The cross spectral density matrix, F , is central to many signal processing techniques. Consider a spatio-temporal scalar field: p(x,t) created by a microphone array. The Fourier transform for each microphone

 is used to create the matrix through ensemble averaging.

The matrix is Hermitian. The elements on the main diagonal, Fii,  are the real valued power spectra. The off diagonal elements are the cross power spectra for each transducer pair.

In many instances, the power spectra is the desired result, however, additional processing on the full matrix can produce the wavenumber-frequency spectra, isolate noise sources through subspace filtering via the Karhunen Loeve expansion, and identify bearing direction of noise sources.