On Some Properties of the Mutual Information Between Extrinsics With Application to Iterative Decoding

Iterative decoding is an efficient error-correction tool based on the exchange of extrinsic probabilities between the constituent decoders. In this paper, the properties of the mutual information between the extrinsic LLR at the output of two constituent decoders are analyzed with application to turbo and LDPC codes. This is a bridge between information-theoretic analysis and practical implementations. It is proved here that the mutual information between extrinsics is a lower bound of the mutual information between each extrinsic and the transmitted message. In addition, an efficient online evaluation is provided in the paper with accuracy validated through numerical experiments. As an application, the mutual information between extrinsics is used for designing efficient stopping criterion and error detection rules at the decoder side. Two online methods for the estimation of optimal scaling factor to be applied to the extrinsic LLR are also derived. In contrast with most references, an analytical expression is obtained that does not require estimation of the actual transmitted bits. All results in the paper are derived for Gaussian distributed LLR with independent mean and variance.