Centered and non-centered principal component analyses in the frequency domain

Given a stationary multidimensional process and the process which is deduced after centering, we wish to study possible links between the principal component analyses, in the frequency domain, of these two processes. It is well known that there is, a priori, no obvious relationship between the centered and non-centered principal component analyses in the temporal domain. Furthermore, we also know that principal component analysis in the frequency domain is reduced to principal component analysis of each spectral component. In this paper, we show the remarkable result that the centered and non-centered principal component analyses in the frequency domain are equal except for a given frequency.