DWT Applications - Choice of phi(t)
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
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
 |
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.
Footer
More about this content: Metadata | Version History
This work is licensed by Phil Schniter under a Creative Commons Attribution License (CC-BY 1.0).
Last edited by Elizabeth Gregory on Jun 9, 2005 12:57 pm GMT-5.