ZCZ阵列偶与互补序列偶理论的研究
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摘要
最佳信号及其设计在现代通信、雷达、空间测控、信号处理、信息加密以及电子对抗等系统的优化设计中,发挥着重要的作用。深入研究各种序列(或阵列)的性质,为工程应用提供更多具有良好相关特性的最佳离散信号,在理论上和应用上都有十分重要的意义。
本文结合了新近提出的零相关区序列阵列理论以及基于阵列偶相关的信号设计思想,主要对ZCZ阵列偶与几类互补序列偶的理论问题进行了研究。具体包括以下几个方面:
(1)提出了ZCZ阵列偶的概念,给出了其等价变换性质以及构造方法,并对构造方法进行了理论证明,同时给出了实例和相关函数。基于最佳周期自相关阵列偶和正交矩阵,采用交织方法得到了几类矩形零相关区ZCZ阵列偶集合。通过选择不同的移位序列,可以生成具有一定体积、阵列偶数以及零相关区的ZCZ阵列偶集。该方法不仅可以拓展ZCZ阵列偶的存在范围,而且可以推广到三元、四元或多元ZCZ阵列偶以及ZCZ阵列的构造中;
(2)提出了二元互补序列偶集的概念,研究了二元互补序列偶集的等价变换性质,提出多种利用已知二元互补序列偶集生成具有更大序列偶数目及长度的二元互补序列偶集的构造方法,并进行了理论证明,同时给出了实例。结果表明该方法可以方便快捷地构造出大量二元互补序列偶集;
(3)提出了非周期相关下二元互补序列偶集的伴集的概念,给出了伴集的定义,利用级联与交织等技术得到了多种伴集集合的构造方法,并进行了理论证明,通过实例表明利用这些方法可以得到大量新的二元互补序列偶集的伴集,较传统的互补序列集有更广阔的存在空间。同时给出了由伴集生成二元互补序列偶集的一些方法。
(4)提出了奇周期二元互补序列偶,研究了其等价变换性质,得到了奇周期二元互补序列偶与非周期和周期二元互补序列偶之间的关系,提出了多种构造合成奇周期二元互补序列偶的方法。通过排除等价类的方法减少搜索数量,搜索得到了若干小体积奇周期二元互补序列偶,通过分析结果得到了一些猜想和结论。提出了奇周期二元互补序列偶集和伴集的概念,并给出了多种构造方法,研究结果表明通过等价变换和合成构造方法可以得到大量的奇周期二元互补序列偶集和伴集,从而扩展了最佳离散信号的选择范围。
以上研究成果对完善阵列偶相关理论,拓展最佳离散信号的选择空间都具有重要的意义。
The perfect discrete signal and its design plays an very important role in modern communications, radar, space ranging and controlling, signal processing, information encryption and electronically countermeasures design optimization. Therefore, an in-depth research on the characters of sequences or arrays and providing more perfect discrete signals with good correlation character for engineering application is of very importance both in theory and applications.
Combined with newly proposed zero correlation zone(ZCZ) sequence and array theory and the signal design idea based on array pair correlation, the problems of ZCZ array pair theory and several kinds of complementary sequence pairs theory is mainly studied in this thesis. The primary results include:
1) The concept of ZCZ array pair is proposed, and it’s properties and construction methods are presented with theory confirmation. By using perfect periodic autocorrelation array pair and orthogonal matrix, several ZCZ array pair sets with rectangle zero correlation zone can be obtained based on inteleaved method. ZCZ array pairs set with certain volume, family size and zero correlation zone can be synthesized through choosing suitable shift sequences. The proposed technique not only can expand the exist scope of ZCZ array pair, but also can be applied to the construction of ternary, quaternary, polyphase ZCZ array pair and ZCZ array.
2) The concept of binary complementary sequence pair sets is defined, the properties and construction methods are studied also. Based on these techniques, binary complementary sequence pair sets with larger sequence pair number and length can be synthesized, the theoretical proof and examples are proposed also. The research results show that a great many binary complementary sequence pair sets can be constructed through these methods.
3) A new concept of binary complementary sequence pair set’s mate is defined under aperiodic correlation condition. Several construction methods for mate are proposed and proved by using concatenation and interleaving techniques. Through these methods a great many binary complementary sequence pair sets and mates can be generated. By using some examples these methods are illustrated. Compared with traditional complementary sequence set, it has wider existence space. Some construction methods for binary complementary sequence pair sets based on set’s mate are proposed also.
4) The concept of odd-periodic binary complementary sequence pair is proposed, its transformation properties and the relationship with aperiodic and periodic binary complementary sequence pair are studied. Several construction methods are presented, and some odd-periodic binary complementary sequence pair with small volume are searched out through the method of eliminating equivalent classes. Some conjectures and conlusions are obtained by the analization of searching results. The concept of odd-periodic binary complementary sequence pair sets and mates is defined, and several construction methodes are proposed. The research results show that lots of odd-periodic binary complementary sequence pair sets and mates can be obtained through these methods.
All the results above are important, for they perfected the theory of array pair and extended the select scope of perfect discrete signal.
本文结合了新近提出的零相关区序列阵列理论以及基于阵列偶相关的信号设计思想,主要对ZCZ阵列偶与几类互补序列偶的理论问题进行了研究。具体包括以下几个方面:
(1)提出了ZCZ阵列偶的概念,给出了其等价变换性质以及构造方法,并对构造方法进行了理论证明,同时给出了实例和相关函数。基于最佳周期自相关阵列偶和正交矩阵,采用交织方法得到了几类矩形零相关区ZCZ阵列偶集合。通过选择不同的移位序列,可以生成具有一定体积、阵列偶数以及零相关区的ZCZ阵列偶集。该方法不仅可以拓展ZCZ阵列偶的存在范围,而且可以推广到三元、四元或多元ZCZ阵列偶以及ZCZ阵列的构造中;
(2)提出了二元互补序列偶集的概念,研究了二元互补序列偶集的等价变换性质,提出多种利用已知二元互补序列偶集生成具有更大序列偶数目及长度的二元互补序列偶集的构造方法,并进行了理论证明,同时给出了实例。结果表明该方法可以方便快捷地构造出大量二元互补序列偶集;
(3)提出了非周期相关下二元互补序列偶集的伴集的概念,给出了伴集的定义,利用级联与交织等技术得到了多种伴集集合的构造方法,并进行了理论证明,通过实例表明利用这些方法可以得到大量新的二元互补序列偶集的伴集,较传统的互补序列集有更广阔的存在空间。同时给出了由伴集生成二元互补序列偶集的一些方法。
(4)提出了奇周期二元互补序列偶,研究了其等价变换性质,得到了奇周期二元互补序列偶与非周期和周期二元互补序列偶之间的关系,提出了多种构造合成奇周期二元互补序列偶的方法。通过排除等价类的方法减少搜索数量,搜索得到了若干小体积奇周期二元互补序列偶,通过分析结果得到了一些猜想和结论。提出了奇周期二元互补序列偶集和伴集的概念,并给出了多种构造方法,研究结果表明通过等价变换和合成构造方法可以得到大量的奇周期二元互补序列偶集和伴集,从而扩展了最佳离散信号的选择范围。
以上研究成果对完善阵列偶相关理论,拓展最佳离散信号的选择空间都具有重要的意义。
The perfect discrete signal and its design plays an very important role in modern communications, radar, space ranging and controlling, signal processing, information encryption and electronically countermeasures design optimization. Therefore, an in-depth research on the characters of sequences or arrays and providing more perfect discrete signals with good correlation character for engineering application is of very importance both in theory and applications.
Combined with newly proposed zero correlation zone(ZCZ) sequence and array theory and the signal design idea based on array pair correlation, the problems of ZCZ array pair theory and several kinds of complementary sequence pairs theory is mainly studied in this thesis. The primary results include:
1) The concept of ZCZ array pair is proposed, and it’s properties and construction methods are presented with theory confirmation. By using perfect periodic autocorrelation array pair and orthogonal matrix, several ZCZ array pair sets with rectangle zero correlation zone can be obtained based on inteleaved method. ZCZ array pairs set with certain volume, family size and zero correlation zone can be synthesized through choosing suitable shift sequences. The proposed technique not only can expand the exist scope of ZCZ array pair, but also can be applied to the construction of ternary, quaternary, polyphase ZCZ array pair and ZCZ array.
2) The concept of binary complementary sequence pair sets is defined, the properties and construction methods are studied also. Based on these techniques, binary complementary sequence pair sets with larger sequence pair number and length can be synthesized, the theoretical proof and examples are proposed also. The research results show that a great many binary complementary sequence pair sets can be constructed through these methods.
3) A new concept of binary complementary sequence pair set’s mate is defined under aperiodic correlation condition. Several construction methods for mate are proposed and proved by using concatenation and interleaving techniques. Through these methods a great many binary complementary sequence pair sets and mates can be generated. By using some examples these methods are illustrated. Compared with traditional complementary sequence set, it has wider existence space. Some construction methods for binary complementary sequence pair sets based on set’s mate are proposed also.
4) The concept of odd-periodic binary complementary sequence pair is proposed, its transformation properties and the relationship with aperiodic and periodic binary complementary sequence pair are studied. Several construction methods are presented, and some odd-periodic binary complementary sequence pair with small volume are searched out through the method of eliminating equivalent classes. Some conjectures and conlusions are obtained by the analization of searching results. The concept of odd-periodic binary complementary sequence pair sets and mates is defined, and several construction methodes are proposed. The research results show that lots of odd-periodic binary complementary sequence pair sets and mates can be obtained through these methods.
All the results above are important, for they perfected the theory of array pair and extended the select scope of perfect discrete signal.
引文
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