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