Peter Strobach

Bi-iteration SVD subspace tracking algorithms

We present a class of fast subspace tracking algorithms that arise from a straightforward extension of Bauers (1957) classical bi-iteration to the sequential processing case. The bi-iteration concept has an unexpected potential in subspace tracking. Our new bi-SVD subspace trackers are well structured and show excellent convergence properties. They outperform the TQR-SVD subspace tracking algorithm. Detailed comparisons confirm our claims. An application to rank and data adaptive signal reconstruction is also discussed.

Square-root QR inverse iteration for tracking the minor subspace

A new algorithm for tracking the eigenvectors associated with the r smallest eigenvalues of an N/spl times/N covariance matrix is introduced. The method is sequential inverse iteration based on a recursive square-root QR factor updating of the covariance matrix with O(N/sup 2/r) operations per time update. The principal operations count of this new tracker is justified by a significantly better performance compared with the fast O(Nr/sup 2/) minor subspace tracker of Douglas et al. (1998).

Two-dimensional equirotational stack subspace fitting with an application to uniform rectangular arrays and ESPRIT

A two-dimensional (2-D) multiple invariance technique for computing signal subspaces for uniform rectangular arrays (URAs) of size M/spl times/N sensors is introduced. The method is based on a multiple maximum overlap configuration of the sensors in the array with m/spl times/n subarrays of (M-m+1)/spl times/(N-n+1) sensors each. We exploit the fact that the stacked subspace of the subarray sensor output signals admits a two-level equirotational stack parametrization. We introduce a TLS-type algorithm for estimating the parameters of this equirotational stack subspace model. Based on this method of equirotational stack subspace fitting, the overall array signal subspace can be estimated with a much higher accuracy than with conventional unstructured SVD and TLS techniques. Detailed experiments validate the theoretical results. We propose a variant of 2-D ESPRIT based on equirotational stack subspace fitting. This 2-D equirotational stack ESPRIT (2-D ES-ESPRIT) algorithm clearly outperforms conventional unstructured variants of 2-D ESPRIT. A detailed comparison with 2-D unitary ESPRIT is presented.

Square Hankel SVD subspace tracking algorithms

Abstract In this paper, we describe two sliding window singular-value decomposition (SVD) subspace tracking algorithms for the serialized (time series) data case. In time series analysis, one often uses a Hankel matrix representation of the data. In sliding window low rank adaptive time-series analysis with overmodeling, we can set both column and row dimensions of this Hankel data matrix equal to the window length. This special case is of particular interest because a square Hankel matrix is symmetrtic and has identical left and right singular subspaces. Thus, we can track the square Hankel SVD using a fast variant of sequential orthogonal iteration. We develop two sliding window square Hankel SVD subspace tracking algorithms for the serialized data case on this basis. These algorithms are studied experimentally in an adaptive filtering context. They have proven particularly useful for enhancement of unknown transients in noise. Further potential application areas are sliding window adaptive frequency estimation and detection for time series data.

Single section least squares adaptive notch filter

A true least squares single section adaptive notch filter is presented. Internal variables of the algorithm are analyzed theoretically. An approximate iterative algorithm is derived from the exact method. >

The recursive companion matrix root tracker

A new sequential O(n/sup 2/) polynomial factorization algorithm that updates all roots of an nth-order polynomial with real time-varying coefficients simultaneously and efficiently in response to coefficient perturbations is introduced. The algorithm is based on a variant of sequential orthogonal iteration and exploits the special structure of the coefficient companion matrix. All internal operations are based on real passive Givens plane rotations and real matrix-vector multiplications. The algorithm is unconditionally stable and requires no initial guess of the root values. Numerical examples are presented to demonstrate the performance of the algorithm. Comparisons are made to the Starer and Nehorai (1991) root tracking algorithm.

Fast recursive low-rank linear prediction frequency estimation algorithms

A class of fast recursive low-rank linear prediction algorithms for the tracking of time-varying frequencies of multiple nonstationary sinusoids in noise is introduced. Realizations with O(Nn) and O(Nn/sup 2/) arithmetic operations per time step are described, where N is the model order and n is the number of independent sinusoids. The key step towards an operations count that depends only linearly on the model order is fast eigensubspace tracking, and the property that the coefficients of a high-order N prediction filter itself constitute a perfectly (or almost perfectly) predictable sequence that can be annihilated using a low-order 2n prediction error filter that carries the desired signal frequency information in its roots. In this concept, root tracking is limited to a low-order filter polynomial, even if the overmodeling factor N/n is much larger than 1 for optimal noise suppression. Extraneous roots are not computed explicitly. Detailed simulation results confirm the tracking capabilities of the new algorithms.

Efficient covariance ladder algorithms for finite arithmetic applications

Abstract This paper is concerned with the problem of constructing numerically robust covariance ladder estimation algorithms for finite arithmetic applications. Conventional least-squares (LS) ladder algorithms suffer from mixed time and order recursive update equations resulting in a poor numerical accuracy when implemented with finite arithmetic. In this paper, a more ‘direct’ and numerically robust computation approach of the ladder update recursions using the most recently introduced algebraic method of generalized residual energies (GREs) is presented. The new algorithms separate time and order recursions in two independent subalgorithms with a highly modular structure. Besides the general framework, five algorithms of this type are presented. Based on these algorithms, a VLSI ladder chip-set is proposed. Fixed-point simulations of the new VLSI structures are performed for several types of input data. The experimental analysis shows that the new algorithms are superior over conventional techniques and can operate at a multiplier wordlength as low as 8 bits.

Fast recursive orthogonal iteration subspace tracking algorithms and applications

A class of fast recursive subspace tracking algorithms based on the orthogonal iteration principle is introduced. Realizations with O(Nr2) and O(Nr) complexity are derived, where N is the model order and r ⩽ N is the rank of the underlying data covariance matrix. Comparisons reveal that our algorithms require fewer operations and offer a better angle performance than the recently introduced TQR-SVD subspace tracker. Complete quasi-code tables of the algorithms are provided. Applications to adaptive frequency estimation and rank adaptive subspace filtering are described in detail.

Bi-iteration recursive instrumental variable subspace tracking and adaptive filtering

In this paper, we propose a class of fast sequential bi-iteration singular value (Bi-SVD) subspace tracking algorithms for adaptive eigendecomposition of the cross covariance matrix in the recursive instrumental variable (RIV) method of system identification. These algorithms can be used for RIV subspace processing of signals in unknown correlated Gaussian noise. Realizations with O(Nr/sup 2/) and O(Nr) operations per time step are described, where N is the input vector dimension, and r is the number of dominant singular values and vectors to be tracked. The algorithms are solely based on passive Givens plane rotations and standard matrix-vector multiplications. The matrix inversion lemma is not used. The application and performance of the algorithms is demonstrated in a low-rank RIV subspace adaptive filtering context.