Noisy-channel coding theorem: Difference between revisions

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{{short description|Limit on data transfer rate}}

{{Information theory}}

{{redirect|Shannon's theorem|text=Shannon's name is also associated with the [[sampling theorem]]}}

In [[information theory]], the '''noisy-channel coding theorem''' (sometimes '''Shannon's theorem''' or '''Shannon's limit'''), establishes that for any given degree of noise contamination of a communication channel, it is possible (in theory) to communicate discrete data (digital [[information]]) nearly error-free up to a computable maximum rate through the channel. This result was presented by [[Claude Shannon]] in 1948 and was based in part on earlier work and ideas of [[Harry Nyquist]] and [[Ralph Hartley]].

The '''Shannon limit''' or '''Shannon capacity''' of a communication channel refers to the maximum [[Code rate|rate]] of error-free data that can theoretically be transferred over the channel if the link is subject to random data transmission errors, for a particular noise level. It was first described by Shannon (1948), and shortly after published in a book by Shannon and [[Warren Weaver]] entitled ''[[The Mathematical Theory of Communication]]'' (1949). This founded the modern discipline of [[information theory]].

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== Mathematical statement ==

[[Image:Noisy-channel coding theorem — channel capacity graph.png|thumb|right|300px|Graph showing the proportion of a channel’s capacity (''y''-axis) that can be used for payload based on how noisy the channel is (probability of bit flips; ''x''-axis)]]

The basic mathematical model for a communication system is the following:

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*{{cite journal|author-link=Claude E. Shannon | doi=10.1002/j.1538-7305.1948.tb01338.x | title=A Mathematical Theory of Communication | year=1948 | last1=Shannon | first1=C. E. | journal=Bell System Technical Journal | volume=27 | issue=3 | pages=379–423 }}

*{{cite book |author-link=Claude E. Shannon |first=C.E. |last=Shannon |title=A Mathematical Theory of Communication |publisher=University of Illinois Press |orig-year=1948 |date=1998 |pages= |url=http://cm.bell-labs.com/cm/ms/what/shannonday/paper.html}}

*{{cite journal |first=J. |last=Wolfowitz |title=The coding of messages subject to chance errors |journal=Illinois J. Math. |volume=1 |issue= 4|pages=591–606 |date=1957 |doi= 10.1215/ijm/1255380682|url=https://projecteuclid.org/download/pdf_1/euclid.ijm/1255380682|doi-access=free }}

{{DEFAULTSORT:Noisy-Channel Coding Theorem}}