The Rihaczek and Margenau-Hill distributions Another possible definition of a time-frequency energy density is given by the Rihaczek distribution. If we consider the interaction energy between a signal $x$ restricted to an infinitesimal interval $\delta_T$ centered on $t$, and $x$ passed through an infinitesimal bandpass filter $\delta_B$ centered on $\nu$, it can be approximated by the following expression: $$\delta_T\ \delta_B\ [x(t)\ X^*(\nu)\ e^{-j2\pi \nu t}].$$ This leads us to interpret the quantity $$R_x(t,\nu)=x(t)\ X^*(\nu)\ e^{-j2\pi \nu t},$$ called the Rihaczek distribution, as a complex energy density at point $(t,\nu)$. This distribution, which corresponds to the element of the Cohen's class for which $f(\xi,\tau)=e^{j\pi \xi \tau}$, verifies many good properties (1-2, 4-11, see section 4.1.1). However, it is complex valued, which can be awkward in practice. It is implemented under the name tfrri.m. The real part of the Rihaczek distribution is also a time-frequency distribution of the Cohen's class ( $f(\xi,\tau)=\cos{(\pi \xi \tau)}$), known as the Margenau-Hill distribution (see the M-file tfrmh.m). It has also numerous interesting properties: 1-5, 8, 10-11. As for the WVD, we can define smoothed versions of the Rihaczek and Margenau-Hill distributions. The file tfrpmh.m computes the pseudo Margenau-Hill distribution. The interference structure of the Rihaczek and Margenau-Hill distributions is different from the Wigner-Ville one: the interference terms corresponding to two points located on $(t_1,\nu_1)$ and $(t_2,\nu_2)$ are positioned at the coordinates $(t_1,\nu_2)$ and $(t_2,\nu_1)$. This can be seen on the following example (see fig. 4.14): >> sig=atoms(128,[32,0.15,20,1;96,0.32,20,1]); >> tfrmh(sig); Figure 4.14: Margenau-Hill distribution of 2 atoms : the position of the interferences is quite different from the one obtained with the WVD Thus, the use of the Rihaczek (or Margenau-Hill) distribution for signals composed of multi-components located at the same position in time or in frequency is not advised, since the interference terms will then be superposed to the signal terms.
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