Subspace estimation using unitary Schur-type methods

This paper presents efficient Schur–type algorithms for estimating the column space (signal subspace) of a low rank data matrix corrupted by additive noise. Its computational structure and complexity are similar to that of an LQ–decomposition, except for the fact that plane and hyperbolic rotations are used. Therefore, they are well suited for a parallel (systolic) implementation. The required rank decision, i.e., an estimate of the number of signals, is automatic, and updating as well as downdating are straightforward. The new scheme computes a matrix of minimal rank which is -close to the data matrix in the matrix 2-norm, where is a threshold that can be determined from the noise level. Since the resulting approximation error is not minimized, critical scenarios lead to a certain loss of accuracy compared to SVD-based methods. This loss of accuracy is compensated by using Unitary ESPRIT in conjunction with the Schur–type subspace estimation scheme. Unitary ESPRIT represents a simple way to constrain the estimated phase factors to the unit circle and provides a new reliability test. Due to the special algebraic structure of the problem, all required factorizations can be transformed into decompositions of real-valued matrices of the same size. The advantages of Unitary ESPRIT dramatically improve the resulting subspace estimates, such that the performance of Unitary Schur ESPRIT is comparable to that of SVD-based methods, at a fraction of the computational cost. Compared to the original Schur method, Unitary Schur ESPRIT yields improved subspace estimates with a reduced computational load, since it is formulated in terms of real-valued computations throughout.