信号的解绕Fourier展开的理论研究以及应用

The aim of the project is both the theoretical study on and practical applications of the unwinding Fourier decomposition that is to decompose signals into components with positive analytic instantaneous frequencies and fast convergence. The high speed convergence rate of unwinding expansion has been observed through experiments, but has not been proved via mathematical methods. One of the topics of the project is to prove, perhaps with some mild conditions, the exponentially convergent rate; or otherwise weaker results based on theoretical analysis. The second topic is the algorithm of the unwinding Fourier expansion, in particular the computation of outer functions. We are to use certain mechanical quadrature methods for the computation. The third topic is higher dimensional analogy of the one-D unwinding expansion. Due to the need of image processing, and higher dimensional system identification, and interesting differences between one and several variables (non-existence of Blaschke product in higher dimensions, etc.), higher dimensional unwinding algorithms would have wide interest and applications. The fourth topic is applications of one and higher dimensional unwinding Fourier expansion. The application part would include biomedical signal (heart and brain electrical signals) and voice and music signal characterization, texture recognition, system identification, etc.

项目的目的是对解绕 Fourier 正频率快速分解进行进一步的理论研究及实际应用。解绕分解收敛之快速性已经实验证实， 但未经理论证明。项目目标之一是证明，或加适当条件后证明，解绕分解具有快如指数阶的收敛速度， 抑或证明其反面，以及什么原因导致达不到有如此快速的收敛速度。目标之二是解绕的实际计算，特别是外函数的计算，拟用机械求积的方法进行计算。目标之三是高维理论。由于图像理论的需要， 以及高维系统辨识的需要，及高维理论与一维相比的相异特征（Blaschke 乘积的不存在性）等， 高维解绕理论具有广泛的兴趣及难度。 目标之四是一维及高维解绕正频率快速分解的应用，主要包括生物医学信号刻画（心电及脑电信号的特征提取）、 语音识别及音乐分类、质料鉴别、控制论（系统辨识）等领域的决定性的应用。在这几个重要应用方向，达到快速逼近及特征提取之目的。

本项目的目的是对解绕 Fourier 正频率快速分解进行进一步的理论研究及实际应用，即研究信号的一种展开方式，解绕展开算法， 此算法目的是将信号展开成具有正的解析相位导数即瞬时频率的信号之和。此项目是近年来调和分析和信号分析的热门题目。我们主要研究一维情况下以及高维情况下信号解绕展开的实际计算。问题的关键是外函数的计算，我们通过使用机械求积的方法进行计算Hilbert变换，进而计算外函数，并在这种情况下得到信号的解绕展开，实验数据表明这种方法使得Hilbert变换以及外函数的计算具有更高的稳定性。