Stability of Algorithms in Signal Processing

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The method of Least Squares has been employed successfully to solve problems in many scientific disciplines, including signal processing. The class of fast Recursive Least Squares Algorithms was introduced in order to address certain desirable performance traits, such as computational complexity, tracking, and misadjustment. Although these algorithms have evolved over many years and are widely used in signal processing, a common approach addressing the issue of algorithmic stability has been lacking. This research provides an analysis that fills this critical void in the research and development of these algorithms. Descriptive definitions of stability are being examined which are consistent with current and previous algorithmic analyses. The effects of the forgetting factor - the extent to which prior information is weighted - with respect to matrix perturbation theory and the control and propagation of arithmetic roundoff errors are being studied. This research is also providing useful and insightful gains in the important areas of filter misadjustment, filter tracking, the conditioning of the auto-correlation matrix, and the usefulness of the filter output. This research program joins the science of Numerical Analysis with the science of Signal Processing, providing tools by which algorithms for signal processing can be better designed and used, so that they will give more accurate results.