Crosspower

Crosspower is a term used in signal processing and related disciplines to describe the cross-power spectrum, a frequency-domain measure of the relationship between two signals. Given two time series x(t) and y(t) with Fourier transforms X(f) and Y(f), the cross-power spectrum Sxy(f) is defined as the expected value of the product X(f) times the complex conjugate of Y(f): Sxy(f) = E{ X(f) Y*(f) }. In practice, Sxy is estimated from data by averaging the product of Fourier transforms across segments, for example using Welch's method.

From Sxy one can derive the cross-power density and, together with the auto-spectral densities Sxx(f) and Syy(f), compute the coherence: gamma^2(f) = |Sxy(f)|^2 / [ Sxx(f) Syy(f) ]. Coherence ranges from 0 to 1 and measures the degree of linear correlation between the signals at each frequency, independent of amplitude.

Applications include assessing synchronization and phase relationships between neurophysiological signals (EEG, MEG), audio and vibration analysis, geophysics, and communications. Cross-power analysis can help estimate time delays (lags) between signals and identify directional coupling in some models.

Notes: crosspower is sometimes used loosely to refer to the cross-spectrum itself or to the product of spectra when combining signals. The term may have other domain-specific meanings or branding in different fields, but in signal processing it most commonly denotes the cross-power spectrum and related coherence measures.