crosscorrelations

Cross-correlation is a statistical measure of similarity between two signals as a function of the displacement of one relative to the other. It is commonly used to detect time delays, align signals, or quantify how well two signals match after shifting.

For continuous-time signals f(t) and g(t), the cross-correlation is defined as R_fg(τ) = ∫_{-∞}^{∞} f(t) g(t+τ) dt. If the signals are complex-valued, the definition uses complex conjugation: R_fg(τ) = ∫ f*(t) g(t+τ) dt. For discrete-time signals x[n] and y[n], the cross-correlation sequence is R_xy[k] = ∑_{n} x[n] y[n+k], with centering or normalization applied when measuring covariance rather than simple correlation.

In the context of stochastic processes, the cross-covariance function between X(t) and Y(t) with means μ_X and μ_Y is C_xy(τ) = E[(X(t)−μ_X)(Y(t+τ)−μ_Y)]. If the processes are jointly stationary, C_xy(τ) depends only on the lag τ. A normalized form, the cross-correlation coefficient r_xy(τ) = C_xy(τ) / (σ_X σ_Y), yields values between −1 and 1.

Cross-correlation is closely related to convolution: it can be computed as a convolution with one sequence reversed. The cross-spectral density S_xy(f), obtained as the Fourier transform of C_xy(τ), provides a frequency-domain view and recovers C_xy(τ) via the inverse transform.

Practical considerations include non-stationarity, finite-sample bias, edge effects, and the need to subtract means or detrend data before computing correlations. Applications span signal processing, time-delay estimation, system identification, geophysics, astronomy, and finance.