The scalogram A similar distribution to the spectrogram can be defined in the wavelet case. Since the continuous wavelet transform behaves like an orthonormal basis decomposition, it can be shown that it preserves energy: $$\int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} \vert T_x(t,a;\Psi)\vert^2\ dt\ \frac{da}{a^2} = E_x$$ where $E_x$ is the energy of $x$. This leads us to define the scalogram of $x$ as the squared modulus of the continuous wavelet transform. It is an energy distribution of the signal in the time-scale plane, associated with the measure $dt\ {da\over a^2}$. As for the wavelet transform, time and frequency resolutions of the scalogram are related via the Heisenberg-Gabor principle: time and frequency resolutions depend on the considered frequency. To illustrate this point, we represent the scalograms of two different signals. The M-file tfrscalo.m generates this representation. The chosen wavelet is a Morlet wavelet of 12points. The first signal is a Dirac pulse at time $t_0=64$: >> sig1=anapulse(128); >> tfrscalo(sig1,1:128,6,0.05,0.45,128,1); Figure 3.19: Morlet scalogram of a Dirac impulse at time $t=64$ : time resolution depends on the considered frequency (or scale) Figure 3.19 shows that the influence of the behavior of the signal around $t=t_0$ is limited to a cone in the time-scale plane: it is "very" localized around $t_0$ for small scales (large frequencies), and less and less localized as the scale increases (as the frequency decreases). The second signal is the sum of two sinusoids of different frequencies (see fig. 3.20): >> sig2=fmconst(128,.15)+fmconst(128,.35); >> tfrscalo(sig2,1:128,6,0.05,0.45,128,1); Figure 3.20: Morlet scalogram of two simultaneous complex sinusoids : frequency resolution depends on the considered frequency (or scale) Here again, we notice that the frequency resolution is clearly a function of the frequency: it increases with $\nu$. The interference terms of the scalogram, as for the spectrogram, are also restricted to those regions of the time-frequency plane where the corresponding auto-scalograms (signal terms) overlap. Hence, if two signal components are sufficiently far apart in the time-frequency plane, their cross-scalogram will be essentially zero.
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