A Sparse Representation of Random Signals

Abstract

Studies of sparse representations of deterministic signals have been well developed. Among other types there exists one called the adaptive Fourier decomposition (AFD) type for the analytic Hardy spaces. This type is recently further extended to the context of Hilbert spaces with a dictionary. Through the Hardy space decomposition of the space of L 2 -signals the AFD type algorithm gives rise to sparse representations of signals of finite energy. To deal with multivariate signals the Hilbert space context comes into play. The multivariate AFD counterpart in Hilbert spaces with a dictionary is called pre-orthogonal AFD (POAFD). In the present study we generalize AFD and POAFD to random analytic signals through formulating stochastic analytic Hardy spaces. To analyze random analytic signals we work on two models, both being called stochastic AFD, or SAFD in brief. The two models are respectively made for (i) expressible as the sum of a deterministic signal and an error term such as a white noise (SAFDI); and for (ii) being random analytic signals divided into different classes of signals obeying certain distributive law (SAFDII). In the second half of the paper we drop the analyticity assumption and generalize the SAFDI and SAFDII to what we call stochastic Hilbert spaces with a dictionary. The generalized methods are named as stochastic pre-orthogonal adaptive Fourier decompositions, SPOAFDI and SPOAFDII. Like the deterministic AFDs and POAFDs, the developed stochastic POAFD algorithms offer powerful tools to analyze random signals.