Continuous wavelet transforms

Here, I give the conventional presentation (i.e. without my preferred scaling, so comparisons can be made by the reader), and application to one of the sample data files. The recent literature about wavelets mentions the inner product of the wavelet and the signal; I prefer the engineering terminology: convolution. I have used the conventional Mexican hat wavelet. Click here for the corresponding Matlab code. Note that the convolution is performed in the Fourier domain --- there is hardly ever a good reason to perform convolutions in the time domain. FFT makes this easy. Note, however, that this implies a periodic wrap-around of the signal; alternatives are well known (zero-padding, etc.), all somewhat arbitrary and affecting the wavelet coefficients near the ends of the signal (see discussion of cone of influence elsewhere in these notes).

The conventional Mexican hat wavelet transform is correct but distorts the relative importance of the coefficients. The problem (as I see it) is that the mathematicians have made a priority of having the wavelets at any scale contain the same energy – so-called L² wavelets for short. I prefer to emphasize the energy content of the signal (rather than the wavelet’s) in terms of energy density per octave: one of the main differences discussed in relation the Mexican hat pages: I reproduce the two figures here just to emphasize my choice: the normalization of the coefficients is the only difference. The justification for my choice is detailed in relation with the Mexican hat transform. The first picture, below, is the orthodox version; the second picture corresponds to my formulae, where the magnitude (color-coded) of the coefficients measures the multiscale decomposition of the signal (relative local contribution per octave, i.e. in logarithmic scale) and, when squared, its energy density (also logarithmic).

The distortion of these relative contributions in the orthodox version is corrected by the appropriate factors in the formulae: tradition is not a very good reason for such complications, in my opinion (matter of taste). The difference is illustrated by looking at the nearly periodic oscillation near time 25, where the period is estimated by the interval of 8 time units between successive maxima of the wavelet transform. The orthodox presentation gives a frequency of .8 Hz, and gives the impression that the higher frequency oscillations (~5 HZ) are somewhat less energetic, and the noise negligible. By contrast, my version shows a correct dominant frequency of the order of .15 Hz, and similar specific energy content (per octave) for all three major components of the signal (main wave, faster oscillations and noise). The information is the same, but I prefer the physical interpretation and simpler formulae in my version. See related pages for software and formulae.