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

578 PERFORMANCE ANALYSIS OF DIGITAL COMMUNICATION SYSTEMS
10.10 OTHER USEFUL PERFORMANCE CRITEIA
The optimum receiver uses the decision strategy that makes the best possible use of the observed
data and any a priori information available. The strategy will also depend on the weights
assigned to various types of error. In this chapter we have thus far assumed that all errors have
equal weight ( or equal cost). This assumption is not justified in all cases, and we may therefore
have to alter the decision rule.
Generalized Bayes Receiver
If we are given a priori probabilities and the cost functions of errors of various types, the
receiver that minimizes the average cost of decision is called the Bayes receiver, and the
decision rule is Bayes' decision rule. Note that the receiver that has been discussed so far
is the Bayes receiver under the condition that all errors have equal cost (equal weight). To
generalize this rule, let
and, as usual,
Ckj = cost of deciding that m = mk when mj was transmitted (10.131)
P(mi lq) = conditional probability that m; was transmitted when q is received
If q is received, then the probability that mj was transmitted is P(mj lq) for allj = 1, 2, ... , M.
Hence, the average cost of the decision m = mk is fh, given by
= L CkjP(mj lq)
j=l
(10.132)
Thus, if q is received, the optimum receiver decides that m = mk if
or
for all i I- k
M
M
L CkjP(mj lq) < L C ij
P(mj lq)
j=l
j=l
Use of Bayes' mixed rule in Eq. (10.133) yields
M
M
L CkjP(mj)P q
(qlmj) < L C ij
P(mj)P q
(qlmj)
j=l
j=l
for all i /. k
for all i I- k
(10.133)
(10.134)
Note that Ckk is the cost of setting m = mk when mk is transmitted. This cost is generally zero.
If we assign equal weight to all other errors, then
k =j
k f. j
(10.135)