 Channel capacity

In electrical engineering, computer science and information theory, channel capacity is the tightest upper bound on the amount of information that can be reliably transmitted over a communications channel. By the noisychannel coding theorem, the channel capacity of a given channel is the limiting information rate (in units of information per unit time) that can be achieved with arbitrarily small error probability.^{[1]} ^{[2]}
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.^{[3]}
Contents
Formal definition
Let X represent the space of signals that can be transmitted, and Y the space of signals received, during a block of time over the channel. Let
be the conditional distribution function of Y given X. Treating the channel as a known statistic system, p_{Y  X}(y  x) is an inherent fixed property of the communications channel (representing the nature of the noise in it). Then the joint distribution
of X and Y is completely determined by the channel and by the choice of
the marginal distribution of signals we choose to send over the channel. The joint distribution can be recovered by using the identity
Under these constraints, next maximize the amount of information, or the message, that one can communicate over the channel. The appropriate measure for this is the mutual information I(X;Y), and this maximum mutual information is called the channel capacity and is given by
Noisychannel coding theorem
The noisychannel coding theorem states that for any ε > 0 and for any rate R less than the channel capacity C, there is an encoding and decoding scheme that can be used to ensure that the probability of block error is less than ε for a sufficiently long code. Also, for any rate greater than the channel capacity, the probability of block error at the receiver goes to one as the block length goes to infinity.
Example application
An application of the channel capacity concept to an additive white Gaussian noise (AWGN) channel with B Hz bandwidth and signaltonoise ratio S/N is the Shannon–Hartley theorem:
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 volts^{2}, so the signaltonoise 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 10^{30 / 10} = 10^{3} = 1000.
Channel capacity in wireless communications
This section^{[4]} focuses on the singleantenna, pointtopoint scenario. For channel capacity in systems with multiple antennas, see the article on MIMO.
AWGN channel
If the average received power is [W] and the noise power spectral density is N_{0} [W/Hz], the AWGN channel capacity is
 [bits/Hz],
where is the received signaltonoise ratio (SNR).
When the SNR is large (SNR >> 0 dB), the capacity is logarithmic in power and approximately linear in bandwidth. This is called the bandwidthlimited regime.
When the SNR is small (SNR << 0 dB), the capacity is linear in power but insensitive to bandwidth. This is called the powerlimited regime.
The bandwidthlimited regime and powerlimited regime are illustrated in the figure.
Frequencyselective channel
The capacity of the frequencyselective channel is given by socalled waterfilling power allocation,
where and is the gain of subchannel n, with λ chosen to meet the power constraint.
Slowfading channel
In a slowfading 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, log _{2}(1 +  h  ^{2}SNR), depends on the random channel gain  h  ^{2}. If the transmitter encodes data at rate R [bits/s/Hz], there is a certain probability that the decoding error probability cannot be made arbitrarily small,
 ,
in which case the system is said to be in outage. With a nonzero probability that the channel is in deep fade, the capacity of the slowfading channel in strict sense is zero. However, it is possible to determine the largest value of R such that the outage probability p_{out} is less than . This value is known as the outage capacity.
Fastfading channel
In a fastfading 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 fastfading channel.
See also
 Bandwidth (computing)
 Bandwidth (signal processing)
 Bit rate
 Code rate
 Error exponent
 Nyquist rate
 Negentropy
 Redundancy
 Sender, Encoder, Decoder, Receiver
 Shannon–Hartley theorem
 Spectral efficiency
 Throughput
Advanced Communication Topics
References
 ^ Saleem Bhatti. "Channel capacity". Lecture notes for M.Sc. Data Communication Networks and Distributed Systems D51  Basic Communications and Networks. http://www.cs.ucl.ac.uk/staff/S.Bhatti/D51notes/node31.html.
 ^ Jim Lesurf. "Signals look like noise!". Information and Measurement, 2nd ed.. http://www.standrews.ac.uk/~www_pa/Scots_Guide/iandm/part8/page1.html.
 ^ Thomas M. Cover, Joy A. Thomas (2006), Elements of Information Theory, John Wiley & Sons, New York
 ^ David Tse, Pramod Viswanath (2005), Fundamentals of Wireless Communication, Cambridge University Press, UK
Categories: Information theory
 Telecommunication theory
 Television terminology
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