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6.450
Principles
of
Digital
Communication
Wednesday,
November
6,
2002
MIT,
Fall
2002
Handout
#32
Due:
Wednesday,
November
13,
2001
Problem
Set
9
Problem
9.1
(Carrierless
modulation)
In
Lecture
13,
we
saw
how
to
modulate
a
base
band
QAM
waveform
to
passband
and
then
demodulate
by
shifting
down
to
passband,
followed
by
filtering
and
sampling
at
baseband.
This
exercise
explores
the
interesting
con
cept
of
completely
eliminating
the
baseband
operations
by
modulating
and
demodulating
directly
at
passband.
(a)
Let
{
a
k
}
be
a
complex
data
sequence
and
let
u
(
t
) =
k
a
k
p
(
t
−
kT
)
be
the
corre
√
sponding
modulated
output.
Let
p
ˆ(
f
)
be
equal
to
T
over
f
∈
[3
/
(2
T
)
,
5
/
(2
T
)]
and
be
equal
to
0
elsewhere.
At
the
receiver,
u
(
t
)
is
filtered
using
p
(
t
)
and
the
output
y
(
t
)
is
then
Tspace
sampled
at
time
instants
kT
.
Show
that
y
(
kT
) =
a
k
for
all
k
∈
Z
.
Don’t
worry
about
the
fact
that
the
transmitted
waveform
u
(
t
)
is
complex.
(b)
For
what
other
brickwall
positive
frequency
filters
P
(
f
)
would
this
scheme
work?
Could
it
work
if
P
(
f
)
was
not
brickwall?
(c)
We
cannot
send
complex
signals
over
real
channels.
Give
a
general
method
for
sending
a
complex
signal
u
(
t
)
whose
spectrum
is
bandlimited
to
a
given
positivefrequency
band
ˆ
[
f
0
, f
1
]
through
a
real
channel
with
a
brick
wall
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 Spring '08
 DanielLee

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