4260 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 57, NO. 11, NOVEMBER 2009 The QS-Householder Sliding Window Bi-SVD Subspace Tracker Peter Strobach Abstract—A fast algorithm for computing the sliding window Bi-SVD subspace tracker is introduced. This algorithm produces, in each time step, a dominant rank- SVD subspace approximant of an rectangular sliding window data matrix. The method is based on the (orthonormal-square) decomposition. It uses two row-Householder transformations for updating and one nonorthogonal Householder transformation for downdating in each time step. The resulting algorithm is long-term stable and shows excellent numerical and structural properties, as known from pure Householder-type algorithms. The dominant com- plexity is multiplications per time update, which is also the lower bound in dominant complexity for an algorithm of this kind. A completely self-contained algorithm summary is provided and a Fortran subroutine of the algorithm is available for download from http://webuser.hs-furtwangen.de/~strobach/qsh- bisvd.for. Index Terms—Householder, QS-Decomposition, singular Vectors, singular value decomposition (SVD), sliding Window Subspace Tracking. I. INTRODUCTION T HE Singular Value Decomposition (SVD) is certainly one of the most important mathematical tools in signal pro- cessing. There are many applications, where one wants to up- date dominant parts of this decomposition, or unitary similar versions thereof, in time upon the arrival of new data. Suppose we have given a rank- SVD approximant of an data ma- trix as follows: (1) where is the dominant orthonormal left singular sub- space basis, is the dominant orthonormal right sin- gular subspace basis, and is an square-root power ma- trix that is two-sided orthonormally or (in the complex case) unitary similar to the diagonal matrix of ordered dominant sin- gular values of . In this paper, we consider the particular case of sliding window updating of this dominant SVD subspace ap- proximant (1) in time upon the arrival of a new data vector on the top of the matrix as follows: (2) Manuscript received January 01, 2009; accepted May 24, 2009. First pub- lished June 30, 2009; current version published October 14, 2009. The associate editor coordinating the review of this manuscript and approving it for publica- tion was Dr. Deniz Erdogmus. The author is with the AST-Consulting Inc., 94133 Röhrnbach, Germany (e-mail: peter_strobach@gmx.de). Digital Object Identifier 10.1109/TSP.2009.2025978 whereas an old snapshot vector is dropped at the bottom of the matrix, and an updated approximant (3) is computed at time step with an overall dominant operations count of multiplications. The algorithm of this paper is a straightforward fast imple- mentation of the sliding window Bi-SVD subspace tracker of Badeau, Richard, and David [1]. Their method was inspired by the exponentially windowed bi-iteration SVD subspace tracker of [2] and the square Hankel SVD subspace tracker of [3]. These algorithms, in turn, can be traced back to the classical “Treppen- iterations” of F. L. Bauer [4]. An accelerated variant of a bi-iteration type algorithm using the so-called “Bi-LS” concept was recently described in [5]. All these methods can be regarded as members of the larger class of sequential orthogonal iteration algorithms. Strobach has recently introduced a unifying theory of fast or- thogonal iteration subspace tracking [6]. This theory is based on the (orthonormal-square) decomposition combined with row-Householder reductions for the effective and efficient up- dating of such -decompositions. This method marks the top in the development of subspace tracking algorithms, because the resulting algorithms are clearly optimal in terms of struc- ture, numerical accuracy and stability, and overall operations count. This comes as no surprise since these algorithms are pure Householder methods and Householder methods are generally known for their superior properties. A first application is the SVD subspace tracker of [7] for the growing-window case with exponential forgetting. In this paper, we take up the basic sliding window Bi-SVD of [1] and show how one can handle this algorithm in terms of the unifying theory of Householder subspace tracking to ar- rive at a highly competitive algorithm with minimum domi- nant operations count. In Section II, we summarize the basic results of [1], which we use as a starting point for our develop- ment. Section III introduces the key “tricks” on which our algo- rithm development will be founded. Sections IV and V present the main steps in the new all-Householder based fast sliding window Bi-SVD subspace tracker. Section VI discusses aspects of practical implementation and summarizes the algorithm in a completely self-contained algorithm summary. A Fortran sub- routine is available for download. Section VII shows the results of various computer experiments using this subroutine in order to visualize the operation of the new algorithm. Section VIII presents the conclusions. 1053-587X/$26.00 © 2009 IEEE