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

580 PERFORMANCE ANALYSIS OF DIGITAL COMMUNICATION SYSTEMS
that if the actual source statistics were known beforehand, one could use Bayes' decision rule
to design a better receiver.
It is apparent that if the source statistics are not known, the maximum likelihood receiver
proves very attractive for a symmetrical signal set. In such a receiver one can specify the error
probability independently of the actual source statistics.
Minimax Receiver
Designing a receiver with a certain decision rule completely specifies the conditional
probabilities P(Clmi). The probability of error is given by
P e M = 1 - P(C)
M
= 1 - L P(m;)P(Clm;)
i=l
Thus, in general, for a given receiver (with some specified decision rule) the error probability
depends on the source statistics P(m;). The error probability is the largest for some source
statistics. The error probability in the worst possible case is [P e M Jmax and represents the upper
bound on the error probability of the given receiver. This upper bound [P e M Jmax serves as an
indication of the quality of the receiver. Each receiver (with a certain decision rule) will have a
certain [P e M Jmax• The receiver that has the smallest upper bound on the error probability, that
is, the minimum [P e M Jmax, is called the minimax receiver.
We shall illustrate the minimax concept for a binary receiver with on-off signaling. The
conditional PDFs of the receiving-filter output sample r at t = T b are p(rl l) and p(rlO). These
are the PDFs of r for the "on" and the "off" pulse (i.e., no pulse), respectively. Figure 10.35a
shows these PDFs with a certain threshold a. If we receive r ::::: a, we choose the hypothesis
"signal present" (1), and the shaded area to the right of a is the probability of false alarm
(deciding "signal present" when in fact the signal is not present). If r < a, we choose the
hypothesis "signal absent" (0), and the shaded area to the left of a is the probability of false
dismissal (deciding "signal absent" when in fact the signal is present). It is obvious that the
larger the threshold a, the larger the false dismissal error and the smaller the false alarm error
(Fig. 10.35b).
We shall now find the minimax condition for this receiver. For the minimax receiver, we
consider all possible receivers (all possible values of a in this case) and find the maximum
Fi g
ure 1 0.35
Explanation of
minimax
concept.
p(rlO)
I
I p(rll)
I
I
(a)
r-;..
False-alarm cost
(b)
a--,..