Complementarity of the Reddi method of source direction estimation with those of Pisarenko and Cantoni and Godara. I.
Derrill J. Bordelon · The Journal of the Acoustical Society of America · 1981
It is shown that the methods of plane-wave signal direction estimation of (a) Reddi, (b) Pisarenko, and (c) Cantoni and Godara (CG), in the case of a nonsingular signal source covariance matrix P, constitute orthogonal projection of the candidate signal direction vectors onto the right eigenspace corresponding to the eigenvalue σ2 of the complex covariance matrix R. Case (a) centralizes on the subspace spanned by the right eigenvectors of R corresponding to eigenvalues larger than σ2, whereas cases (b) and (c) to the orthogonal complement of this subspace being the subspace (right eigenspace) spanned by the right eigenvectors of R corresponding to smallest eigenvalues equal to σ2. It is indicated that Reddi’s and CG’s approach, when restriction is made in the case of CG to equispaced collinear arrays of sensors, are noneconomized versions of Pisarenko’s algorithm. A simplistic beamformer is defined herein which, unlike those derived in the work of Reddi, Pisarenko, and Cantoni and Godara, requires calculation of a minimal eigenvalue, σ2, but no eigenvectors. A maximal linearly independent set of column vectors is selected from the received signal covariance matrix, (R−σ2IN), and these are employed to define a simplistic signal (interference) nulling or preserving beamformer. Finally, it is shown that the case of P non-negative definite, viz., P?0, led CG to a flawed conclusion.