外文翻译FIR滤波器设计技术

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1、外文原文及翻译FIR滤波器设计技术摘要这份报告列举了一些设计FIR滤波器所使用的技术。首先讨论了窗函数法和频率取样法的优点和缺点。FIR数字滤波器也包含了许多优化设计的方法，这些优化技术减少了在频率采样时非采样频率点的误差频率。对于用于设计数字滤波器的技术，例如matlab，进行了简明扼要的探讨。介绍 FIR滤波器的系统函数是一个的多项式，因FIR滤波器的频率响应是频率的实函数，也称其为零相位滤波器。N阶FIR滤波器的系统函数表示为 （1） FIR滤波器是十分重要的，可应用于精确线性相位相应。FIR滤波器的实现方式保证了它是一个稳定的滤波器。 FIR滤波器的设计可分为两部分：（i） 近似问题（

2、ii） 实现问题 解决近似问题，要通过四个步骤找出传递函数：（i） 在频域内找出期望的或最理想的反应（ii） 选择滤波器的阶数（FIR滤波器的长度N）（iii） 选择近似结果中较好的（iv） 选择一种算法寻找最优的滤波器传递函数 选择部分结构处理实现传递函数的形式可能是线路图或程序。 本质上来说，有三种著名的FIR滤波器设计方法：（1） 窗函数法（2） 频率取样法（3） 滤波器的优化设计窗函数法在该方法中，Park87,Rab75, Proakis00从理想的频率响应Hd（w）出发， 一般来说，单位脉冲相应hd（n）的持续时间是无限的，所以在某种程度上说，它必须截断。n=M-1约束着FIR滤波

3、器的长度M。以M-1截断的hd（n）乘以窗函数就得到了滤波器的单位脉冲响应。 矩形窗口的定义为 w(n) = 1 0nM-1 (2)= 0 其它 FIR滤波器的单位脉冲相应为h(n) = hd(n) w(n) (3)= hd(n) 0nM-1= 0 其它 现在,多元化的窗函数w(n)与hd(n)相当于hd(w)与w（w）的卷积，其中，w（w）是窗函数的频域表示。因此Hd(w)与w(w)的卷积为FIR数字滤波器的截断后的频率响应 (4)频率响应也可以利用以下的关系式(5) 由于非均匀收敛的傅里叶级数的不连续性，其自身的波纹前后有一种近似于不连续的频率响应，因此直接截断的hd(n)来获得h(n)将

4、导致吉布斯现象。与此同时，利用（5）得到的频率响应在频域内有波纹的振荡。为了减少波纹，hd(n)不是乘以一个矩形窗口w(n)，而是乘以一个含有圆锥和逐渐衰减到零的窗口。作为主体的序列的hd(n)和w(n)在时域内的卷积相当于其在频域内的乘积，其效果是平滑的。滤波器的窗函数的傅里叶系数对滤波结果的频率响应的影响如下：（i） 一个主要的结果就是过渡带的不连续的两边出现中断（ii） 过渡带的宽度取决于窗函数的频率响应的主瓣宽度（iii） 滤波器的频率响应是通过卷积关系得到的，可以肯定的是，由产生的滤波器绝不是最佳的（iv） 随着M的增加，其主瓣宽度降低从而降低了过渡带的宽度，但是这也过滤掉了更多的脉

5、冲频率响应。（v） 窗函数消除边缘响应引起的效果，并以较低的旁瓣代价增加过渡带的宽度1. Bartlett 三角窗：2. 广义余弦窗3. Kaiser 窗表一从Park87得到的系数WindowabCRetangular100Hanning0.50.50Hamming0.540.460Blackman0.420.50.08 Bartlett 窗函数的设计减少了信息的误差，但其过渡带较宽。Hanning,Hamming 和Blackman 窗的使用会更好，它们可以用于复杂的余弦函数，并可以提供理想的光滑截断的脉冲响应和频率响应。研究的结果表明，最佳的窗函数可能是有一个参数的Kaiser 窗，它可

6、以实现衰减和过渡带宽度的妥协。 窗函数的主要优点是它们比起其它方法更加的简单，且易于使用。事实上，计算窗函数的明确的方程系数就可以成功的使用该方法。 在使用窗函数来设计滤波器时，会遇到三个问题：（i） 该方法只适用于Hd(w)是绝对可积的情况，即只有（2）式可以评估。当Hd(w)是复杂的或不能轻易被评价的闭合形式，写出Hd(n)的数学表达式就变得困难了。（ii） 使用窗函数的灵活性比较差，例如，在低通滤波器的设计中，通频带的边缘频率一般不能用窗口完全掠过不连续区域。因此理想低通滤波器的截止频率，是通带截止频率f1和阻带截止频率f2相关的一个频率响应。（iii） 窗函数法在设计标准滤波器，例如低

7、通、高通、带通，是很有用的。但这也使其在语音、图像处理的程序上的应用是十分有限的。频率取样技术在该方法中，Rad75、Park87、Proakis00是按照前面的方法提供理想的频率响应。现在，是在给定的频率响应中取一系列等间隔的频率，用以得到N的取样。因此，采样频率的响应Hd(w)在其本质上是给了我们Hd(2pnk/N)。因此，利用该滤波器可以计算出下面的公式： (6) 现在使用上述的N阶滤波器响应、连续性频率响应作为计算差值采样频率响应的方法。其近似误差就会完全接近于零点采样频率的误差，并将它们之间的频率有限化。越平滑的频率响应将会越近似，存在于样本点间的差值误差会很小。 一种减小误差的方法

8、是增加频率采样的样本数目Rad75。其它的改善其质量的方法是找出一组无约束变量的制定频率样品。这些无约束变量的值一般都是由计算机来优化的一些简单的函数逼近，其误差接近最小。例如，一个无约束变量可能会选择在低通滤波器的通带于阻带之间的过渡带上的频率响应。有两套不同的频率，可以用于取样。一组样品是fk = k/N 其中 k = 0,1,.N-1其他的均匀间隔的频率样品可以采取fk =(k + 1 / 2)/ N其中k = 0、1、.N-1。 选择第二种时会在指定目标的第二个可能的频率的响应中给我们额外的灵活性。因此一个给定的过渡带边缘频率可能接近表二的频率采样点。在此情况下，依托于表二的设计将采用

9、优化设计程序。 在一张由Rabiner and Gold Rabi70提供的纸上，Rabiner提到了一种技术，即基于采样频率的理念来设计FIR滤波器。Rabiner对于这种方法的步骤的建议如下：(i) 随着样品数量的变化，其提供了相应量级的响应。给定N，设计师决定使用什么样的插值。(ii) 当它被Rabiner发现时，他实验了N从15到256的设计，16N样品的H（w）导致可靠的运算，因此，16到1的插值方法可用。(iii) 给定N值，滤波器的单位样品的响应会被确定，h(N)的计算公式是逆傅里叶变换。(iv) Rabiner建议用两种程序去获得频率响应值。 它们是（a） h(n)是由N/2样

10、品或者（N-1）/2样品去移除其锐利的边缘，然后再15N的冲击响应样品周围的位置都是对称的脉冲响应。（b） h(n)分布在n / 2样品之中，15N 的样品安置在两脉冲响应之间。（c） 零增广序列使用FFT算法转换为插值频率的响应。频率取样技术的优点（i） 与窗函数法不同，该技术可用于任何的响应。（ii） 这种方法在设计非标准滤波器时是非常有用的，它可以处理任何不规则形状的响应。 频率取样法也有一些缺点，即通过插值得到的频率响应只是理想频率的采样点的响应。在其他点，将会出现一些错误。FIR Filter Design TechniquesAbstractThis report deals wi

11、th some of the techniques used to design FIR filters. In the beginning, the windowing method and the frequency sampling methods are discussed in detail with their merits and demerits. Different optimization techniques involved in FIR filter design are also covered, including Rabiners method for FIR

12、filter design. These optimization techniques reduce the error caused by frequency sampling technique at the non-sampled frequency points. A brief discussion of some techniques used by filter design packages like Matlab are also included. Introduction FIR filters are filters having a transfer functio

13、n of a polynomial in z and is an all-zero filter in the sense that the zeroes in the z-plane determine the frequency response magnitude characteristic.The z transform of a N-point FIR filter is given by(1)FIR filters are particularly useful for applications where exact linear phase response is requi

14、red. The FIR filter is generally implemented in a non-recursive way which guarantees a stable filter. FIR filter design essentially consists of two parts (i) approximation problem (ii) realization problem The approximation stage takes the specification and gives a transfer function through four step

15、s. They are as follows:(i) A desired or ideal response is chosen, usually in the frequency domain. (ii) An allowed class of filters is chosen (e.g.the length N for a FIR filters). (iii) A measure of the quality of approximation is chosen. (iv) A method or algorithm is selected to find the best filte

16、r transfer function. The realization part deals with choosing the structure to implement the transfer function which may be in the form of circuit diagram or in the form of a program.There are essentially three well-known methods for FIR filter design namely: (1) The window method (2) The frequency

17、sampling technique (3) Optimal filter design methods The Window Method In this method, Park87, Rab75, Proakis00 from the desired frequency response specification Hd(w), corresponding unit sample response hd(n) is determined using the following relation In general, unit sample response hd(n) obtained

18、 from the above relation is infinite in duration, so it must be truncated at some point say n= M-1 to yield an FIR filter of length M (i.e. 0 to M-1). This truncation of hd(n) to length M-1 is same as multiplying hd(n) by the rectangular window defined as w(n) = 1 0nM-1 (2)0 otherwiseThus the unit s

19、ample response of the FIR filter becomes h(n) = hd(n) w(n) (3) = hd(n) 0nM-1= 0 otherwise Now, the multiplication of the window function w(n) with hd(n) is equivalent to convolution of Hd(w) with W(w), where W(w) is the frequency domain representation of the window function Thus the convolution of H

20、d(w) with W(w) yields the frequency response of the truncated FIR filter (4)The frequency response can also be obtained using the following relation (5)But direct truncation of hd(n) to M terms to obtain h(n) leads to the Gibbs phenomenon effect which manifests itself as a fixed percentage overshoot

21、 and ripple before and after an approximated discontinuity in the frequency response due to the non-uniform convergence of the fourier series at a discontinuity.Thus the frequency response obtained by using (8) contains ripples in the frequency domain. In order to reduce the ripples, instead of mult

22、iplying hd(n) with a rectangular window w(n), hd(n) is multiplied with a window function that contains a taper and decays toward zero gradually, instead of abruptly as it occurs in a rectangular window. As multiplication of sequences hd(n) and w(n) in time domain is equivalent to convolution of Hd(w

23、) and W(w) in the frequency domain, it has the effect of smoothing Hd(w). The several effects of windowing the Fourier coefficients of the filter on the result of the frequency response of the filter are as follows: (i) A major effect is that discontinuities in H(w) become transition bands between v

24、alues on either side of the discontinuity. (ii) The width of the transition bands depends on the width of the main lobe of the frequency response of the window function, w(n) i.e. W(w). (iii) Since the filter frequency response is obtained via a convolution relation , it is clear that the resulting

25、filters are never optimal in any sense. (iv) As M (the length of the window function) increases, the mainlobe width of W(w) is reduced which reduces the width of the transition band, but this also introduces more ripple in the frequency response. (v) The window function eliminates the ringing effect

26、s at the bandedge and does result in lower sidelobes at the expense of an increase in the width of the transition band of the filter. Some of the windows Park87 commonly used are as follows: 1. Bartlett triangular window: 2 Generalized cosine windows3.Kaiser window with parameter : The general cosin

27、e window has four special forms that are commonly used. These are determined by the parameters a,b,c TABLE IValue of coefficients for a,b and c from Park87WindowAbcRetangular100Hanning0.50.50Hamming0.540.460Blackman0.420.50.08The Bartlett window reduces the overshoot in the designed filter but sprea

28、ds the transition region considerably.The Hanning,Hamming and Blackman windows use progressively more complicated cosine functions to provide a smooth truncation of the ideal impulse response and a frequency response that looks better. The best window results probably come from using the Kaiser wind

29、ow, which has a parameter . that allows adjustment of the compromise between the overshoot reduction and transition region width spreading. The major advantages of using window method is their relative simplicity as compared to other methods and ease of use. The fact that well defined equations are

30、often available for calculating the window coefficients has made this method successful.There are following problems in filter design using window method: (i) This method is applicable only if Hd(w) is absolutely integrable i.e only if (2) can be evaluated. When Hd(w) is complicated or cannot easily

31、 be put into a closed form mathematical expression, evaluation of hd(n) becomes difficult. (ii) The use of windows offers very little design flexibility e.g. in low pass filter design, the passband edge frequency generally cannot be specified exactly since the window smears the discontinuity in freq

32、uency. Thus the ideal LPF with cut-off frequency fc, is smeared by the window to give a frequency response with passband response with passband cutoff frequency f1 and stopband cut-off frequency f2. (iii) Window method is basically useful for design of prototype filters like lowpass,highpass,bandpas

33、s etc. This makes its use in speech and image processing applications very limited. The Frequency Sampling Technique In this method, Park87, Rab75, Proakis00 the desired frequency response is provided as in the previous method. Now the given frequency response is sampled at a set of equally spaced f

34、requencies to obtain N samples. Thus , sampling the continuous frequency response Hd(w) at N points essentially gives us the N-point DFT of Hd(2pnk/N). Thus by using the IDFT formula, the filter co-efficients can be calculated using the following formula (6)Now using the above N-point filter respons

35、e, the continuous frequency response is calculated as an interpolation of the sampled frequency response. The approximation error would then be exactly zero at the sampling frequencies and would be finite in frequencies between them. The smoother the frequency response being approximated, the smalle

36、r will be the error of interpolation between the sample points. Now using the above N-point filter response, the continuous frequency response is calculated as an interpolation of the sampled frequency response. The approximation error would then be exactly zero at the sampling frequencies and would

37、 be finite in frequencies between them. The smoother the frequency response being approximated, the smaller will be the error of interpolation between the sample points. One way to reduce the error is to increase the number of frequency samples Rab75. The other way to improve the quality of approxim

38、ation is to make a number of frequency samples specified as unconstrained variables. The values of these unconstrained variables are generally optimized by computer to minimize some simple function of the approximation error e.g. one might choose as unconstrained variables the frequency samples that

39、 lie in a transition band between two frequency bands in which the frequency response is specified e.g. in the band between the passband and the stopband of a low pass filter. There are two different set of frequencies that can be used for taking the samples. One set of frequency samples are at fk =

40、 k/N where k = 0,1,.N-1. The other set of uniformly spaced frequency samples can be taken at fk =(k+1/2)/N for k = 0,1,.N-1. The second set gives us the additional flexibility to specify the desired frequency response at a second possible set of frequencies. Thus a given band edge frequency may be c

41、loser to type-II frequency sampling point that to type-I in which case a type-II design would be used in optimization procedure. In a paper by Rabiner and Gold Rabi70, Rabiner has mentioned a technique based on the idea of frequency sampling to design FIR filters. The steps involved in this method s

42、uggested by Rabiner are as follows:(i) The desired magnitude response is provided along with the number of samples,N . Given N, the designer determines how fine an interpolation will be used. (ii) It was found by Rabiner that for designs they investigated, where N varied from 15 to 256, 16N samples

43、of H(w) lead to reliable computations, so 16 to 1 interpolation was used. (iii) Given N values of Hk , the unit sample response of filter to be designed, h(n) is calculated using the inverse FFT algorithm. (iv) In order to obtain values of the interpolated frequency response two procedures were sugg

44、ested by Rabiner. They are (a) h(n) is rotated by N/2 samples(N even) or (N-1)/2 samples for N odd to remove the sharp edges of impulse response, and then 15N zero-valued samples are symmetrically placed around the impulse response. (b) h(n) is split around the N/2nd sample, and 15N zero-valued samples are placed between the two pieces of the impulse response. (v) The zero augmented sequences are transformed using the FFT algorithm to give the interpolated frequency responses.