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ECE 301, Homework #13, due date: 4/21/2010
http://cobweb.ecn.purdue.edu/
∼
chihw/10ECE301S/10ECE301S.html
Question 1:
[Basic] Find and plot the DTFT of
x
[
n
] =
∑
∞
k
=
∞
δ
[
n

4
k
]. Find and plot
the CTFT of
y
(
t
) =
∑
∞
k
=
∞
δ
(
t

πk
). Observe the similarity between both your answers.
Hint: you should use the generalized DTFT and CTFT formulas. Namely, ﬁnd the
coeﬃcients of DTFS and CTFS ﬁrst and then convert them to DTFT and CTFT by
inspection.
Question 2:
[Basic] Consider a signal
w
(
t
) =
sin(0
.
5
t
)
πt
. Plot
w
(
t
). The
impulsetrain sam
pling
of
w
(
t
) with sample period
T
s
=
π
2
is
z
(
t
) =
w
(
t
)
y
(
t
) =
w
(
t
)
∞
X
k
=
∞
δ
(
t

π
2
k
) =
∞
X
k
=
∞
w
(
π
2
k
)
δ
(
t

π
2
k
)
.
Plot
z
(
t
) for the range

1
.
5
π < t <
1
.
5
π
.
Question 3:
[Basic] Continue from the previous question. Use the multiplication property
of CTFT to plot
Z
(
jω
). Hint: You need to use the observation that “convolving a shifted
impulse is equivalent to shifting the original signal.”
Question 4:
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This note was uploaded on 12/08/2010 for the course ECE 302 taught by Professor Gelfand during the Spring '08 term at Purdue UniversityWest Lafayette.
 Spring '08
 GELFAND

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