# A characterization of sparse nonstationary Gabor expansions

Research output: Working paperResearch

### 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 that occurs when thresholding the frame coefficients for signals belonging to the decomposition space.
Original language English ArXiv 25 Published - Jun 2016

### Fingerprint

Gabor Frames
Decompose
Banach Frames
Approximation Error
Coefficient
Thresholding
Upper bound
Norm

### Keywords

• Time-frequency analysis
• nonstationary Gabor frames
• decomposition spaces
• Banach frames
• nonlinear approximation

### Cite this

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title = "A characterization of sparse nonstationary Gabor expansions",
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 that occurs when thresholding the frame coefficients for signals belonging to the decomposition space.",
keywords = "Time-frequency analysis, nonstationary Gabor frames, decomposition spaces, Banach frames, nonlinear approximation",
author = "Ottosen, {Emil Solsb{\ae}k} and Morten Nielsen",
year = "2016",
month = "6",
language = "English",
publisher = "ArXiv",
type = "WorkingPaper",
institution = "ArXiv",

}

ArXiv, 2016.

Research output: Working paperResearch

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T1 - A characterization of sparse nonstationary Gabor expansions

AU - Ottosen, Emil Solsbæk

AU - Nielsen, Morten

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N2 - 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 that occurs when thresholding the frame coefficients for signals belonging to the decomposition space.

AB - 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 that occurs when thresholding the frame coefficients for signals belonging to the decomposition space.

KW - Time-frequency analysis

KW - nonstationary Gabor frames

KW - decomposition spaces

KW - Banach frames

KW - nonlinear approximation

M3 - Working paper

BT - A characterization of sparse nonstationary Gabor expansions

PB - ArXiv

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