Noisy-channel coding theorem

Noisy-channel coding theorem   (Redirected from Noisy channel coding theorem) Jump to navigation Jump to search Information theory EntropyDifferential entropyConditional entropyJoint entropyMutual informationConditional mutual informationRelative entropyEntropy rateLimiting density of discrete points Asymptotic equipartition propertyRate–distortion theory Shannon's source coding theoremChannel capacityNoisy-channel coding theoremShannon–Hartley theorem vte "Shannon's theorem" redirects here. 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 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 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.

Contents 1 Overview 2 Mathematical statement 3 Outline of proof 3.1 Achievability for discrete memoryless channels 3.2 Weak converse for discrete memoryless channels 3.3 Strong converse for discrete memoryless channels 4 Channel coding theorem for non-stationary memoryless channels 4.1 Outline of the proof 5 See also 6 Notes 7 References 8 External links Overview Stated by Claude Shannon in 1948, the theorem describes the maximum possible efficiency of error-correcting methods versus levels of noise interference and data corruption. Shannon's theorem has wide-ranging applications in both communications and data storage. This theorem is of foundational importance to the modern field of information theory. Shannon only gave an outline of the proof. The first rigorous proof for the discrete case is due to Amiel Feinstein[1] in 1954.

The Shannon theorem states that given a noisy channel with channel capacity C and information transmitted at a rate R, then if {displaystyle RC} , an arbitrarily small probability of error is not achievable. All codes will have a probability of error greater than a certain positive minimal level, and this level increases as the rate increases. So, information cannot be guaranteed to be transmitted reliably across a channel at rates beyond the channel capacity. The theorem does not address the rare situation in which rate and capacity are equal.

The channel capacity {displaystyle C} 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 guarantee that a block of data can be communicated free of error. Advanced techniques such as Reed–Solomon codes and, more recently, low-density parity-check (LDPC) codes and turbo codes, come much closer to reaching the theoretical Shannon limit, but at a cost of high computational complexity. Using these highly efficient codes and with the computing power in today's digital signal processors, it is now possible to reach very close to the Shannon limit. In fact, it was shown that LDPC codes can reach within 0.0045 dB of the Shannon limit (for binary Additive white Gaussian noise (AWGN) channels, with very long block lengths).[2] Mathematical statement The basic mathematical model for a communication system is the following: {displaystyle {xrightarrow[{text{Message}}]{W}}{begin{array}{|c| }hline {text{Encoder}}\f_{n}\hline end{array}}{xrightarrow[{mathrm {Encoded atop sequence} }]{X^{n}}}{begin{array}{|c| }hline {text{Channel}}\p(y|x)\hline end{array}}{xrightarrow[{mathrm {Received atop sequence} }]{Y^{n}}}{begin{array}{|c| }hline {text{Decoder}}\g_{n}\hline end{array}}{xrightarrow[{mathrm {Estimated atop message} }]{hat {W}}}} A message W is transmitted through a noisy channel by using encoding and decoding functions. An encoder maps W into a pre-defined sequence of channel symbols of length n. In its most basic model, the channel distorts each of these symbols independently of the others. The output of the channel –the received sequence– is fed into a decoder which maps the sequence into an estimate of the message. In this setting, the probability of error is defined as: {displaystyle P_{e}={text{Pr}}left{{hat {W}}neq Wright}.} Theorem (Shannon, 1948): 1. For every discrete memoryless channel, the channel capacity, defined in terms of the mutual information {displaystyle I(X;Y)} as {displaystyle C=sup _{p_{X}}I(X;Y)} [3] has the following property. For any {displaystyle epsilon >0} and {displaystyle R

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