Zero and pole The z-transform is most useful when the infinite sum can be expressed in closed form usually a ratio of polynomials in z(or z-1) P(z Zero: The value of z for which X(=)=0 ◆Poe: The value of z for which X(z)=∞ 2021/1/31 Zhongguo Liu_ Biomedical Engineering_shandong Univ
12 2021/1/31 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Zero and pole ( ) ( ) Q(z) P z X z = ◆Zero: The value of z for which X(z) = 0 ◆The z-transform is most useful when the infinite sum can be expressed in closed form, usually a ratio of polynomials in z (or z-1 ). ◆Pole: The value of z for which X(z) =
Example 3.1: Right-sided exponential sequence Determine the z-transform including the roc in Z-plane and a sketch of the pole-zero-plot for sequence xn=aun Solution X()=2a"=∑(az)= n=0 n=0 1-az ROC: az <lor(>al zero: z=0 pole. z=a 2021/1/31 Zhongguo Liu_ Biomedical Engineering_shandong Univ
13 2021/1/31 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Example 3.1: Right-sided exponential sequence ( ) ( ) 1 0 0 n n n n n X z a z az − − = = = = zero : z = 0 pole : z = a xn a un n = ◆Determine the z-transform, including the ROC in z-plane and a sketch of the pole-zero-plot, for sequence: Solution: ROC: 1 az or z a 1 − 1 1 1 z az z a − = = − −
J Z-plane Unit circle Re n]=aun X(z)= o: zeros x poles or z>la gray region: ROc 2021/1/31 Zhongguo Liu_ Biomedical Engineering_shandong Univ
14 2021/1/31 Zhongguo Liu_Biomedical Engineering_Shandong Univ. : zeros : poles Gray region: ROC xn a un n = ( ) z X z z a = − for z a
Ex 3.2 Left-sided exponential sequence Determine the z-transform, including the roc pole-zero-plot, for sequence xn]=au-n Solution X(z)=∑a[-n-]z"=∑a"z n=-0 n=- ROC:z< zero: z=0 pole: z=a 15 2021/1/31 Zhongguo Liu_ Biomedical Engineering_shandong Univ
15 2021/1/31 Zhongguo Liu_Biomedical Engineering_Shandong Univ. Ex. 3.2 Left-sided exponential sequence ( ) 1 1 n n n n n n X z a u n z a z − − − =− =− = − − − = − xn= −a u− n −1 n z a , ( ) 1 1 1 n n n n n a z a z − − = = = − = − ◆Determine the z-transform, including the ROC, pole-zero-plot, for sequence: Solution: ROC: 1 1 1 a z z a z z a − − = − = − − zero z pole z a : 0 : = =
Z-plane Unit circle xIn a ul-n 19e for 16 2021/1/31 Zhongguo Liu_ Biomedical Engineering_shandong Univ
16 2021/1/31 Zhongguo Liu_Biomedical Engineering_Shandong Univ. ( ) z X z z a = − for z a xn= −a u− n −1 n