6 - From Time and Frequency to Time-Frequency

Summary

The simplest and most intuitive time-frequency transforms are the Short-Time Fourier Transform (STFT) and the associated spectrogram (squared STFT), which are in close connection with the Gabor transform when using a Gaussian window. Given a 1D signal, its STFT is a 2D function whose redundancy is controlled by a reproducing kernel. This kernel can be equivalently seen as an ambiguity function, i.e., a 2D time-frequency correlation function that is constrained by uncertainty. Building upon the classical duality which exists between “correlation functions” and “energy distributions,” we end up with the Wigner distribution as a central tool for time-frequency signal analysis. The Wigner distribution happens however to be a member of larger classes (such as Cohen's) to which the spectrogram belongs too.