线性正则变换域卷积理论及其应用研究

Linear canonical transform, as a generalized form of Fourier transforms and fractional Fourier transforms, has become a powerful tool for modern non-stationary signal analysis and processing, and has achieved fruitful research results in recent years. In order to further reduce computation and improve computing efficiency, convolution theory and application based on linear canonical transform has become one of the hot topic research in modern signal processing. Therefore, this project will mainly focus on the research of convolution theory and application based on linear canonical transform which includes the following four parts: (1) Study on the convolution theory for the linear canonical sine transform, linear canonical cosine transform domain. (2) Study on the convolution theory for the quaternion linear canonical transform domain. (3) Study on the convolution theory for multidimensional linear canonical transform domain. (4) Discuss the application in signal sampling, multiplicative filtering and the image encryption based on the achieved convolution and convolution theorem. The project which plays an important role in signal processing, can not only improve and perfect the convolution theory and method of linear canonical transform domain, but also further improve the modern signal processing theory system.

线性正则变换，作为傅里叶变换和分数阶傅里叶变换的广义形式，逐渐成为现代非平稳信号分析与处理强有力的工具之一，近年来，取得了丰硕的研究成果。为了进一步减少计算量和提高运算效率，线性正则变换的卷积理论与应用成为现代信号处理的研究热点之一。由此，本项目将系统地开展基于线性正则变换域卷积理论与应用研究，主要包括：（1）线性正则正（余）弦变换域卷积理论研究。（2）四元数线性正则变换域卷积理论研究。（3）高维线性正则变换域卷积理论研究。（4）探讨所得结果在信号采样、乘性滤波、图像加密中的应用研究。项目研究成果不仅能改进和完善线性正则变换域卷积理论与方法，而且能进一步完善现代信号处理理论体系，具有重要理论意义和应用价值。

本项目开展基于线性正则变换域卷积理论与应用研究，主要研究成果包括：.（1）线性正则正（余）弦变换域卷积理论研究：提出了分数阶正余弦变换卷积理论；提出了分数阶傅里叶正余弦拉普拉斯加权卷积理论；提出了分数傅里叶拉普拉斯卷积；提出了离散分数傅里叶正余弦级数理论；提出了线性正则正余弦卷积理论；并研究了利用卷积理论求解卷积类积分方程组的解。 .（2）四元数线性正则变换域卷积理论研究：定义了加窗四元数分数傅里叶变换，提出了相应的卷积理论及其快速算法；提出了四元数补偿线性正则变换卷积理论；提出了四元数线性正则小波变换的卷积理论；并根据根据卷积运算的性质，求解了一类偏微分方程的解。 .（3）高维线性正则变换域卷积理论研究：提出了n维线性正则小波变换的卷积理论；.（4）探讨所得结果在信号采样、乘性滤波、卷积类积分方程的求解等方面的应用研究：给出了均匀与周期非均匀高阶导数采样方法。得到了相应的带限信号的重构公式；利用所提出的卷积理论设计了一种新的滤波设计方法；利用所得卷积及其相应的卷积定理研究了几类卷积类积分方程，并给出了显式解，讨论了解的计算复杂度。.本项目研究成果不仅能改进和完善线性正则变换域卷积理论与方法，而且能进一步丰富现代非平稳信号处理理论体系。