kruisspectrum

Kruisspectrum, or cross-spectrum, is a frequency-domain representation of the linear relationship between two stochastic time series. It is defined as the Fourier transform of the cross-covariance function R_xy(τ) = E[x(t) y(t+τ)]. Equivalently, for signals with finite energy, the cross-spectrum can be written as S_xy(f) = ∫ R_xy(τ) e^{-i 2π f τ} dτ. In practice, S_xy(f) is estimated from realizations as the expectation of the product X(f) Y*(f), where X(f) and Y(f) are the (possibly windowed) Fourier transforms of x(t) and y(t).

S_xy(f) is complex-valued; its magnitude |S_xy(f)| measures the strength of linear coupling at frequency f, while its phase φ_xy(f) = arg(S_xy(f)) indicates the phase difference between x and y at that frequency. The coherence function C_xy(f) = |S_xy(f)|^2 / (S_xx(f) S_yy(f)) is a related normalized measure that ranges from 0 to 1. The time delay between signals at frequency f can be estimated as Δt = φ_xy(f)/(2π f).

Applications and notes: cross-spectrum is used to study coupling and interactions between signals in fields such as neuroscience (e.g., EEG/MEG), geophysics, engineering, and economics. It complements the auto-spectral densities S_xx(f) and S_yy(f) by revealing how two signals relate across frequencies, not just their individual power spectra.

Estimation considerations: the approach assumes some level of stationarity within the analysis window; spectral leakage and variance depend on windowing and segment averaging methods (e.g., Welch’s method or multitaper). For real-valued signals, S_yx(f) = S_xy*(f). Kruisspectrum is the Dutch term for this concept.