crossphase

Crossphase is a term used in signal processing and related fields to describe the phase relationship between two time-series signals in the frequency domain. It refers to the phase component of the cross-spectral density between the signals, often denoted as S_xy(f). If X(f) and Y(f) are the Fourier transforms of two signals x(t) and y(t), the cross-spectrum is commonly defined as S_xy(f) = E[X*(f) Y(f)], and the crossphase is phi_xy(f) = arg(S_xy(f)). The crossphase captures how the components of one signal are shifted in time relative to the other at a given frequency, with the interpretation of the sign and magnitude depending on the chosen convention.

Estimation and computation typically rely on finite data. One common approach is to segment data, compute the FFT of each segment, and average the product X*(f)Y(f) across segments to obtain an estimate of the cross-spectrum. The crossphase is then the angle of this estimate. Researchers often also compute the coherence, C_xy(f) = |S_xy(f)|^2 / (S_xx(f) S_yy(f)), to quantify the strength of coupling at each frequency and to assess the reliability of the crossphase. Smoothing, multitaper methods, or Welch-type averaging can reduce variance in the estimates, especially in noisy data.

Applications of crossphase analysis span neuroscience (for EEG/MEG connectivity and phase synchronization), physiology, geophysics, and engineering, where understanding the timing relationship between signals is essential. Crossphase should not be interpreted as evidence of causality by itself; for directional influence, complementary methods such as Granger causality or phase-slope index are often used. In optics, cross-phase modulation is a different nonlinear effect and is not the same concept as frequency-domain crossphase.