Grading Correct answer 2 pts Incorrect answer 1 pts No answer 0 pts a y n x n n

# Grading correct answer 2 pts incorrect answer 1 pts

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Grading: Correct answer = +2 pts.; Incorrect answer = -1 pts. No answer = 0 pts. (a) y [ n ] = x [ | n | + n ] L / NL SI / SV C / NC (b) y [ n ] = (0 . 2) | n | log[ x [ n ]] L / NL SI / SV C / NC 2

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(12 Pts.) 3. Suppose that T is a linear and shift-invariant system. For an input x [ n ] depicted below we observe y [ n ] as the output: 1 0 1 2 3 4 n y[n] 2 T 0 1 2 3 4 n x[n] 2 1 (a) ( 3pts.) Find the unit pulse response of the system T . (b) (5pts.) Find the output y [ n ] for n = - 1 , 0 , 1 , 2 produced by the response of the system to the unit step input u [ n ]. (c) (4pts.) Is T a causal system? Justify your answer. 3
(5 Pts.) 4. Calculate the result of the following convolution { 1 , - 4 , 2 , - 1 , 3 , 1 } * {- 1 , 1 , - 1 } . (10 Pts.) 5. Calculate the z -transform and corresponding ROC of the following functions: (a) x [ n ] = 3 - n ( u [ n - 5] - u [ n - 100]) (b) x [ n ] = e - n 2 u [ n - 8] u [ - n + 10] 4

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(6 Pts.) 6. Calculate the inverse z -transform for Y ( z ) = 1 + z - 100 + 1 1 - 5 z - 1 , ROC: | z | > 5 (15 Pts.) 7. The z -transform of x [ n ] is given below: X ( z ) = z z - e j π 3 + z z - 0 . 5 Determine all the valid ROCs of X ( z ) and for each case, calculate the inverse z -transform. 5
(4 Pts.) 8. Calculate the DTFT of x [ n ] = { 1 , 0 , 0 , 2 } (4 Pts.) 9. Assume that the z -transform of x [ n ] is given by X ( z ) = z z - 3 , | z | > 3 .

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