Multi-level modulation codes and multi-stage decoding

Multi-level method is a powerful technique for constructing bandwidth efficient modulation codes. It allows the construction of modulation codes systematically with arbitrary large minimum squared Euclidean distance from component codes in conjunction with proper bits-to-signal mapping. If the component codes are chosen properly, the resultant modulation code not only has good minimum squared Euclidean distance but is also rich in structural properties such as: linear structure, phase invariant property, and trellis structure. A modulation code with linear structure has invariant distance distribution. Phase invariant property is useful in resolving carrier-phase ambiguity and ensuring rapid carrier-phase resynchronization after temporary loss of synchronization. It the component codes have trellis structure, the resultant multi-level modulation code also has trellis structure. Trellis structure allows decoding of a multi-level modulation code with the soft-decision Viterbi decoding algorithm. Furthermore, the multi-level structure allows decoding of a multi-level modulation code with the multi-stage decoding. This type of decoding reduces the decoding complexity. Multi-stage decoding is not optimum even though the decoding of each component is optimum. Based on the analysis and simulation results, the difference in error performance between the optimum decoding of the overall multi-level modulation code and the suboptimum multi-stage decoding of the code is very little, a fraction of dB loss.