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1、译文： 一种基于LMS算法滤除心电图信号中50Hz干扰的改良自适应滤波器 摘 要：本文首先介绍三种传统的数字滤波方法并分析它们滤波的特点，然后将使用一种基于自适应数字滤波和中值滤波的新方法来讨论其缺陷并进行模拟。经论证，新方法不仅成功抑制噪音而且尽可能的保留了有用的信息，其性能优越于其它方法。关键词：自适应滤波，LMS算法，中值滤波1.绪论心电图（ECG）信号表现的是身体某些部分的电信号。一般来说，它们的振幅范围都在10uV-5mV，频率范围在。在采集心电图信号时，可能会发生不同类型的干扰，其中最突出地是50Hz电源线的干扰。因此在心电图信号的采集过程中最主要的任务是如何抑制噪声干扰。通常我们

2、使用平滑滤波器1,陷波滤波器2，以及自适应滤波器3。平滑滤波器主要的缺点是QRS波群的峰值是削减的且信号失真较大。陷波滤波器虽然能过滤固定频率的干扰，但是当频率波动时器过滤能力明显下降。自适应滤波技术已经被证明有效的应用在许多生物医学中。当使用传统自适应滤波器时，我们可能不能确切的了解干扰的频率因为自适应滤波器能自动跟踪其频率变化。现在我们用一种新的基于自适应数字滤波和中值滤波方法来减少误差以及最低限度的流失有用信息（心电图）。2.滤波器设计在测试中使用的心电图数据的采样频率为500Hz、时间间隔为4s、大小单位是uV，如图1所示： 图1 测试的原始心电图信号 图2 200uV，50Hz噪音的

3、心电图信号2.1 平滑滤波器平滑滤波器是一种用于较早的数字滤波的方法，其差分方程是其中x（n）是原始心电图，y（n）是过滤心电图。平滑滤波器的优点是算法简单，处理数率快以及良好的过滤效果，它通常是用在心电图监测中。但是它也有一些缺陷，比如窄带通会影响信号的有用成分4。平滑滤波器滤波结果显示如图3。我们可以看到，这种方法对过滤50Hz噪音具有更好的效果但是对削减高峰QRS波群其变形较大。在图4中我们看到有用信息的损失较大，因此此数字滤波器在诊所诊断是不切实际的。 图3 平滑滤波结果（i=10） 图4 有用信息的损失2.2 陷波滤波器陷波滤波器也被称做简单积分系数带阻滤波器，它是一个全通网减去一个

4、窄带线性相位的FIR滤波器。它们具有想通的传播延迟以及增益，因此我们能得到一个具有边缘特性的陷波滤波器。陷波滤波器具有500Hz的采样频率。转移函数为：差分方程为： 图5 陷波滤波器结果 图6 损失的有用信息与平滑滤波器比较发现，陷波滤波器的失真变得很低。改方法实现了线性相位可以进行实时处理，但是该滤波器的主要缺陷是它具有较大的延迟，且只能过滤有固定频率的干扰。当频率波动时，它的过滤能力明显下降。从图5和图6可以看到不尽人意的过滤效果。2.3 典型自适应滤波器自适应滤波器介绍自适应滤波器的定义是依赖递归算法可以自行设计的系统，这使得当相关的统计数据未知的环境下它能取得令人满意的过滤效果。自适应

5、滤波主要分为两种：线性和非线性。线性自适应滤波器通过应用线性组合的观测值来估算出预期值输入滤波器，自适应滤波被认为是非线性的。字适应滤波还可以分为：（i）监督自适应滤波器，它需要可知的训练序列提供不同的对指定输入信号向量的预期结果。将预期结果与输入信号向量的实际滤波结果比较，产生的误差信号用于调整滤波器的自由参数。该调整参数一直持续到一个稳态条件的确定。（ii） 无监督自适应滤波器，它在没有预期结果下执行自由参数的调整。为了让滤波器执行其功能，其设计包含了一系列规则使它能计算出具体的理想性能的输入输出映射集。在信号处理文献中，无监督自适应滤波通常被称为盲迭代反卷积算法和盲自适应算法。 图7 一

6、种典型自适应滤波器结构如上图所示，我们利用线性神经网络来获得参考信号r(n)表示其与噪音相关性，且与心电图信号无关。 图8 典型自适应滤波器结果输入在地面，与噪音不相关参考信号被引进。我们可以严密的滤除输入的参考信号中的干扰信号，使有用信息能更好的保存。但是从图11我们看到在初始阶段有用信息损失很大。2.4 改良的自适应滤波器为了处理典型自适应滤波器的缺点，我们处理输出信号E（n）用中值滤波提高过滤性能，如图9所示： 图9 改良的自适应滤波器结构 LMS算法LMS算法（最小均方）算法广泛应用于自适应信号的处理，信道均衡，系统识别和模式分类。它最大限度的减少均方误差（MSE）E（n）在目标信号与

7、参考信号之间的迭代技术。最佳权重向量能在基于梯度下降基础上用迭代估计算法。为了退化噪声，我们采用LMS算法。参考输入用于近似的振幅及正弦噪音波段，只需要调整A和。整幅和输入信号波段是可选的，权向量是迭代模式的分型。这就是说在下一刻的重量等于重量当前加一减均方误差梯度，(- E (n )x i (n ).参数是在经验上选择想要的速度产生收敛，其值越大，收敛速度就越快。因此，梯度下降算法可以写成：该参考信号是：滤波输出是：自适应数字滤波器的传递函数： 中值滤波原理我们假设滤波窗口长度W是n=2k+1（单数）或n=2k（偶数）。测量值是N，即观测到的数据是x1，x2，x3，x N，且N n。当前进的

8、W测量值连续，标准的中值滤波输出med(x i )为：其中x（k）是窗口2k+1或2k观察值的最大值。事实上，它是根据观测值的大小排序，那么把中位数认作输出。根据上述定义，输入xi和输出yi方程关系： 我们定义所谓的中值滤波也称为滑动中值滤波，Z显示整数域。如果滤波窗口长度为n=2k+1，脉冲宽度大于或等于k+1时脉冲将在滤波后被保留。但是，当脉冲宽度小于或等于k时，脉冲将在滤波后被滤除。这仅仅是典型的中值滤波去除脉冲噪声和保留信号的详情。中值滤波可以有效的防止突发脉冲干扰，因此，在时间应用中我们必须选择合适的排序周期。如果排序周期数是小的，可能是过滤器不排除干扰。如果更大的延迟，并导致该系统

9、性能恶化。最后输出为： 测试参数当振幅A的相关振幅输入x(t)是10，相位为。则初始权系数都为0，参数因子为，采样时间为。 测试结果我们可以发现辨别输入噪声倒退的调整过程，通过通过适当的未成年人的学习速度使用，LMS算法的输出能够收敛到输入噪声（工频分量心电图的影响可以忽略，因为它不符合相关的噪音）。在最初的阶段振荡的原因是，回归噪音比附加噪声低，如图10所示： 图10 退化噪声（0.4s） 图11 改良自适应滤波器结果 图12 损失的有用信息 图13 损失的有用信息基于典型的自适应滤波和中值滤波的优点，我们处理自适应滤波与中值滤波输出10点，并取得良好效果。从图13我们看到的低损失，新方法不

10、仅新方法不仅有各种各样的自适应滤波器的优点，特别是使用线性回归噪声神经网络，但也有中值滤波特性。因此，它可以确保过滤效果，同时降低了初始误差和心电图的损失。平滑滤波算法简单，易于实时处理，但它有严重的削减高峰。陷波滤波器具有良好的的过滤作用，但它不能用于实时过滤。当频率不定其过滤能力下降明显。对典型的自适应滤波初始误差较大，由于线性神经网络新方法自适应数字滤波和中值滤波的结果是减少初始误差和稳态误差，最小的有用信息（心电图）的损失。参考文献1吴勇程，杨玉华，钟华. 一种从心电图中消除50Hz干扰的数字滤波器新方法 .中华医学仪器. 第一卷. 23 .2龙新民，周静.基于MATLAB及其在心电图

11、信号预处理中的应用的50赫兹带阻滤波器的设计.重庆师范学院学报（自然科学版）.第一卷. 20. 2003年3月3周勇军，路志远，刘中期.自适应滤波器中应用于心电图的初步检测.电磁兼容测试技术.20044孙静霞，白艳强，杨胥新.一种改进Levkov方法f或抑制心电信号50 Hz的干扰.航天医学与医学工程中.第一卷. 13. 20005 殷丽丽，吴粤东.自适应均衡器的MATLAB实现自动基于LMS算法.重庆工学院（机械和电子版）.第一卷. 18. 2004传记作家周润军副教授，在自动化，科学和内蒙古大学技术与工艺专业检测和测量学院系任教。 原文：An Improved Self-Adaptive

12、Filter Based on LMSAlgorithm for Filtering 50Hz Interference in ECG SignalsYuan Weiting Zhou Runjing(Department of Automation, College of Sciences of Technology, Inner Mongolia University, Hohhot 010021 China)Abstract: In this paper, we firstly introduce three conventional digital filtering methods

13、and analyze their qualities of filtering. Then we use a new method based on self-adaptive digital filter and median filter for their defects, and simulate it. It is demonstrated that not only the noises are successfully restrained, but also the useful information is preserved as much as possible, an

14、d the performance of the new method is superior to the others.Keywords: Self-adaptive filter; LMS algorithm; median filter1 IntroductionElectrocardiograph (ECG) signal represents electricity signal from some parts of the body. In general, their amplitudes are in the range of 10uV-5mV and their frequ

15、encies are in the range of 0.05Hz-100Hz. During acquiring ECG signals, different kinds of interferences are likely to happen, and the uppermost one is 50Hz power-line interference in them. But the chief task is how to restrain it in ECG signal processing. Generally we employ smooth filter1, notch fi

16、lter2 and self-adaptive filter3. The primary drawbacks of smooth filter are that the peak of the QRS complex is cut and signal distortion is larger. Notch filter only can filter the interference of fixed frequency. However, when frequency fluctuates, its filtering capabilities descend clearly. Adapt

17、ive filtering technique has been shown to be useful in many biomedical applications. When conventional self-adaptive filter is utilized, we may not know frequency of interference truly because the filter can track variance in frequency automatically. But the initial error of the filter is bigger. No

18、w we use a new method based on self-adaptive digital filter and median filter to reduce error and minimally the losses of the useful information (ECG).2 Filter DesignECG data used in testing is a signal of 500Hz sampling frequency and 4s time span, and its magnitude unit is uV, as shown in Fig 1.Fig

19、 1. Original ECG signal is to be tested.Fig 2. ECG signal with 200uV, 50Hz noise2.1 Smooth FilterSmooth filter is an approach that is used earlier in digital filtering, and its difference equation isWhere x(n) is original ECG , and y(n) is filtered ECG.The advantages of it are simple algorithm, high

20、 processing speed and good filtering effect. It usually is used in ECG monitor. But it also has some flaws, such as narrower pass band and impact on useful component4. Results of smooth filter shows in Figure 3.We can see that this approach can filter 50Hz noise better. However, it cuts the peak of

21、the QRS complex, and its distortion is larger. In Fig 4, we see that the losses of useful information are larger. So this digital filter is not practical in clinic diagnose. Fig 3. Results of smooth filtering (i=10)Fig 4 losses of the useful information2.2 Notch FilterNotch filter is also called sim

22、ple integral coefficient band-stop filter, which is that an all-pass net subtracts a narrowband linear phase FIR filter that has the same propagation delay and gain with it. So we get a filter that has characteristics of edge trapped wave. The notch filter has 500Hz sampling frequency.Transfer funct

23、ion of it is: Its difference equation is: Fig 5. Results of notch filter Fig 6 losses of useful information We compare it with smooth filter, and find that its distortion became low. The method has realized linear phase and can be real-time processing. But the chief flaw of this filter is that it ha

24、s bigger delay and it only can filter the interference of fixed frequency. However, when the frequency fluctuates, its filtering capabilities descend clearly. From Fig5 and Fig6, we can see discontented filtering effects.2.3 Typical Self-adaptive Filter Fig 7. A typical self-adaptive filter structur

25、esAs above figure shows, we utilize linear neural network to acquire reference signal r(n) that is correlated with noise and is uncorrelated with ECG signals. Fig 8. Results of typical self-adaptive filterOn the ground that the reference input uncorrelated with noise is introduced, we may closely co

26、rrelate the filtered interference with the reference input, so the useful information can be better preserved. But we see the larger losses of useful information in initial stage from Fig 11.2.4 Improved Self-adaptive FilterTo handle the drawback of typical self-adaptive filter, we process the outpu

27、t signal E(n) of it with median filter to improve filtering performance, as shown in Fig9. Fig 9. Improved self-adaptive filter structure LMS Algorithm LMS(Least Mean Square) algorithm is widely applied to self-adaptive signal processing, channel balance, system identification and pattern classifica

28、tion. It is an iterative techniques for minimizing the mean square error (MSE) E(n) between the target signal and the reference signal. The optimal weight vector can be estimated using an iterative algorithm based on the gradient descent strategy.For the purpose of regressing noise, we adopt LMS alg

29、orithm. The reference input is used to approximate the amplitude and the phrase of sinusoid noise. We only need to adjust A and . The amplitude and the phrase of the input signal may be optional. The weight vector is iterated by pattern-by-pattern. That is to say that the weight in next moment is eq

30、ual to the weight the current moment add a minus mean square error gradient (- E (n )x i (n ). Parameter is empirically selected to produce convergence at a desired rate; the larger its value, the faster the convergence. Accordingly, the gradient descent algorithm can be written as The reference sig

31、nal isThe filter output isThe transfer function of self-adaptive digital filter is2.4.2 Principles of Median Filter. We assume the length of filter window W is n=2k+1(odd), or n=2k(even). The number of the observed value is N, namely the observed data is x1，x2，x3，x N，, and N n.When W move on the obs

32、erved value sequence, the output of the standard median filter Med（x i） isWhere x(k) is the kth maximum of 2k+1 or 2k observed value in window. In fact, it is to sort according to the size of the observed value, then we regard median as the output. According to above definition, the relation equatio

33、n of input xi and output yi is Where median filter we defined is also called slide median filter, and Z shows integer domain.If the length of filter window is n=2k+1, when pulse width is more than or equal to k+1, the pulse will be preserved after filtered; but when pulse width is less than or equal

34、 to k, the pulse will be removed after filtered. This is just characteristic that median filter removes pulse noise and preserves signal details.Median filter can effectively prevent outburst pulse from interfering. So we must choose proper sorting period in practice. If the number of sorting period

35、 is minor, maybe the filter does not remove interference; if it is greater, it will have big delay, and cause that system performance get worse.The final output isY(n)=med(E(n-k),E(n),E(n+k)n=1,22000. Testing Parameters. The amplitude A of reference input x(t) is 10, and the phase is 0.5. The initia

36、l weight coefficients are both 0. Learning factor is 0.0002. Sampling time is 0.002s. Testing Results. We can find that distinguish adjusting process is how to go along till it has regressed input noise well. Through the use of appropriate minor learning velocity , the output of LMS algorithm is abl

37、e to converge to the input noise(The effects of 50Hz ECG component can be ignored, because it isnt correlated with noise). The reason of the oscillation in initial stage is that the regressed noise is less than the appended noise, as shown in Fig 10.Fig 10. regressed noise(0.4s)Fig 11.Results of imp

38、roved self-adaptive filterFig 12. losses of useful informationFig 13. losses of useful informationBased on merits of typical self-adaptive filter and median filter, we process the output of self-adaptive filter with 10 point median filter, and get well results. We see that there is low lost informat

39、ion in Fig13 from Fig12 and Fig13. The new method not only has all merits of self-adaptive filter, especially regress noise using linear neural network, but also has characteristics of median filter. So it ensures filtering effect, simultaneously reduces the initial error and the losses of ECG.3 Con

40、clusionThe algorithm of smooth filter is simple, and is easy to be real-time processing. But it has serious cutting peak. Notch filter has good filtering effects, but it cannot be real-time filtering. When frequency wanders, its filtering capabilities descend clearly. The initial error of typical se

41、lf-adaptive filter is larger. As a result of linear neural network, a new method based on self-adaptive digital filter and median filter is to reduce initial error and steady state error, and minimally the losses of the useful information (ECG).References1Wu Yongcheng, Yang Yuxing and Hua Zhong: A N

42、ew Digital Filter Method for Eliminating 50Hz Interference from the ECG. Chinese journal of medical instrumentation. Vol. 23. Mar. 19992Long Xingming, Zhou Jing. The Design of 50Hz Bandstop Filter Based on MATLAB and Its Application in ECG Signal Pre-processing. Journal of Chongqing Normal College (

43、Natural Sciences). Vol. 20. Mar. 20033Zhou Yongjun, Lu Zhiyuan and Niu Zhongqi: Primary Application of Self-adaptive Filter in Detect of ECG. EMC Test Technology. 20044Sun Jingxia, Bai Yanqiang and Yang Yuxing: An Improved Levkov Method f or Filtering 50 Hz Interference in ECG Signals. Space medicin

44、e & medical engineerings. Vol. 13. 20005Yin Lili, Wu Yuedong: MATLAB Realization of Automatic Adaptive Equalizer Based on LMS Algorithm. Journal of Chongqing Institute of Technology (Machine and Electronic). Vol. 18. 2004Author BiographyZhou runjing：Associate professor, teaching at Department of Automation, College of Sciences and Technology of Inner Mongolia University and majoring in process detecting and measuring.