Convergence Performance Analysis And Applications Of The Adaptive Least Mean Square (LMS) Algorithm

Adaptive filter theory is an important research topic in the area of adaptive signal processing. It can adapt the filter coefficients according to some criterions and approach optimal filtering. The design of adaptive filter algorithm is an important part within the design of adaptive filter. The performance of the adaptive algorithm decides the performance of the adaptive filter. Least mean square (LMS) algorithm is a classical adaptive algorithm which has simple structure, good stable property, and low computational complexity, thus is easy to be implemented. It has been widely used in many applications such as system identification, noise cancellation, speech signal prediction, adaptive channel equalization, adaptive antenna array and so on.The main drawback of the LMS algorithm is its slow convergence rate, which deteriorates its performance in many applications, In this paper, theoretical performance analysis for several existing modified LMS algorithms has been performed and guidelines for their parameters choice have also been provided. Furthermore, new modified LMS algorithms are proposed in this thesis and supported with theoretical performance analysis. Simulation results show that, as compared with existing modified LMS algorithms, the proposed algorithms perform better in some applications.In this thesis, main research areas of the LMS algorithm are firstly summarized. Then the influences brought by the choices of the step size and tap-length parameters in the LMS algorithm are discussed in detail, which lead to research on variable step size LMS and variable tap-length LMS algorithms. The contents of this thesis are as follows:In the research of variable step size LMS algorithms, existing variable step size LMS algorithms are summarized and discussed. Steady state performance analysis of an approximately optimal variable step size LMS algorithm-Shin's algorithm is given and the theoretical expression of steady state excess mean square error is derived. Simulations are performed to confirm the analysis and show its deteriorated performance under higher noise conditions. Based on the detailed analysis of variable step size LMS algorithms, two new variable step size LMS algorithms are proposed. One is proposed to update the step size based on the gradient in the adaptation. Steady state performance analysis and parameter choice guideline are also provided to facilitate its application. Simulation results confirm the analysis and show that the proposed algorithm performs better under both low noise and high noise conditions. Another new variable step size LMS algorithm is designed for the application with an exponential decay impulse response. An exponential decay matrix is introduced in the step-size iteration, and a variable vector step-size is proposed. Simulation results show that the proposed algorithm has an optimal performance under such application environment.In the research of variable tap-length LMS algorithms, the fractional tap-length LMS algorithm (FTLMS algorithm) is firstly introduced. Because of the nonlinear relationship between the steady state performance and the tap-length, appropriate parameters are difficult to choose while using FTLMS algorithm. Based on reasonable vector division and assumptions, steady sate performance analysis of the FTLMS algorithm is given and the steady-state expression of tap-length is derived. Parameter choice guideline is given accordingly, which also provide theoretical support for its application. Based on the research of the FTLMS algorithm, three new variable tap-length LMS algorithms are proposed, which are combination of variable step size and variable tap-length, variable iteration parameter and variable error width algorithms. Simulation results show that as compared with original FTLMS algorithm, improved variable tap-length algorithms perform better in convergence rate, steady state mean square error and avoidance of the suboptimum problem.Finally, some of the proposed algorithms are applied within adaptive channel equalization and adaptive noise cancellation models. As can be seen from the simulation results, the proposed algorithms have good performances and can be potentially utilized in real applications.