General expressions It is possible to show that if a bilinear time-scale distribution $\Omega_x(t,a)$ is covariant to affine transformations, i.e. $$\Omega_{x_{a',b'}}(t,a)=\Omega_x\left(\frac{t-b'}{a'},\frac{a}{a'}\right),$$ then, it is necessarily parameterized as $$\Omega_x(t,a;\Pi)=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} \Pi\left(\frac{s-t}{a},a \xi\right)\ W_x(s,\xi)\ ds\ d\xi$$ (4.10) where $\Pi(t,\nu)$ is an arbitrary smoothing function. This distribution will also preserve the signal energy provided that $$\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} \Pi(t,\nu)\ dt\ \frac{d\nu}{\vert nu\vert} = 1.$$ The set of such representations defines the affine class, which is the class of time-frequency energy distributions covariant by translation in time and dilation. From expression (4.10), it is straightforward that the Wigner-Ville distribution is an element of the affine class: if we introduce an arbitrary non-zero frequency $\nu_0$, and identify the scale with the inverse of the frequency: $$a=\frac{\nu_0}{\nu},$$ then the WVD corresponds to the element for which $$\Pi(t,\nu)=\delta(t)\ \delta(\nu-\nu_0).$$ A consequence of (4.10) is that the choice of an element in the affine class can be reduced to the choice of an affine correlation kernel $\Pi(t,\nu)$. When $\Pi$ is a two-dimensional low-pass function, it plays the role of an affine smoothing function which tries to reduce the interferences generated by the WVD. Another equivalent expression for a generic element can be found in terms of ambiguity: $$\Omega_x(t,a;\Phi)=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} \Phi(a \xi,\tau/a)\ A_x(\xi,\tau)\ e^{-j2\pi \xi t}\ d\xi\ d\tau,$$ (4.11) where $\Phi(\xi,\tau)$ is the weighting function corresponding to $\Pi$: $$\Phi(\xi,\tau)=\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} \Pi(t,\nu)\ e^{j2\pi(\nu \tau+\xi t)}\ dt\ d\nu,$$ and $A_x(\xi,\tau)$ is the narrow-band ambiguity function already defined in section 4.1.3. Finally, an alternative characterization of the class (4.10) may be given by using the bi-frequency kernel $\Psi(\nu,f)$ $$\Omega_x(t,a;\Pi) = \frac{1}{\vert a\vert}\int\int_{-\infty}^{+\i... ...{a}\right) X^*\left(\frac{f+\frac{\nu}{2}}{a}\right) e^{-j2\pi\nu t/a}\ d\nu df$$ (4.12) with $$\Psi(\nu,f) = \int_{-\infty}^{+\infty} \Pi(t,f)\ e^{-j2\pi\nu t}\ dt,$$ where $X(\nu)$ is the Fourier transform of $x(t)$. We will take advantage of these different (but equivalent) expressions of the affine class in the following.
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