Description
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
