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

subject to the following constraints:
13.5 Channel Capacity of a Continuous Memoryless Channel 759
1 b <p1 (x,p) dx = >.. 1
1 b <p2 (x,p) dx = A2
(13.36)
where A 1 , >..2, ... , Ak are given constants. The result from the calculus of variation states that
the form of p(x) that maximizes I in Eq. (13.35) with the constraints in Eq. (13.36) is found
from the solution of the equation
oF a<p1 o<p2 o<fJk
- + a1 - +a2- + · ·· +ak- = 0
ap op ap ap
(13.37)
The quantities a1, a2, ... , ak are adjustable constants, called undetermined multipliers,
which can be found by substituting the solution of p(x) [obtained from Eq. (13.37)] in
Eq. (13.36). In the present case,
Hence, the solution for p is given by
I
F(p,x) = plog - p
<p1(x, p) =p
<p2(x,p) = x 2 p
or
a ( 1) o 2
- plog - +a1+a2 - xp=0
op p op
Solving for p, we have
Substituting Eq. (13.38) into Eq. (13.34a), we have
1
00
l = e" 1 -I e"2x dx
-00
2
roo 2
= 2e"1 -I
J
o e"zx dx
(13.38)