Closing the gap to the capacity of APSK: Constellation shaping and degree distributions

Constellation shaping is an energy-efficient strategy involving the transmission of lower-energy signals more frequently than higher-energy signals. Previous work has shown that shaping is particularly effective when used with coded amplitude phase-shift keying (APSK), a modulation that has been popularized recently due to its inclusion in the DVB-S2 standard. While shaped APSK can provide significant gains when used with standard off-the-shelf LDPC codes, such as the codes in the DVB-S2 standard, additional non-negligible gains can be achieved by optimizing the LDPC code with respect to the shaped APSK modulation. In this paper, we optimize the degree distributions of the LDPC code used in conjunction with shaped APSK. The optimization process is an extension of the EXIT-chart technique of ten Brink, et al., which has been adapted to account for the shaped APSK modulation. We begin by constraining the code to have the same number of distinct variable-node degrees as the codes in the DVB-S2 standard, and show that the optimization provides 32-APSK systems with an additional coding gain of 0.34 dB at a system rate of R = 3 bits per symbol, compared to shaped systems that use the long LDPC code from the DVB-S2 standard. We then increase the number of allowed variable node degrees by one, and find that an additional 0.1 dB gain is achievable.