M2.The Z-Trasform ad its Properties Readig Material: Page 94-126 of chapter 3 3/22/2011 I. Discrete-Time Sigals ad Systems 1
What did we talk about i MM1? MM1 - Discrete-Time Sigal ad System 3/22/2011 I. Discrete-Time Sigals ad Systems 2
MM1: Discrete-Time Sigals: Sequeces Cotiuous-time sigals are defied alog a cotium of times (aalog sigals) Discrete-time sigals are defied at discrete times Digital sigals are those for which both time ad amplitude are discrete Delayed sequece, y[]=x[- 0 ], 0 is a iteger Discrete-time impulse, Uit step sequece, δ [ ] u [ ] Siusoid sequeces, x[]=acos(ω 0 +φ), - < < 3/22/2011 I. Discrete-Time Sigals ad Systems 3 = = 1, 0, 1, 0, = < 0 0 0 0
A liear system: T{ax1[]+bx2[]}=T{ax1[]}+T{bx2[]}=ay1[]+by2[] A time-ivariat system: T{}: x[] y[], T{}: x[- d ] y[- d ], for ay d A causal system BIBO stability MM1: Discrete-Time Systems x[] LTI system: y[]=σ k=- x[k] h[-k] T{ } Covolutio sum: y[]=x[]*h[] FIR: Fiite-duratio impulse respose systems IIR: Ifiite-duratio impulse respose systems y[] 3/22/2011 I. Discrete-Time Sigals ad Systems 4
How about Exercise Oe? Problem 2.24 o page75 of the textbook; Problem 2.36 o page78 of the textbook. I. Discrete-Time Sigals ad Systems 5
MM2 - The Z-Trasform Readig Material: Page 94-126 of chapter 3 3/22/2011 I. Discrete-Time Sigals ad Systems 6
Review of Laplace Trasform Laplace trasform for the cotiuous-time sigals ad systems: Beefits: X x( t) st ( s) = L( x( t)) = x( t) e dt 0 = 1 2 πj δ + j δ j X( s) e Tur the differetial equatio ito algebraic equatio Deal with discotiuous sigals System s trasiet ad steady-state aalysis st ds, t 0, δ δ 0 3/22/2011 I. Discrete-Time Sigals ad Systems 7
Review of Laplace Trasform (cotiue) Existig Theorem: x(t) is Laplace-trasformable if x(t) is piecewise cotiuous over every fiite iterval x(t) is expoetial order, i.e., x(t) e -at < Properties: Liearity L(ax 1 (t)+bx 2 (t))=ax 1 (s)+bx 2 (s) Shift i time L(x(t-a))=e -as X(s) Complex differetiatio L(tx(t))=-d(X(s))/ds Real differetiatio L(d(x(t))/dt)=sX(s)-X(0) Fial value, iitial value. Applicatios Solve the differetial equatios... 3/22/2011 I. Discrete-Time Sigals ad Systems 8
What s the Z-trasform? The z-trasform for the discrete-time sigals is the couterpart of the Laplace trasform for cotiuous-time sigals The z-trasform is a geeralizatio of the Fourier trasform I the aalysis problems, especially for the discrete-time domai, the z-trasform is more coveiet tha the Fourier trasform 3/22/2011 I. Discrete-Time Sigals ad Systems 9
Defiitio of the Z-Trasform The z-trasform of a sequece x[] is = X ( z ) = x[ ] z Relatio with Fourier trasform, z=re jω X ( re jω ) = = ( x[ ] r The Fourier trasform correspods the z-trasform o the uit circle i the z-plae ) e jω Covergece of the series? 3/22/2011 I. Discrete-Time Sigals ad Systems 10
Regio of Covergece (ROC) For ay give sequece, ROC is a set of values of z for which the z-trasform coverges ROC depeds oly o z, i.e. X(z) <, if = x [ ] z The z-trasform ad all its derivatives are cotiuous fuctios of z withi the ROC ( Lauret series) A ratioal fuctio X(z) is a ratio of two polyomials i z: X(z)=P(z)/Q(z) Zeros of X(z); poles of X(z) 3/22/2011 I. Discrete-Time Sigals ad Systems 11 <
Computatio of the z-trasform Right-sided expoetial sequece (example 3.1 pp.98) a u[] 1/(1-az -1 ), z > a Left-sided expoetial sequece (example 3.2, pp.99) -a u[--1] 1/(1-az -1 ), z < a Sum of two expoetial sequeces x[]=(1/2) u[]+(-1/3) u[] Accordig to defiitio (example 3.3, pp.101) Usig the liearity of z-trasform (example 3.4, pp.102) The ROC is the itersectio of the idividual ROCs 3/22/2011 I. Discrete-Time Sigals ad Systems 12
Computatio of the z-trasform(cotiue) Two-sided expoetial sequece (example 3.5, pp.102) x[]=(-1/3) u[]-(1/2) u[--1] Fiite-legth sequece (example 3.6, pp.103) a 0 N 1 1 x[ ] = X( z) = N 1 0 otherwise z N z a z a The ROC is the etire z-plae except the origi (z=0) The N zeros are z k =ae j(2kπ/n ), k=0,1, N-1 The z-trasform icludes a algebraic expressio ad a ROC 3/22/2011 I. Discrete-Time Sigals ad Systems 13 N
(page104) 3/22/2011 I. Discrete-Time Sigals ad Systems 14
Properties of ROC P1: The ROC is a rig or disk i the z-plae cetered at the origi, i.e., 0 r R z r L P2: The Fourier trasform of x[] coverges absolutely iff the ROC of the z-trasform of x[] icludes the uit circle i the z-plae P3: The ROC ca ot cotai ay poles P4: If x[] is a fiite-duratio sequece, the the ROC is the etire z-plae, except possibly z=0 or z= X ( z ) = N 2 = N 1 x [ ] z If N 1 <0, the exceptio will be z= ; if N 2 >0, the exceptio will be z= 0;if N 2 >0 ad N 1 <0, the exceptio will be both. 3/22/2011 I. Discrete-Time Sigals ad Systems 15
Properties of ROC (cotiue) P5: If x[] is a right-sided sequece, the the ROC exteds outward from the outermost fiite pole to (ad possibly icludig) z= If the sequece has ozero values for egative values of, the ROC will ot iclude z= P6: If x[] is a left-sided sequece, the the ROC exteds iward from the iermost ozero pole to (ad possibly icludig) z= 0 If the sequece has ozero values for positive values of, the ROC will ot iclude z= 0 P7: If x[] is a two-sided sequece, the the ROC will cotai a rig i the z-plae, bouded o the iterior ad exterior by a pole ad does ot cotai ay poles The ROC must be a coected regio 3/22/2011 I. Discrete-Time Sigals ad Systems 16
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Determiig ROC by System Properties The ROC ca be specify through some time-domai properties Example 3.7 pp.110 If the system is stable, the the ROC is. 1/2< z <2 Is the system causal i this case?.. No!! Why If the system is causal, the the ROC is. z >2 Is the system stable i this case?.. No!! Why 3/22/2011 I. Discrete-Time Sigals ad Systems 18-2 Im 1/2 Pole-zero plot for the system fuctio Re
3/22/2011 I. Discrete-Time Sigals ad Systems 19 Z-Trasform Properties Liearity ax 1 []+bx 2 [] ax 1 (z)+bx 2 (z), ROC cotais R 1 R 2 If there is o pole-zero cacellatio, ROC=R 1 R 2 If there is the pole-zero cacellatio, ROC R 1 R 2 possibly Example 3.6, p.103 Time shiftig x[- 0 ] z - 0 X(z), ROC= R x (except for the possible additio or deletio of z=0 or z= ) Expoetial multiplicatio z 0 x[] X(z/z 0 ), ROC= z 0 R x Example 3.15, pp.121 a z a z z z X N u a u a x N N N = = 1 1 ) ( ] [ ] [ ] [ 2 2 1 0 1 0 0 cos 2 1 cos 1 ) ( ] [ ) cos( ] [ + = = z r z r z r z X u r x ω ω ω
Z-Trasform Properties (cotiue 1 ) Differetiatio of X(z) x[] zdx(z)/dz, ROC= R x Example 3.16, pp.122 log( 1 + az 1 ), z > a x[ ] = ( 1) + 1 u[ Cojugatio of a complex sequece x * [] X * (z * ), ROC= R x Time reversal x * [-] X * (1/z * ), ROC=1/ R x a 1] See Example 3.18 pp.124 3/22/2011 I. Discrete-Time Sigals ad Systems 20
Z-Trasform Properties (cotiue 2 ) Covolutio of sequeces x 1 []*x 2 [] X 1 (z)x 2 (z), ROC cotais R 1 R 2 Ca you prove that see page 124 Iitial-value theorem: If x[] is zero for <0 (causal), the x[0]=lim z X(z) Summary of z-trasform properties table 3.2 (pp.126) 3/22/2011 I. Discrete-Time Sigals ad Systems 21
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Exercise Two See the distributed paper 3/22/2011 I. Discrete-Time Sigals ad Systems 23