# How to find a Matched Filter Transfer Function from large signal sample

Lets say I have a system where I have a small sample of a signal with no noise $$\hat{x}(t)$$ and a lot of a similar signal with noise $$y(t) = \hat{x}(t) + n(t)$$, and from $$\hat{x}(t)$$ I want to create a matched filter to 'find' $$\hat{x}(t)$$ in $$y(t)$$

In theory I would go about doing this by first finding $$\hat{X}^*(k) = F^*_k[\hat{x}(t)]$$

So far, no issue, but my signal $$\hat{x}(t)$$ is 10 minutes long sampled at 100Hz and non-stationary, so when I have previously been analysing the power in this signal i have been using Welch's method to find the power in small increments of $$\hat{x(t)}$$ which can be assumed to be stationary and then averaging which helps to remove the noise.

I would like to do something similar to try and have a more accurate transfer function, unfortunately I think I am correct in saying that you can't average small, Fourier transformed, increments in the same way because of the interference caused by the phase difference.

So if I wanted to match $$y(t)$$ with $$\hat{x(t)}$$ would I simply take one small sample of $$\hat{x(t)}$$ and use that to find a transfer function, or are there more clever ways?