Framework: the Wigner-Ville Distribution

The Wigner-Ville distribution: A High Resolution Time-Frequency Distribution:

$$WV(x;t,\omega) = \int x(t+\tau/2)\cdot x^*(t - \tau/2)e^{-j\omega\tau} d\tau$$ (8)

Notice this looks DFT-like.

Despite High resolution, ``cross-terms may lead to an erroneous visual interpretation of the signal's time-frequency structure.'' (See overhead figures.)

So apply lowpass filter to get a Cohen-class Time-Frequency representation:

$$TFR(x;t,\omega) =\int\int \phi_{TF}(u,\Omega) WV(x;t-u,w-\Omega) du \ensuremath{\frac{d\Omega}{2\pi}}$$ (9)

where $\phi_{TF}(u,\Omega)$ acts as a ``two-dimensional lowpass filter," or time-frequency window.

Important concept: This is smoothing over the WV distribution, which itself is DFT-like. So one must remember that we are smoothing over a region in the time-frequency grid.

Now, to do reassignment: find smoothed center of gravity in time and in frequency:

$$\hat{t}(x;t,\omega) = t - \ensuremath{\frac{\int \int u \cdot \phi_{TF}(u,\Omega) WV(x;... ...{TF}(u,\Omega) WV(x;t-u,\omega - \Omega) du \ensuremath{\frac{d\Omega}{2\pi}}}}$$ (10)

$$\hat{\omega}(x;t,\omega) = \omega - \ensuremath{\frac{\int \int \Omega \cdot \phi_{TF}(u,\Om... ...{TF}(u,\Omega) WV(x;t-u,\omega - \Omega) du \ensuremath{\frac{d\Omega}{2\pi}}}}$$ (11)

In our current application, this smoothing will simply be the WV distribution of the familiar time domain windowing function $h(t)$, or:

$$\phi_{TF}(u,\Omega) = WV(h;u;\Omega)$$

NEAT TRICK: when we do this, it can be shown (see Auger and Flandarin) that we can make equations 10 and 11 into equations in terms of just three modified STFTs!

Hand-wavy explanation of how to massage those equations into STFTs: recall Fourier theory:

multiply by the time variable $\rightleftharpoons$ differentiate in frequency
multiply by frequency variable $\rightleftharpoons$ differentiate in time

Since we may arrange the equations around the window $h(t)$ and its transform, we see that these operations need apply only to the window. This leads to a convenient implementation...