Introduction The Bertrand distribution $B_x$ given by (4.15) or (4.16) is in fact covariant by a larger group than the affine group $A$: this group, $G_0$, of transformations $g=(a,b,c)$, where $(a,b)$ is an element of $A$ and $c$ is real, acts on the signal $X$ as: $$X(\nu)\ \rightarrow\ X_g(\nu) = \sqrt{\vert a\vert}\ e^{-j2\pi(\nu b+c \ln(\nu))}\ X(a\nu).$$ The resulting change on $B_x$ is: $$B_x\ \rightarrow\ B_x^g(t,\nu)=B_x\left(\frac{t-b-c/\nu}{a},a\nu\right).$$ Actually, it is possible to generalize this extended covariance property to a sub-class of affine distributions, not only restricted to the Bertrand distribution. It can be shown that the only three-parameter groups, noted $G_k$, including the affine group, are defined as follows: for $k\neq1$, $G_k$ is the group of elements $g=(a,b,c)$ with composition law: $$gg'=(a a',b+a b',c+a^k c').$$ Group $G_1$ has a slightly different composition law: $$gg'=(a a',b+a b'+a \ln(a) c',c+a c').$$ The action of these groups on the analytic signal $X(\nu)$ is then dependent on $k$ according to: $$\begin{eqnarray*} X(\nu)\ \rightarrow\ X_g(\nu)=\sqrt{\vert a\vert}\ X(a\nu) & e... ...}\ X(a\nu) & e^{-j2\pi(\nu b+c\nu \ln(\nu))}\ &\mbox{ for } k=1. \end{eqnarray*}$$ The distributions $P_x^k$ covariant by these three-parameter solvable groups $G_k$, and satisfying the time-reversal invariance ( $Y(\nu)=X^*(\nu)\ \Rightarrow\ P_y^k(t,\nu)=P_x^k(-t,\nu)$), are then found to be: $$P_x^k(t,\nu)=\int_{-\infty}^{+\infty}\nu\ \mu_k(u)\ X(\lambda_k(u)\nu)\ X^*(\lambda_k(-u)\nu)\ e^{j2\pi(\lambda_k(u)-\lambda_k(-u))t\nu}\ du,$$ (4.17) $$\mbox{where }\ \lambda_k(u)=\left({k(e^{-u}-1)\over e^{-ku}-1}\right)^{\frac{1}{k-1}}$$ and $\mu_k(u)$ is a real positive and even function. The definition (4.17) is valid for any real $k$ provided that $\lambda_k(u)$ is defined by continuity for $k=0$ and $k=1$: $$\lambda_0(u)=-\frac{u}{e^{-u}-1} \ \mbox{ and }\ \lambda_1(u)=\exp\left(1+\frac{ue^{-u}}{e^{-u}-1}\right).$$ Expression (4.17) defines the class of affine Wigner distributions. As we will see in the next section, this class, introduced on mathematical considerations, is equivalent to the class of localized bi-frequency kernel distributions (see section 4.2.2). We now investigate special cases of $\mu_k$ leading to distributions satisfying unitarity and/or localization properties.
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