Finite-length sequence Example:x[n]=8[n]+8 [n-5] X(z)=∑x[1z"=∑(on]+δLn-5)z" n= -2z"+2nz”=1+z-1+ ROC:z>0 or entire z-plane,except z=0 Example:y[n]=δ[n]+δ[n+5] Y(z)=1+z,ROC:z<0o or entire z-plane,except z=o 25
25 Finite-length sequence ROC z : 0 or entire z-plane, except z=0 Example: y n n n = + +5 ( ) 5 Y z = +1 z ， or entire z-plane, except z=∞ ROC z : ( ) 2 1 N n N n X z x n z = − = ( ) 5 0 5 n n n n z = − = + − 5 5 1 1 z z = + − =1+ 5 5 0 0 5 n n n n n n z z = = − − = + − Example: x n n n = + − 5
Example 3.6:Finite-length sequence Determine the z-transform,the ROC,pole- zero-plot,for sequence:(e.g.,N=16,0<a<1) -oa冰d a<oo Solution:z-transform Xa)之e-aey,b:+图 <0 n= 1-(az)1 zN-av 1-az-I z-a ROC:lE≠0，or,>0 ROC includes z=a,z=a is not a pole. 26
26 Example 3.6: Finite-length sequence ( ) ( ) 1 1 1 0 0 N N n n n n n X z a z az − − − − = = = = 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 ROC includes z=a, z=a is not a pole. , a 1 , = z az− a ROC z or z : 0, , 0 (e.g., N=16, 0<a<1) , 0 1 0, a n N n x n otherwise − =
with N=16,a is real,and 0<a<1 1z-1z16-a16 215 z-a ROC:z>0 Zeros: 乎m z-plane 15th-order pole let 216-a6=0 Unit circle →Zk=ae2nk16) 8 k61,2,…,15 Re Poles: =0,15th order pole-zero-plot零极点图 27
27 ( ) 1 ROC z : 0 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 16 16 let z a− =0 1,2, ,1 k=0, 5 pk =0 16 16 15 1 z a z z a − = − j k (2 / ) 16 k z ae = ,15th order Zeros: Poles:
z-transform pairs 1.δ[n<>1,ROC:allz ∑8nlz m n=-0 2.网2，Roc林l 3.-刂>2，R0cHl 4.δn-m←→Zm, ∑n-mz” ROC:entire z-plane exceptO(if m>0) or entire z-plane except o(if m< 28
28 z-transform pairs n ROC all z 1, : 1 1 , : 1 1 u n ROC z z − − 1 1 1 , : 1 1 u n ROC z z − − − − − , m n m z − − 1. 2. 3. 4. n n n z =− − - n n n m z =− − ROC excep : 0 entire z pla − ne t (if m0) or entire z plan − e except (i mf 0)
z-transform pairs 5. u ROC: 6.wu-RC 7. 网aeoH4 az -no水>ao0u 8. 29
29 z-transform pairs 1 1 , : 1 n a u n ROC z a az − − 1 1 1 , : 1 n a u n ROC z a az − − − − − ( ) 1 2 1 , : 1 n az na u n ROC z a az − − − ( ) 1 2 1 1 , : 1 n az na u n ROC z a az − − − − − − 5. 6. 7. 8