Studies On The Convergence Performance Of Least Mean Square (LMS) Algorithm And Its Applications

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.