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

Figure 10.21
Determining
optimum
decision regions.
+
10.6 Optimum Receiver for White Gaussian Noise Channels 549
Example 10.2 Binary data is transmitted by using polar signaling over an AWGN channel with noise PSD
N /2. The two signals used are
SJ (t) = -s2(t) = ,✓E cp (t) (10.96)
The symbol probabilities P(m1) and P(m2) are unequal. Design the optimum receiver and
determine the corresponding error probability.
Figure 10.22
Decision regions
for the binary
case in this
example.
'P -
(a)
r(t)
____ __, <p (Tb - t)
(b)
Threshold Decision:
device . N P(m2)
m 1 1f r>-In --
4{£ P(m1)
. N P(m2)
m 2 1f r<-In --
4{£ P(m1)
The two signals are represented graphically in Fig. 10.22a. If the energy of each signal
is E, the distance of each signal from the origin is ,✓E. The distance d between the two
signals is
d = 2,✓E