Channel capacity, in electrical engineering, computer science and information theory, is the tight upper bound on the rate at which information can be reliably transmitted over a communication channel.
Following the terms of the noisy-channel coding theorem, the channel capacity of a given channel is the highest information rate (in units of information per unit time) that can be achieved with arbitrarily small error probability. 
Information theory, developed by Claude E. Shannon during World War II, defines the notion of channel capacity and provides a mathematical model by which one can compute it. The key result states that the capacity of the channel, as defined above, is given by the maximum of the mutual information between the input and output of the channel, where the maximization is with respect to the input distribution. 
The notion of channel capacity has been central to the development of modern wireline and wireless communication systems, with the advent of novel error correction coding mechanisms that have resulted in achieving performance very close to the limits promised by channel capacity.
- 1 Formal definition
- 2 Shannon capacity of a graph
- 3 Noisy-channel coding theorem
- 4 Example application
- 5 Channel capacity in wireless communications
- 6 See also
- 7 External links
- 8 References
The basic mathematical model for a communication system is the following:
- is the message to be transmitted;
- is the channel input symbol ( is a sequence of symbols);
- is the channel output symbol ( is a sequence of symbols);
- is the estimate of the transmitted message;
- is the encoding function for a block of length ;
- is the noisy channel, which is modeled by a conditional probability distribution; and,
- is the decoding function for a block of length .
Let and be modeled as random variables. Furthermore, let be the conditional probability distribution function of given , which is an inherent fixed property of the communication channel. Then the choice of the marginal distribution completely determines the joint distribution due to the identity
which, in turn, induces a mutual information . The channel capacity is defined as
where the supremum is taken over all possible choices of .
Shannon capacity of a graph
If G is an undirected graph, it can be used to define a communications channel in which the symbols are the graph vertices, and two codewords may be confused with each other if their symbols in each position are equal or adjacent. The computational complexity of finding the Shannon capacity of such a channel remains open, but it can be upper bounded by another important graph invariant, the Lovász number.
Noisy-channel coding theorem
The noisy-channel coding theorem states that for any error probability ε > 0 and for any transmission rate R less than the channel capacity C, there is an encoding and decoding scheme transmitting data at rate R whose error probability is less than ε, for a sufficiently large block length. Also, for any rate greater than the channel capacity, the probability of error at the receiver goes to 0.5 as the block length goes to infinity.
C is measured in bits per second if the logarithm is taken in base 2, or nats per second if the natural logarithm is used, assuming B is in hertz; the signal and noise powers S and N are measured in watts or volts2, so the signal-to-noise ratio here is expressed as a power ratio, not in decibels (dB); since figures are often cited in dB, a conversion may be needed. For example, 30 dB is a power ratio of .
Channel capacity in wireless communications
If the average received power is [W] and the noise power spectral density is [W/Hz], the AWGN channel capacity is
When the SNR is large (SNR >> 0 dB), the capacity is logarithmic in power and approximately linear in bandwidth. This is called the bandwidth-limited regime.
When the SNR is small (SNR << 0 dB), the capacity is linear in power but insensitive to bandwidth. This is called the power-limited regime.
The bandwidth-limited regime and power-limited regime are illustrated in the figure.
where and is the gain of subchannel , with chosen to meet the power constraint.
In a slow-fading channel, where the coherence time is greater than the latency requirement, there is no definite capacity as the maximum rate of reliable communications supported by the channel, , depends on the random channel gain , which is unknown to the transmitter. If the transmitter encodes data at rate [bits/s/Hz], there is a non-zero probability that the decoding error probability cannot be made arbitrarily small,
in which case the system is said to be in outage. With a non-zero probability that the channel is in deep fade, the capacity of the slow-fading channel in strict sense is zero. However, it is possible to determine the largest value of such that the outage probability is less than . This value is known as the -outage capacity.
In a fast-fading channel, where the latency requirement is greater than the coherence time and the codeword length spans many coherence periods, one can average over many independent channel fades by coding over a large number of coherence time intervals. Thus, it is possible to achieve a reliable rate of communication of [bits/s/Hz] and it is meaningful to speak of this value as the capacity of the fast-fading channel.
- Bandwidth (computing)
- Bandwidth (signal processing)
- Bit rate
- Code rate
- Error exponent
- Nyquist rate
- Sender, Encoder, Decoder, Receiver
- Shannon–Hartley theorem
- Spectral efficiency
Advanced Communication Topics
- Hazewinkel, Michiel, ed. (2001) , "Transmission rate of a channel", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4
- AWGN Channel Capacity with various constraints on the channel input (interactive demonstration)
- Saleem Bhatti. "Channel capacity". Lecture notes for M.Sc. Data Communication Networks and Distributed Systems D51 -- Basic Communications and Networks. Archived from the original on 2007-08-21.
- Jim Lesurf. "Signals look like noise!". Information and Measurement, 2nd ed.
- Thomas M. Cover, Joy A. Thomas (2006). Elements of Information Theory. John Wiley & Sons, New York.
- Lovász, László (1979), "On the Shannon Capacity of a Graph", IEEE Transactions on Information Theory, IT-25 (1), doi:10.1109/tit.1979.1055985.
- David Tse, Pramod Viswanath (2005), Fundamentals of Wireless Communication, Cambridge University Press, UK
- The Handbook of Electrical Engineering. Research & Education Association. 1996. p. D-149. ISBN 9780878919819.
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