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Faster-than-Nyquist system design for next generation fixed transmission networks Jana, Mrinmoy 2019

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Faster-than-Nyquist System Design for NextGeneration Fixed Transmission NetworksbyMrinmoy JanaM.Tech, Indian Institute of Technology, Kanpur, India, 2010B.E., Jadavpur University, Kolkata, India, 2008A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Electrical and Computer Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)November 2019© Mrinmoy Jana, 2019The following individuals certify that they have read, and recommend to the Facultyof Graduate and Postdoctoral Studies for acceptance, the dissertation entitled:“Faster-than-Nyquist System Design for Next Generation Fixed Trans-mission Networks”submitted by Mrinmoy Jana in partial fulfillment of the requirements for the de-gree of Doctor of Philosophy in Electrical and Computer Engineering.Examining Committee:Lutz Lampe, Electrical and Computer EngineeringSupervisorSudip Shekhar, Electrical and Computer EngineeringSupervisory Committee MemberJulian Cheng, Electrical and Computer EngineeringSupervisory Committee MemberCyril Leung, Electrical and Computer EngineeringUniversity ExaminerEldad Haber, Earth, Ocean and Atmospheric SciencesUniversity ExamineriiAbstractMonumental growth of traffic load in the communication networks has heavily strainedthe existing fixed transmission network infrastructure. Such enormous surge of trafficwarrants enabling higher data rates in these networks, where predominantly opticalfibers and microwave radio links are deployed. With bandwidth becoming an expen-sive resource, and owing to the practical constraints of the electronic components, em-ploying high baud rates alone may be insufficient to accomplish such high throughputsin these optical fiber communication (OFC) and microwave communication (MWC)systems. Hence, increasing the spectral efficiency (SE) is a key requirement for thesenetworks.For this pursuit, this thesis investigates the application of Faster-than-Nyquist(FTN) signaling in fixed transmission networks, with an objective to achieve highSE and data rates. FTN is an enabling technology that offers SE improvements byallowing controlled overlap of the transmitted symbols in time or frequency or both.OFC and MWC systems are suitable platforms for the introduction of FTN signaling,since FTN can moderate the need for higher order modulation formats, which aresensitive to phase noise and fiber nonlinearity. In this thesis, we combine the conceptof FTN signaling with other conventional throughput increasing techniques, suchas polarization multiplexing and multicarrier transmission, to further the data rateimprovements.However, FTN introduces inter-symbol-interference and/or inter-carrier interfer-iiience. Moreover, integrating FTN signaling with polarization multiplexing and mul-ticarrier transmission complicates the realistic implementation. OFC and MWC sys-tems also pose additional practical challenges stemming from the specific commu-nication channel environments and the transceiver components. If not successfullymitigated, all of these impairments and non-idealities significantly deteriorate theperformance of the communication links.In this thesis, we address each of these unique challenges through suitable miti-gation algorithms, to facilitate an efficient FTN transmission. For this, we presentsophisticated system designs equipped with powerful digital signal processing tools.We numerically evaluate the performance of our proposed methods by simulatingrealistic OFC and MWC systems. The simulation results indicate that our proposedspectrally efficient designs offer significant performance advantages over existing com-petitive schemes.ivLay SummaryFixed transmission networks serve as the backbone for the Internet and the mobiledata traffic. Currently, optical fibers and microwave radio constitute majority of thecommunication links in such networks. With bandwidth becoming an increasinglycritical resource, spectrally efficient technologies need to be employed in these net-works to cope with the skyrocketing traffic demands. Faster-than-Nyquist (FTN)signaling is one such befitting technique to accomplish this purpose. However, thebenefits of FTN come at the price of introducing interference. Moreover, practicalOFC and MWC systems present unique complications of their own. In this thesis,we explore the possibility of applying FTN signaling in the next generation fixedtransmission networks. For this, we present powerful signal processing tools to miti-gate the interference and other practical challenges imposed by the realistic OFC andMWC systems. The simulation results we provide in this thesis establish substantialsuperiority of the proposed schemes over state-of-the-art designs.vPrefaceThis thesis is based on original research that I conducted under the supervision ofProfessor Lutz Lampe in the Department of Electrical and Computer Engineering atthe University of British Columbia, Vancouver, Canada.The co-authors in my publications, Dr. Jeebak Mitra and Dr. Ahmed Medra, fromHuawei Technologies, Kanata, ON, Canada, have assisted me towards the problemformulation and provided me valuable suggestions to determine the relevance of thesolutions with respect to the practical intricacies of fixed transmission networks.Below is a list of publications related to the work presented in this thesis. For allof them, I was responsible for reviewing literature, developing solutions, evaluatingthem through simulations, and preparing publication manuscripts. Professor LutzLampe supervised all my work.Publications Related to Chapter 2• M. Jana, A. Medra, L. Lampe, J. Mitra, “Pre-Equalized Faster-Than-NyquistTransmission”, IEEE Trans. Commun., vol. 65, no. 10, pp. 4406–4418, Oct. 2017.• M. Jana, A. Medra, L. Lampe, J. Mitra, “Precoded Faster-than-Nyquist Co-herent Optical Transmission”, in Proc. 42nd European Conf. Opt. Commun.(ECOC), Dusseldorf, Germany, Sep.• M. Jana, J. Mitra, L. Lampe, A. Medra “System and Method for PrecodedFaster-than-Nyquist Signaling”, US Patent 10003390, Date of patent: Jun. 19, 2018.Publications Related to Chapter 3• M. Jana, L. Lampe, J. Mitra, “Dual-Polarized Faster-Than-Nyquist Trans-mission Using Higher Order Modulation Schemes”, IEEE Trans. Commun.,vol. 66, no. 11, pp. 5332–5345, Nov. 2018.• M. Jana, L. Lampe, J. Mitra, “Interference and Phase Noise Mitigation in aDual-Polarized Faster-than-Nyquist Transmission,” Proc. IEEE Int. WorkshopSig. Proc. Adv. Wireless Commun. (SPAWC), Kalamata, Greece, June 2018.• M. Jana, J. Mitra, L. Lampe “Methods and Systems for Interference Mitiga-tion in a Dual-Polarized Communication System”, US Patent 10425256, Dateof patent: Sep. 24, 2019.Publications Related to Chapter 4• M. Jana, L. Lampe, J. Mitra, “Precoded Time-Frequency-Packed MulticarrierFaster-than-Nyquist Transmission,” Finalist, Best Student Paper Award,Proc. IEEE Int. Workshop Sig. Proc. Adv. Wireless Commun. (SPAWC),Cannes, France, July 2019.viiPublications Related to Chapter 5• M. Jana, L. Lampe, J. Mitra, “Interference Cancellation for Time-FrequencyPacked Super-Nyquist WDM Systems”, IEEE Photon. Technol. Lett, vol. 30,no. 24, pp. 2099–2102, Dec. 2018.Publications Related to Chapter 6• M. Jana, L. Lampe, J. Mitra, “Spectrally Efficient Time-Frequency PackedWDM Superchannel Transmission”, to be submitted, 2019.• M. Jana, J. Mitra, L. Lampe “System and Method for Multichannel OpticalTransmission with Interference Mitigation”, Utility patent application beingdrafted, to be submitted, 2019.• M. Jana, J. Mitra, L. Lampe “System and method for Phase Noise mitigationin Coherent Optical Transceivers”, Utility patent application being drafted, tobe submitted, 2019.viiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiLay Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ixList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xviList of Abbreviations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiNotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxviAcknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xxviiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxx1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Background & Motivation . . . . . . . . . . . . . . . . . . . . . . . . 11.1.1 Why Fixed Transmission Networks? . . . . . . . . . . . . . . 11.1.2 Why Faster-than-Nyquist (FTN) Transmission? . . . . . . . 51.2 Enabling Technologies . . . . . . . . . . . . . . . . . . . . . . . . . . 6ix1.2.1 FTN Signaling . . . . . . . . . . . . . . . . . . . . . . . . . . 71.2.2 Time-Frequency Packing . . . . . . . . . . . . . . . . . . . . 71.2.3 Polarization Multiplexing . . . . . . . . . . . . . . . . . . . . 81.2.4 Higher-order Modulation Schemes . . . . . . . . . . . . . . . 81.3 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.3.1 Single Carrier DP FTN OFC Systems . . . . . . . . . . . . . 101.3.2 Single Carrier DP FTN MWC Systems . . . . . . . . . . . . 111.3.3 Multicarrier DP FTN OFC Systems . . . . . . . . . . . . . . 141.4 Contributions of the Thesis . . . . . . . . . . . . . . . . . . . . . . . 151.4.1 Single Carrier DP FTN Transmission for OFC . . . . . . . . 161.4.2 Single Carrier DP FTN HoM Transmission for MWC . . . . . 171.4.3 TFP WDM Superchannel Transmission for OFC . . . . . . . 181.5 Organization of the Thesis . . . . . . . . . . . . . . . . . . . . . . . 192 FTN Transmission for Single-Carrier OFC Systems . . . . . . . . 212.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232.2.1 Precoded FTN . . . . . . . . . . . . . . . . . . . . . . . . . . 232.2.2 Spectral Factorization . . . . . . . . . . . . . . . . . . . . . . 252.3 Non-linear Precoding in FTN Systems . . . . . . . . . . . . . . . . . 262.3.1 THP-precoded FTN . . . . . . . . . . . . . . . . . . . . . . . 262.3.2 Expanded A-priori Demapper (EAD) . . . . . . . . . . . . . 282.3.3 Sliding-Window-EAD (SW-EAD) . . . . . . . . . . . . . . . . 312.3.4 Precoding-loss for FTN-THP Systems . . . . . . . . . . . . . 322.4 Linear Pre-equalization for FTN . . . . . . . . . . . . . . . . . . . . 342.5 Numerical Results and Discussion . . . . . . . . . . . . . . . . . . . 38x2.5.1 Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . . 392.5.2 Performance of FTN-THP with Proposed Demappers . . . . 402.5.3 Computational Complexity Analysis . . . . . . . . . . . . . . 442.5.4 Performance of Proposed FTN-LPE . . . . . . . . . . . . . . 452.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513 FTN Transmission for Single-Carrier MWC Systems . . . . . . . . 523.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 553.3 Adaptive DFE with PN Compensation . . . . . . . . . . . . . . . . . 583.3.1 Combined Phase Noise Tracking (CPNT) . . . . . . . . . . . 593.3.2 Individual Phase Noise Tracking (IPNT) . . . . . . . . . . . . 633.4 XPIC with Precoded FTN . . . . . . . . . . . . . . . . . . . . . . . . 663.5 Numerical Results and Discussion . . . . . . . . . . . . . . . . . . . 693.5.1 Simulation Setup . . . . . . . . . . . . . . . . . . . . . . . . . 693.5.2 Performance with DFE-FTN . . . . . . . . . . . . . . . . . . 713.5.3 Performance with LPE-FTN . . . . . . . . . . . . . . . . . . 763.5.4 Computational Complexity Analysis . . . . . . . . . . . . . . 813.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 824 Multicarrier Faster-than-Nyquist Optical Transmission . . . . . . 834.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 834.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.3 Precoding Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 864.3.1 Joint precoding: 2-D LPE . . . . . . . . . . . . . . . . . . . . 864.3.2 Partial Precoding (PP) . . . . . . . . . . . . . . . . . . . . . 894.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90xi4.4.1 Simulation Parameters . . . . . . . . . . . . . . . . . . . . . . 914.4.2 2-D LPE Gains . . . . . . . . . . . . . . . . . . . . . . . . . . 924.4.3 Feasible Range for 2-D LPE . . . . . . . . . . . . . . . . . . . 944.4.4 PP Gains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.4.5 Computational Complexity . . . . . . . . . . . . . . . . . . . 954.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 965 Towards Terabit-per-second Super-Nyquist Systems . . . . . . . . 975.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 975.2 System model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1005.3 FP WDM Transmission: ICIC . . . . . . . . . . . . . . . . . . . . . 1015.3.1 Linear Equalization . . . . . . . . . . . . . . . . . . . . . . . 1015.3.2 Iterative Equalization: Turbo-PIC . . . . . . . . . . . . . . . 1025.4 TFP WDM Transmission: ISIC & ICIC . . . . . . . . . . . . . . . . 1045.5 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 1045.5.1 Simulation Parameters . . . . . . . . . . . . . . . . . . . . . . 1055.5.2 ISI vs. ICI Trade-off . . . . . . . . . . . . . . . . . . . . . . . 1065.5.3 LE-ICIC vs Turbo-PIC . . . . . . . . . . . . . . . . . . . . . 1075.5.4 TFP Gains . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1085.5.5 Computational Complexity . . . . . . . . . . . . . . . . . . . 1115.6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1126 Flexible Designs for Spectrally Efficient TFP Superchannels . . . 1136.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1136.2 System Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1166.3 Interference Channel Estimation and CPNE . . . . . . . . . . . . . . 1186.3.1 DSP Modules . . . . . . . . . . . . . . . . . . . . . . . . . . . 119xii6.3.2 LMS Update Equations . . . . . . . . . . . . . . . . . . . . . 1196.3.3 Data-aided and Decisions-directed Adaptation . . . . . . . . 1216.4 Iterative PN Estimation (IPNE) . . . . . . . . . . . . . . . . . . . . 1226.4.1 LIPNE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1236.4.2 FGIPNE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1236.5 Interference Cancellation . . . . . . . . . . . . . . . . . . . . . . . . 1246.5.1 Basic Turbo ISIC-ICIC Structure . . . . . . . . . . . . . . . . 1246.5.2 ICIC Scheduling: SPCIC . . . . . . . . . . . . . . . . . . . . 1266.6 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1286.6.1 Simulation Parameters . . . . . . . . . . . . . . . . . . . . . . 1306.6.2 Interference Channel Estimation and Cancellation Gains . . . 1306.6.3 Tolerance to Cascaded ROADMs . . . . . . . . . . . . . . . . 1346.6.4 Tolerance to Laser Linewidth . . . . . . . . . . . . . . . . . . 1376.6.5 Computational Complexity Analysis . . . . . . . . . . . . . . 1396.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1397 Concluding Remarks & Future Directions . . . . . . . . . . . . . . . 1417.1 Summary and Conclusions . . . . . . . . . . . . . . . . . . . . . . . 1417.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1447.2.1 FTN and Probabilistic Shaping . . . . . . . . . . . . . . . . . 1447.2.2 Fiber Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . 1457.2.3 Additional Device Non-idealities and Impairments . . . . . . 145Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163xiiiAppendix A Proofs and Derivations for Chapter 2 . . . . . . . . . . . 164A.1 Proof of Proposition 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . 164A.2 Proof of Proposition 2.2 . . . . . . . . . . . . . . . . . . . . . . . . . 165A.3 PSD And Average Transmit Power with Precoding . . . . . . . . . . 166Appendix B Proofs and Derivations for Chapter 3 . . . . . . . . . . . 170B.1 LMS Update Equations . . . . . . . . . . . . . . . . . . . . . . . . . 170B.1.1 Proof of Lemma 3.1 . . . . . . . . . . . . . . . . . . . . . . . 170B.1.2 Proof of Lemma 3.2 . . . . . . . . . . . . . . . . . . . . . . . 172B.2 LPE-FFF and LPE-FBF Computations . . . . . . . . . . . . . . . . 173Appendix C Proofs and Derivations for Chapter 4 . . . . . . . . . . . 174C.1 Proof of Proposition 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . 174C.2 Proof of Proposition 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . 175C.3 2-D LPE PMD Equalizer LMS Algorithm . . . . . . . . . . . . . . . 175Appendix D Proofs and Derivations for Chapter 5 . . . . . . . . . . . 177D.1 Proof of Lemma 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 177D.2 Proof of Lemma 5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 177Appendix E Proofs and Derivations for Chapter 6 . . . . . . . . . . . 179E.1 FGIPNE Metrics Computation . . . . . . . . . . . . . . . . . . . . . 179xivList of Tables2.1 Computational complexities of the THP-demappers for each bit anditeration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 442.2 Complexity comparison of the demappers per bit per iteration: QPSK,β = 0.3,τ = 0.84 and τ = 0.8. . . . . . . . . . . . . . . . . . . . . . . 443.1 Computational Complexities: CPNT vs. IPNT . . . . . . . . . . . . . 814.1 Complexity, memory and latency, per codeword . . . . . . . . . . . . 955.1 Complexity, memory and latency per codeword . . . . . . . . . . . . 1116.1 Simulation parameters . . . . . . . . . . . . . . . . . . . . . . . . . . 1296.2 Computational Complexity. . . . . . . . . . . . . . . . . . . . . . . . 138xvList of Figures1.1 Example of a typical backhaul and core network infrastructure. Theschematics of the figure are adopted from [1]. . . . . . . . . . . . . . . 21.2 Growth of Internet traffic over the years. . . . . . . . . . . . . . . . . 31.3 Predictions for microwave backhaul capacity per site, according to [4]. 41.4 Summary of thesis contributions: FTN for OFC and MWC fixed trans-mission networks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.1 Baseband system model for a pre-equalized FTN transmission wherethe shaded blocks at the transmitter and the receiver represent theproposed FTN pre-equalizer and symbol demappers respectively. . . . 232.2 FTN pre-equalization with THP and the modulo-equivalent linearstructure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262.3 Linear pre-equalization of FTN ISI. . . . . . . . . . . . . . . . . . . . 342.4 Block diagram of the precoded FTN dual-polarized coherent opticalsimulation setup where the shaded blocks at the transmitter and thereceiver represent the proposed THP/LPE pre-equalizer and symboldemappers respectively. . . . . . . . . . . . . . . . . . . . . . . . . . . 382.5 BER vs. OSNR for FTN-THP with different demappers, illustratingthe performance of the proposed EAD. QPSK, β = 0.3, τ = 0.85 and0.8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40xvi2.6 Auto-correlation of the expanded constellation symbols v for β = 0.3and τ = 0.8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412.7 BER vs. OSNR for FTN-THP with different demappers, illustratingthe performance gains with the proposed SW-EAD over EAD. QPSK,β = 0.3 and τ = 0.8. . . . . . . . . . . . . . . . . . . . . . . . . . . . 422.8 SNR vs. τ in a QPSK FTN-THP system for varying β. . . . . . . . . 432.9 BER vs. OSNR for FTN with LPE precoding. QPSK with τ = 0.8and 16QAM with τ = 0.85, β = 0.3. . . . . . . . . . . . . . . . . . . . 462.10 Normalized PSD of LPE-FTN vs. normalized frequency fT for β =0.3, τ = 0.85. Also included are the PSDs for Nyquist signaling withthe T -orthogonal RRC with β = 0.3 and the τT -orthogonal RRC withβˆ = 0.105. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 472.11 Normalized PSD of LPE-FTN with β = 0.3, τ = 0.78 and Nyquistsignaling with a τT -orthogonal RRC having βˆ = 0.014 vs. normalizedfrequency fT using truncated RRC pulses to illustrate spectral leakage. 482.12 Empirical CCDF of the instantaneous power with average transmitpower = 0 dB, β = 0.3, τ = 0.78. . . . . . . . . . . . . . . . . . . . . 493.1 System model for a DP-FTN transmission. . . . . . . . . . . . . . . . 553.2 Equivalent discrete-time baseband system model for a DP-FTN trans-mission. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 563.3 Detailed Rx-DSP block diagram for the adaptive XPIC and DFE-FTNequalization with CPNT. . . . . . . . . . . . . . . . . . . . . . . . . . 593.4 Joint estimation of the filter tap-weights and PN processes for theDFE-CPNT method. . . . . . . . . . . . . . . . . . . . . . . . . . . . 62xvii3.5 Detailed Rx-DSP block diagram for the adaptive XPIC and DFE-FTNequalization with IPNT. . . . . . . . . . . . . . . . . . . . . . . . . . 633.6 LPE-FTN DSP, where the shaded blocks represent additional signalprocessing compared to a DFE-FTN system. . . . . . . . . . . . . . . 673.7 BER vs. SNR for DP-Nyquist and DP-FTN systems, illustratingthe performance gains of DFE-IPNT over DFE-CPNT, and 256-QAMFTN gains over 1024-QAM Nyquist transmission, respectively. β=0.4,τ=1 (Nyquist) and τ=0.8 (FTN). . . . . . . . . . . . . . . . . . . . 723.8 MSE vs. SNR for 1024-QAM DP-Nyquist systems, illustrating thegains of DFE-IPNT over DFE-CPNT for different XPD values. β=0.4,τ=1 (Nyquist). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733.9 BER vs. SNR for DP-FTN systems, illustrating the performance gainsof LPE-FTN over DFE-FTN. 256 and 1024-QAM, β = 0.3, 0.4, τ = 1(Nyquist) and τ=0.8 (FTN). . . . . . . . . . . . . . . . . . . . . . . 753.10 Spectral efficiency vs. SNR for DP-Nyquist and DP-FTN schemes.256, 512 and 1024-QAM, β = 0.25, 0.3 and 0.4, τ = 1 (Nyquist) andτ=0.8, 0.89 (FTN). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 773.11 Additional SNR required over the respective zero-PN reference systemsto achieve a BER of 10−6, plotted against σ∆. 256, 512, 1024-QAM,β=0.4,τ=1 (Nyquist) and τ=0.8 (FTN). . . . . . . . . . . . . . . . 793.12 Empirical CCDF of the instantaneous power with average transmitpower = 0 dBW. 256-QAM, β = 0.3 and 0.4, τ = 1 (Nyquist) andτ=0.8 (FTN). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 804.1 Precoded-MFTN AWGN system model. . . . . . . . . . . . . . . . . 85xviii4.2 2-D LPE, where the shaded blocks represent additional signal process-ing compared to unprecoded MFTN systems. . . . . . . . . . . . . . . 864.3 Partial precoding, where the shaded blocks represent additional signalprocessing compared to unprecoded MFTN systems. . . . . . . . . . . 884.4 ICI mitigation through PIC. . . . . . . . . . . . . . . . . . . . . . . . 894.5 Simulated MFTN system model: precoded DP TFP WDM opticalsuperchannel transmission. . . . . . . . . . . . . . . . . . . . . . . . . 904.6 BER vs. OSNR, β=0.3, τ=0.85, ξ=0.88. . . . . . . . . . . . . . . . . 924.7 Feasible range of τ, ξ for 2-D LPE. . . . . . . . . . . . . . . . . . . . 934.8 BER vs. OSNR, β=0.3, τ=0.8, ξ=0.9. . . . . . . . . . . . . . . . . . 945.1 Super-Nyquist WDM system model. . . . . . . . . . . . . . . . . . . . 995.2 LE-ICIC, shaded block represents 2-D LMS. . . . . . . . . . . . . . . 1015.3 Turbo-PIC, shown for the X-pol. of the kth SC. . . . . . . . . . . . . 1025.4 Turbo-PIC combined with BCJR-ISIC, shown for the X-pol. of thekth SC. Shaded blocks represent additional processing to perform BCJR.1035.5 400 Gbps system, normalized PSD vs. frequency, with 4 WSSs. . . . 1055.6 1 Tbps system, normalized PSD vs. frequency, with 4 WSSs. . . . . . 1055.7 400 Gbps system, BER vs OSNR for FP WDM systems. . . . . . . . 1065.8 1 Tbps system, BER vs OSNR for FP WDM systems. . . . . . . . . . 1075.9 400 Gbps, ROSNR vs. ξ, with 4 WSSs, illustrating the optimal ξ. . . 1095.10 1 Tbps, ROSNR vs. ξ, with 4 WSSs, illustrating the optimal ξ. . . . 1095.11 400 Gbps, BER vs. OSNR, with 4 WSSs, illustrating additional TFPgains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1105.12 1 Tbps, BER vs. OSNR, with 4 WSSs, illustrating additional TFPgains. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111xix6.1 TFP WDM system model. . . . . . . . . . . . . . . . . . . . . . . . . 1166.2 Jointly estimating PMD filter, TFP interference and PN. . . . . . . . 1186.3 BCJR-ISIC+SPCIC-ICIC, shown for the example of a 3-SC WDMsystem. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1256.4 MSE convergence, 75 kHz LLW, varying τ and ξ. . . . . . . . . . . . 1316.5 BER vs. OSNR, highlighting the benefits of the proposed TFP designover time-only packing. 1040 km fiber, 75 kHz LLW, CPNE+FGIPNE,6% pilot density, varying τ and ξ. . . . . . . . . . . . . . . . . . . . . 1326.6 SE vs. distance, highlighting the benefits of the proposed TFP de-sign over time-only packing and other TFP designs. 75 kHz LLW,CPNE+FGIPNE, 6% pilot density, varying τ and ξ. . . . . . . . . . . 1336.7 BER vs. OSNR, showing tolerance of the proposed scheme to cascadedWSSs. 1040 km fiber, 75 kHz LLW, CPNE+FGIPNE, 6% pilot density,varying τ and ξ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1356.8 ROSNR penalty vs. LLW for Nyquist WDM, showing benefits andlimitations of CPNE, LIPNE and FGIPNE, having varying pilot den-sities. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1366.9 ROSNR penalty vs. LLW, showing benefits and limitations of CPNE,LIPNE and FGIPNE, 6% pilot density. . . . . . . . . . . . . . . . . . 137xxList of Abbreviations4G 4th Generation5G 5th GenerationADC Analog-to-Digital ConverterASE Amplified Spontaneous EmissionAWGN Additive White Gaussian NoiseBCJR Bahl-Cocke-Jelinek-RavivBER Bit Error RateBPS Blind Phase SearchBW BandwidthCCDF Complementary Cumulative Distribution FunctionCD Chromatic DispersionCOSC Coherent Optical Single-CarrierCPNE Coarse Phase Noise EstimationCPNT Combined Phase noise TrackingCPR Carrier Phase RecoveryCSI Channel State InformationDAC Digital-to-Analog ConverterDFE Decision Feedback EqualizerDGD Differential Group DelayDP Dual PolarizationxxiDVB-S2 Second Generation Digital Video Broadcasting Standard for SatelliteEAD Expanded A-priori DemapperFBF Feedback FilterFDE Frequency Domain EqualizerFEC Forward Error CorrectionFFF Feed-forward FilterFG Factor GraphFGIPNE Factor Graph-based Iterative Phase noise EstimationFP Frequency PackingFSE Fractionally Spaced EqualizerFTN Faster-than NyquistHoM Higher-order ModulationI In-phaseICI Inter-carrier InterferenceICIC Inter-carrier Interference CancellationIEEE Institute of Electrical and Electronics EngineersIIR Infinite Impulse ResponseIoT Internet-of-ThingsIP Internet ProtocolIPNE Iterative Phase noise EstimationIPNT Individual Phase noise TrackingIRA Irregular Repeat AccumulateISI Inter-symbol InterferenceISIC Inter-symbol Interference CancellationLDPC Low Density Parity CheckxxiiLE Linear EqualizationLIPNE LMS-based Iterative Phase noise EstimationLLR Log-Likelihood RatioLLW Laser LinewidthLMS Least Mean SquareLO Local OscillatorLPE Linear Pre-equalizationLTI Linear Time-InvariantMAP Maximum A-posteriori ProbabilityMFB Matched-Filter BoundMFTN Multicarrier Faster-than NyquistMIMO Multi-input multiple-outputMSE Mean Squared ErrorMWC Microwave CommunicationMZ Mach-ZehnderOFC Optical Fiber CommunicationOSNR Optical Signal-to-noise RatioPAM Pulse Amplitude ModulationPAPR Peak-to-average Power RatioPCA Principal Component AnalysisPDF Probability Density FunctionPIC Parallel Interference CancellationPLD Peh-Liang-DemapperPMD Polarization Mode DispersionPMF Probability Mass FunctionxxiiiPN Phase NoisePP Partial PrecodingPSD Power Spectral DensityPSP Principal States of PolarizationQ QuadratureQAM Quadrature Amplitude ModulationQPSK Quarternary Phase Shift KeyingROADM Reconfigurable Optical Add-Drop MultiplexerROSNR Required Optical Signal-to-noise RatioRRC Root-Raised-CosineRx-DSP Receiver Digital Signal ProcessingSC Sub-channelSE Spectral EfficiencySIC Successive Interference CancellationSNR Signal-to-Noise RatioSP Single-polarizedSPCIC Serial-and-Parallel Combined Interference CancellationSSMF Standard Single Mode FiberSW-EAD Sliding-Window Expanded A-priori DemapperTFP Time-Frequency PackingTHP Tomlinson Harashima PrecodingWDM Wavelength Division MultiplexingWF Whitening FilterWMF Whitened Matched FilterWSS Wavelength Selective SwitchxxivXPD Cross-polarization DiscriminationXPI Cross-polarization InterferenceXPIC Cross-polarization Interference CancellationxxvNotation∗ Linear convolution Hadamard product or element-wise multiplication|S| Cardinality of a set S|x| Magnitude of the complex number x〈| · |〉 Element-wise magnitudes of the complex scalars‖ · ‖ Vector norm operator(·)∗ Complex conjugation(·)−∗ 1(·)∗{x[j]}N2j=N1The row-vector [x[N1], . . . , x[N2]][·]−1 Matrix inverse[·]H Matrix Hermitian[·]T Matrix transposediag(·, ·, · · · ) Diagonal matrix formed with the inputsE(·) Expectation operatorIm{x} Imaginary part of the complex number xRe{x} Real part of the complex number xσ2x Variance of the signal xx(t) Continuous time analog signal x at any time instant tx[n] = x(nTs) Discrete time counterpart of x(t) sampled with a frequency of1TsVar(·) Element-wise variancexxviZ{·} Z-transformZ−1{·} Inverse Z-transformxxviiAcknowledgmentsI am sincerely grateful to my PhD advisor Professor Lutz Lampe for his constant guid-ance and encouragement throughout the entire duration of these wonderful 4 years.What an incredible journey it has been! His appreciation for a commendable work,and his criticism for not-so-commendable efforts immensely helped me shape myresearch. It has been an absolute pleasure learning from him – not just technicalconcepts, but also, diligence, time management and multi-tasking skills. He will con-tinue to be a constant source of inspiration to me in the years to come. Honestly, Icouldn’t have asked for a better supervisor.I am also thankful to the rest of my thesis advisory committee members: Prof.Sudip Shekhar and Prof. Julian Cheng for their insightful feedbacks and comments.Special thanks go to Prof. Vincent Wong and Prof. Cyril Leung for serving asexaminers for my departmental and final doctoral defense.I am immensely thankful to Prof. Giulio Colavolpe (Dept. of Engineering andArchitecture, University of Parma, PR, Italy) for serving as the external examinerfor my PhD defense and providing me valuable inputs to improve the content of mythesis.I also take this opportunity to thank Dr. Jeebak Mitra (Huawei Technologies,Canada) for his technical guidance towards problem formulation, and his construc-tive criticism on the developed algorithmic designs. I am sincerely thankful for hispersistent questioning of the assumptions, suggestions for considering practical im-xxviiipairments, pointing out the pertinent literature, and thorough reviewing of the pub-lication manuscripts.I convey my special thanks to Dr. Ahmed Medra (Huawei Technologies, Canada)for his mentorship at the beginning of my PhD.My gratitude knows no bounds for my wife Amrita, and my daughter Arianna,for all their infinite love, support, and sacrifices. My eagerness to be with them atthe end of the day got me through the pain and frustration of having uncountablymany bad-simulation-days. I am able to finish my PhD research within the prescribedtime-frame of 4 years, not in spite of them, but because of them.Last, but not the least by any means, I am utterly indebted to my Parents andmy sister, for everything – hoping that the word “everything” has enough breadth toencompass everything. Whatever I have attempted to achieve so far, has always beenintended to make them proud. I thank them from the deepest corner of my heart,for all the endless and unconditional love, and for being the best Parents and sisterin the entire Universe.xxixDedicationTo Mom and Dad, my wife Amrita, my sweet pea Arianna, and my sister Tanu, allof whom collectively form the nucleus of my heart.xxxChapter 1Introduction1.1 Background & Motivation1.1.1 Why Fixed Transmission Networks?Data traffic in the communication networks is increasing at a remarkable rate withevery passing year. Such enormous traffic load is pushing the limits of the existingcore and access network infrastructure, primarily comprised of optical fibers andmicrowave radio links. These fixed transmission networks serve as the backbone forthe Internet and the mobile data traffic across the globe. An example of the fixedtransmission network is pictorially illustrated in Fig. 1.1, where a number of macroand small cell networks are connected to the backhaul and the core networks via acombination of optical fiber and point-to-point microwave links [1].The rapid rise in the Internet-based activities, such as online streaming, video-on-demand, Internet gaming, augmented reality and virtual reality applications, etc.,has contributed towards a massive growth in the global Internet protocol (IP) traffic.According to the Cisco Visual Networking Index [2], the number of devices connectedto the IP networks will be more than three times the global population by 2022.Plethora of those connections will consist of smartphones and Internet-of-Things(IoT) devices. According to the predictions, the global IP traffic will grow nearlythreefold between 2017 and 2022, and will have increased 1500-fold from 2002 to1Core NetworkMicrowaveOptical FiberData center ResidentialHome cellEnterpriseSmall cellPublic AccessSmall cellPublic AccessSmall cellAggregation nodeMacro cellMacro cellCell phonesCell phonesCell phonesCellularbackhaulFigure 1.1: Example of a typical backhaul and core network infrastructure. Theschematics of the figure are adopted from [1].2022. Such a monumental growth of the IP traffic over the years is summarized inFig. 1.2a.Predictions for the growth of mobile data traffic are even more extreme. As aconsequence of the evolving fourth generation (4G) and the developing fifth gener-ation (5G) networks, cellular data rates are expected to grow at a staggering pace.According to the predictions [3], there will be 12.3 billion mobile-connected devicesby 2022, exceeding the world’s projected population of 8 billion at that time. Theaverage smartphone will generate 11 GB of traffic per month by 2022, more thana 4.5-fold increase over the 2017 average of 2 GB per month. Such an abundanceof mobile devices will lead to cellular data traffic to increase at a compound annualgrowth rate of 46 percent from 2017 to 2022, reaching 77 exabytes (1 exabyte= 1018bytes) per month by 2022. The predicted growth of the mobile data traffic per monthis illustrated in Fig. 1.2b, for the 5-year period 2017-2022.2Year Global Internet Traffic1992 100 GB per day1997 100 GB per hour2002 100 GB per second2007 2,000 GB per second2017 46,600 GB per second2022 150,700 GB per second(a) Global IP traffic according to [2]12192941577701020304050607080902017 2018 2019 2020 2021 2022Exabytes/month(b) Mobile IP traffic according to [3]Figure 1.2: Growth of Internet traffic over the years.These trends have imposed an enormous burden on the fixed transmission net-works. Existing fiber-optic and microwave networks are able to facilitate data ratesfrom a few hundreds of Mbps to several hundreds of Gbps. In the recent past, tremen-dous progress has been made to increase the capacity of these networks. For example,since the introduction of coherent processing, single-carrier optical fiber transmissionrates have gone up from 10 Gbps to more than 400 Gbps in the state-of-the-artsystems commercially available today. Keeping up with the similar trend, future380 % of sites20 % of sitesFew percent of sitesMobile broadband150 Mbps300 Mbps1 Gbps2017350 Mbps1-2 Gbps3-10 Gbps2022600 Mbps3-5 Gbps10-20 GbpsTowards 2025Figure 1.3: Predictions for microwave backhaul capacity per site, according to [4].per-carrier data rates are targeted towards 1 Tbps for longhaul optical fiber links 1.Similarly, with the evolution of the next generation cellular standards, the appetitefor microwave backhaul capacity has also increased [4, 6]. The predicted evolution ofthe microwave backhaul capacity is shown in Fig. 1.3. As shown in the figure, it ispredicted that, by 2022, the typical backhaul capacity for a high-capacity microwaveradio site will be in the 1 Gbps range, and increasing to 3 − 5 Gbps by 2025 [4]. Itis also forecast that 80 percent of the next generation sites in an advanced mobilebroadband network will have increased to 600 Mbps by 2025, with peak data ratesexceeding 10 Gbps.Accomplishing such futuristic high throughput targets with the help of the existingtechnologies is a challenging task. Therefore, researchers from both the academiccommunities and the telecommunications industries are actively in the pursuit ofalternative approaches or improved supplements of the existing solutions.1For an information theoretic perspective on the ultimate capacity limits in the fiber networks,interested readers are referred to the very nice invited paper [5].41.1.2 Why Faster-than-Nyquist (FTN) Transmission?One obvious approach to facilitate high data rates in the next generation core net-works is to increase the baud rates of the optical fiber communication (OFC) andthe microwave communication (MWC) systems. However, higher baud rates requirelarger transmission bandwidth (BW), which is becoming an increasingly critical andexpensive resource. Moreover, owing to the practical constraints on even the mostcutting-edge radio-frequency electronic and opto-electronic components with regardsto high BW transmission, it seems unlikely that such high throughputs can be ac-complished by transmitting high baud rates alone. Hence, transmitting more dataper unit time and frequency, i.e., increasing the spectral efficiency (SE), is absolutelycrucial for the future fixed transmission networks.Conventional approaches to achieve such SE improvements are multiplexing multi-ple carriers possibly using spectral shaping with sharp filters, enabling dual-polarized(DP) transmission, and introducing higher-order modulation (HoM) formats. Theimplementation of these known approaches has usually been based under the premisesthat the data symbols are transmitted via waveforms that are orthogonal in time andfrequency. This facilitates symbol detection for transmission over linear time in-variant channels, which is often a good approximation for fixed transmission links.However, it has been shown that Nyquist-rate orthogonal signaling is often restric-tive, and that improvements in terms of SE can be achieved with the so-called FTNsignaling [7–12].FTN signaling is a linear modulation technique, which deliberately relinquishesthe symbols-spacing requirement imposed by the Nyquist criterion. By giving upthis orthogonality condition, theoretically, FTN signaling provides a higher achiev-able rate over a Nyquist transmission [13]. While the basic concept of FTN trans-5mission dates back to the 1970s [7], the actual application of FTN signaling in com-munication systems has been limited primarily due to implementation complexityand silicon feasibility in the years following its proposal. It was only relatively re-cently that the potential for higher SE using FTN has received broader attentionfrom the research community and the telecommunications industry. The benefits ofFTN is well-summarized in [10], which states that FTN “has attracted interest in ourbandwidth-starved world because it can pack 30%-100% more data in the same BWat the same energy per bit and error rate, compared to traditional method”.Transmitting at an FTN rate allows us to approach the capacity of a bandlimitedchannel [13]. From a practical implementation perspective, OFC and MWC systemsare prime candidates for the introduction of FTN as it can moderate the need forHoM formats in such systems, to achieve a target data rate. This is significantlycrucial, since HoM schemes are sensitive to the practical non-idealities such as thefiber-optical nonlinearity and phase noise (PN). In the pursuit of even higher capac-ity in the OFC and MWC links, FTN signaling can also be applied in conjunctionwith the conventional SE and throughput enhancement techniques, such as polariza-tion multiplexing, multicarrier transmission, and HoM formats, to supplement theseknown methods with additional SE benefits.1.2 Enabling TechnologiesIn this section, we briefly revisit some of the SE improvement approaches applicableto the next generation fixed transmission networks. Such techniques do not neces-sarily serve as competitive technologies. Quite the contrary, these approaches can becombined, to reap the aggregate SE benefit by complementing the individual gains.61.2.1 FTN SignalingFTN signaling applies non-orthogonal linear modulation to increase the SE comparedto the well-known orthogonal transmission at Nyquist rate. For a given BW, FTNsignaling translates to a higher baud rate compared to Nyquist systems. On theother hand, FTN transmission leads to the reduction of BW when the baud rate isfixed. When applied to single carrier OFC and MWC systems, the SE improvementsdue to FTN signaling come at a price of introducing inter-symbol interference (ISI).Therefore, enjoying the above benefits of FTN signaling entails successful mitigationof the FTN-induced ISI through sophisticated signal processing.1.2.2 Time-Frequency PackingExtension of FTN signaling to a multi-carrier transmission offers additional SE im-provements by stacking several spectrally overlapping single-carrier channels together.Multi-carrier FTN (MFTN) transmission scheme serves as a research avenue that isbeing actively pursued at present, particularly, in the context of OFC systems. Dueto practical limitations of the opto-electronics to facilitate high-baud-rate single-carrier transmissions [14], and the enhanced impact of the fiber-optic dispersion ona larger transmission BW, an attractive choice to achieve significant data rate im-provements in OFC systems is to employ optical superchannels using super-Nyquist2wavelength division-multiplexing (WDM), also known as the time-frequency pack-ing (TFP) transmission technique [15–21]. In this thesis, we use the terminologies“MFTN” and “TFP” interchangeably. Such a scheme increases the SE through de-liberate reduction of the symbols-spacing, in both time and frequency dimensions,compared to an orthogonal system. However, the SE advantage of TFP systems2The terminology “super-Nyquist”, in general, refers to transmission systems where FTN signal-ing is applied either in time or frequency or both.7comes at the expense of ISI and inter-carrier interference (ICI), which necessitateefficient interference mitigation techniques.1.2.3 Polarization MultiplexingTo further the bandwidth efficiency of the fixed transmission networks, FTN can alsobe combined with antenna polarization multiplexing through a DP transmission.An ideal DP system, where two data-streams are transmitted at the same carrierfrequency by two orthogonal polarizations, offers a doubling of the data rate comparedto a single-polarized (SP) transmission. However, a DP system leads to cross-talkbetween the two polarization data streams, commonly known as polarization modedispersion (PMD) in OFC systems and cross-polarization interference (XPI) in anMWC transmission. While DP optical systems are sufficiently well-investigated, aDP-MWC transmission with XPI cancellation (XPIC) is still being considered as anactive area of research [22–32]. Combining FTN signaling with a DP transmission ismotivated by the fact that FTN signaling can offer additional contribution to the SEimprovement a DP system provides. For example, using an FTN acceleration factorof 0.8 for the two orthogonal data streams offers a 150% increase in SE comparedto an SP Nyquist transmission. To appreciate the true gains of a DP transmission,powerful interference handling techniques should be adopted to counter the PMD orXPI.1.2.4 Higher-order Modulation SchemesAnother obvious and well-known approach for SE improvements in the fixed trans-mission networks is to employ HoM formats. However, due to the nonlinear effectsof the optical channel, employing even moderately high modulation orders is chal-8lenging for OFC systems [11, 33]. Moreover, such systems also suffer from signaldistortion due to PN stemming from the spectral linewidth of the transmitter andreceiver lasers. On the other hand, typical MWC systems use HoM formats. In fact,practical microwave backhaul systems for spectrally efficient transmission are evolv-ing towards adopting very high modulation orders, e.g. 4096-QAM [34]. However,employing extremely high modulation formats makes the communication system vul-nerable to PN that arises due to imperfections in the transmitter and receiver localoscillators (LOs). This makes the OFC and MWC systems suitable for the applica-tion of FTN signaling as it can eliminate the need for very high modulation orders,which are more sensitive to fiber nonlinearity and PN. Therefore, FTN transmission,with powerful interference mitigation techniques, can yield a significant performanceadvantage over a Nyquist system that employs a higher modulation order to achievethe same data rate.1.3 Literature ReviewHaving established the necessary background in the previous section, we now proceedto review the state-of-the-art on the application of FTN signaling in the next gen-eration fixed transmission networks. Based on the current deployment of the OFCand MWC systems in the existing fixed transmission networks, we broadly considerthree application scenarios, namely (a) single carrier DP OFC systems, (b) singlecarrier DP MWC systems, and (c) multicarrier DP OFC systems, for introducingand evaluating the concept of FTN signaling.91.3.1 Single Carrier DP FTN OFC SystemsThe fact that FTN signaling can be an attractive choice for SE improvement hasbeen extensively discussed in the literature, see [10] and references therein. Whilethe original work by Mazo [7] and other early works (e.g. [8, 9, 12]) focused on theminimum distance assuming optimal detection to deal with the FTN-ISI, the devel-opment of sub-optimal equalization methods has received significant attention morerecently. These include reduced-state versions of maximum a-posteriori probability(MAP) symbol equalization based on the Bahl-Cocke-Jelinek-Raviv (BCJR) algo-rithm [35–38] and frequency domain equalization (FDE) [39–41], often operating inan iterative fashion together with forward-error-correction (FEC) decoding. How-ever, the complexity of such turbo-equalization methods is still substantial comparedto the absence of FTN equalization in Nyquist transmission. On the other hand, theperformance of low-complexity linear equalization methods is usually not sufficientespecially when the ISI due to FTN is severe.As an alternative to computationally demanding ISI equalization methods, theexisting FTN literature has also considered transmitter-side pre-equalization tech-niques, which can significantly diminish or completely eliminate the computationalburden from equalization at the receiver. As the FTN introduced interference isperfectly known at the transmitter, pre-equalization does not require the feedbackof the channel state information (CSI) from the receiver to the transmitter. Thisrenders the well-known Tomlinson-Harashima precoding (THP) [42–44] an attractivechoice for pre-equalization. Indeed, THP for FTN has been considered in severalrecent publications in the context of 5G mobile wireless communications [45, 46],MWC [47] and OFC [48–51]. However, the disadvantages of a coded THP systemmanifest themselves in the form of the so-called “modulo-loss” and “precoding-loss”10[44], and a possible increase in the peak-to-average power ratio (PAPR). While theprecoding-loss causes a fixed signal-to-noise ratio (SNR) penalty, the modulo-losscauses an error-rate deterioration by providing inaccurate soft information to theFEC decoder. A few works [52–54] aim to address the modulo-loss problem by im-proving the accuracy of the log-likelihood ratio (LLR) computation. However, thepresented methods are either computationally prohibitive [54] or their performancegains are limited [52, 53]. Accordingly, our research efforts in Chapter 2 are directedtowards facilitating an efficient THP precoded FTN transmission, particularly in acoherent single-carrier longhaul DP OFC framework, by minimizing this loss.Additionally, for the single-carrier and multi-carrier FTN scenarios, we also inves-tigate linear pre-equalization options in Chapter 2 , which bear the potential to offerfurther performance advantages over non-linear precoding techniques. Such a precod-ing method is related to other linear precoding techniques that have been analyzedin the past in conjunction with FTN and partial response signaling (PRS) [55–59].However, these are different, in that, they are either block-based matrix-precodingtechniques or attempt to obtain pre-filter coefficients from optimization problems tomaximize distance properties.1.3.2 Single Carrier DP FTN MWC SystemsThis thesis is the first to present a DP-FTN HoM transmission scheme for im-proved SE in microwave backhaul links. While the polarization cross-talk can beperfectly equalized by linear filters in optical fiber transmission [60], the presence ofFTN-ISI and HoM formats in MWC systems further complicates the system design.DP systems employing XPIC at the receiver have been well investigated in the mi-crowave communication literature for a Nyquist transmission in the context of “syn-11chronous” [22–29] and “asynchronous” [30–32] transmissions. In a synchronous DPtransmission, time and frequency-synchronized received samples from both polariza-tion branches are processed by a two-dimensional (2-D) XPIC filter to remove cross-talk between the two orthogonal polarizations. Alternatively, in an asynchronoustransmission, absence of knowledge about the transmission parameters of the respec-tive other polarization branch precludes the feasibility of performing synchronizationon the interfering data stream. However, the algorithms in previous works for thesesystems do not consider some of the practical challenges encountered in a microwaveradio system. For example, [22–29] describe the XPI mitigation techniques withoutfurnishing sufficient details about the PN compensation algorithms. On the otherhand, [30–32] present XPIC algorithms together with PN mitigation approaches, as-suming an additive white Gaussian noise (AWGN) channel and perfect knowledgeof the XPI channel at the receiver. In practice, a microwave channel can introduceslowly time-varying ISI due to multipath effects [25, 27, 61] and the availability ofa perfect estimate of the XPI channel at the receiver is somewhat unrealistic, par-ticularly in the presence of PN [62]. Moreover, none of the above works considersadditional ISI induced by an FTN transmission.Enjoying the SE benefits of FTN signaling requires successful equalization of theFTN-induced ISI. For this, a significant volume of work considers BCJR based MAPequalization [35–38]. However, it is difficult to apply these methods to a DP-FTNHoM system primarily because their computational complexity becomes intractableas the number of BCJR states increases significantly for very high modulation or-ders. There is also another body of works [39, 40, 63–66] that can be applied tohigher modulation formats without significant increase in complexity. However, theyemploy computationally prohibitive and buffer-space constrained iterative equaliza-12tion schemes, and require explicit channel estimation, which is not computationallytrivial in the presence of PN [62, 67]. Moreover, the above works do not consider anyPN mitigation schemes.Practical microwave systems for spectrally efficient transmission are evolving to-wards adopting very high modulation orders, e.g. 4096-QAM [34], that need robustPN compensation techniques. In light of that, we note the factor-graph (FG) basedmethods for joint FTN and PN mitigation [68, 69], and also the block-based iterativePN compensation techniques [70, 71] in an SP transmission under an AWGN channel.However, the above methods would require additional estimation and equalization al-gorithms for the unknown co-polarization and cross-polarization ISI channels in a DPtransmission. The extension of the above mentioned algorithms to a DP-FTN HoMtransmission under consideration is not straight-forward because channel estimation,FTN and multi-path ISI equalization, and PN compensation tasks are not modu-lar, which invites a joint mitigation approach [62, 67]. Therefore, combining theindividual solutions is challenging under these circumstances, which warrants con-siderable research, and can be subject to future work. In Chapter 3, we consider a2-D adaptive decision feedback equalizer (DFE) to jointly mitigate interference andaccomplish carrier phase recovery in a DP-FTN transmission. We note that 2-DDFE structures without carrier phase recovery have been well studied in the contextof multiple-input multiple-output (MIMO) transmission [72–74], and that previousworks on combining DFE with carrier phase recovery have focused on transmissionswith a single polarization [75–77].131.3.3 Multicarrier DP FTN OFC SystemsIn the TFP WDM literature for OFC, several multicarrier super-Nyquist transmis-sion techniques are considered. One approach adopted in the optical TFP litera-ture [14, 17, 78, 79] considers suppressing the ICI through aggressive transmit-sidefiltering of the individual subchannels (SCs), and the resulting ISI is equalized byBCJR based ISI cancellation (ISIC) methods. Another body of works [16, 18–20]allows only spectral overlap, whereby the ICI is mitigated via linear or nonlinearICI cancellation (ICIC) schemes. However, packing the symbols in only one dimen-sion can be restrictive in achievable rate [13]. While some pioneering works in thesuper-Nyquist literature [13, 80] explored time and frequency packed transmission inAWGN channel scenarios, no proper consideration was given to the practical OFCchannel impairments. For some of these TFP systems, a turbo parallel interferencecancellation (PIC) based ICIC scheme [13, 81], in tandem with the BCJR-ISIC, ispresented under the premises of an AWGN channel. To apply these algorithms to arealistic OFC system, practical fiber-optical impairments need to be taken into ac-count. Moreover, the above PIC based ICIC approach lacks the benefits of sequentialscheduling in a successive interference cancellation (SIC) structure.PN due to the transmitter and receiver laser linewidth (LLW) causes severe sig-nal distortion in WDM systems, and if not successfully mitigated, can significantlyrestrict the performance [82–86]. Conventionally, a feedforward blind phase search(BPS) algorithm is used for carrier-phase recovery (CPR) in Nyquist WDM sys-tems [82]. In BPS, a finite number of test phase angles are evaluated to optimizea cost function, such as the mean squared error (MSE), by making hard symboldecisions of the de-rotated samples. However, the presence of both ISI and ICI inTFP WDM systems precludes the feasibility of making error-free hard symbol deci-14sions preceding the FEC decoder, which renders the BPS algorithm unsuitable forthe considered super-Nyquist transmission. More recently, a CPR algorithm basedon principal component analysis (PCA) has been presented for Nyquist WDM sys-tems employing square constellations [84]. However, to extract the phase informationfrom the principal components, such a method exploits the geometry of the signalconstellation, which gets severely distorted by the ISI and ICI in TFP systems. Forthe same reason, other sophisticated iterative PN compensation algorithms suitablefor Nyquist WDM systems, such as the FG based CPR [70], cannot be directly ap-plied to the super-Nyquist systems without proper consideration of the TFP ISI andICI. The authors of [17] apply FG-based PN cancellation methods for their time-onlypacked systems. However, the CPR method in [17] also needs to be amended beforeapplying to the considered TFP transmission in Chapter 6, because of the absenceof ICI and the restriction to quarternary phase-shift keying (QPSK) in [17].1.4 Contributions of the ThesisIn this dissertation, we aim to achieve data rate enhancements through FTN sig-naling for the fixed transmission systems that use (i) coherent OFC links for long-haul transmission and (ii) point-to-point MWC links. In doing so, we leverage thecommonalities between the optical and the microwave networks to build a commonframework to apply and evaluate the concept of FTN. Practical challenges and im-pairments presented by the OFC and MWC links are taken into consideration whileinvestigating the SE advantages of such FTN systems. The general purpose of ourresearch is (1) the development of effective interference management solutions, and(2) the application and performance assessment of FTN methods. To this end, weadopt signal-processing tools to deal with the impairments present in the practical15July 15, 2019 1FTN for Fixed Transmission Networks  Optical Fiber MicrowaveSingle-Carrier Multi-CarrierSingle-Carrier(Dual Polarized)InterferenceMitigationPNCompensationEqualizerPre-equalizer EqualizerPre-equalizerMAP EqualizerNonlinearLinearLinear/Turbo -Nonlinear/ MAP EqualizerFlexibleTFP DesignsNonlinearEqualizerLinearPrecodingLPE2-D Joint PrecodingPartialPrecoding: Original contribution: Adapted for the considered FTN applicationTx+RxCombinedPN TrackingPN MitigationTx, Rx SeparatePN TrackingFigure 1.4: Summary of thesis contributions: FTN for OFC and MWC fixed trans-mission networks.OFC and MWC systems, together with the intentional interference introduced byFTN, and other implementation issues. The broad contributions of the thesis, asdetailed in the following, are presented schematically in Fig. Single Carrier DP FTN Transmission for OFCA DP single-carrier coherent OFC system is impaired by a number of linear distor-tions, such as chromatic dispersion (CD) and PMD. Therefore, enabling an FTNtransmission for an OFC system requires equalization of these linear impairments,together with the mitigation of the FTN-ISI. In Chapter 2, as an alternative tocomputationally demanding equalization schemes at the receiver, we investigate pre-equalization techniques at the transmitter to pre-mitigate the FTN induced ISI.First, we begin our work by considering THP, and propose techniques to reduce the16modulo-loss. Next, we explore an optimal linear pre-equalization method to com-pletely eliminate the FTN-ISI. The goal of Chapter 2 is to show that the consideredFTN pre-equalization techniques, as an alternative to computationally prohibitivereceiver-side equalization schemes, have the potential to achieve high SE promisedby FTN signaling. Our contributions in Chapter 2 were published in [87, 88].1.4.2 Single Carrier DP FTN HoM Transmission for MWCIn order to increase the throughput of the existing SP microwave links, in Chapter 3,we investigate for the first time a DP-FTN HoM MWC transmission, which suffersfrom ISI, XPI, and PN. Being a phase-only impairment in OFC systems, the po-larization cross-talk can be perfectly equalized by linear filters in such systems [60].However, in MWC systems, XPI manifests itself as a cross-polarization ISI channel,and hence, linear equalization may be restrictive in achieving the desired performance.Moreover, the presence of HoM formats and FTN-ISI further complicates the systemdesign. The direct application of the already existing algorithms for XPIC and PNmitigation to the considered DP-FTN HoM system is not straight-forward, becausethe channel estimation, FTN and multi-path ISI equalization, and the PN compensa-tion tasks are not modular, which invites a joint mitigation approach [62, 67]. There-fore, we investigate joint XPIC and PN compensation techniques for such systems,since combining the individual solutions is challenging under these circumstances.2-D DFE and linear pre-equalization methods coupled with CPR are investigated forthis purpose. The general objective of Chapter 3 is to devise powerful interferencecancellation methods for the existing fixed wireless transmission network, such thata DP-FTN transmission can provide substantial performance improvement over anequivalent DP-Nyquist system that employs a higher modulation order to achieve the17same data rate. The work in Chapter 3 was published in [89, 90].1.4.3 TFP WDM Superchannel Transmission for OFCOptical MFTN systems implemented through time-frequency packed superchannelsoffer additional SE advantages over single-carrier FTN transmission, at the expenseof introducing controlled ISI and ICI. As an alternative to equalization at the re-ceiver, in Chapter 4, we investigate an alternative approach of pre-equalizing theinterference at the transmitter, for the first time in an MFTN system where symbolsare packed in both time and frequency dimensions. Despite offering promising per-formance, the functionality of the precoding techniques in Chapter 4 are limited toa restricted range of time and frequency compression, which renders the precodingsolutions impractical for high data rate systems. In line with the realistic targets ofTbps data rates for the futuristic optical WDM systems, we facilitate Terabit TFPsuperchannels in Chapter 5, where we investigate low-complexity linear and high-performance turbo interference mitigation structures, in the presence of additionalaggressive optical filtering. For this, the ISIC and ICIC algorithms in Chapter 5exploit the known TFP interference channel, without employing additional channelestimation strategies. However, an interference channel estimation approach can offersignificant performance improvement under these circumstances. Such a flexible andspectrally efficient TFP transmission targeting Tbps data rates is presented in Chap-ter 6, together with sophisticated CPR algorithms. This new TFP receiver designenables us to achieve substantial performance and distance improvements comparedto other competitive TFP solutions. Our contributions in Chapter 4-6 have beenpublished in [91–93].181.5 Organization of the ThesisThe organization of the thesis, outlined as follows, encompasses the contributionslisted in the previous section.In Chapter 2, precoded single carrier DP FTN OFC systems are considered. Twosoft demapping algorithms for the nonlinear THP scheme are presented, to reducethe impact of modulo loss. A new linear pre-equalization method is proposed thatyields optimal performance. We provide numerical results for the coded DP FTNOFC system to validate the efficiency of the proposed algorithms.In Chapter 3, we consider for the first time a DP FTN MWC system employingHoM formats. We propose an XPIC and PN mitigation structure, coupled withadaptive DFE or linear precoding, to jointly mitigate interference and accomplishcarrier-phase tracking. We present two PN mitigation strategies based on combinedor separate tracking of the transmitter and receiver PN processes. The effectivenessof the proposed algorithms is demonstrated through computer simulations of a codedDP-FTN microwave communication system in the presence of PN.In Chapter 4, we consider precoding for the first time in an MFTN WDM su-perchannel transmission that enables packing of symbols in both time and frequencydimensions. For this, we propose two precoding solutions, namely a linear 2-D jointprecoding to pre-equalize TFP-ISI and ICI, and a one-dimensional (1-D) linear pre-coding followed by turbo equalization at the receiver. Simulation results for precodedTFP systems are presented the show the benefits of the proposed methods.In Chapter 5, we compare low-complexity linear and high-performance turbo ICICmethods to facilitate Tbps optical TFP WDM superchannels. For more complexstructures, BCJR-ISIC in tandem with PIC-ICIC is employed. Aggressive optical fil-tering is considered in the form of cascades of reconfigurable optical add/drop multi-19plexers (ROADMs) implemented via wavelength selective switches (WSSs). Proposedmethods are validated through numerical results.In Chapter 6, we present flexible designs for Tbps superchannels, where TFP ISIand ICI channels, PMD equalizer coefficients and a coarse PN are jointly estimated.Different scheduling algorithms for turbo-ICIC in conjunction with BCJR-ISIC areinvestigated. Moreover, we propose powerful iterative methods to mitigate PN stem-ming from the laser LLW. Simulation results are presented to confirm the advantagesof the proposed schemes.Finally, in Chapter 7, we provide concluding remarks and future research direc-tions.20Chapter 2Faster-than-Nyquist Transmissionfor Single-Carrier Optical FiberCommunication Systems2.1 IntroductionAs an enabling technology, FTN is advantageous for transmission systems such ascoherent optical communication where the application of HoM formats to increaseSE renders the system more vulnerable to the nonlinear effects of an optical channel[11, 33]. To reap the benefits of an FTN transmission, powerful turbo equalizationtechniques are employed at the receiver to mitigate the FTN induced ISI [35–41].However, the complexity of such turbo-equalization methods is substantial. On theother hand, the low-complexity linear equalization methods are restrictive in achiev-ing the desired performance.We, therefore, turn our attention to pre-equalization techniques which can signifi-cantly diminish or completely eliminate the computational burden from equalizationat the receiver. To this end, the first key observation is that the FTN introducedISI is perfectly known at the transmitter. We exploit that knowledge by consideringnonlinear precoding method THP [42–44] and a linear pre-equalization method, to21pre-mitigate the effects of FTN-ISI at the transmitter.In this chapter, as our first contribution, we propose two computationally efficientdemapping algorithms for an FTN-THP system which outperform the existing mem-oryless demappers from [52, 53] by significant margins. We show that the demapperspresented in this work not only compensate for the modulo-loss but also make THPcompetitive to computationally expensive MAP-based equalization techniques. Hav-ing dealt with the modulo-loss, we then investigate the precoding-loss associated withTHP. For this, we make the second key observation that FTN-ISI stems entirely fromthe transmit pulse-shape and the receive matched filter. The transmit pulse-shapethus contributes partially to the ISI and is a part of the transmitter, whereas, a con-ventional ISI channel in a Nyquist transmission lies outside the transmitter. As aconsequence, the precoding-loss for an FTN-THP transmission over an AWGN chan-nel is different from that in a Nyquist-THP transmission over ISI channels. As oursecond contribution, we derive the analytical expressions for the precoding-loss inan FTN-THP system as a function of the FTN and the pulse-shaping parameters.We show that the precoding-loss of the FTN-THP scheme can be substantial espe-cially when the ISI induced by FTN becomes severe. Motivated by this, we thenturn our focus on the linear precoding options. In particular, we propose a linearpre-equalization (LPE) method to pre-compensate for the FTN-ISI. Due to the factthat FTN is different from classical ISI where the channel lies outside the transmit-ter, linear pre-equalization does not suffer from noise enhancement. It does, however,modify the transmit power spectral density (PSD), and we show that our methodconverts FTN transmission into orthogonal signaling with an equivalent pulse shape.In doing so, the proposed LPE completely eliminates FTN-ISI.The remainder of this chapter is organized as follows. The system model is intro-22LDPC EncoderInterleaverQAM Mapper RRC2𝜏𝑇Optical Front-endDACSSMFCoherent Rx.2x2 MIMO Butterfly PMD Eq.Demapper𝑣′LDPC Decoder𝑎Carrier RecoveryFTN Pre-equalizer𝑟QAMMapperFTNPre-equalizerDACRRC Pulse shapeℎ(𝑡)QAM MapperFTN Pre-equalizerDAC ADCADCWMF+CD Comp.Demapper De-interleaver𝑣′𝑎 𝑟DataInDataOutTransmitter ReceiverWF𝐹(𝑧)WMF+CD Comp.SoftDemapperRx Matched Filterℎ∗(−𝑡)AWGN𝜏𝑇-SamplingTransmitterReceiver𝑎 𝑠𝑟𝑣′RRC2𝜏𝑇InterleaverDe-interleaverFECEncoderFECDecoderData InData OutFigure 2.1: Baseband system model for a pre-equalized FTN transmission wherethe shaded blocks at the transmitter and the receiver represent the proposed FTNpre-equalizer and symbol demappers respectively.duced in Section 2.2. In Section 2.3, we propose two novel demappers for FTN-THPand present the analysis for the precoding loss. The new linear pre-filtering methodfor FTN is proposed in Section 2.4. In Section 2.5, we validate the proposed methodsbased on simulations for a coherent optical transmission setup. Finally, Section 2.6provides concluding remarks.2.2 System Model2.2.1 Precoded FTNWe consider the baseband system model for precoded FTN transmission scheme underan AWGN channel shown in Fig. 2.1. The system model is common for both linearand non-linear pre-equalization methods. As shown in Fig. 2.1, the data bits are firstFEC encoded and then the interleaved and modulated data stream a is precodedwith a discrete-time pre-filter to produce the data symbols r. The precoded symbolsr are pulse-shaped by a T -orthogonal pulse h and then transmitted with an FTNacceleration factor τ < 1. As in [13], the resulting linearly modulated baseband23transmitted signal can be written ass(t) =∑kr[k]h(t− kτT ) . (2.1)For the following, we assume a root-raised-cosine (RRC) pulse-shaping filter h witha roll-off factor β such that∫∞−∞ |h(t)|2dt = 1.At the receiver, the analog received signal, after passing through the matched-filter, is sampled at τT -intervals and then digitally processed by a noise whiteningfilter (WF) to whiten the colored noise due to FTN. Thereafter, the τT sampledsignal v′ is sent to a symbol demapper to produce soft information in the form ofLLRs for the FEC decoder.The overall discrete-time channel impulse response between the precoded symbolsr and the output of the τT -spaced sampling is given byg[n] = (h ∗ f)(nτT ) , (2.2)where f(t) = h∗(−t), ·∗ is complex conjugate, and ∗ denotes the linear convolu-tion. We also introduce G = Z(g), where Z{·} denotes the z-transform. In aNyquist-system (τ = 1), T -orthogonality of the pulse-shape h along with the condi-tion∫∞−∞ |h(t)|2dt = 1 makes G(z) = 1. But for an FTN transmission with τ < 1,G(z) causes ISI across consecutive transmitted symbols.As THP can be seen as a dual to a DFE performed at the receiver [44], we applya spectral factorization to G (consistent with [44, 94]) as detailed in the followingsubsection.242.2.2 Spectral FactorizationTHP requires the implementation of a feed-forward-filter (FFF) F and a feedback-filter (FBF) B to pre-equalize the ISI due to FTN. As the noise samples after theτT -sampler in an FTN system are colored, the purpose of the FFF is then two-fold:to whiten the received noise samples and to shape the end-to-end channel transfer-function into a causal and minimum-phase response [44]. The FBF is then usedas a pre-filter at the transmitter to pre-equalize the overall effective ISI-channel.Computation of FFF and FBF requires the discrete-time spectral factorization [44]G(z) = αQ(z)Q∗(z−∗), (2.3)such that Q(z) is casual, monic and minimum-phase and α > 0 is a scaling factor usedto make Q(z) monic. The necessary and sufficient condition for the realization of theabove spectral factorization (see e.g. [44, 95]) can be written in an FTN transmissionasτT∫ 12τT− 12τT| log (G(ej2pifτT )) |df <∞ . (2.4)Since from (2.2),G(ej2pifτT ) =1τT∞∑k=−∞|Hˆ(f − k/(τT ))|2 , (2.5)where Hˆ is the Fourier-transform of the pulse-shaping filter h. We note thatG(ej2pifτT )in (2.5) is zero in the intervals[− 12τT,−1+β2T]and[1+β2T, 12τT]when the FTN acceler-ation factor τ < 11+β. This causes the condition in (2.4) to fail, which consequentlymakes the spectral factorization (2.3) required for THP unrealizable. Hence, in thefollowing we restrict ourselves to FTN with τ ≥ 11+βfor a given β. Once the fac-torization according to (2.3) is executed, we obtain the FFF and FBF respectively25Modulo2M+-V1(z)-1a[k] x[k]THPf[k]= = +-V1(z)-1v[k] x[k]f[k]a[k]d[k]FTNPre-equalizer≡ +-Modulo2MFBFB(z)-1≡𝑎𝑓𝑟 +-FBFB(z)-1𝑎 𝑟𝑑𝑣≡ +-FBFB(z)-1𝑎𝑓𝑟FTNPre-equalizer≡ +-Modulo2MFBFB(z)-1𝑎𝑓𝑟++Figure 2.2: FTN pre-equalization with THP and the modulo-equivalent linear struc-ture.asF (z) =1αQ∗ (z−∗)and B(z) = Q(z) . (2.6)Using the FFF and FBF computed above, we now proceed to introduce FTN-THPwith an improved demapper in the next section, and a new linear pre-equalizationmethod in Section Non-linear Precoding in FTN SystemsIn this section, we consider non-linear precoding for FTN in the form of THP.2.3.1 THP-precoded FTNSince the effective ISI-channel caused by FTN is a-priori known at the transmitter,the filters from Section 2.2.2 can be computed and applied for THP without anyfeedback from the receiver.Fig. 2.2 depicts the detailed diagram and the associated linear equivalent structureof the block “FTN Pre-equalizer” from Fig. 2.1. The modulo operation in a classicalTHP system as shown in Fig. 2.2 is used to keep the output stable especially for26channels with spectral zeros by bounding it within a well-prescribed range [44]. Theinput symbols a in Fig. 2.2 consist of the modulated symbols and the feedback filter B,as given in (2.6), is a function of the FTN parameter τ and the pulse shape h. As theFTN-ISI is real-valued, without loss of generality, we assume that the symbols a aredrawn from an M -ary 1-D constellation. In the equivalent linear representation, themodulo operation of THP is replaced by an equivalent addition of a unique sequenced to the data symbols a so that precoded symbols r lie in the interval [−M,M).The combination of a and d produces the intermediate signal v, the elements ofwhich are taken from an extended constellation with more than M signal points. Inan ideal noise-free scenario, the signal v′ in Fig. 2.1 is same as v of Fig. 2.2, andthus, to compensate for THP, conventionally a modulo operation is performed onv′ at the receiver. However, for noisy channels and particularly for a relatively lowSNR, this modulo operation is sub-optimal which makes the LLR computation by aconventional soft-demapper unreliable. These inaccurate LLRs are then passed on tothe FEC decoder as shown in Fig. 2.1 and thereby causing a performance degradation,especially in an FEC coded transmission, which is commonly known as the “moduloloss”.To overcome this loss, a modified modulo based demapper for a coded THP systemwas proposed in [52] and its simplified implementation method was also presentedrecently in [53]. However, the residual modulo-loss of these approaches still causes asignificant loss in the bit-error rate (BER). Another near-optimal iterative methodwas shown in [54]. It is based on a quantized-output THP, and its computationalcomplexity is of the order of MAP equalization. In the following, we present two rela-tively simpler soft-demapping algorithms which significantly outperform the demap-per from [52], which we refer to as Peh-Liang-Demapper (PLD), and are competitive27to optimal MAP equalization in terms of BER.2.3.2 Expanded A-priori Demapper (EAD)In order to counter the modulo-loss, we replace the modulo operation of a conventionalTHP demapper with our proposed new demapper, referred to as EAD, which nowforms the module “Soft Demapper” of Fig. 2.1. The proposed EAD is based on the lin-ear equivalent model from Fig. 2.2 and considers the extended constellation of the in-termediate data signal v to compute LLRs. LetA = {aκPAM = ±1,±3, . . . ,±(M − 1)}be the set of M -ary pulse-amplitude modulation (PAM) constellation symbols. Then,the symbols v belong to the extended signal set V = {v[k]} = A + 2MZ. In par-ticular, we note that the probabilities of the signal points v ∈ V are not uniform.Therefore, the EAD computes the LLR value corresponding to the nth bit bn of thekth data symbol a[k] asLLREADk,n = log(Pr (bn = 1|v′[k])Pr (bn = 0|v′[k]))(2.7)= log∑c∈C1,nPr(v′[k]|v[k]=c) Pr(v[k]=c)∑c∈C0,nPr(v′[k]|v[k]=c) Pr(v[k]=c) , (2.8)where Ci,n is the subset of symbols in V corresponding to the nth bit being equal toi ∈ {0, 1} and v′[k] is the kth received sample at the demapper input. The relationbetween v and v′ follows asv′[k] = v[k] + η[k] , (2.9)28where η[k] is a zero-mean AWGN with variance σ2. Note that the colored noisesamples after an FTN-sampler at the receiver are whitened by the FFF, as discussedin Section 2.2.2. Introducing the Gaussian probability density function (PDF) of ηin (2.9), we can simplify (2.8) asLLREADk,n = log∑c∈C1,nPr(v[k]=c) e−|v′[k]−c|22σ2∑c∈C0,nPr(v[k]=c) e−|v′[k]−c|22σ2(2.10)≈ logPr(v[k]= c¯1,n) e− |v′[k]−c¯1,n|22σ2Pr(v[k]= c¯0,n) e− |v′[k]−c¯0,n|22σ2(2.11)= log(α1,nα0,n)+|v′[k]−c¯0,n|2−|v′[k]−c¯1,n|22σ2, (2.12)where (2.11) follows from the nearest neighbor approximation (e.g. [52]), with c¯i,n asthe nearest neighbor to the received sample v′[k] representing the nth bit being equalto i ∈ {0, 1} and αi,n = Pr (v[k] = c¯i,n).The expressions in (2.10) and (2.12) are readily evaluated given the received sam-ples v′[k] and the a-priori probabilities Pr (v[k] = c) for the signal points c ∈ V . Toanalytically compute these probabilities for a given β and τ , we make use of thefollowing proposition.Proposition 2.1. Expanded constellation symbols v[k] ∈ V in Fig. 2.2 have thefollowing probability mass function (PMF):Pr(v[k]=a(κ,i)M,v)=1M[Φ(M+i+aκPAMσf)−Φ(M−i+aκPAMσf)], (2.13)where aκPAM ∈ A, a(κ,i)M,v = aκPAM + 2iM , M+i = (2i+ 1)M , M−i = (2i− 1)M for i ∈ Z,σf is the standard deviation of the signal f and Φ (x) =1√2pi∫ x−∞ e−x22 dx.29Proof. See Appendix A.1.The standard deviation σf in (2.13) can be computed numerically. Simulationresults in Section 2.5 show that EAD can offer substantial gains over PLD especiallywhen the FTN-ISI is less severe. The relation between the LLR calculation by EADand PLD from [52] for severe FTN-ISI is summarized in the following propositionand its corollary.Proposition 2.2. For 2PAM and 4PAM modulations, the LLR expression in (2.12)becomes equivalent to the approximate LLR expression computed by PLD as givenin [52] if the extended constellation symbols of the signal v are assumed to have equalprobabilities.Proof. See Appendix A.2.Corollary 2.2.1. For 2PAM and 4PAM modulations, when the FTN-ISI becomessevere (i.e. τ reduces for a given RRC roll-off β), the LLR expressions computed byEAD and PLD become similar.Proof. For an M -ary PAM constellation, an upper bound on the maximum numberof signal points in V with non-zero probability is given in [44] asVmax = 2⌊M∑P−1k=0 |b[k]|+ 12⌋− 1 , (2.14)where b = Z−1(B) is the THP feedback filter response, P denotes the length of the ISIchannel and the function bxc denotes the largest integer contained in x. Therefore,with large P , V contains more symbols with non-zero probabilities which causes thebell-shaped PMF in (2.13) to flatten and its shape starts resembling closer to that ofa uniform distribution. Then by Proposition 2.2, LLRs computed by EAD becomesimilar to those computed by PLD.30As evaluated above, the gains offered by EAD reduce for decreasing τ . This canbe attributed to the fact that while EAD takes the probabilities of the extendedconstellation symbols into account, it fails to incorporate the auto-correlation ofthe intermediate symbol sequence v into the LLR metric in (2.12). As τ reduces,correlation between successive symbols of v can increase significantly due to severeFTN-ISI. In order to account for this, we present the second demapper design in thefollowing.2.3.3 Sliding-Window-EAD (SW-EAD)The SW-EAD includes L preceding and succeeding observations (corresponding toa sliding window of length 2L + 1) into the computation of LLRs for the currentsymbol. Depending on the severity of the ISI and the observed auto-correlation ofv, a suitable value L is determined. The modified LLR for the nth bit bn of the kthtransmitted symbol a[k] is computed asLLRSW-EADk,n =log(Pr(bn=1|v′[k−L],. . . ,v′[k+L])Pr(bn=0|v′[k−L],. . . ,v′[k+L]))(2.15)= log∑c∈C1,n,v[k−L],...,v[k+L]Pr(#»v′ | #»v c)Pr ( #»v c)∑c∈C0,n,v[k−L],...,v[k+L]Pr(#»v′ | #»v c)Pr ( #»v c) (2.16)= log∑c∈C1,n,v[k−L],...,v[k+L]Pr ( #»v c) e− ‖#»v′− #»v c‖22σ2∑c∈C0,n,v[k−L],...,v[k+L]Pr ( #»v c) e− ‖#»v′− #»v c‖22σ2 , (2.17)31where #»v c = [v[k − L], . . . , v[k] = c, . . . , v[k + L]]T , #»v′ = [v′[k − L], . . . , v′[k + L]]Tand ‖ · ‖ denotes the vector norm operator. The computation of (2.17), using theknown extended constellation symbols v ∈ V and the received samples v′, involvespre-computing and storing the multi-dimensional a-priori probabilities Pr( #»v c). ForL = 0, SW-EAD metric (2.17) reduces to the EAD-computed LLR given in (2.10).Note that, if ∆v = |V| denotes the cardinality of V , then among the ∆2L+1v multi-dimensional sequences, only a small fraction, ρLv number of symbol-vectors can havenon-zero probabilities, depending on the values of M , β and τ . It is sufficient to storeonly these ρLv a-priori probabilities to compute (2.17).The SW-EAD can be used recursively in iterations with an FEC decoder. In thiscase, the extrinsic information provided by the FEC decoder for the coded bits isused to update the a-priori probability Pr( #»v c) in (2.17).2.3.4 Precoding-loss for FTN-THP SystemsIn a THP-precoded Nyquist transmission over an ISI-channel, the precoding operationcauses an increase in the average transmit power which translates into the precodingloss with respect to an equivalent DFE equalization scheme [44, p. 144]. Moreover,an ideal DFE without error propagation can incur an SNR degradation compared tothe matched-filter bound (MFB) depending on the parameter α in (2.3) [44, p. 67-68]. Therefore, the combined SNR loss of a FTN-THP transmission with respect toISI-free orthogonal transmission isSNRFTN-THPLoss = PTHP-DFELoss · SNRDFE-MFBLoss , (2.18)where SNRDFE-MFBLoss = 1/α with α given in (2.3). While the precoding loss PTHP-DFELosshas been well investigated in the literature (e.g. [44]), the situation is slightly differ-32ent for FTN-THP systems, where the transmit power and hence the precoding lossdepend on the ISI channel through the transmit pulse-shape. In order to quantifythe precoding loss, we utilize the results from the following proposition.Proposition 2.3. For an FTN-THP system with the FFF and FBF given in (2.6),the PSD of the transmitted signal is given byΦTHPss (f) = αΦvv(ej2piτfT) Gˆ(f)∑kGˆ(f + kτT) , (2.19)and the average transmitted power isPTHPAvg =ασ2vτT, (2.20)where σ2v and Φvv(ej2piτfT)are the variance and the PSD of the extended constellationsymbols v, respectively, Gˆ(f) = |Hˆ(f)|2, and α > 0 is the constant given in (2.3).Proof. See Appendix A.3.Since (i) a THP and a DFE system with the same FFF and FBF perform iden-tically assuming no modulo loss for THP and no error propagation for DFE and (ii)the transmit power for non-precoded FTN is PAvg =σ2aτT, where σ2a is the variance ofthe M -ary constellation symbol a (e.g. [13]), we have from (2.20) thatPTHP-DFELoss = ασ2vσ2a. (2.21)The overall SNR-loss of the FTN-THP system as compared to the ISI-free transmis-33Modulo2M+-V1(z)-1a[k] x[k]THPf[k]= = +-V1(z)-1v[k] x[k]f[k]a[k]d[k]FTNPre-equalizer≡ +-Modulo2MFBFB(z)-1≡𝑎𝑓𝑟 +-FBFB(z)-1𝑎 𝑟𝑑𝑣≡ +-FBFB(z)-1𝑎𝑓𝑟FTNPre-equalizer≡ +-Modulo2MFBFB(z)-1𝑎𝑓𝑟++Figure 2.3: Linear pre-equalization of FTN ISI.sion follows from (2.18) asSNRFTN-THPLoss =σ2vσ2a. (2.22)2.4 Linear Pre-equalization for FTNFTN-THP not only incurs an SNR loss, but it may also complicate pilot-based chan-nel estimation. Since the THP operation results in an expanded signal constellationv′ after the WF stage shown in Fig. 2.1, the receiver lacks the prior knowledge of theexact representation of pilot symbols introduced into the data stream. To alleviatethis problem, careful attention to the pilot-symbol design is needed [48] or a coarsedetection of the pilot symbol is required before channel estimation [49].These problems warrant the consideration of linear precoding or pre-equalizationmethods. We note that linear precoding is done in PRS transmission, albeit the pur-pose is not pre-equalization but spectral shaping of the transmit signal or the maxi-mization of some performance criteria assuming receiver-side equalization [13, 55–57].Different from this, we propose a linear pre-equalization (LPE) technique to mitigatethe ISI introduced through FTN signaling. More specifically, the pre-equalizationis achieved through a linear pre-filtering method which is derived from the THP34transmitter structure by dropping the modulo operator as shown in Fig. 2.3. Theexclusion of the modulo function renders the overall transmitter of Fig. 2.3 a linearinfinite impulse response (IIR) filtering operation. The minimum-phase property ofthe feedback filter B(z), as discussed in Section 2.2.2, guarantees the stability of theIIR filter.In Nyquist transmission over ISI channels, linear pre-equalization is usually nota preferred choice. In particular, linear pre-equalization to eliminate ISI results inan elevation of the average transmitted power, which creates a similar error-ratedegradation as the noise-enhancement phenomena encountered in a linear zero-forcingequalization [44]. However, as pointed out earlier, in an FTN transmission, the ISIstems from the transmitter pulse-shape and receiver matched filter. In particular,the feedback filter B in Fig. 2.3 is a function of the RRC filter h related through(2.3), (2.5) and (2.6). This leads to the following results for the PSD and the averagetransmit power for FTN-LPE transmission.Proposition 2.4. For an FTN-LPE system in Fig. 2.3 with the FBF given in (2.6),the PSD of the transmitted signal is given byΦLPEss (f) = ασ2aGˆ(f)∑kGˆ(f + kτT) , (2.23)and the average transmit power isP LPEAvg =ασ2aτT, (2.24)where σ2a is the variance of the input constellation symbols a and α > 0 is given in(2.3).Proof. See Appendix A.3.35Corollary 2.4.1. For a Nyquist system, the transmitted PSD becomes ΦNyqss (f) =σ2aTGˆ(f) with an average transmitted power PNyqAvg =σ2aT.Proof. For a Nyquist system, τ = 1, α = 1 from (2.3) and 1T∑kGˆ(f + kT)= 1.Putting these values in (2.23) and (2.24) yields the well-known expressions.Corollary 2.4.2. If τ = 11+β, the PSD of the transmitted signal becomes rectangularwith a bandwidth 1+βT.Proof. The expression∑kGˆ(f + kτT)in (2.23) is the sum of the frequency-shiftedreplicas of Gˆ(f), where the frequency shifts are integral multiples of 1τT. We notethat,Gˆ(f) = 0 ,when |f | > 1 + β2T. (2.25)Therefore, when τ = 11+β, there are no overlaps between the replicas of Gˆ in∑kGˆ(f + kτT).Consequently,∑kGˆ(f + kτT)= Gˆ(f) in the frequency range −1+β2T≤ f ≤ 1+β2T. Thus,from (2.23) we haveΦLPEss (f) = ασ2a, −1+β2T ≤ f ≤ 1+β2T0, otherwise(2.26)To investigate the power-penalty of the FTN-LPE transmission, we use the sameprocedure which was adopted in Section 2.3.4 for the SNR-loss computation of anFTN-THP system. Similar to (2.18), we can write the combined SNR-loss for theLPE asSNRFTN-LPELoss = PLPE-DFELoss · SNRDFE-MFBLoss , (2.27)36where SNRDFE-MFBLoss = 1/α as in (2.18) and following the same reasoning as in Section2.3.4, the precoding loss PLPE-DFELoss can be computed from (2.24) asP LPE-DFELoss = α . (2.28)Hence, the overall SNR-loss of the LPE-THP system as compared to the ISI-freetransmission can be written from (2.27) asSNRFTN-LPELoss = 1 . (2.29)We conclude from (2.29) that FTN-LPE does not suffer from a power penalty dueto channel inversion and achieves the same error-rate performance as an ISI-freetransmission. To do so, linear pre-equalization modifies the transmit PSD accordingto (2.23) that exhibits τT -orthogonality. In fact, a closer inspection of the FBFand FFF filters for LPE reveals that the combination of the LPE pre-filter and theRRC pulse-shape at the transmitter is equivalent to a new τT -orthogonal square-rootNyquist pulse-shaping filter 3. Similarly, at the receiver, the RRC filter, combinedwith the WF, constitutes an equivalent square-root Nyquist matched-filter to thenew transmit pulse-shape. Hence, FTN-LPE with whitened matched-filtering andτT sampling is ISI-free.As an alternative τT -orthogonal signaling scheme, one could directly use a τT -orthogonal RRC filter with roll-off βˆ = τ(1 + β)− 1 for transmit pulse-shaping. Forinstance, an FTN system with T -orthogonal RRC having β = 0.3 and τ = 0.78 resultsin an effective βˆ = 0.014 for the direct τT -orthogonal RRC design. We illustrate inSection 2.5 that due to the stricter roll-off requirement for this new RRC pulse-shape,3Because of this, we remark that our LPE design with FTN signaling can be viewed as a practicalapproach for Brickwall filter implementation.37LDPC EncoderInterleaverQAM Mapper RRC2𝜏𝑇Optical Front-endDACSSMFCoherent Rx.2x2 MIMO Butterfly PMD Eq.Demapper𝑣′LDPC Decoder𝑎Carrier RecoveryFTN Pre-equalizer𝑟QAMMapperFTNPre-equalizerDACRRC Pulse shapeℎ(𝑡)QAM MapperFTN Pre-equalizerDAC ADCADCWMF+CD Comp.Demapper De-interleaver𝑣′𝑎 𝑟DataInDataOutTransmitter ReceiverWF𝐹(𝑧)WMF+CD Comp.SoftDemapperRx Matched Filterℎ∗(−𝑡)AWGN𝜏𝑇-SamplingTransmitterReceiver𝑎 𝑠𝑟𝑣′RRC2𝜏𝑇InterleaverDe-interleaverFECEncoderFECDecoderData InData OutFigure 2.4: Block diagram of the precoded FTN dual-polarized coherent optical sim-ulation setup where the shaded blocks at the transmitter and the r ceiver representthe prop sed THP/LPE pre-equalizer and symbol demappers respectively.the implementation of this filter needs more taps to maintain a given threshold forthe out-of-band power leakage compared to that in the proposed LPE-FTN system.Finally, we remark that the proposed pre-equalization shares similarities with thematrix-based precoded FTN transmission presented in [58, 59]. Similar to FTN-LPE, this method divides the equalization task into pre-filtering at the transmitterand post-filtering at the receiver. However, different from our implementation itis based on block-processing of the transmitted and received symbols and thus itsuffers from inter-block ISI [58] or needs guard intervals and thereby reduces spectralefficiency, in addition to introducing a block delay. An alternate way to minimizethis additional overhead is to increase the block size which requires more elaboratematrix computations.2.5 Numerical Results and DiscussionIn this section, we illustrate and validate the proposed pre-equalization techniquesby way of numerical results, including error-rate simulations for FTN transmission.382.5.1 Simulation SetupFor the simulations, we consider a coherent optical single-carrier (COSC) transmissionsystem. Optical communication systems are a prime candidate for the introductionof FTN as the use of higher-order modulation is challenging in such systems [11,33]. The block diagram of a COSC system with polarization division multiplexingis shown in Fig. 2.4. The precoding algorithms presented hitherto, considering theAWGN system model in Fig. 2.1, are directly applicable to linear optical channelsas impairments such as CD and PMD can be compensated through a proper two-dimensional equalizer [60].In Fig. 2.4, the transmitter and receiver blocks for the discrete-time basebandmodules are same as those in Fig. 2.1 except that the data processing for each ofthe two polarizations is performed separately. For our simulations, we use a low-density parity-check (LDPC) code of rate 0.8, a random bit-interleaver, QPSK and16-ary quadrature amplitude modulation (16QAM) formats, and a fixed baud rateof 32 Gbaud for all values of τ . The RRC pulse-shaping filter is implemented with2-times oversampling having 73 time-domain taps with β = 0.3, and the THP/LPEprecoders are designed using 10-taps for the feedback filter. The baseband analogdata after the digital-to-analog converter (DAC) is processed by the opto-electronicfront-end and transmitted as an optical signal through a 1000 km standard single-mode fiber (SSMF) with CD and mean PMD parameter values of −22.63 ps2/kmand 0.8 ps/√km, respectively, and then is received by the optical coherent receiver.The whitened matched filter (WMF) is combined with the time-invariant frequencydomain CD compensator using overlap-and-add method. For PMD compensation, weused a 13-tap 2×2 butterfly-type fractionally-spaced adaptive LMS equalizer [60, 96].Following carrier recovery, the QAM-demapper computes and passes on LLR values398 9 10 11 12 13 14 1510−610−510−410−310−210−1OSNR [dB]BER  Nyquist (τ = 1)τ = 0.85τ = 0.8CTHP0.9 dBMAPEADPLD1.65 dB4.2 dBFigure 2.5: BER vs. OSNR for FTN-THP with different demappers, illustrating theperformance of the proposed EAD. QPSK, β = 0.3, τ = 0.85 and the LDPC decoder.2.5.2 Performance of FTN-THP with Proposed DemappersWe first compare the performance of FTN-THP using the proposed EAD scheme withrespect to the conventional THP (CTHP) demapper which employs a modulo oper-ation at the receiver and the modulo-based PLD proposed in [52, 53]. Fig. 2.5 showsthe coded BER performance as a function of the optical SNR (OSNR) for differentFTN parameters τ with QPSK modulation. We also include the BER performancewith MAP equalization, which considers 6-taps of the ISI channel and performs 1040−10 −5 0 5 10− (in Symbols)Normalized AutocorrelationFigure 2.6: Auto-correlation of the expanded constellation symbols v for β = 0.3 andτ = 0.8.iterations between the MAP equalizer and LDPC decoder, as a reference. As canbe seen from the figure, when ISI is relatively low with τ = 0.85, EAD achieves aperformance close to that for the computationally demanding MAP equalization andalso to the orthogonal Nyquist-signaling (τ = 1). For this case, EAD outperformsCTHP and PLD by 4.2 dB and 1.65 dB, respectively. When FTN-ISI becomes higherwith τ = 0.8, EAD shows a performance gain of 0.9 dB over PLD which is 0.75 dBless compared to the gain with τ = 0.85. The reduction in gap between EAD andPLD with stronger ISI was predicted in Proposition 2.2.The loss of performance gain by using EAD with τ = 0.8 can partially be at-tributed to the correlation between successive symbols of the extended-constellation418.5 9 9.5 10 10.5 1110−610−510−410−310−210−1OSNR [dB]BER  EAD (L = 0)SW−EAD (L = 1)SW−EAD (L = 1), 10 it.SW−EAD (L = 3)SW−EAD (L = 3), 10 it.MAPNyquist (τ = 1)0.8 dB1.3 dBFigure 2.7: BER vs. OSNR for FTN-THP with different demappers, illustratingthe performance gains with the proposed SW-EAD over EAD. QPSK, β = 0.3 andτ = 0.8.sequence v of Fig. 2.2. The auto-correlation sequence of v is plotted in Fig. 2.6. Thiscorrelation is not taken into account while computing the EAD-LLR metric in (2.12).Fig. 2.7 shows the additional performance gains obtained by SW-EAD over EAD.The different curves represent distinct values of L corresponding to the SW-EADwindow-length (2L + 1) with and without iterations between the demapper and theLDPC decoder. We observe that SW-EAD get improvements of the order of 0.8 dBover EAD which makes it competitive to MAP equalization. With higher values ofL, further improvements for SW-EAD are not expected as only up to 3−4 significanttaps are observed in Fig. 2.6.420.7 0.75 0.8 0.85 0.9 0.95 1−0.500.511.522.533.5τSNR Loss, in dB  AnalyticalMeasuredβ = 0.4β = 0.3β = 0.2Figure 2.8: SNR vs. τ in a QPSK FTN-THP system for varying β.The primary reason for the gap in the BER plots between the SW-EAD and MAPequalization can be ascribed to the SNR loss associated with the THP precoding,which was investigated in Section 2.3.4. In Fig. 2.8, we plot the overall SNR loss(2.22) of a THP-FTN system compared to an ISI-free transmission as a function ofthe FTN parameter τ and for different values of β. For each β, we have consideredonly those values of τ such that τ ≥ 11+β, as explained in Section 2.2.2. We observethat for each β, there exists an optimal τ up to which no SNR loss is experienced.For example, with β = 0.3, this optimal τ is 0.85, which corroborates the BER resultsin Fig. 2.5 where FTN-THP transmission yields BER performance close to that ofNyquist-signaling.43Table 2.1: Computational complexities of the THP-demappers for each bit and iter-ation.Operation EAD/SW-EAD PLD MAP-BCJRAddition/Subtraction ρLv + ∆v + 4ρLv∆v − 2 2M + 2 4NMAP − 2Multiplication ρLv + ∆v M + 2 6NMAPDivision ∆v + 1 M + 3 2NMAP + 1Non-linear (exp. and log.) ρLv + 1 M + 3 4NMAP + 1Table 2.2: Complexity comparison of the demappers per bit per iteration: QPSK,β = 0.3,τ = 0.84 and τ = 0.8.τ Operations EAD PLDSW-EADMAP (6-ISI taps)L = 1 L = 2 L = 30.84ADD. 6 6 52 290 1666 254MUL. 8 4 14 36 132 384DIV. 5 5 5 5 5 129Non-Lin. 5 5 11 33 129 257Total 24 20 82 364 1932 10240.8ADD. 6 6 122 1082 6476 254MUL. 8 4 28 124 502 384DIV. 5 5 5 5 5 129Non-Lin. 5 5 25 121 499 257Total 24 20 180 1332 7482 10242.5.3 Computational Complexity AnalysisIn this section we present an analysis of the computational cost for the proposedTHP-demappers and compare them with the MAP equalization complexity [97]. Toreduce the implementation cost of the LLR metric computations for an M2-ary QAMconstellation, we have taken advantage of the fact that the FTN-ISI is real-valuedand hence, the in-phase (I) and quadrature (Q) components of the received basebandsignals can be individually processed by the demappers and the MAP equalizer.Let P be the number of FTN-ISI taps considered for the MAP-equalization, thenthe complexity of the BCJR algorithm [97] per bit per iteration is O(NMAP), whereNMAP = MP . With the quantities L, ∆v and ρLv as defined in Section 2.3.3, the details44of the mathematical operations, required for implementing the EAD and SW-EADLLR metrics for each bit in each iteration according to (2.10) and (2.17) respectively,are furnished in Table 2.1. To illustrate this analysis with further clarity and easeof comparison, two specific examples are provided in Table 2.2 with τ = 0.84 andτ = 0.8 for β = 0.3 and QPSK modulation.The numbers in Table 2.2 reveal that the implementation complexities of the LLRmetrics computed by EAD in (2.10) and PLD are similar for the FTN transmissionsstudied in this chapter, even though EAD demonstrated substantial performancegains over PLD as shown in Section 2.5.2. Moreover, Table 2.2 shows that whileEAD is significantly more computationally efficient than the MAP equalization, thecomplexity of the SW-EAD rises with increasing window size, especially for low valuesof τ . We recall from Section 2.5.2 that the SW-EAD performance is always limitedeven when the window parameter L increases infinitely, as illustrated in Fig. 2.7. Thisis because SW-EAD can successfully remove the modulo-loss but fails to improve thepower-penalty (SNR-loss) associated with an FTN-THP transmission. Therefore, fora given complexity requirement on the receiver side processing, RRC roll-off β andFTN parameter τ , the window parameter L and the number of iterations betweenthe SW-EAD and the LDPC decoder should be wisely chosen as a desired trade-offbetween performance and complexity.2.5.4 Performance of Proposed FTN-LPEAs described in Section 2.3.4 and shown in Fig. 2.8, an SNR degradation is inherent toTHP precoding for some values of τ and β. Our proposed LPE scheme can overcomethis problem. Fig. 2.9 shows the FTN-LPE BER results for QPSK and 16QAM.We observe that LPE precoding produces an optimal performance, i.e., the BER is458 10 12 14 16 18 20 2210−610−510−410−310−210−1OSNR, dBBER  NyquistLPEMAPEADPLDCTHP16−QAMτ = 0.85QPSKτ = 0.8Figure 2.9: BER vs. OSNR for FTN with LPE precoding. QPSK with τ = 0.8 and16QAM with τ = 0.85, β = 0.3.identical to that of orthogonal signaling. The figure also includes the BER curvesfrom Fig. 2.5 with τ = 0.8 to show the gains offered by LPE over THP. Similarobservations hold true with higher order modulation, such as 16QAM.The optimal BER performance of LPE precoded FTN systems comes at theexpense of transmitted spectral shape modification, as investigated in Section 2.4.Fig. 2.10 plots the normalized analytical transmit PSDs of the LPE precoded FTNsystem, which was derived in (2.23). We also include the normalized PSDs for (non-precoded) Nyquist signaling using the underlying T -orthogonal RRC with β = 0.3and a τT -orthogonal RRC with βˆ = τ(1 + β)− 1 = 0.105 for pulse shaping, respec-tively. For this comparison, all three systems use the same bandwidth, which implies460 0.2 0.4 0.600. PSD (Theoretical)  T-Orthogonal RCLPEAlternate RC (βˆ = 0.105)Figure 2.10: Normalized PSD of LPE-FTN vs. normalized frequency fT for β = 0.3,τ = 0.85. Also included are the PSDs for Nyquist signaling with the T -orthogonalRRC with β = 0.3 and the τT -orthogonal RRC with βˆ = 0.105.that the LPE-FTN system and the Nyquist-system with the RRC with βˆ = 0.105operate at a higher baud rate. We observe that with LPE precoding, the overall PSDbehaves as a τT -orthogonal pulse-shape. That is, the PSD has a odd-symmetry aboutthe normalized frequency fT = 12τ= 0.59 similar to the alternate τT -orthogonal RC.The advantage of the proposed LPE scheme over a direct τT -orthogonal RRCpulse-shaping is illustrated in Fig. 2.11 in terms of the out-of-band emission perfor-mance. Here, the transmit pulse-shaping filters for both the LPE precoded FTNsystem with β = 0.3, τ = 0.78 and the direct τT -orthogonal Nyquist transmission(effective βˆ = 0.014) were implemented by using 73 time domain taps. For both sys-47−1.3 −0.65 0 0.65 1.3−90−80−70−60−50−40−30−20−10010fTNormalized PSD (Unit measure is [dB])  Alternate RRC (βˆ = 0.014)LPE20dBFigure 2.11: Normalized PSD of LPE-FTN with β = 0.3, τ = 0.78 and Nyquistsignaling with a τT -orthogonal RRC having βˆ = 0.014 vs. normalized frequency fTusing truncated RRC pulses to illustrate spectral leakage.tems, transmit PSDs are computed using the twice-oversampled discrete-time samplesbefore the DAC in Fig. 2.4. The normalized PSDs are plotted in Fig. 2.11, as a func-tion of the normalized frequency fT . We observe that LPE transmission results in asignificantly lower (∼ 20 dB) spectral leakage in the side-bands. This improved out-of-band emission performance is advantageous for transmission schemes with strictspectral-emission mask requirements to achieve low interference between adjacentchannels.Precoding may cause a possible increase in the PAPR. We demonstrate the PAPRbehaviour for the precoded FTN techniques by plotting the empirical complementary48−10 −5 0 5 1010−410−310−210−1100Inst. Power, dBProb (Inst. Power > Abscissa)  Nyquist w/o precodingFTN w/o precodingFTN-THPFTN-LPEAlternate RRC (βˆ = 0.014)(a) QPSK−6 −4 −2 0 2 4 6 8 1010−410−310−210−1100Inst. Power, dBProb (Inst. Power > Abscissa)  Nyquist w/o precodingFTN w/o precodingFTN-THPFTN-LPEAlternate RRC (βˆ = 0.014)(b) 16QAMFigure 2.12: Empirical CCDF of the instantaneous power with average transmitpower = 0 dB, β = 0.3, τ = 0.78.cumulative distribution function (CCDF) of the instantaneous power in Fig. 2.12 forQPSK and 16QAM constellations. The modulation parameters are β = 0.3 andτ = 0.78, and the different curves correspond to Nyquist signaling with T -orthogonaland τT -orthogonal pulse-shapes (βˆ = 0.014), unprecoded FTN transmission andFTN employing THP and LPE precoding. All transmission schemes are normalizedto the same average transmitted power of 0 dB. As can be seen from Fig. 2.12a, thePAPR of the FTN-THP system with QPSK modulation is relatively higher than thatfor the LPE precoded FTN system, whereas for 16QAM they perform similarly aspresented in Fig. 2.12b. Furthermore, FTN-LPE transmission yields almost a similarPAPR performance as that of the alternate τT -orthogonal signaling scheme for bothQPSK and 16QAM modulation formats.Finally, we remark that the suitability of the two pre-equalization methods, pro-posed in this chapter, depends on the specific application of FTN. In the currentwork, where FTN-LPE has been shown to outperform the FTN-THP scheme, we49have restricted the application of FTN to two different channel models, (a) an AWGNchannel for the simplicity of the theoretical analysis, and (b) an optical channel as apractical application example to present simulation results. However, the efficiency ofthe proposed FTN-THP can be more pronounced if we consider FTN transmissionsunder different channel models. As the functionality of the proposed THP-demappersdepends only on the a-priori probabilities of the symbols v and not on the actual chan-nel parameters, they can be directly applied under these circumstances, e.g. (1) inFTN transmissions over multi-path ISI channels, THP would be a suitable choiceto pre-equalize the combined ISI due to FTN and the multi-path channel, becauseLPE may exhibit significant power-loss due to channel inversion, as the multi-pathchannel lies outside the transmitter; (2) in the previous example of FTN signalingover multi-path channels, a combination of LPE and THP can also be used at thetransmitter, where LPE can be employed to pre-mitigate the FTN-ISI, whereas, THPcan be applied along with the proposed demappers to pre-equalize the ISI compo-nent, arising only from the multi-path channel; (3) in a multi-user, multi-carrier FTNtransmission scheme, where frequency-packed sub-channels are allocated to differentusers, LPE is not good choice for FTN pre-equalization as it requires joint receiverprocessing in the form of feed-forward filtering, which is not generally a viable optiondue to the geographical separation of the users. However, THP can be employed insuch a scenario with both the feedback and feed-forward filters implemented at thetransmitter. While we have not explored the above mentioned FTN applications indetail in this chapter, they can be considered for possible future works as suitableapplication examples for the proposed FTN pre-equalization methods.502.6 ConclusionsFTN transmission is a non-orthogonal signaling scheme to improve spectral efficiencyat the expense of introducing ISI. As an alternative to computationally demandingequalization at the receiver, in this chapter, we have analyzed pre-equalization tech-niques at the transmitter to mitigate the FTN induced ISI. First, we have consid-ered THP and proposed two new symbol demappers to improve the reliability ofthe computed LLRs by reducing the modulo-loss. Numerical results for a coherentoptical transmission system show that the proposed demappers outperform exist-ing THP demappers by significant margins. Secondly, we have proposed a linearpre-equalization technique which converts the FTN transmission into an orthogonalsignaling at a higher baud rate. LPE precoded FTN systems can thus yield opti-mal ISI-free BER performance. Moreover, the numerical results also suggest thatLPE can cause substantially lower out-of-band emission compared to a direct τT -orthogonal RRC pulse-shaping without significant PAPR penalty. In conclusion, wehave demonstrated that the proposed FTN pre-equalization techniques are effectivemeans to achieve higher spectral efficiency promised by FTN.51Chapter 3Faster-than-Nyquist Transmissionfor Single-Carrier MicrowaveCommunication Systems3.1 IntroductionPoint-to-point microwave radio systems are widely used in cellular backhaul networksdue to their fast and cost-effective deployment. In this chapter, FTN transmissionis employed in a single carrier microwave radio link to increase the SE. To furtherthe SE improvement of such systems, FTN is combined with antenna polarizationmultiplexing and HoM schemes. While adopting HoM for such microwave systemsis a very well-known technique, DP transmission has also attracted a considerableattention in the past few years [22–32]. However, FTN induces ISI, a DP transmissionincurs XPI, and adopting HoM makes the communication system sensitive to PN thatarises due to imperfections in the transmitter and receiver LOs. Different from theDP transmission in OFC systems, for which the PMD manifests itself as a phase-only impairment, and thereby, can be perfectly equalized by linear filters [60], XPIappears as a cross-polarization ISI channel in MWC links. The application of FTNsignaling and HoM formats further complicates system design. Therefore, in order to52fully afford the SE benefits a DP-FTN HoM transmission offers, efficient mitigationtechniques are required to counter these detrimental effects. The effectiveness of theequalization and PN compensation schemes is particularly crucial when FEC codesare employed, which operate in the low to moderate SNR regime.In this chapter, we present for the first time a synchronous4 DP-FTN HoM sys-tem with the objective to increase the data rate of the existing microwave links.Therefore, we assume that timing and frequency synchronization of both polariza-tion data streams are performed prior to the receiver signal processing. To provide asolution for spectrally efficient microwave transmissions with practical impairments,the present work addresses a number of challenges in the form of XPIC in a DPtransmission without the explicit knowledge of the interference channel, FTN andmulti-path ISI equalization, and carrier phase tracking. For this purpose, we proposea simple non-iterative approach, as opposed to computationally demanding iterativeequalization schemes. This makes our solution scalable to very high modulation or-ders, and adaptive to channel variations. The proposed adaptive approach also worksefficiently even in the absence of XPI, corresponding to an SP transmission scenario.The primary challenge to devise a practical adaptive interference mitigation schemelies in designing the pilot symbols required for the training-based equalization due tocorruption of the clean constellation symbols by FTN-induced ISI. To counter this,two possible solutions can be adopted, i.e. either (a) combine the FTN-ISI with themulti-path ISI/XPI, or (b) pre-compensate for the FTN-ISI so that clean pilots canbe used to equalize the residual multi-path ISI/XPI. For the approach (a) above, theoverall interference cancellation problem can be formulated as a DP-Nyquist trans-mission, together with XPIC and PN mitigation for the two polarization branches.4Difference between a synchronous and an asynchronous DP transmission is explained in Sec-tion 1.3.2 of Chapter 1.53As our first contribution, we extend the DFE-based receiver structures from [75, 76]to the DP system of interest. To this end, we derive an adaptive estimator for theaggregate PN, stemming from the transmitter and receiver LOs. However, in thepresence of a significant cross-talk between the two polarizations, PN generated atthe transmitter LO of one orthogonal polarization can significantly influence the de-modulation performance of the other. Motivated by this, as our second contribution,we propose an adaptive technique to track the transmitter and receiver phase noiseprocesses separately for both orthogonal polarizations. The performance gains of-fered by the second method over the first, however, come at a price of slightly morecomputations and storage requirements. Further, exploiting the fact that the ISIinduced by FTN is known at the transmitter, approach (b) mentioned above can alsobe applied to facilitate effective elimination of the residual FTN-ISI. Therefore, asour third contribution, we extend the LPE strategy from Chapter 2 to the DP-FTNsystem. Different from the LPE method in Chapter 2, the linear precoding in thischapter is used in association with the adaptive DFE coupled with PN tracking, toform a combined equalization and PN mitigation structure. Numerical results inSection 3.5.3 of this chapter advocates promising performance gains of this combinedstructure over a DFE-only equalization approach.The remainder of the chapter is organized as follows. The system model is in-troduced in Section 3.2. In Section 3.3, we propose two new adaptive DFE-basedequalization schemes to jointly mitigate ISI, XPI and PN. The pre-equalized FTNtransmission along with XPIC and PN cancellation is presented for a DP-FTN sys-tem in Section 3.4. Section 3.5 demonstrates the benefits of our proposals throughsimulations. Finally, Section 3.6 provides concluding remarks.54September 29, 20171Analog Front-end,Up-conv.H-AntennaV-AntennaInterleaverFECEncoderQAMMapperDACRRC Pulse shape𝑝(𝑡)𝑎1InterleaverFECEncoderQAMMapperDACRRC Pulse shape𝑝(𝑡)𝑎2Bits InBits InAnalog Front-end,Up-conv.(a) TransmitterOctober 25, 20171H-AntennaV-Antenna2-D Adaptive DFE(ISI & XPI)+Phase noise trackingRx Matched Filter𝑝∗(−𝑡)𝜏𝑇-SamplingRx Matched Filter𝑝∗(−𝑡)𝜏𝑇-SamplingQAMDemapperDe-interleaver+FEC DecoderQAMDemapperDe-interleaver+FEC DecoderBits OutBits OutRx DSP𝑢1𝑢2Analog Front-end,Down-conv.Analog Front-end,Down-conv.(b) ReceiverFigure 3.1: System model for a DP-FTN transmission.3.2 System ModelWe consider the transmitter and receiver of a DP-FTN microwave system shown inFig. 3.1. As depicted in the transmitter of Fig. 3.1a, the input data bits for the Hand V polarizations are first FEC encoded and interleaved, followed by modulation.The modulated data streams a1 and a2 are then pulse-shaped by T -orthogonal pulsesp, converted into analog signals, up-converted to a microwave carrier frequency andthen transmitted with an FTN acceleration factor τ < 1 on H and V-polarizations,respectively. The resulting transmitted analog signals for the H and V streams canbe written ass1(t) = Re{ej(2pifct+ϑt1 (t))∑ka1[k]p(t− kτT )}, (3.1)s2(t) = Re{ej(2pifct+ϑt2 (t))∑ka2[k]p(t− kτT )}, (3.2)55September 29, 20171𝑎1 ×ej𝜃𝑡1×ej𝜃𝑟1 𝑛1𝑢1𝑎2 ×ej𝜃𝑡2×ej𝜃𝑟2 𝑛2𝑢2ℎ11ℎ22RxDSP2-DISI ChannelFigure 3.2: Equivalent discrete-time baseband system model for a DP-FTN trans-mission.where fc is the carrier frequency, ϑt1 and ϑt2 are the phase noise impairments, associ-ated with the H and V-transmitter LOs, respectively. For the application of interest,we assume an RRC pulse-shaping filter p with a roll-off factor β.The transmitted signals on both polarizations propagate through a wireless chan-nel to reach the DP-FTN receiver shown in Fig. 3.1b. At the receiver, the matched-filtered and sampled signals u1 and u2 on H and V polarizations, respectively, areprocessed by a receiver discrete signal processing (Rx-DSP) unit, comprising of anadaptive 2-D equalizer and PN tracker, as detailed in Section 3.3. Thereafter, the re-covered H and V polarization signals are demodulated and FEC-decoded to producethe output bits.The equivalent discrete-time baseband model for the DP-FTN system is demon-strated in Fig. 3.2, where the received samples ui[k] at a time instant k, with i = 1, 2for H and V polarization, respectively, can be represented asui[k] = ejθri [k]2∑j=1∑laj[k−l]ejθtj [k−l]hij[l] + ni[k] . (3.3)56In (3.3), {hij} for i, j∈{1, 2} denote the effective co-polarization and cross-polarizationchannel taps, representing the combined effects of the multipath ISI, FTN-ISI andXPI, ni is a zero-mean additive colored (due to FTN-sampling) Gaussian noise samplewith variance σ2ni , and θti and θri represent the sampled transmitter and receiver PNprocesses, respectively. For the application of interest, the PN processes are assumedto be slowly time-varying [30, 32, 75] and can be modeled by Wiener processes [98]asθti [k] = θti [k − 1] + wti [k] , (3.4)θri [k] = θri [k − 1] + wri [k] , (3.5)where wti and wri are the samples of independent zero-mean Gaussian random vari-ables with variances σ2wti and σ2wri, respectively.The equivalent baseband system shown in Fig. 3.2 models the transmitter andreceiver PN processes separately, similar to [22–24]. For an SP transmission in anAWGN channel, we note that the transmitter and receiver PN processes can becombined to model an equivalent sum PN process [70, 71, 99, 100]. Alternatively,when there is a multi-path channel between the transmitter and the receiver, eachreceived symbol includes the contributions from multiple transmitter PN samplesdue to ISI [77]. We note that for an SP transmission, combining the transmitter andreceiver PN distortions can still be considered to be a good approximation of the truesystem model for relatively slow time-variation of the PN processes with respect tothe ISI duration. However, for DP systems, it is important to characterize all fourtransmitter and receiver PN processes separately [22–24] to model the impact of thecross-polarization transmitter PNs. The numerical results presented in Section 3.5.2of this chapter, corresponding to two different PN tracking schemes, stand to further57justify such modeling.With the system model (3.3) at hand, we now proceed to present two new adaptiveequalization and joint PN mitigation techniques in the next section, followed by theprecoded DP-FTN transmission strategy in Section Adaptive DFE with PN CompensationIn this section, we present an adaptive DFE approach to jointly optimize the PN es-timates and equalizer tap co-efficients to mitigate the effects of the two-dimensionalinterference channel and phase noise processes as illustrated in Fig. 3.2. We ex-ploit digital data sharing between the two orthogonal polarizations as in [101], whichenables us to use the past symbol-decisions from both polarization branches in afeedback loop. With the assumption of a slow time-variation of the transmitter andreceiver PN processes [30, 32, 75], the exact knowledge of the PN statistics are notrequired for the proposed algorithms.Moreover, the adaptive equalization approach eliminates the need for an explicitchannel estimation at the receivers, and the proposed methods do not require it-erations5 between the equalizer and the FEC decoder. The main reasons to notconsider iterative decoding and demodulation/equalization in the present work are:(a) the complexity associated with iterative solutions in terms of computations andbuffer space, (b) the simpler scalability with very high modulation orders and adap-tivity with respect to channel variations of non-iterative and thus non-block-basedsolutions [22, 24, 25, 75, 99, 102].5This is not to stipulate that iterative solutions may not have merit for dealing with the consideredproblem, but we argue that it is meaningful to start with computationally simpler and, as our resultsin Section 3.5 show, effective non-iterative solutions. What further gains could be achieved withiterative methods, and under what computational costs and other requirements e.g. with regard tochannel variability, can be subject to future work.58April 3, 2018Rx DSP2-D FFFF+-×e−jෝ𝜑12-D FBFB+-×e−jෝ𝜑2𝑢1𝑢2ො𝑎1ො𝑎2𝑦1𝑦2𝑧1𝑧2To demap. + decod.To demap. + decod.ΣΣFigure 3.3: Detailed Rx-DSP block diagram for the adaptive XPIC and DFE-FTNequalization with CPNT.3.3.1 Combined Phase Noise Tracking (CPNT)The concept of joint equalization and carrier phase recovery presented in [75, 102]for an SP system is extended to the DP-FTN transmission in the following and willbe referred to as the DFE-CPNT method henceforth.Fig. 3.3 shows the Rx-DSP module from Fig. 3.1b in more detail. The receivedH and V polarization sequences u1 and u2 are first de-rotated by the respective PNestimates ϕˆ1 and ϕˆ2, and then fed into an adaptive 2-D DFE. The joint estimationmethod for the PN processes and the DFE filters are detailed later in this section.Each entry of the FFF F and the FBF B has Nf and Nb taps, respectively. The DFEoutput sequences y1 and y2 are provided as inputs to the soft demappers and FEC59decoders. The symbols yi[k], i∈{1, 2}, at the kth symbol interval can be written asyi[k] =2∑j=1(Nf−1∑ν=0fij[ν, k]uj[k − ν]e−jϕˆj [k−ν]−Nb∑µ=1bij[µ, k]aˆj[k − k0 − µ]), (3.6)where fij[ν, k] and bij[µ, k] denote the νth and µth tap at the kth symbol intervalcorresponding to the ith-row and jth-column entries of F and B, respectively, aˆ1 andaˆ2 are the previous symbol-decisions for the H and V-polarization, respectively, andk0 denotes the DFE decision delay [44, 72].For jointly updating the DFE tap-weights and the PN estimates, we use theadaptive least-mean-square (LMS) method [75, 103]. Assuming a slow variation ofthe PN processes and hence, the PN estimates ϕˆi to be practically constant over theduration of Nf symbols corresponding to the FFF length [75], the update algorithmsfor the 2-D XPIC and PN estimates are dictated by the following lemma.Lemma 3.1. The LMS update equations, computed by the stochastic gradient descentalgorithm [103], for the 2-D equalizer tap weights and CPNT estimates are given asf1[k+1] =f1[k]− αP [k]ug[k]E∗1 [k] , (3.7)f2[k+1] =f2[k]− αP [k]ug[k]E∗2 [k] , (3.8)b1[k+1] = b1[k] + δaˆg[k]E∗1 [k] , (3.9)b2[k+1] = b2[k] + δaˆg[k]E∗2 [k] , (3.10)ϕˆ1[k+1] = ϕˆ1[k]− γΥ1[k] , (3.11)ϕˆ2[k+1] = ϕˆ2[k]− γΥ2[k] , (3.12)60where for i ∈ {1, 2}, Ei[k] = yi[k] − aˆi[k − k0] are the error signals, α > 0, δ > 0,γ > 0 are the LMS step-size parameters andfi[k]=[{f ∗i1[m, k]}Nf−1m=0,{f ∗i2[n, k]}Nf−1n=0]T, (3.13)bi[k]=[{b∗i1[m, k]}Nbm=1,{b∗i2[n, k]}Nbn=1]T, (3.14)ug[k]=[{u1[k −m]}Nf−1m=0,{u2[k − n]}Nf−1n=0]T, (3.15)aˆg[k]=[{ˆa1[k−k0−m]}Nbm=1,{ˆa2[k−k0−n]}Nbn=1]T, (3.16)P [k]=diag(e−jϕˆ1[k],..., e−jϕˆ1[k]︸ ︷︷ ︸Nf, e−jϕˆ2[k],..., e−jϕˆ2[k]︸ ︷︷ ︸Nf), (3.17)Υi[k]=cos (ϕˆi[k]) Im(ψi[k])−sin (ϕˆi[k]) Re(ψi[k]) , (3.18)ψi[k] = fH1i [k]ui[k]E∗1 [k] + fH2i [k]ui[k]E∗2 [k] (3.19)ui[k]=[ui[k], . . . , ui[k−Nf +1]]T, (3.20)fij[k]=[f ∗ij[0, k], . . . , f∗ij[Nf−1, k]T, (3.21)where (·)∗, Re(·), Im(·) represent, respectively, the complex conjugate, real and imagi-nary part of a complex scalar, [·]H and [·]T denote the matrix hermitian and transpose,respectively, diag(·) is the diagonal matrix formed with the elements of a vector andthe expression{x[j]}N2j=N1denotes the row-vector [x[N1], . . . , x[N2]].Proof. See Appendix B.1.1.Fig. 3.4 shows the schematics of estimating the DFE tap-weights and PN processesfor the DFE-CPNT method, by using the symbols ui, aˆi and Ei, i∈{1, 2}, accordingto (3.7)-(3.12) of Lemma 3.1. As can be seen from the figure and the above lemma,the estimation of the DFE-FFF and the PN processes are coupled together, similarto the joint estimation approach adopted in [75, 102]. Additionally, the DFE-CPNT61October 24, 2017𝑢1ො𝜑1, ො𝜑2 𝒇1, 𝒇2 𝒃1, 𝒃2.... . .𝑢2 ℰ1 ℰ2 ො𝑎1 ො𝑎2DFE + PN estimatesFigure 3.4: Joint estimation of the filter tap-weights and PN processes for the DFE-CPNT method.scheme uses the symbol decisions to adapt the equalizer filter coefficients and PN esti-mates. However, insertion of known pilot symbols at regular intervals [75, 99] for bothorthogonal polarization transmissions is required for LMS convergence, particularlywhen FEC codes are employed which facilitate lower operating SNRs. Hence, theLMS adaptation for the DFE-CPNT method operates in training mode when knownpilot symbols are transmitted and switches to a decision-directed mode otherwise.The pilot-symbols density is chosen to meet a desired trade-off between performanceand transmission overhead.Due to the cross-talk between the two orthogonal polarizations in a DP systemshown in Fig. 3.2, the CPNT phase estimate ϕˆ1 in (3.11) for the H-polarizationbranch attempts to track the combined PN processes originating in the LOs of theH-polarization transmitter-receiver pair and the V-polarization transmitter. Conse-quently, the accuracy of the PN estimates depends on the level of XPI and hence, onthe cancellation performance of the DFE-based XPIC illustrated in Fig. 3.3. There-62April 3, 2018Rx DSP2-D FFFF+-×e−j෡𝜃𝑟12-D FBFB+-×e−j෡𝜃𝑟2×e−j෡𝜃𝑡1×e−j෡𝜃𝑡2𝑢1𝑢2෤𝑦1෤𝑦2ො𝑎1ො𝑎2𝑧1𝑧2To demap. + decod.To demap. + decod.ΣΣFigure 3.5: Detailed Rx-DSP block diagram for the adaptive XPIC and DFE-FTNequalization with IPNT.fore, the overall performance can be improved by reducing the interdependence be-tween the PN estimation and XPIC. To this end, we present a second joint equaliza-tion and PN tracking method in the following.3.3.2 Individual Phase Noise Tracking (IPNT)The DFE with IPNT method estimates the transmitter and receiver PNs of eachpolarization separately. The detailed block diagram is shown in Fig. 3.5, where thede-rotation of the filtered signal before the slicer-stage of the DFE is highlighted. Thisrequires tracking of two additional PN processes compared to the CPNT method. Thereceiver and transmitter PN estimates for the ith polarization branch, for i∈{1, 2},are denoted by θˆri and θˆti , respectively. Following the 2-D FFF-FBF filtering andphase compensation, the sequences y˜i, i= 1, 2, are passed to the FEC decoding. At63the kth time instant, y˜i[k], i∈{1, 2}, can be written asy˜i[k]=e−jθˆti [k]( 2∑j=1{Nf−1∑ν=0fij[ν, k]uj[k−ν]e−jθˆrj [k−ν]−Nb∑µ=1bij[µ, k]aˆj[k−k0−µ]}). (3.22)Assuming a slow variation of the PN processes as in Section 3.3.1, the LMStracking algorithms for the equalizer and the four PN estimates are given in thefollowing lemma.Lemma 3.2. The LMS update equations for equalizer tap weights and PN estimatesfor the IPNT method are given byf1[k+1]=f1[k]−α˜e−jθˆt1[k]P˜ [k]ug[k]E∗1 [k] , (3.23)f2[k+1] =f2[k]−α˜e−jθˆt2[k]P˜ [k]ug[k]E∗2 [k] , (3.24)b1[k+1] = b1[k]+δ˜e−jθˆt1[k]aˆg[k]E∗1 [k] , (3.25)b2[k+1] = b2[k]+δ˜e−jθˆt2[k]aˆg[k]E∗2 [k] , (3.26)θˆt1 [k+1] = θˆt1 [k]− γ˜tΓt1[k] , (3.27)θˆt2 [k+1] = θˆt2 [k]− γ˜tΓt2[k] , (3.28)θˆr1 [k+1] = θˆr1 [k]− γ˜rΓr1[k] , (3.29)θˆr2 [k+1] = θˆr2 [k]− γ˜rΓr2[k] , (3.30)where α˜ > 0, δ˜ > 0, γ˜t > 0, γ˜r > 0 are the LMS step-size parameters, the remainingvariables for i∈{1, 2} are defined as in Lemma 3.1 and as below:64P˜ [k]=diag(e−jθˆr1[k],...,e−jθˆr1[k]︸ ︷︷ ︸Nf, e−jθˆr2 [k],...,e−jθˆr2 [k]︸ ︷︷ ︸Nf), (3.31)Γti[k]= cos(θˆti [k])Im (ξi[k])− sin(θˆti [k])Re (ξi[k]) , (3.32)ξi[k]=(fHi [k] P˜ [k]ug[k]−bHi [k] aˆg[k])E∗i [k] , (3.33)Γri [k]= Im(e−j(θˆri[k]+θˆt1[k])fH1i [k]ui[k] E∗1 [k]+ e−j(θˆri[k]+θˆt2[k])fH2i [k]ui[k] E∗2 [k]). (3.34)Proof. See Appendix B.1.2.By tracking the transmitter and receiver PN processes independently, IPNT canoutperform CPNT significantly, especially for HoM schemes that are more vulner-able to PN distortions. We validate this claim through numerical simulations inSection 3.5.The adaptive DFE schemes presented in this section, that employ CPNT or IPNTfor PN compensation, equalize the combined ISI due to FTN and multipath reflec-tions. While the ISI induced by the multipath propagation is a-priori unknown, theFTN-ISI stemming from the transmitter pulse-shape and the receiver matched-filteris perfectly known at the transmitters and receivers of both polarizations. There-fore, as an alternative to a combined ISI equalization, a separate static equalizeror pre-equalizer can be employed for FTN-ISI mitigation. In the following, we ex-tend the LPE strategy proposed in Chapter 2 to the DP-FTN transmission underconsideration.We note that the additive noise samples at the input to the adaptive DFE-CPNTor DFE-IPNT presented in Section 3.3 are colored due to FTN signaling [13]. While65the LMS filter tap adaptation algorithm remains the same under colored noise [104,105], the LMS convergence speed may change compared to the white noise scenariodue to the increased eigenvalue spread of the auto-correlation matrix of the equalizerinputs [104–106]. In fact, [104] shows that the LMS algorithm under colored noiseexhibits a directionality of convergence, and hence, the speed of convergence withcolored noise can be faster or slower than that with white noise, depending on theinitialization of the filter tap weights. Moreover, the steady-state MSE with colorednoise can also be either larger or smaller than that with white noise [105]. To thisend, we finally note that one attractive choice in the existing literature to counter theeffects of colored noise due to an FTN transmission is to employ a noise whiteningfilter (WF) at the receiver [35, 107]. This can also be accomplished through LPE asdetailed in Chapter 2, which uses a static FFF at the receiver to whiten the colorednoise samples induced by FTN. This serves as an additional motivation to employLPE in the considered DP-FTN HoM transmission.3.4 XPIC with Precoded FTNPre-equalization of the known FTN-ISI can be performed through linear or non-linear precoding at the transmitter [47, 56]. Since non-linear pre-equalization suchas THP [42, 43] can exhibit significant power-loss in an FTN system as shown inChapter 2, we focus on linear precoding schemes. We consider the LPE methodpresented in Chapter 2 in the context of an SP transmission, and apply it to theDP system considered here to pre-compensate for the FTN-ISI6. The LPE strategypresented in this chapter in the context of a DP-FTN HoM system has the following6We note that LPE FTN transmission corresponds to a spectral shape modification and thus, itcan also be interpreted as using a spectrally more efficient pulse shape as shown in Chapter 2.66September 29, 20171+-LPEFBF+-LPEFBFLPE Precoding𝑎1𝑎2DACDAC෤𝑎2෤𝑎1(a) TransmitterSeptember 29, 20171LPEFFFLPEFFFLPE FFFRx DSP(DFE+ CPNT/IPNT)𝑢1𝑢2෤𝑢2෤𝑢1(b) ReceiverFigure 3.6: LPE-FTN DSP, where the shaded blocks represent additional signalprocessing compared to a DFE-FTN system.differences compared to the LPE proposed in Chapter 2: (a) the inputs to the LPE-FFF at the DP receiver are corrupted with multi-path ISI, XPI and PN, and (b) LPE-FBF and LPE-FFF are used in conjunction with the DFE-CPNT and DFE-IPNTproposed in Section 3.3 of this chapter, which constitutes a combined equalizer andPN cancellation structure for the DP transmission. As later shown in Section 3.5.3,this combined structure not only works efficiently, but also outperforms the DFE-onlyequalization scheme by significant SNR margins.Fig. 3.6 illustrates the additional signal processing performed at the transmitterand receiver of a DP LPE-FTN system compared to an unprecoded transmission. Atthe LPE transmitter of Fig. 3.6a, each of the modulated data symbols a1 and a2 isfiltered by a static LPE-FBF bLPE to produce the sequences a˜1 and a˜2, respectively,before the digital-to-analog conversion and pulse-shaping. Similarly, at the LPEreceiver shown in Fig. 3.6b, the received symbols ui, i=1, 2, are filtered by the staticLPE-FFF fLPE to generate the sequences u˜i. Thereafter, the samples u˜i, i=1, 2, areprocessed by the adaptive 2-D DFE to combat the residual interference and PN.67Since the FTN-ISI is perfectly known at the transmitters and receivers for a givenpair of β and τ , the filters bLPE and fLPE can be computed in advance, without anyfeedback from the receivers. In addition to converting the effective FTN-ISI into aminimum-phase impulse response, fLPE also serves the purpose of whitening the noisesamples [44] at the 2-D DFE input. The computational details of the LPE-FBF andLPE-FFF are relegated to Appendix B.2.Following the LPE-FFF stage at the receiver, the ISI induced by FTN is com-pletely eliminated for each polarization. The residual effects of the multipath ISI, XPIand PN can be subsequently compensated by the LMS-DFE with CPNT or IPNTmethod. Numerical results presented in the following section show that the combi-nation of LPE precoding and an adaptive 2-D DFE at the receiver outperforms aDFE-only equalization approach. However, as mentioned earlier, LPE-FTN modifiesthe spectral shape as shown in the previous chapter.Finally, we remark that for the FTN equalization methods proposed in Sections 3.3and 3.4 in this chapter, we consider DFE and linear precoding, which rely on the spec-tral factorization [44] of the overall FTN-ISI channel, stemming from the transmitterpulse-shape and the receiver matched filter. When τ < 11+β, the presence of uncount-ably many spectral zeros makes such factorization unrealizable as shown in Chapter 2.Therefore, similar to the LPE in Chapter 2 and the precoding method in [47], for ourcurrent work, we assume the following relation between β and τ :τ ≥ 11 + β. (3.35)While this restriction limits τ to be slightly above the Mazo limit [7, 8] correspondingto the same minimum-distance for a given β, the limiting value τ = 11+βby itself issignificantly meaningful as this choice of τ maximizes the FTN capacity [108].683.5 Numerical Results and DiscussionIn this section, we illustrate and validate the proposed methods by way of numer-ical simulations. Due to absence of previous works on DP-FTN HoM systems, webenchmark the error-rate performances of the proposed methods against Nyquisttransmissions in the presence and absence of PN distortions.3.5.1 Simulation SetupFor the simulations, we consider the discrete-time baseband DP-FTN HoM microwavecommunication system shown in Fig. 3.2. FEC coding, modulation and FTN param-eters for both polarization branches are kept identical for evaluating the averageperformance of the DP system. For our simulations, we use an LDPC code7 withrate 0.9 and codeword length 64800 bits, a random bit-interleaver, 256, 512 and1024-QAM formats, and different FTN acceleration factors τ for the DP-FTN trans-missions. As the roll-off factors of the RRC filters in practical microwave systemscan generally vary from 0.25 [25] to 0.5 [24, 27, 111], we have chosen β = 0.25, 0.3and 0.4 for presenting our results. We consider a 23 Mbaud Nyquist symbol rate foreach polarization, which is effectively 23τMbaud with FTN signaling for the samebandwidth [13].To simulate the Wiener PN processes, we have considered equal contributionsof PN due to the transmitter and receiver LOs, such that σwti= σwri , and σ∆ =√σ2wti+σ2wri= 0.13◦ [30, 31], i= 1, 2, corresponding to a PN level of −95 dBc/Hz at100 kHz offset from the center frequency for a 23 Mbaud symbol rate as in [32, 99].7The code is compliant with the second generation digital video broadcasting standard for satel-lite (DVB-S2) applications [109, 110], and this is encoded as an irregular repeat accumulate (IRA)code. LDPC decoding is performed by iterative standard message passing algorithm [110], with themaximum number of LDPC internal iterations set to 50.69The multipath reflections and XPI, which are assumed to be unknown to thetransmitters and receivers, are simulated as a 2×2 ISI channel matrix similar to [27].The matrix elements are modeled by Rummler’s well-known fixed-delay, two-raymodel [61], such that the frequency response of each element of the channel matrixcan be written as a function of frequency f asSij(f)=aij[1−10−dN,ij20 ej2pi(f−fN,ij)τ0], i, j∈{1, 2}. (3.36)In (3.36), dN,ij is the notch-depth set to 5 dB and 3 dB for i=j and i 6=j, respectively,fN,ij is the notch-frequency set to 10 MHz and 7 MHz for i=j and i 6=j, respectively,τ0 = 6.3 ns is a fixed-delay, and aij is a gain constant normalized to produce unitenergy co-polarization channels when i= j and a 15 dB attenuation for i 6= j, suchthat the DP-system has a 15 dB cross-polarization discrimination (XPD) value asin [26, 32].We consider a 15-tap FFF and a 11-tap FBF for the adaptive DFE. An initialamount of pilot symbols are inserted at the beginning of transmission to ensure LMSconvergence, during which the adjustable filter tap-coefficients converge to nearlystationary values as in [75, 102]. Thereafter, four QPSK training symbols, withthe same average power as the data symbols, are transmitted after every 200 datasymbols for each polarization, causing a 2% pilot-transmission overhead [30]. Theadaptive LMS-DFE switches between training and decision-directed modes acrossthe blocks of pilots and data transmission, respectively. Similar to [75], the step-size parameters associated with the DFE filter tap-weights are chosen to be smallerthan those of the PN-estimators to account for faster time-variation of carrier phasesover the multipath channel, and their values, together with the decision delay k0,are optimized to minimize the steady-state MSE. For the DP LPE-FTN system,70additional static FBFs and FFFs with 12 and 15 taps, respectively, are applied at thetransmitters and receivers of both polarizations as described in Section 3.4. Followingequalization and PN mitigation, soft-demapping is performed on the DFE-outputsand the LLRs are passed on to the LDPC decoder.For the subsequent performance analyses in this section, we consider the DP-Nyquist and the DP-FTN transmissions with the same average transmit power. Theerror-rate simulations and the SEs of different systems are evaluated as a function ofSNR, which for the ith polarization data stream, i=1, 2, is computed from (3.3) asSNR =E(|si[k]|2)σ2ni, (3.37)where E(·) denotes the expectation operator and si is the signal component of thereceived samples ui in (3.3), i=1, 2, such thatsi[k] = ejθri [k]2∑j=1∑lrj[k−l]ejθtj [k−l]hij[l] , (3.38)where the sequences rj, j = 1, 2, correspond to the modulated symbols aj for theunprecoded systems, and the precoded symbols a˜j for the LPE precoded FTN trans-missions, respectively.3.5.2 Performance with DFE-FTNWe first investigate the efficiency of the proposed algorithms by conducting the follow-ing two performance comparisons: (a) CPNT vs. IPNT method, and (b) DP-Nyquistvs. DP-FTN transmissions. For this, we consider the adaptive DFE with the PNmitigation techniques presented in Section 3.3. Fig. 3.7 shows the coded BER per-formance measured after the LMS convergence. For the computer simulations, 3007124 26 28 30 32 34 36 38 40 4210−610−510−410−310−210−1100SNR in dB BER  1024QNyquistCPNT vs IPNT     3.2 dB256QNyquist256QFTNCPNT vs IPNT     0.55 dB FTN vs. Nyq   3.3 dBNyq (τ=1), w/o PNNyq (τ=1), FalconerNyq (τ=1), IPNT100Nyq (τ=1), IPNTNyq (τ=1), CPNTFTN, w/o PNFTN, IPNTFTN, CPNTFigure 3.7: BER vs. SNR for DP-Nyquist and DP-FTN systems, illustrating theperformance gains of DFE-IPNT over DFE-CPNT, and 256-QAM FTN gains over1024-QAM Nyquist transmission, respectively. β = 0.4, τ = 1 (Nyquist) and τ = 0.8(FTN).codewords are transmitted in each polarization branch, and the average BER per-formance of both polarization streams is evaluated. For the plots in Fig. 3.7, theRRC roll-off factor is set to β = 0.4, and the DP-FTN transmissions use an FTNacceleration factor τ=0.8. As a reference, we also include the BER performances forthe idealized case that the Nyquist and FTN transmissions are not affected by PNdistortions, labeled as ‘Nyq (τ=1), w/o PN’ and ‘FTN, w/o PN’, respectively, in thefigure. Moreover, we have included another benchmark plot in Fig. 3.7 that serves asan additional reference. The one with the label “Nyq (τ = 1), Falconer” represents7230 31 32 33 34 35 36−36−35−34−33−32−31−30−29−28−27SNR, dBMSE, dB  CPNT, XPD = 10 dBIPNT, XPD = 10 dBCPNT, XPD = 15 dBIPNT, XPD = 15 dBCPNT, XPD = 30 dBIPNT, XPD = 30 dBFigure 3.8: MSE vs. SNR for 1024-QAM DP-Nyquist systems, illustrating the gainsof DFE-IPNT over DFE-CPNT for different XPD values. β=0.4, τ=1 (Nyquist).the implementation of the method presented in [75] for an SP communication systemhaving the same multi-path ISI and PN simulation setting described in Section 3.5.1.A comparison of the DFE-CPNT and DFE-IPNT from Fig. 3.7 shows that theIPNT method outperforms the CPNT technique by 0.55 dB and 3.2 dB for a Nyquisttransmission, employing 256 and 1024-QAM, respectively. This indicates that theIPNT exhibits larger gains over the CPNT, particularly for higher modulation for-mats. Moreover, for both Nyquist and FTN transmissions with 256-QAM, the per-formance with the IPNT scheme can be observed in Fig. 3.7 to be within ∼ 0.5 dBfrom that of a zero-PN system. However, the performance degradation in the pres-ence of PN increases to 1.75 dB with the 1024-QAM Nyquist transmission due to73enhanced vulnerability of higher modulation orders to PN impairments. Addition-ally, we also notice from Fig. 3.7 that when the XPI is negligibly small with anXPD value of 100 dB, the adaptive DFE-IPNT method, indicated by the label “Nyq(τ = 1), IPNT100” is able to achieve similar BER performance as that of the SPtransmission.To perform a more comprehensive comparison between the IPNT and CPNTmethods, we recall from Section 3.3 that the gains of DFE-IPNT over DFE-CPNTincrease for higher modulation orders as the XPI grows stronger, i.e. for smallervalues of XPD [26, 32]. We verify this claim in Fig. 3.8 by plotting the steady-stateMSE averaged over the two polarization data streams as a function of SNR, for a 1024-QAM DP-Nyquist transmission with varying XPD values. As shown in Fig. 3.8, bothIPNT and CPNT yield similar MSEs for milder XPI when XPD = 30 dB. However,as the cross-talk between the two orthogonal polarizations increases, the MSE forIPNT shows significant improvement over CPNT. When the XPI is severe with XPD= 10 dB, the average MSE with the DFE-IPNT scheme can be seen to be 4 dB lowercompared to that of the DFE-CPNT at an SNR of 36 dB.We now proceed to analyze the performance difference between Nyquist and FTNsignaling. For this, we first compare the BER of a 256-QAM Nyquist system withthat of a 256-QAM FTN transmission. Fig. 3.7 shows that employing the samemodulation order and the DFE-IPNT method, the DP-FTN system offers a 25%increase in the data rate, corresponding to an FTN acceleration factor 0.8, over theDP-Nyquist system at the price of a 3.5 dB SNR penalty. Additionally, in Fig. 3.7,we also perform a comparison between a 1024-QAM Nyquist system and a 256-QAMFTN transmission having τ = 0.8, such that both systems achieve the same datarate. For example, with a 23 Mbaud Nyquist symbol rate and a LDPC code rate of7426 28 30 32 34 3610−610−510−410−310−210−1SNR in dB BER  256QNyquist     2.2 dBPrecoding vs. no−precodingβ = 0.31024QNyquist256QLPE256QDFE     3.6 dBFTN vs. Nyq     5.5 dBFTN vs. Nyq256Q Nyq (τ = 1), IPNT256Q LPE, CPNT256Q LPE, IPNT256Q DFE, IPNT256Q DFE, CPNT256Q DFE, IPNT, 30 FFF1024Q Nyq (τ = 1), IPNTFigure 3.9: BER vs. SNR for DP-FTN systems, illustrating the performance gainsof LPE-FTN over DFE-FTN. 256 and 1024-QAM, β = 0.3, 0.4, τ = 1 (Nyquist) andτ=0.8 (FTN).0.9, both DP systems employing different modulation schemes yield a data rate of414 Mbits/sec. Fig. 3.7 highlights a performance gain of 3.3 dB for the 256-QAM FTNsystem over the 1024-QAM Nyquist transmission. This suggests that in the presenceof PN, with the DFE-IPNT method, a DP-FTN system can significantly outperforma DP-Nyquist transmission that uses a higher modulation format to produce the samedata rate.However, the adaptive DFE described in Section 3.3 needs to equalize the com-bined ISI due to multipath propagation and FTN. As we shall observe in the following,the BER performance of the DP-FTN transmission can be further improved by elimi-75nating the residual FTN-ISI by way of LPE precoding at the transmitter as presentedin Section Performance with LPE-FTNFig. 3.9 shows the average BER of the two polarizations for a 256-QAM LPE precodedDP-FTN system with τ = 0.8. Moreover, the RRC roll-off factor is set to β = 0.4,except for the two plots indicated by the label “β = 0.3”. The figure also includesthe Nyquist and DFE-FTN BER curves from Fig. 3.7 to highlight the gains offeredby precoding over unprecoded transmissions. The FTN systems for 256-QAM thatemploy adaptive DFE to equalize the combined ISI due to multipath and FTN-ISIare labeled by ‘256Q DFE, CPNT’ and ‘256Q DFE, IPNT’. The precoded DP-FTNsystems using LPE for pre-mitigating FTN-ISI are indicated by labels ‘256Q LPE,CPNT’ and ‘256Q LPE, IPNT’. We observe that the LPE-FTN transmission providesa performance gain of 2.2 dB over the DFE-FTN DP system. For this, LPE usesan additional static FFF at the receiver with 15-taps before the adaptive 15-tapDFE-FFF for each polarization as described in Section 3.4. For a fair comparisonbetween the precoded and unprecoded systems, we have also plotted the BER ofa DFE-FTN transmission that uses 30 taps for the adaptive DFE, labeled ‘256QDFE, IPNT, 30 FFF’ in Fig. 3.9, in order to account for the additional LPE filteringat the receiver. However, we note an only marginal improvement with the longerDFE filters. Additionally, Fig. 3.9 shows 256-QAM LPE-FTN gains of 5.5 dB and3.6 dB over the 1024-QAM Nyquist systems, with β=0.4 and 0.3, respectively. Thereduction in gain due to a lower roll-off and the same FTN acceleration factor canbe attributed to the stronger FTN-ISI.The benefits of the DP-FTN HoM systems considered in this chapter can be char-7622 24 26 28 30 32 334681012SNR in dBSE (bits/s/Hz/pol.)  Constrained Cap., w/o PN and XPINyquist (τ = 1)DFE−FTNLPE−FTN1024Qβ = 0.41024Qβ = 0.3512Qβ = 0.25256Qβ = 0.4,τ = 0.8β = 0β = 0.3β = 0.4256Q β = 0.3, τ = 0.8β = 0.25256Q β = 0.25, τ = 0.89256Qβ = 0.25256Qβ = 0.4Figure 3.10: Spectral efficiency vs. SNR for DP-Nyquist and DP-FTN schemes. 256,512 and 1024-QAM, β=0.25, 0.3 and 0.4, τ=1 (Nyquist) and τ=0.8, 0.89 (FTN).acterized by the SE improvements they provide. The SE value for the ith polarizationdata stream, i= 1, 2, with the RRC roll-off β, FTN factor τ , modulation order Mi,and a code-rate Ri, can be written asSE =Ri log2(Mi)(1 + β)τbits/sec/Hz/polarization . (3.39)Fig. 3.10 shows the SE achieved, per polarization, by the proposed DP-HoM sys-tems as a function SNR, with different values of β and τ . The required SNR toattain a given SE corresponds to an average BER of 10−6 for the respective systems.In Fig. 3.10, we have also included the normalized constrained capacities [108] corre-77sponding to different roll-off factors in an SP transmission without PN, as a reference.We note that the normalized capacity with β=0 is superior to those of the other RRCpulse-shapes having β > 0 [108]. As observed from the SE values in the figure, forexample, a 256 and a 1024-QAM Nyquist transmission correspond to 5.14 and 6.43bits/sec/Hz/polarization, respectively, with β = 0.4, and LDPC code rate 0.9. TheSE figures improve with decreasing filter bandwidths as shown for the RRC roll-offs0.25 and 0.3. We note that by using the FTN factors 0.8 and 0.89, a 256-QAM FTNsystem can achieve the same SE as a 1024-QAM and a 512-QAM Nyquist trans-mission, respectively. In Fig. 3.10, the comparison between the Nyquist and FTNsystems that yield the same SE shows that a 256-QAM DP-FTN system with τ=0.8,using IPNT PN mitigation method and LPE precoding, can demonstrate a 5.5 dBSNR advantage compared to the 1024-QAM Nyquist system for β = 0.4. Similarly,a 256-QAM DP LPE-FTN system with τ = 0.89 outperforms a 512-QAM Nyquistsystem by an SNR margin of 2.9 dB for β=0.25. Moreover, the 256-QAM precodedFTN systems with different τ values can be seen to offer a 12−25% higher SE thanthe 256-QAM Nyquist signaling with a 0.7−3.2 dB SNR penalty.Next, we show the PN tolerance of different modulation schemes employing theproposed methods with varying PN intensities. In Fig. 3.11, we have plotted theadditional SNR required over the respective ideal systems that are not affected byPN distortions, to attain a coded BER of 10−6, for each value of σ∆ defined inSection 3.5.1. The figure suggests that lower modulation orders offer higher toleranceto PN distortions. For example, when σ∆ =0.2◦, 512 and 1024-QAM Nyquist systemsshow 0.7 dB and 2.9 dB additional SNR penalty from the respective zero-PN referencesystems compared to a 256-QAM system. Moreover, at the same SNR distance, e.g.1 dB from the corresponding zero-PN reference systems, 256-QAM and 512-QAM DP780 0.05 0.1 0.15 0.200.511.522.533.54σ∆, in degreesSNR Difference, dB  0.08° 0.05°2.9 dB0.7 dB1024Q, Nyq (τ = 1)512Q, Nyq (τ = 1)256Q, Nyq (τ = 1)256Q, LPE−FTN  (τ = 0.8)Figure 3.11: Additional SNR required over the respective zero-PN reference systemsto achieve a BER of 10−6, plotted against σ∆. 256, 512, 1024-QAM, β = 0.4,τ = 1(Nyquist) and τ=0.8 (FTN).systems tolerate an additional σ∆ = 1.3◦ and 0.08◦, respectively, over a 1024-QAMDP-Nyquist transmission, using IPNT PN mitigation method. Furthermore, the PNtolerance of the 256-QAM FTN system is observed to be comparable with that ofthe 256-QAM Nyquist transmission.The performance benefits of the LPE precoded FTN systems come at the expenseof a possible increase in PAPR (see chapter 2). We investigate the PAPR behaviorof the precoded and unprecoded 256-QAM FTN systems by plotting the empiricalCCDF of the instantaneous power in Fig. 3.12. With RRC roll-off 0.3 and 0.4, we alsoinclude the PAPR results for the Nyquist transmissions employing 256-QAM for com-79−2 0 2 4 6 810−410−310−210−1100Inst. Power (dBW)Prob (Inst. Power > Abscissa)  Unprecoded Nyq, β = 0.4Unprecoded FTN, β = 0.4LPE−FTN, β = 0.4Unprecoded Nyq, β = 0.3Unprecoded FTN, β = 0.3LPE−FTN, β = 0.3Figure 3.12: Empirical CCDF of the instantaneous power with average transmitpower = 0 dBW. 256-QAM, β=0.3 and 0.4, τ=1 (Nyquist) and τ=0.8 (FTN).parison. All transmission schemes are normalized to the same average transmittedpower of 0 dBW, such that the ‘X-axis’ spread to the right hand side of the X=0 dBWline determines the deviation of the peak power from the average power, i.e. PAPR,with the corresponding probability shown along the ‘Y-axis’. Fig. 3.12 suggests thatFTN signaling can exhibit a 0.75−0.9 dB higher PAPR than the Nyquist transmissionat a CCDF value 10−4. Moreover, PAPR with an FTN transmission increases slightlywith smaller RRC roll-off factors as FTN-ISI grows stronger. However, as seen in thefigure, the LPE-precoded FTN systems yield only a marginally higher PAPR than80Table 3.1: Computational Complexities: CPNT vs. IPNTOperation CPNT IPNTComplex Addition/Subtraction 8Nf +4Nb+4=168 12Nf +8Nb+16=284Complex Multiplication/Division 12Nf +4Nb+16=240 24Nf +8Nb+24=472Total Complex Calculations 408 756Hard Symbol-Decisions 2 2the unprecoded FTN transmissions8.3.5.4 Computational Complexity AnalysisIn this section, we present an analysis of the computational cost for the proposedCPNT and IPNT methods. The number of the mathematical operations neededduring every symbol period of the DP transmission is furnished in Table 3.1 as afunction of Nf and Nb. To illustrate this analysis with further clarity and ease ofcomparison, a specific example, corresponding to our simulation setting Nf =15 andNb =11, is also provided. The numbers in Table 3.1 reveal that the implementationcomplexity to update the IPNT estimates according to (3.23)-(3.30) is slightly higherthan that of the CPNT adaptation given in (3.7)-(3.12), at the cost of a significantlysuperior BER performance demonstrated in Section 3.5.2. Moreover, Table 3.1 showsthat the LMS adaptation in the decision-directed mode performs two hard symboldecisions for the employed QAM constellation. The table entries also reveal that thenumber of complex calculations required for the PN and equalizer-taps adaptationdoes not depend on the modulation order. Therefore, our proposed methods showease of scalability with higher modulation formats.8We remark that the consideration of PAPR reduction schemes, such as [112] for FTN trans-mission, and their operation in tandem with pre-distortion methods usually applied for microwavesystems using HoM, is an interesting extension to mitigate and analyze the effects of PAPR increase.813.6 ConclusionsA synchronous DP-FTN HoM transmission is an attractive choice to increase the SEin fixed wireless backhaul links. However, the SE improvements offered by such asystem comes at the expense of introducing ISI, XPI and vulnerability to PN. Us-ing FTN signaling in a DP transmission can moderate the need for adopting veryhigh modulation orders that are more sensitive to PN distortions. In this chap-ter, DP-FTN HoM systems have been investigated for the first time. In order toequalize interference and recover carrier phase, we proposed a joint XPIC and PNcompensation scheme coupled with an adaptive LMS-DFE. The ISI induced by FTNis mitigated either through the LMS-DFE at the receiver or linear pre-equalization atthe transmitter. Numerical results for a microwave radio transmission show that anFTN signaling with the proposed interference mitigation schemes can exhibit as highas 3−5.5 dB performance improvement over a Nyquist transmission that employs ahigher modulation order to achieve the same data rate. Alternatively, for a givenmodulation scheme, a DP-FTN signaling can offer a 12−25% SE enhancement overa DP-Nyquist signaling with a 0.7−3.2 dB SNR degradation.82Chapter 4Multicarrier Faster-than-NyquistOptical Transmission4.1 IntroductionOFC system is a suitable platform for the introduction of FTN signaling, since con-ventional SE improvement means, such as employing HoM formats, are challengingfor such a system. In Chapter 2, we have considered single carrier optical fiber trans-mission using FTN signaling. However, such systems are restrictive in achieving veryhigh data rates, because the practical limitations of the opto-electronics precludethe feasibility of increasing the baud rate beyond a certain value. Therefore, MFTNtransmission using TFP WDM superchannels [10, 11, 13, 17, 80, 113, 114] is an at-tractive choice to achieve high data rates in OFC systems. Such a benefit of the TFPtransmission comes at the price of introducing ISI and ICI. Therefore, enjoying theSE benefits of the TFP systems entails successful mitigation of such interference.Alternative to the high-complexity equalization strategies, we investigate pre-coding techniques in this chapter, as an extension to the pre-equalization methodpresented in Chapter 2. Since the well-known THP incurs significant precoding lossand modulo-loss [44], particularly with FTN signaling (see Chapter 2), we considerlinear precoding, similar to Chapter 2. However, different from the ISI-only systemsconsidered in Chapter 2, we need to mitigate both ISI and ICI in this chapter. This83thesis is the first to consider precoding in MFTN systems that employ packing ofsymbols in both time and frequency dimensions.For our first contribution, we present a new 2-D linear LPE technique, as anextension of the 1-D LPE proposed in Chapter 2. By orthogonalizing the FTN trans-mission through joint filtering of the constituent SCs of an MFTN system, 2-D LPEyields optimal error-rate performance, which makes it competitive to computation-ally prohibitive and buffer-space constrained BCJR based equalization algorithms.However, such a precoding method is restrictive in terms of the time and frequencycompression achievable in an MFTN transmission. To address this problem, as a sec-ond contribution, we propose a sub-optimal partial precoding strategy, which facili-tates transmitter-side 1-D LPE precoding for the individual SCs of the TFP system,followed by a receiver-side turbo ICI cancellation. We validate the advantages ofour proposed precoding schemes over the existing TFP interference mitigation meth-ods, through numerical simulations of a coherent TFP optical WDM superchanneltransmission, which has attracted significant attention more recently [11, 17, 114].The remainder of the chapter is organized as follows. The system model is intro-duced in Section 4.2. In Section 4.3, we propose two precoding strategies to counterTFP ISI and ICI. Section 4.4 demonstrates the benefits of our proposals throughsimulations. Finally, Section 4.5 provides concluding remarks.4.2 System ModelWe consider the baseband system model for precoded MFTN transmission under anAWGN channel shown in Fig. 4.1. For each kth SC, k∈1, 2, · · · , N , with N being thetotal number of SCs, an LDPC coded and modulated data stream ak is either jointlyor separately precoded by a linear FBF. The precoded signal dk is then frequency84SC NInput bitsLDPCEncQAMMap. DAC......SC 1SC kΣPrecoding(jointorper-SC)RRC......SC 1SC NFreq.ShiftJoint or Per-SCRx DSPMF!"-samplingFreq.Shift......SC 1SC NSC NLDPCDecQAMDemap......SC 1Output bitsSC kAWGNTransmitterReceiver#$ s%$&$'$($)$SC N......SC 1Figure 4.1: Precoded-MFTN AWGN system model.shifted to produce xk, which is converted to an analog signal by digital-to-analogconverters (DAC), and shaped by an RRC pulse h with a roll-off factor β. Thebaseband equivalent aggregate signal s at the MFTN transmitter can be expressedass(t) =∑l∑kxk[l]h(t− lτT )ej2pi(k−N+12 )∆ft , (4.1)where ∆f = ξ 1+βTis the frequency-spacing between the adjacent SCs, with 0<τ ≤ 1and ξ>0 denoting the time and frequency compression ratios, respectively, such thatτ = ξ= 1 corresponds to Nyquist signaling, 1τTis the baud rate per SC, and l is thesymbol index. At the receiver, the RRC matched-filtered and τT -sampled digitalsamples uk of the kth SC are frequency-shifted to produce the signal rk that is jointlyor separately processed by an FFF. Thereafter, the digital samples are sent as inputsto the demapper and the LDPC decoder. For ease of characterization of the precodedTFP systems, we state the following proposition.Proposition 4.1. For the kth SC, k = 1, 2, . . . , N , shown in Fig. 4.1, frequencyshifts of the precoded signal dk at the transmitter and the τT -sampled signal ukat the receiver by an amount −ω0(k − N+12)and ω0(k − N+12), respectively, where85...!"!#.!$2D LPEFFF% .... ...&"&#&$... ...2D LPEFBF '("(#.($ Σ... ...... ..................)")#)$+ -(a) Transmitter!"!#.!$2D LPEFBF% .... ...&"&#&$... ...'"'#.'$2D LPEFFF( .... ...)")#)$... ...(b) ReceiverFigure 4.2: 2-D LPE, where the shaded blocks represent additional signal processingcompared to unprecoded MFTN systems.ω0 =2pi∆fτT , translates the overall TFP channel into a linear time-invariant (LTI)system, and the z-transform H(z) of the corresponding 2-D channel response is aHermitian matrix polynomial.Proof. See Appendix C.1.4.3 Precoding Solutions4.3.1 Joint precoding: 2-D LPESchematics of the 2-D LPE are shown in Fig. 4.2a-4.2b corresponding to the trans-mitter and receiver filtering operations, respectively. In Fig. 4.2a, the modulateddata symbols ak, k = 1, 2, . . . , N from all SCs are jointly processed by a linear 2-D FBF B to produce the precoded symbols dk, translating the overall precodingoperation into an effective IIR filter. The minimum-phase property of the FBF guar-antees the stability of the IIR operation [44]9. At the receiver shown in Fig. 4.2b, thefrequency-shifted symbols rk, k=1, 2, . . . , N from all SCs are further jointly processedby a linear 2-D FFF F , which whitens the colored noise caused by FTN sampling.9We remark that different from a conventional ISI channel, such linear IIR filtering does notinduce a precoding loss in an FTN system (see Chapter 2), since the MFTN “channel” is part ofthe transmitter.86Thereafter, the filtered samples are sent as inputs to a symbol-by-symbol demapper.Inspired by [115], the FFF and the FBF matrix computation from the 2-D channelmatrix H defined in Proposition 4.1 is summarized below.Proposition 4.2. With the Cholesky decomposition of the Hermitian matrix polyno-mial H(z) performed asH(z) = V (z)V H(z−∗), (4.2)where V (z)=∑k≥0Vkz−k is causal and minimum-phase, i.e., Vk=0 for k<0, V (z) isnonsingular for |z| ≥ 1 and V0 is lower triangular, the 2-D LPE FFF and FBF aregiven byF (z) = D−1JV −1(z−∗)J , (4.3)B(z) = D−1JV H (z∗)J , (4.4)respectively, with J being the K×K anti-diagonal identity matrix, K is the number ofTFP-channel taps, (·)∗ represents the complex conjugate of a complex scalar, (·)−∗ =1(·)∗ , [·]H and [·]−1 denote the matrix Hermitian and matrix inverse, respectively, D =diag(v∗0,K,K , · · · , v∗0,1,1), v0,i,j are the ith and jth entries of V0, and diag(· · · ) denotesthe diagonal matrix constructed with the specified elements.Proof. See Appendix C.2.By converting the MFTN transmission into an orthogonal system similar to theLPE method in Chapter 2, 2-D LPE yields optimal error-rate performance withoutperforming any iterations between the equalizer and the LDPC decoder. However,such precoding suffers from the following two drawbacks: (a) for the joint filteringacross SCs, it requires access to the digital samples of all SCs at both the transmitter87!"1D LPE FBF #$%&%".%'1D LPE FFF().... ...*&*"*'... ...1D LPE FFF()1D LPE FFF()PICLDPCDecLDPCDecLDPCDec+&+".+'......Σ!'1D LPE FBF #,Σ!&1D LPE FBF #-Σ...... .......&.".'+ -+ -+ -(a) Transmitter!"1D LPE FBF #$%&%".%'1D LPE FFF().... ...*&*"*'... ...1D LPE FFF($1D LPE FFF(+PICLDPCDecLDPCDecLDPCDec,&,".,'......Σ!'1D LPE FBF #+Σ!&1D LPE FBF #)Σ...... ......-&-"-'+ -+ -+ -(b) ReceiverFigure 4.3: Partial precoding, where the shaded blocks represent additional signalprocessing compared to unprecoded MFTN systems.and the receiver, which precludes the realization of independent SC processing, and(b) 2-D LPE is feasible only for a restricted range of τ, ξ pairs for a given β, sinceorthogonalizing the MFTN transmission requires that the net symbol rate is lowerthan the aggregate TFP bandwidth, and therefore,N(1 + β)τ [(N − 1)ξ + 1]≤1 . (4.5)Moreover, we also note that the computation of B and F requires the factoriza-tion (4.2) that is possible if the following Paley-Wiener condition (see e.g. [44, 116])is satisfiedτT∫ 12τT− 12τT∣∣∣ log det (H (ej2pifτT )) ∣∣∣df <∞ . (4.6)To address such limitations of the 2-D LPE, we propose another precoding strategyas follows.88ICI Est.𝐼+,%,++ Σ--LDPC Decbits out 𝑖𝑡 = 𝑖𝑡234LLR789.𝑖𝑡 < 𝑖𝑡234LLR toSoft SymICI Est.𝐼+,+,%ICI Est.𝐼+,+;%ICI Est.𝐼+;%,+ ......SC 𝑘 − 1SC 𝑘 + 1𝑧+ 𝑤+Figure 4.4: ICI mitigation through PIC.4.3.2 Partial Precoding (PP)PP encapsulates a conceptual combination of transmitter-side pre-equalization andreceiver-side equalization of TFP interference, as shown in Fig. 4.3a-4.3b. In order topre-mitigate the MFTN-ISI, PP employs separate 1-D LPE FBFs and FFFs for eachSC, at the transmitter and receiver, respectively. For this, the FFFs and FBFs arecomputed based on the spectral factorization of the diagonal entries of the channelresponse H(z) (see Chapter 2 and Chapter 3 for the computational details). There-after, the MFTN-ICI is mitigated at the receiver through an iterative PIC approachin a turbo fashion as detailed in Fig. 4.4, similar to [13, 80, 117].As shown in Fig. 4.4, PIC enables the extrinsic LLRs fed back from the LDPCdecoders to estimate and cancel the soft-estimates of the ICI stemming from theadjacent SCs, iteratively. For example, each LDPC iteration uses the extrinsic LLRsfrom the (k−1)th and (k+1)th SCs to compute the soft estimates of the data symbolscorresponding to the neighboring SCs [117]. Next, the soft-estimates Ik−1,k and Ik+1,kare computed and subtracted from the 1-D LPE FFF output symbols zk of the kth89SC NX-pol LDPC EncQAMMap. RRCOpt. Front-endY-pol QAMMap. RRCWDM......SC 1SC kCoh. RxSSMFDACDACIQDACDACIQADCADCIQADCADCIQMF+CD Eq.Other SCs X-pol2D LPE/PartialPrecodingOther SCs Y-pol2D LPE/PartialPrecodingOther SCs X-polOther SCs Y-polFreqShiftFreqShiftSC kMF+CD Eq.2x2PMDFSELDPC EncFreq Shift+1D/2D FFFper Pol.ToLDPCDec!",$!",%&",$&",%Other SCs X-polOther SCs Y-polFigure 4.5: Simulated MFTN system model: precoded DP TFP WDM optical su-perchannel transmission.SC, where Ii,j denotes the ICI from the ith SC to the jth SC, i, j∈1, 2, · · · , N .While PIC is well investigated in the FTN literature as a means to counter ICI, inthis work, we apply it for the first time in tandem with precoding. With this design,PP based TFP systems completely eliminate the MFTN-ISI without performing com-putationally challenging BCJR iterations, and can also offer significant performanceadvantage over unprecoded PIC-only ISI and ICI equalization approach, such as [80].We validate this claim through numerical simulations in Section 4.4. We remark thatfor a given roll-off β, implementation of PP is feasible for the restricted range τ≥ 11+β(see Chapter 2). However, any amount of ξ can be accommodated through the PPimplementation, which allows more flexible precoded TFP design compared to 2-DLPE. However, such benefits come at the expense of sub-optimal performance anditerative detection at the receiver that entails higher complexity and buffering.4.4 Numerical resultsIn this section, we present the simulation results for a precoded TFP WDM super-channel.904.4.1 Simulation ParametersIn this section, we validate the effectiveness of the proposed precoded MFTN de-signs by way of numerical simulations using parameter settings relevant for practicalcoherent optical WDM superchannel transmission, which is a prime candidate forthe introduction of FTN. For the simulations, we consider a DP QPSK 3-SC TFPWDM superchannel having per SC baud rate 40 Gbaud. In the simulation setupshown in Fig. 4.5, the transmitter and receiver blocks for the discrete-time base-band modules are same as those in Fig. 4.1 except that the data processing for eachof the two polarizations is performed separately for each SC. The baseband analogdata after the DAC is processed by the opto-electronic front-end and transmitted asan optical signal through a 1000 km SSMF with CD parameter value −18 ps2/km,PMD 0.5 ps/√km, and then is received by the coherent optical receiver, followed byanalog-to-digital conversion (ADC). LDPC codes from the DVB-S2 standard withrate 0.9 and codeword-length 64800 bits, β = 0.3, 8-tap TFP-ISI for BCJR, 20-tapLPE-FBF, 200-tap LPE-FFF, and maximum iteration count of 10 between the PICand the LDPC decoder are considered for the simulations. Perfect frequency synchro-nization and phase-lock are assumed. Moreover, matched filtering at the receiver iscombined with the time-invariant frequency domain CD compensator using overlap-and-add method, and for PMD compensation, we use a 19-tap 2 × 2 butterfly-typefractionally-spaced adaptive LMS equalizer10.10For the PP TFP systems, independent SC processing enables us to employ the PMD equalizerafter the 1-D LPE FFFs, similar to Chapter 2, where the LMS filter is trained with the knownMFTN-interference induced pilots. However, the matrix filtering operations in the 2-D LPE requiresthe placement of the FFF before the PMD equalizer, for which the LMS update equations need tobe modified compared to a conventional Nyquist WDM transmission [17, 114]. Such modificationis detailed in Appendix C.3.9110.75 11.25 11.75 12.2510−410−310−210−1OSNR, dBBER  Nyquist (τ=ξ=1)2−D LPEPP, 10 it.BCJR−PIC, 10 it.PIC only, 10 it.PP Gain0.6 dB2−D LPEOptimalPerform.Figure 4.6: BER vs. OSNR, β=0.3, τ=0.85, ξ= 2-D LPE GainsWe first show the advantages of the 2-D LPE and PP in Fig. 4.6 by plotting the codedBER performance averaged over both polarizations and all SCs, as a function of theOSNR [17]. For reference, we also add the error rate curves in Fig 4.6 corresponding tothe following three scenarios: (a) Nyquist WDM transmission having the same baudrate and therefore, larger bandwidth, (b) BCJR based ISI equalization in conjunctionwith PIC for ICI mitigation as in [13], denoted by the legend “BCJR-PIC, 10 it.” and(c) PIC based ISI and ICI cancellation as in [80], indicated by the label “PIC only,10 it”. As shown in the figure, 2-D LPE achieves similar performance as that of aNyquist WDM system and an MFTN system employing BCJR-PIC, by successfully920.7 0.75 0.8 0.85 0.9 0.95τξ  β = 0.1β = 0.2β = 0.3β = 0.4Feasible(τ, ξ) PairsFigure 4.7: Feasible range of τ, ξ for 2-D LPE.pre-equalizing the ISI and ICI completely. For this, 2-D LPE relies on simple non-iterative filtering operations, as opposed to the substantially complex and buffer-spaceconstrained BCJR algorithm that is impractical especially for larger constellations.Fig. 4.6 also suggests that PP yields 0.6 dB performance improvement over the PIC-only receiver structure. Moreover, it produces sub-optimal performance compared tothe 2-D LPE for this particular combination of τ and ξ that is well within the rangespecified by the inequality (4.5). However, as shown through the subsequent results,PP is more effective for stricter values of τ and ξ pairs, for which 2-D LPE precodingis infeasible.9310.8 11.7 12.6 13.510−410−310−210−1OSNR, dBBER  Nyquist (τ=ξ=1)PP, 10 it.BCJR−PIC, 10 it.PIC only, 10 it.PP Gain1.1 dBFigure 4.8: BER vs. OSNR, β=0.3, τ=0.8, ξ= Feasible Range for 2-D LPEIn Fig. 4.7, we plot the range of τ and ξ where the spectral factorization (4.2) andthereby, 2-D LPE precoding is infeasible, for varying β. To numerically evaluate suchrange of values that does not satisfy (4.6), we observe the presence of spectral zeros inthe overall TFP channel H(z). Fig. 4.7 indicates that higher values of the RRC roll-offtranslates to a larger range of feasible τ and ξ values for the 2-D LPE. Furthermore,we note that the plots in the figure also correspond to the inequality (4.5). Thismeans that the dimensionality and factorization constraints are equivalent for theconsidered precoded TFP systems.94Table 4.1: Complexity, memory and latency, per codewordMethod Complexity Memory Latency2-D LPE O(N2NfLw) O(NLb) O(Lb)PP O((M+Lc+Nf)NImLw)O(NLbIm) O(LbIm)PIC-Only O((M+Lc+Ls)NImLw)O(NLbIm) O(LbIm)BCJR-PIC O((MLs2 +Lc)NImLw)O(NLbIm) O(LbIm)4.4.4 PP GainsFinally, to show the usefulness of PP in more detail, we deliberately choose a pair oftime and frequency compression ratios in Fig. 4.8 such that (4.5) is violated for β=0.3,and therefore, 2-D LPE can not be employed. Fig. 4.8 shows that the BCJR-PICoutperforms PP by 0.65 dB at the price of significantly higher complexity. However,under such transmission scenarios, PP offers 1.1 dB performance gains over PIC-onlyequalization scheme having similar computational cost.4.4.5 Computational ComplexityThe details of the receiver complexity, latency and memory requirements for the dif-ferent interference mitigation schemes are furnished in Table 4.1, where M , Ls, Lc,Nf , Lb, Im denote the modulation order, truncated ISI and ICI-taps length, 1-D/2-D LPE FFF taps length, LDPC codeword length in bits and the maximum turboiteration count, respectively, and Lw =Lblog2Mis the number of modulated symbolscorresponding to each codeword. Values of the above parameters considered for oursimulations are mentioned at the beginning of this section. Benefits of the proposedprecoded systems can be seen in the performance-complexity trade-off, through suit-able precoding technique selection depending on the MFTN parameters.954.5 ConclusionsIn this chapter, we presented two precoding approaches for the first time in MFTNsystems that enable packing of symbols in both time and frequency dimensions. First,a matrix linear filtering based 2-D LPE precoding is proposed that performs jointprocessing of the SCs to completely eliminate TFP ISI and ICI, and thereby, accom-plishes optimal error rate performance. However, functionality of such precoding islimited to a restricted range of time and frequency compression. Second, we presentedPP that facilitates independent processing of SCs at the transmitter for mitigatingISI, but operates in an iterative fashion with the LDPC decoder, to eliminate ICIat the receiver. Simulation results for a DP QPSK TFP WDM optical superchan-nel suggests up to 1.1 dB performance gains by the proposed precoding techniquesover existing interference mitigation methods having similar or significantly highercomputational cost, buffer space and latency requirement.96Chapter 5Towards Terabit-per-secondSuper-Nyquist Systems5.1 IntroductionPer-carrier data rates of 1 Tbps and more are being targeted in next generationoptical systems to cope with the increasing demands in network traffic[118]. TFPsuperchannel transmission discussed in Chapter 4 is an attractive candidate to ac-complish such a target. TFP offers SE improvements by allowing controlled overlap ofthe SCs in time and frequency, and thereby introduces ISI and ICI. In Chapter 4, weinvestigated precoded MFTN superchannel systems to pre-mitigate the TFP-ISI andICI. However, as shown in that chapter, the functionality of the presented precodingmethods was limited to a restricted range of time and frequency compression11, whichcan enable only a benign packing of symbols. Therefore, they may be insufficient toachieve the desired high SE targeted for the next generation Terabit systems [17].Motivated by this, we consider spectrally more efficient TFP schemes that achieveTbps data rates. For this, we turn our focus to receiver-side high-performance ISICand ICIC schemes, as opposed to the transmitter-side precoding. By doing so, we11While the PP presented in the previous chapter facilitates slightly larger range of frequencycompression compared to the 2-D LPE, PP is restrictive in terms of the amount of time-compressionachievable, depending on the value of the RRC roll-off. This is due to the fact that for each SC, PPemploys the 1-D LPE method presented in Chapter 2, which suffers from such restriction.97ensure that the TFP superchannel systems considered in this chapter are able tofacilitate a wider range of temporal and spectral overlap of the SCs.Moreover, a superchannel signal may be subjected to additional narrow opti-cal filtering due to multiple network elements, such as ROADMs implemented byWSSs [118–120]. Such strict-filtering often leads to significant ISI for the edge-SCs ofa superchannel. Super-Nyquist transmission can be used to reduce such ISI by pack-ing the SCs closer in both time and frequency, and thereby having a lower aggregatesignal BW.In order to restrict the receiver-side complexity, at the beginning, we considerfrequency-packed (FP) transmissions, where the frequency-spacing of the optical car-riers is reduced below the occupied BW of the individual SCs. We apply compu-tationally simple linear equalization (LE)[18] and more powerful turbo PIC [81] tomitigate the ICI. Thereafter, we study TFP transmission to further improve theachievable SE [13]. To counter the additional ISI, Ungerboeck’s modified BCJR al-gorithm [17] is employed for ISIC, in conjunction with the turbo-ICIC method. Wewill show through our numerical results in Section 5.5 that with adequate signal pro-cessing, super-Nyquist systems that are doubly constrained due to the filtering effectsof WSSs exhibit significant OSNR gains over Nyquist transmission, when targetingTerabit data rates.The contributions of this work are summarized as follows. For the first time,a quantitative performance comparison of LE-ICIC and turbo-PIC based receiverstructures for coherent optical transmission is provided. While the application ofLE-ICIC for an optical system has been considered before, e.g. [18], PIC has onlybeen investigated in the TFP-literature under an AWGN channel assumption [13,81]. Here, we apply turbo-PIC to high-baud-rate optical systems targeting 1 Tb/s98SC NX-pol bitsLDPC Enc QAMMap.RRCOpt. Front-endY-pol bitsLDPC Enc QAMMap.RRCWDM......SC 1SC kWSS ...WSS{m-ROADMs Coh. RxSSMFRx DSPD/AD/AIQD/AD/AIQA/DA/DIQA/DA/DIQRRCRRCFigure 5.1: Super-Nyquist WDM system rate in a 150 GHz BW considering practical fiber-optical impairments, and wespecify its performance gains over LE-ICIC. Second, we present an investigation of theperformance degradation due to narrow-filtering effects of cascaded ROADMs that areintegral components of the next-generation flexible-grid optical networks [119, 120].In the context of a TFP transmission, consideration of such severe filtering effects onthe edge SCs of a superchannel is limited in the TFP-literature. In this chapter, toachieve the target data rate under tight WSS filtering, we numerically quantify thepower-efficiency improvements due to TFP transmission over Nyquist WDM systems,as opposed to the existing TFP works [17, 20, 21] that primarily focus on bandwidth-efficiency evaluation, without considering aggressive WSS-filtering effects on the edgeSCs.The remainder of the chapter is organized as follows. The system model is pre-sented in Section 5.2. In Section 5.3, we conduct a performance comparison betweenthe LE-ICIC and the turbo-ICIC methods for the FP systems. We investigate theBCJR-ISIC and the turbo-PIC approaches in Section 5.4. Section 5.5 presents thesimulation results. Finally, in Section 5.6 we provide concluding remarks.995.2 System modelFigure 5.1 shows the block diagram of a DP super-Nyquist WDM superchannel trans-mission. For each polarization branch, i.e. X and Y, of the kth SC, k∈ 1, 2, · · · , N ,where N is the total number of SCs, an LDPC coded and modulated data stream akis pulse-shaped by an RRC filter h having a roll-off factor β. Following DAC conver-sion, the electrical signal is converted to optical domain using individual lasers andMach-Zehnder (MZ) modulators to form the aggregate superchannel transmit signal.The baseband equivalent analog signal for one polarization mode can be expressedass(t) =∑l∑kak[l]h(t− lτT )ej2pi(k−N+12 )∆ft , (5.1)where ∆f = ξ 1+βTis the frequency-spacing between the adjacent SCs, with 0<τ ≤ 1and ξ>0 denoting the time and frequency compression ratios, respectively, such thatτ = ξ= 1 corresponds to Nyquist signaling, 1τTis the baud rate per SC, and l is thesymbol index. The transmitted WDM signal may be filtered by several WSSs whilepropagating through the optical link. At the coherent receiver, the RRC matched-filtered digital samples are fed as inputs to the Rx-DSP after conversion to electricaldomain by an integrated coherent receiver. For ease of characterization of the ISICand ICIC operations that use τT -sampled signals, we state the following lemma.Lemma 5.1. Without optical impairments and noise, RRC matched filtered τT -sampled signal for each polarization of the kth SC, k=1, 2, . . . , N , is given byrˆk[n]=(ak[n] ? g0,0[n])+∑m6=k(bk,m[n] ? g0,m−k[n]) , (5.2)where ? denotes linear convolution, m= 1, 2, . . . , N , bk,m[n] = am[n]e−jω0(m−k)n, ω0 =2pi∆fτT , and gu,v denotes τT samples of fu(t) ? fv(t) with fu(t)=h(t)e−j2piu∆ft.100Freq. Shift"̂#,%Freq. Shift"̂&,%Freq. Shift"̂#,'Freq. Shift"̂&,'.........2)×2) FSEPMD+ICI+ISI (WSS)Comp.LDPC DecLDPC DecLDPC DecLDPC Decbits outbits outbits outbits out"#,%"&,%"#,'"&,'Linear ICI Cancellation{SC 1{SC N...CD Eq.CD Eq.CD Eq.CD Eq.Figure 5.2: LE-ICIC, shaded block represents 2-D LMS.Proof. See Appendix D.1.Following the fiber-optic linear impairments compensation, g0,0 and g0,m−k in theabove lemma represent the known contributions of the TFP-ISI and TFP-ICI, re-spectively.5.3 FP WDM Transmission: ICICIn this section, we consider FP super-Nyquist WDM transmission, together withICIC. The Rx-DSP shown in Fig. 5.1 mitigates the impact of CD, PMD and ICI.We apply LE for ICIC as a natural extension of the linear PMD equalization similarto [18] and a new PIC based ICIC scheme.5.3.1 Linear EqualizationThe LE based ICIC method shown in Fig. 5.2 jointly equalizes PMD and ICI. Forsuch an ICIC scheme, a frequency shift operation is required [18] to align the SCs totheir respective frequency-bins by converting the TFP channel into an LTI system,as summarized in the below Lemma.101CD Eq.?̂?#,+,%?̂?&,+,% 2×2 FSEPMD +ISI (WSS)... 𝑟#,+,%𝑟&,+,% + Σ-- LDPC DecLLR toSoft SymICI Est.𝐼+,%,+?̂?#,+?̂?&,+ 2×2 FSEPMD +ISI (WSS)𝑟#,+𝑟&,++ Σ-- LDPC Dec bits out 𝑖𝑡. = 𝑖𝑡.234LLR789.𝑖𝑡. < 𝑖𝑡.234LLR toSoft SymICI Est.𝐼+,+,%ICI Est.𝐼+,+;%?̂?#,+;%?̂?&,+;% 2×2 FSEPMD +ISI (WSS)𝑟#,+;%𝑟&,+;% + Σ- LDPC DecLLR toSoft SymICI Est.𝐼+;%,+...-... ...Non-linear ICI Cancellation (SIC){SC k{SC 𝑘 − 1{SC 𝑘 + 1CD Eq.CD Eq.CD Eq.CD Eq.CD Eq.Figure 5.3: Turbo-PIC, shown for the X-pol. of the kth SC.Lemma 5.2. The frequency shifts of the discrete time signal (5.2) for the kth SC, k=1, 2, . . . , N , by an amount ω0(k − N+12)render the overall τT -sampled TFP channelan LTI system, with respect to the rotated inputs ak[n]ejω0(k−N+12 ).Proof. See Appendix D.2.The CD compensated and frequency shifted received samples from both polar-ization branches of all SCs are jointly processed by a 2N×2N 2-D adaptive LMSbased fractionally-spaced equalizer (FSE). The outputs of the joint equalizer aresoft-demapped and LDPC decoded to produce bits.5.3.2 Iterative Equalization: Turbo-PICThe proposed ICIC using iterative turbo-PIC for an optical superchannel is shown inFig. 5.3. Different from Fig. 5.2, the CD compensated received samples are processedby a 2×2 adaptive LMS filter to mitigate the impact of PMD only. After polarization-recovery is accomplished, turbo-PIC is subsequently employed for each polarization1022×2 FSEPMD+ISI (WSS)𝑟#,+𝑟&,+SC kIQBCJRBCJRCombineLLRsDemap.+LDPC DecSplitLLRsLLR789.𝑖𝑡 < 𝑖𝑡234IQA-prioriProbabilities+Σ--ICI Est.𝐼+,%,+ICI Est.𝐼+;%,+bits out 𝑖𝑡 = 𝑖𝑡234ICI Est.𝐼+,+,%, 𝐼+,+;%To SC 𝑘 − 1& 𝑘 + 1Figure 5.4: Turbo-PIC combined with BCJR-ISIC, shown for the X-pol. of the kthSC. Shaded blocks represent additional processing to perform stream, wherein the extrinsic LLRs are fed back from the LDPC decoders,and used to cancel the soft-estimates of the ICI stemming from the adjacent SCs,iteratively12. For example, each LDPC iteration uses the extrinsic LLRs from the(k−1)th and (k+1)th SCs to compute the soft symbols [81]. Next, the soft-estimatesIk−1,k and Ik+1,k are computed using (5.2), and subtracted from the received symbolsof the kth SC, where Im,n denotes the ICI from the mth SC to the nth SC, m,n ∈1, 2, · · · , N . Such turbo-PIC scheme enables soft-information exchange across SCssimultaneously at every turbo iteration stage to perform ICIC. Being iterative innature, turbo-PIC requires additional complexity and buffering compared to the LE-ICIC.12For each LDPC iteration, an estimate of the effective noise power is considered by a softdemapping module for the pre-LDPC LLR computation, by treating the residual interference asAWGN [see, e.g., Section III of [17]]1035.4 TFP WDM Transmission: ISIC & ICICTFP WDM systems provide additional SE advantages over FP super-Nyquist systemsby transmitting the symbols at an FTN signaling rate [13]. Thereby, for a fixed baudrate, FTN signaling translates to bandwidth compression of the individual SCs [17],which reduces the amount of ISI introduced in the outer SCs in an FP superchanneldue to narrow WSS filtering. TFP transmission introduces ISI and ICI that areperfectly known a-priori, and therefore, can be mitigated through static equalizers,without the explicit need for channel estimation. To adjust the tap-coefficients of theadaptive 2×2 PMD equalizer, the known TFP-induced interference is incorporatedin the desired signal generation [17] for the LMS algorithm. In this work, we employBCJR equalization based on Ungerboeck’s modeling [17] for TFP-ISIC. We considera truncated K-tap TFP-ISI channel, given by g0,0[n] in Lemma 5.1. Moreover, theBCJR algorithm works in conjunction with turbo-PIC based ICIC as shown Fig. 5.4,to facilitate an efficient TFP transmission. Exploiting the fact that TFP-ISI is realvalued, for a square-constellation, the I and Q components of the baseband receivedsamples are separately processed by the BCJR module to minimize computationalcomplexity. For each LDPC iteration, following the optical channel equalization,and soft ICI cancellation via turbo-PIC, BCJR equalizes TFP-ISI by exchangingextrinsic LLRs with the decoder, similar to [17]. At the final iteration, output bitsare generated from the LDPC decoders.5.5 Results and DiscussionTo show the effectiveness of the proposed TFP designs, we present a 400 Gbps and a1 Tbps superchannel systems, employing QPSK and 16-QAM constellations, respec-tively.104−100 −50 0 50 100−100−80−60−40−20020Frequency (GHz)Normalized PSD (dB)  Before WSSAfter WSSWSS spectrumISIISI3−dBBWNo ICI(a) ξ=1 (Nyq)−100 −50 0 50 100−100−80−60−40−20020Frequency (GHz)Normalized PSD (dB)  Before WSSAfter WSSWSS spectrumless ISI less ISIICI3−dBBW(b) ξ=0.7Figure 5.5: 400 Gbps system, normalized PSD vs. frequency, with 4 WSSs.−100 −50 0 50 100−80−60−40−20020Frequency (GHz)Normalized PSD (dB)  Before WSSAfter WSSWSS spectrumISI No ICI ISI3−dBBW(a) ξ=1 (Nyq)−100 −50 0 50 100−80−60−40−20020Frequency (GHz)Normalized PSD (dB)  Before WSSAfter WSSWSS spectrum3−dBBWless ISI less ISIICI(b) ξ=0.8Figure 5.6: 1 Tbps system, normalized PSD vs. frequency, with 4 WSSs.5.5.1 Simulation ParametersWe consider a dual-carrier DP QPSK 400 Gbps system in a 100 GHz BW with perSC baud rate of 62.5 Gbaud, and a 1 Tbps DP 16-QAM system that uses 3 SCspacked within 150 GHz with a baud rate of 52.09 Gbaud per SC. For the simula-tions, a 1040 km SSMF with CD parameter value −21 ps2/km and PMD parameter10515 20 2510−610−510−410−310−210−1OSNR, dBBER  No WSS1WSS4WSSNo WSSNyqξ = 0.95ξ = 0.9ξ = 0.85ξ = 0.8ξ = 0.7ξ = 0.5(a) LE ICIC15 20 2510−610−510−410−310−210−1OSNR, dBBER  No WSS4WSS1WSSNo WSSNyqξ = 0.95ξ = 0.9ξ = 0.85ξ = 0.8ξ = 0.7ξ = 0.5(b) Turbo-PICFigure 5.7: 400 Gbps system, BER vs OSNR for FP WDM systems.0.5 ps/√km, RRC roll-off β = 0.1, and LDPC codes with rate 0.8 and codeword-length 64800 bits are adopted, with varying τ and ξ. Each WSS is modeled as a6th-order Gaussian filter, whereby 1 and 4 WSS stages correspond to effective 3-dBBWs of 100 GHz and 89.3 GHz for the 400 Gbps system, and 150 GHz and 133.7 GHzfor the 1 Tbps system, respectively. We use 33 T2-spaced taps for the LE-ICIC, and amaximum iteration count of 10 for the turbo-PIC. Perfect frequency synchronizationand phase-lock is assumed for the simulations. Considering 13 spans of SSMFs withan attenuation constant of 0.2 dB/km and span length of 80 km, the launch poweris set as −5 dBm per SC, for which nonlinear distortions are not significant [20].5.5.2 ISI vs. ICI Trade-offFirst, we consider FP super-Nyquist systems. The normalized PSDs of the 400 Gbpssystem and the 1 Tbps WDM system, filtered through 4 WSSs, are shown in Fig. 5.5a-5.5b and Fig. 5.6a-5.6b, respectively, for the Nyquist and FP configurations. Asshown in Fig. 5.5a and Fig. 5.6a, the SCs located at the edges in a Nyquist WDM10616 18 20 22 2410−610−510−410−310−210−1OSNR, dBBER  1WSS4WSSNo WSS Nyqξ = 0.95ξ = 0.9ξ = 0.85(a) LE ICIC16 18 20 22 2410−610−510−410−310−210−1OSNR, dBBER  NoWSS1WSS4WSSNyqξ = 0.95ξ = 0.9ξ = 0.85(b) Turbo-PICFigure 5.8: 1 Tbps system, BER vs OSNR for FP WDM systems.superchannel suffer from significant ISI due to aggressive filtering by the cascadedROADMs. On the other hand, the FP WDM systems in Fig. 5.6b and Fig. 5.6b areprimarily ICI-limited and suffer more from the spectral overlap of the SCs. Therefore,an ISI vs. ICI trade-off exists for the FP systems that is optimized here through theimplemented ICIC methods.5.5.3 LE-ICIC vs Turbo-PICIn Fig. 5.7 and Fig. 5.8, we present the coded BER performance averaged over 150codewords across all SCs, corresponding to the 400 Gbps and the 1 Tbps FP systems,respectively. The following two performance comparisons are made: (1) LE vs. turbo-PIC, and (2) Nyquist vs. FP WDM. For the first comparison, we note that theperformance gains by the turbo-PIC depend on the frequency spacing ξ and thenumber of WSSs that determines the severity of the WSS-ISI. For example, in Fig. 5.8,for the 1 Tbps system with ξ=0.9 and 4 WSSs in the optical link, turbo-PIC shows a1071.4 dB required OSNR (ROSNR)13 gain over the LE-ICIC. For the 400 Gbps systemwith the same configuration, this gain is observed to be 1.7 dB in Fig. 5.7. Moreover,we observe that for FP super-Nyquist systems, an optimal value of ξ exists. This ishighlighted in Fig. 5.9 and Fig. 5.10 for the case of 4 WSSs, corresponding to the400 Gbps and the 1 Tbps systems, respectively, where ROSNR is plotted as a functionof ξ. Comparing the results for the FP super-Nyquist and the Nyquist WDM systemsin Fig. 5.8, we observe moderate gains of 0.6 dB dB for the case of 1 WSS for the1 Tbps transmission. However, with 4 WSSs in the optical link, the FP super-Nyquistsystem having ξ = 0.9 yields substantial performance gain over the Nyquist WDMsystem that exhibits a high BER-floor due to severe WSS-ISI, as shown in Fig. 5.8.For the 400 Gbps system in Fig. 5.7b, an FP gain of 2.95 dB over Nyquist signalingis observed when turbo-ICIC is employed with ξ = 0.9. Further reduction of ξ toreduce the impact of WSS-ISI induces additional FP-ICI that even the turbo-PIC isunable to address.5.5.4 TFP GainsAdditional ROSNR advantages over Nyquist signaling can be obtained by super-Nyquist WDM systems that enable TFP transmission. We illustrate such gains bymeans of error rate plots in Fig. 5.11 and Fig. 5.12 for the 400 Gbps and the 1 TbpsWDM systems, respectively, with 4 WSSs in the optical link. For the ISIC, we usetruncated TFP-ISI channel such that K=3 and 4 for the 1 Tbps system correspondto 64 and 256-state BCJR, respectively. Similarly, for the 400 Gbps transmission, weuse K=3 and 6, and thereby, employing 8 and 64-state BCJR modules, respectively.Our goal is to optimize the average BER performance over different (τ, ξ) pairs. For13We measure the ROSNR in the post-waterfall region of the BER curves with BER < 10−5, sothat practically error-free transmission is achieved.10817.318.319.320.30.7 0.75 0.8 0.85 0.9 0.95 1ROSNR [dB]ξLinear ICICTurbo-PIC𝝃𝐨𝐩𝐭Figure 5.9: 400 Gbps, ROSNR vs. ξ, with 4 WSSs, illustrating the optimal ξ.20.522.524.50.875 0.9 0.925 0.95 0.975ROSNR [dB]ξLinear ICICTurbo-PIC𝝃𝐨𝐩𝐭1.4 dBFigure 5.10: 1 Tbps, ROSNR vs. ξ, with 4 WSSs, illustrating the optimal ξ.performance comparison, we also include the BER plots of the FP super-Nyquistsystems corresponding to the optimal ξ derived through Fig. 5.7-5.10. Additionally,we include the performance of the Nyquist WDM systems without WSS and with10910 15 20 25 3010−610−510−410−310−210−1OSNR, dBBER  6.6dB7.7dB3.65 dB(τ, ξ) pairs4WSSNoWSSOpt. FPNyq, τ = ξ = 1(1,0.9)(0.85,0.8),3taps(0.9,1),3taps(0.8,1),3taps(0.8,0.9),3taps(0.8,0.9),6tapsFigure 5.11: 400 Gbps, BER vs. OSNR, with 4 WSSs, illustrating additional TFPgains.4 WSSs, as a reference. As shown in Fig. 5.12 for the 1 Tbps system, an optimalτ and ξ combination using BCJR-ISIC with K = 3 significantly outperforms theNyquist-WDM transmission that shows high BER floor. Furthermore, such TFPtransmission yields 0.4 dB ROSNR improvement over the optimized FP system thatemploys optimal frequency-spacing. An additional performance gain of 0.7 dB forthe 1 Tbps transmission, corresponding to K = 4, is achieved by the same TFPtransmission at the price of increased BCJR complexity. Similar patterns are alsoobserved for the 400 Gbps system in Fig. 5.11, where as high as 7.7 dB ROSNRgain is achieved by the optimized TFP transmission over the Nyquist WDM systemhaving the same data rate.11017 18 19 20 21 22 2310−610−510−410−310−210−1OSNR, dBBER  Optimized      FP0.7 dB1.1 dB(τ, ξ) pairs4WSSNo WSSNyq, τ = ξ = 1(1,0.9)(0.9,1)(0.95,0.95)(0.98,0.9)(0.9,0.95)(0.9,0.95),K=4Figure 5.12: 1 Tbps, BER vs. OSNR, with 4 WSSs, illustrating additional TFPgains.Table 5.1: Complexity, memory and latency per codewordMetric LE-ICIC Turbo-PIC ICIC BCJR-ISICComplexity O(N2LpLw) O((M+Lc)NImLw)O(MK2 NImLw)Memory O(NLb) O(NLbIm) O(NLbIm)Latency O(Lb) O(LbIm) O(LbIm)5.5.5 Computational ComplexityDetails of the computational complexities, latency and memory requirements forthe proposed ISIC and ICIC schemes for each decoded codeword are furnished inTable 5.1, where M , Lc, Lp, Lb, Im denote the modulation order, truncated ICI-tapslength for the turbo-PIC, 2-D LMS taps length for the LE-ICIC, LDPC codewordlength in bits and the maximum turbo iteration count, respectively, and Lw =Lblog2Mis the number of modulated symbols corresponding to each codeword. Our numericalresults show that at the expense of additional computational cost, TFP transmission111with efficient interference mitigation techniques can significantly improve the super-Nyquist performance in the presence of narrow WSS-filtering, by optimizing the timeand frequency spacing parameters. Our numerical results also suggest that suchimprovements are not achievable by packing the WDM carriers in either time orfrequency dimension alone.5.6 ConclusionsIn this chapter, we developed designs for high data rate TFP superchannel trans-missions that may be subject to aggressive filtering due to WSS. At the expenseof introducing deliberate known interference, super-Nyquist systems can effectivelyreduce the impact of such narrow-filtering by packing the SCs more efficiently. Thisrequires powerful ICI mitigation schemes such as the presented turbo-PIC. Simulationresults show that by employing BCJR based ISIC, together with LE or turbo ICIC,super-Nyquist WDM systems targeting 400 Gbps data rate in a 100 GHz BW, and1 Tbps data rate in a 150 GHz channel, offer significant ROSNR gains over Nyquisttransmission under similar conditions.112Chapter 6Flexible Designs for SpectrallyEfficient Time Frequency PackedSuperchannels6.1 IntroductionWe have seen in the previous chapter that TFP WDM superchannel transmission isan attractive, spectrally efficient technology to achieve Terabit data rates in long-haul OFC systems. In the existing optical super-Nyquist literature, time-only [14,17, 78, 79] and frequency-only [16, 18–20] packing have been considered quite exten-sively, which introduce either ISI or ICI. However, packing the symbols in both timeand frequency dimensions can theoretically provide higher achievable rates [13], atthe price of introducing ISI and ICI simultaneously. For the TFP designs proposedin Chapter 5, the employed powerful ISIC and ICIC algorithms exploited the knowninformation about the TFP ISI and ICI channel, without performing additional chan-nel estimation operation. However, as we will show in this chapter, an interferencechannel estimation approach can offer significant performance improvement for theconsidered TFP transmission targeting Tbps data rates.Another practical challenge that affects WDM transmission is the signal distortion113caused by PN stemming from the spectral linewidth of the transmitter and receiverlaser beams [83, 84]. However, the TFP receiver designs in Chapter 4 assumed perfectCPR, and therefore, no PN compensation algorithm was presented for the consideredTFP systems. It is important to note that the application of HoM formats makes thecommunication systems more sensitive to PN. Moreover, the impact of PN is moresevere for TFP superchannels, since the presence of ISI and ICI precludes the directapplication of off-the-shelf PN mitigation algorithms that are tailored to Nyquisttransmission [70, 82, 84]. Therefore, we need sophisticated signal processing tools tocounter TFP interference together with powerful PN cancellation strategies.In this chapter, we consider both temporal and spectral overlap of the SCs toenable a flexible TFP superchannel transmission. To accomplish this, we presentsignal processing techniques to efficiently mitigate TFP interference, fiber-opticalnon-idealities, and PN. This work encompasses the following three novel contribu-tions.1. A joint ISI and ICI channel estimation method, coupled with PMD equaliza-tion and a coarse PN estimation (CPNE) is proposed. While the TFP-inducedinterference channel is a-priori known [121], such estimation is necessary be-cause (a) estimating the exact number of ISI taps required by the BCJR-ISICand forcing the remaining ISI-taps to zero can lead to significant performanceimprovement [17], and (b) interference stemming from additional sources in theoptical link, such as the electrical/optical filters, can be accounted for. In con-junction with PMD equalization and interference channel estimation, we alsojointly perform a coarse CPR, which is beneficial since (a) it facilitates betterTFP channel estimation by minimizing the overall MSE during the initial pilotstransmission phase, and (b) it offers improved bootstrapping for more sophis-114ticated iterative PN mitigation schemes to clean up the residual PN, especiallywhen LLW is high, and HoM formats are used.2. A novel, iterative, modulation-format-independent PN estimation method, inthe form of post-FEC refined adaptation, is proposed. Moreover, the FG-based solutions presented in [17] and [70] are also properly amended to accountfor the TFP ISI and ICI. While the authors of [17] consider post-FEC harddecisions for the FG metric computations for the case of only TFP-ISI andQPSK transmission, we use soft information to improve robustness againsterror-propagation in HoM systems in the presence of both ISI and ICI.3. A serial-and-parallel combined interference cancellation (SPCIC) based ICICscheme is presented. Different from the existing PIC structures [13, 80, 117],the proposed SPCIC encapsulates a conceptual combination of the SIC andthe PIC paradigms, to offer performance improvements. The proposed ICICsolution in tandem with BCJR-ISIC is shown to exhibit excellent tolerance tohigh LLW and aggressive optical filtering due to cascaded ROADM nodes thatmay be present in the fiber link.The remainder of the chapter is organized as follows. The system model is intro-duced in Section 6.2. In Section 6.3, we propose a joint TFP-interference and a CPNEstrategy. In Section 6.4, we investigate two iterative CPR algorithms. Proposed TFPinterference mitigation techniques are detailed in Section 6.5, followed by numericalresults for optical TFP systems presented in Section 6.6. Finally, Section 6.7 providesconcluding remarks.115SC NX-pol bitsLDPC Enc QAMMap.RRCOpt. Front-endY-pol bitsLDPC Enc QAMMap.RRCWDM......SC 1SC kCoh. RxSSMFDSPD/AD/AIQD/AD/AIQA/DA/DIQA/DA/DIQRRCRRCFigure 6.1: TFP WDM system model.6.2 System ModelIn this work, we consider a DP TFP WDM transmission for longhaul optical fibercommunication [14, 17]. The schematics of such a system are shown in Fig. 6.1. Foreach of the X and Y polarization data streams of the mth SC, m∈1, 2, · · · , N , withN being the total number of SCs, an LDPC coded and modulated data stream xmis shaped by an RRC pulse p with a roll-off factor β. The digital samples are thentransformed into analog signals via digital-to-analog converters (DACs), followed byconversion to the optical domain using individual lasers and MZ modulators. Theequivalent baseband transmitted signal for the X-polarization branch can be writtenassx(t)=∑l∑mxm[l]p(t−lτT )ej(2pi(m−N+12 )∆ft+θ(m)tx (t)), (6.1)where ∆f=ξ 1+βTis the frequency-spacing between the adjacent SCs, τ and ξ are thetime and frequency compression ratios, respectively, such that τ = ξ= 1 correspondsto the Nyquist WDM system, l is the symbol index, 1τTis the per-SC baud rate, andθ(m)tx is the transmitter laser PN corresponding to the mth SC.The transmitted superchannel signal propagates through multiple spans of SSMFs,whereby the optical signal suffers distortion due to fiber-optical impairments, such asthe CD and the PMD [60, 96, 122]. In the presence of CD and the first-order PMD,116the frequency response of the 2×2 DP fiber channel can be written as [96, 122]H(ω) =cosϕ − sinϕsinϕ cosϕejω τd2 00 e−jωτd2 cosϕ sinϕ− sinϕ cosϕ e−jβ22 Lω2 , (6.2)where ω is angular frequency, τd is the differential group delay (DGD), β2 is the CDparameter, L is the fiber length, ϕ is the angle between the reference polarizationsand the principal states of polarization (PSP) of the fiber [96].At the coherent receiver, the received signal is converted to digital samples viaanalog-to-digital converters (ADC). Thereafter, the RRC matched-filtered digitalsamples of the mth SC, distorted by CD, PMD, amplified spontaneous emission (ASE)noise [122], and the receiver laser PN θ(m)r,x , are fed as inputs to the receiver DSP unitas detailed in the next section. The transmitter and the receiver PN for all SCs aremodeled as Wiener processes [123] such that for the mth SC, m∈1, 2, · · · , N ,θ(m)tx [k]= θ(m)tx [k − 1] + ∆(m)tx w(m)tx [k] , (6.3)θ(m)rx [k]= θ(m)rx [k − 1] + ∆(m)rx w(m)rx [k] , (6.4)where k is the sample-index corresponding to the discrete-time baseband model,w(m)tx and w(m)rx are the independent identically distributed standard Gaussian randomvariables, ∆(m)tx = ∆(m)rx =√2pifWTs are the Wiener process standard deviations, withTs being the sampling time in seconds and fW being the LLW, i.e., the full-widthhalf maximum spectral BW of the transmitter and the receiver lasers, respectively,in Hz [123].117CD Eq.mth SC CD Eq.X-polY-pol2x2PMD EqCoh.RxISIICICPNEPMD filt.CD Eq. outputsfrom other SCsKnown pilots{LMS-basedEstimatesXXde-rotationFeedforward loop Feedback loopSIC/PICBCJRBCJRBCJRBCJRLDPC DecLDPC DecLMS/FGIPNEPost-FEC LLRs/Hard-decisionsfrom other SCsde-rotationde-rotationde-rotationPost-FEC LLRsBits out: Blocks After LDPC iterations: Scalar signals: Vector signalsPost-FEC LLRs: Estimation modulesX-polY-polPost-FEC LLRs from other SCSFigure 6.2: Jointly estimating PMD filter, TFP interference and PN.6.3 Interference Channel Estimation and CPNEInspired by the algorithm presented in [17], we propose an adaptive TFP receiver de-sign, where we jointly estimate the TFP-interference channel, the 2×2 PMD equalizertap co-efficients, and perform a coarse estimation of the laser PN. Different from [17],that restricts itself to ISI channel estimation only, in this work, we estimate the ISIand the ICI impulse responses simultaneously, coupled with a coarse CPR. More-over, we also enforce the real-valued constraint on the ISI-taps adaptation algorithmto employ a reduced-complexity BCJR equalization by separately processing the Iand Q components. We will show through the numerical results in Section 6.6 thatsuch an adaptive strategy facilitates flexible superchannel transmission by offeringsignificant performance advantages over non-adaptive TFP design, such as the onepresented in Chapter 5.1186.3.1 DSP ModulesThe receiver DSP design is shown in Fig. 6.2 for the mth SC of the TFP superchannel.As shown in the figure, the received signal in each polarization branch is first processedby a time-invariant CD compensating filter implemented in the frequency domainthrough overlap-and-add method [60, 96, 122]. Thereafter, the CD equalized samplesare filtered by a fractionally-spaced 2 × 2 PMD equalizer to remove the cross-talkbetween the two polarization streams. The estimates of the PMD filter coefficients,TFP-interference channel and a coarse PN are obtained through a pilot symbols-aided LMS-based adaptation algorithm, the details of which are relegated to thenext subsection. Following the polarization recovery, PN cancellation per polarizationbranch for each SC is accomplished by using the CPNEs or the iterative PN estimates.Next, the PN corrected received signal is processed by the turbo ICIC and the BCJR-ISIC modules. After decoding, soft informations in the form of LLRs are fed back fromthe LDPC decoders to mitigate TFP interference and PN, iteratively. The details ofthe iterative PN cancellation structure, together with ICIC and ISIC operations willbe presented in Sections 6.4 and LMS Update EquationsTo derive the update equations for the joint estimation algorithm, we formulate thecolumn vectors a(m)k and u(m)k corresponding to the constellation symbols and theinput samples to the 2×2 PMD equalizer, respectively, by stacking the X and Ypolarized signals of the mth SC at the kth sample time. The error signal is thencomputed as the difference between the phase-rotated PMD filter output and the“desired signal” [17], whereby the desired signal computation incorporates the effectsof the TFP interference into the clean pilot symbols, as shown below. The combined119error signal vector with the X and Y polarization symbols is written as(m)k = [ε(m)k,x, ε(m)k,y]T = z(m)k︸︷︷︸filtered output [e−jθ(m)x,k , e−jθ(m)y,k ]T︸ ︷︷ ︸phase rotation− d(m)k︸︷︷︸desired signal, (6.5)wherez(m)k = [z(m)x,k , z(m)y,k ]T =Nw−1∑i=0W(m)i,k u(m)k−i , (6.6)d(m)k = [δ(m)k,x, δ(m)k,y]T =Ls∑j=−Lsh(m)j,k  a(m)k−j +∑n 6=mLc∑ν=−Lcg(n,m)ν,k  aˆ(n)k−ν . (6.7)In (6.5)-(6.7), ε(m)k,x/y, z(m)k,x/y and δ(m)k,x/y are respectively the error signal, PMD filteredoutput and the desired signal corresponding to the X or the Y polarization (thesubscript x/y means “X respectively Y”) for the mth SC at the kth sample time,W(m)i,k is the ith matrix-tap of the T2-spaced PMD equalizer, Nw is the total numberof PMD filter taps, h(m)j,k and g(n,m)ν,k are the jth and the νth symbol-spaced ISI andICI tap vector, respectively, with Ls and Lc being the total number of ISI and ICIchannel taps, respectively, [· · · ]T denotes the transpose of a vector,  denotes theelementwise vector product, and aˆ(n)k =a(n)k e±j2pi∆fk denotes the rotated constellationsymbol for the nth SC, with ± sign determined from (6.1) depending on the relativepositions of the SCs.From (6.5), we define the MSE asMtot = E(N∑m=1‖(m)k ‖2), (6.8)where E(·) denotes the expectation operator. For ease of formulation of the LMS120update equations based on the gradient decent algorithm [104], we exploit the sym-metry of the TFP channel by enforcing the real and symmetry condition on the ISIimpulse response, and conjugate symmetry on the ICI channel, such thath(m)j,k =(h(m)j,k)∗, m∈ {1, 2, . . . , N},−Ls≤j≤Ls ,h(m)j,k = h(m)−j,k , m∈ {1, 2, . . . , N},−Ls≤j≤Ls ,g(n,m)ν,k =(g(n,m)−ν,k)∗, n∈ {1, 2, . . . , N},−Lc≤ν≤Lc.Finally, computing the gradients ∂Mtot∂W(m)α,k, ∂Mtot∂h(m)β,k, ∂Mtot∂g(m)γ,kand ∂Mtot∂θ(m)x/y,k, where 0 ≤ α ≤Nw − 1, 0 ≤ β ≤ Ls, 0 ≤ γ ≤ Lc, we can write the LMS update equations asW(m)α,k+1 =W(m)α,k − µw([ejθ(m)x,k , ejθ(m)y,k ]T (m)k)(u(m)k−α)H, (6.9)h(m)β,k+1 =h(m)β,k + µhRe[((m)k (a(m)k−β+a(m)k+β)∗)], (6.10)g(n,m)γ,k+1 =g(n,m)γ,k + µg((n)k (ˆa(m)k−γ)∗+((n)k)∗ aˆ(m)k+γ), (6.11)θ(m)x/y,k+1 = θ(m)x/y,k + µθIm[ (δ(m)x/y,k)∗z(m)x/y,ke−jθ(m)x/y,k]. (6.12)where µw>0, µh>0 and µg>0 are the step size parameters, Re[·], (·)∗ and (·)H denotethe real-part, complex conjugation and matrix Hermitian operations, respectively.6.3.3 Data-aided and Decisions-directed AdaptationTo initiate the above adaptive estimation algorithm and accomplish LMS conver-gence, we exploit the continuous pilot symbols transmission during the link-setupphase at the beginning of data transmission [17, 20, 78]. Thereafter, the CPNEsand PMD filter taps are slowly adjusted based on the blocks of Np periodic pilotsymbols inserted uniformly across the data frame after every Nd data symbols, to121continuously track PN and the slow rotation of the PSP [60, 96, 122, 124]. Thepilot symbols density p= NpNp+Ndis chosen to meet a desired trade-off between per-formance and transmission overhead. From the CPNEs obtained during the periodicpilots transmission, interpolation strategies are adopted to account for PN variationover each symbol duration. In this chapter, we apply linear interpolation for suchpurposes. To remove the residual PN, the CPNEs thus obtained in every symbolduration are used to bootstrap more powerful FG-based and LMS-based iterativePN mitigation algorithms presented in the next section. Since the TFP interferencechannel is not likely to change over the course of the transmission, the ISI and ICIimpulse responses can be estimated only once during the link-setup, followed by veryslow adjustments based on the post-LDPC symbol-decisions [17].6.4 Iterative PN Estimation (IPNE)After the LMS-based coarse CPR is accomplished, we employ more involved iterativealgorithms to remove the residual PN. At each LDPC iteration, the a-posteriori LLRsare fed back from the decoders for the purposes of (a) iterative PN estimation andcompensation, and (b) TFP interference cancellation as detailed in Section 6.5. Inthis section, we present two IPNE schemes, namely the low-complexity LMS-basedIPNE (LIPNE) and the high-performance factor graph based IPNE (FGIPNE) [70].The LIPNE, which requires low computational cost and buffer-space, offers decentperformance for small values of LLW. Additionally, the functionality of the LIPNEdoes not depend on the modulation format and the explicit knowledge of the PNstatistics. On the other hand, the FGIPNE shows excellent tolerance to strong PNand severe TFP interference, at the expense of modulation format dependency, highercomplexity, and buffering. Moreover, the computation of the FGIPNE metrics re-122quires an estimate of the aggregate variance of the combined Wiener processes of thetransmitter and receiver laser PN [70]. Depending on the specific TFP applicationscenario, the most suitable IPNE method can be chosen such that a desired trade-offbetween performance and complexity is achieved. An analysis of such trade-off isinvestigated in Section LIPNEFor this scheme, we refine the LMS based CPNEs obtained in (6.12) for each symbolduration, using the LLRs fed back from the LDPC decoders to make hard symbol-decisions. To account for the TFP ISI and ICI, we design the effective pilots byreconstructing the desired signals δ(m)x/y,k according to (6.7) at every iteration, usingthe estimated TFP interference channel and the hard-decisions of symbols from allSCs.While such post-LDPC decision-directed LIPNE is computationally simpler thanother state-space based iterative methods, it may produce sub-optimal performancedue to the error-propagation when hard-symbol decisions are erroneous, especiallywith HoM in the presence of severe ISI and ICI. As shown through the numericalresults in Section 6.6, LPINE for TFP systems incur performance degradation whenLLW is high. Motivated by this, we investigate the more powerful FGIPNE approachpresented in [70], whereby we slightly modify the metric computation to account forthe TFP-ISI and ICI.6.4.2 FGIPNEWe take into account the TFP interference by amending the state-space based FGIPNEalgorithm in [70] to tailor it to our considered super-Nyquist transmission. We re-123mark that the authors of [17] also adopted similar strategies. However, there are thefollowing two main differences of our method compared to theirs: (a) as opposed tothe post-LDPC hard-decisions, we use soft values for the FGIPNE metric computa-tions to reduce the impact of error-propagation, especially when modulation formatslarger than QPSK are employed, and (b) in addition to the TFP-ISI, our super-Nyquist systems also introduce ICI, and therefore, inter-SC data processing has tobe facilitated.Inspired by the technique adopted in [17], we extract the MAP estimates of theindividual PN processes for both polarization streams of all SCs in the superchan-nel. For this, we perform the Tikhonov-parameterization based “forward-backwardrecursive algorithm” [70], whereby we exploit the symbol probabilities derived fromthe LLRs fed back by the LDPC decoders, iteratively. To account for the TFP-ISIand ICI, we properly modify the forward and backward metric computation presentedin [70]. Such modifications together with a relevant review of the FG based algorithmin [70] are shown in Appendix E.1.Following the fine-tuned CPR accomplished through either the LIPNE or theFGIPNE method, the PN mitigated samples are processed by the ISIC and ICICmodules, as detailed in the following.6.5 Interference CancellationIn this section, we describe the interference cancellation operations in detail.6.5.1 Basic Turbo ISIC-ICIC StructureAfter compensating for the PN through CPNE and/or IPNE algorithms, the inputsfrom all SCs are jointly processed by the interference canceler. To mitigate the inter-124ICI gen𝐼",$𝑟&,$ +Σ--LDPCDecbits out 𝐢𝐭 = 𝐢𝐭𝐦𝐚𝐱𝐋𝐋𝐑𝟏 ≤ 𝐢𝐭 < 𝐢𝐭𝒎𝒂𝒙LLR toSoft SymICI gen𝐼$,"ICI gen𝐼$,5ICI gen𝐼5,$SC 2X-pol. BCJR𝟏 ≤ 𝐢𝐭 < 𝐢𝐭𝒎𝒂𝒙𝟏 ≤ 𝐢𝐭 < 𝐢𝐭𝒎𝒂𝒙To SC 1To SC 3𝐢𝐭𝟏 < 𝐢𝐭 < 𝐢𝐭𝒎𝒂𝒙𝐢𝐭𝟏 < 𝐢𝐭 < 𝐢𝐭𝒎𝒂𝒙From SC 1From SC 3𝑟&,$6789:Figure 6.3: BCJR-ISIC+SPCIC-ICIC, shown for the example of a 3-SC WDM sys-tem.ference induced by the TFP transmission, the ISI and ICI impulse responses estimatedin Section 6.3 are provided as inputs to the ISIC and ICIC modules, respectively. ForISIC, the BCJR algorithm based on Ungerboeck’s observation model [125, 126] isemployed. For ICIC, we present a new SPCIC method that involves multiple stagesof PIC and SIC scheduling. Such an ICIC scheme offers significant performancegains over PIC-only ICIC approaches, such as [13, 80] and the method presented inChapter 5, for the TFP superchannels considered in this chapter.The basic operational principles of the turbo ICIC, in conjunction with the BCJR-ISIC, are shown in Fig. 6.3 for the X-polarization of the mth SC, m∈{1, 2, . . . , N}.Following the CD and PMD compensation, and carrier recovery, the estimated ICIchannel and the LLRs fed back from the LDPC decoders are used to compute the softestimates of the ICI In,m, n 6= m, n=1, 2, 3. For the computational details of such softestimates, interested readers are referred to [13, 80, 117]. The soft ICI estimates thus125obtained are then removed from the received samples rX,m of the mth SC, to producethe signal rcleanX,m as shown in Fig. 6.3 when m = 2. Next, at every turbo iteration,BCJR is performed separately on the I and Q branches of both polarization streamsfor each SC, using the estimated ISI taps derived in Section 6.3. At the end of thefinal iteration count itmax, output bits are generated from the LDPC decoders. Thedetails of the SIC and PIC scheduling for the ICIC operation are discussed in thefollowing.6.5.2 ICIC Scheduling: SPCICThe central SCs in a TFP superchannel suffer from stronger ICI compared to the edgeSCs, with the possibility of less reliable LLRs for the first few iterations. Therefore,different from the PIC approach in [13] and the method presented in Chapter 5,we propose to perform ICIC for the central SCs first, prior to initiating the ICICoperations for the edge SCs, until an iteration count threshold is reached, where suchthreshold is a design parameter depending on the values of β, τ and ξ. Based on thenumber of SCs in the superchannel, multiple thresholds corresponding to differentstages of SIC scheduling can be designed. The pseudo-code for the SPCIC operationis provided in Algorithm 1.For the example of a 3 SC superchannel shown in Fig. 6.3, ICIC for the 2nd SC onlyis initiated at the beginning of LDPC iterations, up to an iteration count thresholdit1. Next, ICIC for all 3 SCs is enabled simultaneously via soft-information exchangeacross all of them at every subsequent turbo iteration. When the number of SCs islarger than 3, similar SPCIC scheduling can be adopted as detailed in Algorithm 1,where at the first stage of SIC iterations, ICIC is performed for the 2nd SCs fromboth superchannel edges; for the second stage of iterations, ICIC is conducted on the126Algorithm 1 SPCIC algorithm in conjunction with BCJR-ISIC, shown for the X-polarization of all SCs.At the 0th Iteration:1: for all n ∈ {1, 2, . . . , N} do2: rcleanX,n ← rX,n3: BCJR-ISIC on rcleanX,n (ICI treated as noise)4: end forICIC Initiation:5: for it = 1 : itmax do6: if N ≤ 2 then # No SPCIC7: SPCIC not applicable/PIC-only approach.8: else if N = 3 then # SPCIC, 1 SIC/PIC stage9: rcleanX,2 ← rX,2 − ICI from left SC10: − ICI from right SC11: if it > it1 then12: for all n ∈ {1, 2, 3} do13: rcleanX,n ← rX,n − ICI from left SC14: − ICI from right SC15: end for16: end if17: else # SPCIC, multple SIC/PIC stages18: rcleanX,2 ← rX,2 − ICI from left SC19: rcleanX,N−1 ← rX,N−1 − ICI from right SC20: if it ≤ itbN+12c−1 then21: if it > it1 then22: rcleanX,3 ← rX,3 − ICI from left SC23: rcleanX,N−2 ← rX,N−2−ICI from right SC24: end if25:.........26: if it > itbN+12c−2 then27: if N is odd then28: rcleanX,bN+12c ← rX,bN+12 c−ICI from left SC29: −ICI from right SC30: else31: rcleanX,bN+12c ← rX,bN+12 c−ICI from left SC32: rcleanX,dN+12e ← rX,dN+12 e−ICI from right SC33: end if34: end if12735: else36: for all n ∈ {1, 2, . . . , N} do37: rcleanX,n ← rX,n − ICI from left SC38: − ICI from right SC39: end for40: end if41: end if42: for all n ∈ {1, 2, . . . , N} do43: BCJR-ISIC on rcleanX,n (ICI partially/fully removed)44: end for45: end for3rd SCs from both edges, and so on, until the central SC is reached. Thereafter, ICICfor all SCs are performed simultaneously through a PIC-based scheduling.6.6 Numerical ResultsIn this section, we present numerical results to show the benefits of the proposed TFPsystems. For this, we consider a DP 16-QAM 3-SC WDM superchannel having perSC baud rate 62.5 Gbaud corresponding to a 1.2 Tbps net data rate. We accomplishsuitable BW compression by choosing appropriate values of τ and ξ such that the TFPsuperchannels fit within an aggregate BW not exceeding 175 GHz. We remark thatsuch data rates and BW constraints serve as a realistic target for the next generationoptical networks (see e.g. [120] and references therein). For example, [120] presentsrecent works demonstrating 1000 km fiber transmission of DP 16-QAM dual-carrierNyquist superchannels achieving 400 Gbps data rate packed within a 75 GHz grid, orequivalently, 1.2 Tbps with an aggregate BW of 225 GHz. By way of our spectrallyefficient TFP design in this chapter, our proposed superchannels occupy substantiallylower BW compared to [120]. We also note that spectrally more efficient NyquistWDM systems with similar or higher data rates can be realized employing HoM128Table 6.1: Simulation parametersLDPC parameters ValuesStandard compliance DVB-S2 [109]Block length 64800 bitsCode rate 0.8# Internal iterations 50Transmission parameters ValuesModulation 16-QAMBaud rate per SC 62.5 GbaudN 3β 0.1τ , ξ VaryingLaunch power -5 dBm [20]LLW 10− 400 kHz [85, 127]Fiber parameters ValuesSSMF span length 80 km [128]SSMF number of spans 13β2 −21 ps2/km [128, 129]τd 0.5 ps/√km [128]PSP rotation rate 2.5 kHzFiber amplifier noise-figure 4.5 dB [128–130]Fiber attenuation 0.2 dB/km [128–130]# WSSs 0 & 4 [119]Each WSS 3-dB BW 187.5 GHzRx DSP parameters ValuesNw 19T2-spacedLc 9Ls 3 (64-state BCJR)it1 3itmax 10 [13, 17]Np 10Nd 150 & 300formats, such as [131]. However, such systems have significantly lower transmissionreach due to the application of larger signal constellations.1296.6.1 Simulation ParametersThe simulation parameters used for our numerical evaluation are listed in Table 6.1.The values of the parameters are chosen in alignment with practical optical fiber sys-tems [60, 83, 85, 86, 96, 122, 127–129]. Except for the results in Fig. 6.6, we simulatea fixed 13 spans of the SSMFs corresponding to a fiber length L= 1040 km. For allresults except those in Fig. 6.8 and Fig. 6.9, a fixed LLW of 75 kHz is considered [86].We simulate 30000 symbols at the beginning of transmission for link setup [17, 20],followed by Np = 10, Nd = 150, 300, corresponding to 6% and 3% pilot-densities, re-spectively. Nd = 150 is used for all results except those in Fig. 6.8. A total of 150codewords are transmitted to evaluate the average performance of both polarizationsand all SCs. Moreover, perfect time and frequency synchronization, and no fibernonlinearities14 are assumed.6.6.2 Interference Channel Estimation and CancellationGainsWe first investigate the performance of the proposed joint estimation algorithm byshowing the MSE convergence (see (6.8)) corresponding to the central SC in the 3-SCsuperchannel, as a function of the (pilot) symbol index in Fig. 6.4 during the linksetup phase, for different operating OSNRs and LLW = 75 kHz. As shown in thefigure, for all τ, ξ pairs, the MSEs of both polarization streams settle to stable valuesensuring that the equalizer tap-coefficients and the estimated interference channelsconverge to nearly stationary values.The estimated TFP-ISI and ICI impulse responses are then used by the ISIC and14A low launch power of −5 dBm per SC ensures that the effects of nonlinearity is not signif-icant [20]. Investigation of the benefits of the proposed algorithms in a practical setup, with orwithout nonlinearity compensation techniques, is subject to future work.1300 2000 4000 6000 8000 10000−20−15−10−5SymbolsMSE [dB]  (τ, ξ) pairs(0.9, 0.95), OSNR = 15 dB(0.85, 0.95), OSNR = 20 dB(0.9, 0.9), OSNR = 25 dBX−polY−polFigure 6.4: MSE convergence, 75 kHz LLW, varying τ and ξ.ICIC modules. The PMD filter co-efficients and the CPNEs after the link setup phaseare updated only during the periodic pilots as described in Section 6.3, followed bylinear interpolation of the data-aided PN estimates to obtain per symbol CPNEs,which are later used to bootstrap the LIPNE or the FGIPNE algorithms.Next, we show the benefits of our proposed TFP design over the time-only packedsuper-Nyquist systems, such as [14, 17, 78, 119], in Fig. 6.5. For this, we plot theBER performance averaged over all SCs. For both the time-only packed and the TFPsystems, we use 6% pilots for the CPNE, followed by FGIPNE to mitigate the effectsof PN. For TFP, SPCIC-ICIC in tandem with BCJR-ISIC is applied to mitigate theTFP-interference, while for time-only packing, only BCJR-ISIC is used and no ICIchannel estimation is performed since ξ = 1 for such systems. For a fair comparison,we choose the τ, ξ values for the TFP transmission and the equivalent τ value for thetime-only packing such that both systems achieve the same SE and hence, same BW,13120 21 22 23 2410−610−410−2OSNR, dBBER  (0.9,0.9)(0.84,1)(0.85,0.95)(0.82167,1)169.47 GHz173.25 GHz(τ, ξ) pairs1.1 dB2.1 dBFigure 6.5: BER vs. OSNR, highlighting the benefits of the proposed TFP designover time-only packing. 1040 km fiber, 75 kHz LLW, CPNE+FGIPNE, 6% pilotdensity, varying τ and ξ.with all systems corresponding to 1.2 Tbps data rate. For example, super-NyquistWDM systems with τ = 0.9 and ξ = 0.9 corresponds to τ = 0.84 and ξ = 1 to occupya BW of 173.25 GHz. For this setting, the proposed TFP system yields a perfor-mance improvement of 1.1 dB over the time-only packed transmission, as highlightedin Fig. 6.5. For higher SE values, such gains due to TFP transmission increase. Forexample, a TFP (τ, ξ) combination of (0.85, 0.95) offers 2.1 dB OSNR gain over anequivalent time-packed (0.82167, 1) system that occupies the same BW 169.47 GHz.The results show that instead of aggressively packing the SCs in one dimension, rela-tively benign packing in both the temporal and spectral dimensions can be beneficialwhen the proposed SPCIC-ICIC together with BCJR-ISIC is employed.To further quantify the effectiveness of the proposed TFP design, we show theSE achieved by the proposed TFP systems employing different τ, ξ combinations in1320 500 1000 1500 20006. (km)SE (bits/s/Hz)  (0.87, 1)(0.82167, 1) (0.85, 0.95)(0.84, 1)(0.9, 0.95)(0.9, 0.9)240−km960−km320−km400−km160−kmProposed[17]Chap. 5Chap. 5 with SPCICFigure 6.6: SE vs. distance, highlighting the benefits of the proposed TFP designover time-only packing and other TFP designs. 75 kHz LLW, CPNE+FGIPNE, 6%pilot density, varying τ and ξ.Fig. 6.6, as a function of the transmission distance. For an N -SC DP TFP super-channel with a code rate Rc, modulation size M , and pilot density p%, we computethe SE asSE =2NRc log2Mτ(1 + β) [1 + (N − 1) ξ](1− p100)bits/s/Hz . (6.13)In Fig. 6.6, we also include the SE values of the following two benchmark schemes forreference: (i) the method presented in Chapter 5, employing PIC-ICIC and BCJR-ISIC with the absence of channel estimation, labeled as “Chap. 5” and (ii) time-packed ISI-only systems similar to [17] that achieve the same SE. We observe thata substantial distance improvement of 240− 960 km is achieved by the proposedtechnique over (i) mentioned above. Moreover, to show the benefit of the proposed133SPCIC over the PIC-based ICIC scheme, we also evaluate the performance of (i) byreplacing the PIC-ICIC with SPCIC, indicated by the legend “Chap. 5 with SPCIC”in Fig. 6.6, which increases the transmission distance by 240 km for τ = ξ = 0.9,as highlighted in the figure. Finally, by comparing the proposed TFP systems with(ii) for the same SE values, we observe a link distance improvement by 2−5 spanscorresponding to 160−400 km for different τ , ξ pairs.6.6.3 Tolerance to Cascaded ROADMsWhile propagating through the optical link, a superchannel signal may be subjected toadditional narrow optical filtering due to multiple network elements, such as cascadesof ROADM nodes implemented by WSSs [118–120]. The effect of such aggressivefiltering manifests itself as severe ISI for the edge SCs of the WDM superchannel asdetailed in Chapter 5. The TFP designs proposed in Chapter 5 applied BCJR-ISICfor the deterministic TFP-ISI mitigation without adopting any channel estimationstrategy, while the WSS-ISI was countered through the 2×2 linear PMD equalizer.When a target data rate is intended within a fixed BW, which is dictated by theeffective 3-dB BW of the cascaded WSSs, optimizing the τ, ξ parameters can offerthe best performance under such circumstances, in the form of lowest ROSNR toattain error-free transmission (see Chapter 5). In this section, we perform a similarexercise with the considered 1.2 Tbps TFP WDM systems by applying the proposedinterference and PN estimation and cancellation strategies. Our presented design hasthe added advantage of the ability to estimate the combined interference stemmingfrom the TFP transmission and WSS filtering, both of which can be equalized throughthe BCJR-ISIC used in conjunction with the proposed SPCIC-ICIC.For this investigation, we consider 4 WSSs in the 1040 km fiber link, with each13418 20 22 24 26 2810−610−410−2100OSNR, dBBER  1 dB0 WSS 0.75dB2.65 dB4WSS166.9 GHz1.8 dBNyquistNyquist, Chap. 5(0.9,0.95), Chap. 5(0.9,0.9), Proposed(0.9,0.9), Chap. 5(0.9,0.95), ProposedFigure 6.7: BER vs. OSNR, showing tolerance of the proposed scheme to cascadedWSSs. 1040 km fiber, 75 kHz LLW, CPNE+FGIPNE, 6% pilot density, varying τand ξ.WSS stage modeled as a 6th-order Gaussian filter [118] having a 3-dB BW of 187.5 GHz,producing an effective 3-dB BW of 166.88 GHz for the cascaded structure. We notethat prior to WSS filtering, the aggregate BW of the Nyquist WDM superchannelis 206.25 GHz, which can be reduced by enabling a TFP transmission to moderatethe impact of the narrow filtering in order to achieve the same data rate of 1.2 Tbps.Accounting for the 6% pilot density, spectral restriction by such WSS filtering cor-responds to an SE value of 6.74 bits/s/Hz. In Fig. 6.7, we plot the average BER ofall SCs as a function of OSNR. The figure indicates that significant gains are offeredby the proposed design over the method presented in Chapter 5 for all combinationsof τ and ξ. For example, OSNR gains of 1 dB and 1.8 dB are highlighted in thefigure corresponding to the Nyquist WDM and the τ =0.9, ξ=0.95 TFP system, re-spectively. The primary reason for such performance improvement is estimating the13520 40 60 80 10000. (kHz)ROSNR Penalty (dB)  0.2 dB0.4 dB0.18dB0.15dB0.3 dBNyquist WDMCPNE−only, 6%LIPNE, 10 it., 6%FGIPNE, 10 it., 6%CPNE−only, 3%LIPNE, 10 it., 3%FGIPNE, 10 it., 3%Figure 6.8: ROSNR penalty vs. LLW for Nyquist WDM, showing benefits andlimitations of CPNE, LIPNE and FGIPNE, having varying pilot densities.combined effects of the WSS-ISI and the TFP-ISI together, followed by BCJR-ISICto counter the overall interference, as opposed to the sub-optimal linear equalizationof the WSS-ISI in Chapter 5. We also note that the optimal TFP parameter combina-tion remains the same for the proposed TFP design as that in Chapter 5, i.e. τ = 0.9and ξ = 0.95, which produces an OSNR gain of 2.65 dB over the Nyquist WDM sys-tem under similar conditions. Additionally, we observe that the performance of theTFP system with such optimal τ, ξ combination filtered through 4-WSSs is within0.75 dB of that achieved by the same system without WSS filters in the optical link.13650 100 150 200 250 300 350 400LLW (kHz)00.511.52OSNR Penalty (dB)CPNE-onlyLIPNE"Perfect-decision" CPNEFGIPNENyquist WDM 1 dB0.2 dB(a) Nyquist WDM, i.e. τ = ξ = 1.50 100 150 200 250 300 350 400LLW (kHz)00.511.52OSNR Penalty (dB)CPNE-onlyLIPNE"Perfect-decision" CPNEFGIPNE = 0.9,  = 0.951 dB0.25 dB(b) τ = 0.9, ξ = 0.95.Figure 6.9: ROSNR penalty vs. LLW, showing benefits and limitations of CPNE,LIPNE and FGIPNE, 6% pilot density.6.6.4 Tolerance to Laser LinewidthIn this section, we present results to show the robustness of the proposed TFP designagainst increasing PN levels. First, we study the effect of varying pilot density andLLW on the proposed CPNE, LIPNE and FGIPNE schemes, for the simplistic case ofa Nyquist WDM system. In Fig. 6.8, we plot the ROSNR penalty of such system overthe benchmark transmission that is not impaired with PN, as a function of the LLW ofthe transmitter and the receiver lasers. Both LIPNE and FGIPNE are bootstrappedwith CPNE having 3% and 6% pilot density. Not surprisingly, all PN mitigationmethods produce larger performance degradation with increasing LLW. As shown inthe figure, the gains of LIPNE and FGIPNE over CPNE-only PN mitigation methodalso increase with larger LLWs for both pilot densities. The plots in the figure alsosuggest that LIPNE is performing close to FGIPNE for LLW up to 100 kHz with amaximum performance gap of 0.2 dB. Moreover, FGIPNE performs better by 0.15 dBwhen it is bootstrapped with CPNE using 6% transmission overhead compared to3% pilot density.137Table 6.2: Computational Complexity.Task Item Number of operations(Add, Sub., Mul., Div.) per code symbolEstimationPMD 12NNwISI 14N(Ls + 1)ICI 20N(Lc + 1)CPNE 14NLIPNE 14N itmaxFGIPNE 2N(17M + 11)itmaxEqualizationISIC O(4NMLs2 itmax)ICIC O ((M + Lc)N itmax)After having investigated the performance of the proposed PN cancellation meth-ods in a Nyquist WDM transmission, we now proceed to evaluate their effectivenessin TFP systems with even higher values of LLWs. In Fig. 6.9a-6.9b, we plot theROSNR penalty over the respective zero-PN systems similar to Fig. 6.8, for theNyquist WDM transmission and τ = 0.9,ξ = 0.95 TFP system, respectively. BothIPNE methods are bootstrapped by the CPNE with 6% pilots. In Fig. 6.9, wealso include the “perfect-decision CPNE” scheme corresponding to the genie-assistedknown transmitted symbols, as a reference. The plots in the figures suggest thatwhile LIPNE is performing decently compared to FGIPNE for the Nyquist WDMtransmission, FGIPNE outperforms LIPNE by significant margins in TFP systems,especially when LLW is very high. For example, with 400 kHz LLW, FGIPNE yields1 dB ROSNR improvement over LIPNE with τ = 0.9, ξ = 0.95 as highlighted inFig. 6.9b, whereas such a gain is restricted to only 0.4 dB for the Nyquist trans-mission as shown in Fig. 6.9a. Moreover, FGIPNE is also able to outperform theperfect-decision CPNE, and it offers 1 dB gain over “CPNE-only” PN compensationmethod for both Nyquist WDM and TFP systems.1386.6.5 Computational Complexity AnalysisDetails of the computational complexity for the proposed systems design are furnishedin Table 6.2, where M denotes the modulation order, and the rest of the parametersare already defined in the preceding sections. The numbers in the table correspondto computations required for both polarizations and all SCs in the superchannel.As shown, the LMS based estimation algorithms for PMD filter coefficients, TFPinterference and CPNE scale linearly with the number of SCs. Evidently, BCJR-ISIC constitutes the computationally most challenging module, since the complexitysignificantly magnifies with the length of the truncated TFP-ISI channel and theconstellation size [97]. The computational cost of SPCIC-ICIC, however, increaseslinearly with the modulation order and TFP-ICI taps length. A comparison betweenLIPNE and FGIPNE indicates that FGIPNE exhibits a modulation format depen-dency, and entails slightly more computations compared to LIPNE, with the benefitof substantial performance advantage shown in Section 6.6. Moreover, FGIPNE alsorequires additional buffering to store the forward and backward FG metrics [70].6.7 ConclusionSuperchannel data rates of 1 Tbps and more are being targeted in the next gen-eration optical fiber systems to compete with the increasing demands in networktraffic. To accomplish such target, in this chapter, we proposed flexible designs forspectrally efficient TFP superchannel transmission achieving Tbps data rates withsignificantly higher SE values compared to Nyquist WDM systems. For this, we havepresented sophisticated signal processing tools to efficiently handle TFP interferenceand accomplish CPR. Our simulation results suggest that by employing the proposed139interference channel and PN estimation methods, together with the BCJR-ISIC andthe novel SPCIC-ICIC scheme, the presented TFP WDM systems offer more than2 dB ROSNR gains and 160−400 km transmission distance improvements over state-of-the-art super-Nyquist superchannel transmission techniques. The proposed DSPdesign shows outstanding tolerance to PN with LLW up to 400 kHz. Moreover, oursystem exhibits excellent robustness against additional aggressive optical filtering inthe form of cascades of ROADM nodes comprised of narrow WSS filters that may bepresent in the longhaul fiber link.140Chapter 7Concluding Remarks & FutureDirections7.1 Summary and ConclusionsFTN signaling is a spectrally efficient technology that can facilitate high data ratesin the bandwidth-starved existing fixed transmission networks, which serve as thebackbone for the Internet and the mobile data traffic. The communication links forsuch networks are primarily comprised of optical fibers and point-to-point microwaveradio. In this thesis, we investigated the application of FTN signaling in these OFCand MWC links by taking necessary measures to tackle the interference introducedby FTN, together with the consideration of other practical challenges present in thesesystems. Based on the current deployment of the existing fixed transmission networkinfrastructure, we broadly considered three application scenarios, namely (a) singlecarrier DP OFC systems, (b) single carrier DP MWC systems, and (c) multicarrierDP OFC systems, for introducing and evaluating the concept of FTN signaling.Firstly, we considered FTN transmission of a DP COSC system. Against thebackdrop of the existing literature, which predominantly employs computationallycomplex and buffer-space constrained receiver-side equalization strategies to mitigatethe FTN-ISI, in this thesis, we adopted an alternative approach of pre-equalizing theISI at the transmitter. For this, we began with the application of the well-known non-141linear precoding method THP, and designed novel, cost-efficient soft demappers toreduce the impact of the modulo-loss, which incurs severe performance degradationin COSC systems employing traditional THP-demappers. Moreover, we presenteda linear precoding technique LPE that achieved the optimal BER performance byorthogonalizing the FTN-ISI channel, and thereby, making such precoding schemecompetitive to computationally expensive BCJR-based MAP equalization. Numeri-cal results for a precoded COSC system suggested significant performance and com-plexity gains of the proposed precoding schemes over state-of-the-art methods.Secondly, we applied FTN signaling in a point-to-point microwave link employingHoM formats and polarization multiplexing. In addition to the FTN-ISI, such DPsystems suffer from multiple additional practical challenges, in the form of multipath-ISI, PN and XPI. While the polarization cross-talk in OFC systems can be perfectlyequalized by employing proper linear filters, the application of FTN signaling andHoM formats in MWC systems complicates the XPI mitigation. In this thesis, DP-FTN HoM systems were introduced for the first time. For this, we presented power-ful, and yet computationally simple, signal processing algorithms to jointly equalizeor pre-equalize the aggregate interference stemming from the mutipath reflections,FTN signaling and polarization multiplexing, in tandem with CPR via suitable PNmitigation techniques. Our numerical results for an MWC transmission establishedsignificant performance advantage in favor of FTN signaling over equivalent Nyquistsystems. For example, the newly designed DP-FTN systems empowered with theproposed DSP algorithms offered as high as 5.5 dB performance improvement over aNyquist transmission that employed a higher modulation order to achieve the samedata rate.Thirdly, we considered MFTN OFC systems in the form of TFP WDM superchan-142nels. Since the practical limitations of the opto-electronics preclude the feasibilityof facilitating single carrier transmission with very high baud rates, TFP WDM su-perchannels are useful to achieve high throughputs. We began with the applicationof precoding in TFP superchannels, as an alternative to complicated receiver-sideequalization. For this, we presented pre-equalization schemes at the transmitter tomitigate, jointly or otherwise, the ISI and ICI stemming from the TFP transmission.In this thesis, precoded MFTN systems enabling packing of symbols in both timeand frequency dimensions were introduced for the first time. Simulation results forthe such systems established considerable gains by the proposed methods over com-petitive equalization schemes having similar or higher computational complexity. Tokeep up with the futuristic targets set for the next generation optical networks, wethen considered spectrally efficient TFP superchannels achieving Tbps data rates,packed within a target aggregate BW. For this, we noticed that the functionalityof the proposed precoding methods was limited to a restricted range of time andfrequency compression. Therefore, to achieve higher SE, we turned our focus onthe existing receiver-side turbo equalization methods present in the literature underthe premises of an AWGN channel, and tailored them to the considered OFC sys-tems. Thereafter, we presented more flexible and high-performance DSP designs forthe TFP Terabit-superchannels, by jointly estimating the TFP-ISI and ICI channels,coupled with sophisticated CPR and scheduling algorithms. The presented numericalresults indicated that the proposed spectrally efficient Tbps systems offered signif-icant transmission distance improvement over existing super-Nyquist designs, andexhibited excellent tolerance to high LLWs and aggressive optical filtering stemmingfrom the cascades of ROADM nodes.In conclusion, we have demonstrated that FTN signaling empowered with the143proposed signal processing algorithms in this thesis bears significant merit to meetand exceed the high SE and throughput requirements set for the future generationfixed transmission networks. However, to accomplish the SE improvements overNyquist transmission, FTN systems require additional computational complexity.The numerical results presented in this thesis not only validated the effectiveness ofthe proposed algorithms, but also established substantial superiority of the proposedschemes over state-of-the-art designs.7.2 Future WorkIn the final portion of this dissertation, we present potential avenues for future re-search.7.2.1 FTN and Probabilistic ShapingIn this thesis, we have shown that FTN signaling is an efficient means to obtain higherSE. To achieve further SE improvements in the optical links, other technologies canalso be applied, as an alternative to, or in conjunction with FTN. For example, prob-abilistic shaping [79, 127, 129, 132, 133] employing HoM formats in OFC systems isconsidered to be another interesting research direction that has attracted renewedinterest lately. Such systems provide shaping gains by assigning non-uniform proba-bilities to the signal constellation points, at the price of increased sensitivity to PNand fiber nonlinearity [129, 133]. Investigating whether FTN/TFP systems equippedwith our proposed DSP design serves as a competitive or complementary technol-ogy to probabilistically shaped optical transmission is an attractive future researchavenue worth pursuing.1447.2.2 Fiber NonlinearityFiber nonlinearity stemming from Kerr effects incurs severe performance degradationin practical OFC systems when the launch power is high [134]. Consequently, for suchsystems, nonlinear effects restrict the application of HoM formats, which operate inthe high OSNR regime. In this thesis, the optical launch powers considered for oursimulations were set at sufficiently low values where the effects of fiber nonlinearityare negligible. However, it will be interesting to know how the performance of theproposed TFP transmission is affected in the presence of nonlinearity in a more real-istic OFC link, with or without nonlinear compensation techniques integrated withthe receiver DSP. It is also noteworthy to mention that part of the fiber nonlinearitymanifests itself as a nonlinear PN [133, 134]. Therefore, future research efforts maybe directed towards investigating whether the iterative CPR algorithms we presentedin Chapter 6 of this thesis for TFP systems are capable of eliminating such PN.7.2.3 Additional Device Non-idealities and ImpairmentsIn the numerical evaluation of the proposed algorithms for the OFC and the MWCsystems, we have assumed perfect time and frequency synchronization. However,timing offset and carrier frequency offset introduced by imperfect radio-frequencyand opto-electronic transceiver components (e.g., LOs, lasers, etc.) can severelyrestrict the performance of the communication links. 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(A.1)Hence,Pr (v[k] = aκPAM + 2iM) (A.2)=Pr[(−M+i ≤a[k]−f [k]≤−M−i )∩(a[k]=aκPAM)] (A.3)164=1MPr[(M−i +aκPAM≤f [k]≤M+i +aκPAM)](A.4)=1M[Φ(M+i +aκPAMσf)−Φ(M−i +aκPAMσf)]. (A.5)A.2 Proof of Proposition 2.2In an AWGN channel, the nearest-neighbor approximated LLR, computed by EADfor the nth bit of the kth transmitted symbol is given by (2.12). We can write thenearest neighbors of the received symbol v′[k] for an M -ary PAM asc¯0,n = aκ∗0,nPAM + 2uM , (A.6)c¯1,n = aκ∗1,nPAM + 2vM , (A.7)where aκ∗0,nPAM, aκ∗1,nPAM ∈ A with A being the original M -ary PAM signal set and u, v ∈ Zsuch that |u− v| ≤ 1 for arbitrary bit-mapping.The nearest neighbors of v′[k] remain invariant after the modulo operation andone additional layer of constellation extension applied in PLD [52], if and only if thereexists a w ∈ Z such that−M ≤ v′[k]− 2wM < M , (A.8)− (M + 1) ≤ aκ∗0,nPAM + 2(u− w)M ≤M + 1 , (A.9)− (M + 1) ≤ aκ∗1,nPAM + 2(v − w)M ≤M + 1 . (A.10)If u = v, then w = u = v satisfies (A.8)-(A.10). If |u− v| = 1, which means that c¯0,nand c¯1,n lie on different sides of the modulo boundary before the modulo operation,165(A.8)-(A.10) are satisfied if and only if|v′[k]− c¯i∗,n| ≤ 2 , (A.11)where i∗ ∈ {0, 1} denotes the index for which c¯i∗,n is located across the moduloboundary from v′[k]. It can be easily verified that (A.11) and hence (A.8)-(A.10) aresatisfied for any bit-labeling of the PAM constellation15 when M = 2 or 4.Thus, for 2PAM and 4PAM modulations, PLD computes the LLRs asLLRPLDk,n =|v′[k]−c¯0,n|2−|v′[k]−c¯1,n|22σ2. (A.12)Therefore, under equal probability assumption when α1,n = α0,n, comparing (2.12)and (A.12) yieldsLLREADk,n = LLRPLDk,n . (A.13)A.3 PSD And Average Transmit Power withPrecodingFrom (2.1), the PSD of the FTN transmitted signal s is given by [13, 103]Φss(f) =1τT|Hˆ(f)|2Φrr(ej2pifτT), (A.14)where Φrr is the discrete-time Fourier transform of the auto-correlation of the se-quence r.We assume that the constellation symbol sequence a in Fig. 2.2 and Fig. 2.3 and15For M > 4, the conditions are not satisfied for some labelings. We leave the proof for M > 4as a future work.166the intermediate process v in Fig. 2.2 can be approximated as a zero-mean wide-sense stationary process with auto-correlation sequences σ2aδ and φvv, respectively,where δ is the Kronecker-delta function. Then, from (2.3), the z-transforms of theauto-correlation of r for THP and LPE can be written asΦTHPrr (z)=Φvv(z)1Q(z)Q∗ (z−∗)=αΦvv(z)G(z), (A.15)ΦLPErr (z)=σ2a1Q(z)Q∗ (z−∗)=ασ2aG(z), (A.16)where Φvv is z-transform of φvv.Evaluating (A.15) and (A.16) on the unit circle and using the relation Gˆ(f) =|Hˆ(f)|2, the PSD in (A.14) becomesΦTHPss (f) = αΦvv(ej2pifτT )Gˆ(f)∑kGˆ(f + kτT)(A.17)for FTN-THP andΦLPEss (f) = ασ2aGˆ(f)∑kGˆ(f + kτT)(A.18)for FTN-LPE.In order to compute the average power for the FTN-THP and FTN-LPE systems,we consider an equivalent pulse-shape ψ, such thatΨ(f) = T ′Gˆ(f)∑kGˆ(f + kT ′ ), (A.19)where Ψ is the Fourier transform of ψ and T ′ = τT . It can be easily shown that ψ167satisfies Nyquist’s zero-ISI criterion with respect to the sampling rate T ′ because1T ′∑lΨ(f +lT ′) = 1 . (A.20)Now, to compute the average power of an FTN-THP system from (A.17), we canwrite the autocorrelation of the transmitted signal s, corresponding to a delay τ˜ , asφTHPss (τ˜) =αT ′∑mφmvvψ(τ˜ −mT ′) , (A.21)where φTHPss is the inverse Fourier-transform of the PSD ΦTHPss and φmvv is the au-tocorrelation of v corresponding to a delay m. Therefore, the average power of anFTN-THP system can be written asPTHPAvg =∞∫∞ΦTHPss (f)df (A.22)= φTHPss (0) (A.23)=ασ2vτT, (A.24)where the step (A.23) to (A.24) follows from (A.21), using the fact that ψ is aT ′(= τT )-orthogonal Nyquist-pulse as shown in (A.20) and φ0vv = E(|v|2)= σ2v , withE(·) denoting the expectation operator.To compute the average power of the FTN-LPE system, we note from the T ′-orthogonality of ψ in (A.20) thatψ(0) =∞∫−∞Ψ(f)df = 1 . (A.25)168Therefore, the average power of the FTN-LPE system follows from (A.18) and using(A.25) asP LPEAvg =∞∫∞ΦLPEss (f)df (A.26)=ασ2aT ′∞∫−∞Ψ(f)df (A.27)=ασ2aτT. (A.28)169Appendix BProofs and Derivations forChapter 3B.1 LMS Update EquationsB.1.1 Proof of Lemma 3.1To derive the LMS update equations, we rewrite (3.6) in a more compact way asfollows, with the assumption that the PN estimates ϕˆ1 and ϕˆ2 are practically constantover the duration of Nf symbols, corresponding to the FFF length, due to slow PNvariation [31, 75, 77]:y1[k] = fH1 [k]P [k]ug[k]−bH1 [k] aˆg[k] , (B.1)y2[k] = fH2 [k]P [k]ug[k]−bH2 [k] aˆg[k] . (B.2)Now, the total MSE over the two polarizations can be written asMSETot =E(|E1[k]|2)+E(|E2[k]|2) , (B.3)where Ei[k] = yi[k]−aˆi[k−k0] represents the error signal for the ith branch, with i=1, 2corresponding to the H and the V-polarization, respectively.170The gradient of MSETot with respect to f∗1 [k] follows from (B.1) and (B.3) as∂MSETot∂f ∗1 [k]=∂E(|E1[k]|2)∂f ∗1 [k](B.4)= E[E∗1 [k]∂E1[k]∂f ∗1 [k]](B.5)= E[P [k]ug[k]E∗1 [k]]. (B.6)Similarly, the corresponding gradients of MSETot with respect to other filter-tapweights and the PN estimates can be computed as∂MSETot∂f ∗2 [k]=E[P [k]ug[k]E∗2 [k]], (B.7)∂MSETot∂b∗1[k]=−E[aˆg[k]E∗1 [k]] , (B.8)∂MSETot∂b∗2[k]=−E[aˆg[k]E∗2 [k]] , (B.9)∂MSETot∂ϕˆ1[k]= 2E[Im(e−jϕˆ1[k]ψ1[k])], (B.10)∂MSETot∂ϕˆ2[k]= 2E[Im(e−jϕˆ2[k]ψ2[k])]. (B.11)Now, following the same reasoning as in [75, 102], the minimum MSE with the CPNTmethod can be achieved by jointly adjusting the tap weights and the PN estimatesin proportion to negative values of the respective gradients in (B.6)-(B.11). Usingthe instantaneous values, at a time instant k, as a set of unbiased estimators for thecorresponding gradients [103], we get the LMS update equations (3.7)-(3.12).171B.1.2 Proof of Lemma 3.2For the IPNT method, we can rewrite (3.22) asy˜1[k]=e−jθˆt1 [k](fH1 [k] P˜ [k]ug[k]−bH1 [k] aˆg[k]), (B.12)y˜2[k]=e−jθˆt2 [k](fH2 [k] P˜ [k]ug[k]−bH2 [k] aˆg[k]). (B.13)The overall MSE across both polarizations takes the formMSEIPNT =E(|E˜1[k]|2)+E(|E˜2[k]|2), (B.14)with E˜i[k] = y˜i[k]−aˆi[k−k0], i∈{1, 2}. The expressions for the gradients follow as∂MSETot∂f ∗1 [k]=E[e−jθˆt1 [k]P˜ [k]ug[k]E∗1 [k]], (B.15)∂MSETot∂f ∗2 [k]=E[e−jθˆt2 [k]P˜ [k]ug[k]E∗2 [k]], (B.16)∂MSETot∂b∗1[k]=−E[e−jθˆt1 [k]aˆg[k]E∗1 [k]] , (B.17)∂MSETot∂b∗2[k]=−E[e−jθˆt2 [k]aˆg[k]E∗2 [k]] , (B.18)∂MSETot∂θˆt1 [k]= 2E[Im(e−jθˆt1 [k]ξ1[k])], (B.19)∂MSETot∂θˆt2 [k]= 2E[Im(e−jθˆt2 [k]ξ2[k])], (B.20)∂MSETot∂θˆr1 [k]= 2E[Γr1 [k]], (B.21)∂MSETot∂θˆr2 [k]= 2E[Γr2 [k]]. (B.22)Following a similar argument as in the proof of Lemma 3.1, (B.15)-(B.22) leads to(3.23)-(3.30), which completes the proof.172B.2 LPE-FFF and LPE-FBF ComputationsTo derive the expressions for the LPE-FFF and LPE-FBF as in Chapter 2, we notethat the discrete-time FTN-ISI impulse response for each of the polarization branchescan be written as a function of the transmitter pulse-shape and the receiver matchedfilter asg[n] = (p ∗ q)(nτT ) , (B.23)where q(t) = p∗(−t) and ∗ denotes the linear convolution.Introducing G = Z (g), with Z(·) being the z-transform, we can write the follow-ing spectral factorization [44]:G(z) = λV (z)V ∗(z−∗), (B.24)such that V (z) is causal, monic and minimum-phase, and λ > 0 is a scaling factorused to ensure V (z) is monic. As shown in Chapter 2, the necessary and sufficientcondition for the above spectral factorization is given byτ ≥ 11 + β. (B.25)Now, denoting the z-transforms of bLPE and fLPE by Ψ(z) and ζ(z), respectively,we can writeΨ(z) = V (z) , (B.26)ζ(z) =1λV ∗ (z−∗). (B.27)173Appendix CProofs and Derivations forChapter 4C.1 Proof of Proposition 4.1Projecting the received signal component onto the basis functions h(t−lτT )ej2pi(k−N+12 )∆ftas per [13], the resulting matched filtered analog signal for the kth SC, k=1, 2, . . . , N ,can be written asuˆk(t)=s(t)e−j2pi(k−N+12)∆ft ? h(t) , (C.1)where ? denotes linear convolution. Writing uk[n] as the τT samples of uˆk(t) in (C.1),we getuk[n]=N∑m=1(xm[n]ejω0(m−k)n ? g0,m−k[n]), (C.2)where gu,v denotes τT samples of fu(t) ? fv(t) with fu(t) =h(t)ej2piu∆ft. Multiplyingboth sides of (C.2) by ejω0(k−N+12)n for all k=1, 2, . . . , N , we obtainrk[n]=N∑m=1(dm[n] ? gk−N+12,m−N+12[n]), (C.3)which shows that the frequency-shift operations convert the TFP transmission intoan LTI system, with impulse responses given by gk−N+12,m−N+12, k,m = 1, 2, . . . , N .Denoting by H(z) the z-transform of the 2-D channel, conjugate symmetry of thesequences gk−N+12,m−N+12[n] implies H(z)=HH(z−∗), which completes the proof.174C.2 Proof of Proposition 4.2Following the notations in [115], we define:S(z)=D−1JV H(z∗)J , Σ=DHD, and M (z)=JHH(z∗)J . Considering the conjugatesymmetry of the overall impulse response, we have M (z)=H(z). Therefore, we canwrite:SH(z−∗)ΣS(z) = JV (z−1)V H(z∗)J (C.4)= JH(z−1)J = HH(z−∗) = H(z). (C.5)Based on the above factorization, we obtain, as in [115],F (z) = Σ−1S−H(z−∗), (C.6)B(z) = S(z) . (C.7)Substituting S(z) and Σ defined above produces the result.C.3 2-D LPE PMD Equalizer LMS AlgorithmWe consider a half-symbol spaced LMS equalizer for PMD mitigation [17]. Letus denote the 2 × 2 PMD compensating filter for the ith SC, i = 1, 2, . . . , N , bycxx,i[ν, k] cxy,i[ν, k]cyx,i[ν, k] cyy,i[ν, k], where each entry of the matrix corresponds to the νth fractionally-spaced tap, ν=0, 1, . . . , Nc−1, at the kth time index. Considering Nf static symbol-spaced taps for each filter-entry fij[µ], i, j ∈ {1, 2, . . . , N}, µ=0, 1, . . . , Nf−1, of the2-D LPE static FFF matrix, we can write the X-pol and Y-pol outputs, respectively,for the ith SC, i=1, 2, . . . , N , aswi,X/Y[k]=N∑j=1CHX/Y,j[k] U˜j[k]Υij , (C.8)where the subscript X/Y means “X respectively Y”, wi,X/Y and ui,X/Y are the PMDequalizer input and the frequency-shift output for the ith SC, respectively, shown in175Fig. 4.5. Moreover, in (C.8):CX,j[k]=[{c∗xx,i[m, k]}Nc−1m=0,{c∗xy,i[n, k]}Nc−1n=0]T,CY,j[k]=[{c∗yx,i[m, k]}Nc−1m=0,{c∗yy,i[n, k]}Nc−1n=0]T,U˜j[k]=[{ϑ(κ)j [k]}Nf−1κ=0]Pj ,ϑ(κ)j [k]=[{ui,X[k−2κ−γ]}Nc−1γ=0,{ui,Y[k−2κ−γ]}Nc−1γ=0]T,Pj =diag(e−jωˆ0rjk,..., e−jωˆ0rj(k−2(Nf−1))︸ ︷︷ ︸Nf),ωˆ0 =pi(1 + β)ξτ , rj =j−1−N−12 , Υij =[{fi,j[µ]}Nf−1µ=0]T,where [·]T denotes the matrix transpose and the expression {x[j]}N2j=N1denotes therow-vector [x[N1], . . . , x[N2]]. Writing the error signals asεX/Y,i[k]=ui,X/Y[k]−ai,X/Y[k−k0] , (C.9)with k0 being the decision delay, and ai,X/Y being the modulated symbol for the ithSC and the corresponding polarization, the LMS update equation for the ith SC,i=1, 2, . . . , N , can be written asCX/Y,i[k + 1]=CX/Y,i[k]−αU˜i[k]N∑l=1Υliε∗X/Y,l[k] , (C.10)where α>0 is the step-size parameter.176Appendix DProofs and Derivations forChapter 5D.1 Proof of Lemma 5.1Matched-filtering of the signal component (without noise) at the receiver is estab-lished by projecting (5.2) onto the basis functions h(t− lτT )ej2pi(k−N+12 )∆ft [13], andthe resulting analog signal for the kth SC, k=1, 2, . . . , N , can be written as:rk(t) = s(t)e−j2pi(k−N+12)∆ft ? h(t) (D.1)=N∑m=1∑lam[l]∫sh(s)h(t−s−lτT )ej2pi(m−k)∆f(t−s)ds (D.2)=N∑m=1∑lam[l]e−jω0(m−k)lg0,m−k(t− lτT ). (D.3)Denoting rˆk[n] as the τT samples of rk(t) in (D.3), we getrˆk[n]=N∑m=1(am[n]e−jω0(m−k)n ? g0,m−k[n]), (D.4)which completes the proof of Lemma 5.1.D.2 Proof of Lemma 5.2Multiplying rˆk[n] in (D.4) by e−jω0(m−k)n for all k = 1, 2, . . . , N completes the prooffor Lemma 2. For example, assuming N = 3 and ξ ≥ 0.5, the TFP overall impulseresponse matrix with respect to the inputs ak[n]ejω0(k−N+12 ), k=1, 2, . . . , N , is given177by:Gˆ[n] =g−1,−1[n] g0,−1[n] 0g0,−1[n] g0,0[n] g0,1[n]0 g0,1[n] g1,1[n] . (D.5)178Appendix EProofs and Derivations forChapter 6E.1 FGIPNE Metrics ComputationTo compute the FGIPNE forward and backward metrics, and thereby, obtain theMAP estimates of the laser PN, we refer to the FG shown in Fig. 2 of [70]. Usingsimilar notations as in [17, 70], and focusing on the variable node θk in Fig. 2 of [70],we note that the product of the three incoming messages pd(θk), pf (θk) and pb(θk) isproportional to the conditional probability density function (PDF) p(θk|{rk}), wherek is the symbol index and {rk} is the sequence of the received symbols [17, 135](to familiarize with the concept of FGs and the sum-product algorithm, interestedreaders are referred to [135]). Thereafter, we employ the iterative forward-backwardalgorithm as per Section IV.B of [70]. Owing to space limitation, we will not recallthe algorithm presented in [70] in its entirety. Instead, we will briefly revisit only thecomponents that are relevant to our considered TFP system.Using similar notations as in [70], we note that the iterative algorithm involvesthe computation of the parameters αk and βk according to (28) of [70], to denotethe first and the second-order moments of the transmitted symbols, respectively. Toaccount for the TFP-ISI and ICI in our superchannel transmission, we propose thefollowing modification to the computation of αk and βk. For this, we introduce a new179variableγk = βk −∣∣αk∣∣2 . (E.1)Moreover, to indicate the polarization labeling and the SC index, we denote theseparameters by α(m)k,x/y, β(m)k,x/y and γ(m)k,x/y corresponding to the mth SC and X or Ypolarization, respectively. The modified FGIPNE metrics, stacked as column vectorswith X and Y polarization inputs, are formulated as[α(m)k,x , α(m)k,y ]T =Ls∑j=−Lsh(m)j,k  E(a(m)k−j)+∑n6=mLc∑ν=−Lcg(n,m)ν,k E(aˆ(n)k−ν), (E.2)[γ(m)k,x , γ(m)k,y ]T =Ls∑j=−Ls〈∣∣h(m)j,k ∣∣〉2  Var(a(m)k−j )+∑n6=mLc∑ν=−Lc〈∣∣g(n,m)ν,k ∣∣〉2Var(aˆ(n)k−ν) , (E.3)where Var(·) and 〈∣∣·∣∣〉2 denote element-wise variance and absolute-square operations,respectively. At each LDPC iteration, the expectations and variances in (E.2)-(E.3)are computed for the constellation symbols using the symbol-probabilities obtainedfrom the LLRs fed back by the LDPC decoders. Finally, using the statistical propertyof the Tikhonov PDF, we obtain the MAP estimate of the PN for the correspondingpolarization and SC as [17]θˆMAP = ∠(2rkα∗k2σ2 + γk+ af,k + ab,k), (E.4)where ∠(·) denotes the phase angle of a complex scalar, σ2 is the noise variance perreal dimension, and af,k and ab,k are computed according to (36) and (37) of [70],respectively.180


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