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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:


$\displaystyle \displaystyle
\varphi_{s\tau}(t)$ $\displaystyle \isdef$ $\displaystyle \frac{1}{\sqrt{\vert s\vert}} f^\ast\left(\frac{\tau-t}{s}\right),
\qquad \tau,s,t \in (-\infty,\infty)$  
$\displaystyle X(s,\tau)$ $\displaystyle \isdef$ $\displaystyle \frac{1}{\sqrt{\vert s\vert}} \int_{-\infty}^{\infty} x(t)
f\left(\frac{t-\tau}{s}\right) dt$  

The parameter $ s$ is called a scale parameter (analogous to frequency). The normalization by $ 1/\sqrt{\vert s\vert}$ maintains energy invariance as a function of scale. We call $ X(s,\tau)$ the wavelet coefficient at scale $ s$ and time $ \tau$. The kernel of the wavelet transform $ f(t)$ 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 $ s=2$.
\includegraphics[width=\twidth]{eps/wavelets}

The so-called admissibility condition for a mother wavelet $ \psi(t)$ is

$\displaystyle C_\psi
= \int_{-\infty}^\infty \frac{\vert\Psi(\omega )\vert^2}{\vert\omega \vert}d\omega
< \infty .
$

Given sufficient decay with $ \omega$, this reduces to $ \Psi(0)=0$, that is, the mother wavelet must be zero-mean.

The Morlet wavelet is simply a Gaussian-windowed complex sinusoid:

$\displaystyle \psi(t)$ $\displaystyle \isdef$ $\displaystyle \frac{1}{\sqrt{2\pi}} e^{-j\omega _0 t} e^{-t^2/2}$  
$\displaystyle \;\longleftrightarrow\;\quad
\Psi(\omega )$ $\displaystyle =$ $\displaystyle e^{-(\omega -\omega _0)^2/2}$  

The scale factor is chosen so that $ \left\Vert\,\psi\,\right\Vert=1$. The center frequency $ \omega_0$ is typically chosen so that second peak is half of first:

$\displaystyle \omega _0\eqsp \pi\sqrt{2/\hbox{ln}2} \;\approx\; 5.336
$

In this case, we have $ \Psi(0)\approx 7\times10^{-7}\approx 0$, 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 §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 constant-Q Fourier transform.11.4Before 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 F for related discussion.


Previous: Normalized STFT Basis
Next: Discrete Wavelet Transform

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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.


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