ECE 310 Digital Signal Processing
Homework 3
1. Show that an LSI system with unit pulse response h[n] is causal if and only if h[n] = 0 for n < 0.
2. Show that an LSI system with unit pulse response h[n] is BIBO-stable if and only if P∞
n=−∞ |h[n]|
is bounded (i.e., h[n] is absolutely summable).
3. Determine whether each of the following systems that map input signal {x[n]} to output signal {y[n]}
is BIBO stable.
(a) y[n] = x
5
[n] + 3
(b) y[n] = x[n] ∗ u[n]
(c) y[n] = nx[n]
(d) y[n] = x[n]
x[1]
(e) y[n] = x[n] ∗ h[n], where h[n] =
0 for n < 0
2
(n+1)2
for 0 ≤ n < 100
0.5
n
for n ≥ 100
4. Determine the z-transform and sketch the ROC for each of the following sequences:
(a) x[n] = δ[n + 1] − 2δ[n − 2]
(b) {x[n]} = {−1, 0
↑
, 1, 2, 3}
(c) x[n] =
1
2
n−1
u[n − 2]
(d) x[n] = 2
1
2
n
u[n − 2] + 3
1
3
n−3
u[n + 3]
5. Given the z-transform pair
x[n] ←→ X(z) = 1
1 − (1/3)z−1
, with ROC: |z| > 1/3,
use the z-transform properties to determine the z-transform and ROC of the following sequences
(a) y[n] = x[n − 1]
(b) y[n] = n
2x[n]
(c) y[n] = 2nx[n]
(d) y[n] = cos(πn/4)x[n]
(e) y[n] = (x ∗ u)[n]
(f) y[n] = (x ∗ h)[n] where h[n] = x[n − 2]
