A Note on Probabilistic and Geometric Shaping for the AWGN Channel

When considering coded modulation schemes for the AWGN channel, two main practical limitations prevent achieving channel capacity, namely the need to use a finite constellation and coding inefficiencies. Constellation shaping was given new impetus in recent works by Böcherer et al., which combined probabilistically shaped ASK constellations with LDPC coding. One open question is how far their results are from optimal solutions based on the use of finite constellations. We investigate this issue and show through experimental analysis that most of the loss is due to inefficient coding design and little benefits should be expected by adding two-dimensional geometric shaping.