Need of specific tools for broad-band signals

The WVD, as we have seen in the previous chapter, is a very satisfactory distribution when applied to narrow-band signals. Its use for the description of broad-band signals is also possible, but can lead to surprising images. For example, for an analytic signal whose real part is $\delta(t-t_0)$ , the WVD equals to

$\begin{displaymath}W(t,\nu)=4{\sin(4\pi\nu(t-t_0))\over\pi(t-t_0)}\ U(\nu)\end{displaymath}$

where $U(\nu)$ is the Heaviside function, and thus is not well localized in the neighborhood of

(see fig. 4.24):

     >> sig=anapulse(128);
     >> tfrwv(sig);

**Figure 4.24:** WVD of a Dirac impulse at time
$\begin{figure} \epsfxsize =10cm\epsfysize =8cm \centerline{\epsfbox{figure/en2fig7.eps}}\end{figure}$

Actually, the group of translations in time and frequency (the Weyl-Heisenberg group, see section 3.1.1) on which the WVD is based, and more generally all the Cohen's class, is responsible for these bad localization properties on broad-band signals: since the use of the analytic signal is admitted, the translation in frequency of broad-band signals fails to preserve the frequency support of the signal (the support of its Fourier transform can not be limited to the positive frequency axis). This suggests to replace the WVD by a distribution more fundamentally based on the affine group.

The Doppler effect, which is an important physical phenomenon, provides an additional motivation to use specific methods based on the affine group to analyze broad-band signals. Indeed, it characterizes the fact that a signal returned by a moving target is dilated (or compressed) and delayed compared to the emitted signal. If, for narrow-band emitted signals and low-speed targets (compared to the sound speed in the medium) this phenomenon can be approximated by a translation in time and frequency, for broad-band signals, the dilation of the spectrum has to be taken into account. This is particularly the case in radar and sonar problems where the time-bandwidth product of the emitted signal is important and where the speed of the moving target is often not negligible compared to the wave speed in the medium.

Eric Chassande-Mottin 2005-10-26