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通信工程中英文翻译模板.doc
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1、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 the co
2、nsidered 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 lters
3、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) ada
4、ptive 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: sign
5、 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 step f
6、or 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 approach
7、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 algorit
8、hms, 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
9、 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. Depen
10、ding 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 set
11、 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). In
12、 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 follows f
13、or .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 (difference be
14、tween 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 noise v
15、ariance .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
16、 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 best among them in each i
17、teration 9. Choice of the most appropriate algorithm, in each iteration, reduces to the choice of the best value for the weighting coefcients. The best weighting coefcient is the one that is, at a given instant, the closest to the corresponding value of the Wiener vector. Let be the i th weighting c
18、oefcient for LMS-based algorithm with the chosen parameter q at an instant k. Note that one may now treat all the algorithms in a unied way (LMS: q ,GLMS: q a,SA:q ). LMS-based algorithm behavior is crucially dependent on q. In each iteration there is an optimal value qopt , producing the best perfo
19、rmance of the adaptive al-gorithm. Analyze now a combined adaptive lter, with several LMS-based algorithms of the same type, but with different parameter q.The weighting coefcients are random variables distributed around the ,with and the variance , related by 4, 9:, (4)where (4) holds with the prob
20、ability P(), dependent on . For example, for = 2 and a Gaussian distribution,P() = 0.95 (two sigma rule).Dene the condence intervals for : (5)Then, from (4) and (5) we conclude that, as long as , independently on q. This means that, for small bias, the condence intervals, for different of the same L
21、MS-based algorithm, of the same LMS-based algorithm, intersect. When, on the other hand, the bias becomes large, then the central positions of the intervals for different are far apart, and they do not intersect.Since we do not have apriori information about the ,we will use a specic statistical app
22、roach to get the criterion for the choice of adaptive algorithm, i.e. for the values of q. The criterion follows from the trade-off condition that bias and variance are of the same order of magnitude, i.e.The proposed combined algorithm (CA) can now be summarized in the following steps:Step 1. Calcu
23、late for the algorithms with different from the predened set .Step 2. Estimate the variance for each considered algorithm.Step 3. Check if intersect for the considered algorithms. Start from an algorithm with largest value of variance, and go toward the ones with smaller values of variances. Accordi
24、ng to (4), (5) and the trade-off criterion, this check reduces to the check if (6)is satised, where ,and the following relation holds: .If no intersect (large bias) choose the algorithm with largest value of variance. If the intersect, the bias is already small. So, check a new pair of weighting coe
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