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SC- and SS-wavelet transforms. (English) Zbl 1151.42313
Summary: A. M. Grigoryan [Fourier transform representation by frequency-time wavelets, IEEE Trans. Signal Process. 53 No. 7, 2489–2497 (2005; doi:10.1109/TSP.2005.849180)] has proposed an alternative representation of the Fourier transform, called A-wavelet transform. In that paper, the Cosine and Sine signals defined over one period have been used to develop the Cosine- and Sine-wavelet transforms and using those wavelet transforms the Fourier transform has been represented. For computing the Fourier transform at a given frequency, one does not require to compute the Cosine- and Sine-wavelet transforms at all time points in the time-frequency plane, but at specific time points that are separated out by \(2\pi /\omega , \omega \) is the frequency variable. In this paper, we propose SC- and SS-wavelet transforms that help representing the Fourier transform of a signal in a better way. The SC- and SS-wavelet transforms use the Cosine and Sine signals defined over the smaller intervals (of length \(2\pi /(m\omega ), m \geqslant 1\)) than that (of length \(2\pi /\omega \)) used in the A-wavelet transform. The SC- and SS-wavelet transforms not only give sharper time-frequency localization but also much more information in a better localized form than the A-wavelet transform.
MSC:
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
94A12 Signal theory (characterization, reconstruction, filtering, etc.)
65T50 Numerical methods for discrete and fast Fourier transforms
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