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Spectral effects on the rate of convergence of the LMS adaptive algorithm.

机译：频谱对LMS自适应算法收敛速度的影响。

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An adaptive algorithm can be used to find a least-square-error solution for linearly estimating a random process given observations of a related random process. Such an algorithm can iteratively update a weight vector, an approximation of the optimal Wiener solution, as input data is presented in a streaming way. The Least Mean Square (LMS) algorithm is one of the most popular adaptive algorithms because it is simple and robust, as it is used in many commercial applications of significance. However, the performance of LMS may vary greatly when the autocorrelation matrix of its input has a high eigenvalue spread, so there is a need to be able to predict its performance in practice. The LMS/Newton algorithm is an ideal variation of the LMS algorithm that is immune to the eigenvalue spread problem, and it is commonly used as a theoretical benchmark for adaptive algorithms. In this thesis, we study the performance of LMS relative to LMS/Newton. The analysis is done for stationary and nonstationary signal statistics. In the stationary case, transient behavior results when the adaptive weight vector starts from initial conditions and proceeds toward the Wiener solution, and in steady-state, the adaptive weight vector hovers randomly about the Wiener solution. In the nonstationary case, the Wiener solution varies randomly, and the adapting weight vector is tracking a moving target. In both cases, simple expressions are found for the performance of LMS relative to LMS/Newton. When the input autocorrelation matrix is Toeplitz (as would be the case with an adaptive transversal filter), these expressions are translated to the frequency domain. Our results imply that, if the input power spectrum is similar to the spectrum of the Wiener solution (e.g. a low-pass input and a low-pass Wiener filter), the transient performance of LMS is better than that of LMS/Newton, in spite of a high a eigenvalue spread. If the above spectra are dissimilar (e.g. a low-pass input and a band-pass Wiener filter), LMS/Newton would be faster than LMS. In the nonstationary case, if the input spectrum is similar to the spectrum of the time variations of the Wiener solution, LMS tracks better than LMS/Newton in steady-state. Otherwise, the tracking performance of LMS/Newton would be superior to that of LMS.
机译：给定相关随机过程的观察结果，可以使用自适应算法来找到最小二乘误差解，以线性估计随机过程。当输入数据以流方式呈现时，这种算法可以迭代地更新权重向量（最佳维纳解决方案的近似值）。最小均方（LMS）算法是最流行的自适应算法之一，因为它简单且健壮，因为它被用于许多重要的商业应用中。但是，当LMS输入的自相关矩阵具有较高的特征值分布时，LMS的性能可能会发生很大变化，因此需要在实践中能够预测其性能。 LMS / Newton算法是LMS算法的理想变体，它不受特征值扩散问题的影响，并且通常用作自适应算法的理论基准。在本文中，我们研究了LMS相对于LMS / Newton的性能。进行了静态和非静态信号统计分析。在平稳情况下，当自适应权重矢量从初始条件开始并朝Wiener解进行时，会导致瞬态行为；在稳态下，自适应权重矢量围绕Wiener解随机地徘徊。在非平稳情况下，维纳解决方案随机变化，并且自适应权重向量正在跟踪运动目标。在这两种情况下，都可以找到相对于LMS / Newton而言LMS性能的简单表达式。当输入自相关矩阵为Toeplitz（自适应横向滤波器就是这种情况）时，这些表达式将转换为频域。我们的结果表明，如果输入功率谱与Wiener解决方案的频谱相似（例如，低通输入和低通Wiener滤波器），则LMS的瞬态性能要优于LMS / Newton。尽管特征值分布很高。如果以上光谱不相同（例如低通输入和带通维纳滤波器），则LMS /牛顿会比LMS快。在非平稳情况下，如果输入频谱与维纳解决方案的时间变化频谱相似，则在稳定状态下LMS的跟踪效果优于LMS / Newton。否则，LMS / Newton的跟踪性能将优于LMS。

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