EE113 Digital Signal Processing Homework 4 solution

Description

Problem 1. (15 points) Consider the following periodic signals x[n] and z[n], with period 3 each.
Can you relate the DTFS coefficients of x[n] with the DTFS coefficients of z[n]?
0
𝑥[𝑛] 1 1
0
𝑧[𝑛]
2
1 1
(Hint: Try to find a signal y[n] of period 3 such that z[n] = x[n] ~ y[n].)
Problem 2. (20 points) Determine the DTFTs of the following sequences:
(a). (10 points) x[n] =
1
2

n−3
u[n] +
1
3

n
u[n − 1].
(b). (10 points) x[n] =
1
4

n−1
u[n] + cos
π
3
n


.
Problem 3. (a). (10 points) Determine the IDTFT of the following:
X(Ω) = π
2
· rect 
Ω
π
6

· e
−j2Ω
where we define the rectangular function rect(·) as
rect 
Ω
Ωc

,



1, |Ω| < Ωc
0, Ωc ≤ |Ω| ≤ π
(b). (5 points) Determine the following quantity for the same signal x[n] in (a):
X∞
n=−∞
|x[n]|