Ex 3.3 Sum of two exponential sequences +Determine the z-transform, including the ROC pole-zero-plot for sequence XIn 2 小+-un Solution X(z)=∑ uln+ [n]}z n=-00 ∑ nz“ ∑ []= n=-0 ∑ 2 n=0 n=0 24
24 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 0 1 1 1 1 2 3 n 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:
EXample 3.3 um of two exponential sequences n=0 12 1+ 1+-z RC|>2and1>2→ROC:|> 2 2
25 Example 3.3: Sum of two exponential sequences 1 2 z ( ) 1 0 0 1 1 1 2 3 n n n n X z z z − = − = = + − 1 1 1 2 12 1 1 1 1 2 3 z z z z − − − = − + 1 1 1 1 2 z − = − ROC: 1 3 and z 1 1 1 1 3 z − + + : 1 2 ROC z
m z-plane mz-plane Re 19 2 1+ 2z 12 2 1+-z XIn ]-+|-小 2
26 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 3.4: Sum of two exponential another solution for Ex 3.3 In uIn+ xIn=a u Solution 1-az 2 2 for z>la 113 1+-z ROC. l小n|+ uIn> 2 1+
27 2 1 2 1 1 1 2 1 1 − − , z z u n Z n 3 1 3 1 1 1 3 1 1 + − − , z z u n Z n 2 1 3 1 1 1 2 1 1 1 3 1 2 1 1 1 + + − + − − − , z z z u n u n Z n n xn a un n = ( ) 1 1 1 X z az − = − for z a Solution: ROC: another solution for Ex.3.3 Example 3.4: Sum of two exponential 1 1 2 3 n n x n u n u n = + −
m z-plane z-plane 2 Re 19 2 1+ 2z 12 1+-z XIn ]-+|-小 2
28 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