14.5 Two Sines

In this section we study the distribution of the empirical wavelet coefficients in time and frequency domain.


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The upper two plots represent two sine wave functions with different frequencies.

The lower two plots show the magnitude of wavelet coefficients with respect to their location in time domain ($ x$-axis) and resolution scale ($ y$-axis). The magnitude of wavelet coefficient is proportional to length of the bar.

We see that the character of the distribution of the wavelet coefficients strongly depends on the frequency of the signal. For the low frequency sine wave we have the largest wavelet coefficients allocated at the lower resolution scale. At the same time the largest wavelet coefficients of the high frequency sine wave are allocated at high resolution scales. Thus the wavelet transform reflects well the properties of the input signal in the frequency domain.

The interactive menu provides you with the possibilities to choose the type of wavelet basis (Haar, Daubechies 4, Coiflet 2) and to change the number of the wavelet coefficients at the first resolution scale. Changing the basis you can change the smoothness of the father and the mother functions. This is synonymous to changing the allocation of these functions in frequency domain. This can help to adjust the wavelet basis to the smoothness properties of the signal. For looking at the energy of the signal at a certain resolution scale use the Change level item and see what happens. The combination of these two possibilities leads to a better compression of the signal.


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