1、BEIJING JIAOTONG UNIVERSITYThe Course Group of Signals and Systems, Beijing Jiaotong University. P. R. CHINA. Copyright 2020Signals and Systems Bilateral z-transform and its inversion Definition of bilateral z-transform Properties of bilateral z-transform Inversion of bilateral z-transformDefinition
2、 of bilateral z-transformBilateral z-transform (BZT):X z x k zkk( ) Inversion of bilateral z-transform:k zx Xz zCk)j( d211The range of r=|z| for which the z-transform converges is termed ROC.C denotes a circle in the ROC.( 1) Finite length signalsSolution:|z| 0 or |z| 0X z R k zkNk( ) z kzX xkkNN()
3、12|z| 0, zzzkkN N1= 1 ,1011Definition of bilateral z-transformDetermine the BZT of signal xk=RNk=ukukN and the ROC.ROC includes the entire z-plane except z=0.)z(mI)z(eRCORX z z zkkk k k( ) 2 (2 )001x k zXzkkN() 1ROC lies outside the circle of the pole radius Rx-=2.|z| 2)z(mI)z(eR-xRCORDefinition of
4、bilateral z-transform( 2) right-sided signalsDetermine the BZT of signal xk=2kuk and the ROC.Solution:zzz1 2 2,11If N1 0, its ROC is of the form Rx |z| Rx-x k zXzkkN() 1Definition of bilateral z-transform( 2) right-sided signals )z(mI)z(eR-xRCORzz kX xkkN() 2 X z z zkkk k k k( ) 4 411)z(eR)z(mI+xRCO
5、R zz1 4 1 41,11z 4Definition of bilateral z-transform( 3) left-sided signalsDetermine the BZT of signal xk=-4k u-k-1 and the ROC.Solution:ROC lies inside the circle of the pole radius Rx+=4.zz kX xkkN() 2Definition of bilateral z-transform( 3) left-sided signalsIf N2 0, its ROC is of the form 0 |z|
6、RxIf N2 0, the ROC of a left-sided signal is of the form |z| 2zXz12( ) ,11z 2 ukk2 1Zazakk1 ,11Z azakk1 1 ,11zazaDefinition of bilateral z-transformX(z) does not correspond to xk uniquely, only X(z)+ROC does.magnitude of the polezXz xkkk) ( )z(eR)z( mI42CORzzXz1 2 1 4()1111z24Definition of bilateral
7、 z-transform( 4) two-sided signalsDetermine the BZT of signal xk= 2k uk-4k u-k-1 and the ROC.Solution:, ROC is a ring in z-plane.If the z-transform of a two-sided signal converges, the ROC is a ring in z-plane.Rx |z| Rx+)z(eR)z( mI+xR-xRCORzXz xkkk) ( Definition of bilateral z-transform( 4) two-side
8、d signalsDoes any two sided signal xk correspond to X(z)+ROC? x k u k u kkk =3 5 1z| | 3 z| | 5YesandDefinition of bilateral z-transformz| | 3 z| | 2NOFor examples:and x k u k u kkk =3 2 1Bilateral z-transform and its inversion Definition of bilateral z-transform Properties of bilateral z-transform
9、Inversion of bilateral z-transformProperties signals z-transform ROCLinearity axk+ byk aX(z)+bY(z) Rx RySymmetry x*k X*(z*) RxTime reversal xk X(1/z)Time shift xkn zn X(z) Rx, z=0 or z= exclusiveConvolution xk * yk X(z)Y(z) Rx RyExp. weighting akxk X(z/a) |a| RxLinear weighting kxk Rx , z=0 or z= ex
10、clusive1/ 1/R z Rx x d ( )dz X zzRx=z; Rx-|z|Rx+Z x k X z ( ) Z y k Y z ( ) Ry=z; Ry-|z|Ry+ Properties of bilateral z-transformBilateral z-transform and its inversion Definition of bilateral z-transform Properties of bilateral z-transform Inversion of bilateral z-transformC is a closed circle in ROC
11、 of X(z)Inversion of bilateral z-transformDirect inversion of the z-transform is complicated. We can determine it by partial fraction expansion (PFE).x k X z z zCkj2 = ( ) d11Zakaazzzk ,111Z bkbz z bzk11 1 ,1zazbInversion of bilateral z-transform Partial fraction expansionRight-sided signalsLeft-sid
12、ed signalsmagnitude of the polemagnitude of the polezzXzz( 2)( 3)()2zzXzzz23()23Solution:Example 7.13: Determine xk corresponding to X(z) by PFE.X(z) with different ROCs corresponds to different signals xk.2)z(mI)z(eR3)z(Re)z( Im32ROC)z(Re)z(Im2|z|3 2|z|3 |z|2 Two poles 2, 3zzXzz( 2)( 3)()2zzXzzz23(
13、)23Solution:(1) |z|3 (2) 2|z|3(3) |z|2 x k u k u kkk 2 2 3 3 x k u k u kkk 2 2 3 3 1 x k u k u kkk 2 2 1 3 3 1Example 7.13: Determine xk corresponding to X(z) by PFE.Right-sided signalTwo-sided signalLeft-sided signalX(z) with different ROCs corresponds to different signals xk.2)z(mI)z(eR3)z(Re)z( I
14、m32ROC)z(Re)z(Im2|z|3 2|z|3 |z|2 ukx k u kkk 3 3 2 2 ukx k u kkk 3 3 1 2 2 ukx k u kkk 3 3 1 2 2 1Example 7.13: Determine xk corresponding to X(z) by PFE.zzXzz( 2)( 3)()2zzzz2323AcknowledgmentsMaterials 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.Bilateral z-transform and its inversion