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Category: EE 113

Description

5/5 - (5 votes)

Problem 1. (24 points) z-transform: Determine the z-transforms of the signals given below.

Indicate the ROC for each.

a) (8 points)

x[n] = (

n, n = 0, . . . , 9

0 otherwise.

b) (8 points)

x[n] =

n, n = 0, . . . , 9

10, n ≥ 10

0, otherwise.

c) (8 points)

x[n] =

n, n = 0, . . . , 9

−n + 20, n = 10, . . . , 19

0, otherwise.

Problem 2. (28 points) Inverse z-transform: Determine the inverse z-transform of

X(z) = 2 − 3z

−1

1 − 3z−1 + 2z−2

,

for the following two cases

a) (14 points) The ROC is |z| > 2.

b) (14 points) The ROC is 1 < |z| < 2.

Hint: use partial fraction expansion.

Problem 3. (24 points) An input-output response pair of a relaxed causal and stable LTI system

is given by

x[n] =

1

2

n

u[n], y[n] = n

1

2

n−1

u(n − 1).

a) (8 points) Determine the transfer function of the system and indicate its ROC.

b) (8 points) Determine the poles and zeros of the system.

c) (8 points) Determine a difference equation relating any input sequence x[n] to the corresponding

output sequence y[n].

Problem 4. (24 points) Find the impulse response sequences of the LTI systems with the following

transfer functions:

a) (8 points) H(z) = z

2

(z− 1

2

)(z+ 1

3

)

, |z| >

1

2

.

b) (8 points) H(z) = 1

z

2+ 1

4

, |z| <

1

2

.

c) (8 points) H(z) = z+ 1

3

(z− 1

2

)(z+ 1

4

)

, |z| <

1

4

.

1

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