# Transfer function from experimental time domain data

I have two sets of time domain data. One is the input to a system and the other is its corresponding output, both measured at the same sampling frequency. How to calculate the system's transfer function from this experimental data?

• Generically this falls under system identification theory. Generally it's not easy; but you can do a lot of things. The simplest involves modeling with exponential functions. There is a standard linear algebra way to do it on raw data but some people have found it numerically unstable; when I have done it it has worked fine. But mostly on thermal work where one has to sweep details under the rug; or use FEA. The general approaches also have problems and the old <Y2000 approaches didn't work well but I think I know how to fix it. This is not a shallow subject. – rrogers May 24 '16 at 21:56

## 1 Answer

assuming your sets of time-domain data begin with $n=0$, Z transform both the input data, $x[n]$ and the output data, $y[n]$.

$$X(z) \triangleq \mathcal{Z}\{x[n]\} = \sum\limits_{n=0}^{\infty} x[n] \ z^{-n}$$ $$Y(z) \triangleq \mathcal{Z}\{y[n]\} = \sum\limits_{n=0}^{\infty} y[n] \ z^{-n}$$

divide the Z transform of the output by the Z transform of the input. the result of that division is your transfer function.

$$H(z) \triangleq \frac{Y(z)}{X(z)} = \frac{\sum\limits_{n=0}^{\infty} y[n] \ z^{-n} }{\sum\limits_{n=0}^{\infty} x[n] \ z^{-n} }$$

i cannot guarantee that, if you were to factor numerator and denominator, there will not be identical poles with zeros that could cancel and make the transfer function simpler, but it's a transfer function.

• Well the gcd algorithm is certainly available in all the CAS systems; reduce, axiom, maxim, etc... But I have never tried it in raw data. – rrogers May 24 '16 at 21:50
• @rrogers, of course the top limit of the summations has to be some finite $N$ instead of $\infty$. i dunno how the factoring alg would work. i know MATLAB has one. i think it's factor(). somehow you have to decide if a zero is close enough to a pole to treat them as equal and cancelling each other. – robert bristow-johnson May 25 '16 at 2:54
• "treat them as equal" ; unless it's mathematical based, not due to a physical accident, be very careful with that. In time domain system the mismatch typically introduces a step/offset/slope proportional to the mismatch. The digital people talk about cancelling poles with zeros but I have found it un-robust with part changes and varying environmental conditions. If you must: wrap a feedback loop that actively supports the cancellation; and still be careful because a phase distortion can still haunt you. – rrogers May 25 '16 at 15:12
• But backing up; the gcd() is a stable algorithm whereas straight factoring can be unstable. What is your input signal? Bandwidth, auto-correlation, and such. What is your physical system? Can you logically restrict the upper bandwidth of the output? Does the system load vary. Most physical systems have a varying load that needs to be accommodated. And most of all; for physical systems models are always approximations. I consider them mathematical descriptions; with all the shallowness that the word "description" implies. – rrogers May 25 '16 at 15:13
• I understand this is an old post, but what if the input/out sets are finite and noisy? You probably need to fit an ARMA model and use some metric to minimize the estimation error. There are many algorithms available under System Identification Toolbox in MATLAB. – Ali May 21 '20 at 15:36