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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?

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