Properties Here is a list of the main properties of the WVD [Fla93]. Energy conservation: by integrating the WVD of x all over the time-frequency plane, we obtain the energy of $x$: $$E_x = \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} W_x(t,\nu)\ dt\ d\nu$$ Marginal properties: the energy spectral density and the instantaneous power can be obtained as marginal distributions of $W_x$: $$\begin{eqnarray*} \int_{-\infty}^{+\infty} W_x(t,\nu)\ dt &=& \vert X(\nu)\vert^... ...\int_{-\infty}^{+\infty} W_x(t,\nu)\ d\nu &=& \vert x(t)\vert^2 \end{eqnarray*}$$ Real-valued: $$W_x(t,\nu)\ \in {\cal R},\ \forall\ t, \nu$$ Translation covariance: the WVD is time and frequency covariant: $$\begin{eqnarray*} y(t)=x(t-t_0) &\Rightarrow & W_y(t,\nu)=W_x(t-t_0,\nu)\\ y(t)=x(t) e^{j2\pi \nu_0 t} &\Rightarrow & W_y(t,\nu)=W_x(t,\nu-\nu_0) \end{eqnarray*}$$ Dilation covariance: the WVD also preserves dilations: $$\begin{eqnarray*} y(t)=\sqrt{k}\ x(kt)\ ;\ k>0\ \Rightarrow\ W_y(t,\nu)=W_x(kt,\frac{\nu}{k}) \end{eqnarray*}$$ Compatibility with filterings: it expresses the fact that if a signal $y$ is the convolution of $x$ and $h$ (i.e. the output of filter $h$ whose input is $x$), the WVD of $y$ is the time-convolution between the WVD of $h$ and the WVD of $x$: $$y(t)=\int_{-\infty}^{+\infty} h(t-s)\ x(s)\ ds\ \Rightarrow\ W_y(t,\nu)=\int_{-\infty}^{+\infty} W_h(t-s,\nu)\ W_x(s,\nu)\ ds$$ Compatibility with modulations: this is the dual property of the previous one: if $y$ is the modulation of $x$ by a function $m$, the WVD of $y$ is the frequency-convolution between the WVD of $x$ and the WVD of $m$: $$y(t)=m(t)\ x(t)\ \Rightarrow\ W_y(t,\nu)=\int_{-\infty}^{+\infty} W_m(t,\nu-\xi)\ W_x(t,\xi)\ d\xi$$ Wide-sense support conservation: if a signal has a compact support in time (respectively in frequency), then its WVD also has the same compact support in time (respectively in frequency): $$\begin{eqnarray*} x(t)=0,\ \vert t\vert>T &\Rightarrow & W_x(t,\nu)=0,\ \vert t... ...,\ \vert\nu\vert>B &\Rightarrow & W_x(t,\nu)=0,\ \vert\nu\vert>B \end{eqnarray*}$$ Unitarity: the unitarity property expresses the conservation of the scalar product from the time-domain to the time-frequency domain (apart from the squared modulus): $$\left\vert\int_{-\infty}^{+\infty} x(t)\ y^*(t)\ dt\right\ver... ...y} \int_{-\infty}^{+\infty} W_x(t,\nu)\ W_y^*(t,\nu)\ dt\ d\nu.$$ This formula is also known as the Moyal's formula. Instantaneous frequency: the instantaneous frequency of a signal $x$ can be recovered from the WVD as its first order moment (or center of gravity) in frequency: $$f_x(t)= {\int_{-\infty}^{+\infty} \nu W_{x_a}(t,\nu)\ d\nu \over \int_{-\infty}^{+\infty} W_{x_a}(t,\nu)\ d\nu}$$ where $x_a$ is the analytic signal associated to $x$. Group delay: in a dual way, the group delay of $x$ can be obtained as the first order moment in time of its WVD: $$t_x(\nu)={\int_{-\infty}^{+\infty} t\ W_{x_a}(t,\nu)\ dt\over \int_{-\infty}^{+\infty} W_{x_a}(t,\nu)\ dt}$$ Perfect localization on linear chirp signals: $$x(t)=e^{j2\pi \nu_x(t) t} \mbox{ with } \nu_x(t)=\nu_0+2\beta t\ \Rightarrow\ W_x(t,\nu)=\delta(\nu-(\nu_0+\beta t)).$$
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