Theory, implementation and applications of nonstationary Gabor frames

Signal analysis with classical Gabor frames leads to a fixed time–frequency resolution over the whole time–frequency plane. To overcome the limitations imposed by this rigidity, we propose an extension of Gabor theory that leads to the construction of frames with time–frequency resolution changing o...

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Bibliographic Details
Published in:Journal of computational and applied mathematics 2011-10, Vol.236 (6), p.1481-1496
Main Authors: Balazs, P., Dörfler, M., Jaillet, F., Holighaus, N., Velasco, G.
Format: Article
Language:English
Subjects:
Adaptive representation
Algorithms
Audio signals
Computer Science
Constant-Q transform
Construction
Engineering Sciences
Filter banks
Frames
Invertibility
Mathematical models
Mathematics
Numerical Analysis
Signal and Image Processing
Time–frequency analysis
Transforms
Wavelet transforms
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Summary:Signal analysis with classical Gabor frames leads to a fixed time–frequency resolution over the whole time–frequency plane. To overcome the limitations imposed by this rigidity, we propose an extension of Gabor theory that leads to the construction of frames with time–frequency resolution changing over time or frequency. We describe the construction of the resulting nonstationary Gabor frames and give the explicit formula for the canonical dual frame for a particular case, the painless case. We show that wavelet transforms, constant-Q transforms and more general filter banks may be modeled in the framework of nonstationary Gabor frames. Further, we present the results in the finite-dimensional case, which provides a method for implementing the above-mentioned transforms with perfect reconstruction. Finally, we elaborate on two applications of nonstationary Gabor frames in audio signal processing, namely a method for automatic adaptation to transients and an algorithm for an invertible constant-Q transform.
ISSN:0377-0427
1879-1778
DOI:10.1016/j.cam.2011.09.011