A Characterization of Sparse Nonstationary Gabor Expansions

TL;DR

This paper explores the theoretical foundations of sparse, flexible time-frequency representations using nonstationary Gabor frames, establishing their properties within decomposition spaces and analyzing approximation errors.

Contribution

It introduces a framework linking nonstationary Gabor frames with decomposition spaces, proving Banach frame properties and sparseness characterizations.

Findings

Nonstationary Gabor frames form Banach frames for certain decomposition spaces.

Decomposition space norms are characterized by sparseness of frame coefficients.

Thresholding frame coefficients yields bounded approximation errors for signals in the space.

Abstract

We investigate the problem of constructing sparse time-frequency representations with flexible frequency resolution, studying the theory of nonstationary Gabor frames in the framework of decomposition spaces. Given a painless nonstationary Gabor frame, we construct a compatible decomposition space and prove that the nonstationary Gabor frame forms a Banach frame for the decomposition space. Furthermore, we show that the decomposition space norm can be completely characterized by a sparseness condition on the frame coefficients and we prove an upper bound on the approximation error occurring when thresholding the frame coefficients for signals belonging to the decomposition space.