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.