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Mean power spectrum

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Mean power spectrum

Figure 10: Mean power spectrum obtained by time-averagi ng the energy map at each duration. For non-periodic signals, the mean wavelet spectrum is similar to the Fourier spectrum.

Of course, the energy map can be integrated in time at each duration. The result of this operation (Fig. 10) is to distribute the energy of the signal among the durations - a concept identical to that of the Fourier power spectrum.

Analytically, by integrating the energy density over an integer number of periods, we get

The result is not as sharp as in the Fourier case, where the spike at frequency is due to the exceptionally good match between the analyzing function (a sine wave) and the signal - an occurence that is common in linear system response, but not in non-linear phenomena. The compromise we reached in allowing for single bump localization (something Fourier cannot do) reduces the spectral accuracy.

This can be expressed mathematically as follows. Denote by the Fourier transform of the signal, the Fourier transform of the Mexican hat wavelet. Then, substitutions and simple manipulations of the integrals give the alternative expression of the wavelet transform:

by which the wavelet transform is a band-pass filtered Fourier Transform. The shape of the wavelet

is reflected in the width of its spectrum

; as a narrow wavelet will have a broadband spectrum we see that the compromise between local and spectral localization cannot be avoided.

Jacques Lewalle
Mon Nov 13 10:51:25 EST 1995