= Cross-spectrum
{title2=$P_{UV}(k)$}
{wiki}
= Cross-power spectrum
{synonym}
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 by
$$
\langle U(\mathbf k)V^*(\mathbf k')\rangle=(2\pi)^3\delta_{\rm D}^{(3)}(\mathbf k-\mathbf k')P_{UV}(\mathbf k)
$$
under the <Fourier transform> convention $U(\mathbf x)=\int d^3k\,U(\mathbf k)e^{i\mathbf k\cdot\mathbf x}/(2\pi)^3$. For isotropic fields it depends only on $k=|\mathbf k|$. A deterministic relation $V(\mathbf k)=A(k)U(\mathbf k)$ implies $P_{UV}=A^*P_{UU}$. An angular <cross-power spectrum> is defined analogously from <spherical harmonic> coefficients.
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