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1、IIR Digital Filter Design作者：Sanjit K.Mitra国籍：USA出处：Digital SignalProcessing-A Computer-BasedApproach3eAn important stepin the developmentof adigitalfilter is the determinationof arealizable transfer function G(z) approximating the given frequency response specifications. If an IIR filter is desired,
2、it is also necessary to ensure that G(z) is stable. The process of deriving the transfer function G(z) is called digital filter design. After G(z) has been obtained, the next step is to realize it in the form of a suitable filter structure. In chapter 8,we outlined a variety of basic structures for
3、the realization of FIR and IIR transfer functions. In this chapter,we consider the IIR digital filter design problem. The design of FIR digital filters is treated in chapter 10.First we review some of the issues associated with the filter design problem. A widely used approach to IIR filter design b
4、ased on the conversion of a prototype analog transfer function to a digital transfer function is discussed next. Typical design examples are included to illustrate this approach. We then consider the transformation of one type of IIR filter transfer function into another type, which is achieved by r
5、eplacing the complex variable z by a function of z. Four commonly used transformations are summarized.Finally we consider the computer-aided design of IIR digital filter. To this end, we restrict our discussion to the use of matlab in determining the transfer functions.9.1 preliminary considerations
6、There are two major issues that need to be answered before one can develop the digital transfer function G(z). The first and foremost issue is the development of a reasonable filter frequency response specification from the requirements of the overall system in which the digital filter is to be empl
7、oyed. The second issue is to determine whether an FIR or IIR digital filter is to be designed. In the section ,we examine these two issues first . Next we review the basic analytical approach to the design of IIR digital filters and thenconsider the determination of the filter order that meets the p
8、rescribed specifications. Wealso discuss appropriate scaling of the transfer function.9.1.1 Digital Filter SpecificationsAs in the case of the analog filter,either the magnitude and/or the phase(delay) response is specified for the design of a digital filter for most applications. In some situations
9、, the unit sample response or step response may be specified. In most practical applications, the problem of interest is the development of a realizable approximation to a given magnitude response specification. As indicated in section 4.6.3, the phase response of the designed filter can be correcte
10、d by cascading it with an allpass section. The design of allpass phase equalizers has received a fair amount of attention in the last few years.We restrict our attention in this chapter to the magnitude approximation problem only. We pointed out in section 4.4.1 that there are four basic types of fi
11、lters,whose magnitude responses are shown in Figure 4.10. Since the impulse response corresponding to each of these is noncausal and of infinite length, these ideal filters are not realizable. One way of developing a realizable approximation to these filter would be to truncate the impulse response
12、as indicated in Eq.(4.72) for a lowpass filter. The magnitude response of the FIR lowpass filter obtained by truncating the impulse response of the ideal lowpass filter does not have a sharp transition from passband to stopband but, rather, exhibits a gradual roll-off.Thus, as in the case of the ana
13、log filter design problem outlined in section 5.4.1, the magnitude response specifications of a digital filter in the passband and in the stopband are given with some acceptable tolerances. In addition, a transition band is specified between the passband and the stopband to permit the magnitude to d
14、rop off smoothly. For example, the magnitudeG(ej ) of a lowpass filter may be given as shown in Figure7.1. As indicated in the figure, in the passband defined by 0p, we require that themagnitude approximates unity with an error ofp ,i.e.,1 p G (ej )1 p, for,we require that the magnitude approximates
15、In the stopband, defined byzero with an error of s, i .e.,G(ej )s, forThe frequencies p and s are , respectively, called the passband edge frequency and the stopband edge frequency. The limits of the tolerances in the passband and stopband,p and s, are usually called the peak ripple values. Note tha
16、t the frequency response,and the magnitude response of a.As a result, the digital filtergiven in terms of the lossG(ej ) of a digital filter is a periodic function of real-coefficient digital filter is an even function of specifications are given only for the rangefunction, ( )2010g10 G(ej )Digital
17、filter specifications are often,in dB. Here the peak passband ripple p and theminimum stopband attenuation s are given in dB,i.e., the loss specifications of a digital filter are given byp 2010g10(1 p)dB,s 2010g10( s)dB.9.1 Preliminary ConsiderationsAs in the case of an analog lowpass filter, the sp
18、ecifications for a digital lowpass filter may alternatively be given in terms of its magnitude response, as in Figure 7.2. Here the maximum value of the magnitude in the passband is assumed to be unity, and the maximum passband deviation, denoted as 1/12 ,is given by the minimum value of the magnitu
19、de in the passband. The maximum stopband magnitude is denoted by 1/A.For the normalized specification, the maximum valueof the gain function or themax given bymax2010g10(.12)dBIs called the maximum passband attenuation. For1, as is typically the case, itcan be shown thatmax 2010g10(1 2 p) 2 pThe pas
20、sband and stopband edge frequencies, in most applications, are specified inHz, along with the sampling rate of the digital filter. Since all filter design techniques aredeveloped in terms of normalized angular frequenciesp and s ,the sepcified criticalfrequencies need to be normalized before a speci
21、fic filter design algorithm can be applied.Let FT denote the sampling frequency in Hz, and Fp and Fs denote, respectively,thepassband and stopband edgefrequencies in Hz. Then the normalized angular edgefrequencies in radians are given bypFT2 Fp- 2 FpTFT,2FsT9.1.2 Selection of the Filter TypeThe seco
22、nd issue of interest is the selection of the digital filter type,i.e.,whether anIIR or an FIR digital filter is to be employed. The objective of digital filter design is todevelop a causal transfer function H(z) meeting the frequency response specifications. ForIIR digital filter design, the IIR tra
23、nsfer function is a real rational function of12MPo PiZp2zpMzH(z)=.1.2Nd0 d1zd2z. dNzMoreover, H(z) must be a stable transfer function, and for reduced computationalcomplexity, it must be of lowest order N. On the other hand, for FIR filter design, the FIRtransfer function is a polynomial inminimum v
24、alue of the loss function is therefore 0 dB. The quantityH (z)hnzn0For reduced computational complexity, the degree N of H(z) must be as small as possible.In addition, if a linear phase is desired, then the FIR filter coefficients must satisfy the constraint:hn hn NT here are several advantages in u
25、sing an FIR filter, since it can be designed with exact linear phase and the filter structure is always stable with quantized filter coefficients.However, in most cases, the order N fir of an FIR filter is considerably higher than the order N IIR of an equivalent IIR filter meeting the same magnitud
26、e specifications.Ingeneral, the implementation of the FIR filter requires approximately Nfir multiplicationsper output sample, whereas the IIR filter requires 2N IIR +1 multiplications per output sample. In the former case, if the FIR filter is designed with a linear phase, then the numberof multipl
27、icationsper output sample reduces to approximately (N fir +1)/2.Likewise, most IIR filter designs result in transfer functions with zeros on the unit circle, and the cascade realization of an IIR filter of order N IIR with all of the zeros on the unit circle requires (3 NIIR +3)/2 multiplications pe
28、r output sample. It has been shown that for most practical filter specifications, the ratio N fir /N iir is typically of the order of tens or more and, as a result, the IIR filter usually is computationally more efficientRab75.However ,if the group delay of the IIR filter is equalized by cascading i
29、t with an allpass equalizer, then the savings in computation may no longer be that significant Rab75. In many applications, the linearity of the phase response of the digital filter is not an issue,making the IIR filter preferable because of the lower computational requirements.9.1.3 Basic Approache
30、s to Digital Filter DesignIn the case of IIR filter design, the most common practice is to convert the digital filter specifications into analog lowpass prototype filter specifications, and then to transform it into the desired digital filter transfer function G(z). This approach has beenwidely used
31、 for many reasons:(a) Analog approximation techniques are highly advanced.(b) They usually yield closed-form solutions.(c) Extensive tables are available for analog filter design.(d) Many applications require the digital simulation of analog filters.In the sequel, we denote an analog transfer functi
32、on asHa(s)Pa(s)Da(s)Where the subscript a specifically indicates the analog domain. The digital transferfunction derived form Ha(s) is denoted byG(z)P(z)D(z)The basic idea behind the conversion of an analog prototype transfer functionH a(s) into a digital IIR transfer function G(z) is to apply a map
33、ping from the s-domain tothe z-domain so that the essential properties of the analog frequency response are preserved. The implies that the mapping function should be such that(a) The imaginary(j ) axis in the s-plane be mapped onto the circle of the z-plane.(b) A stable analog transfer function be
34、transformed into a stable digital transfer function.To this end,the most widely used transformation is the bilinear transformation describedin Section 9.2.Unlike IIR digital filter design,the FIR filter design does not have any connectionwith the design of analog filters. The design of FIR filter de
35、sign does not have any connection with the design of analog filters. The design of FIR filters is therefore based ona direct approximation of the specified magnitude response,with the often added requirement that the phase response be linear. As pointed out in Eq.(7.10), a causal FIR transfer functi
36、on H(z) of length N+1 is a polynomial in z -1 of degree N. The correspondingfrequency response is given byH(ej ) hne j n.n0It has been shown in Section 3.2.1 that any finite duration sequence xn of length N+1 iscompletely characterized by N+1 samples of its discrete-time Fourier transfer X(ej ). As
37、aresult, the design of an FIR filter of length N+1 may be accomplished by finding either theimpulse response sequence hn or N+1 samples of its frequency responseH(e j ) . Also,to ensure a linear-phase design, the condition of Eq.(7.11) must be satisfied. Two direct approaches to the design of FIR fi
38、lters are the windowed Fourier series approach and thefrequency sampling approach. We describe the former approach in Section 7.6. The second approach is treated in Problem 7.6. In Section 7.7 we outline computer-based digital filter design methods.作者：Sanjit K.Mitra国籍：USA出处：Digital SignalProcessing
39、-A Computer-Based Approach 3eIIR数字滤波器的设计在一个数字滤波器发展的重要步骤是可实现的传递函数G (z)的接近给定的频率响应规格。如果一个IIR滤波器是理想，它也有必要确保了 G (z)是稳定的。该 推算传递函数G (z)的过程称为数字滤波器的设计。然后 G (z)有所值，下一步就 是实现在一个合适的过滤器结构形式。在第8章，我们概述了为转移的FIR和IIR的各种功能的实现基本结构。在这一章中，我们考虑的 IIR数字滤波器的设计问题。 FIR数字滤波器的设计是在第10章处理。首先，我们回顾与滤波器设计问题相关的一些问题。一种广泛使用的方法来设计IIR滤波器的基
40、础上，传递函数原型模拟到数字的转换传递函数进行了讨论下一步。 典型的设计实例来说明这种方法。 然后，我们考虑到另一种类型，它是由一个函数代 替复杂的变量z达到了一个IIR滤波器的传递函数z的类型转换四种常用的转换进行 了总结。最后，我们考虑的IIR计算机辅助设计数字滤波器。为此，我们限制我们讨 论了 MATLAB在确定传递函数的使用。9.1 初步考虑有两个需要先有一个回答可以发展数字传递函数G (z)的重大问题。首要的问题是一个合理的滤波器的频率响应规格从整个系统中数字滤波器将被雇用的要求发展。第二个问题是要确定的FIR或IIR数字滤波器是设计。在一节中，我们首先检查了这 两个问题。接下来，我
41、们回顾到的IIR数字滤波器设计的基本分析方法，然后再考虑 过滤器的顺序符合规定的规格测定。我们还讨论了传递函数适当的调整。9.1.1 数字过滤器的规格如过滤器的模拟案件，无论是规模和/或相位(延迟)响应对于大多数应用程序指定 一个数字滤波器for the设计。在某些情况下，单位采样响应或阶跃响应可能被指定。在大多数实际应用中，利益问题是一个变现逼近一个给定的幅度响应的规范发展。如第4.6.3所示，所设计的滤波器可以通过级联与全通区段纠正相位响应。全通相位均 衡器的设计接受了最近几年，相当数量的关注。我们在这方面限制的幅度逼近问题的唯一一章我们的注意。我们指出，在第 4.4.1节指出，有四个过滤
42、器，具大小，如图4.10所示的反应基本类型。由于脉冲响应对应于所有这些都是非因果和无限长，这些过滤器是尚未实现的理想。一个发展一个变现 的近似值，这些过滤器的方法是截断的脉冲响应，如式所示。（4.72）为低通滤波器。该FIR低幅度响应滤波器得到截断的理想低通滤波器，从没有一个通带过渡到阻带 尖脉冲响应，而是呈现出逐步 滚降。”因此，正如在模拟滤波器设计5.4.1节中所述的问题情况下，在通带数字滤波器和 阻带幅频响应规格给予一些可接受的公差。止匕外，指定一个过渡带之间的通带和阻带 允许的幅度下降顺利。例如，一个低通滤波器的幅度可能得到如图7.1所示。正如在图中定义的通带0,我们要求的幅度接近同一
43、个，即错误的团结，0在界定的阻带，我们要求的幅度接近零与一的错误。大肠杆菌， 为。的频率，并分别被称为通带边缘频率和阻带边缘频率。在通带和阻带，并且，公差的 限制，通常称为峰值纹波值。请注意，数字滤波器的频率响应是周期函数，以及幅度 响应的实时数字滤波器系数是一个偶函数的。因此，数字滤波规格只给出了范围。数字滤波器的规格，常常给在功能上的损失分贝，。在这里，通带纹波和峰值最小阻 带衰减给出了分贝，也就是说，数字滤波器，给出的损失规格09.1 初步设想正如在一个模拟低通滤波器的情况下，一个数字低通滤波器的规格可能或者给予其规 模在反应方面，如图7.2。在这里，在通带内规模最大的价值被假定为团结，
44、最大通 带偏差，表示为1/,是由通带中的最低值所规模。阻带的最大震级是指由 1/答 对于标准化规格，增益功能或损失函数的最小值最大值，因此。分贝。给予的数量被称为最大通带衰减。1,由于通常情况下，它可以证明通带和阻带边缘频率在大多数应用中，被指定为Hz,随着数字滤波器的采样率。由于所有的过滤器设计技术的规范化发展和角频率来看，临界频率的sepcified之前需要一个特定的过滤器设计算法可以应用于正常化。让表示，在赫兹采样频率，计划生育和Fs分别表示，在通带和阻带的边缘在赫兹频率。然后正常化弧度角频率都是通 过边9.1.2 过滤器类型的选择利息的第二个问题是数字滤波器的类型，即选择，无论是原居民
45、或 FIR数字滤波器 将被雇用。数字滤波器的设计目标是建立一个因果传递函数 H (z)的频率响应规格 会议。对于IIR数字滤波器的设计，即原传递函数是一个真正合理的功能。的H (z)的二此外，高(z)的必须是一个稳定的传输功能，并减少了计算的复杂性，它必须以最 低的全是另一方面，对FIR滤波器的设计，区传递函数是一个多项式：为了降低计算复杂度，n次的H (z)的，必须尽可能的小。止匕外，如果是理想的线 性相位，然后将FIR滤波器系数必须满足的约束：所以采用FIR滤波器的几个优点，因为它可以被设计成精确线性相位滤波器的结构和量化滤波器系数总是与稳定。然而，在大多数情况下，为了 NFIR 一个FI
46、R 滤波器是大大高于同等IIR滤波器会议同样大小的规格为 NIIR高。在一般情况下， FIR滤波器的实现需要每个输出样本约 NFIR乘法，而每IIR滤波器2NIIR 输出示 例乘法要求。在前者情况下，如果 FIR滤波器的设计与线性阶段，那么每个输出的 采样乘法次数减少到大约(NFIR +1 ) / 2。同样，多数IIR滤波器的设计结果与单位 圆上的传递函数零，而级联的IIR滤波器实现秩序与单位圆上的零点都需要(3 +3)/ 2乘法每个输出样本。它已被证明是最实用的过滤器的规格，比NFIR / NIIR 通常 为几十或更多的订单，并作为结果，计算 IIR滤波器通常是更有效Rab75。但是， 如果
47、IIR滤波器的群延迟是由全通均衡器级联与它扳平，然后在计算储蓄可能不再是 显着Rab75。在许多应用中，该数字滤波器的相位响应线性不是问题，使 IIR滤波 器因为较低的计算要求可取。9.1.3 数字滤波器设计的基本方法在IIR滤波器的设计中，最常见的做法是将其转换成模拟低通原型滤波器规格的数字过滤器的规格，然后转换成所需的数字滤波器的传递函数的G (z)的。这种方法已广泛应用于许多原因：(a)模拟技术是非常先进的逼近。(b)他们通常产量封闭形式的解决方案。(c)广泛用于模拟表滤波器设计提供。(d)许多应用需要模拟滤波器数字仿真。在续集中，我们记一个模拟的传递函数为其中，下标 人”明确表示模拟域。数字传递函数导出的形式下(s)是由记背后的传递函数模拟原型哈(s)转换成数字原居民的基本思想传递函数 G (z)是一 个适用于从S -域映射到Z域，使模拟频率的基本属性响应将被保留。在暗示，映射 函数应该是这样的：虚(j)在s平面轴映射到的Z平面圆。一个稳定的信号传递函数转化为一个稳定的数字传输功能。为此，使用最广泛的变革是双线性变换在 9.2节中所述。不像IIR数字滤波器设计，FIR滤波器的设计没有任何的模拟滤波器的设计 连接。