#### 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.11.31.

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*.^{12.5}Before 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.

**Next Section:**

Discrete Wavelet Transform

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Normalized STFT Basis