外文文献及翻译：基于LMS算法的自适应组合滤波器

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

16、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 best among them in ea

17、ch iteration 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 weighti

18、ng coefcient 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 p

19、erformance 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

20、probability 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 sa

21、me lms-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

22、 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 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. c

23、alculate 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. acc

24、ording 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

25、 coefcients or, if that is the last pair, just choose the algorithm with the smallest variance. first two intervals that do not intersect mean that the proposed trade-off criterion is achieved, and choose the algorithm with large variance.step 4. go to the next instant of time.the smallest number of

26、 elements of the set q is l =2. in that case, one of the should provide good tracking of rapid variations (the largest variance), while the other should provide small variance in the steady state. observe that by adding few more between these two extremes, one may slightly improve the transient beha

27、vior of the algorithm.note that the only unknown values in (6) are the variances. in our simulations we estimate as in 4:, (7)for k = 1, 2,. , l and .the alternative way is to estimate as:，for x(i) = 0. （8）expressions relating and in steady state, for different types of lms-based algorithms, are kno

28、wn from literature. for the standard lms algorithm in steady state, and are related ,3. note that any other estimation of is valid for the proposed filter. complexity of the ca depends on the constituent algorithms (step 1), and on the decision algorithm (step 3).calculation of weighting coefficient

29、s for parallel algorithms does not increase the calculation time, since it is performed by a parallel hardware realization, thus increasing the hardware requirements. the variance estimations (step 2), negligibly contribute to the increase of algorithm complexity, because they are performed at the v

30、ery beginning of adaptation and they are using separate hardware realizations. simple analysis shows that the ca increases the number of operations for, at most, n(l1) additions and n(l1) if decisions, and needs some additional hardware with respect to the constituent algorithms.4.illustration of co

31、mbined adaptive filterconsider a system identification by the combination of two lms algorithms with different steps. here, the parameter q is ,i.e. . the unknown system has four time-invariant coefficients,and the fir filters are with n = 4. we give the average mean square deviation (amsd) for both

32、 individual algorithms, as well as for their combination,fig. 1(a). results are obtained by averaging over 100 independent runs (the monte carlo method), with = 0.1. the reference dk is corrupted by a zero-mean uncorrelated gaussian noise with = 0.01 and snr = 15 db, and is 1.75. in the first 30 ite

33、rations the variance was estimated according to (7), and the ca picked the weighting coefficients calculated by the lms with . as presented in fig. 1(a), the ca first uses the lms with and then, in the steady state, the lms with /10. note the region, between the 200th and 400th iteration,where the a

34、lgorithm 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 does not ideally pick up the lms with smaller st

35、ep. the reason is in the statistical nature of the approach. combined adaptive filter achieves even better performance if the individual algorithms, instead of starting an iteration with the coefficient values taken from their previous iteration, take the ones chosen by the ca. namely, if the ca cho

36、oses, in the k-th iteration, the weighting coefficient vector ,then each individual algorithm calculates its weighting coefficients in the (k+1)-th iteration according to: (9)fig. 1. average msd for considered algorithms.fig. 2. average msd for considered algorithms.fig. 1(b) shows this improvement,

37、 applied on the previous example. in order to clearly compare the obtained results,for each simulation we calculated the amsd. for the first lms () it was amsd = 0.02865, for the second lms (/10) it was amsd = 0.20723, for the ca (colms) it was amsd = 0.02720 and for the ca with modification (9) it

38、was amsd = 0.02371.5. simulation resultsthe proposed combined adaptive filter with various types of lms-based algorithms is implemented for stationary and nonstationary cases in a system identification setup.performance of the combined filter is compared with the individual ones, that compose the pa

39、rticular combination.in all simulations presented here, the reference dk is corrupted by a zero-mean uncorrelated gaussian noise with and snr = 15 db. results are obtained by averaging over 100 independent runs, with n = 4, as in the previous section. (a) time varying optimal weighting vector: the p

40、roposed idea may be applied to the sa algorithms in a nonstationary case. in the simulation, the combined filter is composed out of three sa adaptive filters with different steps, i.e. q = , /2, /8; = 0.2. the optimal vectors is generated according to the presented model with ,and with = 2. in the f

41、irst 30 iterations the variance was estimated according to (7), and ca takes the coefficients of sa with (sa1). figure 2(a) shows the amsd characteristics for each algorithm. in steady state the ca does not ideally follow the sa3 with /8, because of the nonstationary problem nature and a relatively

42、small difference between the coefficient variances of the sa2 and sa3. however,this does not affect the overall performance of the proposed algorithm. amsd for each considered algorithm was: amsd = 0.4129 (sa1,), amsd = 0.4257 (sa2,/2), amsd = 1.6011 (sa3, /8) and amsd = 0.2696(comb). (b) comparison

43、 with vs lms algorithm 6: in this simulation we take the improved ca (9) from 3.1, and compare its performance 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, the c

44、omparison of these two algorithms is meaningful. all the parameters for the improved ca are the same as in 3.1. for the vs lms algorithm 6, the relevant parameter values are the counter of sign change m0 = 11,and the counter of sign continuity m1 = 7. figure 2(b)shows the amsd for the compared algor

45、ithms, where one can observe the favorable properties of the ca, especially after the abrupt changes. note that abrupt changes are generated by multiplying all the system coefficients by 1 at the 2000-th iteration (fig. 2(b). the amsd for the vs lms was amsd = 0.0425, while its value for the ca (col

46、ms) was amsd = 0.0323.for a complete comparison of these algorithms we consider now their calculation complexity, expressed by the respective increase in number of operations with respect to the lms algorithm. the ca increases the number of requres operations for n additions and n if decisions.for t

47、he vs lms algorithm, the respective increase is: 3n multiplications, n additions, and at least 2n if decisions.these values show the advantage of the ca with respect to the calculation complexity.6. conclusioncombination of the lms based algorithms, which results in an adaptive system that takes the

48、 favorable properties of these algorithms in tracking parameter variations, is proposed.in the course of adaptation procedure it chooses better algorithms, all the way to the steady state when it takes the algorithm with the smallest variance of the weighting coefficient deviations from the optimal

49、value.acknowledgement. this work is supported by the volkswagen stiftung, federal republic of germany.references1 widrow, b.; stearns, s.: adaptive signal processing. prentice-hall, inc. englewood cliffs, n.j. 07632.2 alexander, s. t.: adaptive signal processing theory and applications. springer-ver

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52、gorithm. ieee trans. assp 34 (1986),309316.7 aboulnasr, t.; mayyas, k.: a robust variable step-size lmstype algorithm: analysis and simulations. ieee trans. sp 45 (1997), 631639.8 mathews, v. j.; cho, s. h.: improved convergence analysis of stochastic gradient adaptive filters using the sign algorit

53、hm. ieee trans. assp 35 (1987), 450454.9 krstajic, b.; stankovic, l. j.; uskokovic, z.; djurovic, i.:combined adaptive system for identification of unknown systems with varying parameters in a noisy enviroment. proc. ieee icecs99, phafos, cyprus, sept. 1999.中文译文基于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l

54、ms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）。lms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

55、适当改进：符号算法（sa）的8，几何平均lms算法（glms）5，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

56、，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和矢量加权系数为权重系数向量计算应根据： （1）其中为算法步长，e是预期值的估计。在中，常数k表式误差，是一个参考信号。根据（1）中不同的预期值估计在，我们可以得出一种各种形式的自适应算法的定义：lms ， ,1,2,5,8 . 变步长lms算法和基本lms算法具有相同的形式，但在适应过程中步长（k）是变化的6，7。正在研究中的自适应滤波问题在于尝试调整权重系数，使系统的输出跟踪参考信号，中是一个零均值与方差的高斯噪声，是最佳权向量（维纳向量）。我们考虑两

57、种情况：是一个常数（固定的情况下），随时间变化（非平稳的情况下）。在非平稳情况下，未知系统参数（即最佳载体）是随时间变化的。我们假设变量可以建立模型为，它是随机独立的零均值，依赖于和自相关矩阵。注意：分析直接服从，如果1，2的条件是满足的，那么加权系数向量收敛于维纳解。定义加权错位系数，13, 。是因为这两个梯度噪声（加权系数的平均值左右的变化）和加权矢量滞后（平均及最佳值的差额）的影响，3。它可以表示为： （2）根据（2），是：（3）是加权系数的偏差，与方差是零均值的随机变量差，它取决于lms的算法类型，以及外部噪声方差。因此，如果噪声方差为常数或是缓慢变化的，为某一特定的基于lms时间不变的算法。在这个意义上说，在后面的分析中我们将假定只依赖算法类型，及其参数。自适应滤波器的一个重要性能衡量标准是其均方差