Applied Science, Faculty of
Electrical and Computer Engineering, Department of
DSpace
UBCV
Askari, Mohammad Taha
Master of Applied Science - MASc
University of British Columbia
The maximum information rate of a communication channel, often referred to as the Shannon capacity, is achieved by optimizing the distribution of the transmitted symbols. Probabilistic constellation shaping is one principled approach to perform this optimization. Its fairly recently developed variant known as Probabilistic Amplitude Shaping (PAS) is of particular interest as it combines constellation shaping and error-correction coding. Conventional shaping methods, including PAS, are typically designed for the Additive White Gaussian Noise (AWGN) channel, or more generally, for linear channels. However, in optical fiber communications, the main bottleneck for increasing data rate is the so-called Kerr nonlinear effect, establishing an overall nonlinear channel. Therefore, the proper understanding of the nonlinear behavior of the optical channel and the development of nonlinearity tolerant PAS have been active areas of research. This thesis revisits the interplay between channel nonlinearity and PAS. We derive an effective linear channel model for the interaction of transmitted symbol energy sequences and the nonlinear distortion. Based on this, we introduce a visualization tool for nonlinearity analysis which permits us to explain several phenomenological observations reported in previous works. Furthermore, we develop a new nonlinearity tolerant PAS scheme, which outperforms state-of-the-art methods by a significant margin.
Interplay between Fiber Nonlinearity andProbabilistic Amplitude ShapingbyMohammad Taha AskariB.Sc., Sharif University of Technology, 2015a thesis submitted in partial fulfillmentof the requirements for the degree ofMaster of Applied Scienceinthe faculty of graduate and postdoctoralstudies(Electrical and Computer Engineering)The University of British Columbia(Vancouver)August 2022© Mohammad Taha Askari, 2022The following individuals certify that they have read, and recommend tothe Faculty of Graduate and Postdoctoral Studies for acceptance, the thesisentitled:Interplay between Fiber Nonlinearity and Probabilistic Am-plitude Shapingsubmitted by Mohammad Taha Askari in partial fulfillment of the re-quirements for the degree of Master of Applied Science in Electricaland Computer Engineering.Examining Committee:Lutz Lampe, Professor, Electrical and Computer Engineering, UBCSupervisorIan Blake, Honorary Professor, Electrical and Computer Engineering, UBCSupervisory Committee MemberLele Wang, Assistant Professor, Electrical and Computer Engineering, UBCSupervisory Committee MemberiiAbstractThe maximum information rate of a communication channel, often referredto as the Shannon capacity, is achieved by optimizing the distribution of thetransmitted symbols. Probabilistic constellation shaping is one principledapproach to perform this optimization. Its fairly recently developed variantknown as Probabilistic Amplitude Shaping (PAS) is of particular interest asit combines constellation shaping and error-correction coding. Conventionalshaping methods, including PAS, are typically designed for the AdditiveWhite Gaussian Noise (AWGN) channel, or more generally, for linear chan-nels. However, in optical fiber communications, the main bottleneck forincreasing data rate is the so-called Kerr nonlinear effect, establishing anoverall nonlinear channel. Therefore, the proper understanding of the non-linear behavior of the optical channel and the development of nonlinearitytolerant PAS have been active areas of research.This thesis revisits the interplay between channel nonlinearity and PAS.We derive an effective linear channel model for the interaction of transmittedsymbol energy sequences and the nonlinear distortion. Based on this, weintroduce a visualization tool for nonlinearity analysis which permits us toexplain several phenomenological observations reported in previous works.Furthermore, we develop a new nonlinearity tolerant PAS scheme, whichoutperforms state-of-the-art methods by a significant margin.iiiLay SummaryOptical fiber communication systems are the backbone of the physical com-munication infrastructure that enables the global Internet. To ensure highthroughput and reliable communication, modern optical fiber communica-tion transceivers apply highly advanced processing techniques. One of themain limitations for further throughput enhancements through existing op-tical links is fiber nonlinearity. This nonlinearity distorts signals and makesit often impossible to retrieve the transmitted data correctly. We introduce anew nonlinearity mitigation method for transmission over optical fiber linksthat helps to overcome this bottleneck for higher data rates.ivPrefaceThis thesis is an original, independent work by the author Mohammad TahaAskari. A version of chapter 4 has been accepted for presentation at the Eu-ropean Conference on Optical Communications (ECOC), September 2022:M.T. Askari, L. Lampe, J. Mitra “Nonlinearity Tolerant Shaping with Se-quence Selection”. A version of Chapters 3 and 4 has been submitted forpublication to a journal. I developed the original ideas, implemented andanalyzed the methods, and wrote the manuscripts. The industry collabo-rator, Dr. Mitra, provided input and guidance on practical aspects of themethods and participated in the writing of the manuscripts. All the researchwas done under the supervision of Professor Lutz Lampe, who helped withdeveloping concepts, formalizing methods, and revising manuscripts.vTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xGlossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiiAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . xvDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Key Contributions . . . . . . . . . . . . . . . . . . . . . . . . 41.3 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Basics of Shaping and Optical Fiber . . . . . . . . . . . . . 62.1 Constellation Shaping . . . . . . . . . . . . . . . . . . . . . . 62.2 Probabilistic Amplitude Shaping . . . . . . . . . . . . . . . . 82.2.1 Working Principle . . . . . . . . . . . . . . . . . . . . 9vi2.3 Amplitude Shaper . . . . . . . . . . . . . . . . . . . . . . . . 112.3.1 Constant Composition Distribution Matching . . . . . 122.3.2 Multiset-Partition Distribution Matching . . . . . . . 152.3.3 Huffman-Coded Sphere Shaping . . . . . . . . . . . . 172.3.4 Enumerative Sphere Shaping . . . . . . . . . . . . . . 192.4 Signal-Space Interpretation of Amplitude Shaping . . . . . . 212.5 Rate-Loss Comparison . . . . . . . . . . . . . . . . . . . . . . 222.6 Overview of Optical Fiber Communication . . . . . . . . . . . 232.6.1 Fiber Loss . . . . . . . . . . . . . . . . . . . . . . . . . 242.6.2 Chromatic Dispersion . . . . . . . . . . . . . . . . . . 252.6.3 Polarization Mode Dispersion . . . . . . . . . . . . . . 262.6.4 Polarization State Drift . . . . . . . . . . . . . . . . . 262.6.5 Fiber Nonlinearity . . . . . . . . . . . . . . . . . . . . 272.6.6 Transceiver DSP . . . . . . . . . . . . . . . . . . . . . 272.6.7 Channel Model . . . . . . . . . . . . . . . . . . . . . . 282.6.8 Split-Step Fourier Method . . . . . . . . . . . . . . . . 302.6.9 Perturbative Model . . . . . . . . . . . . . . . . . . . . 312.6.10 Performance Metrics . . . . . . . . . . . . . . . . . . . 323 Shaping for Optical Fiber Channel . . . . . . . . . . . . . . 343.1 Effects of Nonlinearity on Shaping . . . . . . . . . . . . . . . 343.1.1 Simulation Setup 1 . . . . . . . . . . . . . . . . . . . . 353.1.2 Blocklength . . . . . . . . . . . . . . . . . . . . . . . . 353.1.3 Mapping Strategy . . . . . . . . . . . . . . . . . . . . 363.2 Linear Channel Approximation for Nonlinear Distortion . . . 393.2.1 Linear Channel Model Formulation . . . . . . . . . . . 403.2.2 Filter Properties . . . . . . . . . . . . . . . . . . . . . 423.2.3 Interplay between Filter and PAS blocklength . . . . . 443.2.4 Role of Multidimensional Mapping . . . . . . . . . . . 464 Nonlinearity Tolerant Shaping . . . . . . . . . . . . . . . . . 504.1 Kurtosis-Limited Sphere Shaping . . . . . . . . . . . . . . . . 514.2 Sequence Selection . . . . . . . . . . . . . . . . . . . . . . . . 53vii4.3 Metric for PAS with Sequence Selection . . . . . . . . . . . . 554.3.1 The LSAS Metric . . . . . . . . . . . . . . . . . . . . . 564.3.2 Relation to EDI metric . . . . . . . . . . . . . . . . . 564.4 Performance Results . . . . . . . . . . . . . . . . . . . . . . . 584.4.1 Simulation Parameters . . . . . . . . . . . . . . . . . . 584.4.2 Setup 1: 256 QAM Transmission . . . . . . . . . . . . 594.4.3 Setup 2: 64 QAM transmission . . . . . . . . . . . . . 645 Conclusion and Future Work . . . . . . . . . . . . . . . . . . 675.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 675.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 68Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69A Model Derivation . . . . . . . . . . . . . . . . . . . . . . . . . 78viiiList of TablesTable 3.1 Simulation Parameters for Setup 1 . . . . . . . . . . . . . . 36Table 4.1 Simulation parameters for Setup 2 . . . . . . . . . . . . . . 58ixList of FiguresFigure 2.1 PAS block diagram . . . . . . . . . . . . . . . . . . . . . . 9Figure 2.2 MPDM Huffman tree . . . . . . . . . . . . . . . . . . . . 18Figure 2.3 HCSS Huffman tree . . . . . . . . . . . . . . . . . . . . . 19Figure 2.4 ESS energy trellis . . . . . . . . . . . . . . . . . . . . . . 21Figure 2.5 Signal space . . . . . . . . . . . . . . . . . . . . . . . . . . 22Figure 2.6 Rate-loss comparison . . . . . . . . . . . . . . . . . . . . . 23Figure 2.7 Fiber attenuation . . . . . . . . . . . . . . . . . . . . . . . 25Figure 3.1 Effective SNR and AIR for PAS with nonlinearity . . . . 37Figure 3.2 Mapping strategy . . . . . . . . . . . . . . . . . . . . . . . 38Figure 3.3 Effect of mapping strategy on SNR . . . . . . . . . . . . . 39Figure 3.4 Filter coefficients for SPM and XPM . . . . . . . . . . . . 43Figure 3.5 Filters for different fiber link lengths . . . . . . . . . . . . 44Figure 3.6 Filters for different baud rates . . . . . . . . . . . . . . . 45Figure 3.7 Frequency domain visualization of the effect of shapingblocklength . . . . . . . . . . . . . . . . . . . . . . . . . . 46Figure 3.8 Frequency domain visualization of the effect of mappingstrategy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48Figure 4.1 K-ESS energy trellis . . . . . . . . . . . . . . . . . . . . . 53Figure 4.2 Sequence selection block diagram . . . . . . . . . . . . . . 55Figure 4.3 AIR comparison of sequence selection with different metrics 59Figure 4.4 Effective SNR and AIR for different selection metrics andshaping blocklengths . . . . . . . . . . . . . . . . . . . . . 60xFigure 4.5 Q-factor comparison of different selection metrics . . . . . 61Figure 4.6 Frequency domain visualization of sequence selection us-ing different metrics . . . . . . . . . . . . . . . . . . . . . 62Figure 4.7 AIR comparison of sequence selection with different map-ping strategies . . . . . . . . . . . . . . . . . . . . . . . . 63Figure 4.8 Comparison of K-ESS and sequence selection using LSASfor a short fiber link . . . . . . . . . . . . . . . . . . . . . 64Figure 4.9 Frequency domain visualization of ESS and K-ESS . . . . 66Figure 4.10 Q-factor comparison of K-ESS and sequence selection us-ing LSAS for a long fiber link . . . . . . . . . . . . . . . . 66xiGlossaryAC Arithmetic CodingAIR Achievable Information RateAS Amplitude ShaperASE Amplified Spontaneous EmissionAWGN Additive White Gaussian NoiseBER Bit Error RateB-ESS Band Trellis Enumerative Sphere ShapingBICM Bit-interleaved Coded ModulationBMD Bit Metric DecodingBRGC Binary Reflected Gray CodeBS-CCDM Bisection-based Constant Composition Distribution MatchingCPR Carrier Phase RecoveryCCDM Constant Composition Distribution MatchingCD Chromatic DispersionDGD Differential Group DelayDM Distribution MatchingDSP Digital Signal ProcessingEDI Energy Dispersion IndexEDFA Erbium-Doped Fiber AmplifierxiiEEDI Exponentially-Weighted Energy Dispersion IndexESS Enumerative Sphere ShapingFEC Forward Error CorrectionGS Geometric ShapingHCSS Huffman-Coded Sphere ShapingK-ESS Kurtosis-Limited Enumerative Sphere ShapingKLSS Kurtosis-Limited Sphere ShapingL-CCDM List-encoding Constant Composition Distribution MatcherLDPC Low-Density Parity-CheckLLR Log-Likelihood RatioLSAS Lowpass-filtered Symbol-Amplitude SequenceMB Maxwell-BoltzmannMPDM Multiset-Partition Distribution MatchingMR Multiset RankingNBL Natural-Based LabelingNLI Nonlinear InterferenceNLSE Nonlinear Schro¨dinger EquationPAM Pulse-Amplitude ModulationPAS Probabilistic Amplitude ShapingPMD Polarization Mode DispersionPS Probabilistic ShapingQAM Quadrature-Amplitude ModulationRDS Running Digital SumSOP State of PolarizationSPM Self-Phase ModulationSNR Signal-to-Noise RatioxiiiSpSh Sphere ShapingSR Subset RankingSSFM Split-Step Fourier MethodWDM Wavelength Division MultiplexingXPM Cross-Phase ModulationxivAcknowledgmentsI would like to thank Professor Lutz Lampe for his unsparing help during hissupervision of my master’s studies. I am also grateful to all my colleaguesin Data Communication Group for their assistance throughout the last twoyears. Finally, I am thankful to my family, especially my parents, for theirlimitless support and love, without which I would never be able to completemy master’s studies.xvDedicationI dedicate this thesis to people who know the true meaning of friendship.xviChapter 1IntroductionGod is the Light of the heavens and the earth. The example ofHis light is like a niche within which is a lamp, the lamp iswithin glass, the glass as if it were a pearly star lit from ablessed olive tree, neither of the east nor of the west, whose oilwould almost glow even if untouched by fire. Light upon light.— The Quran, 24:351.1 MotivationOptical fiber communication plays an important role in the global networkfor data communications [1–3]. As the demand for global Internet traffic con-tinues to grow, the capacity of optical communication systems should alsoincrease [3]. One way to accomplish this is by developing communicationsystems with improved spectral efficiency. This has led to the introductionof coherent optical fiber communications, dual-polarized transmission, andthe use of sophisticated Digital Signal Processing (DSP) at transmitter andreceiver to enable transmission with higher-order modulations [4, 5]. Morerecently, the fiber nonlinearity due to the Kerr effect has posed an ulti-mate limit to achievable rates [3]. Methods to shift this limit have focusedon nonlinearity compensation at the receiver site, e.g., using digital back-propagation to revert the interplay between channel linear and nonlineareffects [6, 7]. A complementary approach is to prepare the signal at the1transmitter site to become more nonlinearity tolerant. This opens the doorto the development and use of nonlinearity tolerant constellation shaping.Constellation shaping is a classical method for improving the informa-tion rate of a communication system [8]. It is broadly divided into the twoapproaches of probabilistic and geometric constellation shaping. The formeroptimizes the distribution of signal points, whereas the latter modifies thelocation of signal points in order to increase the information rate, usuallyunder a power constraint. Even though both probabilistic and geometricshaping can provide ultimate shaping gain of 1.53 dB for the AWGN chan-nel [8], Probabilistic Shaping (PS) is superior to Geometric Shaping (GS)specially when Bit Metric Decoding (BMD) is employed [9, 10]. BMD andbinary Forward Error Correction (FEC) are indispensable elements of highthroughput reliable communication systems, especially with the advent ofBit-interleaved Coded Modulation (BICM) [11, 12]. To this end, it is vitalfor a novel communication scheme to be compatible with the commercialbinary FEC codes. Among the various PS methods [13], PAS has emergedas an effective design for integrating shaping and FEC coding [14]. ThePAS architecture consists of an inner FEC encoder concatenated with anouter Amplitude Shaper (AS). The AS maps uniformly distributed inputbits to amplitudes and generates a non-uniform amplitude distribution. Itoperates on finite-length blocks of amplitudes at a time, which results in arate-loss of the Achievable Information Rate (AIR) compared to the infiniteblocklength limit.In addition to the conventional linear shaping gain, shaping also has thepotential to exhibit a nonlinear gain compared to uniform signaling for theoptical fiber channel [15]. Several previous works, e.g., [16–19], attributedthe nonlinear shaping gain to a change in the moments and, in particular,the kurtosis of the amplitudes distribution. For PAS, it has furthermorebeen demonstrated in [20–22] that the effective Signal-to-Noise Ratio (SNR)depends on the blocklength of the AS. As the AS blocklength increases, thenonlinear shaping gain decreases, while the linear shaping gain increases.Since this trend is not observed if a symbol interleaver is applied to shapedsequences, it can be concluded that the temporal properties of the shaped2sequences play a role in the severity of Nonlinear Interference (NLI).Besides the shaping blocklength, the choice of mapping of amplitudesfrom one shaping block to the components of the transmitted signal affectsthe nonlinearity tolerance of PAS transmission. In particular, it has beenreported that mapping shaping blocks across in-phase and quadrature-phasecomponents in single-polarization transmission and also across x- and y-polarization in dual-polarization transmission can increase the effective SNR[23]. This effect can be explained by noting that the optical channel actsas a lowpass filter on symbol-energy sequences causing NLI, and that finiteblocklength PAS produces a spectral dip at frequency zero for such sequences[24].The filter point of view also facilitates understanding the effect of AStogether with other system parameters such as baud rate on NLI [25]. Thesignificance of the properties of symbol-energy sequences for the NLI inPAS transmission is further highlighted by the so-called Energy DispersionIndex (EDI) introduced in [26]. The EDI is a measure for the varianceof the windowed energy of the shaped signal, where the window length isinfinite in the case of the Exponentially-Weighted Energy Dispersion Index(EEDI) [27]. For PAS implemented with Constant Composition DistributionMatching (CCDM) [28], both EEDI and EDI have been demonstrated to begood predictors for effective SNR.The observations of the interplay between PAS and NLI suggest a mod-ification of the traditional AS design that includes the nonlinear shapinggain as an objective. Two types of such modified designs have emerged inthe literature. The first pairs a conventional AS with a selection module,in which the latter chooses from a candidate set of shaped amplitude se-quences generated by the AS. The selection is aided by a criterion thataccounts for the nonlinearity of the optical channel, such as the EDI in theList-encoding Constant Composition Distribution Matcher (L-CCDM) of[29] or the effective SNR obtained from channel emulation using the Split-Step Fourier Method (SSFM) in [30]. The generation of candidate sets canbe integrated into the operation of a conventional AS by inserting addi-tional flipping bits into each information block that is input to the AS. The3second approach for nonlinearity tolerant shaping directly modifies the se-quence generation in the conventional AS. For example, the trellis of theEnumerative Sphere Shaping (ESS) algorithm [31] can be adjusted to obtainsequences with improved nonlinearity tolerance. This has been done basedon the per-sequence kurtosis in K-ESS [32] and considering sequence-energyvariations in Band Trellis Enumerative Sphere Shaping (B-ESS) [33]. Thebisection-based implementation of CCDM (BS-CCDM) presented in [34] ex-ploits the redundancy in CCDM to remove unfavourable CCDM sequencesand thus mitigate fiber nonlinearity.Despite the endeavors to understand the interplay between NLI and PAS,the lack of a comprehensive nonlinearity analysis of PAS is incontrovertible.The understanding gained by such an analysis will determine a road map todesign effective nonlinearity tolerant shaping schemes. This thesis tries tolighten the interaction between nonlinearity and PAS to achieve nonlinearitymitigation via shaping.1.2 Key ContributionsIn this thesis, we revisit the interplay of PAS and fiber nonlinearity usingthe first-order perturbation based model for NLI. First, we develop an effec-tive linear channel model for the conversion of symbol-energy sequences intoNLI. Our model extends the lowpass filter model from [24] and accounts forSelf-Phase Modulation (SPM) and Cross-Phase Modulation (XPM) as wellas intra-polarization and inter-polarization effects. Using the frequency do-main representation of the effective linear channel, we provide insights intothe nonlinearity tolerance of finite-length PAS as a function of system param-eters such as link length and baud rate. This analysis permits us to explainseveral phenomenological observations reported in previous works, such asthe optimization of AS blocklength and mapping strategy to maximize theeffective SNR. As a second contribution, we employ our channel model to in-troduce the Lowpass-filtered Symbol-Amplitude Sequence (LSAS) metric fornonlinearity tolerant PAS using sequence selection. We identify its relationto the EDI, which is interesting in itself as it validates the EDI as a metric4for nonlinearity tolerance of shaped sequences. It furthermore highlightsthat different from the EDI, the derivation of the LSAS metric does not relyon simplifying assumptions for quantifying NLI caused by symbol-energysequences, and it generalizes systematically to XPM and inter-polarizationNLI. The effectiveness of the new metric for PAS with sequence selection isdemonstrated through simulations of single-span and multi-span fiber linksfor single-polarization and dual-polarization setups, which show that theLSAS metric outperforms the EDI metric in terms of effective SNR, AIR,and error rate.Finally, we integrate the different strategies for improved nonlinearitytolerance with PAS. That is, for the first time, we combine (i) multidi-mensional mapping and (ii) nonlinearity tolerant K-ESS with LSAS-basedsequence selection. Our numerical results demonstrate (i) that multidimen-sional mapping is effective in reducing the search space and thus complexityfor PAS with sequence selection and (ii) that the combination of K-ESS andsequence selection yields notable benefits in terms of, e.g., effective SNR forthe nonlinear fiber channel, compared to applying them individually.1.3 OrganizationThe remainder of this thesis is organized as follows. In Chapter 2, we revisitthe system model for probabilistic amplitude shaping and several amplitudeshapers. Moreover, we cover the basics of optical fiber channel. In Chap-ter 3, we consider shaping for optical fiber channel and derive the linearchannel model between symbol-energy sequences and nonlinear interferenceusing perturbation theory. We also discuss the effect of the fiber link andsystem parameters on the channel properties. We then introduce the newnonlinearity tolerant shaping scheme in Chapter 4, and we present quan-titative comparisons of different nonlinearity tolerant methods. Chapter 5concludes the thesis and suggests future work.5Chapter 2Basics of Shaping andOptical FiberThis chapter covers the necessary background about shaping and opticalfiber communication.2.1 Constellation ShapingThe maximum information rate that can be transmitted through a memo-ryless communication channel with an arbitrarily small error probability isknown as channel capacity [35]. It can be expressed asC = maxP (X)I(X;Y ), (2.1)where X and Y are random variables denoting channel input and output,respectively. I(X;Y ) is the mutual information between channel input andoutput, which is defined asI(X;Y ) = H(X)−H(X | Y ), (2.2)6where H(X) is the entropy of symbol’s distribution P (X), which isH(X) = −∑x∈XP (x) log2 P (x). (2.3)In Eq. (2.3), X is the signal constellation from which the symbols are cho-sen. We note that discrete random variables and entropy are consideredas we study digital communication systems throughout this thesis. Similarequations for continuous random variables can be defined.The distribution that maximizes (2.1) is referred to as capacity-achievingdistribution. In addition to transmission with the capacity-achieving dis-tribution, channel coding is required to achieve the capacity rate with anarbitrarily small error in the presence of channel impairments. The capacity-achieving distribution for AWGN channel with average power constraint Pis Gaussian with zero mean and variance equal to the maximum averagepower, i.e., N (0, P ). As digital communication cannot transmit continuousrandom variables, a discrete approximation to Gaussian distribution, knownas Maxwell-Boltzmann (MB) distribution, is usually used. MB distributionis defined asP (x) =exp(−ν|x|2)∑x∈X exp(−ν|x|2), (2.4)where ν is the distribution parameter determining the trade-off betweenaverage energy and entropy [16]. We note that ν = 0 corresponds to theuniform distribution.In a communication system, source coding is performed on raw bits toremove all redundancies [35]. We note that the output of source codingis bits with uniform distribution, which are mapped to uniform symbols.In the previous generations of communication systems, transmission usinga uniform distribution on a uniform constellation has been used, which issub-optimal. The difference between the capacity C and the achieved rateusing uniform distribution is known as shaping gap, which is 0.255 bits perreal channel use (bits/1D) for the AWGN channel [13]. This shaping gap,which is an asymptotic result for an infinite number of channel use andmodulation order, translates into 1.53 dB in extra required SNR [13]. Con-7sidering Eq. (2.1), the mutual information can be maximized by optimizingthe geometric constellation X and the distribution of symbols P (X), whichare known as GS and PS, respectively. Both PS and GS are equally effec-tive in recovering the ultimate shaping gain of 1.53 dB. Hybrid shaping isanother approach in which both constellation points and distribution areoptimized. Several recent works, e.g., [36, 37], have studied joint geometricand probabilistic shaping for an arbitrary channel using end-to-end learning.2.2 Probabilistic Amplitude ShapingFEC is an indispensable element of modern digital communication systems.Coded bits in the output of FEC are approximately uniformly distributed.The uniform coded bits make the integration of binary FEC and PS chal-lenging as shaped symbols have a nonuniform binary representation. Thus,uniform coded bits decrease the shaping gain. Several coding designs, e.g.,[38–41], have been introduced which address this issue. Notably, a bootstrapscheme is proposed as a solution to merge FEC and PS, in which uniformbits are mapped to shaped symbols and then binary FEC generates paritybits [42, 43]. In the bootstrap scheme, also known as chaining construction[44], parity bits are shaped and embedded in the next transmission block,which requires reverse decoding at the receiver. This scheme increases thelatency and may cause error propagation. Recently, PAS has emerged as apopular approach for integrating shaping and FEC coding [14]. In PAS ar-chitecture, uniformly distributed input bits are mapped to amplitudes usingan AS, which generates a non-uniform amplitude distribution. systematicFEC determines the sign bits. PAS has two main advantages. First, itmerges PS with off-the-shelf binary FEC encoders. Second, it provides rategranularity, making it possible to have a continuum of rates without havingseveral FEC blocks with different coding rates. PAS has attracted greatattention in different communications areas because of the advantages men-tioned above.8(a) Transmitter(b) ReceiverFigure 2.1: PAS block diagram for transmission of PAM symbols: (a)transmitter and (b) receiver.2.2.1 Working PrincipleFigure 2.1 shows the operation of PAS at (a) transmitter and (b) receiver,following the description in [14].TransmitterAt the transmitter, the AS maps a block uk of k independent and uniformlydistributed information bits to a block aℓ of ℓ amplitudes of a 2m-ary Pulse-Amplitude Modulation (PAM) constellation. The amplitudes of the PAMsymbols have a marginal distribution Pa. The elements of aℓ are amplitude-to-bit converted into the sequence uℓ(m−1)a of ℓ(m − 1) amplitude bits. Asequence uℓs of ℓ sign bits is obtained from αℓ information bits uαℓ, 0 ≤ α ≤ 1, and (1− α)ℓ bits u(1−α)ℓc generated using a systematic FEC encoder withcoding rate Rc = (m − 1 + α)/m. Each m bits consisting of m − 1 am-plitude bits from uℓ(m−1)a and one sign bit from uℓs are mapped to a PAMsymbol resulting in ℓ symbols sℓ. As sign bits are approximately uniformlydistributed and independent of amplitudes, the symbol’s distribution is com-puted as Ps(s) =12Pa(|s|). In the case of QAM signaling, every two PAMsymbols are mapped to a QAM symbol. We note that amplitude-to-bit con-version and bit-to-symbol conversion are done using the same bit labelingscheme. In this thesis, we use Binary Reflected Gray Code (BRGC) [45],9which is shown to achieve higher BMD rates compared to Natural-BasedLabeling (NBL) [14, 46]. Shaping rate of this scheme isRs =kℓ, (2.5)while the overall transmission rate isR = Rs + α. (2.6)Using an AS with finite length incurs an extra rate-loss to the AIR anddecreases the overall shaping gain. Rate-loss is defined asRloss = H(Pa)−Rs. (2.7)This rate-loss, which is the difference between the distribution’s entropy andshaping rate, diminishes as the blocklength increases.ReceiverFigure 2.1b demonstrates the receiver process for PAS. For each element ofreceived symbols rℓ, m log-likelihood ratios (LLR) are computed based onthe channel conditional probability PR|S(r | s) asL(Bi) = log(∑s∈Si,0 Ps(s)PR|S(r | s)∑s∈Si,1 Ps(s)PR|S(r | s)), (2.8)where Si,u (i ∈ {1, 2, . . .m}) denotes the symbols in constellation S whichlabeling is u (u ∈ {0, 1}) for ith bit. Next, soft bits Λmℓ are decoded to bitsu˜(m+α−1)ℓ, from which u˜αℓ are the recovered information from sign bits andu˜ℓ(m−1)a are the recovered amplitude bits. Amplitude bits are converted toamplitudes using the same labeling as bit-to-amplitude conversion, and thenrecovered shaped amplitudes a˜ℓ are converted back to k information bits.We note that parameter α grants granularity to the rate of PAS. AIR [47–49] is the maximum possible rate for reliable communication using BMD.10AIR for the PAS scheme isAIR = H(Ps)−m∑i=1H(Bi|R)−Rloss. (2.9)2.3 Amplitude ShaperAS is the main shaping blocks of PAS at transmitter. An AS is characterizedby a set of input sequences, a set of output sequences, and an injectivemapping between input sequences and output sequences [13]. As the inputof an AS is a block of uniform bits, the set of input sequences are all binarysequences asSI = {b = (b1, b2, . . . , bk) | bi ∈ {0, 1} ∀i ∈ {1, . . . , k}}, (2.10)where bi are independent and identically distributed (i.i.d) random variableswith Bernoulli distribution Pr(Bi = 1) = 0.5 ∀i ∈ {1, . . . , k}. Each AShas a shaping architecture, which is the design principle behind the set ofoutput sequencesSO = {x = (x1, x2, . . . , xℓ) | xi ∈ A ∀i ∈ {1, . . . , ℓ},1[F(x) = 1]}, (2.11)where 1[.] is the indicator function, which is 1 when its argument is true,A = {a1, a2, . . . , a2m−1} is the set of amplitudes (alphabet set), and F is aspecific property determined by the shaping architecture [13]. The mappingbetween SI and SO is governed by the shaping algorithm. Composition ofsequence x is the number of occurrences of each amplitude of A in sequencex asC(x) = [na1 , na2 , . . . , na2m−1 ], (2.12)wherenai =ℓ∑m=11[xm = ai], (2.13)112m−1∑i=1nai = ℓ. (2.14)We note that the empirical distribution of sequence x is closely relatedto composition as P˜ (x) = C(x)/ℓ. P˜ (x) is also referred to as type ofcomposition C(x) [28]. Moreover, we define the energy of sequence x asE(x) = ∥x∥22 =ℓ∑i=1|xi|2. (2.15)There are two categories of ASs, which are named as nomenclature of Dis-tribution Matching (DM) and Sphere Shaping (SpSh) [13]. DM refers tomethods in which output sequences all come from a set of compositions,which are optimized to have an average empirical distribution close to thecapacity-achieving distribution. EquationFDM(x) = 1[P˜ (x) ∈ PDM](2.16)outlines the shaping architecture of DM, where P is the set of goal distribu-tions. SpSh methods choose the most energy efficient sequences to achievea certain rate, i.e.,FSPSh(x) = 1[E(x) ≤ ESPSh], (2.17)where ESPSh is the maximum energy limit, optimized to achieve the de-sired rate. In the following sections, we study the shaping architecture andalgorithm of several popular ASs for PAS.2.3.1 Constant Composition Distribution MatchingShaping ArchitectureCCDM has been originally employed for PAS [14, 28]. CCDM, which isunder umbrella of DM, maps uniform bits to sequences with a fixed com-position. Thus, all sequences in the output set SCCDMO are permutations of12each other, i.e., all of them have the same number of occurrences for eachsymbol. Thus, all sequences have the same type asPCCDM = {CCCDMℓ}. (2.18)Moreover, number of total unique sequences for CCDM with output block-length ℓ and composition C is determined by multinomial coefficient asMC(C) =ℓ!∏2m−1i=1 nai !. (2.19)Number of input bits k is determined by the number of sequences in (2.19)ask = ⌊log2MC(C)⌋, (2.20)where ⌊.⌋ is the floor function. We note that CCDM has the highest rate-losscompared to other amplitude shaping methods, as the optimal transmissiondoes not require that all sequences with length ℓ have the optimal distribu-tion, still the average distribution over all symbols should be optimal, andas a result, CCDM is not efficiently using the signal-space.Shaping AlgorithmThe easiest approach is indexing the output sequences and using input bitsas an index number via a lookup table. Lookup table requires high stor-age as number of entries increases exponentially with blocklength k, seeEq. (2.19). Another shaping algorithm for CCDM is employing Arith-metic Coding (AC), which is based on considering two intervals for inputand output and then splitting them depending on the distribution of inputbits and output amplitudes [28]. AC encoding process is based on bisectingthe input interval and choosing one sub-interval based on each bit until thechosen input interval lies inside a unique output sub-interval. At this point,the chosen output sub-interval is selected, and its corresponding amplitudeis inserted in the output block. Then, this procedure continues with theupdated composition until the last bit. For the decoding process, input and13output intervals will be swapped.AC is strictly serial in number of input bits for encoding and numberof output amplitudes for decoding, which makes it unattractive for highthroughput applications [28]. Subset ranking (SR) and Multiset Ranking(MR) are alternatives to conventional AC method which do not rely onsequential computation [50]. SR, which only supports binary alphabet, i.e.,A = {a1, a2}, defines an equivalent representation for each binary outputsequence as a subset of indices where one of two alphabets occur. Thecardinality of this subset is called weight w. All subsets with a certainweight can be ranked uniquely using lexicographical ordering. As a result,a one-to-one mapping between binary inputs (equivalently their index) andbinary outputs (equivalently their subset rank) is established. Finding therank of a subset, i.e., counting the number of previous subsets, is calledranking, and mapping a rank to a subset is called unranking. We note thatthe process of shaping and deshaping for CCDM are unranking and rankingfor SR, respectively. A binary-to-non-binary transformer is needed in orderto employ SR for non-binary alphabets. This transformer slightly increasesthe rate-loss. Bit-level DM [51] introduces a procedure to use several binaryAS for non-binary amplitude shaping.Despite SR, MR is not limited to binary alphabets and thus does not addany extra rate-loss. In addition to the target rank, which is the index of in-put sequence, relative rank is defined for each amplitude in the composition,which is the number of sequences having a lower amplitude in a specific posi-tion of the shaped sequence. Relative rank of the lowest amplitude is alwayszero, and relative rank of higher amplitudes is computed cumulatively byadding the relative rank of previous amplitudes with multinomial coefficientof the updated composition by subtracting 1 from the count of one ampli-tude before the corresponding amplitude. For unranking, after computingthe relative rank of all amplitudes, the amplitude with the highest relativerank that does not exceed the total rank is placed in the output block, andthen the corresponding relative rank is subtracted from the total rank. Forranking, the total rank starts from zero, and then at each step relative rankof a given amplitude is added to the total rank, and composition is updated.14MR needs ℓ sequential operation for both mapping and demapping which isan improvement over k sequential operations for AC (usually ℓ ≤ k). More-over, the multinomial coefficients can be calculated and stored in a lookuptable which speeds up the algorithm.2.3.2 Multiset-Partition Distribution MatchingMultiset-partition Distribution Matching (MPDM) is an extension of CCDMwhich lifts the constraint of constant composition [49]. As a result, theoutput set SMPDMO includes sequences from all compositions so that theaverage empirical distribution matches the goal distribution, i.e.,PMPDM =N⋃i=1{Ciℓ}, (2.21)and ∑Ni=1 niCiℓ∑Ni=1 ni= p∗, (2.22)where ni denotes the number of sequences in composition Ci, N is the totalnumber of MPDM compositions, and p∗ is the goal distribution. MPDMis not a single-composition scheme, and as such, it uses the signal-spacemore efficiently compared to CCDM. We note that the total number ofcompositions with length ℓ and alphabets A is (ℓ+|A|−1|A|−1 ), but there is noconstructive algorithm to find compositions Ci, i = 1, . . . , N which satisfyEq. (2.22). Moreover, even if the compositions are found by exhaustivesearch, developing an implementable algorithm to map the input bits toa composition would be challenging. Accordingly, pairwise MPDM is sug-gested, which makes the extra assumption that each composition makes apair with its complementary composition so that the average distributionof the pair equals the goal distribution. This assumption imposes an extrarate-loss on the rate of MPDM. We denote the ith composition pair with(Ci, C¯i), and we note thatCi + C¯i2ℓ= p∗. (2.23)15Based on (2.22), Eq. (2.23) only holds if the number of sequences from eachcomposition in the pair (Ci, C¯i) is the same. As a result, the total numberof sequences for the pair is bounded to two times the minimum number ofsequences of two compositions asM(Ci, C¯i) =2min(MC(Ci),MC(C¯i)) Ci ̸= C¯iMC(Ci) Ci = C¯i , (2.24)where M(Ci, C¯i) is the composition pair size for the case of non-degenerate(Ci ̸= C¯i) and degenerate (Ci = C¯i) pair, respectively. Based on Eq. (2.24)for non-degenerate pair, the composition with more sequences is not fullyutilized, which again increases the rate-loss. In order to devise an algorithmfor determining the composition of an input sequence, two extra constraintsare imposed on the selected compositions so that the size of each compositionpair and the total number of sequences be a power of 2. For the latter, allcomposition pairs are sorted according to their size in descending order, andthen first Npairs pairs are selected so that they sum up to the highest possiblepower of 2. We note that pairwise MPDM discards several sequences basedon Eq. (2.25) and Eq. (2.26) which again translates to an increased rate-loss.ki =⌊log2M(Ci, C¯i)⌋, (2.25)Npairs∑i=12ki = 2k. (2.26)As the input bits are uniform and mapping is an injective function, theprobability distribution of output sequences is also uniform. As a result, theprobability that pair (Ci, C¯i) is chosen is 2ki−k. We note that the distributionof compositions is dyadic, and Eq. (2.26) can be rewritten asNpairs∑i=12ki−k = 1, (2.27)16which is Kraft inequality [35], fulfilled with equality. As a result, a binaryHuffman tree can be employed to address each composition pair using aprefix with length pi = k − ki. We note that the constraint of fixed inputlength is fulfilled. Moreover, compositions inside a non-degenerate pair aredistinguished with 1 bit. To conclude, MPDM shaper uses pi first bits todetermine the composition pair associated with the input sequence, 1 nextbit to determine the composition within the pair, and the remaining bits areused as the payload of a conventional CCDM AS. Figure 2.2 demonstratesan example of MPDM Huffman tree and mapping of an input sequence to acomposition. For deshaping, first, the composition of the shaped sequenceis determined, which determines pi bits, and then ki bits are decoded usinga CCDM deshaper. As MPDM deshaper requires the composition of thesequence before performing the conventional CCDM deshaping, an extralatency is added to the system.2.3.3 Huffman-Coded Sphere ShapingHuffman-coded Sphere Shaping (HCSS) is an SpSh extension of MPDM,which lifts the constraint of composition pairs [50]. Instead of having com-position pairs with an average equal to the goal distribution, all compositionswith energies lower than the maximum energy limit EHCSS are considered,i.e., all points inside an ℓ-sphere in ℓ dimensional space. EHCSS is chosento achieve a given rate for a fixed output blocklength ℓ. Similar to MPDM,number of sequences inside each composition is rounded down to the closestpower of 2, which increases the rate-loss. Moreover, the total number ofsequences is a power of 2 to build a Huffman tree for the dyadic distribu-tion of compositions. In this regard, all compositions inside the ℓ-sphereare sorted according to their size (2ki , i = 1, . . . , N) in descending order andthen Ncomp first compositions are chosen so thatNcomp∑i=12ki = 2k, (2.28)17Figure 2.2: MPDM Huffman tree. Prefix (blue) bits determine thepair, the next bit (red) determines the composition within thepair, and the payload is shaped using CCDM. The blue edgesare chosen based on the prefix bits, and CCDM4 is determinedbased on the next bit. We note that CCDM3 is the degeneratepair.where kℓ is the desired rate. Next, a binary Huffman tree assigns an addresscode with length pi = k − ki bits to each composition. HCSS shaper usesthe first pi bits to determine the corresponding composition, and then con-ventional CCDM shaping operation is performed on the remaining ki bitsusing one of CCDM shaping algorithms, e.g., AC or MR. Figure 2.3 showsan example of Huffman tree generated for HCSS. Deshaping is similar toMPDM in which first the composition is determined, and then conventionalCCDM deshapers are used to recover the information bits. Moreover, asHCSS is composition-based, it is capable of pruning the Huffman tree inorder to decrease the complexity or apply a specific property to the outputset of sequences SO, e.g., discarding the compositions which do not con-tribute much to the total number of sequences, and thus decreasing the sizeof Huffman tree with a small increase in rate-loss.18Figure 2.3: Huffman tree for HCSS method. Prefix (blue) bits deter-mine the composition, and the payload is shaped using CCDM.Blue edges are chosen based on the given prefix.2.3.4 Enumerative Sphere ShapingESS is an SpSh method which uses an energy trellis to index the outputsequences for shaping and deshaping [31, 52]. The energy trellis consists ofℓ+ 1 stages corresponding to the starting node and each output amplitude.Each stage includes several nodes connected to the next stage’s nodes bybranches. Each branch is associated with a specific amplitude, and eachoutput sequence is a path from the first stage to the final stage. The energyof each node at stage n is the accumulated energy of all amplitudes up tostage n, i.e., e =∑ni=1 |ai|2. The total number of paths starting from a nodewith energy e and at stage n to the final stage is T en, which is recursivelycomputed asT en =∑a∈AT e+a2n+1 , (2.29)whereT eℓ =1, e ≤ EESS0, otherwise , (2.30)19and EESS is the maximum energy of sequences, which is determined suchthat the number of sequences meets the desired rate, i.e., k/ℓ. We notethat T 00 is the total number of sequences in the output set SO. Figure 2.4shows an example for energy trellis, which enumerates 19 sequences withalphabets A = {1, 3, 5, 7} and maximum energy 28. Each node is shown bypair (e, T en), as visualized in Figure 2.4b. The index of a sequence a is thecount of sequences that are in lexicographical order before a. Finding theindex of the sequence a can be done at ℓ stages by accumulating the numberof sequences b which are equal to a up to stage n − 1 and their amplitudeat stage n, i.e., bn, is lower than the corresponding amplitude of a, i.e., an.Thus, the index of a based on the trellis isi(a) =ℓ∑n=1∑b∈A,b<anT∑n−1i=1 |ai|2+b2n︸ ︷︷ ︸∆=Tacc(an), (2.31)where Tacc(an) is the number of sequences which are equal a in the firstn − 1 stages, and are lexicographically before a. In this regard, shaping isequivalent to finding the sequence a corresponding to a given index i(a). Ateach stage n, Tacc(an) for an ∈ A is computed and then the amplitude a∗ncorresponding to the maximum Tacc(an), not exceeding i(a), is chosen, i.e.,a∗n = argmaxan∈ATacc(an)s.t. Tacc(an) ≤ i(a). (2.32)Then, the same procedure is repeated at stage n + 1 with updated indexi(a) − Tacc(a∗n). Deshaping is simply finding the index of sequence a usingEq. (2.31). Figure 2.4a demonstrates an example of shaping and deshapingusing the energy trellis.20(a) (b)Figure 2.4: (a) ESS energy trellis for A = {1, 3, 5, 7}, ℓ = 4, andmaximum energy 28. Dashed path corresponds to sequence a =(3, 1, 3, 3) (i(a) = 14) (b) reference node representation.2.4 Signal-Space Interpretation of AmplitudeShapingEach output sequence with length ℓ corresponds to a point in ℓ-dimensionalspace. Because of discrete alphabets A, the ℓ-dimensional space is a reg-ular grid of points, and the set SO is a subset of all points. All pointswith constant energy on the surface of an ℓ-sphere are called an energyshell. Each amplitude shaping architecture constraints the chosen pointsto achieve the desired shaping goal. DM methods choose points which cor-respond to a union of compositions, which translates to one (CCDM) orseveral (MPDM) energy shells in ℓ-dimensional space [13]. SpSh methodschoose the most energy-efficient sequences, which are the points inside anℓ-sphere with a certain radius. Projection of the points of a ℓ-sphere to lowerdimensions indirectly induces a Gaussian distribution as ℓ approaches infin-ity. We note that CCDM output sequences all have the same compositionand thus the same energy ECCDM. As a result, CCDM sequences are onthe surface of an ℓ-sphere, but SCCDMO does not include all sequences insidethe shell as several distinct compositions can possibly have the same energyECCDM. MPDM output sequences correspond to several compositions, so21Figure 2.5: 2-dimensional visualization of signal space for differentASs.signal points are chosen from several shells. HCSS and ESS both choosesignal points inside an ℓ-sphere. Still, their geometric signal-space is not thesame because HCSS discards several sequences from each composition andthe outermost shell to fulfill the constraints for building a Huffman tree. ESSdoes not discard sequences thanks to the energy trellis. Figure 2.5 showsa two dimensional visualization of signal-space for CCDM, MPDM, HCSS,and ESS. To conclude, SpSh methods use the signal-space more efficientlycompared to DM. We note that due to the sphere hardening effect, mostsequences are chosen from shells near the sphere surface as ℓ approachesinfinity, and as a result, for long blocklengths (high dimensions), DM andSpSh methods have the same signal-space.2.5 Rate-Loss ComparisonRate-loss, as defined in Eq. (2.7), is a figure of merit for an AS with in-put blocklength k and output blocklength ℓ. We note that DM methodsshape the amplitudes to achieve the desired distribution, while SpSh meth-ods choose the sequences inside an ℓ-sphere to reach a certain rate. More-over, based on the discussion in Section 2.4, CCDM has the highest rate-losscompared to other ASs, as it does not efficiently use the signal-space. Thus,in order to have a fair rate-loss comparison between different amplitudeshaping methods, we follow the guideline in [13], which is fixing the goaldistribution for CCDM and MPDM, and then using the MPDM shaping22Figure 2.6: Rate-loss comparison for different ASs. 64 QAM(A = {1, 3, 5, 7}) with a goal amplitude distribution of[0.4378, 0.3212, 0.1728, 0.0682].rate for HCSS and ESS. Figure 2.6 shows the rate-loss comparison betweenCCDM, MPDM, HCSS, and ESS for different shaping blocklengths ℓ. We seethat CCDM has the highest rate-loss in comparison with other shapers withan extra rate-loss of about 0.3 bits/amp compared to ESS for ultra-shortblocklengths. Furthermore, Figure 2.6 demonstrates that as blocklengthincreases, the rate-loss decreases. This figure also validates the sphere hard-ening effect as the difference between rate-loss of multi-composition methodsbecomes negligible for blocklengths more than 100. Rate-loss of all meth-ods approaches zero for infinite blocklength. We also see that ESS has thesmallest rate-loss, which is consistent with the discussion in Section 2.4.2.6 Overview of Optical Fiber CommunicationOptical fibers are used for short-haul and long-haul transmission, wherethe former refers to link lengths less than 100 km [53]. Short-haul opticalfiber communication usually uses the intensity-modulation direct-detection23scheme, in which the information is transmitted through the intensity oflight [54, 55]. This scheme is less complex and economical, making it prefer-able for inter data center or intra-city applications. Coherent transceivers,which are used for long-haul optical communication, convey informationthrough both amplitude and phase. Furthermore, as the light wave hastwo orthogonal polarizations, information can be transmitted through bothpolarizations, leading to dual-polarization optical fiber [5]. Moreover, Wave-length Division Multiplexing (WDM) systems have become the predominantscheme to further utilize the optical bandwidth by modulating several carrierfrequencies.As signal experiences several impairments through the fiber, sophisti-cated DSP is needed at transmitter and receiver to enable reliable and highthroughput transmission. In the following sections, we cover the most im-portant impairments in optical fiber and then comment on the commonlyused DSP at transmitter and receiver. We note that other impairments,including carrier phase and frequency offset, also affect the waveform prop-agating through fiber, which are not considered in this thesis in order tofocus on the effect of nonlinearity.2.6.1 Fiber LossData is transmitted by launching an optical waveform into the fiber. Eventhough commercial fibers have loss parameters close to the fundamentallimit of 0.16 dB/km for silica fiber [56], the accumulated power attenuationover long-haul transmission results in practically undetectable signals at re-ceiver [57]. As a result, optical amplifiers are required to realize long-haultransmission. Amplification broadly falls into two groups of lumped and dis-tributed amplification. Lumped amplification refers to the use of an opticalamplifier, e.g., Erbium-Doped Fiber Amplifier (EDFA), at the end of eachspan to compensate for the overall attenuation the waveform experiences inthe span [58, 59]. Distributed amplification, which is outside the scope ofthis thesis, amplifies the waveform continuously throughout the fiber [60].Figure 2.7 shows the signal normalized power profile, propagating through 524Figure 2.7: Normalized power profile for transmission over 5 spanswith span length 20 km, fiber loss parameter 0.2 dB/km, andlumped amplification at the end of each span.spans with lumped amplification. As amplifiers are not ideal, they add Am-plified Spontaneous Emission (ASE) to the input signal, which is modeled asa signal-independent random variable with Gaussian distribution. We notethat both noise and signal waveform attenuate along the fiber, but ASE isgenerated at the end of each span, which decreases the overall effective SNR[53].2.6.2 Chromatic DispersionFiber is a dispersive medium as waveforms with different frequencies propa-gate at different speeds in the fiber. Thus, pulse broadening in the time do-main occurs for the transmitted waveform as it consists of several frequencycomponents [53, 61]. This dispersion effect, which is because of frequency-dependent group velocity, is known as Chromatic Dispersion (CD). CD canbe modeled as an all-pass filter with a frequency-dependent phase shift inthe frequency domain, which translates to different arrival times for variousfrequency components. CD, which is a function of dispersion parameter and25link length, is a deterministic process that can be fully compensated in theoptical or digital domain. Optical compensation is done at the end of eachspan [61, 62], whereas digital compensation (precompensation) is carried outfor the whole fiber link at the receiver (transmitter) [63].2.6.3 Polarization Mode DispersionFiber cable is not perfectly cylindrical symmetric due to manufacturingimperfections and environmental (mechanical and thermal) stresses. Thebirefringence phenomena, which results from imperfections, causes two po-larizations to travel at different speeds [53]. Thus, there is a time sepa-ration between two pulses propagating at orthogonal polarizations, calledDifferential Group Delay (DGD). The time separation leads to dispersionin the received waveform, known as Polarization Mode Dispersion (PMD)[64]. Birefringence and the resulting DGD can occur for any two orthogonalaxes. The orientation is called the birefringence axis, which is a functionof environmental stresses and thus random. DGD is deterministic for shortfiber links, i.e., it grows linearly with fiber length, whereas it is stochasticfor long fiber links due to the random nature of birefringence axes [53].2.6.4 Polarization State DriftLight is a sinusoidal electromagnetic wave with two polarizations, namely xand y, oscillating with the frequency inverse proportional to its wavelength.The orientation of oscillation is called State of Polarization (SOP) [53]. Iftwo sinusoidal electric fields Ex and Ey have the same amplitude and phase,the oscillation will be confined in a plane, which is the case of linear po-larization. SOP is not necessarily linear, as two polarizations can possiblyhave different amplitudes and phases, e.g., case of circular polarization withthe same amplitude and π/2 phase difference [53]. Ideally, light preservesits state of polarization as it propagates through fiber, but due to manu-facturing imperfections and mechanical stresses, the SOP changes, which iscalled SOP drift. SOP drift makes two polarizations not to be orthogonalanymore, and thus energy exchange happens between four dimensions of26a dual-polarization waveform. This effect, which is stochastic due to therandom nature of mechanical and thermal stresses along the fiber, is detri-mental to communications. Synchronization is needed at the receiver toconvey information in a channel that suffers from SOP [53].2.6.5 Fiber NonlinearityFiber is a nonlinear channel which means that the channel behavior is anonlinear function of light intensity. Fiber nonlinearity is classified into twomain effects of stimulated scattering and intensity-dependent nonlinear be-havior (known as Kerr effect) [61]. Scattering effects, including Raman [61,Ch. 8] and Brillouin scattering [61, Ch. 9], refer to any nonlinear intensity-dependent loss or gain in the fiber channel. Kerr nonlinearity refers to in-tensity dependence of refractive index, which results in intensity-dependentphase shift and spectrum broadening in frequency domain [53, 61]. Scatter-ing effects are out of the scope of this thesis, as Kerr effect is the dominantnonlinear effect, especially for long-haul transmission. Nonlinear Kerr ef-fect can be partially compensated at the receiver using digital or opticalcompensation techniques.2.6.6 Transceiver DSPOptical fiber is a dispersive channel that suffers from nonlinear effects andenergy exchange between four dimensions of dual polarization waveform.Thus, in order to successfully convey the information, DSP is needed at thereceiver (transmitter) to compensate (precompensate) the harmful impair-ments the waveform experiences in the fiber. We focus on the receiver DSPchain, as precompensation is out of the scope of this thesis. The optical frontend maps the optical signal to an electric signal at the receiver. Then, ananalog-to-digital converter with a sampling frequency of at least twice thesymbol rate is employed. Next DSP blocks are CD compensation, matchedfiltering with the desired pulse shape, and Carrier Phase Recovery (CPR).The first block compensates for the dispersion due to CD using a filter withfinite [65], or infinite [66] impulse response, and then matched filtering con-27verts the continuous waveform to discrete samples. CPR compensates anyphase rotation due to linear and nonlinear effects in optical fiber [67, 68].Blind phase search and constant phase rotation are two commonly used CPRmethods. We use the former for the simulation results in Chapters 3 and 4.After CPR, decoding, as described in Section 2.2.1, is carried out.2.6.7 Channel ModelMaxwell’s equations, which model the propagation of electromagnetic waves,including optical waves, under certain assumptions, are simplified to theNonlinear Schro¨dinger Equation (NLSE) [61, Ch. 2]. NLSE, which governthe propagation of light through an optical fiber, is written as∂Ex(z, t)∂z=−α2Ex(z, t)− jβ22∂2Ex(z, t)∂t2+ jγEx(z, t)|Ex(z, t)|2, (2.33)where Ex(z, t) is the complex baseband electric field in polarization x, 0 ≤z ≤ L denotes propagation distance along a fiber with length L, t ∈ R de-notes time, α is the attenuation coefficient, β2 is group velocity dispersionparameter, γ is the nonlinearity parameter, and j is imaginary unit. Equa-tion (2.33), which is a partial differential equation with respect to propa-gation distance and time, governs the propagation of optical waveform inpolarization x. We note that a similar equation holds for polarization y.The first, second, and third terms in Eq. (2.33) correspond to fiber atten-uation, CD, and Kerr nonlinear effect, respectively. Even though there isno analytical solution for NLSE in the general form, analytical solution forspecial cases, such as NLSE without dispersion, i.e., β2 = 0, is available asEx(z, t) = Ex(0, t) exp(jγ|Ex(0, t)|2Leff(z)) exp(−αz/2), (2.34)whereLeff(z) =1− exp(−αz)α(2.35)is the effective length that symbols interact due to nonlinearity [53]. Wenote that the exponential decay exp(−αz/2) is because of fiber attenuation.Moreover, the nonlinear term exp(jγ|Ex(0, t)|2Leff(z)) only affects the phase28of transmitted signal and not the amplitude. We note that CD, whichis not included in Eq. (2.34), further increases the channel memory, i.e.,more adjacent symbols interact with each other due to nonlinearity. Opticalamplification and ASE can be embedded in NLSE to derive stochastic NLSE[69] as∂Ex(z, t)∂z=−(α− g(z))2Ex(z, t)− jβ22∂2Ex(z, t)∂t2+ jγEx(z, t)|Ex(z, t)|2 + nx(z, t),(2.36)where g(z) and nx(z, t) are real-valued gain profile function and complex-valued stochastic process representing ASE, respectively. The choice of g(z)and nx(z, t) depends on the amplification scheme, i.e., lumped or distributed.The process nx(z, t) is modeled as a circularly symmetric complex whiteGaussian noise with zero mean [70, 71].Equation (2.36) is extended to dual-polarization propagation as∂Ex(z, t)∂z=−(α− g(z))2Ex(z, t)− jβ22∂2Ex(z, t)∂t2+ jγEx(z, t)(|Ex(z, t)|2 + 23|Ey(z, t)|2)+ jγ13Ey(z, t)2E∗x(z, t) + nx(z, t),(2.37)∂Ey(z, t)∂z=−(α− g(z))2Ey(z, t)− jβ22∂2Ey(z, t)∂t2+ jγEy(z, t)(|Ey(z, t)|2 + 23|Ex(z, t)|2)+ jγ13Ex(z, t)2E∗y(z, t) + ny(z, t),(2.38)which are called coupled-mode NLSEs [72]. In Eq. (2.37) and Eq. (2.38),Ey(z, t) and ny(z, t) are the electric field and ASE noise along the polariza-tion y. We note that nx(z, t) and ny(z, t) are independent and identicallydistributed.Equations (2.37) and (2.38) are valid for an ideal fiber without any bire-fringence. As the effect of birefringence is not negligible for practical fibers,the equations mentioned above need to be corrected accordingly. In thisregard, Manakov equation [73, 74], which averages over rapidly and random29changing polarization state, is∂E(z, t)∂z=−(α− g(z))2E(z, t)−jβ22∂2E(z, t)∂t2+jγ89E(z, t)||E(z, t)||2+n(z, t),(2.39)where E(z, t) = [Ex(z, t), Ey(z, t)]T and n(z, t) = [nx(z, t), ny(z, t)]T . Wenote that both polarizations experience the same nonlinear effects in Eq. (2.39)in contrast with Eq. (2.37) and Eq. (2.38). We use the Manakov equation forsimulation in this thesis as it reasonably approximates propagation throughpractical fibers with birefringence.2.6.8 Split-Step Fourier MethodPropagation equations in Section 2.6.7 do not have any analytical solution;thus, numerical methods are used for simulation and to gain insight intothe channel operation. Split-Step Fourier Method (SSFM) is a numericalapproach for solving nonlinear partial differential equations, which is widelyused for numerical simulation of optical channel [75]. SSFM is based on theassumption that linear and nonlinear parts can be solved separately over asufficiently small spatial step. To this end, each span is divided to NSSFMsteps with length ∆zi (∑NSSFMj=1 ∆zi = Lspan) [53]. Moreover, Eq. (2.39) canbe written as∂E(z, t)∂z= (Dˆ + Nˆ)E(z, t), (2.40)whereDˆ =−(α− g(z))2− jβ22∂2∂t2, (2.41)Nˆ = jγ89||E(z, t)||2, (2.42)show linear and nonlinear effects, respectively. SSFM solves the propagationequation for each step by solving for the linear part (Dˆ) first and then thenonlinear part (Nˆ). The propagation over the entire fiber link is done recur-sively by solving the linear part in the frequency domain and the nonlinearpart in the time domain asE˜(z +∆zi, f) ≈ exp(j2β2π2f2∆zi) exp(−(α− g(z))∆zi/2)E˜(z, f), (2.43)30E(z +∆zi, t) ≈ exp(j89γ||E(z, t)||2∆zi)E(z, t), (2.44)where E˜(z, f) is the Fourier transform of E(z, t). We note that it is crucialto choose a small enough step size ∆zi so that SSFM provides an accuratenumerical solution. The step size is either set to be a small constant valueor is optimized based on a criteria, e.g., the maximum nonlinear phase shiftat each step [76]. The former method decreases the complexity of SSFM.Another aspect of SSFM is the sampling rate, which needs to be more thanthe Nyquist rate to capture the nonlinear effects of optical fiber channel [77].To this end, oversampling with twice the Nyquist rate is usually used forsimulation.2.6.9 Perturbative ModelThe equations in Section 2.6.7 provide an implicit input-output relationshipfor the optical channel by which it is non-trivial to compute the channelcapacity or to perform nonlinearity analysis[78]. Perturbation theory is apopular method to find an approximation to partial differential equations,such that an explicit input-output relationship is derived [79]. The solutionof Eq. (2.39) is written as a power series of the nonlinearity parameter, e.g.,E(z, t) = E0(z, t)+γE1(z, t) for the case of first-order perturbation theory,where E0(z, t) is the unperturbed (linear) solution and E1(z, t) is the first-order perturbed solution. In [80, Eq. (3)], the expression for the nonlinearsignal-signal interactions based on the first-order perturbation analysis of amulti-span dual-polarization coherent WDM systems is provided. Accord-ingly, the input-output relationship for the x-polarization signal in WDM31subchannel c ∈ C after linear equalization can be written asr(c)x (n) = s(c)x (n)+ jγ∑m∈Z∑k∈Z(s(c)x (m+ n)[s(c)x (k +m+ n)]∗s(c)x (k + n)+ s(c)y (m+ n)[s(c)y (k +m+ n)]∗s(c)x (k + n))h(c)spm(m, k)+ jγ∑c′∈C\{c}∑m∈Z∑k∈Z(2s(c′)x (m+n)[s(c′)x (k+m+n)]∗s(c)x (k+n)+ s(c′)y (m+ n)[s(c′)y (k +m+ n)]∗s(c)x (k + n)+ s(c′)x (m+ n)[s(c′)y (k +m+ n)]∗s(c)y (k + n))h(c,c′)xpm (m, k),(2.45)where s(c)p (n) and r(c)p (n) are the nth transmitted symbol and received sam-ple in polarization p ∈ {x, y} and subchannel c ∈ C, respectively. C isthe set of all WDM subchannels. h(c)spm and h(c,c′)xpm contain the perturbationcoefficients for intra-channel and inter-channel NLI, respectively. We notethat intra-channel and inter-channel NLI are also called SPM and XPM,respectively. Perturbation coefficients can be computed using numerical in-tegration [81] or training-based methods, e.g., linear regression [82]. Ananalogous expression to (2.45) can be written for the y-polarization signals.2.6.10 Performance MetricsWe use three standard performance metrics for assessing the performanceof an optical fiber communication system in this thesis. The first is theeffective SNR of symbols after linear equalization:SNReff =E[|S|2]E [|S −R|2] , (2.46)where S and R are random variables denoting transmitted and receivedsymbols, respectively. The second is the Q-factor, which is defined based on32uncoded Bit Error Rate (BER) asQ =√2erfc−1(2BER), (2.47)where erfc denotes the complementary error function. The third metric isAIR, as defined in Eq. (2.9). The effective SNR is useful to highlight thestrength of NLI associated with different shaping schemes. Q-factor andAIR also account for the linear shaping gain.33Chapter 3Shaping for Optical FiberChannelConventional ASs are designed for AWGN channel, which is memorylessand linear. In contrast to AWGN channel, optical fiber is a nonlinear dis-persive channel, i.e., it has finite memory. As amplitude shaping with finiteblocklength imposes dependency on symbols of a shaping block, the finitememory and nonlinearity of optical fiber channel affect PAS.In Section 3.1, we comment on two observations regarding the effect ofnonlinearity on PAS, which highlight the importance of symbols’ temporalstructure for NLI. Next, we introduce a linear distortion model based onperturbation theory in Section 3.2, which sheds light on the interplay be-tween PAS and nonlinearity. The insight provided by this model is used forexplaining the observations in Section 3.1 and for introducing the nonlin-earity tolerant shaping scheme in Chapter 4.3.1 Effects of Nonlinearity on ShapingIn this section, we first describe the simulation setup, adopted from [29], forall the results in this chapter and then discuss two observations regardingthe interplay between PAS and the nonlinearity of optical fiber.343.1.1 Simulation Setup 1A multi-span WDM transmission with 11 channels, 32 GBd baud rate, and50 GHz channel spacing is simulated using the SSFM. The modulation is256 QAM with root raised cosine pulse shaping (roll-off 0.1). Amplitudesare generated using CCDM with a MB distribution to achieve an effectiverate of 2.4 bits/amplitude. The PAS scheme uses a Low-Density Parity-Check (LDPC) code with rate 4/5. The overall rate is thus 5.2 bit/QAM-symbol. The symbols are transmitted through 20 spans of a standard single-mode fiber with 80 km span length, fiber loss 0.2 dB/km, CD parameter17 ps/nm/km, and nonlinearity parameter 1.37 W−1km−1. At the end ofeach span, an EDFA with a 6 dB noise figure is deployed. Both singleand dual polarization transmissions are considered. At the receiver, CD iscompensated, and a mean phase rotation is applied. The central channelis the channel of interest for performance results. Table 3.1 shows the keysimulation parameters for setup 1.3.1.2 BlocklengthBlocklength does not affect the SNR of PAS scheme for transmission overAWGN channel. For a linear channel, as the blocklength increases, rate-loss decreases (see Figure 2.6), and thus AIR increases, i.e., infinite optimalblocklength. Figure 3.1 demonstrates effective SNR and AIR for transmis-sion through optical fiber channel with the setup illustrated in Section 3.1.1.PAS with CCDM as the AS with and without interleaving is considered. In-terleaving is done by shuffling symbols randomly within an FEC block. Wehave also considered uniform signaling and ideal PAS, which is an AS withinfinite blocklength, i.e., drawing symbols randomly from the desired distri-bution. As blocklength increases, effective SNR of PAS decreases, which isan indication of increase in NLI. For ultra-short blocklengths, the effectiveSNR is higher than uniform signaling, while it is lower than its uniformcounterpart for very long blocklengths. Moreover, we observe that effectiveSNR for uniform signaling, ideal PAS, and interleaved PAS is not a functionof blocklength, which is an evidence that the temporal structure of symbols35Table 3.1: Simulation parameters for Setup 1 [29]Parameter ValueModulation 256 QAMAmplitude shaper CCDMShaping rate 2.4 bits/amplitudePolarization Single/DualCenter wavelength 1550 nmSymbol rate 32 GBdWDM spacing 50 GHz# WDM channels 11Pulse shape Root-raised cosinePulse roll-off 0.1Span length 80 km# Spans 20Fiber loss 0.2 dB/kmDispersion parameter 17 ps/nm/kmNonlinearity parameter (γ) 1.37 1/W/kmEDFA noise figure 6 dBOversampling factor 2affects the NLI. We observe that the effective SNR of ideal PAS is lowerthan that of uniform signaling, whereas the AIR of ideal PAS is higher.Thus, amplitude shaping increases the NLI, and the higher AIR of idealPAS is because of linear shaping gain. Figure 3.1b demonstrates that thereis a finite optimal blocklength (around 900) for AIR of PAS with CCDM,which is not the case when interleaving is applied. To conclude, PAS withshort blocklengths enjoys an extra nonlinear shaping gain in addition to theconventional shaping gain. This nonlinear shaping gain is removed wheninterleaving is applied, or blocklength increases.3.1.3 Mapping StrategyWe consider dual-polarized optical fiber transmission using QAM. Thismeans that PAM blocks sℓ are mapped to four-dimensional (4D) symbolsequences, where each symbol has Ix, Qx, Iy, and Qy components for in-36(a) Effective SNR (b) AIRFigure 3.1: (a) Effective SNR and (b) AIR vs. blocklength for trans-mission over optical fiber with a finite length AS.phase and quadrature and x- and y-polarization dimensions, respectively.We differentiate three strategies for this mapping, which we refer to as 1D,2D, and 4D mapping. For the case of 1D mapping, each shaping blocksℓ is mapped to a separate quadrature and polarization dimension. The2D mapping strategy maps each shaping block across the two quadraturecomponents but to a single polarization. Finally, each shaping block ismapped to all four 4D-symbol components in 4D mapping. The differentmapping strategies are illustrated in Figure 3.2 in terms of amplitude blocksaℓ, i.e., we map the shaped amplitude blocks to the signal components andapply the sign bits afterward, as this facilitates the discussion further below.Figure 3.3 shows the effective SNR for dual-polarization transmission us-ing CCDM with 1D, 2D, and 4D mapping. Uniform signaling and ideal ASare shown for reference. We observe that 4D mapping has higher effectiveSNR compared to 1D and 2D mapping for relatively long blocklengths. Onthe other hand, 2D mapping is superior to 1D and 4D mapping for ultra-short blocklengths. The observation for long blocklengths is consistent with[23, Fig 12]. The choice of mapping has an effect on nonlinearity tolerancewith PAS for two reasons. First, the mapping strategy changes the time37(a) 1D(b) 2D(c) 4DFigure 3.2: Three strategies for mapping amplitudes to 4D symbols.a(m)n (m ∈ {1, 2, 3, 4}, n ∈ {1, 2, . . . , ℓ}) is the nth amplitude ofmth shaping block.38Figure 3.3: Effective SNR for PAS with different mapping strategies.Dual polarization transmission.duration during which the total energy of one shaping block is transmit-ted. This can be seen in Figure 3.2 comparing the lengths of the amplitudesequences at the output of the mappers. Thus, the mapping affects theeffective shaping blocklength. Second, 2D and 4D mapping introduce de-pendencies between the different components of the 4D symbols. We willdiscuss the consequences of these interactions between mapping and finite-length PAS in Section 3.2.4 and Chapter 4, respectively.3.2 Linear Channel Approximation for NonlinearDistortionIn this section, we derive the approximation of the nonlinear distortion as theoutput of a linear channel that is excited by the symbol-energy sequences asits input. We discuss the properties of the linear filter representing the chan-nel as a function of fiber-link parameters. We then use this channel modelto explain the effects of finite-length PAS and multidimensional mapping on39nonlinear tolerance of transmission.3.2.1 Linear Channel Model FormulationWe consider the optical fiber channel after compensating linear distortionsat the receiver and focus on nonlinear distortion from signal-to-signal inter-actions. We assume dual-polarization transmission in a WDM system witha set of subchannels C. The transmit symbol in polarization p ∈ P = {x, y}and subchannel c ∈ C at discrete time n ∈ Z is denoted by s(c)p (n), and thesymbol energy isE(c)p (n)∆= |s(c)p (n)|2. (3.1)Then, using the first-order perturbation-based distortion model, as brieflyreviewed in the Appendix A, and only retaining symbol-energy terms, wecan approximate the corresponding received sample asr(c)p (n) ≈ s(c)p (n)[1 + jγ(∑p′∈P∑c′∈C∑m∈ZE(c′)p′ (n+m)h(c,c′)p,p′ (m))︸ ︷︷ ︸∆=D(c)p (n)],(3.2)where γ is the fiber nonlinearity parameter, and h(c,c′)p,p′ (m) ∈ R≥0 are coeffi-cients determined by the fiber channel. We observe that the distortion termcan be written as (∗ denotes convolution)D(c)p (n) =∑p′∈P∑c′∈C(E(c′)p′ ∗ h(c,c′)p,p′)(n), (3.3)i.e., the summation of the linear convolution between the filters h(c,c′)p,p′ andthe symbol-energy sequences E(c′)p′ for all WDM subchannels and both po-larizations. The filters h(c,c′)p,p′ represent intra- (p = p′) and inter-polarization(p ̸= p′) interferences of SPM (c = c′) and XPM (c ̸= c′), respectively.We note that the linear channel model (3.3) is not meant to be sufficientlyaccurate for perturbation-based nonlinearity compensation. However, it issuitable for explaining the interplay between shaping and NLI and guidingthe design of nonlinearity tolerant PAS.40It is insightful to develop (3.2) and (3.3) further, which also allows usto connect to other related work. For this, we introduce the deterministicmean distortion termD¯(c)p =∑p′∈P∑c′∈CE¯(c′)p′∑m∈Zh(c,c′)p,p′ (m) (3.4)and the distortion variation term∆D(c,c′)p,p′ (n) =([E(c)p − E¯(c)p]∗ h(c,c′)p,p′)(n), (3.5)where (E denotes statistical expectation)E¯(c)p = E[E(c)p (n)](3.6)is the mean symbol energy. Then, the total distortion (3.3) decomposes asD(c)p (n) =∑p′∈P∑c′∈C∆D(c,c′)p,p′ + D¯(c)p . (3.7)Applying (3.7) and the first order Taylor series approximation of the expo-nential function to (3.2), we obtain the approximationr(c)p (n) ≈ s(cp (n) exp[jγ(∑p′∈P∑c′∈C∆D(c,c′)p,p′ (n) + D¯(c)p)]. (3.8)The NLI is now approximated as phase noise, which suggests that a CPRmay mitigate the distortion. Since the effectiveness of a CPR depends on thedynamic of the distortion term, the temporal structure of the symbol-energyand thus the shaped amplitude sequence is important. Simulation results in[83, 84] suggest that a powerful CPR can be similarly effective in mitigatingNLI as finite-length PAS, at least in the high SNR regime [83]. But even amean-phase recovery would compensate for the deterministic term D¯(c)p , andtherefore we will exclude it in the design of a selection metric in Section 4.3.Furthermore, the relationship for the filter coefficients h(c,c′)p,p (n) provided41in (A.2) in the Appendix A motivates the approximationh(c,c′)p,p′ ̸=p(n) ≈12h(c,c′)p,p (n)∆= h(c,c′)p (n), ∀c, c′ ∈ C. (3.9)This leads to the simplified approximationD(c)p (n) ≈∑c′∈C([2E(c′)p + E(c′)p′ ̸=p]∗ h(c,c′)p)(n), (3.10)for (3.3), where the nonlinear distortion is approximated as linear filteringof the sum of symbol-energy sequences for the two signal polarizations. Thesimplified formulation (3.10) is equivalent to [24, Eqs. (3) and (4)].3.2.2 Filter PropertiesWe now evaluate the expression for the filters h(c,c′)p,p′ for selected opticalfiber links. If not specified otherwise, the link parameters are as given inTable 3.1.Figures 3.4a and 3.4b show the normalized filter coefficients and theirnormalized magnitude Fourier transforms, respectively, for SPM and forXPM from two adjacent channels for a 1600 km link, i.e., 20 spans of length80 km. We observe that all filters have a lowpass characteristic, and that theXPM filters (c ̸= c′) are more frequency-selective than their SPM counter-parts (c = c′). This suggests that the modulation of the temporal character-istics of amplitude sequences via shaping will be more effective in suppressingNLI from XPM than from SPM. We also observe from the two SPM filters inFigure 3.4a that the intra-polarization coefficients are two times higher thanthe inter-polarization coefficients except for n = 0 (see Appendix A). Thetwo XPM filters for c′ = c+1 and c′ = c−1 in Figure 3.4a are time-reversedfunctions and therefore their magnitude frequency responses in Figure 3.4boverlap completely.Figure 3.5 shows the normalized magnitude frequency response for theintra-polarization SPM filters for fiber links with 1, 10, 20, and 40 spansof 80 km. As the link length increases, the channel memory increases, andas a result, the lowpass filters become more selective. For this, the inset in42(a) (b)Figure 3.4: Normalized filters for a fiber link with length 1600 km,32 GBd, and 50 GHz channel spacing. SPM filter (intra- andinter-polarization) and XPM (intra polarization) filter from twoadjacent channels. (a) Time domain, maximum coefficient nor-malized to 1. (b) Frequency domain (magnitude), maximumnormalized to 0 dB at f = 0. The insets show an intervalaround the origin.Figure 3.5 shows the 3 dB filter bandwidth versus the fiber link length in alog-log plot. We observe that the trend is linear with an approximate slopeof −1. This is consistent with channel memory being linearly proportionalto link length, e.g., [26, Eq. (2)]. Similar observations apply for XPM filters,which we have not shown for brevity. Furthermore, we note the higher mag-nitude of filter frequency responses at zero for longer links, which indicatesa more substantial overall nonlinear distortion.Next, we consider a fiber link with a fixed length of 1600 km and differentbaud rates of 16 GBd, 32 GBd, and 64 GBd. Figure 3.6 shows the normalizedmagnitude frequency response for the intra-polarization SPM filter for thesecases. A larger baud rate and thus signal bandwidth results in a moredistinct lowpass characteristic in the frequency domain. The inset figureshows the 3 dB filter bandwidth versus transmission baud rate in a log-log plot. We observe that the bandwidth decreases as baud rate increaseswith a linear slope of about −1. Again, this is consistent with channel43Figure 3.5: Magnitude frequency response for intra-polarization SPMfilter for a transmission with 32 GBd and 50 GHz channel spac-ing over fibers with various link lengths. Normalized to a max-imum of 0 dB at f = 0. Inset: 3 dB filter bandwidth vs. fiberlink length. The corresponding bandwidth of each filter is shownwith the same color code.memory, measured in time duration, being linearly proportional to baudrate, e.g., [26, Eq. (2)]. We also expect more significant nonlinear SPMeffects with increasing baud rate, which is confirmed by the larger value ofthe frequency response at zero. Based on the observations in Figures 3.5and 3.6, we suggest the relationship BW ∝ 1L·B for the 3 dB bandwidth ofthe SPM intra-polarization filter, fiber link length L, and baud rate B.3.2.3 Interplay between Filter and PAS blocklengthThe linear channel model is an effective tool to provide insights into the roleof PAS in the generation of NLI. In particular, from the channel model (3.3)and the lowpass filter characteristics illustrated in the previous section, weconclude that symbol-energy sequences with weaker low-frequency compo-nents would result in less NLI. Such an argument has already been madein [24] using the simplified approximation (3.10) for comparing multidimen-44Figure 3.6: Magnitude frequency response for intra-polarization SPMfilter for a transmission over a 1600 km fiber link with three dif-ferent baud rates. Normalized to a maximum of 0 dB at f = 0.Inset: 3 dB filter bandwidth vs. baud rate. The correspondingbandwidth of each filter is shown with the same color code.sional mappings in conjunction with finite-length PAS.Figure 3.7 shows the magnitude Fourier transform Fx of the symbol-energy sequence(E(c)x (n)− E¯(c)x)in the x-polarization generated by PASand for 1D mapping. We note that we subtract the mean-energy term assuggested in (3.5). Link and system parameters from Table 3.1 are applied.For PAS, we consider CCDM blocklength 108 and 300, and ideal AS, wherewe directly draw amplitude symbols from the desired distribution. Thecurve for uniform signaling is also included as a reference. Overlaid to thesecurves is the magnitude frequency response of the intra-polarization SPMfilter, with the axis on the right-hand side of the figure. The figure revealsseveral important observations that support previous findings consideringeffective SNR, e.g., [20]. First, ideal AS, which corresponds to practicalAS such as CCDM with an infinite blocklength, causes more severe NLIthan uniform signaling. Second, finite-length CCDM creates sequences witha high-pass characteristic, which becomes more pronounced as blocklength45Figure 3.7: Left: Frequency domain representation of x-polarizationsymbol-energy sequences generated by CCDM with blocklength108 and 300, and by uniform signaling and ideal AS. Thelaunch power is −6.5 dBm per polarization. Right: Normal-ized magnitude frequency response of intra-polarization SPMfilter. Dashed lines mark the 3 dB bandwidth for the filter.decreases. Hence, due to the lowpass channel filter, nonlinear distortion isreduced, and shaped transmission can become more nonlinearity tolerantthan uniform signaling. This is the reason for the nonlinear shaping gainobtained using PAS with short blocklengths. The exact cross-over point forwhen a gain will be observed depends on the filter bandwidth.3.2.4 Role of Multidimensional MappingWe can apply the same methodology as above to explain the role of multi-dimensional mapping of shaped sequences (see Section 3.1.3) for the nonlin-earity tolerance of PAS. However, before we do so, we consider a summarystatistic called Running Digital Sum (RDS), which was suggested in [24], tomotivate the effect of multidimensional mapping on NLI. The RDS is basedon the energy terms 2E(c′)p + E(c′)p′ ̸=p in approximation (3.10). The authorsof [24] only considered the sum E(c′)p + E(c′)p′ ̸=p, which is common for both46polarizations p ∈ {x, y}, and introduced the RDS asλ˜RDS(t) =t−1∑n=0∑c′∈C∑p′∈P(E(c′)p′ (n)− E¯(c′)p′)(3.11)for t ∈ N. We modify this definition by considering all energy terms in(3.10), i.e., we have the RDS for polarization p asλRDS(t) = λ˜RDS(t) +t−1∑n=0∑c′∈C(E(c′)p (n)− E¯(c′)p), (3.12)which is a more accurate predictor for nonlinearity tolerance as we will show.The RDS can be interpreted as a lowpass filtered version of the symbol-energy sequence. A small absolute RDS would thus suggest a weaker nonlin-ear distortion. For example, for CCDM with blocklength ℓ and 1D mapping(see Figure 3.2a), we have λ˜RDS(t) = λRDS(t) = 0 for t = k · ℓ and k ∈ N,i.e., the RDS remains smaller for decreasing shaping blocklength. A similarargument applies to sphere-shaping based AS such as ESS, as most of thesequences are selected from energies near the surface of a sphere. We canextend the discussion to 2D mapping as done in [24]. When CCDM-shapedamplitude sequences are mapped in two dimensions (see Figure 3.2b), thesequence energy is spread over length ℓ/2 and thus λ˜RDS(t) = λRDS(t) = 0for all t = k · ℓ/2, which predicts a further reduction in NLI. However, for4D mapping (see Figure 3.2c), the energy is spread over both polarizations,and while λ˜RDS(t) = 0 for all t = k · ℓ/4, this is not the case for λRDS(t).Based on the behavior of λ˜RDS(t), it was suggested in [24] that PAS with4D and 2D mapping is generally superior to PAS with 2D and 1D mappingin terms of nonlinearity tolerance, respectively. While we concur with theadvantage of 2D over 1D mapping, our refined RDS metric (3.12) does notpermit the conclusion that 4D mapping is generally advantageous.We illustrate our point by plotting the absolute value of the Fouriertransform F sp ofEsp(n) = 2(E(c)p (n)− E¯(c)p)+(E(c)p′ ̸=p(n)− E¯(c)p′ ̸=p), (3.13)47Figure 3.8: Left: Frequency domain representation of aggregatesymbol-energy sequence from (3.13) for x-polarization generatedby CCDM with blocklength 180 and 1D, 2D, and 4D mapping.CCDM with ℓ = 180. Launch power is −6.5 dBm per polariza-tion. Right: Normalized magnitude frequency response of SPMfilter h(c,c′)p from (3.9). Dashed lines mark the 3 dB bandwidthfor the filter.which is the aggregate symbol-energy sequence in (3.10), for p = x in Fig-ure 3.8. We consider CCDM with blocklength 180 and 1D, 2D, and 4D map-ping, and we overlay these curves with the magnitude frequency response ofthe inter-polarization SPM filter. Again, link and system parameters fromTable 3.1 are applied. We observe that the curve for 2D mapping is alwaysbelow that for 1D mapping, which means that 2D mapping always performsbetter than 1D mapping with respect to NLI. The comparison with 4Dmapping is not that simple, as its curve intersects with those for 1D and 2Dmapping. It has a shallower but wider dip around zero frequency than thecurves for 1D and 2D mapping. Hence, 4D mapping does not necessarilyresult in a reduced (or increased) NLI compared to 1D and 2D mapping,but it depends on the optical fiber link, which determines the lowpass filtercurve in Figure 3.8. In particular, we expect 4D mapping to (only) providebenefits for shorter fiber links, which produce wider lowpass filters, and long48shaping blocklengths, which lead to a narrower spectral dip. This conclusionis consistent with the results in [23] and the performance results presentedin Figure 3.3.49Chapter 4Nonlinearity TolerantShapingConventional ASs, as described in Section 2.3, are designed to maximizethe linear shaping gain for AWGN channel and thus are sub-optimal fornonlinear optical fiber channel. Symbols distribution and temporal structureof sequences affect the NLI in optical fiber channel [19]. Kurtosis, definedasµ4 =E[|X − E[X]|4](E[|X − E[X]|2])2 , (4.1)determines the severity of NLI due to sub-optimal distribution. In Eq. (4.1),X is the random variable denoting the symbols in the channel input, andE is the statistical expectation operator. EDI, which is proportional to thevariance of windowed energy of symbols, has been shown to be inversely pro-portional to effective SNR and thus an indicator of NLI due to the temporalstructure of symbols [26]. EDI is defined asΛEDI∆=Var[Gw]E[Gw], (4.2)whereVar[Gw]∆=1ℓℓ−1∑i=0V ar[Gwi ], (4.3)50E[Gw] ∆=1ℓℓ−1∑i=0E[Gwi ]. (4.4)w is window size, and Var denotes statistical variance. Moreover, windowedenergies are defined asGwi∆=i+w/2∑n=i−w/2|Xn|2. (4.5)AWGN optimal distribution has a high kurtosis, which results in non-linear shaping loss for optical fiber channel [19]. Moreover, as described inSection 3.1.2, PAS with long blocklengths suffer from nonlinear loss com-pared to uniform signaling, which is because of the temporal structure ofsymbols, i.e., high windowed energy variations. Thus, design of a nonlin-earity tolerant PAS scheme, which is optimal for optical fiber channel, is ofutmost importance. Nonlinearity tolerant PAS schemes optimize the afore-mentioned factors, i.e., symbols distribution and temporal structure, in or-der to suppress NLI. There are two main families of nonlinearity tolerantschemes: sequence selection and nonlinearity tolerant amplitude shaping.For the latter, the design of conventional ASs are modified to generate morenonlinearity tolerant sequences, i.e., sequences which cause less NLI. Thesequence selection scheme generates several sequences, also known as candi-dates, and then the most nonlinearity tolerant candidate is selected. In thischapter, we explain Kurtosis-Limited Sphere Shaping (KLSS) [32] from thefamily of nonlinearity tolerant ASs, and then we comment on sequence se-lection. Next, we introduce a new nonlinearity tolerant scheme based on themodel derived in Section 3.2.1. We conclude this chapter with extensive nu-merical results to show the superiority of our nonlinearity tolerant schemeto the state-of-the-art nonlinearity tolerant schemes for a wide variety ofsimulation setups.4.1 Kurtosis-Limited Sphere ShapingKLSS is a variant of SpSh methods, which adds the extra constraint ofmaximum kurtosis to the constraint of maximum energy [32]. The output51sequences of KLSS are constrained not to exceed the maximum energy andkurtosis limits, i.e.,FKLSS(x) = 1[E(x) ≤ EKLSS,K(x) ≤ KKLSS], (4.6)whereK(x) = ∥x∥44 =ℓ∑i=1|xi|4 (4.7)is per sequence kurtosis. EKLSS and KKLSS are the maximum energy andkurtosis limits, respectively. K-ESS is an extension of ESS for indexing theoutput sequences of KLSS. The energy trellis is modified such that eachnode at stage n is defined with the accumulated energy and kurtosis ofamplitudes up to stage n, i.e., e =∑ni=1 |ai|2 and k =∑ni=1 |ai|4. Moreover,the total number of paths from a node at stage n with energy e and kurtosisk to the final stage is computed asT (e,k)n =∑a∈AT(e+a2,k+a4)n+1 , (4.8)whereT(e,k)ℓ =1, e ≤ EKLSS, k ≤ KKLSS0, otherwise . (4.9)The pair (EKLSS,KKLSS) is the maximum energy and kurtosis of sequences,which is determined such that the number of sequences meets the desiredrate, i.e., k/ℓ. Shaping and deshaping using the K-ESS trellis is performedsimilar to ESS, as described in Section 2.3. We note that there is not aunique pair (EKLSS,KKLSS) to achieve a certain rate. As KKLSS decreases,more sequences inside ℓ-sphere with energy radius EKLSS are eliminated, andthus EKLSS should be increased to achieve a fixed rate, which causes higheraverage energy. Thus, as KKLSS decreases, nonlinear shaping gain increases,whereas linear shaping gain decreases. In this thesis, we pick the pair withthe lowest kurtosis, i.e., the minimum NLI, among all pairs which achievethe desired shaping rate. Figure 4.1a shows an example of K-ESS energy52(a) (b)Figure 4.1: (a) K-ESS energy trellis for A = {1, 3, 5, 7}, ℓ = 4, maxi-mum energy 36, and maximum kurtosis 324. Dashed path cor-responds to sequence a = (3, 1, 3, 3) (i(a) = 13) (b) referencenode representation.trellis with ℓ = 4, k = 4, and (EKLSS,KKLSS) = (36, 324). Figure 4.1b showsthe reference node, which is defined based on a triplet consisting of energy,kurtosis, and the number of paths to the final stage. We note that the trellisin Figure 4.1a achieves the same rate as the trellis in Figure 2.4a, but witha lower maximum kurtosis and higher maximum energy, i.e., high kurtosissequences such as (1, 5, 1, 1) are eliminated from the set of output sequencesto suppress NLI.4.2 Sequence SelectionWhen the AS of the PAS scheme produces several shaped amplitude blocksfor representing the same block of information bits, the mapping step dis-cussed in Section 3.1.3 is followed by a sequence selector module (see Fig-ure 4.2). The generation of candidate shaping blocks is enabled throughthe use of v redundant bits fv, which are referred to as flipping bits, andwhich are concatenated with k − v information bits uk−v to form the k in-53put bits of the AS. For each combination of flipping bits, a shaping blockof ℓ PAM amplitude symbols aℓf is generated. The amplitudes are mappedto a block of ℓ/d 4D-amplitude symbols according to the (d ∈ {1, 2, 4})-dimensional mapping strategy as shown in Figure 4.2. The sequence selec-tor computes a metric for each 4D candidate and selects the best candidateaˆℓ/d = [aˆℓ/dIx, . . . , aˆℓ/dQy]. We note that the size of the candidate set dependson the mapping strategy. Since 4/d ASs contribute to the generation of4D-amplitude symbols, the selection module computes 24dv × d metrics forproducing ℓ 4D symbols. Hence, computational complexity decreases withthe dimension d of the mapping. After selection, amplitude-to-bit conversionand FEC operations are performed in the same way as for the conventionalPAS scheme shown in Figure 2.1a. When introducing flipping bits the rateloss expression (2.7) generalizes toRloss,v = H(Pa)−Rs,v = H(Pa)− k − vℓ. (4.10)In order to have a fixed effective rate R = k−vℓ +α, k−v should be remainedfixed. Hence, as v increases, k also increases, resulting in lower linear shapinggain for given ℓ. Therefore, sequence selection is a mechanism to trade offlinear with nonlinear shaping gain.L-CCDM, introduced in [29], uses empirical estimation of EDI for se-quence selection. We note that EDI is originally defined for single polar-ization and 1D mapping. EDI for a candidate with length ℓ is empiricallydefined asλEDI∆=σ2GwµGw, (4.11)whereµGw∆=∑ℓ−w/2i=1+w/2 gwiℓ− w , (4.12)µGw∆=∑ℓ−w/2i=1+w/2(gwi − µGw)2ℓ− w + 1 (4.13)are empirical mean and variance of windowed energies, respectively.54(a) 1D (b) 2D(c) 4DFigure 4.2: Sequence selection block diagram for different mappingstrategies: (a) 1D, (b) 2D, and (c) 4D mapping.4.3 Metric for PAS with Sequence SelectionIn this section, we utilize the linear channel model (3.3) to derive a metricfor PAS with sequence selection as introduced in Section 4.2.554.3.1 The LSAS MetricAn effective selection of shaped sequences suppresses the distortion varia-tions defined in (3.5). Hence, the proposed metric isλLSAS =ℓ/d−1∑n=0∑p∈P∣∣∣s(c)p (n)∑p′∈P∑c′∈C˜⊆C∆D(c,c′)p,p′ (n)∣∣∣2, (4.14)where, we recall, ℓ is the blocklength of the AS and d the dimensionalityof the mapping. The metric accounts for nonlinearity expressed in termsof symbol amplitudes and filtering through the fiber channel, and we es-tablished in the previous section that the filters have lowpass characteris-tics. Therefore, we refer to it as the Lowpass-filtered Symbol-AmplitudeSequence (LSAS) metric, as already used in [85] for the special case of 1Dmapping.We note that we consider XPM nonlinearity from only a subset C˜ ofWDM subchannels in the LSAS definition (4.14), because the significantinterference is caused by adjacent subchannels. We perform the selectionof sequences for different WDM subchannels sequentially, i.e., in a greedyfashion, from outer to inner channels of the system. This means that thenumber of candidate sequences and thus selection complexity do not increasecompared to the single-channel case.Since LSAS measures the total NLI from a multidimensional shaped am-plitude sequence with length ℓ/d, the selection will be based on the minimumLSAS metric. In relation to the analysis in Section 3.2.3, we expect thatLSAS-based sequence selection results in a temporal structure of symbol-energy sequences with a deeper and wider spectral dip.4.3.2 Relation to EDI metricIt is insightful to compare the LSAS metric with the EDI metric from [29].For this purpose, we show how the latter can be derived as a simplifiedversion of the former. We specialize in the case of 1D mapping, singlepolarization, and single channel transmission, as EDI has been defined for56this scenario. The EDI is computed based on the empirical average andvariance of symbol-energy sequences. If we approximate the average E¯(c)p in(3.6) with the empirical meanE˜(c)p =1ℓ− wℓ−w/2∑n=1+w/2|s(c)p (n)|2, (4.15)where w is the EDI window length, and introduce the average-free symbol-energy sequence∆E˜(c,c)p,p (n) =w/2∑m=−w/2(E(c)p (n+m)− E˜(c)p ), (4.16)then the EDI metric can be written asλEDI =1(ℓ− w − 1)(w + 1)E˜(c)pℓ−w/2∑n=1+w/2(∆E˜(c,c)p,p (n))2. (4.17)The relation to the LSAS metric follows from noticing that the sequence∆E˜(c,c)p,p (n) is an approximation of the distortion variation ∆D(c,c′)p,p′ (n) from(3.5). In particular, the two quantities would be identical if E¯(c)p = E˜(c)p andh(c,c)p,p (n) =1, if n ∈ {−w2 , . . . , w2 }0, otherwise (4.18)were true. Hence, different from the LSAS metric, the EDI in (4.17) does notinclude the amplitude terms |s(c)p (n)|, uses an approximation of the lowpassfilter of the channel, and it discards w/2 symbols at both ends of the length-ℓ shaped sequence to have meaningful empirical estimates. We note thatE˜(c)p in the denominator of EDI in Eq. (4.17) is approximately constant fordifferent candidates, and thus it does not affect sequence selection. In thenext section we show that for these reasons, LSAS outperforms EDI forsequence selection.57Table 4.1: Simulation parameters for Setup 2 [32]Parameter ValueModulation 64 QAMAmplitude shaper ESS/K-ESSShaping rate 1.5 bits/amplitudePolarization DualCenter wavelength 1550 nmSymbol rate 50 GBdWDM spacing 55 GHz# WDM channels 1/11Pulse shape Root-raised cosinePulse roll-off 0.1Span length 205 km# Spans 1Fiber loss 0.2 dB/kmDispersion parameter 17 ps/nm/kmNonlinearity parameter (γ) 1.30 1/W/kmEDFA noise figure 5 dBOversampling factor 24.4 Performance ResultsSequence selection using EDI and nonlinearity tolerant amplitude shapingusing K-ESS are state-of-the-art schemes for mitigating nonlinear effects inoptical fiber channel. Accordingly, we consider the two simulation setupsfrom the respective references to showcase the validity of the energy-basedmodel and the effectiveness of LSAS for sequence selection.4.4.1 Simulation ParametersThe parameters for the two simulation setups are adopted from [29] and[32], respectively. The first simulation setup is described in Section 3.1.1 andTable 3.1. The main differences of Setup 2 from Setup 1 are the adoptionof 64 QAM transmission, PAS with ESS and K-ESS with a shaping rate of1.5 bits/amplitude and a rate-5/6 FEC, i.e., the overall rate is 4 bit/QAM-symbol, a baud rate of 50 GBd, a channel spacing of 55 GHz for the WDM58Figure 4.3: AIR vs. v for sequence selection using EDI, LSAS withonly SPM components, and LSAS with both SPM and XPMcomponents. Setup 1 with single-polarization transmission and1D mapping.case, and a single-span link of 205 km. The other parameters are identicalor similar (see Tables 3.1 and 4.1).4.4.2 Setup 1: 256 QAM TransmissionWe start with 256 QAM transmission in Setup 1 to compare the effectivenessof the LSAS metric for sequence selection with that of the EDI metric from[29].Sequence SelectionWe first focus on sequence selection with 1D mapping. If not stated other-wise, single-polarization transmission is used. For SNR and AIR results, alaunch-power optimization has been performed for each measurement point.Figure 4.3 shows the AIR as a function of the number of flipping bits vfor PAS with different blocklengths and EDI and LSAS metrics for sequenceselection. EDI uses a window length of w = 100, as suggested in [29]. For59(a) Effective SNR (b) AIRFigure 4.4: (a) Effective SNR and (b) AIR vs. shaping blocklength fordifferent selection metrics with v = 2 redundant bits. Shapingwithout selection (v = 0) shown as a reference. Setup 1 withsingle-polarization transmission and 1D mapping.the LSAS metric, we distinguish two cases: (i) C˜ = {c}, i.e., only SPM isconsidered (“LSAS (SPM)”), and (ii) C˜ = {c − 1, c, c + 1}, i.e., XPM fromadjacent channels is considered (“LSAS (SPM+XPM)”). We observe thatLSAS is superior to EDI for all blocklengths. For ℓ ∈ {180, 300}, we suspectthat discarding w symbols from each shaping block as well as the lowpass-filter approximation affect the EDI metric. Accounting for XPM does notmake a notable difference for short blocklengths, which is due the widespectral dip for CCDM at ℓ ∈ {180, 300} (see Figure 4.6) and the narrowXPM filter (see Figure 3.4b). For ℓ = 900, the LSAS is superior to theEDI metric because of the inclusion of XPM in the selection. Furthermore,we note that there is an optimum value for v, which shifts toward largerv for larger blocklengths. This is because of the trade-off between linearand nonlinear shaping gain in the sequence selection scheme. An initiallyhigher linear shaping gain in the case of larger blocklength provides moreopportunity for increasing the nonlinear shaping gain using more redundantbits. Since the computational complexity of sequence selection grows withv, we choose v = 2 for the following results.60Figure 4.5: Q-factor vs. launch power for different selection metricswith v = 2 redundant bits. Shaping without selection (v = 0)shown as a reference. Setup 1 with dual-polarization transmis-sion, 1D mapping, and shaping blocklength ℓ = 180.Figure 4.4 compares the sequence selection metrics from Figure 4.3 fordifferent shaping blocklengths. Shaping without sequence selection, i.e.,v = 0 is shown as the baseline curve. Selection using EDI converges tothe baseline for short blocklengths, as it requires a blocklength ℓ notablylarger than the window size w to work effectively (see Section 4.3). Asblocklength increases, EDI and LSAS with only SPM converge to the sameperformance. As already noted in Figure 4.3, LSAS with SPM and XPMcomponents outperforms the other selection metrics for larger ℓ, with a gainof about 0.3 dB in SNR and 0.1 bits/QAM-symbol in AIR over shapingwithout selection at ℓ = 900. From the AIR in Figure 4.4b we note thatthe optimal shaping blocklength increases from ℓ = 900 in the case of noselection to ℓ = 1350 for selection using LSAS, which is due to the reducednonlinear impairments. In the remainder of this section, results for LSASrefer to LSAS with only SPM components, which is suitable for blocklengthℓ ≤ 300.61Figure 4.6: Left: Frequency domain representation of aggregatesymbol-energy sequence from (3.13) for x-polarization for shap-ing methods and system setup from Figure 4.5. Launch power is−1 dBm per channel. Right: Normalized magnitude frequencyresponse of SPM filter h(c,c′)p from (3.9). Dashed lines mark the3 dB bandwidth for the filter.Next, we present results for dual-polarization transmission. Figure 4.5shows the Q-factor as a function of launch power for dual-polarization trans-mission and shaping with different selection metrics. In the case of the LSASmetric, we distinguish between (P = {x},P = {y}) and P = {x, y} in (4.14),i.e., a single-polarization and a dual-polarization version. We observe thatthe advantage of the LSAS metric also manifests in the dual-polarizationcase, with some slight additional gain due to performing joint selection overthe two polarization dimensions. We highlight the effect of shaping sequenceselection and the different selection metrics in Figure 4.6, which shows theFourier transforms of F sp of the aggregate symbol-energy sequences definedin (3.13) corresponding to the different shaping methods. We overlay thecurves with the frequency response for the SPM filter h(c,c′)p defined in (3.9),which provides the approximation for the distortion in (3.10). It can beseen that sequence selection makes the spectral dip wider and deeper, which62Figure 4.7: AIR vs. v for sequence selection with LSAS and differentmapping strategies. Setup 1 with dual-polarization transmis-sion.translates to a decrease in NLI considering the lowpass channel filter. More-over, the order of the curves in Figure 4.6 matches the order of the corre-sponding curves in Figure 4.5. This again demonstrates the usefulness of thelinear channel model for nonlinearity analysis. As a final detailed observa-tion, we point out the oscillation of the Fourier transform signal for the EDImetric in Figure 4.6. This results from the rectangular-filter approximationin (4.18), which corresponds to a sinc-function in the frequency domain, andthus frequency components at its nulls do not contribute to the metric.Multidimensional mappingAs the last set of results for Setup 1, we consider sequence selection withdifferent amplitude mapping strategies. For this, we need to adopt the dual-polarization transmission scenario. Furthermore, since the components ofthe 4D signal need to be chosen jointly, only the dual-polarization version ofthe LSAS metric but not the EDI can be applied. Figure 4.7 shows the AIRas a function of v for the three mapping strategies from Section 3.1.3. First,we observe that for shaping without selection (v = 0), the dimensionality63Figure 4.8: Effective SNR vs. launch power for Setup 2. (a) Singlechannel transmission and (b) WDM transmission with 11 chan-nels. Shaping blocklength ℓ = 108 and 4D mapping.of mapping has a notable effect on the system performance. 4D mappingis superior to 1D and 2D mapping strategies only for long blocklengths.This result has been predicted in Section 3.2.4 from the channel model fordistortion and the frequency-domain visualization in Figures 3.7 and 3.8.Second, as the number of selection bits increases, the performances for allthree mapping strategies become identical. Therefore, It is preferable to use4D mapping in conjunction with sequence selection for its lower complexity.4.4.3 Setup 2: 64 QAM transmissionFor Setup 2 from [32], we focus on the comparison of sequence selection usingLSAS with K-ESS, which was proposed for improved nonlinearity tolerancein [32], as well as, for the first time, the combination of the two methods.Figure 4.8 shows the effective SNR for the various shaping schemes for(a) single channel and (b) WDM transmission. PAS with ℓ = 108 and4D mapping are applied, where the latter is beneficial because of the shortlink length. The effective SNR for uniform signaling is included as a ref-erence. We observe that PAS with ESS suffers an SNR loss compared to64uniform signaling, which is due to an increased NLI. A good fraction of thisloss (single-channel case) or even the entire loss (WDM case) is recoveredthrough the use of K-ESS or the addition of sequence selection with LSAS.The combination of K-ESS with LSAS-based selection outperforms all meth-ods with a gain of more than 0.5 dB and 0.3 dB over ESS-based shapingin the single-channel and WDM case, respectively. The SNR-performanceresults can be explained considering the frequency-domain representationof the corresponding symbol-energy signals and the linear channel filter inFigure 4.9. It can be seen that K-ESS is effective in lowering the higher-frequency components, which is a result of generating a distribution withlower kurtosis. The selection-based schemes optimize the temporal structureof symbols to minimize the NLI with the same distribution, which translatesto a wider and deeper spectral dip at frequency zero. As a result, we expectthat the selection-based schemes perform well for a wide variety of transmis-sion setups, while K-ESS is only beneficial for short fiber links correspondingto a wide lowpass filter. This conclusion is consistent with the observationsin [32].Finally, to examine our hypothesis about K-ESS, we consider Setup 2but with a single channel transmission through a long-haul fiber link with 20spans of length 80 km. Figure 4.10 shows the Q-factor curves obtained withESS and K-ESS with ℓ = 108 and with and without LSAS-based sequenceselection. 2D mapping is chosen as it is the best mapping strategy for thislonger link. As suggested by our linear channel model analysis, ESS out-performs K-ESS in this scenario. Furthermore, sequence selection providesa gain when applied to either baseline AS.65Figure 4.9: Left: Frequency domain representation of aggregatesymbol-energy sequence from (3.13) for x-polarization for shap-ing methods from Figure 4.8(b). Launch power is 9 dBm perchannel. Right: Normalized magnitude frequency response ofSPM filter h(c,c′)p from (3.9). Dashed lines mark the 3 dB band-width for the filter.Figure 4.10: Q-factor vs. launch power for Setup 2 but with a singlechannel transmission through 20 spans of 80 km. Shapingblocklength ℓ = 108 and 2D mapping strategy.66Chapter 5Conclusion and Future WorkIn this chapter, we reiterate the main contributions of this thesis and suggestfuture work as next steps to extend this work.5.1 ConclusionIn this thesis, we considered the effects of channel nonlinearity on PAS asconventional ASs are sub-optimal for optical fiber channel, where the sub-optimality is due to AWGN optimal distribution and temporal structureof symbols. We commented on two observations regarding the interplaybetween nonlinearity and shaping, i.e., effect of blocklength and mappingstrategy. Next, we revisited the signal-dependent generation of NLI in opti-cal channel and presented a linear lowpass channel model to approximatelydescribe this process. The model considers symbol-energy sequences as inputand therefore permits the analysis of the interplay between amplitude shap-ing and nonlinear distortion. We showed the evolution of filter coefficientsfor different fiber setups and introduced a frequency domain visualization asa useful tool for analysis and design of PAS schemes for nonlinearity miti-gation. We used the model to explain the observations regarding the effectsof nonlinearity on shaping. Moreover, We applied the model to developthe LSAS metric for PAS with sequence selection. LSAS naturally includesinter-polarization and inter-channel nonlinear effects and extends to mul-67tidimensional mapping. The latter permits the combination of sequenceselection with multidimensional mapping for lower complexity. Throughextensive simulations, we showed the performance benefits resulting fromLSAS, and we confirmed the accuracy of the predictions based on the linearmodel. Finally, we demonstrated that the nonlinear gain from optimizingthe distribution, e.g., via K-ESS, and sequence selection add up, bringingextra nonlinear gain for single span transmission.5.2 Future WorkIn this thesis, we introduced a linear distortion model which explains themechanism for NLI generation due to signal-to-signal interactions. Themodel input is symbols’ energies, but the channel input is symbols. Thus,the effect of signs on the nonlinear performance is missed in the currentmodel. Accordingly, the interaction between symbols’ signs and nonlinear-ity remains as future work. Moreover, signal-noise interactions affect non-linearity, which is missed in this thesis. Finally, the linear distortion modelis based on the assumption of constant phase rotation for CPR, which isnot true for most practical scenarios. 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Rasmussen, “Low complexity digital perturbationback-propagation,” in European Conference on OpticalCommunication (ECOC), 2011, pp. 1–3.77Appendix AModel DerivationIn this appendix, we derive the approximation (3.2) for the linear channelmodel between symbol-energy sequences and nonlinear distortions.We start with the expression (2.45) for the nonlinear distortion term fromsignal-signal interactions based on the first-order perturbation analysis of amulti-span dual-polarization coherent WDM system. To arrive at the repre-sentation used for the distortion model based on symbol-energy sequences,we approximate (2.45) by retaining only the symbol-energy dependent termsin the summations:r(c)x (n) ≈ s(c)x (n)[1+jγ∑m∈Z(|s(c)x (m+ n)|2 + |s(c)y (m+ n)|2)h(c)spm(m, 0)+ jγ∑k∈Z\{0}(|s(c)x (k + n)|2)h(c)spm(0, k)+ jγ∑c′∈C\{c}∑m∈Z(2|s(c′)x (m+n)|2+|s(c′)y (m+n)|2)h(c,c′)xpm (m, 0)].(A.1)Then, noting that h(c)spm(m, 0) = h(c)spm(0,m) [86], and defining filters h(c,c′)p,p′78asc′ = c, p′ = p : h(c,c′)p,p′ (n) =2h(c)spm(n, 0), n ̸= 0h(c)spm(n, 0), n = 0c′ = c, p′ ̸= p : h(c,c′)p,p′ (n) = h(c)spm(n, 0)c′ ̸= c, p′ = p : h(c,c′)p,p′ (n) = 2h(c)xpm(n, 0)c′ ̸= c, p′ ̸= p : h(c,c′)p,p′ (n) = h(c)xpm(n, 0)(A.2)to describe both intra- and inter-channel NLI effects, we obtain (3.2) inSection 3.2.Furthermore, as it can be seen in (A.2), the intra-polarization coefficientsare two times the inter-polarization coefficients, except for n = 0 for SPM.This motivates the approximation of the relation between filter coefficientsin (3.9) in Section 3.2.79
Thesis/Dissertation
eng
Electrical and Computer Engineering
Vancouver : University of British Columbia Library
Attribution-NonCommercial-NoDerivatives 4.0 International
Graduate
Lampe, Lutz
Interplay between fiber nonlinearity and probabilistic amplitude shaping
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