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Ø
6.003
Homework
11
Please
do
the
following
problems
by
Wednesday,
April
28,
2010
.
You
need
not
submit
your
answers:
they
will
NOT
be
graded.
Solutions
will
be
posted.
Problems
1.
Impulsive
Input
Let
the
following
periodic
signal
∞
x
(
t
) =
δ
(
t
−
3
m
) +
δ
(
t
−
1
−
3
m
)
−
δ
(
t
−
2
−
3
m
)
m
=
−∞
be
the
input
to
an
LTI
system
with
system
function
H
(
s
) =
e
s/
4
−
e
−
s/
4
.
Let
b
k
represent
the
Fourier
series
coeﬃcients
of
the
resulting
output
signal
y
(
t
).
Deter
mine
b
3
.
2.
Fourier
transform
Part
a.
Find
the
Fourier
transform
of
x
1
(
t
) =
e
−
t

.
Part
b.
Find
the
Fourier
transform
of
1
x
2
(
t
) =
.
1 +
t
2
Hint:
Try
duality.
3.
Fourier
representations
of
the
pulse
In
this
problem
you
connect
the
four
Fourier
representations,
as
we
did
for
you
in
Lecture
20,
where
we
used
a
triangle
as
the
canonical
function.
Here,
you
use
a
pulse:
f
(
t
) =
î
1
for
−
1
< t <
1;
0
otherwise.
Part
a.
Find
F
(
jω
),
the
continuoustime
Fourier
transform
of
f
(
t
).
State
Parseval’s
theorem
for
the
Fourier
transform,
and
check
that
it
works
by
applying
it
to
f
(
t
) and
F
(
jω
).
Part
b.
Sketch
f
p
(
t
),
a
periodic
version
of
f
(
t
) with
period
T
= 4.
Find
the
Fourier
transform
of
f
p
(
t
) and
compare
it
to
the
Fourierseries
coeﬃcients
f
k
for
f
p
(
t
).
State
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This note was uploaded on 12/14/2011 for the course EECS 6.003 taught by Professor Dennism.freeman during the Spring '11 term at MIT.
 Spring '11
 DennisM.Freeman

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