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1、Combined Adaptive Filter with LMS -BasedAlgorithmsBo vzo Krstaji c,'LJubi SaStankovi c,and Zdravko Uskokovi 'Abstract: 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 compari

2、son of the considered algorithms in the proposed ?lter, we take the ratio between bias and varianee 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 varianee trade-off1. Intro

3、ductionAdaptive ?lters have been applied in signal processing and control, as well as in many practical problems, 1, 2. Performa nee of an adaptive ?lter depe nds ma inly on the algorithm used for updating the ?lter weighting coef?cients. The most commonly used adaptive systems are those based on th

4、e Least Mean Square (LMS) adaptive algorithm and its modi?cations (LMS -based algorithms).The LMS is simple for implementation and robust in a number of applications 1 3However, since it does not always con verge in an acceptable manner, there have bee n many attempts to improve its performanee by t

5、he 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; var

6、ious 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 performanee in the mentioned adaptation

7、phases.We propose a possible approach for the LMS -based adaptive ?lter performanee improvement. Namely, we make a comb in atio n of several LMS -based FIR ?lters with differe nt parameters, a nd provide the criterion for choosing the most suitable algorithm for different adaptation phases.This meth

8、od 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 Secti on 2.Sectio n 3 proposes the criterio n for evaluati on and comb in ati on of adaptive algorithms

9、. Simulation results are presentedin Section 4.2. LMS based algorithmsLet us de? ne the in put sig nal vector X k x(k)x(k 1)L x(k N 1) and vector of weighting coef?cients as Wk W0(k)W1(k)L W 1 (k) T.The weighting coef?cients vector should be calculated according to:W. 1 Wk 2 EekXk(1)where 卩 is the a

10、lgorithm step, E Jis the estimate of the expected value and ek dk WJ X kis the error at the in-stant k,and dk is a reference signal. Depending on the estimation of expected value in (1), onede?nes various forms of adaptive algorithms:the LMS E ekXkekXk，theGLMS E ekXka k 1 a ie X ° a 1i 0k i k i

11、an d the SA E ekX kXkSig n ek ,1,2,5,8 .The VS LMS hasthe sameform as the LMS, butin the adaptation the step 卩(k)s changed 6, 7.The considered adaptive ?ltering problem consists in trying to adjust a set of weightingcoef?cients so that the system output, ykWkTXk, tracks a reference signal, assumed a

12、s,and Wk is_*T _dk W kXk nk ,where nk is a zero mean Gaussian noise with the varianeethe optimal weight vector (Wiener vector). Two caseswill be considered：WkW is a constant(stati onary case) a ndWk*is time-vary ing (non stati onary case). I n non stati onary case the unknown system parameters( i.e.

13、 the optimal vectorWk *)are time variant. It is often assumed that variation of Wk may be modeled asW< 1W<Zk is the zero-mean random perturbation, independenton Xk and nkwith the autocorrelation matrix G E ZkZkT ZI .Note that analysis for the stati onary case directly follows for Z 0 .The weig

14、hti ng coef?cie nt vector con verges to the Wiener one, if the condition from 1, 2 is satis?ed. *De?ne the weighting coef?cientsmisalignment, 1 - 34Wk Wk . It is due to both the effects of gradient noise (weighting coef?cients variations around the average value) and the weightingvector lag (differe

15、nee betweentheaverageand the optimal value), 3. It can be expressedas:VkWEWkEWkWkAccording to (2), the ith element of Vk is:Vi k EWk W* kbias Wk i kWkEWi kwhere bias W k is the weighting2n .Thus,dependson the type of LMS -basedalgorithm, as well ason the external noise varianceis time invariant for

16、a particular LMS -if the noise varianee is constant or slowly -varying.based algorithm. In that sense,in the analysis that follows we will assume thatdependsonlyon the algorithm type, i.e. on its parameters.An important performanee measure for an adaptive ?lter is its mean square deviation (MSD) of

17、weighting eoef?eients. For the adaptive ?lters, it is given by, 3: MSD lim E Vk Vk .k3. 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 iteration 9. Choi

18、ce of the most appropriate algorithm, in each iteration, reducesto the choice of the best value for the weighting coef?cients. The best weighting coef?cient is the one that is, at a given instant, the closest to the corresponding value of the Wiener vector.Let Wi k,q be the i -th weighting coef?cien

19、t for LMS - based algorithm with the chosen parameter q at an instant k. Note that one may now treat all the algorithms in a uni?ed way (LMS: q 三口 ,GLMS:q = a,SA:q = 卩)LMS -basedalgorithm behavior is crucially dependenton q. In each iteration there is an optimal value qopt, producing the best perfor

20、mance of the adaptive algorithm. Analyze now a combined adaptive ?lter, with several LMS -based algorithms of the same type, but with different parameter q.The weighting coef?cients are random variables distributed around the Wi * k ,with bias Wi k, q and the variance2, related by 4, 9:|Wi k,q W k b

21、iasWi k,q | q,where (4) holds with the probability P( k )dependenton k For example, for k= 2 and a Gaussian distribution,P( = 0.95 (two sigma rule).De?ne the con?dence intervals for Wi k,q ,4,9:Di k W k,q 2k q,W k,q 2 qThen, from (4) and (5) we conclude that, as long as bias W k,qq,W* k Di kindepend

22、ently on q. This means that, for small bias, the con?dence intervals, for different q s of the same LMS-based algorithm, of the same LMS -based algorithm, in tersect. When, on the other hand, the bias becomes large, then the central positions of the intervals for different q s are far apart, and the

23、y do not in tersect.Since we do not have apriori information about the biasW k, q ,we will use a speci?c statistical approach 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 varianee are of the same

24、order of magnitude, i.e. pias W k, q | q, 4 .The proposed combined algorithm (CA) can now be summarized in the following steps:Step 1. Calculate Wi k, q for the algorithms with different q sfrom the prede?ned set Qqi,q2,K .2Step 2. Estimate the varianeeq for eachconsidered algorithm.Step 3. Check if

25、 Di k in tersect for the con sidered algorithms. Start from an algorithm with largest value of varianee, and go toward the oneswith smaller values of variances. According to (4), (5) and the trade-off criterion, this check reducesto the check ifWk, qmwk, ql |2qmqlissatis?ed,whereqm , qlQ ,andthefoll

26、owingrelationholds:q ： 2 2qh qmqh2 ql ,qhQ.If no Di k in tersect (large bias) choose the algorithm with largest value of varia nee. If the Di k in tersect, the bias is already small. So, check a new pair of weighti ng coef?cie nts or, if that is the last pair, just choose the algorithm with the smal

27、lest varianee. First two intervals that do not in tersect mea n that the proposed trade-off criteri on is achieved, and choose the algorithm with large varia nee.Step 4. Go to the next instant of time.The smallest number of elements of the setQ is L =2. In that case,one of the q sshould provide good

28、 tracking of rapid variations (the largest varianee), while the other should provide small varianee in the steady state. Observe that by adding few more q s between these two extremes, one may slightly improve the transient behavior of the algorithm.2Note that the only unknown values in (6) are the

29、variances. In our simulations we estimate q asin 4:qmedian Wi k W k 1| / 0.6752,2 2for k = 1,2,. , L and zq .The alternative way is to estimate2n as:21T 2ni 1ei ，for X( i) = 0.(8)22Expressions relatingn and q in steadystate, for different types of LMS-based algorithms, are22known from literature. Fo

30、r the standard LMS algorithm in steady state, n and q are related22q n ,3. Note that any other estimation of q is valid for the proposed filter.Complexity of the CA depends on the constituent algorithms (Step 1), and on the decision algorithm (Step 3).Calculati on of weight ing coefficie nts for par

31、allel algorithms does n ot in crease the calculation time, since it is performed by a parallel hardware realization, thus increasing the hardware requireme nts. The varia nce estimati ons (Step 2), n egligibly con tribute to the in crease of algorithm complexity, becausethey are performed at the ver

32、y beginning of adaptation and they are using separate hardware realizati ons. Simple an alysis shows that the CA in creases the nu mber of operations for, at most, N(L-1) additions and N( L-1 ) IF decisions, and needssome additional hardware with respect to the constituent algorithms.4.1 llustration

33、 of combined adaptive filterCon sider a system ide ntificati on by the comb in ati on of two LMS algorithms with differe nt steps. Here, the parameter q is 卩,i.e. Q q,q2,/10 .The unknown system has four time-invariant coefficients,and the FIR filters are with N = 4. We give the average mea n square

34、deviati on (AMSD) for both in dividual algorithms, as well as for their comb in ati on, Fig. 1(a). Results are obta ined by averagi ng over 100 in depe ndent runs (the Monte Carlo method), with 口 = 0. 1. The referen cedk is corrupted by azero-mea nun correlated Gaussian noise with n = 0. 01 and SNR

35、= 15 dB, and K is 1. 75. In the first 30 iterations the variance was estimated according to (7), and the CA picked the weighting coefficients calculated by the LMS with 口.As presentedin Fig. 1(a), the CA first usesthe LMS with 口 and then,in the steady state, the LMS with (x/10. Note the region, betw

36、een the 200th and 400th iteration,where the algorithm can take the LMS with either stepsize,in different realizations. Here, performance of the CA would be improved by increasing the number of parallel LMS algorithms with steps between these two extrems.Observe also that, in steady state, the CA doe

37、s not ideally pick up the LMS with smaller step. The reasonis in the statistical nature of the approach.Combi ned adaptive filter achieves even better performa nce if the in dividual algorithms, in stead of start ing an iterati on with the coefficie nt values take n from their previous iterati on, t

38、ake the ones chosen by the CA. Namely, if the CA chooses, in the k-th iteration, the weighting coefficient vector WP ,then each individual algorithm calculates its weighting coefficients in the ( k+1) -th iteration according to:2e ecrrcrnwi-Ll/S LktS jju'l 0CoLME-as和4MGOO2004MMOi*s： ijf n uf-we-

39、fb)1:10O0SOXI(9)Fig. 1. Average MSD for considered algorithms.Fig. 2. Average MSD for considered algorithms.Fig. 1(b) shows this improvement, applied on the previous example. In order to clearly compare the obtained results,for each simulation we calculated the AMSD. For the first LMS ( 口 ) it was A

40、MSD = 0. 02865, for the secondLMS (口 /10) it was AMSD = 0. 20723, for the CA (CoLMS) it was AMSD = 0. 02720 and for the CA with modification (9) it was AMSD = 0. 02371.5. Simulation resultsof the combined filter is compared with the individual ones, that compose the particularcomb in ati on.In all s

41、imulations presented here, the referenee dk is corrupted by a zero-mean uncorrelated Gaussian noise with 20.1and SNR = 15 dB. Results are obtained by averaging over 100in depe ndentrun s, with N = 4, asi n the previous sect ion.(a) Time varying optimal weighting vector: The proposed idea may be appl

42、ied to the SAalgorithms in anon stati onary case. I n the simulati on, the comb in ed filter is composed out of three SA adaptive filters with different steps, i.e. Q = 口 , 口 /2, 口 /8; 口 = 0. 2. The optimal vectors is generated according to the presented model with Z0.001,and with K =2.1 n the first

43、 30iterations the varianee was estimated according to (7), and CA takes the coefficients of SA with 口 (SA1).Figure 2(a) shows the AMSD characteristics for each algorithm. In steady state the CA does not ideally follow the SA3 with 口 /8, becauseof the nonstationary problem nature and a relatively sma

44、ll differenee between the coefficient variances of the SA2 and SA3. However,this does not affect the overall performanee of the proposed algorithm. AMSD for each considered algorithm was: AMSD = 0. 4129 (SA1, 口 ), AMSD = 0.4257 (SA2, 口/2), AMSD = 1.6011 (SA3, 口/8) andAMSD = 0. 2696(Comb).(b) Compari

45、son with VS LMS algorithm 6: In this simulation we take the improved CA (9) from 3.1, and compare its performanee with the VS LMS algorithm 6, in the case of abrupt changes of optimal vector. Since the considered VS LMS algorithm6 updates its step size for each weighting coefficient individually, th

46、e comparison of these two algorithms is meaningful.All the parameters for the improved CA are the same as in 3.1. For the VS LMS algorithm6 , the releva nt parameter values are the coun ter of sig n cha ngem0 = 11, and the coun ter of sig n continuity m1 = 7. Figure 2(b)shows the AMSD for the compar

47、ed algorithms, where one can observe the favorable properties of the CA, especially after the abrupt changes. Note that abrupt cha ngesare gen erated by multiplyi ng all the system coefficie nts by -1 at the 2000-th iterati on (Fig. 2(b). The AMSD for the VS LMS was AMSD = 0. 0425, while its value f

48、or the CA (CoLMS) was AMSD = 0. 0323.For a complete comparison of these algorithms we consider now their calculation complexity, expressed by the respective in crease i n nu mber of operati on s with respect to the LMS algorithm. The CA increasesthe number of requres operations for N additions and N

49、 IF decisions.Forthe VS LMS algorithm, the respective in crease is: 3N multiplicati ons, N additi ons, and at least 2N IF decisi ons.These values show the advantageof the CA with respectto the calculation complexity.6. Con clusi onComb in ati on of the LMS based algorithms, which results in an adapt

50、ive system that takes the favorable properties of these algorithms in track ing parameter variatio ns, is proposed .In the course of adaptation procedure it choosesbetter algorithms, all the way to the steady state when it takes the algorithm with the smallest varianee of the weighting coefficient d

51、eviations from the optimal value.Ack no wledgeme nt. This work is supported by the Volkswage n Stift ung, Federal Republic of Germa ny.Refere nces1 Widrow, B.; Stearns, S.: Adaptive Signal Processing.Prentice-Hall, Inc. Englewood Cliffs, N.J. 07632.2 Alexander, S. T.: Adaptive Signal Processing - Th

52、eory and Applications. Springer-Verlag, New York 1986.3 Widrow, B.; McCool, J. M.; Larimore, M. G.; Johnson, C. R.:Stationary and Nonstationary Learning Characteristics of the LMS Adaptive Filter. Proc. IEEE 64(1976) 1151 - 1161.4 Stankovic, L. J.; Katkovnik, V.: Algorithm for the instantaneous freq

53、uency estimation using time-frequency distributions with variable window width. IEEE SP.Letters 5 (1998), 224 - 227. Krstajic, B.; Uskokovic, Z.; Sta nkovic, Lj.: GLMS Adaptive Algorithm in Lin ear Predictio n. Proc. CCECE 97I, pp. 114 - 117,anada,1997. Harris, R. W.; Chabries, D. M.; Bishop, F. A.:

54、 A Variable Step (VS) Adaptive Filter Algorithm. IEEE Trans. ASSP 34 (1986),309- 316.7 Aboulnasr, T.; Mayyas, K.: A Robust Variable Step-Size LMSType Algorithm: Analysis and Simulations. IEEE Trans. SP45 (1997), 631 - 639.8 Mathews, V. J.; Cho, S. H.: Improved Con verge nee An alysis of Stochastic G

55、radie nt Adaptive Filters Using the Sign Algorithm. IEEE Trans. ASSP 35 (1987), 450 - 454.9 Krstajic, B.; Stankovic, L. J.; Uskokovic, Z.; Djurovic, I.:Combined adaptive system for iden tificati on of unknown systems with vary ing parameters in ano isy en virome nt. Proc. IEEE ICECS' 99Phafos, C

56、yprus, Sept. 1999.基于LMS算法的自适应组合滤波器摘要：提出了一种自适应组合滤波器。它由并行 LMS的自适应FIR滤波器和 一个具有更好的选择性的算法组成。作为正在研究中的滤波器算法比较标准， 我们采取偏差和加权系数之间的方差比。仿真结果证实了提出的自适应滤波器 的优点。关键词：自适应滤波器；LMS算法；组合算法；偏差和方差权衡1绪论自适应滤波器已在信号处理和控制，以及许多实际问题 1, 2的解决当中得到 了广泛的应用自适应滤波器的性能主要取决于滤波器所使用的算法的加权系数 的更新。最常用的自适应系统对那些基于最小均方（LMS ）自适应算法及其改 进（基于LMS的算法）。LM

57、S算法是非常简便，易于实施，具有广泛的用途1-3。但是，因为它并 不总是收敛在一个可接受的方式，所以有很多的尝试，以对其性能做适当改进： 符号算法（SA）的8，几何平均LMS算法（GLMS），变步长LMS （最小均方比）算法6，7。每一种基于LMS的算法都至少有一个参数在适应过程（LMS算法和符号算法，加强和GLMS平滑系数，各种参数对变步长 LMS算法的影响）中被预先定义。这些参数的影响关键在两个适应阶段：瞬态和稳态滤波器的输出。这 些参数的选择主要是基于一种算法质量的权衡中所提到的适应性能。我们提出 了一个自适应滤波器的性能改善的方法。也就是说，我们提出了几个基于LMS算法的不同参数的FIR滤波器，并提供不同的适应阶段选择最合适的算法标准。 这种方法可以适用于所有的LMS的算法，虽然我们在这里只考虑其中几个。本文的结构如下，作者认为的LMS的算法概述载于第2节，第3节提出了自 适应算法的改进和组合标准，仿真结果在第 4节。2、基于LMS的算法让我们定义输入信号向量Xk x（k）x（k 1）L x（k N 1）T和矢量加权系数为Wk W0（k）W1（k）L WN 1（k）T权重系数向量计算应根据：Wk1 W； 2 EekXk（ 1）其中卩为算法步长，E 是预期值的估计。在ek dk WkTXk中，常数K 表式误差，dk是一个参考信号。根据（1）中不同的预期值估