8-4B Steady-State Frequency Response of a Linear Discrete-Time System

Transcription

1 8-4B Stady-Stat Frquncy Rspons of a Linar Discrt-im Systm In this sction, w study various proprtis of th discrt-tim Fourir ransform. As w statd arlir, it is th sam as th continuous-tim Fourir ransform of th sampld signal x s( t) n ) δ ( t n ) : X s (jω) 4 W hav also claimd that it can b computing by valuating th Z-transform around th unit circl (providd that th unit circl is insid th ROC) Dfinition of DF: jωn X ( ) X ( z) jω n ) z Not that w us X( ) to indicat th substitution z Lt s first show that ths two dfinitions ar quivalnt. Start with th continuoustim Fourir ransform of th sampld signal: X ( jω ) s n ) δ ( t n ) jωt dt n ) jωt δ ( t n ) dt n ) jωn X ( ) Pag 5-32

2 h primarily rason to study Fourir ransform in continuous-tim linar systm is that if th input to a continuous-tim linar systm is a complx sinusoid of frquncy ω, th output is also a complx sinusoid of frquncy ω with a phas shift and a gain govrnd by H(jω ). jωt t) X(jω) Continuous-im linar systm, H( ) H(jω) ω jωt y( t) H ( j ) Y(jω) H(jω ) ω ω ω his form of analysis, as you know, is calld th stady stat analysis (stady stat as complx sinusoid is not transint). It would b nic if th DF can do th sam for Discrt-tim linar systm or And indd it is tru: jω n Discrt-im linar jωn ( n ) y( n ) H( )? systm, H( ) x y( n ) m jωn m jω ( n m) m n m ) m ) m ) jωm m ) jωn H( ) Bfor w go on, lt s introduc a common rprsntation of th DF basd on normalizd frquncy. X( ) π 4 -ω r ω r 2 4 ω X( ) X( ) r ω maps ω [,π/] to 2π X( ) r [,.5] r normalizd frquncy ωr.5 r 2π Pag 5-33

3 Rmarks. Why calld normalizd frquncy? Givn th sampling frquncy f s / and th frquncy f ω 2π, r can b mor succinctly rprsntd as r ω f 2π f s j2π r Whn using normalizd frquncy r, th DF is writtn as H ( ). 2. Why ignor th ngativ frquncy? For ral-valud n), w can dduc th ngativ frquncy from th positiv frquncy: H( j( ω) ) * jωn [ n ) ] * H ( ) H ( n ) ) jωn (* conjugat) H ( ) & H( ) H ( hus, it is sufficint to show only th positiv part only. h followings ar a list of DF proprtis and common transform pairs. hy can b asily dducd from th Z-transform tabls. ) Pag 5-34

4 Computing DF Just lik th continuous-tim countrpart, it is most common to show th DF in j trms of its amplitud rspons H ( ω j ) and phas rspons H( ω ). Howvr, unlik continuous-tim whr w can draw asymptotic approximation (Bod plot) by ltting ω, w can t do that for DF as it is priodic. Evn though thr ar gomtric tchniqus to draw th frquncy rsponss basd on th locations of pols and zros, thy ar byond th scop of this cours. Hr w sttl with plotting th rspons using Matlab. Pag 5-35

5 Exampl : A simpl dlay H ( z) z o gt th magnitud and phas rspons, w can us th matlab routin frqz. In frqz, th Z-transform is spcifid by th dnominator and numrator cofficints: m b() + b(2) z b( m + ) z H z) ( n a() + a(2) z a( n + ) z >> [h,w] frqz([ ],[]); % Assum >> plot(w,abs(h)) % Givs Amplitud Rspons >> % Givs Phas Rspons; Unwrap rmovs phas jumps >> plot(w,unwrap(angl(h))) Notic that th amplitud is flat (gain ) and th phas is linar. A systm with rspons lik this is calld distortion-lss as thy ssntially kp th input intact. h ngativ slop of th phas, calld th group dlay, masurs th dlay of th frquncy componnt. In this cas, d jω d H ( ) ( ω ), i.. On sampl for all frquncis. dω dω Pag 5-36

6 3 4 Exampl 2: Gnral Linar Phas Systm H ( z).+.2z +.4z +.2z +.z >> [h,w] frqz([ ],[]); >> plot(w,abs(h) >> plot(w,unwrap(angl(h))) h amplitud and phas rsponss ar Amplitud Rspons - -2 Phas Rspons Notic that th phas is linar with group dlay 2 (2 sampls) for all frquncis. It can b shown that th impuls rspons must b symmtric about th middl sampl if it has linar phas. Indd, it is tru for our filtr: Linar-phas filtr is vry important in audio and imag applications bcaus. In audio, th prcption of chords rquirs diffrnt frquncis to rgistr at th sam tim instanc. A non-linar phas filtr dlays thos frquncis by diffrnt amount making th chord prcption disprsd. 2. In imag, color dgs rquir spatial cycls to locat prcisly at a particular location. A non-linar phas filtr distorts th dgs by fattning thm. Can a filtr hav zro dlay (phas)? Ys, but such a filtr MUS BE ACAUSAL! For xampl: Pag 5-37

7 Du to th symmtry rquirmnt, a casual IIR filtr can nvr b linar phas. S th following xampl. +.5z Exampl 3: a simpl zro Y ( z) 2..63z +.42z >> [h,w] frqz([.5],[ ]); >> plot(w,abs(h) >> plot(w,unwrap(angl(h))) Amplitud Rspons Phas Rspons Evn though th amplitud rspons is similar to that of xampl 2, this IIR filtr givs a highly non-linar phas distortion. Exampl 4: A wll-dsignd IIR low-pass filtr can provid closd to linar-phas prformanc at last for th pass-band. Mor in Chaptr 9. >> [b,a] buttr(2,.3); >> [h,w] frqz(b,a); >> plot(w,abs(h)) >> plot(w,unwrap(angl(h))) Amplitud Rspons Phas Rspons Pag 5-38

8 Invrs DF hr ar thr approachs to rcovr n) from X( ). Approach : abl lookup or convrt back to Z-transform using z or z Exampl: H( ) j 2πr.5 Using th substitution: z, w hav H ( z).5z n Applying invrs Z-transform, w gt n ) (.5) u( n ) Approach 2: Explicit Invrs DF Rcall th dfinition of DF: X ( jω ) n ) jωn Compar this with th Fourir sris rprsntation of a priodic signal y(t): k y( t) Y k jkω t hy bar a strong rsmblanc: y(t) is priodic in t with priod 2π/ω X( ) is priodic in ω with priod 2π/ hus, w rcogniz DF is in fact th Fourir sris rprsntation in ω (not t) of X( ) with n) as th Fourir sris cofficints! In tim-domain, w can comput th Fourir cofficints π ω ω jkωt Yk y( t) dt 2 π π ω Similarly, w can comput n) using th following formula: π j jωn n ) X ( ω ) dω π 2π.5 jn 2πr or with normalizd frquncy n ) X ( ) dr Approach 3: Numrical approximation by first sampling th spctrum with N points and thn prforming invrs Discrt Fourir ransform..5 W will discuss this mthod in Chaptr. Pag 5-39

9 Exampl: Find n) whos DF is H ( ) Using th invrs formula w hav.5.5 jn 2πr n ) ( j2π r) dr Intgrating by parts yild r.5 r.5 jn 2πr jn 2πr n ) j2π dr j2πn j2πn.5 r.5 his bcoms jπn n ) n sin( πn) cos( n) n π πn cos( πn) sin( πn) 2 n πn jπn j2πn jπn jπn ( ) Furthr simplification can b don: n cos(πn)/n -/n /n -/n /n -/n /n -/n /n sin(πn)/πn 2 W can s that cos(πn) n n n ( ) n othrwis and sin( πn) 2 δ ( n) πn cos( πn) sin( πn) n ) 2 n πn n n ( ) n othrwis Pag 5-4