The localized bi-frequency kernel distributions A useful subclass of the affine class consists in characterization functions which are perfectly localized on some curve $f=H(\nu)$ in their bi-frequency representation (see (4.12)): $$\Psi(nu,f)=G(\nu)\ \delta(f-H(\nu))\ \Leftrightarrow\ \Phi(\nu,\tau)=G(\nu)\ e^{j2\pi H(\nu) \tau}$$ where $G(\nu)$ is an arbitrary function. The corresponding time-scale distributions, which are referred to as localized bi-frequency kernel distributions, then read $$\Omega_x(t,a;\Pi)=\frac{1}{\vert a\vert}\ \int_{-\infty}^{+\i... ...^*\left(\frac{H(\nu)+\nu/2}{a}\right)\ e^{-j2\pi\nu t/a}\ d\nu.$$ Actually, it has been shown that the only group delay laws on which a localized bi-frequency kernel distribution can be perfectly localized are power laws (i.e. $t_x(\nu)=t_0+c\nu^{k-1}$) and logarithmic laws (i.e. $t_x(\nu)=t_0+c \log{\nu}$). As for the product-kernel distributions, with the formal identification $a=\nu_0/\nu$, we can associate to every time-scale distribution of that kind a time-frequency distribution according to $$C_x(t,\nu;\Phi)=\Omega_x(t,\nu_0/\nu;\Phi).$$ We give in the following particular examples of such distributions. Bertrand distribution If we further impose to these distributions the a priori requirements of time localization and unitarity, we obtain $$G(\nu)=\frac{\nu/2}{\sinh\left(\frac{\nu}{2}\right)}\ \mbox{ and }\ H(\nu)=\frac{\nu}{2}\ \coth\left(\frac{\nu}{2}\right),$$ which leads to the Bertrand distribution, defined as $$B_x(t,a)=\frac{1}{\vert a\vert}\int_{-\infty}^{+\infty} \frac{\nu... ...(\frac{\nu\ e^{-\nu/2}}{2a \sinh\left(\frac{\nu}{2}\right)}\right)\hspace*{3cm}$$ $$\hspace*{1cm}\times X^*\left(\frac{\nu\ e^{+\nu/2}}{2a \sinh\left(\frac{\nu}{2}\right)}\right)\ e^{-j2\pi\nu t/a}\ d\nu$$ (4.15) It validates properties 1. to 7., except the time-marginal property (see page ). Besides, we can show that this distribution is the only localized bi-frequency kernel distribution which localizes perfectly the hyperbolic group delay signals: $$X(\nu)=\frac{e^{j\Phi_x(\nu)}}{\sqrt{\nu}}\ U(\nu)$$ $$\mbox{with }\ \Phi_x(\nu)=-2\pi\left[\nu t_0+\alpha \log\frac... ...ow\ B_x(t,a=\frac{\nu_0}{\nu})=\nu\ \delta(t-t_x(\nu))\ U(\nu)$$ where $t_x(\nu)=-\frac{1}{2\pi}\ \frac{d\Phi_x(\nu)}{d\nu}$ is the group delay. To illustrate this property, consider the signal obtained using the file gdpower.m (taken for $k=0$), and analyze it with the file tfrbert.m (see fig. 4.21): >> sig=gdpower(128); >> tfrbert(sig,1:128,0.01,0.22,128,1); Figure 4.21: Bertrand distribution of an hyperbolic group delay signal Note that the distribution obtained is well localized on the hyperbolic group delay, but not perfectly: this comes from the fact that the file tfrbert.m works only on a subpart of the spectrum, between two bounds $f_{min}$ and $f_{max}$. Note that the larger the frequency bandwidth, the more needed samples, and consequently the longer the computation time. D-Flandrin distribution If we now look for a localized bi-frequency kernel distribution which is real, localized in time and which validates the time-marginal property, we obtain $$G(\nu)=1-(\nu/4)^2\ \mbox{ and }\ H(\nu)=1+(\nu/4)^2.$$ The corresponding distribution then writes: $$\begin{eqnarray*} D_x(t,a)=\frac{1}{\vert a\vert}\ \int_{-\infty}^{+\infty} (1-(... ...X^*\left(\frac{[1+\nu/4]^2}{a}\right)\ e^{-j2\pi\nu t/a}\ d\nu, \end{eqnarray*}$$ which defines the D-Flandrin distribution. It validates properties 1-4., 6. and 7. (see page ), and is the only localized bi-frequency kernel distribution which localizes perfectly signals having a group delay in $\frac{1}{\sqrt{\nu}}$: $$X(\nu)=\frac{e^{j\Phi_x(\nu)}}{\sqrt{\nu}}\ U(\nu)$$ with $$\Phi_x(\nu)=-2\pi[\nu t_0+2\alpha \sqrt{\nu}] \ \Rightarrow\... ...t(t,a=\frac{\nu_0}{\nu}\right)=\nu\ \delta(t-t_x(\nu))\ U(\nu).$$ This can be illustrated using the files gdpower.m with $k=1/2$ and tfrdfla.m, as following (see fig. 4.22): >> sig=gdpower(128,1/2); >> tfrdfla(sig,1:128,0.01,0.22,128,1); Figure 4.22: D-Flandrin distribution of a signal with a group delay in $1/\nu ^{1/2}$ Here again, the distribution is almost perfectly localized. Unterberger distributions Finally, the choice of $$G(\nu)=1\ \mbox{ and }\ H(\nu)=\sqrt{1+\left(\frac{\nu}{2}\right)^2}$$ corresponds to the active Unterberger distribution: $$\begin{eqnarray*} U^{(a)}_x(t,a)=\frac{1}{\vert a\vert}\ \int_0^{+\infty} (1+\fr... ...pha a}\right)\ e^{j2\pi (\alpha-1/\alpha)\frac{t}{a}}\ d\alpha, \end{eqnarray*}$$ which verifies properties 1-4., 6-7. (see page ) except the time-marginal; and the choice of $$G(\nu)=\frac{1}{\sqrt{1+\left(\frac{\nu}{2}\right)^2}}\ \mbox{ and }\ H(\nu)=\sqrt{1+\left(\frac{\nu}{2}\right)^2}$$ corresponds to the passive Unterberger distribution: $$\begin{eqnarray*} U^{(p)}_x(t,a)=\frac{1}{\vert a\vert} \int_0^{+\infty} \frac{2... ...ight)\ e^{j2\pi (\alpha-\frac{1}{\alpha})\frac{t}{a}}\ d\alpha, \end{eqnarray*}$$ which verifies properties 1-3., 6-7. The active Unterberger distribution is the only localized bi-frequency kernel distribution which localizes perfectly signals having a group delay in $1/\nu ^2$: $$X(\nu)=\frac{e^{j\Phi_x(\nu)}}{\sqrt{\nu}}\ U(\nu)$$ with $$\Phi_x(\nu)=-2\pi[\nu t_0-\alpha/\nu] \ \Rightarrow\ U^{(a)}_x(t,a=\nu_0/\nu)=\nu\ \delta(t-t_x(\nu))\ U(\nu).$$ The files gdpower.m, considered for $k=-1$, and tfrunter.m give us (see fig. 4.23): >> sig=gdpower(128,-1); >> tfrunter(sig,1:128,'A',0.01,0.22,172,1); Figure 4.23: Active Unterberger distribution of a signal with a group delay in $1/\nu ^2$ We will go back over these distributions later on (sub-section 4.2.4) in a different context.
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