基于特殊网络结构和分数阶微积分的正交信号构造理论、方法及其在MIMO雷达中的应用研究

Orthogonal signal has an extremely wide application in many domains such as radar signal processing, communication theory, information security and so on. Compared with intelligent algorithms that may only get approximately orthogonal or lower (cross-correlation) autocorrelation signals, the construction method can obtain the completely orthogonal signal as well as the lowest (cross-correlation) autocorrelation signals, which has caught much attention recently. However, the current construction methods can only achieve the orthogonal signals with a specific code length and the NLFM signals by the window function. To address this problem, this project tries to develop a unified construction method and tools to both discrete and continuous orthogonal signals, which may set up new construction theory and method for orthogonal signals. First, through combining the existing Hadamard matrix and the low autocorrelation sequences, the H matrix is obtained by finite orthogonal recursive method to construct the regular bipartite graph with special network structure, and then this method is applied to construct the quasi H matrix; Secondly, based on the self-similarity and fractal dimension of discrete orthogonal signals, fractional signals can be achieved with the fractance circuit and further the NLFM signals can be constructed by fractional calculus theory; Thirdly, the sensitivity and stability of orthogonal signals are examined to provides theoretical support for the practical application of orthogonal signals. Finally, the proposed theory and method will be used in the orthogonal signal design of MIMO radar.

正交信号在雷达信号处理、通信理论、信息安全、等诸多领域有着非常重要和广泛的应用背景。相比智能算法只能得到近似正交或较低自（互）相关信号，用构造法能得到完全正交信号和具有最低自（互）相关信号，因而受到研究者们的极大关注。现有的构造方法只能得到特定码长的正交信号和由窗函数得到的非线性调频信号。本项目拟针对离散和连续型正交信号各自建立统一的构造方法和工具，搭建正交信号的构造理论和方法。首先，将现有Hadamard矩阵和低自相关序列的构造思想相结合，提出了通过有限正交递推方法构造具有特殊网络结构的正则偶图来得到H矩阵，并将此方法推广用于构造拟H矩阵，其次，利用正交信号的自相似性和分数维特征，结合分抗电路可用于实现分数阶信号，提出运用分数阶微积分理论构造非线性调频信号；再次，对正交信号进行灵敏性和稳定性分析，为正交信号的实际应用提供理论支撑；最后，将构造理论和方法应用于MIMO雷达的正交信号设计。

项目背景.正交信号在通信、雷达等中有着极为广泛的应用，相对于智能算法，构造法能得到完全正交信号而受到人们的广泛关注，在其中，构造正交二相码信号是现有研究的核心内容。一组正交信号可视为一正交矩阵的行（列），从而正交信号的构造可转换为Hadamard矩阵的构造问题。.主要研究内容.1..Hadamard矩阵的结构和基本性质特征；.2..基于Parseval关系的Hadamard矩阵构造；.3..低自相关、低互相关序列构造；.4..系统稳定性分析。.重要结果.1..建立了以循环矩阵为矩阵块的Hadamard矩阵构造基本框架。丰富了序列与关联多项式之间的Parseval关系，建立序列、向量序列、矩阵序列、四元数序列等与关联多项式之间的Parseval关系。.2..运用Parseval关系构造出一系列新结构的Hadamard矩阵；.3..结合四元数的Parseval关系，构造出一无穷族的低自相关四元数序列，并给出理论证明；.4..运用Parseval关系，构造出了一类具有低自相关和良好互相关性质的二相码序列；.5..给出了系统网络结构变化对输出的稳定性分析系列结果。.科学意义.1..Hadamard矩阵构造问题是一个世界性的难题。相关问题的解决不仅可以促进相关领域的发展，而且也为现实应用提供巨大支撑。.2..Hadamard矩阵构造和低自相关序列构造是两个不同的研究领域，以循环矩阵为矩阵块的Hadamard矩阵构造，从理论上将二领域联系起来，一方面具有低自相关的序列可以作为循环矩阵块来构造Hadamard矩阵，另一方面，Hadamard矩阵中循环矩阵块的两个约束条件(完美序列族和零负相关)，也为低自相关序列构造提供新的研究方向。.3..Hadamard矩阵的构造也为构造多个连续的MIMO雷达正交信号提供了新的思路和方法；.4..Hadamard矩阵的零负相关性为研究多个系统相互影响的稳定性提供了新的研究方向。