Editing Time–frequency analysis (section)

===Sampling theory===

By the [[Nyquist–Shannon sampling theorem]], we can conclude that the minimum number of sampling points without [[aliasing]] is equivalent to the area of the time–frequency distribution of a signal. (This is actually just an approximation, because the TF area of any signal is infinite.)  Below is an example before and after we combine the sampling theory with the time–frequency distribution:

[[Image:sampling.jpg]]

It is noticeable that the number of sampling points decreases after we apply the time–frequency distribution.

When we use the WDF, there might be the cross-term problem (also called interference). On the other hand, using [[Gabor transform]] causes an improvement in the clarity and readability of the representation, therefore improving its interpretation and application to practical problems.

Consequently, when the signal we tend to sample is composed of single component, we use the WDF; however, if the signal consists of more than one component, using the Gabor transform, Gabor-Wigner distribution function, or other reduced interference TFDs may achieve better results.

The [[Balian–Low theorem]] formalizes this, and provides a bound on the minimum number of time–frequency samples needed.