基于线性正则变换的信号处理理论与关键技术研究

The linear canonical transform, which has received much attention recently, is a new kind signal processing tool, and it can be looked as the generalizations of the Fourier transform and fractional Fourier transform. It is more suitable for nonstationary signal processing because it can realize the extension and rotation of a signal in time frequency plane. This project will perform the search of signal processing theory and key technologies based on the linear canonical transform; it includes the following three parts. (1) The discretization method and fast algorithm of linear canonical transform: the one dimensional and two dimensional discrete linear canonical transform based on the uniform and nonuniform sampling points will be investigated in details. (2) The frequency modulation signal detection and parameter estimation methods in the linear canonical transform domain: carry out the research of parameters optimization method of linear canonical transform based on the properties of the frequency modulation signals in linear canonical transform domain, and then propose new methods for the detection and parameters estimation of frequency modulation signal. (3) Qualitative analysis of signal to noise ration in the linear canonical transform domain in noisy environment: on the basis of the characteristics of noise in the linear canonical transform domain, the quantitative analysis of the signal to noise ratio in the time-frequency plane is investigated, and propose the new methods for the detection and parameter estimation of the signal in noisy environment. This project can not only improve the linear canonical transform theory and the existing methods, but also further enrich the modern non-stationary signal analysis and processing system.

线性正则变换是近年来备受关注的一种新型信号处理工具，是傅里叶变换、分数阶傅里叶变换的广义形式，可以实现信号在时-频平面伸缩、旋转的变化，因此特别适合非平稳信号分析与处理。由此，本项目将系统地开展基于线性正则变换的信号处理理论与关键技术研究，主要内容包括：(1) 线性正则变换的离散化方法与快速算法：研究一维及二维线性正则变换的均匀、非均匀离散化方法及快速算法。(2)线性正则变换域调频信号检测与参数估计：在分析线性正则变换域调频信号特性的基础上，研究线性正则变换最优参数的选取方法，提出调频信号检测与参数估计的新理论与新方法；(3) 噪声环境下线性正则变换域信噪比分析：在研究线性正则变换域噪声特性基础上，给出调频信号在时-频平面上信噪比的定性定量分析，提出噪声环境下调频信号检测与参数估计的新方法。项目研究成果不仅能完善和改进线性正则变换理论与方法，而且还能丰富非平稳信号分析和处理理论体系。

本项目开展了线性正则变换(LCT)的信号处理理论和关键技术研究，主要研究成果包括：.（1）线性正则变换的离散化方法与快速算法研究：提出了存储空间小、计算效率高的分段线性正则变换快速算法；提出了加窗滑动线性正则变换快速算法；提出了具有灵活分辨率的离散线性正则变换算法；提出了基于Goertzel算法的单点LCT快速算法，分析了算法的计算复杂度。.（2）基于线性正则变换的调频信号检测与参数估计研究：定义了角度线性正则希尔伯特变换和四种不同形式的角度线性正则解析信号；建立了四元数线性正则变换的不确定性原理，分别提出了方向性、修正性和多域方差形式不确定性关系；提出了一种基于分数阶傅里叶变换的多项式相位信号的参数估计方法；针对有限长线性调频信号的参数估计问题，分别从时频分析和频谱定义两个角度得到了其简化线性正则变换域频谱特征，并根据此特征提出了基于该变换的参数估计方法。.（3）噪声环境下线性正则变换域信噪比的定性定量分析研究：给出了时间域噪声在离散线性正则变换域的方差矩阵分析，得到了频率域噪声信号模型表述方式；提出了含噪二次调频信号(QFM)信噪比的分数域定性定量分析方法，得到了QFM信号信噪比的解析表达式，提高了QFM信号在分数域的能量聚集程度及参数估计精度。.本项目的研究不仅能完善和改进线性正则变换理论与方法，而且还能丰富非平稳信号分析和处理理论体系。