CS 702
Discrete Mathematics and Probability Theory
Spring 2009
Alistair Sinclair, David Tse
Lecture 18
Inference Example 3: Communication over Physical Channels
Question:
I have one bit of information that I want to communicate over a physical channel. I map the
value of the bit to a voltage signal with amplitude
+
a
or

a
to transmit over the channel, which corrupts it
by additive Gaussian
1
noise. The receiver needs to guess the transmitted bit from the received signal. How
much improvement in reliability do I get by repeating my transmission
n
times?
Comment:
This is similar to the communication problem considered in Lecture Note 16.
There, both
the transmitted symbols and the received symbols are discrete (0 or 1) and the channel randomly flips the
transmitted symbols. (This channel is called the binary symmetric channel.) In the present problem, the
transmitted signal is still discrete (either
+
a
or

a
is transmitted) but the received signals are continuous
valued due to the continuousvalued noise. Most physical channels are analog and so this is a natural model.
In fact, the discrete received symbols considered in the earlier lecture can be viewed as a quantization of
the received analog signal through a 1bit analogtodigital convertor. We are now directly modeling the
underlying analog physical channel.
The parameter
a
is the
amplitude
of the transmitted signal, and is
dictated by the energy or device limitation of the transmitter.
Modeling
The situation is shown in Figure 1.
Let
X
(
= +
a
or

a
) be the value of the voltage signal I want to transmit (to represent the value of the
information bit). Assume that
X
is equally likely to be
+
a
or

a
. The received signal
Y
i
on the
i
th repetition
of
X
is
Y
i
=
X
+
Z
i
,
i
=
1
,
2
,...,
n
with
Z
i
being Gaussian distributed with mean zero and variance
σ
2
.
(This is often abbreviated as
Z
i
∼
N
(
0
,
σ
2
)
.)
The addition is now real addition.
The
Z
i
’s are assumed to be mutually independent across
different repetitions of
X
and also independent of
X
.
The Gaussian distribution is found to be a reasonable model for the noise in many physical channels. One
reason is that the noise is often the aggregation of many small noise sources, and the Central Limit Theorem,
to be discussed in the next lecture, implies that the aggregate should look Gaussian.
Single Transmission
Let’s first solve the special case when the signal
X
is transmitted only once. The problem is: given the
received signal
Y
=
X
+
Z
1
Recall that ”Gaussian” and ”normal” are two equivalent names for the same distribution. In this note, in keeping with standard
practice in the communications community, we will use the term ”Gaussian.” We shall use the term ”normal” in other contexts.
CS 702, Spring 2009, Lecture 18
1
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Figure 1: The system diagram for the communication problem.
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 Spring '08
 PAPADIMITROU
 Probability theory, error probability, n Zi

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