Let xI(t)=(xi1(t),…,xiN(t))T and xJ(t)=(xj1(t),…,xjM(t))T be the multivariate time series for the N-dimensional signals I and the M-dimensional signals J, where T denotes the transpose operator. For instance, these could represent the three-dimensional vector-source activities at two given brain locations, and thus they would have a dimension N = M = 3, or they could indicate two different multivariate time series consisting of the activities of all the sources within a region of interest, and thus they would have a dimension equal to the number of locations belonging to those parcels. Let also XI(f) and XJ(f) be the corresponding vector Fourier transforms. The cross-spectral density matrix between xI(t) and xJ(t) at a given frequency f is defined as
