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

704 DIGITAL COMMUNICATIONS UNDER LINEARLY DISTORTIVE CHANNELS
To determine the optimum power distribution, we would like to maximize the output SNR.
Because the channel input power is limited, the optimization requires
N
max L [H[i][ 2 - P;
{P;:':0} i=l
(12.71)
N
subject to L Pi = P
i=l
with equality if and only if b; = Aa7
N N N
max L [H[i][ 2 • P; = L [H[il[ 4 • L [P; [ 2
(I 2.72a)
{P;:':O} i=l i=l i=I
if
P; = A[H[i][ 2 (12.72b)
Once again, we can invoke the Cauchy-Schwartz inequality
Based on the Cauchy-Schwartz inequality,
Because of the input power constraint I:\: 1 P; = P, the optimum input power distribution
should be
In other words,
N
N
LP; = A · L IH[i]l 2 = P
i=l i=I
(12.73a)
(12.73b)
Substituting Eq. (12.73b) into Eq. (12.72b), we can obtain the optimum channel input power
loading across the N subchannels as
[H[i][ 2
P; = ---- P
L1 [H[i][ 2 (12.74)
This optimum distribution of power in OFDM, also known as power loading, makes very
good sense. When a channel has high gain, it is able to boost the power of its input much more
effectively than a channel with low gain. Hence, the high-gain subchannels will be receiving
higher power loading, while low-gain subchannels will receive much less. No power should
be wasted on the extreme case of a subchannel that has zero gain, since the output of such a
subchannel will make no power contribution to the total received signal power.
In addition to the perspective of maximizing average SNR, information theory can also
rigorously prove the optimality of power loading (known as water pouring) in maximizing the
capacity of frequency-selective channels. This discussion will be presented later (Sec. 13. 7).