论文标题
随机信号的稀疏表示
A Sparse Representation of Random Signals
论文作者
论文摘要
确定性信号稀疏表示的研究已经得到很好的发展。其中存在一种称为自适应傅立叶分解（AFD），该分解（AFD）是通过自适应选择一个在一个复合变量中定义Takeaka-Malmquist系统的参数的选择。 AFD型算法产生了有限能量信号的稀疏表示。 AFD的多元概括是一种称为正交前AFD（POAFD）的概括，后者是由具有词典的Hilbert Space建立的。本研究的目的是将AFD和POAFD概括为随机信号。我们使用两种类型的随机信号。一个是那些表达为具有误差项（例如白噪声）的确定性信号的总和。通常，另一个是遵守某些分布定律的几类随机信号的混合物。在本文的第一部分中，我们通过使用一个复杂变量的分析来开发一种为一维随机信号的AFD类型稀疏表示形式。在第二部分中，没有复杂的分析，我们在随机希尔伯特空间的背景下处理多元随机信号。像确定性信号案例中一样，建立的随机稀疏表示是实践信号分析中的强大工具。
Studies of sparse representation of deterministic signals have been well developed. Amongst there exists one called adaptive Fourier decomposition (AFD) established through adaptive selections of the parameters defining a Takenaka-Malmquist system in one-complex variable. The AFD type algorithms give rise to sparse representations of signals of finite energy. The multivariate generalization of AFD is one called pre-orthogonal AFD (POAFD), the latter being established with the context Hilbert space possessing a dictionary. The purpose of the present study is to generalize both AFD and POAFD to random signals. We work on two types of random signals. One is those expressible as the sum of a deterministic signal with an error term such as a white noise; and the other is, in general, as mixture of several classes of random signals obeying certain distributive law. In the first part of the paper we develop an AFD type sparse representation for one-dimensional random signals by making use analysis of one complex variable. In the second part, without complex analysis, we treat multivariate random signals in the context of stochastic Hilbert space with a dictionary. Like in the deterministic signal case the established random sparse representations are powerful tools in practical signal analysis.
