Editing Array processing (section)

=== 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
 |author1=Jamil Raza |author2=Albert-Jan Boonstra |author3=Alle-Jan van der Veen | date = February 2002
 | title = Spatial Filtering of RF Interference in Radio Astronomy
 | journal = IEEE Signal Processing Letters
 | volume = 9
 | issue = 12
 | pages = 64–67
 | doi = 10.1109/97.991140|bibcode=2002ISPL....9...64R|url=http://resolver.tudelft.nl/uuid:26525a9b-0815-49e2-82c8-b3c69ed4867f }}</ref>