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1、英文原文Combined Adaptive Filter with LMS-Based AlgorithmsBo zo Krstaji c, LJubi sa Stankovi c,and Zdravko UskokoviAbstract: A combined adaptive lter is proposed. It consists of parallel LMS-based adaptive FIR lters and an algorithm for choosing the better among them. As a criterion for comparison of th
2、e considered algorithms in the proposed lter, we take the ratio between bias and variance of the weighting coefcients. Simulations results conrm the advantages of the proposed adaptive lter.Keywords: Adaptive lter, LMS algorithm, Combined algorithm,Bias and variance trade-off1IntroductionAdaptive lt
3、ers 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 coefcients. The most commonly used adaptive systems are those based on the Least Mean Square (LMS)
4、 adaptive algorithm and its modications (LMS-based algorithms).The LMS is simple for implementation and robust in a number of applications 13. However, since it does not always converge in an acceptable manner, there have been many attempts to improve its performance by the appropriate modications:
5、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 dened prior to the adaptation procedure (step for LMS and SA; step and smoothing coefcients for GLMS; various parameters affecting the st
6、ep for VS LMS). These parameters crucially inuence 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 appro
7、ach 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 alg
8、orithms, 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.
9、 LMS based algorithmsLet us dene the input signal vector and vector of weighting coefcients as .The weighting coefcients vector should be calculated according to: （1）where is the algorithm step, E is the estimate of the expected value andis the error at the in-stant k,and dk is a reference signal. D
10、epending on the estimation of expected value in (1), one denes various forms of adaptive algorithms: the LMS，the GLMS,and the SA,1,2,5,8 .The VS LMS has the same form as the LMS, but in the adaptation the step (k) is changed 6, 7.The considered adaptive ltering problem consists in trying to adjust a
11、 set of weighting coefcients so that the system output, tracks a reference signal, assumed as,where is a zero mean Gaussian noise with the variance ,and is the optimal weight vector (Wiener vector). Two cases will be considered: is a constant (stationary case) andis time-varying (nonstationary case)
12、. In nonstationary case the unknown system parameters( i.e. the optimal vector)are time variant. It is often assumed that variation of may be modeled as is the zero-mean random perturbation, independent on and with the autocorrelation matrix .Note that analysis for the stationary case directly follo
13、ws for .The weighting coefcient vector converges to the Wiener one, if the condition from 1, 2 is satised.Dene the weighting coefcientsmisalignment, 13,. It is due to both the effects of gradient noise (weighting coefcients variations around the average value) and the weighting vector lag (differenc
14、e between the average and the optimal value), 3. It can be expressed as:, (2)According to (2), the ith element of is: (3)where is the weighting coefcient bias and is a zero-mean random variable with the variance .The variance depends on the type of LMS-based algorithm, as well as on the external noi
15、se variance .Thus, if the noise variance is constant or slowly-varying, is time invariant for a particular LMS-based algorithm. In that sense, in the analysis that follows we will assume that depends only on the algorithm type, i.e. on its parameters.An important performance measure for an adaptive lter is its mean square deviation (MSD) of weighting coefcients. For the adaptive lters, it is given by, 3:.3. Combined adaptive lterThe basic idea of the combined adaptive lter lies in parallel implementation of two or more adaptive LMS-based algorithms, with the choice of the
