Convergence analysis of turbo decoding of serially concatenated block codes and product codes

The geometric interpretation of turbo decoding has founded a framework, and provided tools for the analysis of parallel-concatenated codes decoding. In this paper, we extend this analytical basis for the decoding of serially concatenated codes, and focus on serially concatenated product codes (SCPC) (i.e., product codes with checks on checks). For this case, at least one of the component (i.e., rows/columns) decoders should calculate the extrinsic information not only for the information bits, but also for the check bits. We refer to such a component decoder as a serial decoding module (SDM). We extend the framework accordingly and derive the update equations for a general turbo decoder of SCPC and the expressions for the main analysis tools: the Jacobian and stability matrices. We explore the stability of the SDM. Specifically, for high SNR, we prove that the maximal eigenvalue of the SDM's stability matrix approaches d - 1, where d is the minimum Hamming distance of the component code. Hence, for practical codes, the SDM is unstable. Further, we analyze the two turbo decoding schemes, proposed by Benedetto and Pyndiah, by deriving the corresponding update equations and by demonstrating the structure of their stability matrices for the repetition code and an SCPC code with 2×2 information bits. Simulation results for the Hamming [(7,4,3)]² and Golay [(24,12,8)]² codes are presented, analyzed, and compared to the theoretical results and to simulations of turbo decoding of parallel concatenation of the same codes.