A Unified Formulation of Gaussian Versus Sparse Stochastic Processes-Part I: Continuous-Domain Theory

    Authors

    M. Unser; P. D. Tafti;Q. Y. Sun

    Comments

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    Abbreviated Journal Title

    IEEE Trans. Inf. Theory

    Keywords

    Sparsity; non-Gaussian stochastic processes; innovation modeling; continuous-time signals; stochastic differential equations; wavelet; expansion; Levy process; infinite divisibility; LINEAR INVERSE PROBLEMS; THRESHOLDING ALGORITHM; SIGNALS; TRANSFORM; ANALOG; Computer Science, Information Systems; Engineering, Electrical &; Electronic

    Abstract

    We introduce a general distributional framework that results in a unifying description and characterization of a rich variety of continuous-time stochastic processes. The cornerstone of our approach is an innovation model that is driven by some generalized white noise process, which may be Gaussian or not (e.g., Laplace, impulsive Poisson, or alpha stable). This allows for a conceptual decoupling between the correlation properties of the process, which are imposed by the whitening operator L, and its sparsity pattern, which is determined by the type of noise excitation. The latter is fully specified by a Levy measure. We show that the range of admissible innovation behavior varies between the purely Gaussian and super-sparse extremes. We prove that the corresponding generalized stochastic processes are well-defined mathematically provided that the (adjoint) inverse of the whitening operator satisfies some L p bound for p > = 1. We present a novel operator-based method that yields an explicit characterization of all Levy-driven processes that are solutions of constant-coefficient stochastic differential equations. When the underlying system is stable, we recover the family of stationary continuous-time autoregressive moving average processes (CARMA), including the Gaussian ones. The approach remains valid when the system is unstable and leads to the identification of potentially useful generalizations of the Levy processes, which are sparse and non-stationary. Finally, we show that these processes admit a sparse representation in some matched wavelet domain and provide a full characterization of their transform-domain statistics.

    Journal Title

    Ieee Transactions on Information Theory

    Volume

    60

    Issue/Number

    3

    Publication Date

    1-1-2014

    Document Type

    Article

    Language

    English

    First Page

    1945

    Last Page

    1962

    WOS Identifier

    WOS:000331902400042

    ISSN

    0018-9448

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