EXample 35: TwO-sided exponential sequence (到响()=an Solution az n<> 1+-z for z< l[-n-1] 2 1+z11-z11+:11)RC.1 12 <|2|<
29 Example 3.5: Two-sided exponential sequence ( ) ( ) 1 1 1 3 2 n n x n u n u n = − − − − ， 1 1 1 , 1 3 1 3 1 3 n Z u n z z − − + 1 1 1 1 , 2 1 1 1 2 2 n Z u z n z − − − − − ( ) 1 1 1 1 1 1 1 1 3 2 X z z z − − = + + − ( ) 1 1 1 X z az − = − for z a 1 n x n a u n = − − − Solution: 1 1 1 2 12 , 1 1 1 1 3 2 z z z z − − − = + − 1 1 3 : 2 ROC z
22 X(=) ROC 1+-z-11 112 2 1+-z 3 rIn un]-[-n-1]19, Z-plane ROC, pole-zero-plot ● Re 12
30 ROC, pole-zero-plot 1 2 1 3 1 − − − x n = − u n u n n n ( ) 1 1 1 1 1 2 1 1 12 : 1 1 1 1 1 1 1 1 3 1 1 3 3 2 2 2 z z X z ROC z z z z z − − − − − = + = + − + −
Finite-length sequence X(z) xIng 12 =N1 Example x]=]+[n-5 X(z)=1+2,ROC: 2>0 or entire z-plane except z=0 31
31 Finite-length sequence xn= n+ n−5 ( ) 5 X z z 1 − = + ， ( ) 2 1 N n N n X z x n z = − = Example : or entire z-plane, except z=0 ROC z : 0
Example 3.6: Finite-length sequence Determine the z-transform the RoC, pole zero-plot, for sequence: (N=16, -1<a<1) xn a",0<n<N-1 0. otherwise <OO Solution : z-transform az<∞ 0 n=0 N ROC:|z≠0 az or >0 N-1 ←- including z=a 32
32 Example 3.6: Finite-length sequence − = , otherwise a , n N x n n 0 0 1 ( ) ( ) 1 1 1 0 0 N N n n n n n X z a z az − − − − = = = = ROC: 1 1 . N N N z a z z a − − = − ( ) 1 1 1 1 N az az − − − = − ◆Determine the z-transform, the ROC, polezero-plot, for sequence: Solution: z-transform including z=a , a 1 , az − z 0 z 0 or , (N=16, -1<a<1)
with N=16, a is real, and o<la<1 ROC: zl>o N-1 J pole-zero-plot z-plane 零极点图 15th-order pole Unit circle Zk ae/(2Tk/N) 8 N=16 k=1,2,L,N-1 Re Pk=0 33
33 ( ) ROC z : 0 1 1 N N N z a X z z z a − − = − with N=16, a is real, and 0<|a|<1 pole-zero-plot 零极点图 (2 / ) , j k N k z ae = 1,2, , 1 k N = − L 0 k p = N=16