University of Minnesota
Dept. of Electrical and Computer Engineering
EE3015 – Signals and Systems
Discussion Session #10: Applications of Fourier representations
1.
Consider the system depicted below. The input signal
x
(
t
) has a Fourier
Transform (FT)
X
(
j
ω
) also shown below.
Signal
)
(
t
x
is multiplied by
)
(
t
p
, which is a periodic signal with period
T
,
fundamental frequency
W
T
2
/
2
0
>
=
π
ω
, and
4
/
1
/
0
=
T
T
.
a.
Find and sketch
P
(
j
ω
), the FT of
p
(
t
)
In class, we showed that the FS coefficients of
p
(
t
) are given by
π
ω
k
T
k
a
t
p
k
FS
)
sin(
)
(
0
0
=
⎯→
←
The FT of
p
(
t
) is then given by
∑
∑
∞
−∞
=
∞
−∞
=
−
=
−
⎯
⎯→
←
k
k
k
FT
k
k
T
k
k
a
t
p
)
(
)
sin(
2
)
(
2
)
(
0
0
0
0
ω
ω
δ
ω
ω
ω
δ
π
Which results in
x
(
t
)
p
(
t
)
h
(
t
)
v
(
t
)
y
(
t
)
ω
1
X
(
j
ω
)
-W
W
t
1
p
(
t
)
…
…
-2T
2
3
T
−
-T
2
T
−
-T
0
T
0
2
T
T
2
3
T
2T

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