The distribution of cycle lengths in graphical models for iterative decoding

This paper analyzes the distribution of cycle lengths in turbo-decoding graphs. The properties of such cycles are of significant interest since it is well-known that iterative decoding can only be proven to converge to the correct posterior bit probabilities (and bit decisions) for graphs without cycles. We estimate the probability that there exist no simple cycles of length less than or equal to k at a randomly chosen node in the graph using a combination of counting arguments and independence assumptions. Simulation results validate the accuracy of the various approximations. For example, for a block length of 64000 a randomly chosen node has a less than 1 % chance of being on a cycle of length less than or equal to 10, but has a greater than 99.9 % chance of being on a cycle of length less than or equal to 20. The effect of the “S-random ” permutation is also analyzed and it is shown that while it eliminates short cycles of length k < 8, it does not significantly affect the overall distribution of cycle lengths. The paper concludes by commenting briefly on how these results may provide insight into the practical success of iterative decoding methods.