Experimental Physics – Can Kramers–Kronig Relation Be Used to Correct Transfer Function Measurements?

Question

In experimental physics, we often make measurements of linear transfer functions; these are complex-valued functions of frequency. If the underlying system is causal, then the transfer function must be analytic, satisfying the Kramers-Kronig relations. Our measurements, however, are corrupted by random (and perhaps systematic) errors.

Is it possible to improve a measurement of a linear transfer function of a causal system in the presence of noise by applying some kind of constraints derived from the Kramers-Kronig relations?

Answer 1

The problem you describe is (mathematically) similar to blind deconvolution. Given a signal which is the result of blurring an image (a linear operation) and adding noise, blind deconvolution tries to estimate the blur and the image.

As described here, the blind deconvolution process consists roughly of:

1. Guess the blurring function (transfer function)
2. Construct an image consistent with your signal and your guess of the transfer function
3. Apply physically reasonable constraints to the constructed image. (For example, non-negativity).
4. Modify your guess of the blurring function to better satisfy these constraints
5. Goto 2.

It sounds like your idea would apply to step 3. I've never seen the K-K relations used this way, but I imagine they'd work just fine.