教案成果讲稿信号2012chapter10ss l

1、10 The Z-Transform 10. The Z- Transform 10.1 The Z- Transform (1) Definition)(jswhere( ) nX zx n z(2) Region of Convergence ( ROC )ROC: Range of z for X(z) to convergeRepresentation: A. Inequality B. Region in Z-plane ( )Zx nX z ROC (3) Relationship between DTFT and Z Transform10 The Z-TransformSimi

2、lar to the relationship between FT and LT nnznxzX)( nnjjenxeX)( nnjjrenxreX)()(jrez Represent z in polar form,When the ROC of X(z) contains the unit circle in the z-plane.10 The Z-Transform)X(reX(z)j jezjX(z)X(e |z-planeThen10 The Z-Transform Example 10.1nuanxn 01nnnnnazznuazX)()(azazzazzX ,)(111(10

3、.9)nuan111 azaz , ZTHow to get the closed form of X(z)? ROC?10 The Z-TransformThe ROC of signals in example 10.1(right-sided signal)Z-plane1 1-1-1a aemunit circleSpecially,some commonly used ZT pairsAre in next slide.10 The Z-Transform(10.9a)nu111 z1 z, ZT(10.9b)cos(nun01 z ZT,)(cos21)(cos121010zzz,

4、)(cos21)(sin21010zzz(10.9c)sin(nun01 z ZT10 The Z-Transform1 1-1-1emunit circleThe ROC of equation 10.9(b),(c)Z-plane10 The Z-Transform Example 10.21 nuanxn11)( 1)(nnnnnazznuazXzazazann11111 )(az ,azazzazzX ,)(111 ZT1 nuan111 azaz ,(10.11)ROC?10 The Z-TransformThe ROC of Example 10.2(a left-sided si

5、gnal)11 ema aa1?Z-plane10 The Z-TransformExample ZTn z01,The ROC is the entire z-plane .Read example 10.34 by yourself!Normally, X(z) is rational in z (or z ?) form.1 nnzn 10.2 The Region of Convergence for Z TransformProperty1: The ROC of X(z) consists of a ring in the z-plane centered about the or

6、igin.Property2: the ROC does not contain any poles.10 The Z-Transformb b-1-1a aemZ-plane10 The Z-TransformProperty3: If xn is of finite duration and is absolutely summable, then the ROC is the entire z-plane ,except possibly z=0 and/or z= .Example ZT1 n zz01,11 zznnn ZT1 n zz0,10 The Z-TransformExam

7、ple 10.611011 azzazazXNNNnnn)( otherwiseNnanxn,010finite durationLength: NazazzNNN 11orROC: z010 a10 The Z-Transformzeros:kNjaezk)(2 121 Nk,.,poles:z=0 (N-1)st order Zeros-pole plot is shown in the figure of next slide.10 The Z-Transformunit circleame(N-1)st orderNote: z=a is cancelled. Z-planePrope

8、rty4: If xn is right sided, and if the circle z =r0 is in the ROC, then all finite values of z for which z r0 will also in the ROC. 10 The Z-TransformZ-planeThe ROC of a right-sided signala aemr0Property5: If xn is left sided, and if the circle z= r0 is in the ROC, then all values of z for which 0z

9、r0 will also in the ROC. 10 The Z-TransformZ-planeThe ROC of a left-sided signalProperty6: If xn is two sided, and if the circle z= r0 is in the ROC, then the ROC will consist of a ring in the z-plane that includes the circle z= r0 .Normally, r1 z r2. (r1 r2)10 The Z-Transformemr1r2r0Z-plane10 The Z

10、-TransformExample 10.70 bbnxn, 10nnnnnnzbzbzX)(nb10 bn.013241How to get the closed form of X(z)?10 The Z-Transformbzb/1,bzbzbz 1111When is not existed. )(,zXb1 Z-planeb bem1/b1/bProperty7: If the Z transform X(z) of xn is rational, then its ROC is bounded by poles or extends to infinity. In addition

11、,no poles of X(z) are contained in the ROC.10 The Z-TransformProperty8: If the Z transform X(z) of xn is rational, and if xn is right sided, then the ROC is the region in the z-plane outside the outmost pole-I.e., outside the circle of radius equal to the largest magnitude of the poles of X(z).Furth

12、ermore,if is causal,then the ROC also includes z= .10 The Z-TransformProperty9: If the Z transform X(z) of xn is rational, and if xn is left sided, then the ROC is the region in the z-plane inside the innermost nonzero pole-I.e., inside the circle of radius equal to the smallest magnitude of the poles of X(z) other than any at z=0 and extending inward to and possibly including z=0.in particular. if xn is