Fourier transform representation by frequency-time wavelets

A new concept of the A-wavelet transform is introduced, and the representation of the Fourier transform by the A-wavelet transform is described. Such a wavelet transform uses a fully scalable modulated window but not all possible shifts. A geometrical locus of frequency-time points for the A-wavelet...

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Bibliographic Details
Published in:IEEE transactions on signal processing 2005-07, Vol.53 (7), p.2489-2497
Main Author: Grigoryan, A.M.
Format: Article
Language:English
Subjects:
Applied sciences
Biomedical signal processing
Detection, estimation, filtering, equalization, prediction
Discrete Fourier transforms
Exact sciences and technology
Fourier transform
Fourier transforms
Frequency
Information, signal and communications theory
Inverse
Optimization
Planes
Representations
short-time Fourier transform
Signal analysis
Signal and communications theory
Signal processing
Signal resolution
Signal, noise
Speech analysis
Telecommunications and information theory
Transforms
Wavelet analysis
wavelet transform
Wavelet transforms
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Description
Summary:A new concept of the A-wavelet transform is introduced, and the representation of the Fourier transform by the A-wavelet transform is described. Such a wavelet transform uses a fully scalable modulated window but not all possible shifts. A geometrical locus of frequency-time points for the A-wavelet transform is derived, and examples are given. The locus is considered "optimal" for the Fourier transform when a signal can be recovered by using only values of its wavelet transform defined on the locus. The inverse Fourier transform is also represented by the A/sup */-wavelet transform defined on specific points in the time-frequency plane. The concept of the A-wavelet transform can be extended for representation of other unitary transforms. Such an example for the Hartley transform is described, and the reconstruction formula is given.
ISSN:1053-587X
1941-0476
DOI:10.1109/TSP.2005.849180