Definition

A time-frequency energy distribution which is particularly interesting is the Wigner-Ville distribution (WVD) defined as:

$\displaystyle W_x(t,\nu)=\int_{-\infty}^{+\infty} x(t+\tau/2)\ x^*(t-\tau/2)\ e^{-j2\pi \nu \tau}\ d\tau,$

(4.5)

or equivalently as

$\begin{displaymath}W_x(t,\nu)=\int_{-\infty}^{+\infty} X(\nu+\xi/2)\ X^*(\nu-\xi/2)\ e^{j2\pi \xi t}\ d\xi.\end{displaymath}$

This distribution satisfies a large number of desirable mathematical properties, as summarized in the next sub-section. In particular, the WVD is always real-valued, it preserves time and frequency shifts and satisfies the marginal properties.

An interpretation of this expression can be found in terms of probability density: expression (4.5) is the Fourier transform of an acceptable form of characteristic function for the distribution of the energy.

Before looking at the theoretical properties of the WVD, let us see what we obtain on two particular synthetic signals.

Example 1: The first signal is the academic linear chirp signal that we already considered. The WVD is available thanks to the M-file tfrwv.m of the Time-Frequency Toolbox (see fig. 4.1).
```
     >> sig=fmlin(256);
     >> tfrwv(sig);
```
Figure 4.1: Wigner-Ville distribution of a linear chirp signal : almost perfect localization in the time-frequency plane
$\begin{figure} \epsfxsize =10cm\epsfysize =8cm \centerline{\epsfbox{figure/en1fig1.eps}}\end{figure}$

If we choose a 3-dimension plot to represent it, we can see that the WVD can take negative values, and that the localization obtained in the time-frequency plane for this signal is almost perfect.
Example 2: When a car goes in front of an observer with a constant speed, the signal heard by this person from the engine changes with time: the main frequency decreases (at a first level of approximation) from one value to another. This phenomenon, known as the doppler effect, expresses the dependence of the frequency received by an observer from a transmitter on the relative speed between the observer and the transmitter. The corresponding signal can be generated thanks to the M-file doppler.m of the Time-Frequency Toolbox. Here is an example of such a signal (see fig. 4.2):
```
     >> [fm,am,iflaw]=doppler(256,50,13,10,200);
     >> sig=am.*fm;
     >> tfrwv(sig);
```
Figure 4.2: WVD of a doppler signal : many interference terms are present, due to the bilinearity of the distribution
$\begin{figure} \epsfxsize =10cm\epsfysize =8cm \centerline{\epsfbox{figure/en1fig2.eps}}\end{figure}$

Looking at this time-frequency distribution, we notice that the energy is not distributed as we could expect for this signal. Although the signal term is well localized in the time-frequency plane, numerous other terms (the interference terms, due to the bilinearity of the WVD) are present at positions in time and frequency where the energy should be null. We will see earlier how to get rid of these terms.

Eric Chassande-Mottin 2005-10-26