  This paper is devoted to the characterization of an extended family of CARMA
(continuous-time autoregressive moving average) processes that are solutions of
stochastic differential equations driven by white Levy 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 non-stationary, depending on the location of the poles
in the complex plane. The most basic non-stationary representatives (with a
single pole at the origin) are the Levy 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.
