Time-frequency signal representations

This invited paper – of a tutorial and review character – presents an overview of two classes of time-frequency signal representations. The first class, in which the signal arises linearly, deals with the windowed Fourier transform and its sampled version (also known as the Gabor transform) and the inverse of the latter: Gabor’s signal expansion. We will show how Gabor’s signal expansion and the windowed Fourier transform are related and how they can benefit from each other. The second class, in which the signal arises quadratically (or bilinearly, as it is often called), is based on the Wigner distribution. We will show some examples of the Wigner distribution and discuss some of its important properties. Being a bilinear signal representation, the Wigner distribution shows artifacts in the case of multi-component signals. To reduce these artifacts, a large class of bilinear signal representations has been constructed, known as the shift-covariant Cohen class. We will consider this class and we will see how all its members can be considered as properly averaged versions of the Wigner distribution.