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

778 INTRODUCTION TO INFORMATION THEORY
signals of the same PSD shape, the stronger signal with larger power has an unfair advantage
and costs more to transmit. Thus, a fair approach to channel capacity maximization should
limit the total input signal power to a transmitter power constraint P x . Finding the best input
PSD under the total power constraint is known as the problem of maximum capacity power
loading.
The PSD that achieves the maximum capacity power loading is the solution to the
optimization problem of:
oo
max f
log ( 1 + IH (J ) l 2 Sx lf)
) df
(13.84)
Sx (f)
-00 Sn (J)
subject to 1_: Sx (j) df S P x
To solve this optimization problem, we again partition the channel ( of bandwidth B) into K
narrow flat channels centered at {f;, i = 1, 2, ... , K} of bandwidth !::,.f = B/K. By denoting
Ni = Sn (j;) t,.f
H1 = H(j;)
S1 = Sx (f;) t,.j
the optimization problem becomes a discrete problem of
K
(
IHi l 2 Si
) !::,.f
N-l
max "'log 1 + --
{S }
, i=l
(13.85a)
subject to
L S1 = P
i=l
(13.85b)
The problem of finding the N optimum power values {S1} is the essence of the optimum power
loading problem.
This problem can be dealt by introducing a standard Lagrange multiplier ).. to form a
modified objective function
(13.86)
Taking a partial derivative of G(S1 , ... , S K ) with respect to S j
and setting it to zero, we have
j = 1, 2, ... , K
We rewrite this optimality condition into
j = l, 2, ... , K