{{S^{{m_0}}},{S^{{m_1}}}}}(\tau ) = 0$

    情况2:${m_0} \ne {m_1}$,存在以下两种子情况:

    (1) ${l_0} = {l_1}$, ${g_0} \ne {g_1}$, $0 \le \tau \le T - 1$。当${\tau _0} = {\tau _1} \!=\! 0$时,有${C_{{S^{{m_0}}},{S^{{m_1}}}}}(0) \!=\! {R_{{a^{{\theta _0}}},{a^{{\theta _1}}}}}(0)\displaystyle\sum {_{t = 0}^{T - 1}} b_t^{{g_0}}{(b_t^{{g_1}})^*}$,又因为$B$为正交序列集,所以${C_{{S^{{m_0}}},{S^{{m_1}}}}}(\tau ) = 0$。当${\tau _1} \!=\! 0$${\tau _0} \!\ne \!0$时,可得${\theta _2} \!=\! {\theta _3} \!=\! [{l_1}{( - 1)^{{t_1}}} - {l_0}{\left( { - 1} \right)^{{t_1}}} + {\tau _0}]_{od L}$,所以${\theta _2} \ne 0$,即${\theta _0} \ne {\theta _1}$,根据正交矩阵的性质,有${R_{{a^{{\theta _0}}},{a^{{\theta _1}}}}}(0) = 0$,即${C_{{S^{{m_0}}},{S^{{m_1}}}}}(\tau ) = 0$。当${\tau _1} \ne 0$时,与情况1类似,根据正交矩阵的性质,有${R_{{a^{{\theta _0}}},{a^{{\theta _1}}}}}(0) = 0$,即${C_{{S^{{m_0}}},{S^{{m_1}}}}}(\tau ) = 0$

    (2) ${l_0} \ne {l_1}$, $0 \le \tau \le Z - 1$。当${\tau _0} = {\tau _1} = 0$时,已知${l_0} \ne {l_1}$,则${\theta _3} \ne 0$,即${\theta _0} \ne {\theta _1}$,根据正交矩阵的性质,有${R_{{a^{{\theta _0}}},{a^{{\theta _1}}}}}(0) = 0$,即${C_{{S^{{m_0}}},{S^{{m_1}}}}}(\tau ) = 0$。当${\tau _0} = 0$$0 < {\tau _1} \le Z - 1$时,即${\tau _1} \ne 0$,与情况1类似,有等式${C_{{S^{{m_0}}},{S^{{m_1}}}}}(\tau ) = 0$成立。证毕

    推论1 已知$m = lT + g$,序列集${S^m}$可以表示为${S^m} = {S^{(l,g)}}$, $0 \le l \le L - 1$, $0 \le g \le T - 1$$S$具有如下性质:

    (1) 任意序列${S^m}$具有理想自相关特性,相关函数满足

    CSm(τ)=0,0<|τ|T
    (8)

    (2) 根据参数$l$的取值将序列集进行分组,组内不同互补序列${S^{({l_0},{g_0})}}$${S^{({l_1},{g_1})}}$具有理想互相关特性, 即满足组内互补(Intra-Group Complementary, IaGC)特性,相关函数满足

    CS(l0,g0),S(l1,g1)(τ)=0,0|τ|T1
    (9)

    其中,$0 \le {l_0} = {l_1} \le L - 1$, $0 \le {g_0} \ne {g_1} \le T - 1$

    (3) 不同组的互补序列${S^{({l_0},{g_0})}}$${S^{({l_1},{g_1})}}$具有长度为$Z$的零相关区,相关函数满足

    CS(l0,g0),S(l1,g1)(τ)=0,0|τ|Z1
    (10)

    其中,$0 \le {l_0} \ne {l_1} \le L - 1$$0 \le {g_0},{g_1} \le T - 1$

    证明 略。由定理1的证明过程即可得证。

    定理2 序列集$S$的参数$(LT,Z){\rm{ACS}}_N^T$达到理论界限,是最优的零相关区非周期互补序列集。

    证明 本文所构造的序列集$S$参数为$(LT,Z) {\rm{ACS}}_N^T$,其中$L = N/Z$, $K = T/Z$。设${M_0}$表示零相关区非周期互补序列集中序列数目的理论上界,有

    M0=NT/Z=NKZ/Z=NK=LZK=LT
    (11)

    可知,序列集$S$的参数达到理论界限,是最优的零相关区非周期互补序列集。证毕

    步骤 1 设序列$u = (u(0),u(1), ··· ,u(N - 1))$为整数集上的多电平完备序列。

    步骤 2 构造含有$N$个序列的序列集$U = \{ {U^0}, {U^1}, ··· ,{U^{N - 1}}\}$,每个序列${U^n} = (U_0^n,U_1^n, ··· ,U_{N - 1}^n)$的长度为$N$,具体表示为

    Unt=u(n+t)odN
    (12)

    步骤 3 构造序列集$H = \{ {h^0},{h^1}, ··· ,{h^{N - 1}}\} $,其中${h^n} = (h_0^n,h_1^n, ··· ,h_{N - 1}^n)$。根据$N$的奇偶分如下两种情况构造,具体表示为

    N为偶数时,令$N = 2N'$,则

    hnt={(a+bj)U2nt+(c+dj)U2n+1t,0nN/21(a+bj)U2nN+1t+(c+dj)U2nNt,N/2nN1
    (13)

    N为奇数时,令$N = 2N' + 1$,则

    hnt={(a+bj)Unt+(c+dj)Un+N+1t,0nN1(a+bj)Unt+(c+dj)Unt,n=N(a+bj)Unt+(c+dj)UnN1t,N+1nN1
    (14)

    其中,$0 \le n,t \le N - 1$, $a,b,c,d$都是整数且$ac + bd = 0$

    定理3 序列集$H$是一个高斯整数正交序列集。

    证明 不妨设${n_0} < {n_1}$,取$H$的任意两行${h^{{n_0}}}$, ${h^{{n_1}}}$,分以下两种情况讨论:

    情况1:当$N$为奇数时,有$N = 2N' + 1$,具体讨论如下。

    (1) 当$0 \le {n_0} \ne {n_1} \le N' - 1$时,${h^{{n_0}}}$, ${h^{{n_1}}}$的相关函数计算如式(15)。

    Rhn0,hn1(0)=N1t=0[(a+bj)Un0t+(c+dj)Un0+N+1t][(a+bj)Un1t+(c+dj)Un1+N+1t]=N1t=0[(a2+b2)Un0t(Un1t)+(ac+bd+bcjadj)Un0t(Un1+N+1t)+(ac+bd+adjbcj)Un0+N+1t(Un1t)+(c2+d2)Un0+N+1t(Un1+N+1t)]
    (15)

    ${n_0} \ne {n_1}$, ${n_0} \ne {n_1} + N' + 1$, ${n_1} \ne {n_0} + N' + 1$,所以${R_{{h^{{n_0}}},{h^{{n_1}}}}}(0) = 0$

    (2) 当$0 \le {n_0} \le N' - 1$, ${n_1} = N'$时,${h^{{n_0}}}$, ${h^{{n_1}}}$的相关函数计算如式(16)。

    Rhn0,hn1(0)=N1t=0[(a+bj)Un0t+(c+dj)Un0+N+1t][(a+bj)Un1t+(c+dj)Un1t]=N1t=0[(a2+b2)Un0t(Un1t)+(ac+bd+bcjadj)Un0t(Un1t)+(ac+bd+adjbcj)Un0+N+1t(Un1t)+(c2+d2)Un0+N+1t(Un1t)]
    (16)

    ${n_0} \ne {n_1}$, ${n_1} \ne {n_0} + N' + 1$,所以${R_{{h^{{n_0}}},{h^{{n_1}}}}} (0) = 0$

    (3) 当$0 \le {n_0} \le N' - 1$, $N' + 1 \le {n_1} \le N - 1$时,${h^{{n_0}}}$, ${h^{{n_1}}}$的相关函数计算如式(17)。

    Rhn0,hn1(0)=N1t=0[(a+bj)Un0t+(c+dj)Un0+N+1t][(a+bj)Un1t+(c+dj)Un1N1t]=N1t=0[(a2+b2)Un0t(Un1t)+(ac+bd+bcjadj)Un0t(Un1N1t)+(ac+bd+adjbcj)Un0+N+1t(Un1t)+(c2+d2)Un0+N+1t(Un1N1t)]
    (17)

    ${n_0} = {n_1} - N' - 1$,则${n_1} = {n_0} + N' + 1$,式(17)的中间两项可化简为$2(ac + bd){E_u} = 0$。又${n_1} \ne {n_0}$, ${n_0} + N' \!+\! 1 \ne {n_1} \!-\! N' \!-\! 1$,所以${R_{{h^{{n_0}}},{h^{{n_1}}}}}(0) \!=\! 0$;若${n_0} \ne {n_1} -\! N' \!-\! 1$,则${n_1} \!\ne\! {n_0} \!+\! N' \!+\! 1$,则${R_{{h^{{n_0}}},{h^{{n_1}}}}} (0) = 0$

    (4) 当${n_0} = N'$, $N' + 1 \le {n_1} \le N - 1$时,与情况$0 \le {n_0} \le N' - 1$, ${n_1} = N'$同理,可得${R_{{h^{{n_0}}},{h^{{n_1}}}}} (0) = 0$

    (5) 当$N' + 1 \le {n_0} \ne {n_1} \le N - 1$时,与情况$0 \le {n_0} \ne {n_1} \le N' - 1$同理,可得${R_{{h^{{n_0}}},{h^{{n_1}}}}}(0) = 0$

    情况2:$N$为偶数时,与$N$为奇数时同理,故省略。

    综合情况1和情况2可知,序列集$H$是一个高斯整数正交序列集。证毕。

    例1 取周期为6的三电平完备序列$u = (1, - 1, 1, - 1,1,2)$,令$a = 1,\,b = 1,\,c = 1,\,d = - 1$,由定理3的构造方法可以得到高斯整数正交序列集$B$

    B=[2j,2j,2j,2j,3j,3+j;2j,2j,3j,3+j,2j,2j;3j,3+j,2j,2j,2j,2j;2j,2j,2j,2j,3+j,3j;2j,2j,3+j,3j,2j,2j;3+j,3j,2j,2j,2j,2j]

    取8阶Hadamard矩阵A和上述6阶高斯整数正交序列集$B$,设$Z = 2$,则$L = 4,K = 3$。令参数$q = 0$,可以得到序列集$S = \{ {S^0},{S^1}, ··· ,{S^{23}}\} $,部分序列表示如下:

    S0=S(0,0)=(2j,2j,2j,2j,3j,3+j;2j,2j,2j,2j,3j,3+j;2j,2j,2j,2j,3+j,3j;2j,2j,2j,2j,3+j,3j;2j,2j,2j,2j,3j,3j;2j,2j,2j,2j,3j,3j;2j,2j,2j,2j,3+j,3+j;2j,2j,2j,2j,3+j,3+j)
    S1=S(0,1)=(2j,2j,3j,3+j,2j,2j;2j,2j,3+j,3j,2j,2j;2j,2j,3j,3+j,2j,2j;2j,2j,3+j,3j,2j,2j;2j,2j,3j,3j,2j,2j;2j,2j,3+j,3+j,2j,2j;2j,2j,3j,3j,2j,2j;2j,2j,3+j,3+j,2j,2j)
    S6=S(1,0)=(2j,2j,2j,2j,3j,3+j;2j,2j,2j,2j,3+j,3j;2j,2j,2j,2j,3+j,3+j;2j,2j,2j,2j,3j,3j;;2j,2j,2j,2j,3j,3j;2j,2j,2j,2j,3+j,3+j;2j,2j,2j,2j,3+j,3j;2j,2j,2j,2j,3j,3+j)
    S7=S(1,1)=(2j,2j,3j,3+j,2j,2j;2j,2j,3j,3+j,2j,2j;2j,2j,3+j,3+j,2j,2j;2j,2j,3+j,3+j,2j,2j;2j,2j,3j,3j,2j,2j;2j,2j,3j,3j,2j,2j;

    序列集$S$为零相关区非周期互补序列集,参数表示为$(24,2){\rm{ACS}}_8^6$。序列数目的理论上界${M_0} = 8\left\lfloor {{6/2}} \right\rfloor = 24$,可知,序列集$S$的参数达到理论界限,是最优的高斯整数零相关区非周期互补序列集。

    表1对不同文献构造ZACS的构造基础、结果序列集参数、是否达到最优等方面进行了对比。文献[10]构造了组间互补序列集,ZCZ长度等于初始PC序列子序列个数L,不可灵活选择。文献[12]构造的ZACS组内具有理想的互相关性能,但是序列组间的ZCZ长度被限制为2的整数次幂,即$\{ 1,2,4, 8, ··· , {2^{n - 1}}\}$。文献[13]基于四元ZACS,利用迭代的方法构造了两类四元ZCZ非周期互补序列集,分别扩展了序列集的个数和ZCZ长度,但ZCZ长度由初始序列决定,为固定值Z或2Z。文献[14]同样基于四元ZACS进行构造,参数受初始序列集限制,在满足$T = M\left\lfloor {{N/Z}} \right\rfloor $时,序列集的参数可达最优。文献[8]、文献[9]及本文均以正交矩阵或正交序列集为初始序列,与PC和ZACS相比正交矩阵的数量更多,方法受限更小,但文献[8]的构造方法基于有限域${\rm{GF}}(p)$${\rm{GF}}({p^n})$,正交矩阵阶数$N$与零相关区$Z$的比值被限制为素数$p$${p^n}$。文献[9]构造了具有低列序列PMEPR的 ZACS,但只有初始Golay序列具有理想自相关性能时才能得到最优的ZACS。本文基于正交序列集构造了一类参数达到理论界限的零相关区非周期互补序列集,序列组内具有理想的互相关特性,即在序列长度内互相关函数全为零,并且在满足Z|N的条件下不同组序列间零相关区长度可以灵活设定。通过引入参数q,本文方法可以得到多个不同的零相关区非周期互补序列集,使其数量得以扩展。

    表  1  零相关区非周期互补序列集参数比较
    文献定理构造基础结果序列参数是否达到最优ZCZ选择是否灵活
    文献[8]方法1正交矩阵${{A}_{{Q} \times {Q}}}$和${{H}_{{L} \times {L}}}$, $Q = pZ$,$L = NZ$$(pNZ,Z){\rm{ACS}}_Q^L$
    文献[9]格雷序列和正交矩阵${{U}_{{V} \times {V}}}$$(KL,Z){\rm{ACS}}_N^{KZ}$$L = N$时最优
    文献[10]完备互补序列集${\rm{PC}}(M,L)$和正交矩阵${{A}_{{P} \times {P}}}$$([M,P],L){\rm{IGC}}_M^{LP}$
    文献[12]完备互补序列集${\rm{PC}}(M,L)$$([{2^n},{2^n}M],Z){\rm{IaGC}}_{{2^n}M}^{{2^n}L}$组内最优
    文献[13]方法1ZACS序列集参数$(M,Z){\rm{ACS}}_P^N$$(2M,Z){\rm{ACS}}_{2P}^{2N}$
    文献[13]方法2ZACS序列集参数$(M,Z){\rm{ACS}}_{2P}^N$$(M,2Z){\rm{ACS}}_{2P}^{2N}$$M = 2P\left\lfloor {{N/Z}} \right\rfloor $时最优
    文献[14]ZACS序列集参数$(T,Z){\rm{ACS}}_M^N$$(T,Z){\rm{ACS}}_M^N$$T = M\left\lfloor {{N/Z}} \right\rfloor $时最优
    本文正交矩阵${A_{{N} \times {N}}}$和正交序列集$B$, $N = LZ$, $T = KZ$$(LT,Z){\rm{ACS}}_N^T$
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    本文方法以正交序列集作为基序列构造ZACS,为满足实际需求,可以选择不同类型的正交序列集。若以二元正交矩阵作为基序列,ZCZ长度会受到限制;若以多相正交矩阵如DFT矩阵作为基序列,其阶数N可为任意整数,因此,ZCZ长度可以灵活选择;以高斯整数正交序列集作为基序列,ZCZ长度受限小。虽然复数序列在扩频的实现难度上大于二元序列,但高斯整数序列可以同时利用载波的幅度和相位传输信息,能实现较高的传输速率和频谱利用率。

    本文首先基于正交序列集提出了零相关区非周期互补序列集的构造方法,序列集的参数均可达到最优。本文构造的零相关区非周期互补序列集有如下性质:(1)非周期互补序列集有理想自相关性能;(2)同一组内的非周期互补序列具有理想的互相关特性;(3)不同组的序列具有零相关区,且在满足Z|N条件下零相关区长度的选择灵活,可以为MC-CDMA系统、MIMO信道估计提供大量可用序列。其次,基于多电平完备序列,给出了高斯整数正交序列集的构造方法,可以为ZACS的构造提供初始序列。

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