ULV and generalized ULV subspace tracking adaptive algorithms

Traditional adaptive filters assume that the effective rank of the input signal is the same as the input covariance matrix or the filter length N. Therefore, if the input signal lives in a subspace of dimension less than N, these filters fail to perform satisfactorily. In this paper, we present two new algorithms for adapting only in the dominant signal subspace. The first of these is a low-rank recursive-least-squares (RLS) algorithm that uses a ULV decomposition (Stewart 1992) to track and adapt in the signal subspace. The second adaptive algorithm is a subspace tracking least-mean-squares (LMS) algorithm that uses a generalized ULV (GULV) decomposition, developed in this paper, to track and adapt in subspaces corresponding to several well-conditioned singular value clusters. The algorithm also has an improved convergence speed compared with that of the LMS algorithm. Bounds on the quality of subspaces isolated using the GULV decomposition are derived, and the performance of the adaptive algorithms are analyzed