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Lecture 18
November 12, 2009
1
6.003: Signals and Systems
CT Fourier Transform
November 12, 2009
Midterm Examination #3
Wednesday, November 18, 7:309:30pm, Walker Memorial
(this exam is
after
drop date).
No recitations on the day of the exam.
Coverage: cumulative with more emphasis on recent material
lectures 1–18
homeworks 1–10
Homework 10 will not collected or graded. Solutions will be posted.
Closed book: 3 page of notes (
8
1
2
×
11
inches; front and back).
Designed as 1hour exam; two hours to complete.
Review sessions during open oﬃce hours.
Conﬂict? Contact freeman@mit.edu by Friday, November 13, 2009.
CT Fourier Transform
Representing signals by their frequency content.
X
(
jω
)=
Ú
∞
−∞
x
(
t
)
e
−
jωt
dt
(“analysis” equation)
x
(
t
)=
1
2
π
Ú
∞
−∞
X
(
jω
)
e
jωt
dω
(“synthesis” equation)
•
generalizes Fourier series to represent aperiodic signals.
•
equals Laplace transform
X
(
s
)

s
=
ω
if ROC includes
jω
axis.
→
inherits properties of Laplace transform.
•
complexvalued function of
real
domain
ω
.
•
simple ”inverse” relation
→
more general than tablelookup method for inverse Laplace.
→
“duality.”
•
ﬁltering.
•
applications in physics
Visualizing the Fourier Transform
Fourier transform is a function of a single variable: frequency
ω
.
Time representation:
−
1
1
x
1
(
t
)
1
t
Frequency representation:
2
π
X
1
(
jω
) =
2 sin
ω
ω
ω
Check Yourself
Signal
x
2
(
t
)
and its Fourier transform
X
2
(
jω
)
are shown below.
−
2
2
x
2
(
t
)
1
t
b
ω
0
X
2
(
jω
)
ω
Which is true?
1.
b
= 2
and
ω
0
=
π/
2
2.
b
= 2
and
ω
0
= 2
π
3.
b
= 4
and
ω
0
=
π/
2
4.
b
= 4
and
ω
0
= 2
π
5. none of the above
Check Yourself
Stretching time compresses frequency.
Find a general scaling rule.
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 Spring '11
 DennisM.Freeman

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