View source for Hitachi DX07

The '''BCJR algorithm''' is an algorithm for [[maximum a posteriori]] decoding of [[error correcting code]]s defined on trellises (principally [[convolutional code]]s).  The algorithm is named after its inventors: Bahl, Cocke, [[Frederick Jelinek|Jelinek]] and Raviv.<ref name="bcjr">L.Bahl, J.Cocke, F.Jelinek, and J.Raviv, "Optimal Decoding of Linear Codes for minimizing symbol error rate", IEEE Transactions on Information Theory, vol. IT-20(2), pp.284-287, March 1974.</ref>  This algorithm is critical to modern iteratively-decoded error-correcting codes including [[turbo code]]s and [[low-density parity-check code]]s.

== Steps involved ==
Based on the [[convolutional code|trellis]]:
* Compute Forward probabilities  <math>\alpha</math>
* Compute Backward probabilities <math>\beta</math>
* Compute smoothed probabilities based on other information (i.e. noise variance for [[AWGN]], bit crossover probability for [[Binary symmetric channel]])

== Variations ==

=== SBGT BCJR ===
Berrou, Glavieux and Thitimajshima Simplification.<ref>Sichun Wang and François Patenaude, "A Systematic Approach to Modified BCJR MAP Algorithms for Convolutional Codes," ''EURASIP Journal on Applied Signal Processing'', vol. 2006, Article ID 95360, 15 pages, 2006. {{doi|10.1155/ASP/2006/95360}}</ref>

===Log-Map BCJR ===
<ref>P. Robertson, P. Hoeher and E. Villebrun, "Optimal and Sub-Optimal Maximum A Posteriori Algorithms Suitable for Turbo Decoding", European Transactions on Telecommunications, Vol. 8, 1997.
</ref>

===Max-Log-Map BCJR ===

==See also==
* [[Forward-backward algorithm]]
* [[Maximum a posteriori|Maximum a posteriori (MAP) estimation]]
* [[Hidden Markov model]]

==References==

<references/>

==External links==
* [http://www.inference.phy.cam.ac.uk/mackay/itila/ The on-line textbook: Information Theory, Inference, and Learning Algorithms], by [[David J.C. MacKay]], discusses the BCJR algorithm in chapter 25.

{{DEFAULTSORT:Bcjr Algorithm}}
[[Category:Error detection and correction]]