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

574 PERFORMANCE ANALYSIS OF DIGITAL COMMUNICATION SYSTEMS
Solution of these two equations yields
and
Both signal sets (Fig. 10.32a and b) have the same error probability, but the latter has
a smaller mean energy. If E and E' are the respective mean energies of the two sets, then
and
-
2 2
E' = P(m1 )k1 + P(m2)k2
= P(m1 )P 2 (m2)d 2 + P(m2)P 2 (m1)d 2
= P(mi)P(m2)d 2
Note that for P(m1) + P(m2) = 1, the product P(m1 )P(m2) is maximum when P(m1) =
P(m2) = 1/2, in which case
and consequently
Therefore,
and for the case of equiprobable signals,
d 2
E' < ­ - 4
E
E' < -
- 2
In this case,
- E
E' = - 2
d
E = - 2
and 2
d 2
E' = -
The signals in Fig. 10.32b are called antipodal signals when k1 = k2. The error probability
of the signal set in Fig. 10.32a (and 10.32b) is equal to that in Fig. 10.22a and can be found
from Eq. (10.97a).