Cross-spectrum ($P_{UV}(k)$)

A cross-spectrum relates the Fourier transforms of two random fields or two stochastic processes. The cross-spectrum of two stationary time series gives the discrete-time case.

For statistically homogeneous random fields, the three-dimensional cross-power spectrum is defined byunder the Fourier transform convention $U(\mathbf{x})=\int d^{3}kU(\mathbf{k})e^{i\mathbf{k}\cdot \mathbf{x}}/(2\pi {)}^3$. For isotropic fields it depends only on $k=|\mathbf{k}|$. A deterministic relation $V(\mathbf{k})=A(k)U(\mathbf{k})$ implies $P_{UV}=A^{\ast }{P}_{UU}$. An angular cross-power spectrum is defined analogously from spherical harmonic coefficients.