Discrete-TimeIIRFilterDesignfromContinuous-TimeFilters

Click to edit Master title style,Click to edit Master text styles,Second level,Third level,Fourth level,Fifth level,Copyright (C) 2005 Gner Arslan,*,Discrete-Time IIR Filter Design from Continuous-Time Filters,Quote of the Day,Experience is the name everyone gives to their mistakes.,Oscar Wilde,Content and Figures are from Discrete-Time Signal Processing, 2e by Oppenheim, Shafer, and Buck, 1999-2000 Prentice Hall Inc.,Filter Design Techniques,Any discrete-time system that modifies certain frequencies,Frequency-selective filters pass only certain frequencies,Filter Design Steps,Specification,Problem or application specific,Approximation of specification with a discrete-time system,Our focus is to go from spec to discrete-time system,Implementation,Realization of discrete-time systems depends on target technology,We already studied the use of discrete-time systems to implement a continuous-time system,If our specifications are given in continuous time we can use,D/C,x,c,(t),y,r,(t),C/D,H(e,j,),Filter Specifications,Specifications,Passband,Stopband,Parameters,Specs in dB,Ideal passband gain =20log(1) = 0 dB,Max passband gain = 20log(1.01) = 0.086dB,Max stopband gain = 20log(0.001) = -60 dB,Butterworth Lowpass Filters,Passband is designed to be maximally flat,The magnitude-squared function is of the form,Chebyshev Filters,Equiripple in the passband and monotonic in the stopband,Or equiripple in the stopband and monotonic in the passband,Filter Design by Impulse Invariance,Remember impulse invariance,Mapping a continuous-time impulse response to discrete-time,Mapping a continuous-time frequency response to discrete-time,If the continuous-time filter is bandlimited to,If we start from discrete-time specifications T,d,cancels out,Start with discrete-time spec in terms of,Go to continuous-time,= /T and design continuous-time filter,Use impulse invariance to map it back to discrete-time = T,Works best for bandlimited filters due to possible aliasing,Impulse Invariance of System Functions,Develop impulse invariance relation between system functions,Partial fraction expansion of transfer function,Corresponding impulse response,Impulse response of discrete-time filter,System function,Pole s=s,k,in s-domain transform into pole at,Example,Impulse invariance applied to Butterworth,Since sampling rate T,d,cancels out we can assume T,d,=1,Map spec to continuous time,Butterworth filter is monotonic so spec will be satisfied if,Determine N and,c,to satisfy these conditions,Example Contd,Satisfy both constrains,Solve these equations to get,N must be an integer so we round it up to meet the spec,Poles of transfer function,The transfer function,Mapping to z-domain,Example Contd,Filter Design by Bilinear Transformation,Get around the aliasing problem of impulse invariance,Map the entire s-plane onto the unit-circle in the z-plane,Nonlinear transformation,Frequency response subject to warping,Bilinear transformation,Transformed system function,Again T,d,cancels out so we can ignore it,We can solve the transformation for z as,Maps the left-half s-plane into the inside of the unit-circle in z,Stable in one domain would stay in the other,Bilinear Transformation,On the unit circle the transform becomes,To derive the relation between, and ,Which yields,Bilinear Transformation,Example,Bilinear transform applied to Butterworth,Apply bilinear transformation to specifications,We can assume T,d,=1 and apply the specifications to,To get,Example Contd,Solve N and,c,The resulting transfer function has the following poles,Resulting in,Applying the bilinear transform yields,Example Contd,