1、10.1 The z-Transform10. The z-Transform 10.1 The z-TransformLTI (1) Definition10.1 The z-Transform10.1 The z-Transformz-planeUnit circle10.1 The z-Transforma. Especially, when z=ej， above equation becomes The Fourier transform of signal xn:So, the relationship between the Fourier transform and the z

2、-transform is:(2) The relationship between Z-transform and the Fourier transform of xnb. On the other hand, 10.1 The z-Transform(2) Region of Convergence ( ROC )ROC: Range of z for X(z) to converge Representation: A. Inequality B. Region in z-plane10.1 The z-TransformExample 10.1 Determine the z-Tra

3、nsform of xn and its ROC.10.1 The z-TransformSolution:10.1 The z-TransformFigure 10.2ReImUnit circleExample 10.2 Determine the z-Transform of xn and its ROC.10.1 The z-TransformSolution:10.1 The z-TransformFigure 10.310.1 The z-Transformandhave same Z- transform representation, but their ROC is diff

4、erent.ZZNote: for a signal xn, we must give out the z-transform with its ROC.10.1 The z-Transform(3) The pole-zero plot of X(z)X(z) can be represented the ratio of two polynomials the numerator polynomial;the denominator polynomial;10.1 The z-TransformDefinition:The zeros of X(z): the roots of the n

5、umerator polynomial N(z) is called the zeros of X(z). The poles of X(z): the roots of the denominator polynomial D(z) is called the poles of X(z).10.1 The z-TransformThe representation of X(z) through its poles and zeros in the z-plane is referred to the pole-zero plot of X(z).Definition:In the z-pl

6、ane, use “X” to indicate the poles of X(z); and use “O” to indicate the zeros of X(z); On the other hand, If MN, z, X(z) , X(z) have (M-N) poles at infinity. If Mr0 will also in the ROC. 10.2 The ROC of the z-TransformFigure 10.7 right-sided sequence xn.10.2 The ROC of the z-TransformaROC of a right

7、-sided sequence: Property 5: If xn is left-sided sequence, and if the circle |z|=r0 is in the ROC, then all values of z for which 0|z|r0 will also be in the ROC. 10.2 The ROC of the z-Transform10.2 The ROC of the z-TransformbReROC of left-sided sequence:Property 6: If xn is two sided, and if the cir

8、cle |z|=r0 is in the ROC, then the ROC will consist of a ring in the z-plane that includes the circle |z|=r0 . 10.2 The ROC of the z-Transform10.2 The ROC of the z-TransformabROC of two-sided sequence:Example 10.6 Determine the z-transform of the following signals.10.2 The ROC of the z-Transform10.2

9、 The ROC of the z-TransformSolution:Zeros of X(z):N-1 poles of X(z):pole of X(z):這個(gè)極點(diǎn)與k=0時(shí)的零點(diǎn)抵消了。10.2 The ROC of the z-TransformFigure 10-9Example 10.7Determine the z-transform of the following signals.10.2 The ROC of the z-Transform10.2 The ROC of the z-Transform0b1Property 7: If the z-transform X(

10、z) of xn is rational, then its ROC is bounded by poles or extends to infinity. 10.2 The ROC of the z-TransformProperty 8: 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 t

11、o the largest magnitude of the poles of X(z), Furthermore, if xn is causal (i.e., if it is right sided and equal to 0 for n0) , then the ROC also includes z=0.10.2 The ROC of the z-TransformExample 10.8Consider all of the possible ROCS of X(z).Figure 10.1210.2 The ROC of the z-Transform 10.3 The Inv

12、erse z-Transform10.3 The inverse z-TransformShow:10.3 The inverse z-TransformThe calculation for inverse z-transform X(z):(1) Integration of complex function by equation.(2) using fraction expansion 10.3 The inverse z-Transform(3)Long division (Taylors series) 長除法（泰勒級(jí)數(shù)展開法） Appendix Partial Fraction

13、ExpansionConsider a fraction polynomial:10.3 The inverse z-Transform即，X(z)是z的有理分式。把X(z)表示成z-1的兩個(gè)多項(xiàng)式之比形式。10.3 The inverse z-TransformDiscuss two cases of D(z-1)=0, for distinct roots, and same roots.我們這里對(duì)X(z) 以z-1進(jìn)行部分分式展開。10.3 The inverse z-TransformCase 1: Distinct roots:thus10.3 The inverse z-Trans

14、formCalculate A1 :Generally10.3 The inverse z-TransformUsing the following relationships to obtain xn.10.3 The inverse z-Transform10.3 The inverse z-TransformExample : Compute the inverse z-transform of X(z).Solution:10.3 The inverse z-Transform10.3 The inverse z-TransformCase 2: Same root:So,10.3 T

15、he inverse z-TransformFor first order poles:10.3 The inverse z-TransformMultiply two sides by (1-p1z-1)r : For r-order poles:10.3 The inverse z-TransformSo 10.3 The inverse z-Transform10.3 The inverse z-TransformusingWe can obtain xn.10.3 The inverse z-TransformOr usingWe can obtain xn.10.3 The inve

16、rse z-Transform10.3 The inverse z-TransformExample:Determine the inverse z-transform.Solution:10.3 The inverse z-Transform10.3 The inverse z-Transform10.3 The inverse z-TransformExample 10.9 10.10 10.11 Determine the inverse z-transform of X(z).10.3 The inverse z-Transform(3).10.3 The inverse z-Tran

17、sformIf X(z) is not rational , compute xn by the following relationshipsLong division (Taylors series) 長除法（泰勒級(jí)數(shù)展開法）(a)(b)Example 10.12 10.14Determine the inverse z-transform of X(z).10.3 The inverse z-Transform(a)(b)Example 10.13Determine the inverse z-transform of X(z) by long division.10.3 The inv

18、erse z-Transform(a)10.3 The inverse z-TransformSolution:11-az-11+ az-1+a2z-2+1-az-1az-1az-1-a2z-2a2z-2(b)10.3 The inverse z-TransformSolution:1-az-1+1- a-1z-a-2z2-1-a-1za-1za-1z-a-2z2a-2z210.5 Properties of the z-Transform(1) Linearity10.5 properties of the z-Transform10.5 properties of the z-Transf

19、orm線性性質(zhì): 線性組合后的收斂域R是線性組合前兩個(gè)信號(hào)的收斂域R1與R2的公共區(qū)域. 如果在線性組合過程中出現(xiàn)零點(diǎn)與極點(diǎn)相抵消的情況,則收斂域可能會(huì)擴(kuò)大.10.5 properties of the z-TransformExample:(2) Time shifting10.5 properties of the z-Transform10.5 properties of the z-Transform(3) Scaling in the z-domain可見：z平面上的尺度展縮，等效于xn乘以指數(shù)序列。當(dāng)z0為復(fù)指數(shù)時(shí)， z平面上的尺度展縮對(duì)應(yīng)于Z平面上的點(diǎn)沿角度方向進(jìn)行旋轉(zhuǎn)，沿徑向方

20、向伸張或壓縮。(4) Time Reversal10.5 properties of the z-Transform(5) Time expansion10.5 properties of the z-Transform(6) Conjugation (7) Convolution property10.5 properties of the z-TransformFirst difference:Accumulation:Example 10.15 10.1610.5 properties of the z-Transform(8) Differentiation in the z-domain10.5 properties of the z-TransformExample 10.17 10.18 Find the inverse z- transform of X(z).(a)(b)10.5 properties of the z-Transform(9)