ECEN 661 Modulation Theory
Spring 2008
Midterm Exam #2
Problem 1.
(40 points)
A certain 8ary quadrature amplitude modulation scheme sends one of the following signals:
,
,
,
where
,
and
is a constant.
The signal is received over an AWGN
channel so the the received signal is
,
where
is zeromean Gaussian noise with a (twosided) PSD of
.
(a) Describe how the ML receiver works.
That is, what processing does it do on the received sig
nal and how does it make a decision.
(b) Evaluate the exact probability of error of the ML receiver.
Express your answer in terms of
.
(c) Derive a union bound on the probability of error of the ML receiver.
Express your answer in
terms of
.
(d) Find the asymptotic efficiency of this signalling format.
Solution:
Choose as basis functions:
and
.
Based on these
basis functions the vector representation of the signals is
.
The signal
space diagram is shown below.
s
i
t
( )
A
i
ω
c
t
(
)
cos
B
i
ω
c
t
(
)
sin
+
=
0
t
T
s
<
≤
i
1 2
…
8
,
,
,
=
A
i
a
±
3
a
±
,
{
}
∈
B
i
a
±
{
}
∈
a
r t
( )
s
i
t
( )
n t
( )
+
=
n t
( )
N
o
2
⁄
E
b
N
o
⁄
E
b
N
o
⁄
f
1
t
( )
2
T
s

ω
c
t
(
)
cos
=
f
2
t
( )
2
T
s

ω
c
t
(
)
sin
=
s
i
A
i
2
T
s
2

B
i
2
T
s
2

,
=
*
*
*
*
*
*
*
*
r
1
r
2
b
3
b
b
–
3
b
–
b
b
–
b
a
2
T
s
2

=
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(a) Project the received signal onto the basis functions to form the received vector:
,
where
,
.
Then choose which signal was sent based on the decision
regions as shown in the above figure.
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 Spring '08
 Miller
 Pallavolo Modena, Quadrature amplitude modulation

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