Wavelet Transform, Wavelets.

Cuthbert Nyack

Fourier analysis provides a description of signals in terms of sinusoids and is ideal for signals whose frequency content does not change with time. One way of applying Fourier analysis to signals with time varying spectra is to use the STDT (Short Time Fourier Transform) where the signal is broken up into short segments and Fourier Analysis applied to each section. This will be done in DSP using the FFT.

Here a different approach is used. In the frequency domain, a sinusoid occupies 1 point on the frequency axis and in the time domain it spreads out from -¥ to + ¥. One way to do analysis of signals is to use a function which extends over a finite range of frequency and time. The continuous wavelet transform is such an approach. Unfortunately some lack of mathematical clarity exists here and the result has been a vast literature addressing questions like :- what is an acceptable wavelet, types of wavelets, how to construct wavelets, orthogonality, relation of wavelets to other concepts, reconstruction of signal from wavelet etc. Here we introduce 2 wavelets, the Mexican Hat(MH) and the Gabor(GB). The MH wavelet is suited for analysis of fine scale transients in signals while the GB wavelet is suited for signals with sections looking "sinusoidal" like audio. After introducing the wavelets, the GB wavelets is used to produce Scalograms of some basic signals.
The equation describing the MH wavelet is shown below:-

This wavelet has 2 variables t and w, the width of the wavelet.

The equation for the GB wavelet is shown below. It has 3 variables t, w(width) and n(frequency). This wavelet is complex and is one of a class of non orthogonal wavelets. Since the Fourier Transform of a gaussian is also a gaussian, then it is reasonable to use a gaussian for a function required for a time frequency representation of a signal. The GB wavelet in the frequency domain is a gaussian translated along the frequency axis.

From the basic wavelet, scaled wavelets can be produced using the equation below:-

The applet below show the MH and GB wavelets and indicates how the parameters s, w and n affect both the time and frequency domain description of the wavelet.
Both s and w have a similar effect. Increasing both of them widens the wavelet in the time domain and narrows it in the frequency domain. In the frequency domain, the center frequency is also reduced. The opposite happens when s and w are reduced. The parameter n applies to the GB wavelet and changes the period of the oscillations in the time domain while translating the spectrum along the frequency axis without affecting its width. The spectrum of the GB wavelet is also a gaussian.

When enabled the following gif file show how the applet should appear:-

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