# How to Find the Frequency Response of a Communication Channel from Input and Output Symbols in MATLAB?

I want to find the channel frequency response of a digital communications system. I have the functions of the input symbol (a triangle) and the output symbol - a distorted triangle: $$\dfrac{1}{1 + \left( \frac{2t}{T} \right)^2 }$$ I want to plot the frequency response of the channel as if it was a digital filter.

How can I do this in MATLAB?

I tried using symbolic variables, computing Fourier transforms of the input and the output signal using fourier command, and then finding the numerical values using subs and double commands, with no success.

## 1 Answer

I'm not sure about the symbols.
Yet if you have a function in time as the input of the channel and another function in time which is the output of the channel and assuming the channel doesn't have zeros in the spectrum and the input function as well, the channel Transfer Function is given by:

$$\frac{{X}_{Out}(f)}{{X}_{In}(f)}$$

In MATLAB, just apply FFT on each, and do element by element division.
This will give you the DFT of the transfer function.
Apply IFFT to see the time response function.