Interference structure The interference structure of the affine Wigner distributions can be determined thanks to the following geometric argument: two points $(t_1,\nu_1)$ and $(t_2,\nu_2)$ belonging to the trajectory on which a distribution is localized interfere on a third point $(t_i,\nu_i)$ which is necessarily located on the same trajectory. Consequently, using the result of section 4.2.2 which says that the localized bi-frequency kernel distributions are localized on power law group delays of the form $t_x(\nu)=t_0+c\nu^{k-1}$, one can show that the coordinates $(ti,\nu_i)$ are determined by the relation ([GF92]) $$\omega_i=\left({\omega_2^k-\omega_1^k\over k(\omega_2-\omega_1)}\right) ^{\frac{1}{k-1}}$$ where $\omega=\nu$ or $\omega=(t-t_0)^{\frac{1}{k-1}}$. These "mid-point" coordinates can be computed using the M-file midpoint.m of the Time-Frequency Toolbox. Figure 4.26 represents the location of interference point corresponding to two points of the time-frequency plane $(t_1,f_1)$ and $(t_2,f_2)$, for different values of $k$. Figure 4.26: Locus of the interferences between 2 points for the affine Wigner distributions (parameterized by $k$). For $k=2$, which corresponds to the Wigner-Ville distribution, we obtain the geometric mid-point In particular, for $k=2$, corresponding to the Wigner-Ville distribution, we obtain the geometric mid-point. To illustrate this interference geometry, let us consider the case of a signal with a sinusoidal frequency modulation: >> [sig,ifl]=fmsin(128); The file plotsid.m allows one to construct the interferences of an affine Wigner distribution perfectly localized on a power-law group-delay (specifying $k$), for a given instantaneous frequency law (or the superposition of different instantaneous frequency laws). For example, if we consider the case of the Bertrand distribution ($k=0$) (see fig. 4.27), >> plotsid(1:128,ifl,0); Figure 4.27: Theoretical diagram of the interferences of the Bertrand distribution for a sinusoidal frequency modulation we obtain an interference structure completely different from the one obtained for the Wigner-Ville distribution ($k=2$) (see fig. 4.28): >> plotsid(1:128,ifl,2); Figure 4.28: Theoretical diagram of the interferences of the Wigner-Ville distribution for a sinusoidal frequency modulation For the active Unterberger distribution ($k=-1$), the result is the following (see fig. 4.29): >> plotsid(1:128,ifl,-1); Figure 4.29: Theoretical diagram of the interferences of the active Unterberger distribution for a sinusoidal frequency modulation We can notice the presence of an inflexion point (corresponding to the intersection of an infinite number of lines joining two symmetric points from the sinusoid) in the case of the WVD distribution, which disappears in the other distributions.
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