B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

538 PERFORMANCE ANALYSIS OF DIGITAL COMMUNICATION SYSTEMS
Figure 1 O. 16
Binary communication
in the
presence of
noise.
er
(a)
(b)
when s1 and sk are widely separated in signal space. In this case, there is virtually no overlap
between the two scattered regions. If either mj or mk is transmitted, the received signal will lie
in one of the two scatter regions. From the position of the received signal, one can decide with
a very small probability of error whether mj or mk was transmitted. In Fig. 10. 16a, the received
signal r is much closer to Sk than to Sj , It is therefore more likely that mk was transmitted.
Note that theoretically each scatter extends to infinity, although the probability of observing
the received signal diminishes rapidly as a point is scattered away from the center. _Hence,
there will always be some overlap between the two scatter sets, resulting in a nonzero error
probability. Thus, even though the received r is much closer to Sk in Fig. 10. 16a, it may still
be generated by Sj plus channel noise.
Figure 10. 16b illustrates the case of stronger noise. In this case, there is a considerable
overlap between the two scattered regions. Because the received signal r is closer to Sj than
to Sk , it is more likely that mj was transmitted. But in this case there is also a considerable
probability that mk may have been transmitted. Hence in this situation, there will be a much
higher probability of error in any decision scheme.
The optimum receiver must decide, from a knowledge of r, which message has been
transmitted. The signal space must be divided into M nonoverlapping, or disjoint, decision
regions R 1 , R2, ..., RM , corresponding to the M messages m 1 , m2, ..., mM. If r falls in the
region R k , the decision is m k . The problem of designing the receiver then reduces to choosing
the boundaries of these decision regions R1 , R2, ..., RM to minimize the probability of error
in decision making.
To recapitulate: A transmitter sends a sequence of messages from a set of M
messages m1 , m2, ..., mM . These messages are represented by finite energy waveforms
s1 (t), s2 (t), ... , SM (t). One waveform is transmitted every T 0 = TM seconds. We assume that
the receiver is time-synchronized with the transmitter. The waveforms are corrupted during
transmissions by an AWGN of PSD N /2. Knowing the received waveform, the receiver must
decide which waveform was transmitted. The merit criterion of the receiver is the minimum
probability of error in making this decision.
10.6.2 Dimensionality of the Detection Signal Space
Let us now discuss the dimensionality of the signal space in our detection problem. If there was
no noise, we would be dealing with only M waveforms s1 (t), s2(t), ... , SM (t). In this case
a signal space of, at most, M dimensions would suffice. This is because the dimensionality