A Theory for Multiresolution Signal Decomposition: The Wavelet Representation
IEEE Transactions on Pattern Analysis and Machine Intelligence
Ten lectures on wavelets
Representation of the Fourier Transform by Fourier Series
Journal of Mathematical Imaging and Vision
Fourier transform representation by frequency-time wavelets
IEEE Transactions on Signal Processing
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Grigoryan [Fourier transform representation by frequency-time wavelets, IEEE Trans. Signal Process. 53 (7) (2005) 2489-2497] 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@p/@w, @w 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@p/(m@w), m=1) than that (of length 2@p/@w) 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.