Implementing the Belief Propagation Algorithm in MATLAB

For an excellent introduction and mathematical treatise of modern iterative decoding theory, we refer to [4]. Worth mentioning is also the survey paper [2] on factor graphs and the sum-product algorithm, the superclass that contains belief propagation. This current technical note provides Matlab code to implement the dynamical system formulation of the belief propagation algorithm and a few related concepts, as detailed in [6]. More conventional implementations —that is, from a coding perspective— exist and some are publicly available [3]. The Matlab code examples detailed in this report can be found, along with the most up-to-date version of this report itself, at [5]. Our presentation differs also in another aspect from the standard ones: Unlike the information theory convention, where messages and codewords are represented by row vectors, we throughout use column vectors as this is standard in dynamical systems. Of course this does not lead to differences other than representational ones. This report is organized as follows: In Section 2 we give a simple example on how one can generate a very basic random parity-check matrix and compute a corresponding generator matrix. Section 3 details the channel transmission and Section 4 provides code to implement the belief propagation algorithm as a dynamical system. The output trajectories obtained using this Matlab code can then be plotted using the routine in