From the WVD to the Bertrand distribution Now that we defined the symmetric wide-band ambiguity function, it would be interesting to obtain an expression equivalent to the one linking the WVD and the narrow-band ambiguity function, but replaced in the affine context. This can be done by applying a Fourier transform to the delay variable of the symmetric WAF, and a Mellin transform to the $\alpha$ variable: $$B_x(t,\nu) = \int_0^{+\infty} \int_{-\infty}^{+\infty} \Xi_x^{(s)}(\alpha,\tau)\ e^{-j2\pi\nu \tau}\ \alpha^{j2\pi t-1}\ d\tau\ d\alpha$$ $$=\nu \int_{-\infty}^{+\infty} \frac{u/2}{\sinh\left(\frac{u}{2}\r... ...nu\ u\ e^{+u/2}}{2 \sinh\left(\frac{u}{2}\right)}\right)\ e^{-j2\pi\nu ut}\ du$$ (4.16) which corresponds to the Bertrand distribution, already introduced in section 4.2.2 (the equivalence between formula (4.15) and (4.16) is obtained by identifying $\nu$ as the inverse of the scale: $\nu={\nu_0\over a}$ with $\nu_0=1$Hz).
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