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1、IIR Digital Filter Design Sanjit K.Mitra国籍：USA出处：Digital Signal Processing -A Computer-Based Approach 3e An important step in the development of a digital filter is the determination of a realizable transfer function G(z) approximating the given frequency response specifications. If an IIR filter is

2、 desired,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 struct

3、ures for 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 filte

4、r design based 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 ach

5、ieved by replacing 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 co

6、nsiderations 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

7、is to be employed. 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 then consider the determination of the filter order th

8、at meets the prescribed specifications. We also discuss appropriate scaling of the transfer function. Digital Filter Specifications As 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

9、 situations, 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 , the phase response of the designed filter can be co

10、rrected 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 that there are four basic types of f

11、ilters,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 a

13、nalog filter design problem outlined in section , 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 drop

14、 off smoothly. For example, the magnitude of a lowpass filter may be given as shown in Figure 7.1. As indicated in the figure, in the passband defined by 0, we require that the magnitude approximates unity with an error of ,i.e., .In the stopband, defined by ,we require that the magnitude approximat

15、es zero with an error of .e., for .The frequencies and are , respectively, called the passband edge frequency and the stopband edge frequency. The limits of the tolerances in the passband and stopband, and , are usually called the peak ripple values. Note that the frequency response of a digital fil

16、ter is a periodic function of ,and the magnitude response of a real-coefficient digital filter is an even function of . As a result, the digital filter specifications are given only for the range . Digital filter specifications are often given in terms of the loss function, in dB. Here the peak pass

17、band ripple and the minimum stopband attenuation are given in dB,i.e., the loss specifications of a digital filter are given by , . 9.1 Preliminary Considerations As in the case of an analog lowpass filter, the specifications for a digital lowpass filter may alternatively be given in terms of its ma

18、gnitude 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/,is given by the minimum value of the magnitude in the passband. The maximum stopband magnitude is denoted by 1/A. For the normalized s

19、pecification, the maximum value of the gain function or the minimum value of the loss function is therefore 0 dB. The quantity given by Is called the maximum passband attenuation. For 1, as is typically the case, it can be shown that The passband and stopband edge frequencies, in most applications,

20、are specified in Hz, along with the sampling rate of the digital filter. Since all filter design techniques are developed in terms of normalized angular frequencies and ,the sepcified critical frequencies need to be normalized before a specific filter design algorithm can be applied. Let denote the

21、sampling frequency in Hz, and FP and Fs denote, respectively,the passband and stopband edge frequencies in Hz. Then the normalized angular edge frequencies in radians are given by Selection of the Filter Type The second issue of interest is the selection of the digital filter type,i.e.,whether an II

22、R or an FIR digital filter is to be employed. The objective of digital filter design is to develop a causal transfer function H(z) meeting the frequency response specifications. For IIR digital filter design, the IIR transfer function is a real rational function of . H(z)=Moreover, H(z) must be a st

23、able transfer function, and for reduced computational complexity, it must be of lowest order N. On the other hand, for FIR filter design, the FIR transfer function is a polynomial in : For reduced computational complexity, the degree N of H(z) must be as small as possible. In addition, if a linear p

24、hase is desired, then the FIR filter coefficients must satisfy the constraint: T here are several advantages in using 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 NFIR o

25、f an FIR filter is considerably higher than the order NIIR of an equivalent IIR filter meeting the same magnitude specifications. In general, the implementation of the FIR filter requires approximately NFIR multiplications per output sample, whereas the IIR filter requires 2NIIR +1 multiplications p

26、er output sample. In the former case, if the FIR filter is designed with a linear phase, then the number of multiplications per output sample reduces to approximately (NFIR+1)/2. Likewise, most IIR filter designs result in transfer functions with zeros on the unit circle, and the cascade realization

27、 of an IIR filter of order with all of the zeros on the unit circle requires (3+3)/2 multiplications per output sample. It has been shown that for most practical filter specifications, the ratio NFIR/NIIR is typically of the order of tens or more and, as a result, the IIR filter usually is computati

28、onally more efficientRab75. However ,if the group delay of the IIR filter is equalized by cascading it 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,m

29、aking the IIR filter preferable because of the lower computational requirements. Basic Approaches to Digital Filter Design In 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 trans

30、form it into the desired digital filter transfer function G(z). This approach has been widely used 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 r

31、equire the digital simulation of analog filters.In the sequel, we denote an analog transfer function as ,Where the subscript a specifically indicates the analog domain. The digital transfer function derived form Ha(s) is denoted by The basic idea behind the conversion of an analog prototype transfer

32、 function Ha(s) into a digital IIR transfer function G(z) is to apply a mapping from the s-domain to the 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 map

33、ped onto the circle of the z-plane.(b) A stable analog transfer function be transformed into a stable digital transfer function.To this end,the most widely used transformation is the bilinear transformation described in Section 9.2. Unlike IIR digital filter design,the FIR filter design does not hav

34、e any connection with the design of analog filters. The design of FIR filter design does not have any connection with the design of analog 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 pha

35、se response be linear. As pointed out in Eq.(7.10), a causal FIR transfer function H(z) of length N+1 is a polynomial in z-1 of degree N. The corresponding frequency response is given by .It has been shown in Section that any finite duration sequence xn of length N+1 is completely characterized by N

36、+1 samples of its discrete-time Fourier transfer X(). As a result, the design of an FIR filter of length N+1 may be accomplished by finding either the impulse response sequence hn or N+1 samples of its frequency response . Also, to ensure a linear-phase design, the condition of Eq.(7.11) must be sat

37、isfied. Two direct approaches to the design of FIR filters are the windowed Fourier series approach and the frequency 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 metho

38、ds. Sanjit K.Mitra国籍：USA出处：Digital Signal Processing -A Computer-Based Approach 3eIIR数字滤波器的设计 在一个数字滤波器开展的重要步骤是可实现的传递函数Gz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Gz是稳定的。该推算传递函数Gz的过程称为数字滤波器的设计。然后Gz有所值，下一步就是实现在一个适宜的过滤器结构形式。在第8章，我们概述了为转移的FIR和IIR的各种功能的实现根本结构。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

39、忆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滤波器的根底上，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到达了一个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答复可以开展数字传递函数Gz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开展。第二个问题是要确定的FIR或IIR数字滤波器是

40、设计。在一节中，我们首先检查了这两个问题。接下来，我们回忆到的IIR数字滤波器设计的根本分析方法，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。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标准开展。如第所示，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，在第节指出，

41、有四个过滤器，其大小，如图4.10所示的反响根本类型。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。一个开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。此外，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个，即错误的团

42、结， 。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，。9.1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响方面，如图7.2。在这里，在通带内规模最大的价值被假定为团结，最大通带偏差，表示为

43、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标准化开展和角频率来看，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，方案生育和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，无论是原居民或FIR数字滤波器将被雇用。数字滤波器

44、的设计目标是建立一个因果传递函数Hz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Hz的=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Hz的，必须尽可能的小。此外，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FIR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

45、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一输出例如乘法要求。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

46、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。数字滤波器设计的根本方法在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Gz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“一明确表示模拟域。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根本思想传递函数Gz是一个适用于从S -域映射到Z域，使模拟频率的根本属性响应将被保存。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。