000198001 001__ 198001
000198001 005__ 20190213064553.0
000198001 022__ $$a0018-9448
000198001 02470 $$2ISI$$a000331902400042
000198001 0247_ $$2doi$$a10.1109/Tit.2014.2298453
000198001 037__ $$aARTICLE
000198001 245__ $$aA Unified Formulation of Gaussian Versus Sparse Stochastic Processes-Part I: Continuous-Domain Theory
000198001 269__ $$a2014
000198001 260__ $$aPiscataway$$bIeee-Inst Electrical Electronics Engineers Inc$$c2014
000198001 300__ $$a18
000198001 336__ $$aJournal Articles
000198001 520__ $$9eng$$a We introduce a general distributional framework that results in a unifying description and characterization of a rich variety of continuous-time stochastic processes. The cornerstone of our approach is an innovation model that is driven by some generalized white noise process, which may be Gaussian or not (e.g., Laplace, impulsive Poisson, or alpha stable). This allows for a conceptual decoupling between the correlation properties of the process, which are imposed by the whitening operator L, and its sparsity pattern, which is determined by the type of noise excitation. The latter is fully specified by a Lévy measure. We show that the range of admissible innovation behavior varies between the purely Gaussian and super-sparse extremes. We prove that the corresponding generalized stochastic processes are well-defined mathematically provided that the (adjoint) inverse of the whitening operator satisfies some $ L _{ p } $ bound for p ≥ 1. We present a novel operator-based method that yields an explicit characterization of all Lévy-driven processes that are solutions of constant-coefficient stochastic differential equations. When the underlying system is stable, we recover the family of stationary continuous-time autoregressive moving average processes (CARMA), including the Gaussian ones. The approach remains valid when the system is unstable and leads to the identification of potentially useful generalizations of the Lévy processes, which are sparse and non-stationary. Finally, we show that these processes admit a sparse representation in some matched wavelet domain and provide a full characterization of their transform-domain statistics.
000198001 6531_ $$aSparsity
000198001 6531_ $$anon-Gaussian stochastic processes
000198001 6531_ $$ainnovation modeling
000198001 6531_ $$acontinuous-time signals
000198001 6531_ $$astochastic differential equations
000198001 6531_ $$awavelet expansion
000198001 6531_ $$aLevy process
000198001 6531_ $$ainfinite divisibility
000198001 700__ $$0240182$$aUnser, Michael$$g115227$$uEcole Polytech Fed Lausanne, Biomed Imaging Grp, CH-1015 Lausanne, Switzerland
000198001 700__ $$aTafti, Pouya D.$$uEcole Polytech Fed Lausanne, Biomed Imaging Grp, CH-1015 Lausanne, Switzerland
000198001 700__ $$aSun, Qiyu
000198001 773__ $$j60$$k3$$q1945-1962$$tIeee Transactions On Information Theory
000198001 8564_ $$uhttp://bigwww.epfl.ch/publications/unser1401.html$$zURL
000198001 8564_ $$uhttp://bigwww.epfl.ch/publications/unser1401.pdf$$zURL
000198001 8564_ $$uhttp://bigwww.epfl.ch/publications/unser1401.ps$$zURL
000198001 909C0 $$0252054$$pLIB$$xU10347
000198001 909CO $$ooai:infoscience.tind.io:198001$$pSTI$$pGLOBAL_SET$$particle
000198001 917Z8 $$x115226
000198001 937__ $$aEPFL-ARTICLE-198001
000198001 973__ $$aEPFL$$rREVIEWED$$sPUBLISHED
000198001 970__ $$aunser1401/LIB
000198001 980__ $$aARTICLE