Obtaining an expression for the transfer function when using FFT

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:

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.

• 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. Commented Mar 19, 2014 at 21:15
• 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?). Commented Mar 19, 2014 at 22:07
• 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). Commented Mar 20, 2014 at 8:19