Editing Array processing (section)

=== 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>