# How to estimate noise by eigendecomposition of the variance covariance matrix?

I'm new here so I will try to be as clear as possible. I am trying to apply some techniques from signal processing framework to denoise financial time series.

I would like to know if what I am trying to do makes sense or if I am committing some theoretical mistake;

I have a time series that I assume to be composed by two sources (non observable), one a genuine signal while the other is just white noise. $$Y(t)=X(t)+e(t)$$

I also assume that the white noise is additive and is uncorrelated with the signal. From here what I am planning to do is to apply the STFT to the observable data.

For every frequency I construct a Hankel matrix (a sort of trajectory matrix for the Fourier coefficients). Then I obtain the variance-covariance matrix by multiplying the Hankel matrix by its hermitanian.

Here is where things become confusing: I want to make the eigenvector decomposition of this matrix, such that I'll get something like:

$$U(\Lambda+\sigma^2I)U'\,\text,$$ assuming that the error term is white noise, and there is no overlap between windows, and $$\Lambda$$ being the diagonal matrix containing the eigenvalues of $$X$$ on its diagonal.

If I further assume that the signal $$X$$ has a smaller dimension, i.e. has some 0 eigenvalues I could estimate $$\sigma^2$$ (PSD of the noise term) for this given frequency.

Is there any way to obtain the signal $$X$$ from here?

I don't really know if what I'm doing is ok or not, and I would really appreciate if you could give me some insights.