Digital Signal Processing Reference
In-Depth Information
compared to what would be required for transmission with ideal channel coding,
which would in fact achieve error probability arbitrarily close to zero.
6.4 Capacity of frequency-dependent channel
Imagine we have a channel with transfer function
H
(
f
) and additive Gaussian
noise with a power spectrum
S
qq
(
f
)
,
not necessarily flat.
denote the
frequency band where
H
(
f
) is nonzero. With
S
xx
(
f
) denoting the input power
spectrum, the signal-to-noise ratio in a very narrow band of width Δ
f
centered
around a frequency
f
k
is given by
Let
F
|
2
Δ
f
S
qq
(
f
k
)Δ
f
S
xx
(
f
k
)
|
H
(
f
k
)
.
(6
.
12)
If we only had this narrow band available for transmission, then the channel
capacity would be
1+
S
xx
(
f
k
)
.
|
H
(
f
k
)
|
2
C
once
=log
(6
.
13)
2
S
qq
(
f
k
)
This is essentially obtained by using Eq. (6.3). However there is one difference.
It is assumed now that the signal and noise, limited to the bandwidth Δ
f,
can be
complex. In this case we have to add the capacities from the real and imaginary
paths, so the factor of 0
.
5 disappears.
2
Since the channel can be used Δ
f
times
per second, the capacity for a
subband
of bandwidth Δ
f
is
1+
S
xx
(
f
k
)
|
2
|
H
(
f
k
)
C
Δ
f
=Δ
f
log
2
bits per second.
(6
.
14)
S
qq
(
f
k
)
By adding up these capacities of the individual subbands we get the total ca-
pacity. The sum becomes an integral in the limit, so the total capacity is
1+
S
xx
(
f
)
df
=
|
2
|
H
(
f
)
C
log
bits per second,
(6
.
15)
2
S
qq
(
f
)
F
where
F
is the frequency band where
H
(
f
) is nonzero. The total channel input
power is
p
0
=
F
S
xx
(
f
)
df .
(6
.
16)
For fixed
p
0
we can allocate the power across the frequency band intelligently
by shaping the power spectrum
S
xx
(
f
)
.
The problem therefore is to maximize
the integral (6.15) subject to the power constraint (6.16). This can be done by
setting up an appropriate Lagrangian and setting its derivative to zero.
2
Actually, in the complex case the Gaussian noise is assumed to be zero-mean circularly
symmetric, and the signal achieving capacity is also zero-mean circularly symmetric Gaussian.
We will discuss these details in Secs. 6.6 and 6.7.
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