# 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 $T$.

• Hi: If you convert that frequency band component of the signal to the equivalent time domain signal, then, once you're in the time domain, estimating the covariance matrix is straightforward. I'm not sure how you convert that frequency band component of signal to the equivalent time domain signal ? maybe inverse DFT ? I – mark leeds Feb 25 '18 at 16:34
• I just happened to be reading this when I realized that I don't understand how a multi-dimensional series is represented in the frequency domain. Is A) each dimension treated separately and then they are added up at each time t or B) are they viewed as a linear combination so that the fourier transform becomes univariate ? if someone knows of a text or reference, it's appreciated. Or maybe it's so obvious that it doesn't even need explanation ???? thanks. – mark leeds May 26 '18 at 23:11