For linear time-frequency representation, the resolution is limited by Gabor limit. However, if interference is allowed, the resolution may be higher. What is the possible resolution in such a case? In "Electroencephalography Feature Extraction Using High Time-Frequency Resolution Analysis ", they stated

We all know that the Wigner-Ville distribution achieves to the lower bound of the uncertainty principle and have the highest resolution in all time-frequency distributions

Why is this statement true?


The statement is true if you restrict it to all TF distributions in the Cohen Class. That is rather easy to see, because the Wigner distribution generates all other distribution in that class by convolution with a non-negative time frequency kernel and therefore attains the minimal uncertainty.

Beyond the Cohen class things get much more complicated. First of all, the very simple concept of uncertainty in the Cohen class is not necessarily useful for non-bilinear distributions. So it's not obvious how to even generalise the question. So statements that attempt to be more general than the Cohen class first need some clarification about mathematical terminology and possibly even concepts. Outside the Cohen class you will also find distributions without time-frequency translation invariance. If they're bilinear they can have a TF uncertainty that is just as good as the Wigner-Ville distribution. Dropping bilinearity too, you can even beat WV with reassignment distributions of all sorts.

So summing up, the statement is true with the (possibly implcit) restriction to the Cohen class, which is the most common theoretical framework for TF distributions. Without that restriction it is not accurate.

  • $\begingroup$ Thanks! I am curious what is the lower bound referred by the authors. $\endgroup$
    – Kattern
    Jun 30 '14 at 8:07

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