Digital Signal Processing Reference
In-Depth Information
6
Channel capacity
6.1
Introduction
The fundamentals of digital communications were developed many decades ago
in the ground breaking work of Shannon [1948, 1949]. Central to the capability
of a communication channel is the channel capacity, which is studied in great
detail in many communication texts [Gallager, 1968], [McEliece, 1977], [Cover
and Thomas, 1991], [Proakis, 1995]. The channel capacity is the upper bound on
the rate at which information can be communicated through a noisy channel with
arbitrarily small error probability. The capacity can be achieved, in principle,
by appropriate channel coding. We summarize here some of the results that are
central to our discussions. For details the reader should consult one of the above
references.
6.2
Ideal lowpass channel
Figure 6.1(a) shows a channel with transfer function
H
(
f
) and additive Gaussian
noise
q
(
t
)
.
(For convenience we have used
f
instead of
ω
for frequency.) Assume
that
H
(
f
) is an ideal bandlimited channel with total bandwidth 2
B
Hz (one-
sided bandwidth of
B
):
H
(
f
)=
1
−
B
≤
f<B
(6
.
1)
0
otherwise.
Since the receiver filter can be restricted to this band, the noise is also bandlim-
ited. Assume the noise has a flat spectrum within this band as shown in Fig.
6.1(b). Let the total signal power be
p
0
. Assume this is uniformly distributed
within the band as shown in Fig. 6.1(c). The total noise power in the channel
band is the integral of the noise spectrum:
σ
q
=
N
0
B.
(6
.
2)
216
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