IIR Digital Filter DesignAn 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 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 structures 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 filter 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 achieved 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 considerationsThere 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 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 that meets the prescribed specifications. We also 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, the unitsample 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 corrected 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 filters,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 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 analog 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 drop off smoothly. For example, the magnitude )(ωj e G of a lowpass filter may be given as shown in Figure 7.1. As indicatedin the figure, in the passband defined by 0p ωω≤≤, we require that the magnitude approximates unity with an error of p δ±,i.e.,p p j p for e G ωωδδω≤+≤≤-,1)(1.In the stopband, defined by πωω≤≤s ,we require that the magnitude approximates zero with an error of i s ,δ.e.,,)(s j e G δω≤for πωω≤≤s .The frequencies p ω and s ωare , respectively, called the passband edge frequency and the stopband edge frequency. The limits of the tolerances in the passband and stopband, p δands δ, are usually called the peak ripple values. Note that the frequency response )(ωj e G of a digital filter is a periodic function ofω,and the magnitude response of a real-coefficient digital filter is an even function ofω. As a result, the digital filter specifications are given only for the range πω≤≤0.Digital filter specifications are often given in terms of the loss function,)(log 20)(10ωωζj e G -=, in dB. Here the peak passband ripple p α and the minimum stopband attenuation s α are given in dB,i.e., the loss specifications of a digital filter are given bydB p p )1(log 2010δα--=,dB s s )(log 2010δα-=.9.1 Preliminary ConsiderationsAs in the case of an analog lowpass filter, the specifications 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 1/21ε+,is given by the minimum value of the magnitude in the passband. The maximum stopband magnitude is denoted by 1/A.For the normalized specification, the maximum value of the gain function or the minimum value of the loss function is therefore 0 dB. The quantity max α given by dB )1(log 20210max εα+=Is called the maximum passband attenuation. For p δ<<1, as is typically the case, it can be shown thatp p αδα2)21(log 2010m ax ≅--≅The passband and stopband edge frequencies, in most applications, 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 p ωand s ω,thesepcified critical frequencies need to be normalized before a specific filter design algorithm can be applied.Let T F denote the sampling frequency in Hz, and F P and F s denote, respectively,thepassband and stopband edge frequencies in Hz. Then the normalized angular edge frequencies in radians are given byT F F F F p T p T p p ππω22==Ω= T F F F F s Ts T s s ππω22==Ω= 9.1.2 Selection of the Filter TypeThe second issue of interest is the selection of the digital filter type,i.e.,whether an IIR 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 1-z . H(z)=NMdNz z d z d d pMz z p z p p ------++++++++......2211022110 Moreover, H(z) must be a stable 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 1-z : ∑=-=Nn n z n h z H 0][)(For 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:][][N n h n h -±=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 N FIR of an FIR filter is considerably higher than the order N IIR of an equivalent IIR filter meeting the same magnitude specifications. In general, the implementation of the FIR filter requires approximately N FIR multiplications per output sample, whereas the IIR filter requires 2N IIR +1 multiplications per output sample. In theformer case, if the FIR filter is designed with a linear phase, then the number of multiplications per 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 IIR N with all of the zeros on the unit circle requires [(3IIR N +3)/2] multiplications per 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 efficient[Rab75]. 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,making the IIR filter preferable because of the lower computational requirements.9.1.3 Basic Approaches 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 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 require the digital simulation of analog filters. In the sequel, we denote an analog transfer function as)()()(s D s P s H a a a =, Where the subscript "a" specifically indicates the analog domain. The digital transfer function derived form H a (s) is denoted by)()()(z D z P z G = The basic idea behind the conversion of an analog prototype transfer function H a (s) into a digital IIR transfer function G(z) is to apply a mapping from the s-domain to thez-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 Ω) axis in the s-plane be mapped 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 have 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 phase 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∑=-=Nn n j j e n h e H 0][)(ωω. It has been shown in Section 3.2.1 that any finite duration sequence x[n] of length N+1 is completely characterized by N+1 samples of its discrete-time Fourier transfer X(ωj e ). As a result, the design of an FIR filter of length N+1 may be accomplished by finding either the impulse response sequence {h[n]} or N+1 samples of its frequency response )H(e j ω. Also, to ensure a linear-phase design, the condition of Eq.(7.11) 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 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：SanjitK.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。