Liliana Beltran

Jessica Martinez

Jose Luis Martinez

Daniel Villavicencio

all their support, for their friendship, and for their motivation. I thank all my colleagues in the department for their help, and for making my stay at Bell Labs a very pleasant one. It

3

"I have heard articulate speech by sunlight! I have heard a ray of the sun laugh and cough and sing! I have been able to hear a shadow and I have even perceived by ear the passage of a cloud across the sun's disk. You are the grandfather of the Photophone and I want to share my delight at my success".

- Alexander Graham Bell, in a letter to his father Alexander Melville Bell, dated February

ABSTRACT

Global data traffic is expected to reach up to 4.3 ZB per year by 2020. This growth will be driven by the rise of two main technological fields: the creation of a new cloud integrated network of seemingly infinite capacity, and the Internet of Things (IoT) that will lead to a massive digitization and connection of devices. By 2021, the number of IP connected devices is expected to be more than three times the global population.

This amount of data traffic will challenge the core IP/optical infrastructure over which these services will flow. With most of the global communications being eventually transported on submarine point-to-point fiber-optic systems, different cutting-edge technologies have been under research during the past years to cope with the exponential increase of the global network-traffic.

With fiber Kerr nonlinear effects limiting the performance of current fiber-optic transmission systems, intense research has been done in the development of techniques searching to mitigate their effects. Approaches based on advanced digital signal processing (DSP) algorithms have taken the lead over optical techniques, as they can be more easily implemented in already deployed systems. In this sense, two main algorithms have been proposed in the literature: digital backpropagation (DBP), which is the most efficient algorithm in terms of performance improvement although its high computational complexity; and perturbative nonlinear compensation (PNLC), which presents a reduced complexity at the expense of lower achievable gains. Although, nonlinear compensation (NLC) is still not present in commercial systems due to its high complexity, it has been used in most of the recent "hero experiments" for distances above 6,000 km. However, with the continuous increase of processing capabilities of integrated circuits, DSP-based NLC techniques will be most likely implemented in future generation systems.

Besides nonlinear compensation techniques, the use of multi-level modulation formats beyond 16QAM and high symbol-rates have gained momentum in the last years. One of the breakthroughs has been the introduction of QAM-based probabilistic constellation shaping (PCS-QAM), which has proven to outperform regular QAM formats. Not only the achievable rates approach the linear Shannon capacity as the constellation adopts a Gaussian like distribution, but rate adaptation is also possible while keeping the low complexity non-iterative decoding bit interleaved coded modulation scheme.

This thesis focuses on the evaluation of performance and achievable benefits of advanced modulation formats as PCS-QAM and digital nonlinear equalization techniques to increase the performance and throughput of long-haul optical fiber systems.

In the first part, the performance and achievable gains of low complexity nonlinear compensation techniques based on filtered DBP (FDBP), and PNLC are evaluated for longhaul submarine optical fiber systems. First, semi-analytical formulae are obtained for the optimization of the parameters involved in the FDBP algorithm, making complex and timeconsuming exhaustive optimization unnecessary. Second, the impact of polarization mode dispersion (PMD) and polarization dependent loss (PDL) on the performance of FDBP and PNCL is experimentally addressed. Finally, a numerical study showing the practical limits of single channel nonlinear compensation versus algorithm complexity is carried out for transoceanic fiber-optic transmission systems

In the second part, the design and achievable rate increase brought by PCS-QAM over regular formats is addressed. First, the performance comparison of four advanced modulation formats for transoceanic coherent systems, i.e., 32QAM, 64QAM, 64APSK, and PCS-64QAM, is performed experimentally and numerically considering the theoretical gap to the linear Shannon capacity, nonlinear performance, back-to-back implementation penalties, and DSP challenges. Then, the design and optimization of a truncated-PCS (TPCS) to maximize the throughput of trans-Pacific optical fiber links is presented.

In the third part, PCS-QAM and NLC techniques are combined to demonstrate several transoceanic transmission records. First, the highest spectral efficiency for distances above 10,000 km using C-band EDFA only amplification is demonstrated. Then, cutting-edge CMOS digital-to-analog convertor's technology and wideband receivers together with PCS-QAM and NLC are used to achieve the highest per channel bit-rate up to date for trans-Atlantic and trans-Pacific distances. Finally, a record 65 Tb/s using C+L bands EDFA only amplification over 6,600 km employing PCS-64QAM, multi-rate FECs, and NLC is shown.

The final part of this thesis is devoted to the conclusions.

RESUME

Il est estimé que d'ici 2021 le trafic mondial de données atteindra 4.3 ZB par an, et que le nombre d'appareils IP connectés sera plus de trois fois supérieur à la population mondiale.

Cette quantité de trafic de données mettra au défi l'infrastructure des réseaux IP/optiques qui représente l'épine dorsale des systèmes de télécommunications. En effet, la majeure partie des communications mondiales est transportée par des systèmes transocéaniques à fibre optique. Afin de faire face à cette demande croissante du trafic de données, différentes technologies sont actuellement étudiées pour augmenter la capacité de ces systèmes de transmission très longue distance.

La capacité des systèmes de transmission optiques modernes est fortement limitée par les effets non-linéaires de type Kerr dans la fibre. Dans ce contexte, différentes techniques de traitement de signal numérique (DSP) à complexité relativement élevée ont été proposées pour mitiger ces effets, notamment la rétropropagation numérique filtrée (FDBP), et la compensation non linéaire perturbative (PNLC). Avec le développement rapide des circuits intégrés à haute vitesse, les techniques de compensation non linéaire (NLC) basées sur DSP seront très probablement mises en oeuvre dans les systèmes de future génération.

En outre, l'utilisation des formats de modulation multi-niveaux à haute efficacité spectrale au-delà de 16QAM a pris de l'ampleur pour augmenter la capacité de transmission de ces systèmes optiques. En particulier l'introduction du format QAM avec mise en forme probabiliste (PCS-QAM), démontré plus performant que les formats QAM classiques, a permis non seulement de s'approcher de la limite fondamentale de Shannon en régime linéaire, car la constellation adopte une distribution de type Gaussien, mais aussi d'adapter le débit du canal tout en conservant un schéma de codage à faible complexité basé sur la modulation codée avec entrelacement de bits (BICM).

Dans ce contexte, ce travail de thèse est axé sur l'étude de la performance et des gains réalisables par des techniques DSP à faible complexité pour mitiger les effets non-linéaires de type monocanal, ainsi que l'évaluation de la performances et mise en oeuvre des formats PCS-QAM pour augmenter l'efficacité spectrale (SE) des systèmes pour les distances transocéaniques.

Dans le premier chapitre, les principaux aspects d'un système de communication numérique et des systèmes cohérents optiques longue distance basés sur le multiplexage en longueur d'onde (WDM) sont traités. D'abord, nous décrivons les principaux éléments de base de la transmission numérique, tels que : les taux réalisables dans les systèmes BICM, les codes de correction d'erreurs (FEC), les formats de modulation de cardinalité élevée incluant PCS-QAM, et la mise en forme des impulsions. Ensuite, nous décrivons les différents blocs d'un système cohérent optique long-distance, incluant : l'émetteur optique, les principaux phénomènes physiques affectant un signal optique pendant la propagation, l'architecture du récepteur cohérent, suivi par les principaux blocs DSP utilisés pour la récupération de données transmises. Enfin, nous abordons l'évaluation de la performance du système en termes du rapport signal à bruit (SNR), y compris toutes les sources de bruit comme : le bruit des amplificateurs optiques, le bruit lié au transmetteur et au récepteur, et le bruit non-linéaire lié à l'effet Kerr. Ce dernier est calculé à partir des modèles analytiques basés sur la théorie de perturbation de premier ordre.

Le deuxième chapitre est basé sur l'évaluation de la performance des algorithmes FDBP et PNLC utilisés pour mitiger les effets non-linéaires de type monocanal. Dans la première partie, nous étudions la dépendance des paramètres impliqués dans l'algorithme FDBP avec les différents paramètres du système, tels que : nombre de canaux WDM, puissance optique transmise, débit de modulation symbole et coefficients physiques de la fibre. Nous montrons numériquement et expérimentalement que les paramètres FDBP dépendent principalement de l'élargissement temporel des impulsions par étape FDBP, ce qui nous permet d'obtenir des formules semi-analytiques simples pour leur calcul, et d'éviter une optimisation exhaustive complexe. Dans la deuxième partie, nous étudions expérimentalement l'impact des effets de dispersion modale de polarisation (PMD) et de pertes dépendant de la polarisation (PDL) sur la performance des algorithmes FDBP et PNLC. Nous montrons que des valeurs de PMD aussi grandes que 5 fois la durée du symbole conduisent à une dégradation maximale observable de seulement ~0.1 dB dans le gain réalisable par la FDBP et la PNLC; tandis que la dégradation du gain correspondant à des valeurs PDL jusqu'à 9dB est de seulement ~0.2dB. Ces observations expérimentales montrent la robustesse de ces algorithmes contre les effets PDL et PMD. Enfin, dans la troisième partie, nous évaluons numériquement les gains pratiques réalisables par la FDBP et la PNLC en relation avec la complexité liée à son implémentation.

Nous montrons que dans le cas idéal sans bruit linéaire d'amplification, la FDBP travaillant à une étape par tronçon de fibre peut atteindre 90% du gain maximum théorique, tandis que pour la PNLC il est réduit à ~75%. En considérant le bruit linéaire d'amplification, ces gains réalisables sont encore diminués, avec une dégradation plus élevée pour les distances de transmission plus longues. Le bruit linéaire dégrade la performance de la FDBP et de la PNLC en même proportion indépendamment de la complexité de l'algorithme. Pour les régimes OSNR des systèmes transatlantiques, la PNLC peut atteindre des gains compris entre 0.5dB et 0.7dB pour les systèmes utilisant des canaux modulés à 32GBd, tandis que pour les systèmes avec des canaux modulés à 64GBd, les gains augmentent de 0.7 dB jusqu'à 1.1 dB.

Le troisième chapitre est basé sur la conception et l'évaluation de la performance du format PCS-QAM par rapport aux formats de modulation réguliers en amplitude et en quadrature La dernière partie de cette thèse est consacrée aux conclusions.

LIST OF ACRONYMS AND SYMBOLS

COHERENT OPTICAL FIBER

TELECOMMUNICATIONS SYSTEMS

In this chapter, we first discuss the current status and trends of transoceanic submarine optical fiber systems. Then, we present the fundamental notions of digital communications and information theory over the linear Gaussian channel. Finally, we present the operation principles of coherent optical fiber systems, where linear and nonlinear impairments of signal transmission and the different techniques for their mitigation are also discussed.

TRANSOCEANIC SUBMARINE OPTICAL FIBER SYSTEMS STATUS AND TRENDS

The first deployment of a trans-Atlantic cable was done in 1858 relying North America and Europe. It consisted of several cooper wires and was used for telegraph communications. Even though the cable only served for three weeks, it represents an heroic event in the history of the human technical revolution. Almost a century later, in 1956, the first transatlantic telephone cable system (TAT-1) was inaugurated, initially carrying 36 telephone channels. The evolution of submarine cables was slow in the following 38 years, and by 1994, the maximum number of voice channels was increased to 10,500 [START_REF] Bray | Innovation and the Communications Revolution: From the Victorian Pioneers to Broadband Internet[END_REF][START_REF] Guarneri | The Conquest of the Atlantic[END_REF][START_REF] Chesnoy | Undersea Fiber Communication Systems[END_REF].

On the other hand, the first erbium doped fiber amplifier (EDFA) for optical fiber systems was shown in 1987, where optical gains up to 26 dB at a wavelength of 1530 nm were demonstrated [START_REF] Hecht | The Evolution of Optical Amplifiers[END_REF][START_REF] Mears | High-gain rare-earth-doped fiber amplifier at 1.54 μm[END_REF]. This experiment led to a worldwide effort on developing high gain EDFAs.

In 1989, an experimental demonstration using EDFAs to amplify multiple wavelength signals without cross-talk paved the way to amplified wavelength division multiplexing (WDM) systems [START_REF] Desurvire | Saturatlon-lnduced crosstalk in high-speed erbium-doped fiber amplifiers at λ = 1.53 μm[END_REF]. The advent of WDM with optical amplification started a revolution for the development of high-capacity optical fiber systems. In 1996, the first trans-Atlantic optical cable making use of EDFA amplification and WDM technology (TAT-12/13) was deployed, initially carrying 2 WDM channels modulated at 5 Gb/s [START_REF] Hecht | The Evolution of Optical Amplifiers[END_REF] using intensity modulation with direct detection (IM-DD).

The advances in high speed analog-to-digital converters (ADC), digital-to-analog converters (DAC), and high-speed application-specific integrated circuits (ASIC) in the early 2000s, together with the increased demand on data traffic renewed the interest for coherent technologies which were extensively studied during the 1980s. Since then, coherent detection has been the milestone technology for long haul optical fiber transmission systems. Its adoption has not only allowed the use of more complex higher order modulation formats, but has also enabled the use of advance digital signal processing (DSP) techniques enabling electronic equalization of deterministic system impairments. Coherent detection has become the de-facto standard for current commercial long-haul optical fiber systems. Modern commercial transoceanic cables can carry up to 20 Tb/s, equivalent to 312 million voice channels.

The transmission rates allowed by submarine optical cables are several tens of thousands of times more than can be achieved by satellites. In consequence, transoceanic optical fiber systems have become the backbone of international communications. It is commonly said that 99% of the total voice and data traffic is at present carried worldwide by submarine cables [START_REF] Chesnoy | Undersea Fiber Communication Systems[END_REF]. The evolution of system throughput, spectral efficiency (SE), and channel bit-rate in research lab demonstrations for trans-Atlantic distances (> 6,000 km) since 1999 is shown in Fig. 1.2 [START_REF] Bergano | 640 Gb/s Transmission of Sixty-four 10 Gb/s WDM Channels Over 7200km With 0.33[END_REF][START_REF] Davidson | 1800 Gb/s Transmission of One Hundred and Eighty 10 Gb/s WDM Channels over 7,000 km using the Full EDFA C-Band[END_REF][START_REF] Vareille | 3Tbit/s (300x11.6Gbit/s) Transmission Over 7380 km Using C+L Band with 25GHz Channel Spacing and NRZ Format[END_REF][START_REF] Sugahara | km transmission of 32[END_REF][START_REF] Charlet | One hundred WDM channel transatlantic transmission experiment at 43 Gbit/s using Raman repeaters with large 65 km spacing[END_REF][START_REF] Charlet | WDM transmission at 6 Tbit/s capacity over transatlantic distance, using 42.7Gb/s Differential Phase-Shift Keying without pulse carver[END_REF][START_REF] Masuda | Tb/s (135 × 111-Gb/s/ch) no-guard-interval coherent OFDM transmission over 6,248 km using SNR maximized secondorder DRA in the extended L-band[END_REF][START_REF] Salsi | 155×100Gbit/s coherent PDM-QPSK transmission over 7,200km[END_REF][16][START_REF] Salsi | 31 Tb/s transmission over 7,200 km using 46 Gbaud PDM-8QAM[END_REF][START_REF] Ghazisaeidi | 52.9 Tb/s Transmission over Transoceanic Distances using Adaptive Multi-Rate FEC[END_REF][START_REF] Ghazisaeidi | Transoceanic Transmission Systems Using Adaptive Multirate FECs[END_REF][START_REF] Zhang | Capacity-approaching transmission over 6375 km at spectral efficiency of 8.3 bit/s/Hz[END_REF][START_REF] Ghazisaeidi | 65Tb/s Transoceanic Transmission Using Probabilistically-Shaped PDM-64QAM[END_REF][START_REF] Cai | 70.4 Tb/s Capacity over 7,600 km in C+L Band Using Coded Modulation with Hybrid Constellation Shaping and Nonlinearity Compensation[END_REF][START_REF] Rios-Müller | Optimized spectrally efficient transceiver for 400-Gb/s single carrier transport[END_REF][START_REF] Fernandez De Jauregui Ruiz | Record 560 Gb/s Single-Carrier and 850 Gb/s Dual-Carrier Transmission over Transoceanic Distances[END_REF]. Before the arrival of coherent detection, the maximum achievable system throughput was 6 Tb/s using WDM channels modulated at 10 Gb/s reaching SEs of 0.8 b/s/Hz.

After 2008, a non-stopping increase have been made possible thanks to the synergy of different cutting-edge technological advances.

Great advances in CMOS technology have led to high speed DACs and ADCs. Their sampling rates have increased from 22 GSamples/s in 2006 [START_REF] Schvan | A 22GS/s 5b ADC in 0.13μm SiGe BiCMOS[END_REF] to 100 GSamples/s in 2016 [START_REF] Schuh | 100 GSa/s BiCMOS DAC Supporting 400 Gb/s Dual Channel Transmission[END_REF]. These great improvements have allowed the use of Nyquist pulse shaping techniques to further increase the SE. Together with coherent detection, they have also allowed the possibility to use powerful DSP algorithms to compensate for system impairments. Compensation of chromatic dispersion (CD), polarization mode dispersion (PMD), as well as transmitter/receiver imperfections such as time skew and imbalances of quadrature components and polarization tributaries are common practices in current coherent optical systems. Moreover, a great increase in the system SE has been possible by the use of multi-level modulation formats and high symbol-rate channels. Systems working at 40 Gb/s with binary phase shift keying (BPSK) were rapidly updated to 100 Gb/s by the use of quadrature phase shift keying (QPSK) [START_REF] Masuda | Tb/s (135 × 111-Gb/s/ch) no-guard-interval coherent OFDM transmission over 6,248 km using SNR maximized secondorder DRA in the extended L-band[END_REF][START_REF] Salsi | 155×100Gbit/s coherent PDM-QPSK transmission over 7,200km[END_REF], and to 200 Gb/s -400 Gb/s using 16-quadrature amplitude modulation (16QAM) [START_REF] Rios-Müller | Optimized spectrally efficient transceiver for 400-Gb/s single carrier transport[END_REF][START_REF] Zhang | 200 Gb/s and dual-wavelength 400 Gb/s transmission over transpacific distance at 6 b/s/Hz spectral efficiency[END_REF][START_REF] Salsi | 38.75 Tb/s transmission experiment over transoceanic distance[END_REF]. The net system throughput passed from 15 Tb/s in 2009, to 54,2 Tb/s by 2016, [START_REF] Salsi | 155×100Gbit/s coherent PDM-QPSK transmission over 7,200km[END_REF][START_REF] Ghazisaeidi | Submarine transmission systems using digital nonlinear compensation and adaptive rate forward error correction[END_REF]. Higher order modulation formats beyond 16QAM have gained momentum in the last couple of years. Experimental demonstrations using 32QAM and 64QAM

have been presented [START_REF] Zhang | Capacity-approaching transmission over 6375 km at spectral efficiency of 8.3 bit/s/Hz[END_REF][START_REF] Ghazisaeidi | 65Tb/s Transoceanic Transmission Using Probabilistically-Shaped PDM-64QAM[END_REF][START_REF] Ghazisaeidi | Advanced C+L Band Transoceanic Transmission Systems Based on Probabilistic Shaped PDM-64QAM[END_REF][START_REF] Cai | 64QAM based coded modulation transmission over transoceanic distance with >60 Tb/s capacity[END_REF]. However, one of the recent breakthroughs has been the introduction of constellation shaping techniques, which have proven to outperform regular QAM formats. Geometric constellation shaping based on 64-amplitude phase shift keying (64PSK) was used in [START_REF] Zhang | Capacity-approaching transmission over 6375 km at spectral efficiency of 8.3 bit/s/Hz[END_REF] to achieve a record C-band SE of 8.3 b/s/Hz (net 34.9 Tb/s) after 6,375 km, while 4D-PS-9/12-56APSK was used in [START_REF] Cai | 70.4 Tb/s Capacity over 7,600 km in C+L Band Using Coded Modulation with Hybrid Constellation Shaping and Nonlinearity Compensation[END_REF] [START_REF] Cho | Trans-Atlantic Field Trial Using Probabilistically Shaped 64-QAM at High Spectral Efficiencies and Single-Carrier Real-Time 250-Gb/s[END_REF].

Hard-decision forward error correction codes (HD-FEC) have been replaced with the more powerful soft-decision FECs (SD-FEC). Specially, high performance FEC codes based on spatially-coupled low-density parity-check (SC-LDPC) have been introduced [START_REF] Schmalen | A generic tool for assessing the soft-FEC performance in optical transmission experiments[END_REF][START_REF] Schmalen | Spatially coupled soft-decision error correction for future lightwave systems[END_REF][START_REF] Schmalen | On the design of capacity-approaching unit-memory spatially coupled LDPC codes for optical communications[END_REF]. Adaptive multi-rate FEC was used in [START_REF] Ghazisaeidi | 52.9 Tb/s Transmission over Transoceanic Distances using Adaptive Multi-Rate FEC[END_REF][START_REF] Ghazisaeidi | Transoceanic Transmission Systems Using Adaptive Multirate FECs[END_REF][START_REF] Ghazisaeidi | Submarine transmission systems using digital nonlinear compensation and adaptive rate forward error correction[END_REF][START_REF] Ghazisaeidi | Advanced C+L Band Transoceanic Transmission Systems Based on Probabilistic Shaped PDM-64QAM[END_REF], to achieve up to 20% capacity increase over trans-Atlantic and trans-Pacific distances with respect a single FEC code rate.

With fiber Kerr nonlinear effects limiting the performance of current fiber-optic transmission systems, different digital nonlinear compensation (NLC) techniques have been proposed. The most efficient algorithm in terms of performance improvement is digital backpropagation (DBP) [START_REF] Ip | Compensation of Dispersion and Nonlinear Impairments Using Digital Backpropagation[END_REF][START_REF] Li | Electronic post-compensation of WDM transmission impairments using coherent detection and digital signal processing[END_REF]; however, its hardware implementation is challenging due to its high computational complexity. Different approaches to reduce the complexity of NLC have been proposed, as the low-pass filter DBP (FDBP) [START_REF] Du | Improved single channel backpropagation for intra-channel fiber nonlinearity compensation in long-haul optical communication systems[END_REF][START_REF] Gao | Assessment of Intrachannel Nonlinear Compensation for 112 Gb/s Dual-Polarization 16QAM Systems[END_REF], or perturbative nonlinear compensation (PNLC) [START_REF] Ghazisaeidi | Calculation of Coefficients of Perturbative Nonlinear Pre-Compensation for Nyquist Pulses[END_REF]. DSPbased NLC has been used in all recent transmission experiments for distances above 6,000 km [18-20, 22, 29, 30, 41].

Finally, novel optical components as optical fibers with ultra-low loss (<0.16 dB/km), large effective area (≥150μm 2 ), and high chromatic dispersion (>20ps/nm/km), together with wideband amplification based on C+L bands (~70nm) have been commonly adopted in laboratory experiments; increasing the SE by reducing the span loss, increasing tolerance to fiber nonlinearities, and practically doubling the useful bandwidth per optical cable [START_REF] Ghazisaeidi | 52.9 Tb/s Transmission over Transoceanic Distances using Adaptive Multi-Rate FEC[END_REF][START_REF] Ghazisaeidi | Transoceanic Transmission Systems Using Adaptive Multirate FECs[END_REF][START_REF] Zhang | Capacity-approaching transmission over 6375 km at spectral efficiency of 8.3 bit/s/Hz[END_REF][START_REF] Ghazisaeidi | 65Tb/s Transoceanic Transmission Using Probabilistically-Shaped PDM-64QAM[END_REF][START_REF] Cai | 70.4 Tb/s Capacity over 7,600 km in C+L Band Using Coded Modulation with Hybrid Constellation Shaping and Nonlinearity Compensation[END_REF][START_REF] Ghazisaeidi | Submarine transmission systems using digital nonlinear compensation and adaptive rate forward error correction[END_REF][START_REF] Ghazisaeidi | Advanced C+L Band Transoceanic Transmission Systems Based on Probabilistic Shaped PDM-64QAM[END_REF][START_REF] Cai | 64QAM based coded modulation transmission over transoceanic distance with >60 Tb/s capacity[END_REF].

While high performance optical fibers are starting to be commercially deployed, C+L band amplification is likely to be adopted by the industry in the foreseeable future.

Compared to optical components, novel DSP techniques present an advantage for increasing the throughput of optical cables. With the wet plant lying on the seabed with twentyfive years' life span and almost no possible upgrade capability [START_REF] Chesnoy | Undersea Fiber Communication Systems[END_REF][START_REF] Pecci | Experimental characterization of submarine "open cable" using Gaussian-Noise model and OSNRWET parameter[END_REF], new optical components can only be adopted for optical cables to be laid in the future, and are not viable for already deployed systems. On the contrary, novel DSP techniques can be relatively easily implemented in the submarine line terminal equipment, leading to a simpler and smoother upgrade of already deployed systems.

In this work, we focus on the evaluation of novel DSP techniques to increase the throughput and spectral efficiency of transoceanic optical fiber systems. In particular DSP techniques for the mitigation of fiber nonlinear distortions will be investigated, as well as on the use of capacity approaching modulation formats for next generation optical fiber systems.

DIGITAL TELECOMMUNICATION SYSTEMS

The main objective of a digital transmission system is to transmit information from a given source to a given destination without errors despite the presence of noise and other signal perturbations. Fig. 1.3 shows a schematic representation of the main building blocks of a digital telecommunication system. The process of transmitting information can be described as follows. This error control scheme process is also known as forward error correction (FEC).

The resulting coded bits are then mapped to symbols drawn from a given constellation alphabet, also referred as a modulation format. The module which maps coded bits to symbols is called the mapper. For an equiprobable alphabet of size 𝑀, the number of coded bits per symbol is 𝑚 = 𝑙𝑜𝑔 2 (𝑀) . For example, lets 𝐴 be the constellation alphabet with complex constellation points, such that 𝐴 = {𝑎 1 , 𝑎 2 , … , 𝑎 𝑀 }. The mapper will map m-bit blocks at the output of the FEC, to a unique member of A. This process will generate a symbol sequence represented by 𝒙 = (𝑥 1 , 𝑥 2 , … , 𝑥 𝑘 , … , ) with 𝑥 𝑘 ∈ 𝐴.

In practice, information is transmitted over a physical medium in the form of an electromagnetic wave. Therefore, the discrete symbols at the mapper output have to be transformed to a signal waveform appropriate for transmission. This process is known as pulseshaping. In this sense, while the modulation format determines the maximum information each symbol can carry, pulse shaping determines the spectral width occupied by the signal.

After pulse shaping, different kind of equalization techniques to compensate for channel impairments can be applied. Thanks to the availability of high resolution DACs, equalization can be performed at the transmitter side in the digital domain by means of DSP. This is referred as pre-equalization.

The resulting signal is then modulated into a carrier wave and transmitted through the physical channel. In the case of an optical fiber system, the signal is modulated into a laser source by means of an electro-optic IQ-modulator (see Sec. 1.3.1). After transmission, the received-signal coherently detected (see Sec. 1.3.3). Digital post-equalization can also be performed by means of DSP.

After post-equalization, the received symbols 𝒚 = (𝑦 1 , 𝑦 2 , … , 𝑦 𝑘 , … , ) have to be converted back to a sequence of bits. To do so, a module called the channel decoder estimates the transmitted coded bits and performs error correction. Finally, the information bits are decompressed and the message is recovered.

We recall that in the frame of information theory, all elements between the channel encoder and the channel decoder are referred as the communication channel. That is to say, all elements working at the symbol level as shown in Fig. 1.3.

Achievable Information Rates

Channel Capacity

The maximum information rate that can be transmitted over a given communication channel was first formulated by Shannon in [START_REF] Shannon | A mathematical theory of communication[END_REF], and is referred as the channel capacity, 𝐶. Therefore, reliable communication can be achieved only if the information rate, 𝑅, is smaller than or equal to the channel capacity, i.e., 𝑅 ≤ 𝐶.

The channel capacity is determined by the statistical properties of the channel. The simplest additive white Gaussian noise (AWGN) channel is given by: where and are complex random variables representing the channel input and output respectively, and Z is a complex Gaussian random variable of zero mean accounting for noise.

In this case, the AWGN channel will transform into following the channel conditional probability:

with 𝑥 and 𝑦 realizations of and , and 𝑁 0 the complex noise variance. The quantity that measures the maximum amount of information that can be transported over the channel is referred as the mutual information, 𝕀, and is obtained by [START_REF] Essiambre | Capacity Limits of Optical Fiber Networks[END_REF]:

where 𝑝 𝑋𝑌 (𝑥, 𝑦) is the joint probability, and 𝑝 𝑋 (𝑥) and 𝑝 𝑌 (𝑦) are the marginal probability density functions of and . The channel capacity is the maximum mutual information, where the maximization is performed over all possible input distributions 𝑝 𝑋 (𝑥), i.e., 𝐶 = max 𝑝 𝑋 (•) 𝐼( ; ). [START_REF] Shannon | A mathematical theory of communication[END_REF] that for the AWGN channel, the optimum source distribution that maximizes C is itself complex and Gaussian, of the form: with P the signal power. The channel capacity is therefore given by the celebrated Shannon capacity formula:

Shannon showed in

𝐶 = 𝑙𝑜𝑔 2 (1 + 𝑆𝑁𝑅) ( 1.5 )
and is measured in bits per channel use, where 𝑆𝑁𝑅 = 𝑃 𝑁 0 ⁄ is the signal to noise ratio. The channel capacity can be seen as the maximum achievable information rate at which information can be transmitted over the channel for a given SNR.

= + 𝑍

( 1.1 )

𝑝 𝑌|𝑋 (𝑦|𝑥) = 1 𝜋𝑁 0 𝑒 -|𝑦-𝑥| 2 𝑁 0 ( 1.2 ) 𝕀( ; ) = ∬ 𝑝 𝑋𝑌 (𝑥, 𝑦) • 𝑙𝑜𝑔 2 𝑝 𝑌|𝑋 (𝑦|𝑥) 𝑝 𝑌 (𝑦) 𝑑𝑥𝑑𝑦 = ∬ 𝑝 𝑋𝑌 (𝑥, 𝑦) • 𝑙𝑜𝑔 2 𝑝 𝑋𝑌 (𝑥, 𝑦) 𝑝 𝑋 (𝑥) • 𝑝 𝑌 (𝑦) 𝑑𝑥𝑑𝑦 ( 1.3 ) 𝑝 𝑋 (𝑥) = 1 𝜋𝑃 𝑒 -|𝑥| 2 𝑃
( 1.4 )

Constrained Capacity

The above formulation has been obtained considering that the input and output alphabets 𝐴, and 𝐵 have an infinite number of elements, such that the input and output distributions 𝑝 𝑋 (𝑥)

and 𝑝 𝑌 (𝑦) are continuous Gaussian functions. In practical systems, 𝐴 and 𝐵 are finite, and their distributions are typically not Gaussian. The resulting information rates are denoted as modulation constrained capacity and are bounded away from the Shannon capacity.

Considering a finite constellation alphabet 𝐴 = {𝑎 1 , 𝑎 2 , … , 𝑎 𝑖 , … 𝑎 𝑀 } of size M, the constrained capacity can be estimated by Monte-Carlo time averaging for long-enough N as:

where 𝑥 𝑘 and 𝑦 𝑘 are the k-th symbol of the transmitted and received symbol sequence of size N. We define the source entropy 𝐻( ) as:

Eq. ( 1.6 ) corresponds to the maximum information rate that can be achieved for a given constellation. In practice, this limit is only attainable in systems where the channel coding and the mapper are jointly designed. This scheme is referred as coded modulation.

Generalized Mutual Information

In order to reduce system complexity associated to coded modulation, another coding scheme referred as bit-interleaved coded modulation (BICM) is typically preferred.

BICM allows to separate coding and mapping, resulting in a higher flexibility in system design. Within this scheme, the generated coded bits at the output of the FEC are interleaved to sparse possible burst errors during transmission. The resulting coded interleaved bits are then partitioned into blocks of length 𝑚 = 𝑙𝑜𝑔 2 (𝑀), and mapped to modulation symbols. At the receiver side, soft information about the coded bits is propagated from the demodulator to the decoder in the form of bit-wise a posteriori probabilities or log-likelihood ratios. This decoding scheme is known as bit-metric decoding (BMD).

Due to the interleaving, the BICM system can be regarded as 𝑚 parallel memoryless and independent binary channels. The maximum information rate of BICM is usually referred as the generalized mutual information (GMI) defined by [START_REF] Cho | Normalized generalized mutual information as a forward error correction threshold for probabilistically shaped QAM[END_REF][START_REF] Böcherer | Achievable rates for probabilistic shaping[END_REF]: where 𝑞 𝑌|𝑋 is the auxiliary channel conditional probability, 𝑏 𝑘,𝑖 is the i-th bit of the k-th transmitted symbol, and 𝐴 𝑏 𝑘,𝑖 is the set of the constellation symbols whose i-th bit is 𝑏 𝑘,𝑖 .

𝕀 ̂( ; ) = 1 𝑁 ∑
Employing a BICM scheme results in a capacity loss with respect to the coded modulation constrained capacity. However, this loss is very small assuming Gray-mapping.

Forward Error Correction Codes

As already mentioned, the main goal of channel coding, or FEC, is to add redundancy to the binary information sequence at the output of the source encoder. After transmission over the noisy channel, the FEC decoder exploits this redundancy to fully recover the source information. In general, the binary channel encoder will introduce redundancy bits in form of parity checks. The code rate, 𝑟 𝑐 , is defined by:

𝑟 𝑐 = 𝐾 𝑐 𝑁 𝑐 ( 1.9 )
The FEC encoder takes a block of 𝐾 𝑐 information bits, and adds 𝑁 𝑐 -𝐾 𝑐 redundant bits. The FEC overhead is defined by:

𝑂𝐻 = 𝑁 𝑐 -𝐾 𝑐 𝐾 𝑐 = 1 -𝑟 𝑐 𝑟 𝑐 ( 1.10 )
The Shannon capacity and the GMI derivations in Seq. 1.2.1 assumed ideal FEC codes with block lengths 𝑁 𝑐 tending to infinity. The information rates of real systems with practical FEC codes will be bounded away from capacity.

A useful metric to quantify the performance of digital communication system is the bit-errorrate (BER), defined as the ratio between the number of received erroneous bits and the total number of transmitted bits. FEC coding schemes are usually characterized by their coding gain, [START_REF] Schmalen | Spatially coupled soft-decision error correction for future lightwave systems[END_REF][START_REF] Leven | Status and Recent Advances on Forward Error Correction Technologies for Lightwave Systems[END_REF]. SC-LDPC codes can achieve net coding gains close to theoretical limits, operating with less than 1 dB SNR gap to capacity. SC-LDPC will be use as the chosen coding scheme within this work.

which

Modulation Formats

A constellation alphabet having M points can carry a maximum of 𝑚 = 𝑙𝑜𝑔 2 (𝑀) bits per symbol. This maximum is achieved only when the probability of occurrence of the constellation points is uniform, i.e., 𝑃 𝑋 ( = 𝑎 𝑘 ) = 1 𝑀 ⁄ , such that the constellation entropy is defined as:

𝐻( ) = 𝑚 = 𝑙𝑜𝑔 2 (𝑀) ( 1.11 )
The The average power associated with a constellation is given by:

𝑃 = 1 𝑀 ∑|𝑎 𝑘 | 2 𝑀 𝑘=1
( 1.12 )

For a constant output power of the modulator, constellations with higher values of M will have their points more closely packed together, leading to higher BER for the same SNR, and therefore requiring stronger FECs, i.e., lower code rates. For a given modulation format, BER can be minimized by the use of a proper bit-to-symbol mapping. In general, the probability of symbol error is higher for adjacent symbols, such that the minimum BER is found when adjacent symbols have only one bit difference. The mapping having this property is called Gray mapping.

As observed in Table 1, the distribution of the discrete constellations points is not Gaussian, and therefore suboptimal for the AWGN channel. The resulting achievable information rates will be bounded away from the Shannon capacity. Modulation formats with "Gaussian-like" distributions can be used to approach capacity.

These formats rely on applying some kind of shaping to a constellation with a discrete number of points. The shaping technique can either be geometrical, where equiprobable constellation points are arranged in the complex plane to make the constellation to appear Gaussian, it can be probabilistic, where the constellation points are visited with nonuniform probability densities approximating the Gaussian distribution; or it can be a combination of both geometrical and probabilistic techniques. These techniques are well known in the literature [START_REF] Cover | Elements of Information Theory[END_REF][START_REF] Forney | Multidimensional constellations-Part I: Introduction, figures of merit, and generalized cross constellations[END_REF][START_REF] Forney | Multidimensional constellations-Part II: Voronoi constellations[END_REF], but have been recently introduced for fiber-optic telecommunications systems, and have shown to outperform regular square QAM formats [20-22, 30, 41, 50-52].

A type of PCS based on QAM constellation was recently proposed in [START_REF] Böcherer | Bandwidth efficient and rate-matched low-density parity-check coded modulation[END_REF], presenting two main design advantages. First, it can be applied to any square QAM constellation which are desirable due to their simplicity, and for which the BICM scheme can be employed with almost no penalty thanks to the use of symbol Gray mapping; and second, FEC implementation is independent of shaping, such that no change in FEC engine is required for different shaping modes, allowing for rate adaptation with fine granularity [START_REF] Buchali | Rate adaptation and reach increase by probabilistically shaped 64-QAM: An experimental demonstration[END_REF]. In terms of performance, PCS based on 64QAM was shown to achieve higher shaping gains compared to geometric shaping based on 64-amplitude phase shift-keying (64APSK) [START_REF] Ghazisaeidi | Advanced C+L Band Transoceanic Transmission Systems Based on Probabilistic Shaped PDM-64QAM[END_REF], while at the same time being more tolerant to frequency offsets and phase noise resulting in lower implementation penalties for low operating signal-to-noise ratio (SNR) regimes [START_REF] Fernandez De Jauregui Ruiz | Performance comparison of advanced modulation formats for transoceanic coherent systems[END_REF]. For the above reasons, PCS-QAM has become one of the main subjects of research and is becoming an interesting candidate for implementation in future coherent optical commercial systems.

In PCS-QAM, the probability mass function (PMF) of the QAM constellation points are set according to a discrete Maxwell-Boltzmann distribution with free parameter 𝜈 ≥ 0, according to:

𝑃 𝑋 (𝑥) = exp(-𝜈|𝑥| 2 ) ∑ exp(-𝜈|𝑥| 2 ) 𝑥∈𝐴 ( 1.13 )
The free parameter 𝜈 can be optimized to change the source distribution and minimize the gap to the Shannon capacity for a given target SNR. The entropy, 𝐻 , of the resulting constellation can then be obtained as per ( 1.7 ) .Considering a given FEC code rate, 𝑟 𝑐 , the ideal transmission rate, R, is calculated by [START_REF] Buchali | Rate adaptation and reach increase by probabilistically shaped 64-QAM: An experimental demonstration[END_REF]:

𝑅 = 𝐻 -(1 -𝑟 𝑐 ) • 𝑚 ( 1.14 )
where 𝑚 is the number of bits per symbol of the QAM constellation at which PCS is applied, i.e. 𝑚 = 6 for 64QAM.

In practice, the DSP module which sets the PMF of the constellation points for a given 𝜈 is called the distribution matcher (DM). At the transmitter, the DM is concatenated with a systematic binary encoder for FEC. This scheme is referred to as probabilistic amplitude shaping (PAS). At the receiver side, an inverse DM is placed after the de-mapper and the FEC decoder. Please note that the de-mapper should be modified to account for the symbols prior probabilities. The design of the DMs is an actual research topic and it is out of the scope of this work. The PCS-QAM can be seen as an instance of BICM with bit-metric mismatched decoding, for which the correct metric to measure the achievable rates is the GMI. A detail description of the PAS scheme is found is in [START_REF] Böcherer | Bandwidth efficient and rate-matched low-density parity-check coded modulation[END_REF]. Unlike QAM constellations, APSK constellations usually do not support Gray mapping and require complex iterative FEC decoding. In [START_REF] Liu | APSK constellation with gray mapping[END_REF], a 64APSK constellation supporting Gray mapping was introduced, and further used in [START_REF] Zhang | Capacity-approaching transmission over 6375 km at spectral efficiency of 8.3 bit/s/Hz[END_REF] for transoceanic distances, and it was shown to outperform regular QAM. In particular, 64APSK is composed of 4 concentric rings, each with 16 uniformly spaced PSK points as shown in Fig. 1.5c. On the other hand, hybrid constellations make use of coded modulation with complex iterative decoding, being unattractive for practical implementation in commercial systems and will not be considered in this work.

.3.1 System Performance Metrics

As already mentioned, the post-FEC BER in actual coherent optical communication systems is typically <10 -15 . In general, such low values make post-FEC BER evaluation practically impossible by means of numerical simulations and experimental off-line processing.

In order to relax this requirement, another performance metric referred as the FEC limit is typically used. It is defined as the maximum BER at the input of the FEC decoder for which the decoder successfully decodes the received block. It general, this approach works well when employing HD-FECs, however, for SD-FECs it is less accurate [START_REF] Cho | Analysis of soft-decision FEC on non-AWGN channels[END_REF]. In this sense, it has been recently shown that for BICM systems the GMI can be used to make a more robust prediction of the post-FEC BER when QAM formats are used [START_REF] Cho | Normalized Generalized Mutual Information as a Forward Error Correction Threshold for Probabilistically Shaped QAM[END_REF][START_REF] Alvarado | Replacing the Soft-Decision FEC Limit Paradigm[END_REF][START_REF] Yoshida | Performance Metrics for Systems with Soft-Decision FEC and Probabilistic Shaping[END_REF][START_REF] Alvarado | Information rates and post-FEC BER prediction in optical fiber communications[END_REF].

In this work, we focus on three main performance metrics, i.e., SNR, pre-FEC BER (pre-FEC Q 2 -factor), and the GMI. The pre-FEC BER can be calculated from the transmitted and decided received symbols as:

𝐵𝐸𝑅 = 1 𝑁 • 𝐻 ∑ 𝑑 𝐻 (ℳ(𝑥 𝑘 ), ℳ(𝑥 ̂𝑘)) 𝑁 𝑘=1 ( 1.15 )
where 𝑥 𝑘 is the k'th transmitted symbol, N is the sequence length, ℳ(•) accounts for symbolto-bit mapping, 𝑑 𝐻 (•) is the Hamming distance, and 𝑥 ̂𝑘 is the k'th decided symbol obtained using maximum a posteriori (MAP) detection as:

𝑥 ̂𝑘 = argmin 𝑥∈𝐴 [|𝑦 𝑘 -𝑥| 2 - 𝑝 𝑋 (𝑥) 𝑆𝑁𝑅 ] ( 1.16 )
BER is usually converted to another performance metric called Q 2 -factor by:

𝑄 2 = 20𝑙𝑜𝑔 10 [√2 • 𝑒𝑟𝑓𝑐 -1 (2 • 𝐵𝐸𝑅)]
( 1.17 )

with 𝑒𝑟𝑓𝑐 -1 the inverse complementary error function.

While the GMI and the pre-FEC Q 2 -factor are performance metrics of a digital communication system depending on the modulation format and decoding schemes, the SNR is a direct measure of signal distortions. As it will be pointed out in Sec. 1.4, in modern optical long-haul optical fiber systems as the ones considered in this work, all sources of noise can be well approximated by Gaussian noise statistics, and can be correctly accounted for by SNR calculation. It can be analytically shown that the SNR can be estimated from the transmitted and received symbol sequences as:

𝑆𝑁𝑅 ̂= [ |𝒙| 2 |𝒚| 2 |𝒙𝒚 † | 2 -1] -1 ( 1.18 )
where † is the Hermitian conjugate.

Pulse Shaping

After choosing the modulation format, the signal has to be adapted to cope with the physical properties of the medium. Rectangular pulses are not suitable for transmitting data over band limited channels, as they will require infinite bandwidth to avoid distortion. However, limiting the signal bandwidth will produce pulses to spread over time, such that overlapping between different pulses will occur. At the receiver, the waveform will be converted back to the digital domain by means of the ADC. This process is realized by sampling the received waveform with a given sampling frequency. Without proper sampling and proper pulse shape, the sampled signal will suffer from inter-symbol interference (ISI).

The signal waveform can be written as:

𝑥(𝑡) = ∑ 𝑥 𝑘 𝑔(𝑡) 𝑘 ( 1.19 )
where 𝑥 𝑘 corresponds to the k'th transmitted constellation symbol. Each symbol will be multiplied by a given pulse shape, 𝑔(𝑡). For a given symbol-rate, the pulse shape 𝑔(𝑡) that minimizes the signal bandwidth is the 𝑠𝑖𝑛𝑐 function, as its Fourier transform is the rectangular function such that:

𝑔(𝑡) = 𝑠𝑖𝑛𝑐(𝑡) ( 1.20 ) ℱ{𝑔(𝑡)} = 𝑔 ̃(𝑓) = 𝑟𝑒𝑐𝑡(𝑓 • 𝑇 𝑠 ) ( 1.21 )
where 𝑇 𝑠 is the symbol duration. The bandwidth of the resulting complex signal is therefore 𝑊 = 𝑅 𝑠 , where 𝑅 𝑠 = 1/𝑇 𝑠 is the symbol-rate. One particular property of the 𝑠𝑖𝑛𝑐 function is that it is zero at all sampling instants 𝑡 = 𝑘𝑇 𝑠 except for 𝑘 = 0, i.e:

𝑔(𝑘𝑇) = { 1; 𝑘 = 0 0; 𝑘 ≠ 0 ( 1.22 )
such that perfect sampling at instances 𝑘𝑇 𝑠 allows recovering the transmitted data symbols without ISI. However, the main disadvantage of sync pulses is that the amplitude decays slowly such that unperfect sampling will lead to significant ISI. Another pulse shape which satisfies ( 1.22 ) is the raise-cosine (RC) pulse defined as:

𝑔 ̃(𝑓) = { 1, |𝑓| ≤ (1 -𝜌) 2𝑇 𝑠 1 2 [1 + 𝑐𝑜𝑠 ( 𝜋𝑇 𝑠 𝜌 [|𝑓| - (1 -𝜌) 2𝑇 𝑠 ])] , (1 -𝜌) 2𝑇 𝑠 < |𝑓| ≤ (1 + 𝜌) 2𝑇 𝑠 0, 𝑜𝑡ℎ𝑒𝑟𝑤𝑖𝑠𝑒 ( 1.23 )
where 𝜌 is the roll-off factor. RC pulses decay faster than the 𝑠𝑖𝑛𝑐 pulses when 𝜌 > 0, and are therefore more tolerant to imperfect sampling. However, the required spectrum is increased to

𝑊 = 𝑅 𝑠 (1 + 𝜌).
Another and most commonly used pulse shape is the root-raised cosine (RRC), which is obtained by applying the square root to the frequency domain version of ( 1.23 ). A priori, RRC pulses do not satisfy ( 1.22 ) and therefore exhibit ISI. However, when the received signal is corrupted by noise, the optimum receiver that maximizes the SNR for AWGN channels is called the matched-filter receiver. The match filter function equals 𝑔(𝑡), such that having an RRC pulse and an RRC matched filter will lead to an overall pulse shape with RC spectrum.

Throughout this work we will focus only on RRC pulse shaping.

COHERENT OPTICAL COMMUNICATIONS SYSTEMS

Up to know, an overview of a general digital telecommunication system, where information is transmitted from source to destination using a given physical channel, has been presented.

For the particular case of transoceanic optical systems, the signal has to be adapted to the fiber channel. Fig. 1.8 shows the main structure of a coherent long-haul optical communication system based on WDM technology, consisting of three main blocks: the optical transmitter, the

0 T s 2T s 3T s -T s -2T s -3T s 𝑔(𝑡) 2 𝑔(𝑓) 2 0 R s /2 R s -R s /2 -R s 𝜌 = 0 𝜌 = 1 (a) (b)
optical link, and the coherent optical receiver. In the following, an overview of each of the main blocks is presented.

Optical Transmitter

The main function of the optical transmitter is to convert an electrical waveform into the optical domain for further transmission through the optical link. For typical high-speed systems, this process is achieved by modulating the light of a laser source with an external optical modulator, which is driven by an electrical signal containing the data to be transmitted.

To maximize the spectral efficiency, all possible physical degrees of freedom of light should be considered. Modern coherent systems make use of amplitude, phase and polarization for data modulation. Therefore, any kind of multi-level modulation formats as the ones presented in Sec. 1.2.3 can be employed. The main structure of a WDM amplitude, phase and polarization diversity transmitter is sketched in Fig. 1.9.

After pulse shaping and pre-equalization, the two discrete waveforms to be transmitted over the two orthogonal polarizations 𝐻 and 𝑉 of light are divided into their real and imaginary parts BiCMOS technology working at 100 GS/s have been demonstrated [START_REF] Schuh | 100 GSa/s BiCMOS DAC Supporting 400 Gb/s Dual Channel Transmission[END_REF].

For high-speed long-haul transmission, MZ modulators are preferred over electroabsorption modulators (EML) due to their higher electro-optical bandwidth (~40 GHz), small insertion losses (≤4 dB), high extinction ratio (≥20 dB), and lower wavelength dependency.

Optical Link

The WDM signal at the output of the transmitter is then propagated through the optical link.

For long haul applications, the link consists of a concatenation of single mode fiber spans, and optical amplifiers as sketched in Fig. 1.8.

The most common type of amplification is the erbium doped fiber amplifier (EDFA). The output power of an EDFA can exceed 25 dBm, they present relative flat gain spectrum, can achieve noise figures (NF) below 5 dB, and amplification bandwidths of ~35 nm. Moreover, they can be designed to work on C or L bands, such that WDM systems with a useful bandwidth of ~9.7 THz have been demonstrated [START_REF] Cai | 70.4 Tb/s Capacity over 7,600 km in C+L Band Using Coded Modulation with Hybrid Constellation Shaping and Nonlinearity Compensation[END_REF]. Other amplification schemes based on Raman amplification, or hybrid Raman/EDFA also exist. However, due to their high pump power requiring high electrical power feeding, their use have been limited to terrestrial systems.

Recently, large bandwidth amplification covering continuous 100 nm optical bandwidth using

Digital

Microwave Optical Driver Amplifier 𝑥 𝐻𝐼 (𝑡)

𝑥 𝐻𝑄 (𝑡) 𝑥 𝑉𝐼 (𝑡) 𝑥 𝑉𝑄 (𝑡) π/2
semiconductor optical amplifiers (SOA) has also been demonstrated for terrestrial distances [START_REF] Renaudier | First 100-nm Continuous-Band WDM Transmission System with 115Tb/s Transport over 100km Using Novel Ultra-Wideband Semiconductor Optical Amplifiers[END_REF].

In legacy WDM systems, fiber CD was compensated either by introducing optical dispersion compensation units (DCU) in the link, or by the combination of fiber spans with positive and negative dispersion coefficients. These systems are typically referred as dispersion managed (DM). CD is now compensated in the digital domain thanks to the use of coherent detection and DSP, avoiding the use of the highly lossy DCUs, and simplifying the design of optical links.

These systems are referred as dispersion unmanaged (DU) and have become the standard for current optical coherent systems. In this work, we focus only in DU long-haul transmission systems based on EDFA amplification (Fig. 1.8). Furthermore, we consider the case where all spans are identical (homogeneous spans).

During propagation, the signal will be impaired by fiber linear and nonlinear effects, as well as by amplified spontaneous emission (ASE) noise introduced at each EDFA stage. These impairments can be analytically represented as following.

Let's consider the scalar optical field of a given WDM channel labeled by 𝑠, and with central carrier angular frequency 𝜔 𝑠 , be defined by:

𝐸 𝑠 (𝑧, 𝑡) = ∑ 𝑥 𝑘 𝑔 𝑘,𝑠 (𝑧, 𝑡) 𝑘 𝑒𝑥𝑝(-𝑖𝜔 𝑠 𝑡 + 𝑖𝛽(𝜔)𝑧) ( 1.24 )
where 𝑧 represents distance, 𝑡 represents time, 𝑥 𝑘 correspond to the k'th complex dimensionless modulation format symbols defining the message (Sec. 1.2.3), 𝑔 𝑠 (0, 𝑡) is the pulse shape of the 𝑠 'th optical channel waveform (Sec. 1.2.4), and 𝛽 is the propagation constant. The total optical field considering all WDM channels can then be expressed as:

𝐸(𝑧, 𝑡) = 𝐸 0 (𝑧, 𝑡) + ∑ 𝐸 𝑠 (𝑧, 𝑡) 𝑠 ( 1.25 )
where the channel of interest (COI) corresponding to 𝑠 = 0 has been singled out from the summation. In this work, we suppose that 𝑔 0 (0, 𝑡) = 𝑔 𝑠 (0, 𝑡) for all s, such that all channels have the same pulse shaping.

We define the pulse energy as:

𝜀 = ∫ |𝑔 0 (0, 𝑡)| 2 𝑑𝑡 ∞ -∞ ( 1.26 )
and the normalized optical field 𝑈(𝑧, 𝑡) at a given propagation distance z as:

𝑈(𝑧, 𝑡) = 𝐸(𝑧, 𝑡) 𝑓(𝑧)√𝜀 ( 1.27 )
where 𝑓(𝑧) is the normalized power profile along 𝑧 of the system shown in Fig. 1.8. The evolution of the scalar optical field along the optical fiber can be described by a normalized version of the nonlinear Schrödinger equation expressed by [START_REF] Ghazisaeidi | A Theory of Nonlinear Signal-Noise Interactions in Wavelength Division Multiplexed Coherent Systems[END_REF][START_REF] Mecozzi | Nonlinear Shannon Limit in Pseudolinear Coherent Systems[END_REF]:

𝜕𝑈 𝜕𝑧 = -𝑖 𝛽 2 2 𝜕 2 𝑈 𝜕𝑡 + 𝛽 3 6 𝜕 3 𝑈 𝜕𝑡 3 ⏟ 𝑑𝑖𝑠𝑝𝑒𝑟𝑠𝑖𝑜𝑛 + 𝑖𝛾𝜀𝑓(𝑧)|𝑈| 2 𝑈 ⏟ 𝐾𝑒𝑟𝑟 𝑛𝑜𝑛𝑙𝑖𝑛𝑒𝑎𝑟𝑖𝑡𝑖𝑒𝑠 + 𝑛(𝑧, 𝑡) √𝜀𝑓(𝑧) ⏟ 𝐴𝑆𝐸 𝑛𝑜𝑖𝑠𝑒 ( 1.28 )
where 𝛽 2 is the group velocity dispersion (GVD) corresponding to the acceleration of the spectral components of the pulse, 𝛽 3 is the GVD slope accounting for the variation of GVD as a function of the angular frequency , 𝛾 = 2𝜋𝑛 2 /𝜆 0 𝐴 𝑒𝑓𝑓 is the nonlinear coefficient, 𝑛 2 is the fiber nonlinear Kerr refractive index, 𝐴 𝑒𝑓𝑓 is the effective area, 𝜆 0 is the COI wavelength, and 𝑛(𝑧, 𝑡)

is the ASE noise source. Please note that we assume 𝛽 2 , 𝛽 3 and 𝛾 to be 𝑧 independent, such that all spans are made of the same fiber. For lumped amplification, and considering that the EDFA exactly compensates for the span loss, 𝑓(𝑧) can be expressed as:

𝑓(𝑧) = 𝑒𝑥𝑝 (-𝛼(𝑧 -𝑧 𝑛-1 )) 𝑧 𝑛-1 ≤ 𝑧 ≤ 𝑛=1…𝑁 𝑠𝑝𝑎𝑛 𝑧 𝑛 ( 1.29 )
where 𝛼 is the attenuation coefficient of the fiber.

In the case of dual polarization, ( 1.28 ) is divided into two coupled NLSE equations for which the Manakov approximation is commonly used to account for the random birefringence along the optical fiber. The normalized Manakov equation is expressed by:

𝜕𝑼 𝜕𝑧 = -𝑖 𝛽 2 2 𝜕 2 𝑼 𝜕𝑡 + 𝛽 3 6 𝜕 3 𝑼 𝜕𝑡 3 + 8 9 𝛾𝜀̅ 𝑓(𝑧)𝑼 † 𝑼𝑼 + 𝒏(𝑧, 𝑡) √𝜀̅ 𝑓(𝑧) ( 1.30 )
where 𝑼 = [𝑈 𝑉 (𝑧, 𝑡), 𝑈 𝐻 (𝑧, 𝑡)] 𝑇 , 𝒏(𝑧, 𝑡) = [𝑛 𝑉 (𝑧, 𝑡), 𝑛 𝐻 (𝑧, 𝑡)] 𝑇 , and 𝜀̅ = 𝜀 2 ⁄ . The superscript 𝑇 stands for matrix transpose operation and the superscript † stands for Hermitian conjugation.

In the following we assume a scalar field for simplicity of explanation.

Fiber Dispersion

When light propagates through a dielectric medium as the fiber, its speed will be decreased proportional to the dielectric constant of the medium. For an optical fiber, the phase velocity, 𝑣 𝑝 , is:

𝑣 𝑝 = 𝑐 𝑛(𝜔) ( 1.31 )
where 𝑐 is the speed of light in vacuum, and 𝑛(𝜔) is the refractive index of the fiber which is in general frequency dependent. The propagation constant 𝛽 is related to 𝑣 𝑝 by the following expression:

𝛽(𝜔) = 𝜔 𝑣 𝑝 ( 1.32 )
The frequency dependence of 𝛽 means that different spectral components of the field will travel with different velocities during propagation. For normal dispersive fibers, this leads to a temporal broadening of the optical pulses within each channel, and a walk-off between pulses belonging to different WDM channels.

Eq. ( 1.32 ) can be expanded in a Taylor series with respect to the COI central frequency as:

𝛽(𝜔) = 𝛽 0 + 𝛽 1 (𝜔 -𝜔 0 ) + 1 2 𝛽 2 (𝜔 -𝜔 0 ) 2 + 1 6 𝛽 3 (𝜔 -𝜔 0 ) 3 + ⋯ ( 1.33 )
where 𝛽 𝑖 is the 𝑖'th derivative of 𝛽 with respect to 𝜔. For a perfectly cylindrical fiber, 𝛽 0 and 𝛽 1 do not lead to any direct signal degradation during transmission. On the other hand, the group velocity dispersion 𝛽 2 is responsible for pulse broadening, and is linked to the dispersion coefficient 𝐷 through the following expression:

𝐷 = - 2𝜋𝑐 𝜆 2 𝛽 2 ( 1.34 )
The third order dispersion 𝛽 3 is related to the dispersion slope parameter 𝑆, by:

𝑆 = 4𝜋𝑐 𝜆 3 (𝛽 2 + 𝜋𝑐 𝜆 𝛽 3 ) ( 1.35 )
For DU systems working away from the zeroth dispersion point as the ones considered in this work, 𝛽 3 is very small compared to 𝛽 2 and can be neglected 1 . In this case, considering only the group velocity dispersion 𝛽 2 , and neglecting fiber nonlinearities (i.e., 𝛾 = 0 ), ( 1.28 ) can be solved straightforward, leading to:

𝑢(𝑧, 𝑡) = 𝐷 ̂𝑧[𝑢(0, 𝑡) + 𝑢 𝐴𝑆𝐸 (𝑧, 𝑡)] ( 1.36 )
where

𝐷 ̂𝑧[𝑢(0, 𝑡)] = ℱ -1 {𝑒𝑥𝑝 (-𝑖 𝛽 2 2 𝜔 2 𝑧) ℱ{𝑢(0, 𝑡)}}
( 1.37 ) 1 When working with high symbol-rate channels this approximation is less accurate, such that 𝛽 3 has to be considered for CD compensation.

is the dispersion operator, 𝑢(0, 𝑡) is the optical field at the input of the link, and 𝑢 𝐴𝑆𝐸 (𝑧, 𝑡)

accounts for the accumulated ASE noise at a given distance 𝑧 . As fiber dispersion is deterministic, it can be efficiently compensated for in the digital domain by applying the adjoint of the operator 𝐷 ̂𝑧 to the received signal. This technique will be addressed in Sec.1.3.4.2.

Consider a DU system composed of an optical fiber with dispersion coefficient 𝐷 = 20.6 ps/nm•km (i.e., Corning Vascade EX3000), after 6,000 km transmission, the accumulated dispersion reaches 1.23x10 5 ps/nm. An optical pulse corresponding to a WDM channel modulated at 32 GBd will broaden almost 1000 times its width.

Fiber Kerr Nonlinearities

The last term in ( 1.28 ) corresponds to nonlinear interactions due to the Kerr effect, which causes variations of the refractive index of the fiber proportional to the power of the optical field.

Fiber nonlinearities have been widely studied over the years, and their effects on optical pulse propagation have been commonly described within the frame of DM systems, where the pulses do not drastically change their shape during propagation. Within this frame, nonlinear effects were divided into categories depending on their specific way of impacting the optical signal,

self-phase modulation (SPM), cross-phase modulation (XPM), non-degenerated four-wave mixing (FWM), and degenerated FWM.

In this frame, since 2010 intense research has been made on developing analytical models to describe nonlinear interactions in PM WDM coherent systems. These models have proved by means of numerical simulations and experiments to be quite accurate on predicting system performance for DU links working in the pseudolinear regime. The majority of these models rely on solving ( 1.28 ) by applying first order regular perturbation (RP1) theory.

The first group of these models rely on a frequency domain approach, where the optical spectrum is sliced into spectral components and nonlinear interactions are analytically expressed similar to classical four wave mixing (FWM) formulation. The most well-known model is called the Gaussian Noise (GN) model and was formally introduced in [START_REF] Poggiolini | Analytical Modeling of Nonlinear Propagation in Uncompensated Optical Transmission Links[END_REF][START_REF] Carena | Modeling of the Impact of Nonlinear Propagation Effects in Uncompensated Optical Coherent Transmission Links[END_REF][START_REF] Poggiolini | The GN-Model of Fiber Non-Linear Propagation and its Applications[END_REF][START_REF] Johannisson | Perturbation Analysis of Nonlinear Propagation in a Strongly Dispersive Optical Communication Systems[END_REF]. The key underlying hypothesis of the GN model is that the transmitted signal statistically behaves as a stationary random Gaussian process, and that fiber nonlinearities lead to a nonlinear interference (NLI) which manifest itself as an additive white Gaussian noise (AWGN). These assumptions seem to hold in long-haul DU systems, where large values of accumulated dispersion make the power spectral density (PSD) of the signal tend to a circular complex Gaussian distribution independently of the modulation format. It is important to note that early attempts of the "GN model" tend back to 1993 [START_REF] Splett | Ultimate transmission capacity of amplified optical fiber communication systems taking into account fiber nonlinearities[END_REF], however the signal Gaussianity assumption didn't hold for the DM systems used at that time and thus preventing widespread research.

The second group of these models utilize a time-domain analysis, where fiber nonlinearities are modeled by the FWM interaction between signal pulses at different time instances. The first study was introduced in [START_REF] Mecozzi | System impact of intrachannel nonlinear effects in highly dispersed optical pulse transmission[END_REF] and retaken in [START_REF] Mecozzi | Nonlinear Shannon Limit in Pseudolinear Coherent Systems[END_REF] where a rigorous analysis of the RP1 method was carried out considering a multi span amplified link with coherent detection. Contrary to the frequency domain analysis, no signal Gaussianity has to be assumed. In [START_REF] Dar | Properties of nonlinear noise in long dispersionuncompensated fiber links[END_REF][START_REF] Dar | Accumulation of nonlinear interference noise in fiber-optic systems[END_REF][START_REF] Dar | Inter-Channel Nonlinear Interference Noise in WDM Systems: Modeling and Mitigation[END_REF][START_REF] Dar | Pulse Collision Picture of Inter-Channel Nonlinear Interference in Fiber-Optic Communications[END_REF], a proper formulation considering modulation format dependency was developed, leading also to an updated version of the GN model called enhanced-GN (EGN) [START_REF] Carena | EGN model of non-linear fiber propagation[END_REF]. Later on, a rigorous theory based on [START_REF] Mecozzi | Nonlinear Shannon Limit in Pseudolinear Coherent Systems[END_REF][START_REF] Dar | Accumulation of nonlinear interference noise in fiber-optic systems[END_REF], but including nonlinear signal-noise interactions (NSNI) as first proposed in [START_REF] Serena | Nonlinear SignalNoise Interaction in Optical Links With Nonlinear Equalization[END_REF], was developed in [START_REF] Ghazisaeidi | A Theory of Nonlinear Signal-Noise Interactions in Wavelength Division Multiplexed Coherent Systems[END_REF].

Finally, a third approach also in time domain was developed in [START_REF] Serena | An Alternative Approach to the Gaussian Noise Model and its System Implications[END_REF][START_REF] Serena | A Time-Domain Extended Gaussian Noise Model[END_REF], where the autocorrelation function of the nonlinear distortions was propagated along the link in a similar way as the split step Fourier (SSFM) method. However, contrary to the SSFM where the link is modeled by a single path composed of a concatenation of linear (𝐷 ̂) and nonlinear (𝑁 ̂) blocks, this method used several parallel paths each of them containing only one 𝑁 ̂ element, such that parallel computing could be performed.

For a suitable explanation to describe the different types of nonlinear interactions, we make use the first kind of models, where nonlinear interactions are modeled as a FWM process between four frequency components of the optical spectrum [START_REF] Caren | Modeling of the Impact of Nonlinear Propagation Effects in Uncompensated Optical Coherent Transmission Links[END_REF]. Let's assume for the moment the propagation through an optical fiber of only one WDM channel with a given spectral bandwidth 𝑊 . The FWM approach consists of slicing up its spectrum into an infinitesimal number of spectral components as shown in Fig. 1.10, and then to analyze their nonlinear beating using classical FWM formulation. Let the single WDM channel be represented as the sum of different discrete spectral tones of the form:

𝑈(𝑧, 𝑡) = ∑ 𝜁 𝑖 (𝑧) 𝑖 𝑒𝑥𝑝(-𝑖𝜔 𝑖 𝑡) ( 1.38 ) ω 𝑈(𝑧, 𝜔) 2
where 𝜁 𝑖 (𝑧) represents the spectrum of the spectral tone 𝜔 𝑖 , and is considered to be a zeromean random variable whose statistics depends on the transmitted symbols [START_REF] Dar | Properties of nonlinear noise in long dispersionuncompensated fiber links[END_REF]. Inserting ( 1.38 ) into the nonlinear term of ( 1.28 ), leads to:

𝑖𝛾𝜀𝑓(𝑧)|𝑈| 2 𝑈 = 𝑖𝛾𝜀𝑓(𝑧) ∑ ∑ ∑ 𝜁 𝑖 𝜁 𝑗 * 𝜁 𝑘 exp (-𝑖(𝜔 𝑖 -𝜔 𝑗 + 𝜔 𝑘 )) 𝑘 𝑗 𝑖 ( 1.39 )
From ( 1.39 ) the fiber nonlinearities produce nonlinear beatings between any three spectral components of the optical field, giving rise to a nonlinear interference (NLI) tone located at 𝜔 𝑙 = 𝜔 𝑖 -𝜔 𝑗 + 𝜔 𝑘 . The strength of the NLI will depend on system parameters as: modulation format, link power profile, dispersion coefficient, etc. The above process can be generalized to multiple WDM channels, and nonlinear distortions can be classified depending on the spectral position of the interfering frequency tones 𝜔 as shown in Fig. 1.11. The first type of nonlinear interactions corresponds to the case where all spectral tones rely within the COI. This process can be cataloged as self-channel interference (SCI). It induces phase variations which are proportional to the instantaneous power of the COI itself, and which are transformed into amplitude variations by the coupling between dispersion and nonlinearities.

The second type can be classified as cross channel interference (XCI), which corresponds to the case where NLI arises from the interaction of the COI with only one any other channel.

0 𝜔 1 𝜔 (a) COI NLI 0 𝜔 2 𝜔 3 𝜔 1 𝜔 (b) COI NLI 0 𝜔 1 𝜔 2 𝜔 3 𝜔 (c) COI NLI 0 𝜔 1 𝜔 2 𝜔 3 𝜔 (d) COI NLI 𝜔 2 𝜔 3
XCI induces a phase shift on the COI proportional to the instantaneous power of the interfering channels, which will also be translated into amplitude variations by the interplay with fiber dispersion. Moreover, a polarization dependent NLI will induce a different phase-shift onto each polarization component of the field. As the change on SOP will depend on the instantaneous power of the co-propagating channels, it results in a noise-like SOP scattering and, hence, depolarization.

The third and fourth type can be classified as multi-channel interference (MCI), which arise from the interaction between the COI and two or three other channels. The strength of MCI in most relevant fibers with channel symbol-rates above 25 GBd, is of second importance and can be safely neglected.

The previous types of NLI can be further classified into two general groups: intra-channel nonlinearities containing SCI only, and inter-channel nonlinearities containing XCI and MCI. A priori, nonlinear interactions are deterministic and can be compensated for in the digital domain, if the information of all WDM channels is available. However, since joint processing of multiple WDM channels is currently considered to be prohibitively complex for commercial systems, inter-channel NLI is commonly treated as noise.

For the rest of this work, we will follow the time-domain analysis of [START_REF] Ghazisaeidi | A Theory of Nonlinear Signal-Noise Interactions in Wavelength Division Multiplexed Coherent Systems[END_REF] and [START_REF] Mecozzi | Nonlinear Shannon Limit in Pseudolinear Coherent Systems[END_REF] to describe NLI, as this method will be used to evaluate the NLI variance for performance analysis in Sec.

1.4.3, and to calculate the inter-channel NLI for DSP compensation in Sec. 1.3.5.2.

Eq. ( 1.28 ) can be solved using RP1 theory assuming that fiber nonlinearity acts as a small perturbation to the field. In the following, a fast description of the derivation of nonlinear distortions is presented. For a detail derivation, the reader is invited to [START_REF] Ghazisaeidi | A Theory of Nonlinear Signal-Noise Interactions in Wavelength Division Multiplexed Coherent Systems[END_REF], and [START_REF] Mecozzi | Nonlinear Shannon Limit in Pseudolinear Coherent Systems[END_REF]. In the frame of RP1 theory, we can expand 𝑈 with respect to 𝛾 as:

𝑈(𝑧, 𝑡) = 𝑢 (0) (𝑧, 𝑡) + ∆𝑢 𝑁𝐿 ( 1.40 )
where ∆𝑢 𝑁𝐿 (𝑧, 𝑡) = 𝛾𝑢 (1) (𝑧, 𝑡), and 𝑢 (𝑛) (𝑧, 𝑡) is the n'th order perturbation of the normalized field 𝑈(𝑧, 𝑡). By substituting ( 1.40 ) into ( 1.28 ), and neglecting 𝛽 3 we obtain the zeroth and first order terms of the form:

𝜕𝑢 (0) (𝑧, 𝑡) 𝜕𝑧 = -𝑖 𝛽 2 2 𝜕 2 𝑢 (0) (𝑧, 𝑡) 𝜕𝑡 + 𝑛(𝑧, 𝑡) √𝜀𝑓(𝑧) ( 1.41 ) 𝜕∆𝑢 𝑁𝐿 (𝑧, 𝑡) 𝜕𝑧 = -𝑖 𝛽 2 2 𝜕 2 ∆𝑢 𝑁𝐿 (𝑧, 𝑡) 𝜕𝑡 + 𝑖𝛾𝜀𝑓(𝑧)|𝑢 (0) (𝑧, 𝑡)| 2 𝑢 (0) (𝑧, 𝑡) ( 1.42 )
Whose solutions are:

𝑢 0 (𝑧, 𝑡) = 𝐷 ̂𝑧[𝑢 0 (0, 𝑡) + 𝑢 𝐴𝑆𝐸 (𝑧, 𝑡)] ( 1.43 ) ∆𝑢 𝑁𝐿 (𝑧, 𝑡) = 𝑖𝛾𝜀𝐷 ̂𝑧 ∫ 𝑓(𝑧 ′ )𝑑𝑧′×𝐷 ̂𝑧′ † 𝑧 0 [|𝑢 (0) (𝑧 ′ , 𝑡)| 2 𝑢 (0) (𝑧′, 𝑡)] ( 1.44 )
where 𝐷 ̂𝑧 † is the adjoint of 𝐷 ̂𝑧. For an optical link of length L, the total field before detection (𝑧 = 𝐿) may be conveniently written as:

𝑢(𝐿, 𝑡) = 𝐷 ̂𝐿[𝑢 (0) (0, 𝑡) + 𝑢 𝐴𝑆𝐸 (𝐿, 𝑡)] + ∆𝑢 𝑁𝐿 (𝐿, 𝑡) ( 1.45 )
The resulting signal is the sum of the dispersed transmitted pulses and ASE noise, plus nonlinear distortions coming from the Kerr effect. At the receiver, the signal is first filtered to extract the COI, and then mixed with a local oscillator (LO) to perform balanced coherent detection (Seq. 1.3.3). Assuming the total modulated WDM waveform at the input be expressed by:

𝑠(0, 𝑡) = ∑ 𝑥 𝑘 𝑢 𝑘 (0) (0, 𝑡) 𝑘 + ∑ 𝑥 𝑘,𝑠 𝑢 𝑘,𝑠 (0) (0, 𝑡 -𝛿𝑇 𝑠 ) 𝑘,𝑠≠0
𝑒𝑥𝑝(-𝑖𝛺 𝑠 𝑡 + 𝑖𝜙 𝑠 (0))

( 1.46 )
where 𝛺 𝑠 , 𝜙 𝑠 , and 𝛿𝑇 𝑠 are the frequency, phase and time offsets of the 𝑠'th channel with respect to the COI. After sampling and match filtering, the sampled photocurrent can be expressed as:

𝐼 𝑘 = 𝐴 𝐿0 * [𝑥 𝑘 + ∆𝑥 𝑘,𝐴𝑆𝐸 + ∆𝑥 𝑘,𝑁𝐿 ] ( 1.47 )
where 𝐴 𝐿0 * is the amplitude of the LO, 𝑥 𝑘 are the transmitted symbols on the COI, ∆𝑥 𝑘,𝐴𝑆𝐸 is a linear perturbation due to ASE noise, and ∆𝑥 𝑘,𝑁𝐿 is a nonlinear perturbation term of the form:

∆𝑥 𝑘,𝑁𝐿 = 𝑖𝛾𝜀 ∫ 𝑑𝑧𝑓(𝑧)|𝑢 (0) (𝑧, 𝑡)| 2 𝑢 (0) (𝑧, 𝑡)𝑢 𝑘 (0) * (𝑧, 𝑡) 𝐿 0 ( 1.48 )
The previous equation represents the total nonlinear distortions using RP1 theory. By inserting ( 1.46 ) into ( 1.48 ) and neglecting NSNI, ∆𝑥 𝑘,𝑁𝐿 can be written as:

∆𝑥 𝑘,𝑁𝐿 = ∑ 𝑥 𝑚+𝑘 𝑥 𝑛+𝑘 𝑥 𝑝+𝑘 * 𝐶 𝑚,𝑛,𝑝 (0,0) 𝑚,𝑛,𝑝 + 2 ∑ ∑ 𝑥 𝑚+𝑘 𝑥 𝑛+𝑘,𝑠 𝑥 𝑝+𝑘,𝑠 * 𝐶 𝑚,𝑛,𝑝 (0,𝑠) 𝑚,𝑛,𝑝 𝑠 + ∑ ∑ 𝑥 𝑚+𝑘,𝑠 𝑥 𝑛+𝑘,𝑠 ′ 𝑥 𝑝+𝑘,𝑠+𝑠 ′ * 𝐶 𝑚,𝑛,𝑝 (𝑠,𝑠 ′ ) 𝑚,𝑛,𝑝 𝑠,𝑠 ′ 𝑠≠𝑠 ′ ( 1.49 )
where the term 𝐶 𝑚,𝑛,𝑝 corresponds to the system kernel weighting the nonlinear interactions between different signal symbols, and is expressed as:

𝐶 𝑚,𝑛,𝑝 (𝑠,𝑠 ′ ) = 𝑖𝛾𝜀 ∫ 𝑑𝑧𝑓(𝑧)𝑒 𝑖[𝜙 𝑠 (𝑧)+𝜙 𝑠 ′ (𝑧)-𝜙 𝑠+𝑠 ′ (𝑧)] 𝐿 0 × ∫ 𝑑𝑡 𝑢 0 (0) * (𝑧, 𝑡)𝑢 𝑚,𝑠 (0) (𝑧, 𝑡 -𝑡 𝑠 )𝑢 𝑛,𝑠 ′ (0) (𝑧, 𝑡 -𝑡 𝑠 ′ )𝑢 𝑝 (0) * (𝑧, 𝑡 -𝑡 𝑠+𝑠 ′ ) ( 1.50 )
The first, second and third sums in ( 1.49 ) correspond to intra-channel (SCI), degenerate inter-channel (XCI) and non-degenerate inter-channel (MCI) FWM terms. As stated before, for long-haul DU links with channel symbol-rates above 25 GBd, the MCI term is much smaller than SCI and XCI, and can be safely neglected.

ASE Noise

The second term of ( 1.47 ) corresponds to ASE noise added to the 𝑘'th symbol of the COI, and can be modeled as a zero-mean Gaussian random variable of variance 𝜎 𝐴𝑆𝐸 2 . For an EDFA, the PSD of ASE over the two signal polarizations is given by:

𝑁 𝐴𝑆𝐸 = 2𝑛 𝑠𝑝 ℏ𝜔 0 (𝐺 -1) ( 1.51 )
where 𝑛 𝑠𝑝 is the spontaneous emission factor, ℏ is the Plank's constant divided by 2π, 𝜔 0 is the reference angular frequency, and 𝐺 is the gain of the amplifier. It is more practical to link the quantity 𝑛 𝑠𝑝 to the most commonly used noise factor, 𝐹, defined as the ratio between the optical signal-to-noise ratio (OSNR) at the input and at the output of the amplifier by:

𝐹 = 𝑂𝑆𝑁𝑅 𝑖𝑛 𝑂𝑆𝑁𝑅 𝑜𝑢𝑡 = 1 𝐺 + 2𝑛 𝑠𝑝 (𝐺 -1) 𝐺 ( 1.52 )
In this case, 𝑁 𝐴𝑆𝐸 can be expressed as:

𝑁 𝐴𝑆𝐸 = ℏ𝜔 0 (𝐺𝐹 -1) ( 1.53 )
The ASE PSD is assumed constant over a given reference bandwidth 𝐵 𝑟𝑒𝑓 , which is conventionally chosen to be 0.1 nm (12.5 GHz). In the case of homogeneous spans, the total ASE power over the reference bandwidth can be simply defined as:

𝑃 𝐴𝑆𝐸 = 𝑁 𝐴𝑆𝐸 𝐵 𝑟𝑒𝑓 𝑁 𝑠𝑝𝑎𝑛𝑠 ( 1.54 )
For a wavelength around 1550 nm and assuming that optical amplifiers completely compensate for span loss, ( 1.54 ) can be approximated to:

𝑃 𝐴𝑆𝐸 𝑑𝐵𝑚 ≈ -58 𝑑𝐵𝑚 + 𝑁𝐹 + 10𝑙𝑜𝑔10(𝑁 𝑠𝑝𝑎𝑛𝑠 ) + 𝛼𝐿 𝑠𝑝𝑎𝑛 ( 1.55 )
where 𝑁𝐹 is the amplifier noise figure in dB, i.e., 𝑁𝐹 = 10𝑙𝑜𝑔10(𝐹) .

Polarization Mode Dispersion

Real optical fibers present unintentional variations in the core shape due to stresses or deformations during the manufacturing process. This circular asymmetry leads to fiber birefringence. Therefore, an input pulse to the fiber will split into two orthogonal fast and slow polarization modes, which will acquire a relative time delay due to a difference in their group velocities. This relative time delay is referred as differential group delay (DGD), and is expressed by:

∆𝑡 = 𝜏 𝐷𝐺𝐷 = 𝐿. ∆𝛽 ( 1.56 )
where ∆𝛽 is the difference in group velocity between the fast and slow modes, and 𝐿 is the propagation distance. DGD and is sketched in Fig. 1.12a.

However, fiber birefringence is not constant but changes randomly along the fiber. These In PM WDM systems, PMD leads to coherent crosstalk between the two orthogonal multiplexed polarizations (𝐻, 𝑉), and to changes in the relative polarization state of the different wavelength channels as they propagate along the optical fiber. When operating in the linear regime, the first effect can be efficiently compensated by adaptive equalization algorithms as the one described in Sec. 1.3.4.3, while the second is of no importance.

Considering the fiber nonlinear regime, stochastic nonlinear interactions will take place due to the random polarization changes between different spectral components. In this case, the variance of the NLI term in ( 1.47 ) is not constant, and has been shown to statistically follow a

Gaussian distribution [START_REF] Rossi | Stochastic Nonlinear Interference in Dispersion Managed Coherent Optical Links[END_REF]. While the effects of PMD on system performance in DU systems are very small due to the large accumulated CD, it becomes a fundamental limitation for nonlinear compensation techniques.

Polarization Dependent Loss

Besides PMD, another polarization effect is polarization dependent loss (PDL). It results from the asymmetry in the insertion loss or gain of an optical element (i.e., amplifiers, isolators, re-configurable optical add-drop multiplexers (ROADMs), etc.).

The transfer function of a PDL element can be modeled as:

𝑯 𝑃𝐷𝐿 = [ cos (𝛼) sin (𝛼) -sin (𝛼) cos (𝛼) ] [ 1 0 0 √𝛤 ] [ cos (𝛼) -sin (𝛼) sin (𝛼) cos (𝛼) ]
( 1.60 )

𝛤 𝑑𝐵 = -10𝑙𝑜𝑔 10 (𝛤 ) ( 1.61 )
where 𝛤 𝑑𝐵 ≥ 0 is the PDL coefficient in dB, and 𝛼 corresponds to a mismatch between the polarization states of the incident signal and the principal polarization states (PPS) of the PDL component.

Fig. 1.13: PDL effect: the reference signal without PDL propagates through a PDL element aligned with (a) its polarization axes, and (b) at an angle of θ=45° with respect to its polarization axes.

Due to fiber birefringence, PDL translates into signal power fluctuations depending on the random evolution of the states of polarization (SOP), leading to OSNR penalty and break of the orthogonality between the polarization tributaries. Similar to PMD, PDL is a random process whose statistics follow a Maxwellian distribution when expressed in dB as shown in [START_REF] Mecozzi | The Statistics of Polarization-Dependent Loss in Optical Communication Systems[END_REF].

Contrary to the unitary processes as CD or PMD, the power fluctuations caused by PDL cannot be undone by digital equalization. System margins are typically considered in order to account for PDL-induced penalties. Other techniques aiming to mitigate PDL effects as Polarization-Time coding [START_REF] Awwad | Polarization-Time Coding for PDL Mitigation in Long-Haul PolMux OFDM Systems[END_REF], or Pairwise coding [START_REF] Zhu | Improved Polarization Dependent Loss Tolerance for Polarization Multiplexed Coherent Optical Systems by Polarization Pairwise Coding[END_REF] have been suggested.

Optical Coherent Receiver

After transmission, the COI is optically filtered and detected. The main function of the coherent receiver is to convert the incoming optical signal back to the electrical domain.

Assuming intradyne coherent detection, the optical signal is coherently combined with the optical field of a continuous wave free-running LO before optical-electrical conversion. Fig. 1.14

shows the structure of a full coherent polarization diverse receiver.

At its input, the signal is divided into its two orthogonal polarizations components by the use of polarization beam splitters (PBS). Each polarization enters a free-space mixer structure whose second input is fed by the unmodulated light coming from a LO tuned at the same wavelength of the signal. Within each mixer, the LO is separated into two 90°-phase shifted parts that interfere with the incident signal. This process produces two pairs of outputs in quadrature for each polarization. The 8 resulting optical fields are then detected individually using PIN photodiodes. Balanced detection is used to suppress the DC components corresponding to direct detection. The photocurrents after balanced detection are described as: The main advantage of coherent detection is evident from ( 1.62 ). First, as the generated photocurrents are proportional to 𝐴 𝐿𝑂 , by making the power of the LO higher than that of the signal, the receiver sensitivity can be greatly improved. Moreover, 𝐼 and 𝑄 signal components can be fully recovered; thus, a complete reconstruction of the field can be made and advanced post-processing algorithms can be applied to compensate for system impairments. Finally, the two orthogonal polarizations are detected independently, such that information can be coded into each polarization.

(

Standard Digital Signal Processing Blocks

The effects that are compensated for in the digital domain by means of DSP in current coherent optical systems are: CD, PMD, frequency and phase time-varying offsets, and transmitter/receiver 𝐼𝑄 imbalances. The main DSP clocks compensating for these effects are sketched in Fig. 1.15, and will be discussed in the following.

Signal Conditioning

According to the Shannon-Nyquist criterion, the received signal has to be sampled at a rate of at least twice its bandwidth. The first step of DSP is to resample the signal at 2 samples per symbol (sps), followed by a digital filter to suppress out-of-band noise.

At this stage, any residual DC component of the signal can also be removed to compensate for signal components that are artifacts of the receiver structure and not present in the optical spectrum. The signals are also normalized in power, removing any power imbalance within the receiver, and simplifying much of the signal processing performed later.

Chromatic Dispersion Compensation

As presented in Sec. 1.3.2, mitigation of fiber CD can be performed by applying the adjoint of the dispersion operator 𝐷 ̂𝑧 .For a total propagation distance 𝐿, CD compensation can be written as: The operator 𝐷 ̂𝐿 † can be efficiently implemented using overlap-and-save methods and fast Fourier transforms (FFT) [START_REF] Xu | Frequency-Domain Chromatic Dispersion Equalization Using Overlap-Add Methods in Coherent Optical System[END_REF].

𝒚 ̂𝐻 = 𝐷 ̂𝐿 † [𝒚 𝐻 ] ( 1.

Adaptive Equalization

A key aspect of a polarization diversity coherent receiver is to recover the two orthogonal polarization tributaries. Due to fiber birefringence, the two polarizations will be coupled and ISI will occur. As presented in Sec. 1.3.2.4, fiber birefringence can be modeled as a concatenation of birefringent sections whose axes change randomly along the fiber.

The total concatenation of birefringent sections can be modelled into an equivalent channel impulse response. The task of the equalizer is to estimate the inverse of the impulse response to reverse the effects of polarization coupling and PMD. The equalizer performs the following matrix operation:

[ 𝑦 ̂𝑘 𝐻 𝑦 ̂𝑘 𝑉 ] = [ 𝒉 𝑘 𝐻𝐻 𝒉 𝑘 𝑉𝐻 𝒉 𝑘 𝐻𝑉 𝒉 𝑘 𝑉𝑉 ] [ 𝒚 𝑘 𝐻 𝒚 𝑘 𝑉 ] ( 1.65 )
where 𝑦 𝑘 and 𝑦 ̂𝑘 are the 𝑘-th symbols at the input and output of the equalizer respectively. We define 𝒚 𝑘 as a 2𝑁 𝑡 +1 long vector (𝒚 𝑘 = (𝑦 𝑘-𝑁 𝑡 , … 𝑦 𝑘 , . . 𝑦 𝑘+𝑁 𝑡 ) 𝑇 ), and the 𝑁 𝑡 -tap equalizer filters 𝒉 𝑘 = (ℎ 𝑘,-𝑁 𝑡 , … ℎ 𝑘 , . . ℎ 𝑘,𝑁 𝑡 ) . The problem of adaptive equalization is finding the equalizer filters 𝒉 as a function of time (𝑘). A common algorithm for constant amplitude signals is the constant modulus algorithm (CMA) which tries to minimize the following error function:

𝜀 𝑘 𝐻 = 1 -|𝑦 ̂𝑘 𝐻 | 2
( 1.66 )

𝜀 𝑘 𝑉 = 1 -|𝑦 ̂𝑘 𝑉 | 2
In the case of multi-level modulation formats, another algorithm referred as multi-modulus algorithm (MMA) can be used. In this case, the error function takes the form

𝜀 𝑘 𝐻 = 𝑟̂2 -|𝑦 ̂𝑘 𝐻 | 2
( 1.67 )

𝜀 𝑘 𝑉 = 𝑟̂2 -|𝑦 ̂𝑘 𝑉 | 2
where 𝑟̂ is the closest constellation amplitude to the received symbol. The estimation can be improved by sending a periodic sequence of known pilot symbols. In this case, the error function can be modified to:

𝜀 𝑘 𝐻 = (1 -𝑝 𝑘 ) • 𝑟̂2 + 𝑝 𝑘 • |𝑥 𝑘 𝐻 | 2 -|𝑦 ̂𝑘 𝐻 | 2
( 1.68 )

𝜀 𝑘 𝑉 = (1 -𝑝 𝑘 ) • 𝑟̂2 + 𝑝 𝑘 • |𝑥 𝑘 𝐻 | 2 -|𝑦 ̂𝑘 𝑉 | 2
where 𝑥 𝑘 is the k-th transmitted pilot symbol, and 𝑝 𝑘 is k-th element of a binary gating function indicating the position of the pilot symbols (𝑝 𝑘 ∈ {0,1}). Considering a pilot rate 𝑟 𝑝 = 𝑁 𝑝 𝐾 𝑝 ⁄ (i.e., sending 𝑁 𝑝 consecutive pilot symbols at the beginning of a sequence block of length 𝐾 𝑝 ), the gating function is:

𝑝(𝑘) = 𝑟𝑒𝑐 ( 𝑘 𝑁 𝑝 ) ⨂ ∑ 𝛿(𝑘 -(𝑛 -1)𝐾 𝑝 -1) (2𝑁+1) 𝐾 𝑝 ⁄ 𝑛=1 ( 1.69 )
where 𝑟𝑒𝑐(•) is the rectangular function, and 𝛿(•) the Dirac-delta function. The equalizer filters 𝒉 can then be updated by stochastic gradient descent [START_REF] Savory | Digital filters for coherent optical receivers[END_REF]. In addition to polarization dependent impairments, this adaptive equalizer can also be used to implement the matched filter, find the optimal sampling instant and compensate residual CD [START_REF] Hauske | Optical Performance Monitoring in Digital Coherent Receivers[END_REF].

Carrier Estimation

As seen in Seq. 1.3.3, the use of a free running LO will lead to frequency and time-varying phase offsets. Therefore, the problem of carrier estimation can be divided into a coarse estimation of the frequency offset, followed by a fine estimation of the slowly varying phase and residual frequency offset.

Assuming that CD, DGD, and polarization demultiplexing have been completely compensated for, and ignoring for NL effects, the signal of one of the polarization tributaries at the output of the equalizer can be written as:

𝑦 𝑘 = 𝑥 𝑘 𝑒𝑥𝑝(𝑖𝜁 𝑘 𝑘 + 𝑖𝜉 𝑘 ) + 𝑛 𝑘 ( 1.70 )
where 𝑥 𝑘 are the transmitted symbols, 𝜁 𝑘 is the time-discrete frequency offset between the COI and the LO defined as 𝜁 = 𝛺 𝐿𝑂 𝑇 𝑠 , with 𝑇 𝑠 the symbol period, 𝜉 𝑘 is a discrete time-varying phase, and 𝑛 𝑘 is an additive zero-mean circularly-symmetric complex-valued Gaussian noise.

Frequency estimation can be performed based on the maximization of periodogram as [START_REF] Selmi | Block-Wise Digital Signal Processing for PolMux QAM/PSK Optical Coherent Systems[END_REF]:

𝜁 ̂𝑘 = 1 4 𝑎𝑟𝑔 max 𝜁 | 1 𝑁 ∑ (𝑦 𝑘+𝑛 ) 4 • 𝑒 -𝑖𝜁𝑘 𝑁 𝑛=-𝑁 | 2 ( 1.71 )
with 2𝑁 + 1 the number of available samples. If 𝑁 is a factor of 2, the maximization can be efficiently done through the computation of FFT.

Once the frequency estimation is done, the estimation of the phase 𝜉 has to be performed.

The time-varying phase can be modeled as a Wiener process in which 𝜉 𝑘+1 = 𝜉 𝑘 + 𝑏 𝑘 , where 𝑏 𝑘 is a random variable with zero mean and variance 2𝑊 𝑝 𝑇 𝑠 , and 𝑊 𝑝 is the sum of linewidths of the signal and LO lasers. Phase estimation can be done using the well-known Blind Phase Search (BPS) algorithm as:

𝜉 ̂𝑘 = 𝑎𝑟𝑔 min 𝜉 ∑ |𝑦 𝑘+𝑛 -𝑥 ̂𝑘+𝑛 𝑒 -𝑖𝜉 )| 2 𝑁 0 -𝑙𝑜𝑔 10 (𝑃(𝑥 ̂𝑘+𝑛 )) 𝑁 𝑛=-𝑁 ( 1.72 )
where 𝑁 is the averaging length, 𝑥 ̂𝑘 is the decided symbol, and 𝑃(𝑥 ̂𝑘) corresponds to the prior probabilities. Please note that for equiprobable constellations ( 1.72 ) is reduced to:

𝜉 ̂𝑘 = 𝑎𝑟𝑔 min 𝜉 ∑ |𝑦 𝑘+𝑛 -𝑥 ̂𝑘+𝑛 𝑒 -𝑖𝜉 )| 2 𝑁 𝑛=-𝑁 ( 1.73 )
As the phase varies typically much slowly that the symbol duration, phase estimation can be performed by averaging over many symbol intervals 𝑁 . Increasing 𝑁 will improve the estimation by averaging out Gaussian noise, however fast variations of the phase will be not able to be tracked. On the contrary, choosing low values of 𝑁 will allowed tracking fast phase variations, but the estimation will be poor as additive noise is not sufficiently averaged.

Therefore, 𝑁 has to be optimized depending on the system SNR and on the phase variation speed. Typical linewidth values of external cavity lasers (ECL) are around 100 kHz and can go up to a few MHz for distributed feedback (DFB) lasers.

The minimization in ( 1.72 ) cannot be performed analytically, and there is typically more than one phase that minimizes this metric depending on the rotational symmetry of the constellation, 𝑆. Therefore, ( 1.72 ) can be solved by testing equally spaced candidate phases, evaluate the metric and choose the phase that minimizes it. However, the metric has ambiguities over the range between 0 to 2𝜋, such that we can test phases between 0 and 2𝜋/𝑆, and then apply a phase unwrapper that removes discontinuities.

After phase estimation, the phase ambiguity has to be corrected for. This can be performed using a pilot-aided approach similar to the one described in Sec. 1.3.4.3, where 𝑁 𝑝 pilot symbols are sent within a block length 𝐾 𝑝 . The phase ambiguity is calculated and averaged over the 𝑁 𝑝 pilots within each symbol block, and the resulting estimated phase ambiguity is then applied to the entire block. This process is known as pilot-aided cycle slip removal.

Post Equalization

The last DSP stage is a post-equalizer which compensates for any phase/gain mismatch between 𝐼 and 𝑄 components of the received signal. These 𝐼𝑄 imbalances may arise from an improper biasing of the PM-IQ modulator, imperfections in the coherent receiver front-end, or timing mismatches (delay skew) due to different physical path lengths in the circuit trace. The penalties associated to 𝐼𝑄 imbalances grow as the order of the modulation and symbol-rate increases [START_REF] Faruk | Digital Signal Processing for Coherent Transceivers Employing Multilevel Formats[END_REF].

Nonlinear Digital Signal Processing

After efficient mitigation of the previous impairments, the performance of actual coherent long-haul transmission systems is limited by fiber nonlinearities. As seen in Sec. In OPC the propagating signal is phase-conjugated at the middle of the optical link, such that transmission of the conjugated signal through the second half of the link cancels out CD and NL. However, to be efficient, this approach requires a fully symmetric transmission line which is difficult to achieve in practical systems. In addition, it requires the transmission link to be modified by the insertion of one or several optical phase conjugator elements [START_REF] Petropoulos | Multi-channel all-optical signal processing based on parametric effects[END_REF][START_REF] Yoshima | Nonlinearity mitigation through optical phase conjugation in a deployed fibre link with full bandwidth utilization[END_REF][START_REF] Sackey | Kerr Nonlinearity Mitigation: Mid-Link Spectral Inversion Versus Digital Backpropagation in 5×28-GBd PDM 16-QAM Signal Transmission[END_REF]. On the other hand, coherent superposition in PSA is based on co-propagating two phase-conjugated waves (signal and idler), and then coherently superimposing them at the receiver side by means of a PSA. This process can also be implemented in the digital domain, which is better known as phase-conjugated twin waves (PCTW). The main drawback of PSA and PCTW is the reduction by half of the SE, making them incompatible with high capacity optical transmission systems [START_REF] Liu | Phase-conjugated twin waves for communication beyond the Kerr nonlinearity limit[END_REF][START_REF] Olsson | Phase-Sensitive Amplified Transmission Links for Improved Sensitivity and Nonlinearity Tolerance[END_REF][START_REF] Eriksson | Achievable rates comparison for phase-conjugated twin-waves and PM-QPSK[END_REF][START_REF] Tian | Demonstration of digital phase-sensitive boosting to extend signal reach for long-haul WDM systems using optical phase-conjugated copy[END_REF]. In contrast to the idea of having large processing bandwidth capabilities, both OPC and PSA have been only demonstrated for a reduced number of optical channels with relative low gains. Optical NLC techniques are deemed to be too expensive and complex to be adopted by the industry at least in the near future.

The main advantage of DSP-based NLC is that it does not require the transmission link to be modified, making it more flexible and less costly for implementation in already deployed systems. In this sense, the most well-known technique is digital backpropagation (DBP) [START_REF] Ip | Compensation of Dispersion and Nonlinear Impairments Using Digital Backpropagation[END_REF][START_REF] Li | Electronic post-compensation of WDM transmission impairments using coherent detection and digital signal processing[END_REF], which is the most efficient algorithm in terms of performance improvement, and it is often used as a reference for all other nonlinear compensation techniques. The main drawback of DBP is its high computational complexity, making its hardware implementation unfeasible.

Different approaches to reduce its complexity have been proposed [START_REF] Du | Improved single channel backpropagation for intra-channel fiber nonlinearity compensation in long-haul optical communication systems[END_REF][START_REF] Li | Implementation efficient nonlinear equalizer based on correlated digital backpropagation[END_REF][START_REF] Rafique | Compensation of intrachannel nonlinear fibre impairments using simplified digital backpropagation algorithm[END_REF][START_REF] Ip | Complexity versus performance tradeoff for fiber nonlinearity compensation using frequency-shaped, multi-subband backpropagation[END_REF][START_REF] Napoli | Reduced Complexity Digital Back-Propagation Methods for Optical Communication Systems[END_REF], specially filtered DBP (FDBP). Besides DBP, another novel approach based on perturbation theory called perturbative nonlinear compensation (PNLC) has proven to achieve considerable NLC gains with reduced complexity [START_REF] Ghazisaeidi | Calculation of Coefficients of Perturbative Nonlinear Pre-Compensation for Nyquist Pulses[END_REF][START_REF] Tao | Multiplier-Free Intrachannel Nonlinearity Compensating Algorithm Operating at Symbol Rate[END_REF][START_REF] Gao | Reducing the Complexity of Nonlinearity Precompensation Using Symmetric EDC and Pulse Shaping[END_REF]. Approaches based on 3rd order frequency domain Volterra series have also been studied [START_REF] Bakhshali | Frequency-Domain Volterra-Based Equalization Structures for Efficient Mitigation of Intrachannel Kerr Nonlinearities[END_REF]. In combination with the continuous increase of processing capabilities and lower power consumptions of integrated circuits, DSP-based NLC techniques are the most promising candidates for implementation in future generation systems.

In this work, we center our study on the performance of FDBP and PNLC. In the following, we briefly describe their implementation principles.

Digital Backpropagation

DBP involves calculating a numerical solution of ( 1.28 ) by means of the split step Fourier method (SSFM), where the fiber is treated as a series of linear and nonlinear sections, assuming that their joint interaction is independent within a small fiber section ℎ. The linear DBP step is based on applying the adjoint dispersion operator 𝐷 ̂ℎ † over the length 𝑧 = ℎ, while the DBP nonlinear step is defined by the nonlinear operator based on the Manakov equation by:

𝑁 ̂ℎ𝑒𝑓𝑓 𝐻 [𝒚 𝐻 ] = 𝒚 𝐻 • 𝑒 𝑖𝑃 8 9 𝛾𝜅ℎ 𝑒𝑓𝑓 (|𝒚 𝐻 | 2 +|𝒚 𝑉 | 2 ) ( 1.74 ) 𝑁 ̂ℎ𝑒𝑓𝑓 𝑉 [𝒚 𝑉 ] = 𝒚 𝑉 • 𝑒 𝑖𝑃 8 9 𝛾𝜅ℎ 𝑒𝑓𝑓 (|𝒚 𝐻 | 2 +|𝒚 𝑉 | 2 ) ( 1.75 )
where the parameter 𝜅 is a constant of proportionality that has to be optimized, 𝒚 𝑉 and 𝒚 𝐻 are DBP input signal samples over 𝐻 and 𝑉 polarizations, |𝒚 𝐻 | 2 + |𝒚 𝑉 | 2 is the normalized received signal power such that its mean equals the unity, and 𝑃 is the optical power per polarization at the position of the fiber where 𝑁 ̂ is applied. For DBP steps equal or higher than the span length 𝐿 𝑠𝑝𝑎𝑛 , the effective DBP step length is defined as: ℎ 𝑒𝑓𝑓 = 𝐿 𝑒𝑓𝑓 /𝑁 𝑠𝑡𝑝𝑠 , with 𝐿 𝑒𝑓𝑓 = 1 -exp (-𝛼𝐿 𝑠𝑝𝑎𝑛 ) 𝛼 ⁄ being the effective span length, α the fiber attenuation coefficient, and

𝑁 𝑠𝑡𝑝𝑠 = 𝐿 𝑠𝑝𝑎𝑛 ℎ
⁄ the number of DBP steps per span.

In order to decrease the complexity of the standard DBP, it has been shown that low-pass filtering |𝒚 𝐻 | 2 + |𝒚 𝑉 | 2 helps decreasing 𝑁 𝑠𝑡𝑝𝑠 without sacrificing performance, this technique is known as FDBP [START_REF] Du | Improved single channel backpropagation for intra-channel fiber nonlinearity compensation in long-haul optical communication systems[END_REF]. In this case, 𝑁 ̂ can be expressed by:

𝑁 ̂ℎ𝑒𝑓𝑓 𝐻 [𝒚 𝐻 ] = 𝒚 𝐻 • 𝑒 𝑖𝑃 8 9 𝛾𝜅ℎ 𝑒𝑓𝑓 ( 𝔉 -1 {𝔉{|𝒚 𝐻 | 2 +|𝒚 𝑉 | 2 }•𝐻(𝑓)}) ( 1.76 ) 𝑁 ̂ℎ𝑒𝑓𝑓 𝑉 [𝒚 𝑉 ] = 𝒚 𝑉 • 𝑒 𝑖𝑃 8 9 𝛾𝜅ℎ 𝑒𝑓𝑓 ( 𝔉 -1 {𝔉{|𝒚 𝐻 | 2 +|𝒚 𝑉 | 2 }•𝐻(𝑓)}) ( 1.77 )
where 𝔉 {•} and 𝔉 -1 {•} stand for Fourier and inverse Fourier transform, and 𝐻(𝑓) is the frequency response of the nonlinear phase correction filter. DBP/FDBP outputs after each step ρ can be expressed by:

𝒚 𝜌+1 𝐻 = 𝑁 ̂ℎ𝑒𝑓𝑓 𝐻 [𝐷 ̂ℎ † [𝒚 𝜌 𝐻 ]]
( 1.78 )

𝒚 𝜌+1 𝑉 = 𝑁 ̂ℎ𝑒𝑓𝑓 𝑉 [𝐷 ̂ℎ † [𝒚 𝜌 𝑉 ]] ( 1.79 )
In general, the computational complexity of FDBP per step is higher than that of standard DBP, requiring two extra n-point real number FFT, and one extra real number multiplication.

However, FDBP allows reducing the total number of steps per link, resulting in an overall reduced complexity. As mentioned, the linear step is performed in the frequency domain by means of FFT, while the nonlinear step is performed in the time domain, both working typically at 2 samples per symbol.

Perturbative Nonlinear Compensation

As presented in Seq. 1.3.2.2, ( 1.28 ) can be solved using a time-domain RP1 theory, where fiber NL is treated as a small perturbation to the optical field. Expressions for the nonlinear distortion ∆𝑥 𝑘,𝑁𝐿 can be obtained following ( 1.49 ). The idea behind PNLC is to pre-calculate these distortions and subtract them from the signal. Contrary to DBP, PNLC operates at the symbol level and requires a single compensation stage per link. Therefore, its complexity is greatly reduced compared to DBP.

Considering the simplest scheme where the transmitter and receiver have only access to the COI, the information of other WDM channels is unknown and cannot be used for NLC.

Therefore, assuming only intra-channel NLC, the compensated 𝑘 'th symbol in 𝐻 and 𝑉 polarizations of the received noisy signals 𝑦 ̂𝑘 𝐻 and 𝑦 ̂𝑘 𝑉 after PNLC can be written as [START_REF] Ghazisaeidi | Calculation of Coefficients of Perturbative Nonlinear Pre-Compensation for Nyquist Pulses[END_REF]: where 𝜅 is a scaling parameter to account for channel power uncertainty that has to be optimized, and 𝐶 𝑚,𝑛 is a complex matrix of size 2𝐵×2𝐵 containing all the pre-calculated intrachannel perturbative coefficients. The second terms of ( 1.80 ) and ( 1.81 ) are similar to ( 1.49 ) when only intra-channel nonlinearities are considered, and where 𝑝 = 𝑚 + 𝑛. The 𝐶 𝑚,𝑛 coefficients are functions of the power profile, dispersion map and pulse shape. For the special case of Nyquist pulses DU systems, they can be expressed as [START_REF] Ghazisaeidi | Calculation of Coefficients of Perturbative Nonlinear Pre-Compensation for Nyquist Pulses[END_REF]:

𝑦
𝐶 𝑚,𝑛 = 𝑖 8 9 𝛾𝑃 0 𝐿 𝑑 ∫ 𝑑𝜂𝑓(𝜂) 𝑒𝑥𝑝(𝑖𝑚𝑛/𝜂) 2𝜋𝜂 × (1 - |𝑚| + |𝑛| 2𝜋𝜂 ) 𝐿 𝑡𝑜𝑡 𝐿 𝑑 ⁄ |𝑚|+|𝑛| 2𝜋 ( 1.82 )
where 𝛾 is the fiber nonlinear coefficient, 𝑃 0 is the transmitted channel optical power, Ld is the dispersion length defined by 𝐿 𝑑 = 𝛽 2 𝑇 𝑠

PERFORMANCE ANALYSIS

As mentioned in Sec. 1.2, different metrics for system performance evaluation are generally used, among which we find the MI, GMI, pre-FEC BER, and Q 2 -factor. For an AWGN channel, all the above metrics are functions of the SNR. While the first are metrics of performance of a digital system depending on the modulation format and decoding schemes, the SNR is the conventional physical measure of transmission. It is the reciprocal of the normalized mean squared error between the transmitted and received symbols, and therefore is a suitable measure of signal distortions independent of the modulation format. All impairments following Gaussian statistics can be accurately accounted for by SNR evaluation. In the following, system performance evaluation in terms of SNR is described.

AWGN Channel

As mentioned in Sec. 1.3.2, the use of several optical amplifiers in the link will degrade the OSNR of the system due to accumulation of ASE noise. The definition of OSNR is:

𝑂𝑆𝑁𝑅 = 𝑃 𝑐ℎ 𝑃 𝐴𝑆𝐸 ( 1.84 )
where 𝑃 𝑐ℎ is the total average channel signal power over the two polarizations, and 𝑃 𝐴𝑆𝐸 is the power of ASE noise as per ( 1.55 ). ASE noise is defined over a reference bandwidth, 𝐵 𝑟𝑒𝑓 , which is commonly taken to be 12.5 GHz or equivalently 0.1 nm. Contrary to the OSNR, the SNR accounts only for the noise inside the bandwidth of the signal, inside which the PSD of ASE is assumed to be locally white. Under this assumption, the OSNR and SNR for a polarization-multiplexed signal are related by:

𝑂𝑆𝑁𝑅 = 𝑅 𝑠 𝐵 𝑟𝑒𝑓 𝑆𝑁𝑅 ( 1.85 )
The SNR accounting for ASE noise can then be expressed as:

𝑆𝑁𝑅 = 𝑃 𝑐ℎ 𝜎 𝐴𝑆𝐸 2 ( 1.86 )
where the ASE variance 𝜎 𝐴𝑆𝐸 2 over the COI is expressed as:

𝜎 𝐴𝑆𝐸 2 = 𝑃 𝐴𝑆𝐸 𝑅 𝑠 𝐵 𝑟𝑒𝑓 ( 1.87 )
The above definition of SNR corresponds exactly to the SNR that can be measured on the received electrical signal constellation at the input of the FEC.

Back-to-back (B2B) Penalties

An ideal ADC presents a perfectly linear response and simplify quantizes the incoming signal by 2 𝑁𝑂𝐵 number of discrete levels, where NOB corresponds to the number of bits associated with the ADC. Therefore, an 8-bit ADC will have 256 quantization levels.

Considering a noiseless sinusoidal wave at the input of an ideal ADC, the SNR at its output is obtained by: 𝑆𝑁𝑅 𝑑𝐵 = 6.02 • 𝑁𝑂𝐵 + 1.76

( 1.88 )
However, a real ADC will add some extra noise and distortions to the signal that will contribute to further degrade the SNR. We can then define the effective number of bits (ENOB), as the resolution of an ideal ADC that would have the same SNR as the real ADC under study.

The ENOB of state-of-the art 8 bit-ADC/DACs is <6 bits. The limited resolution of ADC/DACs will limit the maximum achievable SNR of the system.

Additional SNR penalties will arise from high signal pre-emphasis to overcome 𝐼𝑄 modulator limited bandwidth, from an increased peak-to-average power ratio (PAPR) due to the use higher order modulation formats, from added ASE noise due to optical amplification of the signal at the transmitter and receiver side, and DSP penalties due to unperfect signal equalization. In this work, we model B2B penalties by introducing two variables into ( 1.86 ), 𝜅 𝑇𝑅𝑋 accounting for TX-RX impairments, and η accounting for DSP penalties. The presence of 𝜅 𝑇𝑅𝑋 will impose a maximum achievable SNR independent on signal power, while η will be translated into a horizontal translation of the curve. Under these constraints, ( 1.86 ) can be rewritten as:

𝑆𝑁𝑅 = 𝑃 𝑐ℎ 𝜂𝜎 𝐴𝑆𝐸 2 + 𝜅 𝑇𝑅𝑋 𝑃 𝑐ℎ ( 1.89 )
The proposed modeling of B2B SNR is also depicted in Fig. 1.17 showing to be sufficient to match experimental results. In practice, the values of 𝜂 and 𝜅 𝑇𝑅𝑋 have to be obtained from experimental curves.

Nonlinear Variance

All the three types of perturbative models described in Seq. 1.3.2.2 can be used to calculate the variance of NLI distortions. It is important to notice that all three approaches converge to similar results, and can be used to accurately predict the performance of modern DU long haul optical coherent systems. In this work, the time-domain model of [START_REF] Ghazisaeidi | A Theory of Nonlinear Signal-Noise Interactions in Wavelength Division Multiplexed Coherent Systems[END_REF] is preferred due to its reduced computational complexity. The impact of NSNI on the overall performance for fully loaded WDM systems has been shown to be negligible. Moreover, when NLC is applied, its impact is only important for a large number of compensated WDM [START_REF] Ghazisaeidi | A Theory of Nonlinear Signal-Noise Interactions in Wavelength Division Multiplexed Coherent Systems[END_REF]. Therefore, NSNI will not be considered in this work.

The nonlinear distortions are treated as a NLI noise term. For signal-signal interactions, its variance is given by:

𝜎 𝑁𝐿𝐼 2 = 𝑎 𝑁𝐿𝐼 𝑃 𝑐ℎ 3 ( 1.90 )
where 𝑎 𝑁𝐿𝐼 is a proportionality coefficient independent of signal power accounting for both intrachannel and inter-channel nonlinear distortions, i.e.

𝑎 𝑁𝐿𝐼 = 𝑎 𝑁𝐿𝐼-𝑖𝑛𝑡𝑟𝑎 + 𝑎 𝑁𝐿𝐼-𝑖𝑛𝑡𝑒𝑟 ( 1.91 )
In general, 𝑎 𝑁𝐿𝐼 is a function of all system parameters, i.e., modulation format, symbol-rate, fiber type, propagation distance, channel count, etc. The estimation of 𝑎 𝑁𝐿𝐼 is reduced to calculate:

𝑎 𝑁𝐿𝐼 = 𝑉𝑎𝑟(∆𝑥 𝑘,𝑁𝐿 )/𝑃 𝑐ℎ 3 ( 1.92 )
Following [START_REF] Ghazisaeidi | A Theory of Nonlinear Signal-Noise Interactions in Wavelength Division Multiplexed Coherent Systems[END_REF], 𝑎 𝑁𝐿𝐼-𝑖𝑛𝑡𝑟𝑎 and 𝑎 𝑁𝐿𝐼-𝑖𝑛𝑡𝑒𝑟 can be expressed as are given by: where 𝜇 𝑛 the 𝑛'th moment of the constellation with symbols 𝑎 𝑘 defined by:

𝑎 𝑁𝐿𝐼-𝑖𝑛𝑡𝑟𝑎 =
𝜇 𝑛 = 〈|𝑎 𝑘 | 𝑛 〉 ( 1.95 )
and where the various coefficients 𝒳 are functions of the power profile, pulse shaping and dispersion map (see Appendix of [START_REF] Ghazisaeidi | A Theory of Nonlinear Signal-Noise Interactions in Wavelength Division Multiplexed Coherent Systems[END_REF]) for a given WDM channel denoted by 𝑠. The COI corresponds to the case of 𝑠 = 0 and is dropped from the notation.

Theory of Nonlinear Channel

Considering all three sources of noise (ASE, B2B, and NLI) to be uncorrelated, the total system SNR can finally be written as:

𝑆𝑁𝑅 =
𝑃 𝑐ℎ 𝜂𝜎 𝐴𝑆𝐸 2 + 𝜅 𝑡𝑟𝑥 𝑃 𝑐ℎ + 𝑎 𝑁𝐿𝐼 𝑃 𝑐ℎ 3

( 1.96 )

The power which maximizes the SNR, also known as nonlinear threshold (NLT), can be obtained by finding the 𝑃 satisfying 𝑑𝑆𝑁𝑅/𝑑𝑃 = 0, and is given by:

𝑃 𝑁𝐿𝑇 = ( 𝜂𝜎 𝐴𝑆𝐸 2 2𝑎 𝑁𝐿𝐼 ) 1/3 ( 1.97 )
and the corresponding optimum SNR can be then expressed as:

𝑆𝑁𝑅 𝑁𝐿𝑇 = 1 (𝜂𝜎 𝐴𝑆𝐸 2 ) 2/3 ( 27 4 𝑎 𝑁𝐿𝐼 ) 1/3 + 𝑘 𝑡𝑟𝑥 ( 1.98 )
Fig. 1.18 shows the performance of an arbitrary system in terms of channel power versus SNR generated as per ( 1.96 ), where 𝑃 𝑁𝐿𝑇 and 𝑆𝑁𝑅 𝑁𝐿𝑇 are also shown.

In [START_REF] Dar | Pulse Collision Picture of Inter-Channel Nonlinear Interference in Fiber-Optic Communications[END_REF], the dynamics of inter-channel NLI was studied following a pulse collision approach.

It was interestingly pointed out that inter-channel NLI has two types of noise contributions:

Theoretical Limits to Nonlinear Compensation

The possibility to analytically evaluate the nonlinear variance have not only served to address system performance evaluation, but also to estimate the ultimate potential benefits of NLC. The evaluation of NLC benefit is typically made in terms of the improvement in the peak SNR described by ( 1.98 ). A typical system performance with and without NLC is shown in Fig.

1.19.

A reduction in 𝑎 𝑁𝐿𝐼 is translated into an improvement on the maximum SNR by:

∆𝑆𝑁𝑅 𝑁𝐿𝑇 [𝑑𝐵] = 1 3 ∆𝑎 𝑁𝐿𝐼 [𝑑𝐵]
( 1.99 )

Meaning that a 3 dB improvement in 𝑎 𝑁𝐿𝐼 will only be translated into 1 dB improvement in system SNR. As mentioned in Sec. 1.4.3, the total nonlinear variance 𝑎 𝑁𝐿𝐼 is composed of two terms, accounting for intra-channel ( 𝑎 𝑁𝐿𝐼-𝑖𝑛𝑡𝑟𝑎 ) and inter-channel ( 𝑎 𝑁𝐿𝐼-𝑖𝑛𝑡𝑒𝑟 ) nonlinear distortions. A reduction in 𝑎 𝑁𝐿𝐼 will therefore depend on the number of considered WDM channels for NLC, which will depend on the electronic processing bandwidth of the TX or RX, and the corresponding complexity on DSP.

Usually, the bandwidth of the state-of-the-art RX technology takes a lead on the TX technology, such that more than one WDM channel can be detected by a single wideband receiver. Therefore, multi-channel NLC could be in principle accomplished either by DBP or The computational complexity of multi-channel NLC is too high compared to single channel NLC techniques. In the case of DBP, increasing the back-propagated bandwidth requires an increase on the number of DPB steps applied to maintain the same accuracy [START_REF] Liga | On the performance of multichannel digital backpropagation in high-capacity long-haul optical transmission[END_REF]. In the case of PNLC, besides the intra-channel 𝐶 𝑚,𝑛 matrix, one inter-channel 𝐾 𝑚,𝑛 matrix must be computed for each compensated channel. Multichannel PNLC has been recently

experimentally demonstrated [START_REF] Frey | Single-step Perturbationbased Nonlinearity Compensation of Intra-and Inter-Subcarrier Nonlinear Interference[END_REF]. In this work we mainly focus on single channel NLC 2 . Therefore, considering only intrachannel NLC, the maximum achievable gain is given by:

∆𝑆𝑁𝑅 𝑁𝐿𝑇 [𝑑𝐵] < 10𝑙𝑜𝑔 10 ( (𝑎 𝑁𝐿𝐼-𝐼𝑛𝑡𝑟𝑎 + 𝑎 𝑁𝐿𝐼-𝐼𝑛𝑡𝑒𝑟 ) 1/3 + 𝜉 𝑎 𝑁𝐿𝐼-𝐼𝑛𝑡𝑒𝑟 1/3 + 𝜉 ) ( 1.100 )
where 𝜉 is defined by:

𝜉 = 𝜅 𝑡𝑟𝑥 (𝜂𝜎 𝐴𝑆𝐸 2 ) 2/3 ( 27 4 ) 1/3
( 1.101 ) 2 An experimental demonstration using multi-channel NLC based on DBP is presented in Sec. 4.2

SUMMARY

This chapter dealt with the main aspects of a digital communication system, and optical long-haul coherent systems based on wavelength division multiplexing. We first described the main building blocks and key aspects of digital transmission, such as: achievable rates in BICM systems, forward error correction codes, higher order modulation formats including novel probabilistic constellation shaping, and pulse-shaping.

We then described the different blocks of an optical long-haul coherent system. We reviewed the architecture of an optical transmitter capable of generating arbitrary modulation formats for data transmission. We reviewed the key physical phenomena impairing an optical signal during propagation as: amplifier noise, chromatic dispersion, fiber nonlinearities, polarization mode dispersion, and polarization dependent loss. We also presented the architecture of a coherent receiver able to map the optical field into the electrical domain, followed by all the digital signal processing blocks for mitigation of linear and nonlinear impairments.

Moreover, we described the metrics for performance evaluation as the generalized mutual information, and the signal to noise ratio. We addressed the system performance in terms of SNR including all sources of noise as: optical amplifier noise, back-to-back noise, and nonlinear noise. Finally, we presented analytical models based on perturbation theory to evaluate the maximum nonlinear gain provided by nonlinear compensation algorithms.

All these concepts will be used in the following chapters. Chapter 2 will be on the performance evaluation of the nonlinear digital signal processing algorithms based on digitalback propagation and perturbative nonlinear compensation. Chapter 3 will deal with the design and performance evaluation of probabilistic constellation shaping versus regular modulation formats. Finally, Chapter 4 will present some transmission hero experiments using the techniques described before.

LOW-COMPLEXITY DIGITAL NONLINEAR COMPENSATION

In this chapter, we evaluate the performance and gain benefits of low-complexity nonlinear compensation algorithms, i.e., filtered digital-backpropagation (FDBP) and perturbative nonlinear compensation (PNLC). In the first part, we derive by means of numerical simulations semi-analytical formulae to calculate the parameters involved within the FDBP algorithm avoiding complex exhaustive optimization. In the second part, we investigate the impact of PMD and PDL on the performance of FDBP and PNLC to assess their robustness to these effects when used to compensate single channel nonlinear distortions. Finally, we show the practical achievable gains of FDBP and PNLC for different algorithm complexities for transoceanic distances.

FILTERED DIGITAL BACKPROPAGATION OPTIMIZATION

As stated in Sec. 1.3.5.1, DBP relies on solving the inverse NLSE or Manakov equation using the well-known SSFM. In a real fiber, interaction between dispersion and nonlinearities takes place during propagation. However, within the frame of the SSFM, the dispersive and nonlinear effects alternate over a small propagation step ℎ. This approximation leads to an error term dependent on the size of ℎ [START_REF] Sinkin | Optimization of the Split-Step Fourier Method in Modeling Optical-Fiber Communications Systems[END_REF]. For the SSFM to approach the exact solution of the NLSE or Manakov equations, the step size should tend to zero. A large step size leads to the creation of numerical artifacts reducing the accuracy of the numerical solution [START_REF] Matera | Sideband instability induced by periodic power variation in long-distance fiber links[END_REF][START_REF] Forghieri | Modeling of wavelength multiplexed lightwave systems[END_REF][START_REF] Bosco | Suppression of Spurious Tones Induced by the Split-Step Method in Fiber Systems Simulation[END_REF].

Different criteria to properly choose the value ℎ for fiber transmission simulations can be found in the literature [START_REF] Sinkin | Optimization of the Split-Step Fourier Method in Modeling Optical-Fiber Communications Systems[END_REF].

While choosing small step values leads to a higher accuracy, it also leads to high computational requirements, as the number of computations grows proportional to the number of steps applied. For practical NLC, choosing small step sizes is unattractive as it makes highresolution DBP too complex for commercial implementation. In general, increasing ℎ will lead to an overestimation of nonlinear distortions, as the nonlinearity of each step is lumped at a single point and the phase mismatch between different frequency components induced by CD is ignored within each step. Overestimation of fiber nonlinearities will then lead to the creation of spurious frequency components [START_REF] Du | Improved single channel backpropagation for intra-channel fiber nonlinearity compensation in long-haul optical communication systems[END_REF][START_REF] Bosco | Suppression of Spurious Tones Induced by the Split-Step Method in Fiber Systems Simulation[END_REF]. The constant of proportionality, 𝜅, (also referred as nonlinear scaling factor) present in ( 1.74 ) and ( 1.75 ), has been introduced to overcome this overestimation when large values of ℎ are used. Typically, it has to be optimized depending on system parameters and DBP step sizes [START_REF] Ip | Compensation of Dispersion and Nonlinear Impairments Using Digital Backpropagation[END_REF][START_REF] Gao | Assessment of Intrachannel Nonlinear Compensation for 112 Gb/s Dual-Polarization 16QAM Systems[END_REF][START_REF] Ip | Nonlinear Compensation Using Backpropagation for Polarization-Multiplexed Transmission[END_REF].

In [START_REF] Du | Improved single channel backpropagation for intra-channel fiber nonlinearity compensation in long-haul optical communication systems[END_REF], a modification of the standard DBP was proposed, where the intensity waveform within the nonlinear step was low-pass filtered to reduce overestimation arising from high frequency components. As mentioned in Sec. 1.3.5.1, this technique is referred as FDBP, and it allows reducing the number of SSFM steps per span, 𝑁 𝑠𝑡𝑝𝑠 , without sacrificing performance.

This approach requires first choosing the nonlinear phase correction low-pass filter (LPF) type, and then the joint optimization of 𝜅, and the 3-dB LPF bandwidth 𝛣 𝐹𝐷𝐵𝑃 . This joint optimization is a complex problem since, a priori, both 𝛣 𝐹𝐷𝐵𝑃 and 𝜅 are functions of eight independent variables, i.e., 𝑁 𝑠𝑡𝑝𝑠 , fiber dispersion coefficient 𝐷, Kerr nonlinear coefficient 𝛾, attenuation coefficient 𝛼, channel launched power 𝑃 𝑐ℎ , symbol-rate 𝑅 𝑠 , span length 𝐿 𝑠𝑝𝑎𝑛 , and channel count 𝑆 𝑐ℎ . The joint optimization of 𝛣 𝐹𝐷𝐵𝑃 and 𝜅 has to be in principle performed for every system configuration under study [START_REF] Du | Improved single channel backpropagation for intra-channel fiber nonlinearity compensation in long-haul optical communication systems[END_REF][START_REF] Gao | Assessment of Intrachannel Nonlinear Compensation for 112 Gb/s Dual-Polarization 16QAM Systems[END_REF].

In the following, we study the dependency of 𝛣 𝐹𝐷𝐵𝑃 and 𝜅 on all the above-mentioned variables by means of numerical simulations, and we provide semi-analytical formulae to estimate them in the case of DU systems. It consists of a first linear sub-step compensating for CD, followed by a nonlinear sub-step compensating for fiber nonlinearities. For a given link composed of 𝑁 𝑠𝑝𝑎𝑛𝑠 , this process is repeated a total of 𝑀 𝑡𝑜𝑡 times depending on the applied FDBP 𝑁 𝑠𝑡𝑝𝑠 as 𝑀 𝑡𝑜𝑡𝑎𝑙 = 𝑁 𝑠𝑝𝑎𝑛𝑠 • 𝑁 𝑠𝑡𝑝𝑠 .

FDBP Parameter Optimization

To reduce complexity, we are interested in 𝑁 𝑠𝑡𝑝𝑠 ≤ 1. 𝐻(𝑓) corresponds to the frequency response of the LPF involved in the nonlinear sub-step.

In [START_REF] Rafique | Compensation of intrachannel nonlinear fibre impairments using simplified digital back-propagation algorithm[END_REF], several filters types were investigated leading to similar performances, although the performance of a Gaussian LPF was marginally better, which was then used in [START_REF] Gao | Assessment of Intrachannel Nonlinear Compensation for 112 Gb/s Dual-Polarization 16QAM Systems[END_REF] and [START_REF] Gao | Nonlinearity Compensation of 224 Gb/s Dual-Polarization 16-QAM Transmission Over 2700 km[END_REF].

Therefore, in the following we assume a super-Gaussian (SG) LPF with transfer function:

𝐻(𝑓) = 𝑒𝑥𝑝 [-ln (2) | 𝑓 𝛣 𝐹𝐷𝐵𝑃 | 2𝑛 ] ( 2.1 )
where 𝑛 is the SG order. The filter is normalized such that: ∫|𝐻(𝑓)| 2 𝑑𝑓 = 1/𝑑𝑡 and 𝐻(𝛣 𝐹𝐷𝐵𝑃 ) = 0.5 • 𝐻(0), where 𝑑𝑡 = 1/(𝑠𝑝𝑠 • 𝑅 𝑠 ). Moreover, we assume that the signal processed by FDBP algorithm has been sampled at 2 sps.

For numerical simulations, de-correlated binary de Bruijn sequences of length 2 13 are used to generate 49 GBd PM-16QAM symbols. The signal is then digitally shaped using a frequencydomain RRC filter with roll-off factor 0.01. We do both single-channel and WDM 7-channel simulations. Laser linewidths are set to zero. In order to correctly account for nonlinear distortions, the total simulated bandwidth is three times the WDM signal bandwidth.

The transmission line consists of 30 spans of 100 km standard single mode fiber (SSMF)

with 𝐷 = 16.54 [ps/nm/km], 𝛾 = 1.3 [1/W/km] and 𝛼 = 0.2 [dB/km]. Span loss is perfectly compensated at the span end by a noiseless EDFA, while the total noise is loaded at the receiver side. Fiber propagation is simulated by SSFM, where the step size is updated using At the receiver side, the COI corresponding to the center channel is matched filtered and sampled at 2 sps. These samples are then processed either by CD compensation block or by FDBP block. Finally, carrier phase estimation is performed using BPS algorithm. We do not consider any polarization effect as PMD or PDL. Therefore, adaptive equalization is not performed In this section, we measure the system performance in terms of Q 2 -factor. The FDBP gain is then defined as the difference between the Q 2 -factor when using FDBP and when using CD compensation at a given power. Q 2 -factor is calculated from BER using Monte Carlo where 40 different random noise seeds were applied for noise loading to count at least 2000 errors for each simulated point. Under this configuration, the uncertainty of the estimated Q 2 -factor is found to be ±0.06 dB assuming a confidence interval of 99%.

X -1 |•| 2 X H( f ) -1 X 𝑃 𝑐ℎ 𝛾𝜅ℎ 𝑒𝑓𝑓 e ( i•) X 𝑦 𝜌 𝐻 x M tot 𝑒 𝑖 1 2 2 2 ℎ X -1 |•| 2
In a first instance, we show the joint optimization of 𝛣 𝐹𝐷𝐵𝑃 and 𝜅. To do so, we perform single channel simulations and we set the channel optical power to 𝑃 𝑐ℎ = 2 dBm, which corresponds to the NLT when no NLC is applied. For FDBP, we set 𝑁 𝑠𝑡𝑝𝑠 = 0.5, which corresponds to performing one FDBP step every 2 spans, while the SG LPF order is set to 𝑛 = 0.5. The FDBP parameters 𝜅 and 𝛣 𝐹𝐷𝐵𝑃 are independently swept in steps of 0.025 and 0.2 GHz respectively. The uncertainty on Q 2 -factor translates to equivalent uncertainties in 𝜅 and 𝛣 𝐹𝐷𝐵𝑃 of ±0.025 and ±0.6 GHz respectively. Fig. 2.2a illustrates the contour plot of FDBPcompensated Q 2 -factor showing that 𝜅 and 𝛣 𝐹𝐷𝐵𝑃 should be jointly optimized to achieve the maximum FDBP gain. This joint optimization will be performed in the following for each system parameter under study.

P ch = 2 dBm
Following the above joint optimization procedure, we now study the dependency of the FDBP gain on the SG LPF order. For the same system configuration, we sweep 𝑛 according to 𝑛 = {0.125, 0.25, 0.5, 1, 2, 3}. The corresponding FDBP gains are shown in Fig. 2.3a using the optimal values of 𝜅 and 𝛣 𝐹𝐷𝐵𝑃 for each case. It is observed that the FDBP gain dependence on 𝑛 is weak but an optimum point is found when 𝑛 = 0.5. This agrees with [START_REF] Rafique | Compensation of intrachannel nonlinear fibre impairments using simplified digital back-propagation algorithm[END_REF], where the FDBP gain depended slightly for different LPF functions. From now on we fix 𝑛 = 0.5. We now investigate the dependency of 𝜅 and 𝛣 𝐹𝐷𝐵𝑃 with channel optical power and number of WDM channels. We consider channel optical powers ranging from 2 dBm to 5 dBm, which corresponds to NLT to NLT+3dBm, and we consider 1-channel or 7-channels spaced at 50

GHz. At the transmitter side, each channel is decorrelated by applying random time delays and random polarization rotations with respect to the COI. The results are presented in Fig. 2.3b.

It is observed that neither channel count nor launched power have an impact on optimizing 𝜅 and 𝛣 𝐹𝐷𝐵𝑃 .

In previous works, [START_REF] Du | Improved single channel backpropagation for intra-channel fiber nonlinearity compensation in long-haul optical communication systems[END_REF][START_REF] Gao | Assessment of Intrachannel Nonlinear Compensation for 112 Gb/s Dual-Polarization 16QAM Systems[END_REF][START_REF] Gao | Nonlinearity Compensation of 224 Gb/s Dual-Polarization 16-QAM Transmission Over 2700 km[END_REF], it has been shown that 𝜅 and 𝛣 𝐹𝐷𝐵𝑃 depend strongly on the value of 𝑁 𝑠𝑡𝑝𝑠 . Changing 𝑁 𝑠𝑡𝑝𝑠 translates into changing the accumulated dispersion per step (𝛽 2 ℎ) and the nonlinear phase rotation (𝛾ℎ 𝑒𝑓𝑓 𝑃 𝑐ℎ ), implying that the dependency of 𝜅 and 𝛣 𝐹𝐷𝐵𝑃 on system parameters might be uniquely through these products. To test this hypothesis, we do exhaustive joint optimization considering two cases: 2) we fix 𝑁 𝑠𝑡𝑝𝑠 = 1 and perform propagation simulations using hypothetical fiber parameters where 𝛽 2 and 𝛾 are modified such that the products 𝛽 2 ℎ and 𝛾ℎ 𝑒𝑓𝑓 𝑃 𝑐ℎ coincide with those in the first case for various values 𝑁 𝑠𝑡𝑝𝑠 .

We recall that the dispersion coefficient 𝐷 is related to 𝛽 2 by ( 1.34 ). In the following we use 𝐷 instead of 𝛽 2 as it has more practical meaning. The results of these two previous cases are presented in Fig. 2 These expressions are the main results of this section. The dimension of the numerator in ( 2.4 ) is [s], such that 𝜅 is dimensionless. Please note that these expressions are only valid when considering a SG LPF of order 𝑛 = 0.5 normalized such that: ∫|𝐻(𝑓)| 2 𝑑𝑓 = 1/𝑑𝑡 and 𝐻(𝛣 𝐹𝐷𝐵𝑃 ) = 0.5 • 𝐻(0), and working at 2 sps. However, the above results show that simple expressions can be obtained for a given filter type depending mainly on the dispersion-induced pulse broadening per FDBP step.

Experimental Validation

In order to validate the above formulae, we perform experimental measurements using the test-bed shown in Fig. 2

IMPACT OF STOCHASTIC POLARIZATION EFFECTS ON NLC ALGORITHMS

In the absence of any stochastic effect, the NLSE or the Manakov equation governing the propagation of light in the optical fiber are fully reversible by zero forcing equalization. In this case, applying full-field high-resolution DBP with a large number of 𝑁 𝑠𝑡𝑝𝑠 will fully compensate fiber nonlinearities. However, in the presence of stochastic effects as NSNI, PMD, and/or PDL, the fiber channel is not fully reversible as the stochastic interplay between these effects and nonlinearity is not taken into account for NLC. The presence of these stochastic effects becomes a fundamental limitation to completely mitigate fiber nonlinear effects.

The impact of PMD in the case of ideal high-resolution DBP has been extensively analyzed in [START_REF] Liga | On the performance of multichannel digital backpropagation in high-capacity long-haul optical transmission[END_REF][START_REF] Ip | Nonlinear Compensation Using Backpropagation for Polarization-Multiplexed Transmission[END_REF][START_REF] Gao | Influence of PMD on fiber nonlinearity compensation using digital back propagation[END_REF][START_REF] Goroshko | Fundamental Limitation of Digital Back Propagation due to Polarization Mode Dispersion[END_REF][START_REF] Liga | Ultra-Wideband Nonlinearity Compensation Performance in the Presence of PMD[END_REF]. It was shown that the penalty on the DBP achievable gain grows as the backpropagated bandwidth is increased. This degradation is explained by the fact that frequency components located far from each other will suffer from a higher induced depolarization due to PMD, leading to a higher mismatch between the true nonlinear interactions and the ones considered within the DBP algorithm. To this extent, different modified DBP algorithms trying to take into account PMD have been recently proposed [START_REF] Goroshko | Overcoming Performance Limitations of Digital Back Propagation due to Polarization Mode Dispersion[END_REF][START_REF] Czegledi | Polarization-Mode Dispersion Aware Digital Backpropagation[END_REF][START_REF] Czegledi | Modified Digital Backpropagation Accounting for Polarization-Mode Dispersion[END_REF]. The impact of PDL on NLC has never been addressed to the best of our knowledge.

From a practical point of view, performing high-resolution and multi-channel DBP is still extremely challenging and not yet viable for commercial implementation. If NLC techniques are to be adopted in the near future, they will be limited to single channel compensation and low complexity architectures (as PNLC with reduced LUT sizes or FDBP with low 𝑁 𝑠𝑡𝑝𝑠 ). In this case, the impact of PMD/PDL on NLC will be greatly reduced. First, because the backpropagated bandwidth is decreased, and second, because the penalty brought by the low complexity NLC implementation will dominate the achievable performance.

With this practical implementation of NLC in mind, we investigate the impact brought by PMD and PDL on the performance of FDBP and PNLC in the special case of transoceanic DU systems. We focus on PM-16QAM WDM channels modulated at 32 GBd and 50 GHz spacing corresponding to a practical study case.

Experimental Results

We start by performing transmission experiments for which the experimental set-up is depicted in Fig. 2 In general, PMD is an effect that is distributed along the optical fiber. While legacy deployed fibers can have large amounts of PMD, modern fibers used for long haul applications have a very small PMD coefficient typically below 0.05 ps/√km. This is the case of our Corning Vascade EX3000 fiber used in our laboratory, such that an external PMD element must be introduced in the link to emulate PMD effects. In [START_REF] Bertolini | Do's and Dont's for a Correct Nonlinear PMD Emulation in 100Gb/s PDM-QPSK Systems[END_REF], it was shown that the correct emulation of PMD depends on the characteristics of the PMD emulator, and on its position in the experimental setup to correctly account for PMD-nonlinear interactions. For DU systems, it was shown that placing a polarization maintaining fiber (PMF) section at the transmitter side was enough to correctly emulate the effects of PMD; while for DM systems, at least one PMF section was required at each recirculating-loop round trip. In this sense, in order to emulate PMD in our experimental set-up, a PMF fiber of 𝜏 𝑒 𝐷𝐺𝐷 = 20 ps was inserted at the end of each loop and should be enough to correctly account for PMD-nonlinear interactions.

Contrary to PMD, PDL is a lumped effect introduced by optical elements. The major contribution is done by optical WSS used for channel equalization and add/drop capabilities, and which are typically placed after sever fiber spans. Therefore, PDL emulation is performed In order to study the impact of PDL and PMD on NLC gain, we fix the optical power to 18 dBm which corresponds to the optimum Q 2 -factor when NLC is applied. Moreover, we focus only on FDBP with 𝑁 𝑠𝑡𝑝𝑠 = 1, 𝑁 𝑠𝑡𝑝𝑠 = 0.25, and PNLC, as we are only interested in low complexity NLC architectures. At 18 dBm, FDBP 𝑁 𝑠𝑡𝑝𝑠 = 1, 𝑁 𝑠𝑡𝑝𝑠 = 0.25, and PNLC provide 1 dB, 0.8 dB and 0.7 dB of gain respectively.

The characterization of the nonlinear gain provided by FDBP and PNLC is done under four different scenarios:

1) No PDL -No PMD,
2) PMD only,

3) PDL only, and 4) PDL + PMD.

For each case, we record 600 different waveforms and processed them offline. The PDL and PMD values can be estimated based on the adaptive equalizer butterfly filter coefficients 𝒉 [START_REF] Hauske | Optical Performance Monitoring in Digital Coherent Receivers[END_REF][START_REF] Bosco | Joint DGD, PDL and Chromatic Dispersion Estimation in Ultra-Long-Haul WDM Transmission Experiments with Coherent Receivers[END_REF], whose transfer function is:

𝑯(𝑓) = [ 𝔉{𝒉 𝐻𝐻 (𝑡)} 𝔉{𝒉 𝑉𝐻 (𝑡)} 𝔉{𝒉 𝐻𝑉 (𝑡)} 𝔉{𝒉 𝑉𝑉 (𝑡)} ] ( 2.7 )
Normalizing 𝑯(𝑓) by the square root of its determinant gives:

𝑼(𝑓) = 𝑯(𝑓) 𝑑𝑒𝑡(√𝑯(𝑓)) ( 2.8 )
where 𝑼(𝑓) is a unitary matrix accounting for the inverse channel DGD, which can be obtained by [START_REF] Hauske | Optical Performance Monitoring in Digital Coherent Receivers[END_REF]:

𝜏̂𝐷 𝐺𝐷 = 2√𝑑𝑒𝑡 ( 1 2𝜋 𝑑𝑼(𝑓) 𝑑𝑓 )| 𝑓=0 ( 2.9 )
whereas PDL can be estimated as [START_REF] Bosco | Joint DGD, PDL and Chromatic Dispersion Estimation in Ultra-Long-Haul WDM Transmission Experiments with Coherent Receivers[END_REF]:

𝛤 ̂𝑑𝐵 = |10𝑙𝑜𝑔 10 ( 𝜆 1 (𝑓) 𝜆 2 (𝑓) )| 𝑓=0 ( 2.10 )
with 𝜆 1 and 𝜆 2 the eigenvalues of 𝑯 † (𝑓) • 𝑯(𝑓). We now characterize the system performance without NLC. For each of the four cases, Fig.

2.13a

shows the Q 2 -factor probability density Pr(𝑄 2 ) , while Fig. 2.13b shows the probability of the Q 2 -factor loss (𝑄𝐿) defined as 𝑄𝐿 = 𝑄 𝑚𝑎𝑥 2 -𝑄 2 [dB]. We observe that system performance is unaffected in case of PMD-only, as observed in [START_REF] Serena | Nonlinear Penalty Reduction Induced by PMD in 112 Gbit/s WDM PDM-QPSK Coherent Systems[END_REF]. On the other hand, in the case of PDLonly the mean Q 2 -factor decreases by ~0.5 dB, while the maximum observable 𝑄𝐿 is increased by ~1.4 dB. Adding PMD to PDL helps to decrease 𝑄𝐿 by 0.3 dB compared to the PDL-only case, while the mean value is unchanged. Similar results were obtained by means of numerical simulations in [START_REF] Liu | Intra-symbol frequency-domain averaging based channel estimation for coherent optical OFDM[END_REF]. The previous results are in agreement with other works studying the impact of PMD and PDL on the performance of coherent optical fiber systems. In the linear regime, the PMDinduced effects can be effectively compensated by digital adaptive equalization, such that even high values of PMD cause no system penalty as long as a sufficiently large impulse response of the butterfly filters is considered. Contrary to PMD, PDL is a non-unitary linear transformation and its effects cannot be compensated by DSP. PDL induces fluctuations on the system SNR by random polarization dependent OSNR degradation, an effect that cannot be compensated for by any means [START_REF] Duthel | Impact of Polarisation Dependent Loss on Coherent POLMUX-NRZ-DQPSK[END_REF][START_REF] Vassilieva | Impact of Polarization Dependent Loss and Cross-Phase Modulation on Polarization Multiplexed DQPSK Signals[END_REF].

In the nonlinear regime, PMD reduces intra-channel nonlinear distortions due to the induced depolarization and decorrelation between the signal tributaries, leading to an improvement in system performance. However, this improvement is almost negligible in DU systems, as decorrelation due to large values of CD dominates over PMD [START_REF] Serena | Nonlinear Penalty Reduction Induced by PMD in 112 Gbit/s WDM PDM-QPSK Coherent Systems[END_REF][START_REF] Bertran-Pardo | Demonstration of the benefits brought by PMD in polarization-multiplexed systems[END_REF][START_REF] Serena | Nonlinear Phase Noise Mitigation by Polarization Mode Dispersion in Dispersion Managed coherent PDM-QPSK Systems[END_REF]. In the case of PDL, the loss of orthogonality induces instantaneous peak-to-peak optical power variations that will lead to an increase in nonlinear signal distortions. However, in DU systems, the large values of accumulated CD induces strong signal fluctuations that dominate over those induced by PDL [START_REF] Vassilieva | Systematic investigation of interplay between nonlinear and polarization dependent loss effects in coherent polarization multiplexed systems[END_REF][START_REF] Tao | Interaction between PDL and intra-channel nonlinearity in dual polarization systems[END_REF][START_REF] Vassilieva | Statistical analysis of the interplay between nonlinear and PDL effects in coherent polarization multiplexed systems[END_REF][START_REF] Vassiliev | Experimental investigation of the statistics of the interplay between nonlinear and PDL effects in coherent polarization multiplexed systems[END_REF][START_REF] Rossi | Polarization-Dependent Loss Impact on Coherent Optical Systems in Presence of Fiber Nonlinearity[END_REF].

We now investigate on the impact of PMD and PDL on NLC. We apply FDBP with

Q 2 Factor [dB]
A possible explanation of the higher sensitivity of PNLC to DGD compared to FDBP is the following. While FDBP and PNLC are based on different approaches to compensate for fiber nonlinear distortions, another characteristic which distinguish them is their relative position within the DSP chain. While FDBP is placed at the very input, PNLC is placed after all other compensating blocks. A time delay between the two signal polarizations due to DGD will be compensated by the CMA, such that the relative position between the two polarizations after propagation is lost for PNLC. On the contrary, for FDBP the backpropagated signals is still impaired by DGD. Therefore, as DGD increases a higher impact on PNLC with respect to FDBP is expected. Even though, the decrease in PNLC achievable gain due to PMD is kept below 0.1 dB for values up to 150 ps of DGD. The maximum observable ∆𝑄𝐿 𝑁𝐿𝐶 is kept below 0.15 dB for both PNLC and FDBP. When PMD is added to the link (Fig. 2.15b) the maximum ∆𝑄𝐿 𝑁𝐿𝐶 is basically unchanged for both cases of FDBP, while for PNLC is increased by 0.1 dB as stated previously.

Removing the PMD element and introducing PDL into the link (Fig. 2.15c) shows no further decrease on the mean ∆𝑄 𝑁𝐿𝐶 2 for all algorithms types. However, ∆𝑄𝐿 𝑁𝐿𝐶 is greatly increased, being FDBP the most affected one with values ~3x higher than when no PDL/PMD is perturbing the system. On the contrary, PNLC shows to be more robust with only ~2 x ∆𝑄𝐿 𝑁𝐿𝐶 increases Results presented in Fig. 2.15 show that PNLC seems to be more sensitive to PMD than FDBP. On the contrary, PNLC is more robust than FDBP to PDL, as ∆𝑄𝐿 𝑁𝐿𝐶 is increased by ~0.15 dB in comparison to 0.25 dB for FDBP. Furthermore, in the presence of both PDL and PMD, while the overall ∆𝑄𝐿 𝑁𝐿𝐶 is reduced, PNLC is still more robust than FDBP.

The reason of the apparent higher robustness of PNLC to PDL compared to FDBP is still an open problem which must be further studied.

Simulation Results

In order to validate the results shown experimentally, we now evaluate the impact of PMD and PDL on FDBP and PNLC by means of numerical simulations.

At the transmitter side, de-correlated binary de Bruijn sequences of length 2 13 For each of the studied experimental cases (PMD-only, PDL-only, and PMD+PDL) we perform 600 different transmission simulations using NVIDIA Tesla K80 GPU cards, where the SSFM algorithm has been optimized for GPU compatibility. For each loading noise seed, we apply FDBP 𝑁 𝑠𝑡𝑝𝑠 = 1, 𝑁 𝑠𝑡𝑝𝑠 = 0.25, PNLC with a 300x300 LUT, and only CD compensation (uncompensated). Then the gain of FDBP and PNLC is calculated and averaged over all noise seeds. The process is repeated for all 600 transmitted waveforms and all studied cases. Fig. 2. 16 shows Pr{∆𝑄 𝑁𝐿𝐶 2 } and Pr{∆𝑄𝐿 𝑁𝐿𝐶 } for the above cases. Please note that the NLC achievable gains are higher due to the reduced number of transmitted channels, and so the fluctuations in the gain due to PMD and PDL. The first case corresponds to a typical current system configuration, while the second reflects the industry trend to continue increasing the per-channel bit rate using high symbolrate signals. The modulation format is PM-16QAM.

At the transmitter side, the COI is synthetized using binary de Bruijn sequences of length 2 14 . In order to avoid sequence correlations, all adjacent WDM channels are generated using independent random sequences also of length 2 14 . To correctly account for the spectral broadening due to fiber nonlinearities, the total simulated bandwidth is three times the WDM signal bandwidth (i.e., 3x4.1 THz). The generated signal for each channel is then digitally shaped using a frequency-domain RRC filter with roll-off factor 0.01. Adjacent channels with respect to the COI are further decorrelated by randomly rotating their input state of polarization.

All laser linewidths are set to 0 Hz, and we assume equal optical power for all channels. Fiber transmission is simulated using the SSFM applied to the Manakov equation and executed using NVIDIA Tesla K80 GPU cards. Lumped amplification at the end of each span completely compensates for span loss, while noise loading is performed at the receiver side At the receiver side, the COI is optically filtered with an ideal rectangular filter with 1.05xRs bandwidth, and sampled at 2 sps. After CD compensation, a block performing match filtering is applied. Then, carrier phase estimation based on BPS is performed, with the averaging length chosen to equal the sequence length such that only the global phase is compensated for. Finally, SNR is calculated following ( 1.18 ).

TX n = 1 n = N span … Optical
NLC is performed either by DBP/FDBP or by PNLC. In the first case, the CD compensation block is replaced by the DBF/FDBP block Fig. 2.17b. In the case of PNLC, it is applied between the phase estimation block and SNR calculation. The different simulation set-up as well as the DSP blocks are sketched in Fig. 2.17.

Numerical Results

When evaluating the gain of NLC techniques by means of numerical simulations it is common to calculate the gain from the well-known bell-curves (cf. Fig. 2.18), where the transmitted optical power is swept over a given value range and the performance is obtained for each point with and without NLC. As nonlinear transmission simulations and NLC processing have to be performed for each optical power, this technique requires high computational resources when the total simulated bandwidth is high.

As seen in Sec. 1.4.3, in the frame of RP1 theory, fiber nonlinearities are treated as an AWGN process whose variance is completely characterized by: 𝑃 𝑁𝐿𝐼 = 𝑎 𝑁𝐿𝐼 𝑃 3 , and where the NLI term 𝑎 𝑁𝐿𝐼 is independent on signal power. Under this assumption, the optimum SNR gain brought by NLC is given by the difference in the NLI term before and after NLC as per ( 1.99 ).

Therefore, as long as the system operates within the validity of RP1 theory, it is sufficient to calculate the 𝑎 𝑁𝐿𝐼 term with and without NLC to obtain the NLC gain indifferently of the optical power of the transmitted signal.

This approach is simpler as only one numerical simulation has to be performed at a fixed optical power. As all other sources of noise apart from nonlinearities are exactly known, the 𝑎 𝑁𝐿𝐼 term after SNR calculation can be obtained by:

𝑎 𝑁𝐿𝐼 ̂= ( 𝑃 𝑆𝑁𝑅 ̂-𝑃 𝐴𝑆𝐸 𝑇𝑜𝑡𝑎𝑙 𝑅 𝑠 𝐵 𝑟𝑒𝑓 ) 𝑃 3 ⁄ ( 2.11 )
where 𝑆𝑁𝑅 ̂ is the estimated SNR calculated from the transmitted and received symbols as per ( 1.18 ), and where we have assumed a perfect transmitter and receiver. 𝑃 𝐴𝑆𝐸 𝑇𝑜𝑡𝑎𝑙 corresponds to the total equivalent ASE noise accumulated over all fiber spans calculated as per ( 1.55 ).

In the absence of ASE noise (i.e., 𝑃 𝐴𝑆𝐸 𝑇𝑜𝑡𝑎𝑙 = 0), 𝑎 𝑁𝐿𝐼 can be computed straightforward from ( 2.11 ) avoiding the numerical uncertainty in 𝑃 𝐴𝑆𝐸 .

We perform a preliminary set of numerical simulations to validate the system performance prediction and NLC gain using the 𝑎 𝑁𝐿𝐼 approach. In order to reduce computation time, we consider 5 channels modulated at 64 GBd with 75 GHz spacing.

In the first set, we perform SSFM simulations for different optical transmitted powers ranging from 0 to 3 dBm. Noise loading is performed at the receiver side, and 𝑆𝑁𝑅 ̂ is computed with and without NLC averaged over 15-seed ASE noise loadings. When NLC is applied, noise loading is performed after the DBP block to avoid degradation of the gain due to self-induced NSNI within the algorithm.

In the second set, a single SSFM simulation is performed for a fixed transmitted power corresponding to the system NLT. Without adding ASE noise, 𝑎 𝑁𝐿𝐼 is obtained as per ( 2.11 ) with and without NLC. Then, SNR vs optical power curves are obtained analytically following ( 1.96 ) using 𝑎 𝑁𝐿𝐼 ̂.

Fig. 2.18 shows the SNR vs channel optical power for the two previous described cases after 7,150 km, where the theoretical performance based on the perturbative model of [START_REF] Ghazisaeidi | A Theory of Nonlinear Signal-Noise Interactions in Wavelength Division Multiplexed Coherent Systems[END_REF] is also included. NLC is performed by high resolution DBP with Nstps = 20, and with FDBP with Nstps = 1. A good match between analytical model and SSFM simulations is found with and without NLC. High resolution DBP achieves the maximum achievable gain, while it is reduced for FDBP. As expected, obtaining the system performance from the SSFM computation of 𝑎 𝑁𝐿𝐼 ̂

gives the same results as performing independent SSFM for varying optical powers. From now on, system performance and NLC gains are obtained from SSFM computation of 𝑎 𝑁𝐿𝐼 . Fig. 2.18: System performance with and without NLC for a 5x32 GBd WDM channels spaced at 75 GHz after 7,150 km EX3000 transmission. (Lines) complete analytical model, (filled markers) SSFM simulations, and (blanked markers), semi-analytical model using SSFM estimated 𝑎 𝑁𝐿𝐼 . 5 channels 64 GBd @ 75 GHz 7,150 km EX3000 fiber 91

Filtered Digital Backpropagation

We now move to full C-band simulation. We start by evaluating the NLC gain versus FDBP algorithm complexity. At a first instance, we neglect ASE noise, such that the signal entering the FDBP algorithm is only impacted by deterministic fiber nonlinearities. This case can be seen as the upper limits of FDBP, where the achievable gain is only due to the associated algorithm accuracy. Considering the case of 32 GBd channels at 37.5 GHz spacing, Fig.

92

We note that the DBP/FDBP gain keeps growing with the transmitted distance, as the intrachannel nonlinear variance accumulates much faster along distance with respect to interchannel nonlinearities [START_REF] Dar | Inter-Channel Nonlinear Interference Noise in WDM Systems: Modeling and Mitigation[END_REF]. The maximum achievable gain provided by high resolution DBP lies in the region between 1 to 1.2 dB for all considered distances. Performing FDBP with Nstps = 1 leads to a reduction of ~0.2 dB in achievable gain, which is further decreased for lower Nstps values. For distances above 6,000 km, considerable gains above 0.6 dB are attainable with FDBP steps as low as Nstps = 1/10. To better compare the achievable gain vs FDBP complexity, Fig. 2.19b shows the FDBP gain percent with respect to high resolution DBP vs

Nstps. Performing FDBP with Nstps = 1 achieves almost 90% of the maximum gain. For transoceanic distances, 50% of the maximum gain can be achieved with Nstps = 1/16. Doubling the channel symbol-rate and reducing the number of WDM channels by half, leads to an increase of ~0.2 dB on the achievable gains. For transoceanic distances, the maximum achievable gain is ~1.5 dB, while Nstps = 1/30 is enough to achieve gains above 0.5 dB. These results show that increasing the channel symbol-rate not only leads to an increase in the achievable NLC gain, but that Nstps can be greatly reduced while still achieving considerable gains.

FDBP with Nstps = 1 leads to 90% of the maximum achievable gain, while lower values of Nstps lead to similar results compared to the 32 GBd case. From Fig. 2.19 and Fig. 2.20, it is observed that lower gains are achieved for decreasing propagation distances, with a larger impact for the 64 GBd system. This can be due to the use of not optimum values of FDBP coefficients. However, performing exhaustive optimization of 𝜅 and 𝛣 𝐹𝐷𝐵𝑃 leads to the same results. This behavior needs to be further verified. However, the achievable gain percent is relatively the same for transoceanic distances.

The above results were obtained considering a noiseless system, such that the FDBP algorithm only deals with the deterministic nonlinear impairments and is not affected by ASE noise. However, in real systems ASE noise is added at each amplification point, leading to NSNI during propagation. NSNI not only impacts the system performance, but also the effectiveness of a nonlinear equalizer placed at the receiver side. First because NSNI taking place during propagation is not compensated due to its stochastic nature, and also because self-induced NSIN will take place within the NLC algorithms [START_REF] Ghazisaeidi | A Theory of Nonlinear Signal-Noise Interactions in Wavelength Division Multiplexed Coherent Systems[END_REF][START_REF] Serena | Nonlinear SignalNoise Interaction in Optical Links With Nonlinear Equalization[END_REF][START_REF] Beygi | On nonlinearly-induced noise in single-channel optical links with digital backpropagation[END_REF][START_REF] Rafique | Impact of signal-ASE four-wave mixing on the effectiveness of digital backpropagation in 112 Gb/s PM-QPSK systems[END_REF]. A type of DBP referred as stochastic DBP taking into account ASE noise have also been proposed [START_REF] Irukulapati | Stochastic Digital Backpropagation[END_REF].

However, due to its increased complexity it is not considered in this work.

To study the impact of noise in the effectiveness of FDBP, we now load ASE noise at the receiver side. Thus, NSNI taking place during forward propagation is neglected. The FDBP gain is obtained for different values of loaded ASE noise. We take as a reference a system with amplifiers having NF = 5 dB, for which the reference OSNR is computed (OSNRref). Then we degrade the received OSNR by a given value 𝛥𝑂𝑆𝑁𝑅 = {1, 2, 3, 4, 5} dB. The resulting system OSNR is therefore 𝑂𝑆𝑁𝑅 = 𝑂𝑆𝑁𝑅 𝑟𝑒𝑓 -𝛥𝑂𝑆𝑁𝑅 Fig. 2.21a shows the FDBP gain vs distance for Nstps = 1, and different values of 𝛥𝑂𝑆𝑁𝑅 considering the system configuration at 32GBd The maximum achievable gain with noiseless high resolution DBP and noiseless FDBP are also shown for comparison. As found in [START_REF] Serena | Nonlinear SignalNoise Interaction in Optical Links With Nonlinear Equalization[END_REF],

after a certain distance the FDBP gain decreases due to NSNI. Therefore, there exists an optimal link length for which DBP is maximally effective. The penalty in FDBP gain increases for lower values of received OSNR. For a fixed distance of 6,600 km, Fig. 2.21b shows the FDBP gain vs 𝛥𝑂𝑆𝑁𝑅 for different values of Nstps. It is observed that the gain is equally degraded for all values of Nstps, where similar results were found for other distances. Therefore, the impact brought by NSIN within the FDBP algorithm is independent on Nstps.

111 channels 32 GBd @ 37.5 GHz 111 channels 32 GBd @ 37.5 GHz 6,600 km 0.9 Similar results for the 64GBd system configuration are shown in Fig. 2.21c and Fig. 2.21d., However, it is observed that the NLC gain degradation due to linear noise is higher than that of the 32GBd case. This is explained by the fact that doubling the symbol-rate will also double the in-band noise entering the FDBP algorithm, leading to higher induced NSNI.

Perturbative Nonlinear Compensation

We now evaluate the achievable gains provided by PNLC. A noiseless system is considered first as in the previous section, such that the achievable PNLC gains are only due to the efficiency in inverting deterministic nonlinear distortions. The complexity of the algorithm comes from the number of 𝐶 𝑚,𝑛 coefficients considered for nonlinear compensation, where the optimum value depends on the channel memory induced by chromatic dispersion.

PNLC gain percent [dB]

For the 32 GBd configuration, Fig. 2.22 shows the PNLC gain percent with respect to the maximum achievable gain provided by high resolution DBP versus the one-sided Cmn LUT size for three propagation distances. The scaling parameter present in ( 1.80 ) and ( 1.81 ) has been optimized for each point. It is observed that the maximum gain provided by PNLC is around 70% of the maximum achievable gain. Moreover, the gain saturates at ~50% of the ideal LUT size depending on the channel memory. In order to further reduce complexity, all Cmn terms below a given threshold with respect to C0,0 can be discarded. GBd configuration. Contrary to the previous case, we only consider a maximum propagation distance of 6,600 km, as the number of Cmn elements grows quadratically with the channel symbol-rate. Therefore, for each distance, the optimum LUT size for the 64 GBd configuration is 4 times bigger than the 32 GBd case. The achievable gains are ~80% of the maximum achievable gain provided by high resolution DBP. As for 32 GBd channels, the optimum Cmn coefficient discard threshold at which the gain saturates is ~35 dB. Now we investigate on the impact of noise on the performance of PNLC. As it was done for FDBP, the OSNR at the receiver side is degraded from its reference value by a given amount 𝛥𝑂𝑆𝑁𝑅, and PNLC is applied. Shows the PNLC gain versus LUT size for a fixed distance of 6,600 km, where the maximum gain achievable gain for noiseless DBB and PNLC are also shown for comparison. As observed in Fig. 2.23, the impact of noise on PNLC gain does not depend on the considered LUT size. This was the same findings as for FDBP, where noise impact was independent with the applied Nstps. From Fig. 2.23a, it is observed that increasing 𝛥𝑂𝑆𝑁𝑅 from 0 dB to 5 dB leads to a gain degradation of ~0.1 dB. This is the same degradation found at 6,600km for FDBP as shown in Fig. 2.21b. The above process has been repeated for the other distances considered in Fig. 2.22, (i.e., 2750 km,10450 km for 32GBd, and 2750 km for 64 GBd). It was found that the degradation in PNLC gain versus OSNR penalty closely match the gain degradation of the FDBP. Therefore, we can conclude that both FDBP and PNLC are equally affected by linear noise, and the gain degradation is independent of the algorithm complexity.

Besides the NLC degradation due to a decreased algorithm complexity and linear noise impact, both FDBP and PNLC can achieve gains above 0.5 dB for transoceanic distances with low-complexity architectures for the two studied system configurations

SUMMARY

In this chapter we evaluated the performance and practical achievable gains of lowcomplexity single-channel NLC algorithms. In the first part, we studied the dependency of the parameters involved within the FDBP algorithm on different system parameters as: channel count, launched power, symbol-rate, and fiber coefficients. We showed numerically and experimentally that the FDBP parameters depend mainly on the pulse broadening per FDBP step, allowing us to derive simple semi-analytical formulae for their computation, and avoiding complex exhaustive optimization.

In the second part, we experimentally addressed the impact of PMD and PDL on the performance of FDBP and PNLC. We showed that PMD values as large as 5 times the symbol duration lead to a maximum observable degradation of only ~0.1 dB in the achievable gain of FDBP and PNLC; while the corresponding gain degradation due to PDL values as large as 9 dB is only ~0.2 dB. These experimental observations show the robustness of these algorithms to PDL and PMD.

Finally, in the third part, we numerically evaluated the practical achievable gains provided by FDBB and PNLC for different algorithm complexities for two transoceanic system configurations, i.e., 32 GBd channels at 37.5 GHz spacing and 64 GBd channels at 75 GHz spacing. For an ideal noiseless case, the maximum achievable gain provided by highresolution DBP is above 1 dB for distances above 6,000 km. FDBP working at one step per span can achieve 90% of this maximum gain, while for PNLC with optimum LUT size it is

HIGH-CAPACITY MODULATION FORMATS

The great advances in high-speed high-resolution DAC/ADC's have not only allowed the use of powerful DSP algorithms to compensate for system impairments in coherent optical fiber systems, but also the possibility to generate higher spectral efficiency signals using higher order modulation formats with arbitrary pulse shaping.

Modulation formats beyond 16QAM have been recently explored by a few research groups to significantly increase the SE of submarine systems [START_REF] Zhang | Capacity-approaching transmission over 6375 km at spectral efficiency of 8.3 bit/s/Hz[END_REF][START_REF] Cai | 64QAM based coded modulation transmission over transoceanic distance with >60 Tb/s capacity[END_REF]. In [START_REF] Zhang | Capacity-approaching transmission over 6375 km at spectral efficiency of 8.3 bit/s/Hz[END_REF], 32QAM was used to achieve 7.9 b/s/Hz over 6,800 km, while in [START_REF] Cai | 64QAM based coded modulation transmission over transoceanic distance with >60 Tb/s capacity[END_REF], a SE of 7.1 b/s/Hz was achieved after 5,380 km using 64QAM with coded modulation and iterative decoding.

Other formats besides regular QAM have also been explored. As presented in Sec. 1.2.3, according to Shannon channel coding theorem, the optimum source distribution for the AWGN channel is complex-circular and Gaussian. In practice, one can use formats with a discrete number of constellation points, and then apply some kind of constellation shaping to make the source distribution closer to Gaussian, which helps decrease the gap between the constrained and the linear Shannon capacity. Ref. [START_REF] Zhang | Capacity-approaching transmission over 6375 km at spectral efficiency of 8.3 bit/s/Hz[END_REF] showed a record C-band SE of 8.3 b/s/Hz by using geometric constellation shaped 64APSK. Later on, in [START_REF] Ghazisaeidi | Advanced C+L Band Transoceanic Transmission Systems Based on Probabilistic Shaped PDM-64QAM[END_REF], the novel probabilistic constellation shaping based on 64QAM constellation (PCS-64QAM) was employed for the first time over 6,600 km to achieve a SE of 7.3 b/s/Hz employing C+L bands with EDFA only amplification. This experiment was followed by a field trial over a 5,523 km in service C-band EDFA-only trans-Atlantic Facebook cable to demonstrate a SE of 7.46 b/s/Hz [START_REF] Cho | Trans-Atlantic Field Trial Using Probabilistically Shaped 64-QAM at High Spectral Efficiencies and Single-Carrier Real-Time 250-Gb/s[END_REF].

Finally, hybrid formats employing both geometrical and probabilistic shaping have also been experimentally demonstrated. The 4D-PS-9/12-56APSK was used in [START_REF] Cai | 70.4 Tb/s Capacity over 7,600 km in C+L Band Using Coded Modulation with Hybrid Constellation Shaping and Nonlinearity Compensation[END_REF] to achieve 7.2 b/s/Hz over 7,600 km, using C+L bands EDFA-only amplification, while 4D-PS-7/12-40APSK was demonstrated in [START_REF] Cai | 51.5 Tb/s Capacity over 17,107 km in C+L Bandwidth Using Single Mode Fibers and Nonlinearity Compensation[END_REF] to achieve 5.29 b/s/Hz after 17,107 km. However, these hybrid formats are based on coded modulation with complex iterative decoding, therefore unattractive from the industry point of view.

In the first part of this chapter, we experimentally compare the performance and achievable rates of different higher order modulation formats employing low complexity BICM with BM decoding (i.e. 32QAM, 64QAM, 64APSK, PCS-64QAM) for transoceanic distances, together with some DSP challenges associated with their practical implementation. Then, we focus on the design of a truncated version of PCS-64QAM (TPCS-6AQAM) optimized to maximize the SE of trans-Pacific optical links.

MODULATION FORMATS COMPARISON

As seen in Sec. 1.2.3 ,"Gaussian-like" constellations decrease the gap between the constrained and the linear Shannon capacity, outperforming regular QAM formats. However, in the nonlinear regime, the fourth and sixth moments of the constellation become important in computing the total noise variance as presented in Sec. 1.3.2.2. Moreover, the constellation shaping naturally results in more circular symmetric constellation patterns, which imposes some challenges in the DSP chain, and might lead to additional implementation penalties. Moreover, generating and receiving complex constellations requires higher resolution of DAC and ADC's.

Therefore, a fair comparison of the performance of higher order formats for coherent optics requires addressing at least four issues: theoretical gap to the linear Shannon capacity, nonlinear performance at optimum launch power, B2B implementation penalties due to limited DAC and ADC resolutions, and DSP challenges.

In this section, we experimentally compare the performance of four modulation formats for transoceanic distances, i.e., 32QAM, 64QAM, 64APSK, and PCS-64QAM, addressing all the above-mentioned issues.

Theoretical Performance

We start by evaluating the theoretical performance of the four mentioned formats. We adopt two performance metrics: the SNR, and the GMI, which is the correct measure to be used for systems employing BICM as presented in Sec. 1.2.1.3. We consider a PCS-64QAM with a source entropy, H, of 5.4 b/symb/pol, which was shown in [START_REF] Ghazisaeidi | Advanced C+L Band Transoceanic Transmission Systems Based on Probabilistic Shaped PDM-64QAM[END_REF] to minimize the gap to the linear Shannon capacity for a SNR target of 12 dB. Fig. 3.1 shows the GMI for all the above formats over the SNR region of interest. According to ( 1.96 ), the total SNR after transmission through the optical fiber is inversely proportional to the nonlinear noise variance arising from the fiber Kerr nonlinear effects, with the nonlinear noise variance being directly proportional to 4'th, µ4, and 6'th , µ6, constellation moments according to ( 1.93 ) and ( 1.94 ). As observed in Fig. 3.2a, µ4 and µ6 increase with the constellation order, with the Gaussian constellation maximizing both coefficients. Therefore, it is expected that Gaussian-like constellations will present a higher nonlinear penalty after fiber transmission. Fig. 3.2b shows the theoretical SNR vs optical power obtained as per ( 1.96 ) for a system composed of 61 channels at 49 GBd with 50 GHz spacing, where the nonlinear variance is calculated following ( 1.93 ) and ( 1.94 ). The transmission distance is set to 6,600 km composed of 55 km EX3000 fiber spans, and we consider the same B2B penalties for all formats. The SNR is then transformed to GMI following the curves of Fig. From the above results, it is clear that PCS-64QAM outperforms all other formats in terms of achievable GMI assuming all formats have the same B2B SNR performance. However, the combination of high symbol-rate, Gaussian-like constellations, and low SNR working regimes will lead to DSP challenges, which if are not well addressed will lead to extra implementation penalties.

The first of these challenges was addressed in [START_REF] Ghazisaeidi | Advanced C+L Band Transoceanic Transmission Systems Based on Probabilistic Shaped PDM-64QAM[END_REF], where it was experimentally shown that the Gaussian-like constellation associated to PCS-64QAM and 64APSK cause the polarization demultiplexer to converge more difficulty, such that a pilot-assisted MMA has to be used after pre-convergence by a CMA block. In order to address this issue 4 between the estimated polarization rotation angle, 𝛼 ̂, and the real one, 𝛼, by:

𝑀𝑆𝐸 = 𝔼[(𝛼 ̂-𝛼) 2 ]
( 3.4 )

For theoretical analysis we focus only on QAM formats, as 64APSK is less attractive for commercial implementation and is less performant than PCS-64QAM. We also include 16QAM for better comparison.

For simplicity, we assume a constant polarization rotation angle over the whole sequence length. Fig. 3.3 shows the MSE of the ML estimation versus the observation window size N, for two different SNR working regimes, (i.e., 16 dB and 12 dB).

Due to the constant rotation angle, the MSE continuously decrease versus the averaging length as AWGN is averaged out from the estimation. However, it is observed that the ML estimation converges more slowly for PCS-64QAM, with a higher penalty when passing from 16 dB to 12 dB compared to the other QAM formats. This simple analysis shows that the polarization tracking speed of the polarization demultiplexer is greatly reduced for PCS-64QAM, leading to a poor channel estimation and to possible performance degradation. A similar analysis can also be performed for the carrier phase estimation DSP block 5 [START_REF] Rios-Müller | Maximum Likelihood Carrier Phase Estimation Based on Monte Carlo Integration[END_REF].

Considering a channel model as per ( 1.70 ) where the time-varying phase is modeled as a Wiener process, and assuming a zero-residual frequency offset, the ML phase estimation can be performed following the BPS algorithm based on ( 1.72 ). For the case of QAM formats which present a rotational symmetry of π/2, it is sufficient to test equally spaced phases between 0 and π/2, followed by unwrapping. For all formats, we test 64 phases and perform fully data aided cycle slip removal for MSE calculation. When the SNR decreases to 12 dB (Fig. 3.4 bottom row), 64QAM behaves better than 32QAM and PCS-64QAM. Please note that in the case of QAM constellations, the constellation points with the highest energy contribute the most to the ML estimation. These points are not present in the case of 32QAM, and they have the lowest probability of occurrence in the case of PCS-64QAM. Therefore, it is well expected that the phase estimation of the two last mentioned formats will be degraded. 5 The theoretical study of the convergence of the phase estimation for different modulation formats is also an unpublished and ongoing work. From Fig. 3.4 we can conclude that for low SNR regimes, PCS-64QAM requires higher averaging window lengths compared to the other considered formats, being more sensitive to phase noise.

Experimental Performance

After comparing the theoretical performance of all formats, and the challenges associated to some DSP blocks, we now proceed to compare them experimentally. We start by comparing the performance in B2B configuration. To do so, for each format we generate random signal sequences of length 2 15 with 49 GBd RRC pulses with roll-off 0.01. The generation is performed offline using the standard random number generator MatLab function, which in the case of PCS-

1.E-05

1.E-04

1.E-03

1.E-02

1.E-01

1.E+00 As found theoretically, for low SNR regimes the performance of PCS-64QAM and 64APSK

is degraded due to a poor channel estimation. As shown in [START_REF] Ghazisaeidi | Advanced C+L Band Transoceanic Transmission Systems Based on Probabilistic Shaped PDM-64QAM[END_REF], this degradation can be overcome by the use of pilot symbols for DSP convergence. Increasing the pilot rate will lead to an improvement of the signal SNR, but will reduce the effective GMI as the pilot rate should be subtracted from the net information rate. Therefore, the optimum pilot rate is the one that maximizes the effective GMI. Fig. 3.5 shows the pilot-aided DSP blocks used for 64APSK and PCS-64QAM. and PCS-64QAM with 0.8 dB, and finally 64APSK with 1.2 dB. In the case of 64APSK, there is a manifest degradation of performance for low OSNRs, such that OSNRs below 17 dB cannot be processed with a 2% pilot rate.

The implementation penalties shown in Fig. 3.7a will be translated into penalties in GMI. Fig. 3.7b shows the experimental GMI corrected for pilot use versus OSNR curves for all formats (solid markers), where the ideal curves are also shown for comparison (blank markers). The implementation penalties of 64QAM shown in Fig. 3.7a annihilate its superior theoretical SE with respect to 32QAM, making both formats having the same performance over the OSNR region of interest. This is the same case for 64APSK, which due to its poor performance for low OSNR values and the higher DSP pilot-aided rate, the achievable SE is the same as that of 32QAM, and 64QAM. Therefore, PCS-64QAM achieves a gain of ~0.3 bits/symbol/polar with respect to all other formats at 19 dB OSNR. In terms of SNR, PCS-64QAM presents an increased penalty with optical power with respect to 64QAM and 32QAM. This penalty corroborates the higher nonlinear penalty due to the Gaussian-like constellation of PCS-64AM, as in the linear regime PCS-64QAM, 32QAM, and 64QAM formats have comparable performance. Regarding 64APSK, its poor linear SNR is translated into a degradation in the linear regime of Fig. 3.8a, being of ~0.6 dB in SNR compared to the other formats at an optical power of 13 dBm. It is observed that 32QAM and 64QAM achieve the same maximum SE of 3.7 bits/symb/pol, showing that the use of 64QAM presents no benefit compared to 32QAM. At the optimum power, PCS-64QAM has a gain of ~0.25 bits/symb/pol compared to both 32QAM and 64QAM.

The gain provided by 64APSK is only 0.06 bits/symb/pol compared to 32QAM and 64QAM.

From the above results, we can conclude that PCS-64QAM achieves the best performance considering all types of implementation penalties, making it a good candidate for future generation systems.

TRUNCATED PROBABILISTIC CONSTELLATION SHAPING

Due to the superior performance of PCS-64QAM over other formats, we now extend its use to trans-Pacific distances. As presented in Sec. 1.2.3, the PMF of the PCS-QAM constellation points are set according to a discrete Maxwell-Boltzmann distribution with free parameter 𝛎.

This parameter should be optimized to change the source distribution and minimize the gap to the Shannon capacity for a given target SNR. In the last section, the SNR target was around 12 dB corresponding to the SNR after a trans-Atlantic distance of 6,600 km. For trans-Pacific distances (>10,000 km) the source entropy of PCS-64QAM has to be re-optimized. In this section, we address the design of a truncated PCS-64QAM constellation (TPCS-64QAM), and its use for trans-Pacific distances.

The design process of PCS-64QAM is performed by means of numerical simulations and is discussed next. First, we generate random symbol sequences of length 2 15 for different PCS-64QAM source entropies, H. Then, AWGN is added to each sequence to achieve a given SNR, for which we obtain the information rate, R, by two means: first, we consider an ideal FEC, i.e., a capacity achieving infinite block length FEC for which the correct measurement of R corresponds to the calculation of the GMI; and second, we actually apply our offline homemade FEC decoder to the noisy simulated sequences as described in [ [51] Sec. VI.1]. Finally, we calculate the gap to capacity; i.e., the difference between the Shannon capacity and R for a given SNR and H.

By construction, PCS-64QAM is compatible only with FEC code rates, r, larger than or equal to 2/3 [START_REF] Böcherer | Bandwidth efficient and rate-matched low-density parity-check coded modulation[END_REF]. Therefore, we have used a SC-LDPC FEC code family of 25 rates ranging from 0.67 to 0.91 with steps of 0.01. For more information about SC-LDPC codes please refer to [START_REF] Schmalen | A generic tool for assessing the soft-FEC performance in optical transmission experiments[END_REF][START_REF] Schmalen | Spatially coupled soft-decision error correction for future lightwave systems[END_REF][START_REF] Schmalen | On the design of capacity-approaching unit-memory spatially coupled LDPC codes for optical communications[END_REF].

As stated previously, the PCS-64QAM employed in [START_REF] Ghazisaeidi | Advanced C+L Band Transoceanic Transmission Systems Based on Probabilistic Shaped PDM-64QAM[END_REF] and presented in the last section, was optimized for a SNR target of 12 dB for a target distance of 6,600 km, resulting in H = 5.4 b/symb/pol. Considering the minimum code rate r = 2/3, this PCS-64QAM is not feasible below 3.4 b/symb/pol, corresponding to ~10.5 dB SNR including FEC penalty.

For a target distance of 10,285 km (see Sec. 4.1), the optimum operation point lies in the SNR region between 9 dB to 10 dB, such that the PCS-64QAM of [START_REF] Ghazisaeidi | Advanced C+L Band Transoceanic Transmission Systems Based on Probabilistic Shaped PDM-64QAM[END_REF] cannot be used.

Moreover, for practical implementation concerns, SNR margins up to 1 dB accounting for system end-of-life (EOL) conditions must be considered. Therefore, the minimum required SNR supported by the system should be 8 dB. The optimum PCS-64QAM should minimize the gap to capacity over the region of interest, while still considering the minimum required SNR. Fig. 3.9 shows the minimum supported SNR that can be decoded by FEC with minimum rate r = 0.67 versus PCS-64QAM source entropy, for an ideal FEC (GMI), and for our SC-LDPC codes. Considering a minimum required SNR of 8 dB, the maximum source entropy H when the ideal FEC is considered is 4.8 b/symb/pol, which is reduced to 4.6 b/symb/pol when practical SC-LDPC FEC decoding is applied. Therefore, source entropies bigger than 4.6 b/symb/pol are not supported for our system. Fig. 3.9: Minimum supported SNR for a FEC rate r = 0.67. Fig. 3.10a shows the gap to capacity considering a SNR of 9 dB. For the ideal FEC (square markers) the H that minimizes the gap to capacity is 4.6 b/symb/pol, however we observe a negligible difference for values between 4.3 to 5 b/symb/pol, all of them having a gap to capacity lower than 0.05 b/symb/pol. A different optimum value of H is obtained when actual SC-LDPC FEC decoding is applied (circle markers), which can be explained as follows. For higher values of H, lower FEC code rates are required to successfully decode the sequences for a SNR of 9 dB (Fig. 3.10b). The lower the code rate, our SC -LDPC codes present a higher gap to the maximum achievable rate (GMI) (cf Fig. 3.11b). b/symb/pol compared to the GMI. The slight apparent improvement of TPCS-64QAM with respect to PCS-64QAM when SC-LDPC codes are applied might be to some numerical difference in signal power normalization. However, this discrepancy is less than 0.1 dB in SNR and can be neglected within the frame of this work.

Finally, Fig. 3.12 compares the GMI for the designed TPCS-64QAM with the PCS-64QAM used in [START_REF] Ghazisaeidi | Advanced C+L Band Transoceanic Transmission Systems Based on Probabilistic Shaped PDM-64QAM[END_REF], as well as for some standard formats, 64QAM, 32QAM and 16QAM. In the interval

SNR 16dB SNR 9dB

TPCS-64QAM from 7 dB to 10.5 dB SNR, the proposed TPCS-64QAM outperforms all other formats, having a SNR penalty below 0.2 dB with respect to the Shannon limit. Fig. 3.12: GMI for various modulation formats.

SUMMARY

In the first part of this chapter, we theoretically and experimentally compared the performance of PCS-64QAM, 64APSK, 64QAM, and 32QAM, considering linear implementation penalties and nonlinear penalties after 6,600 km. We showed that while 32QAM and 64QAM formats are simple to generate and receive, this last one suffers from more implementation penalties, and achieves the same SE as 32QAM despite its theoretical superior performance. We also showed that both PCS-64QAM and 64APSK present new DSP challenges, and require pilot-assisted polarization demultiplexing. Regarding 64APSK, it was experimentally shown that it requires a higher pilots-aided DSP rate, and its performance degrades rapidly at low values of SNRs, making its implementation challenging. Taking into account linear, nonlinear, and DSP impairments, PCS-64QAM outperforms 32QAM, 64QAM and 64APSK for coherent transoceanic system applications.

In the second part, we showed the design of a truncated version of PCS-64QAM (TPCS-6AQAM) optimized to maximize the SE of trans-Pacific optical links. The TPCS64QAM minimizes the gap to capacity over the SNR region of 7-11.5 dB, while considering FEC implementation penalties corresponding to our SC-LDPC codes. removal, and least-mean square adaptive post equalization. To process the PCS signals, a preconvergence CMA followed by 1% pilot aided MMA are used for polarization demultiplexing as shown in Sec. 3.1. After DSP processing, the SNR and GMI (corrected for pilot overheads) are computed for each waveform. Then SC-LDPC FEC decoding is carried out.

We start by characterizing the system performance versus channel launched power in terms of both SNR and GMI at 6,209 km and 10,285 km. We consider 64QAM, PCS-64QAM of [START_REF] Ghazisaeidi | Advanced C+L Band Transoceanic Transmission Systems Based on Probabilistic Shaped PDM-64QAM[END_REF] and our customized TPCS-64QAM. Pre-emphasis is performed by varying the power of 8 channels situated in the middle of the band, and measuring SNR and GMI for the center channel (1550.92nm). Fig. 4.2a shows the pre-emphasis curves at 6,209 km for 64QAM and PCS-64QAM of [START_REF] Ghazisaeidi | Advanced C+L Band Transoceanic Transmission Systems Based on Probabilistic Shaped PDM-64QAM[END_REF], while Fig. 4.2b corresponds to 10,285 km for 64QAM and our designed TPCS-64QAM. Fig. 4.2: SNR and GMI versus channel pre-emphasis for 64QAM, PCS-64QAM of [START_REF] Ghazisaeidi | Advanced C+L Band Transoceanic Transmission Systems Based on Probabilistic Shaped PDM-64QAM[END_REF] and TPCS-64QAM, at (a) 6,209 km and (b) 10,285 km. Fig. 4.2 shows that the operating point of our system is 1 dB below the NLT. However, increasing the EDFAs output power above 16.6 dBm is not possible due to their limited output power range. As will be seen, this will limit the achievable gain provided by NLC algorithms. At 6,209 km, PCS-64QAM of [START_REF] Ghazisaeidi | Advanced C+L Band Transoceanic Transmission Systems Based on Probabilistic Shaped PDM-64QAM[END_REF] outperforms 64QAM in terms of GMI by 0.25 b/symb/pol, which is in agreement with Fig. 3.12. In terms of SNR, both formats achieve the same performance.

At 10,285 km, our optimized TPCS-64QAM outperforms 64QAM by 0.25 b/symb/pol. Please note again that this gain is decreased in the nonlinear regime, as the contribution of the fourth and sixth moments of the more Gaussian like TPCS-64QAM to nonlinear variance is higher than that of 64QAM.

Considering a pre-emphasis of 0 dB, we then investigate the performance over different distances ranging from 1,500 km to 10,285 km. In terms of SNR, all three formats achieve similar results. In terms of GMI, PCS-64QAM outperformed 64QAM and TPCS-64QAM up to a distance of 6,200 km (11 dB to 16 dB SNR region). After this distance, the GMI for PCS-64QAM decreased below 3.4 b/symb/pol which is its limit of operation considering a minimum allowed FEC rate of 2/3. For trans-Pacific distances (>10,000km), TPCS-64QAM outperforms 64QAM with a mean GMI of 3.2 b/symb/pol.

Next, we fix the distance at 10,285 km and perform transmission of all 84 channels. Fig. 4.4a shows the measured SNR ranging from 8.9 dB to 9.9 dB. The average value is found to be 9.3 dB. In the same manner, Fig.

throughput. Fig. 4.4c shows the 7 different FEC code rates used to decode each channel. We also investigate the achievable system throughput considering EOL conditions. To do so, we degrade the received OSNR by 1dB by adding ASE noise at the transmitter side as shown in Please note that the measured OSNR does not account for fiber Kerr nonlinear distortions, such that the relation between OSNR and SNR is nonlinear. A degradation of the system OSNR will lead to a lower degradation in SNR near the NLT.

Finally, we study the achievable gains provided by NLC based on FDBP. In this section, we consider only DBP/FDBP applied at the receiver side by replacing the CD compensation block. As already mentioned, the power operating point of our transmission line is 1 dB below the optimum when no DBP is applied, and 2 dB below the optimum point when DBP is applied.

This imposes a strong limitation on the achievable gain provided by DBP. At the operating point (0 dB pre-emphasis) the achievable gain is limited to ΔSNR0dB = 0.4 dB (ΔGMI0dB = 0.11 b/symb/pol), while the peak-to-peak gain is ΔSNR0dB = 1 dB (ΔGMI0dB = 0.3 b/symb/pol). Please note that the already mentioned achievable gains are slightly over-estimated, as the preemphasis curves are obtained by varying the power of only 8 channels within the WDM comb.

In order to study the achievable NLC gain of the entire system versus the low-complexity FDBP algorithm, we now process all 84 channels with different number of FDBP steps. The nominal channel optical power is -2.64 dBm (84 channels with 16.6 dBm total EDFA output power); however, to account for the non-flat WDM power profile and power uncertainties, we optimize the channel power within the FDBP algorithm. Fig. 4.5b shows the total throughput and the spectral efficiency versus the total FDBP steps. The maximum increase in system throughput is found to be 830 Gb/s for a total of 25.48 Tb/s, corresponding to a spectral efficiency of 6.06 b/s/Hz. Finally, we compare the obtained results with already installed commercial systems based on 8QAM tributaries. We transmit 84 real time 8QAM channels at 150 Gb/s using industrial tributaries with a channel spacing of 50 GHz over the 10,285 km deployed testbed. All channels were decoded error free with 0.5 dB margin above our 25% SD-FEC Q 2 -factor threshold (5.4 dB). The achieved net throughput was 12.6 Tb/s corresponding to 3 b/s/Hz spectral efficiency.

Compared to commercial 8QAM tributaries, the solution based on 49 GBd channels employing TPCS-64QAM and multi-rate FEC have the potential of doubling the throughput of existing trans-Pacific networks [START_REF] Kamalov | FASTER Open Submarine Cable[END_REF].

HIGH SYMBOL-RATE TRANSOCEANIC TRANSMISSION

There is a clear trend in the industry to increase the per-channel bit rate of PM-WDM coherent transmission systems. The main drive for this trend is coping with the ever-increasing throughput requirements while curbing the cost per bit via reducing the component count.

Thanks to the coherent receiver technology, the feasible bit rate per wavelength of single-carrier (SC) transceivers over transoceanic distances has dramatically increased from 40 Gb/s (0.8 b/s/Hz) in 2008 to 400 Gb/s (6 b/s/Hz) in 2014 [START_REF] Rios-Müller | Optimized spectrally efficient transceiver for 400-Gb/s single carrier transport[END_REF][START_REF] Charlet | Transmission of 81 channels at 40Gbit/s over a transpacific-distance erbium-only link, using PDM-BPSK modulation, coherent detection, and a new large effective area fibre[END_REF][START_REF] Charlet | 72×100Gb/s transmission over transoceanic distance, using large effective area fiber, hybrid Raman-Erbium amplification and coherent detection[END_REF][START_REF] Salsi | WDM 200Gb/s single-carrier PDM-QPSK transmission over 12,000km[END_REF]. As the bandwidth of the state-ofthe-art RX technology usually takes a lead on the TX technology, dual-carrier (DC) TX design detected by a single wideband receiver may be employed to reduce the component count at the RX side. DC 400 Gb/s transmission over 9,200 km was demonstrated in [START_REF] Zhang | 200 Gb/s and dual-wavelength 400 Gb/s transmission over transpacific distance at 6 b/s/Hz spectral efficiency[END_REF].

In this section, we employ the capacity achieving PCS-QAM and digital NLC based on DBP, together with the cutting-edge CMOS technology available for DAC, and an advanced 70 GHz wideband receiver, to demonstrate new per-channel transmission records over transoceanic distances beyond 400 Gb/s using high symbol-rate channels above 60 GBd.

Single-carrier Experiments

Our first goal is to increase the per-channel bit rate using a SC configuration. To do so, we first need to optimize the PCS-64QAM source entropy according to the SNR region of interest.

We target a trans-Atlantic distance of 6,600 km for which the SNR region is ~12 dB, and a trans-Pacific distance of 13,200 km with a target SNR of ~9 dB.

In Sec. 3.2, we designed the TPCS-64QAM with source entropy of 4.3 b/symb/pol for a target SNR of 9 dB, including FEC implementation penalties. However, the PCS-64QAM designed in [START_REF] Ghazisaeidi | Advanced C+L Band Transoceanic Transmission Systems Based on Probabilistic Shaped PDM-64QAM[END_REF] and used in the last sections targeting a SNR of 12 dB was optimized to reduce the gap to capacity in terms of GMI, and FEC penalties were not taken into consideration.

Therefore, we reoptimized PCS-64QAM targeting a SNR region of ~11-13 dB as done for TPCS-64QAM. The resulting optimum PCS-64QAM source entropy is found to be H = 4.9 b/symb/pol. The minimum supported SNR considering FEC implementation penalties for the PCS-64QAM with H = 4.9 b/symb/pol is 9.5 dB and minimizes the gap to capacity in the SNR region between 11.5 dB and 13.5 dB.

Dual-carrier Experiments

As increasing the SC symbol-rate beyond 74 GBd degrades the overall system performance due to TX impairments, we now switch to a DC TX architecture to completely fill out the RX bandwidth. We design a DC test super-channel consisting of two 68 GBd PCS-64QAM subcarriers at 69.4 spacing. Fig. 4.10 illustrates the measured signal spectra at the TX output, as well as the amplitude and the phase response of our wideband RX. For the reference, the SC is also shown.

It is observed that the amplitude and phase response of the scope present oscillations after a 30 GHz bandwidth, which will be translated into some degradation in performance for the DC configuration. Taking advantage of the large RX bandwidth allowing us to simultaneously recover the two 68 GBd subcarriers for joint DSP processing, we employ DBP to jointly compensate for intersubcarrier, and intra-subcarrier nonlinearities. The optimum total launched power with joint DBP is found to be 13 dBm. We measure only the central super channel at various distances. Fig. 4.12 illustrates the achievable net bit rate of the DC super-channel vs. distance with joint DBP (diamonds), and without NLC (squares). The number of DBP steps per span was set to 20 as in the SC measurements.

It is observed that the use of joint DBP, allows reaching net 850 Gb/s at 6,600 km, and 1 Tb/s transmission over 3,960 km, corresponding to the highest reported transmission using terabit-class super-channels. The joint DBP results in a transmission reach increase of ~20%.

Experimental Results

We perform transmission of all 179 channels. We apply high-resolution DBP with 10 steps per span, where the reference channel power within the DBP algorithm has been optimized to account for the non-flat spectrum power profile. In this sense, channels lying at C-band left edge operate more deeply in the nonlinear regime than the other channels, but the power-adaptive DBP provides more gain in processing C-band left edge, such that the compensated SNR distribution is more uniform than the uncompensated SNR.

Finally, we apply our family of SC-LDPC codes to all channels, and compute the net system throughput corrected for pilot use. The WDM channels are divided into a given number of groups depending on their performance. The optimal set of FEC rates that maximize the throughput are found using the optimization algorithm presented in [START_REF] Ghazisaeidi | Submarine transmission systems using digital nonlinear compensation and adaptive rate forward error correction[END_REF].

SUMMARY

In the first part of this chapter, we demonstrated the achievable benefits of PCS, NLC and multi-rate SC-LDPC codes to increase the system throughput of current trans-Pacific optical fiber links. We employed the truncated PCS-64QAM (TPCS-64QAM) designed for a target SNR to achieve a net throughput of 24.6 Tb/s and a spectral efficiency of 5.9 b/s/Hz after 10,285 km straight line. Moreover, we investigated the gain provided by low-complexity FDBP. Since we operated in the linear regime, the throughput increase due to NLC was limited to ~4%, leading to 25.4 Tb/s and a spectral efficiency of 6.06 b/s/Hz.

In the second part we demonstrated new per channel-rate transmission records for transoceanic distances thanks to cutting-edge DAC's technology and wideband receivers, optimum transceiver design leveraging PCS-64QAM and digital NLC. In particular, we demonstrated single-carrier 550 Gb/s, over 6,600 km, and 430 Gb/s over 13,200 km. We also demonstrated dual-carrier 850 Gb/s after 6,600 km, and dual-carrier 1Tb/s after 3,960 km.

Finally, in the third part we used PCS-64QAM, together with DBP and adaptive multi-rate SC-LDPC FEC codes to achieve a record of 65 Tb/s net system throughput after 6,600 km fiber transmission, with spectral efficiency of 7.3 b/s/Hz.

All the above demonstrations correspond to transmission records at the corresponding time of their realizations.

FDBP step, and we have derived simple semi-analytical formulae for their computation. The obtained results were then validated by a trans-Atlantic experiment, where the maximum achievable gains were also compared to theoretical predictions based on perturbation theory.

Moreover, the aforementioned semi-analytical formulae were further validated along this thesis,

showing that even when a more rigorous theoretical study can be performed to completely understand the dependency of these parameters on system variables, they are valid over a wide range of practical system configurations and can be used to avoid complex and time consuming exhaustive optimization. where it has been shown that the maximum theoretical gain is around 0.5 dB, submarine systems enjoy more NLC gain in excess of 1 dB; making NLC more interesting for ultra-longhaul applications. However, these maximum theoretical gains are degraded due to the considered low-complexity NLC architectures, and also due to linear noise impacting their performance when placed at the receiver side. For practical OSNR regimes, PNLC can achieve gains up to 0.7 dB for systems employing 32 GBd channels, and up to 1.1 dB when 64 GBd channels are used. In the case of FDBP, the maximum gains are increased to ~1 dB and ~1.3 dB respectively. It was also shown that linear noise equally affects both FDBP and PNLC, and that the degradation is independently of the algorithm complexity. These results show that considerable gains can be achieved by low-complexity NLC in transoceanic systems.

Besides NLC techniques, the spectral efficiency of optical fiber systems can be increased by the use of more sophisticated modulation formats aiming to reduce the gap between the constrained and the Shannon capacity. In the third part of this work, I investigated the achievable gains of the novel PCS-QAM over regular QAM formats. First, I experimentally compared the performance of four modulation formats for transoceanic distances, i.e., 32QAM, 64QAM, 64APSK, and PCS-64QAM, addressing four issues: theoretical gap to the linear Shannon capacity, nonlinear performance at optimum launch power, B2B implementation penalties due to limited DAC and ADC resolutions, and DSP challenges. Even when PCS-64QAM achieves theoretical values of GMI close to capacity and outperforms all other formats, it presents new DSP challenges that if are not well addressed can lead to higher implementation penalties. Compared to regular QAM formats, the convergence of the polarization demultiplexing and phase estimation algorithms is greatly reduced for PCS-64QAM when working in low SNR regimes. In practice, in order to improve algorithm convergence, a pilotassisted polarization demultiplexing is required, and the pilot rate has to be properly optimized.

In the nonlinear fiber regime, the "Gaussian-like" constellation of PCS-64QAM results in higher penalties compared to regular QAM formats due to an enhanced nonlinear noise. Despite the nonlinear penalties and considering an optimized DSP, PCS-64QAM can achieve practical gains of ~0.25 bits/symb/pol with respect to regular QAM formats, making it a good candidate for future generation systems. Afterwards, following the industry trend to increase the per-channel bit rate to reduce the cost per bit, new per-channel bit rate using single-carrier (74 GBd) and dual-carrier (2x68 GBd) configurations together with PCS-QAM and NLC were demonstrated. The first single-carrier 550 Gb/s over 6,600 km, and 430 Gb/s over 13,200 km were shown, while the first dual-carrier 850 Gb/s after 6,600 km, and 1Tb/s after 3,960 km were also demonstrated.

One key point to maximize the throughput of submarine systems is to employ both C + L bands to double the throughput per single-mode optical fiber cable. In the last part of this work, a record of 65 Tb/s over 6,600 km employing the novel PCS-64QAM was demonstrated for the first time. Together with DBP and adaptive multi-rate SC-LDPC FECs, a spectral efficiency of 7.3 b/s/Hz over both C+L bands with EDFA-only amplification was achieved.

Perspectives

In this work, I presented the practical achievable gains of single channel NLC for transoceanic transmission systems. It was shown that considerable gains can be obtained with relative low-complexity NLC architectures. The effectiveness of NLC techniques relies on the level of accuracy to correctly model the optical fiber channel. The presence of stochastic effects and link uncertainties will directly impact the achievable gains of NLC techniques. While PMD, PDL, and ASE noise were already studied in this work, the impact of in-line filtering, and uncertainties in the link power profile and fiber coefficients could still be a subject of research.

Moreover, adaptive NLC approaches to blindly estimate the channel parameters could also be further investigated.

It is worth mentioning that even with a perfect knowledge of the channel, the maximum achievable gains are still limited by the number of WDM available for NLC. For the transoceanic systems studied in this work, the maximum achievable gains considering ideal single channel NLC can reach up to ~1.5 dB (~35% reach increase). Multi-channel NLC could be used to further increase performance. However, the main problem behind multi-channel NLC is that apart from being more impacted by stochastic effects, it is still uncertain if the extra gain justifies the associated increase in computational complexity. The study of the practical achievable gains of multi-channel NLC versus added complexity could be performed.

Regarding PCS-QAM, it was shown that the gain in the achievable transmission rate is accompanied by different DSP challenges that need to be addressed to avoid implementation penalties. A more profound study on the convergence speed of the polarization demultiplexer and phase estimation needs to be carried out.

3. 2

 2 TRUNCATED PROBABILISTIC CONSTELLATION SHAPING ..................................... 3.3 SUMMARY ...................................................................................................... 4 TRANSOCEANIC TRANSMISSION RECORDS ........................................ 4.1 25.4 TB/S OVER TRANS-PACIFIC DISTANCES USING TPCS-64QAM ................ 4.2 HIGH SYMBOL-RATE TRANSOCEANIC TRANSMISSION .........................................

4. 4

 4 SUMMARY ...................................................................................................... 5 CONCLUSIONS AND PERSPECTIVES .................................................... REFERENCES ...................................................................................................................... PUBLICATIONS .................................................................................................................... AS FIRST AUTHOR .................................................................................................... AS CO-AUTHOR ........................................................................................................

(

  QAM). Dans la première partie, nous comparons théoriquement et expérimentalement le débit de données atteignable par PCS-64QAM, 64APSK, 64QAM et 32QAM, en considérant des pénalités linéaires et non linéaires après 6600 km. Nous montrons que les formats PCS-64QAM et 64APSK présentent de nouveaux défis pour l'implémentation des algorithmes DSP, en nécessitant l'utilisation de symboles pilotes. En ce qui concerne le 64APSK, nous montrons expérimentalement que sa performance se dégrade rapidement à de faibles valeurs de SNR, ayant besoin d'un taux de pilotes plus élevé dans la chaine DSP. En tenant en compte les dégradations linéaires, non linéaires et les défis d'implémentation du DSP, le PCS-64QAM atteint des SE plus élevées par rapport à 32QAM, 64QAM et 64APSK pour les distances transocéaniques. Dans la deuxième partie, nous montrons la conception d'une version de PCS-64QAM nommée PCS-64QAM tronqué (TPCS-6AQAM) optimisée pour maximiser la SE des liens optiques transpacifiques. Le TPCS-64QAM proposé maximise la capacité sur la région de SNR entre 7 et 11.5 dB, tout en prenant en compte les pénalités liées au FEC. Dans le quatrième chapitre, nous combinons la NLC avec les formats PCS-64QAM et TPCS-64QAM pour démontrer différents records de transmission transocéaniques. Dans la première partie, nous utilisons le TPCS-64QAM, la FDBP et des codes FEC basés sur la technologie SC-LDPC pour atteindre un record de 25.4 Tb/s avec une SE de 6.06 b/s/Hz après 10285 km. Nous montrons que le TPCS-64QAM apporte un gain de 0.25 b/symb/pol par rapport au 64QAM. De plus, nous montrons que l'augmentation de débit due à la FDBP est limitée à ~ 4% à cause de la puissance de sortie limitée de nos amplificateurs optiques. Dans la deuxième partie, nous présentons des nouveaux records de débit de données par canal pour les distances transocéaniques. Grâce à l'utilisation d'un DAC à haute vitesse, à un récepteur large bande, au PCS-64QAM et à la NLC, nous démontrons 550Gb/s par canal sur 6600km et 430 Gb/s sur 13200km. De plus, nous démontrons également 850 Gb/s à double porteuse sur 6600 km et 1Tb/s sur 3960 km. Enfin, dans la troisième partie, nous utilisons PCS-64QAM, ainsi que des codes SC-LDPC et la NLC pour atteindre un débit net de 65Tb/s sur 6600 km, avec une SE de 7.3 b/s/Hz. Toutes les démonstrations ci-dessus correspondaient à des records de transmission au moment de leurs réalisations.

Fig. 1 .

 1 Fig. 1.1 shows the map of optical submarine cables laid worldwide by 2017, stretching over 1.1 million km [7].

Fig. 1 . 2 :

 12 Fig. 1.2: (a) Net system throughput and spectral efficiency (SE) evolution in main research experiments over the years, (b) increase in channel bit rate since the arrival of coherent detection.

Fig. 1 . 3 :

 13 Fig. 1.3: Basic elements of a digital communication system.

Fig. 1 .

 1 4 shows the GMI in b/channel use for different formats, as well as the Shannon capacity.

Fig. 1 . 4 :

 14 Fig. 1.4: Soft-decision generalized mutual information for QPSK, 8/16/32/64QAM. Insets: 32QAM constellation for different SNR values.

Fig. 1 .

 1 5a and Fig. 1.5b show the constellations and the PMFs of the in-phase or quadrature components for the standard 64QAM and PCS based on 64QAM with an arbitrary value of 𝜈.

Fig. 1 . 5 :

 15 Fig. 1.5: Constellations and PMFs of 𝐼 or 𝑄 components for (a) 64QAM and (b) PCS-64QAM formats. (c) 64APSK constellation.

Fig. 1 .Fig. 1 . 6 :

 116 Fig. 1.6 shows the GMI for 64QAM, 64APSK and PCS-64QAM, where the last one is optimized for a SNR value of 12 dB. It is observed that PCS outperforms other formats and approaches the Shannon capacity. A performance comparison between regular QAM formats, 64APSK, and PCS-64QAM is presented in Sec. 3.1.

Fig. 1 . 7 :

 17 Fig. 1.7: (a) Sinc pulses in time domain, and (b) raised cosine spectrum with various roll-offs factors.

Fig. 1 . 8 :

 18 Fig. 1.8: General block diagram of long haul optical transmission systems.

Fig. 1 .

 1 Fig. 1.10: WDM channel decomposed on a finite number of spectral tones.

Fig. 1 . 11 :

 111 Fig. 1.11: Different types of FWM nonlinear interactions inducing a NLI tone at the channel of interest.

  Fig.1.12: (a) DGD over a fiber section with constant birefringence, and (b) schematic representation of a real fiber as a concatenation of infinitesimal birefringent sections randomly coupled.

  PMD can be modeled by the concatenation of independent randomly oriented birefringent components given by:𝐻 𝑃𝑀𝐷 = ∏ 𝑅 𝑖 𝐷 𝑖 𝑅 𝑖 -1 𝑁 𝑖=1 ( 1.57 ) 𝑅 𝑖 = [ cos (𝛼 𝑖 ) sin (𝛼 𝑖 ) -sin (𝛼 𝑖 ) cos (𝛼 𝑖 ) ] ( 1.58 ) 𝐷 𝑖 = [ 𝑒𝑥𝑝(-𝑖(𝜔∆𝑡 𝑖 + 𝜃 𝑖 )/2) 0 0 𝑒𝑥𝑝(𝑖(𝜔∆𝑡 𝑖 + 𝜃 𝑖 )/2) ] ( 1.59 ) Each birefringent element 𝐷 𝑖 introduces a constant phase shift. The rotation matrix 𝑅 𝑖 accounts for the random axes orientation of each PMD element.

Fig. 1 .

 1 Fig. 1.14: Full coherent polarization diverse coherent detection. PBS: polarization beam splitter, PD: photodiode.

Fig. 1 .

 1 Fig. 1.15: Standard DSP blocks for linear impairments mitigation.

  63 )𝒚 ̂𝑉 = 𝐷 ̂𝐿 † [𝒚 𝑉 ]( 1.[START_REF] Carena | Modeling of the Impact of Nonlinear Propagation Effects in Uncompensated Optical Coherent Transmission Links[END_REF]

  1.3.2, signalsignal fiber nonlinear interactions arising from the Kerr effect are a priori deterministic and could be compensated by DSP. Different DSP based nonlinear compensation (NLC) techniques have been under research during the past years. Optical NLC techniques have also been studied. The main interest behind optical NLC techniques is their large processing bandwidth capabilities that are forbidden by DSP. Two main approaches have been proposed in the literature: mid-link optical phase conjugation (OPC), and coherent superposition in phase sensitive amplification (PSA). Both techniques are based on transmitting a phase-conjugated version of the signal to cancel out fiber nonlinearities (NL).

2 ⁄,

 2 𝐿 𝑡𝑜𝑡 is the total link length, η is the normalized propagation distance with respect to 𝐿 𝑑 , and 𝑓(•) is the power profile. Considering homogeneous spans with EDFA-only amplification compensating completely for the span loss, 𝑓(•) is expressed as:𝑓(𝜂) = 𝑒𝑥𝑝 (-𝛼mod(𝐿 𝑑 𝜂, 𝐿 𝑠𝑝𝑎𝑛 )) ( 1.83 )with α the fiber attenuation coefficient and mod(•) the modulus operator. The term 𝑚 = 𝑛 = 0 is excluded from ( 1.82 ). The complexity and performance of PNLC are determined by the size of the 𝐶 𝑚,𝑛 matrix. When more terms are considered, a higher performance is achieved at the cost of a higher complexity. In general, the optimum size of the 𝐶 𝑚,𝑛 matrix will depend on the channel memory induced by chromatic dispersion. Fig.1.16 shows an example of the normalized amplitude in dB of the 𝐶 𝑚,𝑛 , coefficients, i.e. 20𝑙𝑜𝑔 10 (𝐶 𝑚,𝑛 𝐶 0,0 ⁄ ), for an arbitrary system. An approach to reduce the complexity of PNLC is simply to discard all terms below a given amplitude threshold ξth, while keeping B fix[START_REF] Tao | Multiplier-Free Intrachannel Nonlinearity Compensating Algorithm Operating at Symbol Rate[END_REF].

Fig. 1 . 16 :

 116 Fig. 1.16: Magnitude of the (a) complete and (b) truncated 𝐶 𝑚,𝑛 look-up-table (LUT) for an arbitrary system normalized to 𝐶 0,0 . For the truncated case, all valued below -20 dB have been discarded.

Fig. 1 .

 1 Fig. 1.17: Typical B2B performance showing implementation penalties.

Fig. 1 .

 1 Fig. 1.18: SNR versus average channel launched power for an arbitrary system.

  practice, multichannel NLC is challenging and limited success has been demonstrated experimentally.

Fig. 1 .

 1 Fig. 1.19: SNR versus average channel launched power for an arbitrary system.

Fig. 2 .

 2 Fig. 2.1a shows the position of the FDBP block in the DSP chain. As fiber CD is also compensated within FDBP, it replaces the CD compensation block. Fig. 2.1b shows the schematic representation of FDBP based on ( 1.78 ) and ( 1.79 ).

Fig. 2 . 1 :

 21 Fig. 2.1: (a) FDBP position in DSP chain, and (b) FDBP block-diagram.

  the nonlinear phase-rotation method to keep the nonlinear phase within each step below 1 mrad. No change in results were observed for lower values.

Fig. 2 . 2 :

 22 Fig. 2.2: Joint optimization of 𝜅 and 𝛣 𝐹𝐷𝐵𝑃 for single channel transmission considering FDBP with Nstps = 0.5 and SG LPF order n = 0.5. κ

Fig. 2 . 3 :

 23 Fig. 2.3: (a) FDBP gain vs SG LPF order, and (b) optimum values of 𝜅 and 𝛣 𝐹𝐷𝐵𝑃 for different optical powers considering single and 7 channels propagation.

1 )

 1 we perform 30x100 km SSMF simulation and optimize 𝜅 and 𝛣 𝐹𝐷𝐵𝑃 for different values of 𝑁 𝑠𝑡𝑝𝑠 , and

Fig. 2 .

 2 Fig.2.9 shows the Q 2 -factor vs. power curves after 4,800 km transmission both with and without FDBP compensation. In the case of FDBP compensation, three values of 𝑁 𝑠𝑡𝑝𝑠 are examined: 0.25, 0.5 and 1, and 𝜅 and 𝛣 𝐹𝐷𝐵𝑃 are found as per ( 2.3 ) and ( 2.4 ).

  DBP with 𝑁 𝑠𝑡𝑝𝑠 = 10 achieves a maximum gain of 0.8 dB in optimum Q 2factor, which is in line with the value predicted by the theoretical model. On the other hand, FDBP with 𝑁 𝑠𝑡𝑝𝑠 = 1, 𝑁 𝑠𝑡𝑝𝑠 = 0.25 and PNLC provide a gain in optimum Q 2 -factor of 0.7 dB, 0.55 dB and 0.45 dB respectively.

Fig. 2 .Fig. 2 .

 22 Fig.2.12a shows the probability distribution Pr(𝜏̂𝐷 𝐺𝐷 ) of the estimated DGD (𝜏̂𝐷 𝐺𝐷 ) for the PMD-only and PDL+PMD cases obtained as per ( 2.9 ). As expected, they follow Maxwellian distributions with a mean value of 55 ps, in good agreement with the theoretical expected value of 58 ps. By using ( 2.10 ), Fig.2.12b shows the probability distribution Pr(𝛤 ̂𝑑𝐵 ) of the estimated PDL (𝛤 ̂𝑑𝐵 ) for the PDL-only and PDL+PMD cases. In the same manner, they follow Maxwellian distributions with a mean value of ~4 dB, in close agreement with the theoretical expected value of 4.6 dB. Please note that ( 2.10 ) tends to underestimate the true PDL value, as the equalizer response depends on the SNR[START_REF] Hauske | Optical Performance Monitoring in Digital Coherent Receivers[END_REF].From Fig.2.12 it is observed that the maximum PDL value corresponds to ~9 dB, while the maximum DGD is ~160 ps, corresponding to ~5 times the symbol duration. Both values well in excess of the ones found in current commercial systems.

Fig. 2 .

 2 Fig. 2.13: (a) Uncompensated Q 2 -factor probability density and (b) Q 2 -factor loss (QL) probability for each study case.

𝑁 𝑠𝑡𝑝𝑠 = 1 ,

 1 𝑁 𝑠𝑡𝑝𝑠 = 0.25 and PNLC for each of the four different study cases. For the special case of PMD-only, Fig. 2.14 shows the uncompensated Q 2 -factor and the NLC gain (∆𝑄 𝑁𝐿𝐶 2 ) versus the estimated DGD. As observed previously, the uncompensated Q 2 -factor remains constant for all values of DGD. On the other hand, the gain provided by FDBP with 𝑁 𝑠𝑡𝑝𝑠 = 1 and 𝑁 𝑠𝑡𝑝𝑠 = 0.25 remains basically constant, while PNLC shows a higher degradation q

Fig. 2 .Fig. 2 . 2 -∆𝑄 𝑁𝐿𝐶 2 [

 2222 Fig. 2.14: Experimental Q 2 -factor and nonlinear compensation gain (∆𝑄 𝑁𝐿𝐶 2 ) vs estimated DGD.

Fig. 2 . 15 :

 215 Fig. 2.15: Experimental nonlinear gain (∆𝑄 𝑁𝐿𝐶 2 ) probability distribution and nonlinear gain loss (∆𝑄𝐿 𝑁𝐿𝐶 ) for (a) No PDL nor PMD, ((b) PMD-only, (c) PDL-only, and (d) PDL + PMD.

FDBP, N stps = 1 FDBP 85 Finally

 185 , when both PDL and PMD are added into the link (Fig.2.15d) the mean ∆𝑄 𝑁𝐿𝐶 2 is further decreased by ~0.1 dB for all algorithms, while the maximum observable ∆𝑄𝐿 𝑁𝐿𝐶 is reduced by ~0.1 dB compared to the PDL-only case.

  are used to generate 49 GBd PM-16QAM symbols. We use RRC pulses with roll-off factor 0.01. Laser linewidths are set to zero. In order to reduce computation time, we consider only 3 WDM channels. The channel power is set to 0 dB corresponding to optimum power when NLC is applied. The transmission line is equal to the experimental one, where a PMD (𝜏 𝑒 𝐷𝐺𝐷 = 20 ps) and/or PDL (𝛤 𝑒 𝑑𝐵 =1.6 dB) element are placed after each 12 fiber spans (equivalent of 1 loop).The polarization of the signal at the input and output of both elements is randomly rotated following ( 1.57 ) and ( 1.60 ). The amplifier NF is set such that the uncompensated Q 2 -factor is ~5 dB to match the experimental performance working regime. Noise is loaded at the receiver side.The DSP blocks consist on match filtering, CD compensation, polarization demultiplexing based on CMA using 35 taps, and carrier phase correction based on BPS. BER is computed from 20-seed ASE noise loading, and transformed into Q 2 -factor. FDBP and PNLC are optionally applied.

Fig. 2 . 16 :

 216 Fig. 2.16: Numerical nonlinear gain (∆Q NLC 2 ) probability distribution and nonlinear gain loss (∆QL NLC ) for (a) No PDL nor PMD, ((b) PMD-only, (c) PDL-only, and (d) PDL + PMD.

Fig. 2 . 17 :

 217 Fig. 2.17: Numerical set-up for a WDM system composed 55xNspan identical fiber spans of Corning EX3000 fiber for (a) an uncompensated system, (b) FDBP based post-NLC, and (c) PNLC based post-NLC. (d) Lumped RX noise loading scheme.

  Lumped RX noise loading as shown in Fig.2.17d. Amplifiers are set in gain mode; therefore, signal depletion by ASE is not considered. The amplifiers NF is set to 5 dB.

  with N stps = 20 FDBP with N stps = 1 Uncompensated Lines: analytical model Filled markers: SSFM Empty markers: semi-analytical using SSFM estimated a NLI SNR [dB] Channel optical power [dBm]

  Fig. 2.19: (a) Noiseless FDBP gain versus propagation distance for different Nstps values. (b) Noiseless FDBP gain percent relative to high resolution DBP versus Nstps for different propagation distances for a system composed of 111 channels at 32 GBd and 37.5 GHz spacing.

Fig. 2 .

 2 Fig. 2.20: (a) Noiseless FDBP gain versus propagation distance for different Nstps values. (b) Noiseless FDBP gain percent relative to high resolution DBP versus Nstps for different propagation distances for a system composed of 55 channels at 64 GBd and 75 GHz spacing.

Fig. 2 .

 2 Fig. 2.20 shows the same curves but considering 64 GBd channels with 75 GHz spacing.

Fig. 2 .

 2 Fig. 2.21: (a),(d) NLC gain versus propagation distance for noiseless DBP with Nstps = 20, noiseless FDBP with Nstps = 1, and FDBP with Nstps = 1 for different received OSNR regimes. (b),(c) FDBP gain versus OSNR penalty for different Nstps .

Fig. 2 .

 2 Fig. 2.22: PNLC gain percent versus (a),(c) Cmn LUT size, and (b),(d) Cmn discard threshold.One-side LUT size [coefficients]Threshold level[dB]

Fig. 2 .

 2 22b shows the PNLC gain percent considering a 50% LUT versus the threshold value used to discard terms. The maximum gain percent corresponds to the ones of Fig. 2.22a, where saturation is achieved for threshold values above 35 dB. Similarly, Fig. 2.22c and Fig. 2.22d shows the achievable PNLC gain percent for the 64

Fig. 2 .

 2 Fig. 2.23: PNLC gain versus Cmn LUT size for different working OSNR regimes at 6,600 km for (a) 32 GBd and (b) 64 GBd system configurations.

Fig. 3 . 1 :

 31 Fig. 3.1: GMI for 32QAM, 64QAM, 64APSK, and PCS-64QAM with H = 5.4 bi/symb/pol.

3 . 1 .Fig. 3 . 2 :

 3132 Fig. 3.2: (a) Normalized 4'th and 6'th constellation moment, and (b) theoretical SNR/GMI vs optical power for 32QAM, 64QAM, 64APSK, and PCS-64QAM.

Fig. 3 . 3 :

 33 Fig. 3.3: MSE of the ML polarization rotation angle estimation for 32QAM, 64QAM, 64APSK, and PCS-64QAM at (a) 15 dB and (b) 12 dB SNR.

Fig. 3 .

 3 Fig.3.4 top row shows the MSE of the phase estimation angle considering no laser linewidth, and a 100 kHz linewidth for a SNR of 16 dB. When the laser linewidth is set to 0, the MSE continuously decreases for increasing values of N, as AWGN is averaged out from the estimation. When a 100 kHz laser linewidth is considered, an optimum value of N is found, as low values of N are not enough to average out AWGN noise, while high values of N are not able to track the time-varying phase. It is observed that PCS-64QAM behaves similar to 32QAM, with 64QAM being the format that converges more slowly.

Fig. 3 . 4 :

 34 Fig. 3.4: MSE of the ML phase estimation for 16QAM, 32QAM, 64QAM, 64APSK, and PCS-64QAM considering a zero, and a 100 kHz laser linewidth, for two working SNR regimes.

  to emulate the DM. The generated sequences are loaded to an 88 GSamples/s DAC, which drives a PM IQ-MOD modulating a single TLS. The generated optical signal is passed through a double stage EDFA, where a variable optical attenuator (VOA) is placed in the middle to achieve a desired OSNR. The signal is then filtered and detected by an 80 GSamples/s scope receiver with 33 GHz electrical bandwidth. Sampled waveforms are then processed off-line by standard DSP blocks. For 32QAM and 64QAM formats, DSP blocks consisting of polarization de-multiplexing using CMA, carrier frequency and phase recovery using BPS, pilot-based cycle-slip removal using 1% pilots followed by LMSE adaptive postequalization. Finally, SNR and GMI corrected for pilot use are computed for each waveform following ( 1.8 ) and( 1.18 ).

Fig. 3 . 5 :

 35 Fig. 3.5: Pilot-aided DSP for PCS-64QAM and 64APSK.

Fig. 3 .Fig. 3 . 6 :

 336 Fig.3.6 shows the GMI and SNR versus the pilot percentage for the two last mentioned formats for a B2B OSNR of 19 dB corresponding to the end-of-link OSNR of our transmission line after 6,600 km. An optimum value of 1% is found for PCS-64QAM, while it is increased to 2% for 64APSK. In terms of SNR performance, PCS-64QAM and 64APSK working at optimum pilot rate present a degradation of ~0.3 compared to the achievable SNR using 100% pilots.

Fig. 3 .

 3 Fig.3.7a shows the experimental B2B SNR using the optimized DSP pilot rates. At an OSNR of 19 dB, 32QAM has the lowest implementation penalty of 0.5 dB in SNR, followed by 64QAM

Fig. 3 . 7 :

 37 Fig. 3.7: Experimental (a) SNR and (b) effective GMI performance in B2B for 32QAM, 64QAM, PCS-64QAM, and 64APSK.

Fig. 3 .

 3 Fig.3.8a shows the experimental performance (markers) comparison in terms of SNR vs the total optical launched power. The theoretical predictions (lines) are also shown, where the performance in the linear regime is obtained by matching the B2B curves presented in Fig.3.7a and by the characterization of the experimental linear noise of our test-bed.

Fig. 3 . 8 :

 38 Fig. 3.8: Experimental (a) SNR and (b) effective GMI performance after 6,600 km for all studied modulation formats.

Fig. 3 .

 3 Fig.3.8b shows the performance in terms of effective GMI vs total optical launched power.

Fig. 3 .

 3 Fig. 3.10c shows the gap to capacity in the SNR range between 7 dB to 11 dB for H = 4.6 b/symb/pol and H = 4.3 b/symb/pol, for an ideal FEC and that of our SC-LDPC respectively. As found before, when an ideal FEC is considered, the optimum H is 4.6 b/symb/pol. For practical SC-LDPC codes, the optimum PCS-64QAM mode is H = 4.3 b/symb/pol, which minimizes the gap to capacity over the SNR region of interest, and support SNR values down to 7 dB for successfully FEC decoding.

67 rFig. 3 .

 673 Fig. 3.10: (a) gap to Shannon capacity at SNR = 9 dB, (b) optimum r for successfully FEC decoding at SNR = 9 dB as a function of PCS source entropy, H. (c) Gap to capacity for H = 4.3 b/symb/po,l and H = 4.6 b/symb/pol as a function of SNR.

Fig. 3 .

 3 Fig. 3.11b illustrates R as a function of SNR for the optimized PCS-64QAM (blank markers), and TPCS-64QAM (filled markers). TPCS-64QAM does not lead to any meaningful difference compared to PCS-64QAM. At 9 dB SNR, our SC-LDPC codes present a penalty of <0.2

  markers: TPCS-64QAM (H=4.33 b/symb/pol) Blank markers: PCS-64QAM (H=4.3 b/symb/pol)

Fig. 4 . 1 :

 41 Fig. 4.1: (a) Experimental set-up, (b) optical spectrum at the transmitter and after 10,285km. DAC: digital-to-analog convertor, IQ-mod: IQ-modulator, TLS: tunable laser source, WS: wave shaper, CSF: coherent submarine fiber.

Fig. 4 .

 4 Fig. 4.3: (a) SNR and (b) GMI versus distance for 64QAM, PCS-64QAM and TPCS-64QAM.

Fig. 4 . 4 :

 44 Fig. 4.4: Performance of the 84 TPCS-64QAM channels, (a) SNR, (b) GMI and SE, (c) necessary FEC code rate to achieve error free decoding.

Fig. 4 .

 4 Fig. 4.1a. The results are shown in Fig. 4.4 in blank markers. The SNR is degraded 0.6 dB in average, which corresponds to 6.7% decrease on system throughput which reached 23.1 Tb/s.

Fig. 4 .

 4 Fig. 4.5a shows the TPCS-64QAM pre-emphasis curve of Fig. 4.2b but with and without high resolution DBP (Nstps = 20).

Fig. 4 .

 4 Fig. 4.6 illustrates the information rate, R, in terms of the GMI (solid lines), and after applying SC-LDPC FEC codes (dashed lines), of the two custom shaped constellations optimized for trans-Atlantic and trans-Pacific distances.

Fig. 4 . 6 :

 46 Fig. 4.6: Information rate, R, of the two custom shaped constellations.

  Fig. 4.8: (a) Transmitter set-up for single-carrier configuration, and (b) single-carrier transmitted spectrum.

Fig. 4 .Fig. 4 . 9 :

 449 Fig.[START_REF] Hecht | The Evolution of Optical Amplifiers[END_REF].9a shows the five SNR values per channel acquisition (circles), the average information rate (squares) based on the GMI, and the one achieved after FEC decoding (diamonds) for SC transmission over 6,600 km. The total optical power is set to 13 dBm which was found to be the optimum power in the presence of DBP. Contrary to the previous sections where we used low-complexity NLC, here we employ high-resolution DBP with 20 steps per span in order to achieve the maximum rate after transmission. It is observed that all channels transport more than 560 Gb/s net bit rate. Fig.4.9b illustrates similar results while transmitting over 13,200 km. In this case all channels transport more than 430 Gb/s net bit rate.

Fig. 4 .Fig. 4 .

 44 Fig. 4.10: Single-carrier 74 GBd and dual carrier 68 GBd measured signal spectra at TX output, and scope amplitude and phase responses.

Fig. 4 .Frequency

 4 Fig. 4.11: B2B curves for (a) single-carrier, and (b) dual-carrier configurations.

Fig. 4 .Fig. 4 . 13 :

 4413 Fig. 4.12: Dual-carrier net rate versus distance with (diamonds) and without (squares) joint DBP.

  /S LEVERAGED BY PCS-64QAMIn this section, we use the PCS-64QAM shown in Sec. 3.1 with a source entropy of 5.4 b/symb/pol to demonstrate a full C+L-band 6,600 km transoceanic transmission achieving a record net data rate of 65 Tb/s, with a SE of 7.3 b/s/Hz. The average per-channel net bit rate is 363.1 Gb/s[START_REF] Ghazisaeidi | Advanced C+L Band Transoceanic Transmission Systems Based on Probabilistic Shaped PDM-64QAM[END_REF].

Fig. 4 .

 4 Fig. 4.14a shows the C-band transmitter setup. We employ a single TLS as a test channel modulated with a dedicated PM IQ-mod, and surrounded by 87 loading channels divided into even and odd rails, modulated with distinct modulators. Each modulator. Each modulator is driven by a dedicated DAC loaded with different randomly-generated sequences operating at 88 Gsamples/s. The sequence length is 36492 symbols. Pulse shaping is performed using RRC pulses with roll-off 0.01. Loading channels are coupled through a WSS to the measured channel. A polarization scrambler is placed at the WSS output to further decorrelate wavelength channels. The same configuration is used for the L-band transmitter, although the number of loading channels is 90, leading to a total number of 179 C+L channels. The recirculating loop sketched in Fig. 4.14b. It consists of 12 spans of 55 km ultra low loss Corning Vascade EX3000 fiber, with C and L-band EDFAs completely compensating for span loss. The output power of all EDFAs is set to 19 dBm corresponding to NLT+1 dB to achieve maximum gain after NLC. The transmitted signal is received after 10 loops by a real-time scope with 33 GHz bandwidth, sampling at 80 Gsamples/s. DSP blocks are the same as in Chap. 3, where a 1% pilot-aided MMA is employed for PCS-64QAM signals.

Fig. 4 .

 4 Fig. 4.14: (a) C-band transmitter, and (b) recirculating loop. DAC: digital-to-analog convertor, PM IQmod: polarization multiplexed IQ-modulator, TLS: tunable laser source, DFB: distributed feedback laser, PS: polarization scrambler, WSS: wavelength selective switch, A.O: acousto optic switch.

Fig. 4 .

 4 15a illustrates the measured SNRs and GMIs before and after applying DBP. The mean SNR over C + L bands after NLC is ~12 dB, leading to a mean GMI of ~8 b/symb.

Fig. 4 .

 4 Fig. 4.15: (a) PCS-64QAM SNR and GMI of all channels, with and without NLC, and (b) DBP channel power deviation from nominal (up), and DBP SNR gain vs wavelength.

Fig. 4 .Fig. 4 .

 44 Fig. 4.16: (a) Net throughput vs. number of optimized code rates, and (b) summary of C+L-band transoceanic demonstrations at the time of this experiment.

region of 8 -

 8 10 dB, and experimentally compared it with standard 64QAM format. The proposed solution achieved a gain of 0.25 b/symb/pol over 64QAM. We reported a C-band EDFA-only trans-Pacific transmission using 84 channels of 49 GBd TPCS-64QAM and 7 SC-LDPC codes

I

  then studied the impact of stochastic effects as PMD and PDL on the performance of FDBP and the novel PNLC algorithm. If DSP-based NLC techniques are to be adopted in commercial systems, they need to be robust against stochastic effects that are typically not considered in their implementation. While it has been shown that stochastic effects severely reduce the performance of fine resolution multi-channel DBP, it was demonstrated that low-complexity single-channel NLC algorithms which are more practical and close to implementation are less impacted by stochastic effects. The impact of PMD and PDL on FDBP and PNLC was experimentally studied by introducing PMD and PDL elements into our submarine transmission line. I performed massive transmission experiments processing 2,400 different waveforms to correctly account for PMD and PDL statistics. It was found that for a practical fully loaded Cband system composed of 32 GBd PM-16QAM channels, DGD values as large as 150 ps barely impact the performance of FDBP, while a small degradation of ~0.1 dB was found for PNLC.Considering PDL, it was found that values as large as 9 dB lead to a degradation in achievable gain of ~0.2 dB. As the PMD and PDL values introduced into the link are well in excess of modern commercial systems, it can be concluded that both FDBP and PNLC are robust to the studied stochastic effects.With practical implementation in mind, I then studied the achievable gains provided by FDBB and PNLC for different algorithm complexities for transoceanic distances by means of numerical simulations. Two system configurations were considered; the first one based on 32 GBd PM-16QAM channels with 37.5 GHz spacing, and the second one based on 64 GBd channels at 75 GHz spacing. Thanks to the use of GPU cards, a total WDM signal bandwidth of 4 THz corresponding to a fully-loaded C-band system was simulated. Contrary to terrestrial systems

Then I moved to

  the design of a PCS-64QAM targeting trans-Pacific distances. The resulting format was referred as TPCS-64QAM since it was created using a truncated Maxwell-Boltzman distribution. The TPCS-64QAM had an entropy of 4.3 b/symb/pol and minimized the gap to capacity over the SNR region of 7 to 11 dB, while at the same time considering FEC implementation penalties.Finally, in the last part of this work, PCS-QAM, and NLC were employed with state-of-theart DACs, wide-band receivers and multi-rate FECs to demonstrate new transoceanic transmission records. First, the achievable benefits in terms of system throughput increase by applying the previously designed TPCS-64QAM, multi-rate SC-LDPC codes, and NLC based on FDBP were demonstrated. In particular, a spectral efficiency of 5.9 b/s/Hz (net 24.6 Tb/s) after 10,285 km straight-line transmission was demonstrated. The achievable gains provided by FDBP for different algorithm complexities were also addressed, showing that the spectral efficiency can be increased to 6.06 b/s/Hz (net 25.4 Tb/s). Compared to actual systems operating with 150 Gb/s 8QAM tributaries, the proposed solution based on TPCS-64QAM has a promising potential of doubling the throughput of existing trans-Pacific networks.

  .4 PERFORMANCE ANALYSIS ................................................................................ .1 FILTERED DIGITAL BACKPROPAGATION OPTIMIZATION ........................................ 2.1.1 FDBP Parameter Optimization ................................................................ 2.1.2 Experimental Validation .......................................................................... 2.2 IMPACT OF STOCHASTIC POLARIZATION EFFECTS ON NLC ALGORITHMS .............. 2.2.1 Experimental Results .............................................................................. 2.2.2 Simulation Results .................................................................................. 2.3 PRACTICAL LIMITS OF NLC ALGORITHMS ........................................................... 2.3.1 Simulation Set-up .................................................................................... 2.3.2 Numerical Results ...................................................................................2.4 SUMMARY ........................................................................................................

26, 1880 1 COHERENT OPTICAL FIBER TELECOMMUNICATIONS SYSTEMS ....... 1.1 TRANSOCEANIC SUBMARINE OPTICAL FIBER SYSTEMS STATUS AND TRENDS ..... 1.2 DIGITAL TELECOMMUNICATION SYSTEMS .......................................................... 1.2.1 Achievable Information Rates ................................................................. 1.2.2 Forward Error Correction Codes ............................................................. 1.2.3 Modulation Formats ................................................................................ 1.2.4 Pulse Shaping ......................................................................................... 1.3 COHERENT OPTICAL COMMUNICATIONS SYSTEMS .............................................. 1.3.1 Optical Transmitter .................................................................................. 1.3.2 Optical Link ............................................................................................. 1.3.3 Optical Coherent Receiver ...................................................................... 1.3.4 Standard Digital Signal Processing Blocks ............................................. 1.3.5 Nonlinear Digital Signal Processing ........................................................ 11.4.1 AWGN Channel ....................................................................................... 1.4.2 Back-to-back (B2B) Penalties ................................................................. 1.4.3 Nonlinear Variance.................................................................................. 1.4.4 Theory of Nonlinear Channel .................................................................. 1.4.5 Theoretical Limits to Nonlinear Compensation ....................................... 1.5 SUMMARY ........................................................................................................ LOW-COMPLEXITY DIGITAL NONLINEAR COMPENSATION ................. 23 HIGH-CAPACITY MODULATION FORMATS .............................................. 3.1 MODULATION FORMATS COMPARISON ............................................................. 3.1.1 Theoretical Performance ....................................................................... 3.1.2 Experimental Performance ...................................................................

Table 1 :

 1 Some modulation formats used for in coherent optical fiber systems.

	two real-valued components of the constellation symbols Re(𝑎 𝑘 ), and Im(𝑎 𝑘 ) are
	referred as the in-phase (𝐼) and quadrature (𝑄) components respectively, with Re and Im
	standing for real and imaginary parts. The most well-known modulation formats for coherent
	optical systems (sketched in Table 1) are: quadrature phase shift keying: (QPSK), and M -
	quadrature amplitude modulation (QAM), i.e., 8QAM, 16QAM, 32QAM, 64QAM.

  (𝐼 and 𝑄 components), and transformed to an electrical analog waveform by means of four DACs. The four resulting signals are denoted as 𝑥 𝑉𝐼 , 𝑥 𝑉𝑄 , 𝑥 𝐻𝐼 , and 𝑥 𝐻𝑄 .

	multiplexer. The resulting field is finally amplified and sent to the fiber link. Throughout this work
	all modulation formats are considered to be dual polarization.
					xN span
	TX ω 1						RX ω 1
	TX ω 2 ….	MUX	EDFA	Fiber	EDFA	DEMUX	RX ω 2 ….
	TX ω Sch						RX ω Sch
	Optical Transmitter	Optical Link	Coherent Receiver
	Fig. 1.9: Polarization division multiplexing IQ (PM-IQ) WDM transmitter.
	In practice, the bandwidth and resolution of DACs are the key factors to achieve higher data
	rate transmissions. High-speed state-of-the-art DACs based on 28nm CMOS technology
	allowing sampling rates of 92 GS/s are commercially available [59], while DACs based on
	An optical carrier with central frequency 𝜔 𝑆 is splitted into its two orthogonal polarizations,
	each of them feeding two nested Mach-Zehnder (MZ) modulators with 𝜋/2 shift between their
	outputs. Each MZ is driven by the aforementioned electrical waveform signals. After
	modulation, the two orthogonal polarizations are recombined by the use of a polarization beam
	coupler (PBC). In the case of WDM transmission, the same process is performed for the rest
	Sch -1 optical channels, which outputs are then optically multiplexed by means of a WDM

  .4, confirming that the optimum values of 𝜅 and 𝛣 𝐹𝐷𝐵𝑃 are equal for both cases. This allows us to fix 𝑁 𝑠𝑡𝑝𝑠 and modify the fiber coefficients (𝐷, 𝛼, 𝛾 ) to study their individual impact on 𝜅 and 𝛣 𝐹𝐷𝐵𝑃 . 𝜅 and 𝛣 𝐹𝐷𝐵𝑃 are jointly optimized by exhaustive search as previously presented, and their optimum values are obtained.Fig. 2.5a shows the optimum values of 𝛣 𝐹𝐷𝐵𝑃 versus 𝛼 for the particular case of 𝛾 =1.3 [1/W/km] for different values of 𝐷, while Fig. 2.5b shows 𝛣 𝐹𝐷𝐵𝑃 versus 𝛾 for 𝛼 = 0.2 [dB/km] also for different values of 𝐷 . It is observed that the optimum 𝛣 𝐹𝐷𝐵𝑃 presents negligible dependency on 𝛾 and 𝛼, but strongly depend on the value of 𝐷. The same behavior was of the FDBP parameters. Therefore, it is only the product 𝛽 2 ℎ which has to be taken into account.

	(a)		8		@ γ =1.3 [1/W/km]	(b)	8		@ α = 0.2 [dB/km]
			6		D * =2				6		D*=2
	B FDBP [GHz]		4		D=5			B FDBP [GHz]	4		D=5
		4									0.6
	B FDBP [GHz]	2 3	D=10•D SSMF γ= 10•γ SSFM 0.16 2 0.18 0	D=17 *: D unit is [ps/nm/km] D=2•D SSMF γ= 2 • γ SSFM 0.2 0.22	2 0	0.5	D=D SSMF γ= γ SSFM 0.7 0.9	D=17 1.1	0.4 0.5 1.3	1.5
		1		α [dB/km]						γ [1/W/km]	0.3
		0									0.2
											κ
		-1		D=5•D SSMF					Squares: case 1	0.1
				γ= 5•γ SSFM						Triangles: case 2
		-2									0
			0.1	0.2	0.3	0.4	0.5	0.6		0.7	0.8	0.9	1
							N stps
						∆𝑇 =	2𝐷ℎ𝜆 0 𝑅 𝑠 𝑐			( 2.2 )
						𝛣 𝐹𝐷𝐵𝑃 =	1.48 ∆𝑇			( 2.3 )
						𝜅 = √ 44.9×10 -12 ∆𝑇			( 2.4 )

Fig. 2.4: Dependency of 𝜅 and 𝛣 𝐹𝐷𝐵𝑃 with 𝑁 𝑠𝑡𝑝𝑠 (squares) and 𝛽 2 ℎ, 𝛾ℎ 𝑒𝑓𝑓 𝑃 products (triangles) fixing 𝑁 𝑠𝑡𝑝𝑠 = 1, for 1 channel with 𝑛 = 0.5, P 𝑐ℎ =2 dBm. We now investigate on the individual dependency of 𝜅 and 𝛣 𝐹𝐷𝐵𝑃 on fiber parameters. To reduce computation time, we fix 𝑁 𝑠𝑡𝑝𝑠 = 0.5 , and sweep 𝐷, 𝛼 and 𝛾 in the range: 𝐷 = {2, 5, 7, 11, 15, 17, 20 } ps/nm/km, 𝛾 = {0.5, 0.8, 1.3, 1.5} 1/W/km and 𝛼 = {0.16, 0.18, 0.20, 0.22} dB/km, which are in the range of actual fiber parameters. For each of the resulting 112 combinations, observed for all possible combinations of 𝐷, 𝛼 and 𝛾, as well as for the optimum 𝜅. These results agree with Fig. 2.3b, where 𝜅 and 𝛣 𝐹𝐷𝐵𝑃 did not present any variation when changing the channel power, implying that the product 𝛾ℎ 𝑒𝑓𝑓 𝑃 𝑐ℎ does not play a role in the determination Fig. 2.5: (a) 𝛣 𝐹𝐷𝐵𝑃 dependency on 𝛼 and 𝐷 for 𝛾 = 1.3 1/W/km, and (b) 𝛣 𝐹𝐷𝐵𝑃 dependency on 𝛾 and 𝐷 for α = 0.2 dB/km. Considering the unique dependency of 𝜅 and 𝛣 𝐹𝐷𝐵𝑃 on 𝐷, we fix 𝛾 =1.3 1/W/km and 𝛼 =0.22 dB/km, and study their dependency with 𝐷 for different symbol-rates. The results are presented in Fig. 2.6. Instead of showing the dependency of 𝜅 and 𝛣 𝐹𝐷𝐵𝑃 with 𝐷, we use the dispersioninduced pulse broadening per FDBP step defined as: which takes into consideration 𝐷, ℎ and the channel symbol-rate 𝑅 𝑠 . 𝜆 0 is the COI central wavelength, and 𝑐 is the speed of light. We consider symbol-rates of 𝑅 𝑠 = {24, 32, 43, 49} GBd and PM-16QAM signals. It is observed that the optimum values of 𝜅 and 𝛣 𝐹𝐷𝐵𝑃 depend mainly on ∆𝑇, for which the following expressions can be found by numerical fitting. *: D unit is [ps/nm/km] Fig. 2.6: 𝜅 and 𝛣 𝐹𝐷𝐵𝑃 dependency on the pulse broadening per FDBP step at 24, 32, 43 and 49 GBd, and for 𝑛 = 0.5.

  All channels are synthetized by decorrelated binary De Bruijn sequences of length 215 with RRC pulse shaping and 0.01 with roll-off. An optical fiber piece with 80 ps of chromatic dispersion is added after the loading channels for further decorrelation of about 3 symbols between adjacent channels.

	.10. On the transmitter side, a WDM loading comb of 63 C-band DFB lasers
	is modulated with a PM IQ-MOD driven by an 88 GSamples/s DAC. The modulated DFB lasers

are then passed through a WSS who cuts a spectral band of 50 GHz exactly at the middle of the DFB comb (1545.72 nm). A single TLS used as measurement channel at 1545.72 nm is modulated by a second PM IQ-MOD and coupled to the rest of the WDM channels. Fig. 2.10 : Experimental set-up. DAC: digital-to-analog convertor, PM I/Q-mod: polarization multiplexing I/Q modulator, TLS: tunable laser source, DFB: distributed feedback laser, MUX: multiplexer, AO: acousto-optic switch, WSS: wavelength selective switch, PS: synchronous polarization scrambler. The recirculating loop consists of 12 spans of 55 km Corning® Vascade® EX3000 fiber (𝐷=20 ps/nm/km, 𝛾=0.62 1/W/km and 𝛼=0.16 dB/km), with EDFAs exactly compensating for span loss, leading to a total transmission distance of 6,600 km. At the receiver side, the COI is filtered and detected by a standard coherent receiver with a 33 GHz real-time scope working at 80 GS/s. Sampled waveforms are processed off-line by standard coherent receiver DSP blocks described in Sec. 1.3.4. NLC is optionally applied to the received waveforms either by FDBP or PNLC.

  filter

					RX Front-end		DSP
					Bw= 1.05 𝑅 𝑠 t=	𝑘𝑇 𝑠 2 ⁄	CD Matched Filter t= 𝑘𝑇 𝑠	BPS	SNR/ 𝑎 𝑁𝐿𝐼
					RX Front-end			DSP
	TX	n = 1	…	n = N span	Optical filter Bw= 1.05 𝑅 𝑠 t=	𝑘𝑇 𝑠 2 ⁄	DBP/ FDBP	Matched Filter t= 𝑘𝑇 𝑠	BPS	SNR/ 𝑎 𝑁𝐿𝐼
					RX Front-end			DSP
	TX	n = 1	…	n = N span	Optical filter Bw= 1.05 𝑅 𝑠 t=	𝑘𝑇 𝑠 2 ⁄	CD Matched Filter t= 𝑘𝑇 𝑠	BPS	PNLC	SNR/ 𝑎 𝑁𝐿𝐼
					P ASE Total
	TX	n = 1	…	n = N span	+	RX Front-end	DSP

  𝑥 𝑘 is the k'th sent symbol over 𝐻 and 𝑉 polarizations, 𝑛 is an AWGN of variance 𝑁 0 /2, 𝛼 is a random polarization rotation angle, and 𝑦 𝑘 is the k'th received corrupted symbol. The polarization demultiplexer can be modeled as a ML estimator of 𝛼 of the form: where we have considered only one signal polarization for simplicity. Following ( 3.2 ), the estimator will choose the angle 𝛼 ̂ that maximizes the probability distribution function 𝑃(|𝑦 𝑘+𝑛 𝐻 ||𝛼 𝑏 ) between all possible test angles 𝛼 𝑏 . For a complex-circular AWGN channel, the signal amplitude probability follows a Rice distribution, such that 𝑃(|𝑦 𝑘+𝑛 𝐻 ||𝛼 𝑏 ) can

	1 2 3 4 5 6 7 8 9 10 be written as: (a) 𝑃(|𝑦 𝑘+𝑛 𝐻 ||𝛼 𝑏 ) 32QAM 𝜇 4 ⁄ 𝜇 2 𝜇 6 𝜇 2 ⁄ = ∑ ( 2 • |𝑦 ̃𝑘+𝑛 64QAM 2 3 𝐻 | 𝑁 0 ) 𝑀 𝑚=1 Normalized Constellation Moment where 𝑎 𝑚 is the m'th complex constellation point, 𝑃(𝑎 𝑚 ) is its corresponding a priori probability, 64APSK PCS-64QAM Gaussian SNR [dB] (b) 3.6 3.8 4 4.2 4.4 4.6 10 10.5 11 11.5 12 12.5 32QAM GMI [b/symb/pol] 61ch @ 50GHz 6,600 km EX3000 exp ( -(|𝑦 ̃𝑘+𝑛 𝐻 | 2 + |𝑎 𝑚 | 2 ) 𝑁 0 ) 𝐼 0 ( 2 • |𝑦 ̃𝑘+𝑛 𝐻 | 2 • |𝑎 𝑚 | 2 𝑁 0 ) 𝑃(𝑎 𝑚 ) ( 3.3 ) 3.2 3.4 9 64QAM 9.5 PCS-64QAM 64APSK 𝐼 0 (•) is the modified Bessel function of the first kind with order zero, and 𝑦 ̃𝑘+𝑛 𝐻 = 𝑦 𝑘+𝑛 𝐻 • cos 𝛼 𝑏 +
	𝑦 𝑘+𝑛 𝑉	0 • sin 𝛼 𝑏 . Following ( 3.2 ) and ( 3.3 ), we can calculate the minimum square error (MSE) 1 2 3 4 5 6 Total power [dBm] Constellation format 13 15 17 19
								, we can study the polarization
	demultiplexer performance dependency on modulation format by considering a simple fiber
	channel model of the form:
		[	𝑦 𝑘 𝐻 𝑦 𝑘 𝑉 ] = [	cos (𝛼) sin (𝛼) -sin (𝛼) cos (𝛼)	] [	𝑥 𝑘 𝐻 𝑥 𝑘 𝑉 ] + [	𝐻 𝑛 𝑘 𝑛 𝑘 𝑉 ]	( 3.1 )
	where 𝛼 ̂𝑘 = argmax 𝛼 𝑏	𝑁 𝑛=-𝑁 ∏ 𝑃(|𝑦 𝑘+𝑛 𝐻 ||𝛼 𝑏 )	( 3.2 )
	4 The theoretical study of the convergence of the polarization demultiplexer for different modulation
	formats is an unpublished and ongoing work.

  operate at 49 GBd with 50 GHz channel spacing to reflect the industry trend towards increasing the channel symbol-rate. Fig. 4.1a illustrates the experimental set-up. The test channel consists of a single TLS which is modulated with a dedicated PM I/Q-mod. It is surrounded by 83 loading channels modulated by a second distinct modulator. Each modulator is driven by a dedicated DAC operating at 90 Gsamples/s, and loaded with a different randomly generated sequence of length 35680. Nyquist pulse shaping based on root-raised cosine with 0.01 roll-off is used to generate 49 GBd signals. The resulting WDM comb is amplified and a noise source is added to emulate the OSNR at the EOL conditions. The transmission line consists of a straight-line of 188 spans of 110 µm² effective area CSF with 54.4 km average span length. Eight equalization spans using shape equalizers are used to flatten the gain shape, followed by EDFAs. In total, 197 C-band EDFAs with 34 nm bandwidth and 16.6 dBm average output power are used, for a total transmission distance of 10,285 km.

		4.1
				TPCS-64QAM
		3.9		PCS-64QAM [ref. 7] 30]
				64QAM
	GMI [bits/symb/pol]	3.1 3.3 3.5 3.7		32QAM 16QAM
		2.9
		2.7
		8	8.5	9	9.5	10	10.5	11	11.5	12
					SNR [dB]

This is an ongoing and unpublished work.

ACKNOWLEDGEMENTS

I would like to start by thanking all the members of the jury for the precious time they have taken to read and evaluate my report, as well as for their fruitful advices and comments.

It gives me immense pleasure to thank Amirhossein Ghazisaeidi for all his support and continuous guidance during the practical phase of this thesis. I thank him for his astute criticism and for encouraging me to always give my best to present high-quality and meaningful results.

I thank him for his patience and for transmitting me part of his immense knowledge.

I would also like to express my sincere gratitude to Gabriel Charlet for encouraging me to perform this thesis after my Master's internship. I thank him for all the practical discussions and for teaching me how to present results in an original and relevant way. I thank him for his guidance, for his trust, and for all his support. I thank Jeremie Renaudier for all his suggestions and help during the final part of this thesis, and which who I would have liked to work more closely since the beginning of my work. I also thank Patrice Tran, Patrick Brindel, Dylan LeGac and Christian Simonneau for having introduced me to the various techniques and tricks in the lab, but specially for your infinite patience on all those hours setting-up, optimizing, and performing all the experiments. I thank also Christian Doriz for all the fruitful discussions and work during the Optoplan project. I specially thank Alexis Carbo, Elie Awwad, Rafael Rios-Müller and Kaoutar Benyahya for

It is observed that when only PMD is applied both FDBP with 𝑁 𝑠𝑡𝑝𝑠 = 1 and PNLC are equally impacted. When only PDL is added to the link, PNLC shows to be more robust compared to FDBP as found experimentally, while adding PMD+PDL reduces the gain fluctuations compared to the PDL-only case. The trends found numerically are closed in line to the experimental observations. Considering the experimental results, both FDBP and PNLC show to be robust against PMD and PDL impairments for full C-band transoceanic systems when only intra-channel nonlinearities are compensated for.

PRACTICAL LIMITS OF NLC ALGORITHMS

The availability of analytical models predicting the performance of optical fiber transmission systems have also led to evaluate the ultimate upper bounds limits of NLC [START_REF] Poggiolini | The GN-Model of Fiber Non-Linear Propagation and its Applications[END_REF][START_REF] Dar | Inter-Channel Nonlinear Interference Noise in WDM Systems: Modeling and Mitigation[END_REF][START_REF] Serena | Nonlinear SignalNoise Interaction in Optical Links With Nonlinear Equalization[END_REF][START_REF] Dar | Nonlinear Interference Mitigation: Methods and Potential Gain[END_REF][START_REF] Dar | On the Limits of Digital Back-Propagation in Fully Loaded WDM Systems[END_REF]. Theoretically speaking, these upper bounds are only attainable with ideal NLC which completely cancels out the NLI generated by all frequency components inside the considered compensated bandwidth.

From the practical point of view, NLC algorithms with low computational complexity as FDBP and PNCL are preferred due to their lower requirements on DSP. However, this reduction in algorithm complexity is accompanied by a decrease in their efficiency to cancel out NLI, and therefore to a decrease in achievable NLC gain.

In this section we investigate the practical limits of single-channel NLC based on FDBP and PNLC versus different algorithm complexities in fully-loaded C-band submarine systems 3 .

Simulation Set-up

We focus our study on DU systems with homogeneous spans employing lumped EDFA amplification. The link under study corresponds to a typical high-capacity submarine system composed of 55 km Corning Vascade EX3000 fiber spans ( 𝐷= 20.6 ps/nm/km, γ = 0.62 1/W/km, α = 0.157 dB/km), where the full C-band is filled with WDM channels (~4.1 THz signal bandwidth). We consider two system configurations:

1. 111 WDM channels at 32 GBd spaced at 37.5 GHz, and 2. 55 WDM channel at 64 GBd spaced at 75 GHz. reduced to ~75%. For the noisy case, these attainable gains are further decreased, with a higher degradation for longer transmission distances, leading to an optimal link length for which NLC is maximum. Linear noise equally affects both FDBP and PNLC, independently of the algorithm complexity. For practical OSNR working regimes and trans-Atlantic distances, PNLC can achieve gains between 0.5dB and 0.7 dB for systems employing 32GBd channels, while for 64 Gbd systems it is increased to 0.7dB up to 1.1 dB.

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TRANSOCEANIC TRANSMISSION RECORDS

In this chapter, we show how the use of PCS-64QAM, novel DSP techniques as NLC, and multi-rate FECs, cutting-edge CMOS technology available for DACs, and advanced wideband receivers can be used to increase the throughput and channel bit-rate of submarine systems.

In the first section we employ TPCS-64QAM to demonstrate 25.4 Tb/s after 10,285 km. The gain provided by TPCS-64QAM over regular 64QAM is investigated over trans-Atlantic and trans-Pacific distances. Moreover, the potential benefits of NLC to increase the net system throughput is addressed. In the second section, we use TPCS-64QAM and PCS-64QAM together with NLC, and advanced wideband transmitters and receivers to demonstrate the following new per-channel bit-rate records: single-carrier 550 Gb/s after 6,600 km, 430 Gb/s single-carrier after 13,200 km, and 850Gb/s dual-carrier after 3,960 km. Finally, we present a trans-Atlantic transmission record using C + L band EDFA-only amplification leveraging 65 Tb/s thanks to the use of PCS-64QAM, NLC, and multi-rate FECs.

25.4 TB/S OVER TRANS-PACIFIC DISTANCES USING TPCS-64QAM

In Sec. 3.2, the TPCS-64QAM was optimized to approach capacity in the SNR region corresponding to trans-Pacific distances. In this section, we experimentally analyze its achievable benefits in terms of system throughput. To better compare the prospect TPCS-64QAM achievable gains for already deployed systems, we employ a straight-line testbed composed of coherent submarine fiber (CSF) and C-band EDFA-only amplification, which corresponds to a typical configuration of long-haul commercial submarine systems. We opt to

Once the source entropy of the PCS-64QAM formats have been set, we now optimize the symbol-rate. We fix the format to PCS-64QAM with source entropy of 4.9 b/symb/pol, and fix the OSNR to 19 dB which corresponds to the end-of-link OSNR of our transmission line after 6,600 km. We then sweep the channel symbol-rate and calculate the achievable net bit rate. We observed that for our 88 Gsamples/s DAC, the SC symbol-rate that maximizes the net bit rate is 74 GBd. While the ideal bit rate linearly increases with symbol-rate, practical implementation penalties coming from TX impairments also increase with symbol-rate such that the performance is degraded beyond 74 GBd.

Having optimized the modulation formats and symbol-rate, we proceed to perform transmission measurements. The WDM signal is composed of three rails: one test group consisting of a single TLS, one loading group consisting of 7 TLS, and a final group consisting Mitigation of fiber linear effects and TX-RX impairments is a common practice in current commercial systems, such that system performance rests limited by fiber Kerr nonlinear effects.

While single channel NLC is still not present in commercial systems, it will be most likely implemented in future generation systems. On the other hand, capacity approaching modulation formats as PCS-QAM have been recently introduced for the fiber channel, being a good candidate for commercial implementation.

The first part of this thesis was devoted on discussing the current status and trends of transoceanic submarine optical fiber systems, followed by describing the fundamental notions of digital telecommunication systems and transoceanic optical fiber systems.

In the second part of this thesis, I investigated the performance and achievable benefits of low-complexity DSP-based NLC techniques for transoceanic distances. The FDBP algorithm has been proposed in the literature to reduce the high computational complexity of the wellknown DBP. Either its lower computation complexity, FDBP involves an exhaustive optimization of two parameters which have to be a priori optimized for every link under study. Following this, I first studied by means of numerical simulations the dependency of these two parameters on different system variables as: channel count, launched power, symbol-rate, and fiber coefficients. I showed that the FDBP parameters depend mainly on the pulse broadening per

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