Definitions and interpretation The idea of the continuous wavelet transform (CWT) is to project a signal $x$ on a family of zero-mean functions (the wavelets) deduced from an elementary function (the mother wavelet) by translations and dilations: $$T_x(t,a;\Psi)=\int_{-\infty}^{+\infty} x(s)\ \Psi^*_{t,a}(s)\ ds\,: \mbox{\it Continuous Wavelet Transform}$$ where $\Psi_{t,a}(s)=\vert a\vert^{-1/2}\ \Psi\left({s-t\over a}\right)$. The variable $a$ corresponds now to a scale factor, in the sense that taking $\vert a\vert>1$ dilates the wavelet $\Psi$ and taking $\vert a\vert<1$ compresses $\Psi$. By definition, the wavelet transform is more a time-scale than a time-frequency representation. However, for wavelets which are well localized around a non-zero frequency $\nu_0$ at scale $a = 1$, a time-frequency interpretation is possible thanks to the formal identification $\nu={\nu_0\over a}$. Figure 3.9: Time-scale atoms. The CWT is a projection of the analyzed signal on such atoms whose time duration is inversely proportional to the central frequency The basic difference between the wavelet transform and the short-time Fourier transform is as follows: when the scale factor $a$ is changed, the duration and the bandwidth of the wavelet are both changed but its shape remains the same. And in contrast to the STFT, which uses a single analysis window, the CWT uses short windows at high frequencies and long windows at low frequencies. This partially overcomes the resolution limitation of the STFT: the bandwidth $B$ is proportional to $\nu$, or $$\frac{B}{\nu}=Q\,: \mbox{ constant}.$$ We call it a constant-Q analysis. The CWT can also be seen as a filter bank analysis composed of band-pass filters with constant relative bandwidth.
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