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

Abstract

Abstract-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 continuousdomain model.

MMSE denoising of sparse and non-Gaussian AR(1) processes

“…In this section, we first present our extended class of nonGaussian (or sparse) continuous-time AR(1) processes using the innovation model proposed in [13] and [14]. The model boils down to the linear filtering of a non-Gaussian white noise (innovation) by using a suitable (1st order) shaping filter.…”

Sparse Stochastic Processes and Discretization of Linear Inverse Problems

“…To that end, the first step is to formulate a discrete version of the continuousdomain innovation model (14). Since, in practical applications, we are only given the samples s[k] k∈ of the signal, we obtain the discrete-domain innovation model by applying to them the discrete counterpart L d of the whitening operator L. The fundamental requirement for our formulation is that the composition of L d and L −1 results in a stable, shift-invariant operator whose impulse response is well localized [23] …”

“…We are also relying on Lebesgue's dominated convergence theorem to move the derivative with respect to ω inside the integral that defines f β ∨ L (ω). In particular, (23) implies that f β ∨ L is continuous and vanishes at the origin. The last step is to establish its conditional positive definiteness which is achieved by interchanging the order of summation.…”

Interpretation of continuous-time autoregressive processes as random exponential splines

“…Among the family of CAR models, the Gaussian ones are by far the most popular and the easiest to specify [1]. The family can also be extended to allow for non-Gaussian sparse models [2], [3], [4].…”