In electrical engineering and computer science, channel capacity is the tightest upper bound on the amount of information that can be reliably transmitted over a communications channel. By the noisy-channel 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]} Electrical Engineers design power systems… … and complex electronic circuits. ... Computer science, or computing science, is the study of the theoretical foundations of information and computation and their implementation and application in computer systems. ... The ASCII codes for the word Wikipedia represented in binary, the numeral system most commonly used for encoding computer information. ... Channel, in communications (sometimes called communications channel), refers to the medium used to convey information from a sender (or transmitter) to a receiver. ... In information theory, the noisy channel coding theorem establishes that however contaminated with noise interference a communication channel may be, it is possible to communicate digital data (information) error-free up to a given maximum rate through the channel. ... Channel, in communications (sometimes called communications channel), refers to the medium used to convey information from a sender (or transmitter) to a receiver. ... Claude Shannon In information theory, the Shannon entropy or information entropy is a measure of the uncertainty associated with a random variable. ...

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. A bundle of optical fiber. ... Claude Elwood Shannon (April 30, 1916 - February 24, 2001) has been called the father of information theory, and was the founder of practical digital circuit design theory. ... Combatants Allied powers: China France Great Britain Soviet Union United States and others Axis powers: Germany Italy Japan and others Commanders Chiang Kai-shek Charles de Gaulle Winston Churchill Joseph Stalin Franklin Roosevelt Adolf Hitler Benito Mussolini Hideki Tōjō Casualties Military dead: 17,000,000 Civilian dead: 33,000... In probability theory and, in particular, information theory, the mutual information, or transinformation, of two random variables is a quantity that measures the mutual dependence of the two variables. ...

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 Image File history File links This is a lossless scalable vector image. ...

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 Given two jointly distributed random variables X and Y, the conditional probability distribution of Y given X (written Y | X) is the probability distribution of Y when X is known to be a particular value. ...

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 In probability theory, given two jointly distributed random variables X and Y, the marginal distribution of X is simply the probability distribution of X ignoring information about Y, typically calculated by summing or integrating the joint probability distribution over Y. For discrete random variables, the marginal probability mass function can...

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 In probability theory and, in particular, information theory, the mutual information, or transinformation, of two random variables is a quantity that measures the mutual dependence of the two variables. ...

Noisy channel coding theorem

The channel 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. In information theory, the noisy-channel coding theorem establishes that however contaminated with noise interference a communication channel may be, it is possible to communicate digital data (information) error-free up to a given maximum rate through the channel. ... A bundle of optical fiber. ...

Example application

An application of the channel capacity concept to an additive white Gaussian noise channel with B Hz bandwidth and signal-to-noise ratio S/N is the Shannon–Hartley theorem: In information theory, the Shannon–Hartley theorem is an application of the noisy channel coding theorem to the archetypal case of a continuous-time analog communications channel subject to Gaussian noise. ...

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 10^{30 / 10} = 10³ = 1000. The decibel (dB) is a logarithmic unit of measurement that expresses the magnitude of a physical quantity (usually power) relative to a specified or implied reference level. ...

References

See also

• Bit rate
• Code rate
• Shannon–Hartley theorem
• Hartley's law
• Sender, Encoder, Decoder, Receiver
• Negentropy
• Exformation
• Redundancy
• Error exponent
• Throughput
• Link and system spectral efficiency