An Adaptive Algorithm Based On Discrete Wavelet Transform

This paper focuses on the principle of adaptive filtering and properties of LMS algorithm.Based on the analysis of the traditional adaptive filtering algorithm, the wavelet is introduced into the adaptive filtering. Adaptive filtering based on wavelet analysis is discussed and analyzed in detail. Based on adaptive filtering algorithm and based on wavelet transform (WTLMS), a novel algorithm is presented in this paper-a adaptive filtering algorithm with variable step size based on wavelet transform, because a adaptive LMS algorithm with fixed step size can't solve the contradiction between convergence speed and steady state error. Theoretical analysis and computer simulation demonstrate that the novel algorithm has the better convergence property than the modified ones.Although classical LMS algorithm has simple complexity and is easy to be implemented, its convergence speed is very slow. In order to speedup the convergence process, one can increase the step size, but at the same time, the steady state error will be large and even the algorithm may become unstable.Fixed step size cannot result in fast convergence speed and low residual error simultaneously. In order to solve this problem, we can build a nonlinear function relationship between step size factor and the error signal, that is, we can adopt an adaptive LMS algorithm with a variable step size. At the same time, the eigenvalue spread, that is, the condition number of auto-correlation matrix of the input signal, mainly affects the convergence rate of LMS algorithm. The larger the condition number is, the slower the algorithm converges, and vice versa. The method that a wavelet is introduced into the adaptive filtering can solve this problem very well because the wavelet has the good capacity to decrease the correlation property of the input vector. The condition number of the auto-correlation matrix of the input signal of the equalizer is reduced by wavelet transform so that the convergence rate of the algorithm is faster. Now in order to improve the convergence speed and the stability of traditional LMS further, a novel algorithm based on WTLMS algorithm is presented in this paper. A variable step size LMS and transform domain LMS are immerged into one, which is applied to adaptive filtering system. The excellent properties of the novel algorithm are confirmed by theoretical analysis andcomputational complexity is very slow. Especially, another new non-linear functional relationship between step size factor and error signal (e(n)) is adopted in this new algorithm. The functional relationship is not only simpler than Sigmoid functional relationship, but also has the property of slight change e(n) near to zero. Therefore it is superior to Sigmoid functional relationship in the process of step size change of adaptive steady state.Computer simulation results and theoretical analysis show the algorithm performance is better than that of former algorithms.Firstly, the paper discusses the application background of adaptive filtering technology and the great meaning of applying adaptive filtering technology to the communication system. Besides the merits and demerits of some adaptive filtering algorithms are analyzed and summarized, and so are the problems existed in algorithms. Meanwhile, wavelet analysis applied to adaptive filtering is also discussed. Present conditions and development prospect of adaptive filtering technology is also introduced in this paper.Secondly, the basic principle of LMS algorithm, the convergence range of step size factor in algorithm is discussed in detail. Theoretical analysis and computer simulations show that fixed step size cannot result in fast convergence speed and low residual error simultaneously. In order to solve this problem, the LMS with transient step size is also discussed in this paper, which is compared with traditional LMS algorithm. The paper has proved from theory that the convergence rate of traditional LMS is sensitive to the eignvalue spread of auto-correlation matrix of the input vector. If the distribution of eignvalue spread is too scattered, the spectrum dynamic range of convergence step size will be very small and the convergence speed will be very slow.Although the wavelet is a new and developing discipline, it is developing very rapidly and its mathematical theory has been used widely in many fields. The paper discusses in detail the basic theory of wavelet analysis and the basic concepts of wavelet transform. At the same time, the paper has taken advantage of the character of orthogonal wavelet with locality in time-frequency domains.Besides, Mallat fast algorithm and multiple resolution analysis are also introduced in this paper.Fourthly, the wavelet theory is introduced into the adaptive filtering.The filtering is represented by a set of orthogonal wavelets and the corresponding coefficients. The paper also gives the adaptive algorithms. Theoretical analysis and simulations show that the WTLMS converges faster than the conventional FIRLMS based adaptive filtering while the increase in the computational complexity is very slow. The spectrum dynamic range of auto-correlation matrix of the input vector is decreased so that the convergence speed of the LMS algorithm is improved. Meanwhile, the paper discusses by means of experiments the differences of the same wavelet transform under the different scales so that the optimum value of scale is obtained.At last, based on the WTLMS algorithm, a novel method is proposed in this paper, that is, a adaptive filtering algorithm with variable step size of the modified Sigmoid function based on wavelet transform. Put another way, the algorithm with transient step size and the LMS algorithm transformed in another domain are merged into the adaptive filtering system. Therefore, the WTLMS algorithm is improved. Experimental results demonstrate that the new algorithm has not only better convergence property but also the same stable mean square errors than the former algorithms.To sum up, the event of the wavelet theory provides a novel way for filtering design so that the application of LMS is pushed forward. Moreover the strategy can be implemented very easily. Therefore doing some research on the adaptive filtering technology based on wavelet analysis is very meaningful in both theory and engineering.