Some examples Two special families of affine Wigner distributions can be determined by imposing constraints on $P_x^k$. The first one is unitarity (see page , property 5.), which is satisfied if $\mu_k$ is given by $$\mu_k(u)=\sqrt{\lambda_k(u)\ \lambda_k(-u)\ {d(\lambda_k(u)-\lambda_k(-u))\over du}}.$$ The second one is time-localization (property 4.), which implies that $$\mu_k(u)=\sqrt{\lambda_k(u)\ \lambda_k(-u)}\ {d(\lambda_k(u)-\lambda_k(-u))\over du}.$$ $k=0$: the Bertrand distribution The choice of $k=0$ under one or the other (or both) constraints leads to the Bertrand distribution, already defined in sections 4.2.2 and 4.2.3: $P_x^0(t,\nu)=B_x(t,\nu)$. In fact, it is the only affine Wigner distribution which satisfies simultaneously the unitarity and the time localization. $k=2$: the Wigner-Ville distribution The unitary affine Wigner distribution corresponding to $k=2$ is the Wigner-Ville distribution (see section 4.1.1) provided that $x$ is analytic: $P_x^2(t,\nu)=W_x(t,\nu)$. $k=1/2$: The D-Flandrin distribution The time-localization constraint together with the choice $k=1/2$ leads to the D-Flandrin distribution, already defined in section 4.2.2: $P_x^{1/2}(t,\nu)=D_x(t,\nu)$. $k=-1$: The active Unterberger distribution Another known example of time-localized distribution is obtained for $k=-1$: it corresponds to the active Unterberger distribution (see section 4.2.2). While this form is non-unitary, it cooperates with its passive form to produce an isometry-like relation: $$\int_{-\infty}^{+\infty} \int_0^{+\infty} U^{(a)}_x(t,\nu)\ U... ...eft\vert\int_{-\infty}^{+\infty} x(u)\ y^*(u)\ du\right\vert^2.$$ $k\ \rightarrow\ \pm\infty$: The Margenau-Hill distribution Finally, under the unitarity constraint, it is interesting to consider the two distributions obtained for $k\rightarrow -\infty$ and $k \rightarrow +\infty$: if we note respectively $P_-$ and $P_+$ these two distributions and take their arithmetic mean, we obtain exactly the Margenau-Hill distribution (see section 4.1.4): $${P_x^+(t,\nu)+P_x^-(t,\nu)\over 2}=\Re\left\{R_x(t,\nu)\right\}.$$
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