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

542 PERFORMANCE ANALYSIS OF DIGITAL COMMUNICATION SYSTEMS
The decision procedure to maximizing the probability of correct decision P(C), thereby
minimizing the probability of error, is now clear. Once we receive q = q, we evaluate all M
a posteriori probability functions {P(mj l q = q) }. Then we make the decision in favor of that
message for which the a posteriori probability is highest-that is, the receiver decides that
m = mk if
for allj I- k (10.81)
Thus, the detector that minimizes the error probability is the maximum a posteriori
probability (MAP) detector.
We can use Bayes' rule (Chapter 8) to determine the a posteriori probabilities. We have
(10.82)
Hence, the receiver decides m = mk if the decision function
P(mi)P q
(qlmi)
P q
(q)
i = 1, 2, ... , M
is maximum for i = k.
Note that the denominator p q
(q) is common to all decision functions and is not effected
by the decision. Hence, it may be ignored during the decision. Thus, the receiver sets m = mk
if the decision function
i = 1, 2, ... , M (10.83)
is maximum for i = k. Thus, once q is obtained, we compute the decision function [Eq. (10.83)]
for all messages m1, m2, ... , mM and decide that the message for which the function is
maximum is the one most likely to have been sent.
We now turn our attention to finding the decision functions. The a priori probability P(mi)
represents the probability that the message mi will be transmitted. These probabilities must be
known if the criterion discussed is to be used.* The term p q
(qlmi) represents the PDF of q
when the transmitter sends s(t) = si(t). Under this condition,
and
q = Si+ n
n = q -Si
The point s i is constant, and n is a random point. Obviously, q is a random point with the same
distribution as n but centered at the points Si.
Alternatively, the probability density at q = q (given m = mi) is the same as the probability
n = q - Si. Hence [Eq. (10.78a)],
1
P (q lm ·) =p (q -s·) = e -11 q-s;11 2 ;N
q , n , (rrN)N/2
(10.84)
* In case these probabilities are unknown, one must use other merit criteria, such as maximum likelihood or
minimax, as will be discussed later.