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Say I have two signals, a(t) and b(t) where the former is the input and the latter in the output. These signals are both recorded by sampling at every 0.01s. The Fast Fourier Transform was applied to both of these providing me with two large arrays of complex numbers. The FFT of b was divided by the FFT of a to provide me with a new array of complex numbers. This was plotted as can be seen below:

enter image description here

Where x axis is real any y axis is imaginary.

How would one go about producing a closed form type polynomial for the transfer function of such a system?

Edit: Inverse FFT of the above. Complex components were of the order e^-14 and thus rounded to 0.

enter image description here

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    $\begingroup$ The complex FFT{b}/FFT{a} is an estimate for the frequency response of the (supposed) system with input a and output b. What do you mean by 'closed form type polynomial'? If you want an FIR approximation, you could - as a first try - do an inverse FFT of the complex transfer function. This gives you the coefficients of an FIR filter approximating this function. $\endgroup$ – Matt L. Mar 19 '14 at 21:15
  • $\begingroup$ Plotted the inverse and added it to the question. I think the reflection indicates the impulse response is linear. It appears to be shifted (but by a massive amount though?). $\endgroup$ – user3394391 Mar 19 '14 at 22:07
  • $\begingroup$ I'm not sure if your FIR coefficients are OK. The good thing is that they are real-valued (up to numerical errors), but what puzzles me is that they are symmetrical (which would imply that FFT{b}/FFT{a} is real-valued). $\endgroup$ – Matt L. Mar 20 '14 at 8:19

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