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

l 0. 10 Other Useful Performance Criteria 579
and the decision rule in Eq. (10.134) reduces to the rule in Eq. (10.83), as expected. The
generalized Bayes receiver for M = 2, assuming C1 1 = C22 = 0, sets m = mi if
Otherwise, the receiver decides that m = mz.
Maximum Likelihood Receiver
The strategy used in the Bayes receiver discussed in the preceding subsection is general, except
that it can be implemented only when the a priori probabilities P(m1 ), P(m2), ... , P(mM) are
known. Frequently this information is not available. Under these conditions various possibilities
exist, depending on the assumptions made. When, for example, there is no reason to expect
any one signal to be more likely than any other, we may assign equal probabilities to all the
messages:
1
P(m1 ) = P(m2) = · · · = P(mM ) = - M
Bayes' rule [Eq. (10.83)] in this case becomes: set m = mk if
for all i i= k (10.136)
Observe that pq (qlmk) represents the probability of observing q when mk is transmitted. Thus,
the receiver chooses that signal which, when transmitted, will maximize the likelihood (probability)
of observing the received q. Hence, this receiver is called the maximum likelihood
receiver. Note that the maximum likelihood receiver is a Bayes receiver for the cost of
Eq. (10.135) under the condition that the a priori message probabilities are equal. In terms
of geometrical concepts, the maximum likelihood receiver decides in favor of that signal
which is closest to the received data q. The practical implementation of the maximum likelihood
receiver is the same as that of the Bayes receiver (Figs. 10.18 and 10.19) under the
condition that all a priori probabilities are equal to 1 / M .
If the signal set is geometrically symmetrical, and if all a priori probabilities are equal
(maximum likelihood receiver), then the decision regions for various signals are congruent.
In this case, because of symmetry, the conditional probability of a correct decision is the same
no matter which signal is transmitted, that is,
P(Clmi) = constant
for all i
Because
M
P(C) = L P(mi)P(Clmi)
i=l
in this case
P(C) = P(Clmi) (10.137)
Thus, the error probability of the maximum likelihood receiver is independent of the actual
source statistics P(mi) for the case of symmetrical signal sets. It should, however, be realized