【正文】 英文原文 Combined Adaptive Filter with LMSBased Algorithms Abstract: A bined adaptive ?lter is proposed. It consists of parallel LMSbased adaptive FIR ?lters and an algorithm for choosing the better among them. As a criterion for parison of the considered algorithms in the proposed ?lter, we take the ratio between bias and variance of the weighting coef?cients. Simulations results con?rm the advantages of the proposed adaptive ?lter. Keywords: Adaptive ?lter, LMS algorithm, Combined algorithm,Bias and variance tradeoff 1． Introduction Adaptive ?lters have been applied in signal processing and control, as well as in many practical problems, [1, 2]. Performance of an adaptive ?lter depends mainly on the algorithm used for updating the ?lter weighting coef?cients. The most monly used adaptive systems are those based on the Least Mean Square (LMS) adaptive algorithm and its modi?cations (LMSbased algorithms). The LMS is simple for implementation and robust in a number of applications [1–3]. However, since it does not always converge in an acceptable manner, there have been many attempts to improve its performance by the appropriate modi?cations: sign algorithm (SA) [8], geometric mean LMS (GLMS) [5], variable stepsize LMS(VS LMS) [6, 7]. Each of the LMSbased algorithms has at least one parameter that should be de?ned prior to the adaptation procedure (step for LMS and SA。 step and smoothing coef?cients for GLMS。 various parameters affecting the step for VS LMS). These parameters crucially in?uence the ?lter output during two adaptation phases:transient and steady state. Choice of these parameters is mostly based on some kind of tradeoff between the quality of algorithm performance in the mentioned adaptation phases. We propose a possible approach for the LMSbased adaptive ?lter performance improvement. Namely, we make a bination of several LMSbased FIR ?lters with different parameters, and provide the criterion for choosing the most suitable algorithm for different adaptation phases. This method may be applied to all the LMSbased algorithms, although we here consider only several of them. The paper is organized as follows. An overview of the considered LMSbased algorithms is given in Section 3 proposes the criterion for evaluation and bination of adaptive algorithms. Simulation results are presented in Section 4. 2. LMS based algorithms Let us de?ne the input signal vector Tk NkxkxkxX )]1()1()([ ???? ?and vector of weighting coef?cients as TNk kWkWkWW )]()()([ 110 ?? ?.The weighting coef?cients vector should be calculated according to: }{21 kkkk XeEWW ???? （ 1） where 181。 is the algorithm step, E{} is the estimate of the expected value and kTkkk XWde ?? is the error at the instant k,and dk is a reference signal. Depending on the estimation of expected value in (1), one de?nes various forms of adaptive algorithms: the LMS ? ?? ?kkkk XeXeE ? ， the GLMS ? ? ? ?? ??? ?? ???? ki ikikikk aXeaaXeE 0 10,1, and the SA ? ? ? ?? ?kkkk es ig nXXeE ? ,[1,2,5,8] .The VS LMS has the same form as the LMS, but in the adaptation the step 181。(k) is changed [6, 7]. The considered adaptive ?ltering problem consists in trying to adjust a set of weighting coef?cients so that the system output, kTkk XWy ? , tracks a reference signal, assumed as kkTkk nXWd ?? * ,where kn is a zero mean Gaussian noise with the variance 2n? ,and *kW is the optimal weight vector (Wiener vector). Two cases will be considered: WWk ?* is a constant (stationary case) and *kW is timevarying (nonstationary case). In nonstationary case the unknown system parameters( . the optimal vector *kW )are time variant. It is often assumed that variation of *kW may be modeled as Kkk ZWW ??? ** 1 is the zeromean random perturbation, independent on kX and kn with the autocorrelation matrix ? ? IZZEG ZTkk 2??? .Note that analysis for the stationary case directly follows for 02?Z? .The weighting coef?cient vector converges to the Wiener one, if the condition from [1, 2] is satis?ed. De?ne the weighting coef?cientsmisalignment, [1–3], *kkk WWV ?? . It is due to both the effects of gradient noise (weighting coef?cients variations around the average value) and the weighting vector lag (difference between the average and the optimal value), [3]. It can be expressed as: ? ?? ? ? ?? ?*kkkkk WWEWEWV ???? , (2) According to (2), the ith element of kV is: