摘要
最佳离散信号及其设计在现代通信、雷达、声纳、制导、空间测控、以及电子对抗等系统中，扮演着越来越重要的角色。深入地研究各种序列的性质，无论在理论上还是在应用上都有非常重要的意义。经过了几十年，人们在最佳离散信号的研究上已取得了大量的重要成果，目前仍在做更深入地研究。
最佳离散信号一般以循环相关、非循环相关、并元相关等为设计准则。本文的研究主要涉及到循环相关和并元相关，所以主要阐述在循环相关和并元相关准则下的最佳信号设计。本文对国内外分别以循环相关和并元相关为准则的最佳离散信号设计的研究现状，进行了综合分析，并在前人的基础上对最佳离散信号做了进一步研究和发扬。
第一部分是以循环相关函数为准则对最佳离散信号进行设计。首先，介绍了序列偶的概念和序列偶自相关函数的性质，为以下的理论研究奠定了基础。其次，提出了几乎最佳自相关序列偶的理论，它是几乎最佳自相关序列的深入和发展，探讨了它的性质，谱特性，存在的必要条件。最后，在几乎最佳自相关序列偶的基础上，给出了几乎最佳周期互补二元序列偶族的概念，进一步深入了几乎最佳自相关序列的研究，给出并证明了几乎最佳周期互补二元序列偶族的构造方法。
第二部分是以并元相关函数为准则对最佳离散信号进行设计。因为二进制并元码和Bent函数是彼此完全等价的，所以本文将对二进制并元码的研究转化为对Bent函数的研究。首先，提出了Bent互补函数偶族的概念。其次，深入地研究了它的性质，特别是与Hadamard互补矩阵偶族、并元互补序列偶族的等价关系。然后，研究了利用列正交阵列、并元最佳阵列偶而得到的Bent互补函数偶族的构造方法。最后，提出了Bent函数偶侣的概念，扩充了Bent互补函数偶族的构造方法。
The perfect discrete single and its design method play an important part in the optimizing design in the area of modernistic communication, radar, sonar, navigation, space ranging and controlling and electronically antagonism systems. It is of theoretical and application importance to study all kinds of sequence properties. An abundance of research results were published through several decades and further researches have been carried out regarding these theories.
The design rules of the perfect discrete single usually include periodic correlation and aperiodic correlation, dyadic correlation, etc. The goal of this paper is to study the perfect discrete singles basing on periodic correlation and dyadic correlation. After analyzing and integrating the actuality of the kind of perfect singles in internal and external, this paper studies it further.
The designs of the perfect discrete single based on the rule of periodic correlative function are discussed in the first part of the paper. Firstly, the concept of sequence pairs and the some properties of the sequence pairs are introduced, which lays the foundation for later work. Secondly, according to almost perfect autocorrelation sequences, the theory of almost perfect auto- correlation sequence pairs is presented. Some characters and Fourier spectrum characters of the almost perfect autocorrelation sequence pairs are discussed, and the necessary conditions of existences of the almost perfect auto- correlation sequence pairs are given. Lastly, the concept of families of almost perfect periodic complementary sequence pairs is given on the basis of the almost perfect autocorrelation sequence pairs. The method on constructing families of almost perfect periodic complementary sequence pairs is given and proved.
The designs of the perfect discrete single based on the rule of dyadic correlative function are discussed in the secondly part of the paper. The study of Bent functions takes the place of the study binary dyadic codes, because they are equipollent. Firstly, the concept of the families of Bent complemen- tary function pairs is defined, and several transform properties are given and proved. The equivalent relation between the families of Bent complementary function pairs and the families of dyadic complementary sequence pairs is discussed, so are the families of Hadmard complementary array pairs. Then, these constructions of the families of Bent complementary function pairs that are based on some special arrays, for example orthogonal column arrays and dyadic perfect arrays, are studied. In the end, Bent function pair mates are given, which increase the constructions of the families of Bent complementary function pairs.
引文
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