<title>Simultaneous subspace tracking and rank estimation</title>

Abstract

Jacobi-type singular value decomposition (SVD) is an exceptionally suitable method for recursive SVD updating. It has a low computational complexity per update, O(n<SUP>2</SUP>), where n is the problem size. Due to its parallel structure, it is very attractive for real-time applications on systolic arrays. In frequency and angle of arrival tracking problems, SVD can be used to track the signal subspace. Typically, only a few, r, dominant eigencomponents need to be tracked, where r is less than n. In this paper we show how to modify the Jacobi-type SVD to track only the r-dimensional signal subspace by forcing the (n - r)-dimensional noise subspace to be spherical. Thereby, the computational complexity is brought down from O(n<SUP>2</SUP>) to O(nr). Furthermore, we show how to exploit the structure of the Jacobi-type SVD to estimate the signal subspace dimension, r, in addition to tracking the subspace itself. Most available computationally efficient subspace tracking algorithms rely on off-line estimation of the singal subspace dimension. This acts as a bottleneck in real-time parallel implementations. The Jacobi-type spherical subspace tracking algorithm presented in this paper is thus a subspace tracking method of computational complexity O(nr), capable of tracking both the signal subspace and its dimension simultaneously. Thereby, the parallelism of the algorithm is not destroyed, as demonstrated in a systolic array implementation. Simulation results are presented to show the applicability of the algorithm in adaptive frequency tracking problems.