We introduce a class of adaptive filters based on sequential adaptive eigendecomposition (subspace tracking) of the data covariance matrix. These new algorithms are completely rank revealing, and hence, they can perfectly handle the following two relevant data cases where conventional recursive least squares (RLS) methods fail to provide satisfactory results: (1) highly oversampled "smooth" data with rank deficient of almost rank deficient covariance matrix and (2) noise-corrupted data where a signal must be separated effectively from superimposed noise. This paper contradicts the widely held belief that rank revealing algorithms must be computationally more demanding than conventional recursive least squares. A spatial RLS adaptive filter has a complexity of O(N/sup 2/) operations per time step, where N is the filter order. The corresponding low-rank adaptive filter requires only O(Nr) operations per time step, where r/spl les/N denotes the rank of the data covariance matrix. Thus, low-rank adaptive filters can be computationally less (or even much less) demanding, depending on the order/rank ratio N/r or the compressibility of the signal. Simulation results substantiate our claims. This paper is devoted to the theory and application of fast orthogonal iteration and bi-iteration subspace tracking algorithms.

Low-rank adaptive filters

This paper introduces a fast implementation of the power iteration method for subspace tracking, based on an approximation that is less restrictive than the well-known projection approximation. This algorithm, referred to as the fast approximated power iteration (API) method, guarantees the orthonormality of the subspace weighting matrix at each iteration. Moreover, it outperforms many subspace trackers related to the power iteration method, such as PAST, NIC, NP3, and OPAST, while having the same computational complexity. The API method is designed for both exponential windows and sliding windows. Our numerical simulations show that sliding windows offer a faster tracking response to abrupt signal variations.

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Fast approximated power iteration subspace tracking

This book provides an introduction to narrowband array signal processing, classical and subspace-based direction of arrival (DOA) estimation with an extensive discussion on adaptive direction of arrival algorithms. The book begins with a presentation of the basic theory, equations, and data models of narrowband arrays. It then discusses basic beamforming methods and describes how they relate to DOA estimation. Several of the most common classical and subspace-based direction of arrival methods are discussed. The book concludes with an introduction to subspace tracking and shows how subspace tracking algorithms can be used to form an adaptive DOA estimator. Simulation software and additional bibliography are given at the end of the book.

Narrowband Direction of Arrival Estimation for Antenna Arrays

In this paper, we develop a general approach for reducing the computational complexity of any direction finding method implemented with forward/backward (FB) averaging. We develop simplified FB data matrices in a manner paralleling previous work related to centrohermitian (correlation) matrices. Based on these simplified data matrices, efficient construction and updating of the FB correlation matrix is developed. In addition, efficient FB FFT, FB beamspace, FB EVD updating, FB SVD, and FB SVD updating methods are derived. In most cases, FB-based direction-finding methods can be simplified so that the computational complexity is reduced below that of an analogous forward only implementation. Thus, effectively twice the amount of data is processed with less total computation. >

Efficient direction-finding methods employing forward/backward averaging

We present a class of fast subspace tracking algorithms that arise from a straightforward extension of Bauer's (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.

Bi-iteration SVD subspace tracking algorithms

The transposed VR (TQR) iteration is a square root version of the symmetric QR iteration. The TQR algorithm converges directly to the singular value decomposition (SVD) of a matrix and was originally derived to provide a means to identify and reduce the effects of outliers for robust SVD computation. The paper extends the TQR algorithm to incorporate complex data and weighted norms, formulates a TQR-iteration based adaptive SVD algorithm, develops a real time systolic architecture, and analyzes performance. The applications of high resolution angle and frequency tracking are developed and the updating scheme is so tailored. A deflation mechanism reduces both the computational complexity of the algorithm and the hardware complexity of the systolic architecture, making the method ideal for real time applications. Simulation results demonstrate the performance of the method and compare it to existing SVD tracking schemes. The results show that the method is exceptional in terms of performance to cost ratio and systolic implementation. >

A TQR-iteration based adaptive SVD for real time angle and frequency tracking

The transposed QR (TQR) iteration is a square root version of the symmetric QR iteration and defines the TQR algorithm. The authors review the TQR algorithm and extend it to incorporate weighting schemes and complex data. Geometrically, the algorithm breaks each QR iteration into least square regression fit followed by a rotation to the regression hyperplane. This basic insight leads to a rapidly converging adaptive algorithm for tracking the singular values and right singular vectors of an exponentially weighted and downward growing data matrix. The applications of high resolution angle and frequency tracking are developed using subspace averaging based deflation to reduce computation. Simulation results demonstrate the performance of the method, and it is compared to other SVD tracking schemes. >

L.P. Ammann

Papers

A TQR-iteration based adaptive SVD for real time angle and frequency tracking

An adaptive TQR-SVD for angle and frequency tracking