Stability of Algorithms in Signal Processing

最小二乘法已成功地应用于解决许多科学学科中的问题，包括信号处理。 快速递归最小二乘算法的引入是为了解决某些期望的性能特征，如计算复杂度，跟踪和失调。 虽然这些算法已经发展了很多年，并广泛用于信号处理，一个共同的方法来解决算法的稳定性问题一直缺乏。 这项研究提供了一个分析，填补了这些算法的研究和开发中的这一关键空白。 正在检查稳定性的描述性定义，其与当前和先前的算法分析一致。 目前正在研究遗忘因子-先验信息加权的程度-对矩阵扰动理论的影响以及算术舍入误差的控制和传播。 这项研究还提供了有用的和有见地的增益的重要领域的滤波器失调，滤波器跟踪，自相关矩阵的条件，和有用的滤波器输出。 该研究计划将数值分析科学与信号处理科学相结合，提供了可以更好地设计和使用信号处理算法的工具，以便它们能够给出更准确的结果。

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.