Adaptive eigenvalue decomposition with applications to frequency or direction estimation and tracking

Summary form only given. Computation of the eigensystem of the updated matrix based on the knowledge of the eigensystem of the original covariance matrix was studied for the case of additive rank-k modification. This corresponds to adding blocks of data to and deleting them from the data matrix. Note that k is usually small compared to the dimensions of the Hermitian covariance matrix. An efficient parallel algorithm was developed for the rank-k modification problem. It makes use of a generalized spectrum slicing theorem relating the location of the modified eigenvalues to the location of the eigenvalues of the original matrix. At each time update, the set of eigenvalues can be solved in parallel by using efficient nonlinear search procedures. The eigenvector can be computed explicitly in terms of an intermediate vector that is a solution of a Hermitian homogeneous system much reduced in size (k×k). Further reduction in computational efficiency can be achieved in signal processing applications by making use of the multiplicities of the noise eigenvalues. These ideas were applied to eigenbased techniques for frequency or angle-of-arrival estimation