CrossPowerSpectrum
Cross power spectrum, or cross-spectral density S_xy(f), is a frequency-domain measure of how two signals x(t) and y(t) co-vary across frequencies. For stationary processes, S_xy(f) is the Fourier transform of their cross-covariance function R_xy(τ) = E[x(t) y*(t+τ)]. Equivalently, if X(f) and Y(f) are the Fourier transforms of x and y, then S_xy(f) = E{X(f) Y*(f)}, and in practice it is often estimated by averaging product spectra from multiple data segments.
A common estimation approach uses the Welch method: split the data into overlapping segments, apply a window
Interpretation and related quantities: the phase of S_xy(f) indicates the phase difference between x and y at
Units: if x and y have units Ux and Uy, then S_xy(f) has units Ux·Uy per Hz;
Applications span signal processing, system identification, neuroscience (neural coherence), geophysics, and audio engineering, where frequency-specific relationships