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Wavelet Transform

  • The wavelet transform (WT) is a method to search for transient trains of rhythms in non-stationary EEG records. The WT provides a compromise between time-domain and frequency-domain localization of a train. As FT, the WT is a decomposition of EEG into a sum of orthogonal signals, these special signals now being not sine waves, but a family of short oscillatory trains of various duration and frequency content, the so called wavelets.

As it was stated earlier in the section describing the FT, any continuos signal can be decomposed into a sum of mutually orthogonal signals. For classical FT these signals are sine waves of different frequencies, amplitudes and phases. It is possible to use, however, other sets of mutually orthogonal signals, since this class of signals is wide. What particular kind of signals to chose as a basis for decomposition depends on the kind of elementary components that are expected to be the most interesting part of the analyzed EEG. If the components of interest are sustained rhythms that exist during comparatively long time (comparable with the analysis epoch) or if the time of appearance of transient rhythms inside the analyzed epoch is not the point of interest, then the choice of sine waves (and hence classical FT) is reasonable. If the EEG signal is supposed to be essentially non-stationary and comprised of many transient rhythmic trains, and if, plus to this, we are interested not only in the frequency of these transient oscillations but also in the position of rhythmic trains on a time scale, then the family of wavelets is a choice.

FT provides a detailed frequency information on a signal in a given time interval, but completely looses information on how the frequency content changes with time inside the interval. In contrast, WT finds a compromise between time-domain and frequency domain localization of rhythmic trains and hence gives the knowledge on both time and frequency structure of a signal inside the analysed window, although with some limited resolution. A series of FT windows (the windowed FT) may also provide this dual information but in this case the resolution in both time and frequency domain is fixed, does not depend on the analyzed frequency, but depends entirely on the width of the window. One of the advantages of WT upon windowed FT is that WT’s time resolution is variable and depends on the frequency of a component — for high frequencies the time resolution is better than for low frequencies.

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Figure. Three different wavelet bases,   Morlet, Paul,  Mexican hat (respectively). The plots give the real part (solid) and imaginary part (dashed) for the wavelets in the time domain. All other wavelets in a set are compressed/dilated and translated copies of a mother wavelet.

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A full set of wavelet functions, used for decomposition, consists of a so called mother wavelet and a set of its time shifted and compressed/dilated copies. A mother wavelet is some zero mean function localized both in frequency and time domains, e.g., oscillatory signal amplitude modulated by a bell function. The particular examples of mother wavelets are Morelet wavelet, Mexican hat function (Figure 3), etc (Torrence and Compo 1998). Any compressed/dilated wavelet is characterized by a scale parameter (s) which is actaully a dilation coefficient and is an inverse of frequency (larger scales correspond to dilated signals and smaller scales correspond to compressed signals). As it was mentioned earlier, the certain sets of such functions may serve as full orthogonal bases for decomposition of arbitrary signal defined in a given time interval (this may be mathematically strictly proved (Daubechies 1992)).

The computation of wavelet transform consists of the follows steps:

  1. The mother wavelet is chosen to serve as a prototype for all other waveltets in the process.
  2. A definite set of scale parameters s is found.
  3. For every s, a definite set of time shifts t is found.
  4. For every given s and for every given t , the scalar product of the corresponding wavelet with the given realization of EEG is calculated (a scalar product is a sum of count-by-count products of these functions).
  5. The procedure is repeated for every s and for every t . The result is a (descrete) function of two variables — s and t — defined on st plane. Since s corresponds to (inverse) frequency and t corresponds to time delay, this function describes the time-frequency characteristics of the original EEG trial. The high value of this function at some s and t indicates that a rhythmic train appeared at approximately this frequency (defined by s) and at approximately this time.

It should be noted, that

  • at any scale s the wavelet function has not one frequency, but a band of frecuencies, and the bandwidth is inverse proportional to s. That means, that the finer the resolution in time domain is, the less is resolution in frequency domain, and vice versa;
  • s-1 is not exactly the frequency, at which the spectrum of wavelet function reaches it’s maximum value, and the relation between this two values depends on the type of mother wavelet;
  • the choice of different mother wavelet results in different values of wavelet transform but qualitative results are similar.

In EEG studies the WT makes it possible to describe time-frequency characteristics of both transients in spontaneous EEG and time-varying rhythms in event related brain activity (Schiff et al 1994). The increase of time resolution of the method with frequency makes it especially worth-while for the analysis of high-frequency (gamma) EEG band (Tallon-Baudry et al 1996).

 

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