In this paper, we present a connection between the discrete Gabor expansion and the evolutionary spectral theory. Including a scale parameter in the Gabor expansion, we obtain a new representation for deterministic signals that is analogous to the Wold-Cramer decomposition for non-stationary processes. The energy distribution resulting from the expansion is easily calculated from the Gabor coee...

Parametric modeling and estimation of complex valued homogeneous random fields with mixed spectral distributions is a fundamental problem in two-dimensional (2-D) signal processing. The parametric model under consideration results from the 2-D Wold-type decomposition of the random field. The same model naturally arises as the physical model in problems of space-time adaptive processing of airbo...

Perturbations (r τt)t≥0 of the semigroup of shifts (τt)t≥0 on L(R+) are studied under the assumption that r τt−τt belongs to a certain Schatten–von Neumann class Sp with p ≥ 1. It is shown that, for the unitary component in the Wold– Kolmogorov decomposition of the cogenerator of the semigroup (r τt)t≥0, any singular spectral type may be achieved by S1-perturbations. An explicit construction is...

This paper considers the problem of the achievable accuracy in jointly estimating the parameters of a complex-valued two-dimensional (2-D) Gaussian and homogeneous random field from a single observed realization of it. Based on the 2-D Wold decomposition, the field is modeled as a sum of purely indeterministic, evanescent, and harmonic components. Using this parametric model, we first solve a k...

Kurtosis of a time signal has been a popular tool for detecting nongaussianity. Recently, kurtosis as a function frequency defined in spectral domain has been successfully used in the fault detection of induction motors, machine bearings. A link between the nongaussianity and nonstationaity has been established through Wold-Cramer’s decomposition of a nonstationary signal, and the properties of...

Evanescent random fields arise as a component of the 2-D Wold decomposition of homogenous random fields. Besides their theoretical importance, evanescent random fields have a number of practical applications, such as in modeling the observed signal in the space time adaptive processing (STAP) of airborne radar data. In this paper we derive an expression for the rank of the low-rank covariance m...

We study a class of left-invertible operators which we call weakly concave operators. It includes the and some subclasses expansive strict $m$ -isometries with $m > 2$ . prove Wold-type decomposition for also obtain Berger–Shaw-type theorem analytic finitely cyclic The proofs these results rely heavily on spectral dichotomy provides fairly close relationship, written in terms reciprocal auto...

This article presents extensions of the Cramér-Wold theorem to measures that may have infinite mass near the origin. Corresponding results for sequences of measures are presented together with examples showing that the assumptions imposed are sharp. The extensions build on a number of results and methods concerned with injectivity properties of the Radon transform. Using a few tools from distri...

In 1948 H. Wold introduced an isometric isomorphism between a Hilbert (linear) space formed from the weighted shifts of a numerical sequence and a suitable Hilbert space of values of a second order stochastic sequence. Recently W.A. Gardner has resurrected the idea in the context of numerical cyclostationary sequences and processes , but without complete details. In this paper we present the de...

In 1948 H. Wold introduced an isometric isomorphism between a Hilbert (linear) space formed from the weighted shifts of a numerical sequence and a suitable Hilbert space of values of a second order stochastic sequence. Motivated by a recent resurrection of the idea in the context of cyclostationary sequences and processes, we present the details of the Wold isomorphism between cyclostationary s...