
In the present (
Hilbert space) setting, we can now easily define the
continuous wavelet transform in terms of its
signal basis set:

The parameter

is called a
scale parameter
(analogous to
frequency). The normalization by

maintains energy
invariance as a function of scale. We call

the
wavelet coefficient
at scale

and time

. The
kernel of the wavelet transform

is called the
mother
wavelet, and it typically has a
bandpass spectrum. A
qualitative example is shown in Fig.
11.31.
Figure:
Typical qualitative appearance
of first three wavelets when the scale parameter is
.
![\includegraphics[width=0.8\twidth]{eps/wavelets}](http://www.dsprelated.com/josimages_new/sasp2/img2341.png) |
The so-called
admissibility condition for a mother wavelet

is
Given sufficient decay with

, this reduces to

, that
is, the mother wavelet must be zero-mean.
The
Morlet wavelet is simply a
Gaussian-windowed complex
sinusoid:
The scale factor is chosen so that

. The
center
frequency 
is typically chosen so that second peak is half of
first:
 |
(12.119) |
In this case, we have

, which is close enough to
zero-mean for most practical purposes.
Since the scale parameter of a wavelet transform is analogous to
frequency in a
Fourier transform, a wavelet transform display is often
called a
scalogram, in analogy with an
STFT ``
spectrogram''
(discussed in §
7.2).
When the mother wavelet can be interpreted as a windowed
sinusoid
(such as the Morlet wavelet), the wavelet transform can be interpreted
as a
constant-Q Fourier transform.
12.5Before the theory of wavelets, constant-Q Fourier transforms (such as
obtained from a classic third-octave
filter bank) were not easy to
invert, because the
basis signals were not
orthogonal. See Appendix
E
for related discussion.
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