1、Sichuan University Ch 10 The z-Transform1The Z-TransformThe primary focus of this chapter will be on:1. The z-Transform and the Region of Convergence for the z-Transform2. The Inverse z-Transform3. Geometric Evaluation of the Fourier Transform from the Pole-Zero Plot4. Properties of the z-Transform
2、and some Common z-Transform Pairs5. Analysis and Characterization of LTI Systems Using z-Transform6. System Function Algebra and Block Diagram Representations7. The Unilateral z-TransformSichuan University Ch 10 The z-Transform210.0 INTRODUCTIONl z-transform is the discrete-time counterpart of the L
3、aplace transform, the motivation for and properties of the z-transform closely parallel those of the Laplace transform. However, they have some important distinctions that arise from the fundamental differences between continuous-time and discrete-time signals and systems. l z-transform expand the a
4、pplication in which Fourier analysis can be used. Sichuan University Ch 10 The z-Transform310.1 The z-TransformThe z-transform of a general discrete-time signal xn is defined asnnznxzX)(where z is a complex variable. We will denote the transform relationship between xn and X(z) as ( )Zx nX zThe defi
5、nation of the z-transform :The relationship between xn and X(z) Sichuan University Ch 10 The z-Transform4The relationship between the z-transform and the discrete-time Fourier transformExpressing the complex variable z in polar form as,jreznnjjrenxreX)()(nnjnernx)(jreX is the Fourier transform of xn
6、 multiplied by a real exponential So, the z-transform is an extension of the DTFT.nrFor r = 1, or equivalently, |z| = 1, z-transform equation reduces to the Fourier transform. )()(nxeXzXjezjFSichuan University Ch 10 The z-Transform5The z-transform reduces to the Fourier transform for values of z on
7、the unit circle.z-plane 1ImReUnit circlejezDifferent from the continuous-time case, the z-transform reduces to the Fourier transform on the contour in the complex z-plane corresponding to a circle with a radius of unity . The z-transform reduces to the discrete-time Fourier transformThe z-transform
8、reduces to the discrete-time Fourier transformSichuan University Ch 10 The z-Transform6In general, the z-transform of a sequence has associated with it a range of values of z for which X(z) converges, and this range of values is referred to as the region of convergence (ROC). For convergence of the
9、z-transform, we require that the Fourier transform of converge. For any specific sequence xn, it is this convergence for some value of r.nrnxIf the ROC includes the unit circle, then the Fourier transform also converges. The region of convergence ( ROC )depends only on r= |z|, just like the ROC in s
10、-plane only depends on Re(s).Sichuan University Ch 10 The z-Transform7.)(01nnnnnzznuzXFor convergence of X(z), we require that nnz01Consequently, the region of convergence is the range of values of z for which 11.zor zThen zzzzzzXnn,11)(101Unit circlez-planeIm a 1 RePole-zero plot and region of conv
11、ergence for Example 10.1 for 0 1Example 10.1 Consider the signal.nunxnSichuan University Ch 10 The z-Transform8( )( )x nu n101( )1nnX zzz1z The ROC does not include the unit circle, consequently, it is impossible to obtain the Fourier transform from ,( )X zzjeImReZ plate1 11()(2)1jjkX eke ,11Zzu nzz
12、 Now Consider the step signalSichuan University Ch 10 The z-Transform9Example 10.2 Determine the z-transform of .1nuanxnnnnznuazX 1)(If , this sum converges andzazX1111)(11zaUnit circlez-planeIma 1 RePole-zero plot and region of convergence for Example 10.2 for 0 11nnnza1nnnza01)(1nnzaazazzaz,111Sic
13、huan University Ch 10 The z-Transform10Example 10.3 Consider a signal that is the sum of two real exponentials: .216317nununxnnThe z-transform is then nnnnnnznuznuzX216317)(13117z21zIm 1/3 1/2 1 3/2 Re12116z121131123111zzz213123zzzzSichuan University Ch 10 The z-Transform11Example 10.4. Consider the
14、 signal:1( )( )( )2(1)2nnx nu nun 10111( )( )221111 212nnnnnnX zzzzz1ROC:22z Generally，the ROC of consists of a ring in the z-plane centered about the origin.( )X z2 21/21/2Z plateImReUnit circleSichuan University Ch 10 The z-Transform12Example 10.5 Consider the signal .4sin31nunnxn312131214/4/nuejn
15、uejnxnjnjThe z-transform of this signal isnnjnnjzejzejzX014/014/31213121)(,|z| 1/3 Im 1/3 1 Re4/314/31231)(jjezezzzX14/3114/3111211121zejzejjjSichuan University Ch 10 The z-Transform13()( )( )( )()iippzzN zX zMD zzz( )X z if the z-transform is rational, its numerator and denominator polynomial can b
16、e factarized. so, the z-transform is chacterized by all its poles and zeros except a constant factor . ( )X zMThe geometric representation of the z-transformthe Pole-Zero plot：( )X zSichuan University Ch 10 The z-Transform14 xn can be only determined by all poles and zeros of X(z) and the ROC of the
17、 X(z). The pole-zero plot, illustrate all poles and zeros of the z-transform in z plane, is the geometric representation of the z-transform .( )X z( )X z The pole-zero plot is especially useful for describing and analyzing the properties of the discrete-time LTI system.Sichuan University Ch 10 The z
18、-Transform1510.2 The Region of Convergence for the z-TransformProperties of the ROC for z-transform：Property 1 The ROC of X(z) consists of a ring in the z-plane centered about the origin.Property 2 The ROC does not contain any poles.Property 3 If xn is of finite duration, then the ROC is the entire
19、z-plane, except possibly z = 0 and/or z = . nnx n rConvergence is dependent only on and not on .rzSichuan University Ch 10 The z-Transform16Example 10.6 Consider the unit sample signal n. 1Znnnn z with an ROC consisting of the entire z-plane, including z = 0 and z = . On the other hand, 111Znnnnzz t
20、he ROC consists of the entire z-plane, including z = but excluding z = 0. Similarly, 11Znnnnzz the ROC consists of the entire z-plane, including z = 0 but excluding z = . Sichuan University Ch 10 The z-Transform17Property 4 If xn is a right-sided sequence, and if the circle is in the ROC, then all f
21、inite values of z for which will also be in the ROC.0rz 0rz Imz-plane ReSichuan University Ch 10 The z-Transform185) Property If xn is a left-sided sequence, and if the circle is in the ROC, then all values of z for which will also be in the ROC.00rz 0rz Imz-plane Re6) Property If xn is two sided, a
22、nd if the circle is in the ROC, then the ROC will consist of a ring in the z-plane that includes the circle . Imz-plane Re0rz 0rz Sichuan University Ch 10 The z-Transform19Sichuan University Ch 10 The z-Transform2011101( )1()NNNNNnnNna zzaX za zazzzapoles：za（order 1）0z （order N-1）zeros：2jkNzae(0,11)
23、kN jIm z Re z(8)N aa0 0(1)N At z=a, the zero cancel the pole. Consequently, there are no poles other than at the origin.Example 10.7 Conside the finite duration xn( )x n ,01,nanN0a 0,othernSichuan University Ch 10 The z-Transform21Example 10.8 Consider a two sided sequence.0,bbnxn 1nubnubnxnn11 ,1Zn
24、b u nzbbz 11111,.1Znb u nzb zb For b 1, there is no common ROC, and thus the sequence will not have a z-transform.For b 1, the ROCs overlap, and thus the z-transform for the composite sequence isbzbbzbzzbbzbbzzX1,11111)(12111Unit circleImb 1/b ReSichuan University Ch 10 The z-Transform22 Property 8
25、If the z-transform X(z) of xn is rational, and if xn is right sided, then the ROC is the region in the z-plane outside the outermost pole i.e., outside the circle of radius equal to the largest magnitude of the poles of X(z). Furthermore, if xn is causal, then the ROC also includes z = . Property 9
26、If the z-transform X(z) of xn is rational, and if xn is left sided, then the ROC is the region in the z-plane inside the innermost nonzero pole i.e., inside the circle of radius equal to the smallest magnitude of the poles of X(z) other than any at z = 0 and extending inward to and possibly includin
27、g z = 0. In particular, if xn is anticausal, then the ROC also includes z = 0. Property 7 If the z-transform X(z) of xn is rational, then its ROC is bounded by poles or extends to infinity. Sichuan University Ch 10 The z-Transform23Example 10.9 Consider all of the possible ROCs that can be connected
28、 with the function.2111)(1131zzzX ReUnit circleImThe ROC if xn is left sided The ROC if xn is right sided The ROC if xn is two sided and is the only one of the three for which the Fourier transform converges.zeros：121,23zz0z （order 2）poles：Sichuan University Ch 10 The z-Transform24()( ) isinsidethe
29、ROC.jnj nnjnX rex n r ex n rzreF2Applying the inverse Fourier transform to both sides 1 ( )()2njj nx n rX reed21( )()2jnj nx nX rer ed一一.The expression of the inverse z-transform：10.3. The Inverse z-TtansformMultiplying both sides by , we obtain nrChanging the variable of integration from to z with
30、r fixed and varying over a interval: jzddjredzjThus, the basic inverse z-transform equation is:dzzzXjnxn 1)(212Sichuan University Ch 10 The z-Transform25The formal evaluation of the integral for a general X(z) requires the use of contour integration in the complex plane. The symbol denotes integrati
31、on around a counterclockwise circular contour centered at the origin and with radius r. There are two alternative procedures for obtaining a sequence from its z-transform: one is partial-fraction expansion, the other is power-series expansion. 二二. The procedures for obtaining a sequence from its z-t
32、ransform：Sichuan University Ch 10 The z-Transform26The partial-fraction expansion1( )1iiiAX za zFor rathional z-transform X(z), determine all poles of it. Then, it can be expanded by the method of partial-fraction, and we obtain expression of the z-transform as a linear combination of simpler terms:
33、Now specify the ROC associated with each term.Finally, determine the inverse z-transform of each of these indivial terms based on the ROC associated with each.So that the inverse transform of X(z) equals the sun of the inverse transforms of the individual terms in the equation.Sichuan University Ch
34、10 The z-Transform27Example 10.10 Consider the z-transform .3141,113)(131141165zzzzzXThere are two poles, one at z=1/3 and one at z=1/4. Performing the partial-fraction expansion, we obtain 1311411211)(zzzX1411411 ,41nZu nzz 131131221,31nZunzz 131241nununxnnSichuan University Ch 10 The z-Transform28
35、Power-series expansionThis procedure is motivated by the observation that the definition of the z-transform can be interpreted as a power series involving both positive and negative powers of z. The coefficients in this power series are, in fact, the sequence values xn. 212( ) 2 1012nnX zx n zxzxzxx
36、zxzSichuan University Ch 10 The z-Transform29Example 10.11 Consider the z-transform .0,324)(12zzzzXFrom the power-series definition of the z-transform, we can determine the inverse transform of X(z) by inspection:otherwisennnnx, 01, 30, 22, 4That is,13224nnnnx00nZnnz Some useful ZT pairs:Sichuan Uni
37、versity Ch 10 The z-Transform30Example 10.10 Consider the z-transform .,11)(1azazzX Then performing long division: 222211112211111zazaazazazazzaazFrom the ROC, we can conclude that the corresponding sequence xn is right-sided, so that we arrange the numerator polynomial and the denominator polynomia
38、l with a order of the power of z decreasing (or a order of the power of increasing).1z12210111nnnaza zaza z nuanxnSichuan University Ch 10 The z-Transform31 If the ROC is |z|, before performing long division, we arrange the numerator polynomial and the denominator polynomial with a order of the powe
39、r of z increasing (or a order of the power of decreasing).1zzazaazzaza1112211111221111nnna za zaza z 1nuanxn11( ),.1X zzaaz Then performing long division: Sichuan University Ch 10 The z-Transform32For two-sided sequence 111536( )11(1)(1)43zX zzz1143zperforming the long division to the two terms resp
40、ectively.1112( )111143X zzz1ROC2ROC1ROC :| 1/4z 2ROC :| 1/3z Sichuan University Ch 10 The z-Transform33the Residue Theorem11( )( )2 ncx nX z zdzj1( )Res( ), niix nX z zz is a pole outside the contour C 。iz0n ，1( )Res( ), niix nX z zz is a pole inside the contour C。iz0n ，Contour integrationSichuan Un
41、iversity Ch 10 The z-Transform34Example 10.11 Consider the z-transform . 1,5 . 0112)(23zzzzzzzXkkpznpznkkzzzzzszzXsnx22315 . 0112Re)(ReFrom the definition of the inverse z-transform,dzzzXjnxn 1)(21This contour integration can be evaluated through using the Residue Theorem, thusSince the ROC is |z|1,
42、 so the corresponding sequence is right sided.Sichuan University Ch 10 The z-Transform352nFor , has only two first-order poles: 1)(nzzX5 . 0, 121zzThennznznzzzzzzzznx5 . 01381125 . 0125 . 022312231nFor , has three first-order poles: 1)(nzzX0, 5 . 0, 1321zzzThen5 . 325 . 68nx0nFor , has two first-ord
43、er poles:and a second-order pole:1)(nzzX, 5 . 0, 121zz03zSichuan University Ch 10 The z-Transform3613112)(Re, 85 . 012)(Re5 . 02235 . 01122311zznzznzzzzzzXszzzzzzXs65 . 0112!121)(Re0222301zznzzzzzzdzdzzXs16138nxThenConsequently, 25 . 0138 15 . 3nunnnxnSichuan University Ch 10 The z-Transform37 In th
44、e discrete-time case, the Fourier transform can be evaluated geometrically by considering the pole and zero vectors in the z-plane. Since in this case the rational function is to be evaluated on the contour |z| = 1, we consider the vectors from the poles and zeros to the unit circle. Consider a firs
45、t-order causal discrete-time system with a impulse response:nuanhnIts z-transform is azazzazzH,11)(110.4. Geometric Evaluation of The Fourier Transform From The Pole-Zero PlotSichuan University Ch 10 The z-Transform38For |a| 1, the ROC includes the unit circle, and consequently, the Fourier transfor
46、m of hn converges and is equal to H(z) for .The frequency response for the first-order system is jez 1()1jjH eae 12()/jH eVV the pole-zero plot for H(z), including the vectors from the pole (at z = a) and zero (at z = 0) to the unit circle. Sichuan University Ch 10 The z-Transform39Magnitude of the
47、frequency response for a = 0.95 and a = 0.5 Phase of the frequency response for a = 0.95 and a = 0.5 the magnitude of the frequency response will be maximum at =0 and will decreases monotonically as increases from 0 to .for 01aSichuan University Ch 10 The z-Transform40for 10 aa a1V2V jeRe zjIm z1 1(
48、)jH e0.5a0.95a0 020.5a0.95aMagnitude of the frequency response for a = -0.95 and a = -0.5 Phase of the frequency response for a = -0.95 and a = -0.5 Sichuan University Ch 10 The z-Transform411) LinearityNote: ROC is at least the intersection of R1 and R2, which could be empty, also can be larger tha
49、n the intersection. For sequence with rational z-transform, if the poles of aX1(z)+bX2(z) consist of all of the poles of X1(z) and X2(z) (if there is no pole-zero cancellation), then the ROC will be exactly to the overlap of the individual ROC. If the linear combinition is such that some zeros are i
50、ntroduced that cancel poles, then ROC may be larger.Ifand 111 ( ),Zx nXzwith ROCR 222 ( ),Zx nXzwith ROCR then121212 ( )( ),Zax nbx naXzbXzwith ROC containing RR 10.5. Properties of The z-TransformSichuan University Ch 10 The z-Transform42Except for the possible addition or deletion of the origin or
