BCJR algorithm

The BCJR algorithm is an algorithm for maximum a posteriori decoding of error correcting codes defined on trellises (principally convolutional codes). The algorithm is named after its inventors: Bahl, Cocke, Jelinek and Raviv.^[1] This algorithm is critical to modern iteratively-decoded error-correcting codes including turbo codes and low-density parity-check codes.

Steps involved

Based on the trellis:

Compute forward probabilities $\alpha$
Compute backward probabilities $\beta$
Compute smoothed probabilities based on other information (i.e. noise variance for AWGN, bit crossover probability for binary symmetric channel)

Berrou, Glavieux and Thitimajshima simplification.^[2]

Susa framework implements BCJR algorithm for forward error correction codes and channel equalization in C++.

^ 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.
^ 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
^ 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.