Renyi information Another interesting information that one may need to know about an observed non-stationary signal is the number of elementary signals composing this observation. This also leads us to the following question: how much separation between two elementary signals must one achieve in order to be able to conclude that there are two signals present rather than one ? A solution to this problem is given by applying an information measure to a time-frequency distribution of the signal. Unfortunately, the well known Shannon information, defined as $$\begin{eqnarray*} I_x = -\int_{-\infty}^{+\infty} f(x)\ \log_2 f(x)\ dx \end{eqnarray*}$$ where $f(x)$ is the probability density function of $x$, can not be applied to some time-frequency distributions due to their negative values. The generalized form of information, which admits negative values in the distribution, will then be used. This information, known as Renyi information, is given by $$\begin{eqnarray*} R_x^{\alpha} = \frac{1}{1-\alpha}\ log_2\left\{\int_{-\infty}^{+\infty} f^{\alpha}(x)\ dx\right\} \end{eqnarray*}$$ in the continuous case, where $\alpha$ is the order of the information. First order Renyi information ($\alpha=1$) reduces to Shannon information. Third order Renyi information, applied to a time-frequency distribution $C_x(t,nu)$, is defined as $$\begin{eqnarray*} R_C^3 = -\frac{1}{2}\ log_2\left\{\int_{-\infty}^{+\infty}\int_{-\infty}^{+\infty} C_x^3(t,\nu)\ dt\ d\nu\right\}. \end{eqnarray*}$$ The result produced by this measure is expressed in bits: if one elementary signal yields zero bit of information ($2^0$), then two well separated elementary signals will yield one bit of information ($2^1$), four well separated elementary signals will yield two bits of information ($2^2$), and so on. This can be observed by considering the WVD of one, two and then four elementary atoms, and then by applying the Renyi information on them. The file renyi.m computes this information measure: >> sig=atoms(128,[64,0.25,20,1]); >> [TFR,T,F]=tfrwv(sig); >> R1=renyi(TFR,T,F) ------> -0.2075 >> sig=atoms(128,[32,0.25,20,1;96,0.25,20,1]); >> [TFR,T,F]=tfrwv(sig); >> R2=renyi(TFR,T,F) ------> 0.779 >> sig=atoms(128,[32,0.15,20,1;96,0.15,20,1;... 32,0.35,20,1;96,0.35,20,1]); >> [TFR,T,F]=tfrwv(sig); >> R3=renyi(TFR,T,F) ------> 1.8029 We can see that if R is set to 0 for one elementary atom by subtracting R1, we obtain a result close to 1 for two atoms (R2-R1=0.99) and close to 2 for four atoms (R3-R1=2.01). If the components are less separated in the time-frequency plane, the information measure will be affected by the overlapping of the components or by the interference terms between them (see [WBI91] for more details on this analysis). In particular, it is possible to show that the Renyi information measure provides a good indication of the time separation at which the atoms are essentially resolved, with a better precision than with the time-bandwidth product.
© GdR ISIS - Contact