Chapter 6
Optimal Demodulation
From
Introduction to Communication Systems
Copyright by Upamanyu Madhow, 2008-2011
As we saw in the chapter on digital modulation, we can send bits over a channel by choosing
one of a set of waveforms to send. For example, when sending a single 16QAM symbol, we are
choosing one of 16 passband waveforms:
s
b
c
,b
s
=
b
c
p
(
t
) cos 2
πf
c
t
−
b
s
p
(
t
) sin 2
πf
c
t
where
b
c
,
b
s
each take values in
{±
1
,
±
3
}
.
We are thus able to transmit log
2
16 = 4 bits of
information.
In this chapter, we establish a framework for recovering these 4 bits when the
received waveform is a noisy version of the transmitted waveform. More generally, we consider
the fundamental problem of
M-ary signaling in additive white Gaussian noise (AWGN):
one of
M
signals,
s
1
(
t
)
,...,s
M
(
t
) is sent. The received signal equals the transmitted signal plus white
Gaussian noise (to be defined later in this chapter). At the receiver, we are faced with a
hypothesis
testing
problem: we have
M
possible hypotheses about which signal was sent, and we have to
make our “best” guess as to which one holds, based on our observation of the received signal. For
our application, we are typically interested in finding a guessing strategy, more formally termed
a
decision rule,
which minimizes the
probability of error
(i.e., the probability of making a wrong
guess). We can now summarize the goals of this chapter as follows.
Goals:
We wish to design optimal receivers when the received signal is modeled as follows:
H
i
:
y
(
t
) =
s
i
(
t
) +
n
(
t
)
, i
= 1
,,,.M
where
H
i
is the
i
th hypothesis, corresponding to signal
s
i
(
t
) being transmitted, and where
n
(
t
)
is white Gaussian noise. We then wish to analyze the performance of such receivers, to see how
performance measures such as the probability of error depend on system parameters. It turns
out that, for the preceding AWGN model, the performance depends only on the received signal-
to-noise ratio (SNR) and on the “shape” of the signal constellation
{
s
1
(
t
)
,...,s
M
(
t
)
}
. Underlying
both the derivation of the optimal receiver and its analysis is a geometric view of signals and
noise as vectors, which we term
signal space concepts.
Once we have this background, we are in
a position to discuss elementary power-bandwidth tradeoffs. For example, 16QAM has higher
bandwidth efficiency than QPSK, so it makes sense that it has lower power efficiency; that is,
it requires higher SNR, and hence higher transmit power, for the same probability of error. We
will be able to quantify this intuition based on the material in this chapter. We will also be able
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