Identifying a signal by its power spectrum

Question

Background: There is a method in optics for determining the electric field of light (both intensity and phase) via a three step process:

1. Add a known phase shift to the light (called the "diversity phase").
2. Perform a Fourier transform (via a lens) and measure the intensity.
3. Repeat this for several different diversity phases (at least two), and use certain algorithms to infer the light field.

The inference process is typically done numerically. However, I've found an analytical method for solving the problem, but it only works in 1D (whereas optics is 2D). This has me wondering: Are there situations in signal processing where one would like to identify e.g. a time-domain signal from only it's power spectrum? If so, do there already exist good algorithms or analytic solutions for this problem?

I am a physicist by training, so I have little background in signal processing, but I would appreciate any references or context that seasoned signal analysts can provide.

Answer 1

Are there situations in signal processing where one would like to identify e.g. a time-domain signal from only it's power spectrum?

Well, your described optical algorithm doesn't "only" depend on the power spectrum, but on the power spectrum under different phase shifts – which is a way of estimating the Fourier transform of the time (or spatial) domain.

By the way, I'd say the method you've described is a signal processing method. The fact it's not done mainly with a computer doesn't really change that!

But sure, things like that do happen. Similarly to the optical question you're answering with your method, there's cases where you can only observe the magnitude (or power) of a signal's Fourier transform, but not the signal itself, I'm sure. (I must admit I don't have a problem at hand, right now, because usually, you actually measure the current that some electrical phenomenon induces, and that allows for phase estimation.)

A related thing, which also happens to have applications in optics, are the Kramers-Kronig relations, which can be used to estimate a complex signal (or equivalently, a system) that's sufficiently envelope-bounded from just either real or imaginary part; an application are Kramers-Kronig Receivers:

Following direct detection of the total field intensity, the complex-valued electric field of the signal is extracted from the measured photocurrent thanks to the Kramers–Kronig relation linking the phase of the total field to its intensity.

I.e. you can use a "simple" photodiode to just detect intensity, and still get full info on the complex E-field; this means you get twice as much useful info across.