bassoon
t
6.003:
Signals
and
Systems
6.003:
Signals
and
Systems
Fourier
Representations
March
30,
2010
Fourier
Representations
Fourier
series
represent
signals
in
terms
of
sinusoids
.
→
leads
to
a
new
representation
for
systems
as
ﬁlters
.
Lecture
14
March
30,
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.
Fourier
Series
Representing
signals
by
their
harmonic
components.
ω
ω
0
2
ω
0
3
ω
0
4
ω
0
5
ω
0
6
ω
0
0
1
2
3
4
5
6
←
harmonic
#
DC
→
fundamental
second
harmonic
third
fourth
ﬁfth
sixth
Musical
Instruments
Harmonic
content
is
natural
way
to
describe
some
kinds
of
signals.
Ex:
musical
instruments
(http://theremin.music.uiowa.edu/MIS)
piano
cello
bassoon
t
t
t
oboe
horn
altosax
t
t
t
violin
t
1
252
seconds
1
Musical
Instruments
Harmonic
content
is
natural
way
to
describe
some
kinds
of
signals.
Ex:
musical
instruments
(http://theremin.music.uiowa.edu/MIS)
piano
k
cello
k
bassoon
k
oboe
k
horn
k
altosax
k
violin
k
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²
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6.003:
Signals
and
Systems
Lecture
14
March
30,
2010
Musical
Instruments
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
Harmonics
Harmonic
structure
determines
consonance
and
dissonance.
octave
(D+D’)
ﬁfth
(D+A)
D+E
±
time(periods of "D")
harmonics
0123456789101112
0123456789101112
–1
0
1
D
D'
D
A
D
E
0123456789
01234567
Harmonic
Representations
Harmonic
Representations
What
signals
can
be
represented
by
sums
of
harmonic
components?
Is
it
possible
to
represent
ALL
periodic
signals
with
harmonics?
What
about
discontinuous
signals?
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 Summer '10
 staff
 Fourier Series, Square wave, Fourier Series Properties

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