Combined Adaptive Filter with LMS-Based AlgorithmsBo v zo Krstaji c, LJubi v sa Stankovi c,and Zdravko UskokoviAbstract : A combined adaptive ?lter is proposed. It consists of parallel LMS-based adaptiveFIR ?lters and an algorithm for choosing the better among them. As a criterion for comparison 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 trade -off 1. IntroductionAdaptive ?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 commonly used adaptive systems are those based on the Least Mean Square (LMS) adaptive algorithm and its modi?cations (LMS -based algorithms).The LMS is simple for impleme ntati on and robust i n a nu mber of applicati ons 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 step-size LMS(VS LMS) 6, 7.Each of the LMS-based 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 trade-off between the quality of algorithm performance in the mentioned adaptation phases.We propose a possible approach for the LMS-based adaptive ?lter performance improvement.Namely, we make a combination of several LMS- based 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 LMS-based algorithms, although we here consider only several of them.The paper is organized as follows. An overview of the considered LMS-based algorithms is given in Section 2.Section 3 proposes the criterion for evaluation and combination of adaptive algorithms. Simulation results are presented in Section 4.2. LMS based algorithmsLet us de? ne the in put sig nal vector Xk x(k)x(k 1) x(k N 1) and vector of weighting coef?cientsas Wk Wo(k)W (k) Wn.The weighting coef?cients vectorshould be calculated accord ing to:Wk!Wk2 EekXkwhere is the algorithm step, E is the estimate of the expected value and ekdk WkXkis the error at the in-sta nt k,a nd dk is a reference sig nal. Depe nding on the estimati on of expectedvalue in (1), one de?nes various forms of adaptive algorithms:k a io1 a ekiXki， a 1the LMS EekXkekXk, the GLMS E ekXkand the SA E ekX k X ksign ek ,1,2,5,8 .The VS LMS has the same form as the LMS, but in theadaptation the step (k) is changed 6, 7.The considered adaptive ?ltering problem consists in trying to adjust a set of weighting coef?cients sothat the system output, yk WkTXk, tracks a reference signal, assumedasdk W kXknk ,where nk is a zero mean Gaussian noise with the varianee2 n ,andW is the optimal weight vector (Wiener vector). Two cases will be considered:WkW is a*constant (stationary case) and Wk is time-varying (nonstationary case). In nonstationary case the*unknown system parameters( i.e. the optimal vector Wk )are time variant. It is often assumed that1 * * * -=-=-variation of Wk may be modeled as Wk 1 Wk ZK is the zero-mean random perturbation,nk ,where nk is a zero mean Gaussian noise with the varianeeT 2in depe ndent on Xk a nd n k with the autocorrelati on matrix G E ZkZk Z I .Note that.- 一. 一 一.一 c2.一 _analysis for the stationary case directly follows for Z 0 .The weighting coef?cient vectorcon verges to the Wiener on e, if the con diti on from 1,2 is satis?ed.De?ne the weighti ng coef?cie ntsmisalig nment, 1 3, Vk Wk- Wk . It is due to both the effects of gradie nt no ise (weighti ng coef?cie nts variati ons around the average value) and the weighti ng vector lag (differe nee betwee n the average and the optimal value), 3. It can be expressed as:VkWkEWkEWkWk*,According to (2), the ith element of Vk is:Vi k EWk bias Wk i kW- k W k EWkwhere bias Wi k is the2weighting coef?cient bias and i k is a zero-mean random variable with the varianee.Thevariance depends on the type of LMS-based algorithm, as well as on the external noise variance2 2_ . .