12.15 Dwt applications - choice of phi(t)

Transforms are signal processing tools that are used to give a clear view of essential signal characteristics. Fouriertransforms are ideal for infinite-duration signals that contain a relatively small number of sinusoids: one cancompletely describe the signal using only a few coefficients. Fourier transforms, however, are not well-suited to signals ofa non-sinusoidal nature (as discussed earlier in the context of time-frequency analysis ). The multi-resolution DWT is a more general transform that is well-suited to a larger class ofsignals. For the DWT to give an efficient description of the signal, however, we must choose a wavelet $\psi (t)$ from which the signal can be constructed (to a good approximation) using only a few stretched and shifted copies.

We illustrate this concept in using two examples. On the left, we analyze a step-like waveform, while on the right we analyze a chirp-likewaveform. In both cases, we try DWTs based on the Haar and Daubechies db10 wavelets and plot the log magnitudes of the transform coefficients $$

c k d k d k − 1 d k − 2 … d 1

Observe that the Haar DWT yields an extremely efficient representation of the step-waveform: only a few of thetransform coefficients are nonzero. The db10 DWT does not give an efficient representation: many coefficients are sizable. This makessense because the Haar scaling function is well matched to the step-like nature of the time-domain signal. In contrast, theHaar DWT does not give an efficient representation of the chirp-like waveform, while the db10 DWT does better. This makes sense because the sharp edges of theHaar scaling function do not match the smooth chirp signal, while the smoothness of the db10 wavelet yields a better match.