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An application of the channel capacity concept to an additive white Gaussian noise (AWGN) channel with B Hz bandwidth and signal-to-noise ratio S/N is the Shannon–Hartley theorem: C = B log 2 ( 1 + S N ) {\displaystyle C=B\log _{2}\left(1+{\frac {S}{N}}\right)\ }
If the requirement is to transmit at 50 kbit/s, and a bandwidth of 10 kHz is used, then the minimum S/N required is given by 50000 = 10000 log 2 (1+S/N) so C/B = 5 then S/N = 2 5 − 1 = 31, corresponding to an SNR of 14.91 dB (10 x log 10 (31)). What is the channel capacity for a signal having a 1 MHz bandwidth, received with a SNR of −30 dB ?
The channel capacity can be calculated from the physical properties of a channel; for a band-limited channel with Gaussian noise, using the Shannon–Hartley theorem. Simple schemes such as "send the message 3 times and use a best 2 out of 3 voting scheme if the copies differ" are inefficient error-correction methods, unable to asymptotically ...
The Shannon capacity of a graph G is bounded from below by α(G), and from above by ϑ(G). [5] In some cases, ϑ(G) and the Shannon capacity coincide; for instance, for the graph of a pentagon, both are equal to √ 5. However, there exist other graphs for which the Shannon capacity and the Lovász number differ. [6]
Shannon's diagram of a general communications system, showing the process by which a message sent becomes the message received (possibly corrupted by noise) This work is known for introducing the concepts of channel capacity as well as the noisy channel coding theorem. Shannon's article laid out the basic elements of communication:
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