Wavelet analysis as a wideband generalization of time-frequency analysis
Gerald Kaiser · The Journal of the Acoustical Society of America · 1998
The wavelet transform and the windowed Fourier transform are compared. Although these two look quite different in the time domain, they are very similar in the frequency domain. The only difference is that the windowed Fourier transform divides the frequency domain into bands of equal width, whereas the wavelet transform uses frequency bands of constant ratio. This is just how frequency bands occur in practice (in logarithmic rather than linear scale). Since low-frequency bands are narrow while high-frequency bands are wide in this scheme, the Nyquist sampling rates for low-frequency components of the signal are low while those for high-frequency components are high. The sampling rate in wavelet analysis is thus automatically adapted to the frequency range being analyzed. When both the signal and the analyzing wavelet are narrow band, it follows easily that the wavelet transform reduces to the windowed Fourier transform, the analyzing window for the latter being derived from the analyzing wavelet for the former. Therefore wavelet analysis may be viewed as a wideband generalization of ordinary time-frequency analysis, to which it reduces in the narrow-band limit. This view is especially useful in sonar.