Nonstationary Gabor frames - approximately dual frames and reconstruction errors

Abstract

Nonstationary Gabor frames, recently introduced in adaptive signal analysis, represent a natural generalization of classical Gabor frames by allowing for adaptivity of windows and lattice in either time or frequency. Due to the lack of a complete lattice structure, perfect reconstruction is in general not feasible from coefficients obtained from nonstationary Gabor frames. In this paper it is shown that for nonstationary Gabor frames that are related to some known frames for which dual frames can be computed, good approximate reconstruction can be achieved by resorting to approximately dual frames. In particular, we give constructive examples for so-called almost painless nonstationary frames, that is, frames that are closely related to nonstationary frames with compactly supported windows. The theoretical results are illustrated by concrete computational and numerical examples.

Approximately dual frames of nonstationary Gabor frames for l2(Z) and reconstruction errors

“…12 State-of-the-art results on NSG frames for L 2 (R) are collected in previous studies. [12][13][14][15][16] In Balazs et al, 13 the authors determined a simple sufficient condition for an NSG system with compactly supported window functions to constitute a frame for L 2 (R). The condition guarantees that the frame operator is a multiplication operator and thus easily inverted.…”

“…The painless NSG frames have been used in realizing various time-or frequency-adaptive transforms. 13,[17][18][19][20][21] The theory of NSG frames beyond the painless case was developed in Dörfler and Matusiak, 14,15 where NSG frames with fast decaying window functions were considered and approximately dual frames were derived from sufficiently close known NSG frames. Note that the existence of a dual frame with the same structure is not guaranteed for general NSG frames.…”

“…Due to the lack of a complete lattice structure, perfect reconstruction is in general not feasible from NSG frame coefficients. Inspired by Dörfler and Matusiak, 15 we resort to approximately dual frames of NSG frames for l 2 (Z) and estimate the reconstruction errors. We provide different approximately dual frames of NSG frames under some conditions and give the reconstruction error bounds (see Theorems 3.1 and 3.2).…”

Nonstationary Gabor frames for

“…State‐of‐the‐art results on NSG frames for are collected in other studies . In Ref., the authors determined a simple sufficient condition for an NSG system with compactly supported window functions to constitute a frame for .…”

“…12 State-of-the-art results on NSG frames for L 2 (R) are collected in other studies. [12][13][14][15] In Ref., 13 the authors determined a simple sufficient condition for an NSG system with compactly supported window functions to constitute a frame for L 2 (R). The condition guarantees that the frame operator is a multiplication operator and thus easily inverted.…”