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1、 FIR Digital Filter Design In chapter 9 we considered the design of IIR digital filters. For such filters, it is also necessary to ensure that the derived transfer function G(z) is stable. On the other hand, in the case of FIR digital filter design,the stability is not a design issue as the transfer
2、 function is a polynomial in z-1 and is thus always guaranteed stable. In this chapter, we consider the FIR digital filter design problem. Unlike the IIR digital filter design problem, it is always possible to design FIR digital filters with exact linear-phase. First ,we describe a popular approach
3、to the design of FIR digital filters with linear-phase. We then consider the computer-aided design of linear-phase FIR digital filters. To this end, we restrict our discussion to the use of matlab in determining the transfer functions. Since the order of the FIR transfer function is usually much hig
4、her than that of an IIR transfer function meeting the same frequency response specifications, we outline two methods for the design of computationally efficient FIR digital filters requiring fewer multipliers than a direct form realization. Finally, we present a method of designing a minimum-phase F
5、IR digital filter that leads to a transfer function with smaller group delay than that of a linear-phase equivalent. The minimum-phase FIR digital filter is thus attractive in applications where the linear-phase requirement is not an issue. 10.1 preliminary considerations In this section,we first re
6、view some basic approaches to the design of FIR digital filters and the determination of the filter order to meet the prescribed specifications. 10.1.1 Basic Approaches to FIR Digital Filter DesignUnlike IIR digital filter design, FIR filter design does not have any connection with the design of ana
7、log filters. The design of FIR filters is therefore based on a direct approximation of the specified magnitude response,with the often added requirement that the phase response be linear. Recall a causal FIR transfer function H(z) of length N+1 is a polynomial in z-1 of degree N: (10.1)The correspon
8、ding frequency response is given by (10.2)It has been shown in section 5.3.1 that any finite duration sequence xn of length N+1 is completely characterized by N+1 samples of its discrete-time Fourier transform X. As a result, the design of an FIR filter of length N+1 can be accomplished by finding e
9、ither the impulse response sequence hn or N+1 samples of its frequency response H. Also ,to ensure a linear-phase design, the condition ,must be satisfied. Two direct approaches to the design of FIR filters are the windowed Fourier series approach and the frequency sampling approach. We describe the
10、 former approach in Section 10.2. The second approach is treated in Problems 10.31 and 10.32. In section 10.3, we outline computer-based digital filter design methods.10.1.2 Estimation of the Filter Order After the type of the digital filter has selected, the next step in the filter design process i
11、s to estimate the filter order should be the smallest integer greater than or equal to the estimated value.FIR Digital Filter Order Estimation For the design of lowpass FIR digital filters, several authors have advanced formulas for estimating the minimum value of the filter order N directly from th
12、e digital filter specifications: normalized passband edge angular frequency , normalizef stopband edge angular frequency , peak passband ripple ,and peak stopband ripple . We review three such formulas.Kaisers Formula. A rather simple formula developed by Kaiser Kai74 is given by .We illustrate the
13、application of the above formula in Example 10.1.Bellangers Formula. Another simple formula advanced by Bellanger is given by Bel8110.1 Preliminary Considerations .Its application is considered in Example 10.2.Hermanns Formula. The formula due to Hermann et al.Her73 gives a slightly more accurate va
14、lue for the order and is given by ,Where ,And ,With a1=0.005309, a2=0.07114 ,a3=-0.4761,a4=0.00266, a5=0.5941, a6=0.4278,b1=11.01217, b2=0.51244.The formula given in Eq.(10.5) is valid for . If , then the filter order formula to be used is obtained by interchanging and in Eq.(10.6a) and (10.6b). For
15、 small values of and , all of the above formulas provide reasonably close and accurate results. On the other hand, when the values of and are large, Eq.(10.5) yields a more accurate value for the order.A Comparison of FIR Filter Order FormulasNote that the filter order computed in Examples 10.1, 10.
16、2 and 10.3, using Eqs.(10.3),(10.3),and (10.5),Respectively ,are all different. Each of these three formulas provide only an estimate of the required filter order. The frequency response of the FIR filter designed using this estimated order may or may not meet the given specifications. If the specif
17、ications are not met, it is recommended that the filter order be gradually increased until the specifications are met. Estimation of the FIR filter order using MATLAB is discussed in Section 10.5.1. An important property of each of the above three formulas is that the estimated filter order N of the
18、 FIR filter is inversely proportional to the transition band width () and does not depend on the actual location of the transition band. This implies that a sharp cutoff FIR filter with a narrow transition band would be of very high order, whereas an FIR filter with a wide transition band will have
19、a very low order. Another interesting property of Kaisers and Bellangers formulas is that the order depends on the product . This implies that if the values of and are interchanged, the order remains the same. To compare the accuracy of the the above formulas, we estimate using each formula the orde
20、r of three linear-phase lowpass FIR filters of known order, bandedges, and ripples. The specifications of the three filters are as follows: Filter No.1: Filter No.2: Filter No.3: .The results are given in Table 10.1. Each one of the three formulas given above can also be used to estimate the order o
21、f highpass, bandpass, and bandstop FIR filters. In the case of the bandpass and bandstop filters, there are two transition bands. It has been found that here the filter order basically depends on the transition band with the smallest width. We illustrate the use of the Kasiers formula in estimating
22、the order of a linear-phase bandpass FIR filter in Example 10.4.作者：Sanjit K.Mitra国籍：USA出处：Digital Signal Processing -A Computer-Based Approach 3eFIR数字滤波器的设计 在第9章，我们考虑了IIR数字滤波器的设计。对于这样的过滤器，它也必须确保派生传递函数G（z）是稳定的。另一方面，在FIR数字滤波器设计的情况下，稳定是不是设计问题，因为传递函数是一个在z-1的多项式，因而始终保证稳定。在这一章中，我们考虑的FIR数字滤波器的设计问题。 不同的是IIR
23、数字滤波器设计问题，它总是可以设计一种精确的FIR线性相位数字滤波器。首先，我们描述了发展与线性相位FIR数字滤波器设计流行的方法。然后，我们考虑线性相位FIR数字滤波器的计算机辅助设计。为此，我们限制我们讨论了MATLAB在确定传递函数的使用。自区传递函数顺序通常比转移的IIR会议相同的频率响应规格功能还高，我们概述了计算效率比直接的FIR需要较少的乘法器实现形式的数字滤波器设计的两种方法。最后，我们提出一个设计最低FIR数字滤波器的相位，导致一个比一个更小的线性相位延迟相当于该组的传递函数方法。最小相位FIR数字滤波器因此，在应用中的线性相位的要求是没有问题的吸引力。10.1初步考虑在本节
24、中，我们第一次审查的FIR数字滤波器的设计和定阶滤波器，以满足规范规定的一些基本方法。10.1.1基本途径FIR数字滤波器设计不像IIR数字滤波器设计，FIR滤波器设计没有任何的模拟滤波器的设计连接。FIR滤波器设计的基础上，因此在指定的幅度响应直接逼近，与经常补充规定，即相位响应是线性的。记得有因果区传递函数H（z）的长度为N +1是在Z - 1的n次多项式： （10.1）相应的频率响应，给出了 （10.2）它已被证明在第5.3.1节，任何有限的时间序列x长度为n的N +1的特点是完全由N +1其离散时间傅里叶变换的样本，结果十，一个FIR滤波器的设计长度为N +1可以通过寻找或脉冲响应序列
25、 n的或N +1其频率响应阁下也样本，以确保线性相位设计，条件 ，必须得到满足。两个的FIR滤波器的设计方法是直接的窗口Fourier级数法，频率抽样方法。我们在10.2节描述了前一种方法。第二种方法是治疗中存在的问题10.31和10.32。在10.3节，我们列出了基于计算机的数字滤波器的设计方法。10.1.2估算过滤器顺序后的数字滤波器有选择的类型，在滤波器设计过程的下一步是评估筛选顺序应该是最小的整数大于或等于估计价值。FIR数字滤波器的阶的估计对于低通FIR数字滤波器的设计，一些作者拥有先进的公式估算的数字滤波器规格的过滤器阶数N直接最小值：归通带边缘角频率，角频率normalizef阻
26、带的边缘，峰值通带纹波，阻带峰值纹波。我们回顾三个这样的公式。Kaiser的公式。一个相当简单的公式由Kaiser Kai74发展是给予 。我们说明了上述公式中的应用实例10.1。贝兰杰的公式。另一个简单的公式贝兰杰先进为Bel8110.1初步设想 。它的应用被认为是在例10.2。Hermann的公式。由于该公式赫尔曼等人。Her73给出了更精确的顺序稍有价值，给予 ，凡 ，和 ，随着a1 = 0.005309，2= 0.07114，a3的=- 0.4761，A4纸= 0.00266，A5的= 0.5941，A6的= 0.4278，B1的= 11.01217，B2的= 0.51244。式中给出
27、的公式。（10.5）是有效的。如果，那么滤波器阶公式将要采用通过交换和式获得。（10.6a）和（10.6b）。 对于小值和所有上述公式，并提供准确的结果相当接近。另一方面，当和值大，情商。（10.5），得到一个更精确的值的顺序。FIR滤波器的阶公式比较请注意，滤波器的阶在例10.1，10.2和10.3计算，使用均衡器。（10.3），（10.3）和（10.5），分别是各不相同。这三个每个公式只提供所需要的滤波器的阶的估计。在频率响应的FIR滤波器的设计采用了这个估计顺序可能或可能不符合给定的规格。如果不符合规范，建议，该滤波器秩序逐步增加，直到符合规格要求。FIR滤波器的阶的估计是利用MATLA
28、B节中讨论10.5.1。 作者：上述三个公式每一个重要的特点就是估计滤波器阶FIR滤波器的N是成反比的过渡频带宽度（）和不依赖于过渡乐队的实际位置。这意味着，一个尖锐的截止区与窄过渡带滤波器将是非常高的顺序，而有广泛的FIR带通滤波器的过渡将有一个非常低的顺序。 另一个Kaiser的和贝兰杰的公式有趣的特性是在产品上的顺序而定。这意味着，如果和价值互换，订单保持不变。 比较了上述公式的准确性，我们估计使用每个公式三线性相位低通已知秩序，bandedges，和涟漪FIR滤波器秩序。这三个过滤器的规格如下： 过滤器一： 过滤二： 过滤三：。结果如表10.1。 每个给予上述三个公式之一，也可以用来估计高通，带通秩序，带阻FIR滤波器。在带通和带阻滤波器的情况下，有两个过渡频带。人们已经发现，这里的过滤器顺序基本上与最小宽度过渡带而定。我们说明了Kasier的公式估计一个线性相位的FIR带通滤波器的阶在例10.4使用。作者：Sanjit K.Mitra国籍：USA出处：Digital Signal Processing -A Computer-Based Approach 3e
