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.

Distribution matching (DM) transforms independent and Bernoulli(1/2) distributed bits into a sequence of output symbols with a desired distribution. A fixed-to-fixed length, invertible DM architecture based on shell mapping is presented. It is shown that shell mapping for distribution matching (SMDM) is the optimum DM for the informational divergence metric and that finding energy optimal sequences is a special case of divergence minimization. Additionally, it is shown how to find the required shell mapping weight function to approximate arbitrary output distributions. SMDM is combined with probabilistic amplitude shaping (PAS) to operate close to the Shannon limit. SMDM exhibits excellent performance for short blocklengths as required by ultra-reliable low-latency (URLLC) applications. SMDM outperforms constant composition DM (CCDM) by 0.6dB when used with 64-QAM at a spectral efficiency of 3bits/channel use and a 5G low-density parity-check code with a short blocklength of 192bits.

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.

Probabilistic shaping for intensity modulation and direct detection (IM/DD) links is discussed and a peak power constraint determined by the limited modulation extinction ratio (ER) of optical modulators is introduced. The input distribution of 4-ary unipolar pulse amplitude modulation (PAM) symbols is optimized for short-reach transmission links without optical amplification nor in-line dispersion compensation. The resulting distribution is symmetric around its mean allowing to use probabilistic amplitude shaping (PAS) to generate symbols that are protected by forward error correction (FEC) and that have the optimal input distribution. The numerical analysis is confirmed experimentally for both an additive white Gaussian noise (AWGN) channel and a fiber channel, showing gains in transmission reach and transmission rate, as well as rate adaptability.

Fast probabilistic shaping is proposed using the probabilistic amplitude shaping (PAS) architecture to combine SC-LDPC codes with Streaming Distribution Matching (sDM). The parallelization factor of sDM is 100 times higher than for Constant Composition Distribution Matching (CCDM), at similar performance.

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.