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6.003:
Signals
and
Systems
Lecture
15
April
1,
2010
6.003:
Signals
and
Systems
Fourier
Series
April
1,
2010
Midterm
Examination
#2
Wednesday,
April
7,
7:309:30pm
.
No
recitations
on
the
day
of
the
exam.
Coverage:
Lectures
1–15
Recitations
1–15
Homeworks
1–8
Homework
8
will
not
collected
or
graded.
Solutions
will
be
posted.
Closed
book:
2
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.
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
piano
t
k
violin
violin
t
k
bassoon
bassoon
t
k
π
T
π
T
Last
Time:
Fourier
Series
Determining
harmonic
components
of
a
periodic
signal.
a
k
=
1
x
(
t
)
e
−
j
2
kt
dt
(“analysis”
equation)
T
T
∞
²
2
x
(
t
)=
x
(
t
+
T
)=
a
k
e
j
kt
(“synthesis”
equation)
k
=
−∞
We
can
think
of
Fourier
series
as
an
orthogonal
decomposition
.
π
T
π
T
π
T
π
T
Orthogonal
Decompositions
Vector
representation
of
3space:
let
r
¯
represent
a
vector
with
components
{
x
,
y
,
and
z
}
in
the
{
x
ˆ
,
y
ˆ
,
and
z
ˆ
}
directions,
respectively.
x
=¯
r
·
x
ˆ
y
r
·
y
ˆ
(“analysis”
equations)
z
r
·
z
ˆ
r
¯ =
xx
ˆ +
yy
ˆ +
zz
ˆ
(“synthesis”
equation)
Fourier
series:
let
x
(
t
)
represent
a
signal
with
harmonic
components
{
a
0
,
a
1
,
...
,
a
k
}
for
harmonics
{
e
j
0
t
,
e
j
2
t
,
,
e
j
2
kt
}
respectively.
a
k
=
1
x
(
t
)
e
−
j
2
kt
dt
(“analysis”
equation)
T
T
∞
²
2
x
(
t
)=
x
(
t
+
T
a
k
e
j
kt
(“synthesis”
equation)
k
=
−∞
π
T
π
T
π
T
π
T
π
T
π
T
Orthogonal
Decompositions
Integrating
over
a
period
sifts
out
the
k
th
component
of
the
series.
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