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

10.6 Optimum Receiver for White Gaussian Noise Channels 545
Figure 1 O. 18
Optimum M-ary
receiver:
(a) matched filter
detector;
(b) correlation
detector.
r(t)
Comparator
(select
largest)
I_ - - - - - - - - - - - - - - - - - - T - - - - - - - - - - - -
---0------1
Sample at I = T M
(a)
r(t)
Comparator
(select
largest)
Sample at I - T M
(b)
The term <q, s; > is computed according to this equation by first generating ri and then computing
the sum of rjS i j (remember that the Sij are known), as shown in Fig. 10.19a. The M correlator
detectors in Fig. 10.18b can be replaced by N filters matched to cp, (t), cpz(t), ... , fPN (t), as
shown in Fig. 10.19b. These types of optimum receiver (Figs. 10.18 and 10.19) perform identically.
The choice will depend on the hardware cost. For example, if N < M and signals { IP.i (t)}
are easier to generate than {sj(t) }, then the design of Fig. 10. 19 would be chosen.
10.6.4 Decision Regions and Error Probability
To compute the error probability of the optimum receiver, we must first determine decision
regions in the signal space. As mentioned earlier, the signal space is divided into M
nonoverlapping, or disjoint, decision regions R1, R 2 , ... , RM , corresponding to M messages.
If q falls in the region R k , the decision is that m k was transmitted. The decision regions
are chosen to minimize the probability of error in the receiver. In light of this geometrical
representation, we shall now try to interpret how the optimum receiver sets these decision
regions.