In Chapter 2, we discussed the analysis of discrete-time systems to obtain their output due to a given input sequence in the time domain, using recursive algorithm, convolution, and the *z*-transform technique. In Chapter 3, we introduced the concept of their response in the frequency domain, by deriving the DTFT or the frequency response of the system. These two chapters and Chapter 1 were devoted to the analysis of DT systems. Now we discuss the synthesis of these systems, when their transfer functions or their equivalent models are given. If we are given the input–output sequence, it is easy to find the transfer function *H*(*z*) as the ratio of the *z* transform of the output to the *z* transform of the input. If, however, the frequency response of the system is specified, in the form of a plot, such as when the passband and stopband frequencies along with the magnitude and phase over these bands, and the tolerances allowed for these specifications, are specified, finding the transfer function from such specifications is based on approximation theory. There are many well-known methods for finding the transfer functions that approximate the specifications given in the frequency domain. In this chapter, we will discuss a few methods for the design of IIR filters that approximate the magnitude response specifications for lowpass, highpass, bandpass, and bandstop filters. Usually the specifications for a digital filter are given ...

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