Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.341:DISCRETE-TIME SIGNAL PROCESSING OpenCourseWare 2006 Lecture 8 DT Filter Design:IIR.Filters Reading:Section 7.1 in Oppenheim,Schafer Buck (OSB). In the last lecture we studied various forms of filter realizations.Today we will take one step back,focusing our attention on determining the actual transfer functions to be realized. Given specifications of the desired properties of the system,we approximate the specifications using a causal discrete time system.In this lecture,we will show how IIR systems can be approximated by rational functions of z.Next time we will look at FIR system approximation using polynomials of z. So far,we have tended to make reference only to ideal filters.In practical designs,ideal sys- tems can not be realized exactly,since the impulse responses are infinitely long and non-causal. The idea is to approximate the desired frequency response within certain error tolerances.As such,specifications are usually given as tolerance levels in different frequency bands in the range 0≤w≤π.OSB Figure7.2 gives a sample tolerance scheme.The transition band is a“don't care"region in which the filter gain can be any finite value.Notice that no requirements have been specified for the phase response of the system.Typical filter design procedures focus only on magnitude approximation: Hideal((e")=lH(e)leit）→lHef(e“l≈Hideal(=lHe“l. Nonetheless,specifications can involve both magnitude and phase(or group delay).Such gen- eralized approximation is a harder problem,but may be desired in specific applications.In particular,integer or fractional delays can only be achieved with FIR filters. Most of our following discussions will be phrased for piecewise constant LPF's.Keep in mind,however,that much applies more generally,since specific transformations can convert LPF's to HP,BP,or notch filters. IIR vs.FIR Given a set of specifications,first we need to decide if the desired filter should be IIR or FIR. The following table summarizes different factors that could be considered when making this decision: 1Massachusetts Institute of Technology Department of Electrical Engineering and Computer Science 6.341: Discrete-Time Signal Processing OpenCourseWare 2006 Lecture 8 DT Filter Design: IIR Filters Reading: Section 7.1 in Oppenheim, Schafer & Buck (OSB). In the last lecture we studied various forms of filter realizations. Today we will take one step back, focusing our attention on determining the actual transfer functions to be realized. Given specifications of the desired properties of the system, we approximate the specifications using a causal discrete time system. In this lecture, we will show how IIR systems can be approximated by rational functions of z. Next time we will look at FIR system approximation using polynomials of z. So far, we have tended to make reference only to ideal filters. In practical designs, ideal sys￾tems can not be realized exactly, since the impulse responses are infinitely long and non-causal. The idea is to approximate the desired frequency response within certain error tolerances. As such, specifications are usually given as tolerance levels in different frequency bands in the range 0 ≤ ω ≤ π. OSB Figure 7.2 gives a sample tolerance scheme. The transition band is a “don’t care” region in which the filter gain can be any finite value. Notice that no requirements have been specified for the phase response of the system. Typical filter design procedures focus only on magnitude approximation: Hideal(ejω) = |H(ejω) ejθ(ω) |Heff(ejω) Hideal(ejω) | = H(ejω | → | ≈ | | |) . Nonetheless, specifications can involve both magnitude and phase (or group delay). Such gen￾eralized approximation is a harder problem, but may be desired in specific applications. In particular, integer or fractional delays can only be achieved with FIR filters. Most of our following discussions will be phrased for piecewise constant LPF’s. Keep in mind, however, that much applies more generally, since specific transformations can convert LPF’s to HP, BP, or notch filters. IIR vs. FIR Given a set of specifications, first we need to decide if the desired filter should be IIR or FIR. The following table summarizes different factors that could be considered when making this decision: 1