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ECE 5670 : Digital Communications (Spring 2011)
Homework 3
Due February 17 in class
Instructor: Salman Avestimehr
Oﬃce 325 Rhodes Hall
1. In this exercise we will improve the energy eﬃciency of the position modulation scheme
introduced in Lecture 4. We do this by being more careful in evaluating the unreli
ability level, as compared to the simple union bound we used in the lecture. In the
lecture we have seen that as long as the energy eﬃciency (also called energy per bit)
E
b
>
4
σ
2
log
e
2
,
(1)
we have arbitrarily reliable communication. We will now improve this by a factor of
0.5. Speci±cally we show that as long as the energy eﬃciency
2
σ
2
log
e
2
<
E
b
<
4
σ
2
log
e
2(
2
)
we have reliable communication using position modulation.
(a) Show that the exact probability of error of communicating using the position
with ML decoding at the receiver is
P
[
E
]=1
−
Z
∞
−∞
f
±
a
−
√
B
E
b
σ
²
(1
−
Q
(
a
))
2
B
−
1
da
(3)
=
Z
∞
−∞
f
±
a
−
√
B
E
b
σ
²
³
1
−
(1
−
Q
(
a
))
2
B
−
1
´
da.
(4)
Here the function
f
(
·
) is simply the histogram of the Gaussian with zero mean
and unit variance:
f
(
a
)
def
=
1
√
2
π
e
−
a
2
2
,
∀
a.
(5)
Hint
: It is easy to calculate the probability of being correct ±rst.
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This note was uploaded on 10/02/2011 for the course ECE 5670 taught by Professor Scaglione during the Spring '11 term at Cornell University (Engineering School).
 Spring '11
 SCAGLIONE

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