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Cross-correlation spectroscopy with spatial interferometry, is possible by simply substituting a signal with voltage <math>V_Y(t)</math> in equation {{EquationNote|Eq.II}} to produce the cross-correlation <math>R_{\text{XY}}(\tau)</math> and the cross-spectrum <math>S_{\text{XY}}(f)</math>.
== Example: Spatial Filtering ==
In radio astronomy, RF interference must be mitigated to detect and observe any meaningful objects and events in the night sky.
[[File:Tele_Array.jpg|thumb|An array of radio-telescopes with an incoming wave]]
=== Projecting Out The Interferer ===
For an array of Radio Telescopes with a spatial signature of the interfering source <math>\mathbf{a}</math> that is not a known function of the direction of interference and its time variance, the signal covariance matrix takes the form:
<math>\mathbf{R} = \mathbf{R}_v + \sigma_s^2 \mathbf{a} \mathbf{a}^{\dagger} + \sigma_n^2 \mathbf{I}</math>
Where <math>\mathbf{R}_v</math> is the visibilities covariance matrix (sources), <math>\sigma_s^2</math> is the power of the interferer, and <math>\sigma_n^2</math> is the noise power, and <math>\dagger</math> denotes the Hermitian transpose. One can construct a projection matrix <math>\mathbf{P}_a^{\perp}</math>, which, when left and right multiplied by the signal covariance matrix, will reduce the interference term to zero.
<math>\mathbf{P}_a^{\perp} = \mathbf{I} - \mathbf{a}(\mathbf{a}^{\dagger} \mathbf{a})^{-1} \mathbf{a}^{\dagger}</math>
So the modified signal covariance matrix becomes:
<math>\tilde{\mathbf{R}} = \mathbf{P}_a^{\perp} \mathbf{R} \mathbf{P}_a^{\perp} = \mathbf{P}_a^{\perp} \mathbf{R}_v \mathbf{P}_a^{\perp} + \sigma_n^2 \mathbf{P}_a^{\perp}</math>
Since <math>\mathbf{a}</math> is generally not known, <math>\mathbf{P}_a^{\perp}</math> can be constructed using the eigen-decomposition of <math>\mathbf{R}</math>, in particular the matrix containing an orthonormal basis of the noise subspace, which is the orthogonal complement of <math>\mathbf{a}</math>. The disadvantages to this approach include altering the visibilities covariance matrix and coloring the white noise term.<ref>{{cite journal
| author = Jamil Raza, Albert-Jan Boonstra, Alle-Jan van der Veen
| date = February 2002
| title = Spatial Filtering of RF Interference in Radio Astronomy
| pii = S 1070-9908(02)03406-5}}</ref>
=== Spatial Whitening ===
This scheme attempts to make the interference-plus-noise term spectrally white. To do this, left and right multiply <math>\mathbf{R}</math> with inverse square root factors of the interference-plus-noise terms.
<math>\tilde{\mathbf{R}} = (\sigma_s^2 \mathbf{a} \mathbf{a}^{\dagger} + \sigma_n^2 \mathbf{I})^{-{\frac{1}{2}}} \mathbf{R}(\sigma_s^2 \mathbf{a} \mathbf{a}^{\dagger} + \sigma_n^2 \mathbf{I})^{-{\frac{1}{2}}}</math>
The calculation requires rigorous matrix manipulations, but results in an expression of the form:
<math>\tilde{\mathbf{R}} = (\cdot)^{-{\frac{1}{2}}} \mathbf{R}_v(\cdot)^{-{\frac{1}{2}}} + \mathbf{I}</math>
This approach requires much more computationally intensive matrix manipulations, and again the visibilities covariance matrix is altered.<ref>{{cite journal
| author = Amir Leshem, Alle-Jan van der Veen
| date = August 16, 2000
| title = Radio astronomical imaging in the presence of strong radio interference
| doi = 10.1109/18.857787
|arxiv = astro-ph/0008239}}</ref>
=== Subtraction of Interference Estimate ===
Since <math>\mathbf{a}</math> is unknown, the best estimate is the dominant eigenvector <math>\mathbf{u}_1</math> of the eigen-decomposition of <math>\mathbf{R} = \mathbf{U} \mathbf{\Lambda} \mathbf{U}^{\dagger}</math>, and likewise the best estimate of the interference power is <math>\sigma_s^2 \approx \lambda_1 - \sigma_n^2</math>, where <math>\lambda_1</math> is the dominant eigenvalue of <math>\mathbf{R}</math>. One can subtract the interference term from the signal covariance matrix:
<math>\tilde{\mathbf{R}} = \mathbf{R} - \sigma_s^2 \mathbf{a} \mathbf{a}^{\dagger}</math>
By right and left multiplying <math>\mathbf{R}</math>:
<math>\tilde{\mathbf{R}} \approx (\mathbf{I} - \alpha \mathbf{u}_1 \mathbf{u}_1^{\dagger})\mathbf{R}(\mathbf{I} - \alpha \mathbf{u}_1 \mathbf{u}_1^{\dagger}) = \mathbf{R} - \mathbf{u}_1 \mathbf{u}_1^{\dagger} \lambda_1(2 \alpha - \alpha^2)</math>
Where <math>\lambda_1(2 \alpha - \alpha^2) \approx \sigma_s^2</math> by selecting the appropriate <math>\alpha</math>. This scheme requires an accurate estimation of the interference term, but does not alter the noise or sources term.<ref>{{cite journal
| author = Amir Leshem, Albert-Jan Boonstra, Alle-Jan van der Veen
| date = November, 2000
| title = Multichannel Interference Mitigation Techniques in Radio Astronomy
| doi = 10.1086/317360
|arxiv = astro-ph/0005359}}</ref>
== Summary ==
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