Different aspects of probabilistic shaping for a multi-span optical communication system are studied. First, a numerical analysis of the additive white Gaussian noise (AWGN) channel investigates the effect of using a small number of input probability mass functions (PMFs) for a range of signal-to-noise ratios (SNRs), instead of optimizing the constellation shaping for each SNR. It is shown that if a small penalty of at most 0.1 dB SNR to the full shaping gain is acceptable, just two shaped PMFs are required per quadrature amplitude modulation (QAM) over a large SNR range. For a multi-span wavelength division multiplexing (WDM) optical fiber system with 64QAM input, it is shown that just one PMF is required to achieve large gains over uniform input for distances from 1,400 km to 3,000 km. Using recently developed theoretical models that extend the Gaussian noise (GN) model and full-field split-step simulations, we illustrate the ramifications of probabilistic shaping on the effective SNR after fiber propagation. Our results show that, for a fixed average optical launch power, a shaping gain is obtained for the noise contributions from fiber amplifiers and modulation-independent nonlinear interference (NLI), whereas shaping simultaneously causes a penalty as it leads to an increased NLI. However, this nonlinear shaping loss is found to have a relatively minor impact, and optimizing the shaped PMF with a modulation-dependent GN model confirms that the PMF found for AWGN is also a good choice for a multi-span fiber system.
An LDPC coded modulation scheme with probabilistic shaping, optimized interleavers and noniterative demapping is proposed. Full-field simulations show an increase in transmission distance by 8% compared to uniformly distributed input.
The interplay of shaped signaling and fiber nonlinearities is reviewed in the asymptotic and finite-length regime. We present explanations and discuss implications of an optimum shaping length of just a few hundred symbols.
A new geometric shaping method is proposed, leveraging unsupervised machine learning to optimize the constellation design. The learned constellation mitigates nonlinear effects with gains up to 0.13 bit/4D when trained with a simplified fiber channel model.
Probabilistic amplitude shaping (PAS) can flexibly vary the spectral efficiency (SE) of fiber-optic systems. In this paper, we demonstrate the application of PAS to bit-wise hard decision decoding (HDD) of product codes (PCs) by finding the necessary conditions to select the PC component codes. We show that PAS with PCs and HDD yields gains up to $2.7$ dB and SE improvement up to approximately $1$ bit/channel use compared to using PCs with uniform signaling and HDD. Furthermore, we employ the recently introduced iterative bounded distance decoding with combined reliability of PCs to improve pe…
This paper studies the impact of probabilistic shaping on effective signal-to-noise ratios (SNRs) and achievable information rates (AIRs) in a back-to-back configuration and in unrepeated nonlinear fiber transmissions. For back-to-back, various shaped quadrature amplitude modulation (QAM) distributions are found to have the same implementation penalty as uniform input. By demonstrating in transmission experiments that shaped QAM input leads to lower effective SNR than uniform input at a fixed average launch power, we experimentally confirm that shaping enhances the fiber nonlinearities. Howeve…
We generalize probabilistic amplitude shaping (PAS) with binary codes to the case of non-binary codes defined over prime finite fields. Firstly, we introduce probabilistic shaping via time sharing where shaping applies to information symbols only. Then, we design circular quadrature amplitude modulations (CQAM) that allow to directly generalize PAS to prime finite fields with full shaping.
The geometry of dual-polarization four-dimensional constellations is optimized in the optical fiber channel using a recent nonlinear interference model. A 0.27 bit/4D rate gain and 13% reach increase are attained compared to polarization-multiplexed formats.
We show that a 0.9 dB SNR improvement can be obtained via short-blocklength enumerative sphere shaping for single-span transmission at 56 GBd. This gain vanishes for higher symbol rates and a larger number of spans.
A new probabilistic shaping distribution that outperforms Maxwell-Boltzmann is studied for the nonlinear fiber channel. Additional gains of 0.1 bit/symbol MI or 0.2 dB SNR for both DP-256QAM and DP-1024QAM are reported after 200 km nonlinear fiber transmission.
We transmit probabilistic enumerative sphere shaped dual-polarization 64-QAM at 350Gbit/s/channel over 1610km SSMF using a short blocklength of 200. A reach increase of 15% over constant composition distribution matching with identical blocklength is demonstrated.
We show that separating the in-phase and quadrature component in optimized, machine-learning based demappers of optical communications systems with geometric constellation shaping reduces the required computational complexity whilst retaining their good performance.
In this paper, an unsupervised machine learning method for geometric constellation shaping is investigated. By embedding a differentiable fiber channel model within two neural networks, the learning algorithm is optimizing for a geometric constellation shape. The learned constellations yield improved performance to state-of-the-art geometrically shaped constellations, and include an implicit trade-off between amplification noise and nonlinear effects. Further, the method allows joint optimization of system parameters, such as the optimal launch power, simultaneously with the constellation shap…
Constellation shaping is a practical and effective technique to improve the performance and the rate adaptivity of optical communication systems. In principle, it could also be used to mitigate the impact of nonlinear effects, possibly increasing the information rate beyond the current limit dictated by fiber nonlinearity. However, this appealing idea is frustrated by the difficulty of designing an effective shaping strategy that takes into account the nonlinearity and long memory of the fiber channel, as well as the possible interplay with other nonlinearity mitigation strategies. As a result…
Probabilistic shaping of quadrature amplitude modulation (QAM) is used to enhance the sensitivity of an optical communication system. Sensitivity gains of 0.43 dB and 0.8 dB are demonstrated in back-to-back experiments by shaping of 16QAM and 64QAM, respectively. Further, numerical simulations are used to prove the robustness of probabilistic shaping to a mismatch between the constellation used and the signal-to-noise ratio (SNR) of the channel. It is found that, accepting a 0.1 dB SNR penalty, only four shaping distributions are required to support these gains for 64QAM.
Probabilistic shaping based on constant composition distribution matching (CCDM) has received considerable attention as a way to increase the capacity of fiber optical communication systems. CCDM suffers from significant rate loss at short blocklengths and requires long blocklengths to achieve high shaping gain, which makes its implementation very challenging. In this paper, we propose to use enumerative sphere shaping (ESS) and investigate its performance for the nonlinear fiber optical channel. ESS has lower rate loss than CCDM at the same shaping rate, which makes it a suitable candidate to…
Pulse shaping for coherent optical fiber communication has been an active area of research for the past decade. Most of the early schemes are based on classic Nyquist pulse shaping that was originally intended for linear channels. The best known classic scheme, the split digital back-propagation (DBP), uses joint pre-distortion and post equalization and hence, a nonlinear transmitter (TX); it, however, suffers from spectral broadening on the fiber due to the Kerr-effect. With the advent of deep learning in communications, it has been realized that an Autoencoder can learn to communicate effici…
GFDM is a new modulation format whose advantages compared to OFDM reportedly make it a preferable modulation format for 5G. However, the non-orthogonal nature of GFDM with matched filtering (MF) receiver for pulses with good time-frequency localization is one of its disadvantages, leading to the proposal of WCP-COQAM, employing offset quadrature amplitude modulation (OQAM). In this paper, we prove that a pulse satisfying orthogonality conditions for OQAM-OFDM will also satisfy orthogonality with WCP-COQAM, thus the pulse design methods developed for OQAM-OFDM can also be used with WCP-COQAM. T…
We propose two novel techniques to implement sequence selection (SS) for fiber nonlinearity mitigation, demonstrating a nonlinear shaping gain of 0.24 bits/s/Hz, just 0.1 bits/s/Hz below the SS capacity lower bound.
Quadrature amplitude modulation (QAM), deployed in billions of communication devises, exhibits a shaping-loss of $π\mathrm{e}/6$ ($\approx 1.53$ dB) compared to the Shannon-Hartley theorem. With inspiration gained from special (leaf, flower petal, and seed) packing arrangements (so called spiral phyllotaxis) found among plants, we have designed a shape-versatile, circular symmetric, modulation scheme, \textit{the Golden angle modulation (GAM)}. Geometric- and probabilistic-shaping-based GAM schemes are designed that practically overcome the shaping-loss of 1.53 dB. Specifically, we consider mu…
Discrete-modulation continuous-variable quantum key distribution with probabilistic amplitude shaping over a linear quantum channel
Scuola Superiore Sant’Anna（圣安娜高等学院）；National Laboratory of Photonic Networks, CNIT（光子网络国家实验室，CNIT）；Palacký University（帕拉茨基大学）；Università degli Studi di Milano（米兰大学）；INFN, Sezione di Milano（意大利国家核物理研究所，米兰分部）
Lower bounds on mutual information (MI) of long-haul optical fiber systems for hard-decision and soft-decision decoding are studied. Ready-to-use expressions to calculate the MI are presented. Extensive numerical simulations are used to quantify how changes in the optical transmitter, receiver, and channel affect the achievable transmission rates of the system. Special emphasis is put to the use of different quadrature amplitude modulation formats, channel spacings, digital back-propagation schemes and probabilistic shaping. The advantages of using MI over the prevailing $Q$-factor as a figure…
Signal shaping is vital to approach Shannon's capacity, yet it is challenging to implement at very high speeds. For example, probabilistic shaping often requires arithmetic coding to realize the target distribution. Geometric shaping requires look-up tables to store the constellation points. In this paper, we propose a four-dimensional amplitude coding (4D-AC) geometrical shaper architecture. The proposed architecture can generate in real time geometrically shaped 4D formats via simple logic circuit operations and two conventional quadrature amplitude modulation (QAM) modulators. This paper de…
Machine learning is used to compute achievable information rates (AIRs) for a simplified fiber channel. The approach jointly optimizes the input distribution (constellation shaping) and the auxiliary channel distribution to compute AIRs without explicit channel knowledge in an end-to-end fashion.
In optical communication systems, short blocklength probabilistic enumerative sphere shaping (ESS) provides both linear shaping gain and nonlinear tolerance. In this work, we investigate the performance and complexity of ESS in comparison with fiber nonlinearity compensation via digital back propagation (DBP) with different steps per span. We evaluate the impact of the shaping blocklength in terms of nonlinear tolerance and also consider the case of ESS with a Volterra-based nonlinear equalizer (VNLE), which provides lower complexity than DBP. In single-channel transmission, ESS with VNLE achi…
In this work, we construct energy-efficient variable-to-fixed length (V2F), fixed-to-variable length (F2V), and variable-to-variable length (V2V) prefix-free codes, which are optimal (or near-optimal) in the sense that no (or few) other codes with the size can achieve a smaller energy per code letter for the same entropy rate. Under stringent constraints of 4096 entries or below per codebook, the constructed codes yield an energy per code letter within a few tenths of a dB of the unconstrained theoretic lower bound, across a wide range of entropy rates with a very fine granularity. We also pro…
In this paper, we propose a geometric shaping (GS) strategy to design 8, 16, 32 and 64-ary modulation formats for the optical fibre channel impaired by both additive white Gaussian (AWGN) and phase noise. The constellations were optimised to maximise generalised mutual information (GMI) using a mismatched channel model. The presented formats demonstrate an enhanced signal-to-noise ratio (SNR) tolerance in high phase noise regimes when compared with their quadrature amplitude modulation (QAM) or AWGN-optimised counterparts. By putting the optimisation results in the context of the 400ZR impleme…
We consider probabilistic amplitude shaping (PAS) as a means of increasing the spectral efficiency of fiber-optic communication systems. In contrast to previous works in the literature, we consider probabilistic shaping with hard decision decoding (HDD). In particular, we apply the PAS recently introduced by Böcherer \emph{et al.} to a coded modulation (CM) scheme with bit-wise HDD that uses a staircase code as the forward error correction code. We show that the CM scheme with PAS and staircase codes yields significant gains in spectral efficiency with respect to the baseline scheme using a st…
A pragmatic coded modulation system is presented that incorporates signal shaping and exploits the excellent performance and efficient high-speed decoding architecture of staircase codes. Reliable communication within 0.62 bits/s/Hz of the estimated capacity (per polarization) of a system with L=2000 km is provided by the proposed system, with an error floor below 1E-20. Also, it is shown that digital backpropagation increases the achievable spectral efficiencies---relative to linear equalization---by 0.55 to 0.75 bits/s/Hz per polarization.
We introduce a trainable coded modulation scheme that enables joint optimization of the bit-wise mutual information (BMI) through probabilistic shaping, geometric shaping, bit labeling, and demapping for a specific channel model and for a wide range of signal-to-noise ratios (SNRs). Compared to probabilistic amplitude shaping (PAS), the proposed approach is not restricted to symmetric probability distributions, can be optimized for any channel model, and works with any code rate $k/m$, $m$ being the number of bits per channel use and $k$ an integer within the range from $1$ to $m-1$. The propo…
Hamburg University of Technology（汉堡工业大学）；Nokia Bell Labs（诺基亚贝尔实验室）
The choice of constellations largely affects the performance of communication systems. When designing constellations, both the locations and probability of occurrence of the points can be optimized. These approaches are referred to as geometric and probabilistic shaping, respectively. Usually, the geometry of the constellation is fixed, e.g., quadrature amplitude modulation (QAM) is used. In such cases, the achievable information rate can still be improved by probabilistic shaping. In this work, we show how autoencoders can be leveraged to perform probabilistic shaping of constellations. We de…
In this paper, achievable information rates (AIR) for fiber optical communications are discussed. It is shown that AIRs such as the mutual information and generalized mutual information are good design metrics for coded optical systems. The theoretical predictions of AIRs are compared to the performance of modern codes including low-parity density check (LDPC) and polar codes. Two different computation methods for these AIRs are also discussed: Monte-Carlo integration and Gauss-Hermite quadrature. Closed-form ready-to-use approximations for such computations are provided for arbitrary constell…
Quadrature amplitude modulation (QAM) exhibits a shaping-loss of $π\mathrm{e}/6$, ($\approx1.53$ dB) compared to the AWGN Shannon capacity. With inspiration gained from special (leaf, flower petal, and seed) packing arrangements (spiral phyllotaxis) found among plants, a novel, shape-versatile, circular symmetric, modulation scheme, the Golden Angle Modulation (GAM) is introduced. Disc-shaped, and complex Gaussian approximating bell-shaped, GAM-signal constellations are considered. For bell-GAM, a high-rate approximation, and a mutual information optimization formulation, are developed. Bell-G…
Capacity is the eternal pursuit for communication systems due to the overwhelming demand of bandwidth hungry applications. As the backbone infrastructure of modern communication networks, the optical fiber transmission system undergoes a significant capacity growth over decades by exploiting available physical dimensions (time, frequency, quadrature, polarization and space) of the optical carrier for multiplexing. For each dimension, stringent orthogonality must be guaranteed for perfect separation of independent multiplexed signals. To catch up with the ever-increasing capacity requirement, i…
The stationary statistical properties of independent, identically distributed (i.i.d.) input symbols provide insights on the induced nonlinear interference (NLI) during fiber transmission. For example, kurtosis is known to predict the modulation format-dependent NLI. These statistical properties can be used in the design of probabilistic amplitude shaping (PAS), which is a popular scheme that relies on an amplitude shaper for increasing spectral efficiencies of fiber-optic systems. One property of certain shapers used in PAS -- including constant-composition distribution matchers -- that is of…
In this work, targeting, e.g., future generation cellular, microwave-links, or optical fiber systems, we propose a new geometric shaping design for golden angle modulation (GAM) based on a (double) truncated Gaussian input distribution. The design improves the mutual information (MI), and the peak-to-average power ratio, over the full signal-to-noise ratio (SNR) range relative to two key GAM schemes introduced in [1],[2]. Inspired by the proposed geometric shaping, a simpler, SNR-dependent, design is also suggested. The performance is numerically evaluated with respect to MI and compared with …
In this paper, probabilistic shaping is numerically and experimentally investigated for increasing the transmission reach of wavelength division multiplexed (WDM) optical communication system employing quadrature amplitude modulation (QAM). An optimized probability mass function (PMF) of the QAM symbols is first found from a modified Blahut-Arimoto algorithm for the optical channel. A turbo coded bit interleaved coded modulation system is then applied, which relies on many-to-one labeling to achieve the desired PMF, thereby achieving shaping gain. Pilot symbols at rate at most 2% are used for …
In this Letter we analyze the benefit of digital compensation of fiber nonlinearity, where the digital signal processing is divided between the transmitter and receiver. The application of the Gaussian noise model indicates that, where there are two or more spans, it is always beneficial to split the nonlinearity compensation. The theory is verified via numerical simulations, investigating transmission of single channel 50 GBd polarization division multiplexed 256-ary quadrature amplitude modulation over 100 km standard single mode fiber spans, using lumped amplification. For this case, the ad…
Nokia（诺基亚）；Aston University（阿斯顿大学）；Delft University of Technology（代尔夫特理工大学）
We present a novel end-to-end autoencoder-based learning for coherent optical communications using a "parallelizable" perturbative channel model. We jointly optimized constellation shaping and nonlinear pre-emphasis achieving mutual information gain of 0.18 bits/sym./pol. simulating 64 GBd dual-polarization single-channel transmission over 30x80 km G.652 SMF link with EDFAs.
In order to realize probabilistically shaped signaling within the probabilistic amplitude shaping (PAS) framework, a shaping device outputs sequences that follow a certain nonuniform distribution. In case of constant-composition (CC) distribution matching (CCDM), the sequences differ only in the ordering of their constituent symbols, whereas the number of occurrences of each symbol is constant in every output block. Recent results by Amari \textit{et al.} have shown that the CCDM block length can have a considerable impact on the effective signal-to-noise ratio (SNR) after fiber transmission. …
The nondispersive per-sample channel model for the optical fiber channel is considered. Under certain smoothness assumptions, the problem of finding the minimum amount of noise energy that can render two different input points indistinguishable is formulated. This minimum noise energy is then taken as a measure of distance between the points in the input alphabet. Using the machinery of optimal control theory, necessary conditions that describe the minimum-energy noise trajectories are stated as a system of nonlinear differential equations. It is shown how to find the distance between two inpu…
We propose a new four-dimensional orthant-symmetric 128-ary modulation format (4D-OS128) with a spectral efficiency of 7bit/4D-sym. The proposed format fills the gap between polarization-multiplexed 8- and 16-ary quadrature-amplitude modulation (PM-8QAM and PM-16QAM).Numerical simulations show that 4D-OS128 outperforms two well-studied 4D geometrically-shaped modulation formats: 128SP-16QAM and 7b4D-2A8PSK by up to 0.65dB for bit-interleaved coded modulation at the same spectral efficiency. These gains are experimentally demonstrated in a 11$\times$233Gbit/s wavelength division multiplexing (W…
In wireless communication systems, there are many stages for signal transmission. Among them, mapping and demapping convert a sequence of bits into a sequence of complex numbers and vice versa. This operation is performed by a system of constellations~ -- by a set of labeled points on the complex plane. Usually, the geometry of the constellation is fixed, and constellation points are uniformly spaced, e.g., the same quadrature amplitude modulation (QAM) is used in a wide range of signal-to-noise ratio (SNR). By eliminating the uniformity of constellations, it is possible to achieve greater val…
