Typs of Transfr Typs of Transfr Th tim-domain classification of an LTI digital transfr function squnc is basd on th lngth of its impuls rspons: - Finit impuls rspons (FIR) transfr function - Infinit impuls rspons (IIR) transfr function In th cas of digital transfr functions with frquncy-slctiv frquncy rsponss, thr ar two typs of classifications () Classification basd on th shap of th magnitud function H ( j ω) () Classification basd on th th form of th phas function θ(ω) Classification Basd on agnitud Charactristics On common classification is basd on an idal magnitud rspons A digital filtr dsignd to pass signal componnts of crtain frquncis without distortion should hav a frquncy rspons qual to on at ths frquncis, and should hav a frquncy rspons qual to zro at all othr frquncis Th rang of frquncis whr th frquncy rspons taks th valu of on is calld th passband Th rang of frquncis whr th frquncy rspons taks th valu of zro is calld th stopband 3 4 5 Frquncy rsponss of th four popular typs of idal digital filtrs with ral impuls rspons cofficints ar shown blow: Lowpass Highpass Bandpass Bandstop 6 Lowpass filtr: Passband - Stopband - Highpass filtr: Passband - Stopband - ω ω c < ω π ω c ω π ω < ω c Bandpass filtr: Passband - ωc ω ωc Stopband - ω < ωc and ω c < ω π Bandstop filtr: Stopband - ωc < ω < ωc Passband - ω ωc and ω c ω π ω c
7 ω ω c Th frquncis c, ωc, and ar calld th cutoff frquncis An idal filtr has a magnitud rspons qual to on in th passband and zro in th stopband, and has a zro phas vrywhr 8 Earlir in th cours w drivd th invrs DTFT of th frquncy rspons H ( j LP ω ) of th idal lowpass filtr: sin ω n h n c LP [ ], < n < πn W hav also shown that th abov impuls rspons is not absolutly summabl, and hnc, th corrsponding transfr function is not BIBO stabl 9 Also, h LP [n] is not causal and is of doubly infinit lngth Th rmaining thr idal filtrs ar also charactrizd by doubly infinit, noncausal impuls rsponss and ar not absolutly summabl Thus, th idal filtrs with th idal brick wall frquncy rsponss cannot b ralizd with finit dimnsional LTI filtr To dvlop stabl and ralizabl transfr functions, th idal frquncy rspons spcifications ar rlaxd by including a transition band btwn th passband and th stopband This prmits th magnitud rspons to dcay slowly from its maximum valu in th passband to th zro valu in th stopband orovr, th magnitud rspons is allowd to vary by a small amount both in th passband and th stopband Typical magnitud rspons spcifications of a lowpass filtr ar shown blow A causal stabl ral-cofficint transfr function H( is dfind as a boundd ral (BR) transfr function if H ( ) for all valus of ω Lt x[n] and y[n] dnot, rspctivly, th input and output of a digital filtr charactrizd by a BR transfr function H( with X ( ) and Y ( ) dnoting thir DTFTs
Thn th condition H ( ) implis that Y ( ) Intgrating th abov from π to π, and applying Parsval s rlation w gt n y[ n] X ( n ) x[ n] Thus, for all finit-nrgy inputs, th output nrgy is lss than or qual to th input nrgy implying that a digital filtr charactrizd by a BR transfr function can b viwd as a passiv structur If H ( ), thn th output nrgy is qual to th input nrgy, and such a digital filtr is thrfor a losslss systm 3 4 A causal stabl ral-cofficint transfr function H( with H ( ) is thus calld a losslss boundd ral (LBR) transfr function Th BR and LBR transfr functions ar th kys to th ralization of digital filtrs with low cofficint snsitivity Exampl Considr th causal stabl IIR transfr function H( K, < α < αz whr K is a ral constant Its squar-magnitud function is givn by H( K ) H( H( z ) z ( + α ) αcosω 5 6 Th maximum valu of H( ) is obtaind whn α cosω in th dnominator is a maximum and th minimum valu is obtaind whn α cosω is a minimum For α >, maximum valu of αcosω is qual to α at ω, and minimum valu is α at ω π Thus, for α >, th maximum valu of H( ) is qual to K /( α) at ω and th minimum valu is qual to K /( + α) at ω π On th othr hand, for α <, th maximum valu of αcosω is qual to α at ω π and th minimum valu is qual to α at ω 7 8 3
Hr, th maximum valu of H( ) is qual to K /( α) at ω π and th minimum valu is qual to K /( α) at ω Hnc, th maximum valu can b mad qual to by choosing K ± ( α), in which cas th minimum valu bcoms ( α) /( + α) Hnc, K H( αz < α < is a BR function for K ± ( α) Plots of th magnitud function for α ±.5 with valus of K chosn to mak H( a BR function ar shown on th nxt slid, 9 agnitud.8.6.4 K ±.5, α.5...4.6.8 Lowpass Filtr agnitud.8.6.4 K ±.5, α -.5...4.6.8 Highpass Filtr Dfinition An IIR transfr function A( with unity magnitud rspons for all frquncis, i.., A( ), for all ω is calld an allpass transfr function An-th ordr causal ral-cofficint allpass transfr function is of th form + d + d z d z z A z +... + + ( ) ± + + d z +... + d z + d z 3 If w dnot th dnominator polynomials of A ( as D (: + D ( + d z +... + d z + d z thn it follows that A ( can b writtn as: z D ( z ) A ( ± D ( Not from th abov that if z r jφ is a pol of a ral cofficint allpass transfr function, thn it has a zro at φ z j r 4 Th numrator of a ral-cofficint allpass transfr function is said to b th mirrorimag polynomial of th dnominator, and vic vrsa ~ W shall us th notation D ( to dnot th mirror-imag polynomial of a dgr- polynomial D (, i.., ~ D ( z D ( z ) 4
5 Th xprssion z D ( z ) A ( ± D ( implis that th pols and zros of a ralcofficint allpass function xhibit mirrorimag symmtry in th z-plan. +.8z +.4z + z 3 A3 ( +.4z +.8z.z 3 Imaginary Part.5.5 -.5 - -.5-3 Copyright Ral 5, Part S. K. itra 6 To show that A ( ) w obsrv that ( ) ( z D z A z ) ± D ( z ) Thrfor A Hnc A ( ( ) ( ) ( ) ( z D z z D z z A z ) D ( D ( z ) z j ω ) A ( A ( z ) Now, th pols of a causal stabl transfr function must li insid th unit circl in th z-plan Hnc, all zros of a causal stabl allpass transfr function must li outsid th unit circl in a mirror-imag symmtry with its pols situatd insid th unit circl Figur blow shows th principal valu of th phas of th 3rd-ordr allpass function. +.8z +.4z + z 3 A3 ( +.4z +.8z.z 3 Not th discontinuity by th amount of π in th phas θ(ω) Phas, dgrs 4 - Principal valu of phas 7 8-4..4.6.8 9 If w unwrap th phas by rmoving th discontinuity, w arriv at th unwrappd phas function θ c (ω) indicatd blow Not: Th unwrappd phas function is a continuous function of ω Phas, dgrs - -4-6 -8 Unwrappd phas -..4.6.8 3 Th unwrappd phas function of any arbitrary causal stabl allpass function is a continuous function of ω Proprtis ()A causal stabl ral-cofficint allpass transfr function is a losslss boundd ral (LBR) function or, quivalntly, a causal stabl allpass filtr is a losslss structur 5
3 ()Th magnitud function of a stabl allpass function A( satisfis: <, for z > A(, for z >, for z < (3)Lt τ(ω) dnot th group dlay function of an allpass filtr A(, i.., d dω c τ( ω) [ θ ( ω)] 3 Th unwrappd phas function θ c (ω) of a stabl allpass function is a monotonically dcrasing function of ω so that τ(ω) is vrywhr positiv in th rang < ω < π Th group dlay of an -th ordr stabl ral-cofficint allpass transfr function satisfis: π τ( ω) dω π 33 A Simpl Application A simpl but oftn usd application of an allpass filtr is as a dlay qualizr LtG( b th transfr function of a digital filtr dsignd to mt a prscribd magnitud rspons Th nonlinar phas rspons of G( can b corrctd by cascading it with an allpass filtr A( so that th ovrall cascad has a constant group dlay in th band of intrst 34 G( A( Sinc A( ), w hav G( ) A( ) G( ) Ovrall group dlay is th givn by th sum of th group dlays of G( and A( Exampl Figur blow shows th group dlay of a 4 th ordr lliptic filtr with th following spcifications: ωp. 3π, δ p db, 35 db Group dlay, sampls 5 5 δ s Original Filtr..4.6.8 Figur blow shows th group dlay of th original lliptic filtr cascadd with an 8 th ordr allpass sction dsignd to qualiz th group dlay in th passband Group dlay, sampls 3 5 5 5 Group Dlay Equalizd Filtr..4.6.8 35 36 6
Classification Basd on Phas Charactristics A scond classification of a transfr function is with rspct to its phas charactristics In many applications, it is ncssary that th digital filtr dsignd dos not distort th phas of th input signal componnts with frquncis in th passband Zro-Phas Transfr On way to avoid any phas distortion is to mak th frquncy rspons of th filtr ral and nonngativ, i.., to dsign th filtr with a zro phas charactristic Howvr, it is not possibl to dsign a causal digital filtr with a zro phas 37 38 Zro-Phas Transfr Zro-Phas Transfr 39 For non-ral-tim procssing of ral-valud input signals of finit lngth, zro-phas filtring can b vry simply implmntd by rlaxing th causality rquirmnt On zro-phas filtring schm is sktchd blow x[n] H( v[n] u[n] H( w[n] u[ n] v[ n], y[ n] w[ n] 4 It is asy to vrify th abov schm in th frquncy domain Lt X ( ), V ( j ω), U ( j ω), W ( j ω), and Y ( ) dnot th DTFTs of x[n], v[n], u[n], w[n], andy[n], rspctivly From th figur shown arlir and making us of th symmtry rlations w arriv at th rlations btwn various DTFTs as givn on th nxt slid Zro-Phas Transfr Zro-Phas Transfr 4 x[n] H( v[n] u[n] H( w[n] ( j ω u[ n] v[ n], y[ n] w[ n] V ) H ( ) X ( ), W ) H ( ) U ( ) ( j ω j ω ( j ω ( j ω j ω U ) V*( ), Y ) W*( ) Combining th abov quations w gt Y ( ) W*( ) H *( ) U *( ) H ( j ) V ( ) H*( ) H( ) X( * ω H ( ) X ( ) 4 ) Th function filtfilt implmnts th abov zro-phas filtring schm In th cas of a causal transfr function with a nonzro phas rspons, th phas distortion can b avoidd by nsuring that th transfr function has a unity magnitud and a linar-phas charactristic in th frquncy band of intrst 7
43 Zro-Phas Transfr Th most gnral typ of a filtr with a linar phas has a frquncy rspons givn by H ( ) D which has a linar phas from ω to ω π Not also H ( ) τ ( ω) D 44 Th output y[n] of this filtr to an input x[ n] A j ωn is thn givn by y[ n] A D n A ( n D) If x a (t) and y a (t) rprsnt th continuoustim signals whos sampld vrsions, sampld at t nt,arx[n] and y[n] givn abov, thn th dlay btwn x a (t) and y a (t) is prcisly th group dlay of amount D 45 If D is an intgr, thn y[n] is idntical to x[n], but dlayd by D sampls If D is not an intgr, y[n], bing dlayd by a fractional part, is not idntical to x[n] In th lattr cas, th wavform of th undrlying continuous-tim output is idntical to th wavform of th undrlying continuous-tim input and dlayd D units of tim 46 If it is dsird to pass input signal componnts in a crtain frquncy rang undistortd in both magnitud and phas, thn th transfr function should xhibit a unity magnitud rspons and a linar-phas rspons in th band of intrst 47 Figur blow shows th frquncy rspons if a lowpass filtr with a linar-phas charactristic in th passband 48 Sinc th signal componnts in th stopband ar blockd, th phas rspons in th stopband can b of any shap Exampl-Dtrmin th impuls rspons of an idal lowpass filtr with a linar phas rspons: H ( LP ) no,, < ω < ωc ω ω π c 8
49 Applying th frquncy-shifting proprty of th DTFT to th impuls rspons of an idal zro-phas lowpass filtr w arriv at sin ω n n h n c( o) LP [ ], < n < π( n n ) As bfor, th abov filtr is noncausal and of doubly infinit lngth, and hnc, unralizabl o 5 By truncating th impuls rspons to a finit numbr of trms, a ralizabl FIR approximation to th idal lowpass filtr can b dvlopd Th truncatd approximation may or may not xhibit linar phas, dpnding on th valu of chosn n o n o If w choos N/ with N a positiv intgr, th truncatd and shiftd approximation ^ sin ω n N h n c( / ) LP[ ], n N π( n N / ) will b a lngth N+ causal linar-phas FIR filtr Figur blow shows th filtr cofficints obtaind using th function sinc for two diffrnt valus of N Amplitud.6.4. N Amplitud.6.4. N 3 5 5 -. 4 6 8 Tim indx n -. 4 6 8 Tim indx n 53 Zro-Phas Rspons Bcaus of th symmtry of th impuls rspons cofficints as indicatd in th two figurs, th frquncy rspons of th truncatd approximation can b xprssd as: ^ H LP N ^ j n LP LP n ( ) h [ n] ω N/ H whr H LP(ω), calld th zro-phas rspons or amplitud rspons, is a ral function of ω ( ω) 54 inimum-phas and aximum- Considr th two st-ordr transfr functions: bz+ z z b, H ( < < + +, H (, a b a z+ a Both transfr functions hav a pol insid th unit circl at th sam location z a and ar stabl But th zro of H ( is insid th unit circl at z b, whras, th zro of H ( is at z situatd in a mirror-imag symmtry b 9
inimum-phas and aximum- Figur blow shows th pol-zro plots of th two transfr functions inimum-phas and aximum- Howvr, both transfr functions hav an idntical magnitud function as H( H( z ) H( H( z ) Th corrsponding phas functions ar H ( z ) H ( sin ω sin ω tan b+ cosω a+ cosω arg[ H( )] tan bsin ω sin ω tan + bcosω a+ cosω arg[ H( )] tan 55 56 inimum-phas and aximum- inimum-phas and aximum- 57 Figur blow shows th unwrappd phas rsponss of th two transfr functions for a.8 and b. 5 Phas, dgrs - - -3 H ( H ( -4..4.6.8 58 From this figur it follows that H ( has an xcss phas lag with rspct to H ( Th xcss phas lag proprty of H ( with rspct to H ( can also b xplaind by obsrving that w can writ bz + z + b bz + H ( z + a z + a z + b H ( z ) A ( z ) 59 inimum-phas and aximum- whr A ( ( bz + )/( z + b) is a stabl allpass function Th phas functions of H ( and H ( ar thus rlatd through arg[ H( )] arg[ H( )] + arg[ A( )] As th unwrappd phas function of a stabl first-ordr allpass function is a ngativ function of ω, it follows from th abov that H ( has indd an xcss phas lag with rspct to H( 6 inimum-phas and aximum- Gnralizing th abov rsult, lt H m ( b a causal stabl transfr function with all zros insid th unit circl and lt H( b anothr causal stabl transfr function j satisfying ( ω H ) Hm( ) Ths two transfr functions ar thn rlatd through H ( Hm( A( whr A( is a causal stabl allpass function
inimum-phas and aximum- inimum-phas and aximum- Th unwrappd phas functions of H m ( and H( ar thus rlatd through arg[ H ( )] arg[ Hm( )] + arg[ A( )] H( has an xcss phas lag with rspct to H m ( A causal stabl transfr function with all zros insid th unit circl is calld a minimum-phas transfr function A causal stabl transfr function with all zros outsid th unit circl is calld a maximum-phas transfr function A causal stabl transfr function with zros insid and outsid th unit circl is calld a mixd-phas transfr function 6 6 inimum-phas and aximum- Exampl Considr th mixd-phas transfr function ( +. 3z )(. 4 z ) H ( (. z )( +. 5z ) W can rwrit H( as ( +. 3z )(. 4z H ( (. z )( +. 5z ). 4 z ). 4z 63 inimum-phas function Allpass function