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| {{Distinguish|TurboCoder Digital Medical Coding Software and Publications}}
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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 in 1993, which were the first practical codes to closely approach the [[Shannon–Hartley theorem|channel capacity]], a theoretical maximum for the [[code rate]] at which reliable communication is still possible given a specific noise level. Turbo codes are finding use in [[3G]] mobile communications and ([[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 are nowadays competing with [[LDPC code]]s, which provide similar performance.
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| ==History==
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| Prior to turbo codes, the best constructions were serial [[concatenated code]]s based on an outer [[Reed-Solomon error correction]] code combined with an inner [[Viterbi algorithm|Viterbi-decoded]] short constraint length [[convolutional code]], also known as RSV codes.
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| In 1993, turbo codes were introduced by [[Claude Berrou|Berrou]], [[Alain Glavieux|Glavieux]], and [[Punya Thitimajshima|Thitimajshima]] (from Télécom Bretagne, former [[École Nationale Supérieure des Télécommunications de Bretagne|ENST Bretagne]], France) in their paper: "''Near Shannon Limit Error-correcting Coding and Decoding: Turbo-codes''" published in the Proceedings of IEEE International Communications Conference.<ref>{{Citation|url=http://www-elec.enst-bretagne.fr/equipe/berrou/Near%20Shannon%20Limit%20Error.pdf|first=Claude|first2=Alain|first3=Punya|last=Berrou|last2=Glavieux|last3=Thitimajshima|title=Near Shannon Limit Error – Correcting|accessdate=11 February 2010}}</ref> 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=http://www-elec.enst-bretagne.fr/equipe/berrou/com_mag_berrou.pdf}}</ref>
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| 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 and repeat-accumulate codes. Iterative turbo decoding methods have also been applied to more conventional FEC systems, including Reed-Solomon corrected convolutional codes.
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| ==An example encoder==
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| There are many different instances of turbo codes, using different component encoders, input/output ratios, interleavers, and puncturing patterns. This example encoder implementation describes a classic turbo encoder, and demonstrates the general design of parallel turbo codes.
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| 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 convolutional 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]].
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| Hardware-wise, this turbo-code encoder consists of two identical RSC coders, С<sub>1</sub> and C<sub>2</sub>, as depicted in the figure, which are connected to each other using a concatenation scheme, called ''parallel concatenation'':
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| [[Image:turbo encoder.svg]]
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| In the figure, ''M'' is a memory register. The delay line and interleaver force input bits d<sub>k</sub> to appear in different sequences.
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| At first iteration, the input sequence ''d''<sub>k</sub> appears at both outputs of the encoder, ''x''<sub>k</sub> and'' y''<sub>1k</sub> or ''y''<sub>2k</sub> due to the encoder's systematic nature. If the encoders ''C''<sub>1</sub> and ''C''<sub>2</sub> are used respectively in ''n''<sub>1</sub> and ''n''<sub>2</sub> iterations, their rates are respectively equal to
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| :<math>\begin{align}
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| ~R_1 &= \frac{n_1 + n_2}{2n_1 + n_2}\\
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| ~R_2 &= \frac{n_1 + n_2}{n_1 + 2n_2}
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| \end{align}</math>
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| ==The decoder== | |
| The decoder is built in a similar way to the above encoder. Two elementary decoders are interconnected to each other, but in serial way, not in parallel. The <math>\scriptstyle DEC_1</math> decoder operates on lower speed (i.e., <math>\scriptstyle R_1</math>), thus, it is intended for the <math>\scriptstyle C_1</math> encoder, and <math>\scriptstyle DEC_2</math> is for <math>\scriptstyle C_2</math> correspondingly. <math>\scriptstyle DEC_1</math> yields a [[Turbo code#Soft decision approach|soft decision]] which causes <math>\scriptstyle L_1</math> delay. The same delay is caused by the delay line in the encoder. The <math>\scriptstyle DEC_2</math>'s operation causes <math>\scriptstyle L_2</math> delay.
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| [[Image:turbo decoder.svg]]
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| An interleaver installed between the two decoders is used here to scatter error bursts coming from <math>\scriptstyle DEC_1</math> output. ''DI'' block is a demultiplexing and insertion module. It works as a switch, redirecting input bits to <math>\scriptstyle DEC_1</math> at one moment and to <math>\scriptstyle DEC_2</math> at another. In OFF state, it feeds both <math>\scriptstyle y_{1k}</math> and <math>\scriptstyle y_{2k}</math> inputs with padding bits (zeros).
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| Consider a memoryless [[additive white Gaussian noise|AWGN]] channel, and assume that at ''k''-th iteration, the decoder receives a pair of random variables:
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| :<math>\begin{align}
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| ~x_k &= (2d_k - 1) + a_k\\
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| ~y_k &= 2( Y_k - 1) + b_k
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| \end{align}</math>
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| where <math>\scriptstyle a_k</math> and <math>\scriptstyle b_k</math> are independent noise components having the same variance <math>\scriptstyle \sigma^2</math>. <math>\scriptstyle Y_k</math> is a ''k''-th bit from <math>\scriptstyle y_k</math> encoder output.
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| Redundant information is demultiplexed and sent through ''DI'' to <math>\scriptstyle DEC_1</math> (when <math>\scriptstyle y_k \;=\; y_{1k}</math>) and to <math>\scriptstyle DEC_2</math> (when <math>\scriptstyle y_k \;=\; y_{2k}</math>).
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| <math>\scriptstyle DEC_1</math> yields a soft decision; i.e.:
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| :<math>\Lambda(d_k) = \log\frac{p(d_k = 1)}{p(d_k = 0)}</math>
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| and delivers it to <math>\scriptstyle DEC_2</math>. <math>\scriptstyle \Lambda(d_k)</math> is called the ''logarithm of the likelihood ratio'' (LLR). <math>\scriptstyle p(d_k \;=\; i),\, i \,\in\, \{0,\, 1\}</math> is the ''a posteriori probability'' (APP) of the <math>\scriptstyle d_k</math> data bit which shows the probability of interpreting a received <math>\scriptstyle d_k</math> bit as <math>\scriptstyle i</math>. Taking the ''LLR'' into account, <math>\scriptstyle DEC_2</math> yields a hard decision; i.e., a decoded bit.
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| It is known that the [[Viterbi algorithm]] is unable to calculate APP, thus it cannot be used in <math>\scriptstyle DEC_1</math>. Instead of that, a modified [[BCJR algorithm]] is used. For <math>\scriptstyle DEC_2</math>, the [[Viterbi algorithm]] is an appropriate one.
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| However, the depicted structure is not an optimal one, because <math>\scriptstyle DEC_1</math> uses only a proper fraction of the available redundant information. In order to improve the structure, a feedback loop is used (see the dotted line on the figure).
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| ==Soft decision approach==
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| The decoder front-end produces an integer for each bit in the data stream. This integer is a measure of how likely it is that the bit is a 0 or 1 and is also called ''soft bit''. The integer could be drawn from the range [−127, 127], where:
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| * −127 means "certainly 0"
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| * −100 means "very likely 0"
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| * 0 means "it could be either 0 or 1"
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| * 100 means "very likely 1"
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| * 127 means "certainly 1"
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| * etc.
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| This introduces a probabilistic aspect to the data-stream from the front end, but it conveys more information about each bit than just 0 or 1.
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| For example, for each bit, the front end of a traditional wireless-receiver has to decide if an internal analog voltage is above or below a given threshold voltage level. For a turbo-code decoder, the front end would provide an integer measure of how far the internal voltage is from the given threshold.
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| To decode the {{nowrap|''m'' + ''n''}}-bit block of data, the decoder front-end creates a block of likelihood measures, with one likelihood measure for each bit in the data stream. There are two parallel decoders, one for each of the {{frac|''n''|2}}-bit parity sub-blocks. Both decoders use the sub-block of ''m'' likelihoods for the payload data. The decoder working on the second parity sub-block knows the permutation that the coder used for this sub-block.
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| ==Solving hypotheses to find bits== | |
| The key innovation of turbo codes is how they use the likelihood data to reconcile differences between the two decoders. Each of the two convolutional decoders generates a hypothesis (with derived likelihoods) for the pattern of ''m'' bits in the payload sub-block. The hypothesis bit-patterns are compared, and if they differ, the decoders exchange the derived likelihoods they have for each bit in the hypotheses. Each decoder incorporates the derived likelihood estimates from the other decoder to generate a new hypothesis for the bits in the payload. Then they compare these new hypotheses. This iterative process continues until the two decoders come up with the same hypothesis for the ''m''-bit pattern of the payload, typically in 15 to 18 cycles.
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| An analogy can be drawn between this process and that of solving cross-reference puzzles like [[crossword]] or [[sudoku]]. Consider a partially completed, possibly garbled crossword puzzle. Two puzzle solvers (decoders) are trying to solve it: one possessing only the "down" clues (parity bits), and the other possessing only the "across" clues. To start, both solvers guess the answers (hypotheses) to their own clues, noting down how confident they are in each letter (payload bit). Then, they compare notes, by exchanging answers and confidence ratings with each other, noticing where and how they differ. Based on this new knowledge, they both come up with updated answers and confidence ratings, repeating the whole process until they converge to the same solution.
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| ==Performance==
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| Turbo codes perform well due to the attractive combination of the code's random appearance on the channel together with the physically realisable decoding structure. Turbo codes are affected by an [[error floor]].
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| ==Practical applications using turbo codes==
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| Telecommunications:
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| * Turbo codes are used extensively in [[3G]] and [[4G]] mobile telephony standards; e.g., in [[High Speed Packet Access|HSPA]], [[EV-DO]] and [[3GPP Long Term Evolution|LTE]].
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| * [[MediaFLO]], terrestrial mobile television system from [[Qualcomm]].
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| * The [[return link|interaction channel]] of [[satellite communication]] systems, such as [[DVB-RCS]].<ref>[http://www.etsi.org/deliver/etsi_en/301700_301799/301790/01.05.01_60/en_301790v010501p.pdf Digital Video Broadcasting (DVB); Interaction channel for Satellite Distribution Systems], ETSI EN 301 790, V1.5.1, May 2009.</ref>
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| * New [[NASA]] missions such as [[Mars Reconnaissance Orbiter]] now use turbo codes, as an alternative to RS-[[Viterbi decoder|Viterbi]] codes.
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| * Turbo coding such as block turbo coding and convolutional turbo coding are used in [[IEEE 802.16]] ([[WiMAX]]), a wireless metropolitan network standard.
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| ==Bayesian formulation==
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| From an [[artificial intelligence]] viewpoint, turbo codes can be considered as an instance of loopy [[belief propagation]] in [[Bayesian network]]s.<ref>{{Citation
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| | author1=McEliece, Robert J. | author1-link=Robert McEliece
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| | author2=MacKay, David J. C. | author2-link=David McKay{{dn|{{subst:DATE}}|date=February 2013}}
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| | author3=Cheng, Jung-Fu
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| | title=Turbo decoding as an instance of Pearl's “belief propagation” algorithm
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| | journal=IEEE Journal on Selected Areas in Communications
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| | volume=16
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| | issue=2
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| | pages=140–152
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| | year=1998
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| | issn=0733-8716
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| | doi=10.1109/49.661103
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| | postscript=.
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| }}</ref>
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| ==See also==
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| * [[Convolutional code]]
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| * [[Viterbi algorithm]]
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| * [[Soft-decision decoding]]
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| * [[Interleaver]]
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| * [[BCJR algorithm]]
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| * [[Low-density parity-check code]]
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| * [[Turbo equalizer]]
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| ==References==
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| {{reflist}}
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| ==External links==
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| * [http://www.spectrum.ieee.org/computing/software/closing-in-on-the-perfect-code "Closing In On The Perfect Code"], IEEE Spectrum, March 2004
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| * [http://www.csee.wvu.edu/~mvalenti/documents/valenti01.pdf "The UMTS Turbo Code and an Efficient Decoder Implementation Suitable for Software-Defined Radios"] (''International Journal of Wireless Information Networks'')
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| * {{Citation | author=Dana Mackenzie | title=Take it to the limit | journal=New Scientist | volume=187 | issue=2507 | year=2005 | pages=38–41 | postscript=. | issn=0262-4079 }} ([http://www.newscientist.com/article.ns?id=mg18725071.400 preview], [http://geilenkotten.homeunix.org/TC_NS_09072005.pdf copy])
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| * [http://www.sciencenews.org/articles/20051105/bob8.asp "Pushing the Limit"], a ''[[Science News]]'' feature about the development and genesis of turbo codes
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| * [http://www-turbo.enst-bretagne.fr/ International Symposium On Turbo Codes]
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| * [http://www.iterativesolutions.com/Matlab.htm Coded Modulation Library], an open source library for simulating turbo codes in matlab
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| * [http://www.ifp.uiuc.edu/~singer/journalpapers/tuchler_2002a.pdf "Turbo Equalization: Principles and New Results"], an ''[[IEEE Transactions on Communications]]'' article about using convolutional codes jointly with channel equalization.
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| * [http://www.kn-s.dlr.de/turbo/Turbo%20Sim.pdf "PDF Slideshow illustrating the decoding process"] A PDF Slideshow illustrating the decoding process
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| * [http://itpp.sourceforge.net IT++ Home Page] The [[IT++]] is a powerful C++ library which in particular supports turbo codes
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| * [http://www.inference.phy.cam.ac.uk/mackay/CodesTurbo.html Turbo codes publications by David MacKay]
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| * [http://www.newae.com/tiki-index.php?page=Turbo Extended CML Code including extensive Powerpoint Presentation on Turbo Codes]
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| {{CCSDS|state=collapsed}}
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| {{Use dmy dates|date=September 2010}}
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| {{DEFAULTSORT:Turbo Code}}
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| [[Category:Error detection and correction]]
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| [[Category:Capacity-approaching codes]]
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| {{Link GA|ru}}
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