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Continuous Wavelet Transform
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.
10.32.
Figure:
Typical qualitative appearance
of first three wavelets when the scale parameter is .

The socalled admissibility condition for a mother wavelet
is
Given sufficient decay with
, this reduces to
, that
is, the mother wavelet must be zeromean.
The Morlet wavelet is simply a Gaussianwindowed complex sinusoid:
The scale factor is chosen so that
. The
center
frequency is typically chosen so that second peak is half of
first:
In this case, we have
, which is close enough to
zeromean 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 §6.2).
When the mother wavelet can be interpreted as a windowed sinusoid
(such as the Morlet wavelet), the wavelet transform can be interpreted
as a constantQ Fourier transform.^{11.4}Before the theory of wavelets, constantQ Fourier transforms (such as
obtained from a classic thirdoctave filter bank) were not easy to
invert, because the basis signals were not orthogonal. See Appendix F
for related discussion.
Previous: Normalized STFT BasisNext: Discrete Wavelet Transform
About the Author: Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at
Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See
http://ccrma.stanford.edu/~jos/ for details.