信号与系统信号与系统信号与系统 (18).pdf
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 typicle signalsProperties of unilateral z-transformInversion of unilateral z-transformx kX z zzkj 2()d1c1C)z(eR)z(mIInversion of unilateral z-transformDirect inversion of the z-transform is complicated.We can determine it by using the one-to-one relation between a signal and its unilateral z-transform.Where the symbol denotes integration around a circle of radius|z|=r in a counter-clockwise direction in ROC.c Power Series Expansion(PSE)Partial Fraction Expansion(PFE)Inversion of unilateral z-transformPower Series Expansion(PSE)zzX zx k zxxxkk()012120When X(z)is expressed as a power series in z1,the values of the signal xk are then given by the coefficient associated with zk.x kxxx 0,1,2,Inversion of unilateral z-transformThis inversion method is limited to find the finite values of xk.zzX zzzz0.50.5(),120.522Solution:We use long division to express X(z)as a power series in z1.x k 2,0.5,1.25,zz20.5 2zz0.50.522zz2 12 z0.51zz0.50.250.251 z1.250.251zz1.250.6250.62512 z0.51z1.252PSE method may not lead to a closed-form expression for xk.obtainingExample 7.10:Determine xk corresponding to X(z)+ROC by PSE.That iszzX z()20.51.2512A za za zX zB zbb zb znnmm()1()()101Inversion of unilateral z-transformPartial Fraction Expansion(PFE)In most cases,we frequently encounter z-transforms that are a ratio of polynomials in z or in z1.We can use the partial-fraction expansion to express X(z)as a sum of terms.Z Zazzaa u kzazk1,11Z Zazzaka u kzaazazk1(),1221Inversion of unilateral z-transformPartial Fraction Expansion(PFE)zzX zzzz0.50.5(),120.522X zzzzzzzzXzzAB0.5()(1)(0.5)1()20.51AzXzz(1)()111BzX zz(0.5)()110.5 x ku kk 1(0.5)X(z)is an improper rational fraction,its poles are distinct.zzX zzzz10.5(),|1If possible,factoring z in the numerator polynomial,then PFE Solution:Example 7.11:Determine x(t)corresponding to X(z)+ROC by PFE.ProperrationalfractionzzX zzz(1)(1)(),12X zzzzzzXzABC1()1(1)()21BzXzz(1)()0.5112 zAX z zzd()(1)|0.25d112CzXzz(1)()0.2511 x ku kkk424(1)11X(z)is a proper rational fraction,the pole(1)is repeated twice.zzzX zzzz1(1)1()0.250.50.252Solution:Example 7.12:Determine x(t)corresponding to X(z)+ROC by PFE.zzX zzz712(),42122zzXzzzXzzAB24()22(3)(4)3()14231 AzXzz(3)()1913BzX zz(4)()3314x kku kkk 2 33 419 3 111X(z)is an improper rational fraction,its poles are z1=3，z2=4.Firstly we need to get the part of the proper rational fraction.zzX z34()21933Z ZzaAau kzaAk1,1Solution:Example 7.12:Determine x(t)corresponding to X(z)+ROC by PFE.AcknowledgmentsMaterials used here are accumulated by authors for years with helpfrom colleagues,media or other sources,which,unfortunately,cannotbe noted specifically.We gratefully acknowledge those contributors.Inversion of unilateral z-transform
