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1、BEIJING JIAOTONG UNIVERSITYSignals and Systems Bilateral z-transform and its inversionDefinition of bilateral z-transformProperties of bilateral z-transformInversion of bilateral z-transformDefinition of bilateral z-transformBilateral z-transform(BZT):X zx k zkk()Inversion of bilateral z-transform:k
2、zxX zzCk)j(d2 11The 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 zRk zkNk()zk zXxkkNN()12|z|0,zzzkkNN1=1,1011Definition of bilateral z-transformDetermine the BZT of signal xk=RNk=ukukN and the ROC.ROC incl
3、udes the entire z-plane except z=0.)z(mI)z(eRCORX zzzkkkkk()2(2)001x k zX zkkN()1ROC lies outside the circle of the pole radius Rx-=2.|z|2)z(mI)z(eR-xRCORDefinition of bilateral z-transform（2）right-sided signalsDetermine the BZT of signal xk=2kuk and the ROC.Solution:zzz122,11If N1 0,its ROC is of t
4、he form Rx|z|Rx-x k zX zkkN()1Definition of bilateral z-transform（2）right-sided signals)z(mI)z(eR-xRCORzzkXxkkN()2X zzzkkkkkk()4411)z(eR)z(mI+xRCOR zz14141,1111z4Definition of bilateral z-transform（3）left-sided signalsDetermine the BZT of signal xk=-4ku-k-1 and the ROC.Solution:ROC lies inside the c
5、ircle of the pole radius Rx+=4.zzkXxkkN()2Definition of bilateral z-transform（3）left-sided signalsIf N2 0,its ROC is of the form 0|z|RxIf N20,the ROC of a left-sided signal is of the form|z|2zX z12(),11z2 ukk21Z azakk1 ,11Z azakk11 ,11zazaDefinition of bilateral z-transformX(z)does not correspond to
6、 xk uniquely,only X(z)+ROC does.magnitude of the polezX zx kkk)()z(eR)z(mI42CORzzX z1214()1111z24Definition of bilateral z-transform（4）two-sided signalsDetermine the BZT of signal xk=2kuk-4ku-k-1 and the ROC.Solution:,ROC is a ring in z-plane.If the z-transform of a two-sided signal converges,the RO
7、C is a ring in z-plane.Rx|z|Rx+)z(eR)z(mI+xR-xRCORzX zx kkk)(Definition of bilateral z-transform（4）two-sided signalsDoes any two sided signal xk correspond to X(z)+ROC?x ku kukkk=3 51z|3z|5YesandDefinition of bilateral z-transformz|3z|2NOFor examples:and x ku kukkk=3 21Bilateral z-transform and its
8、inversionDefinition of bilateral z-transformProperties of bilateral z-transformInversion of bilateral z-transformPropertiessignalsz-transformROCLinearityaxk+bykaX(z)+bY(z)Rx RySymmetryx*kX*(z*)RxTime reversalxkX(1/z)Time shiftxknznX(z)Rx,z=0 or z=exclusiveConvolutionxk*ykX(z)Y(z)Rx RyExp.weightingak
9、xkX(z/a)|a|RxLinear weightingkxk Rx,z=0 or z=exclusive1/1/RzRxxd()dzX zzRx=z;Rx-|z|Rx+Z Zx kX z ()Z Zy kY z ()Ry=z;Ry-|z|Ry+Properties of bilateral z-transformBilateral z-transform and its inversionDefinition of bilateral z-transformProperties of bilateral z-transformInversion of bilateral z-transfo
10、rmC is a closed circle in ROC 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 kX z zzCkj 2 =()d11Z akaazzzk ,111Z bkbzzbzk111 ,1zazbInversion of bilateral z-transform Partial fraction expansionRight-
11、sided signalsLeft-sided signalsmagnitude of the polemagnitude of the polezzX zz(2)(3)()2zzX zzz23()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 2,3zzX zz(2)(
12、3)()2zzX zzz23()23Solution:(1)|z|3(2)2|z|3(3)|z|2 x ku ku kkk 2 2 3 3 x ku kukkk 2 2 3 31 x kukukkk 2 21 3 31Example 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(Im32RO
13、C)z(Re)z(Im2|z|3 2|z|3|z|2 u kx ku kkk 3 3 2 2 ukx ku kkk 3 31 2 2 ukx kukkk 3 31 2 21Example 7.13:Determine xk corresponding to X(z)by PFE.zzX zz(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