Separable smoothing The problem with the previous smoothing function $\Pi(s,\xi)=W_h(s,\xi)$ is that it is controlled only by the short-time window $h(t)$. If we add a degree of freedom by considering a separable smoothing function $$\Pi(t,\nu)=g(t)\ H(-\nu)$$ (where $H(\nu)$ is the Fourier transform of a smoothing window $h(t)$), we allow a progressive and independent control, in both time and frequency, of the smoothing applied to the WVD. The obtained distribution $$SPW_x(t,\nu)=\int_{-\infty}^{+\infty} h(\tau)\ \int_{-\infty}... ...-t)\ x(s+\tau/2)\ x^*(s-\tau/2)\ ds\ e^{-j2\pi \nu \tau}\ d\tau$$ is known as the smoothed-pseudo Wigner-Ville distribution (noted smoothed-pseudo-WVD or SPWVD). The previous compromise of the spectrogram between time and frequency- resolutions is now replaced by a compromise between the joint time-frequency resolution and the level of the interference terms: the more you smooth in time and/or frequency, the poorer the resolution in time and/or frequency. Note that if we only consider a smoothing in frequency i.e. if $g(t)=\delta(t)$, we obtain the pseudo-WVD. * Example: The signal that we consider here is composed of two components: the first one is a complex sinusoid (normalized frequency 0.15) and the second one is a Gaussian signal shifted in time and frequency: >> sig=fmconst(128,.15) + amgauss(128).*fmconst(128,0.4); If we display the WVD, the pseudo-WV and the smoothed-pseudo-WVD of this signal (see fig. 4.8, fig. 4.9 and fig. 4.10), >> tfrwv(sig); >> tfrpwv(sig); >> tfrspwv(sig); Figure 4.8: WVD of a signal composed of a gaussian atom and a complex sinusoid. Interferences are present between the two components Figure 4.9: Pseudo-WVD of the same signal : the frequency smoothing done by the pseudo-WVD degrades the frequency resolution without really attenuating the interferences Figure 4.10: Smoothed-pseudo-WVD of the same signal : the time-smoothing carried out by the smoothed-pseudo-WVD considerably reduces these interferences we can make the following remarks: from the WVD, we can see the two signal terms located at the right positions in the time-frequency plane, as well as the interference terms between them. As these interference terms oscillate globally perpendicularly to the time-axis, the frequency smoothing done by the pseudo-WVD degrades the frequency resolution without really attenuating the interferences. On the other hand, the time-smoothing carried out by the smoothed-pseudo-WVD considerably reduces these interferences; and as the time resolution is not of fundamental importance here, this representation is suitable for this signal. An interesting property of the smoothed-pseudo WVD is that it allows a continuous passage from the spectrogram to the WVD, under the condition that the smoothing functions $g$ and $h$ are gaussian. The time-bandwidth product then goes from 1 (spectrogram) to 0 (WVD), with an independent control of the time and frequency resolutions. This is clearly illustrated by the function movsp2wv.m, which considers different transitions, on a signal composed of four atoms. To visualize these snapshots, load the mat-file movsp2wv (obtained by running movsp2wv.m; but as it takes a long time to run, we saved the result in a mat file) and run movie (see fig. 4.11): >> load movsp2wv >> clf; movie(M,10); Figure 4.11: Different transitions from the spectrogram to the WVD, using the smoothed-pseudo-WVD. The signal is composed of 4 gaussian atoms This movie shows the effect of a (time/frequency) smoothing on the interferences and on the resolutions: the WVD gives the best resolutions (in time and in frequency), but presents the most important interferences, whereas the spectrogram gives the worst resolutions, but with nearly no interferences; and the smoothed-pseudo WVD allows to choose the best compromise between these two extremes.
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