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38 pages, 549 KiB

Open AccessArticle

Extending Fibre Nonlinear Interference Power Modelling to Account for General Dual-Polarisation 4D Modulation Formats

by Gabriele Liga, Astrid Barreiro, Hami Rabbani and Alex Alvarado

Entropy 2020, 22(11), 1324; https://0-doi-org.brum.beds.ac.uk/10.3390/e22111324 - 20 Nov 2020

Cited by 20 | Viewed by 2355

In optical communications, four-dimensional (4D) modulation formats encode information onto the quadrature components of two arbitrary orthogonal states of polarisation of the optical field. Many analytical models available in the optical communication literature allow, within a first-order perturbation framework, the computation of the [...] Read more.

In optical communications, four-dimensional (4D) modulation formats encode information onto the quadrature components of two arbitrary orthogonal states of polarisation of the optical field. Many analytical models available in the optical communication literature allow, within a first-order perturbation framework, the computation of the average power of the nonlinear interference (NLI) accumulated in coherent fibre-optic transmission systems. However, all such models only operate under the assumption of transmitted polarisation-multiplexed two-dimensional (PM-2D) modulation formats, which only represent a limited subset of the possible dual-polarisation 4D (DP-4D) formats. Namely, only those where data transmitted on each polarisation channel are mutually independent and identically distributed. This paper presents a step-by-step mathematical derivation of the extension of existing NLI models to the class of arbitrary DP-4D modulation formats. In particular, the methodology adopted follows the one of the popular enhanced Gaussian noise model, albeit dropping most assumptions on the geometry and statistic of the transmitted 4D modulation format. The resulting expressions show that, whilst in the PM-2D case the NLI power depends only on different statistical high-order moments of each polarisation component, for a general DP-4D constellation, several other cross-polarisation correlations also need to be taken into account. Full article

(This article belongs to the Special Issue Information Theory of Optical Fiber)

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16 pages, 941 KiB

Open AccessArticle

Back-to-Back Performance of the Full Spectrum Nonlinear Fourier Transform and Its Inverse

by Benedikt Leible, Daniel Plabst and Norbert Hanik

Entropy 2020, 22(10), 1131; https://0-doi-org.brum.beds.ac.uk/10.3390/e22101131 - 06 Oct 2020

Cited by 3 | Viewed by 1828

Abstract

In this paper, data-transmission using the nonlinear Fourier transform for jointly modulated discrete and continuous spectra is investigated. A recent method for purely discrete eigenvalue removal at the detector is extended to signals with additional continuous spectral support. At first, the eigenvalues are [...] Read more.

In this paper, data-transmission using the nonlinear Fourier transform for jointly modulated discrete and continuous spectra is investigated. A recent method for purely discrete eigenvalue removal at the detector is extended to signals with additional continuous spectral support. At first, the eigenvalues are sequentially detected and removed from the jointly modulated received signal. After each successful removal, the time-support of the resulting signal for the next iteration can be narrowed, until all eigenvalues are removed. The resulting truncated signal, ideally containing only continuous spectral components, is then recovered by a standard NFT algorithm. Numerical simulations without a fiber channel show that, for jointly modulated discrete and continuous spectra, the mean-squared error between transmitted and received eigenvalues can be reduced using the eigenvalue removal approach, when compared to state-of-the-art detection methods. Additionally, the computational complexity for detection of both spectral components can be decreased when, by the choice of the modulated eigenvalues, the time-support after each removal step can be reduced. Numerical simulations are also carried out for transmission over a Raman-amplified, lossy SSMF channel. The mutual information is approximated and the eigenvalue removal method is shown to result in achievable rate improvements. Full article

(This article belongs to the Special Issue Information Theory of Optical Fiber)

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27 pages, 3573 KiB

Open AccessArticle

Hierarchical Distribution Matching for Probabilistic Amplitude Shaping

by Stella Civelli and Marco Secondini

Entropy 2020, 22(9), 958; https://0-doi-org.brum.beds.ac.uk/10.3390/e22090958 - 30 Aug 2020

Cited by 12 | Viewed by 2592

Abstract

Probabilistic amplitude shaping—implemented through a distribution matcher (DM)—is an effective approach to enhance the performance and the flexibility of bandwidth-efficient coded modulations. Different DM structures have been proposed in the literature. Typically, both their performance and their complexity increase with the block length. [...] Read more.

Probabilistic amplitude shaping—implemented through a distribution matcher (DM)—is an effective approach to enhance the performance and the flexibility of bandwidth-efficient coded modulations. Different DM structures have been proposed in the literature. Typically, both their performance and their complexity increase with the block length. In this work, we present a hierarchical DM (Hi-DM) approach based on the combination of several DMs of different possible types, which provides the good performance of long DMs with the low complexity of several short DMs. The DMs are organized in layers. Each upper-layer DM encodes information on a sequence of lower-layer DMs, which are used as “virtual symbols”. First, we describe the Hi-DM structure, its properties, and the encoding and decoding procedures. Then, we present three particular Hi-DM configurations, providing some practical design guidelines, and investigating their performance in terms of rate loss and energy loss. Finally, we compare the system performance obtained with the proposed Hi-DM structures and with their single-layer counterparts: a

0.19 d B

SNR gain is obtained by a two-layer Hi-DM based on constant composition DMs (CCDM) compared to a single-layer CCDM with same complexity; a

0.12 d B

gain and a significant complexity reduction are obtained by a Hi-DM based on minimum-energy lookup tables compared to a single-layer DM based on enumerative sphere shaping with same memory requirements. Full article

(This article belongs to the Special Issue Information Theory of Optical Fiber)

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25 pages, 924 KiB

Open AccessFeature PaperArticle

On the Capacity of Amplitude Modulated Soliton Communication over Long Haul Fibers

by Yu Chen, Iman Tavakkolnia, Alex Alvarado and Majid Safari

Entropy 2020, 22(8), 899; https://0-doi-org.brum.beds.ac.uk/10.3390/e22080899 - 15 Aug 2020

Cited by 8 | Viewed by 2767

Abstract

The capacity limits of fiber-optic communication systems in the nonlinear regime are not yet well understood. In this paper, we study the capacity of amplitude modulated first-order soliton transmission, defined as the maximum of the so-called time-scaled mutual information. Such definition allows us [...] Read more.

The capacity limits of fiber-optic communication systems in the nonlinear regime are not yet well understood. In this paper, we study the capacity of amplitude modulated first-order soliton transmission, defined as the maximum of the so-called time-scaled mutual information. Such definition allows us to directly incorporate the dependence of soliton pulse width to its amplitude into capacity formulation. The commonly used memoryless channel model based on noncentral chi-squared distribution is initially considered. Applying a variance normalizing transform, this channel is approximated by a unit-variance additive white Gaussian noise (AWGN) model. Based on a numerical capacity analysis of the approximated AWGN channel, a general form of capacity-approaching input distributions is determined. These optimal distributions are discrete comprising a mass point at zero (off symbol) and a finite number of mass points almost uniformly distributed away from zero. Using this general form of input distributions, a novel closed-form approximation of the capacity is determined showing a good match to numerical results. Finally, mismatch capacity bounds are developed based on split-step simulations of the nonlinear Schr

\ddot{o}

dinger equation considering both single soliton and soliton sequence transmissions. This relaxes the initial assumption of memoryless channel to show the impact of both inter-soliton interaction and Gordon–Haus effects. Our results show that the inter-soliton interaction effect becomes increasingly significant at higher soliton amplitudes and would be the dominant impairment compared to the timing jitter induced by the Gordon–Haus effect. Full article

(This article belongs to the Special Issue Information Theory of Optical Fiber)

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15 pages, 833 KiB

Open AccessArticle

Optimization of Probabilistic Shaping for Nonlinear Fiber Channels with Non-Gaussian Noise

by Henrik Enggaard Hansen, Metodi P. Yankov, Leif Katsuo Oxenløwe and Søren Forchhammer

Entropy 2020, 22(8), 872; https://0-doi-org.brum.beds.ac.uk/10.3390/e22080872 - 08 Aug 2020

Cited by 6 | Viewed by 3249

Abstract

Probabilistic constellation shaping is investigated in the context of nonlinear fiber optic communication channels. Based on a general framework, different link types are considered—1. dispersion-managed channels, 2. unrepeatered transmission channels and 3. ideal distributed Raman amplified channels. These channels exhibit nonlinear effects to [...] Read more.

Probabilistic constellation shaping is investigated in the context of nonlinear fiber optic communication channels. Based on a general framework, different link types are considered—1. dispersion-managed channels, 2. unrepeatered transmission channels and 3. ideal distributed Raman amplified channels. These channels exhibit nonlinear effects to a degree that conventional probabilistic constellation shaping strategies for the additive white Gaussian (AWGN) noise channel are suboptimal. A channel-agnostic optimization strategy is used to optimize the constellation probability mass functions (PMFs) for the channels in use. Optimized PMFs are obtained, which balance the effects of additive amplified spontaneous emission noise and nonlinear interference. The obtained PMFs cannot be modeled by the conventional Maxwell-Boltzmann PMFs and outperform optimal choices of these in all the investigated channels. Suboptimal choices of constellation shapes are associated with increased nonlinear effects in the form of non-Gaussian noise. For dispersion-managed channels, a reach gain in 2 spans is seen and across the three channel types, gains of >0.1 bits/symbol over unshaped quadrature-amplitude modulation (QAM) are seen using channel-optimized probablistic shaping. Full article

(This article belongs to the Special Issue Information Theory of Optical Fiber)

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15 pages, 1157 KiB

Open AccessArticle

Bounds on the Transmit Power of b-Modulated NFDM Systems in Anomalous Dispersion Fiber

by Shrinivas Chimmalgi and Sander Wahls

Entropy 2020, 22(6), 639; https://0-doi-org.brum.beds.ac.uk/10.3390/e22060639 - 09 Jun 2020

Cited by 15 | Viewed by 2808

Abstract

The performance of various nonlinear frequency division multiplexed (NFDM) fiber-optic transmission systems has been observed to decrease with increasing signal duration. For a class of NFDM systems known as b-modulators, we show that the nonlinear bandwidth, signal duration, and power are coupled [...] Read more.

The performance of various nonlinear frequency division multiplexed (NFDM) fiber-optic transmission systems has been observed to decrease with increasing signal duration. For a class of NFDM systems known as b-modulators, we show that the nonlinear bandwidth, signal duration, and power are coupled when singularities in the nonlinear spectrum are avoided. When the nonlinear bandwidth is fixed, the coupling results in an upper bound on the transmit power that decreases with increasing signal duration. Signal-to-noise ratios are consequently expected to decrease, which can help explain drops in performance observed in practice. Furthermore, we show that there is often a finite bound on the transmit power of b-modulators even if spectral singularities are allowed. Full article

(This article belongs to the Special Issue Information Theory of Optical Fiber)

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19 pages, 594 KiB

Open AccessArticle

Hard-Decision Coded Modulation for High-Throughput Short-Reach Optical Interconnect

by Bin Chen, Yi Lei, Gabriele Liga, Chigo Okonkwo and Alex Alvarado

Entropy 2020, 22(4), 400; https://0-doi-org.brum.beds.ac.uk/10.3390/e22040400 - 31 Mar 2020

Cited by 4 | Viewed by 3499

Abstract

Coded modulation (CM), a combination of forward error correction (FEC) and high order modulation formats, has become a key part of modern optical communication systems. Designing CM schemes with strict complexity requirements for optical communications (e.g., data center interconnects) is still challenging mainly [...] Read more.

Coded modulation (CM), a combination of forward error correction (FEC) and high order modulation formats, has become a key part of modern optical communication systems. Designing CM schemes with strict complexity requirements for optical communications (e.g., data center interconnects) is still challenging mainly because of the expected low latency, low overhead, and the stringent high data rate requirements. In this paper, we propose a CM scheme with bit-wise hard-decision FEC and geometric shaping. In particular, we propose to combine the recently introduced soft-aided bit-marking decoding algorithm for staircase codes (SCCs) with geometrically-shaped constellations. The main goal of this CM scheme is to jointly boost the coding gain and provide shaping gain, while keeping the complexity low. When compared to existing CM systems based on M-ary quadrature-amplitude modulation (MQAM,

M = 64, 128, 256

) and conventional decoding of SCCs, the proposed scheme shows improvements of up to

0.83

dB at a bit-error rate of

10^{- 6}

in the additive white Gaussian noise channel. For a nonlinear optical fiber system, simulation results show up to

24 %

reach increase. In addition, the proposed CM scheme enables rate adaptivity in single-wavelength systems, offering six different data rates between 450 Gbit/s and 666 Gbit/s. Full article

(This article belongs to the Special Issue Information Theory of Optical Fiber)

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Review

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31 pages, 649 KiB

Open AccessReview

Path Integral Approach to Nondispersive Optical Fiber Communication Channel

by Aleksei V. Reznichenko and Ivan S. Terekhov

Entropy 2020, 22(6), 607; https://0-doi-org.brum.beds.ac.uk/10.3390/e22060607 - 29 May 2020

Cited by 4 | Viewed by 2113

Abstract

In the present paper we summarize the methods and results of calculations for the theoretical informational quantities obtained in our works for the nondispersive optical fiber channel. We considered two models: the per-sample model and the model where the input signal depends on [...] Read more.

In the present paper we summarize the methods and results of calculations for the theoretical informational quantities obtained in our works for the nondispersive optical fiber channel. We considered two models: the per-sample model and the model where the input signal depends on time. For these models we found the approach for the calculation of the mutual information exactly in the nonlinearity parameter but for the large signal-to-noise power ratio. Using this approach for the per-sample model we found the lower bound of the channel capacity in the intermediate power range. Full article

(This article belongs to the Special Issue Information Theory of Optical Fiber)

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