This tutorial paper provides a foundation for integrating probabilistic shaping (PS) and forward error correction (FEC). A layered PS-FEC architecture consisting of a PS encoder and an FEC encoder is introduced, of which probabilistic amplitude shaping is a practical instance. Achievable PS encoding rates and achievable FEC decoding rates are derived using information-theoretic arguments. The developed tools are applied to the design and performance assessment of optical transponders based on measurements from optical transmission experiments.

The capacity-achieving input distribution of the discrete-time, additive white Gaussian noise (AWGN) channel with an amplitude constraint is discrete and seems difficult to characterize explicitly. A dual capacity expression is used to derive analytic capacity upper bounds for scalar and vector AWGN channels. The scalar bound improves on McKellips' bound and is within 0.1 bit of capacity for all signal-to-noise ratios (SNRs). The 2-D bound is within 0.15 bits of capacity provably up to 4.5 dB; numerical evidence suggests a similar gap for all SNRs. As the SNR tends to infinity, these bounds are accurate and match with a volume-based lower bound. For the 2-D complex case, an analytic lower bound is derived by using a concentric constellation and is shown to be within 1 bit of capacity.

Finite precision approximations of discrete probability distributions are considered, applicable for distribution synthesis, e.g., probabilistic shaping. Two algorithms are presented that find the optimal M-type approximation Q of a distribution P in terms of the variational distance II Q - PII ₁ and the informational divergence D( QIIP). Bounds on the approximation errors are derived and shown to be asymptotically tight. Several examples illustrate that the variational distance optimal approximation can be quite different from the informational divergence optimal approximation.

We successfully tested the suitability of probabilistically shaped constellations in a German nationwide fiber ring of the R&D field test network of Deutsche Telekom. In this ring, eight German cities can be reached by add/drop nodes implemented by passive optical add/drop multiplexers, offering 12 bands of 400 GHz widths in the extended C-band. Multiple cascaded add/drop multiplexers are reducing the available bandwidth. We transmitted a 1 Tb/s four-carrier super-channel with an extended family of probabilistically shaped constellations using 16QAM, 36QAM, and 64QAM and variable bandwidth. We verified in the field environment by two examples that probabilistic shaping is outperforming standard 64QAM and 16QAM formats in reach. We experimentally tested two typical 1 Tb/s working paths with 419 and 951 km, demonstrating spectral efficiencies of 6.75 and 5 bit/s/Hz, respectively. The maximum reach of about 1500 km was extrapolated for a 1 Tb/s probabilistically shaped super-channel transmitted in a 200 GHz bandwidth. For protection paths with longer distances than 1500 km, the spectral efficiency has to be reduced with the net-bitrate in steps of 100 Gb/s. For the longest bidirectional path of the test network, i.e., a protection path of 2159 km, the maximum achieved net-bitrate was 800 Gb/s.

High speed optical transmission systems suffer from intersymbol interference (ISI) and colored noise induced by nonlinear bandwidth limited optical and electrical components. As a countermeasure, this article investigates deep neural network soft-demappers. In particular, we propose a bidirectional recurrent neural network soft-demapper (BRNN-SD) and benchmarked its performance against a time delay neural network soft-demapper (TDNN-SD) and a reference digital signal processing (DSP) scheme consisting of a Volterra nonlinear equalizer accompanied by a symbol-spaced whitening filter and a BCJR detector. On coherent 92GBd dual polarization (DP)-32QAM back-to-back measurements, the proposed soft-demapper matches the performance of the reference DSP. In 800 Gb/s 96GBd DP-32QAM 32-channel dense wavelength division multiplexing (DWDM) transmission over a 600 km fiber link the proposed approach outperforms the reference DSP.

A new approach for combining non-binary lowdensity parity-check (NB-LDPC) codes with higher-order modulation and probabilistic amplitude shaping (PAS) is presented. Instead of symbol-metric decoding (SMD), a bit-metric decoder (BMD) is used so that matching the field order of the nonbinary code to the constellation size is not needed, which increases the flexibility of the coding scheme. Information rates, density evolution thresholds and finite-length simulations show that the flexibility comes at no loss of performance if PAS is used.