基于LMS算法的自适应组合滤波器的设计与开发中英文翻译.docx

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1、Combined Adaptive Filter with LMS-Based AlgorithmsAbstract: A combined adaptive filter is proposed. It consists of parallel LMS-basedadaptive FIR filters and an algorithm for choosing the better among them. As acriterion for comparison of the considered algorithms in the proposed filter, we takethe

2、ratio between bias and variance of the weighting coefficients. Simulations resultsconfirm the advantages of the proposed adaptive filter.Keywords: Adaptive filter, LMS algorithm, Combined algorithm,Bias and variancetrade-off1. IntroductionAdaptive filters have been applied in signal processing and c

3、ontrol, as well as in manypractical problems, 1, 2. Performance of an adaptive filter depends mainly on thealgorithm used for updating the filter weighting coefficients. The most commonlyused adaptive systems are those based on the Least Mean Square (LMS) adaptivealgorithm and its modifications (LMS

4、-based 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 beenmany attempts to improve its performance by the appropriate modifications: signalgorithm (SA) 8, geometric mean LMS (G

5、LMS) 5, variable step-size LMS(VSLMS) 6, 7.Each of the LMS-based algorithms has at least one parameter that should be definedprior to the adaptation procedure (step for LMS and SA; step and smoothingcoefficients for GLMS; various parameters affecting the step for VS LMS). Theseparameters crucially i

6、nfluence the filter output during two adaptation phases:transientand steady state. Choice of these parameters is mostly based on some kind of trade-offbetween the quality of algorithm performance in the mentioned adaptation phases.We propose a possible approach for the LMS-based adaptive filter perf

7、ormanceimprovement. Namely, we make a combination of several LMS-bascd FIR filters withdifferent parameters, and provide the criterion for choosing the most suitablealgorithm for different adaptation phases. This method may be applied to all theLMS-based algorithms, although we here consider only se

8、veral of them.The paper is organized as follows. An overview of the considered LMS-basedalgorithms is given in Section 2.Section 3 proposes the criterion for evaluation andcombination of adaptive algorithms. Simulation results are presented in Section 4.2. LMS based algorithmsLet us define the input

9、 signal vector Xk = x(k)x(k - 1)x(k - N +1) and vector ofweighting coefficients as Wk =叫(攵)W(%) %_1(2) .The weighting coefficientsvector should be calculated according to:Wk+l=Wk2EekXk(1)where p is the algorithm step, E- is the estimate of the expected valueand =dk -WjX.is the error at the in-stant

10、k,and dk is a reference signal.Depending on the estimation of expected value in ，one defines various forms ofadaptive algorithms:the LMS (郎 % = ekXk), the GLMS 隰 Xk=工 Q -O 。 1),and the SA(eX,= Xksign(ek),! ,2,5,8 .The VS LMS has the same form as theLMS, but in the adaptation the step p(k) is changed

11、 6, 7.The considered adaptive filtering problem consists in trying to adjust a set ofweighting coefficients so that the system output, yk = WkrXk. tracks a reference signal,assumed as dk = W *1 Xk + nk,where nk is a zero mean Gaussian noise with thevariance cr； ,and Wk is the optimal weight vector (

12、Wiener vector). Two cases will beconsidered: Wk = W is a constant (stationary case) and Wj is time-varying(nonstationary case). In nonstationary case the unknown system parameters( i.e. theoptimal vector )are time variant. It is often assumed that variation of may bemodeled as W： = Wj + ZK is the ze

13、ro-mean random perturbation, independent onXk and nk with the autocorrelation matrix G = EZkZ= / .Note that analysisfor the stationary case directly follows for b； = 0 .The weighting coefficient vectorconverges to the Wiener one, if the condition from 1,2 is satisfied.Define the weighting cocfficien

14、tsmisalignment, 1-3,= Wk - W； . It is due to boththe effects of gradient noise (weighting coefficients variations around the averagevalue) and the weighting vector lag (difference between the average and the optimalvalue), 3. It can be expressed as:Q)匕=(叽-陀)+,(叽)-窿),According to (2), the ith element

15、 of Vk is:匕(攵)=侬叱k)_叱*&)+ Mk)_凤哂)=bias(W. (k) + pi (k )(3)where bias(Wi(Z:) is the weighting coefficient bias and pi (k) is a zero-meanrandom variable with the variance cr2 .The variance depends on the type ofLMS-based algorithm, as well as on the external noise variance a .Thus, if the noisevariance is constant or slowly-varying, cr2 is time invariant for a particularLMS-based algorithm. In that sense, in the analysis that follows we will assumethat cr2 depends only on the algorithm type, i.e. on its parameters.An important performance measure for an adaptive filter is its mean square