The affine smoothed pseudo Wigner distribution: separable kernel One way to overcome the trade-off between time and frequency resolutions of the scalogram is, as for the smoothed-pseudo-WVD, to use a smoothing function which is separable in time and frequency. The resulting distribution is called the affine smoothed pseudo Wigner distribution (noted ASPWD), and writes $$ASPW_x(t,a) = \frac{1}{a} \int\int_{-\infty}^{+\infty} h\left(\fr... ...t(\frac{s-t}{a}\right)\ x(s+\frac{\tau}{2})\ x^*(s-\frac{\tau}{2})\ ds\ d\tau.$$ (4.14) It allows a flexible choice of time and scale resolutions in an independent manner through the choice of the windows $g$ and $h$. Properties 1. and 3. (see page ) are satisfied by this distribution provided that $g$ is real and $h$ is hermitian. As for the SPWVD (see section 4.1.2), the ASPWD allows a continuous passage from the scalogram to the WVD, under the condition that the smoothing functions g and h are gaussian. The time-bandwidth product then goes from 1 (scalogram) to 0 (WVD), with an independent control of the time and frequency resolutions. This is illustrated by the function movsc2wv.m, which considers different transitions, on a signal composed of four atoms. To visualize these snapshots, load the mat-file movsc2wv and run movie (see fig. 4.19): >> load movsc2wv >> clf; movie(M,10); Figure 4.19: Different transitions between the scalogram and the WVD thanks to the ASPWD. The analyzed signal is composed of 4 gaussian atoms Here again, the WVD gives the best resolutions (in time and in frequency), but presents the most important interferences, whereas the scalogram gives the worst resolutions, but with nearly no interferences; and the ASPWD allows to choose the best compromise between these two extremes. To summarize, we have seen that on one hand, the spectrogram is a time-frequency distribution obtained from the WVD by smoothing, and that on the other hand, the scalogram is a time-frequency distribution obtained from the WVD by affine smoothing. The WVD is therefore at the intersection of both classes of time-frequency and time-scale distributions. Besides, it is possible to construct a continuous transition from the spectrogram to the scalogram via the WVD, by changing the smoothing function $\Pi$ acting on the WVD. The equivalent area of such function $\Pi$ will vary from zero (we then obtain the "unsmoothed" WVD) to a limit fixed by the Heisenberg-Gabor uncertainty principle (spectrogram and scalogram). This choice corresponds to using the SPWVD's or the ASPWD's with gaussian smoothing functions. The time-bandwidth product then runs from 0 (WVD) to 1 (spectrogram or scalogram) and truly controls both transitions. Figure 4.20 illustrates different transitions between the spectrogram and the scalogram on a synthetic signal composed of three gaussian atoms, for different values of $BT$. Figure 4.20: From the spectrogram to the scalogram via the WVD This analysis brings us to the conclusion that, instead of looking at the two extreme representations (spectrogram and scalogram) separately, a deeper insight can be gained by considering a whole continuum between the two extremes, with the WVD as a necessary intermediate step. Moreover, the transition allows a trade-off between joint resolutions and interferences reduction.
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