Cross-spectral density (CSD)

The cross-spectral density generalizes the PSD to a pair of signals: it is the Fourier transform of their cross-covariance, a complex-valued function of frequency whose magnitude measures how much power the two channels share at each frequency and whose phase measures their average timing (phase lag) there. It is the fundamental building block of every frequency-domain connectivity measure, including coherence, phase-locking and Granger-style measures, and, arranged across channels, it forms the cross-spectral matrix used by beamformers.

Like the PSD, the raw cross-periodogram is a high-variance estimator and must be smoothed by multitaper or Welch averaging before its magnitude and phase are trustworthy. A crucial subtlety for EEG and MEG is that instantaneous, zero-lag correlation from volume conduction inflates the real part of the CSD, so measures built on its imaginary part are designed to suppress that artifact.