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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.11.31.
Figure: Typical qualitative appearance of first three wavelets when the scale parameter is $ s=2$ .
\includegraphics[width=0.8\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:

$\displaystyle \omega _0\eqsp \pi\sqrt{2/\hbox{ln}2} \;\approx\; 5.336$ (12.119)

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 §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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Discrete Wavelet Transform
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Normalized STFT Basis