wordIIR Digita 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 plex variable z by a function of z. Four monly used transformations are summarized. Finally we consider the puter-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 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 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. 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 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 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 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 , 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 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 approximates 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 filter 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 passband 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, t