基于LMS算法的自适应组合滤波器中英文翻译

1、combined adaptive filter with lms-based algorithmsabstract: 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 considered algorithms in the proposed lter, we take the rat

2、io 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 have been applied in signal processing and control, as we

3、ll 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) adaptive algorithm and its modications (lms-based algorithms

4、).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 algorithm (sa) 8, geometric mean lms (glms) 5, variable

5、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 for vs lms). these parameters crucially inuence the lter o

6、utput 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 for the lms-based adaptive lter performance improvement.

7、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 algorithms, although we here consider only several of them.the p

8、aper 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 based algorithmslet us dene the input signal vector and

9、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. depending on the estimation of expected value in (1), one dene

10、s 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 of weighting coefcients so that the system output, track

11、s 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 nonstationary case the unknown system parameters( i.e. t

12、he 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 for .the weighting coefcient vector converges to the wiene

13、r 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 between the average and the optimal value), 3. it can be ex

14、pressed 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 variance .thus, if the noise variance is constant or slowl

15、y-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 coefcien

16、ts. 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 iteration 9. choice of the most appropriate algorithm, in

17、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 coefcient for lms-based algorithm with the chosen paramete

18、r 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 performance of the adaptive al-gorithm. analyze now a combined

19、 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 probability p(), dependent on . for example, for = 2 and a ga

20、ussian 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 lms-based algorithm, of the same lms-based algorithm, inte

21、rsect. 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 approach to get the criterion for the choice of adaptive alg

22、orithm, 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. calculate for the algorithms with different from the predened

23、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. according to (4), (5) and the trade-off criterion, this check re

24、duces 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 coefcients or, if that is the last pair, just choose the alg

25、orithm 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 elements of the set q is l =2. in that case, one of the shou

26、ld 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 behavior of the algorithm.note that the only unknown values in (6

27、) 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 known from literature. for the standard lms algorithm in steady

28、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 coefficients for parallel algorithms does not increase the calculation t

29、ime, 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 very beginning of adaptation and they are using separate hardw

30、are 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 combined adaptive filterconsider a system identification by the

31、 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 individual algorithms, as well as for their combination,fig.

32、 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 iterations the variance was estimated according to (7), and the

33、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 algorithm can take the lms with either stepsize,in different r

34、ealizations. 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 step. the reason is in the statistical nature of the approach.

35、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 chooses, in the k-th iteration, the weighting coefficient vector

36、 ,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, applied on the previous example. in order to clearly compare

37、 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 was amsd = 0.02371.5. simulation resultsthe proposed combined

38、 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 particular combination.in all simulations presented here, the r

39、eference 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 proposed idea may be applied to the sa algorithms in a nonstat

40、ionary 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 first 30 iterations the variance was estimated according to (7

41、), 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 small difference between the coefficient variances of the sa2

42、 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 with vs lms algorithm 6: in this simulation we take the impr

43、oved 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 comparison of these two algorithms is meaningful. all the para

44、meters 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 algorithms, where one can observe the favorable properties of the

45、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 (colms) was amsd = 0.0323.for a complete comparison of these algo

46、rithms 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 the vs lms algorithm, the respective increase is: 3n multiplic

47、ations, 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 favorable properties of these algorithms in tracking paramet

48、er 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 value.acknowledgement. this work is supported by the volkswag

49、en stiftung, federal republic of germany.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

50、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l

51、ms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）

52、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

53、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（msd）的加权系数。对于自适应滤波器，它被赋值，3：3、组合自适应滤波器合并后的自适应滤波器的基本思想是在两个或两个以上自适应lms算法并行实现与每个迭代之间的最佳选择，9。在每次迭代中选择最合适的算法，选择最佳的加权系数值。最好的加权系数是1，即在给定的时刻，向相应的维纳矢量值最接近。让是以基本lms算法为基础的第i个

54、加权系数，在瞬间选择参数q和系数k。注意，现在我们可以在一个统一的处理方式（lms: q ,glms: q a,sa:q ）下。基于lms算法的行为主要依赖于q，在每个迭代中有一个最佳值，生产的最佳表现的自适应算法。现在分析最小均方与一些基于相同类型的算法相结合的自适应滤波器，但参数q是不同的。加权系数周围分布随机变量和和方差，相关4，9：。 （4）（4）中的概率p()依赖的值. 例如= 2的高斯分布，p()= 0.95（两个规则）。置信区间的定义 （5）接着，从（4）式到（5）式我们认为只要关于独立q，这意味着，对于小偏差，置信区间对同一的lms的算法是不同的，而对同一的lms的算法则相交。

55、另一方面，当偏置变大，然后中央位置的不同间隔距离很大,而且他们不相交。由于我们对有关信息没有先验知识，我们将使用一种特定的统计学方法得到的标准,即自适应算法选择的q值问题。这个标准的平衡状态,从或同一个数量级的,即。提出的联合算法(ca)现在可以被总结为下面的步骤:第1步：从不同预定义设置中为算法计算。第2步：估计每个算法的方差。第3步：检查是否相交对于算法。从一个最大的差异值算法走向与差异较小的值。根据（4），（5）和取舍的标准，如果下式成立那么将会减少这个检查： （6）当和以下关系成立：如果没有相交（大偏差）选择具有最大的方差的值算法。如果相交，偏差已经很小。因此，检查了一对新的加权系数，或者，如果是最后一对，只选择具有最小方差的算法。首先两个区间不相交意味着实现了取舍标准，并选择最大方差算法。第4步：转到下一个瞬间。元素的集合q中最小的数l= 2。在这种情况下，应提供良好的跟踪快速变化（最大的差异），而其他应提供小的方差的稳定状态。通过增加更多的观察,这两个极端之间,我们可以稍微改进算法的瞬态行为。需要注意的是,只有未知值(6)的差异。在仿真中我们估计4式： （7）当k = 1，2，. ，l和替代的方法是估计为 （8）有关表达式和在稳定状态为lms算法的不同类型，从已知文献中可以看出。对于标准的lms算法在稳定状态，和是相关的。,3. 需要注意的是，任何其他估