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DWT Applications - Choice of phi(t)

Module by: Phil Schniter. E-mail the author

Transforms are signal processing tools that are used to give a clear view of essential signal characteristics. Fourier transforms are ideal for infinite-duration signals that contain a relatively small number of sinusoids: one can completely describe the signal using only a few coefficients. Fourier transforms, however, are not well-suited to signals of a 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 of signals. For the DWT to give an efficient description of the signal, however, we must choose a wavelet ψt ψ 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 Figure 1 using two examples. On the left, we analyze a step-like waveform, while on the right we analyze a chirp-like waveform. 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 T d k T d k 1 T d k 2 T d 1 T c k d k d k 1 d k 2 d 1 .

Figure 1
Figure 1 (step_chirp_wave.png)

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

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