2110:. 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.
822:
265:
137:. Iterative turbo decoding methods have also been applied to more conventional FEC systems, including ReedâSolomon 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.
2094:
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
124:
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
2051:
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
518:
140:
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.
109:". 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,
1157:
2089:
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
128:
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
70:
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
1619:
340:
121:, 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.
2118:
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
2065:-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
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1022:
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75:(LDPC) codes, which provide similar performance. Until the patent for turbo codes expired, the patent-free status of LDPC codes was an important factor in LDPC's continued relevance.
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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.
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2334:"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."
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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
2829:
2024:
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
2016:
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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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
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35:(FEC) codes developed around 1990â91, but first published in 1993. They were the first practical codes to closely approach the maximum channel capacity or
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The decoder is built in a similar way to the above encoder. Two elementary decoders are interconnected to each other, but in series, not in parallel. The
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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
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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.
2048:
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.
513:{\displaystyle {\begin{aligned}~R_{1}&={\frac {n_{1}+n_{2}}{2n_{1}+n_{2}}}\\~R_{2}&={\frac {n_{1}+n_{2}}{n_{1}+2n_{2}}}\end{aligned}}}
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183:. This example encoder implementation describes a classic turbo encoder, and demonstrates the general design of parallel turbo codes.
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There are many different instances of turbo codes, using different component encoders, input/output ratios, interleavers, and
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The fundamental patent application for turbo codes was filed on 23 April 1991. The patent application lists
2628:. 9th International Symposium on Turbo Codes and Iterative Information Processing (ISTC). pp. 151â5.
2585:"20 years of turbo coding and energy-aware design guidelines for energy-constrained wireless applications"
86:
has argued the term turbo code is a misnomer since there is no feedback involved in the encoding process.
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1984:
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1152:{\displaystyle {\begin{aligned}~x_{k}&=(2d_{k}-1)+a_{k}\\~y_{k}&=2(Y_{k}-1)+b_{k}\end{aligned}}}
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at which reliable communication is still possible given a specific noise level. Turbo codes are used in
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and P. Hoeher, who, in the late 80s, highlighted the interest of probabilistic processing." He adds "
2212:), a wireless metropolitan network standard, uses block turbo coding and convolutional turbo coding.
256:, as depicted in the figure, which are connected to each other using a concatenation scheme, called
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2252:
2151:
2143:
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An interleaver installed between the two decoders is used here to scatter error bursts coming from
32:
2741:
2685:"The UMTS Turbo Code and an Efficient Decoder Implementation Suitable for Software-Defined Radios"
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block is a demultiplexing and insertion module. It works as a switch, redirecting input bits to
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An analogy can be drawn between this process and that of solving cross-reference puzzles like
98:
as the sole inventor of turbo codes. The patent filing resulted in several patents including
2778:(A Fast Forward Error Correction Toolbox) for high speed turbo codes simulations in software
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Digital Video
Broadcasting (DVB); Interaction channel for Satellite Distribution Systems
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This encoder implementation sends three sub-blocks of bits. The first sub-block is the
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Brejza, M.F.; Li, L.; Maunder, R.G.; Al-Hashimi, B.M.; Berrou, C.; Hanzo, L. (2016).
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Battail, GĂ©rard (1998). "A conceptual framework for understanding turbo codes".
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Hardware-wise, this turbo code encoder consists of two identical RSC coders,
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GarzĂłn-BohĂłrquez, Ronald; Nour, Charbel Abdel; Douillard, Catherine (2016).
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article about using convolutional codes jointly with channel equalization.
2622:
Improving Turbo codes for 5G with parity puncture-constrained interleavers
2840:
Parallel
Concatenated Convolutional Coding: Turbo Codes (MatLab Simulink)
2513:"Turbo decoding as an instance of Pearl's "belief propagation" algorithm"
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to appear in different sequences. At first iteration, the input sequence
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parity bits for the payload data, computed using a recursive systematic
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is a memory register. The delay line and interleaver force input bits d
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delay. The same delay is caused by the delay line in the encoder. The
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In a later paper, Berrou gave credit to the intuition of "G. Battail,
2902:
2730:
2103:
1614:{\displaystyle \Lambda (d_{k})=\log {\frac {p(d_{k}=1)}{p(d_{k}=0)}}}
64:
40:
107:
Near
Shannon Limit Error-correcting Coding and Decoding: Turbo-codes
2161:
2157:
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is a powerful C++ library which in particular supports turbo codes
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3081:
2874:
2225:
viewpoint, turbo codes can be considered as an instance of loopy
2176:
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data bit which shows the probability of interpreting a received
1011:-th iteration, the decoder receives a pair of random variables:
821:
264:
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However, the depicted structure is not an optimal one, because
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3047:
2738:, an open source library for simulating turbo codes in matlab
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2209:
2147:
2346:"Iterative Decoding of Binary Block and Convolutional Codes"
3055:
3042:
2897:
2775:
2410:
2392:
2344:
Hagenauer, Joachim; Offer, Elke; Papke, Luiz (March 1996).
2187:
2125:
2099:-bit pattern of the payload, typically in 15 to 18 cycles.
2077:-bit parity sub-blocks. Both decoders use the sub-block of
52:
1220:
are independent noise components having the same variance
2582:
235:
of the payload data is carried out by a device called an
144:
Prior to turbo codes, the best constructions were serial
2727:
feature about the development and genesis of turbo codes
1314:
Redundant information is demultiplexed and sent through
306:
due to the encoder's systematic nature. If the encoders
2139:
2135:
48:
44:
2696:
International
Journal of Wireless Information Networks
2471:
The ten-year-old turbo codes are entering into service
1988:
1946:
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is unable to calculate APP, thus it cannot be used in
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2028:. The integer could be drawn from the range , where:
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190:-bit block of payload data. The second sub-block is
1763:{\displaystyle \textstyle p(d_{k}=i),\,i\in \{0,1\}}
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2008:
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334:iterations, their rates are respectively equal to
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2742:"Turbo Equalization: Principles and New Results"
2564:IEEE Journal on Selected Areas in Communications
2520:IEEE Journal on Selected Areas in Communications
2781:
16:High-performance forward error correction codes
2876:Consultative Committee for Space Data Systems
2860:
1893:yields a hard decision; i.e., a decoded bit.
3015:Space Communications Protocol Specifications
2444:
2319:
2160:, terrestrial mobile television system from
1756:
1744:
3109:Networking, interoperability and monitoring
2827:Estimate Turbo Code BER Performance in AWGN
2592:IEEE Communications Surveys & Tutorials
721:
105:The first public paper on turbo codes was "
2867:
2853:
1690:{\displaystyle \textstyle \Lambda (d_{k})}
2811:
2701:
2019:
1737:
2771:Turbo codes publications by David MacKay
2126:Practical applications using turbo codes
936:at another. In OFF state, it feeds both
285:appears at both outputs of the encoder,
2561:
2406:"Near Shannon Limit Error â Correcting"
2353:IEEE Transactions on Information Theory
2268:Serial concatenated convolutional codes
2216:
1471:{\displaystyle \textstyle y_{k}=y_{2k}}
1391:{\displaystyle \textstyle y_{k}=y_{1k}}
563:decoder operates on lower speed (i.e.,
131:serial concatenated convolutional codes
3134:
2731:International Symposium On Turbo Codes
2647:
2313:
1242:{\displaystyle \textstyle \sigma ^{2}}
2848:
2194:use turbo codes as an alternative to
2142:mobile telephony standards; e.g., in
174:
2494:, ETSI EN 301 790, V1.5.1, May 2009.
2134:Turbo codes are used extensively in
198:(RSC code). The third sub-block is
2994:Spacecraft Monitoring & Control
2754:IEEE Transactions on Communications
2038:0 means "it could be either 0 or 1"
13:
2550:
2467:
2009:{\displaystyle \textstyle DEC_{1}}
1967:{\displaystyle \textstyle DEC_{2}}
1928:{\displaystyle \textstyle DEC_{1}}
1886:{\displaystyle \textstyle DEC_{2}}
1667:
1652:{\displaystyle \textstyle DEC_{2}}
1525:
1508:{\displaystyle \textstyle DEC_{1}}
1426:{\displaystyle \textstyle DEC_{2}}
1346:{\displaystyle \textstyle DEC_{1}}
1000:inputs with padding bits (zeros).
929:{\displaystyle \textstyle DEC_{2}}
894:{\displaystyle \textstyle DEC_{1}}
855:{\displaystyle \textstyle DEC_{1}}
781:{\displaystyle \textstyle DEC_{2}}
713:{\displaystyle \textstyle DEC_{1}}
649:{\displaystyle \textstyle DEC_{2}}
556:{\displaystyle \textstyle DEC_{1}}
31:) are a class of high-performance
14:
3158:
2641:
2412:, vol. 2, pp. 1064â70,
1699:logarithm of the likelihood ratio
993:{\displaystyle \textstyle y_{2k}}
961:{\displaystyle \textstyle y_{1k}}
216:bits of data with a code rate of
2650:"Closing In On The Perfect Code"
2447:"CLOSING IN ON THE PERFECT CODE"
2322:"CLOSING IN ON THE PERFECT CODE"
1825:{\displaystyle \textstyle d_{k}}
1796:{\displaystyle \textstyle d_{k}}
1304:{\displaystyle \textstyle y_{k}}
1271:{\displaystyle \textstyle Y_{k}}
1213:{\displaystyle \textstyle b_{k}}
1184:{\displaystyle \textstyle a_{k}}
820:
810:{\displaystyle \textstyle L_{2}}
746:{\displaystyle \textstyle L_{1}}
678:{\displaystyle \textstyle C_{2}}
614:{\displaystyle \textstyle C_{1}}
592:), thus, it is intended for the
585:{\displaystyle \textstyle R_{1}}
263:
102:, which expired 29 August 2013.
51:mobile communications (e.g., in
39:, a theoretical maximum for the
2978:Proximity-1 Space Link Protocol
2821:3GPP LTE Turbo Reference Design
2555:
2085:Solving hypotheses to find bits
117:(from Télécom Bretagne, former
3142:Error detection and correction
2497:
2485:
2461:
2438:
2386:
2337:
2295:
2113:
1935:. Instead of that, a modified
1731:
1712:
1683:
1670:
1605:
1586:
1578:
1559:
1541:
1528:
1515:yields a soft decision; i.e.:
1129:
1110:
1067:
1045:
523:
1:
3116:Service-oriented architecture
2798:(4). scholarpedia.org: 6496.
2445:Erico Guizzo (1 March 2004).
2320:Erico Guizzo (1 March 2004).
2288:
2263:Low-density parity-check code
2196:ReedâSolomon error correction
150:ReedâSolomon error correction
3028:Telemetry modulation systems
2648:Guizzo, Erico (March 2004).
1847:{\displaystyle \textstyle i}
1007:channel, and assume that at
152:code combined with an inner
7:
3019:Performance Enhancing Proxy
2236:
2192:Mars Reconnaissance Orbiter
160:, also known as RSV codes.
10:
3163:
3147:Capacity-approaching codes
2666:10.1109/MSPEC.2004.1270546
2604:10.1109/COMST.2015.2448692
2035:â100 means "very likely 0"
89:
3120:Message Abstraction Layer
3108:
3064:
3027:
3007:
2986:
2965:
2928:
2882:
2813:10.4249/scholarpedia.6496
2634:10.1109/ISTC.2016.7593095
2511:; Cheng, Jung-Fu (1998),
2041:100 means "very likely 1"
2973:Command Loss Timer Reset
2966:Telemetry command uplink
2747:27 February 2009 at the
2736:Coded Modulation Library
2702:Mackenzie, Dana (2005).
2253:Forward error correction
2032:â127 means "certainly 0"
1772:a posteriori probability
202:parity bits for a known
156:short constraint length
73:low-density parity-check
33:forward error correction
2832:1 February 2019 at the
2690:20 October 2016 at the
2418:10.1109/ICC.1993.397441
2223:artificial intelligence
2173:satellite communication
2044:127 means "certainly 1"
1978:is an appropriate one.
135:repeat-accumulate codes
2918:Adaptive Entropy Coder
2704:"Take it to the limit"
2273:Soft-decision decoding
2020:Soft decision approach
2010:
1968:
1929:
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1003:Consider a memoryless
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258:parallel concatenation
27:(originally in French
2011:
1969:
1930:
1896:It is known that the
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901:at one moment and to
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2402:Thitimajshima, Punya
2217:Bayesian formulation
2130:Telecommunications:
1985:
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788:'s operation causes
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341:
2999:Beacon mode service
2804:2010SchpJ...5.6496K
2782:Kerouédan, Sylvie;
2719:"Pushing the Limit"
2680:on 11 October 2009.
2509:MacKay, David J. C.
2505:McEliece, Robert J.
2169:interaction channel
1624:and delivers it to
181:puncturing patterns
125:signal processing.
100:US Patent 5,446,747
2987:Telemetry downlink
2944:Concatenated codes
2474:, Bretagne, France
2248:Convolutional code
2227:belief propagation
2006:
2005:
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196:convolutional code
175:An example encoder
158:convolutional code
148:based on an outer
146:concatenated codes
21:information theory
3129:
3128:
3008:Telemetry general
2939:Binary Golay code
2576:10.1109/49.661112
2532:10.1109/49.661103
2457:on 23 April 2023.
2365:10.1109/18.485714
2332:on 23 April 2023.
2283:Viterbi algorithm
2231:Bayesian networks
2190:missions such as
2175:systems, such as
1976:Viterbi algorithm
1898:Viterbi algorithm
1609:
1089:
1027:
685:correspondingly.
504:
431:
423:
350:
3154:
2929:Error Correction
2883:Data compression
2869:
2862:
2855:
2846:
2845:
2817:
2815:
2776:AFF3CT Home Page
2715:
2681:
2676:. Archived from
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2517:
2501:
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2468:Berrou, Claude,
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2453:. Archived from
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2390:
2384:
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2367:. Archived from
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2328:. Archived from
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1647:
1646:
1620:
1618:
1617:
1612:
1610:
1608:
1598:
1597:
1581:
1571:
1570:
1554:
1540:
1539:
1514:
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1511:
1506:
1503:
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1477:
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1474:
1469:
1466:
1465:
1450:
1449:
1432:
1430:
1429:
1424:
1421:
1420:
1397:
1395:
1394:
1389:
1386:
1385:
1370:
1369:
1352:
1350:
1349:
1344:
1341:
1340:
1311:encoder output.
1310:
1308:
1307:
1302:
1299:
1298:
1277:
1275:
1274:
1269:
1266:
1265:
1248:
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1219:
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1208:
1207:
1190:
1188:
1187:
1182:
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1178:
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1150:
1148:
1144:
1143:
1122:
1121:
1099:
1098:
1087:
1082:
1081:
1060:
1059:
1037:
1036:
1025:
999:
997:
996:
991:
988:
987:
967:
965:
964:
959:
956:
955:
935:
933:
932:
927:
924:
923:
900:
898:
897:
892:
889:
888:
861:
859:
858:
853:
850:
849:
824:
816:
814:
813:
808:
805:
804:
787:
785:
784:
779:
776:
775:
752:
750:
749:
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741:
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719:
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708:
707:
684:
682:
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676:
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655:
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620:
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612:
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591:
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583:
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550:
519:
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501:
486:
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473:
461:
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450:
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429:
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407:
394:
393:
392:
380:
379:
369:
360:
359:
348:
267:
230:
215:
3162:
3161:
3157:
3156:
3155:
3153:
3152:
3151:
3132:
3131:
3130:
3125:
3104:
3099:
3085:
3060:
3023:
3003:
2982:
2961:
2924:
2878:
2873:
2834:Wayback Machine
2749:Wayback Machine
2692:Wayback Machine
2644:
2625:
2587:
2558:
2553:
2551:Further reading
2548:
2547:
2515:
2502:
2498:
2490:
2486:
2477:
2475:
2466:
2462:
2443:
2439:
2430:
2428:
2398:Glavieux, Alain
2391:
2387:
2377:
2375:
2374:on 11 June 2013
2371:
2348:
2342:
2338:
2318:
2314:
2307:
2301:
2300:
2296:
2291:
2278:Turbo equalizer
2239:
2219:
2200:Viterbi decoder
2128:
2116:
2087:
2073:
2067:
2066:
2056:
2022:
1999:
1995:
1986:
1983:
1982:
1957:
1953:
1944:
1941:
1940:
1918:
1914:
1905:
1902:
1901:
1876:
1872:
1863:
1860:
1859:
1837:
1834:
1833:
1815:
1811:
1808:
1805:
1804:
1786:
1782:
1779:
1776:
1775:
1719:
1715:
1706:
1703:
1702:
1677:
1673:
1664:
1661:
1660:
1642:
1638:
1629:
1626:
1625:
1593:
1589:
1582:
1566:
1562:
1555:
1553:
1535:
1531:
1523:
1520:
1519:
1498:
1494:
1485:
1482:
1481:
1458:
1454:
1445:
1441:
1438:
1435:
1434:
1416:
1412:
1403:
1400:
1399:
1378:
1374:
1365:
1361:
1358:
1355:
1354:
1336:
1332:
1323:
1320:
1319:
1294:
1290:
1287:
1284:
1283:
1261:
1257:
1254:
1251:
1250:
1232:
1228:
1225:
1222:
1221:
1203:
1199:
1196:
1193:
1192:
1174:
1170:
1167:
1164:
1163:
1146:
1145:
1139:
1135:
1117:
1113:
1100:
1094:
1090:
1084:
1083:
1077:
1073:
1055:
1051:
1038:
1032:
1028:
1021:
1019:
1016:
1015:
980:
976:
973:
970:
969:
948:
944:
941:
938:
937:
919:
915:
906:
903:
902:
884:
880:
871:
868:
867:
845:
841:
832:
829:
828:
800:
796:
793:
790:
789:
771:
767:
758:
755:
754:
736:
732:
729:
726:
725:
703:
699:
690:
687:
686:
668:
664:
661:
658:
657:
639:
635:
626:
623:
622:
604:
600:
597:
594:
593:
575:
571:
568:
565:
564:
546:
542:
533:
530:
529:
526:
507:
506:
497:
493:
481:
477:
476:
469:
465:
456:
452:
451:
449:
442:
436:
432:
426:
425:
416:
412:
403:
399:
395:
388:
384:
375:
371:
370:
368:
361:
355:
351:
344:
342:
339:
338:
333:
326:
319:
312:
305:
298:
291:
284:
277:
270:In the figure,
255:
248:
217:
207:
177:
154:Viterbi-decoded
92:
17:
12:
11:
5:
3160:
3150:
3149:
3144:
3127:
3126:
3124:
3123:
3112:
3110:
3106:
3105:
3103:
3102:
3097:
3093:
3088:
3083:
3079:
3074:
3068:
3066:
3062:
3061:
3059:
3058:
3053:
3050:
3045:
3040:
3035:
3031:
3029:
3025:
3024:
3022:
3021:
3011:
3009:
3005:
3004:
3002:
3001:
2996:
2990:
2988:
2984:
2983:
2981:
2980:
2975:
2969:
2967:
2963:
2962:
2960:
2959:
2954:
2951:
2946:
2941:
2936:
2932:
2930:
2926:
2925:
2923:
2922:
2921:
2920:
2912:
2911:
2910:
2905:
2900:
2895:
2886:
2884:
2880:
2879:
2872:
2871:
2864:
2857:
2849:
2843:
2842:
2837:
2824:
2818:
2784:Berrou, Claude
2779:
2773:
2768:
2761:IT++ Home Page
2758:
2739:
2733:
2728:
2716:
2714:(2507): 38â41.
2699:
2682:
2643:
2642:External links
2640:
2639:
2638:
2616:
2580:
2570:(2): 245â254.
2557:
2554:
2552:
2549:
2546:
2545:
2526:(2): 140â152,
2496:
2484:
2460:
2437:
2394:Berrou, Claude
2385:
2359:(2): 429â445.
2336:
2312:
2293:
2292:
2290:
2287:
2286:
2285:
2280:
2275:
2270:
2265:
2260:
2255:
2250:
2245:
2243:BCJR algorithm
2238:
2235:
2218:
2215:
2214:
2213:
2203:
2184:
2165:
2155:
2127:
2124:
2115:
2112:
2086:
2083:
2055:To decode the
2046:
2045:
2042:
2039:
2036:
2033:
2021:
2018:
2002:
1998:
1994:
1991:
1960:
1956:
1952:
1949:
1937:BCJR algorithm
1921:
1917:
1913:
1910:
1879:
1875:
1871:
1868:
1858:into account,
1842:
1818:
1814:
1789:
1785:
1758:
1755:
1752:
1749:
1746:
1743:
1740:
1736:
1733:
1730:
1727:
1722:
1718:
1714:
1711:
1697:is called the
1685:
1680:
1676:
1672:
1669:
1645:
1641:
1637:
1634:
1622:
1621:
1607:
1604:
1601:
1596:
1592:
1588:
1585:
1580:
1577:
1574:
1569:
1565:
1561:
1558:
1552:
1549:
1546:
1543:
1538:
1534:
1530:
1527:
1501:
1497:
1493:
1490:
1464:
1461:
1457:
1453:
1448:
1444:
1419:
1415:
1411:
1408:
1384:
1381:
1377:
1373:
1368:
1364:
1339:
1335:
1331:
1328:
1297:
1293:
1264:
1260:
1235:
1231:
1206:
1202:
1177:
1173:
1160:
1159:
1142:
1138:
1134:
1131:
1128:
1125:
1120:
1116:
1112:
1109:
1106:
1103:
1101:
1097:
1093:
1086:
1085:
1080:
1076:
1072:
1069:
1066:
1063:
1058:
1054:
1050:
1047:
1044:
1041:
1039:
1035:
1031:
1024:
1023:
986:
983:
979:
954:
951:
947:
922:
918:
914:
911:
887:
883:
879:
876:
848:
844:
840:
837:
803:
799:
774:
770:
766:
763:
739:
735:
706:
702:
698:
695:
671:
667:
642:
638:
634:
631:
607:
603:
578:
574:
549:
545:
541:
538:
525:
522:
521:
520:
500:
496:
492:
489:
484:
480:
472:
468:
464:
459:
455:
448:
445:
443:
439:
435:
428:
427:
419:
415:
411:
406:
402:
398:
391:
387:
383:
378:
374:
367:
364:
362:
358:
354:
347:
346:
331:
324:
317:
310:
303:
296:
289:
282:
275:
253:
246:
176:
173:
91:
88:
68:communications
15:
9:
6:
4:
3:
2:
3159:
3148:
3145:
3143:
3140:
3139:
3137:
3121:
3117:
3114:
3113:
3111:
3107:
3101:
3094:
3092:
3089:
3087:
3080:
3078:
3075:
3073:
3070:
3069:
3067:
3063:
3057:
3054:
3051:
3049:
3046:
3044:
3041:
3039:
3036:
3033:
3032:
3030:
3026:
3020:
3016:
3013:
3012:
3010:
3006:
3000:
2997:
2995:
2992:
2991:
2989:
2985:
2979:
2976:
2974:
2971:
2970:
2968:
2964:
2958:
2955:
2952:
2950:
2947:
2945:
2942:
2940:
2937:
2934:
2933:
2931:
2927:
2919:
2916:
2915:
2913:
2909:
2906:
2904:
2901:
2899:
2896:
2894:
2891:
2890:
2888:
2887:
2885:
2881:
2877:
2870:
2865:
2863:
2858:
2856:
2851:
2850:
2847:
2841:
2838:
2835:
2831:
2828:
2825:
2822:
2819:
2814:
2809:
2805:
2801:
2797:
2793:
2789:
2785:
2780:
2777:
2774:
2772:
2769:
2766:
2762:
2759:
2756:
2755:
2750:
2746:
2743:
2740:
2737:
2734:
2732:
2729:
2726:
2725:
2720:
2717:
2713:
2709:
2708:New Scientist
2705:
2700:
2697:
2693:
2689:
2686:
2683:
2679:
2675:
2671:
2667:
2663:
2659:
2655:
2654:IEEE Spectrum
2651:
2646:
2645:
2635:
2631:
2624:
2623:
2617:
2613:
2609:
2605:
2601:
2597:
2593:
2586:
2581:
2577:
2573:
2569:
2565:
2560:
2559:
2541:
2537:
2533:
2529:
2525:
2521:
2514:
2510:
2506:
2500:
2493:
2488:
2473:
2472:
2464:
2456:
2452:
2451:IEEE Spectrum
2448:
2441:
2427:
2423:
2419:
2415:
2411:
2407:
2403:
2399:
2395:
2389:
2370:
2366:
2362:
2358:
2354:
2347:
2340:
2331:
2327:
2326:IEEE Spectrum
2323:
2316:
2304:
2298:
2294:
2284:
2281:
2279:
2276:
2274:
2271:
2269:
2266:
2264:
2261:
2259:
2256:
2254:
2251:
2249:
2246:
2244:
2241:
2240:
2234:
2232:
2228:
2224:
2211:
2207:
2204:
2201:
2197:
2193:
2189:
2185:
2182:
2178:
2174:
2170:
2166:
2163:
2159:
2156:
2153:
2149:
2145:
2141:
2137:
2133:
2132:
2131:
2123:
2121:
2111:
2109:
2105:
2100:
2098:
2093:
2082:
2080:
2070:
2063:
2059:
2053:
2049:
2043:
2040:
2037:
2034:
2031:
2030:
2029:
2027:
2017:
2000:
1996:
1992:
1989:
1979:
1977:
1958:
1954:
1950:
1947:
1939:is used. For
1938:
1919:
1915:
1911:
1908:
1899:
1894:
1877:
1873:
1869:
1866:
1857:
1854:. Taking the
1840:
1816:
1812:
1787:
1783:
1774:(APP) of the
1773:
1753:
1750:
1747:
1741:
1738:
1734:
1728:
1725:
1720:
1716:
1709:
1700:
1678:
1674:
1643:
1639:
1635:
1632:
1602:
1599:
1594:
1590:
1583:
1575:
1572:
1567:
1563:
1556:
1550:
1547:
1544:
1536:
1532:
1518:
1517:
1516:
1499:
1495:
1491:
1488:
1479:
1462:
1459:
1455:
1451:
1446:
1442:
1417:
1413:
1409:
1406:
1382:
1379:
1375:
1371:
1366:
1362:
1337:
1333:
1329:
1326:
1317:
1312:
1295:
1291:
1282:-th bit from
1281:
1262:
1258:
1233:
1229:
1204:
1200:
1175:
1171:
1140:
1136:
1132:
1126:
1123:
1118:
1114:
1107:
1104:
1102:
1095:
1091:
1078:
1074:
1070:
1064:
1061:
1056:
1052:
1048:
1042:
1040:
1033:
1029:
1014:
1013:
1012:
1010:
1006:
1001:
984:
981:
977:
952:
949:
945:
920:
916:
912:
909:
885:
881:
877:
874:
865:
846:
842:
838:
835:
825:
823:
818:
801:
797:
772:
768:
764:
761:
737:
733:
724:which causes
723:
722:soft decision
704:
700:
696:
693:
669:
665:
640:
636:
632:
629:
621:encoder, and
605:
601:
576:
572:
547:
543:
539:
536:
498:
494:
490:
487:
482:
478:
470:
466:
462:
457:
453:
446:
444:
437:
433:
417:
413:
409:
404:
400:
396:
389:
385:
381:
376:
372:
365:
363:
356:
352:
337:
336:
335:
330:
323:
316:
309:
302:
295:
288:
281:
273:
268:
266:
261:
259:
252:
245:
240:
238:
234:
228:
224:
220:
214:
210:
205:
201:
197:
193:
189:
184:
182:
172:
170:
166:
161:
159:
155:
151:
147:
142:
138:
136:
132:
126:
122:
120:
119:ENST Bretagne
116:
115:Thitimajshima
112:
108:
103:
101:
97:
96:Claude Berrou
87:
85:
81:
80:turbocharging
76:
74:
69:
66:
62:
58:
54:
50:
46:
42:
38:
37:Shannon limit
34:
30:
26:
22:
2948:
2795:
2792:Scholarpedia
2791:
2788:"Turbo code"
2752:
2724:Science News
2722:
2711:
2707:
2695:
2678:the original
2660:(3): 36â42.
2657:
2653:
2621:
2595:
2591:
2567:
2563:
2556:Publications
2523:
2519:
2499:
2487:
2476:, retrieved
2470:
2463:
2455:the original
2450:
2440:
2429:, retrieved
2409:
2388:
2376:. Retrieved
2369:the original
2356:
2352:
2339:
2330:the original
2325:
2315:
2297:
2220:
2129:
2117:
2101:
2096:
2091:
2088:
2078:
2068:
2061:
2057:
2054:
2050:
2047:
2025:
2023:
1980:
1895:
1855:
1771:
1698:
1623:
1480:
1315:
1313:
1279:
1161:
1008:
1002:
863:
826:
819:
527:
328:
321:
320:are used in
314:
307:
300:
293:
286:
279:
271:
269:
262:
257:
250:
243:
241:
226:
222:
218:
212:
208:
199:
191:
187:
185:
178:
165:J. Hagenauer
162:
143:
139:
127:
123:
106:
104:
93:
77:
28:
24:
18:
3065:Frequencies
2949:Turbo codes
2598:(1): 8â28.
2478:11 February
2431:11 February
2258:Interleaver
2206:IEEE 802.16
2120:error floor
2114:Performance
2052:threshold.
524:The decoder
237:interleaver
233:permutation
204:permutation
169:R. Gallager
25:turbo codes
3136:Categories
2957:LDPC codes
2303:US 5446747
2289:References
61:deep space
59:) and in (
29:Turbocodes
2903:JPEG 2000
2836:(MatLab).
2540:0733-8716
2104:crossword
1742:∈
1668:Λ
1551:
1526:Λ
1398:) and to
1230:σ
1124:−
1062:−
720:yields a
84:Hagenauer
65:satellite
41:code rate
3052:Proposed
3017:(SCPS):
2953:Proposed
2908:122.0.B1
2830:Archived
2786:(2010).
2745:Archived
2688:Archived
2674:21237188
2612:12966388
2426:17770377
2404:(1993),
2378:20 March
2237:See also
2221:From an
2181:DVB-RCS2
2162:Qualcomm
2158:MediaFLO
2026:soft bit
862:output.
111:Glavieux
3034:Current
2935:Current
2889:Images
2800:Bibcode
2186:Recent
2177:DVB-RCS
2072:⁄
1832:bit as
1770:is the
1701:(LLR).
817:delay.
656:is for
90:History
3091:K band
3077:S band
3072:X band
2672:
2610:
2538:
2424:
2308:
2202:codes.
2108:sudoku
1974:, the
1433:(when
1353:(when
1162:where
1088:
1026:
430:
349:
231:. The
113:, and
3048:OQPSK
2914:Data
2751:, an
2670:S2CID
2626:(PDF)
2608:S2CID
2588:(PDF)
2516:(PDF)
2422:S2CID
2372:(PDF)
2349:(PDF)
2210:WiMAX
2148:EV-DO
1278:is a
3100:band
3086:band
3056:GMSK
3043:QPSK
3038:BPSK
2898:JPEG
2893:ICER
2765:IT++
2763:The
2721:, a
2536:ISSN
2480:2010
2433:2010
2380:2014
2188:NASA
2179:and
2167:The
2150:and
2144:HSPA
2138:and
1191:and
1005:AWGN
968:and
327:and
313:and
249:and
133:and
55:and
53:UMTS
2808:doi
2712:187
2662:doi
2630:doi
2600:doi
2596:918
2572:doi
2568:916
2528:doi
2414:doi
2361:doi
2229:in
2171:of
2152:LTE
2106:or
1856:LLR
1548:log
1478:).
1318:to
299:or
292:and
200:n/2
192:n/2
57:LTE
19:In
3138::
2806:.
2794:.
2790:.
2710:.
2706:.
2668:.
2658:41
2656:.
2652:.
2606:.
2594:.
2590:.
2566:.
2534:,
2524:16
2522:,
2518:,
2507:;
2449:.
2420:,
2408:,
2400:;
2396:;
2357:42
2355:.
2351:.
2324:.
2233:.
2146:,
2140:4G
2136:3G
2122:.
2060:+
1659:.
1316:DI
1249:.
864:DI
304:2k
297:1k
260::
239:.
225:+
221:/(
211:+
82:.
63:)
49:4G
45:3G
23:,
3122:)
3118:(
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2632::
2614:.
2602::
2578:.
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2530::
2416::
2382:.
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2198:-
2183:.
2164:.
2154:.
2097:m
2092:m
2079:m
2074:2
2069:n
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2058:m
2001:1
1997:C
1993:E
1990:D
1959:2
1955:C
1951:E
1948:D
1920:1
1916:C
1912:E
1909:D
1878:2
1874:C
1870:E
1867:D
1841:i
1817:k
1813:d
1788:k
1784:d
1757:}
1754:1
1751:,
1748:0
1745:{
1739:i
1735:,
1732:)
1729:i
1726:=
1721:k
1717:d
1713:(
1710:p
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1675:d
1671:(
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1600:=
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1500:1
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1456:y
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1443:y
1418:2
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1410:E
1407:D
1383:k
1380:1
1376:y
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1367:k
1363:y
1338:1
1334:C
1330:E
1327:D
1296:k
1292:y
1280:k
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1259:Y
1234:2
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1172:a
1141:k
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1108:2
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1096:k
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1071:+
1068:)
1065:1
1057:k
1053:d
1049:2
1046:(
1043:=
1034:k
1030:x
1009:k
985:k
982:2
978:y
953:k
950:1
946:y
921:2
917:C
913:E
910:D
886:1
882:C
878:E
875:D
847:1
843:C
839:E
836:D
802:2
798:L
773:2
769:C
765:E
762:D
738:1
734:L
705:1
701:C
697:E
694:D
670:2
666:C
641:2
637:C
633:E
630:D
606:1
602:C
577:1
573:R
548:1
544:C
540:E
537:D
499:2
495:n
491:2
488:+
483:1
479:n
471:2
467:n
463:+
458:1
454:n
447:=
438:2
434:R
418:2
414:n
410:+
405:1
401:n
397:2
390:2
386:n
382:+
377:1
373:n
366:=
357:1
353:R
332:2
329:n
325:1
322:n
318:2
315:C
311:1
308:C
301:y
294:y
290:k
287:x
283:k
280:d
276:k
272:M
254:2
251:C
247:1
244:C
229:)
227:n
223:m
219:m
213:n
209:m
188:m
47:/
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