Research on the Constructions of Gaussian Integer Zero Correlation Zone Sequence Set
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摘要: 该文提出两类高斯整数零相关区(ZCZ)序列集的构造方法。方法1以ZCZ序列集为基础,利用插零滤波法构造高斯整数ZCZ序列集,并给出了所构造的高斯整数ZCZ序列集度的计算方法。方法2提出了两种高斯整数正交矩阵的构造方法,进而基于正交矩阵构造最优高斯整数ZCZ序列集。该文所构造的高斯整数ZCZ序列集可以应用于准同步码多分址(QS-CDMA)、正交频分复用(OFDM)和多输入多输出(MIMO)等多种通信系统中,在抑制干扰的同时,提高系统的频谱效率。Abstract: Two constructions of Gaussian integer Zero Correlation Zone (ZCZ) sequence set are researched. In Construction I, the method of zero padding is implemented on the ZCZ sequence set, and then the Gaussian integer ZCZ can be obtained by the filtering operation. Furthermore, the degree of the Gaussian integer ZCZ sequence set is calculated in this paper. In Construction II, two constructions of Gaussian integer orthogonal matrix are proposed. In addition, the optimal Gaussian integer ZCZ sequence sets are constructed based on the orthogonal matrix. The two classes of Gaussian integer ZCZ sequence sets presented in this paper can be applied to many communication systems such as Quasi-Synchronous Code Division Multiple Access (QS-CDMA), Orthogonal Frequency Division Multiplexing (OFDM) and Mutiple-Input Multiple-Output (MIMO) system to suppress the interference and improve the spectrum efficiency.
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Key words:
- Signal processing /
- Gaussian integer /
- Zero Correlation Zone (ZCZ) sequence /
- Degree /
- Orthogonal matrix
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表 1
${c\,^0}$ 和${c\,^1}$ 序列元素$\begin{aligned}{c\,^0} =& [6 + 11j,6 + 2j, - 6 + 10j,6 + 11j, - 6 - 2j,\\& \quad 6 - 10j, - 6 - 11j, - 6 - 2j, - 6 + 10j,6 + 11j, - 6 - 2j,\\& \quad - 6 + 10j,6 + 11j,6 + 2j, - 6 + 10j, 6 + 11j,6 + 2j,\\& \quad 6 - 10j, - 6 - 11j,6 + 2j, - 6 + 10j,6 + 11j, - 6 - 2j,\\& \quad - 6 + 10j, - 6 - 11j,6 + 2j, - 6 + 10j, - 6 - 11j,6 + 2j,\\& \quad 6 - 10j,- 6 - 11j,6 + 2j,6 - 10j,6 + 11j, - 6 - 2j,\\& \quad 6 - 10j, - 6 - 11j,6 + 2j, - 6 + 10j, - 6 - 11j,- 6 - 2j,6 - 10j,\\& \quad - 6 - 11j, - 6 - 2j,{6 - 10j,6 + 11j, - 6 - 2j,6 - 10j]}\end{aligned} $ $\begin{aligned}{c\,^1} =& [ - 6 - 11j, - 6 - 2j, - 6 + 10j, - 6 - 11j, - 6 - 2j,6 - 10j,\\& \quad 6 + 11j, - 6 - 2j,6 - 10j, - 6 - 11j, - 6 - 2j,6 - 10j,6 + 11j,\\& \quad 6 + 2j,6 - 10j,6 + 11j, - 6 - 2j, - 6 + 10j, - 6 - 11j, - 6 - 2j,\\& \quad - 6 + 10j,6 + 11j,6 + 2j, - 6 + 10j,6 + 11j, - 6 - 2j, - 6 + 10j,\\& \quad {6 + 11j,6 + 2j,6 - 10j,6 + 11j,6 + 2j, - 6 + 10j, - 6 - 11j,}\\& \quad { - 6 - 2j, - 6 + 10j, - 6 - 11j,6 + 2j,6 - 10j, - 6 - 11j,6 + 2j,}\\& \quad { - 6 + 10j, - 6 - 11j,6 + 2j,6 - 10j,6 + 11j,6 + 2j,6 - 10j]}\end{aligned}$ 表 2 正交矩阵的构造实例
矩阵 构造方法 参数 矩阵实例 ${{\text{H}}_2}$ 基于多电平 $\begin{array}{l} a = (1 + j) \\ b = (1 - j) \\ \end{array} $ ${{\text{H}}_2} = \left[ {\begin{array}{*{20}{c}} 2&{2j} \\ {2j}&2 \end{array}} \right]$ ${{\text{H}}_3}$ 基于DFT变换 $\begin{array}{l} {m_i} = (18,1,6) \\ {a_i} = ( - 1, - 1, - 3) \\ {b_i} = (3, - 4,3) \\ \end{array} $ ${{\text{H}}_3} = \left[ {\begin{array}{*{20}{c}} {18 + 6j,18 - 6j,18} \\ {1 - 8j,1 + j,1 + 7j} \\ {6 + 6j,6 - 12j,6 + 6j} \end{array}} \right]$ ${{\text{H}}_5}$ 基于多电平 $\begin{array}{l} a = (1 + j, - 1 + j) \\ b = (1 - j, - 1 - j) \\ c = (2 + j) \\ \end{array} $ ${{\text{H}}_5} = \left[ {\begin{array}{*{20}{c}} {1 + 5j,1 - 5j, - 4, - 4, - 4,} \\ {5 + j, - 5 + j, - 4j, - 4j, - 4j} \\ {4,4, - 1 + 5j, - 1 - 5j,4} \\ { - 4j, - 4j, - 5 + j,5 + j, - 4j} \\ { - 4 - 2j, - 4 - 2j, - 4 - 2j, - 4 - 2j,6 + 3j} \end{array}} \right]$ ${{\text{H}}_6} = {{\text{H}}_2} \otimes {{\text{H}}_3}$ Kronecker积 同${{\text{H}}_2}$和${{\text{H}}_3}$的参数 ${{\text{H}}_6} = \left[ \begin{array}{l} 36 + 12j,36 - 12j,36, - 12 + 36j,12 + 36j,36j; \\ 2 - 16j,2 + 2j,2 + 14j,16 + 2j, - 2 + 2j, - 14 + 2j; \\ 12 + 12j,12 - 24j,12 + 12j, - 12 + 12j,24 + 12j, - 12 + 12j; \\ - 12 + 36j,12 + 36j,36j,36 + 12j,36 - 12j,36; \\ 16 + 2j, - 2 + 2j, - 14 + 2j,2 - 16j,2 + 2j,2 + 14j; \\ - 12 + 12j,24 + 12j, - 12 + 12j,12 + 12j,12 - 24j,12 + 12j \\ \end{array} \right]$ 表 3 高斯整数ZCZ序列集参数比较
文献定理 构造基础 序列集参数 $\eta $ 文献[10] 定理3 2元正交矩阵 ${\rm{ZCZ}}(L,N,Z - 1),\gcd (N,Z) = 1$
${\rm{ZCZ}}(L,N,Z - 2),\gcd (N,Z) > 1$$\eta \le 1$ 文献[11] 定理1 多元${\rm{ZCZ}}(L,N,K)$序列集 ${\rm{ZCZ}}(L,N,K)$ $\eta \le 1$ 文献[14] 定理1 移位序列和完备高斯整数序列 ${\rm{ZCZ}}(2N,2M,Z)$ $\eta \le 1$ 文献[19] 定理1 2元伪随机序列 ${\rm{ZCZ}}(2N,2M,Z),N = {2^n} - 1$ $\eta \le 1$ 本文 定理1 ${\rm{ZCZ}}(N,M,Z)$序列集和完备高斯整数序列 ${\rm{ZCZ}}(NK,M,ZK)$ $\eta \le 1$ 本文 构造法2 高斯整数正交矩阵 ${\rm{ZCZ}}(NK,N,K)$ $\eta = 1$ -
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