6.003: Signals and Systems
Lecture 20
November 19, 2009
1
6.003: Signals and Systems
Relations among Fourier Representations
November 19, 2009
Fourier Representations
We’ve seen a variety of Fourier representations:
•
CT Fourier series
•
CT Fourier transform
•
DT Fourier series
One more today:
DT Fourier transform
... and relations among all four representations.
DT Fourier transform
Representing aperiodic DT signals as sums of complex exponentials.
DT Fourier transform
X
(
e
j
Ω
)=
∞
Ø
n
=
−∞
x
[
n
]
e
−
j
Ω
n
(“analysis” equation)
x
[
n
]=
1
2
π
Ú
<
2
π>
X
(
e
j
Ω
)
e
j
Ω
n
d
Ω
(“synthesis” equation)
Comparison to DT Fourier Series
From periodic to aperiodic.
DT Fourier Series
a
k
=
a
k
+
N
=
1
N
Ø
n
=
<N>
x
[
n
]
e
−
j
Ω
0
kn
; Ω
0
=
2
π
N
(“analysis” equation)
x
[
n
]=
x
[
n
+
N
] =
Ø
k
=
<N>
a
k
e
j
Ω
0
kn
(“synthesis” equation)
DT Fourier transform
X
(
e
j
Ω
)=
∞
Ø
n
=
−∞
x
[
n
]
e
−
j
Ω
n
(“analysis” equation)
x
[
n
]=
1
2
π
Ú
<
2
π>
X
(
e
j
Ω
)
e
j
Ω
n
d
Ω
(“synthesis” equation)
DT Fourier Series and Transform
Example of series.
Let
x
[
n
]
represent the following periodic DT signal.
n
x
[
n
] =
x
[
n
+ 8]
−
1 1
−
8
8
a
k
=
1
N
Ø
n
=
<N>
x
[
n
]
e
−
j
Ω
0
kn
; Ω
0
=
2
π
N
=
1
8
±
1 +
1
2
e
−
j
2
π
8
k
+
1
2
e
j
2
π
8
k
²
=
1 + cos
πk
4
8
k
a
k
−
1 1
−
8
8
1
4
DT Fourier Series and Transform
Example of transform.
Let
x
[
n
]
represent the aperiodic base of the previous signal.
n
x
[
n
]
−
1 1
H
(
e
j
Ω
) =
∞
Ø
n
=
−∞
x
[
n
]
e
−
j
Ω
n
=
±
1 +
1
2
e
−
j
Ω
+
1
2
e
j
Ω
²
= 1 + cos Ω
Ω
X
(
e
j
Ω
)
2
π
−
2
π
2
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View Full Document6.003: Signals and Systems
Lecture 20
November 19, 2009
2
DT Fourier Series and Transform
Similarities.
DT Fourier series
k
a
k
−
1 1
−
8
8
1
4
DT Fourier transform
Ω
X
(
e
j
Ω
)
2
π
−
2
π
2
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
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 Spring '11
 DennisM.Freeman
 Fourier Series, CT Fourier Transform

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