Pseudo-WVD

The definition (4.5) requires the knowledge of the quantity

$\begin{displaymath}q_x(t,\tau)=x(t+\tau/2)\ x^*(t-\tau/2)\end{displaymath}$

from $\tau=-\infty$ to $\tau=+\infty$ , which can be a problem in practice. That is why we often replace $q_x(t,\tau)$ in (4.5) by a windowed version of it, leading to the new distribution:

$\begin{displaymath}PW_x(t,\nu)=\int_{-\infty}^{+\infty} h(\tau)\ x(t+\tau/2)\ x^*(t-\tau/2)\ e^{-j2\pi \nu \tau}\ d\tau\end{displaymath}$

where

is a regular window. This distribution is called the pseudo Wigner-Ville distribution (noted pseudo-WVD or PWVD in the following). This windowing operation is equivalent to a frequency smoothing of the WVD since

$\begin{displaymath}PW_x(t,\nu) = \int_{-\infty}^{+\infty} H(\nu-\xi)\ W_x(t,\xi)\ d\xi\end{displaymath}$

where $H(\nu)$ is the Fourier transform of

. Thus, because of their oscillating nature, the interferences will be attenuated in the pseudo-WVD compared to the WVD. However, the consequence of this improved readability is that many properties of the WVD are lost: the marginal properties, the unitarity, and also the frequency-support conservation; the frequency-widths of the auto-terms are increased by this operation.

* Example: The M-file tfrpwv.m calculates the pseudo-WVD of a signal, with the possibility to change the length and shape of the smoothing window. If we consider a signal composed of four gaussian atoms (obtained thanks to atoms.m), each localized at a corner of a rectangle,

     >> sig=atoms(128,[32,.15,20,1;96,.15,20,1;...
                       32,.35,20,1;96,.35,20,1]);

and compute its WVD (see fig. 4.4)

     >> tfrwv(sig);

**Figure 4.4:** WVD of 4 gaussian atoms : many interferences are present
$\begin{figure} \epsfxsize =10cm\epsfysize =8cm \centerline{\epsfbox{figure/en1fig4.eps}} \end{figure}$

we can see the four signal terms, along with six interference terms (two of them are superimposed). If we now compute the pseudo-WVD (see fig. 4.5),

     >> tfrpwv(sig);

**Figure 4.5:** The frequency-smoothing operated by the pseudo-WVD attenuates the interferences oscillating perpendicularly to the frequency axis
$\begin{figure} \epsfxsize =10cm\epsfysize =8cm \centerline{\epsfbox{figure/en1fig5.eps}}\end{figure}$

we can note the important attenuation of the interferences oscillating perpendicularly to the frequency axis, and in return the spreading in frequency of the signal terms.

Eric Chassande-Mottin 2005-10-26