advantage over a XF correlators in that the computational complexity is O(N2). Therefore, FX correlators are more efficient for larger arrays. Correlation spectrometers like the Michelson interferometer vary the time lag between signals obtain the power spectrum of input signals. The power spectrum of a signal is related to its autocorrelation function by a Fourier transform: where the autocorrelation function for signal X as a function of time delay is Cross-correlation spectroscopy with spatial interferometry, is possible by simply substituting a signal with voltage in equation to produce the cross-correlation and the cross-spectrum . Example: spatial filtering In radio astronomy, RF interference must be mitigated to detect and observe any meaningful objects and events in the night sky. Projecting out the interferer For an array of Radio Telescopes with a spatial signature of the interfering source that is not a known function of the direction of interference and its time variance, the signal covariance matrix takes the form: where is the visibilities covariance matrix (sources), is the power of the interferer, and is the noise power, and denotes the Hermitian transpose. One can construct a projection matrix , which, when left and right multiplied by the signal covariance matrix, will reduce the interference term to zero. So the modified signal covariance matrix becomes: Since is generally not known, can be constructed using the eigen-decomposition of , in particular the matrix containing an orthonormal basis of the noise subspace, which is the orthogonal complement of . The disadvantages to this approach include altering the visibilities covariance matrix and coloring the white noise term. Spatial whitening This scheme attempts to make the interference-plus-noise term spectrally white. To do this, left and right multiply with inverse square root factors of the interference-plus-noise terms. The calculation requires rigorous matrix manipulations, but results in an expression of the form: This approach requires much more computationally intensive matrix manipulations, and again the visibilities covariance matrix is altered. Subtraction of interference estimate Since is unknown, the best estimate is the dominant eigenvector of the eigen-decomposition of , and likewise the best estimate of the interference power is , where is the dominant eigenvalue of . One can subtract the interference term from the signal covariance matrix: By right and left multiplying : where by selecting the appropriate . This scheme requires an accurate estimation of the interference term, but does not alter the noise or sources term.