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

13.7 Frequency-Selective Channel Capacity 779
By defining a new variable W = (Jc ln 2) - 1 , we ensure that the optimum power allocation
among the K subchannels is
i = 1, 2, ... , K
(13.87a)
such that
(13.87b)
The optimum power loading condition of Eq. (13.87) is not quite yet complete because some
Si may become negative if no special care is taken. Therefore, we must further constrain the
solution to ensure that S; :=:: 0 via
S· = max (w
1 • L'\f - !!.!_ o)
. IHi l 2 '
such that
L St = P
i = 1, 2, ... , K
(13.88a)
(13.88b)
The two relationships in Eq. (13.88) describe the solution of the power loading optimization
problem. We should note that there remains an unknown parameter W that needs to be specified.
By enforcing the total power constraint LS; = P, we can finally determine the unknown
parameter W.
Finally, we take the limit as M - 0 and K - oo. Since S; = S x (f;)L'\f and N; =
S n (f;)L'\f, the optimum input signal PSD becomes
.
S n lf)
Sx (f ) = max ( W - ) IH(f)l
2 , 0
(13.89a)
We note again that there is no closed-form solution given for the optimum constant W. Instead,
the optimum W is obtained from the total input power constraint
or
1_: Sx (f) df = P
P= [ J{f: W-S n (f)/IH(f)l 2 >0}
(w - S n lf) )df
IH (f)l 2 (13.89c)
(13.89b)
Substituting the optimum PSD Eq. (13.89) into the capacity formula will lead to the maximum
channel capacity value of
[ ( W IH(f) 1 2 ) .
Cmax = log ---- dj
{f: W-S n (f)/IH(/)1 2 >0} Sn (f)
(13.90)
Water-Pouring Interpretation of Optimum Power Loading
The optimum channel input PSD must satisfy the power constraint Eq. (13.89c). Once the
constant W has been determined, the transmitter can adjust its transmission PSD to Eq. ( 13 .89a),
which will maximize the channel capacity. This optimum solution to the channel input PSD
optimization problem is known as the water-filling or water-pouring solution. 5