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

546 PERFORMANCE ANALYSIS OF DIGITAL COMMUNICATION SYSTEMS
Figure 1 O. 19
Another form of
optimum M-ary
receiver:
(a) correlator;
(b) matched filter.
Compute
N
L rtu
j= 1
< r, S2 >
Comparator
(select
largest)
m
I
I
Sample at t = T M
(a)
r(t)
I
I
I
I
:
I
Compute
N
L ri;J
j= l
Comparator
(select
largest)
m
I
Sample at t = T M
(b)
The decision function is given by Eq. (10.86). The optimum receiver sets 111 = mk if the
decision function
is maximum for i = k. This equation defines the decision regions.
Geometric Interpretation in Signal Space
For simplicity, let us first consider the case of equiprobable messages, that is, P(mi) = 1/M
for all i. In this case, the first term in the decision function is the same for all i and, hence,
can be dropped. Thus, the receiver decides that 111 = mk if the term - 1 lq - sd 1 2 is largest
(numerically the smallest) for i = k. Alternatively, this may be stated as follows: the receiver
decides that 111 = mk if the decision function liq - s;ll 2 is minimum for i = k. Note that
liq - s i II is the distance of point q from point s i. Thus, the decision procedure in this case has a