SC- and SS-wavelet transforms

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 2π/ω, ω 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 2π/(mω), m⩾1) than that (of length 2π/ω) 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.