Editing Array processing (section)

== 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:Telescope array.png|thumb|An array of radio telescopes with an incoming radio wave and RF interference]]

=== 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>

=== 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
 |author1=Amir Leshem |author2=Alle-Jan van der Veen | date = August 16, 2000
 | title = Radio astronomical imaging in the presence of strong radio interference
 | journal = IEEE Transactions on Information Theory
 | volume = 46
 | issue = 5
 | pages = 1730–1747
 | doi = 10.1109/18.857787
 |arxiv = astro-ph/0008239|s2cid=4671806 }}</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
 |author1=Amir Leshem |author2=Albert-Jan Boonstra |author3=Alle-Jan van der Veen | date = November 2000
 | title = Multichannel Interference Mitigation Techniques in Radio Astronomy
 | journal = Astrophysical Journal Supplement Series
 | volume = 131
 | issue = 1
 | pages = 355–373
 | doi = 10.1086/317360
 |arxiv = astro-ph/0005359|bibcode=2000ApJS..131..355L|s2cid=50311217 }}</ref>