  An energy-efficient approach is presented for shaping a bit-interleaved
low-density parity-check (LDPC) coded amplitude phase-shift keying (APSK)
system. A subset of the interleaved bits output by a binary LDPC encoder are
passed through a nonlinear shaping encoder whose output is more likely to be a
zero than a one. The "shaping" bits are used to select from among a plurality
of subconstellations, while the unshaped bits are used to select the symbol
within the subconstellation. Because the shaping bits are biased, symbols from
lower-energy subconstellations are selected more frequently than those from
higher-energy subconstellations. An iterative decoder shares information among
the LDPC decoder, APSK demapper, and shaping decoder. Information rates are
computed for a discrete set of APSK ring radii and shaping bit probabilities,
and the optimal combination of these parameters is identified for the additive
white Gaussian noise (AWGN) channel. With the assistance of
extrinsic-information transfer (EXIT) charts, the degree distributions of the
LDPC code are optimized for use with the shaped APSK constellation. Simulation
results show that the combination of shaping, degree-distribution optimization,
and iterative decoding can achieve a gain in excess of 1 dB in AWGN at a rate
of 3 bits/symbol compared with a system that does not use shaping, uses an
unoptimized code from the DVB-S2 standard, and does not iterate between decoder
and demodulator.
