In telecommunication, forward error correction (FEC) is a system of error control for data transmission, whereby the sender adds redundant data to its messages, which allows the receiver to detect and correct errors (within some bound) without the need to ask the sender for additional data. The advantage of forward error correction is that retransmission of data can often be avoided, at the cost of higher bandwidth requirements on average, and is therefore applied in situations where retransmissions are relatively costly or impossible. Copy of the original phone of Alexander Graham Bell at the Musée des Arts et Métiers in Paris Telecommunication is the transmission of signals over a distance for the purpose of communication. ... System (from Latin systēma, in turn from Greek sustēma) is a set of entities, real or abstract, comprising a whole where each component interacts with or is related to at least one other component. ... In information theory and coding error control is divided in two main categories: Error Detection Redundancy allows a receiver to check whether received data has been corrupted during transmission. ... Data transmission is the conveyance of any kind of information from one space to another. ... Redundancy in information theory is the number of bits used to transmit a message minus the number of bits of actual information in the message. ...

FEC devices are often located close to the receiver of an analog signal, in the first stage of digital processing after a signal has been received. That is, FEC circuits are often an integral part of the analog-to-digital conversion process. Many FEC coders can also generate a bit-error rate (BER) signal which can be used as feedback to fine-tune the analog receiving electronics. Many FEC algorithms, such as the Viterbi algorithm, can take (quasi-) analog data in, and generate digital data on output. In telecommunication, an error ratio is the ratio of the number of bits, elements, characters, or blocks incorrectly received to the total number of bits, elements, characters, or blocks sent during a specified time interval. ... The Viterbi algorithm, named after its developer Andrew Viterbi, is a dynamic programming algorithm for finding the most likely sequence of hidden states – known as the Viterbi path – that result in a sequence of observed events, especially in the context of hidden Markov models. ...

The maximum fraction of errors that can be corrected is determined in advance by the design of the code, so different forward error correcting codes are suitable for different conditions.

How it works

FEC is accomplished by adding redundancy to the transmitted information using a predetermined algorithm. Each redundant bit is invariably a complex function of many original information bits. The original information may or may not appear in the encoded output; codes that include the unmodified input in the output are systematic, while those that do not are nonsystematic. Redundancy in information theory is the number of bits used to transmit a message minus the number of bits of actual information in the message. ...

An extremely simple example would be an analog to digital converter that samples three bits of signal strength data for every bit of transmitted data. If the three samples are mostly all zero, the transmitted bit was probably a zero, and if three samples are all one, the transmitted bit was probably a one. The simplest example of error correction is for the receiver to assume the correct output is given by the most frequently occurring value in each group of three.

This allows an error in any one of the three samples to be corrected by "democratic voting". In practice, this is a very poor FEC, but it does illustrate the principle. In practice, FEC codes typically examine the last several dozen, or even the last several hundred, previously received bits to determine how to decode the current small handful of bits (typically in groups of 2 to 8 bits).

Averaging noise to reduce errors

FEC could be said to work by "averaging noise"; since each data bit affects many transmitted symbols, the corruption of some symbols by noise usually allows the original user data to be extracted from the other, uncorrupted received symbols that also depend on the same user data. This is somewhat analogous to the way that insurance companies and mutual funds manage and spread risk.

• Because of this "risk-pooling" effect, digital communication systems that use FEC tend to work perfectly above a certain minimum signal-to-noise ratio and not at all below it.
• This all-or-nothing tendency becomes more pronounced as stronger codes are used that more closely approach the theoretical limit imposed by the Shannon limit.

Signal-to-noise ratio (often abbreviated SNR or S/N) is an electrical engineering concept defined as the ratio of a signal power to the noise power corrupting the signal. ... In information theory, the Shannon-Hartley theorem states the maximum amount of error-free digital data (that is, information) that can be transmitted over a communication link with a specified bandwidth in the presence of noise interference. ...

Types of FEC

The two main categories of FEC are block coding and convolutional coding. The Railway Block Code is a system of bells used to send simple messages about train operations from one signalbox to another. ... In telecommunication, a convolutional code is a type of error-correcting code in which (a) each m-bit information symbol (each m-bit string) to be encoded is transformed into an n-bit symbol, where m/n is the code rate (n ≥ m) and (b) the transformation is a function...

• Block codes work on fixed-size blocks (packets) of bits or symbols of predetermined size.
• Convolutional codes work on bit or symbol streams of arbitrary length.
• A convolutional code can be turned into a block code, if desired.
• Convolutional codes are most often decoded with the Viterbi algorithm, though other algorithms are sometimes used.

There are many types of block codes, but the most notable is Reed-Solomon coding because of its widespread use on the Compact disc, the DVD, and in computer hard drives. Golay, BCH and Hamming codes are other examples of block codes. The Viterbi algorithm, named after its developer Andrew Viterbi, is a dynamic programming algorithm for finding the most likely sequence of hidden states – known as the Viterbi path – that result in a sequence of observed events, especially in the context of hidden Markov models. ... Reed-Solomon error correction is a coding scheme which works by first constructing a polynomial from the data symbols to be transmitted and then sending an over-sampled plot of the polynomial instead of the original symbols themselves. ... A Compact Disc or CD is an optical disc used to store digital data, originally developed for storing digital audio. ... DVD (commonly known as Digital Versatile Disc or Digital Video Disc) is an optical disc storage media format that can be used for data storage, including movies with high video and sound quality. ... A Golay code can be binary or ternary. ... A BCH (Bose, Ray-Chaudhuri, Hocquenghem) code is a much studied code within the study of coding theory and more specifically error-correcting codes. ... In telecommunication, a Hamming code is a linear error-correcting code named after its inventor, Richard Hamming. ...

Hamming ECC is commonly used to correct NAND flash memory errors. This provides single-bit error correction and 2 bit error detection. Hamming codes are only suitable for more reliable single level cell (SLC) NAND. More dense multi level cell (MLC) NAND requires stronger multi-bit correcting ECC such as BCH or Reed-Solomon. A USB Flash Memory Device Flash memory is a form of EEPROM (Electrically-Erasable Programmable Read-Only Memory) that allows multiple memory locations to be erased or written in one programming operation. ...

Nearly all block codes apply the algebraic properties of finite fields. In abstract algebra, a finite field or Galois field (so named in honor of Évariste Galois) is a field that contains only finitely many elements. ...

Concatenate FEC codes to reduce errors

Block and convolutional codes are frequently combined in concatenated coding schemes in which the convolutional code does most of the work and the block code (usually Reed-Solomon) "mops up" any errors made by the convolutional decoder.

• This has been standard practice in satellite and deep space communications since Voyager 2 first used the technique in its 1986 encounter with Uranus.

Voyager Project redirects here. ... Atmospheric characteristics Atmospheric pressure 120 kPa Hydrogen 83% Helium 15% Methane 1. ...

Turbo codes

The most recent (early 1990s) development in error correction is turbo coding, a scheme that combines two or more relatively simple convolutional codes and an interleaver to produce a block code that can perform to within a fraction of a decibel of the Shannon limit. Turbo codes are a class of recently-developed high-performance error correction codes finding use in deep-space satellite communications and other applications where designers seek to achieve maximal information transfer over a limited-bandwidth communication link in the presence of data-corrupting noise. ... Interleaving in computer science is a way to arrange data in a non-contiguous way in order to increase performance. ... In information theory, the Shannon-Hartley theorem states the maximum amount of error-free digital data (that is, information) that can be transmitted over a communication link with a specified bandwidth in the presence of noise interference. ...

• One of the earliest commercial applications of turbo coding was the CDMA2000 1x (TIA IS-2000) digital cellular technology developed by Qualcomm and sold by Verizon Wireless, Sprint, and other carriers.
• The evolution of CDMA2000 1x specifically for Internet access, 1xEV-DO (TIA IS-856), also uses turbo coding. Like 1x, EV-DO was developed by Qualcomm and is sold by Verizon Wireless, Sprint, and other carriers (Verizon's marketing name for 1xEV-DO is Broadband Access, Sprint's consumer and business marketing names for 1xEV-DO are Power Vision and Mobile Broadband, respectively.).

CDMA2000 is a family of third-generation (3G) mobile telecommunications standards that use CDMA, a multiple access scheme for digital radio, to send voice, data, and signalling data (such as a dialed telephone number) between mobile phones and cell sites. ... Qualcomm (NASDAQ: QCOM) is a wireless telecommunications research and development company based in San Diego, California. ... Verizon Wireless owns and operates the second-largest wireless telecommunications network in the United States and the largest wireless telecommunications in Puerto Rico along with Puerto Rico Telephone (PRT), based on total wireless customers. ... Sprint Nextel Corporation (NYSE: S) is one of the largest telecommunications companies in the world. ... Evolution-Data Optimized, abbreviated as EV-DO or EVDO and often EV, is a wireless radio broadband data standard adopted by many CDMA mobile phone service providers in United States, Canada, Mexico, Europe, Asia, Russia, Brazil, and Australia. ... Qualcomm (NASDAQ: QCOM) is a wireless telecommunications research and development company based in San Diego, California. ... Verizon Wireless owns and operates the second-largest wireless telecommunications network in the United States and the largest wireless telecommunications in Puerto Rico along with Puerto Rico Telephone (PRT), based on total wireless customers. ... Sprint Nextel Corporation (NYSE: S) is one of the largest telecommunications companies in the world. ...

See also

• Low-density parity-check code

A low-density parity-check code (LDPC code) is an error correcting code, a method of transmitting message over a noisy transmission channel. ...

References

• Clark, George C., Jr., and J. Bibb Cain. Error-Correction Coding for Digital Communications. New York: Plenum Press, 1981. ISBN 0-306-40615-2.
• Lin, Shu, and Daniel J. Costello, Jr. "Error Control Coding: Fundamentals and Applications". Englewood Cliffs, N.J.: Prentice-Hall, 1983. ISBN 0-13-283796-X.
• Mackenzie, Dana. "Communication speed nears terminal velocity". New Scientist 187.2507 (9 July 2005): 38–41. ISSN 0262-4079.
• Wicker, Stephen B. Error Control Systems for Digital Communication and Storage. Englewood Cliffs, N.J.: Prentice-Hall, 1995. ISBN 0-13-200809-2.
• Wilson, Stephen G. Digital Modulation and Coding, Englewood Cliffs, N.J.: Prentice-Hall, 1996. ISBN 0-13-210071-1.

External links

• Raptor Forward Error-Correction Technology. Retrieved on 26 April 2007.
• Forward Error-Correction Coding. Crosslink - The Aerospace Corporation magazine of advances in aerospace technology. The Aerospace Corporation (Volume 3, Number 1 (Winter 2001/2002)). Retrieved on 05 March 2006.
• How Forward Error-Correcting Codes Work. Crosslink - The Aerospace Corporation magazine of advances in aerospace technology. The Aerospace Corporation. Retrieved on 05 March 2006.
• Morelos-Zaragoza, Robert (2004). The Error Correcting Codes (ECC) Page. Retrieved on 05 March 2006.
• Forget Sudoku and smile for the camera (2006). Retrieved on 5 December 2006.