ECE 310 Digital Signal Processing –
Homework 1
1. Evaluate and represent your final answer in both Cartesian and polar forms.
(a) 3e
−jπ/4 + 4e
jπ/4
(b) (1 + j)
2
1 − j
2. Determine all the roots of 2z
3 + 3 = 0 on the complex plane.
3. Sketch the following signals:
(a) sin
π
3
n
δ[n − 2]
(b) n(u[n] − u[n − 8])
(c) u[−n + 3]u[n + 5]
where u[n] is the unit step signal in the discrete-time variable n.
4. Express the sequence {x[n]} = {· · · , 0, −1, 0, 3
↑
, 0, 7, 0, · · · } in terms of the unit pulse (impulse) signal
δ[n]. Here · · · denotes zeros.
5. Sketch (by hand) the magnitude and phase of G(ω) = sin(ω/2) over the interval ω ∈ [−π, π]. Label
your plots.
6. Derive closed-form expressions for the magnitude and phase of the function G(ω) = 1 − e
−jω of the
real variable ω. Sketch (by hand) the magnitude and phase over the interval ω ∈ [−π, π]. Label your
plots.
7. Consider the following discrete-time system
y[n] = x[n] + 2x[n − 2].
Determine if the system is: 1) linear; 2) time-invariant.
8. Consider the following discrete-time system
y[n] = 10x[n] cos(0.25πn).
Determine if the system is: 1) linear; 2) time-invariant
