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

752 INTRODUCTION TO INFORMATION THEORY
Figure 13.3
Plot of p (z).
p(z)
z-
If we define
1 1
z z
p(z) = z log - + z log -=-
with z = I
z, then
l(x; y) = p(aPe + iiPe) - p(Pe) (13.23)
The function p(z) vs. z is shown in Fig. 13.3 It can be seen that p(z) is maximum at
z = ½- (Note that we are interested in the region O < z < l only.) For a given Pe, p(Pe)
is fixed. Hence from Eq. (13.23) it follows that /(x; y) is maximum when p(aPe + iiPe)
is maximum. This occurs when
or
This equation is satisfied when
aPe + iiPe = 0.5
aPe + (I - a)(l - Pe) = 0.5
a= 0.5
(13.24)
For this value of a, p(aPe + iiPe) = l and
C s = max /(x;y) = 1 - p(Pe)
P(x;)
= 1 - [Pe log _2._ + (1 - Pe) log ( )]
Pe 1 Pe
(13.25)