# Can the Kramers–Kronig relation be used to correct transfer function measurements?

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?

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 I'm sure the answer is "yes", but I'm very interested how you would actually make such an improvement. Good question +1 – Keenan Pepper Mar 10 '11 at 21:46 I wonder whether this would be equivalent to least-squares fitting an analytic model (trying various numbers of poles and zeroes) à la VECTFIT? – nibot Mar 10 '11 at 22:40

## 1 Answer

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.

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Thanks for the link; that looks like a good lead! I like the idea of an iterative technique, but I wonder whether it is really so straight-forward to simply "apply" the K-K relations (i.e. integral relations on sampled data). – nibot Mar 10 '11 at 22:38
Possibly naive, but: Fourier transform your transfer function to give an impulse response. Truncate the part of the impulse response that is acausal, inverse transform, and you have your 'constrained' transfer function. – Andrew Mar 10 '11 at 22:41
Interesting! Sounds very much like the Fienup iterative phase retrieval algorithm. But I would also like to introduce some additional information: I have estimates of the standard error on each of the measurement points (via a coherence measurement). Unclear how to mix this in. – nibot Mar 10 '11 at 22:48
Maybe this iteration: 1. In time domain, truncate acausal portion; 2. In freq domain, nudge towards measured data, with nudging weighted by error estimate, 3. repeat. ? – nibot Mar 10 '11 at 22:51
Also brings up the problem of how to inverse fourier transform an irregularly sampled spectrum (Lomb-Scargle?). – nibot Mar 10 '11 at 22:52
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