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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 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
.
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.
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