Inside Collection (Course): Digital Signal Processing (Ohio State EE700)

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.

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