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Fourier transform representation by frequency-time wavelets

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1 Author(s)
A. M. Grigoryan ; Dept. of Electr. Eng., Univ. of Texas, San Antonio, TX, USA

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*-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.

Published in:

IEEE Transactions on Signal Processing  (Volume:53 ,  Issue: 7 )