Right-sided Ex.3.3 Sum of two exponential sequences Determine the z-transform,including the ROC pole-zero-plot,for sequence: 24+([n Solution: x(e之(4a =24x"+(4z 22少+ 18
18 Ex. 3.3 Sum of two exponential sequences ( ) ( ) 1 1 2 3 n n x n u n u n = + − ( ) ( ) ( ) 1 1 2 3 n n n n X z u n u n z − =− = + − ( ) ( ) 0 1 1 0 1 1 n 2 3 n n n z z − − = = = + − ( ) ( ) 1 1 2 3 n n n n n n u n z u n z − − =− =− = + − ◆Determine the z-transform, including the ROC, pole-zero-plot, for sequence: Solution: Right-sided
Ex.3.3:Sum of two exponential sequences 24+立(4 xIn]=a"uln] 2+ X(②1-az ROCa 2z 12 1 1-+ ROC: 19
19 Ex.3.3: Sum of two exponential sequences 1 2 ROC: z 1 3 and z : 1 2 ROC z ( ) ( ) 0 1 1 0 1 1 2 3 n n n n z z − − = = = + − 1 1 1 1 2 z − = − 1 1 1 1 3 z − + + ( ) ( )( ) 1 1 1 2 12 1 1 1 1 2 3 z z z z − − − = − + n x n a u n = ROC z a : ( ) 1 1 1 X z az = − − ( ) ( ) ( ) 1 1 2 3 n n n n n n X z u n z u n z − − =− =− = + −
gm 3-plane Fm z-plane 1 1 Re 1 Re 1-2 1 1+ 21 73 1-2 22) -3z1+3 1 e (3
20 1 1 1 1 2 z − − 1 1 1 1 3 z − + ( ) ( )( ) 1 1 1 2 12 1 1 1 1 2 3 z z z z − − − − + ( ) ( ) 1 1 2 3 n n x n u n u n = + − 1 2 z 1 3 z 1 2 z
Example 3.5: Two-sided exponential sequence 网=(--小 xn=aun] Solution:Right-sided Left-sided X2片1-az 女时 ROC:>a x[n-4-n-刂 形寸好 X(}Iazt ROC:a X(2F 2-12 1 22可c 23
23 Example 3.5: Two-sided exponential sequence ( ) ( ) 1 1 1 3 2 n n x n u n u n = − − − − ， ( ) 1 1 1 , 3 1 1 3 1 3 n Z u n z z − − + ( ) 1 1 1 1 , 2 1 1 2 1 2 n Z u n z z − − − − − ( ) 1 1 1 X z az = − − ROC: z a 1 n x n a u n =− − − Solution: 1 1 3 : 2 ROC z Right-sided Left-sided ( ) ( )( ) 1 1 1 2 12 , 1 1 1 1 3 2 z z z z − − − = + − ( ) 1 1 1 1 1 1 1 1 3 2 X z z z − − = + + − n x n a u n = ROC z a : ( ) 1 1 1 X z az = − −
正g gm z-plane 网-（网份n-可 Two-sided exponential sequence Re 1 ROC,pole-zero-plot Compared with Ex.3.3:Sum of two exponential sequences 24
24 ROC, pole-zero-plot ( ) ( ) 1 1 1 3 2 n n x n u n u n = − − − − ( ) ( )( ) 1 1 1 2 12 , 1 1 1 1 3 2 z z z z − − − = + − ( ) 1 1 1 1 1 1 1 1 3 2 X z z z − − = + + − 1 1 3 : 2 ROC z Two-sided exponential sequence Compared with Ex.3.3: Sum of two exponential sequences