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1、BEIJING JIAOTONG UNIVERSITYThe Course Group of Signals and Systems,Beijing Jiaotong University.P.R.CHINA.Copyright 2020Signals and Systems Complex frequency-domain analysis for signalsz-domain representation for D-T signalsUnilateral z-transform of typical signalsProperties of unilateral z-transform

2、Inversion of unilateral z-transformProperties of unilateral z-transformLinearityTime shiftConvolutionSummationExponential weightingz-domain differentiationInitial&final value theoremsLinearity propertyZ Zax kbx kaX zbXz ()()1212Z Zx kXz(),11Z Zx kXz(),22zRx1zRx2zRRxxmax(,)12Properties of unilateral

3、z-transformTime shift propertyZ Zx kn u knzX zn()zRxZ Zx k u kX z (),zRx for noncausal sequence(n0)Z Zx kn u kzX zx k zknnk ()1zRxProperties of unilateral z-transform for causal sequence(n0)Time shift propertyZ Zx kn u kzX zx k zzRknxnk (),1kxukkx ku kx1 1 1 1-1-2-1 x k u k Z ZuzkXxxkz 1()1 1Propert

4、ies of unilateral z-transformx ku k1 1kxk0 1 kxk0-1-2Z Zx ku k 2 z X zz xx2()112Time shift propertykxkx ku kx ku kx2 1 1 2 2 2Z Zx kn u kzX zx k zzRknxnk (),1Properties of unilateral z-transform x k u k x ku k2 2kxk02 kxk0Solution:Linearity:NX zzzz()11111Nzz111As RNk is a finite-length sequence,the

5、ROC is|z|0Z Zzu kz1,111Z Zzu kNzzN1,11Time shift:xk=RNk=ukukNExample 7.3:Determine the unilateral z-transform of xk.no poles in the X(z)xk is represented asx kkkk 24According to the time shift property，zX zzz1()11224z 1x kknnknn 1,2,0,1,2,0,21,0,1,2,Solution:Example 7.4:Determine the unilateral z-tr

6、ansform of xk.This is a geometric series of infinite length in the ratio(z2).magnitude of the poleConvolution in time-domain;multiplication in z-domainConvolution propertyZ Zx k u kx k u kX zXz ()()1212zRRxxmax(,)12Z Zx k u kXz (),11Z Zx k u kXz (),22zRx1zRx2Properties of unilateral z-transformSolut

7、ion:ZZZZZZZZx ku ku ku ku kkkkk 2 3 2 3 Z Zzzu kzk1222,11Z Zzzu kzk1333,11zzzzX zz(3),(1 2)(1 3)(2)()1112z3z2z3By the convolution propertyx ku ku kkk 2 3 Example 7.5:Determine the unilateral z-transform of xk.magnitude of the poley ku ku kiik ikkk(3)(4)(3)*(4)0By the convolution property zzY zz(3)(4

8、)()2z4signal yk can be represented asy kiik ik(3)(4)0Example 7.6:Determine the unilateral z-transform of yk.Solution:Summation propertyZ Zx k u kX z (),ifzRxZ Zzx nX znk1(),101thenzRxmax(,1)Properties of unilateral z-transform xk is a causal sequenceBy the convolution property,we can obtain Z Zzu k1

9、,11x ku kx n u knx nnnk*0ZZZZZZkxknxzuX znk,1()01Z Zx kX z(),LetzRxmax(1,)z1zRxcausalAsProperties of unilateral z-transformExponential weighting property(z-domain scaling property)Z Zx k u kX z (),zRxZ Za x k u kX z ak (/),za RxifthenProperties of unilateral z-transformSolution:By the exponential we

10、ighting propertyx kk u kk cos()0Z Z zzk u kzk12(/)cos(/)cos()1(/)cos012001Z Z zzk u kz,12coscos()1cos012001zz1zzz,12cos1cos122100Example 7.7:Determine the unilateral z-transform of xk.The unilateral z-transform for cos0kuk isZ Z zkx kzX zd d()z-domain differentiation property(Linear weighting proper

11、ty)Z Zx k u kX z (),zRxzRxifthenProperties of unilateral z-transformMultiplication by k in the time domain corresponding to differentiation with respect to z and multiplication of the result by z in the z-domain.Solution:By the z-domain differentiation propertyZ Zazzaa u kzk1,11Z Z zzaka u kzzkd dza

12、zaazaazzaz)(1,(21 21Example 7.8:Determine the unilateral z-transform and ROC.xk=kakuk xzX zzlim(1)()1If the ROC of(z1)X(z)includes the unit circle in z-plane,thenZ Zx k u kX z ()zRxifxX zz0lim()thenProperties of unilateral z-transform Initial and final value theoremsX(z)=1/(1-a z-1),|z|a|Solution:xa

13、zX zzz 101lim()lim11If|a|1,the ROC of(z1)X(z)includes the unit circle in z-plane,thenxazzX zzzz01lim(1)()lim1111Example 7.9:Determine the initial and final value of xk.xkx klNlN 01And the z-transform of finite length signal x1k is X1(z),Then we can derive the z-transform of xNk asZkXzz zXzxNlNNl1),(

14、)110z1If the causal sequence xNk can be represented asProperties of unilateral z-transformAcknowledgmentsMaterials used here are accumulated by authors for years with help from colleagues,media or other sources,which,unfortunately,cannot be noted specifically.We gratefully acknowledge those contributors.Properties of unilateral z-transform