1 s of Objetive - Transform a given lowpass digital transfer funtion G L ( to another digital transfer funtion G D ( that ould be a lowpass, highpass, bandpass or bandstop filter z has been used to denote the unit delay in the prototype lowpass filter G L ( and z to denote the unit delay in the transformed filter G D ( to avoid onfusion 2 s of Unit irles in z- and z -planes defined by jω, ω z e z e j Transformation from z-domain to z -domain given by z F( Then G ( G { F( } D L s of From z F(, thus z F(, hene > if z > 1 F( z ) if z 1 < if z < 1 Reall that a ausal stable allpass funtion A( satisfies the ondition 4 s of < if z > 1 A( if z 1 > if z < 1 Therefore 1/ F( must be a ausal stable allpass funtion whose general form is 1 L 1 α < 1 ( ) α* ± l z, l F z l 1 z αl 5 To transform a lowpass filter G L ( with a utoff frequeny ω to another lowpass filter G D ( with a utoff frequeny ω, the transformation is 1 1 λ z z F( z λ where λ is a funtion of the two speified utoff frequenies 6 On the unit irle we have jω jω e e λ jω λe From the above we get jω jω jω e λ e m1 e m1 m1 (1 ± λ) jω jω λe λe Taking the ratios of the above two expressions tan( ω 1+ λ tan( ω 1 λ 1
7 sin( ( ω ω ) λ sin( ( ω + ω ) Solving we get Example- Consider the lowpass digital filter ( 662(1 + z ) (1 259z )(1 676z + 917z 2 ) whih has a passband from d to 25π with a 5 db ripple Redesign the above filter to move the passband edge to 5π G L 8 Here sin( 5π) λ 194 sin( π) Hene, the desired lowpass transfer funtion is G ( z ) G ( z + 194 D -1-2 - L G L ( 1+ 194 z G D ( -4 2 4 6 8 1 ω/π z 9 The lowpass-to-lowpass transformation 1 1 λ z z F( z λ an also be used as highpass-to-highpass, bandpass-to-bandpass and bandstop-tobandstop transformations 1 Desired transformation z +λ z 1+ λz The transformation parameter λ is given by os( ( ω + ω ) λ os( ( ω ω ) where ω is the utoff frequeny of the lowpass filter and ω is the utoff frequeny of the desired highpass filter 11 Example-Transform the lowpass filter 662(1 + z ) (1 259z )(1 676z + 917z G ( L 2 with a passband edge at 25π to a highpass filter with a passband edge at 55π Hereλ os( 4π) / os( 15π) 468 The desired transformation is z 468 z 468z ) 12 The desired highpass filter is GD( z ) G( z 468 z 468 z 2 4 6 8 2π 4π 6π 8π π Normalized frequeny 2
The lowpass-to-highpass transformation an also be used to transform a highpass filter with a utoff at ω to a lowpass filter with a utoff at ω and transform a bandpass filter with a enter frequeny at ω o to a bandstop filter with a enter frequeny at ω o to-bandpass Desired transformation z 2 2λρ ρ z z + ρ + 1 ρ + 1 ρ 2 2λρ z z + 1 ρ + 1 ρ + 1 1 14 15 to-bandpass The parameters λ and ρ are given by os( ( ω2 + ω 1) λ os( ( ω2 ω1) ρ ot( ( ω2 ω1) tan( ω where ω is the utoff frequeny of the lowpass filter, and ω1 and ω 2 are the desired upper and lower utoff frequenies of the bandpass filter 16 to-bandpass Speial Case - The transformation an be simplified if ω ω2 ω1 Then the transformation redues to z λ z z λ z where λ os( ω with ω o ) o denoting the desired enter frequeny of the bandpass filter 17 to-bandstop Desired transformation 2 2λ ρ z z + 1+ ρ 1+ ρ z ρ 2 2λ z z + 1+ ρ 1+ ρ 1 18 to-bandstop The parameters λ and ρ are given by os( ( ω2 + ω 1) λ os( ( ω2 ω1) ρ tan( ( ω2 ω1) tan( ω ω where is the utoff frequeny of the lowpass filter, and ω1 and ω 2 are the desired upper and lower utoff frequenies of the bandstop filter
19 Funtion Using The allpass funtion needed for the spetral transformation from a speified lowpass transfer funtion to a desired highpass or bandpass or bandstop transfer funtion an be generated using 2 Funtion Using -Highpass Transformation Basi form: [AllpassNum,AllpassDen] allpasslp2hp(wold,wnew) where wold is the speified angular bandedge frequeny of the original lowpass filter, and wnew is the desired angular bandedge frequeny of the highpass filter 21 Funtion Using -Bandpass Transformation Basi form: [AllpassNum,AllpassDen] allpasslp2bp(wold,wnew) where wold is the speified angular bandedge frequeny of the original lowpass filter, and wnew is the desired angular bandedge frequeny of the bandpass filter 22 Funtion Using -Bandstop Transformation Basi form: [AllpassNum,AllpassDen] allpasslp2bs(wold,wnew) where wold is the speified angular bandedge frequeny of the original lowpass filter, and wnew is the desired angular bandedge frequeny of the bandstop filter 2 Funtion Using -Highpass Example wold 25π, wnew 55π The statement [APnum, APden] allpasslp2hp(25, 55) yields the mapping z + 468 z 468z + 1 24 Using The pertinent M-files are iirlp2lp, iirlp2hp, iirlp2bp, and iirlp2bs -Highpass Example 66( 1+ z ) G LP ( 2 95z + 5669z 115z Passband edge wold 25π Desired passband edge of highpass filter wnew 55π 4
25 Using The ode fragments used are b 66*[1 1]; a [1. -95 5669-115]; [num,den,apnum,apden] iirlp2hp(b,a,25,55); The desired highpass filter obtained is 218( z ) G HP ( 2 521z + 661z 29z 26 Order Estimation - For IIR filter design using bilinear transformation, statements to determine the order and bandedge are: [N, Wn] buttord(wp, Ws, Rp, Rs); [N, Wn] heb1ord(wp, Ws, Rp, Rs); [N, Wn] heb2ord(wp, Ws, Rp, Rs); [N, Wn] ellipord(wp, Ws, Rp, Rs); 27 Example- Determine the minimum order of a Type 2 Chebyshev digital highpass filter with the following speifiations: Fp 1kHz, Fs 6 khz, F T 4 khz, α p 1dB, α s 4 db Here, Wp 2 1/ 4 5, Ws 2 6/ 4 Using the statement [N,Wn] heb2ord(5,,4); we get N 5 and Wn 224 28 Filter Design - For IIR filter design using bilinear transformation, statements to use are: [b,a] butter(n,wn) [b,a] heby1(n,rp,wn) [b,a] heby2(n,rs,wn) [b,a] ellip(n,rp,rs,wn) 29 The form of transfer funtion obtained is B( b(1) + b(2) z + L+ b( N + 1) z G( N A( 1+ a(2) z + L+ a( N + 1) z The frequeny response an be omputed using the M-file freqz(b,a,w) where w is a set of speified angular frequenies It generates a set of omplex frequeny response samples from whih magnitude and/or phase response samples an be omputed N Example-Design an ellipti IIR lowpass filter with the speifiations: F p 8 khz, Fs 1 khz, FT 4 khz, α p 5dB, α s 4 db Here, ωp 2πFp / FT 4π, ωs 2πFs / FT 5π Code fragments used are: [N,Wn]ellipord(4,5,5,4; [b,a]ellip(n,5,4,wn); 5
Gain response plot is shown below: Ellipti IIR Lowpass Filter -1 Passband Details -2-4 -2 - -4-6 2 4 6 8 1 ω/π -5 1 2 4 ω/π 1 6