GEORGIA INSTITUTE OF TECHNOLOGY
SCHOOL of ELECTRICAL and COMPUTER ENGINEERING
ECE 2025
Fall 2008
Problem Set #13
Assigned:
17Nov08
Due Date:
Week of 1Dec08
Reading: In
SP First
, Chapters 11 and 12 on
Fourier Transform and its Applications
.
Please check
tsquare.gatech.edu
often. All official course announcements are posted there.
ALL
of the
STARRED
problems should be turned in for grading. A solution will be posted to
the web. Some problems have solutions similar to those found on the CDROM.
Your homework is due in recitation at the beginning of class.
After the beginning of your
assigned recitation time, the homework is considered late and will be given a zero. Please follow
the format guidelines (cover page, etc.) for homework.
All students should turn this in on Wednesday 12/3 or Thursday 12/4; this will be during recitation
or lab, depending on your particular schedule.
PROBLEM 13.1
*:
(a) Determine the
forward
Fourier Transform
X
(
jω
), given
x
(
t
):
(i)
x
(
t
) = 3 cos
2
(17
t
)
(ii)
x
(
t
) =
δ
(
t
+ 1)

2
e

t
u
(
t

1)
(iii)
x
(
t
) =
d
dt
sin(17
π
(
t

2))
2
πt

4
π
(iv)
x
(
t
) =
u
(
t

1)

u
(
t

8)
(v)
x
(
t
) = 7 + 10 cos(4
t

π/
3)
(b) Determine the
inverse
Fourier Transform
x
(
t
), given
X
(
jω
):
(i)
X
(
jω
) = cos
2
(3
ω
)
(ii)
X
(
jω
) =
jω

1
2 +
jω
(iii)
X
(
jω
) =
j
13
{
δ
(
ω

3
π
)

δ
(
ω
+ 3
π
)
}
(iv)
X
(
jω
) =
u
(
ω
+
π
)
u
(
π

ω
)
e

jω/
10
(v)
X
(
jω
) =
1
1 +
j
(
ω

1)
+
1
1 +
j
(
ω
+ 1)
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PROBLEM 13.2
*:
The impulse response of an LTI system is given by
h
(
t
) =
sin(8
πt

4
π
)
2
t

1
(a) First, make a detailed and accurately labeled sketch of
h
(
t
) over the time interval

1
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 Spring '08
 JUANG
 Digital Signal Processing, Signal Processing, LTI system theory

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