Turbo code: Difference between revisions

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In [[information theory]], '''turbo codes''' (originally in French ''Turbocodes'') are a class of high-performance [[forward error correction]] (FEC) codes developed around 1990–91, but first published in 1993. They were the first practical codes to closely approach the maximum channel capacity or [[Shannon–Hartley theorem|Shannon limit]], a theoretical maximum for the [[code rate]] at which reliable communication is still possible given a specific noise level. Turbo codes are used in [[3G]]/[[4G]] mobile communications (e.g., in [[UMTS]] and [[LTE (telecommunication)|LTE]]) and in ([[Deep Space Network|deep space]]) [[satellite]] [[telecommunication|communications]] as well as other applications where designers seek to achieve reliable information transfer over bandwidth- or latency-constrained communication links in the presence of data-corrupting noise. Turbo codes compete with [[Low-density parity-check code|low-density parity-check]] (LDPC) codes, which provide similar performance. Until the patent for turbo codes expired,<ref>{{cite patent |url=https://www.google.com/patents/US5446747 |country=US |number=5446747}}</ref> the patent-free status of LDPC codes was an important factor in LDPC's continued relevance.<ref name="Closing">{{cite journal |author=Erico Guizzo |title=CLOSING IN ON THE PERFECT CODE |journal=IEEE Spectrum |date=Mar 1, 2004 |url=https://spectrum.ieee.org/closing-in-on-the-perfect-code|archive-url=https://archive.today/20230423205925/https://spectrum.ieee.org/closing-in-on-the-perfect-code|url-status=dead|archive-date=23 April 2023}} "Another advantage, perhaps the biggest of all, is that the LDPC patents have expired, so companies can use them without having to pay for intellectual-property rights."</ref>

The name "turbo code" arose from the feedback loop used during normal turbo code decoding, which was analogized to the exhaust feedback used for engine [[turbocharging]]. [[Joachim Hagenauer|Hagenauer]] has argued the term turbo code is a misnomer since there is no feedback involved in the encoding process.<ref>{{cite webjournal |url=http://www.ima.umn.edu/csg/bib/bib16.0429hage.pdf |first1=Joachim |last1=Joachim Hagenauer |display-authors=etal |title=Iterative Decoding of Binary Block and Convolutional Codes |accessdate=20 March 2014 |url-status=dead |archiveurl=https://web.archive.org/web/20130611235418/http://www.ima.umn.edu/csg/bib/bib16.0429hage.pdf |archivedate=11 June 2013 |first2=Elke |last2=Offer |first3=Luiz |last3=Papke |volume=42 |issue=2 |date=March 1996 |journal=IEEE Transactions on Information Theory|pages=429–445 |doi=10.1109/18.485714 }}</ref>

==History==

The fundamental patent application for turbo codes was filed on 23 April 1991. The patent application lists [[Claude Berrou]] as the sole inventor of turbo codes. The patent filing resulted in several patents including [https://wwwpatents.google.com/patentspatent/US5446747 US Patent 5,446,747], which expired 29 August 2013.

The first public paper on turbo codes was "''Near Shannon Limit Error-correcting Coding and Decoding: Turbo-codes''".<ref>{{Citation|chapter-url=https://www.researchgate.net/publication/3604275 |firstfirst1=Claude |first2=Alain |first3=Punya |lastlast1=Berrou |author1-link=Claude Berrou|last2=Glavieux |author2-link=Alain Glavieux |last3=Thitimajshima |titleauthor3-link=Punya Thitimajshima |chapter=Near Shannon Limit Error – Correcting |accessdate=11 February 2010 |date=1993 |volume=2 |pages=1064–70 |doi=10.1109/ICC.1993.397441 |title=Proceedings of IEEE International Communications Conference|s2cid=17770377 }}</ref> This paper was published 1993 in the Proceedings of IEEE International Communications Conference. The 1993 paper was formed from three separate submissions that were combined due to space constraints. The merger caused the paper to list three authors: Berrou, [[Alain Glavieux|Glavieux]], and [[Punya Thitimajshima|Thitimajshima]] (from Télécom Bretagne, former [[École Nationale Supérieure des Télécommunications de Bretagne|ENST Bretagne]], France). However, it is clear from the original patent filing that Berrou is the sole inventor of turbo codes and that the other authors of the paper contributed material other than the core concepts.{{Synthesis inline|date=February 2021|sure=yes}}

Turbo codes were so revolutionary at the time of their introduction that many experts in the field of coding did not believe the reported results. When the performance was confirmed a small revolution in the world of coding took place that led to the investigation of many other types of iterative signal processing.<ref>{{cite journal |author=Erico Guizzo |title=CLOSING IN ON THE PERFECT CODE |journal=IEEE Spectrum |date=Mar 1, 2004 |url=https://spectrum.ieee.org/closing-in-on-the-perfect-code|archive-url=https://archive.today/20230423205925/https://spectrum.ieee.org/closing-in-on-the-perfect-code|url-status=dead|archive-date=23 April 2023}}</ref>

The first class of turbo code was the parallel concatenated convolutional code (PCCC). Since the introduction of the original parallel turbo codes in 1993, many other classes of turbo code have been discovered, including serial versions [[serial concatenated convolutional codes]] and [[repeat-accumulate code]]s. Iterative turbo decoding methods have also been applied to more conventional FEC systems, including Reed-SolomonReed–Solomon corrected convolutional codes, although these systems are too complex for practical implementations of iterative decoders. Turbo equalization also flowed from the concept of turbo coding.

In addition to turbo codes, Berrou also invented recursive systematic convolutional (RSC) codes, which are used in the example implementation of turbo codes described in the patent. Turbo codes that use RSC codes seem to perform better than turbo codes that do not use RSC codes.

Prior to turbo codes, the best constructions were serial [[concatenated code]]s based on an outer [[Reed-SolomonReed–Solomon error correction]] code combined with an inner [[Viterbi algorithm|Viterbi-decoded]] short constraint length [[convolutional code]], also known as RSV codes.

In a later paper, Berrou gave credit to the intuition of "G. Battail, [[Joachim Hagenauer|J. Hagenauer]] and P. Hoeher, who, in the late 80s, highlighted the interest of probabilistic processing." He adds "[[Robert G. Gallager|R. Gallager]] and M. Tanner had already imagined coding and decoding techniques whose general principles are closely related," although the necessary calculations were impractical at that time.<ref>{{Citation|first=Claude|last=Berrou|title=The ten-year-old turbo codes are entering into service|___location=Bretagne, France|accessdate=11 February 2010|url=https://www.researchgate.net/publication/3199004}}</ref>

==An example encoder==

There are many different instances of turbo codes, using different component encoders, input/output ratios, interleavers, and [[Punctured code|puncturing patterns]]. This example encoder implementation describes a classic turbo encoder, and demonstrates the general design of parallel turbo codes.

This encoder implementation sends three sub-blocks of bits. The first sub-block is the ''m''-bit block of payload data. The second sub-block is ''n/2'' parity bits for the payload data, computed using a recursive systematic [[convolutional code]] (RSC code). The third sub-block is ''n/2'' parity bits for a known [[permutation]] of the payload data, again computed using an RSC code. Thus, two redundant but different sub-blocks of parity bits are sent with the payload. The complete block has {{nowrap|''m'' + ''n''}} bits of data with a code rate of {{nowrap|''m''/(''m'' + ''n'')}}. The [[permutation]] of the payload data is carried out by a device called an [[interleaver]].

Hardware-wise, this turbo code encoder consists of two identical RSC coders, С''C''₁ and ''C''₂, as depicted in the figure, which are connected to each other using a concatenation scheme, called ''parallel concatenation'':

[[File:turbo encoder.svg]]

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==Bayesian formulation==

From an [[artificial intelligence]] viewpoint, turbo codes can be considered as an instance of loopy [[belief propagation]] in [[Bayesian network]]s.<ref>{{Citation

| author1last1=McEliece, |first1=Robert J. | author1-link=Robert McEliece

| author2last2=MacKay, |first2=David J. C. | author2-link=David J. C. MacKay

| author3last3=Cheng, |first3=Jung-Fu

| title=Turbo decoding as an instance of Pearl's "belief propagation" algorithm

| journal=IEEE Journal on Selected Areas in Communications

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==See also==

* [[BCJR algorithm]]

* [[Convolutional code]]

* [[ViterbiForward algorithmerror correction]]

* [[Soft-decision decoding]]

* [[Interleaver]]

* [[BCJR algorithm]]

* [[Low-density parity-check code]]

* [[Serial concatenated convolutional codes]]

* [[Soft-decision decoding]]

* [[Turbo equalizer]]

* [[ForwardViterbi error correctionalgorithm]]

==References==

{{reflistReflist}}

==Further reading==

===Publications===

* {{cite journal |last=Battail, |first=Gérard. "|title=A conceptual framework for understanding turbo codes." |journal=IEEE Journal on Selected Areas in Communications 16.|volume=916 |issue=2 (|date=1998): |pages=245–254|doi=10.1109/49.661112 }}

* {{cite journal |last1=Brejza |first1=M.F. |last2=Li |first2=L. |last3=Maunder |first3=R.G. |last4=Al-Hashimi |first4=B.M. |last5=Berrou |first5=C. |last6=Hanzo |first6=L. |url=https://eprints.soton.ac.uk/378161/1/tutorial.pdf |doi=10.1109/COMST.2015.2448692

* Brejza, Matthew F., et al. "|title=20 years of turbo coding and energy-aware design guidelines for energy-constrained wireless applications." |journal=IEEE Communications Surveys & Tutorials 18.|volume=918 |issue=1 (|date=2016): |pages=8–28.|s2cid=12966388 }}

*{{cite conference |last1=Garzón-Bohórquez, |first1=Ronald, |first2=Charbel Abdel |last2=Nour, and |first3=Catherine |last3=Douillard. "|title=Improving Turbo codes for 5G with parity puncture-constrained interleavers." |conference=9th International Symposium on Turbo Codes and Iterative Information Processing (ISTC), |date=2016 9th|pages=151–5 International|url=https://hal.science/hal-01421989/file/Final%20Manuscript.pdf Symposium on|doi=10.1109/ISTC. IEEE, 2016.7593095}}

==External links==

* [{{cite journal |first=Erico |last=Guizzo |url=https://spectrum.ieee.org/computing/software/closing-in-on-the-perfect-code "|archive-url=https://web.archive.org/web/20091011113149/http://www.spectrum.ieee.org/computing/software/closing-in-on-the-perfect-code |url-status=dead |archive-date=11 October 2009 |title=Closing In On The Perfect Code"], |journal=IEEE Spectrum, |date=March 2004 |volume=41 |issue=3 |pages=36–42 |doi=10.1109/MSPEC.2004.1270546 |s2cid=21237188 |url-access=subscription }}

* [http://www.csee.wvu.edu/~mvalenti/documents/valenti01.pdf "The UMTS Turbo Code and an Efficient Decoder Implementation Suitable for Software-Defined Radios"] {{Webarchive|url=https://web.archive.org/web/20161020193559/http://www.csee.wvu.edu/~mvalenti/documents/valenti01.pdf |date=20 October 2016 }} (''International Journal of Wireless Information Networks'')

* {{Citationcite |journal author|first=Dana |last=Mackenzie | title=Take it to the limit | journal=New Scientist | volume=187 | issue=2507 | year=2005 | pages=38–41 | postscripturl=. | issn=0262-4079 }} ([https://www.newscientist.com/article.ns?id=mg18725071.400 preview], [https://web.archive.org/web/20060902150939/http://geilenkotten.homeunix.org/TC_NS_09072005.pdf copy])}}

* [httphttps://www.sciencenews.org/articlesarticle/20051105/bob8.asppushing-limit "Pushing the Limit"], a ''[[Science News]]'' feature about the development and genesis of turbo codes

* [http://www-turbo.enst-bretagne.fr/ International Symposium On Turbo Codes]

* [http://www.iterativesolutions.com/Matlab.htm Coded Modulation Library], an open source library for simulating turbo codes in matlab

* [http://www.ifp.uiuc.edu/~singer/journalpapers/tuchler_2002a.pdf "Turbo Equalization: Principles and New Results"] {{Webarchive|url=https://web.archive.org/web/20090227062216/http://www.ifp.uiuc.edu/~singer/journalpapers/tuchler_2002a.pdf |date=27 February 2009 }}, an ''[[IEEE Transactions on Communications]]'' article about using convolutional codes jointly with channel equalization.

* [http://itpp.sourceforge.net IT++ Home Page] The [[IT++]] is a powerful C++ library which in particular supports turbo codes

* [http://www.inference.phy.cam.ac.uk/mackay/CodesTurbo.html Turbo codes publications by David MacKay]

* [https://aff3ct.github.io AFF3CT Home Page] (A Fast Forward Error Correction Toolbox) for high speed turbo codes simulations in software

* [http://www.scholarpedia.org/article/Turbo_codes{{cite journal |title=Turbo code] by Dr. |first1=Sylvie |last1=Kerouédan and [[|author2-link=Claude Berrou|Dr. |first2=Claude |last2=Berrou]] (|journal=Scholarpedia |date=2010 |volume=5 |issue=4 |page=6496 |publisher=scholarpedia.org)|doi=10.4249/scholarpedia.6496 |doi-access=free |bibcode=2010SchpJ...5.6496K }}

* [https://www.intel.com/content/dam/www/programmable/us/en/pdfs/literature/an/an505.pdf 3GPP LTE Turbo Reference Design].

* [https://www.mathworks.com/help/comm/ug/estimate-turbo-code-ber-performance-in-awgn.html Estimate Turbo Code BER Performance in AWGN] {{Webarchive|url=https://web.archive.org/web/20190201172029/https://www.mathworks.com/help/comm/ug/estimate-turbo-code-ber-performance-in-awgn.html |date=1 February 2019 }} (MatLab).

* [https://www.mathworks.com/help/comm/examples/parallel-concatenated-convolutional-coding-turbo-codes.html Parallel Concatenated Convolutional Coding: Turbo Codes (MatLab Simulink)]

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[[Category:Error detection and correction]]

[[Category:Capacity-approaching codes]]

[[Category:French inventions]]