Copyright 2001, S. K. MitraDesign Other Types of Analog FiltersHP,BP,BS analog filters are all based on their corresponding prototype LP filters design. 1Copyright 2001, S. K. MitraDesign of HighpassAnalog Filters2Specifications for HP analog filterHP filterTransfer function HHP()Specifications LP analog filterAnalog FilterTransfer function HLP(s)(1)(2)(3)?Copyright 2001, S. K. MitraDesign of HighpassAnalog Filters3Copyright 2001, S. K. MitraDesign of Other Types ofAnalog Filters4Copyright 2001, S. K. MitraDesign of Other Types ofAnalog Filters(low-pass to high-pass)5Copyright 2001, S. K. MitraDesign of HighpassAnalog Filters6Specifications for HP analog filter ( , ,)HP filterTransfer function HHP()Specifications LP analog filter ( , ,)Analog FilterTransfer function HLP(s)(1)(2)(3)Copyright 2001, S. K. MitraAnalog Highpass Filter7Copyright 2001, S. K. MitraAnalog Highpass Filter利用转化后的指标设计低通滤波器低通高通将高通滤波器的指标转化成低通滤波器的指标8Copyright 2001, S. K. MitraDigital Filter Design: Basic Approaches9Specifications in terms of DTFT (wp,ws,A)IIR digital filterTransfer function G(z)Specifications in terms of CTFT (p, s,A)Analog FilterTransfer function Ha(s)(1)(2)(3)?Copyright 2001, S. K. MitraIIR Digital Filter Design: Basic IdeaBasic idea behind the conversion of into is to apply a mapping from the s-domain to the z-domain so that essential properties of the analog frequency response are preserved10Copyright 2001, S. K. MitraRequirements of the MappingThus mapping function should be such that:Imaginary axis in the s-plane be mapped onto the unit circle of the z-planeA stable analog transfer function be mapped into a stable digital transfer function11Copyright 2001, S. K. MitraIIR Digital Filter Design: Bilinear Transformation MethodBilinear transformation Above transformation maps a single point in the s-plane to a unique point in the z-plane and vice-versa12Copyright 2001, S. K. MitraBilinear TransformationRelation between G(z) and is then given byIn general, the parameter T has no effect on G(z) and T = 2 is chosen for convenience13Copyright 2001, S. K. MitraBilinear TransformationInverse bilinear transformation for T = 2 isFor Thus, 14Copyright 2001, S. K. MitraBilinear TransformationMapping of s-plane into the z-plane15Copyright 2001, S. K. MitraIIR Digital Filter Design Using Bilinear TransformationExample - ConsiderApplying bilinear transformation to the above we get the transfer function of a first-order digital lowpass Butterworth filter16Copyright 2001, S. K. MitraIIR Digital Filter Design Using Bilinear TransformationRearranging terms we getwhere17Copyright 2001, S. K. MitraDigital Filter Design: Basic Approaches18Specifications in terms of DTFT (wp,ws,A)IIR digital filterTransfer function G(z)Specifications in terms of CTFT (p, s,A)Analog FilterTransfer function Ha(s)(1)(2)(3)?Bilinear mappingCopyright 2001, S. K. MitraBilinear TransformationFor with T = 2 we have or 19Copyright 2001, S. K. MitraBilinear TransformationMapping is highly nonlinear Complete negative imaginary axis in the s-plane from to is mapped into the lower half of the unit circle in the z-plane from to Complete positive imaginary axis in the s-plane from to is mapped into the upper half of the unit circle in the z-plane from to 20Copyright 2001, S. K. MitraBilinear TransformationNonlinear mapping introduces a distortion in the frequency axis called frequency warpingEffect of warping shown below21Copyright 2001, S. K. MitraSteps in the design of a digital filter22Specifications in terms of DTFT (wp,ws,A)IIR digital filterTransfer function G(z)Specifications in terms of CTFT (p, s,A)Analog FilterTransfer function Ha(s)(1)(2)(3)Bilinear mapping必考！必考！Copyright 2001, S. K. MitraBilinear TransformationSteps in the design of a digital filter:(1) Prewarp to find their analog equivalents by (2) Design the analog filter(3) Design the digital filter G(z) by applying bilinear transformation toTransformation does not preserve phase response of analog filter23Copyright 2001, S. K. MitraIIR Lowpass Digital Filter Design Using Bilinear TransformationExample - Design a lowpass Butterworth digital filter with , , dB, and dBThus24No ripples in passbandCopyright 2001, S. K. MitraIIR Lowpass Digital Filter Design Using Bilinear TransformationPrewarping we getThe inverse transition ratio isThe inverse discrimination ratio is25Copyright 2001, S. K. MitraIIR Lowpass Digital Filter Design Using Bilinear TransformationThusWe choose N = 3To determine we use 26Copyright 2001, S. K. MitraIIR Lowpass Digital Filter Design Using Bilinear TransformationWe then get3rd-order lowpass Butterworth transfer function for is27Copyright 2001, S. K. MitraIIR Lowpass Digital Filter Design Using Bilinear TransformationApplying bilinear transformation to we get the desired digital transfer functionMagnitude and gain responses of G(z) shown below:28Copyright 2001, S. K. MitraSpectral Transformations of IIR Digital FiltersObjective - Transform a given lowpass digital transfer function to another digital transfer function that could be a lowpass, highpass, bandpass or bandstop filter has been used to denote the unit delay in the prototype lowpass filter and to denote the unit delay in the transformed filter to avoid confusion29Copyright 2001, S. K. MitraLowpass-to-Highpass Spectral TransformationDesired transformationThe transformation parameter is given by where is the cutoff frequency of the lowpass filter and is the cutoff frequency of the desired highpass filter30Copyright 2001, S. K. MitraLowpass-to-Highpass Spectral TransformationExample - Transform the lowpass filter with a passband edge at to a highpass filter with a passband edge at HereThe desired transformation is31Copyright 2001, S. K. MitraLowpass-to-Highpass Spectral TransformationThe desired highpass filter is32Copyright 2001, S. K. MitraLowpass-to-Highpass Spectral TransformationThe lowpass-to-highpass transformation can also be used to transform a highpass filter with a cutoff at to a lowpass filter with a cutoff at and to transform a bandpass filter with a center frequency at to a bandstop filter with a center frequency at33Copyright 2001, S. K. MitraSteps in the design of a highpass IIR digital filter (1)34Specifications for highpass IIR digital filter (wp,ws,A)HP IIR digital filter Transfer function GHP(z)Specifications for highpass analog filter( , ,A)LP Analog FilterTransfer function HLP(s)(1)(2)(3)Bilinear mappingSpecifications for prototype LP analog filter ( , ,A)HP Analog FilterTransfer function HHP()(4)(5)Copyright 2001, S. K. MitraSteps in the design of a highpass IIR digital filter (2)35Specifications for highpass IIR digital filter (wp,ws,A)HP IIR digital filter Transfer function GHP()Specifications for highpass analog filter( , ,A)LP Analog FilterTransfer function HLP(s)(1)(2)(3)Bilinear mappingSpecifications for prototype LP analog filter ( , ,A)LP digital FilterTransfer function GLP(z)(4)(5)Copyright 2001, S. K. MitraLowpass-to-Lowpass Spectral TransformationTo transform a lowpass filter with a cutoff frequency to another lowpass filter with a cutoff frequency , the transformation iswhere a is a function of the two specified cutoff frequencies 36Copyright 2001, S. K. MitraLowpass-to-Lowpass Spectral TransformationOn the unit circle we have From the above we get Taking the ratios of the above two expressions37Copyright 2001, S. K. MitraLowpass-to-Lowpass Spectral TransformationSolving we getExample - Consider the lowpass digital filterwhich has a passband from dc to with a 0.5 dB rippleRedesign the above filter to move the passband edge to38Copyright 2001, S. K. MitraLowpass-to-Lowpass Spectral TransformationHereHence, the desired lowpass transfer function is39Copyright 2001, S. K. MitraLowpass-to-Lowpass Spectral TransformationThe lowpass-to-lowpass transformationcan also be used as highpass-to-highpass, bandpass-to-bandpass and bandstop-to-bandstop transformations40Copyright 2001, S. K. MitraLowpass-to-Bandpass Spectral TransformationDesired transformation 41Copyright 2001, S. K. MitraLowpass-to-Bandpass Spectral TransformationThe parameters and are given by where is the cutoff frequency of the lowpass filter, and and are the desired upper and lower cutoff frequencies of the bandpass filter42Copyright 2001, S. K. MitraLowpass-to-Bandpass Spectral TransformationSpecial Case - The transformation can be simplified if Then the transformation reduces towhere with denoting the desired center frequency of the bandpass filter43Copyright 2001, S. K. MitraLowpass-to-Bandstop Spectral TransformationDesired transformation44Copyright 2001, S. K. MitraLowpass-to-Bandstop Spectral TransformationThe parameters and are given by where is the cutoff frequency of the lowpass filter, and and are the desired upper and lower cutoff frequencies of the bandstop filter45