Continuous wavelet transforms
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Therefore, we see that some spectral resolution is achieved
by selection of the wavelet size, and some temporal resolution
follows from the location of the wavelet relative to the
signal (Fig. 6).
Figure 6: Contour line representation of the wavelet transform for
a near-continuum of durations.
We can identify the individual peaks and troughs
of the cosine wave as well as their frequency: this compromise
between the temporal domain and the spectral domain is the
hallmark of wavelet transforms. The cost of this compromise
is the need to map the signal as a function of both time
The normalization factor is introduced for later convenience.
In the case of our cosine function, cos(at) in general, the transform can be calculated
exactly, and is
from which we can see the basic cosine echoing the original
signal, the power-law showing a steep decrease in
magnitude toward larger 's (short durations) and the
exponential drop-off toward small 's (long durations)
Of course, the time () and
duration () axis are discretized in
actual data processing. The result, a wavelet map, is shown
on Fig. 6.
Mon Nov 13 10:51:25 EST 1995