Joint-smoothings of the WVD The following distributions correspond to particular cases of the Cohen's class for which the parameterization function depends only on the product of the variables $\tau$ and $\xi$: $$f(\xi,\tau)=\Phi(\tau\xi)$$ (4.9) where $\Phi$ is a decreasing function such that $\Phi(0)=1$ (the Rihaczek and Margenau-Hill distributions are particular elements of this class). A direct consequence of this definition is that the marginal properties will be respected. Besides, since $\Phi$ is a decreasing function, $f$ is a low-pass function, and according to (4.8), this parameterization function will reduce the interferences. That is why these distributions are also known as the Reduced Interference Distributions. The Choi-Williams distribution One natural choice for Phi is to consider a gaussian function: $$f(\xi,\tau)=\exp{\left[-\frac{(\pi \xi \tau)^2}{2\sigma^2}\right]}.$$ The corresponding distribution, $$CW_x(t,\nu)=\sqrt{\frac{2}{\pi}} \int\int_{-\infty}^{+\infty}... ...tau}{2})\ x^*(s-\frac{\tau}{2})\ e^{-j2\pi \nu \tau}\ ds\ d\tau$$ is the Choi-Williams distribution. Note that when $\sigma\ \longrightarrow\ +\infty$, we obtain the WVD. Inversely, the smaller $\sigma$, the better the reduction of the interferences. This distribution verifies properties 1-5, 10-11, and can be computed with the M-file tfrcw.m. The "cross"-shape of the parameterization function of the Choi-Williams distribution implies that the efficiency of this distribution strongly depends on the nature of the analyzed signal. For instance, if the signal is composed of synchronized components in time or in frequency, the Choi-Williams distribution will present strong interferences. This can be observed on the following example: we analyze four gaussian atoms positioned at the corners of a rectangle rotating around the center of the time-frequency plane (see fig. 4.15): >> load movcw4at >> clf; movie(M,5); Figure 4.15: Choi-Williams distribution of 4 atoms rotating around the middle of the time-frequency plane : when the time/frequency supports of the atoms overlap, strong interferences appear on the overlap support When the time/frequency supports of the atoms overlap, some AF-interference terms are not completely attenuated (those present around the axes of the ambiguity plane), and the efficiency of the distribution is quite poor. The Born-Jordan and Zhao-Atlas-Marks distributions If we impose to the distributions defined by (4.9) the further condition to preserve time- and frequency- supports, the simplest choice for $f$ is then: $$f(\xi,\tau)={\sin{(\pi \xi \tau)}\over\pi \xi \tau}$$ which defines the Born-Jordan distribution: $$BJ_x(t,\nu)=\int_{-\infty}^{+\infty} \frac{1}{\vert\tau\vert}... ...t/2} x(s+\tau/2)\ x^*(s-\tau/2)\ ds\ e^{-j2\pi \nu \tau} d\tau.$$ Properties 1-5, 8, 10-11 are verified by this distribution, and the corresponding M-file of the Time-Frequency Toolbox is tfrbj.m. If we smooth the Born-Jordan distribution along the frequency axis, we obtain the Zhao-Atlas-Marks distribution, defined as $$ZAM_x(t,\nu)=\int_{-\infty}^{+\infty} \left[\ h(\tau)\ \int_{... ...\tau/2)\ x^*(s-\tau/2)\ ds\right]\ e^{-j2\pi \nu \tau}\ d\tau.$$ This distribution, also known as the Cone-Shaped Kernel distribution, validates properties 3-4 and 8 (only for time) (see the M-file tfrzam.m for its computation).
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