Cross-spectrum of two stationary time series ($f_{XY}(\lambda )=\frac{1}{2\pi }{\sum }_h\mathbb{E}[X_t{Y}_{t+h}]e^{-ih\lambda }$)

For zero-mean real weakly stationary processes whose cross-covariances are summable, the cross-spectrum is the Fourier series of their lagged covariance, with the displayed convention. Reality gives $f_{XY}(-\lambda )=\overline{f_{XY}(\lambda )}$, but the cross-spectrum need not be real or even. Exchanging the two series gives $f_{YX}(\lambda )=\overline{f_{XY}(\lambda )}$. The entries form a Hermitian spectral density of a stationary process matrix.