A Solution to the Recursive Generalized Eigenvalue Problem

Abstract

Dr. Matthew Bromberg is an independent consultant. For the last 6 years Dr. Bromberg has been involved in the research and development of array processing algorithms for reuse enhancement for wireless communication systems and for interference mitigation for both commercial and military applications while he was at Radix Technologies. Dr. Bromberg was a key inventor of the technology that led to the formation and funding of Beam Reach Networks and the formation of Protean Radio Networks. He has authored several papers and patents in this area. Abstract Previously, we have discussed a recursive solution to the vector normal equation utilizing the recursive Cholesky and Modified Gram-Schmidt Orthogonalization (MGSO) algorithms. Previously, we have also discussed a blind adaptive approach for detection and extraction of signals of interest in the presence of noise and interference without relying on preamble or training sequences. The heart of the blind adaptive algorithm is based upon solving the recursive generalized eigenvalue problem, the solution of which is discussed in this paper based on the recursive Cholesky or QR factors and the Householder and QL algorithm with implicit shifts. Even though, the solution to the recursive generalized eigenvalue problem is slightly more efficient than the solution to the direct generalized eigenvalue when all the eigenvalues and eigenvectors are required, the improvement is more noticeable when only one eigenvalue and the corresponding eigenvector are required. Therefore, the solution that is proposed here serves well for recursive generalized eigenvalue problems that require only one eigenvalue and the corresponding eigenvector