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For 200Gbit/s net rates, uniform PAM-4, 6 and 8 are experimentally compared against probabilistic shaped PAM-8 cap and cup variants. In back-to-back and 20km measurements, cap shaped 80GBd PAM-8 outperforms 72GBd PAM-8 and 83GBd PAM-6 by up to 3.50dB and 0.8dB in receiver sensitivity, respectively, Comment: Originally published in ECOC-2021. We have updated Figure 3. The change also affects the overall outcome. In contrast to the published version, compared to uniform PAM-8 72 GBd, PS-PAM-8 80 GBd performance is updated to 3.50 dB instead of 5.17 dB, while for PAM-6 83 GBd the gain becomes 0.8 dB instead of 2.17 dB. The changes are adapted in all sections except the experimental setup and DSP section

We report an industry leading optical dense wavelength division multiplexing (DWDM) field trial with line rates per channel exceeding 1.66 Tb/s using 130 GBaud dual-polarization probabilistic constellation shaping 256-ary quadrature amplitude modulation (DP-PCS256QAM) in a high capacity data center interconnect (DCI) scenario. This research trial was performed on 96.5 km of field-deployed standard single mode G.652 fiber infrastructure of Deutsche Telekom in Germany employing Erbium-doped fiber amplifier (EDFA)-only amplification. A total of 34 channels were transmitted with 150 GHz spacing for a total fiber capacity of 56.51 Tb/s and a spectral efficiency higher than 11bit/s/Hz. In the single-channel transmission scenario 1.71 Tb/s was achieved over the same link. In addition, we successfully demonstrate record net bitrates of 1.88 Tb/s in back-to-back (B2B) using 130 GBaud DP-PCS400QAM., Comment: This work has been submitted to the IEEE Photonics Technology Letters (PTL) for possible publication. Copyright may be transferred without notice, after which this version may no longer be accessible

Several applications in communication, control, and learning require approximating target distributions to within small informational divergence (I-divergence). The additional requirement of invertibility usually leads to using encoders that are one-to-one mappings, also known as distribution matchers. However, even the best one-to-one encoders have I-divergences that grow logarithmically with the block length in general. To improve performance, an encoder is proposed that has an invertible one-to-many mapping and a low-rate resolution code. Two algorithms are developed to design the mapping by assigning strings in either a most-likely first or least-likely first order. Both algorithms give information rates approaching the entropy of the target distribution with exponentially decreasing I-divergence and with vanishing resolution rate in the block length., Comment: 13 pages, 6 figures

The probabilistic shaping scheme from Honda and Yamamoto (2013) for polar codes is used to enable power-efficient signaling for on-off keying (OOK). As OOK has a non-symmetric optimal input distribution, shaping approaches that are based on the concatenation of a distribution matcher followed by systematic encoding do not result in optimal signaling. Instead, these approaches represent a time sharing scheme where only a fraction of the codeword symbols is shaped. The proposed scheme uses a polar code for joint distribution matching and forward error correction which enables asymptotically optimal signaling. Numerical simulations show a gain of 1.8 dB compared to uniform transmission at a spectral efficiency of 0.25 bits/channel use for a blocklength of 65,536 bits., Comment: accepted for publication in IEEE Communications Letters

An unsupervised learning approach based on expectation maximization is proposed to obtain the parameters of a soft decision forward error correction decoding metric for probabilistic shaping. The algorithm depends only on the channel observations and does not require transmitted data., Comment: Accepted for ECOC 2018 (Rome), Th1H.3

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., Comment: Invited paper at CISS 2018, special session "Optimal Communications with Discrete Input Signals"

The rate flexibility and probabilistic shaping gain of $4$-dimensional signaling is experimentally tested for short-reach, unrepeated transmission. A rate granularity of 0.5 bits/QAM symbol is achieved with a distribution matcher based on a simple look-up table., Comment: Presented at OFC'18, San Diego, CA, USA

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 0.6 dB when used with 64-QAM at a spectral efficiency of 3 bits/channel use and a 5G low-density parity-check code with a short blocklength of 192 bits, Comment: accepted for IEEE Wireless Communication Letters

Product distribution matching (PDM) is proposed to generate target distributions over large alphabets by combining the output of several parallel distribution matchers (DMs) with smaller output alphabets. The parallel architecture of PDM enables low-complexity and high-throughput implementation. PDM is used as a shaping device for probabilistic amplitude shaping (PAS). For 64-ASK and a spectral efficiency of 4.5 bits per channel use (bpcu), PDM is as power efficient as a single full-fledged DM. It is shown how PDM enables PAS for parallel channels present in multi-carrier systems like digital subscriber line (DSL) and orthogonal frequency-division multiplexing (OFDM). The key feature is that PDM shares the DMs for lower bit-levels among different sub-carriers, which improves the power efficiency significantly. A representative parallel channel example shows that PAS with PDM is 0.93 dB more power efficient than conventional uniform signaling and PDM is 0.35 dB more power efficient than individual per channel DMs., Comment: Including 7 diagrams and 5 figures with simulation results

Distribution matching is the process of invertibly mapping a uniformly distributed input sequence onto sequences that approximate the output of a desired discrete memoryless source. The special case of a binary output alphabet and one-to-one mapping is studied. A fixed-length distribution matcher is proposed that is optimal in the sense of minimizing the unnormalized informational divergence between its output distribution and a binary memoryless target distribution. Upper and lower bounds on the unnormalized divergence are computed that increase logarithmically in the output block length $n$. It follows that a recently proposed constant composition distribution matcher performs within a constant gap of the minimal achievable informational divergence., Comment: 5 pages, 1 figure; Lemma 6 updated; This work will be presented at ISIT 2017

In this work, the design of robust, protograph-based low-density parity-check (LDPC) codes for rate-adaptive communication via probabilistic shaping is considered. Recently, probabilistic amplitude shaping (PAS) by B\"ocherer et al. has been introduced for capacity approaching and rate-adaptive communication with a bitwise-demapper and binary decoder. Previous work by the authors considered the optimization of protograph based LDPC codes for PAS and specific spectral efficiencies (SEs) to jointly optimize the LDPC code node degrees and the mapping of the coded bits to the bit-interleaved coded modulation (BICM) bit-channels. We show that these codes tend to perform poor when operated at other rates and propose the design of robust LDPC codes by employing a min-max approach in the search for good protograph ensembles via differential evolution. The considered design uses a single 16 amplitude-shift-keying (ASK) constellation and a robust 13/16 rate LDPC code to operate between 0.7 to 2.7 bits per channel use. For a blocklength of 16224 bits and a target frame error rate of 1e-3 the proposed code operates within 1.32 dB of continuous AWGN capacity for 0.7 to 1.3 bpcu and within 1.05 dB for 1.3 bpcu to 2.7 bpcu., Comment: Accepted for Publication in the Proceedings of 2016 9th International Symposium on Turbo Codes and Iterative Information Processing (ISTC), Invited paper for the special session on "Recent Advances in Coding for Higher Order Modulation"

We implemented a flexible transmission system operating at adjustable data rate and fixed bandwidth, baudrate, constellation and overhead using probabilistic shaping. We demonstrated in a transmission experiment up to 15% capacity and 43% reach increase versus 200 Gbit/s 16-QAM., Comment: Accepted for presentation as postdeadline paper at ECOC 2015

Distribution matching transforms independent and Bernoulli(1/2) distributed input bits into a sequence of output symbols with a desired distribution. Fixed-to-fixed length, invertible, and low complexity encoders and decoders based on constant composition and arithmetic coding are presented. Asymptotically in the blocklength, the encoder achieves the maximum rate, namely the entropy of the desired distribution. Furthermore, the normalized divergence of the encoder output and the desired distribution goes to zero in the blocklength., Comment: 5 pages, 5 figures

A new coded modulation scheme is proposed. At the transmitter, the concatenation of a distribution matcher and a systematic binary encoder performs probabilistic signal shaping and channel coding. At the receiver, the output of a bitwise demapper is fed to a binary decoder. No iterative demapping is performed. Rate adaption is achieved by adjusting the input distribution and the transmission power. The scheme is applied to bipolar amplitude shift keying (ASK) constellations with equidistant signal points and it is directly applicable to two-dimensional quadrature amplitude modulation (QAM). The scheme is implemented by using the DVB-S2 low-density parity-check (LDPC) codes. At a frame error rate of 1e-3, the new scheme operates within less than 1 dB of the AWGN capacity 0.5log2(1+SNR) at any spectral efficiency between 1 and 5 bits/s/Hz by using only 5 modes, i.e., 4-ASK with code rate 2/3, 8-ASK with 3/4, 16-ASK and 32-ASK with 5/6 and 64-ASK with 9/10., Comment: 13 pages, 11 figures, 10 tables