# Question related to Deconvolution and Fourier Transform

Currently I am completely new to Signal Processing and I am about to kick off a project related to the system identification.

The general idea is given a pair of sent and received signals, I have to find the impulse response in it, i.e. the system. And I'm currently thinking if I can use the technique in deconvolution to handle this problem, so I am doing research on it and got completely messed. Here are some question I would like to ask.

As far as I got from some website, or posts here, I found that if there is no zeros in the fourier transform of the input, I can use direct division on H(w) = Y(w) / X(w), but the question is, if I received the signal datas, the dimensions on X and Y are different, How can I do the the point-to-pont division? or is there any other way to do this? or is it wrong?

Some say that I have to do some filtering/ regularization on the data, but the question is pretty much the same.

It will be much appreciated if you could anwer my question, and many thanks if you could provide me with some readings or books to tackle this problem.

There could be 2 reasons why your 2 time series have different lengths :

# 1. They have a different sample rate

## The situation

The sample rate of your signal is the frequency at which samples have been converted from the analog domain to the digital domain. If 2 signals of the same duration have been sampled at different rates, the resulting time series will have different length( the longer being the one for the fastest sample rate).

## The solution

The only solution to that problem is to resample one of your signals. For this, you will need an interpolation filter. In Matlab, the whole process is very simple. However, if you need to implement that in low-level code it's going to be more complex (lots of things to take in account). I won't go into technical details here, there's a lot you can find here and on the rest of the internet.

# 2. They have different durations

## The situation

It's very much possible that your output signal has been recorded for a longer time than the input signal. In that case, you end up with 2 signals sampled at the same rate but with different lengths.

## The solution

If that's the case, you're quite lucky as the solution is trivial. You just have to do something "zero-padding" which consists simply in adding as many zeroes at the end of the shortest signal as needed for it to have the same size at the other.

Deconvolution is most often asked for signals with the same sampling. If you have $x[k]$, and $y[k]$ in the (same) sampled time domain, and they are of different lengths, this is quite normal, as the filtered signal $h\ast x$ is generally longer, which is taken care of in acquiring the output signal $y[k]$.