A Systematic Approach to Modified BCJR MAP Algorithms for Convolutional Codes
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A Systematic Approach to Modified BCJR MAP Algorithms for Convolutional Codes Sichun Wang1 and Franc¸ois Patenaude2 1 Defence
Research and Development Canada – Ottawa, Ottawa, ON, Canada K1A 0Z4 Research Centre Canada, Ottawa, ON, Canada K2H 8S2
2 Communications
Received 19 November 2004; Revised 19 July 2005; Accepted 12 September 2005 Recommended for Publication by Vincent Poor Since Berrou, Glavieux and Thitimajshima published their landmark paper in 1993, different modified BCJR MAP algorithms have appeared in the literature. The existence of a relatively large number of similar but different modified BCJR MAP algorithms, derived using the Markov chain properties of convolutional codes, naturally leads to the following questions. What is the relationship among the different modified BCJR MAP algorithms? What are their relative performance, computational complexities, and memory requirements? In this paper, we answer these questions. We derive systematically four major modified BCJR MAP algorithms from the BCJR MAP algorithm using simple mathematical transformations. The connections between the original and the four modified BCJR MAP algorithms are established. A detailed analysis of the different modified BCJR MAP algorithms shows that they have identical computational complexities and memory requirements. Computer simulations demonstrate that the four modified BCJR MAP algorithms all have identical performance to the BCJR MAP algorithm. Copyright © 2006 S. Wang and F. Patenaude. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
1.
INTRODUCTION
In 1993, Berrou et al. [1] introduced new types of codes, called turbo codes, which have demonstrated performance close to the theoretical limit predicted by information theory [2]. In the iterative decoding strategy for turbo codes, a soft-input soft-output (SISO) MAP algorithm is used to perform the decoding operation for the two constituent recursive systematic convolutional codes (RSC). The SISO MAP algorithm presented in [1], which is called the BGT MAP algorithm in [3], is a modified version of the BCJR MAP algorithm proposed in [4]. The BGT MAP algorithm formally appears very complicated. Later, Pietrobon and Barbulescu derived a simpler modified BCJR MAP algorithm [5], which is called the PB MAP algorithm [3]. However, the PB MAP algorithm is not a direct simplification of the BGT MAP algorithm, even though they share similar structures. In [3], the BGT MAP algorithm is directly simplified to obtain a new modified BCJR MAP algorithm that keeps the structure of the BGT MAP algorithm but uses simpler recursive procedures. This new modified BCJR MAP algorithm is called the SBGT MAP algorithm in [3]. The main difference between the SBGT and BGT MAP algorithms lies in the fact
that for the BGT MAP algorithm, the forward and backward recursions (cf. [1, equations (21) and (22)]) are formulated in such a way
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