ChapterThezTransformPPT教學(xué)課件

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頁/共98頁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頁/共98頁nReviewhnxn=znyn=H(z)znkkzkhzH)( 10.1 The z-Transform(assuming it converges)(*)(zHzkhzzkhzkhknxnhnxnynkknkknk第3頁/共98頁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頁/共98頁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頁/共98頁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頁/共98頁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頁/共98頁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頁/共98頁 Example 10.1nuanxn 01)()(nnnnnazznuazXazzazzX111)(| ,111azaznuaZnaz 第9頁/共98頁Rez-planeIm1-1a0a11| ,111zznuZSpecially,The ROC of signals in example 10.1(right-sided signal)unit circle-1z-planeImRe1unit circle第10頁/共

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

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

9、jjZ3/ 1| ,z第18頁/共98頁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頁/共98頁converges slower than10Nnnrnx11Nnnrnx第23頁/共98頁P(yáng)roperty5: 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頁/共98頁P(yáng)roperty6: 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頁/共98頁P(yáng)roperty7: 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頁/共98頁P(yáng)roperty8: 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頁/共98頁P(yáng)roperty9: 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頁/共98頁Example 10.8)21)(311 (1)(11zzzXImReUnit circlez-planeHow may possible ROCs in this Figure?第34頁/共98頁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頁/共98頁From)()(nxrDTFTreXzXnj)(jnreXIDTFTnxrdereXjnj2)(21derreXnxnjwj)(212 10.3 The Inverse Z-Transform第36頁

16、/共98頁SoLet Integration around a counterclockwise closed circular contour centered at the origin and with radius r.dzzzXjnxn 1)(21jrez dzjzd1 第37頁/共98頁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頁/共98頁P(yáng)artial 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頁/共98頁31,)311 (2)411 (1)(11zzzzXExample 10.931311411653111 zzzzzX,)()(Then第40頁/共98頁we have known411nunxn41 z, Z

18、T)(14111 z3122nunxnZT)(13112 z31 z,)()(nununxnxnxnn3124121 So313112411111 zzzzX,)()()(第41頁/共98頁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頁/共98頁When ROC41z 1)31(2 1)41(nununxnn313112411111 zzzzX,)()()(第4

19、3頁/共98頁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頁/共98頁Example 10.13Consider 111 azzX)(az ROC1:)11 az111 az1 az221 zaaz22 za. 2211zaaz111 az. 2211zaaz,.,21aanx long division第45頁/共98頁az ROC2:)11 a

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

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

22、A2Bmeunit circle*11BExample second order systems 第54頁/共98頁0| )(|jeHr =0.95r =0. 75第55頁/共98頁10.5.1 Linearity)(11zXnxZTR1R2),()(2121zbXzaXnbxnaxZTR1R2IfThenNote: R1R2 may be larger than R1 or R210.5 The Properties of Z-Transform)(22zXnxZT第56頁/共98頁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頁/共98頁10.5.3 Scaling in z-DomainRIf)/(00zzXnxzZTnThen|z0|R)(00zeXnxejZTnjSpecially,RR)() 1(zXnxZTnif)(zXnxZT0第58頁/共98頁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頁/共98頁10.5.5 Time ExpansionRIf)()(kZTkzXnxRk/1Then)(zXnxZTkofmultipleanotisnifkofmultipleaisnifknxnxk, 0,/)( mmnknnnkkzkmxznxznxzX/)(第60頁/共98頁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頁/共98頁10.5.7 The Convolution Property)()(2121zXzXnxnxZTR1 R2IfThenNote: if R1R2=, is not existed.)()(zXzX21)(11zXnxZTR1R2)(22zXnxZT第62頁/共98頁 Examplenunxkxngnk ZT)(zGRIfThen)(zG111)(zzXR|z|1)(z

26、XnxZT第63頁/共98頁10.5.8 Differentiation in the z-DomainRIfR)(zXdzdznnxZTThenExampleThen nunanZT,)(2111111 azazazdzdzaz )(zXnxZTnuan111 azaz ,ZTFrom: 第64頁/共98頁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頁/共98頁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頁/共98頁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頁/共98頁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頁/共98頁Example 10.2081412223 zzzzzzH)(not causal systemExample 10.211525222150152222111 zzzzzzzzH.)(.(.)(2 z,causal systemROC:1rz 第69頁/共98頁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頁/共98頁1067458em.013254n.n140.n0.n140.n0.第71頁/共98頁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頁/共98頁Example 10.2511211311 zzzH)(131121 nxnxnyny)()(12131211 nununhnn21zROC1:We can getROC2:21z)()(nununhnn 12131121第73頁/共98頁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頁/共98頁111 10 ,.23nny nau nais real)/(nunxn611 Solution: )()()(zXzYzH11 111111 1 (1), | 1/6610 , | 1/2111123zTzTx nzzay nzzz)()()(11113112116113510 zzzzaaAnd So 第75頁/共98頁For 47) 1(H9 a21zROC:2311

33、613261165 nxnxnxnynyny)()(111131121161121 zzzzzH212161651316131zzzznnx)(12 nny)(1472 Then 第76頁/共98頁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頁/共98頁Temr02/1unit circle-1-11 1)()(2)()(2zHdzdzzHzHdzdzzG(e) is the impulse response of a stable system. nhnhnng 第78頁/共98頁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頁/共98頁)(zH1)(zH2)(zX)(zY)()()()(zHzHzHzH2111 Feed-back第80頁/共98頁)(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頁/共98頁Example 10.28nxnyny 141)()()(zXzYzzH 14111)()(1zYzzWLet)(41)()(zWzXzYThen1 nynw)()(41)(1zXzYzzY第82頁/共98頁Equiva

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