The scalogram A first example of affine distribution is given by the scalogram (see section 3.4.2). Indeed, it is possible to express it as a smoothed version of the WVD: $$\vert T_x(t,a;\Psi)\vert^2 = \int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} W_x(s,\xi)\ W_{\Psi}\left(\frac{s-t}{a},a \xi\right)\ ds\ d\xi.$$ (4.13) Thus, the scalogram corresponds to the distribution of the affine class for which $\Pi(t,\nu)=W_{\Psi}(t,\nu)$. Expression (4.13), to be compared with expression (4.7), shows that the scalogram is the affine counterpart of the spectrogram. The scalogram validates properties 1. and 3. and is always positive. To illustrate the importance of the smoothing operated by $\Pi$ on the WVD of $x$, let us consider the case of a Morlet wavelet $\Psi$. If we note $\delta_T$ and $\delta_B$ the respectively time and frequency widths of the smoothing operated by the spectrogram of window $\Psi$ ($\delta_T$ and $\delta_B$ are constant values), these widths become variable with the frequency in the case of the scalogram: $$\delta_T(\nu)=\nu_0\ \delta_T/\nu\ ;\ \delta_B(\nu)=\nu\ \delta_B/\nu_0$$ ($\nu_0$ is the central frequency of the wavelet). This result, already made out in the context of the wavelet transform analysis, is a characteristic of any constant-Q analysis (see section 3.2.1): at a high frequency, since the signal changes rapidly, a short analysis window is sufficient, whereas at a low frequency, a large window is necessary to identify correctly the pulsation of the signal which changes slowly. However, the importance of the joint smoothing operated by the scalogram is still equivalent to the one of the spectrogram: $$\delta_T(\nu)\ \delta_B(\nu)\ =\ \delta_T\ \delta_B.$$ Besides, the trade-off between time and frequency resolutions, following from the Heisenberg-Gabor inequality and which applies to the spectrogram, is also valid for the scalogram. So as to see the effect of this frequency-dependent smoothing, we analyze with the scalogram (Morlet wavelet) a signal composed of two gaussian atoms, one with a low central frequency, and the other one with a high one (see fig. 4.18): >> sig=atoms(128,[38,0.1,32,1;96,0.35,32,1]); >> tfrscalo(sig); Figure 4.18: Morlet scalogram of 2 atoms : the time- and frequency- resolutions depend on the frequency (or scale) By default, the file tfrscalo.m uses an interactive mode in which you have to specify, from the plot of the spectrum, the approximate lower and higher frequency bounds, as well as the number of samples you wish in frequency (you should indicate here a lower frequency lower than 0.05 and a higher frequency greater than 0.4). The result obtained brings to the fore dependency, with regard to the frequency, of the smoothing applied to the WVD, and consequently of the resolutions in time and frequency.
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