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Spectrally efficient waveforms for coherent optical fiber transmission

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Spectrally efficient waveforms for coherent optical fiber transmission

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Content Spectrally Efficient Waveforms for Coherent Optical Fiber Transmission by Bishara Shamee A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (ELECTRICAL ENGINEERING) May 2021 Copyright 2021 Bishara Shamee Dedication To my children Fareed, Muna and Pierre To my wife Pierrette and to my mother Mariam and father Farid. ii Acknowledgements Returning to pursue a PhD in optical communications after many years in industry as a communications systems engineer is a delightful journey few are fortunate enough to undertake. Mine encountered many unusual twists and turns and I would not be writing the dissertation without my adviser and mentor, Professor Alan E. Willner. He gave me the confidence to continue and went out of his way to guide me. His vision and insight will always inspire me to overcome a wide variety of challenges with creative out of the box solutions. His sense of humor is all but superficial when he referenced the ’Toll Booth’ scene from the ’Blazing Saddles’ movie. To Professor Willner, I am grateful and thankful for your trust, support, encouragement and wisdom. I would like to thank members of my committee, Professor Alexander Sawchuk and Professor Stephan Haas for their unconditional gracious support and constructive feedback. Special thanks to Diane Demetras, the graduate adviser at USC for her delightful spirit, kindness, and the advice that transformed the lives of many graduate students. I am fortunate to be one of them. I would like to thank the Optical Communications Laboratory (OCLab) group for their motivation, drive and engaging discussions. In particular, I wish to thank Dr. Scott Nuccio, Dr. Lou Christen, Dr. Nisar Ahmed, Dr. Ahmed Almaiman, Dr. Morteza Ziyadi, Dr. Terry Lewis, Dr. Amir Mohajerin- Ariaei, Ahmad Falahpour and Dr. Yinwen Cao. I would also like to acknowledge the Space Park community at TRW for giving me the time away from work to start my studies. In particular, Tom Zeiller, Diane Legler, Earl Kofler and Greg Dell. iii The Space and Airborne Systems community at the Raytheon Company provided me with the freedom to continue my studies. In particular, Dr. Michelle Hauer and Dr. Todd Clatterbuck. Raytheon is an amazing place to work, grow and contribute at many levels across a wide variety of disciplines. I am honored to acknowledge my friend Dr. Steven Wilkinson, whose leadership at Raytheon is truly one of a kind. Whether solving relativistic time corrections or pursuing perfection, he remains an inspiring team leader. He accumulated about 200; 000 bicycle miles as of this year affirming that the tour is the reward! My mother in-law, Mona is the most generous person I had ever encountered. Her old fashioned unconditional and continuous love for her family is uniquely remarkable. I am grateful for her kindness, patience, and of course, the delicious Lebanese cuisine. My children Muna, Pierre, and Fareed have been forgiving of the time I took away from them and generously shared their insights and allowed me to witness their innocence, delightful curiosity and growth. My wife Pierrette encouraged me to take the first step in the thousand mile journey. Her unwavering faith and strength have been and continue to be the keystone of our family. I must have done something very good in my past life to have her in my present life. Finally, I wish to thank my parents Farid and Mariam for their selfless love and sacrifices. My father ignited my passion for engineering and the desire to build systems when he explained a closed loop feedback control system to me at an early age. I am grateful for their care and guidance. iv Table of Contents Dedication ii Acknowledgements iii List Of Tables vii List Of Figures viii Abstract xi Chapter 1: Introduction 1 1.1 Polarization Division Multiplexing and 4-D Signal Space . . . . . . . . . . . . . . . . 4 1.2 Optical Transmission Band Extension . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.3 Multi Level Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.4 Spatial Division Multiplexing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 1.5 Waveform Shaping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22 1.6 Waveform Drivers in a Communications System . . . . . . . . . . . . . . . . . . . . . 25 Chapter 2: Spectrally Efficient Waveforms 29 2.1 Raised Cosine Waveform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.2 Generalized Nyquist Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.3 Generalized Raised Cosine Waveform . . . . . . . . . . . . . . . . . . . . . . . . . . 40 2.4 Weighted Raised Cosine Waveform . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.5 Conjugate Root Pulse Waveform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 2.6 Improved Nyquist Waveform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 Chapter 3: Optical Fiber Channel Waveform Degradation Mechanisms 65 3.1 Loss in Single Mode Fiber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 3.2 Dispersion in Single Mode Fiber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69 3.3 Self Phase Modulation Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.4 Cross Phase Modulation Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . 72 3.5 Four Wave Mixing Nonlinearity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 Chapter 4: Digital Processing for Coherent Communications 78 4.1 MZM Modulator Linearization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 4.2 Dispersion Compensation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83 4.3 Forward and Backward Propagation Algorithms . . . . . . . . . . . . . . . . . . . . . 87 4.3.1 NonLinear Linear Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . 92 v 4.3.2 Linear Nonlinear Propagation . . . . . . . . . . . . . . . . . . . . . . . . . . 94 4.3.3 Linear Nonlinear Linear Propagation . . . . . . . . . . . . . . . . . . . . . . 94 4.3.4 Nonlinear Linear Nonlinear Propagation . . . . . . . . . . . . . . . . . . . . . 95 4.3.5 Propagation Models Numerical Comparison . . . . . . . . . . . . . . . . . . . 96 Chapter 5: Weighted Raised Cosine Transmission Performance 99 5.1 Error Vector Magnitude Metric (EVM) . . . . . . . . . . . . . . . . . . . . . . . . . . 99 5.2 Peak to Average Power Ratio Metric . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 5.3 Bandwidth Occupancy Percentile Metric . . . . . . . . . . . . . . . . . . . . . . . . . 103 5.4 Weighted Raised Cosine Waveform Performance . . . . . . . . . . . . . . . . . . . . 103 5.5 Waveform Transmission Performance . . . . . . . . . . . . . . . . . . . . . . . . . . 110 References 113 Appendix A Mathematical Supplement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 A.1 Cauchy-Schwartz Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 A.1.1 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 A.1.2 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124 A.1.3 Inner Product Spaces and Cauchy-Schwartz Inequality . . . . . . . . . . . . . 125 Appendix B Frequently Used Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 B.1 Trigonometric Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128 B.2 Wavelength and Frequency Conversion . . . . . . . . . . . . . . . . . . . . . . . . . . 129 B.3 Binary and Decimal Prefixes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 Appendix C Coherent Receiver Matrix Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 Appendix D Mach-Zehnder Modulator Mathematical Representation . . . . . . . . . . . . . . . . . . . . 136 vi List Of Tables 1.1 WDM Recommendation ITU-T G.694.1 Spectral grids. . . . . . . . . . . . . . . . . . 3 1.2 Number of bits per nSOP-mQAM PDM symbol for few cases . . . . . . . . . . . . . 4 1.3 Communication systems waveform parameters impact summary. . . . . . . . . . . . . 28 3.1 Two Tone FWM intermodulation products . . . . . . . . . . . . . . . . . . . . . . . . 73 3.2 Three Tone FWM intermodulation products . . . . . . . . . . . . . . . . . . . . . . . 75 4.1 Summary of Numerical Propagation Algorithmic Errors . . . . . . . . . . . . . . . . . 97 5.1 Square constellation PAPR for various orders with rectangular shaping . . . . . . . . . 102 B.1 Typical wavelength bands to frequency mapping at 1550 nm . . . . . . . . . . . . . . 129 B.2 Decimal prefix conversion factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 B.3 Binary prefix conversion factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 vii List Of Figures 1.1 Historical global Internet daily traffic as reported by Cisco [19]. . . . . . . . . . . . . 2 1.2 The five mechanisms to increase the data capacity. Adapted from [118]. . . . . . . . . 3 1.3 4-D signal Polarization modulator [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.4 SSMF loss across the optical bands associated with the EDFA band [26]. . . . . . . . . 8 1.5 Detector and quantization converter SNR and sensitivity. . . . . . . . . . . . . . . . . 12 1.6 Doubling the QAM data rate squares the order of the constellation. . . . . . . . . . . . 13 1.7 QAM modulation BER efficiency relative to the Shannon bound. . . . . . . . . . . . . 15 1.8 MCF fiber representation used to increase the data capacity [129] . . . . . . . . . . . . 17 1.9 FMF mode multiplexing for increased tranmission capacity. . . . . . . . . . . . . . . . 18 1.10 Single fiber supporting FMF, MCF with mode multiplexing and de-multiplexing [76] . 18 1.11 Fiber spatial LP modes in FMF obtained from the generalized Manakov model. . . . . 19 1.12 Simplified communications system with crosstalk treated as a MIMO system. . . . . . 20 1.13 MIMO decoupling of the transfer matrix H. . . . . . . . . . . . . . . . . . . . . . . . 22 1.14 Shaped and unshaped spectrum comparison. . . . . . . . . . . . . . . . . . . . . . . . 23 1.15 ITU spacing of 100 and 50 GHz shaped and unshaped spectra. . . . . . . . . . . . . . 24 1.16 Impulse response of unshaped and shaped waveforms. . . . . . . . . . . . . . . . . . 25 1.17 Multi-Wavelength fiber transmission coherent system with MCF and FMF. . . . . . . . 26 1.18 Eye diagrams illustrating the peak growth due to waveform shaping. . . . . . . . . . . 28 viii 2.1 RCW time and frequency responses ( = 0; 0:6; 1:0). . . . . . . . . . . . . . . . . . . 31 2.2 RCW spectral response parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.3 A general communications link for Generalized Nyquist Criterion analysis . . . . . . . 34 2.4 ISI behavior of RCW and RRCW cascade. . . . . . . . . . . . . . . . . . . . . . . . . 42 2.5 ISI free RRCW shaping filter frequency response and compensation phase. . . . . . . 46 2.6 GRCW impulse response. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 2.7 GRRCW and Matched GRRCW for = 0:35. . . . . . . . . . . . . . . . . . . . . . . 49 2.8 GRRCW and Matched GRRCW for = 1:0. . . . . . . . . . . . . . . . . . . . . . . 49 2.9 WRCW construction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 2.10 Phase functions for Inter Symbol Interference (ISI) free waveform . . . . . . . . . . . 54 2.11 First order conjugate waveforms for different excess bandwidth. . . . . . . . . . . . . 56 2.12 Fourth order conjugate waveforms for different excess bandwidth. . . . . . . . . . . . 57 2.13 ISI free illustration of the flipped-exponential waveform. . . . . . . . . . . . . . . . . 61 2.14 Time and frequency domain of the flipped exponential. . . . . . . . . . . . . . . . . . 62 2.15 Time and frequency domain of the flipped hyperbolic secant waveforms. . . . . . . . . 62 2.16 ISI free flipped hyperbolic and inverse hyperbolic secant waveforms. . . . . . . . . . . 63 2.17 Time domain and frequency domain of the flipped inverse hyperbolic secant waveforms. 64 3.1 Pulse propagation through dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3.2 Two Tone FWM graphical representation . . . . . . . . . . . . . . . . . . . . . . . . 74 3.3 Three equally spaced tones FWM intermodulation terms . . . . . . . . . . . . . . . . 75 3.4 Three unequally spaced tones FWM intermodulation terms . . . . . . . . . . . . . . . 75 3.5 Graphical Three Tone FWM Representation . . . . . . . . . . . . . . . . . . . . . . . 76 4.1 Digital Pre-distortion of the Mach-Zehnder Amplitude Modulator. . . . . . . . . . . . 82 ix 4.2 High and low gain MZM predistortion . . . . . . . . . . . . . . . . . . . . . . . . . . 82 4.3 Raised Cosine dispersion and compensation illustration. . . . . . . . . . . . . . . . . . 85 4.4 Weighted Raised Cosine dispersion and compensation illustration. . . . . . . . . . . . 86 4.5 Nonlinear Linear and dispersion SSFM over a spatial interval . . . . . . . . . . . . 93 4.6 Linear Nonlinear distributed dispersion SSFM spatial interval . . . . . . . . . . . . 94 4.7 Linear Nonlinear Linear distributed dispersion SSFM over a spatial interval . . . . . 95 4.8 Nonlinear Linear Nonlinear distributed nonlinearity SSFM a spatial interval . . . . . 96 5.1 Maximum response of RCW and WRCW . . . . . . . . . . . . . . . . . . . . . . . . 104 5.2 RCW and WRCW peak power comparison . . . . . . . . . . . . . . . . . . . . . . . 105 5.3 RCW and WRCW waveform responses for excess bandwidth parameters 0:2; 0:6. . . . 106 5.4 PAPR of RCW and WRCW comparison. . . . . . . . . . . . . . . . . . . . . . . . . . 107 5.5 99% Occupied bandwidth RCW and WRCW comparison. . . . . . . . . . . . . . . . 108 5.6 Back to back communications assessment link model. . . . . . . . . . . . . . . . . . . 109 5.7 RCW, WRCW and rectangular waveforms EVM. . . . . . . . . . . . . . . . . . . . . 110 5.8 RCW and WRCW BER with data converters and dispersion compensation . . . . . . . 111 5.9 80 km, 64-QAM transmission with linear and nonlinear compensation. . . . . . . . . . 112 C.1 Coherent Inphase and Quadrature Detector . . . . . . . . . . . . . . . . . . . . . . . . 134 D.1 A general depiction of the Mach Zehnder Modulator controlled by two independent voltages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 D.2 Genral MZM operation with two independent voltage and bias drives . . . . . . . . . 139 D.3 MZM Amplitude Modulator bias and response . . . . . . . . . . . . . . . . . . . . . . 140 x Abstract Compared to the available Radio Frequency bandwidth, the optical bandwidth over fiber is larger by orders of magnitude. However, with the rapid increase in bandwidth demand [19] due the global in- frastructure connectivity, the fiber capacity is reaching its limitations and new methods are needed to meet the expected demand [65, 118]. Meeting the demand is a multi-dimensional challenge driven by practical solutions leading the technology development. The expected demand challenge response has been formulated along the lines of packing more bits per symbol, increasing the number of channels per medium to include polarization, multiple fiber modes communication, multiple orbital angular momentum modes, multiple core fiber, advancing the technology to utilize more optical bands in fiber, and finally waveform shaping to increase the trans- mission spectral efficiency to increase the capacity in a given bandwidth [91, 118]. Early fiber transmissions, utilized bandwidth inefficient amplitude waveforms such as ON/OFF keying at a one bit per symbol. Later on, optical communications leveraged the advances of the digital Radio Frequency communication and utilized the two dimensional carrier amplitude and phase to in- crease the number of bits per symbol. Early two dimensional waveforms and the ON/OFF keying were unshaped and required bandwidth much larger than the symbol rate due to the complexity of more efficient waveforms. Channel spectral efficiency is defined in terms of the channel capacity in bits per second per Hertz and for a given signal to noise ratio, the efficiency is largest at the symbol rate or the Nyquist rate xi as given by Shannon Capacity Theorem discussed in section 1.3. As will be discussed in Chapter 1 increasing the capacity is a multi-dimensional challenge. The challenges span the optical and digital signal processing to the development of new fiber technologies and the corresponding amplifiers. In this thesis, the focus will be on the digital waveform shaping to increase the spectral efficiency of the transmission by reducing the occupied spectral bandwidth while considering the end to end link as a whole. The simulation and analysis will compare the proposed waveform to a reference waveform. Section 1.6 identifies the waveform shaping performance drivers as the utilized bandwidth approaches the symbol rate or the Nyquist bandwidth. The efficiency increase, waveform shaping increases the fiber penalties due to the fiber nonlinearity [44] as well as increasing the linear operating range [75,117] of the transmitter Digital to Analog Converter (DAC), modulator drivers and receiver Analog to Digital Converter (ADC). With advances in the digital technology [16], signal processing may play a significant role in waveform shaping and mitigation of fiber impairment. xii Chapter 1 Introduction While spectral efficiency in the Radio Frequency (RF) domain received much attention due to limited frequency band allocations and increased demand, the optical fiber communications spectral efficiency received similar attention when the demand started to increased at a rapid rate. According to Cisco Systems Inc. [19], the daily internet traffic in 1992 was around 100 Giga Bytes (GB) and is expected to exceed 1 10 Exa Bytes (EB) or 10 7 Tera Bytes (TB) in 2022 as summarized in Figure 1.1 on page 2. This chapter reviews the current approaches to meet the expected demand and highlights our specific research area to increase the spectral efficiency. The expected magnitude of the data transfer demand identifies a capability shortfall and surveys the methodologies identified in the literature to resolve the upcoming challenge. The approaches include Polarization Division Multiplexing (PDM) and 4-D signal representation in section 1.1, expanding the utilized optical bands in section 1.2, increasing the number of modulation levels or states in section 1.3, utilizing Space Division Multiplexing (SDM) in section 1.4, waveform shaping in section 1.5, and section 1.6 describes a model of the fiber link to identify and assess the factors impacting the spectral efficiency. In terms of fiber capacity, the expected traffic demand will exceed the capacity of a single mode fiber specified by the International Telecommunication Union (ITU) recommendation ITU-T G694.1. 1 The decimal prefixes instead of the binary prefixes to remain consistent with the Cisco [19]. Please refer to appendix B.3 for description of the binary and the decimal prefixes and their relationship. 1 1992 1997 2002 2007 2017 2022 10 1 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 Daily Internet Traffic (Tera Bytes) Figure 1.1: Historical global Internet daily traffic as reported by Cisco [19]. The capacity of a single mode fiber per ITU recommendation is summarized in Table 1.1 on page 3 for Wave Division Multiplexing (WDM) channel spacing of 12:5; 25:0; 50:0 and 100 GHz. The table identifies the specific grid frequencies and the total number of channels at the corresponding spacing. To illustrate the capacity challenge, consider occupying the 916 channels at 12:5 GHz spacing without any guard bands, results in available bandwidth of about 0:012 Exa Hz (EH) using the mapping of appendix B.2 on page 129. Assuming transmissions at 6 bits per symbol with dual polarization, then the ITU grid will support at most 0:018 EB which is three orders of magnitude less than the expected demand of 10 EB in 2022. Furthermore, the expected capability would be required to allocate operating margins to meet or exceed the expected demand. Figure 1.2 on page 3 depicts the methodologies to extend the fiber capacity. Nakazawa [64, 65] ad- vocated the ’3M’ technologies consisting of Multi Level Modulation (MLM), Multi Core Fiber (MCF) and Multi Division Multiplexing (MDM) as the mechanisms to increase the capacity by effectively increasing the number of simultaneous data streams. Willner [113] demonstrated Orbital Angular Mo- mentum (OAM) multiplexing to increase the capacity in free space and over short distances in Few 2 Table 1.1: WDM Recommendation ITU-T G.694.1 Spectral grids. Spacing Central Frequencies Channels Wavelength (GHz) (THz) (nm) 12:5 184:5 193:1 + 0:0125n 195:9375 916 1530:0413 1624:8914 n =688;; 227 25:0 184:5 193:1 + 0:0250n 195:9250 458 1530:1389 1624:8914 n =344;; 113 50:0 184:5 193:1 + 0:0500n 195:9000 229 1530:3341 1624:8914 n =172;; 56 100:0 184:5 193:1 + 0:1000n 195:9000 115 1530:3341 1624:8914 n =86;; 28 Mode Fiber (FMF). Winzer [115, 118, 119] identified five dimensions to extend the capacity namely: polarization, frequency, time, SDM, MLM, and pulse shaping. Figure 1.2: The five mechanisms to increase the data capacity. Adapted from [118]. In the following, each methodology will be summarized and underline the specific research con- ducted within this thesis, namely the waveform shaping aspect. 3 1.1 Polarization Division Multiplexing and 4-D Signal Space Over the same wavelength, Polarization Division Multiplexing (PDM) provides orthogonal and inde- pendent transmission streams so the total data rate is the sum of individual data rates. Existing polar- ization multiplexing schemes utilize two orthogonal polarizations and independent streams to double the net rate. A mechanism to multiplex the polarization proposed by B¨ ulow [14] increases the num- ber of states associated with a symbol without increasing the receiver sensitivity by adding a State Of Polarization (SOP) to the symbol alphabet. Ifn is the number of SOP states andm is the number of states in a Quadrature Amplitude Modulation (QAM) constellation, then the bits per symbol increases by log 2 (n) so that annSOP-mQAM PDM modulation symbol has log 2 (n) +log 2 (m) as summarized in Table 1.2. Table 1.2: Number of bits per nSOP-mQAM PDM symbol for few cases mQAM 2 4 8 16 32 64 128 256 512 1024 nSOP 1 1.00 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00 2 2.00 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00 11.00 3 2.58 3.58 4.58 5.58 6.58 7.58 8.58 9.58 10.58 11.58 4 3.00 4.00 5.00 6.00 7.00 8.00 9.00 10.00 11.00 12.00 5 3.32 4.32 5.32 6.32 7.32 8.32 9.32 10.32 11.32 12.32 6 3.58 4.58 5.58 6.58 7.58 8.58 9.58 10.58 11.58 12.58 7 3.81 4.81 5.81 6.81 7.81 8.81 9.81 10.81 11.81 12.81 8 4.00 5.00 6.00 7.00 8.00 9.00 10.00 11.00 12.00 13.00 To double the bit rate or capacity for a givenmQAM order, the number of SOP states would have to increase exponentially. For example, doubling the bit rate for Quadrature Phase Shift Keying (QPSK) requires 4SOP or 4 additional SOP states so that the total number of states is 16 or 4 4. On the other hand doubling the bit rate for 64QAM, would require 64SOP states. The optimal symbol format depends on the transmission distance, modulator and demodulator signal processing capability of the channel by trading the SOP and the QAM states. ThenSOP-mQAM PDM symbol mapping provides 4 an additional degree of freedom. We point out that the increased sensitivity of thenSOP-mQAM was obtained using Reed-Solomon (RS) coding due to the introduced burst errors in the encoding [14]. At 28 Gbuad and 6SOP-4QAM, a 1:7 dB sensitivity improvement over dual polarization PDM-QPSK. Welti [111] analyzed the theoretical 4-D modulation format based on four independent channels and devised three classes of 4-D modulation with specified minimum distance and an expected performance to exceed the QAM modulation in two independent channels. Starting with M data symbols, say m i ;i = 1; 2;;M, an extendedM 4 channel symbol set is defined to map theM data symbols. For large M, the 4-D modulation outperformed the two dimensional modulations with higher efficiency and lower power. The Signal to Noise Ratio (SNR) gain for M = 256, ranged around 1 dB [111]. Cusani [22] proposed a spectrally efficient optical multilevel system using the 4-D signal space that outperformed the traditional two dimensional and two orthogonal polarization QAM. Coherent systems with dual polarizations and two quadratures waveform result in a 4-D signal space. For example, a QPSK on each polarization results in a 4-D signal space with a total of 8 levels or phase angles which is distinct from the polarization shift keying as proposed by Benedetto [8]. The 8 levels correspond to the four states of the QPSK constellation times the two polarization states. Karls- son [47] determined that a 4-D signal with 8-level format has a 1:76 dB gain over Binary Phase Shift Keying (BPSK) for uncoded optical transmission for Additive White Gaussian Noise (AWGN) since the signal space can be used more effectively by increasing the number of levels without increasing the required power [14]. Agrell [2] analyzed and compared the power efficient modulation formats in coherent systems and presented a polarization modulator realizing a 4-D signal depicted in Figure 1.3. The incoming laser with the dual polarization states 2 is split and phase modulated to realize the 4-D states. To illustrate the number of states through the architecture, the total number of states or levels is the number of states on 2 The polarizations are the perpendicular s, from the German senkrecht, and the parallel p, to the incidence plane. 5 the parallel polarization phase times the number of states on the perpendicular polarization phase times two for the polarization states. Figure 1.3 illustrates that the parallel phase modulates both polarization states equally and the first Polarization Beam Splitter (PBS) separates the polarization with the parallel phase. The phase modulator with the differential phase signal s p cancels out the parallel p phase signal so that only the perpendicular phase signal, s , is combined on the perpendicular polarization axis of the second PBS. Figure 1.3: 4-D signal Polarization modulator [2]. In terms of experimental 4-D modulation demonstration, B¨ ulow [14] discussed earlier, attained a 1:7 dB higher sensitivity with 6SOP and QPSK than polarization division multiplexing and QPSK. The 1:7 dB increase compares with a gain of 1:8 dB theoretical gain derived from Table 1.2 by taking the ratio of the corresponding QPSK entries for 6SOP and 2SOP. Nelson [67] experimentally compared coherent polarization-switched QPSK to polarization mul- tiplexed QPSK to realize a 0:9 dB sensitivity improvement at 10 3 BER with 1:6 dB higher launch power tolerance for 10 100 km transmission. Sj¨ odin [97] performed a numerical and experimental transmission comparison between polariza- tion switched and polarization multiplexed as a function of the fiber span length. Both the numerical 6 and experimental results indicate that the polarization switched format outperforms the polarization multiplexed format. Fischer [29] investigated an experimental WDM utilizing six polarization state with QPSK to sup- port that the 4-D signal transmission at 28 GBaud with a net bit rates of 112 Giga bit per second (Gbps) and 126 Gbps respectively. The 4-D waveform resulted in a sensitivity gain of 1:5 dB which extended the transmission distance from 3360 km to 4320 km. 1.2 Optical Transmission Band Extension The second mechanism to increase the infrastructure capacity is to extend the optical wavelength trans- mission band as much as possible into the ’O’, ’E’, and ’S’, beyond the currently utilized ’C’ and ’L’ bands 3 . While the Erbium Doped Fiber Amplifier (EDFA) provides low noise amplification over the range of 1530 nm to 1620 nm and drives the telecommunications infrastructure [26] in the C and L bands of the spectrum, future capacity demands would ultimately require amplification beyond the bandwidth of current EDFA technology which remains a challenge. In this section, a review of the attained milestones towards widening the transmission bandwidth amplification will be presented. Ito [45] demonstrated a Gain Shifted Thulium doped Fiber Amplifier (GS-TDFA) as a promising candidate to amplify the S-band over the range 1478 nm to 1512 nm where the silica fiber loss remains low as depicted in Figure 1.4. The transmission demonstrated a 10 Tera bit per second (Tbps) over 117 km utilizing an EDFA for the C and L bands and the GS-TDFA for the S-band. 3 The ITU-T approved optical band assignment is O (Original) E (Extended) S (Short) C (Conventional) L (Long) 12601360 nm 13601460 nm 14601530 nm 15301565 nm 15651625 nm 100 nm 100 nm 70 nm 35 nm 60 nm 7 Figure 1.4: SSMF loss across the optical bands associated with the EDFA band [26]. Emori [25] demonstrated a 100 nm Raman amplifier ranging 1520 nm to 1620 nm with a maximum of 1:6 dB gain variation. Yeh [124] demonstrated a hybrid S to L band amplifier with 120 nm bandwidth covering the range from 1480 to 1600 nm. Boubal [10] demonstrated a 4:16 Tbps over 135 km in the S, C and L bands using a distributed Raman pre-amplification over continuous bandwidth of 104 nm. Fukuchi [31] reviewed the amplification technologies at the corresponding wavelength range for high capacity WDM transmission. Eventhough amplification research has slowed down recently, it is expected that the increased capacity demand would re-intensify the research efforts [3]. In addition to amplification, other factors include linear effects such as disperion and non-linear effects such as Self Phase Modulation (SPM), Cross Phase Modulation (XPM) and Four Wave Mixing (FWM) discussed in Schneider [82] and summarized in chapter 3 need to be considered as well. The mitigation of the linear and nonlinear effects are discussed in chapter 4. 8 1.3 Multi Level Modulation The third dimension to increase the capacity of a communications system is to increase number of levels or transitions associated with a symbol. The increase can be accomplished by increasing the number of bits representing a given symbol or by adding states as discussed earlier in section 1.1 with the polarization switching methodology. Ifb denotes the number of bits per symbol at a given symbol rate,R s , then doubling the bit rate of the communications channel is equivalent to doubling the number of bits per symbol as shown below: R b =bR s ) 2R b = 2 (bR s ) = (2b)R s : (1.1) A symbol withb bits, transitions toM b levels whereM b = 2 b since each bit can be in two states. Equivalently, a symbol withM b transitions is associated withb = log 2 (M b ) bits. The number of levels to double the bit rate is the square of the number of levels as shown below: M 2b = 2 2b = 2 b 2 =M 2 b : (1.2) Due to the limited dynamic range of the hardware, increasing the number of levels associated with a symbol reduces the separation between levels and effectively decreasing the signal to noise ratio as detailed below. The SNR at the receiver or the transmitter is limited by the resolution of the converter due to the finite full scale of the converter. Suppose the converter full scale is Full Scale (FS), then the resolution of the converter,q, is the FS divided by the number of levels, that is: q = FS 2 b : (1.3) 9 so that the input to an ADC or the output of a DAC, ^ x, is an integer multiple of the resolution: ^ x =lq; for somel2 [0;M 1] (1.4) where M = 2 b 1. The corresponding quantization errorjxlqj is assumed to have a uniform distribution over a single resolution stepq. This assumption simplifies the analysis to a large degree eventhough it is not true since the error depends on the signalx as pointed out by Widrow [112]. Under the uniform assumption, the quantization noise power, 2 q , is: 2 q = 1 q Z q=2 q=2 2 d (1.5) = 1 12 q 2 = 1 12 FS 2 b 2 : (1.6) As the number of bits increase, the quantization noise power is reduced for a fixedFS. Assuming a converter tone periodT and full scale peak to peak amplitude, the tone power,P s , is P s = 1 T Z T=2 T=2 FS 2 2 sin 2 2 T t dt (1.7) = 1 T FS 2 2 Z T=2 T=2 1 2 1 cos 2 T t dt (1.8) = 1 T FS 2 2 T 2 (1.9) = 1 8 FS 2 : (1.10) 10 The corresponding quantization SNR: SNR q = P s 2 q (1.11) = 1 8 FS 2 1 12 FS 2 b 2 = 3 2 2 2b : (1.12) The quantization noiseSNR q in decibel yields the well known 6 dB per bit SNR: SNR q (dB) = 10 log 10 3 2 2 2b = 1:76 + 6:02b: (1.13) The detection noise consisting of the analog front end to include the detector as well as the converter thermal noise is assumed to have a Gaussian distribution with zero mean and variance or power over the detection bandwidth of 2 d . The detection SNR is given by: SNR d = P s 2 d = FS 2 8 2 d (1.14) with combined detector and quantization SNR written in reciprocal relationship: 1 SNR = 1 SNR d + 1 SNR q : (1.15) As the number of bits increase to support the shaping, two conditions need to be considered.The first is when the detection noise is dominant and the second when the quantization noise is dominant. When the detector noise is dominant and occupying the full scale of the converter dynamic range, the SNR of the converter is controlled by the detector noise as can be seen from Figure 1.5a. As the detection noise is decreased from full scale, the SNR improvement is a linear with a slope of 1 dB SNR 11 (a) Detector plus qantization converter SNR (b) Single bit converter SNR sensitivity Figure 1.5: Detector and quantization converter SNR and sensitivity. improvement per 1 dB of noise reduction. The SNR improvement continues until the quantization noise dominates at which the SNR levels as defined by equation (1.13). For example, the 4 bit converter SNR, levels off at 25:76 dB and the 10 bit converter levels off at 61:96 dB. As the number of bits of the converter increase, there is minor improvement when the converter detection noise is dominant as illustrated in Figure 1.5b at higher noise loading. For example, at detection noise of40 dB, increasing the number of bits from 7 to 8 bits, the SNR gain is 1 dB instead of the 6 dB. As the detector noise drops, the single bit sensitivity levels at 6 dB as expected due to the quantization noise which limits the SNR. However, attaining the quantization limits, requires that the detection noise drops below the quantization noise which may be very difficult for non-specialized detectors as illustrated by the one bit sensitivity curve in Figure 1.5b. Increasing the QAM modulation levels imposes additional requirements on the timing recovery and the residual phase error due to the two dimensional nature of the format as illustrated in Figure 1.6 due to the reduced separation between the constellation points. The angular separation between the closest adjacent constellation points is about 4:7 for 64-QAM [66] and reduced to 2 for 256-QAM. 12 (a) QPSK constellation (b) 16-QAM constellation (c) 256-QAM constellation Figure 1.6: Doubling the QAM data rate squares the order of the constellation. Beppu [9] demonstrated a 2048-QAM waveform over 150 km transmission following the earlier work of Koizumi [50,51] that demonstrated 512-QAM and 1024-QAM waveforms over 240 and 150 km respectively. To mitigate the timing error at these higher order constellations, the demonstration utilized a frequency stabilized fiber laser with 4 kHz linewidth at 3 GBaud with a pilot tone for carrier tracking with an optical phase locked loop. The level increase drives an increase in channel capacity in bits per second per unit bandwidth. To quantify the capacity increase or the spectral efficiency, a short review of the channel capacity is presented and the reader is referred to [21, 74] for a thorough discussion. If a channel transmits a symbol everyT s seconds, then the corresponding symbol rate isR s = 1=T s Baud. The associated bit rate,R b can be determined from the symbol rate and the number of bits per symbol, that is: R b bit sec =b bit symbol R s symbol sec (1.16) The signal power to noise power can be written as P s NoW = E s R s N o W = E b bR b =b N o W = E b R b N o W (1.17) 13 Using Shannon’s capacity formula in white Gaussian noise [21, 74]: C bits sec =W log 2 1 + P s N o W =W log 2 1 + E b N o R b W ; (1.18) the spectral efficiency measures the transmission capacity in one hertz of bandwidth as a function of SNR. Transmitting at the channel capacity, ie,R b =C, the spectral efficiency in bits/sec/Hz is C W = log 2 1 + E b N o C W : (1.19) For a given SNR, the spectral efficiency given in quation (1.19) identifies two limiting cases. The first is the case where the energy per bit to the noise density,E b =N o , approaches infinity and the second is the case when spectral efficiencyC=W approaches zero. Solving forE b =N o : E b N o = 2 C=W 1 C=W ; (1.20) The limiting case as (C=W )!1 is: lim (C=W)!1 E b N o = lim (C=W)!1 2 C=W 1 C=W =1 (1.21) to imply that larger energy per bit would be required to attain larger spectral efficiency. The second case occurs whenC=W tends to 0 which corresponds to power efficient transmission as the transmitted rate approaches zero with fixed bandwidth and a limit 4 of: lim (C=W)!0 E b N o = lim (C=W)!0 2 C=W 1 C=W = ln 2 =1:59 [dB]: (1.22) 4 To remind the reader, ify = 2 x , then lny = x ln2 and the derivativedy=dx can be obtained by differentiating both sides to yield(dy=dx)=y = ln2 so thatdy=dx = 2 x ln2. The limit is obtained by using L’Hopitals rule. 14 (a) BER for QAM constellations. (b) QAM and FSK spectral efficiency Figure 1.7: QAM modulation BER efficiency relative to the Shannon bound. The error free QAM constellation is determined [7, 74] by the product of two error free orthogonal Pulse Amplitude Modulation (PAM) symbols each with p M transitions. The Symbol Error Rate (SER) is then the complement of the error free transmission: SER M = 1 1 SER p M 2 (1.23) where SER p M is the SER for a PAM transmission of p M transitions given by SER p M = 2 1 1 p M Q 3 M 1 E s N o : (1.24) whereQ() is the Marcum Q-function. The constellation is Gray coded so that adjacent symbols differ only by a single bit and thus a single symbol error correspond to a single bit error with Bit Error Rate (BER): BER M = 2 1 1 p M Q 3 M 1 kE b N o (1.25) 15 Figure 1.7a depicts the BER for QAM constellations as a function of the energy per bit over noise density. For a representative error rate of 10 4 , the spectral efficiency measured in bits/sec/Hz in- creases as the energy per bit increases as shown in Figure 1.7b along the dotted points with BER of 1:0 4 . Furthermore, Figure 1.7b illustrates that doubling the two bit per symbol corresponding to QPSK modulation to 4 bits per symbol or 16-QAM modulation, the signal power would have to increase by 3:8 dB. Doubling from 4 bit per symbol modulation to 8 bit per symbol modulation or 256-QAM, the signal power would increase by at least 9 dB to maintain the same BER. To double from 6 bit per symbol or 64-QAM modulation to 12 bit per symbol or 2048-QAM, the signal power would increase by at least 16 dB to maintain the same BER. Increasing the efficiency by increasing the number of bits per symbol while maintaining a constant BER requires an increase in the signal power impacting the transmitter and the receiver realizations discussed in section 1.6. 1.4 Spatial Division Multiplexing The fourth dimension to increase the capacity is the number of communication streams or data channels within a single fiber [64, 65, 115, 118, 119]. Applying the Wave Division Multiplexing (WDM) on multiple cores in a Multi Core Fiber (MCF) or the fiber modes in Few Mode Fiber (FMF). Hayashi [37] published one of the early MCF design and fabrication papers with ultra-low loss of 0:12 dB/km and 30 dB of crosstalk 5 suppression consisting of seven cores with WDM depicted in Figure 1.8a and 1.8b on page 17 and demonstrated by Zhu [129] with 112 Tbps transmission over a 76.8-km link. The MCF depicted in Figure 1.8a shows each fiber core along with the associated wavelength utilized for the 5 Crosstalk is the power that couples into a core from adjacent cores. If the coupling coefficient between thei-th andj-th core ish ij , then the crosstalk power at thei-th core due to the adjacent cores is i = X i6=j h ij P j 16 transmission [37]. The wavelengths separation ranged from 26 nm between core 1 and core 6 to 2 nm between core 2 and core 7. (a) Seven core MCF and the wavelength assignment (b) MCF used in112 Tbps demonstration Figure 1.8: MCF fiber representation used to increase the data capacity [129] Recently, Turukhin [107] and Igarashi [41] demonstrated 105:1 Tbps and 1:03 Exa bit per second (Ebps) transmission over 14; 350 km and 7; 326 km using 12 and 7 cores MCF and WDM respectively. Sakaguchi [78] demonstrated a 305 Tbps space division multiplexed transmission over 19-core fiber with 100 WDM. In addition to the MCF technology, multiple modes in a fiber can be used to increase the trans- mission capacity whereby each mode transports independent data as depicted in the FMF transmission depicted in Figure 1.9. As the modulated carriers are launched into the fiber modes, energy couples across the modes due to fiber propagation and degrade the performance. To reduce the coupling de- grading effects, utilizing a low crosstalk FMF at the transmitter and the Multiple Input Multiple Output (MIMO) signal processing technology at the receiver as depicted in Figure 1.13 on page 22. MCF and FMF provide two multiplexing schemes to increase data capacity and could be applied concurrently with WDM. Takara [102] demonstrated a 1:01 Peta bit per second (Pbps) over 12-core fiber and 222 WDM link over 52 km. Ryf [77] demonstrated mode division multiplexing over 96 km of FMF using a 6 6 MIMO processing. Shibahara [94] demonstrated a 12-core with 3 modes transmis- sion over 527 km and MIMO processing. Recently, Soma [98] demonstrated 10:16 Pbps transmission 17 Figure 1.9: FMF mode multiplexing for increased tranmission capacity. using dense spatial division multiplexing, 6 modes and 19-cores of fiber over the entire C and L band for 11:3 km. Riesen [76] constructed a 4-core FMF with cross extinction of 25 dB that supported 3 Linearly Polarized (LP) modes, namely LP 01 LP 11a and LP 11b and illustrated in Figure 1.10. The same fiber can be used as a mode multiplexer and demultiplexer alleviating complex structures to launch the modes. Lu´ ıs [55] utilized the FMF fiber to demonstrate 3:37 km transmission with 256-QAM to attain a 1:2 Pbps link with Low Density Parity Check (LDPC) Forward Error Correction (FEC). Figure 1.10: Single fiber supporting FMF, MCF with mode multiplexing and de-multiplexing [76] For completeness, the LP modes are spatial modes that can be understood from a generalized Man- akov multi-mode propagation equation studied by Essiambre [28] and Mumtaz [63] for step-index and graded-index fiber and illustrated in Figure 1.11. 18 Figure 1.11: Fiber spatial LP modes in FMF obtained from the generalized Manakov model. Huang [40] demonstrated free space optical transmission using beams with multiplexed orthogonal OAM modes [123]. In addition to mode division multiplexing using fiber modes with FMF, spatial multiplexing can be attained by using orthogonal OAM modes. Two OAM beams with two different modes [113] can be expressed mathematically by: U 1 (r;;z) =A 1 (r;z) exp (jl 1 ) U 2 (r;;z) =A 2 (r;z) exp (jl 2 ) (1.26) wherer refers to the radial positionz is the propagation distance,l 1 andl 2 are the azimuth indexes for the two OAM modes representing the number of 2 changes per one spatial 2 turn and can be realized by a phase plate or using spatial light modulators. Two OAM beams are orthogonal if: Z 2 0 U 1 (r;;z)U 2 d = 8 > > > < > > > : 0; ifl 1 6=l 2 A 1 A 2 ; ifl 1 =l 2 (1.27) 19 Utilizing the orthogonality of the OAM modes, Xie [122] demonstrated 200 Gbps by multiplexing two modes. Yue [126] demonstrated a 1:6 Tbps using two OAM modes over specialized V ortex fiber and 10 WDM channels. Finally, the MIMO processing algorithm is sufficiently general to decouple modes encountered in MCF provided the coupling characteristics are known or can be estimated. A simplified summary of the MIMO algorithm assumes a communication system with N transmitters and N receivers where each transmitter transmits one mode to an intended receiver. However, the channel couples unwanted multiplexed signals into receivers as depicted in Figure 1.12 to be mitigated by the MIMO algorithm. Figure 1.12: Simplified communications system with crosstalk treated as a MIMO system. Mathematically, denote the transmitters byx 1 ;x 2 ;;x N and the corresponding receivers byy 1 ;y 2 ;;y N , then thek-th received signal,y k consists [64, 65, 77] of the weighted sum of all the transmitted signals through the channel response expressed as follows: y k =h k1 x 1 +h k2 x 2 +h kN x N ; k = 1; 2; ;N (1.28) whereh kk is the desired energy fromx k toy k , whileh kj ;j6= k;j = 1;;N denote the undesired energy from the other unwanted channles along the the mode associated withx k to be suppressed by 20 the MIMO signal processing. In general, thex k -th transmitter associated with they k -th receiver, can be written as a system of equations in matrix form as shown in equation (1.29): y 1 = h 11 x 1 + h 12 x 2 + h 1N x N + w 1 y 2 = h 21 x 1 + h 22 x 2 + h 2N x N + w 2 . . . . . . . . . . . . . . . y N = h M1 x 1 + h M2 x 2 + h MN x N + w N 9 > > > > > > > > > > = > > > > > > > > > > ; ) y = Hx + w: (1.29) The resulting parameters H, x; y and w are the FMF system matrix, transmitter, receiver and noise vectors respectively [65]. Solving for the transmitted data, x, is equivalent to transforming the system matrix H into a diagonal matrix [39] by the Singular Value Decomposition (SVD): H = UV y (1.30) where y is the transpose conjugate operator and is a diagonal matrix of the eigenvalues ofH y H. The left matrix U in equation (1.30) and the right matrix V are singular and unitary, ie, U y U =V y V = I: (1.31) The MIMO processing algorithm used in wireless communications [13] for minimizing ISI trans- forms the transmitted signal x to Vx to decouple the received signal as shown below: y = Hs + w = UV y Vx + w = U x + w: (1.32) 21 Multiplying the received signal y by U y , derives an estimate ^ x of the transmitted signal x: ^ x =U y y =U y U x +U y w = x +U y w (1.33) The recovered data is decoupled since the singular matrix is diagonal matrix and the noise power remains the same since it is scaled by a unitary matrix: E U y w y U y w =E n w y UU y w o =E n w y w o (1.34) In summary, the input data vector consisting of multiple data streams is transformed by a right singular matrix V to be modulated and transmitted. The receiver decouples the mode cross-talk by applying a left singular matrixU y as illustrated in Figure 1.13. V H = UV y + U y x ^ x w Figure 1.13: MIMO decoupling of the transfer matrix H. 1.5 Waveform Shaping The fifth dimension to increase the channel capacity is the effective utilization of the available spectrum by reducing the unnecessary bandwidth and the guard bands between adjacent channels. Shaping the spectrum can be accomplished with various degrees depending on the system constraints while the un- shaped spectrum may interfere with adjacent channels due to the wide spectral occupancy. Figure 1.14 on page 23 depicts an unshaped Non Return to Zero (NRZ) spectrum associated at a given symbol rate R s with significant sidelobes that will interfere with adjacent channels. 22 Figure 1.14: Shaped and unshaped spectrum comparison. With controlled shaping, the same data rate bandwidth would occupy less bandwidth depending on the spectral control of the waveform as illustrated by comparing spectral control that ranges between R s and 2R s as illustrated in Figure 1.14. As the occupied bandwidt is reduced, the spectral efficiency increases accordingly as indicated by equation (1.19). To illustrate the importance of the spectrum shaping and control in addressing the future expected demand, the frequency plan depicted in Figure 1.15a depicts a plan at the 100 GHz spacing complying to the ITU recommendation summarized in Table 1.1 on page 3. The unshaped transmission requires a relatively wide guard band between adjacent channels to reduce the Adjacent Channel Interference (ACI) and occupies a significant bandwidth. The shaped spectra depicts a tighter frequency plan with less adjacent channel interference at the 100 GHz spacing. A tight spectral plan can be devised to maximize the bandwidth utilization is illustrated in Figure 1.15b at the 50 GHz spacing. 23 (a)100 GHz spacing unshaped and shaped spectra (b)50 GHz spacing unshaped and shaped spectra Figure 1.15: ITU spacing of 100 and 50 GHz shaped and unshaped spectra. Controlled spectral shaping increases the overall channel capacity at the cost of driving the hardware and processing subsystems as identified in section 1.6. To illustrate the characteristics of the shaping, Figure 1.16 on page 25 depicts three waveforms illustrating the temporal response of an unshaped waveform and two shaped waveforms with distinct spectral efficiency. The unshaped waveform is a rectangular pulse of one symbol duration not requiring any signal processing for transmission with the lower spectral efficiency as can be seen from the corresponding plot in Figure 1.15. The remaining waveforms depict the two temporal response of the waveforms with spectral occupancy of 1:015R s and 1:75R s where R s is the symbol rate. Is is sufficient for now to point out that shaping requires temporal control supported by digital signal processing. The waveforms illustrated in Figure 1.16 correspond to the spectral response depicted in Fig- ure 1.15a and Figure 1.15b. To occupy the 50 GHz spacing, the spectral efficiency was set to a higher value resulting in almost 100% bandwidth utilization while the lower efficient waveform is suited for the 100 GHz spacing. The required unshaped spectrum requires spacing so that the sidelobes drop below a level with minimal ACI. 24 Figure 1.16: Impulse response of unshaped and shaped waveforms. 1.6 Waveform Drivers in a Communications System Each of the five dimensions outlined in Figure 1.2 on page 3 include PDM in section 1.1, Wavelength band expansion in section 1.2, Multi-Level Modulation in section 1.3, SDM in section 1.4, and wave- form shaping in section 1.5 extends the communications capacity of the underlying channels. Figure 1.17 on page 26 describes an optical communications link over a fiber channel illustrating the capacity increase mechanisms identified in Figure 1.2 on page 3. The transmitter maps the bits to symbols and digitally shapes the symbols. The pulse shaping constructs a waveform utilizing the bandwidth effectively as described in section 1.5. The DAC converts the digital waveform to its analog equivalent with sufficient bits to maintain the temporal fidelity and the spectral efficiency. The analog waveform is amplified and imparted onto the optical carrier by utilizing an optical Mach Zehnder Modulator (MZM). 25 Figure 1.17: Multi-Wavelength fiber transmission coherent system with MCF and FMF. 26 The optical channel consisting of the fiber that may consist of single mode, few mode fiber or multiple cores encounters the fiber transmission impairments such as Chromatic Dispersion (CD), SPM, XPM or FWM to limit the performance of the system. The receiver de-multiplexes the received channels and separates the polarization into its compo- nents. The separated polarizations are mixed with a local laser to detect the inphase and quadrature components. The detected components are digitized using an photonic [90,95,108] or electronic ADC with sufficient resolution and sample rate to prevent aliasing at a low quantization with SNR exceeding the threshold. The Digital Signal Processing (DSP) perform the coherent digital signal processing to recover the data,and mitigate channel impairments to minimize the bit error. The first consideration of the waveform shaping is the increased peak power of the shaped waveform as depicted in Figure 1.18b compared to the unshaped waveform in Figure 1.18a. The increased peak power of the waveform requires wider linear dynamic range to produce the DAC output and the elec- tronic driver and the MZM with minimal distortion to maintain the desired waveform shape. Chapter 4 discuss the methodologies to minimize the effects of the increased peak power. The second consideration is the timing accuracy at the decision sampling point as illustrated in Figure 1.18a and Figure 1.18b. The unshaped waveform holds the same value for a wider duration of the symbol versus the shaped waveform which requires tighter timing at the sampling instant to reduce ISI. Finally, the performance through the fiber channel will also be affected due to shaping as the instantaneous power increases. Table 1.3 summarizes the degradation mechanisms affected by the waveform along with the corresponding system elements. In this work, the focus will be on the waveform shaping dimension to investigate methods to uti- lize the optical and the electrical hardware effectively. while reducing the necessary peak power with minimal impact to the spectral efficiency and timing accuracy to increase the system capacity. 27 (a) Unshaped64-QAM eye diagram (b) Shaped64-QAM eye diagram Figure 1.18: Eye diagrams illustrating the peak growth due to waveform shaping. Table 1.3: Communication systems waveform parameters impact summary. Element Degradation Mechanism DAC Non-linear compression and limited dynamic range RF Driver Non-linear compression and limited dynamic range MZM Non-linear compression and limited dynamic range Fiber CD, SPM, XPM, FWM Detectors Limited frequency and dynamic range ADC Non-linear compression and limited dynamic range 28 Chapter 2 Spectrally Efficient Waveforms Chapter 2 reviews the pulse shaping literature and proposes a new bandwidth efficient waveform with reduced Peak to Average Power Ratio (PAPR). As stated in section 1.6, the peak power of the waveform drives the transmitter, receiver and the fiber degradation caused by high instantaneous power presented in Chapter 3. Designing a wavefor for reduced peak power without affecting the spectral efficiency is the focus of this study. The Nyquist Criterion (NC) is a critical criterion underlying the spectral efficiency of the waveform defined by the smallest bandwidth required to reconstruct the sampled waveform. The Raised Cosine Waveform (RCW) is the classical bandwidth efficient waveform and the reference waveform in this study. Gibby [33] generalized the NC and introduced the Generalized Nyquist Criterion (GNC) to derive general conditions for zero ISI enabling a class of new bandwidth efficient waveforms such as the Generalized Raised Cosine Waveform (GRCW) and Conjugate Root Pulse Waveform (CRPW). Other bandwidth efficient waveforms include Improved Nyquist Waveform (INW) and the proposed Weighted Raised Cosine Waveform (WRCW). 29 2.1 Raised Cosine Waveform The Raised Cosine Waveform (RCW) is the reference bandwidth efficient pulse shaping waveform satisfying the Nyquist zero ISI [4, 7, 74, 101]. The modulated RCW bandwidth is controlled by the excess bandwidth or roll-off parameter, denoted by that quantifies the bandwidth ratio in excess of the Nyquist symbol rate,R s =2. Excess bandwidth is between 0 and 1 with 0 denotes that the occupied bandwidth is the symbol rate while a unity excess bandwidth denotes an occupied bandwidth of 2R s as illustrated in Figure 2.2. Using normalized time, = t=T , whereT is the symbol time, the RCW impulse and frequency response [74] can be expressed as: h () = 8 > > > > > > > > > > > < > > > > > > > > > > > : 1; = 0 4 sin (=(2)) =(2) ; =1=(2) sin () cos () (1 2) (1 + 2) ; elsewhere (2.1) H (f) = 8 > > > > > > > > > > > < > > > > > > > > > > > : 1; 0jfj (1)R s 2 1 2 1 + cos jfj (1)R s 2 R s ; (1)R s 2 jfj (1 +)R s 2 0; jfj (1 +)R s 2 (2.2) Figure 2.1 illustrates the impulse and frequency response of the raised cosine for minimal excess bandwidth of 0, nominal excess bandwidth of 0:6 and full excess bandwidth 1. As the excess bandwidth tends towards 0, the time response increases except at symbol instants. The off symbol value drives the peak increase due to the summation of overlapping symbols. 30 (a) RCW time domain impulse response (b) RCW frequency response Figure 2.1: RCW time and frequency responses ( = 0; 0:6; 1:0). The occupied bandwidth of the modulated waveform is (1 +)R s and ranges fromR s to 2R s as the excess bandwidth parameter,, varies from zero to unity. The lower values of correspond to the spectral efficient modulations with higher peak values. The two special cases of the RCW of zero and full excess bandwidth cases correspond to = 0 and = 1 respectively determine bounds on spectral efficiency and peak value. The first case for = 0 is the sinc() waveform described in equation( 2.3) has a brick frequency response of widthR s as shown in Figure 2.1b. R s (1 +)R s =2 R s =2 (1)R s =2 (1)R s =2 R s =2 (1 +)R s =2 R s # 0 # 0 " 1 " 1 # 0 # 0 " 1 " 1 f H(f) Figure 2.2: RCW spectral response parameters. 31 h (; = 0) = 8 > > > > < > > > > : 1; = 0 sin () ; elsewhere. (2.3) In addition to the sharp cutoff frequency response of bandwidth equal to the symbol rate, the = 0 case has the highest peak value through the channel and the highest timing error ISI of the RCW waveforms. Both the peak value and the timing error ISI are determined by the response of the wave- form around the symbol sampling instant corresponding to the zero crossing of the waveform. The zero crossings, , of the waveform are separated by a symbol period as can be examined from equa- tion (2.3): sin ( ) = 0 ! =k; k =1;2; (2.4) The second case of = 1 described by equation (2.5) reduces to the product of a sinc() and a cos(). The frequency response bandwidth is twice the symbol rate with lower peak value through the channel and lowest timing error ISI of the RCW waveforms. h (; = 1) = 8 > > > > > > > > > > < > > > > > > > > > > : 1; = 0 1 2 ; = 1 2 sin () cos () (1 2) (1 + 2) ; elsewhere (2.5) In the second case, = 1, the zero crossings occur at the symbol instances and at the zeros of the cos() term in equation (2.5): cos ( ) (1 2 ) (1 + 2 ) = 0 ) =k + 1 2 ; k =1;2; (2.6) 32 The zero crossings occur at twice the symbol rate and may affect the clock recovery for some data based clock recovery mechanisms. For non-zero excess bandwidth, the zero crossings depend on the excess bandwidth as described by equation (2.2). In practice, the RCW may be partitioned into two filters by taking the square root of the frequency response to be discussed in section 2.3. The resulting Root Raised Cosine Waveform (RRCW) filters shape both the transmitted and received waveforms forming a matched filter pair that maximizes the SNR. However, the zero condition ISI is met when the combined transmitter and receiver constitute an RCW. As indicated earlier, the GNC conditions permit the construction of waveforms meeting the zero ISI condition without the matching as required by the RCW. 2.2 Generalized Nyquist Criterion Gibby [33] extended the Nyquist cirterion by relaxing the constraint that the spectrum of the waveform vanishing outside an interval to a decay requirement instead of a sharp cutoff. In this section, the GNC developed in [33] will be presented to realize spectrally efficient waveforms discussed in section 2.3. Denoting the transmitted symbols bya k at a symbol rateR s with a corresponding symbol time of T = 1=R s , the modulated waveform,s(t), at time instantt, can be expressed as the sum of delayed and weighted shaping pulses,h(t), at time instantt: s(t) = 1 X k=1 a k h(tkT ): (2.7) 33 The zero ISI condition states that at the sampling instant, the adjacent symbols do not contribute to the sampled signal rendering the signal interference free. Mathematically, in the time domain or in the frequency domain, the zero ISI condition is: h(kT ) = 8 > > > < > > > : 1; k = 0 0; k =1;2;;1: (2.8) 1 = X n H (f +nR s ) (2.9) A general linear transmission system shown in Figure 2.3 can be characterized by its impulse re- sponse [33] when an impulse is transmitted though a shaping filter,U(f), transmission channel,V (f) and a receive filterW (f). The received signal given byr(t) can be described by the inverse Fourier transform of the transmission shaping filter, the channel and the receive filter: r(t) =F 1 fW (f)V (f)U(f)g F 1 fH(f)g (2.10) whereH(f) is the cascade of theW (f)V (f)U(f). The GNC relaxes the constraint on the spectrum from being identically zero outside a region to a uniform convergence criterion requiring that the spec- trum falls assymptotically to zero better than or equal to a second order decay. To derive the GNC, the impulse response of the tranmission link is characterized in terms of an amplitude and a phase with necessary conditions for zero ISI [33]. Shaping,U(f) Channel,V (f) Receiver,W (f) r(t) (t) Figure 2.3: A general communications link for Generalized Nyquist Criterion analysis . 34 The receivedr(t) is the inverse transform of the system described in Figure 2.3: r(t) = Z 1 1 H(f) exp(j2ft)df (2.11) Writing the inverse transform over disjoint frequency segments of width 1=T , the impulse response at thek-th instant,r(kT ) is: r(kT ) = Z 1 1 H(f) exp(j2fkT )df (2.12) = 1 X n=1 Z (2n+1) 2 Rs (2n1) 2 Rs H (f) exp (j2fkT ) df (2.13) = 1 X n=1 Z Rs 2 Rs 2 H(f +nR s ) exp(j2kTf)df (2.14) whereR s = 1=T is the symbol rate and the substitutionf! f +nR s changed the integral limits in equation (2.14). The summation and integration can be exchanged provided that the P 1 n=1 H (f +nR s ) exp(j2kTf) is uniformly convergent as follows: r(kT ) = 1 X n=1 Z Rs 2 Rs 2 H(f +nR s ) exp(j2kTf)df (2.15) = Z Rs 2 Rs 2 1 X n=1 H (f +nR s ) exp (j2kTf) df (2.16) Z Rs 2 Rs 2 G (f) exp (j2kTf) df (2.17) whereG(f) is defined to be the summation of the system frequency response over the entire frequency line in segments ofR s : 35 G (f) 1 X n=1 H (f +nR s ): (2.18) Equation (2.17) is a Fourier series 1 coefficient expansion of G(f) and can be expressed as the summation of a periodic complex exponentials: G(f) = X k r(kT ) exp (j 2kTf): (2.20) Meeting the zero ISI condition in equation (2.8) implies that only the zeroth Fourier coefficient is not zero while all other coefficients are zero. Hence, G(f) =r(0) constant; for all f: (2.21) ExpandingH(f) in terms of its amplitude and phase,H(f) =A(f) exp (j(f)) in equation (2.18) and using the zero ISI condition in equation (2.21), the GNC reduces to: r(0) =G(f) = 1 X n=1 H (f +nR s ) = 1 X n=1 A (f +nR s ) exp (j (f +nR s )): (2.22) The uniform convergence requires that A(f) ! f q for q 2 instead of the more restrictive A(f) = 0 for some frequency and beyond. Waveforms that satisfy the Nyquist criterion vanish outside a frequency interval, are uniformly convergent and comply with the GNC. Waveforms that comply with the GNC are not necessarily compliant with the Nyquist criterion. 1 The Fourier series of a periodic functiong(t) is set of coeficientsc k such that g(t) = 1 X k=1 c k exp(j2kt) $ c k = Z g(t)exp(j2kt) dt (2.19) 36 The sum in equation (2.22) is over an infinite number of segments of the spectrum which can ex- tremely difficult to solve for an arbitrary spectrum in order to attain the GNC for zero ISI. Constraining the spectrum yields practical solutions, and in particular if the spectrum vanishes outsidejfj>R s , r(0) = 1 X n=1 A (f +nR s ) exp (j (f +nR s )): (2.23) Using the even amplitude symmetry and odd phase symmetry, the positive images are redundant and can be dropped so that the GNC condition for zero ISI [33] is: A (f) exp (j (f)) +A (fR s ) exp (j (fR s )) = Real Constant: (2.24) Expanding out equation (2.24) in terms of the real and imaginary components, the constrained GNC citerion is a system of equations: A(f) cos ( (f)) +A(f) cos ( (fR s )) =K (2.25) A(f) sin ( (f)) +A(f) sin ( (fR s )) = 0: (2.26) Written in matrix form, the amplitude at the frequency segments can be determined in terms of the phase at the corresponding segments: 0 B B @ cos ( (f)) cos ( (fR s )) sin ( (f)) sin ( (fR s )) 1 C C A 0 B B @ A (f) A (fR s ) 1 C C A = 0 B B @ K 0 1 C C A (2.27) 37 which can be solved for the amplitude segments by taking the inverse: 0 B B @ A (f) A (fR s ) 1 C C A = 1 0 B B @ sin ( (fR s )) cos ( (fR s )) sin ( (f)) cos ( (f)) 1 C C A 0 B B @ K 0 1 C C A (2.28) where = sin ( (fR s ) (f)) is the determinant. Multiplying out the right hand side, the equalized amplitude responses to yield zero ISI are: A(f) = K sin ( (fR s )) sin ( (fR s ) (f)) (2.29) A (fR s ) = K sin ( (f)) sin ( (fR s ) (f)) : (2.30) Similarly, the phase over the frequency segments can be determined in terms of the amplitude at the corresponding frequency segments. Using the trigonometric relationship, sin() = p 1 cos 2 (), the matrix terms of the constraint equation (2.27) can be rewritten with only cos terms: 0 B B @ cos ( (f)) cos ( (fR s )) p 1 cos 2 ( (f)) p 1 cos 2 ( (fR s )) 1 C C A 0 B B @ A (f) A (fR s ) 1 C C A = 0 B B @ K 0 1 C C A (2.31) Equation (2.31) is nonlinear and can be solved by multiplying out the terms to write the cos terms and squaring the second row to derive the terms shown in equation (2.33) below: cos ( (f))A (f) + cos ( (fR s )) A (fR s ) =K (2.32) 1 cos 2 ( (f)) A 2 (f) 1 cos 2 ( (fR s )) A 2 (fR s ) = 0: (2.33) 38 Equations (2.34) and (2.35) are the same equation with slight reordering of the right and left hand sides to simplify the reductions with equation (2.36) as follows: A(f) cos((f)) =KA(fR s ) cos((fR s )) (2.34) KA(f) cos((f)) =A(fR s ) cos((fR s )) (2.35) A 2 (f) (1 cos 2 ((f))) =A 2 (fR s ) (1 cos 2 ((fR s ))) (2.36) Squaring equation (2.34) and adding to equation (2.36, the phase term atfR s : A 2 (f) =K 2 2KA(fR s ) cos((fR s )) +A 2 (fR s ) (2.37) Similarly, squaring equation (2.35) and adding to equation (2.36) yields the phase atf: A 2 (fR s ) =K 2 2KA(f) cos((f)) +A 2 (f) (2.38) Simplifying the equations yields the phase equalization conditions: (f) = cos 1 K 2 +A 2 (f)A 2 (fR s ) 2KA(f)) (2.39) (fR s ) = cos 1 K 2 A 2 (f) +A 2 (fR s ) 2KA(fR s )) (2.40) Thus phase response of the zero ISI waveform can be determined from the magnitude response or the magnitude response can be determined from the phase response. 39 2.3 Generalized Raised Cosine Waveform Alagha [4] introduced the Generalized Raised Cosine Waveform (GRCW) to provide ISI free transmis- sion based on the GNC discussed in section 2.2 using a phase compensation technique that extended the generalized approach of Xia [121] illustrated in section 2.5 includes the root raised cosine waveform. The phase compensation scheme decomposes any Nyquist filter into two cascaded Nyquist filters. In particular, applying the scheme to the root raised cosine waveform may preclude the matched filtering at the receiver in order to simplify the system design [4] at increased signal strength penalty. Matched filtering maximizes the received signal power and consequently the SNR prior to symbol detection [69, 71, 106]. Suppose a finite energy waveform given by s(t) passes through a combined transmission channel and receiver filter,h(t), then the received signal,r(t) is: r(t) = Z 1 1 s()h(t)d (2.41) = Z 1 1 S(f)H(f) exp (j2ft)df (2.42) The first equation (2.41) is the time domain representation expressed as a convolution while the sec- ond equation (2.41) is the equivalent representation expressed by the corresponding inverse Fourier transform. The received signal at a symbol time,r(T ), is [71]: r(T ) = Z 1 1 s()h(T)d (2.43) = Z 1 1 S(f)H(f) exp (j2fT )df: (2.44) 40 Applying Cauchy-Schwartz Inequality (CSI) described in appendix A, equation (A.10), the received value at the symbol instant can be bounded: jr(T )j 2 = Z 1 1 S(f)H(f) exp (j2fT )df 2 (2.45) Z 1 1 S(f)S (f)df Z 1 1 H(f) exp (j2fT )H (f) exp (j2fT )df (2.46) = Z 1 1 jS(f)j 2 df Z 1 1 jH(f)j 2 df (2.47) with a maximum Matched value attained whenS(f) equals the conjugate ofH(f) exp (j2fT ): S(f) =H (f) exp (j2fT ): (2.48) Denoting the transmit, receive and the combined shaping filters byG t (f),G r (f) andG(f) respectively, the amplitude and phase matching conditions of equation (2.48) are: G t (f) =G r (f) exp(j2fT ) () 8 > > > < > > > : jG t (f)j =jG r (f)j 6 G t (f) =2fT 6 G r (f): (2.49) The combined transmit and receive filters result in an ISI free waveform when the overall response is Nyquist. Using the matching conditions in equation (2.49), the transmit and receive filters amplitude and phase requirements for matching and ISI free transmission can be derived. The amplitude of each element can be determined to be: G(f) =G t (f)G r (f) () jG(f)j =jG t (f)jjG r (f)j =jG t (f)j 2 =jG r (f)j 2 : (2.50) 41 The amplitude of both the transmit and receive is the square root of the combined Nyquist filter: jG t (f)j =jG r (f)j = p jG(f)j: (2.51) The transmit and receive phase cancel out with a constant residual delay of a symbol time,T : 6 G(f) = 6 G t (f) + 6 G r (f) =2fT 6 G r (f) + 6 G r (f) =2fT: (2.52) To recap, the matching and ISI free transmission result in the conditions summarized by equations (2.51) and (2.52). Partitioning a Nyquist filter into transmit and receive matching filters may render each as a non Nyquist even though the cascade is matched and ISI free. (a) RCW versus RRCW impulse response. (b) RCW cascade versus RRCW cascade response. Figure 2.4: ISI behavior of RCW and RRCW cascade. For example, the Nyquist RCW waveform stated in equation (2.2) and the RRCW illustrated in Figure 2.4a shows that value of the RCW is zero at symbol epochs to provide an ISI free waveform while the RRCW is not zero at symbol epochs and thus is not ISI free waveform. Figure 2.4b shows that the cascading two RCWs does not result in an ISI free transmission while cascading the RRCW results in ISI free transmission. 42 Utilizing the GNC discussed in section 2.2, the phase relationship detailed in equation (2.39) and (2.40) is established to decompose the Nyquist filter into ISI free segments [4, 121]. To illustrate the approach, a generalize method to decompose into square root Nyquist compliant by compensating the phase. Restating the GNC criterion [33] from section 2.2 for zero ISI. A (f) exp (j (f)) +A (fR s ) exp (j (fR s )) =K; (2.53) whereK is a real constant,A(f) is the amplitude and(f) is the phase response. Equation (2.53) is a repeat of equation (2.24) derived in section 2.2 and solved for both the phase and amplitude of the ISI free transmission with solution stated in equations (2.39) and (2.40) repeated below: (f) = cos 1 K 2 +A 2 (f)A 2 (fR s ) 2KA(f)) (2.54) (fR s ) = cos 1 K 2 A 2 (f) +A 2 (fR s ) 2KA(fR s )) : (2.55) A Nyquist filter with a known amplitude response and zero phase, can be transformed into a square root Nyquist filter by compensating the phase [4]. LetH(f) be a Nyquist filter with a zero phase, then applying the GNC relation given in equation (2.53) with(f) = 0 toH(f) to yield which will be used to construct the root Nyquist waveform: H(f) +H(fR s ) =K: (2.56) 43 Since H(f) has zero phase, then it is real and non-zero, then the root Nyquist filter, G(f), can be constructed by adding a phase compensation to the square root ofH(f): G(f) = p H(f) exp(j (f)); (2.57) where (f) can be determined from equation (2.54): (f) = cos 1 K 2 +jG (f)j 2 jG (fR s )j 2 2KjG (f)j ! (2.58) = cos 1 K 2 +H (f)H (fR s ) 2K p H (f) ! ; 0fR s ; (2.59) where we used the fact thatjG(f)j 2 = H(f). Using equation (2.56) and setting K 2 = T , the compensating phase (f) can be simplified to: (f) = cos 1 K 2 +H (f)H (fR s ) 2K p H (f) ! (2.60) = cos 1 T +H (f)T +H (f) 2 p T p H (f) ! (2.61) = cos 1 2H (f) 2 p T p H (f) ! (2.62) = cos 1 p H (f) p T ! ; 0fR s : (2.63) Given the Nyquist filter with zero phase, H(f), the root Nyquist filter G(f) is given by equa- tion (2.57) where the compensating phase (f) is given in equation (2.63). It is interesting to note here, that by considering the RCW, a relation of the phase form can be derived and to synthesize the GRCW initially presented by Xia [121] and later expanded by Alagha [4]. 44 Rewriting the RCW defined by equation (2.2) in terms of cos 2 for convenience and to derive the compensating phase to realize the Generalized Root Raised Cosine Waveform (GRRCW) and will be distinguished from the traditional RRCW. Setting the symbol time T to unity for convenience, the traditional RCW is: H (f) = 8 > > > > > > > > > > > < > > > > > > > > > > > : 1; 0jfj (1) R s 2 cos 2 jfj (1) R s 2 2R s ; (1) R s 2 jfj (1 +) R s 2 0; jfj (1 +) R s 2 : (2.64) The root of RCW is the traditional RRCW is not a Nyquist waveform. However, cascading two traditional RRCW yields a matched Nyquist waveform with maximum signal as discussed earlier in this section: p H (f) = 8 > > > > > > > > > > > < > > > > > > > > > > > : 1; 0jfj (1) R s 2 cos jfj (1) R s 2 2R s ; (1) R s 2 jfj (1 +) R s 2 0; jfj (1 +) R s 2 : (2.65) Using equation (2.63) the compensating phase can be described in equation (2.67) and extended outside the interval (0 f R s ) while maintaining an odd phase, ie, (f) = (f). Figure 2.5a and 2.5b show frequency and phase response of the ISI free RRCW. The RRCW filter spectral oc- cupancy can be controlled by the excess bandwidth parameter, as illustrated in Figure 2.5a. As approaches 0, the spectral occupancy is 2R s =2 and as approaches 1, the spectral occupancy is 2R s . 45 R s (1 +)R s =2 R s =2 (1)R s =2 (1)R s =2 R s =2 (1 +)R s =2 R s # 0 # 0 " 1 " 1 # 0 # 0 " 1 " 1 f p H(f) (a) RRCW frequency response derived form RCW. R s (1 +)R s =2 R s =2 (1)R s =2 (1)R s =2 R s =2 (1 +)R s =2 R s =2 =4 f (f) (b) RRCW compensation phase for ISI free response. Figure 2.5: ISI free RRCW shaping filter frequency response and compensation phase. As stated earlier, the phase compensation assures ISI free filter and matching maximizes the signal strength. The unmatched loss can be estimated using CSI summarized by equation (A.5) or (A.11). The illustrated case of the RCW demonstrates an approach to generalize the design of RCW by constructing phase compensation profiles with common properties to be discussed next. 46 (f) = cos 1 p H (f) ; 0fR s (2.66) = 8 > > > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > > > : 0; (1) R s 2 f (1) R s 2 f (1) R s 2 2R s ; (1 +) R s 2 f (1) R s 2 f + (1) R s 2 2R s ; (1) R s 2 f (1 +) R s 2 0; jfj (1 +) R s 2 (2.67) Alagha [4] presented phase compensations to generate GRCW derived from the RCW by utilizing the Xia [121] phase functions discussed in section 2.5 and equation (2.24) satisfying the GNC for ISI free shaping. The phase functions denoted by n (x) are: n (x) = 8 > > > > > > > > < > > > > > > > > : 1; x<1 p n (x) 1x 1 1 x> 1: (2.68) wherep n are polynomials of order (2n 2) continuous functions given by p n (x) = k=n X k=1 a k x 2k1 = n Z x 0 1u 2 n1 du; (2.69) n = 1 R 1 0 (1u 2 ) n1 dx = 1 p n (1) (2.70) 47 The GNC amplitude and phase relationship for an ISI waveform is: A(f) =T cos 2 ((f)) (2.71) A p = p T cos((f)) exp(j(f)) (2.72) (f) = 4 V 2T f 1 2T 4 (2.73) V (x) 8 > > > > > > > > < > > > > > > > > : 1 x<1 p n (x) jfj 1 1 x> 1 (2.74) (a) GRCW for = 0:35;P 1 ;P 5 (b) GRCW for = 1:0;P 1 ;P 5 Figure 2.6: GRCW impulse response. Figure 2.6a, 2.6b depict the GRCW for excess bandwidth 0:35 and 1:0. The zero ISI is met since the samples are zero at the symbol times with controlled peak values. The root of the GRCW denoted by GRRCW meets the zero ISI as depicted in Figures 2.7 and 2.8. The GRRCW is not even and does have higher signal extremes than the GRCW. 48 (a) GRRCW for = 0:35;P 1 ;P 5 (b) Matched GRRCW for = 0:35;P 1 ;P 5 Figure 2.7: GRRCW and Matched GRRCW for = 0:35. (a) GRRCW for = 1:0;P 1 ;P 5 (b) Matched GRRCW for = 1:0;P 1 ;P 5 . Figure 2.8: GRRCW and Matched GRRCW for = 1:0. 49 2.4 Weighted Raised Cosine Waveform The Weighted Raised Cosine Waveform (WRCW) is constructed by multiplying the time domain RCW by a kernel with compact support [89, 92, 93] so that the overlapping segments are reduced without introducing a waveform discontinuity and cause spectral growth to reduce the spectral efficiency. A suitableC 1 kernel [46],k(;), with compact support that reduces the Peak to Average Power Ratio (PAPR) over the excess bandwidth range is: k(;) = 8 > > > > < > > > > : exp 1 1jj p() jj< 1 0; jj 1; (2.75) where is the normalized time andp() is an index function of the excess bandwidth. The kernel is defined over the interval (1; 1). As the index mappingp() increases, the kernel is reduced over the overlapping region without affecting the values at the symbol instants. The index function dependency on the excess bandwidth reduces the overlapping region for smaller values of the excess bandwidth. Figure 2.9a defines excess bandwidth index mapping,p() while Figure 2.9b illustrates the kernel as the index increases over the values of 2; 2:5; 4:0 corresponding to excess bandwidth of 0; 0:6 and 1. Weighted Raised Cosine Waveform (WRCW) reduces the overlapping region for smaller excess band- width more than the larger excess bandwidth with a piece-wise linear excess bandwidth dependency, p(), that is: p() = 8 > > > > > < > > > > > : 5 6 + 2 0 < 6 10 25 10 + 15 4 6 10 6 10 1: (2.76) The next step in constructing the WRCW is to extend the kernel over adjacent symbols to control the nearest symbol regions. By adding a shifted version of the kernel window and limiting to unity 50 (a) Kernel power as a function of. (b) WRCW kernel. (c) WRCW window). (d) RCW, weighting window and the WRCW. Figure 2.9: WRCW construction for all symbols away from2 to form a ”W” shaped kernel stated in equation (2.77) and depicted in Figure 2.9c. For regions beyond a larger number of symbols,N, the waveform is dropped gradually to 0 to bound the window energy by a suitable unit step restriction. The weighting windoww() is then: w () = 8 > > > > > > > > > > > > > < > > > > > > > > > > > > > : k ( 2) +k () +k ( + 2) 2 [2; 2] 1 (N;2)[ (2;N) u(N)k(N) [N;1) u( +N)k( +N) (1;N]: (2.77) 51 To illustrate the construction of the waveform, Figure 2.9a shows the kernel for various power values. As noted earlier, the kernel decreases for smaller excess bandwidth to reduce the peak of the smaller excess bandwidth waveform more than the higher excess bandwidth waveform. Once the kernel is constructed, a weight window in the shape ofW is constructed by a limiting the summation of a shifted kernel at2 as depicted in Figure 2.9c. Figure 2.9d depicts the raised cosine with the shaping window and the resulting weighted raised cosine for excess bandwidth of = 0:6. Please note that the WRCW is less than the RCW over the region (1; 0) and (1; 0) with smaller modulated peak waveform value given by equation (2.7). 2.5 Conjugate Root Pulse Waveform The RCW described in section 2.1 is the reference waveform satisfying the Nyquist criterion for ISI free transmission. To maximize the received signal to noise ratio, the received signal is matched to the transmitted waveform leading to a square root raised cosine transmission shaping. If G(!) is the RCW, then the decomposition of p G(!) yield an ISI free waveform with maximum signal to noise ratio provided the received signal is matched with the transmitter. Xia [121], presented a family of waveforms satifying the ISI free criterion that without matched filtering. The waveforms reduce the hardware complexity by attaining the ISI free condition without matched filtering which may be applicable in some environments at the loss of the matched filtering signal to noise ratio gain. Let(x) be an even, convex and continuous function [121], such that: (x) +(1x) =(x) +(1 +x) = 1 for all x2 (1;1): (2.78) 52 then over the interval (0; 1), the function will be defined to meet equation (2.78) and outside the interval (0; 1), the continuous function is set to a constant value as shown below: (x) = 8 > > > < > > > : 0; x 0 1; x 1: (2.79) Examples of such functions [121] satisfying equations (2.79) and (2.78) are 1 and 4 given in equa- tions (2.80) and (2.81) illustrated in Figure 2.10. To show the conditions in equation (2.78) are met, consider forx < 0, then 1 (x) = 0 and 1 (1x) = 1 since 1x > 1. Forx > 1, 1 (x) = 1 while 1 (1x) = 0 since 1x< 0. In the region 0<x< 1, the sum 1 (x)+ 1 (1x) = 1 since 1 (x) =x and 1 (1x) = 1x. Similarly, the case can be proved for 4 (x). These shaping pulses defined [121] applicable for the RCW with an illustrated case of = 1=3 that can be easily generalized to arbitrary excess bandwidth. 1 (x) = 8 > > > > > > > > < > > > > > > > > : 0; x 0; x; 0x 1 1 x 1 (2.80) 4 (x) = 8 > > > > > > > > < > > > > > > > > : 0; x 0 x 4 (35 84x + 70x 2 20x 3 ); 0x 1 1 x 1 (2.81) 53 Tan [103] presented a raised cosine with the property that the transform is conjugate symmetric. LetG(!) be a conjugate symmetric transform, ie,G(!) =G (!) then using the properties stated in equation (2.78), the GRCW can be Figure 2.10: Phase functions for ISI free waveform G(!) = 8 > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > : 1; j!j (1) T 1 2 1 + exp j 1 !T 2 1 2 ; (1) T <!< (1 +) T 1 2 1 exp j 1 !T + 2 2 1 2 ; (1 +) T <!< (1) T 0 j!j (1 +) T (2.82) The property that the waveform is conjugate symmetric, ie, 54 G(!) =G (!) (2.83) and summarized in theorem 2.5.1 and illustrated in [121] for the case of = 1=3. The underlying principle of this result is that the matched receiver waveform, the transfer function would beG (!)G(!) andG(!) and ISI free. Theorem 2.5.1. The pulse shaping filterg(t) defined by the inverse transform of equation (2.82) is (i) real valued, (ii) ISI free, and (iii) the matched system is also ISI free. Proof. To show that g(t) is real valued, it is sufficient to show 2 that G(!) = G (!). Evaluating the conjugate of G(!) at! over the region (1 +) T < ! < (1) T , and using (x) = 1(1x) stated in equation (2.78): G (!) = 1 2 1 exp j 1 !T + 2 2 1 2 (2.84) = 1 2 1 exp j 1 1 1 !T + 2 2 1 2 (2.85) = 1 2 1 + exp j 1 1 !T + 2 2 1 2 (2.86) = 1 2 1 + exp j 1 + !T 2 1 + 1 2 2 (2.87) = 1 2 1 + exp j 1 !T 2 1 2 (2.88) =G(!): (2.89) 2 A quick reminder that the transform of a real waveform is conjugate symmetric is shown herein: G (!) = Z g(t)exp(j!t)dt = Z g (t)exp(j!t)dt = Z g(t)exp(j!t)dt =G(!) 55 To prove that G(!) has the ISI free property, it is sufficient to prove that G(!) satisfies the Nyquist criterion, namely: X n G(! + 2n=T ) = 1: (2.90) Which can be established by expanding G(!) given in equation(2.82) over consecutive segments in integer multiples of 2. (a) Time domain waveform,g(t) (b) Freqnecy response,G(!) (c) Time domain waveform,h(t) =g(t)g(t) (d) Frequency response ofH(!) =G(!)G(!) Figure 2.11: First order conjugate waveforms for different excess bandwidth. Figure 2.11a and 2.12a depict the first and fourth order conjugate waveform satisfying the Nyquist criterion while Figure 2.11b and 2.12b depicts the corresponding Fourier transforms. Please note that zeros of the waveform occur at symbol time to comply with the ISI free condition. The matched 56 waveform consist of the waveform g(t) and its corresponding matched filter shown in Figure 2.11c and 2.12c meets that ISI free condition with the zeros occurring at symbol times. The corresponding transforms shown in Figure 2.11b, 2.11d, 2.12b and 2.12d. The fourth order waveform has smoother frequency roll-off due to the phase shaping function defined earlier. Adding multiple 2 shifted copies of the frequency response, illustrates the ISI free criterion stated in equation (2.90). (a) Time domain waveform,g(t) (b) Freqnecy response,G(!) (c) Time domain waveform,h(t) =g(t)g(t) (d) Frequency response ofH(!) =G(!)G(!) Figure 2.12: Fourth order conjugate waveforms for different excess bandwidth. 57 Demeechai [23] generalized the waveform presented in [121]. IfP (!) is even and vanishes outside an interval, thenQ(!), defined below satisfies the ISI free criterion whereQ(!) is defined by: Q(!) = 8 > > > < > > > : P (!)j p (1P (!))P (!); j!j 0 P (!)j p (1P (!))P (!); j!j< 0 (2.91) In particular, if P (!) = G(!)G (!) where G(!) is the real waveform defined in equation (2.82), thenQ(!) is ISI free to generalize the waveform introduced by Xia [121] and studied by Tan [104]. ExaminingQ (!)Q(!) reduces toP (!) which is equal to ISI free waveformG (!)G(!). 2.6 Improved Nyquist Waveform The Nyquist waveform or the raised cosine presented in section 2.1 is the most utilized waveform for its spectral efficieny and control. Beaulieu [6] introduced a Better Than Nyquist Waveform (BTNW) with better eye opening and reduced ISI in the presence of timing error for the same excess bandwidth. Assalini [5] introduced Improved Nyquist Waveform (INW) which are a variant on the Better Than 58 Nyquist Waveform (BTNW) and corrected the typographical error waveform in [6]. Equation (2.92) de- fines the corrected waveform,S 2 (f), denoted [5] by Flipped Exponential Improved Nyquist Waveform (FEINW), to indicate the nature of the frequency response excess bandwidth as depicted in Figure 2.13: S 2 (f) = 8 > > > > > > > > > > > > > < > > > > > > > > > > > > > : 1; 0jfj 1 2T (1) exp (1) 2T jfj (1) 2T jfj 1 2T 1 exp jfj (1 +) 2T 1 2T jfj (1 +) 2T 0; (1 +) 2T jfj; (2.92) where 2Tln(2)= and T is the symbol time where the periodic extension satisfies the Nyquist Criterion. The time domain impulse response,s 2 (t) of the flipped-exponential transform,S 2 , given in equation (2.93), is real since it is the inverse of a real and symmetric transform given in equation (2.92) and depicted in Figure 2.14: s 2 (t) = sinc(t=T ) 2T 4 t sin (t=T ) + 2 2 cos (t=T ) 2 4 2 t 2 + 2 : (2.93) 59 The conjugate of any real and symmetric transform is real can be directly established by examining the inverse transform. LetS(f) be a real symmetric transform, then the impulse responses(t) can be shown to be real. Starting with the conjugate ofs(t) and show that it is identical tos(t): s (t) = Z 1 1 S(f) exp (j2ft)df (2.94) = Z 1 1 S (f) exp (j2ft)df (2.95) = Z 1 1 S(f) exp (j2ft)df (2.96) =s(t): (2.97) The ISI free property of the flipped-exponential can be established by considering the infinite sum of the shifted copies of the spectrum equals to a unity expressed in equation (2.90) and illustrated in Figure 2.13 for the case of excess bandwidth equal to 0:35. The second INW is the Flipped Hyperbolic Secant Improved Nyquist Waveform (FHSINW) spec- tral response given by [5]: S 3 (f) = 8 > > > > > > > > > > > > > < > > > > > > > > > > > > > : 1; 0jfj 1 2T (1) sech jfj (1) 2T (1) 2T <jfj 1 2T 1 sech (1 +) 2T jfj 1 2T <jfj (1 +) 2T 0; (1 +) 2T <jfj; (2.98) where 2Tln(2 + p 3)=. The time domain impulse response of the FHSINW is the inverse transform of a real and symmetric transform given in equation (2.98) and depicted in Figure 2.15b. The time domain impulse response is a real waveform given in equation (2.99) and depicted in Figure 2.15: 60 Figure 2.13: ISI free illustration of the flipped-exponential waveform. s 3 (t) = sinc(t=T ) T 8t sin (t=T )F 1 (t) + 2 cos (t=T ) (1F 2 (t)) + 4F 3 (t) 1 4 2 t 2 + 2 (2.99) F 1 (t) = 1 X k=0 (1) k (2k + 1) ((2k + 1) ) 2 + (2t) 2 (2.100) F 2 (t( = 1 X k=0 (1) k (2t) 2 ((2k + 1) ) 2 + (2t) 2 (2.101) F 3 (t) = 1 X k=0 (1) k (2t) 2 ((2k + 1) ) 2 + (2t) 2 exp ( (2k + 1) =(2T )) (2.102) The limiting behavior of the flipped hyperbolic secant as the excess bandwidth aproaches zero, or the most spectrally efficient waveform, is the sinc(t=T )=T . 61 (a) Flipped-exponential time impulse response (b) Flipped-exponential frequency response Figure 2.14: Time and frequency domain of the flipped exponential. (a) Flipped-hyperbolic secant time impulse response (b) Flipped-hyperbolic secant frequency response Figure 2.15: Time and frequency domain of the flipped hyperbolic secant waveforms. The third improved Nyquist waveform is the flipped-inverse hyperbolic secant (farcsech) waveform with secant spectral response given by [5]: 62 S 4 (f) = 8 > > > > > > > > > > > > > < > > > > > > > > > > > > > : 1; 0jfj 1 2T (1) 1 T archsech T (1 +) 2T jfj (1) 2T <jfj 1 2T T archsech T jfj (1) 2T 1 2T <jfj (1 +) 2T 0; (1 +) 2T <jfj (2.103) The time response of the flipped-inverse hyperbolic secant is difficult to obtain in closed form and is normally derived by converting the frequency domain to the time domain by the taking inverse Fourier transform via the Fast Fourier Transform (FFT). Finally, for completeness, the ISI free property of the flipped-hyperbolic and the flipped-inverse hyperbolic secant can be established by considering the infinite sum expressed in equation (2.90) and illustrated in Figure 2.16. (a) Flipped hyperbolic secant frequency response (b) Flipped Inverse hyperbolic secant frequency re- sponse Figure 2.16: ISI free flipped hyperbolic and inverse hyperbolic secant waveforms. Figures 2.14a, 2.15a, and 2.17a depict the time domain impulse response of the flipped exponential, flipped hyperpolic secant and the flipped inverse hyperbolic secant waveforms for the cases of excess 63 bandwidth of = 0; 0:35; 1:0 which represent the least spectrally to the most spectrally efficient wave- forms along with a typical value of 0:35. As expected, the all these waveforms satisfy the ISI free criterion by having a zero value at the symbol time epochs. The main difference is the extent of the ISI that is introduced as a result of timing errors and non-linear loading of the electronics and optical components which will be studied in later chapters. Figures 2.14b, 2.15b, and 2.17b depict the flipped exponential, flipped hyperpolic secant and the flipped inverse hyperbolic secant waveforms frequency response. The main difference is the rolloff to zero amplitude characteristics and the behaviour at zero frequency. All these waveforms exhibit similar behavior to the sinc() at zero excess bandwidth 3 . (a) Flipped inverse hyperbolic secant time impulse re- sponse (b) Flipped inverse hyperbolic secant frequency re- sponse Figure 2.17: Time domain and frequency domain of the flipped inverse hyperbolic secant waveforms. 3 To avoid ambiguity thesinc(x) function issin(x)=(x). 64 Chapter 3 Optical Fiber Channel Waveform Degradation Mechanisms As multiple channels propagate over a fiber channel, both linear and nonlinear effects degrade the trans- mitted waveforms. In this chapter, the waveform degrading mechanisms will be explored to assess the degradation due to the linear such as loss and dispersion and nonlinear such as Self Phase Modulation (SPM), Cross Phase Modulation (XPM) and Four Wave Mixing (FWM). In some applications, the fiber effects can be used constructively, for example Yilmaz [125] used dispersion for true time delays and Willner [114] used the nonlinear effects for Optical Signal Processing. In general, fiber channel effects on communications links is mostly undesirable since the ideal channel is an all pass linear channel. A fiber optic channel consists of optical fibers, optical amplifiers, add-drop multiplexers among other network elements. The analysis in this study is restricted to transmissions through a single mode fiber without any other network elements as illustrated in Figure 1.17 on page 26. First, a brief review of the optical transmission channel based on the standard textbooks [1,79] described by the Generalized Nonlinear Schr¨ odinger Equation (GNLSE) as stated in [1] followed by a description of the transmission mechanisms affecting the analysis in this study. A model of the fiber transmission will be constructed to simulate the waveform degradation of the proposed waveform and compared to the reference waveform. 65 Agrawal [1] presented the GNLSE describing a propagating 1 electric field,E(z;t) in thez direction at time instantt. In the analysis that follows, only few dominant terms of the propagation equation will be kept. Starting with then-th order equation at the operating frequency! o : @ @z + 1 2 (! o ) +j d (! o ) d! @ @t j 1 X n=1 j n n (! o ) n! @ n @t n ! E(z;t) = j (! o ) +j d d! @ @t E(z;t) Z 1 0 R()E (z;t)E(z;t)d (3.1) where(! o )is the propagating media absorption coefficient, and the Taylor series expansion of the wave number(!) around the operating optical frequency! o : (!) = 0 + (!! o ) 1 + 1 2! (!! o ) 2 2 + 1 3! (!! o ) 3 3 + (3.2) n (! o ) d n (! o ) d! n : (3.3) The Raman intra-pulse scatteringR(t) affects pulse to pulse scattering energy transfer, the nonlinear parameter (! o ) in equation (3.1) is a function of the nonlinear Kerr parametern 2 and the effective mode area,A eff defined by: (! o ) = n 2 ! o cA eff (3.4) A eff = RR 1 1 E (x;y)E(x;y)dxdy 2 RR 1 1 (E (x;y)E(x;y)) 2 dxdy (3.5) Assuming that the attenuation and non-linearity change with frequency is relatively small that is, d (! o ) d! = d d! = 0 (3.6) 1 Please refer to equation (2.3.36) of the Agrawal [1]. 66 then the GNLSE reduces to the Nonlinear Schr¨ odinger Equation (NLSE): @E(z;t) @z + 1 2 (! o )E(z;t)j 1 X n=1 j n n (! o ) n! @ n @t n E(z;t) = j (! o ) E(z;t) Z 1 0 R()E (z;t)E(z;t)d (3.7) Another simplification, is when the Raman scatteringR(t) is assumed to be of short duration [1]: R(t) = (1f R )(t) +f R h R (t)(t) (3.8) wheref R is the delayed Raman response. The reduced NLSE is: @E(z;t) @z + 1 2 (! o )E(z;t)j 1 X n=1 j n n (! o ) n! @ n E(z;t) @t n =j (! o )kE(z;t)k 2 E(z;t) (3.9) Dispersion terms higher than third order can be dropped to simplify the propagation equation to: @ @z + 1 2 (! o ) + 1 @ @t j 2 @ 2 @t 2 +j 3 @ 3 @t 3 E(z;t) =j (! o )kE(z;t)k 2 E(z;t) (3.10) To simplify the equation, let to be a moving time frame at the group velocity [81]: =t z v g t 1 z (3.11) then the propagation equation (3.10) will be stationary in: 0 B B @ u 1 C C A = 0 B B @ 1 0 1 1 1 C C A 0 B B @ z t 1 C C A (3.12) 67 Then the propagation equation transforms according to 0 B B @ @E @z @E @t 1 C C A = 0 B B @ @u @z @ @z @u @t @ @t 1 C C A 0 B B @ @E @u @E @ 1 C C A = 0 B B @ 1 1 0 1 1 C C A 0 B B @ @E @u @E @ 1 C C A (3.13) substituting into the left hand side, the group velocity term vanishes: @E @z + 2 E + 1 @E @t +j 2 2 @ 2 E @t 2 3 6 @ 3 E @t 3 =j kEk 2 E @E @u 1 @E @ + 2 E + 1 @E @ +j 2 2! @ 2 E @ 2 3 3! @ 3 E @ 3 =j kEk 2 E @E @u + 2 E +j 2 2! @ 2 E @ 2 3 3! @ 3 E @ 3 =j kEk 2 E (3.14) Replacing the propagation distance variableu byz and the moving time frame byt results in the propagation equation used in the waveform study reduces to [1]: @E(z;t) @z + 2 E(z;t) +j 2 2! @ 2 E(z;t) @t 2 3 3! @ 3 E(z;t) @t 3 =j kE(z;t)k 2 E(z;t) (3.15) Equation (3.15) describes the fiber propagation model to be used in this work. The equation de- scribes the fiber effects to include linear of loss and dispersion to a third order and non-linear effects to include the Kerr non-linearity. 3.1 Loss in Single Mode Fiber For the case of loss only, with 6= 0; = 0; 2 = 3 = 0, then the resulting propagation equation reduces to a linear equation with constant coefficients that can be solved in closed form: @E(z;t) @z + 2 E(z;t) = 0 (3.16) 68 and the solution is determined to be: E(z;t) =E (0;t) exp 2 z (3.17) is an exponential decay due to the absorption loss in the fiber. Standard Silica fiber have a nominal loss of 0:2 dB/km around 1550 nm. 3.2 Dispersion in Single Mode Fiber Setting the loss and the nonlinear coefficient to zero, = 0; = 0 with non-zero dispersion, 2 6= 0; 3 6= 0 in equation (3.15) to evaluate the transmission dispersion only effects: @E(z;t) @z +j 2 2! @ 2 E(z;t) @t 2 3 3! @ 3 E(z;t) @t 3 = 0 (3.18) Taking the Fourier transform of the propagation equation results in a spatial differential equation : @ ^ E @z (z;!) +j 2 2! (j!) 2 ^ E(z;!) 3 3! (j!) 3 ^ E(z;!) = 0 @ ^ E @z (z;!)j 2 2! ! 2 +j 3 3! ! 3 ^ E(z;!) = 0 @ ^ E @z (z;!)j 2 2! ! 2 3 3! ! 3 | {z } Frequency Dispersion ^ E(z;!) = 0 (3.19) Which can be solved as a differential equation for the transform of the propagating field, ^ E(z;!) = ^ E(0;!) exp j 2 2! 3 3! ! ! 2 z : (3.20) 69 Dispersion in fiber is the dependence of the group velocity on the wavelength that degrades the propagating field as illustrated in the propagation of the narrow pulse in Figure 3.1. z z Figure 3.1: Pulse propagation through dispersion The group delay 2 is essentially the delay variation: 1 2 d d! 2 2! 3 3! ! ! 2 z = 2 ! 3 2 ! 2 z (3.21) indicating that propagation in fiber result in the higher frequencies lagging the lower frequencies since the square of higher frequencies is higher than the lower frequencies. 3.3 Self Phase Modulation Nonlinearity Considering the non-linear terms in the propagation equation (3.15) repeated herein for convenience, @E(z;t) @z + 2 E(z;t) +j 2 2! @ 2 E(z;t) @t 2 3 3! @ 3 E(z;t) @t 3 =j kE(z;t)k 2 E(z;t) (3.22) Assuming = 0; 2 = 3 = 0; 6= 0, then the propagation equation reduces to: @E(z;t) @z =j kE(z;t)k 2 E(z;t): (3.23) 2 The transformX(f)exp(j2) is the Fourier transform ofx(t) so that the delay is. 70 The solution given in [62] has the form: E(z;t) =E(0;t) exp j kE(z;t)k 2 z ; (3.24) satisfies the propagation equation (3.23) : @E(z;t) @z = @ E(0;t) exp j kE(z;t)k 2 z @z =j kE(z;t)k 2 E(0;t) exp j kE(z;t)k 2 z =j kE(z;t)k 2 E(z;t): (3.25) Assuming the initial distribution is a unit impulse, that isE(0;t) =(t), then the Fourier transform of the solution is ^ E(z;!) = 1 p 2 Z 1 1 E(0;t) exp j kE(z;t)k 2 z exp(j!t)dt = 1 p 2 Z 1 1 (t) exp j kE(z;t)k 2 z exp(j!t)dt = 1 p 2 exp j kE(z; 0)k 2 z (3.26) The phase term when the initial temporal distribution is an impulse function is a function of the propa- gation distance reflecting a change of frequency. More generally, the phase of the nonlinear case only solution given by (3.24) is kEk 2 z which depends on the field power distribution and the propagation distance. The SPM effects arise from the variation of the fiber index of refraction which causes phase shift [35, 48, 99, 100], namely n =n o +n 2 E E SPM =n 2 k 0 L eff E E (3.27) 71 wheren o is the linear refractive index of the material andn 2 is the non-linear refractive index. For a signal propagating a distance ofz, the effective phase change of the signal is The linear phase shift (z;t) = 2n o z (3.28) The nonlinear phase shift (z;t) = 2n 2 E E ^ z (3.29) 3.4 Cross Phase Modulation Nonlinearity Extending the SPM to multi-channel trasnmission, XPM is the phase shift due to intensity variation of adjacent channels [82]. The non-linear phase shift affecting thei-th channel in anN-channel system is: i (z;t) = 2n 2 E i E i + N X j6=i E j E j ! ^ z (3.30) where i (z;t) is the induced cross phase modulation due to adjacent users into thei-th channel at timet and propagation distancez. 3.5 Four Wave Mixing Nonlinearity Fiber non-linearity stems from the third order susceptibility, 3 , effect and in the following we shall present a review of the general case as described in [11] in order to summarize the communications link nonlinear impact on the signal of interest. In summary, the 3 third order effect is given by: 72 o (3) E 3 (t) (3.31) where o is the permittivity of free space and E(t) is the input field. For the case of a single frequency,! 1 = 2f 1 , the FWM results [11] in Third Harmonic Generation (THG) : o (3) E 3 (t) = o (3) Ee j! 1 t +E e j! 1 t 3 = o (3) E 3 e j3! 1 t +E 3 e 3j! 1 t + 3E 2 E e 2j! 1 t e j! 1 t + 3EE 2 e j! 1 t e 2j! 1 t = o (3) E 3 e j3! 1 t +E 3 e 3j! 1 t + 3E 2 E e j! 1 t + 3EE 2 e j! 1 t (3.32) That is the resulting frequencies are at the fundamental and its third harmonic, namely 3f 1 . In the case of two frequencies,f 1 andf 2 with separation of =f 2 f 1 , then similar analysis can be applied by expanding the terms of the third order effects. Adding a corresponding phase ' ij to the analysis yields the total combinations summarized in Table 3.1: Table 3.1: Two Tone FWM intermodulation products f i f j f ij ' ij 1 1 f 11 = 2f 1 f 1 =f 1 ' 11 =' 1 1 2 f 12 = 2f 1 f 2 =f 1 (f 2 f 1 ) =f 1 ' 12 = 2' 1 ' 2 2 1 f 21 = 2f 2 f 1 =f 2 + (f 2 f 1 ) =f 2 + ' 21 = 2' 2 ' 1 2 2 f 22 = 2f 2 f 2 =f 2 ' 22 =' 2 Figure 3.2 shows the third order effect intermodulation product and the fundamental tones. In this case, all the tones are equally spaced by the separation =f 2 f 1 . 73 Frequency f 1 f 2 f 12 f 21 ' 1 ' 2 2' 1 ' 2 2' 2 ' 1 Figure 3.2: Two Tone FWM graphical representation Given three input frequencies representing three center frequencies, f i ;f j ;f k , then a fourth fre- quencyf ijk is generated by the third order (3) nonlinearity in fiber channel [17, 38, 56, 82, 105]. The generated fourth frequency is determined by the relationship [82]: f ijk =f i +f j f k : (3.33) Since any of the three input frequenciesi;j;k can be any of three values 1; 2 or 3 leading to a total of 27 possibilities. The first two indexes are summed, with unchanged resulting frequencyf ijk reducing the 27 possibilities to 3 2 3 = 18. However, since the first two frequencies add and the addition is the commutative, the total number is further divided by two to arrive at 9 possibilities. Including the original three tones, we arrive at a total of 12 tones listed in Table 3.2. Figure 3.3 and Figure 3.4 depicts the third order (3) intermodulation products for the equal and unequal tone spacing respectively [38]. 74 Table 3.2: Three Tone FWM intermodulation products i j k f ijk ' ijk 1 1 2 3 f 123 =f 1 +f 2 f 3 =f 213 ' 123 =' 1 +' 2 ' 3 =' 213 2 2 1 3 3 1 3 2 f 132 =f 1 +f 3 f 2 =f 312 ' 132 =' 1 +' 3 ' 2 =' 312 4 3 1 2 5 2 3 1 f 231 =f 2 +f 3 f 1 =f 321 ' 231 =' 2 +' 3 ' 1 =' 321 6 3 2 1 7 1 1 2 f 112 =f 1 +f 1 f 2 = 2f 1 f 2 ' 112 =' 1 +' 1 ' 2 = 2' 1 ' 2 8 1 1 3 f 113 =f 1 +f 1 f 3 = 2f 1 f 3 ' 113 =' 1 +' 1 ' 3 = 2' 1 ' 3 9 2 2 1 f 221 =f 2 +f 2 f 1 = 2f 2 f 1 ' 221 =' 2 +' 2 ' 1 = 2' 2 ' 1 10 2 2 3 f 223 =f 2 +f 2 f 3 = 2f 2 f 3 ' 223 =' 2 +' 2 ' 3 = 2' 2 ' 3 11 3 3 1 f 331 =f 3 +f 3 f 1 = 2f 3 f 1 ' 331 =' 3 +' 3 ' 1 = 2' 3 ' 1 12 3 3 2 f 332 =f 3 +f 3 f 2 = 2f 3 f 2 ' 332 =' 3 +' 3 ' 2 = 2' 3 ' 2 f f 1 ' 1 f 223 ' 223 f 2 ' 2 f 132 ' 132 f 3 ' 3 f 221 ' 221 f 123 ' 123 f 112 ' 112 f 231 ' 231 f 332 ' 332 f 113 ' 113 f 331 ' 331 Figure 3.3: Three equally spaced tones FWM intermodulation terms f f 1 ' 1 f 2 ' 2 f 3 ' 3 f 123 ' 123 f 132 ' 132 f 231 ' 231 f 112 ' 112 f 113 ' 113 f 221 ' 221 f 223 ' 223 f 331 ' 331 f 332 ' 332 Figure 3.4: Three unequally spaced tones FWM intermodulation terms Figure 3.5 is a general look up tool to determine the location of the third order intermodulation frequencies for any relative spacing. The tones aref 1 ;f 2 andf 3 with the vertical axis representing the offset location of f 2 from f 1 and f 3 . For example if f 2 is in the middle of f 1 and f 3 , then there is a 75 zero offset as illustrated by the dashed horizontal line. The intersection of the horizontal line with the intermodulation lines, represents the location of the specific intermodulation term. As illustrated, for the equally spaced, case, the first intersection isf 113 , followed byf 112 andf 123 and so forth. Also, we note that in this casef 112 andf 123 are equal. The result is coincident with Figure 3.3. Similarly, if the horizontal dashed line is moved to 0:8, then the intersection with the intermodulation lines coincides with Figure 3.4. If the frequency f 2 coincides with f 3 , then the horizontal line will shift to 1 and the intersection with the intermodulation lines will coincide with Figure 3.2 resulting in 4 unique frequencies. Figure 3.5: Graphical Three Tone FWM Representation 76 The general expression forN channels [82, 105] through a third order non-linearity can be derived by choosing the first non-repeating three frequencies to be N (N 1) (N 2). Of these non- repeating three, the first two can be switched without changing the value of the third harmonic since the first two add to reduce the number of outputs by half. Once the first three non-repeating has been selected, the remaining term include when the first two indexes are identical leading toN1(N1) choices. The total number of possible third order termsM is then M = 1 2 (N (N 1) (N 2)) +N (N 1) = 1 2 [(N (N 1) (N 2)) + 2N (N 1)] = 1 2 N (N 1) ((N 2) + 2) = 1 2 N (N 1) (N) = 1 2 N 2 (N 1) (3.34) The first order third order harmonic terms are bounded The highest frequency if f NN1 = f N + (f N f 1 ) while the lowest frequency isf 11N = f 1 (f N 1). The highest and lowest frequencies determine the FWM frequency extent. The FWM intermodulation terms will reside in-band of adjacent channels to degrade a multi-channel system depending on the power of each channel. 77 Chapter 4 Digital Processing for Coherent Communications Chapter 4 presents the digital signal processing used to mitigate the fiber channel effects on the wave- form. The processing is based on the Nonlinear Schr¨ odinger Equation (NLSE) to include the simulation approach using classical algorithms based on the Split Step Fourier Method (SSFM). Since the closed form answers to nonlinear propagation are difficult to obtain, the algorithm parameters were analyzed by simulation to use the most accurate scheme to reduce numerical errors in waveform assessment. As will be discussed, forward propagation determines the degraded receiver waveform while the backward propagation corrects for the degradation and determines an estimate of the initial transmitted waveform In general, a coherent optical link modulates the phase and amplitude of the transmitted laser beam though optical fiber or free space. The recent developments of signal processing technology and al- gorithms provided the capability to extend the trasnmission distance and improve the bit error perfor- mance BER of optical communications linksby compensating for the linear and nonlinear effects. In particular, coherent optical communications utilized the complex signal processing to enhance its ad- vantages over Intensity Modulation Direct Detection (IMDD) even though the later detection sensitivity is independent of the carrier phase or SOP [49]. Signal processing algorithms play a significant role to mitigate the waveform and fiber degradation identified in Table 1.3 and chapter 3. In addition, signal processing is a key element in performing the 78 receiver phase recovery and tracking, equalize the received signal for imbalances and ISI, decouple and orthogonalize the polarization, perform the MIMO processing, and reduce transmission degradation through the fiber channel. This study evaluates the signal processing algorithms pertinent to the factors that affect the waveform shaping and mitigation and to that effect only these signal processing elements will be addressed: • Digital to Analog Converter Loading: Will be addressed in the waveform shaping section. • MZM Linearization: Predistortion algorithm to extend the linear dynamic range of the modula- tion as discussed in section 4.1. • Dispersion Compensation: Signal processing algorithms to minimize the dispersion effects at the receiver in order to evaluate the link residual dispersion. • Forward Propagation: Implemented a fiber propagation model to propagate the transmission through the fiber. • Back Propagation: Implemented the back propagation algorithm in order to mitigate the propa- gation effects through the fiber. Other elements of the receiver signal processing will not be implemented are: • Carrier Tracking: Will not be implemented since the transmitter laser is a narrow linewidth stable laser model. • Symbol Timing Recovery: Will not be implemented since reference ADC sampling clock model is a stable with low phase noise. One approach to mitigate dispersion and fiber nonlinear effects utilizes Digital Back Propagation (DBP) where the received signal is back propagated in software to estimate the original transmission 79 [24,44,52,54,54,72,83,96]. With the advent of high speed signal processing technology[27], coherent detection and signal processing provide the means to extend the reach of an optical link by compen- sating for the optical channel to include transmission fiber, electrical front end and optical modulator non-linear effects. 4.1 MZM Modulator Linearization The optical MZM modulator is a nonlinear device that imparts the informatio bits onto the optical carrier. The MZM nonlinearity compresses the higher order QAM constellation formats and reduces the spectral efficiency due to the spectral growth. Existing methods of compensation include compensation at the transmitter by predistorting [73] the waveform to restore the constellation at the receiver. Other techniques include receiver post-compensation [68]. Essiambre [27] utilized digital predistortion to mitigate fiber nonlinearities and showed that the digital predistortion is significantly more susceptible to fiber nonlinearity. In this study, the predistortion is restrcited to the MZM nonlinearity to assess the compensation effectiveness of the waveform. The modulation transfer function of the MZM is summarized in appendix D equation (D.8) with a bias ofV =2: E o = cos V (u +b) E i (4.1) = cos V u V 2 E i (4.2) = sin V u E i : (4.3) 80 The linear element of the MZM transfer function can be extracted from the first order term of the Taylor 1 expansion around the shifted bias point of 0, that is: sin V u sin (0) + V cos (0)u = V u (4.4) Pre-distorting the MZM is reduced to determining the pre-distortion mappingg(u) so that applying the signalg(u) results in a net linear transfer function through the MZM, i.e.: sin V g(u) = V u (4.5) whereu is the applied voltage. Solving equation 4.5 for the pre-distortion function,g(u): g(u) = V sin 1 V u ; wherejuj V 0:38V : (4.6) The pre-distortion mapping through the MZM modulator is a linear map shown below: E o = sin V g(u) E i = sin V V sin 1 V u = V u E i (4.7) Figure 4.1 depicts the digital pre-distortion for the MZM nonlinear transfer function with a output signalE o . The input signal ranges fromV =2 toV =2 to be scaled and pre-distorted so that the output is a linear mapping as verified by equation (4.7). Figure 4.2a depicts the uncompensated MZM field response following the sinusoidal response given in equation (4.3). The pre-distortion maps the input signal so the output of the MZM is linear. In this 1 Expandingf(u) is a Taylor series around0 yields: f(u) =f(0)+f 0 (0)u+ 1 2! f 00 (0)u 2 + f(0)+f 0 (0)u 81 u V sin 1 () V V sin () E o Modulator Pre-Distortion Mach-Zehnder Modulator Figure 4.1: Digital Pre-distortion of the Mach-Zehnder Amplitude Modulator. particular pre-distortion realization, the slope or the gain of the compensated linear curve is equal to the slope or gain of the MZM at the bias point. As such, the linear response attains maximum amplitude at the reduced input voltage ofV = given in equation (4.6). (a) high gain (b) low gain Figure 4.2: High and low gain MZM predistortion Alternatively, if the gain or the slope of the compensated response is reduced so that the maximum response is attained at the maximum input value, then the input dynamic range is unchanged. Setting the slope of the resulting linear response to 2=V , the pre-distortion condition becomes: sin V g(u) = 2 V u (4.8) 82 The low gain pre-distortion mapping is determined by inverting equation (4.8): g(u) = V sin 1 2 V u ; wherejuj V 2 0:5V : (4.9) Figure 4.2b depicts the uncompensated and the compensated MZM field response described in equation (4.9). The low gain supports a wider range than the higher gain given in equation (4.6). 4.2 Dispersion Compensation LetE(z;t) be the envelope of an optical pulse propagating [1, 79] in a single mode fiber as a function of time t and position z along the fiber as described earlier in chapter 3. The homogeneous NLSE propagation equation is given by: 2 E(z;t) + @E(z;t) @z + 1 @E(z;t) @t +j 2 2 @ 2 E(z;t) @t 2 3 6 @ 3 E(z;t) @t 3 = 0 (4.10) where the propagation parameters and equation has been described earlier in chapter 3. To assess the dispersion compensation algorithm on the proposed waveform, we summarize the dispersion com- pensation algorithm and present the dispersion compensation results to be combined in the end to end simulation. 83 Taking the Fourier transform of the propagation equation with respect to the moving time frame transforms the equation into a differential equation into an algebraic equation [1]: 0 = @ ^ E @z (z;!) + 2 ^ E(z;!) +j 2 2! (j!) 2 ^ E(z;!) 3 3! (j!) 3 ^ E(z;!) = @ ^ E @z (z;!) + 2 j 2 2! ! 2 +j 3 3! ! 3 ^ E(z;!) = @ ^ E @z (z;!) + 2 ^ E(z;!) | {z } Attenuation j 2 2! ! 2 3 3! ! 3 ^ E(z;!) | {z } Frequency Dispersion (4.11) The field at a distancez = L can then be determined by solving the equation (4.11) as first order differential equation inz with constant coefficients under an assumption that!; 2 ; 3 ; do not depend on propagation distancez. Assuming an initial field ^ E(0;!), then the field atz =L is: ^ E(L;!) = ^ E(0;!) exp 2 L exp j 2 2! 3 3! ! ! 2 L (4.12) ^ E(0;!)g(;L)H(!; 2 ; 3 ;L); (4.13) where the attenuation and dispersion transfer functions,g andH are defined respectively: g(;L) = exp 2 L (4.14) H(!; 2 ; 3 ;L) = exp j 2 2! 3 3! ! ! 2 L : (4.15) The initial distribution can be determined by amplifying the attenuation and inverting the dispersion transfer functions, that is: ^ E(0;!) =g()H (!; 2 ; 3 ) ^ E(L;!): (4.16) 84 where the attenuation and dispersion compensation transfer functions are: g(;L) = exp 2 L (4.17) H (!; 2 ; 3 ;L) = exp j 2 2! 3 3! ! ! 2 L : (4.18) The transmission compensation consists of the gain scalingg() and the frequency dispersion com- pensationH(!; 2 ; 3 ) depending on the propagation distanceL. Normally, the gain term is lumped into the overall system gain and can be scaled at the receiver. Figure 4.3a illustrates the impact of the dispersion through fiber on the RCW waveform and the corresponding dispersion compensated waveform in Figure 4.3b. (a) RCW without dispersion compensation (b) RCW with dispersion compensation Figure 4.3: Raised Cosine dispersion and compensation illustration. Ip and Kahn [44] proposed a dispersion and nonlinear compensation algorithm utilizing backprop- agation summarized in sections 4.3 and 4.3.5. 85 (a) WRCW dispersion uncompensated (b) WRCW disperion compensated Figure 4.4: Weighted Raised Cosine dispersion and compensation illustration. 86 4.3 Forward and Backward Propagation Algorithms Solving the propagation equation analytically is difficult in general and one resort to a numerical tech- niques. The methodology for the propagation is also applicable to the undoing the deterministic effects of the fiber propagation due to dispersion and SPM. The propagation equation [1] stated in (3.15) and repeated below for continuity [18, 43, 59, 60, 62, 80, 86, 109, 127, 128]: @E(z;t) @z + 2 E(z;t) +j 2 2! @ 2 E(z;t) @t 2 3 3! @ 3 E(z;t) @t 3 =j kE(z;t)k 2 E(z;t) (4.19) Rearranging equation (4.19) and defining a linear and non-linear differential operators, the prop- agation equation can be rewritten as the sum of a linear and non-linear operatorsL(E) andN (E)E respectively: @E(z;t) @z = 2 E(z;t)j 2 2! @ 2 E(z;t) @t 2 + 3 3! @ 3 E(z;t) @t 3 +j kE(z;t)k 2 E(z;t) L(E) +N (E) (4.20) where L(E) 2 E(z;t)j 2 2! @ 2 E(z;t) @t 2 + 3 3! @ 3 E(z;t) @t 3 (4.21) N (E)j kE(z;t)k 2 E(z;t) (4.22) A common method to solving the propagation equation determines the field in fiber at a positionz and timet is to utilize the Split Step Fourier Method (SSFM) by propagating the field one small distance at a time. The size of the distance step is a function of the fiber parameters, signal power and the desired accuracy. In this study we shall assume that the step size has been determined and will not 87 optimize its selection. Assuming thekE(z;t)k 2 is slowly varying over a the step , the propagation equation (4.20) can be expanded out in a Taylor series around the pointE(z;t): E (z + ;t) =E(z;t) + 1! @E(z;t) @z + 2 2! @ 2 E(z;t) @z 2 + = 1 X k=0 k k! @ k E(z;t) @z k (4.23) The terms of the series can be stated in terms of the propagation equation (4.20): @E(z;t) @z = (L +N )E(z;t) (4.24) which is the original equation. Expanding the second order term: @ 2 E(z;t) @z 2 = @ @z @E(z;t) @z = @ @z [(L +N )E(z;t)] = (L +N ) @ @z E(z;t) = (L +N ) (L +N )E(z;t) = (LL +LN +NL +NN )E(z;t) (4.25) sinceL is independent ofz andN (E) has slow variation over the step . The linear and non-linear operator productsLN andNL do not commute and are applied in order asserting that dispersion and 88 non-linearity are coupled while propagating in fiber. To simplify the expression for the development define fL +Ng (L +N ) (4.26) fL +Ng 2 =fL +NgfL +Ng (LL +LN +NL +NN ) (4.27) (4.28) to express the Taylor expansion as: @E(z;t) @z =fL +NgE(z;t) (4.29) @ 2 E(z;t) @z 2 =fL +Ng 2 E(z;t) (4.30) Similarly, expanding the third order term of the Taylor series, @ 3 E(z;t) @z 3 = @ @z @ 2 E(z;t) @z 2 = @ @z fL +Ng 2 E(z;t) =fL +Ng 2 @E(z;t) @z =fL +Ng 2 fL +NgE(z;t) =fL +Ng 3 E(z;t) (4.31) Using the same approach, thek-th partial derivative of the Taylor expansion equation (4.23) is @ k E(z;t) @z k =fL +Ng k E(z;t) (4.32) 89 which can be used to write the solution at positionz + to be E (z + ;t) =E(z;t) + 1! fL +NgE(z;t) + + k k! fL +Ng k E(z;t) + (4.33) = 1 X k=0 k k! fL +Ng k E(z;t) (4.34) = exp ( (L +N ))E (z;t) (4.35) that agrees with the extact solution stated in Agrawal [1]. Please note thatfg maintains the order of the fiber dispersion and fiber non-linear operators as defined earlier in equation (4.26). While the Taylor expansion constructed the solution atz+, it may not be suitable to numerical evaluation and in particular to digital signal processing due to its slow convergence as each term is an order of magnitude smaller than the previous term and the dominant term is a first order term. The exact solution given by equation (4.35) is an operator expression consisting of the sum of the linear and the non-linear operators. Using Zassenhaus Formula [15, 57, 110] to expand the exact solution in equation (4.35): exp ( (L +N )) = exp (L) exp (N ) exp 2 C 2 exp 3 C 3 exp ( n C n ) (4.36) whereC 2 , andC 3 are defined by the operator commutator: C 2 = 1 2 [L;N ] 1 2 (LNNL) (4.37) C 3 = 1 3 [N; [L;N ]] + 1 6 [L; [L;N ]] (4.38) 90 and any terms higher thanC 3 can be computed recursively [15]. Ignoring terms on the order of 2 and higher, the resulting operator expansion reduces to the product of the linear and the non-linear operator terms in agreement with Agrawal [1]: exp ( (L +N )) exp (L) exp (N ) (4.39) The SSFM approach to solving the propagation equation (4.19) proceeds by solving the equation with the linear and the non-linear segments separately and then combining the results. Setting the linear segmentL(E) = 0, and denoting the non-linear segment solution by E n , the non-linear differential equation associated with the non-linear operator is: @E(z;t) @z =N (E) =j kE(z;t)k 2 E(z;t) (4.40) Assuming that the variation in power, or change tokE(z;t)k 2 , is relatively small over thez incre- ment , the equation can be solved as a first order equation: E n (z + ;t) =E(z;t) exp j kE(z;t)k 2 (4.41) Setting the non-linear operatorN (E) = 0, the differential equation reduces to: @E n (z;t) @z =L(E n ) = 2 E n (z;t)j 2 2! @ 2 E n (z;t) @t 2 + 3 3! @ 3 E n (z;t) @t 3 (4.42) 91 4.3.1 NonLinear Linear Propagation Taking the Fourier transform of equation (4.42) results in a first order differential equation and trans- forms the temporal component into an algebraic equation. First, the Fourier transformation of the operator is expanded: F @E n (z;t) @z = Z 1 1 @E n (z;t) @z e j!t dt = Z 1 1 L (E n (z;t)) e j!t dt @ ^ E n (z;!) @z = 2 ^ E n (z;!)j 2 2! (j!) 2 ^ E n (z;!) + 3 3! (j!) 3 ^ E n (z;!) = 2 ^ E n (z;!) +j 2 2! ! 2 ^ E n (z;!)j 3 3! ! 3 ^ E n (z;!) = 2 ^ E n (z;!) + j 2 2! ! 2 j 3 3! ! 3 ^ E n (z;!) = 2 +jD (!; 2 ; 3 ) ^ E n (z;!) (4.43) where is the attenuation,D is the dispersion as a function of !; 2 and 3 and ^ E n (z;!) is the Fourier transform along the temporal dimension of the field E n (z;t). Solving the differential equa- tion (4.43) over a single step yields the Fourier transform response atz + : ^ E n (z + ; !) = ^ E n (z; !) exp 2 +jD (!; 2 ; 3 ) (4.44) 92 Taking the inverse transform to arrive atE n (z + ;t) and substituting the non-linear solution: E n (z + ; t) =F 1 n ^ E n (z + ; !) o =F 1 n ^ E n (z; !) exp 2 +jD (!; 2 ; 3 ) o =F 1 n exp 2 +jD (!; 2 ; 3 ) F E(z;t) exp j kE(z;t)k 2 o (4.45) While the SSFM outlined in equation (4.45) assumes to solve the non-linear component first and then the linear or dispersion component consistent with the traditional solution recommendations [1] as the second order and higher terms capturing the commutative effects were dropped. A numerical simulation was conducted to evaluate the numerical precision of the effects of interchanging the order of non-linearity and dispersion. The results will be shown section 4.3.5. j kk 2 expfg Ffg F 1 fg 2 +jD (!; 2 ; 3 ) E(z + ;t) E(z;t) Figure 4.5: Nonlinear Linear and dispersion SSFM over a spatial interval 93 4.3.2 Linear Nonlinear Propagation Ffg F 1 fg () exp j kk 2 2 +jD (!; 2 ; 3 ) E(z + ;t) E(z;t) Figure 4.6: Linear Nonlinear distributed dispersion SSFM spatial interval 4.3.3 Linear Nonlinear Linear Propagation To attain greater accuracy, a common approach is to apply propagated through half the dispersion, followed by the non-linearity and then apply the remaining dispersion half [1]. Mathematically, the propagation equation can be stated in equation (4.46) with solution depicted in Figure 4.7: @E(z;t) @z = 1 2 L + N + 1 2 L E(z;t) (4.46) 94 Ffg F 1 fg () exp j kk 2 Ffg F 1 fg 2 +jD (!; 2 ; 3 ) 2 +jD (!; 2 ; 3 ) 2 2 E(z + ;t) E(z;t) Figure 4.7: Linear Nonlinear Linear distributed dispersion SSFM over a spatial interval 4.3.4 Nonlinear Linear Nonlinear Propagation An alternative suggested by Mollenauer [62] is to apply half the non-linearity followed by dispersion and then the remaining half of the non-linearity. Mathematically, the equation is @E(z;t) @z = 1 2 N + L + 1 2 N E(z;t) (4.47) with the corresponding propagation block diagram depicted in Figure 4.8. Half of the non-linearity is first applied to propagate the phase followed by the Fourier transform to propagate the field under the linear dispersion and followed by transforming to the time domain by taking the inverse Fourier transform. The time domain field propagates under the ramining half of the non-linearity to derive the propagated field by an increment of for all the time. 95 () exp 2 j kk 2 Ffg F 1 fg () exp 2 j kk 2 2 +jD (!; 2 ; 3 ) E(z + ;t) E(z;t) Figure 4.8: Nonlinear Linear Nonlinear distributed nonlinearity SSFM a spatial interval 4.3.5 Propagation Models Numerical Comparison The various propagation methodologies presented in section 4.3 may be used to propagate the field in fiber depending on the operating constraints at hand. The LN and NL have the same computational complexity less than the LNL and NLN at a cost of accuracy. To evaluate and compare the numerical precision of the propagation algorithms, a pulse is propagated forward and backward a distance to be compared with the original waveform. The difference is the measure of the numerical precision of the propagation algorithm. The initial pulse [62] at launch point withz = 0 is a Gaussian pulse of the formE(z = 0;t) is E(0;t) = 8 > > > < > > > : exp (t 2 =2) fort2 [T;T ] 0 elsewhere (4.48) 96 where T is a sufficiently large time duration that the field amplitude is well below the precision of the simulation processor. Denote the returned pulse to the launch point by ~ E(0;t), then the error between the transmitted pulse and the returned pulse is the integrated error over the entire distance: = Z T T E(0;t) ~ E(0;t) 2 dt (4.49) Table 4.1: Summary of Numerical Propagation Algorithmic Errors Disp Disp Comp NL NL Comp Algorithm 2, Error Error dB Group I 0 0 0 0 LN +2.45e-11 -106.12 0 0 0 0 NL +2.45e-11 -106.12 0 0 0 0 LNL +5.30e-11 -102.76 0 0 0 0 NLN +2.45e-11 -106.12 Group II 1 0 0 0 LN +5.26e+03 +37.21 1 0 0 0 NL +5.26e+03 +37.21 1 0 0 0 LNL +5.57e+03 +37.46 1 0 0 0 NLN +5.26e+03 +37.21 Group III 1 1 0 0 LN +4.56e-11 -103.41 1 1 0 0 NL +4.56e-11 -103.41 1 1 0 0 LNL +8.80e-11 -100.56 1 1 0 0 NLN +4.56e-11 -103.41 Group IV 1 1 1 0 LN +7.36e+03 +38.67 1 1 1 0 NL +7.37e+03 +38.67 1 1 1 0 LNL +7.35e+03 +38.66 1 1 1 0 NLN +7.36e+03 +38.67 Group V 1 1 1 1 LN +8.97e-09 -80.47 1 1 1 1 NL +4.41e-08 -73.56 1 1 1 1 LNL +4.57e-08 -73.40 1 1 1 1 NLN +2.45e-08 -76.11 Table 4.1 summarizes the cases investigated to evaluate the numerical precision of the propagation algorithms divided into five groups, I, II,III, IV and V . Group I launches the pulse forward and backward to the initial location with error on the order of 10 11 . Group I denotes the numerical errors in the system since there were no dispersion or non-linear effects. 97 Group II denotes the transmission error due to dispersion without compensation. The no dispersion compensation error is on the order of 5 10 +03 indicating that no bits could be recovered. Group III denotes a transmission with dispersion followed by dispersion compensation with a net error on the order of 4 10 11 which is comparable to Group I identifying that the dispersion compen- sation recovered the signal as expected. Group IV introduces dispersion and fiber nonlinearity with only dispersion compensation. Since there were no nonlinear compensation, the error was on the order of 7 10 +03 rendering the signal unrecoverable. Furthermore, the error in Group IV is higher than Group II identifying the nonlinear error plus a residual dispersion error. Group V introduces dispersion and its compensation along with nonlinearity and its compensation by DBP using the linear and nonlinear propagation algorithms presented in sections 4.3.1, 4.3.2, 4.3.3 and 4.3.4, The error in the nonlinear followed by the linear and followed by the nonlinear yielded the most accurate result on the order of 2 10 08 when dispersion and nonlinearity were introduced and back propagated. The waveform assessment will be based on the most accurate forward and backward propagation. 98 Chapter 5 Weighted Raised Cosine Transmission Performance Chapter 5 presents compares the proposed waveform Weighted Raised Cosine Waveform (WRCW) to the reference Raised Cosine Waveform (RCW) waveform and assesses the link performance through fiber to include the dispersion compensation, nonlinear compensation and modulator pre-distortion. Firstly, the error metrics to assess the waveform are introduced followed by the assessment of the waveform. 5.1 Error Vector Magnitude Metric (EVM) The Error Vector Magnitude (EVM) is a common figure of merit used to evaluate the quality of a com- munications link [32,36] and to estimate the BER [87] or SNR [58] of the underlying communications system. The two dimensional constellationC(p;q) is: C(p;q)C I (p) +jC Q (q) = (2p 1 p M) +j (2q 1 p M); p;q = 1; 2;; p M (5.1) The received signal requires scaling so it can be compared to the reference constellation. A received voltage ofN symbols is 99 v(k) =v I (k) +jv Q (k); k = 1; 2;;N (5.2) The average power of the ideal constellation and the received constellation denoted byP c andP v are defined to be: P c = X p X q C 2 I (p) +C 2 Q (q) P v = N X n=1 v 2 I (n) +v 2 Q (n) (5.3) Both the constellation and the received signal are normalized to a common reference and then are compared. The normalization factor is determined so that mean power is unity [61]. Dividing the constellation,C pq , by the constellation powerP c , to arrive at normalized constellation symbol: ~ C(p;q) = C(p;q) p P c A c C(p;q) ~ v(k) = v(k) p P v A v v(k) (5.4) The normalized power of the constellation is P ~ C = X p X q ~ C 2 (p;q) =A 2 c X p X q C(p;q) = 1 P c P c = 1 P ~ v =A 2 v P v =A 2 v N X n=1 v 2 I (n) +v 2 Q (n) = 1 P v P v = 1 (5.5) The normalized signal and constellation can be compared to compute the EVM as follows: EVM = 1 P c 1 N N X k=1 j~ v(k) ~ c(k)j 2 (5.6) 100 5.2 Peak to Average Power Ratio Metric Palicot [70] generalized the PAPR to the Power Ratio (PR) to relate the PAPR versions existing in the literature by introducing the (i) infinite continuous PR, (ii) the finite continuous PR, (iii) the finite sampled PR and (iv) the infinite sampled PR. The definition of interest to the discussion in this paper is the finite sampled definition of the power ratio. The finite sampled power ratio denoted by the PAPR N defined by: PAPR N = Peak Power of N samples Average Power of N samples = Max k2[0;N1] s (k)s(k) 1 N P N1 0 s (k)s(k) (5.7) where we require that the number of samples,N, be sufficiently large so that the finite approxima- tion is close to the limiting infinite case so that: PAPR PAPR N PAPR 1 = lim N!1 PAPR N : (5.8) The pulse shaping response directly affects the PAPR and is normally difficult to obtain in closed form for arbitrary pulse shapes. In this paper, we present the PAPR for the rectangular pulse shap- ing [120] to serve as a baseline and provide a simulation based comparative measure to demonstrate the effectiveness of the proposed weighted waveform. The peak power of the square constellation occurs at the corners where the inphase and quadrature indexp;q are equal to p M: P peak =I 2 p M +Q 2 p M = 2 2 p M 2 1 ! 2 = 2 p M 1 2 : (5.9) 101 The average power can be obtained by averaging over the constellation points over theM symbols: P avg = 1 M X p X q (I(p) +jQ(q)) (I(p) +jQ(q)) (5.10) = 1 M X p X q (I 2 (p) +Q 2 (q)) (5.11) = 4 p M 0 @ 1 2 + 3 2 + 5 2 + + 2 p M 2 ! 1 ! 2 1 A (5.12) = 2 (M 1)=3; (5.13) where we used symmetry to reduce the sum of elements over the positive indexes and summa- tion over the square of odd integers [12] resulting in the PAPR [120] shown in equation (5.14) and summarized in Table 5.1 for some modulation orders. PAPR = 3 p M 1 p M + 1 (5.14) Table 5.1: Square constellation PAPR for various orders with rectangular shaping M 4 16 64 256 1024 1 PAPR 1.0 1.80 1.33 2.65 2.82 3.00 PAPR (dB) 0.0 2.55 3.68 4.23 4.50 4.77 Without any timing error, the RCW has zero ISI with higher PAPR at the lower roll-off factors corresponding to better bandwidth efficiency. It is our goal to compare the PAPR and the bandwidth efficiency of weighted raised cosine waveform to the raised cosine. First we present the construction of the weighted raised cosine followed by the comparison with the raised cosine and the rectangular pulse shaping. 102 5.3 Bandwidth Occupancy Percentile Metric The Bandwidth Occupancy metric of a system, is the bandwidth B of the system’s response, H(f), that contains the specified percentile of the total power. Ifp denotes the percentile power within the bandwidthB, , then the Bandwidth Occupancy percentile,p, is given by: p = 100 R B=2 B=2 H (f)H(f)df R 1 1 H (f)H(f)df (5.15) The occupancy metric is a figure of merit used to define the effective bandwidth of a system typically ranging from 90% to 98% depending on the specific link. 5.4 Weighted Raised Cosine Waveform Performance The PAPR for the weighted raised cosine given in equation (2.7) is the sum of delayed shaping pulses with peak values at a normalized time where derivative of s() vanishes and the corresponding maximum value is s( ). The derivative of the shaped waveform s() is non-linear and difficult to solve analytically since it is an infinite sum of weighted raised cosines. Alternatively, an upper bound can be determined by bounding the sum at as follows: s() = 1 X k=1 a k w(k)h (k) 1 X k=1 p M 1 sign [w( k)h ( k)]w( k)h ( k) p M 1 1 X k=1 kw( k)kkh ( k)k p M 1 1 X k=1 kh ( k)k; (5.16) 103 where we utilized that the absolute value of real number is given by the sign of the number multi- plied by the number, ie,jxj= sign(x)x. Equation (5.16) forms an upper bound for the weighted raised cosine. Figure 5.1b illustrates the construction of the peak value of the weighted raised cosine and the raised cosine to illustrate the smaller peak of the weighted raised cosine. The maximum can be shown with an input data sequence matching the sign of the raised cosine as in equation (5.16). In particular, symbol at = 0 is positive with positive contribution over (1; 0) so the adjacent symbol at = 1 is negative with positive value over (1; 0) and thus constructively add with the symbol at 0. Similarly, the symbol at =1 is positive so that contribution also adds constructively and continuing on in this manner justifies the maximum peak construction as depicted in figure 5.1a. Numerically computing the upper bound for all excess bandwidth over its range of (0; 1), we il- lustrate that the weighted raised cosine peak power value is less than the raised cosine for all excess bandwidth as shown in figure 5.2. For the lower excess bandwidths the decrease in peak power is exceeds 2:2 dB. (a) The input data sequence matching the sign of the impulse response to maximize the waveform within (1;0), (b) At0:5, the weighted raised cosine peak is less than the raised cosine Figure 5.1: Maximum response of RCW and WRCW 104 Figure 5.2: RCW and WRCW peak power comparison The effects on the PAPR can be observed by examining the modulated waveform eye diagram shown in figure 5.3. For excess bandwidth of 0:2, figure 5.3a depicts the raised cosine with excess bandwidth of = 0:2 while figure 5.3c depicts the raised cosine for = 0:6. As stated earlier, the smaller the excess bandwidth, the better the spectral utilization at a cost of higher PAPR. The higher PAPR leads to spectral regrowth [85] and thus it is desired to reduce the PAPR while maintaining the spectral efficiency. figure 5.3b and figure 5.3d depict the weighted raised cosine for excess bandwidth = 0:2 and = 0:6 respectively. Both of these cases illustrate that the weighted raised cosine peaks are reduced with reduced PAPR. The weighted raised cosine exhibit a non-uniform PAPR reduction over the excess bandwidth range where the larger excess bandwidths have a lower peak and a smaller PAPR. Figure 5.4 shows the PAPR of the raised cosine as well as the weighted raised cosine for all excess bandwidths for 64-QAM modulation. The largest peak of the raised cosine is around 8:5 dB compared to 7 dB for the weighted 105 (a) RCW time domain response with amplitudes at lower excess bandwidths with higher PAPR and ISI. (b) WRCW time domain response with amplitudes at lower excess bandwidths (c) RCW time domain response with amplitudes at higher excess bandwidths with lower PAPR and ISI. (d) WRCW time domain response with amplitudes at higher excess bandwidths Figure 5.3: RCW and WRCW waveform responses for excess bandwidth parameters 0:2; 0:6. raised cosine around excess bandwidth of 0:1. Furthermore, the figure illustrates that both the raised cosine and the weighted raised cosine have comparable PAPR for excess bandwidth near 1. The figure 106 also identifies the PAPR of the or rectangular shaped waveform is better than either shaped raised cosine. Figure 5.4: PAPR of RCW and WRCW comparison. Figure 5.5 depicts the 99% occupied power normalized bandwidth for the raised cosine, the weighted raised cosine and the rectangular shaped waveforms. As expected the raised cosine has the best spectral utilization compared to the weighted raised cosine and the rectangular shaped. The rectangular shaping normalized bandwidth is 20 and is independent of the excess bandwidth. For small excess bandwidth the weighted raised cosine normalized bandwidth is about 1:7 compared to 1 for the raised cosine. Figure 5.6 illustrates the block diagram of the back to back communications link model used to assess the EVM of the weighted raised cosine and raised cosine waveform. The model consists of a bit source followed by a symbol mapper driving a pulse shaping filter with two control parameter: the excess bandwidth and the power function defined in equation (2.76). Once the excess bandwidth is specified to the shaping filter, the weighted raised cosine shaping pulse determined by equation (2.75) 107 Figure 5.5: 99% Occupied bandwidth RCW and WRCW comparison. drives the Digital to Analog Converter (DAC) and its amplifier which modulates an optical carrier with a MZM in the linear range biased at quadrature. The back to back simulated transmission depicted in figure 5.6 compared the EVM of the weighted raised cosine waveform against both the rectangular and the raised cosine waveforms. EVM is a com- mon figure of merit that can assess the quality of the modulated waveform [61] and is used to predict the bit error rates and its relationship to the BER and SNR [30,32,87,88]. EVM is the average power of the error between the received symbols and the ideal constellation grid or reference grid after both the received and the ideal grid are normalized to a common scale [61]. Denoting the receivedN symbols byR(k) fork = 1; 2;;N, then the normalized received symbols ~ R(k) are scaled according to: ~ R(k) =AR(k) (5.17) 108 whereA is the received symbols scale factor andA 2 is the inverse average power per received symbol: A 2 = 1 1 N P N k=1 R (k)R(k) = N P N k=1 R (k)R(k) : (5.18) Bit Source Tx Laser MZM Hybrid Symbol Map WRC DAC Symbol Map WRC DAC LPF LPF Receiver EVM Rx Laser p p Figure 5.6: Back to back communications assessment link model. The average power of the normalized received symbol power is unity since 1 N N X k=1 ~ R (k) ~ R(k) =A 2 1 N N X k=1 ~ R (k) ~ R(k) = 1 (5.19) Finally, figure 5.7 depicts the EVM for the weighted raised cosine, raised cosine and the rectangular shaping where all the waveforms show the same EVM of about 0:015% implying that the waveform shaping is essentially identical for all shaping waveforms and the proposed methodology does not impact the EVM. A new weighted raised cosine pulse shaping that maintains the same EVM at a reduced PAPR with slightly increased bandwidth penalty compared to the raised cosine but much less than the rectangular pulse shaping. The PAPR improvement is on the order of 1:5 dB for the lower excess bandwidth. Due 109 Figure 5.7: RCW, WRCW and rectangular waveforms EVM. to the reduced PAPR, the RF drive to the optical modulators can be increased to improve the signal to noise ratio of the transmitted waveform. 5.5 Waveform Transmission Performance The WRCW has lowered the PAPR as designed and shown in Figure 5.4 relative to the RCW over all the excess bandwidth ranging from 0 to 1. Compared to the RCW, the increased bandwidth ranged from 1:7 at very low excess bandwidth to about 1:08 for excess bandwidth near unity as shown in Figure 5.5. For completeness, the unshaped waveform had the lowest PAPR with a relatively 20 times increased bandwidth. To evaluate the waveform in a transmission channel, an 80 km fiber link was simulated and com- pared against the common RCW with the same excess factor. The constructed model included: • Waveform shaping 110 • A Nonlinear Linear Nonlinear fiber channel propagation model based on the forward propagation model discussed in section 4.3 • A coherent receiver with signal processing capability to include dispersion compensation, non- linear compensation utilizing the Nonlinear Linear Nonlinear back propagation model • Evaluation metrics to include BER and EVM • Phase recover and tracking The first simulation case models a 64-QAM with excess bandwidth of 0:3 with raised cosine and weighted raised cosine to include the data converters, dispersion and dispersion compensation over a fiber span of 80 km. The weighted raised cosine has a gain of 1:5 dB over the raised cosine due to lower PAPR. An error floor resulting from the residual dispersion compensation is at 2 10 5 , well below the FEC threshold as depicted in Figure 5.8. Figure 5.8: RCW and WRCW BER with data converters and dispersion compensation The second simulation case included the data converters at a loading factor of 25% and the fiber nonlinearity with waveform shaping of the root raised cosine and weighted root raised cosine. In this 111 model, noiseless detector applied in order to assess the nonlinearity and data converters on the wave- form. The results depicted in Figure 5.9a show that the weighted raised cosine has a wider nonlinearity tolerance than the raised cosine. It is important to point out that expected performance is will be limited by the noise and the gain will not exceed the 1:5 dB. The third simulation case is an EVM simulation to include data converters, nonlinearity and non- linearity compensation with noiseless detectors. The EVM of the weighted raised cosine outperforms the raised cosine at the nonlinearity threshold and below. Both waveforms performs similarly at higher launch power. (a) RCW and WRCW BER metric (b) RCW and WRCW EVM metric Figure 5.9: 80 km, 64-QAM transmission with linear and nonlinear compensation. In conclusion the weighted raised cosine outperforms the raised cosine in the presence of nonlin- earities due to fiber, data converters and dispersion below the nonlinear launch power threshold. As the launch power increases, both waveforms behave similarly. The expected gain is at most 1:5 dB limited by the noise in the system. 112 References [1] Govind Agrawal. Front matter. In Nonlinear Fiber Optics (Fifth Edition), Optics and Photonics. Academic Press, Boston, fifth edition edition, 2013. [2] Erik Agrell and Magnus Karlsson. Power-efficient modulation formats in coherent transmission systems. J. Lightwave Technol., 27(22):5115–5126, Nov 2009. [3] Erik Agrell, Magnus Karlsson, A R Chraplyvy, David J Richardson, Peter M Krummrich, Pe- ter Winzer, Kim Roberts, Johannes Karl Fischer, Seb J Savory, Benjamin J Eggleton, Marco Secondini, Frank R Kschischang, Andrew Lord, Josep Prat, Ioannis Tomkos, John E Bowers, Sudha Srinivasan, Ma¨ ıt´ e Brandt-Pearce, and Nicolas Gisin. Roadmap of optical communica- tions. Journal of Optics, 18(6):063002, may 2016. [4] Nader Sheikholeslami Alagha and Peter Kabal. Generalized raised-cosine filters. IEEE trans- actions on Communications, 47(7):989–997, 1999. [5] Antonio Assalini and Andrea Tonello. Improved nyquist pulses. Communications Letters, IEEE, 8:87 – 89, 03 2004. [6] Norman C Beaulieu, Christopher C Tan, and Mohamed Oussama Damen. A” better than” nyquist pulse. IEEE Communications Letters, 5(9):367–368, 2001. [7] Sergio Benedetto and Ezio Biglieri. Principles of digital transmission: with wireless applica- tions. Springer Science & Business Media, 1999. [8] Sergio Benedetto and Pierluigi Poggiolini. Theory of polarization shift keying modulation. IEEE Transactions on communications, 40(4):708–721, 1992. [9] Shohei Beppu, Keisuke Kasai, Masato Yoshida, and Masataka Nakazawa. 2048 qam (66 gbit/s) single-carrier coherent optical transmission over 150 km with a potential se of 15.3 bit/s/hz. Optics express, 23(4):4960–4969, 2015. [10] F Boubal, E Brandon, L Buet, S Chernikov, V Havard, C Heerdt, A Hugbart, W Idler, L Labrunie, P Le Roux, et al. 4.16 tbit/s (104/spl times/40 gbit/s) unrepeatered transmission over 135 km in s+ c+ l bands with 104 nm total bandwidth. In Proceedings 27th European Conference on Optical Communication (Cat. No. 01TH8551), volume 1, pages 58–59. IEEE, 2001. [11] Robert W Boyd. Nonlinear optics. Elsevier Science, 2013. [12] Ilja N Bronshtein, Konstantin A Semendyayev, Gerhard Musiol, and Heiner Muehlig. Tables. In Handbook of Mathematics, pages 1007–1091. Springer, 2004. [13] Tim Brown, Persefoni Kyritsi, and Elizabeth De Carvalho. Practical guide to MIMO radio channel: With MATLAB examples. John Wiley & Sons, 2012. 113 [14] Henning B¨ ulow. Polarization qam modulation (pol-qam) for coherent detection schemes. In Optical Fiber Communication Conference and National Fiber Optic Engineers Conference, page OWG2. Optical Society of America, 2009. [15] Fernando Casas, Ander Murua, and Mladen Nadinic. Efficient computation of the zassenhaus formula. Computer Physics Communications, 183(11):2386–2391, 2012. [16] X. Chen, S. Chandrasekhar, S. Randel, G. Raybon, A. Adamiecki, P. Pupalaikis, and P. J. Winzer. All-electronic 100-ghz bandwidth digital-to-analog converter generating pam signals up to 190 gbaud. Journal of Lightwave Technology, 35(3):411–417, Feb 2017. [17] A. R. Chraplyvy. Limitations on lightwave communications imposed by optical-fiber nonlinear- ities. Journal of Lightwave Technology, 8(10):1548–1557, Oct 1990. [18] JM Christian, GS McDonald, and P Chamorro-Posada. Helmholtz-manakov solitons. Physical Review E, 74(6):066612, 2006. [19] Cisco. Cisco visual networking index: Forecast and trends, 2017–2022. https: //www.cisco.com/c/en/us/solutions/collateral/service-provider/visual-networking-index-vni/ white-paper-c11-741490.html, 2019. [20] International Electrotechnical Commission. International Standard:Letter symbols to be used in electrical technology – Part 2: Telecommunications and electronics. International Electrotech- nical Commission, 4.0 edition, 2019. [21] T.M. Cover and J.A. Thomas. Elements of Information Theory. Wiley, 2012. [22] Roberto Cusani, Eugenio Iannone, Antonio Maria Salonico, and Maria Todaro. An efficient multilevel coherent optical system: M-4q-qam. Journal of lightwave technology, 10(6):777– 786, 1992. [23] Tanee Demeechai. Pulse-shaping filters with isi-free matched and unmatched filter properties. IEEE transactions on communications, 46(8):992, 1998. [24] Liang B Du, Danish Rafique, Antonio Napoli, Bernhard Spinnler, Andrew D Ellis, Maxim Kuschnerov, and Arthur J Lowery. Digital fiber nonlinearity compensation: toward 1-tb/s trans- port. IEEE signal processing magazine, 31(2):46–56, 2014. [25] Y Emori, K Tanaka, and S Namiki. 100 nm bandwidth flat-gain raman amplifiers pumped and gain-equalised by 12-wavelength-channel wdm laser diode unit. Electronics Letters, 35(16):1355–1356, 1999. [26] R. Essiambre, G. Kramer, P. J. Winzer, G. J. Foschini, and B. Goebel. Capacity limits of optical fiber networks. Journal of Lightwave Technology, 28(4):662–701, Feb 2010. [27] R-J Essiambre, Peter J Winzer, Xun Qing Wang, Wonsuck Lee, Christopher A White, and Ells C Burrows. Electronic predistortion and fiber nonlinearity. IEEE photonics technology letters, 18(17):1804–1806, 2006. [28] Ren´ e-Jean Essiambre, Robert W Tkach, Roland Ryf, I Kaminow, T Li, and AE Willner. Fiber nonlinearity and capacity: Single-mode and multimode fibers. In Optical Fiber Telecommuni- cations VI B, pages 1–37. Academic Press, 2013. 114 [29] Johannes Karl Fischer, Saleem Alreesh, Robert Elschner, Felix Frey, Christian Meuer, Lutz Molle, Carsten Schmidt-Langhorst, Takahito Tanimura, and Colja Schubert. Experimental in- vestigation of 126-gb/s 6polsk-qpsk signals. Optics express, 20(26):B232–B237, 2012. [30] St´ ephane Forestier, Philippe Bouysse, Raymond Quere, Alain Mallet, J-M Nebus, and Luc Lapierre. Joint optimization of the power-added efficiency and the error-vector measurement of 20-ghz phemt amplifier through a new dynamic bias-control method. IEEE Transactions on Microwave Theory and Techniques, 52(4):1132–1141, 2004. [31] Kiyoshi Fukuchi. Wideband and ultra-dense wdm transmission technologies toward over 10-tb/s capacity. In Optical Fiber Communication Conference, page ThX5. Optical Society of America, 2002. [32] Khaled M Gharaibeh, Kevin G Gard, and Michael B Steer. Accurate estimation of digital com- munication system metrics-snr, evm and/spl rho/in a nonlinear amplifier environment. In Mi- crowave Measurements Conference, Fall 2004. 64th ARFTG, pages 41–44. IEEE, 2004. [33] RA Gibby and JW Smith. Some extensions of nyquist’s telegraph transmission theory. The Bell System Technical Journal, 44(7):1487–1510, 1965. [34] Vagn Lundsgaard Hansen. Functional Analysis, Entering Hilbert Space. World Scientific, 2006. [35] Akira Hasegawa. Theory of information transfer in optical fibers: A ˆ A tutorial review. Optical Fiber Technology, 10(2):150 – 170, 2004. [36] Roland Hassun, Michael Flaherty, Robert Matreci, and Mitch Taylor. Effective evaluation of link quality using error vector magnitude techniques. In Wireless Communications Conference, 1997., Proceedings, pages 89–94. IEEE, 1997. [37] Tetsuya Hayashi, Toshiki Taru, Osamu Shimakawa, Takashi Sasaki, and Eisuke Sasaoka. Ultra- low-crosstalk multi-core fiber feasible to ultra-long-haul transmission. In National Fiber Optic Engineers Conference, page PDPC2. Optical Society of America, 2011. [38] KO Hill, DC Johnson, BS Kawasaki, and RI MacDonald. cw three-wave mixing in single-mode optical fibers. Journal of Applied Physics, 49(10):5098–5106, 1978. [39] Bengt Holter. On the capacity of the mimo channel: A tutorial introduction. In Proc. IEEE Norwegian Symposium on Signal Processing, pages 167–172, 2001. [40] Hao Huang, Guodong Xie, Yan Yan, Nisar Ahmed, Yongxiong Ren, Yang Yue, Dvora Rogawski, Moshe J Willner, Baris I Erkmen, Kevin M Birnbaum, et al. 100 tbit/s free-space data link en- abled by three-dimensional multiplexing of orbital angular momentum, polarization, and wave- length. Optics letters, 39(2):197–200, 2014. [41] Koji Igarashi, Takehiro Tsuritani, Itsuro Morita, Yukihiro Tsuchida, Koichi Maeda, Masateru Tadakuma, Tsunetoshi Saito, Kengo Watanabe, Katsunori Imamura, Ryuichi Sugizaki, et al. 1.03-exabit/skm super-nyquist-wdm transmission over 7,326-km seven-core fiber. In 39th Eu- ropean Conference and Exhibition on Optical Communication (ECOC 2013), pages 1–3. IET, 2013. 115 [42] Bureau international des poids et mesures. The International System of Units (SI). Bureau international des poids et mesures, 2019. [43] Ezra Ip. Nonlinear compensation using backpropagation for polarization-multiplexed transmis- sion. Journal of Lightwave Technology, 28(6):939–951, 2010. [44] Ezra Ip and Joseph M Kahn. Compensation of dispersion and nonlinear impairments using digital backpropagation. Journal of Lightwave Technology, 26(20):3416–3425, 2008. [45] T Itoi, Kiyoshi Fukuchi, and Tadashi Kasamatsu. Enabling technologies for 10 tb/s transmission capacity and beyond. In Proceedings 27th European Conference on Optical Communication (Cat. No. 01TH8551), volume 4, pages 598–601. IEEE, 2001. [46] S. G. Johnson. Saddle-point integration of C1 “bump” functions. ArXiv e-prints, August 2015. [47] Magnus Karlsson and Erik Agrell. Which is the most power-efficient modulation format in optical links? Optics express, 17(13):10814–10819, 2009. [48] Kazuro Kikuchi. Electronic post-compensation for nonlinear phase fluctuations in a 1000-km 20-gbit/s optical quadrature phase-shift keying transmission system using the digital coherent receiver. Optics Express, 16(2):889–896, 2008. [49] Kazuro Kikuchi. Digital coherent optical communication systems: Fundamentals and future prospects. IEICE Electronics Express, 8(20):1642–1662, 2011. [50] Yuki Koizumi, Kazushi Toyoda, Tatsunori Omiya, Masato Yoshida, Toshihiko Hirooka, and Masataka Nakazawa. 512 qam transmission over 240 km using frequency-domain equalization in a digital coherent receiver. Optics express, 20(21):23383–23389, 2012. [51] Yuki Koizumi, Kazushi Toyoda, Masato Yoshida, and Masataka Nakazawa. 1024 qam (60 gbit/s) single-carrier coherent optical transmission over 150 km. Optics express, 20(11):12508–12514, 2012. [52] Xiaoxu Li, Xin Chen, Gilad Goldfarb, Eduardo Mateo, Inwoong Kim, Fatih Yaman, and Guifang Li. Electronic post-compensation of wdm transmission impairments using coherent detection and digital signal processing. Optics Express, 16(2):880–888, 2008. [53] Piotr Mikusinski Lokenath Debnath. Introduction to Hilbert Spaces with Applications. Aca- demic Press, 2005. [54] Arthur J Lowery. Fiber nonlinearity mitigation in optical links that use ofdm for dispersion compensation. IEEE Photonics Technology Letters, 19(19):1556–1558, 2007. [55] Ruben S Lu´ ıs, Georg Rademacher, Benjamin J Puttnam, Tobias A Eriksson, Hideaki Furukawa, Andrew Ross-Adams, Simon Gross, Michael Withford, Nicolas Riesen, Yusuke Sasaki, et al. 1.2 pb/s transmission over a 160mu m cladding, 4-core, 3-mode fiber, using 368 c + l band pdm- 256-qam channels. In 2018 European Conference on Optical Communication (ECOC), pages 1–3. IEEE, 2018. 116 [56] M. W. Maeda, W. B. Sessa, W. I. Way, A. Yi-Yan, L. Curtis, R. Spicer, and R. I. Laming. The effect of four-wave mixing in fibers on optical frequency-division multiplexed systems. Journal of Lightwave Technology, 8(9):1402–1408, Sep 1990. [57] Wilhelm Magnus. On the exponential solution of differential equations for a linear operator. Communications on pure and applied mathematics, 7(4):649–673, 1954. [58] Hisham A Mahmoud and Huseyin Arslan. Error vector magnitude to snr conversion for nondata- aided receivers. IEEE Transactions on Wireless Communications, 8(5), 2009. [59] VG Makhan’kov and OK Pashaev. Nonlinear schr¨ odinger equation with noncompact isogroup. Theoretical and Mathematical Physics, 53(1):979–987, 1982. [60] Dietrich Marcuse, CR Manyuk, and PKA Wai. Application of the manakov-pmd equation to studies of signal propagation in optical fibers with randomly varying birefringence. Journal of Lightwave Technology, 15(9):1735–1746, 1997. [61] Michael D McKinley, Kate A Remley, Maciej Myslinski, J Stevenson Kenney, Dominique Schreurs, and Bart Nauwelaers. Evm calculation for broadband modulated signals. In 64th ARFTG Conf. Dig, pages 45–52, 2004. [62] Linn F Mollenauer and James P Gordon. Solitons in optical fibers: fundamentals and applica- tions. Academic Press, 2006. [63] Sami Mumtaz, Ren´ e-Jean Essiambre, and Govind P Agrawal. Nonlinear propagation in mul- timode and multicore fibers: generalization of the manakov equations. Journal of Lightwave Technology, 31(3):398–406, 2012. [64] Masataka Nakazawa. Extremely advanced transmission with 3m technologies (multi-level mod- ulation, multi-core & multi-mode). In Optical Fiber Communication Conference, pages OTu1D– 1. Optical Society of America, 2012. [65] Masataka Nakazawa. Exabit optical communication explored using 3m scheme. Japanese Jour- nal of Applied Physics, 53(8S2):08MA01, 2014. [66] Masataka Nakazawa, Kazuro Kikuchi, and Tetsuya Miyazaki. High spectral density optical communication technologies, volume 6. Springer Science & Business Media, 2010. [67] LE Nelson, X Zhou, Naoise Mac Suibhne, AD Ellis, and P Magill. Experimental comparison of coherent polarization-switched qpsk to polarization-multiplexed qpsk for 10 100 km wdm transmission. Optics express, 19(11):10849–10856, 2011. [68] T. Nguyen, T. B. Nguyen, T. H. Bui, N. Trang, and T. T. Le. Novel technique for nonlinearity postcompensation of mach-zehnder modulator in an analog-optical link. In 2014 International Conference on Advanced Technologies for Communications (ATC 2014), pages 149–152, Oct 2014. [69] D. O. North. An analysis of the factors which determine signal/noise discrimination in pulsed- carrier systems. Proceedings of the IEEE, 51(7):1016–1027, July 1963. 117 [70] Jacques Palicot and Yves Lou¨ et. Power ratio definitions and analysis in single carrier modula- tions. In Signal Processing Conference, 2005 13th European, pages 1–4. IEEE, 2005. [71] A. Papoulis. Maximum response with input energy constraints and the matched filter principle. IEEE Transactions on Circuit Theory, 17(2):175–182, May 1970. [72] C Par´ e, NJ Doran, A Villeneuve, and P-A B´ elanger. Compensating for dispersion and the non- linear kerr effect without phase conjugation. Optics letters, 21(7):459–461, 1996. [73] G. Paryanti and D. Sadot. Predistortion of mach–zehnder modulator using symmetric imbalance. Journal of Lightwave Technology, 35(13):2757–2768, 2017. [74] John G Proakis. Digital communications fourth edition, 2001. McGraw-Hill Companies, Inc., New York, NY , 2000. [75] Peter S Rha and Sage Hsu. Peak-to-average ratio (par) reduction by pulse shaping using a new family of generalized raised cosine filters. In Vehicular Technology Conference, 2003. VTC 2003-Fall. 2003 IEEE 58th, volume 1, pages 706–710. IEEE, 2003. [76] Nicolas Riesen, Simon Gross, John D. Love, Yusuke Sasaki, and Michael J. Withford. Mono- lithic mode-selective few-mode multicore fiber multiplexers. Scientific Reports, 7(1), 2017. [77] Roland Ryf, Sebastian Randel, Alan H Gnauck, Cristian Bolle, Alberto Sierra, Sami Mumtaz, Mina Esmaeelpour, Ellsworth C Burrows, Ren´ e-Jean Essiambre, Peter J Winzer, et al. Mode- division multiplexing over 96 km of few-mode fiber using coherent 6, 6 mimo processing. Journal of Lightwave technology, 30(4):521–531, 2012. [78] Jun Sakaguchi, Benjamin J Puttnam, Werner Klaus, Yoshinari Awaji, Naoya Wada, Atsushi Kanno, Tetsuya Kawanishi, Katsunori Imamura, Harumi Inaba, Kazunori Mukasa, et al. 305 tb/s space division multiplexed transmission using homogeneous 19-core fiber. Journal of Lightwave Technology, 31(4):554–562, 2013. [79] Bahaa EA Saleh and Malvin Carl Teich. Fundamentals of photonics. Fundamentals of Pho- tonics: Second Edition, by Bahaa EA Saleh and Malvin Carl Teich. ISBN 978-0-471-35832-9. Published by John Wiley & Son, Hoboken, NJ, USA, 2007., 2007. [80] JM Sanz-Serna. Methods for the numerical solution of the nonlinear schr¨ odinger equation. mathematics of computation, 43(167):21–27, 1984. [81] Seb J Savory, Giancarlo Gavioli, Robert I Killey, and Polina Bayvel. Electronic compensation of chromatic dispersion using a digital coherent receiver. Optics express, 15(5):2120–2126, 2007. [82] Thomas Schneider. Nonlinear optics in telecommunications. Springer Science & Business Me- dia, 2013. [83] Marco Secondini, Domenico Marsella, and Enrico Forestieri. Enhanced split-step fourier method for digital backpropagation. In 2014 The European Conference on Optical Commu- nication (ECOC), pages 1–3. IEEE, 2014. [84] Matthias Seimetz. High-order modulation for optical fiber transmission, volume 143. Springer, 2009. 118 [85] John F Sevic and Michael B Steer. On the significance of envelope peak-to-average ratio for estimating the spectral regrowth of an rf/microwave power amplifier. IEEE transactions on microwave theory and techniques, 48(6):1068–1071, 2000. [86] A Shabat and V Zakharov. Exact theory of two-dimensional self-focusing and one-dimensional self-modulation of waves in nonlinear media. Soviet physics JETP, 34(1):62, 1972. [87] Rishad Ahmed Shafik, Md Shahriar Rahman, and AHM Razibul Islam. On the extended rela- tionships among evm, ber and snr as performance metrics. In Electrical and Computer Engi- neering, 2006. ICECE’06. International Conference on, pages 408–411. IEEE, 2006. [88] Rishad Ahmed Shafik, Md Shahriar Rahman, AHM Razibul Islam, and Nabil Shovon Ashraf. On the error vector magnitude as a performance metric and comparative analysis. In Emerging Technologies, 2006. ICET’06. International Conference on, pages 27–31. IEEE, 2006. [89] B Shamee, M Ziyadi, A Mhajerin-Ariaei, A Almaiman, Y Cao, N Ahmed, SR Wilkinson, and AE Willner. Modified raised cosine waveform shaping with reduced peak to average power ratio. In Laser Communication and Propagation through the Atmosphere and Oceans V, volume 9979, page 99790G. International Society for Optics and Photonics, 2016. [90] Bishara Shamee, Nisar Ahmed, Morteza Ziyadi, Amirhossein Mohajerin-Ariaei, Ahmed S. Al- maiman, Yinwen Cao, Steven R. Wilkinson, and Alan E. Willner. Oam based analog to digi- tal converter (adc). In Third International Conference on Optical Angular Momentum, paper OAM01-102. Society of Photo-Instrumentation Engineers (SPIE), 2015. [91] Bishara Shamee, Louis Christen, Scott Nuccio, Jeng-Yuan Yang, and Alan Willner. Gaussian minimum shift keying for spectrally efficient and dispersion tolerant optical communications. In CLEO/QELS: 2010 Laser Science to Photonic Applications, pages 1–2. IEEE, 2010. [92] Bishara Shamee, Amirhossein Mohajerin-Ariaei, Ahmed Almaiman, Yinwen Cao, Fatemeh Al- ishahi, and Alan Willner. Weighted raised cosine waveform with reduced peak to average power ratio for optical transmission. In Optical Modeling and Performance Predictions X, volume 10743, page 107430J. International Society for Optics and Photonics, 2018. [93] Bishara Shamee, Morteza Ziyadi, Amirhossein Mohajerin-Ariaei, Ahmed Almaiman, Yinwen Cao, Nisar Ahmed, Steven R. Wilkinson, and Alan E. Willner. Smoothly clipped root raised cosine waveforms for an effective loading of a coherent optical m-qam modulator. In Ad- vanced Modulation Format and DSP I, paper 9774-9. Society of Photo-Instrumentation En- gineers (SPIE) Photonics West, 2016. [94] Kohki Shibahara, Doohwan Lee, Takayuki Kobayashi, Takayuki Mizuno, Hidehiko Takara, Ak- ihide Sano, Hiroto Kawakami, Yutaka Miyamoto, Hirotaka Ono, Manabu Oguma, et al. Dense sdm (12-core3-mode) transmission over 527 km with 33.2-ns mode-dispersion employing low-complexity parallel mimo frequency-domain equalization. Journal of Lightwave Technol- ogy, 34(1):196–204, 2016. [95] Barry L Shoop. Photonic analog-to-digital converters. Wiley Encyclopedia of Telecommunica- tions, 2003. 119 [96] Oleg V Sinkin, Ronald Holzlohner, John Zweck, and Curtis R Menyuk. Optimization of the split-step fourier method in modeling optical-fiber communications systems. Journal of light- wave technology, 21(1):61–68, 2003. [97] Martin Sj¨ odin, Ben J Puttnam, Pontus Johannisson, Satoshi Shinada, Naoya Wada, Peter A Andrekson, and Magnus Karlsson. Transmission of pm-qpsk and ps-qpsk with different fiber span lengths. Optics express, 20(7):7544–7554, 2012. [98] Daiki Soma, Yuta Wakayama, Shohei Beppu, Seiya Sumita, Takehiro Tsuritani, Tetsuya Hayashi, Takuji Nagashima, Masato Suzuki, Masato Yoshida, Keisuke Kasai, Masataka Nakazawa, Hidenori Takahashi, Koji Igarashi, Itsuro Morita, and Masatoshi Suzuki. 10.16-peta- b/s dense sdm/wdm transmission over 6-mode 19-core fiber across the c+l band. J. Lightwave Technol., 36(6):1362–1368, Mar 2018. [99] M. Stern, J. P. Heritage, R. N. Thurston, and S. Tu. Self-phase modulation and dispersion in high data rate fiber-optic transmission systems. Journal of Lightwave Technology, 8(7):1009–1016, Jul 1990. [100] RH Stolen and Chinlon Lin. Self-phase-modulation in silica optical fibers. Physical Review A, 17(4):1448, 1978. [101] Gordon L St¨ uber. Principles of mobile communication. Springer Science & Business Media, 2011. [102] Hidehiko Takara, Akihide Sano, Takayuki Kobayashi, Hirokazu Kubota, Hiroto Kawakami, Aki- hiko Matsuura, Yutaka Miyamoto, Yoshiteru Abe, Hirotaka Ono, Kota Shikama, et al. 1.01-pb/s (12 sdm/222 wdm/456 gb/s) crosstalk-managed transmission with 91.4-b/s/hz aggregate spec- tral efficiency. In European Conference and Exhibition on Optical Communication, pages Th–3. Optical Society of America, 2012. [103] C. C. Tan and N. C. Beaulieu. An investigation of transmission properties of xia pulses. In 1999 IEEE International Conference on Communications (Cat. No. 99CH36311), volume 2, pages 1197–1201 vol.2, June 1999. [104] C. C. Tan and N. C. Beaulieu. Transmission properties of conjugate-root pulses. IEEE Transac- tions on Communications, 52(4):553–558, April 2004. [105] RW Tkach, AR Chraplyvy, Fabrizio Forghieri, AH Gnauck, and RM Derosier. Four-photon mixing and high-speed wdm systems. Journal of Lightwave Technology, 13(5):841–849, 1995. [106] George L Turin. An introduction to digitial matched filters. Proceedings of the IEEE, 64(7):1092–1112, 1976. [107] A Turukhin, OV Sinkin, HG Batshon, H Zhang, Y Sun, M Mazurczyk, CR Davidson, J-X Cai, MA Bolshtyansky, DG Foursa, et al. 105.1 tb/s power-efficient transmission over 14,350 km using a 12-core fiber. In 2016 Optical Fiber Communications Conference and Exhibition (OFC), pages 1–3. IEEE, 2016. [108] George C Valley. Photonic analog-to-digital converters. Optics express, 15(5):1955–1982, 2007. 120 [109] JAC Weideman and BM Herbst. Split-step methods for the solution of the nonlinear schr¨ odinger equation. SIAM Journal on Numerical Analysis, 23(3):485–507, 1986. [110] George H Weiss and Alexei A Maradudin. The baker-hausdorff formula and a problem in crystal physics. Journal of Mathematical Physics, 3(4):771–777, 1962. [111] G. Welti and Jhong Lee. Digital transmission with coherent four-dimensional modulation. IEEE Transactions on Information Theory, 20(4):497–502, July 1974. [112] B. Widrow and I. Koll´ ar. Quantization Noise: Roundoff Error in Digital Computation, Signal Processing, Control, and Communications. Cambridge University Press, Cambridge, UK, 2008. [113] A. E. Willner, H. Huang, Y . Yan, Y . Ren, N. Ahmed, G. Xie, C. Bao, L. Li, Y . Cao, Z. Zhao, J. Wang, M. P. J. Lavery, M. Tur, S. Ramachandran, A. F. Molisch, N. Ashrafi, and S. Ashrafi. Optical communications using orbital angular momentum beams. Adv. Opt. Photon., 7(1):66– 106, Mar 2015. [114] Alan E. Willner, Salman Khaleghi, Mohammad Reza Chitgarha, and Omer Faruk Yilmaz. All- optical signal processing. J. Lightwave Technol., 32(4):660–680, Feb 2014. [115] P. J. Winzer. Modulation and multiplexing in optical communications. In 2009 Conference on Lasers and Electro-Optics and 2009 Conference on Quantum electronics and Laser Science Conference, pages 1–2, June 2009. [116] P. J. Winzer and R. . Essiambre. Advanced optical modulation formats. Proceedings of the IEEE, 94(5):952–985, 2006. [117] Peter J Winzer. High-spectral-efficiency optical modulation formats. Journal of Lightwave Technology, 30(24):3824–3835, 2012. [118] Peter J. Winzer. Making spatial multiplexing a reality. Nature Photonics, page 345, 2014. [119] Peter J. Winzer, David T. Neilson, and Andrew R. Chraplyvy. Fiber-optic transmission and networking: the previous 20 and the next 20 years. Opt. Express, 26(18):24190–24239, Sep 2018. [120] Robert Wolf, Frank Ellinger, and Ralf Eickhoff. On the maximum efficiency of power amplifiers in ofdm broadcast systems with envelope following. In MOBILIGHT, pages 160–170. Springer, 2010. [121] Xiang-Gen Xia. A family of pulse-shaping filters with isi-free matched and unmatched filter properties. IEEE Transactions on Communications, 45(10):1157–1158, Oct 1997. [122] Guodong Xie, Yongxiong Ren, Yan Yan, Hao Huang, Nisar Ahmed, Long Li, Zhe Zhao, Changjing Bao, Moshe Tur, Solyman Ashrafi, and Alan E. Willner. Experimental demonstration of a 200-gbit/s free-space optical link by multiplexing laguerre–gaussian beams with different radial indices. Opt. Lett., 41(15):3447–3450, Aug 2016. [123] Alison M Yao and Miles J Padgett. Orbital angular momentum: origins, behavior and applica- tions. Advances in Optics and Photonics, 3(2):161–204, 2011. 121 [124] Chien-Hung Yeh, Chien-Chung Lee, and Sien Chi. 120-nm bandwidth erbium-doped fiber am- plifier in parallel configuration. IEEE Photonics Technology Letters, 16(7):1637–1639, 2004. [125] Omer F. Yilmaz, Lior Yaron, Salman Khaleghi, M. Reza Chitgarha, Moshe Tur, and Alan Will- ner. True time delays using conversion/dispersion with flat magnitude response for wideband analog rf signals. Opt. Express, 20(8):8219–8227, Apr 2012. [126] Yang Yue, Nenad Bozinovic, Yongxiong Ren, Hao Huang, Moshe Tur, Poul Kristensen, Sid- dharth Ramachandran, and Alan E Willner. 1.6-tbit/s muxing, transmission and demuxing through 1.1-km of vortex fiber carrying 2 oam beams each with 10 wavelength channels. In Optical Fiber Communication Conference, pages OTh4G–2. Optical Society of America, 2013. [127] V . E. Zakharov. The Inverse Scattering Method, pages 243–285. Springer Berlin Heidelberg, Berlin, Heidelberg, 1980. [128] Vladimir Evgen’evich Zakharov and Alexey Borisovich Shabat. A scheme for integrating the nonlinear equations of mathematical physics by the method of the inverse scattering problem. i. Functional analysis and its applications, 8(3):226–235, 1974. [129] B Zhu, TF Taunay, M Fishteyn, X Liu, S Chandrasekhar, MF Yan, JM Fini, EM Monberg, and FV Dimarcello. 112-tb/s space-division multiplexed dwdm transmission with 14-b/s/hz aggregate spectral efficiency over a 76.8-km seven-core fiber. Optics Express, 19(17):16665– 16671, 2011. 122 Appendix A Mathematical Supplement A.1 Cauchy-Schwartz Inequality In this appendix, a review of the Cauchy-Schwartz Inequality (CSI) to be used as the basis of Matched Filtering theory and summarizes the results from Complex Vector Spaces and its the associated norm leading to the CSI and its proof [34, 53] along with its applicability to Matched filtering for ISI free shaping. The summary is included in order to refresh the reader’s memory in Fields, Vector Spaces, and Inner Product Spaces and the CSI. A.1.1 Fields A field is a setF with two operations + and called addition and multiplication that mapsFF into F. For all elementsu;v;w in the fieldF, the addition and multiplication conform to: a) Associativity:u + (v +w) = (u +v) +w b) Commutativity:u +v =v +u anduv =vu c) There exists an additive identity, 0 such thatu + 0 = 0 +u =u d) There exists a multiplicative identity, 1 such thatu 1 = 1u =u 123 e) There exists a additive inverse,u such thatu + (u) = 0 f) There exists a multiplicative inverse,u 1 such thatuu 1 =u 1 u = 1 andu6= 0. g) Multiplicative distribution over additionu (v +w) = (uv) + (uw) Essentially, a fieldF is closed set under addition and multiplication. Examples of a field are real and complex number systems denoted byR andC with the familiar multiply and addition operations. A.1.2 Vector Spaces A vector space,V, is a set with two operations defined by: a) Vector Addition:VV!V given by (u +v)2V b) Scalar Multiplication:FV!V given by (u), where2F,u2V and (u)2V with the following conditions satisfied for all elementsu;v;w2V and scalars;2F: a) Commutativity:u +v =v +u b) Associativity: (u +v) +w =u + (v +w) c) For any two elements,u;v2V, there exists an elementw2V such thatu +w =v d) (u) = ()u e) ( +)u = (u) + (u) f) (u +v) =u +v g) 1u =u 124 In terms of notation, elements or the vector spaceV are called vectors and elements of the fieldF are called scalars. Furthermore, if the field of scalars, F, is the real number set, thenV is called a real vector space. If the field of scalars,F, is the complex number set, then the vector spaceV is called the Complex Vector Space. A.1.3 Inner Product Spaces and Cauchy-Schwartz Inequality Given a complex vector spaceV, then an inner producth;i is a mapping fromVV intoC with the following defining characteristics: a)hu; vi =hv; ui, where the bar denotes complex conjugate, b)hu; vi =hu; vi = hu;vi, (scaling) c)hu +v;wi =hu;wi +hv;wi, (linearity) d)hu; ui 0 and if equal to 0, thenu = 0. A vector space with an inner product is called a Inner Product Space. Where a norm satisfies the following properties: a)kuk 0 b)kuk = 0 =)u = 0 c)kuk =jjkuk for every2F andu2V d)ku +vkkuk +kvk for everyu;v2V 125 A mapping fromVV into The inner product induces a normkk given by kuk 2 =hu;ui =hu;ui (A.1) kuk = p hu;ui (A.2) (A.3) Theorem 1 (Cauchy-Schwartz Inequality). For any two elementsu;v of a complex inner product space, V, the following holds[53]: jhu;vijkukkvk or equivalently jhu;vij 2 kuk 2 kvk 2 =hu;uihv;vi; with equality if and only if the elementsu is a scalar multiple ofv oru =v for some2F. Applying CSI to the vector space ofL 2 functions with a an inner product defined by hf;gi = Z 1 1 f(x)g(x)dx; (A.4) wheref andg are any two functions with finite square integral, the CSI can be written as: jhf;gij 2 hf;fihg;gi (A.5) Z 1 1 f(x)g(x)dx 2 Z 1 1 f(x)f(x)dx Z 1 1 g(x)g(x)dx; (A.6) with equality whenf(x) =g(x) for some2C. Replacingg withg, CSI reduces to: 126 jhf;gij 2 hf;fihg;gi (A.7) Z 1 1 f(x)g(x)dx 2 Z 1 1 f(x)f(x)dx Z 1 1 g(x)g(x)dx (A.8) Z 1 1 f(x)g(x)dx 2 Z 1 1 f(x)f(x)dx Z 1 1 g(x)g(x)dx (A.9) Z 1 1 f(x)g(x)dx 2 Z 1 1 f(x)f(x)dx Z 1 1 g(x)g(x)dx (A.10) jhf;gij 2 hf;fihg;gi (A.11) and with equality whenf(x) = g(x) for some. Either form of the CSI stated in equation (A.6) or (A.10) could be used as needed. In terms of inner products, either equation (A.5) or (A.11) can be used. 127 Appendix B Frequently Used Equations In this appendix we list some of the most common trigonometric identities B.1, the methodology to convert wavelength bands to frequency bands in free space B.2 and both the unit prefixes in both binary and decimal B.3. B.1 Trigonometric Identities sin ( +) = sin cos + cos sin (B.1) cos ( +) = cos cos sin sin (B.2) 2 sin() sin() = cos() cos( +) (B.3) 2 cos() cos() = cos() + cos( +) (B.4) 2 sin() cos() = sin( +) sin() (B.5) sin (2) = 2 sin cos (B.6) cos (2) = cos 2 sin 2 (B.7) 128 B.2 Wavelength and Frequency Conversion This section summarizes a useful relationship between wavelength bands and the corresponding fre- quency bands at a specific wavelength in free space. Starting with the total variation of the wavelength frequency product, =c ) + = 0 (B.8) wherec 300 10 6 m/s is the speed of light in free space. By collecting terms and substituting the values at the common wavelength of 1550 nm, the wavelength to occupied bandwidth is: = = c (B.9) = 300 10 6 1550 10 9 1 GHz 10 9 (B.10) = 300 10 6 1550 1 GHz (B.11) = 193:548 10 3 GHz (B.12) = 193:548 10 3 1550 GHz (B.13) 125 GHz (nm) (B.14) Table B.1 shows some representative optical band conversion from wavelength to frequency Table B.1: Typical wavelength bands to frequency mapping at 1550 nm (nm) 1 0:5 0:4 0:3 0:2 0:1 (GHz) 125 65:5 50:0 37:5 25:0 12:5 129 B.3 Binary and Decimal Prefixes The decimal prefixes of a base unit were defined by the Syst` eme International d’Unit´ es (SI) and are tabulated in the Bureau International des Poids et Mesures brochure [42]. Starting with multiples of 10 3 , a summary of the utilized prefixes is shown in Table B.2 as cross reference matrix: Table B.2: Decimal prefix conversion factors 1 kilo mega giga tera peta exa zeta yotta 1 k M G T P E Z Y 1 10 0 10 3 10 6 10 9 10 12 10 15 10 18 10 21 10 24 kilo k 10 +3 10 0 10 3 10 6 10 9 10 12 10 15 10 18 10 21 mega M 10 +6 10 +3 10 0 10 3 10 6 10 9 10 12 10 15 10 18 giga G 10 +9 10 +6 10 +3 10 0 10 3 10 6 10 9 10 12 10 15 tera T 10 +12 10 +9 10 +6 10 +3 10 0 10 3 10 6 10 9 10 12 peta P 10 +15 10 +12 10 +9 10 +6 10 +3 10 0 10 3 10 6 10 9 exa E 10 +18 10 +15 10 +12 10 +9 10 +6 10 +3 10 0 10 3 10 6 zeta Z 10 +21 10 +18 10 +15 10 +12 10 +9 10 +6 10 +3 10 0 10 3 yotta Y 10 +24 10 +21 10 +18 10 +15 10 +12 10 +9 10 +6 10 +3 10 0 While the SI prefixes define multiples of 10, the binary representation is more conveniently repre- sented in multiples of 2 which were developed by the International Electrotechnical Commission (IEC) [20] and derived from the SI prefixes. Each binary prefix is constructed by retaining the first two letters of the corresponding SI decimal prefix and adding the word ”bi” for binary. The binary prefix symbol is derived from the corresponding SI symbol and adding the letter ’i’ as tabulated in cross reference Table B.3. 130 Table B.3: Binary prefix conversion factors 1 kibi mebi gibi tebi pebi exbi zebi yobi 1 Ki Mi Gi Ti Pi Ei Zi Yi 1 2 0 2 10 2 20 2 30 2 40 2 50 2 60 2 70 2 80 kibi Ki 2 +10 2 0 2 10 2 20 2 30 2 40 2 50 2 60 2 70 mebi Mi 2 +20 2 +10 2 0 2 10 2 20 2 30 2 40 2 50 2 60 gibi Gi 2 +30 2 +20 2 +10 2 0 2 10 2 20 2 30 2 40 2 50 tebi Ti 2 +40 2 +30 2 +20 2 +10 2 0 2 10 2 20 2 30 2 40 pebi Pi 2 +50 2 +40 2 +30 2 +20 2 +10 2 0 2 10 2 20 2 30 exbi Ei 2 +60 2 +50 2 +40 2 +30 2 +20 2 +10 2 0 2 10 2 20 zebi Zi 2 +70 2 +60 2 +50 2 +40 2 +30 2 +20 2 +10 2 0 2 10 yobi Yi 2 +80 2 +70 2 +60 2 +50 2 +40 2 +30 2 +20 2 +10 2 0 The expansions in (B.15) through (B.18) illustrate the relative magnitude between the binary and the decimal prefixes for some cases: 1 kibi = 2 10 = (2 10 ) 1 = 1000 + 24 = 1 k + 24 (B.15) 1 mebi = 2 20 = (2 10 ) 2 = (k + 24) 2 = 1 M + 2 24 k + 24 2 (B.16) 1 gebi = 2 30 = (2 10 ) 3 = (k + 24) 3 = 1 G + 3 24 M + 3 24 k + 24 3 (B.17) 1 tebi = 2 40 = (2 10 ) 4 = (k + 24) 4 = 1 T + 4 24 G + 6 24 2 M + 4 24 3 k + 24 4 (B.18) At higher prefix multipliers the difference between the two designations increases and will impact the data rate estimates. For completeness, the general expression relating the binary and decimal pre- fixes can be written as: 2 10n = 2 10 n = (k + 24) n = n X m=0 n m 24 m k nm : (B.19) 131 Appendix C Coherent Receiver Matrix Representation Coherent hybrid and detectors are used in a coherent optical system to derive the inphase and quadrature components of the received waveform relative to a local laser. The hybrid consist of four 90 couplers followed by a set of detectors as depicted in Figure C.1. The coherent hybrid has the reference laser denoted byL and the incoming received signalS. Each of the four couplers, C 00 ; C 01 ; C 10 and C 11 can be represented by a 22 matrix shown below: C 00 = C 01 = C 10 = C 11 = 1 p 2 0 B B @ 1 j j 1 1 C C A : (C.1) The first pair of couplers, C 00 ;C 10 introduce a 90 shift on each element so that remaining 90 shift allow for the four quadrants projection of the signal and the laser. The first stage is a partitioned matrix,C: C = 2 6 6 4 C 00 0 0 C 10 3 7 7 5 : 132 The output of the first coupler set is C x where x is the laser and signal 4 dimensional column vector, x = (S; 0; 0;L) T . The permutation P routes the appropriate output of couplerC 00 andC 10 to the remaining couplers as depicted in Figure C.1. The permutation matrix, P, is: P = 0 B B B B B B B B B B @ +1 0 0 0 0 0 +1 0 0 +1 0 0 0 0 0 j 1 C C C C C C C C C C A (C.2) Finally, the last coupler stage is identical to the first stage, so that coherent hybrid transfer matrix is CPC transforming the input vector x is: CPC x = 1 2 0 B B B B B B B B B B @ +1 j j 1 j 1 +1 j j +1 j +1 1 j +1 +j 1 C C C C C C C C C C A 0 B B B B B B B B B B @ S 0 0 L 1 C C C C C C C C C C A = 1 2 0 B B B B B B B B B B @ SL j(S +L) LjS S +jL 1 C C C C C C C C C C A : (C.3) To illustrate the phase partition of the coherent hybrid, consider the case when S = 0, then the reference laser output of the coherent hybrid is: CPC 0 B B B B B B B B B B @ 0 0 0 L 1 C C C C C C C C C C A = 1 2 0 B B B B B B B B B B @ +1 j j 1 j 1 +1 j j +1 j +1 1 j +1 +j 1 C C C C C C C C C C A 0 B B B B B B B B B B @ 0 0 0 L 1 C C C C C C C C C C A = 1 2 0 B B B B B B B B B B @ L jL L jL 1 C C C C C C C C C C A : (C.4) 133 j i @ @ @ @ @ @ @ @ - - @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ P 0 P 2 () 2 0 L 0 j P 3 I Q S P 1 Coherent Hybrid C 00 C 11 C 10 C 01 S= p 2 jL= p 2 jS= p 2 () 2 () 2 () 2 L= p 2 jL= p 2 (LjS)=2 (S +jL)=2 j(S +L)=2 (SL)=2 Figure C.1: Coherent Inphase and Quadrature Detector The hybrid output aligns to the;=2; 0; =2 which are =2 apart around the unit circle. Denoting the output of the coherent hybrid by y = CPC x, then y y = CPC x CPC x (C.5) = x C P C CPC x (C.6) = 1 4 x x (C.7) since the matrix CPC is unitary 1 . The output power of the coherent hybrid denoted byP 0 ,P 1 ,P 2 and P 3 respectively are: 1 Please note: The factor of4 can be dropped for this discussion without affecting the result 134 P 0 = (SL) (SL)=4 = (S S 2 Re (S L) +L L)=4 (C.8) P 1 =j (S +L) j(S +L)=4 = (S S + 2 Re (S L) +L L)=4 (C.9) P 2 = (LjS) (LjS)=4 = (L L 2 Im (S L) +S S)=4 (C.10) P 3 = (jLS) (jL +S)=4 = (S S + 2 Im (S L) +L L)=4: (C.11) The power of each coherent hybrid output projects the signalS onto theIQ plane of the reference laser to provide the inphase and quadrature components depicted in Figure C.1. Finally, the power differenceP 1 P 0 andP 3 P 2 are the received inphase and quadrature components: IP 1 P 0 = Re (S L) (C.12) QP 3 P 2 = Im (S L) (C.13) 135 Appendix D Mach-Zehnder Modulator Mathematical Representation A Mach-Zehnder Modulator is an optical device that divides the input laser into two paths where each path may be is independently phase controlled by an external source as shown in Figure D.1. The two separate paths are combined to produce an output field that depends on the relative voltage difference driving the relative phase between the two paths. When the phase difference between the two path is 0 or 2, the interference is constructive resulting in the output equal to the input assuming no path loss. If the phase difference is 180 , the interference is destructive resulting in a null output. In effect, the controlling voltage modulates the input optical carrier suitable for transmission. Figure D.1: A general depiction of the Mach Zehnder Modulator controlled by two independent volt- ages Depending on the input voltage, the MZM can be operated in different modes to support various modulation formats [84]. The following is a brief summary of the MZM mathematical description of 136 the output and biasing the modulator to derive the push-pull amplitude modulator. Denote the input field byE i , then the field in along the first path is E i 2 exp j V v 1 ; (D.1) wherev 1 is applied phase control voltage andV is the voltage to produce a 180 phase shift in the field of the corresponding path. Similarly, ifv 2 is the controlling phase along the second path, then the combined field at the output of the interferometer is the sum of the two fields, namely: E o = E i 2 exp j V v 1 + E i 2 exp j V v 2 : (D.2) Simplifying the output field y transformingv 1 andv 2 according to: v 1 v 1 2 + v 1 2 + v 2 2 v 2 2 v 2 v 2 2 + v 2 2 + v 1 2 v 1 2 The first path can be written as: E i 2 exp j V v 1 = E i 2 exp j V v 1 2 + v 1 2 + v 2 2 v 2 2 = E i 2 exp j V v 1 2 + v 2 2 exp j V v 1 2 v 2 2 = E i 2 exp j 2V (v 1 +v 2 ) exp j 2V (v 1 v 2 ) : (D.3) 137 Performing similar manipulation to the second arm and substituting into equation D.2: E o = E i 2 exp j 2V (v 1 +v 2 ) exp j 2V (v 1 v 2 ) + exp j 2V (v 1 v 2 ) = exp j 2V (v 1 +v 2 ) cos 2V (v 1 v 2 ) E i (D.4) where the cos term resulted from adding a complex exponential and its conjugate. The output field amplitude is a sinusoidal term as a function of voltage difference across the two arms or paths of the MZM. The output field instantaneous power is the product of the field and its conjugate: E o E o = cos 2 2V (v 1 v 2 ) E i E i = 1 2 1 + cos V (v 1 v 2 ) E i E i : (D.5) The power transfer function is the ratio of the output power to the input power given by: T E o E o E i E i = 1 2 1 + cos V (v 1 v 2 ) : (D.6) Depending on the applied input voltages, v 1 and v 2 , the MZI can be biased and operated in dif- ferently to support a wide variety of modulation [116]. Considering each input voltage to consist of a bias and a time varying component and amplitude as illustrated in Figure D.2. Letv 1 = u 1 +b 1 and v 2 =u 2 +b 2 , then the output field can be written as: E o = exp j 2V (u 1 +u 2 ) exp j 2V (b 1 +b 2 ) cos 2V ((u 1 u 2 ) + (b 1 b 2 )) E i (D.7) 138 Figure D.2: Genral MZM operation with two independent voltage and bias drives For a push pull operation with differential voltage drive, the bias and the input voltages are adjusted so thatu 1 =u 2 =u andb 1 =b 2 =b as illustrated in Figure D.3a. The resulting output field of the MZM is: E o = cos 2V (2u + 2b) E i = cos V (u +b) E i (D.8) The power output is given by E o E o = cos 2 V (u +b) E i E i = 1 2 1 + cos 2 V (u +b) E i E i (D.9) with a power transfer function given by: T E o E o E i E i = 1 2 1 + cos 2 V (u +b) (D.10) To illustrate, suppose the biasb is set to 0, then the output field is given by E o = cos V u E i (D.11) 139 describing an amplitude modulator as a function of the input signal u. The modulator is a non- linear modulator due to the cos term and may constrain the linear dynamic range of the modulation as illustrated in Figure D.3b. If the bias is shifted toV =2, then the modulator will have a positive slope at the operating bias and dynamic range ofV with a small linear range around the bias point. (a) Push Pull MZM amplitude modulator (b) Output field and power Figure D.3: MZM Amplitude Modulator bias and response 140

Abstract (if available)

Abstract Compared to the available Radio Frequency bandwidth, the optical bandwidth over fiber is larger by orders of magnitude. However, with the rapid increase in bandwidth demand due the global infrastructure connectivity, the fiber capacity is reaching its limitations and new methods are needed to meet the expected demand. Meeting the demand is a multi-dimensional challenge driven by practical solutions leading the technology development. ❧ The expected demand challenge response has been formulated along the lines of packing more bits per symbol, increasing the number of channels per medium to include polarization, multiple fiber modes communication, multiple orbital angular momentum modes, multiple core fiber, advancing the technology to utilize more optical bands in fiber, and finally waveform shaping to increase the transmission spectral efficiency to increase the capacity in a given bandwidth. ❧ Early fiber transmissions, utilized bandwidth inefficient amplitude waveforms such as ON/OFF keying at a one bit per symbol. Later on, optical communications leveraged the advances of the digital Radio Frequency communication and utilized the two dimensional carrier amplitude and phase to increase the number of bits per symbol. Early two dimensional waveforms and the ON/OFF keying were unshaped and required bandwidth much larger than the symbol rate due to the complexity of more efficient waveforms. ❧ In this thesis, the focus will be on the digital waveform shaping to increase the spectral efficiency of the transmission by reducing the occupied spectral bandwidth while considering the end to end link as a whole. The simulation and analysis will compare the proposed waveform to a reference waveform and identifies the waveform shaping performance drivers as the utilized bandwidth approaches the symbol rate or the Nyquist bandwidth. Waveform shaping increases the penalties due to the fiber nonlinearity as well as increasing the linear operating range of the transmitter DAC, modulator drivers and receiver ADC. With advances in the digital technology, signal processing may play a significant role in waveform shaping and mitigation of fiber impairment.

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Creator Shamee, Bishara Farid (author)

Core Title Spectrally efficient waveforms for coherent optical fiber transmission

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Electrical Engineering

Degree Conferral Date 2021-08

Publication Date 05/21/2021

Defense Date 05/20/2021

Publisher Los Angeles, California (original), University of Southern California (original), University of Southern California. Libraries (digital)

Tag error vector magnitude,fiber transmission,generalized Nyquist criterion,optical communications,optical modulator compensation,peak to average ratio,spectral efficiency,spectral occupancy,waveform shaping,weighted raised cosine waveform

Format theses (aat)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Willner, Alan (committee chair), Haas, Stephan (committee member), Sawchuk, Alexander (committee member)

Creator Email bishara.shamee@gmail.com,bishara.shamee@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-oUC13012632

Unique identifier UC13012632

Identifier etd-ShameeBish-9638.pdf (filename)

Legacy Identifier etd-ShameeBish-9638

Document Type Dissertation

Format theses (aat)

Rights Shamee, Bishara Farid

Internet Media Type application/pdf

Type texts

Source 20210611-usctheses-batch-839 (batch), University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

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