6.003: Signals and Systems
Lecture 16
November 5, 2009
1
6.003: Signals and Systems
Fourier Series
November 5, 2009
Last Time: Describing Signals by Frequency Content
Harmonic content is natural way to describe some kinds of signals.
Ex: musical instruments (http://theremin.music.uiowa.edu/MIS)
piano
t
piano
k
violin
t
violin
k
bassoon
t
bassoon
k
Last Time: Fourier Series
Determining harmonic components of a periodic signal.
a
k
=
1
T
T
x
(
t
)
e
−
j
2
π
T
kt
dt
(“analysis” equation)
x
(
t
)=
x
(
t
+
T
) =
∞
k
=
−∞
a
k
e
j
2
π
T
kt
(“synthesis” equation)
Separating harmonic components
Underlying properties.
1.
Multiplying two harmonics produces a new harmonic with the
same fundamental frequency:
e
jkω
0
t
×
e
jlω
0
t
=
e
j
(
k
+
l
)
ω
0
t
.
Closure: the set of harmonics is closed under multiplication.
2.
The integral of a harmonic over any time interval with length
equal to a period
T
is zero unless the harmonic is at DC:
t
0
+
T
t
0
e
jkω
0
t
dt
≡
T
e
jkω
0
t
dt
=
0
, k
= 0
T, k
= 0
=
Tδ
[
k
]
Separating harmonic components
Underlying properties.
1.
Multiplying two harmonics produces a new harmonic with the
same fundamental frequency:
e
jkω
0
t
×
e
jlω
0
t
=
e
j
(
k
+
l
)
ω
0
t
.
Closure: the set of harmonics is closed under multiplication.
2.
The integral of a harmonic over any time interval with length
equal to a period
T
is zero unless the harmonic is at DC:
t
0
+
T
t
0
e
jkω
0
t
dt
≡
T
e
jkω
0
t
dt
=
0
, k
= 0
T, k
= 0
=
Tδ
[
k
]
Orthogonality: harmonics are orthogonal (
⊥
) to each other.
Fourier Series as Orthogonal Decompositions
Analogy with vectors in 3space.
Let
ˆ
x
,
ˆ
y
, and
ˆ
z
represent direction vectors in 3space.
This preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
This is the end of the preview.
Sign up
to
access the rest of the document.
 Spring '11
 DennisM.Freeman
 Fourier Series, Frequency, Human voice, Vowel, vocal cords

Click to edit the document details