  Parallel, additive white Gaussian noise (AWGN) channels with an average sum
power constraint are considered. It is shown how the waterfilling Shannon
capacity can be approached by higher order modulation and probabilistic
amplitude shaping (PAS). This is achieved by a new distribution matching
approach called product distribution matching (PDM). The asymptotic performance
of PDM is analyzed by achievable rates. A heuristic for optimizing the input
distribution is proposed, which enables signaling at a target spectral
efficiency with a fixed-rate forward error correction (FEC) code, while the
optimal power allocation is ensured by mercury-waterfilling and a simple
bit-loading strategy. Finite blocklength simulation results with 5G low-density
parity-check codes show power savings of around 1 dB compared to a conventional
scheme with uniform input distributions.
