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Analysis of a cosine wave

Next: Continuous wavelet transforms Up: No Title Previous: A wavelet

Analysis of a cosine wave

Take a particularly simple signal: a cosine wave. It is immaterial for this introduction that a Fourier analysis would, objectively, be better in this case (If you only have periodic signals in your lab, you don't need wavelets).

Now, we are going to stretch and shrink the wavelet along the time axis (we assume the data is in the time domain), and at each time compare the shape of the signal to that of the wavelet. The duration of the wavelet can be measured by the width of its trough, or similar measures: for mathematical reasons, we have adopted the distance between the minimum of and its first zero-crossing as the unit duration. To scan a full cycle of an oscillation will require approximately 4 durations of the Mexican hat (Fig. 3 shows two wavelets offset by 2 durations).

Figure 3: The wavelet (dotted line) is translated rela tive to the signal (solid line).

Let us in fact take this unit duration as one quarter of the period of the cosine wave. As we slide the wavelet over the signal, we have a pretty good match (but with the wrong signs) between the bumps around time zero, then a deterioration of the match as the signal and the wavelet are `out-of-phase', then a good match again (with the correct signs, this time), and so on (Fig. 4). Note that the matching at one time involves a window around that time: the localization is approximate.

Let us do this analytically. Let be the signal, and the wavelet at the approximately matching duration is . We can measure the `match' between the wavelet and the signal by the integral

with a preponderance of negative contributions initially.

To examine the function at a different time T, we have

giving successively increasing values, then decreasing, then increasing again, as T increases.

Figure 4: The dashed line plots the `match' between the wavelets (dotted line) and the signal (solid line) as a function of time.

Now, let us repeat this process with a wavelet of similar shape, but of much shorter or longer duration (Fig 5). This wavelet does not replicate the features of the signal very well. In fact, as the duration becomes shorter and shorter, the signal will be more nearly constant over the duration of the wavelet, and the integral will decrease in absolute value. A similar trend occurs if the wavelet has a much longer duration, say ): several oscillations of the signal will be included in the trough of the wavelet, with little net result. Thus we see that one of the results of the admissibility condition (section 3) is that the wavelet automatically subtracts the local mean value of the signal.

Figure 5: The `match' of the previous figure is explored for wavelets of different durations (dotted lines). The wavelet transform is shown in dashed lines. Maxima and minima are located correctlym but the largest values of the transform are obtained when the wavelet duration matches the period of the signal.

Next: Continuous wavelet transforms Up: No Title Previous: A wavelet

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