Publications
English only   BIG > Publications > Gaussian vs. Stochastic

CONTENTS
Home Page
News & Events
People
Publications
Tutorials and Reviews
Research
Demos

PDF
Postscript
All BibTeX References

# A Unified Formulation of Gaussian versus Sparse Stochastic Processes—Part II: Discrete-Domain Theory

## M. Unser, P.D. Tafti, A. Amini, H. Kirshner

### IEEE Transactions on Information Theory, vol. 60, no. 5, pp. 3036-3051, May 2014.

This paper is devoted to the characterization of an extended family of continuous-time autoregressive moving average (CARMA) processes that are solutions of stochastic differential equations driven by white Lévy innovations. These are completely specified by: 1) a set of poles and zeros that fixes their correlation structure and 2) a canonical infinitely divisible probability distribution that controls their degree of sparsity (with the Gaussian model corresponding to the least sparse scenario). The generalized CARMA processes are either stationary or nonstationary, depending on the location of the poles in the complex plane. The most basic nonstationary representatives (with a single pole at the origin) are the Lévy processes, which are the non-Gaussian counterparts of Brownian motion. We focus on the general analog-to-discrete conversion problem and introduce a novel spline-based formalism that greatly simplifies the derivation of the correlation properties and joint probability distributions of the discrete versions of these processes. We also rely on the concept of generalized increment process, which suppresses all long range dependencies, to specify an equivalent discrete-domain innovation model. A crucial ingredient is the existence of a minimally supported function associated with the whitening operator L; this B-spline, which is fundamental to our formulation, appears in most of our formulas, both at the level of the correlation and the characteristic function. We make use of these discrete-domain results to numerically generate illustrative examples of sparse signals that are consistent with the continuous-domain model.

@ARTICLE(http://bigwww.epfl.ch/publications/unser1402.html,
AUTHOR="Unser, M. and Tafti, P.D. and Amini, A. and Kirshner, H.",
TITLE="A Unified Formulation of {G}aussian {\textit{versus}} Sparse
Stochastic Processes---{P}art {II}: {D}iscrete-Domain Theory",
JOURNAL="{IEEE} Transactions on Information Theory",
YEAR="2014",
volume="60",
number="5",
pages="3036--3051",
month="May",
note="")


© 2014 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from IEEE.
This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder.