The distribution of loop lengths in graphical models for turbo decoding

This correspondence analyzes the distribution of loop lengths in graphical models for turbo decoding. The properties of such loops are of significant interest in the context of iterative decoding algorithms based on belief pr opagation. We estimate the probability that there exist no loops of length less than or equal to c at a randomly chosen node in the acyclic directed g raphical (ADG) model for turbo decoding, using a combination of counting ar guments and approximations. When K, the number of information bits, is larg e, this probability is approximately e(-)2(c-1)4/K, for c greater than or e qual to 4, where nodes for input information bits are Ignored for convenien ce. The analytical results are validated by simulations. For example, for t urbo codes with K = 64 000, a randomly chosen node has a less than 1 % chan ce of being on a loop of length less than or equal to 10, but has a greater than 99.9 % chance of being on a loop of length less than or equal to 20.