零相关区互补序列及小整数集上完备序列设计与分析
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摘要
扩频序列的相关特性(包括周期/非周期相关特性和自相关/互相关相关特性)主要决定了码分多址(CDMA)通信系统的抗多址干扰(MAI)、多径干扰(MI)和邻区干扰(ACI)的能力,从而对该系统的性能和容量产生直接影响。扩频序列的研究工作主要集中在扩频序列理论界、扩频序列设计与扩频序列应用等方面。本博士论文则专注于探讨具有零相关区(ZCZ)的互补序列和小整数集上的完备序列,具体研究如下内容：(1)序列长度为奇数的非周期二进制Z-互补对零相关区长度上界,(2)非周期二进制Z-互补对和非周期四相Z-互补对的存在性,(3)非周期四相和四电平Z-互补集及其伴的构造,(4)小整数集上完备序列和具有少量零元素三进制完备序列。
把零相关区的概念移植到互补序列,研究了非周期二进制Z-互补序列、非周期四相Z-互补序列和非周期四电平Z-互补序列。给出并证明了序列长度为奇数的非周期二进制Z-互补对零相关区长度上界,即对于序列长度N为奇数的非周期二进制Z-互补对,它的零相关区长度上界为(N+1)/2。此外,本文还分别给出了序列长度为奇数和偶数的非周期二进制Z-互补对存在的必要条件。
提出了一种Z-互补对递归构造法,即从序列长度较短的非周期Z-互补对开始,交错使用满足一定条件的两对序列对,不断构造越来越长的非周期Z-互补对。用此方法证明了零相关区长度Z=2、3、4、5和6的非周期二进制Z-互补对的存在性,也用此方法证明了零相关区长度Z=2、3、4的非周期四相Z-互补对的存在性。
提出了非周期Z-互补对的初等变换和等价类代表概念。对于固定长度的非周期Z-互补对,可以用此非周期Z-互补对等价类代表的集合表示。经过计算机搜索,分别得到了长度N≤9的非周期四相Z-互补对和非周期四电平Z-互补对的代表、最大零相关区长度与白相关函数和。分析表明：在序列长度相同时,非周期四电平Z-互补对和非周期四相Z-互补对的最大零相关区几乎相同,但都比非周期二进制Z-互补对的最大零相关区大,并且它们的数目均比二进制Z-互补对的数目多得多。
通过对非周期四相互补集及其伴的构造方法进行改进,提出了非周期四相Z-互补集及其伴的构造,这些构造方法也适用于非周期四电平Z-互补集及其伴的构造。对于非周期四相Z-互补对的伴与零相关区长度的关系,一般来说,零相关区长度越短,非周期四相Z-互补对伴的数目就越多。
用矩阵方法研究完备序列,揭示了多电平循环Hadamard矩阵和多电平完备序列之间的关系,即一个多电平完备序列的充要条件是：一个多电平序列是完备序列当且仅当这个序列的循环矩阵是多电平循环Hadamard矩阵。得到了整数集上完备序列的必要条件,即奇数长度的整数集上完备序列的能量一定是一个整数平方数。并且讨论了字符集{±1,±2)和{±1,±3}上完备序列的存在性。给出了整数集上完备序列的充分条件,得到了长度为3、4、6和7的两电平完备序列。
研究了具有少量零元素三进制完备序列,并且论证了它的存在性。提出了三进制完备序列等价类代表的概念。给出了含有k个零元素三进制完备序列的必要条件。证明了长度为奇数的含有一个零元素或两个零元素三进制完备序列是不存在的。证明了任意长度的含有且仅含有两个及其以上相邻零元素三进制完备序列是不存在的。提出了一个三进制完备序列搜索算法,搜索结果表明：长度小于49的含有一个零元素的三进制完备序列是不存在的。除一个长度为6的含有两个零元素三进制完备序列之外,其他长度小于37的含有两个零元素三进制完备序列是不存在的。
The correlation (including periodic/aperiodic correlation, and auto-correlation/cross-correlation correlation) function of the spreading sequences which are employed in code-division multiple-access (CDMA) communication systems play a critical role on the capability of reducing multiple access interference (MAI), the multipath interference (MI), and adjacent cell interference (ACI), therefore directly influence the performance and capacity of the systems. Existing research on spreading sequences mainly focuses on three aspects, i.e. the theoretic sequence bounds, the sequence design and the sequence applications respectively. In this thesis, complementary sequences with zero correlation zone (ZCZ) and perfect sequences over small integers are investigated concretely, that is,(1) upper bound on zero correlation zone width of aperiodic binary Z-complementary pairs of odd length,(2) existence of aperiodic Z-complementary pairs of binary and quadriphase sequences,(3) constructions of aperiodic quadriphase and four-level Z-complementary sets and their mates, and (4) perfect sequences over small integers and ternary perfect sequences with a few zero elements.
By extending the ZCZ concept to complementary sequences, aperiodic Z-complementary sequences (including binary/quadriphase/four-level) are investigated. It is shown that an upper bound on zero correlation zone width of aperiodic Z-complementary pairs of binary sequences of odd length N is (N+1)/2. Moreover, necessary conditions for aperiodic Z-complementary pairs of binary sequences are given.
A new recursive construction of aperiodic Z-complementary pairs is proposed. That is, aperiodic Z-complementary pairs of longer length are constructed from aperiodic Z-complementary pairs of shorter one by using two pairs of sequences that satisfied certain conditions. It is proved that there exist aperiodic Z-complementary pairs of binary sequences of any length with ZCZ widths2,3,4,5and6. It is also proved that there exist aperiodic Z-complementary pairs of quadriphase sequences of any length with ZCZ widths2,3and4.
Elementary operations on aperiodic Z-complementary pairs and the representatives of the equivalence class of them are proposed. The set of Z-complementary pairs of some fixed lengths is determined by the set of inequivalent representatives. Based on computer search, a specific representative for each aperiodic quadriphase (and four-level) Z-complementary pair of length N≤9, its maximum ZCZ width, and Summed ACF are given. Binary Z-complementary pairs are compared with quadriphase and four-level Z-complementary ones. It is shown that the aperiodic quadriphase and four-level Z-complementary pairs are normally better than aperiodic binary ones of the same length, in terms of the number of Z-complementary pairs and the maximum ZCZ width. Besides, the aperiodic quadriphase Z-complementary pairs are nearly the same as aperiodic four-level Z-complementary pairs in terms of the maximum ZCZ width.
By modifying or improving the original methods of constructing complementary sets and their mates, constructions of aperiodic quadriphase (and four-level) Z-complementary sets and their mates are given. Generally speaking, the shorter ZCZ width, the more mates of an aperiodic quadriphase Z-complementary pair.
Perfect sequences are studied by using matrix method. The relation between multilevel cyclic Hadamard matrices and multilevel perfect sequences is investigated. A sequence is a multilevel perfect sequence if and only if the cyclic matrix associated with the sequence is a multilevel cyclic Hadamard matrix. Necessary conditions for perfect sequences over integers are proposed. It is shown that, the sum of squares of all elements of perfect sequence of odd length over integers is a square number. The existence of perfect sequences over the alphabets{±1,±2} and {±1,±3} is also investigated. A sufficient condition for perfect sequences over integers is given, and two-level perfect sequences of length3,4,6and7are obtained.
Ternary perfect sequences with a few zero elements are studied, and their existence is also investigated. Necessary conditions for ternary perfect sequences with k zero elements are given. It is proved that there exist no ternary perfect sequences of odd lengths with one zero element or two zero elements. It is also proved that there exist no ternary perfect sequences of any length with two and above adjacent zero elements. An efficient search algorithm for ternary perfect sequences is given. The result concerning the non-existence of ternary perfect sequences of lengths less than49with one zero element are obtained. The non-existence of ternary perfect sequences of lengths less than37with two zero elements is determined, but excluding the case of length6.
把零相关区的概念移植到互补序列,研究了非周期二进制Z-互补序列、非周期四相Z-互补序列和非周期四电平Z-互补序列。给出并证明了序列长度为奇数的非周期二进制Z-互补对零相关区长度上界,即对于序列长度N为奇数的非周期二进制Z-互补对,它的零相关区长度上界为(N+1)/2。此外,本文还分别给出了序列长度为奇数和偶数的非周期二进制Z-互补对存在的必要条件。
提出了一种Z-互补对递归构造法,即从序列长度较短的非周期Z-互补对开始,交错使用满足一定条件的两对序列对,不断构造越来越长的非周期Z-互补对。用此方法证明了零相关区长度Z=2、3、4、5和6的非周期二进制Z-互补对的存在性,也用此方法证明了零相关区长度Z=2、3、4的非周期四相Z-互补对的存在性。
提出了非周期Z-互补对的初等变换和等价类代表概念。对于固定长度的非周期Z-互补对,可以用此非周期Z-互补对等价类代表的集合表示。经过计算机搜索,分别得到了长度N≤9的非周期四相Z-互补对和非周期四电平Z-互补对的代表、最大零相关区长度与白相关函数和。分析表明：在序列长度相同时,非周期四电平Z-互补对和非周期四相Z-互补对的最大零相关区几乎相同,但都比非周期二进制Z-互补对的最大零相关区大,并且它们的数目均比二进制Z-互补对的数目多得多。
通过对非周期四相互补集及其伴的构造方法进行改进,提出了非周期四相Z-互补集及其伴的构造,这些构造方法也适用于非周期四电平Z-互补集及其伴的构造。对于非周期四相Z-互补对的伴与零相关区长度的关系,一般来说,零相关区长度越短,非周期四相Z-互补对伴的数目就越多。
用矩阵方法研究完备序列,揭示了多电平循环Hadamard矩阵和多电平完备序列之间的关系,即一个多电平完备序列的充要条件是：一个多电平序列是完备序列当且仅当这个序列的循环矩阵是多电平循环Hadamard矩阵。得到了整数集上完备序列的必要条件,即奇数长度的整数集上完备序列的能量一定是一个整数平方数。并且讨论了字符集{±1,±2)和{±1,±3}上完备序列的存在性。给出了整数集上完备序列的充分条件,得到了长度为3、4、6和7的两电平完备序列。
研究了具有少量零元素三进制完备序列,并且论证了它的存在性。提出了三进制完备序列等价类代表的概念。给出了含有k个零元素三进制完备序列的必要条件。证明了长度为奇数的含有一个零元素或两个零元素三进制完备序列是不存在的。证明了任意长度的含有且仅含有两个及其以上相邻零元素三进制完备序列是不存在的。提出了一个三进制完备序列搜索算法,搜索结果表明：长度小于49的含有一个零元素的三进制完备序列是不存在的。除一个长度为6的含有两个零元素三进制完备序列之外,其他长度小于37的含有两个零元素三进制完备序列是不存在的。
The correlation (including periodic/aperiodic correlation, and auto-correlation/cross-correlation correlation) function of the spreading sequences which are employed in code-division multiple-access (CDMA) communication systems play a critical role on the capability of reducing multiple access interference (MAI), the multipath interference (MI), and adjacent cell interference (ACI), therefore directly influence the performance and capacity of the systems. Existing research on spreading sequences mainly focuses on three aspects, i.e. the theoretic sequence bounds, the sequence design and the sequence applications respectively. In this thesis, complementary sequences with zero correlation zone (ZCZ) and perfect sequences over small integers are investigated concretely, that is,(1) upper bound on zero correlation zone width of aperiodic binary Z-complementary pairs of odd length,(2) existence of aperiodic Z-complementary pairs of binary and quadriphase sequences,(3) constructions of aperiodic quadriphase and four-level Z-complementary sets and their mates, and (4) perfect sequences over small integers and ternary perfect sequences with a few zero elements.
By extending the ZCZ concept to complementary sequences, aperiodic Z-complementary sequences (including binary/quadriphase/four-level) are investigated. It is shown that an upper bound on zero correlation zone width of aperiodic Z-complementary pairs of binary sequences of odd length N is (N+1)/2. Moreover, necessary conditions for aperiodic Z-complementary pairs of binary sequences are given.
A new recursive construction of aperiodic Z-complementary pairs is proposed. That is, aperiodic Z-complementary pairs of longer length are constructed from aperiodic Z-complementary pairs of shorter one by using two pairs of sequences that satisfied certain conditions. It is proved that there exist aperiodic Z-complementary pairs of binary sequences of any length with ZCZ widths2,3,4,5and6. It is also proved that there exist aperiodic Z-complementary pairs of quadriphase sequences of any length with ZCZ widths2,3and4.
Elementary operations on aperiodic Z-complementary pairs and the representatives of the equivalence class of them are proposed. The set of Z-complementary pairs of some fixed lengths is determined by the set of inequivalent representatives. Based on computer search, a specific representative for each aperiodic quadriphase (and four-level) Z-complementary pair of length N≤9, its maximum ZCZ width, and Summed ACF are given. Binary Z-complementary pairs are compared with quadriphase and four-level Z-complementary ones. It is shown that the aperiodic quadriphase and four-level Z-complementary pairs are normally better than aperiodic binary ones of the same length, in terms of the number of Z-complementary pairs and the maximum ZCZ width. Besides, the aperiodic quadriphase Z-complementary pairs are nearly the same as aperiodic four-level Z-complementary pairs in terms of the maximum ZCZ width.
By modifying or improving the original methods of constructing complementary sets and their mates, constructions of aperiodic quadriphase (and four-level) Z-complementary sets and their mates are given. Generally speaking, the shorter ZCZ width, the more mates of an aperiodic quadriphase Z-complementary pair.
Perfect sequences are studied by using matrix method. The relation between multilevel cyclic Hadamard matrices and multilevel perfect sequences is investigated. A sequence is a multilevel perfect sequence if and only if the cyclic matrix associated with the sequence is a multilevel cyclic Hadamard matrix. Necessary conditions for perfect sequences over integers are proposed. It is shown that, the sum of squares of all elements of perfect sequence of odd length over integers is a square number. The existence of perfect sequences over the alphabets{±1,±2} and {±1,±3} is also investigated. A sufficient condition for perfect sequences over integers is given, and two-level perfect sequences of length3,4,6and7are obtained.
Ternary perfect sequences with a few zero elements are studied, and their existence is also investigated. Necessary conditions for ternary perfect sequences with k zero elements are given. It is proved that there exist no ternary perfect sequences of odd lengths with one zero element or two zero elements. It is also proved that there exist no ternary perfect sequences of any length with two and above adjacent zero elements. An efficient search algorithm for ternary perfect sequences is given. The result concerning the non-existence of ternary perfect sequences of lengths less than49with one zero element are obtained. The non-existence of ternary perfect sequences of lengths less than37with two zero elements is determined, but excluding the case of length6.
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