信号与系统信号与系统信号与系统 (17).pdf
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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
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