Relationship between analytic signal and analytic function definitions

Question

Is there a relationship between definitions of analytic signals and analytic functions? I have understand that analytic signal real and imaginary parts are related by Hilbert transform and analytic function are differentiable at every point. Is these definitions some way related? Or is it just coincidence that both are 'analytic'?

Answer 1

I have not studied rigorous complex analysis, but here is what I found in Wikipedia:

This page indicates that if a complex function $f$ of a complex variable is analytic (in the complex analysis sense) in the closed upper half-plane and satisfies some additional requirement, then $\mathrm{Im} (f) = \mathcal H\{\mathrm{Re} (f)\}$, where $\mathcal H$ is the Hilbert transform.

On the other hand, this page says that if $s$ is a real function of a real variable, then $s+j\mathcal H\{s\}$ is an analytic signal. (That is, it has no negative frequency components.)

Therefore, the two definitions seem to be related.

Answer 2

First, an analytic signal is a signal which only has positive frequencies. I believe one can show that if a signal has only positive frequency parts then the signals real and imaginary parts related by Hilbert transform. That is to say, it is the positive frequency restriction which is crucial to the definition.

Finally, the relationship between analytic signals and analytic functions is known as the Paley-Wiener Theorem. The theorem implies that any function with only positive frequency components is an analytic function in the upper half plane (in the time-domain).