2 D ISCRETE T IME S IGNAL P ROCESSING O PPENHEIM S CHAFFER PHI 2003 3 D IGITAL

2 d iscrete t ime s ignal p rocessing o ppenheim s

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2. D ISCRETE T IME S IGNAL P ROCESSING , O PPENHEIM & S CHAFFER , PHI, 2003. 3. D IGITAL S IGNAL P ROCESSING , S. K. M ITRA , T ATA M C -G RAW H ILL , 2 ND E DITION , 2004.
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Digital Signal Processing 10EC52 Dept., of ECE/SJBIT Page 142 UNIT - 8 D ESIGN OF IIR FILTERS FROM ANALOG FILTERS (B UTTERWORTH AND C HEBYSHEV ) 8.1 Introduction A digital filter is a linear shift-invariant discrete-time system that is realized using finite precision arithmetic. The design of digital filters involves three basic steps: The specification of the desired properties of the system. The approximation of these specifications using a causal discrete-time system. The realization of these specifications using _nite precision arithmetic. These three steps are independent; here we focus our attention on the second step. The desired digital filter is to be used to filter a digital signal that is derived from an analog signal by means of periodic sampling. The speci_cations for both analog and digital filters are often given in the frequency domain, as for example in the design of low pass, high pass, band pass and band elimination filters. Given the sampling rate, it is straight forward to convert from frequency specifications on an analog _lter to frequency speci_cations on the corresponding digital filter, the analog frequencies being in terms of Hertz and digital frequencies being in terms of radian frequency or angle around the unit circle with
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Digital Signal Processing 10EC52 Dept., of ECE/SJBIT Page 143 the point Z=-1 corresponding to half the sampling frequency. The least confusing point of view toward digital filter design is to consider the filter as being specified in terms of angle around the unit circle rather than in terms of analog frequencies. Figure 7.1: Tolerance limits for approximation of ideal low-pass filter A separate problem is that of determining an appropriate set of specifications on the digital filter. In the case of a low pass filter, for example, the specifications often take the form of a tolerance scheme, as shown in Fig. 4.1 Many of the filters used in practice are specified by such a tolerance scheme, with no constraints on the phase response other than those imposed by stability and causality requirements; i.e., the poles of the system function must lie inside the unit circle. Given a set of specifications in the form of Fig. 7.1, the next step is to and a discrete time linear system whose frequency response falls within the prescribed tolerances. At this point the filter design problem becomes a problem in approximation. In the case of infinite impulse response (IIR) filters, we must approximate the desired frequency response by a rational function, while in the finite impulse response (FIR) filters case we are concerned with polynomial approximation.
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