Multi-stage Decoding for Multi-level Block Modulation Codes

In this paper, we investigate various types of multi-stage decoding for multi-level block modulation codes, in which the decoding of a component code at each stage can be either soft-decision or hard-decision, maximum likelihood or bounded-distance. Error performance of codes is analyzed for a memoryless additive channel based on various types of multi-stage decoding, and upper bounds on the probability of an incorrect decoding are derived. Based on our study and computation results, we find that, if component codes of a multi-level modulation code and types of decoding at various stages are chosen properly, high spectral efficiency and large coding gain can be achieved with reduced decoding complexity. In particular, we find that the difference in performance between the suboptimum multi-stage soft-decision maximum likelihood decoding of a modulation code and the single-stage optimum decoding of the overall code is very small, only a fraction of dB loss in SNR at the probability of an incorrect decoding for a block of 10~ . Multi-stage decoding of multi-level modulation codes really offers a way to achieve the best of three worlds, bandwidth efficiency, coding gain and decoding complexity.