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Parallel, additive white Gaussian noise (AWGN) channels with an average sumpower constraint are considered. It is shown how the waterfilling Shannoncapacity can be approached by higher order modulation and probabilisticamplitude shaping (PAS). This is achieved by a new distribution matchingapproach called product distribution matching (PDM). The asymptotic performanceof PDM is analyzed by achievable rates. A heuristic for optimizing the inputdistribution is proposed, which enables signaling at a target spectralefficiency with a fixed-rate forward error correction (FEC) code, while theoptimal power allocation is ensured by mercury-waterfilling and a simplebit-loading strategy. Finite blocklength simulation results with 5G low-densityparity-check codes show power savings of around 1 dB compared to a conventionalscheme with uniform input distributions.

Approaching Waterfilling Capacity of Parallel Channels by Higher Order Modulation and Probabilistic Amplitude Shaping