Iterative decoding and graphical code representations

Citation

Xu, Meina (1999) Iterative decoding and graphical code representations. Dissertation (Ph.D.), California Institute of Technology. doi:10.7907/q731-9q50. https://resolver.caltech.edu/CaltechETD:etd-02262008-093428

Abstract

Since the invention of turbo codes, there has been an explosion of interest in iterative decoding and graphical representation of codes. This thesis examines the iterative decoding of codes defined on graphs with cycles, which appears to be an efficient means of achieving the Shannon limit. Much of this analysis is on the iterative min-sum decoding of tail-biting codes and cycle codes. We have identified the pseudocodeword as the cause of the suboptimal performance of the iterative decoder, and we have obtained a union bound for the performance of the iterative decoder on both AWGN and BSC channels. Using the union bound argument, for cycle codes, we have shown that the performance of the iterative decoder is asymptotically as good as that of the ML decoder. As for tail-biting codes, the same thing is true if the lowest weight pseudocodeword is at least the minimum weight of the code. Unfortunately, the analysis of tail-biting codes and cycle codes does not extend to turbo codes and low density parity check codes in general. Our next approach is to determine the average behavior of message passing algorithms by studying the evolution of their "message" densities. For the class of "repeat and accumulate" serially concatenated turbo-like codes, we have devised an algorithm for determining their "threshold" values. When the signal-to-noise ratio is larger than the threshold value, the error probabilities of message passing algorithms approach zero, whereas if the signal-to-noise ratio is less than the threshold value, the error probability stays bounded away from zero. Some message passing algorithms and graphical representation of codes are efficient means of devising ML or MAP decoding algorithms. We have proposed a junction tree representation for linear block codes, and we have shown that the minimum junction tree can be less complex than the minimal trellis.

Thesis Files