Interferences As the WVD is a bilinear function of the signal $x$, the quadratic superposition principle applies: $$W_{x+y}(t,\nu)\ =\ W_x(t,\nu)\ +\ W_y(t,\nu)\ +\ 2\Re{\{W_{x,y}(t,\nu)\}}$$ where $$W_{x,y}(t,\nu)\ =\ \int_{-\infty}^{+\infty} x(t+\tau/2)\ y^*(t-\tau/2)\ e^{-j2\pi \nu \tau}\ d\tau$$ is the cross-WVD of $x$ and $y$. This can be easily generalized to $N$ components, but for the sake of clarity, we will only consider the two-component case. Unlike the spectrogram interference terms, the WVD interference terms will be non-zero regardless of the time-frequency distance between the two signal terms. These interference terms are troublesome since they may overlap with auto-terms (signal terms) and thus make it difficult to visually interpret the WVD image. However, it appears that these terms must be present or the good properties of the WVD (marginal properties, instantaneous frequency and group delay, localization, unitarity ...) cannot be satisfied. Actually, there is a trade-off between the quantity of interferences and the number of good properties. o Interference geometry The rule of interference construction of the WVD can be summarized as follows: two points of the time-frequency plane interfere to create a contribution on a third point which is located at their geometrical midpoint. Besides, these interference terms oscillate perpendicularly to the line joining the two points interfering, with a frequency proportional to the distance between these two points. This can be seen on the following example: we consider two atoms in the time-frequency plane, analyzed by the WVD, whose relative distance is increasing from one realization to the other, and then decreasing. The WVDs were calculated and saved on the file movwv2at.mat. We load them and run the sequence using the function movie (see fig. 4.3): >> load movwv2at >> clf; movie(M,10); Figure 4.3: Structure of the interferences between 2 components with different locations in time and frequency : we can notice the change in the direction of the oscillations, as well as the change in the period of these oscillations We can notice, from this movie, the evolution of the interferences when the distance between the two interfering terms changes, and in particular the change in the direction of the oscillations.
© GdR ISIS - Contact