- Forward error correction
In telecommunication, information theory, and coding theory, forward error correction (FEC) or channel coding is a technique used for controlling errors in data transmission over unreliable or noisy communication channels. The central idea is the sender encodes their message in a redundant way by using an error-correcting code (ECC). The American mathematician Richard Hamming pioneered this field in the 1940s and invented the first error-correcting code in 1950: the Hamming (7,4) code.
The redundancy allows the receiver to detect a limited number of errors that may occur anywhere in the message, and often to correct these errors without retransmission. FEC gives the receiver the ability to correct errors without needing a reverse channel to request retransmission of data, but at the cost of a fixed, higher forward channel bandwidth. FEC is therefore applied in situations where retransmissions are costly or impossible, such as when broadcasting to multiple receivers in multicast. FEC information is usually added to mass storage devices to enable recovery of corrupted data.
FEC processing in a receiver may be applied to a digital bit stream or in the demodulation of a digitally modulated carrier. For the latter, FEC is an integral part of the initial analog-to-digital conversion in the receiver. The Viterbi decoder implements a soft-decision algorithm to demodulate digital data from an analog signal corrupted by noise. 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.
The maximum fractions of errors or of missing bits that can be corrected is determined by the design of the FEC 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. A redundant bit may be a complex function of many original information bits. The original information may or may not appear literally in the encoded output; codes that include the unmodified input in the output are systematic, while those that do not are non-systematic.
A simplistic example of FEC is to transmit each data bit 3 times, which is known as a (3,1) repetition code. Through a noisy channel, a receiver might see 8 versions of the output, see table below.
Triplet received Interpreted as 000 0 (error free) 001 0 010 0 100 0 111 1 (error free) 110 1 101 1 011 1
This allows an error in any one of the three samples to be corrected by "majority vote" or "democratic voting". The correcting ability of this FEC is:
- Up to 1 bit of triplet in error, or
- up to 2 bits of triplet omitted (cases not shown in table).
Though simple to implement and widely used, this triple modular redundancy is a relatively inefficient FEC. Better 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.
- Because of this "risk-pooling" effect, digital communication systems that use FEC tend to work well above a certain minimum signal-to-noise ratio and not at all below it.
- This all-or-nothing tendency — the cliff effect — becomes more pronounced as stronger codes are used that more closely approach the theoretical Shannon limit.
- Interleaving FEC coded data can reduce the all or nothing properties of transmitted FEC codes when the channel errors tend to occur in bursts. However, this method has limits; it is best used on narrowband data.
Most telecommunication systems used a fixed channel code designed to tolerate the expected worst-case bit error rate, and then fail to work at all if the bit error rate is ever worse. However, some systems adapt to the given channel error conditions: hybrid automatic repeat-request uses a fixed FEC method as long as the FEC can handle the error rate, then switches to ARQ when the error rate gets too high; adaptive modulation and coding uses a variety of FEC rates, adding more error-correction bits per packet when there are higher error rates in the channel, or taking them out when they are not needed.
Types of FEC
- Block codes work on fixed-size blocks (packets) of bits or symbols of predetermined size. Practical block codes can generally be decoded in polynomial time to their block length.
- Convolutional codes work on bit or symbol streams of arbitrary length. They are most often decoded with the Viterbi algorithm, though other algorithms are sometimes used. Viterbi decoding allows asymptotically optimal decoding efficiency with increasing constraint length of the convolutional code, but at the expense of exponentially increasing complexity. A convolutional code can be turned into a block code, if desired, by "tail-biting".
There are many types of block codes, but among the classical ones the most notable is Reed-Solomon coding because of its widespread use on the Compact disc, the DVD, and in hard disk drives. Other examples of classical block codes include Golay, BCH, Multidimensional parity, and Hamming codes.
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. Denser multi level cell (MLC) NAND requires stronger multi-bit correcting ECC such as BCH or Reed–Solomon[dubious ].
Classical block codes are usually implemented using hard-decision algorithms, which means that for every input and output signal a hard decision is made whether it corresponds to a one or a zero bit. In contrast, soft-decision algorithms like the Viterbi decoder process (discretized) analog signals, which allows for much higher error-correction performance than hard-decision decoding.
Nearly all classical block codes apply the algebraic properties of finite fields.
Concatenated FEC codes for improved performance
Classical (algebraic) block codes and convolutional codes are frequently combined in concatenated coding schemes in which a short constraint-length Viterbi-decoded convolutional code does most of the work and a block code (usually Reed-Solomon) with larger symbol size and block length "mops up" any errors made by the convolutional decoder. Single pass decoding with this family of error correction codes can yield very low error rates, but for long range transmission conditions (like deep space) iterative decoding is recommended.
Concatenated codes have been standard practice in satellite and deep space communications since Voyager 2 first used the technique in its 1986 encounter with Uranus. The Galileo craft used iterative concatenated codes to compensate for the very high error rate conditions caused by having a failed antenna.
Low-density parity-check (LDPC)
Low-density parity-check (LDPC) codes are a class of recently re-discovered highly efficient linear block codes. They can provide performance very close to the channel capacity (the theoretical maximum) using an iterated soft-decision decoding approach, at linear time complexity in terms of their block length. Practical implementations can draw heavily from the use of parallelism.
LDPC codes were first introduced by Robert G. Gallager in his PhD thesis in 1960, but due to the computational effort in implementing encoder and decoder and the introduction of Reed–Solomon codes, they were mostly ignored until recently.
LDPC codes are now used in many recent high-speed communication standards, such as DVB-S2 (Digital video broadcasting), WiMAX (IEEE 802.16e standard for microwave communications), High-Speed Wireless LAN (IEEE 802.11n), 10GBase-T Ethernet (802.3an) and G.hn/G.9960 (ITU-T Standard for networking over power lines, phone lines and coaxial cable). Other LDPC codes are standardized for wireless communication standards within 3GPP MBMS (see fountain codes).
Turbo coding is an iterated soft-decoding 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. Predating LDPC codes in terms of practical application, they now provide similar performance.
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. It is also used for the evolution of CDMA2000 1x specifically for Internet access, 1xEV-DO (TIA IS-856). 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.).
Local decoding and testing of codes
Sometimes it is only necessary to decode single bits of the message, or to check whether a given signal is a codeword, and do so without looking at the entire signal. This can make sense in a streaming setting, where codewords are too large to be classically decoded fast enough and where only a few bits of the message are of interest for now. Also such codes have become an important tool in computational complexity theory, e.g., for the design of probabilistically checkable proofs.
Locally decodable codes are error-correcting codes for which single bits of the message can be probabilistically recovered by only looking at a small (say constant) number of positions of a codeword, even after the codeword has been corrupted at some constant fraction of positions. Locally testable codes are error-correcting codes for which it can be checked probabilistically whether a signal is close to a codeword by only looking at a small number of positions of the signal.
List of error-correcting codes
- AN codes
- BCH code
- Constant-weight code
- Convolutional code
- Group codes
- Golay codes, of which the Binary Golay code is of practical interest
- Goppa code, used in the McEliece cryptosystem
- Hadamard code
- Hagelbarger code
- Hamming code
- Latin square based code for non-white noise (prevalent for example in broadband over powerlines)
- Lexicographic code
- Long code
- Low-density parity-check code, also known as Gallager code, as the archetype for sparse graph codes
- LT code, which is a near-optimal rateless erasure correcting code (Fountain code)
- m of n codes
- Online code, a near-optimal rateless erasure correcting code
- Raptor code, a near-optimal rateless erasure correcting code
- Reed–Solomon error correction
- Reed–Muller code
- Repeat-accumulate code
- Repetition codes, such as Triple modular redundancy
- Tornado code, a near-optimal erasure correcting code, and the precursor to Fountain codes
- Turbo code
- Walsh–Hadamard code
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