A Mean-Square Stability Analysis of the Least Mean Fourth Adaptive Algorithm
A Mean-Square Stability Analysis of the Least Mean Fourth Adaptive Algorithm
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DOI：
10.1109/tsp.2007.894423
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发表时间：
2007-08
影响因子：
5.4
通讯作者：
P. I. Hübscher;J. Bermudez;V. Nascimento
中科院分区：
文献类型：
--
作者：
P. I. Hübscher;J. Bermudez;V. Nascimento
本文在均方意义下对最小均四分之一（LMF）自适应算法进行了一种新的收敛性分析。该分析改进了以前的结果，因为它是有效的非高斯噪声分布，并明确显示算法的稳定性依赖于初始条件的权重。解析表达式推导出的步长，初始权重误差向量，均方稳定性之间的关系。该分析假设白色零均值高斯参考信号和独立同分布（i.i.d.）测量噪声与任何偶概率密度函数（pdf）。它已经由Nascimento和Nascimdez ["Probability of Divergence for the Least-Mean Fourth（LMF）Algorithm，" IEEE Transactions on Signal Processing，vol 54，no. 4，pp. 1376 - 1385，Apr.2006]，LMF算法对于其pdf具有无限支持的参考信号不是均方稳定的。然而，作为步长值的函数的发散概率仅在其移动超过给定阈值时趋于突然上升。我们的分析提供了一个简单的（但精确的）估计的区域迅速上升的分歧的概率。因此，目前的分析是有用的预测算法的不稳定性在大多数实际应用中。
This paper presents a new convergence analysis of the least mean fourth (LMF) adaptive algorithm, in the mean square sense. The analysis improves previous results, in that it is valid for non-Gaussian noise distributions and explicitly shows the dependence of algorithm stability on the initial conditions of the weights. Analytical expressions are derived presenting the relationship between the step size, the initial weight error vector, and mean-square stability. The analysis assumes a white zero-mean Gaussian reference signal and an independent, identically distributed (i.i.d.) measurement noise with any even probability density function (pdf). It has been shown by Nascimento and Bermudez ["Probability of Divergence for the Least-Mean Fourth (LMF) Algorithm," IEEE Transactions on Signal Processing, vol 54, no. 4, pp. 1376-1385, Apr. 2006] that the LMF algorithm is not mean-square stable for reference signals whose pdfs have infinite support. However, the probability of divergence as a function of the step size value tends to rise abruptly only when it moves past a given threshold. Our analysis provides a simple (and yet precise) estimate of the region of quick rise in the probability of divergence. Hence, the present analysis is useful for predicting algorithm instability in most practical applications.