New interpretation of the Cohen's class The dual expression of the Cohen's class formulation (expression (4.6)) in terms of AF writes $$C_x(t,\nu;f) = \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} f(\xi,\tau)\ A_x(\xi,\tau)\ e^{-j2\pi(\nu \tau+\xi t)}\ d\xi\ d\tau$$ (4.8) (recall that $f$ is the two-dimensional Fourier transform of $\Pi$). This expression is very instructive about the role played by the parameterization function $f(\xi,\tau)$. Indeed, $f$ acts as a weighting function that tries to let the signal terms unchanged, and to reject the interference terms. Actually, the change from the time-frequency plane to the ambiguity plane allows a precise characterization of the weighting function $f$, and thus of the smoothing function $\Pi(t,\nu)$. For example, the WVD corresponds to a constant parameterization function: $f(\xi,\tau)=1,\ \forall\ \xi,\ \tau$: no difference is made between the different regions of the ambiguity plane. For the spectrogram, $f(\xi,\tau)=A_h^*(\xi,\tau)$: the ambiguity function of the window $h$ determines the shape of the weighting function. And for the smoothed-pseudo-WVD, we have $f(\xi,\tau)=G(\xi)\ h(\tau)$: the weighting function is separable in time and frequency, which is very useful to adapt it to the shape of the AF-signal terms. We will end this section by presenting other energy distributions that are members of the Cohen's class.
© GdR ISIS - Contact