BCJR algorithm

The Bahl-Cocke-Jelinek-Raviv (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. This algorithm is critical to modern iteratively-decoded error-correcting codes, including turbo codes and low-density parity-check codes.

Based on the trellis:

• Compute forward probabilities ${\displaystyle \alpha }$
• Compute backward probabilities ${\displaystyle \beta }$
• 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.

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

• Forward-backward algorithm
• Maximum a posteriori (MAP) estimation
• Hidden Markov model

• The online textbook: Information Theory, Inference, and Learning Algorithms, by David J.C. MacKay, discusses the BCJR algorithm in chapter 25.
• The implementation of BCJR algorithm in Susa signal processing framework