Boston University
Department of Electrical and Computer Engineering
ENG EC515 Digital Communication (
Fall 2009
)
Solution to Problem set 4
Posted:
Mon 12 Oct
Suggested reading:
Classnotes, handouts, and Ch. 4 textbook
Notation:
Q
(
x
) denotes the Qfunction deﬁned as
Q
(
x
)
=
1
√
2
π
R
∞
x
exp(

t
2
/
2)d
t
and SNR in deci
bels (dB)
=
10 log
10
(SNR).
Problem 4.1
(
16
ary QAM)
[10pts] One of 16 equally likely messages is to be transmitted over a
vector (zeromean) AWGN channel with noise variance
σ
2
. The transmitter uses the 16ary QAM
signal constellation where the points have coordinates
{
(
d
(2
i

1)
2
,
d
(2
j

1)
2
)
}
,
i
,
j
=

1
,
0
,
1
,
2.
(a)
[4pts] Find the
exact expression
for the minimum probability of error in terms of the
Q
function using the method which involves enumerating all types of decoding regions. Com
pare your answer with the method discussed in the class which is based on the idea of two
independent 4ary PAM decoders.
(b)
[3pts] Is it possible to modify this signal constellation to reduce the probability of error
without changing
E
max
? If yes, explain how. If no, explain why.
(c)
[3pts] Plot the graph of log
10
(
P
16

PAM
e
) versus SNR
16

PAM
in dB. Also plot the graph of
log
10
(
P
16

QAM
e
) versus SNR
16

QAM
in dB. For small values of error probability, roughly how
much more SNR in decibels is needed to get the same probability of error? What insight
does this o
ﬀ
er?
Solution:
[10pts]
√
E
max
d
d
φ
1
φ
2
(0
,
0)
decision regions
boundary
4
corner
decision regions
8
non

corner
1
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[4pts] There are 3 types of decoding regions:
I
) 4 corners,
II
) 4 squares, and
III
) 8 non
corner boundary rectangular strips. Let
P
I
c

m
denote the probability of correct decoding given
that message
M
=
m
belonging to region
I
was transmitted. Similarly, let
P
II
c

m
and
P
III
c

m
denote
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 Fall '09
 VenkateshSaligrama
 Quadrature amplitude modulation, Phaseshift keying, Emax, constellation diagram, signal constellation

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