Discrete Vector-Valued Nonuniform Gabor Frames

TL;DR

This paper studies discrete vector-valued nonuniform Gabor frames, establishing conditions for their existence, stability, and relationships with window sequences, expanding the understanding of nonuniform time-frequency analysis in signal processing.

Contribution

It introduces necessary and sufficient conditions for DVNUG Bessel sequences and frames, and explores their stability and structural properties in nonuniform signal spaces.

Findings

Characterization of DVNUG frames in nonuniform spaces

Stability of DVNUG frames under small perturbations

Arithmetic mean sequences form nonuniform Gabor frames

Abstract

Gabor frames have interested many mathematicians and physicists due to their potential applications in time-frequency analysis, in particular, signal processing. A Gabor system is a collection of vectors which is obtained by applying modulation and shift operators to non-zero functions in signal spaces. In many applications, for example, signal processing related to Gabor systems, the corresponding shifts may not be uniform. That is, the set associated with shifts may not be a group under usual addition. We analyze discrete vector-valued nonuniform Gabor frames (DVNUG frames, in short) in discrete vector-valued nonuniform signal spaces, where the indexing set associated with shifts may not be a subgroup of real numbers under usual addition, but a spectrum which is based on the theory of spectral pairs. First, we give necessary and sufficient conditions for the existence of DVNUG Bessel sequences in discrete vector-valued nonuniform signal spaces in terms of Fourier transformations of the modulated window sequences. We provide a characterization of DVNUG frames in discrete vector-valued nonuniform signal spaces. It is shown that DVNUG frames are stable under small perturbation of window sequences associated with given discrete vector-valued nonuniform Gabor systems. We observed that the arithmetic mean sequences associated with window sequences of a given DVNUG frame collectively constitutes a discrete nonuniform Gabor frame. Finally, we discuss an interplay between window sequences of DVNUG systems and their corresponding coordinates.