ChapterThezTransformPPT教学课件

1、The z-TransformAnother transform, any end?Motivation: analogous to LT in CTSo, the definition and properties of the z-Transform closely parallel those of the Laplace TransformPay attention to the relation and difference between LT and zT第1页/共98页ContentsThe z-Transform (bilateral and unilateral) and

2、the inverse z-transformThe Region of Convergence for the zTProperties of the z-TransformAnalysis DT LTI system using zT第2页/共98页nReviewhnxn=znyn=H(z)znkkzkhzH)( 10.1 The z-Transform(assuming it converges)(*)(zHzkhzzkhzkhknxnhnxnynkknkknk第3页/共98页Definition:nnznxzX)(The (Bilateral)z-Transform ( )Zx nX

3、z When z=ej (unit magnitude), H(ej) corresponds to the FT of hnWhen z=rej(complex), H(z) corresponds to the zT of h(t)hnxn=znyn=H(z)znkkzkhzH)(第4页/共98页l Relationship between DTFT and zT(Similar to FT and LT)nnjjezenxeXzXj)()()()(jeXzXUnit circlejez 1ImRez-planeUnit circle (r= 1) in the ROC DTFT X(ej

4、) exists第5页/共98页xnr-nlROC=z=rej at which )()(zXeXj)()()(nnnjnnnjjrnxDTFTernxrenxreXnnrnxdepends only on r=|z|, just like the ROC in s-plane only depends on Re(s)第6页/共98页l Relationship between LT and ZTIf),( :Re),()(21ssXtxLand sampled signalnTpnTtnTxttxtx)()()()()(nnsTpLenxsX)()(supposennZznxzXnTxnx

5、)()(We can getsTez (1)(2)(1) = (2)Then, z-transform as DT version of Laplace transform with z=esT第7页/共98页s-plane - z-plane relationship j axis in s-plane(s=j) |z|=|esT|=1 a unit circle in z-plane LHP in s-plane, Re(s)0 |z|=|esT|0 |z|=|esT|1, outside the |z|=1 circle. Special case, Re(s)=+|z|=. A ver

6、tical line in s-plane, Re(s)=constant |esT|=constant, a circle in z-plane.z=esT第8页/共98页 Example 10.1nuanxn 01)()(nnnnnazznuazXazzazzX111)(| ,111azaznuaZnaz 第9页/共98页Rez-planeIm1-1a0a11| ,111zznuZSpecially,The ROC of signals in example 10.1(right-sided signal)unit circle-1z-planeImRe1unit circle第10页/共

7、98页| ,111azaznuaZnROC outside |z|=1 unstableinclude |z|=1 stable第11页/共98页| ,111azaznuaZnROC outside |z|=1 unstableinclude |z|=1 stable第12页/共98页 Example 10.21 nuanxn11)( 1)(nnnnnazznuazXazazzazzX ,)(111zazazann11111)(az | ,11 11azaznuaZn第13页/共98页The ROC of Example 10.2(a left-sided signal)11-aImRe0a1

8、z-plane第14页/共98页| ,11 11azaznuaZnROC include |z|=1 stableinside |z|=1 unstable第15页/共98页ExampleznZ0 , 1The ROC is the entire z-plane .1nnzn第16页/共98页Example 10.3216317nununxnn1121163117zzZ21| ,z第17页/共98页Example 10.44sin31nunnxn)31)(31(231)(4/4/jjZezezzzX3121312144nuejnuejnjnj1414311121311121)(zejzejzX

9、jjZ3/ 1| ,z第18页/共98页Rational z-Transformslxn = linear combination of DT exponentialsX(z) is rational)()()(zDzNzXpolynomials in z characterized (except for a prefactor) by its poles and zeros. sometimes, it is convenient for X(z) to be expressed in terms of polynomials in z-1 (nr0 will also in the RO

10、C. z-planeThe ROC of a right-sided signalProver0第22页/共98页converges slower than10Nnnrnx11Nnnrnx第23页/共98页Property5: If xn is left sided, and if the circle |z|=r0 is in the ROC, then all values of z for which 0|z|r0 will also in the ROC. The ROC of a left-sided signalz-planebr0ImRe第24页/共98页Property6: I

11、f xn is two sided, and if the circle |z|= r0 is in the ROC, then the ROC will consist of a ring in the z-plane that includes the circle |z|= r0 .Normally, r1|z|r2. (r1 1 No z-transform for b|n|.第30页/共98页Property7: If the z transform X(z) of xn is rational, then its ROC is bounded by poles or extends

12、 to infinity. In addition,no poles of X(z) are contained in the ROC.xxImReUnit circlez-planexx第31页/共98页Property8: If the z-transform X(z) of xn is rational if xn is right sided, then the ROC is the region in the z-plane outside the outmost pole i.e., outside the circle of radius equal to the largest

13、 magnitude of the poles of X(z).Furthermore, if xn is causal, then the ROC also includes z=.第32页/共98页Property9: If the z-transform X(z) of xn is rational, 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 s

14、mallest magnitude of the poles of X(z) other than any at z=0 and extending inward to and possibly including z=0. in particular. if xn is anticausal, then the ROC also includes z=0. 第33页/共98页Example 10.8)21)(311 (1)(11zzzXImReUnit circlez-planeHow may possible ROCs in this Figure?第34页/共98页There are t

15、hree possible ROCs shown in Figure before.2 z231 z31 zROC1ROC2ROC3left sidedright sidedtwo sidedImReUnit circlez-planeImReUnit circlez-planeImReUnit circlez-planeHomework: 10.2 10.3 10.6 10.7第35页/共98页From)()(nxrDTFTreXzXnj)(jnreXIDTFTnxrdereXjnj2)(21derreXnxnjwj)(212 10.3 The Inverse Z-Transform第36页

16、/共98页SoLet Integration around a counterclockwise closed circular contour centered at the origin and with radius r.dzzzXjnxn 1)(21jrez dzjzd1 第37页/共98页The calculation for inverse z-Transform:(1) Integration of complex function by equation.(2) Compute by PFE ( Partial Fraction Expansion ).(3) power-se

17、ries expansion 第38页/共98页Partial Fraction Expansion of rational X(z)Normally, miiizaAzX111)(nuaAniiThe ROC is outside z=aithe inverse ZT isThe ROC is inside z=aithe inverse ZT is 1nuaAnii第39页/共98页31,)311 (2)411 (1)(11zzzzXExample 10.931311411653111 zzzzzX,)()(Then第40页/共98页we have known411nunxn41 z, Z

18、T)(14111 z3122nunxnZT)(13112 z31 z,)()(nununxnxnxnn3124121 So313112411111 zzzzX,)()()(第41页/共98页Example 10.10the same X(z) in example 10.9, ROC3141 z ZT)(13112 z31 z,411nunxn41 z, ZT)(14111 z)(13122 nunxn 1)31(2)41(nununxnnSoBecause 第42页/共98页When ROC41z 1)31(2 1)41(nununxnn313112411111 zzzzX,)()()(第4

19、3页/共98页Example 10.12 zzzzX032412,)(othersnnnnx, 01, 30, 22, 4From the power-series definition 13224 nnnnx orZT0nn zzn00,( ) nX zx n zcoefficient of z-n第44页/共98页Example 10.13Consider 111 azzX)(az ROC1:)11 az111 az1 az221 zaaz22 za. 2211zaaz111 az. 2211zaaz,.,21aanx long division第45页/共98页az ROC2:)11 a

20、z1za1 221zaaz 22za. 221zazaza11 0 ,.,12aanx111 az. 221zaza long divisionHomework:10.9 10.10 10.24第46页/共98页10.4 Geometric Evaluation of The FT From the Pole-zero Plot)(1111)()()()()()(jjNkkjMrrjNkkMrreeHpezepzzzzHkrjkkjjrrjeBpeeAzeNkkMrrjBAeH11)(NkkMrr11)(第47页/共98页RezImzj1p2pje1z2z1122NkkMrrjBAeH11)(

21、NkkMrr11)()()()(jjeeHzH第48页/共98页lowpasshighpass1pje1p1zje)(jeH)(jeH00第49页/共98页bandpass)(jeH01p2pje1p2p1zje)(jeH0bandstop第50页/共98页all-pass1p2prrr1r11z2z)(jeHT2T20)(jeHjeje第51页/共98页Example first order systems nuanhn azzazzH 111)(aeeeHjjj )(11A1Bmeunit circle1|a第52页/共98页0| )(|jeHa=0.95a=0. 5第53页/共98页11

22、A2Bmeunit circle*11BExample second order systems 第54页/共98页0| )(|jeHr =0.95r =0. 75第55页/共98页10.5.1 Linearity)(11zXnxZTR1R2),()(2121zbXzaXnbxnaxZTR1R2IfThenNote: R1R2 may be larger than R1 or R210.5 The Properties of Z-Transform)(22zXnxZT第56页/共98页10.5.2 Time ShiftingRIf)(00zXznnxnZTR, except the possi

23、ble addition or deletion of the origin or infinityThennuanazzaz 111az ,ZTExampleThen 11 nuanZTazazz 1111az ,)(zXnxZTFrom: 第57页/共98页10.5.3 Scaling in z-DomainRIf)/(00zzXnxzZTnThen|z0|R)(00zeXnxejZTnjSpecially,RR)() 1(zXnxZTnif)(zXnxZT0第58页/共98页10.5.4 Time ReversalRIf)1(zXnxZT1/RThennuan111 azaz ,ZT n

24、uan 111111 zaazaz)/(1 az,ZTExampleThen and )(zXnxZTFrom:1 nuanZT1 az,111 az第59页/共98页10.5.5 Time ExpansionRIf)()(kZTkzXnxRk/1Then)(zXnxZTkofmultipleanotisnifkofmultipleaisnifknxnxk, 0,/)( mmnknnnkkzkmxznxznxzX/)(第60页/共98页10.5.6 ConjugateRIfRThen)(*zXnxZT)(zXnxZTNote: If xn is real, Xz = X*(z*). Thus

25、if X(z) has a pole (or zero) at z=z0, it must have a pole (or zero) at the complex conjugate point z=z0*.第61页/共98页10.5.7 The Convolution Property)()(2121zXzXnxnxZTR1 R2IfThenNote: if R1R2=, is not existed.)()(zXzX21)(11zXnxZTR1R2)(22zXnxZT第62页/共98页 Examplenunxkxngnk ZT)(zGRIfThen)(zG111)(zzXR|z|1)(z

26、XnxZT第63页/共98页10.5.8 Differentiation in the z-DomainRIfR)(zXdzdznnxZTThenExampleThen nunanZT,)(2111111 azazazdzdzaz )(zXnxZTnuan111 azaz ,ZTFrom: 第64页/共98页10.5.9 The Initial-(and Final-)Value Theorems)(0limzXxzIf xn=0 for n r1 .),()(limlimzXzxnxzn11 The ROC of (z-1)X(z) contains |z|=1第65页/共98页10.6 S

27、ome ZT PairsPage 776 Table 10.2Example), 3(|:| ,) 3() 1()2(753)(23246zzzzzzzzXnxZT?,0 xxWe can get1)(0limzXxz?xnot existHomework: 10.11 10.16 10.17 10.31第66页/共98页Consider a LTI system:hnH(z)xnyn=xn*hnX(z)Y(z)=X(z)H(z)(nzzH)(nz10.7 Analysis and Characterization of LTI systems Using ZT第67页/共98页10.7.1

28、Causality(1) A causal system.),(00 nnh(|z| r1 )ROC:),(zH(2) For rational ,)()()(zDzNzH ROC:(including infinity)exterior of a circlea. exterior of a circle outside the outmost pole(r1 )(|z| r1 ),(zHb. The order of the numerator N(z) cannot be greater than the order of the D(z) denominator.A causal sy

29、stem第68页/共98页Example 10.2081412223 zzzzzzH)(not causal systemExample 10.211525222150152222111 zzzzzzzzH.)(.(.)(2 z,causal systemROC:1rz 第69页/共98页10.7.2 Stability(1) A stable system|z|=1(the unit circle),(zH(2) A causal stable system with rational )()()(zDzNsHAll poles lies inside the unit circle of

30、z-planeROC:Includes Example 10.2224 Read by yourself!第70页/共98页1067458em.013254n.n140.n0.n140.n0.第71页/共98页10.7.3 LTI Systems Characterized by Linear Constant-Coefficient Difference Equations Z-transform: MkkkNkkkzXzbzYza00)()()()()(zXzYzazbzHNkkkMkkk 00(rational) Usually, a practical system is causal

31、 and stable.MkkNkkknxbknya00第72页/共98页Example 10.2511211311 zzzH)(131121 nxnxnyny)()(12131211 nununhnn21zROC1:We can getROC2:21z)()(nununhnn 12131121第73页/共98页10.7.4 Examples Relating System Behavior to the System FunctionExample 10.26 Suppose that we are given the following information about an LTI s

32、ystem:1.111 10 ,.23nny nau nais real)/(nunxn611 2.nnx)(12 nny)(1472 Determine the system function H(z).第74页/共98页111 10 ,.23nny nau nais real)/(nunxn611 Solution: )()()(zXzYzH11 111111 1 (1), | 1/6610 , | 1/2111123zTzTx nzzay nzzz)()()(11113112116113510 zzzzaaAnd So 第75页/共98页For 47) 1(H9 a21zROC:2311

33、613261165 nxnxnxnynyny)()(111131121161121 zzzzzH212161651316131zzzznnx)(12 nny)(1472 Then 第76页/共98页Example 10.27 a stable,causal system H(z) (rational) contains a pole z=1/2 and a zero somewhere on the unit circle. Other zeros and poles are unknown.Whether can we definitely say that it is true or fa

34、lse each of following statements?(a) convergence.)(nhFn21(b) for somewhere.0)(jeHTT(c) hn has finite duration.(d) hn is real.FInsufficient information第77页/共98页Temr02/1unit circle-1-11 1)()(2)()(2zHdzdzzHzHdzdzzG(e) is the impulse response of a stable system. nhnhnng 第78页/共98页10.8.1 System Functions

35、for Interconnections of LTI Systems)(zH1)(zH2)(zX)(zY)()()(zHzHzH21 )(zH1)(zH2)()()(zHzHzH21 )(zX)(zYParallelSeries(cascade)10.8 System Function Algebra and Block Diagram Representations第79页/共98页)(zH1)(zH2)(zX)(zY)()()()(zHzHzHzH2111 Feed-back第80页/共98页)(zX1)(zX2)()(zXzX21 Basic elements:adder)(zaX)(

36、zXa)(zXz1 )(zX1 zunit delaymultiplication by a coefficient10.8.2 Block Diagram Representation for causal LTI Systems Described by Difference Equations and Rational System Functions 第81页/共98页Example 10.28nxnyny 141)()()(zXzYzzH 14111)()(1zYzzWLet)(41)()(zWzXzYThen1 nynw)()(41)(1zXzYzzY第82页/共98页Equiva

37、lent representationnxny41/1-znw1 z41/ nxnynw)()(1zYzzW)(41)()(zWzXzY第83页/共98页Example 10.29)()()(zXzYzzzH 1141121)()()()(zXzYzzzH 11214111)()()(zYzVz 121Let)()(zXzzV14111 12 nvnvny第84页/共98页nxnvnyns1-z2-1-z41/nw1 z41/nx2 ny 1nvnsnwNote: the number of integrator =the order of the difference equationcanonic)()(zXzzV14111 )()()(zYzVz 121第85页/共98页Example 10.3021118141114112111 zzzzzH)()(nxnynyny 281141(1) direct-form)()(zYzzF1 Let)()()(zYzzFzzE21 第86页/共98页nx1-znf41/-81/ny1-zneThen)()()()(zEzFzXzY8141 )()(zYzzF1 )()()(zYzzFzzE21 第87页/共98页