Connexions module: m0098 1
Capacity of a Channel
∗
Don Johnson
This work is produced by The Connexions Project and licensed under the
Creative Commons Attribution License
†
Abstract
Shannon derived the maximum datarate of a source coder's output that can be transmitted through
a bandlimited additive white noise channel with no error.
In addition to the Noisy Channel Coding Theorem and its converse
1
, Shannon also derived the capacity
for a bandlimited (to
W
Hz) additive white noise channel. For this case, the signal set is unrestricted, even
to the point that more than one bit can be transmitted each "bit interval." Instead of constraining channel
code eciency, the revised Noisy Channel Coding Theorem states that some error-correcting code exists such
that as the block length increases, error-free transmission is possible if the source coder's datarate,
B(A)R
,
is less than capacity.
C=Wlog2(1 + SNR)
bits/s (1)
This result sets the maximum datarate of the source coder's output that can be transmitted through the
bandlimited channel with no error.
2
Shannon's proof of his theorem was very clever, and did not indicate
what this code might be; it has never been found. Codes such as the Hamming code work quite well in
practice to keep error rates low, but they remain greater than zero. Until the "magic" code is found, more
important in communication system design is the converse. It states that if your data rate exceeds capacity,
errors will overwhelm you no matter what channel coding you use. For this reason, capacity calculations are
made to understand the fundamental limits on transmission rates.
Exercise 1
(Solution on p. 3.)
The rst denition of capacity applies only for binary symmetric channels, and represents the
number of bits/transmission. The second result states capacity more generally, having units of
bits/second. How would you convert the rst denition's result into units of bits/second?
Example 1
The telephone channel has a bandwidth of 3 kHz and a signal-to-noise ratio exceeding 30 dB (at
least they promise this much). The maximum data rate a modem can produce for this wireline
channel and hope that errors will not become rampant is the capacity.
C= 3 ×103log21 + 103
=
29.901 kbps (2)
Thus, the so-called 33 kbps modems operate right at the capacity limit.
∗
Version 2.13: Jul 15, 2007 5:43 pm GMT-5
†
http://creativecommons.org/licenses/by/1.0
1
"Noisy Channel Coding Theorem" <http://cnx.org/content/m0073/latest/>
2
The bandwidth restriction arises not so much from channel properties, but from spectral regulation, especially for wireless
channels.
http://cnx.org/content/m0098/2.13/
