Factored Spherical Subspace Tracking

Keywords: subspace tracking, Givens rotations, systolic arrays
1 Introduction
Tracking the singular value decomposition (SVD) of a time-varying data matrix is an important component of several algorithms for engineering applications. Examples include subspace algorithms for high resolution direction nding of narrow-band signal sources using antenna arrays 1] and subspace algorithms for system identi cation 2]. The rst step in these algorithms consists in determining the dominant subspace of a recursively de ned data matrix
X k] 2 Rk
M
2 3 X k?1] 5 X k] = 4
xTk]
This work was supported in part by the ESPRIT BRA 6632 project of the EU. Filiep Vanpoucke is a research assistant with the N.F.W.O. Marc Moonen is a research associate with the N.F.W.O. (Belgian National Fund for Scienti c Research). e-mail: f liep.vanpoucke,marc.mooneng@esat.kuleuven.ac.be.
1
where the scalar < 1 is an exponential weighting factor and x k] 2 RM is the data vector1 at the kth sampling time tk . The singular value decomposition of X k] is de ned by
Abstract
Tracking the dominant subspace of a measured data matrix is an essential part of many signal processing algorithms. We present a modi cation to the so-called spherical subspace tracking algorithm. This algorithm has a low computational complexity, but su ers from accumulation of numerical errors. We show that this numerical instability can be circumvented by using a minimal orthogonal parameterization of the subspace basis matrix. The resulting factored spherical SVD updating algorithm then consists exclusively of rotation operations. In view of implementing the algorithm in real-time applications with high data rates, a linear systolic architecture is derived.
X k] = U n ] 4 k
k]
V T] k
s k]
3 3 2 s V kT ] 5 5 4 n n V k]T
k]
(1)
T where U k] 2 Rk M and V k] 2 RM M are orthogonal matrices (U T ] U k] = V k] V k] = IM ) k and k] 2 RM M is a diagonal matrix with positive real numbers in non-increasing order, the so-called singular values. When partitioning the matrices according to the set of D largest (in sk] ) and M ? D smallest (in nk] ) singular values, the dominant subspace S is de ned as s the column space of the matrix V k]. In the direction nding application D would typically be the number of narrow-band emitters in the frequency band of interest, and M is the number of antennas in the array. The accuracy of the signal parameter estimates ultimately depends on the accuracy of the signal subspace estimate. Several applications, e.g. in communication systems or radar systems, require tracking of the dominant subspace in real-time at data rates in the range of 100 ? 1000 ksample/sec. However, tracking the SVD exactly is computationally expensive. Therefore, in the literature several algorithms have been developed which trade accuracy for lower complexity 3, 4, 5, 6]. In this paper we propose a modi cation to the spherical subspace tracking algorithm of DeGroat 3], which will be reviewed in section 2. It is a non-iterative algorithm with a low s complexity, O(M D). This is attained by keeping track of the matrix V k] instead of the full matrix V k], and of two averaged singular levels instead of the exact singular values. Although the latter approximation may seem crude, it can be shown that the algorithm still provides a consistent estimate of the signal subspace in a stationary environment 7]. Recently the algorithm has been generalized to multiple levels 8]. Here, we concentrate on the two-level algorithm. The generalization to the multi-level algorithm does not pose any additional di culty. The spherical subspace tracking algorithm has one major drawback. In nite precision s arithmetic the subspace tracking matrix V k] gradually loses its orthogonality due to error accumulation. Current solutions to stabilize the algorithm are based on re-orthogonalization by means of pair-wise Gram-Schmidt transformations 4] or renormalization 9]. However, they only keep the deviation from orthogonality bounded and are di cult to implement on parallel architectures. As an alternative, in section 3 we introduce a minimal parameterization of V s ] in terms of a sequence of Givens rotations, each characterized by a single rotation angle. k By computing with the rotation angles instead of the explicit matrix V s ], the orthogonality is k preserved at each time instant. This parameterization has already been applied successfully to stabilize a related Jacobi algorithm for SVD updating 10]. Due to its regularity, this factored spherical subspace tracking algorithm is well suited for implementation on application speci c parallel hardware. Starting from the signal ow graph, in section 4 a linear systolic array is derived in by a placement and scheduling step. With D + 1 processors a pipelining period of 2M ? 1 cycles is obtained.
Factored Spherical Subspace Tracking
Filiep Vanpoucke and Marc Moonen Katholieke Universiteit Leuven Dept. of Electrical Engineering, ESAT K. Mercierlaan 94, 3001 Leuven, Belgium