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]