# How to estimate covariance matrix using Fourier representation?

So, I have multidimensional time-series $X \in R^{(d \times T)}$, and I want to determine the covariance matrix of that signal in a specific frequency band.

I might filter the signal to that specific frequency band and do the usual $XX'$. However, if I assume the signal to be stationary, I might use the Fourier representation of the signal for a much efficient implementation, according to the Plancherel theorem

However, using only a subset of the Fourier coefficients (those corresponding to my frequency band of interest) for computing the covariance matrix, I get a super ill-conditioned matrix.

Does anybody have an idea about how to get around this (that does not involve regularization of the matrix)? or may any other tips on how to approach the problem from a different perspective? Thanks! :)

Well, when you filter the signal to a specific fequency band in the frequency domain, you project your signal to a subspace of $C^T$ . Hence you obtain the a covariance with rank less than $T$.