Digital Signal Processing Reference
In-Depth Information
no power
allocated here
S
(
f
)
S
(
f
)
qq
xx
water level
f
0
Figure 6.2
. Demonstration of the water-filling power allocation.
The solution is well known from Shannon's work (see Sec. 12-2-1 of Proakis
[1995]), and is similar to the water-filling solution discussed in Sec. 22.3 later:
we simply take the input power spectrum to be
⎧
⎨
1
λ
−
S
qq
(
f
)
if this is non-negative
S
xx
(
f
)=
|
H
(
f
)
|
2
(6
.
17)
⎩
0
otherwise.
This allocation is demonstrated in Fig. 6.2. Note that the effective channel noise
spectrum which enters this crucial equation is
S
qq
(
f
)=
S
qq
(
f
)
|
(6
.
18)
H
(
f
)
|
2
that is, the spectrum of the noise
q
(
t
)shapedby1
/H
(
f
). Here
λ
is a constant
that arises from setting up a Lagrangian for the constrained optimization prob-
lem. Its value can be found by using the total power constraint (6.16). While
the frequency shaping of power may not be practicable as prescribed by (6.17),
a similar
power allocation
can readily be made by splitting the channel into a
finite number of subbands as done in discrete multitone (DMT) systems (Chap.
7).
6.5 Splitting the channel into subbands
Consider the additive Gaussian noise channel of Fig. 6.1(a) with an ideal lowpass
filter
H
(
f
) as in Eq. (6.1), but with noise spectrum as shown in Fig. 6.3(b).
Unlike in Fig. 6.1(b) the noise spectrum is different in different parts of the
passband. If the signal power is uniformly distributed as in Fig. 6.3(c) then the
total signal power is
p
0
and noise power is
σ
q
=
(
N
0
+
N
1
)
B
.
(6
.
19)
2
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