【作者】 谷源涛；

【导师】 唐昆；

【作者基本信息】 清华大学 ， 通信与信息系统， 2003， 博士

【摘要】 论文以稳定的最大步长为切入点对加快LMS收敛速度的方法进行了全面地研究。说明增大步长取值范围是加速LMS收敛的根本途径，提出研究的三个步骤顺序是：输入信号功率谱平坦化，滤波器长度控制和叠代步长控制；并具体地研究了每个步骤，包括：(1)基于理想分析滤波器组和输入信号功率谱分段平坦的假设，分析子带加权自适应滤波器(WSAF)的工作性能：从均值意义上证明WSAF和LMS的收敛过程存在等价关系，同时解释前者稳态误差大的原因；从均方意义上分析该算法的收敛性能，得到时间常数，稳态误差，最大步长等描述自适应算法的一系列重要指标；(2)在最小均方误差准则下研究自适应滤波器长度对LMS收敛速度的影响，针对具有指数衰减包络的未知系统推导出保证LMS收敛速度最快的最优滤波器长度序列；证明当待辨识未知系统的冲激响应严格单调减时，滤波器长度的代价函数具有”简单的形状”，并通过引入对代价函数抽取的概念，将满足在抽取点上具有“简单的形状”的未知系统冲激响应的要求放松到分块能量单调减，从而极大地提高了理论的适用范围，随后构造出一种梯度搜索算法动态地寻找并跟踪最优滤波器长度；(3)基于最小均方误差准则，研究最优步长及其性质：提出最优步长定理和步长选择定理，推导出最优步长序列及变步长LMS算法收敛速度的理论极限，在此基础上提出最优变步长模型的概念；随后构造了两种实现算法：直接计算最优步长算法和平行滤波器组算法；利用前面提出的两个定理，证明了平行滤波器组算法具有逼近最优变步长模型的收敛性能。在分别研究每个步骤的基础上，将三者有机结合起来，提出一种实用的联合控制滤波器长度和步长的多延时分块LMS算法，具有收敛跟踪速度快，计算量小等优点。 此外，论文还研究非独立假设下LMS算法的收敛性能：通过分析基于独立理论推导均方叠代过程产生误差的原因，提出一个精确描述大步长情况下LMS算法均方收敛性能修正公式。 最后通过回声抵消和信道均衡等应用验证论文中提出的多种算法的实用性。更多还原

【Abstract】 Adaptive signal processing is used extensively in many applications of communication system and electronics engineering. The Least Mean Square (LMS) algorithm is one of the most popular adaptive algorithms because of its simplicity, stability, and ease of implementation. However, conventional LMS suffers from low convergence rate, especially when the input signal is highly correlated or the adaptive filter has many taps. Therefore, study on the convergence performance and the practical algorithm with improved convergence rate is not only a significant topic related to LMS, but also one of the frontiers in the field of adaptive signal processing.In present dissertation, theoretical studies on the convergence behavior of LMS were performed. Based on the discussion of stable maximum stepsize, the research on convergence improvement is divided into there steps: whitening input signal, controlling filter length and controlling stepsize. Then the later chapters discuss these points in details.Weighted Subband Adaptive Filter is a successful algorithm on whitening the input signal. We prove that the mean behavior of WSAF with correlated input is identical to that of LMS with white input. Further, the mean square convergence of WSAF is a single exponential that is similar to LMS driven by white signal, which explains the outstanding performance of WSAF. The analysis is based on the assumption of ideal analysis filter banks and piecewise flat input power spectrum. However, the experiments with some realistic analysis filter banks and input with continuous power spectrum also show that the proposed conclusion is valid.Filter length control is also an important method to improve the potential maximum stepsize. Under the criteria of minimizing mean square error, the decimated cost function of filter length is proved to be in "plain shape", if the unknown system response has a decayed envelop. Then a novel gradient length control scheme is proposed, which is based on the stochastic gradient descent method.The discussion above should be considered as the preparation of stepsizecontrol, which is essential to the improvement of LMS performance. The optimal step-size theorem and superior step-size theorem are proposed, followed the optimal stepsize sequence and theoretical convergence limits. According to the theoretical analysis, two practical algorithms are constructed. Especially, we prove that the proposed parallel-filter-banks algorithm has the best performance which may approach the optimal one in terms of convergence rate.Besides the research on accelerating convergence, study on exactly describing the mean square behavior of LMS is also presented. Based on the theory derived by independence assumption, we propose a modifier formula, which can exactly predict the track of MSE descending in convergence process. Then with the derivation of the original formula, we analyze the limitation of independence theory and provide a qualitative explanation to prove the validity of the proposed modification.Finally, the applications on echo cancellation and channel equalization demonstrate the practical significance of the presented theory and algorithm.更多还原