Transformation between distributions in time–frequency analysis

In the field of time–frequency analysis, several signal formulations are used to represent the signal in a joint time–frequency domain.

There are several methods and transforms called "time-frequency distributions" (TFDs), whose interconnections were organized by Leon Cohen. The most useful and popular methods form a class referred to as "quadratic" or bilinear time–frequency distributions. A core member of this class is the Wigner–Ville distribution (WVD), as all other TFDs can be written as a smoothed or convolved versions of the WVD. Another popular member of this class is the spectrogram which is the square of the magnitude of the short-time Fourier transform (STFT). The spectrogram has the advantage of being positive and is easy to interpret, but also has disadvantages, like being irreversible, which means that once the spectrogram of a signal is computed, the original signal can't be extracted from the spectrogram. The theory and methodology for defining a TFD that verifies certain desirable properties is given in the "Theory of Quadratic TFDs".

The scope of this article is to illustrate some elements of the procedure to transform one distribution into another. The method used to transform a distribution is borrowed from the phase space formulation of quantum mechanics, even though the subject matter of this article is "signal processing". Noting that a signal can recovered from a particular distribution under certain conditions, given a certain TFD ρ₁(t,f) representing the signal in a joint time–frequency domain, another, different, TFD ρ₂(t,f) of the same signal can be obtained to calculate any other distribution, by simple smoothing or filtering; some of these relationships are shown below. A full treatment of the question can be given in Cohen's book.

If we use the variable ω=2πf, then, borrowing the notations used in the field of quantum mechanics, we can show that time–frequency representation, such as Wigner distribution function (WDF) and other bilinear time–frequency distributions, can be expressed as

...
Wikipedia