±
6.003:
Signals
and
Systems
Lecture
20
April
22,
2010
6.003:
Signals
and
Systems
Relations
among
Fourier
Representations
April
22,
2010
Midterm
Examination
#3
Wednesday,
April
28,
7:309:30pm.
No
recitations
on
the
day
of
the
exam.
Coverage:
Lectures
1–20
Recitations
1–20
Homeworks
1–11
Homework
11
will
not
collected
or
graded.
Solutions
will
be
posted.
Closed
book:
3
pages
of
notes
(
8
1
2
×
11
inches;
front
and
back).
Designed
as
1hour
exam;
two
hours
to
complete.
Review
sessions
during
open
oﬃce
hours.
Fourier
Representations
We’ve
seen
a
variety
of
Fourier
representations:
•
CT
Fourier
series
•
CT
Fourier
transform
•
DT
Fourier
series
•
DT
Fourier
transform
Today:
relations
among
the
four
Fourier
representations.
π
N
π
N
π
T
π
T
Four
Fourier
Representations
We
have
discussed
four
closely
related
Fourier
representations.
DT
Fourier
Series
DT
Fourier
transform
∞
a
k
=
a
k
+
N
=
1
x
[
n
]
e
−
j
2
kn
X
(
e
j
Ω
)=
x
[
n
]
e
−
j
Ω
n
N
n
=
<N>
n
=
−∞
x
[
n
]=
x
[
n
+
N
a
k
e
j
2
kn
x
[
n
1
X
(
e
j
Ω
)
e
j
Ω
n
d
Ω
k
=
<N>
2
π
<
2
π>
CT
Fourier
Series
CT
Fourier
transform
a
k
=
1
±
x
(
t
)
e
−
j
2
kt
dt
X
(
jω
±
∞
x
(
t
)
e
−
jωt
dt
T
T
−∞
∞
±
∞
x
(
t
x
(
t
+
T
a
k
e
j
2
kt
x
(
t
1
X
(
)
e
dω
2
π
k
=
−∞
−∞
Four
Types
of
“Time”
discrete
vs.
continuous
(
±
)
and
periodic
vs
aperiodic
(
↔
)
DT
Fourier
Series
DT
Fourier
transform
n
n
CT
Fourier
Series
CT
Fourier
transform
t
t
Four
Types
of
“Frequency”
discrete
vs.
continuous
(
↔
)
and
periodic
vs
aperiodic
(
±
)
DT
Fourier
Series
DT
Fourier
transform
2
π
k
N
Ω
CT
Fourier
Series
CT
Fourier
transform
2
π
k
ω
T
1
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View Full Document6.003:
Signals
and
Systems
Lecture
20
April
22,
2010
Relation
between
Fourier
Series
and
Transform
Relations
among
Fourier
Representations
Diﬀerent
Fourier
representations
are
related
because
they
apply
to
signals
that
are
related.
DTFS
(discretetime
Fourier
series):
periodic
DT
DTFT
(discretetime
Fourier
transform):
aperiodic
DT
CTFS
(continuoustime
Fourier
series):
periodic
CT
CTFT
(continuoustime
Fourier
transform):
aperiodic
CT
periodic
DT
DTFS
aperiodic
DT
DTFT
periodic
CT
CTFS
aperiodic
CT
CTFT
N
→∞
periodic
extension
T
periodic
extension
interpolate
sample
interpolate
sample
Relation
between
Fourier
Series
and
Transform
A
periodic
signal
can
be
represented
by
a
Fourier
series
or
by
an
equivalent
Fourier
transform.
x
(
t
)=
x
(
t
+
T
∞
k
=
−∞
a
k
e
jω
0
kt
Fourier
Transform
0
ω
0
k
Fourier
Series
01
a
0
a
1
a
−
1
a
2
a
−
2
a
3
a
−
3
a
4
a
−
4
↔
A
periodic
signal
can
be
represented
by
a
Fourier
series
or
by
an
equivalent
Fourier
transform.
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 Summer '10
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 Digital Signal Processing, Fourier Series, CT Fourier Transform

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