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-l) X(z) Q(2 Zero: The value of z for which X(2)=0 ◆Poe: The value of z for which X(z)=∞ 19
19 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 K Determine the z-transform, including the ROC in z-plane and a sketch of the pole-zero-plot, for sequence: a|<1 a>1? rIn Solution a"ulr DTFT no DTFT X(=)=∑q"="=(a az n=0 0 ROC: az<1or(2>lo ze:2= 0, pole: z=a
20 Example 3.1: Right-sided exponential sequence ( ) ( ) 1 0 0 n n n n n X z a z az − − = = = = zero z pole z a : 0 : = = ， 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 o1 r z a − ( ) 1 1 1 1 az az − − − = − a 1 DTFT z z a = − 0 a 1? no DTFT
Z-plane Unit circle Re o: zeros 1-az x: poles or z>la Gray region: ROC 21
21 : zeros : poles Gray region: ROC xn a un n = ( ) z X z z a = − for z a 1 1 1 az − = −
Ex 3.2 Left-sided exponential sequence Determine the z-transform, including the roc pole-zero-plot, for sequence x]=-a"l-n-1 Solution X(z) anul-n 1=-00 1=-00 ∑a"z"=∑(a2z) a z-a z n=1 i-a az|<1 ROC zero: z=0 pole: z=a 22
22 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: z z a = − zero z pole z a : 0 : = = ( ) 1 1 1 1 a z a z a z − − − − = − − 0 -1 a z 1
gm z-plane x]=-al-n-1] Unit circle X(z)= for z< a Re 比较 rn=a un o: zeros X(z)= forz>a X: poles Gray region: ROC 23
23 ( ) z X z z a = − for z a xn= −a u− n −1 n for z a xn a un n = ( ) z X z z a = − ， : zeros : poles Gray region: ROC 比较 :