Capacity and error exponent analysis of multilevel coding with multistage decoding

The capacity and random coding error exponent of multilevel coding (MLC) with multistage decoding (MSD) are analyzed. General discrete memoryless channels with arbitrary input distributions are considered. The capacities of MLC with maximum likelihood decoding and with MSD are calculated, and it is shown that using MLC may result in loss in the achievable rate for reliable communication. Necessary and sufficient conditions for the rate loss to be zero are derived. A new random coding error exponent is derived for MLC with MSD. For the special case of uniform inputs, the new error exponent can be easily calculated by its inverse function, which is given by the sum of the inverse error exponents of the conditional sub-channels.