5 Transform Analysis of LTI Systems

Transcription

1 5 Transform Analysis of LTI Systms ² For an LTI systm with input x [n], output y [n], and impuls rspons h [n]: Fig. 48-F1 ² Nots: 1. y [n] = h [n] x [n]. 2. Y ( jω ) = H ( jω ) X ( jω ). 3. From th Convolution of Squncs proprty of th z-transform, Y (z) = H (z) X (z). 1. h [n] and H (z) form a uniqu z-transform pair. 2. H (z) is calld th systm function of th LTI systm, which compltly charactrizs th systm. 3. H (z) is rlatd to th frquncy rspons H ( jω ) by z! jω. 5.1 Th Frquncy Rspons of LTI Systms ² Th frquncy rspons H ( jω ) is in gnral complx and can b dscribd altrnativly by its magnitud rspons jh ( jω )j and phas rspons \H ( jω ). Y jω = H jω X jω ) Y jω = H jω {z } magnitud rspons, or gain, or magnitud distortion X jω \Y jω = \H jω + \X jω. {z } phas rspons, or phas shift, or phas distortion ² Th principal valu of \H ( jω ) is dnotd by ARG[H ( jω )] and d nd to tak valu in th rang π < ARG[H jω ] π. Thus, \H ( jω ) can b rprsntd as \H jω = ARG[H jω ] + 2πr(ω) 1

2 whr r(ω) is an intgr function of ω. Sinc ARG[H ( jω )] taks th modulo-2π valu of \H ( jω ) in (π, π], it wraps th phas rspons on a circl and is thus calld th wrappd phas of \H ( jω ). W also us arg[h ( jω )] to dnot a continuous (unwrappd) phas function of \H ( jω ). Fig. 5.1 Ex: For an idal ltr, \H ( jω ) = 0, 8ω. ² Th group dlay of an LTI systm with H ( jω ) is d nd as τ(ω) grd[h jω ], d ª arg H jω (continuous and di rntiabl phas rspons) dω = d dω = d dω ² Ex: Idal Dlay Systm Now, w hav ² Ex: Idal Lowpass Filtr (LPF) Now, w hav \H jω ª (xcpt jump points) ARG[H jω ] ª. (xcpt jump points) h id [n] = δ[n n d ] ) H id jω = jωn d jh id jω j = 1 \H id jω = ωn d for jωj < π τ id (ω) = n d for jωj < π. h ilp [n] = sin ω cn πn ) H ½ ilp jω 1, jωj < ωc = 0, ω c < jωj π. jh ilp jω j = 1 for jωj < ω c \H ilp jω = 0 for jωj < ω c τ ilp (ω) = 0 for jωj < ω c. 2

3 It has zro phas rspons. For most applications, a linar phas rspons can b tolratd sinc th phas rspons can b compnsatd for by introducing dlay in othr parts of a largr systm. In th cas, w can also accpt an idal LPF with linar phas rspons as which has h lp [n] = sin ω c(n n d ) π(n n d ) ) H ½ lp jω jωn d, jωj < ωc = 0, ω c < jωj π jh lp jω j = 1 for jωj < ω c \H lp jω = ωn d for jωj < ω c τ lp (ω) = n d for jωj < ω c. ² Ex: Idal highpass ltr (HPF) H ½ ihp jω 0, jωj < ωc = 1, ω c < jωj π = 1 H ilp jω Now, w hav ² Nots: ) h ihp [n] = δ [n] h ilp [n]. jh ihp jω j = 1 for ω c < jωj π \H ihp jω = 0 for ω c < jωj π τ ihp (ω) = 0 for ω c < jωj π. 1. Th impuls rsponss for idal ltrs (with idal cuto ) is not computationally µ ralizabl sinc thy xtnd from 1 to 1. limitd in tim domain Ã! unlimitd in frquncy domain unlimitd in tim domain Ã! limitd in frquncy domain 2. Rcall that Thus, δ [n n d ] F Ã! jωn d Fig.49-F1 i.., all th input sampls su r th sam amount of dlay n d, calld group dlay. 3

4 3. If \H ( jω ) = α ωn d, thn τ(ω) = n d rprsnts th systm s group dlay to th input squnc. Thrfor, th group dlay rprsnts a convnint masur of th linarity of th phas rspons. 4. Rad Subsction (pp ) for th illustration of cts of group dlay and attnuation. 5.2 LTI Systms Charactrizd by LCCD Equations ² Considr an LTI systm with input x [n] and output y [n] charactrizd by th N th-ordr LCCD quation NX a k y [n k] = k=0 MX b k x [n k], a 0 6= 0. k=0 Taking z-transform both-sidd, NX a k z k Y (z) = k=0 ) H (z) = Y (z) X (z) = MX b k z k X (z) k=0 P M k=0 b kz k P N k=0 a kz k = b 0 a 0 Q M (1 c kz 1 ) Q N (1 d kz 1 ) whr fc k g M is th st of zros and fd kg N is th st of pols. ² Ex: Scond-Ordr Systm Considr th LTI systm charactrizd by th scond-ordr LCCD quation y [n] y [n 1] 3 y[n 2] = x [n] + 2x[n 1] + x [n 2] 8 ) H (z) = 1 + 2z1 + z z1 3 = (1 + z 1 ) 2 8 z2 (1 1 2 z1 ) z1 4 which has two rst-ordr pols at 1 and 3, and on scond-ordr zro 2 4 at 1. Nots: (1) Th ROC is not spci d yt! (2) Th LCCD quation can not uniquly dtrmin an LTI systm! 4

5 5.2.1 Stability and Causality ² A su cint and ncssary (i ) condition for (BIBO) stabl LTI systms is 1X jh [n]j < 1. (absolutly summabl) Now, for jzj = 1, n=1 jh (z)j 1X n=1 h [n] z n = 1X n=1 jh [n]j. Hnc, whthr P 1 n=1 jh [n]j is nit or not dtrmins whthr th ROC of H (z) includs unit circl or not. Thus, an LTI systm is stabl i th ROC of H (z) includs unit circl. Not that stabl systms hav a frquncy rspons. ² A systm is causal i h [n] = 0 for n < 0. Thus, H (z) = P 1 n=0 h [n] zn xists for jzj approaching in nity. This implis that a su cint and ncssary condition for h [n] bing causal is that th ROC of H (z) should b outsid th pol with th largst magnitud. ² Ex: Considr th LTI systm with input and output rlatd by y [n] 5 y [n 1] + y[n 2] = x [n] 2 1 ) H (z) = z1 + z = 1 2 (1 1 2 z1 ) (1 2z 1 ) = z 2 (z 1 ) (z 2). 2 Th pol-zro plot for H (z) is indicatd in Fig. 5.7 Thr ar thr possibl choics for th ROC: 1. If ROC = fzjjzj < 1 g, th systm is nithr causal nor stabl If ROC = fzj 1 < jzj < 2g, th systm is stabl but not causal If ROC = fzjjzj > 2g, th systm is causal but not stabl. ² Nots: 1. Stability and Causality ar not ncssarily compatibl! 2. If an LTI systm is stabl and causal, its ROC should tak th form fz : jzj > jz 0 jg with jz 0 j < 1. 5

6 5.2.2 Invrs Systms ² Th invrs systm of an LTI systm with H (z) has systm function H i (z) such that Fig. 51-F1 ( ) This implis H i (z) = 1 H (z) and h [n] h i [n] = δ [n]. Not: Not vry systm has an invrs. For xampl, th idal lowpass systm dos not hav an invrs. Fig. 51-F2 ² Considr a rational systm function H (z) = µ b0 a 0 with a 0 6= 0 and b 0 6= 0. Its invrs is H i (z) = µ a0 b 0 Q M (1 c kz 1 ) Q N (1 d kz 1 ) Q N (1 d kz 1 ) Q M (1 c kz 1 ). For ( ) to hold, th ROC s of H (z) and H i (z) must ovrlap. xampl, if H (z) is causal, its ROC is For jzj > max jd k j. k Thn, any appropriat ROC for H i (z) that ovrlaps th abov rgion is a valid ROC for H i (z). On th othr hand, H i (z) is causal i th ROC of H i (z) is givn by jzj > max jc k j. k If max k jc k j < 1 (i.., all th zros of H (z) locat insid th unit circl), thn H i (z) is stabl as wll. Thrfor, both H (z) and H i (z) ar stabl and causal i both th pols and th zros of H (z) locat insid th unit circl. Such systms ar rfrrd to as minimum-phas systms. 6

7 Ex: H (z) = 1 0.5z1, jzj > z 1 ) H i (z) = 1 0.9z1, jzj > z 1 Both systms ar causal and stabl. Thus, H (z) is a minimum-phas systm. Ex: H (z) = z1 0.5 = ( z 1 2 ) 1 2z1, jzj > z 1 ) H i (z) = 1 0.9z1 z = (2)1 0.9z1 1 2z 1 Two ROC s jzj > 2 and jzj < 2 for H i (z) ar allowabl. 1. jzj > 2: H i (z) is causal but unstabl. 2. jzj < 2: H i (z) is stabl but noncausal. This indicats that th invrs systm for H (z) may not b uniqu! Also, H (z) is not a minimum-phas systm Impuls Rspons for Rational Systm Functions ² Considr a rational H (z) with only rst-ordr pols, xprssd as H (z) = M XN r=0 B r z r + NX A k 1 d k z 1 with M N and di rnt d k s. If w want H (z) to b causal, its ROC should b jzj > max jd k j; in this cas k h [n] = M XN r=0 B r δ [n r] + Thr ar two classs of LTI systms: NX A k d n ku [n]. First, at last on nonzro pol is not cancld by a zro. h [n] has at last on trm of th form A k d n ku [n] and thus of in nit lngth. Such systms ar IIR systms. 7

8 Ex: For a 6= 0, H (z) = 1 1 az 1, jzj > jaj ) h [n] = an u [n]. Scond, if a rational H (z) has only zros, i.., H (z) has th form H (z) = MX b k z k k=0 thn its impuls rspons has th form h [n] = MX b k δ [n k]. k=0 Such systms ar FIR systms. ² An LCCD quation of th form y [n] = MX b k x [n k] k=0 can d nitly d n an FIR systm. Howvr, an LCCD quation of th form NX MX a k y [n k] = b k x [n k] k=0 k=0 dos not ncssarily spcify an IIR (or FIR) systm. Ex: y [n] = x [n] + x [n 1] is an FIR systm with h[n] = δ [n] + δ [n 1]. Not that this systm can also b xprssd as ) y [n] y [n 1] = x [n] x [n 2]. 5.3 Frquncy Rspons for Rational Systm Functions ² For a stabl LTI systm with H (z) = P M k=0 b kz k P N k=0 a kz k, 8

9 it has a frquncy rspons (sinc unit circl 2 ROC) H jω = P M k=0 b k jωk P N k=0 a k jωk ) H jω = µ b0 a 0 Q M (1 c k jω ) Q N (1 d k jω ). ² Dfns: 1. jh ( jω )j is th magnitud rspons for H ( jω ), givn by H jω = b 0 a 0 Q M j1 c k jω j Q N j1 d k jω j. 2. jh ( jω )j 2 is th magnitud-squard rspons for H ( jω ), givn by H jω 2 = b 0 a 0 2 Q M (1 c k jω ) (1 c k jω ) Q N (1 d k jω ) (1 d k jω ) log 10 jh ( jω )j is th log magnitud rspons, or gain in db, for H ( jω ). 20 log 10 H jω = 20 log 10 b 0 a 0 + MX 20 log 10 j1 c k jω j NX 20 log 10 j1 d k jω j log 10 jh ( jω )j is calld attnuation in db. 5. \H ( jω ) is th phas rspons for H ( jω ), givn by \H jω = \ µ b0 a 0 + MX \ 1 c k jω NX \ 1 d k jω. 9

10 6. ARG [H ( jω )] is th principal phas rspons for H ( jω ), with π < ARG [H ( jω )] π and givn by ARG H jω b0 MX = ARG + ARG 1 c k jω a 0 NX ARG 1 d k jω + 2πr (ω). Not: \H ( jω ) = ARG [H ( jω )] + 2πr (ω) with r (ω) an intgr function of ω. 7. arg [H ( jω )] is th continuous phas rspons for H ( jω ), givn by arg H jω b0 MX = arg + arg 1 c k jω a 0 Nots: (a) (b) NX arg 1 d k jω. Fig.5.1 ARG H jω µ Im fh ( = tan 1 jω )g. R fh ( jω )g 8. grd [H ( jω )] is th group dlay for H ( jω ), givn by grd H jω = NX d arg 1 dk jω dω MX d arg 1 ck jω dω = d dω arg H jω NX jd k j 2 Rfd k jω g = 1 + jd k j 2 2 Rfd k jω g MX jc k j 2 Rfc k jω g 1 + jc k j 2 2 Rfc k jω g. 10

11 Nots: (a) ARG 1 a jω = ARG 1 R a jωª j Im a jωª µ Im fa = tan 1 jω g. 1 R fa jω g ) d dω ARG 1 a jω = jaj 2 R fa jω g 1 + jaj 2 2 R fa jω g sinc d dx tan1 x = 1 1+x 2. (Show this yourslf.) (b) Excpt at jumps from π to π or π to π, d ª arg H jω = d ª ARG H jω dω dω = d dω tan Frquncy Rspons of First-Ordr Systms ² Considr a factor 1 r jθ jω with 0 < r < 1: µ Im fh ( jω )g. R fh ( jω )g log magnitud = 20 log 10 1 r jθ jω = 10 log r 2 2r cos (ω θ) principal phas = ARG 1 r jθ jω µ r sin (ω θ) = tan 1 1 r cos (ω θ) 3. Excpt at discontinuitis, 4. Nots: group dlay = d dω tan1 = µ r sin (ω θ) 1 r cos (ω θ) r 2 r cos (ω θ) 1 + r 2 2r cos (ω θ) 11

12 (a) Log magnitud, phas, and group dlay ar all priodic with 2π in ω. Thus, it su cs to considr th frquncy rang 0 ω < 2π for ths functions. (b) maximum log magnitud (occurs at ωθ = π) = 20 log 10 (1 + r) minimum log magnitud (occurs at ωθ = 0) = 20 log 10 j1 rj For r ¼ 1, th dip is sharp! Fig. 54-F1 Fig. 5.11(a) (c) Whn ω = θ and ω = θ + π, phas rspons = 0. Fig. 54-F2 For r ¼ 1, thr is a jump at ω = θ. (in nity slop) Fig. 5.11(b) (d) Whn ω = θ, thr is a dip for group dlay. For r ¼ 1, th dip is sharp! Fig. 54-B1 Fig. 5.11(c) () At ω = θ + π, w hav maximum magnitud and zro phas rspons. ² Gomtric Construction: In infrring th frquncy rspons charactristics from pol-zro plots of LTI systms, th associatd vctor diagram in th complx plan ar hlpful. Now, considr th LTI systm with rst-ordr systm function H (z) = z rjθ z (It has H jω = 1 r jθ jω ) with r < 1 (a stabl systm). Th rlativ position of a point on unit circl (i.., jω ) from th pols/zros of a systm function carris important information of H ( jω ), i.., th frquncy rspons of th LTI systm. Fig whr th thr vctors ar d nd by v 1 = jω, v 2 = r jθ, and v 3 = v 1 v 2. Th thr vctors hav th following intrprtation: 12

13 pol vctor = v p = zro vctor = v o = jω {z} a point on unit circl jω {z} a point on unit circl {z} 0 = jω = v 1 pol {z} r jθ = v 3. zro v 2 = r jθ vctor from origin to zro at r jθ. Now, th rsponss of H ( jω ) can b intrprtd in trms of vctor rlationship as follows: 1. Th magnitud rspons at ω is H jω = 1 r jθ jω = Sinc jv p j = 1, jh ( jω )j = jv o j is minimum as ω = θ. On th othr hand, jh ( jω )j is maximum as ω = θ + π. As ω incrass from θ to θ + π, jv o j / jv p j incrass from minimum to maximum. As ω incrass from θ + π to 0, and to θ, jv o j / jv p j dcrass from maximum to minimum. As r approachs 1, th magnitud dips sharply as ω is around θ. Fig. 55-B1 Fig Th phas rspons is \H ( jω ) = \v o \v p = φ 3 ω which is zro as ω = θ and ω = θ + π (as w hav larnd prviously). Fig. 56-F1 Fig As r = 1, \H ( jω ) = \v o \v p has a jump. 3. Th group dlay is v o v p. d ª ARG H jω = grd H jω dω Fig. 56-F3 13

14 Not: If thn H jω = 1 1 r jθ jω µ H (z) = z z r jθ v 2 = r jθ, v o = jω, v p = jω r jθ Fig. 56-B1 H jω = jv oj jv p j \H jω = \v o \v p. Thn, all th prvious discussion (for r < 1) on log magnitud, phas, and group dlay will apply with signs rvrsd. ² Exampls of Gomtric Construction with Multipl Pols and Zros Ex 5.8: Considr th stabl LTI systm with H (z) = (1 + z1 ) ( z 1 + z 2 ) ( z 1 ) ( z z 2 ) This systm has pols at z = 0.683, j zros at z = 1, j Fig Fig Th systm has µ 1 log magnitud at ω = π, , 2π phas jumps At ω ¼ 0.22π and 2π 0.22π, thr ar phas jumps du to th us of th principal valu in plotting. S Exs 5.6 & 5.7 for slf-study. 14

15 5.4 Rlationship Btwn Magnitud and Phas ² Considr th magnitud squar of th frquncy rspons H jω 2 = H jω H jω = H (z) H (z)j z= jω = H (z) H µ 1 z z= jω whr w hav d nd = C (z)j z= jω C (z), H (z) H µ 1 z. For an LTI systm spci d by an LCCD quation, its H (z) has th form H (z) = b Q M 0 (1 c kz 1 ) Q a N 0 (1 d kz 1 ) which yilds C (z) = b 0 a 0 2 Q M (1 c kz 1 ) (1 c k z) Q N (1 d kz 1 ) (1 d k z). Thus, a pol d k of H (z) givs two pols d k and (d k )1 of C (z); similarly, a zro c k of H (z) givs two zros c k and (c k )1 of C (z). Fig. 57-B1 For a stabl and causal H (z), all pols ar locatd insid th unit circl, and thus th pols of C (z) can uniquly spcify th pols of H (z); but this dos not apply to zros. Not: C (z)j z= jω carris only th information of th magnitud rspons of H ( jω ). ² Ex: Di rnt Systm With th Sam C (z) Considr two stabl systms with H 1 (z) = H 2 (z) = 2(1 z 1 )( z1 ) (1 0.8 jπ/4 z 1 )(1 0.8 jπ /4 z 1 ) (1 z 1 )(1 + 2z 1 ) (1 0.8 jπ/4 z 1 )(1 0.8 jπ /4 z 1 ) 15

16 which can b dscribd by th pol-zro plots Both systms hav th sam C (z) C (z) = Fig. 5.16(a)-(b) 4(1 z 1 )( z1 )(1 z)( z) (1 0.8 jπ /4 z 1 )(1 0.8 jπ/4 z 1 )(1 0.8 jπ /4 z)(1 0.8 jπ /4 z) dscribd by Fig. 5.16(c) ² Ex: Dtrmination of H(z) From C(z) Givn C (z) with th pol-zro plot Fig Th conjugat rciprocal pairs of pols and zros ar (p 1, p 4 ), (p 2, p 5 ), (p 3, p 6 ) (z 1, z 4 ), (z 2, z 5 ), (z 3, z 6 ). For H (z) to b causal and stabl, w can only choos p 1, p 2, p 3 as pols of H (z). Howvr, thr ar 8 possibl choics for zros, i.., (z 1, z 2, z 3 ), (z 1, z 2, z 6 )... If, furthrmor, w want th co cints in LCCD quation to b ral (ral input squnc yilds ral output squnc in this cas); thn zros ar ithr ral or in complx conjugat pairs. In this cas, w hav th following choics for zros: (z 3, z 1, z 2 ), (z 3, z 4, z 5 ), (z 6, z 1, z 2 ), (z 6, z 4, z 5 ). ² Considr th all-pass systm Not that H ap (z) H ap H ap (z) = z1 a 1 az 1. Fig. 58-B1 µ 1 = z1 a z 1 az z a 1 1 a z = 1. 16

17 Thn, C (z) = H (z) H µ 1 z = H (z) H ap (z) H µ 1 z H ap µ 1. z Th sam C (z) rsults! This implis that th numbr of pols and zros of C (z) dos not uniquly spcify th numbr of pols and zros of H (z). Sinc th pol and zro of H ap (z) ar canclld in C (z), th numbr of pols and zros of H (z) is not rstrictd by th numbr of pols and zros of C (z). 5.5 Allpass Systms ² Considr a stabl all-pass systm H ap (z) = z1 a 1 az 1 with jaj < 1 and jzj > jaj. It has a magnitud rspons that is indpndnt of ω sinc jh ap (z)jj z= jω = jω a ( 1 a jω = jω a 1 a jω 2 ) 1 2 ½ ( jω a ) ( jω a) = (1 a jω ) (1 a jω ) = 1. A systm for which th frquncy-rspons magnitud is a constant is rfrrd to as an all-pass systm. Thus, an all-pass systm passs all of th frquncy componnts of its input with constant gain or attnuation. Bcaus H ap (z) H ap ¾ 1 2 µ 1 = z1 a z 1 az z a 1 1 a z = 1 holds for arbitrary nonzro a valus, an all-pass systm can b gnrally d nd as th systm with H ap (z) satisfying H ap (z) H ap 1 z = A with A > 0. 17

18 Not that an all-pass systm can b dscribd by a product of multipl rational z-functions in th abov form of H ap (z), prhaps with di rnt a valus, i.., H ap (z) = C Q z 1 a i i 1 a i z 1 with C a constant. Hr, a i s can b rstrictd to ja i j < 1 in ordr to d n a stabl and causal all-pass systm. ² Considr a ral systm with a rational H (z). Now, if h [n] is ral, thn H jω = H jω (pair 7 of Tabl 2.1) ) H (z) = H (z ) for z = jω. Furthr, if H (z) is rational, thn H (z) = b Q M 0 (1 c kz 1 ) Q a N 0 (1 d kz 1 ) = b 0 = b 0 a 0 Q M (1 c k z1 ) Q N (1 d k z1 ). a 0 Q M 1 ck (z ) 1 Q N 1 dk (z ) 1 This implis that (1) ( b 0 a 0 ) = ( b 0 a 0 ) ; (2) c k s (and d k s) ar ithr ral or in a complx conjugat pair, in gnral. Thus, allpass systms with ral impuls rspons hav a gnral form of YM r z 1 d k YM c (z 1 ρ k H ap (z) = ) (z1 ρ k ) ~ 1 d k z 1 (1 ρ k z 1 ) (1 ρ k z1 ) whr d k s ar ral and ρ k s ar complx. If th allpass systm is stabl and causal, thn jd k j < 1 and jρ k j < 1 for all k. Not: In an all-pass systm with ral impuls rspons, ach pol is paird with a complx conjugat rciprocal zro. Fig

19 ² Th phas rspons for H ap (z) = z1 a with a = r jθ is 1az 1 jω r jθ \ = \ jω r jθ \ 1 r jθ jω 1 r jθ jω = ω + \ 1 r jω jθ \ 1 r jθ jω µ r sin (ω θ) = ω + tan 1 1 r cos (ω θ) µ r sin (θ ω) tan 1 1 r cos (θ ω) = ω 2 tan 1 µ r sin (ω θ) 1 r cos (ω θ) Likwis, th phas rspons for a systm with ~ is \ H ap jω in ~ XM r µ = (M r + 2M c ) ω 2 tan 1 dk sin ω 1 d k cos ω XM c µ 2 tan 1 rk sin (ω θ k ) 1 r k cos (ω θ k ) µ + tan 1 rk sin (ω + θ k ) 1 r k cos (ω + θ k ) whr w lt ρ k = r k jθ k. ² Ex: First- and Scond-Ordr All-Pass Systms Fig ² Proprtis of Stabl and Causal Allpass systms: Sinc an allpass systm consists of a product of z1 a, it su cs to 1az 1 study th phas rspons and group dlay for z1 a in ordr to charactriz thos for th allpass 1az 1 systms.. 1. grd [H ap ( jω )] 0 for a stabl and causal allpass systm. 19

20 Pf: grd jω a 1 a jω = grd a=r jθ jω r jθ 1 r j(θω) = d µ r sin (ω θ) ω 2 tan 1 dω 1 r cos (ω θ) (xcpt thos on jumps of 2π) 1 r 2 = j1 r jθ jω 2 (you show it!) j 0 (sinc jaj < 1) QED 2. arg [H ap ( jω )] 0, 0 ω < π, for a ral-valud (which has an odd phas rspons, i.., arg [H ap ( jω )] = arg [H ap ( jω )]), stabl and causal allpass systm. Pf: Sinc grd H ap jω = d dω arg H ap jω ) arg H Z ω ap jω = grd hh ap ³ jω0 i dω 0 + arg H ap j0 0 for 0 ω < π. Now, (a) arg H ap j0 = ~ arg (b) grd [H ap ( jω )] 0. (" Mr Y Thus, arg [H ap ( jω )] 0. QED 1 d k 1 d k # " Mc Y #) j1 ρ k j 2 j1 ρ k j 2 = 0. ² Thus, th phas and group dlay rsponss of a stabl and causal allpass systm can b dsignd to nabl di rnt applications. Th dsign of phas rspons and group dlay rspons may b di rnt for various applications, for xampl, compnsator for phas distortion or compnsator for group dlay, transforming th phas rspons or group dlay rspons of a ltr into anothr form without changing th magnitud rspons, and in th thory of minimum-phas systms. 20

21 5.6 Minimum-Phas Systms ² A minimum-phas systm H min (z) is th on that both its H (z) as wll as 1/H (z) (th systm function of its invrs systm xists!) should hav pols and zros insid unit circl. That is, both th systm and its invrs ar causal and stabl. ² Not: For a minimum-phas systm with H min (z), C (z) = H min (z) H min (1/z ) uniquly dtrmins H min (z), which will consist of all th pols and zros of C (z) that li insid th unit circl Minimum-Phas and All-Pass Dcomposition ² Any rational systm function can b xprssd as H (z) = H min (z) H ap (z). Pf: First, for a minimum-phas systm, it is trivial (sinc H ap (z) = 1). Nxt, considr a nonminimum-phas systm with Q Md H (z) = H 1 (z) (1 c kz 1 ) Q Mn (1 d kz 1 ) whr H 1 (z) is minimum-phas and c k and d k ar outsid unit circl. Thn Q Md 1 (c H (z) = H 1 (z) k ) 1 z 1 Q Md Q Mn 1 (d k ) 1 z 1 (1 c kz 1 ) 1 (d k )1 z 1 Q Mn {z } (1 d kz 1 ) 1 (c k )1 z 1. (S) {z } H min (z) This provs th proprty. QED Nots: H ap(z) 1. A nonminimum-phas systm can b formd from a minimumphas systm from (S), and vic vrsa. 2. Th corrsponding minimum-phas systm of a nonminimum-phas systm is obtaind by r cting th zros (pols) lying outsid unit circl to thir conjugat rciprocal locations insid. Fig. 61-F1 By such transformation, both minimum-phas and nonminimumphas systms hav th sam magnitud rspons but di rnt phas rsponss. 21

22 ² Ex: Minimum-Phas/All-Pass Dcomposition Considr two stabl and causal systms with H 1 (z) = 1 + 3z z1 H 2 (z) = ( jπ/4 z 1 )( jπ/4 z 1 ) z1. Both systm functions ar nonminimum-phas. Now, w want to convrt thm into th corrsponding minimum-phas systm function. For H 1 (z), w r ct th zro z = 3 to z = 1/3 by th all-pass systm H ap (z) = z z1 (which has a zro at z = 3) to giv th minimum-phas countrpart as H 1,min (z) = z z1 through th rlation H 1 (z) = H 1,min (z)h ap (z) = z z1 z z1. For H 2 (z), w r ct th zros z = 3 2 jπ/4 to z = 2 3 jπ /4 by th all-pass systm H ap (z) = (z jπ/4 )(z jπ /4 ) ( jπ /4 z 1 )( jπ /4 z 1 ) (which has complx conjugat zros at z = 3 2 jπ /4 ) to giv th minimumphas countrpart as H 2,min (z) = 9 4 through th rlation H 2 (z) = H 2,min (z)h ap (z) = 9 4 ( jπ/4 z 1 )( jπ /4 z 1 ) z1 ( jπ/4 z 1 )( jπ/4 z 1 ) z1 (z jπ /4 )(z jπ /4 ) ( jπ /4 z 1 )( jπ /4 z 1 ). 22

23 5.6.2 Frquncy-Rspons Compnsation of Non-Minimum-Phas Systms ² W want to compnsat for th distortion introducd by a distorting systm with H d (z) and rcovr back th original signal s[n] to som xtnt through a stabl and causal compnsating systm with H c (z). Thus, considr Fig If prfct compnsation is achivd, thn s c [n] = s[n]. Howvr, if th distorting systm is stabl and causal and w rquir th compnsating systm to b stabl and causal, thn prfct compnsation is possibl only if H d (z) is a minimum-phas systm (so that it has a stabl and causal invrs). In this cas, G (z) = H d (z) H c (z) = 1. ² Whn H d (z) is a nonminimum-phas systm, a stabl and causal H c (z) is not availabl to achiv G (z) = 1. In th cas, our goal is modi d to dsign H c (z) so that jg (z)j = 1, 8z, i.., th frquncy-magnitud rspons of H d (z) is compnsatd. Now, considr th distorting systm with a nonminimum-phas systm function If w st H d (z) = H d,min (z) H ap (z). H c (z) = 1 H d,min (z) thn th whol systm bcoms an all-pass systm with systm function G (z) = H d (z) H c (z) = H ap (z). Thus, jg (z)j = 1 and \G (z) = \H ap (z). ² Ex: Compnsation of an FIR Systm Considr th ral-valud FIR distorting systm with H d (z) = (1 0.9 j0.6π z 1 )(1 0.9 j0.6π z 1 ) ( j0.8π z 1 )( j0.8π z 1 ). Fig Fig

24 which is a nonminimum phas systm. Now, r ct th zros z = 1.25 j0.8π to z = 0.8 j0.8π, w hav H d,min (z) = (1.25) 2 (1 0.9 j0.6π z 1 )(1 0.9 j0.6π z 1 ) (1 0.8 j0.8π z 1 )(1 0.8 j0.8π z 1 ) Fig and can rprsnt H d (z) = H d,min (z) H ap (z) with H ap (z) = (z1 0.8 j0.8π )(z j0.8π ) (1 0.8 j0.8π z 1 )(1 0.8 j0.8π z 1 ) (which has complx conjugat zros at z = 1.25 j0.8π ). Fig This indicats that th stabl and causal compnsating systm achiving jg (z)j = 1 is 1 H c (z) = H d,min (z) Proprtis of Minimum-Phas Systms ² W considr ral-valud stabl and causal systms (which hav odd phas rsponss, i.., \H ( jω ) = \H ( jω )) and shall dvlop intrsting and important proprtis of stabl and causal minimum-phas systms with H min (z) rlativ to all othr stabl and causal systms (with all pols locatd insid th unit circl) having th sam frquncyrspons magnitud and H (z) = H min (z) H ap (z) = H min (z) Q Mn Q Mn (1 c kz 1 ) 1 (c k ) 1 z 1 whr th zros in H ap (z) ar locatd outsid th unit circl. Not that H ap (z) hr has pols locating insid th unit circl and ar assumd to b stabl and causal. (1) Minimum Phas-Lag (Minus Phas Dlay) Proprty 24

25 Any nonminimum-phas stabl and causal systm with H ( jω ) has an unwrappd phas dscribd by arg H jω = arg H min jω + arg H ap jω {z } stabl & causal allpass systms ) arg [H ap ( j0 )] = 0 and arg [H ap ( jω )] 0 for 0 ω < π arg H min jω (not corrct without th assumption arg[h ap ( j0 )] = 0) for 0 ω < π. Thus, th r ction of zros of H min (z) from insid th unit circl to conjugat rciprocal locations outsid th unit circl always dcrass th (unwrappd) phas or incrass th ngativ of th phas (noting that th ngativ of th phas is usually calld th phaslag function). Thus, th causal and stabl systm having jh min ( jω ) j as its magnitud rspons and also has all its zros and pols insid th unit circl has th minimum phas-lag function (for 0 ω < π) of all th stabl and causal systms having that sam magnitud rspons. Thus, th stabl and causal systm with H min ( jω ) is calld th minimum-phas or minimum-phas-lag systm. Usual Assumption: It is commonly assumd that, for all systm functions, H ( j0 ) = P n h[n] > 0, i.., arg [H (j0 )] = 0. This assumption is ncssary to nsur arg [H ap ( jω )] 0 for 0 ω < π (s th proof in Sction 5.5). (2) Minimum Group-Dlay (Positiv Dlay) Proprty Sinc grd [H ap ( jω )] 0 for a ral-valud stabl and causal allpass systm, ) grd H jω = grd H min jω + grd H ap jω grd H min jω for a stabl and causal ral-valud H ( jω ). (3) Minimum Enrgy-Dlay Proprty A minimum-phas systm has th minimum nrgy-dlay than any othr stabl and causal ral systms that hav th sam magnitud function. 25

26 (a) First, w s that 1X n=0 jh [n]j 2 = 1 2π (b) Scond, d ning w hav E [n] = Z π π Z π = 1 2π π 1X = jh min [n]j 2 n=0 H jω 2 dω (Parsval s thorm) H min jω 2 dω nx jh [m]j 2, E min [n] = m=0 E [n] E min [n]. (S Figs and 5.28) Pf: Lt Now, obtain (Not: Fig. 62-B1 nx jh min [m]j 2 m=0 H min (z) = Q (z) 1 z k z 1, jz k j < 1. H (z) = H min (z) z1 z k 1 z k z = Q (z) z 1 z 1 k. H jω = Q jω jω z k = Q 1 jω z k jω = Hmin ) jω Taking invrs z-transform, w hav h [n] = q [n 1] z kq [n] h min [n] = q [n] z k q [n 1] whr q [n] = 0 for n < 0 (sinc h min [n] is causal). D n nx nx ɛ= jh min [m]j 2 jh [m]j 2. m=0 m=0 26

27 W hav for n > 0 that nx ɛ = jq [m] zk q [m 1]j 2 jq [m 1] z k q [m]j2ª = = m=0 nx fjq [m]j 2 + jz k j 2 jq [m 1]j 2 q [m] z k q [m 1] m=0 q [m] z k q [m 1] jq [m 1]j 2 jz k j2 jq [m]j 2 +q [m 1] z kq [m] + q [m 1] z k q [m]g nx 1 jzk j 2 jq [m]j 2 1 jz k j 2 jq [m 1]j 2ª m=0 = 1 jz k j 2 jq [n]j 2 ( * causal) 0 ( * jz k j < 1) QED (c) jh [0]j < jh min [0]j. Pf: For causal squncs, h [0] = lim H (z) z!1 = lim H min (z) H ap (z). z!1 Now, jh [0]j = lim H min (z) H ap (z) lim H min (z) lim H ap (z). z!1 z!1 z!1 Sinc ) KY z 1 a k H ap (z) = 1 a k z, ja kj < 1 (for a stabl and causal systm) 1 KY lim H ap (z) = (a KY z!1 k) = ja kj < 1. Thus, jh [0]j < lim H min (z) = jh min [0]j. z!1 QED (d) From (a), (b) and (c), w show that a minimum-phas systm has minimum nrgy-dlay proprty. Figs

28 5.7 Linar Systms With Gnralizd Linar Phas ² A dsird digital ltr is th on with constant in-band gain and zro phas rspons as wll. In th cass whr zro-phas is not attainabl, linar phas rspons is dsirabl sinc linar phas rsults in a constant group dlay Systms With Linar Phas ² Considr an LTI systm with H id ( jω ) = jωα, jωj < π with α a ral constant. ) Hid jω = 1, \Hid jω = ωα, grd H id jω = α for jωj < π and h id [n] = Sa (π (n α)) 8n Rcall: y [n] = 1X Fig. 64-F1 If α = intgr = n d, y [n] = x [n n d ]. x [k] Sa (π (n k α)) If α 6= intgr, y [n] = x c ((n α) T ). (prvious discussion) Fig Thus, αt is th amount of dlay th systm h id [n] introducs to th input x c (t)! In gnral, an LTI systm with H ( jω ) = jh ( jω )j jωα, jωj < π rshaps th magnitud of x [n] by jh ( jω )j and dlays th output by α. Fig On xampl is th linar-phas idal lowpass ltr with H ½ lp jω = jωα, jωj < ω c 0, ω c < jωj π h lp [n] = sin(ω c(n α)). π(n α) Ex: Idal Lowpass With Linar Phas 28

29 1. Lt α = n d, an intgr. Thn, h lp [2n d n] = h lp [n]. In th cas, w can d n a zro-phas systm with bh lp jω = H lp jω jωn d = jh lp jω j b hlp [n] = sin(ω cn) πn = b h lp [n] Th vn squnc b h lp [n] that has zro phas can b simply obtaind by shifting h lp [n] to th lft by n d sampls. Fig. 5.32(a) 2. Lt 2α = n d b an odd intgr. Thn, h lp [2α n] = h lp [n]. Fig. 5.32(b) Sinc α is not an intgr, it is not possibl to shift h lp [n] to obtain an vn squnc that has zro phas. 3. Thr ar cass without symmtry in squnc. Fig. 5.32(c) ² In gnral, a ral linar-phas systm (which has constant group dlay) has frquncy rspons H jω = jh jω j jωα. If 2α is an intgr, th corrsponding ral h[n] has vn symmtry about α, i.., h[2α n] = h[n] (You can show this by invrs Fourir transform); othrwis, h[n] will not hav symmtry Gnralizd Linar Phas ² An LTI systm with H ( jω ) is said to hav a gnralizd linar phas rspons (with constant group dlay) if H jω = A jω jωα+jβ whr α, βar constants and A ( jω ) is a ral (possibly bipolar) function of ω. Not: arg H jω = β ωα, jωj < π grd H jω = α 29

30 ² Now, H ( jω ) has a gnralizd linar phas rspons ) H jω = A jω j(βωα) = A jω cos (β ωα) + ja jω sin (β ωα) 1X = h [n] jωn = n=1 1X n=1 h [n] cos (ωn) j µ P 1 ) tan 1 n=1 h [n] sin (ωn) P 1 n=1 h [n] cos (ωn) ) ) ) 1X n=1 1X n=1 1X n=1 1X n=1 h [n] sin (ωn) = tan 1 µ A ( jω ) sin (β ωα) A ( jω ) cos (β ωα) h [n] A jω [sin (β ωα) cos (ωn) + cos (β ωα) sin (ωn)] = 0 h [n] sin (β ωα+ ωn) A jω = 0 h [n] sin (ω (n α) + β) = 0 8ω which is a ncssary condition for gnralizd linar phas ral-valud systm. ² A class of gnralizd linar-phas systms ar thos with β = 0 or π 2α = M an intgr h[2α n] = h[n] (i.., h[n] has vn symmtry about α) For this class of systms, (I) bcoms 1X n=1 h [n] sin (ω (n α)) = 0 8ω. ² Anothr class of gnralizd linar-phas systms ar thos with β = π/2 or 3π/2 2α = M an intgr h[2α n] = h[n] (i.., h[n] has odd symmtry about α) (I) 30

31 For this class of systms, (I) bcoms 1X h [n] cos (ω (n α)) = 0 8ω. n= Causal Gnralizd Linar Phas FIR Systms ² If a gnralizd linar phas systm is causal, thn 1X h [n] sin (ω (n α) + β) = 0 8ω. n=0 ² Lt us considr ral-valud FIR systms, with h [n] = 0 for n < 0 or n > M, blow. 1. Typ I FIR Linar Phas Systms: h [n] = h [M n] for 0 n M with M an vn intgr. S F ig. 5.33(a). Its frquncy rspons is of th form H jω MX = h [n] jωn = = n=0 M/21 X n=0 h [n] jωn + X M/2+M/2 M/21 X n=0 = jω M 2 +h n=m/2 = jω M 2 {z } linar phas M/2+M/2 X n=m/2 h [M n] jωn 1 M/2 X h h [M n] jωn = h n 0i jωm jωn0 A n 0 =Mn n 0 =0 h [n] jωn + jω(mn) + h M/21 X n=0 M jω M 2 2 M 2 jω M 2 h [n] ³ jω(n M 2 ) + jω( n) M 2 {z } 2 cos(ω(n M 2 )) M/2 X a [k] cos (ωk) k=0 31

32 whr M a [0] = h 2 M a [k] = 2h 2 k, k = 1,..., M 2. P Not: M/2 k=0 a [k] cos (ωk) is ral, which implis that th systm H ( jω ) is a gnralizd linar phas systm. 2. Typ II FIR Linar-Phas Systms: h [n] = h [M n] for 0 n M with M an odd intgr. S F ig. 5.33(b). Similarly, its frquncy rspons is of th form H jω (M+1)/2 X µ µ = jω M 2 b [k] cos ω k 1 2 whr M + 1 b [k] = 2h k, k = 1, 2,..., M Typ III FIR Linar-Phas Systms: h [n] = h [M n] for 0 n M with M an vn intgr and h M 2 = 0. S F ig. 5.33(c). Its frquncy rspons is of th form with H jω = M/21 X n=0 = jω M 2 +h M 2 = j jω M 2 h [n] jωn jω(mn) + h M/21 X n=0 jω M 2 M jω M 2 2 h [n] ³ jω(n M 2 ) jω( n) M 2 M/2 X c [k] sin (ωk) M c [k] = 2h 2 k, k = 1, 2,..., M 2. 32

33 4. Typ IV FIR Linar Phas Systms: h [n] = h [M n] for 0 n M with M an odd intgr. S F ig. 5.33(d). Its frquncy rspons is of th form with H jω = jω M 2 (M +1)/2 X M + 1 d [k] = 2h k 2 ² Ex: Typ I FIR Linar Phas Systm ½ 1, 0 n 4 h [n] = 0, othrwis H jω j2ω sin(5ω/2) = sin(ω/2) Fig ² Ex: Typ II FIR Linar Phas Systm ½ 1, 0 n 5 h [n] = 0, othrwis H jω j5ω/2 sin(3ω) = sin(ω/2) Fig ² Ex: Typ III FIR Linar Phas Systm h [n] = δ[n] δ[n 2] H jω = (j2 sin ω) jω µ µ d [k] sin ω k 1 2, k = 1, 2,..., M Fig ² Ex: Typ IV FIR Linar Phas Systm h [n] = δ[n] δ[n 1] H jω = (j2 sin(ω/2)) jω/2 33 Fig. 5.37

34 ² Zro Locations for Causal, Ral, Linar Phas FIR Systms: For a ral and causal FIR systm with h [n] = 0 for n < 0 or n > M, thr ar on pol of ordr M at z = 0 and M zros. Morovr, zros ar ithr ral or in a complx conjugat pair. 1. For typ I and II systms, H (z) = MX h [n] z n = n=0 = z M H z 1. MX h [M n] z n = n=0 MX h [n] z n z M n=0 Thus, H (z 0 ) = 0 ) z 0 M H z 0 1 = 0. This mans that Fig. 67-F1 Fig. 5.38(a)-(b) (a) If z 0 is complx and not on unit circl, z 1 0, z 0, and (z 0) 1 ar also zros. (b) If z 0 is complx and on unit circl, z 0 is also zro. (z 1 0 = z 0) (c) If z 0 is ral and not on unit circl, z 1 0 is also zro. (d) If z 0 = 1 (or z 0 = 1), thn thr is no othr nw zro d nd by it. () If z = 1 and M is odd, H (1) = (1) M H (1) = H (1) ) H (1) = 0. In othr words, 1 has to b a zro for a typ II systm. 2. For typ III and IV systms, MX H (z) = h [n] z n = n=0 = z M H z 1 MX MX h [M n] z n = h [n] z n z M n=0 n=0 Thus, H (z 0 ) = 0 ) z M 0 H z 1 0 = 0. Thrfor, (a), (b), (c), (d) in 1. apply.. If z = 1, H (1) = H (1) ) H (1) = 0, i.., 1 has to b a zro for both typ III and IV systms 34

35 f. If z = 1 and M is vn, H (1) = (1) M H (1) = H (1) ) H (1) = 0, i.., 1 has to b a zro for a typ III systms. Nots: 1. Th abov constraints on th zro locations of ral and causal linar phas FIR systms impos limitations on th magnitud rspons that can b dsignd. 2. FIR systms can b asily dsignd to hav xact linar phas (or gnralizd linar phas). Howvr, IIR systms ar mor cint in th systm dsign of obtaining dsird magnitud rspons (s Chap. 7) Rlation of FIR Linar-Phas Systms to Minimum-Phas Systms ² Bcaus all ral FIR linar-phas systms hav M zros ithr on th unit circl or at conjugat rciprocal locations, thir systm functions can b factord into H(z) = H min (z)h uc (z)h max (z) whr H min (z) is a minimum-phas trm with M i zros locating insid th unit circl, H max (z) a maximum-phas trm d nd by H max (z) = H min (z 1 )z M i with M i zros locating outsid th unit circl, and H uc (z) a trm containing only M o zros on th unit circl. Thus, M = 2M i + M o. ² Ex: Dcomposition of a Linar-Phas Systms Considr th minimum-phas trm with M i = 4 zros H min (z) = (1.25) 2 (1 0.9 j0.6π z 1 )(1 0.9 j0.6π z 1 ) (1 0.8 j0.8π z 1 )(1 0.8 j0.8π z 1 ). Th corrsponding maximum-phas systm is H max (z) = (0.9) 2 ( j0.6π z 1 )( j0.6π z 1 ) ( j0.8π z 1 )( j0.8π z 1 ). 35

36 Thus, th composit systm H(z) = H min (z) H max (z) has linar phas and th following rsponss sinc jh min ( jω )j = jh max ( jω )j, 20 log 10 jh( jω )j = 40 log 10 jh min ( jω )j \H( jω ) = ωm i sinc \H max ( jω ) = ωm i \H min ( jω ), and grd[h( jω )] = M i. Fig Fig Fig