ORTHOGONAL FREQUENCY-DIVISION MULTIPLEXING FOR OPTICAL COMMUNICATIONS

A DISSERTATION SUBMITTED TO THE DEPARTMENT OF ELECTRICAL ENGINEERING AND THE COMMITTEE ON GRADUATE STUDIES OF STANFORD UNIVERSITY IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY

Daniel Jose Fernandes Barros September 2011

© 2011 by Daniel Jose Fernandes Barros. All Rights Reserved. Re-distributed by Stanford University under license with the author.

This work is licensed under a Creative Commons Attribution- Noncommercial 3.0 United States License. http://creativecommons.org/licenses/by-nc/3.0/us/

This dissertation is online at: http://purl.stanford.edu/yz748tf7178

ii I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Joseph Kahn, Primary Adviser

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

John Gill, III

I certify that I have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of Doctor of Philosophy.

Bernard Widrow

Approved for the Stanford University Committee on Graduate Studies. Patricia J. Gumport, Vice Provost Graduate Education

This signature page was generated electronically upon submission of this dissertation in electronic format. An original signed hard copy of the signature page is on file in University Archives.

iii Abstract

ABSTRACT

The drive towards higher spectral efficiency and maximum power efficiency in optical systems has generated renewed interest in the optimization of optical transceivers. In this work, we study the different optical applications: Wide Area Networks (WANs), Metropolitan Area Networks (MANs), Local Area Networks (LANs) and Personal Area Networks (PANs).

In WANs or long-haul systems, orthogonal frequency-division multiplexing (OFDM) can compensate for linear distortions, such as group-velocity dispersion (GVD) and polarization-mode dispersion (PMD), provided the cyclic prefix is sufficiently long. Typically, GVD is dominant, as it requires a longer cyclic prefix. Assuming coherent detection, we show how to analytically compute the minimum number of subcarriers and cyclic prefix length required to achieve a specified power penalty, trading off power penalties from the cyclic prefix and from residual inter- symbol interference (ISI) and inter-carrier interference (ICI). We derive an analytical expression for the power penalty from residual ISI and ICI. We also show that when nonlinear effects are present in the fiber, single-carrier with digital equalization outperforms OFDM for various dispersion maps. We also study the impairments of electrical to optical conversion when using Mach-Zehnder (MZ) modulators. OFDM has a high peak-to-average ratio (PAR), which can result in low optical power efficiency when modulated through a Mach-Zehnder (MZ) modulator. In addition, the nonlinear characteristic of the MZ can cause significant distortion on the OFDM signal, leading to in-band intermodulation products between subcarriers. We show that a quadrature MZ with digital pre-distortion and hard clipping is able to overcome the previous impairments. We consider quantization noise and compute the minimum number of bits required in the digital-to-analog converter (D/A). Finally, we discuss a dual-drive MZ as a simpler alternative for the OFDM modulator, but our results show that it requires a higher oversampling ratio to achieve the same performance as the quadrature MZ.

iv Abstract

In MANs, we discuss the use OFDM for combating GVD effects in amplified direct-detection (DD) systems using single-mode fiber. We review known direct- detection OFDM techniques, including asymmetrically clipped optical OFDM (ACO- OFDM), DC-clipped OFDM (DC-OFDM) and single-sideband OFDM (SSB-OFDM), and derive a linearized channel model for each technique. We present an iterative procedure to achieve optimum power allocation for each OFDM technique, since there is no closed-form solution for amplified DD systems. For each technique, we minimize the optical power required to transmit at a given bit rate and normalized GVD by iteratively adjusting the bias and optimizing the power allocation among the subcarriers. We verify that SSB-OFDM has the best optical power efficiency among the different OFDM techniques. We compare these OFDM techniques to on-off keying (OOK) with maximum-likelihood sequence detection (MLSD) and show that SSB-OFDM can achieve the same optical power efficiency as OOK with MLSD, but at the cost of requiring twice the electrical bandwidth and also a complex quadrature modulator. We compare the computational complexity of the different techniques and show that SSB-OFDM requires fewer operations per bit than OOK with MLSD.

In LANs, we compare the performance of several OFDM schemes to that of OOK in combating modal dispersion in multimode fiber links. We review known OFDM techniques using intensity modulation with direct detection (IM/DD), including DC-OFDM, ACO-OFDM and pulse-amplitude modulated discrete multitone (PAM-DMT). We describe an iterative procedure to achieve optimal power allocation for DC-OFDM, and compare analytically the performance of ACO-OFDM and PAM- DMT. We also consider unipolar M-ary pulse-amplitude modulation (M-PAM) with minimum mean-square error decision-feedback equalization (MMSE-DFE). For each technique, we quantify the optical power required to transmit at a given bit rate in a variety of multimode fibers. For a given symbol rate, we find that unipolar M-PAM with MMSE-DFE has a better power performance than all OFDM formats. Furthermore, we observe that the difference in performance between M-PAM and OFDM increases as the spectral efficiency increases. We also find that at a spectral efficiency of 1 bit/symbol, OOK performs better than ACO-OFDM using a symbol

v Abstract rate twice that of OOK. At higher spectral efficiencies, M-PAM performs only slightly better than ACO-OFDM using twice the symbol rate, but requires less electrical bandwidth and can employ analog-to-digital converters at a speed only 81% of that required for ACO-OFDM.

In PANs, we evaluate the performance of the three IM/DD OFDM schemes in combating multipath distortion in indoor optical wireless links, comparing them to unipolar M-PAM with MMSE-DFE. For each modulation method, we quantify the received electrical SNR required at a given bit rate on a given channel, considering an ensemble of 170 indoor wireless channels. When using the same symbol rate for all modulation methods, M-PAM with MMSE-DFE has better performance than any OFDM format over a range of spectral efficiencies, with the advantage of M-PAM increasing at high spectral efficiency. ACO-OFDM and PAM-DMT have practically identical performance at any spectral efficiency. They are the best OFDM formats at low spectral efficiency, whereas DC-OFDM is best at high spectral efficiency. When ACO-OFDM or PAM-DMT are allowed to use twice the symbol rate of M-PAM, these OFDM formats have better performance than M-PAM. When channel state information is unavailable at the transmitter, however, M-PAM significantly outperforms all OFDM formats. When using the same symbol rate for all modulation methods, M-PAM requires approximately three times more computational complexity per processor than all OFDM formats and 63% faster analog-to-digital converters, assuming oversampling ratios of 1.23 and 2 for ACO-OFDM and M-PAM, respectively. When OFDM uses twice the symbol rate of M-PAM, OFDM requires 23% faster analog-to-digital converters than M-PAM but OFDM requires approximately 40% less computational complexity than M-PAM per processor.

vi Dedication

DEDICATION

Esta dissertação é dedicada

à minha irmã Bárbara Barros.

vii Acknowledgments

ACKNOWLEDGMENTS

This thesis would not be possible without the help and support of many people.

I would like to express my gratitude and appreciation for my principal adviser, Prof. Joseph Kahn, for his continuous support and guidance during my doctoral program at Stanford. Prof. Kahn always paid close attention to the work of his students and always aimed to perform exceptional work. I am fortunate to have benefited from his knowledge, and his pursuit for excellence serves as an example for me.

I would also like to thank my associate adviser, Prof. Bernard Widrow, for his supervision and availability. I had the opportunity to work with Prof. Widrow in a medical project where we studied diaphragm electromyography (EMG) signals from patients with amyotrophic lateral sclerosis (ALS). I learned immensely from Prof. Widrow and I am very grateful for all his help and advice.

I would like to thank Prof. Robert Twiggs from the Aeronautics and Astronautics department for all of his support. I had the opportunity to work in his team where we develop a cube satellite. The purpose of this project was to gather temperature, acceleration and GPS data at 30000 feet and transmit the data in real-time through a radio to the base station in the ground. The cube satellite was launched on a rocket at White Sands missile range, New Mexico.

I would also like to thank the members of my Oral Exam committee, who were Prof. Joseph Kahn, Prof. Bernard Widrow, Prof. Donald C. Cox and Prof. John Gill for their willingness to be part of my committee. In particular, I want to thank Prof. John Gill for his promptness to serve both as my dissertation reader and the chair of my oral exam.

I also would like to thank Stanford University for the Stanford Graduate Fellowship (SGF) and the Portuguese Foundation for Science and Technology for the scholarship SFRH/BD/22547/2005.

viii Acknowledgments

I would like to express my sincere appreciation to my wonderful girlfriend, Rita Lopez, for all her understanding and support in the most difficult moments of my program. I am also grateful for all the fun memories we shared together and without her help this thesis would have not been possible.

I am also very grateful for all the support and guidance of my family, in particular my parents, Maria Luisa Fernandes and Jose Manuel de Barros. My parents were always there for me and supported me in all the decisions I made. I also want to thank my in-laws Maria Teresa de Oliveira Braga and Rui Fernando Lopez.

Special mention must go to all members of the Optical Communications Group of Prof. Kahn. I am thankful to Alan Lau, Ezra Ip and Jeff Wilde for all the ideas we shared during our weekly group meetings. In addition, I also want to thank Rahul Panicker, Mahdieh Shemirani, Tarek Abouzeid, Gwang-Hyun Gho, Reza Mahalati, Daulet Askarov and Dany Ly-Gagnon for all the fun and memories.

Last but not least, I also want to thank all my great friends at Stanford: Hugo Louro, Rita Oliveira, Hugo Caetano, Rita Fragoso, Isaac and Martha Martinez, Josh and Kelly Alwood, João Vicente, João Rodrigues, Sabina Alistar, Kristiaan de Greve, Sandra Beleza, Irina Weissbrot, Rinki Kapoor, Serena Faruque, Henrique Miranda, Francisco Santos and many others. We shared great and fun moments together which made life at Stanford much more enjoyable.

ix Table of Contents

TABLE OF CONTENTS

Abstract ...... iv Dedication ...... vii Acknowledgments ...... viii 1 Introduction ...... 1 2 Wide Area Networks ...... 7 2.1 Introduction ...... 7 2.2 WAN and OFDM Review ...... 9 2.2.1 WANs ...... 9 2.2.2 OFDM Review ...... 11 2.3 Group-Velocity Dispersion ...... 15 2.3.1 Theory ...... 15 2.3.2 Simulation Results ...... 17 2.4 Polarization-Mode Dispersion ...... 23 2.4.1 Theory ...... 23 2.4.2 Dual-Polarization Receiver ...... 25 2.4.3 Single-Polarization Receiver ...... 27 2.5 Fiber Nonlinearity ...... 28 2.5.1 Theory ...... 28 2.5.2 Simulation Results ...... 30 2.6 Computational Complexity ...... 33 2.7 Optical Modulator ...... 35 2.7.1 PAR and MZ Review ...... 37 2.7.1.1 Peak-to-Average Power Ratio ...... 37 2.7.1.2 Mach-Zehnder Modulator ...... 37 2.7.2 Quadrature MZ Optimization ...... 39 2.7.2.1 Quadrature MZ Optimization ...... 39 2.7.2.2 Clipping Simulation Results ...... 42

x Table of Contents

2.7.2.3 Quantization Effects ...... 46 2.7.3 Dual-Drive MZ Optimization ...... 48 2.7.3.1 Dual-Drive MZ Modulator ...... 48 3 Metropolitan Networks ...... 56 3.1 Introduction ...... 56 3.2 Metropolitan Networks and Power Allocation Review ...... 58 3.2.1 Metropolitan Networks ...... 58 3.2.2 Power and Bit Allocation Review ...... 59 3.2.2.1 Gap Approximation ...... 59 3.2.2.2 Optimum Power Allocation ...... 61 3.3 OFDM System Model for Metro Links ...... 63 3.4 Analysis of Direct-Detection OFDM Schemes ...... 64 3.4.1 DC-Clipped OFDM ...... 64 3.4.2 Asymmetrically Clipped Optical OFDM ...... 66 3.4.3 Single-Sideband OFDM ...... 67 3.4.4 Effects of Amplifier Noise ...... 70 3.5 Comparison of Direct-Detection Modulation Formats ...... 71 3.5.1 Computational Complexity ...... 80 4 Local Area Networks ...... 82 4.1 Introduction ...... 82 4.2 Local Area Networks and Discrete Bit Allocation Review ...... 83 4.2.1 Local Area Networks ...... 83 4.2.2 Discrete Bit Allocation Review ...... 84 4.3 System Model for LANs ...... 85 4.3.1 Overall System Model ...... 85 4.3.2 Multimode Fiber Model ...... 86 4.3.3 Performance Measures ...... 88 4.4 Analysis of IM/DD OFDM Schemes ...... 90 4.4.1 DC-Clipped OFDM ...... 90 4.4.2 Asymmetrically Clipped Optical OFDM ...... 91

xi Table of Contents

4.4.3 PAM-Modulated Discrete Multitone ...... 92 4.5 Comparison of IM/DD Modulation Formats ...... 95 5 Personal Area Networks ...... 105 5.1 Introduction ...... 105 5.2 Personal Area Networks ...... 106 5.3 System Model and Performance Measures...... 107 5.3.1 Overall System Model ...... 107 5.3.2 Optical Wireless Channel ...... 108 5.3.3 Performance Measures ...... 110 5.3.4 Ceiling-Bounce Model ...... 111 5.4 Comparison of IM/DD Modulation Formats ...... 112 5.4.1 OOK and OFDM Performance ...... 114 5.4.2 Unipolar 4-PAM and OFDM Performance ...... 119 5.4.3 Outage Probability ...... 122 5.5 Computational Complexity ...... 125 6 Conclusions ...... 127 6.1 Conclusions ...... 127 6.2 Future Work ...... 130 A Appendix ...... 131 References ...... 135

xii List of Tables

LIST OF TABLES

Table 1.1 Characteristics of the different optical applications...... 3 Table 1.2 Characteristics of the different optical applications...... 4 Table 2.1 Number of subcarriers and cyclic prefix required to achieve an overall power penalty less than 1 dB. We consider 4-QAM subcarriers, R = 26.7 GHz, D = 17 ps/(nm·km) with 98% inline optical dispersion compensation...... 22 Table 2.2 Computational complexity in real operations per bit for the various modulation formats. We consider 4-QAM subcarriers, R = 26.7 GHz, D = 17 ps/(nm·km) with 98% inline optical dispersion compensation...... 35 Table 3.1 Electrical and optical bandwidths required to transmit at a bit rate R...... 72

Table 3.2 OFDM parameters for the various dispersion indexes γ. N is the DFT size, Nu is the number of used subcarriers and ν is the cyclic prefix...... 78 Table 3.3 Memory required for various values of the dispersion index γ for OOK with MLSD [3]...... 81 Table 3.4 Number of real operations required per bit for SSB-OFDM and OOK with MLSD for the various dispersion indexes γ...... 81 Table 5.1 System parameters for the various modulation formats for different bit rates and symbol rates. N is the DFT size, Nu is the number of used subcarriers, ν is the cyclic prefix, Nf is the number of taps of the feedforward filter, Nb is the number of taps of the feedback filter and Be is the required electrical bandwidth in MHz...... 116 Table 5.2 Computational complexity in real operations per bit for the various modulation formats...... 125

xiii List of Figures

LIST OF FIGURES

Fig. 1.1 OFDM and single-carrier waveforms...... 2 Fig. 2.1 DWDM system with 50 GHz channel spacing...... 10

Fig. 2.2 Long-haul link with NA spans...... 10 Fig. 2.3 OOK modulation in traditional optical systems...... 11 Fig. 2.4 Digital implementation of OFDM transmitter and receiver...... 12 Fig. 2.5 OFDM symbol, including the cyclic prefix and windowing...... 13

Fig. 2.6 OFDM spectrum, assuming rectangular windowing (Nwin = 0)...... 13 Fig. 2.7 Pulse spreading due to fiber GVD...... 15 Fig. 2.8 OFDM symbol with insufficient cyclic prefix...... 16 Fig. 2.9 Probability of symbol error in presence of GVD for L = 80 km, D = 17 ps/(nm.km),

Nc = 64, Nu = 52, Npre = 5 and R = 26.7 GHz.OFDM symbol with insufficient cyclic prefix. .. 18 −4 Fig. 2.10 ISI + ICI power penalty at PS = 10 for L = 80 km, D = 17 ps/(nm.km), Nc = 64, Nu = 52 and R = 26.7 GHz...... 19 −4 Fig. 2.11 Overall power penalty at PS = 10 for L = 80 km, D =17 ps/(nm.km), Nc = 64, Nu = 52 and R = 26.7 GHz...... 19 Fig. 2.12 Overall power penalties at R = 26.7 GHz for 2% undercompensated GVD lengths of a) 1000 km b) 2000 km c) 3000 km d) 4000 km and e) 5000 km. The dashed lines are the analytical values and the solid lines are the simulations results with insufficient cyclic prefix...... 21 Fig. 2.13 Optimum number of used subcarriers and cyclic prefix length as a function of the fiber length for R = 26.7 GHz when no optical dispersion compensation is used...... 23 Fig. 2.14 Polarization-mode dispersion (PMD...... 24 Fig. 2.15 OFDM polarization-multiplexed system...... 25 Fig. 2.16 Combined PMD and GVD equalizer for an OFDM polarization-multiplexed system...... 26 Fig. 2.17 Probability of symbol error in presence of first-order PMD using a single- polarization receiver, assuming τ = 21 ps, a| = |b| = 0.5, Nu = 52, Nc = 64, Npre = 1 and R = 26.7 GHz...... 28 Fig. 2.18 Impact of nonlinearity on the transmission of BPSK: (a) without nonlinearity, (b) with nonlinearity...... 30 Fig. 2.19 Phase noise variance for OFDM and SC transmissions with 100% dispersion compensation per span: a) for 20 spans, as a function of launched power, and b) as function of the number of spans...... 32 Fig. 2.20 Phase noise variance for OFDM and SC transmissions for 20 spans, as a function of launched power: a) 90% dispersion compensation per span and b) 0% dispersion compensation per span...... 33

xiv List of Figures

Fig. 2.21 Model for single-drive MZ modulator, corresponding to one phase of a quadrature MZ modulator...... 38 Fig. 2.22 Quadrature MZ modulator...... 38 Fig. 2.23 OFDM transmitter using quadrature MZ modulator...... 39 Fig. 2.24 Pre-distortion transfer characteristic for quadrature MZ modulator...... 41 Fig. 2.25 OFDM transmitter including hard clipping, pre-distortion and electrode frequency response compensation...... 41 Fig. 2.26 Electrical and optical MZ waveforms. The solid and dotted lines represent the waveforms with and without hard clipping, respectively...... 42 Fig. 2.27 MZ electrode frequency response...... 43

Fig. 2.28 Optical power efficiency for Nc = 64, Nu = 52 and R = 29.66 GHz...... 44 −4 Fig. 2.29 Receiver sensitivity penalty at PS = 10 , Nc = 64, Nu = 52 and R = 29.66 GHz...... 44 Fig. 2.30 Optical power efficiency for different number of subcarriers...... 45 Fig. 2.31 Receiver sensitivity penalty for different number of subcarriers...... 46

Fig. 2.32 Optical power efficiency for different number of bits in the D/A, Nc = 64, Nu = 52 and R = 29.66 GHz...... 47

Fig. 2.33 Receiver sensitivity penalty for different number of bits in the D/A, Nc = 64, Nu = 52 and R = 29.66 GHz...... 47 Fig. 2.34 Model for dual-drive MZ modulator...... 49 Fig. 2.35 Dual-drive MZ plane...... 49 Fig. 2.36 OFDM transmitter using the dual-drive MZ with hard-clipping, pre-distortion and electrode frequency response compensation...... 50 Fig. 2.37 OFDM span regions for a fixed clipping level. The dotted and solid regions correspond to the quadrature MZ and the dual-drive MZ, respectively...... 50 Fig. 2.38 Real component of the dual-drive MZ output electric field...... 51 Fig. 2.39 Output electric field of the dual-drive MZ for the cases of no trajectory optimization and with trajectory optimization before the super-Gaussian filter. The green and blue lines represent the output electric field and the correct OFDM waveform, respectively...... 52

Fig. 2.40 Dual-drive MZ receiver sensitivity penalty with trajectory optimization for Ms = 2.5,

Nu = 52 and R = 29.66 GHz...... 53

Fig. 2.41 Dual-drive MZ optical power efficiency with trajectory optimization for Ms = 2.5, Nu = 52 and R = 29.66 GHz...... 54 Fig. 2.42 Dual-drive and quadrature MZ receiver sensitivity penalties as a function of the oversampling ratio with no frequency loss compensation and with trajectory optimization for

CR = 2.5, drive voltages range between ±Vπ and R = 29.66 GHz...... 54 Fig. 3.1 Metro network diagram...... 58 Fig. 3.2 Metro link with an optical filter...... 59 Fig. 3.3 Intensity modulation (IM) by direct modulation of the laser current...... 59 Fig. 3.4 Achievable bit rates as a function of SNR for various values of the gap Γ...... 60 Fig. 3.5 Optimal power allocation for OFDM...... 62 Fig. 3.6 OFDM system model for metro links...... 63

xv List of Figures

Fig. 3.7 Block diagram of a DC-OFDM transmitter...... 64 Fig. 3.8 Equivalent transfer function for DC-OFDM for R = 20 Gbit/s, L = 100 km and D = 17 ps/nm/km, corresponding to γ = 0.87...... 66 Fig. 3.9 Block diagram of an ACO-OFDM transmitter...... 67 Fig. 3.10 Block diagram of an SSB-OFDM transmitter...... 68 Fig. 3.11 PSD of the detected signal in SSB-OFDM...... 69 Fig. 3.12 PSD of the different modulation formats...... 72

Fig. 3.13 Flow chart of DC-OFDM and SSB-OFDM optimization using the bias ratio (BR). . 74 Fig. 3.14 Normalized optical SNRs required for the different OFDM formats for γ = 0.25 to −3 achieve Pb = 10 . The number of used subcarriers, Nu, is indicated for each curve...... 75 Fig. 3.15 Subcarrier power distribution for DC-OFDM, ACO-OFDM, SSB-OFDM and to −3 achieve a Pb = 10 for γ = 0.25. The number of used subcarriers is 416, 416 and 208 for DC-

OFDM (BR = 1.1), ACO-OFDM and SSB-OFDM (BR = 1.0), respectively...... 76 −3 Fig. 3.16 Normalized optical SNR required for DC-OFDM to achieve Pb = 10 for γ = 0.25.

The number of used subcarriers is Nu = 832...... 77 Fig. 3.17 Minimum normalized optical SNR required for various dispersion indexes γ for the different OFDM formats...... 78 −3 Fig. 3.18 OSNR values (over 0.1 nm) required to obtain Pb = 10 at 10.7 Gbit/s for OFDM and for OOK with MLSD [3]...... 79 −3 Fig. 3.19 OSNR values (over 0.1 nm) required to obtain Pb = 10 at 10.7 Gbit/s for OFDM and for OOK with MLSD [3]. In this case, the OFDM signal occupies the full channel bandwidth (B0 = 35 GHz)...... 80 Fig. 4.1 LAN block diagram...... 84 Fig. 4.2 LAN link...... 84 Fig. 4.3 OFDM system model for LANs...... 85 Fig. 4.4 (a) Mode power distribution, (b) mode delays and (c) frequency response for fiber 1183. We consider a 1-km length in computing the delays and frequency response...... 87 Fig. 4.5 3-dB bandwidth distribution of the multimode fibers simulated. All fibers have 1-km length...... 88 Fig. 4.6 Rms delay spread (D) distribution of the multimode fibers simulated. All fibers have 1-km length...... 90 Fig. 4.7 Block diagram of a PAM-OFDM transmitter...... 92 −4 Fig. 4.8 Receiver electrical SNR required to obtain 10 Gbit/s at Pb = 10 for ACO-OFDM and OOK for fibers with 1-km length. The bit allocation granularity is β = 0.25 and ACO-OFDM OFDM has the same symbol rate as OOK ( Rs = Rs = 10 GHz)...... 96 Fig. 4.9 Optical spectra of M-PAM and OFDM. The symbol rate for OFDM is twice that for OFDM M-PAM, Rs = 2Rs...... 97 −4 Fig. 4.10 Receiver electrical SNR required to achieve Pb = 10 at 10 Gbit/s for different modulations formats in fibers with 1-km length. The bit allocation granularity is β = 0.25. The OFDM symbol rate for all OFDM formats is twice that for OOK, Rs = 2Rs = 20 GHz...... 98

xvi List of Figures

−4 Fig. 4.11 Receiver electrical SNR required for DC-OFDM to achieve Pb = 10 at 10 Gbit/s for different values of the bias ratio in fiber 295...... 99 Fig. 4.12 Receiver electrical SNR required for various OFDM formats with continuous and −4 discrete bit allocations to achieve Pb = 10 at 10 Gbit/s for fibers of 1-km length. The symbol OFDM rate for all OFDM formats is Rs = 20 GHz...... 100 Fig. 4.13 Subcarrier power distribution for ACO-OFDM with continuous bit allocation and −4 with discrete bit loading (with granularity β = 0.25) for 10 Gbit/s at Pb = 10 in fiber 10. The OFDM symbol rate is Rs = 20 GHz...... 101 Fig. 4.14 Cumulative distribution function (CDF) of the required receiver electrical SNR to −4 obtain Pb = 10 at 10 Gbit/s for different modulation formats. The symbol rate for OOK is Rs = 10 GHz and the symbol rate for OFDM is the same or twice that for OOK, as indicated in the figure...... 102 −4 Fig. 4.15 Receiver electrical SNR required to obtain 20 Gbit/s at Pb = 10 for the different modulations formats for fibers of 1-km length. The bit allocation granularity is β = 0.25 and the symbol rate for 4-PAM is Rs = 10 GHz. The symbol rate for ACO-OFDM is the same or twice that for 4-PAM, as indicated in the figure...... 102 Fig. 4.16 Cumulative distribution function (CDF) of the required receiver electrical SNR to −4 obtain 20 Gbit/s at Pb = 10 for different modulation formats. The symbol rate for 4-PAM is

Rs = 10 GHz and the symbol rate for ACO-OFDM is the same or twice that for 4-PAM, as indicated in the figure...... 103 Fig. 5.1 Indoor optical wireless transmission...... 107 Fig. 5.2 OFDM system model for LANs...... 108 Fig. 5.3 Impulse response of an exemplary non-directional, non-LOS (diffuse) channel. This channel has no LOS component h(0)(t). The contributions of the first five reflections, h(1)(t),…, h(5)(t), are shown...... 110 −4 Fig. 5.4 Electrical SNR required to achieve Pb = 10 vs. normalized delay spread DT at bit rates of 50, 100 and 300 Mbit/s (spectral efficiency of 1 bit/symbol) for ACO-OFDM and OOK. The bit allocation granularity is β = 0.25 and the symbol rates for ACO-OFDM are the same as those for OOK, as indicated...... 114 −4 Fig. 5.5 Electrical SNR required to achieve Pb = 10 vs. normalized delay spread DT at bit rates of 50, 100 and 300 Mbit/s (spectral efficiency of 1 bit/symbol) for different modulations formats. The dashed lines correspond to the SNR requirement predicted using the ceiling- bounce (CB) model. The bit allocation granularity is β = 0.25 and the symbol rates for ACO- OFDM are twice those for OOK, as indicated...... 117

Fig. 5.6 Cumulative distribution function (CDF) of the electrical SNR required to achieve Pb = 10−4 at bit rates of 50, 100 and 300 Mbit/s (spectral efficiency of 1 bit/symbol) for different modulation formats. The symbol rates for OFDM are the same as or twice those for OOK, as indicated...... 119 −4 Fig. 5.7 Electrical SNR required to achieve Pb = 10 vs. normalized delay spread DT at bit rates of 100, 200 and 600 Mbit/s (spectral efficiency of 2 bit/symbol) for different modulations formats. The bit allocation granularity is β = 0.25 and the symbol rates for ACO-OFDM are the same as or twice those for 4-PAM, as indicated...... 120

xvii List of Figures

Fig. 5.8 Cumulative distribution function (CDF) of the electrical SNR required to achieve Pb = 10−4 at bit rates of 100, 200 and 600 Mbit/s (spectral efficiency of 2 bit/symbol) for different modulation formats. The symbol rates for OFDM are the same as or twice those for 4-PAM, as indicated...... 120 Fig. 5.9 Outage probability for OOK and ACO-OFDM with coding and various bit allocations averaged over all channels. All modulation formats use the code RS(127,107) over GF(8). The information bit rate is 300 Mbit/s (spectral efficiency of 1 bit/symbol) and the symbol rates for OOK and OFDM are 356 MHz and 600 MHz, respectively. The bit allocation granularity is β = 1, i.e., integer bit allocation...... 124 Fig. A-1 DWDM system with 50 GHz channel spacing ...... 134

xviii Introduction

1 INTRODUCTION

Research in optical communication systems has received ever-increasing interest in recent years. The available spectrum in fiber is being rapidly populated with continued growth of internet traffic driven by bandwidth-hungry applications, such as video and music sharing, so there is an increasing demand to increase spectral efficiency transmission while maintaining high SNR efficiency, i.e., minimizing the required average transmitted energy per bit. A major contributing factor in the development of new optical systems is the advance in very large scale integration (VLSI) technology, which enables digital signal processing-based receivers at GHz clock speeds. When the outputs of an analog optoelectronic downconverter are sampled at Nyquist rate, the digitized waveform retains the full information of the received optical electric field, enabling compensation of transmission impairments by a digital signal processor (DSP). A digital receiver is highly advantageous because adaptive algorithms can be used to compensate time-varying transmission impairments. Forward error-correction codes with soft-decision decoding can also be implemented. Moreover, digitized signals can be delayed, split and amplified without degradation in signal quality. Digital receivers are already widely used in wireless and DSL systems that operate at comparatively lower data rates.

Optical networks can be divided into four classes: wide area networks (WANs), metropolitan area networks (MANs), local area networks (LANs) and personal area networks (PANs). There are also data center networks (DCNs) which are very similar to LANs. There have been several experiments showing the increased performance of using single-carrier modulation with digital equalization for the different optical applications [1,2,3]. However, with the advances in digital receivers for optical systems, one can now employ now more advanced modulations formats, such as orthogonal frequency division multiplexing (OFDM). OFDM is a multi-carrier modulation format that divides the total bandwidth into N orthogonal slots called subcarriers, as shown in Fig. 1.1. Among all multicarrier decompositions, OFDM with

1 Introduction cyclic prefix is practical and converges to the optimum performance [4]. Furthermore, the power of the OFDM subcarriers can be optimized according to the application.

Single-Carrier OFDM

f f Bandwidth Bandwidth

Time Time

1 t t -1

Ts TOFDM Fig. 1.1 OFDM and single-carrier waveforms.

OFDM is currently used in wireless and DSL systems for its optimized performance in frequency-selective channels. Hence, it is important to know if OFDM can achieve higher performance than single-carrier with equalization for the next generation optical systems.

This thesis focuses on the analysis and optimization of the OFDM performance for the different optical applications. The requirements and constrains of the different optical applications are summarized in Table 1.1.

WAN MAN LAN PAN Applications (>1000 km) (50 – 500 km) (0.3 – 5 km) (10 m) Goal 100-1000 Gbit/s 10-100 Gbit/s 10-100 Gbit/s 300 Mbit/s Single-Mode Single-Mode Multimode Medium Free Space Fiber Fiber Fiber Electric Field Electric Field Modulation Intensity Intensity (Complex) or Intensity Direct Direct Detection Coherent Direct Detection Detection Detection Method (Electric field) (Intensity) (Intensity) (Intensity) Dominant Chromatic/Polar. Chromatic Multimode Multipath Dispersion (Linear) (Nonlinear) (Linear) (Linear)

2 Introduction

Ambient Dominant Amplifier Amplifier Thermal Light Noise (AWGN) (Chi-squared χ2) (AWGN) (Shot noise) Noise Type Signal-Indep. Signal-Dep. Signal-Indep. Signal-Indep. DC-OFDM OFDM DC-OFDM DC-OFDM OFDM ACO-OFDM Formats ACO-OFDM ACO-OFDM SSB-OFDM Other Fiber Clipping Clipping Clipping Noise Effects Nonlinearity Noise Noise Table 1.1 Characteristics of the different optical applications.

Long-haul systems (i.e., WANs) typically use single-mode fiber (SMF) and the dominant linear impairments are group-velocity dispersion, i.e., different frequencies travel at different speeds, and polarization-mode dispersion, i.e., different polarizations arrive at the receiver with different delays (Chapter 2). The different fiber types are summarized in Table 1.2. Next generation WANs will employ coherent detection, i.e., detect the phase and amplitude of the optical electric field. In WANs, the optical amplifiers have sufficient high gain so that its amplified spontaneous emission (ASE) is dominant over thermal and shot noises. The ASE noise is modeled as complex additive white Gaussian noise (AWGN). When the input power is high, fiber nonlinear effects become dominant and degrade the system performance.

Multimode Fiber (MMF) Single-Mode Fiber (SMF) Large core (62.5 μm) Small core (9 μm) Supports many propagation modes Supports only a pair of degenerate modes

Refractive index changes with Modal dispersion: different modes o o frequency (GVD) travel with different propagation Fiber asymmetry can cause constants o polarization-mode dispersion (PMD) o Large delay spreads o Small delay spreads Low bandwidth-distance product High bandwidth-distance product (5 Gbit/s·km) (200 Tbit/s·km)

3 Introduction

Used in local area networks (LANs) Used in wide area networks (WANs) Table 1.2 Characteristics of the different optical applications.

Metro systems (MANs) also use SMF but normally use direct-detection at the receiver, i.e., detect the fiber instantaneous power (or intensity). When direct-detection is used, the transmitted waveforms have to be non-negative since only the intensity is detected (Chapter 3). However, the transmitter in metro systems can either modulate the electric field or the intensity. The amplifier ASE noise is no longer Gaussian due to the nonlinear photodetection process and becomes Chi-squared χ2 distributed. Furthermore, in the detection process, the amplifier noise interacts with the received signal and becomes signal-dependent.

LANs typically use multi-mode fiber (MMF) and employ intensity modulation with direct detection. The main impairment is modal dispersion, i.e., different modes travel with different propagation constants (Chapter 4). Optical amplifiers are not used in LANs and the dominant noises are thermal noise or shot noise from the receiver. The DCNs also use MMF over short distances as LANs and typically operate at 10 Gbit/s.

Indoor optical wireless systems (i.e. PANs) use free-space (i.e., air) and also employ intensity modulation with direct detection. Usually, optical wireless systems transmit either in the infrared or visible-light spectra. The main impairment is multipath distortion, i.e., different reflections arrive at the receiver with different delays (Chapter 5). The dominant noises are ambient light (shot noise) or thermal noise from the receiver.

This thesis is organized as: in Chapter 2, we review OFDM modulation and the major types of impairments in long-haul systems using single-mode fiber and coherent detection. When only linear effects are present, we derive analytical expressions for the inter-symbol interference, and inter-carrier interference incurred when an insufficient cyclic prefix is used. We use these expressions to compute power penalties for representative examples, comparing the results to simulations. We also present the required number of subcarrier and cyclic prefix length for different fiber lengths.

4 Introduction

Afterwards, we include fiber nonlinear effects in our analysis and compare the performance of OFDM to single-carrier systems. We also present two feasible optical modulators to generate an optical OFDM signal. We show how to minimize the nonlinear distortion of the modulators for best performance. We also compute the required number of bits for the digital-to-analog (D/A) converters and sampling frequencies for the two optical modulators.

In Chapter 3, we review the fundamentals of metro systems. We also review methods for power and bit allocation for multicarrier systems and describe the optimal water-filling solution. Furthermore, we introduce the different OFDM formats that can be used with direct-detection and derive equivalent linear channel models for each one, assuming only GVD is present in the SMF. In addition, we discuss the effects of amplifier noise in direct-detection systems and present how to optimize the different OFDM formats in metro systems. We conclude the chapter by comparing the optical power required to transmit at a given bit rate for the different optimized OFDM formats and for OOK with MLSD, making use of previously published results for the latter.

In Chapter 4, we introduce the multimode fiber model and review different OFDM formats that can be used with intensity modulation. We then compare the receiver electrical SNR required to transmit at a given bit rate for the different OFDM formats and for unipolar PAM with MMSE-DFE equalization at different spectral efficiencies when the transmitter has channel state information (CSI).

In Chapter 5, we review the indoor optical wireless model for infrared or visible light communication. We use the same OFDM formats as in chapter 4. We first assume that the transmitter has CSI and compare the receiver electrical SNR required to transmit at a given bit rate for the different OFDM formats and for unipolar PAM with MMSE-DFE equalization at different spectral efficiencies. We then compare the receiver electrical SNR for the different modulation formats with error correcting codes when there is no transmitter CSI. We conclude the chapter by comparing the computation complexity of the different modulation formats.

5 Introduction

In Chapter 6, we give the concluding remarks of this thesis and discuss directions for future work.

6 Wide Area Networks

2 WIDE AREA NETWORKS

2.1 Introduction

Orthogonal frequency-division multiplexing (OFDM) is a multi-carrier modulation that has been extensively investigated and deployed in wireless and wireline communications [5], [6]. It is receiving increased interest in the fiber-optic research community for its robustness against inter-symbol interference (ISI), since the symbol period of each subcarrier can be made long compared to the delay spread caused by group-velocity dispersion (GVD) and polarization-mode dispersion (PMD) [7], [8]. Although single-carrier and multi-carrier systems using coherent detection have fundamentally the same power and spectral efficiencies for a given modulation format in the presence of unitary impairments such as GVD and PMD, there may be differences due to practical constraints.

Experiments have been performed demonstrating the potential of OFDM with coherent detection in optical systems [8], [9], but very little closed-form performance analysis has been performed to date. In principle, the impulse response due to GVD has infinite duration, so the proper choice of cyclic prefix length is not obvious. If the cyclic prefix is too short relative to the impulse response duration, ISI and inter-carrier interference (ICI) will occur. On the other hand, if the cyclic prefix is excessively long relative to the number of subcarriers, the sampling rate must be increased significantly, and a large fraction of the transmitted energy is wasted in cyclic prefix samples, leading to a substantial power penalty. For a given cyclic prefix length, these penalties can be reduced to an arbitrary degree by increasing the number of subcarriers.

In practice, however, it may be desirable to minimize the number of subcarriers employed. For example, it is known that the peak-to-average power (PAR) ratio is proportional to Nc, the number of subcarriers [5]. OFDM signals are modulated using Mach-Zehnder (MZ) modulators having nonlinear, peak-limited transfer

7 Wide Area Networks

characteristics [10], [11], so minimizing Nc will help maximize optical power efficiency in the modulator. As another example, laser phase noise destroys the orthogonality between subcarriers, causing ICI. It has been shown that for a given laser linewidth, the variance of ICI is proportional to Nc [12], so minimizing Nc can help in combating laser phase noise. Hence, in this chapter, assuming coherent detection, we find the minimum number of subcarriers and cyclic prefix length that achieve low power and sampling penalties in the presence of GVD and PMD.

In this chapter, we also study the MZ modulator. Since the MZ has a nonlinear, peak-limited transfer characteristic, the high peaks of the OFDM signal can degrade the system performance. A conventional solution to the PAR problem is to reduce the operating range in the MZ to accommodate the OFDM peak. However, this solution results in a significant power efficiency penalty, which may require the use of optical amplification at the transmitter to boost the signal level. An alternative solution would be using peak-reduction algorithms studied for wireless systems [5] but all these algorithms present a computational burden to the transmitter [13], which might be prohibitive at the speeds of optical systems. In this chapter, we present hard clipping with pre-distortion as a simple and effective approach to combat the nonlinearity in the quadrature MZ and increase the optical power efficiency. In addition, we study the combined effects of having a finite number of bits in the D/A and the MZ nonlinearity. We then extend our study to the dual-drive MZ for generating an optical OFDM signal since it requires less hardware than the quadrature MZ.

This chapter is organized as follows. In section 2.2, we review the fundamentals of wide area networks (WANs) and multi-carrier systems. We introduce the canonical OFDM model and determine the minimum oversampling ratio required to avoid aliasing. In section 2.3, we focus on GVD, and derive analytical expressions for the ISI and ICI incurred when an insufficient cyclic prefix is used. We use these expressions to compute power penalties for representative examples, comparing the results to simulations. We compare the performance of OFDM to single-carrier systems. In section 2.4, we consider first-order PMD, and discuss how to extend the analysis to arbitrary-order PMD. We consider both single- and dual-polarization

8 Wide Area Networks receivers. In section 2.5, we present the fiber nonlinearity model and compare the performance of OFDM to single-carrier systems when both linear and nonlinear effects are present. We calculate the computational complexities of the different modulation formats in section 2.6. In section 2.7, we review PAR fundamentals in multi-carrier systems and the quadrature drive MZ as an optical modulator. In addition, we introduce the MZ canonical model including the electrode frequency response. We then focus on the quadrature MZ, and study through simulations the optical power efficiency (OPE) and system performance gains when pre-distortion and hard clipping are used. Moreover, we consider quantization noise from the D/A and study through simulations the minimum number of bits required and the optimum clipping level. Afterwards, we introduce the dual-drive MZ as a simpler option for the OFDM modulator. We then analyze the oversampling required to achieve low performance degradation in the dual-drive MZ.

2.2 WAN and OFDM Review

2.2.1 WANs

Modern long-haul optical systems (or WANs) use single-mode fiber (SMF) and operate around 1.55 μm since it is the region with the lowest fiber attenuation. The total available bandwidth at 1.55 μm is around 8 THz (C-band). The total available bandwidth is partitioned into several channels, typically with 100 or 50 GHz bandwidth, and is referred as dense wavelength-division multiplexing (DWDM). Fig. 2.1 shows a block diagram of a DWDM system with 50 GHz channel spacing.

9 Wide Area Networks

λ1 TX 1 RX 1 λ2 TX 2 SMF RX 2 … … AWG AWG

λN −1 TX N−1 RX N−1 λN TX N RX N

50 GHz …

λ λ λ λ λ 1 2 N −1 N Fig. 2.1 DWDM system with 50 GHz channel spacing.

Long-haul systems transmit over long distances, often well beyond 1000 km. The long-haul link is divided into spans, typically 80 to 100 km long. An optical amplifier is placed at the end of each span to compensate the SMF loss. Many long- haul systems use inline optical dispersion compensation. In such systems, a dispersion-compensating fiber (DCF) is also used on each span to cancel some of the dispersion introduced by the SMF. The DCF usually compensates between 90% and 98% to the SMF dispersion, depending on the dispersion map. A second amplifier is placed after the DCF in order to compensate the additional attenuation introduced by the DCF. Fig. 2.2 shows a block diagram of a long-haul link.

SMF DCF SMF DCF

TX RX

Span 1 Span NA Fig. 2.2 Long-haul link with NA spans.

Traditional optical systems used binary formats such as on-off keying (OOK) for each DWDM channel. In OOK, the transmitter turns the laser ON and OFF to represent bits ‘1’ and ‘0’ as shown in Fig. 2.3.

10 Wide Area Networks

TX RX 0 101011 0

Laser Electrical Fiber LPF

Decision

Encoder 0 1 0 1 0 1 1 0 Bits 0 1 0 1 0 1 1 0

Fig. 2.3 OOK modulation in traditional optical systems.

Given practical constraints on filters for DWDM, the binary formats using direct-detection (DD) can only achieve spectral efficiencies around 0.8 b/s/Hz per polarization; over the usable fiber bandwidth of about 10 THz, this corresponds to an aggregate capacity of about 8 Tbit/s [14]. Until recently, this was considered much larger than that required by real world applications. However, with the continued growth of internet traffic driven by video and music sharing, the available spectrum in fiber is being rapidly populated, so there is renewed interest in high spectral efficiency transmission. The most promising detection technique for achieving high spectral and power (SNR) efficiency is coherent detection with polarization multiplexing. Coherent detection with polarization multiplexing detects the amplitude and phase of the optical electrical field in the two fiber polarizations. Hence, higher spectral efficiencies can be achieved with coherent detection since the information can encoded in all the available degrees of freedom. Coherent systems use lasers with very low linewidths (e.g. < 300 KHz) to alleviate carrier phase and frequency recovery at the receiver.

2.2.2 OFDM Review

In OFDM, the inverse discrete Fourier transform (IDFT) and DFT are used to modulate and demodulate the data constellations on the subcarriers, as shown in Fig. 2.4. The IDFT and DFT replace the banks of analog I/Q modulators and demodulators that would otherwise be required. To show the equivalence between OFDM and analog multi-carrier systems, we can write the OFDM signal as

11 Wide Area Networks

Nc −1 2π dtqfj , ofdm ()= ∑∑ ,kq ()− sym ekTtbXtx (2.1) k q=0 where Xq,k denotes the qth subcarrier constellation symbol transmitted on the kth

OFDM symbol, b(t) is a pulse shape, Tsym = (Nc + Npre + Nwin)Tc = Tofdm + Tpre + Twin is the OFDM symbol period and fd = 1/(NcTc) is the frequency separation between subcarriers. We define Tc as the sample period, and Nc, Npre and Nwin are integers.

hchannel(t) = p(t) × hfiber(t) × v(t)

p(t) v(t) X I'(t) Y X0,k 0,k I(t) 0,k x Pulse Anti- D/A A/D Remove X1,k Add Prefix Shaping Aliasing h (t) Prefix H (0) (N ) . fiber Coherent . eq pre (N ) . . + P/S laser MZ Fiber O/E Down- S/P pre . Y1,k X1,k . . + x . . Windowing conversion . IDFT . . Windowing DFT . (Nwin) Pulse Anti- H (1) . . D/A A/D (Nwin) . eq Shaping LO Aliasing . Q(t) Q'(t) . XN-2,k p(t) v(t) Y X x (t) = I(t)+jQ(t) N-1,k N-1,k XN-1,k ofdm x

Heq(N-1) Fig. 2.4 Digital implementation of OFDM transmitter and receiver.

If Eq. (2.1) is sampled every sample period, we have

2πqn Nc −1 j Nc . ofdm ()nTx c = ∑∑ , ()ckq − sym ekTnTbX (2.2) k q=0

An OFDM symbol corresponds to Nc + Npre + Nwin samples, as shown in Fig.

2.5. The block of Nc samples in Eq. (2.2) corresponds to the IDFT. The remaining Npre and Nwin terms are a periodic extension of the OFDM signal known as the cyclic prefix, and pulse shaping known as windowing. The cyclic prefix is used so that the sequence of received samples in one symbol is equivalent to one period of a circular convolution between the transmitted OFDM symbol xofdm(t) and the sample-rate samples of the channel impulse response hchannel(t). In the frequency domain, this corresponds to the multiplication of subcarrier q by the corresponding sample of the channel frequency response Hchannel(ω). Thus, a single-tap equalizer on subcarrier q can be used to invert any amplitude and phase distortion introduced by the channel,

−1 i.e., eq = channel π NqfHqH cs )/2()( . For our system, the channel frequency response

Hchannel(ω) is given by

H channel ω = ω fiber ω VHP ω)()()()( , (2.3)

12 Wide Area Networks

where P(ω) and V(ω) are the pulse-shaping and anti-aliasing filters and Hfiber(ω) represents the fiber frequency response. The cyclic prefix samples (Npre = Npos + Nneg) are appended to the signal after the IDFT. Ideally, the cyclic prefix length should be no smaller than the duration of the channel impulse response hchannel(t).

sym = ofdm + pre + TTTT win

N win Npos Nc Nneg

… ■ □ ● ○ … ■ □ ● ○ …

Effective Tx time (Tofdm) Fig. 2.5 OFDM symbol, including the cyclic prefix and windowing.

The additional Nwin samples are used to control the OFDM spectrum by introducing additional pulse shaping. Common choices for the window functions include rectangular and raised-cosine pulses [5], [6]. The rectangular window corresponds to Nwin = 0. Assuming a rectangular window and uncorrelated transmitted * symbols, E[xn,k xl,m ] = 0 for n ≠ l, the power spectrum of xofdm(t) is given by

T N c −1 ⎛ T ⎞ S ω = sym P sinc 2 ⎜ sym − 2πω qf ⎟ , xx () 2 ∑ q ⎜ ()d ⎟ (2.4) Nc q=0 ⎝ 2π ⎠ 2 where Pq = E[|Xq| ] is the average power per symbol on sub-carrier q. A plot of Eq. (2.4) is shown in Fig. 2.6, assuming all subchannels have equal powers.

2πfd Sxx()ω

…

2π ω T sym ⎛ 2 ⎞ BW ⎜()12 fN dc +−π≈ω ⎟ Tsym ⎝ ⎠ Fig. 2.6 OFDM spectrum, assuming rectangular windowing (Nwin = 0).

We note in Fig. 2.6 that the OFDM spectrum resembles an ideal rectangular shape and has a bandwidth approximately given by

13 Wide Area Networks

⎛ 2 ⎞ πω ⎜ )1(2 fN +−≈ ⎟ π +≤ )1(2 fN , BW ⎜ dc ⎟ dc (2.5) ⎝ Tsym ⎠ To achieve a desired symbol rate R, the frequency separation between subcarriers (fd=1/NcTc) must be equal to fd = (Nc+Npre)/Nc×R/Nc. Thus, Eq. (2.5) can be rewritten as

N c + )1( c + NN pre )( BW ≈ 2πω R . (2.6) N c N c In the case of the rectangular window, the OFDM spectrum has significant sidelobes, as shown in Fig. 2.6. These can be reduced by using alternate windows, such as the raised-cosine, which require Nwin > 0. Alternately, the sidelobes can be reduced by using a pulse-shaping filter p(t) after the digital- to-analog converter (D/A).

The symbols extensions required for the cyclic prefix (Npre) and windowing

(Nwin) represent penalties, since they do not carry useful information and are discarded at the receiver. They represent a power penalty because a portion of the energy is wasted on these samples and also a sampling penalty because the sample rate 1/Tc, which is equal to the sampling frequency, has to be higher to maintain the desired symbol rate R, i.e., fs = 1/Tc = R×(Nc+Npre+Nwin)/Nc.

For typical values used in OFDM systems, the OFDM bandwidth given by

Eq. (2.6) is very close to the Nyquist bandwidth, i.e., ωBW ≈ ωNy= 2π×R. The confined spectrum of OFDM is a practical advantage over single-carrier systems where, the

90% power bandwidth is typically of order ωsc ≈ π × 22 R [15]. Another important advantage of OFDM is in the minimum required oversampling ratio. In OFDM, oversampling is performed by not modulating the edge subcarriers in Fig. 2.4, i.e., by inserting zero subcarriers. Assuming that only Nu of the Nc subcarriers are modulated, the oversampling ratio is Ms = Nc/Nu. In OFDM, it is possible to employ arbitrary rational oversampling ratios, unlike single-carrier transmission, where this may require complex signal processing. Since the OFDM spectrum falls off more rapidly than a single-carrier spectrum, lower oversampling ratios can be employed. We have found

14 Wide Area Networks that when the pulse-shaping and anti-aliasing filters are chosen properly, an oversampling ratio Ms = 1.2 is sufficient to avoid aliasing, while single-carrier systems typically require an oversampling ratio Ms = 1.5 or 2 [16]. Finally, we note that the expressions defined previously are also valid when oversampling is used, provided that Npre and Nwin are scaled to reflect the oversampling ratio, i.e., Npre(Ms) =

Ms×Npre(Ms = 1) and Nwin(Ms) = Ms×Nwin(Ms = 1) and the sampling frequency is scaled by the oversampling ratio, i.e., fs = 1/Tc = Ms×R×(Nc+ Npre(Ms)+ Nwin(Ms))/Nc.

2.3 Group-Velocity Dispersion

2.3.1 Theory

The refractive index of the fiber changes with frequency so the different frequencies of a pulse propagate at different speeds. This effect is called group- velocity dispersion (GVD). Fiber GVD spreads the transmitted symbols, causing ISI that degrades the error probability, as shown in Fig. 2.7.

Input

SMF Output

t t Fig. 2.7 Pulse spreading due to fiber GVD.

The fiber frequency response in the presence of GVD is given by

β − ω 2 2 Lj 2 (2.7) H fiber ω)( = e , where β2 is the fiber GVD parameter and L is the fiber length. Although the receiver employs a single-tap equalizer on each subcarrier to invert the channel, ISI can occur when the cyclic prefix is insufficient, so that symbols in a neighboring block overlap with the symbol of interest. In order to avoid ISI, ideally, the cyclic prefix duration

Tpre should no smaller than the duration of the impulse response hchannel(t). However, GVD leads to an infinite-duration impulse response, and the tails of the impulse response not covered by the cyclic prefix lead to residual ISI, as shown in Fig. 2.8.

15 Wide Area Networks

hchannel(t) … … t … … Npos Nneg Npos Nneg Symbol k Symbol k+1 Fig. 2.8 OFDM symbol with insufficient cyclic prefix.

Referring to Fig. 2.8, note that ISI on symbol k comes both from symbols k−1 and k+1, since the channel impulse response is two-sided. If Npre = Npos + Nneg samples are used for the cyclic prefix and if the channel impulse response has positive and negative lengths Lp−1 and Ln−1, respectively, the residual ISI on the nth time- domain sample in the kth OFDM symbol interval can be written as

Lp −1 k = ∑ k−1()( prec pos −+++ channel rhrNNNnxnISI )() Nr += 1 pos . (2.8) Nneg −− 1 + ∑ k+1 ( negc −−− channel uhuNNnx )() Lu n +−= 1 In Appendix A, we show that if the transmitted symbols are uncorrelated and if

Nc > max(Lp−Npos−1, Ln−Nneg−1), after the DFT, the ISI variance on subcarrier q is given by

Lp−1 Nneg −− 1 ⎛ 2 ⎞ 2 q)( = σσ 2 ⎜ qH )( + qH )( 2 ⎟ , ISI s ⎜ p ∑∑ n ⎟ (2.9) ⎝ Np pos += 1 Ln n +−= 1 ⎠ 2 2 where σs = E[|x(n)| ] is the mean power per sample in the time-domain waveform, and Hp(q) and Hn(q) are the Nc-point DFTs of the positive and negative tails of the channel impulse response, respectively. They can be written as

2πvq L p −1 − j Nc p )( = ∑ channel )( evhqH = pv . 2πvq (2.10) c 1−− nN − j Nc n )( = ∑ channel )( evhqH −= LNv nc Moreover, since the linear convolution with the channel can no longer be considered as one period of a circular convolution, inter-carrier interference (ICI) will occur within the kth symbol. In Appendix A, we show that the ICI has the same

16 Wide Area Networks variance as the ISI. Thus, the variance of the total interference on subcarrier q is given by

L −1 N −− 1 ⎛ p 2 neg ⎞ σ 2 q = 2)( σ 2 ⎜ qH )( + qH )( 2 ⎟ . + ICIISI s ⎜ p ∑∑ n ⎟ (2.11) ⎝ Np pos += 1 Ln n +−= 1 ⎠ In order to compute the probability of symbol error, we must also take into account the power penalty of the cyclic prefix. If we assume that only Nu of the Nc subcarriers are used, the probability of symbol error for 4-QAM-modulated subcarriers can be written as

Nu −1 ⎛ ⎛ ⎞ P ⎞ 1 ⎜ 1 Nu ⎛ q ⎞ ⎟ P = erfc ⎜ ⎟⎜ ⎟ , (2.12) s N ∑ ⎜ 2 ⎜ + NN ⎟⎜ +σσ 22 q)( ⎟ ⎟ u q=0 ⎝ ⎝ upre ⎠⎝ 0 +ICIISI ⎠ ⎠ where Nu/(Nu+Npre) is the extra energy wasted on the cyclic prefix samples, Pq is the 2 average power per symbol of subcarrier q, σ0 is the variance of sampled additive 2 2 white Gaussian noise (σ0 = N0R), and σ ISI+ICI(q) is the variance of the total interference on subcarrier q.

2.3.2 Simulation Results

We assume that fiber nonlinearity and laser phase noise effects are either negligible or have been compensated, so the fiber may be modeled as a linear channel. Therefore, the only impairments in the system are GVD and PMD. In order to minimize the sampling penalty, at the transmitter, we use a rectangular window, and perform pulse shaping using a fifth-order Butterworth lowpass filter having a 3-dB cutoff frequency equal to half the bandwidth of the OFDM signal, given by Eq. (2.6). At the receiver, the anti-aliasing filter is an identical Butterworth lowpass filter. We assume transmission of 53.4 Gbit/s in one polarization. The subcarriers are modulated using 4-QAM and a FEC code with an overhead of 255/239, so the symbol rate is R = 26.7 GHz.

As an initial example, we consider a fiber length of 80 km with a dispersion D

= 17 ps/(nm⋅km). The FFT size is 64 and the oversampling ratio is Ms = 1.2, i.e. only

52 subcarriers are used. The minimum required oversampling ratio of Ms = 1.2 was

17 Wide Area Networks determined by adding zero subcarriers until noise aliasing became negligible [5]. The cyclic prefix length, referred to an oversampling ratio Ms = 1, is 5 samples. A plot of the symbol-error probability for this example is shown in Fig. 2.9.

-1 10 Ideal Cyclic Prefix

-2 Cyclic Prefix + GVD 10 Simulation

s -3

P 10

-4 10

-5 10 6 7 8 9 10 11 12 13 14 15 16 E /N (dB) s o Fig. 2.9 Probability of symbol error in presence of GVD for L = 80 km, D = 17 ps/(nm.km), Nc = 64, Nu = 52, Npre = 5 and R = 26.7 GHz.OFDM symbol with insufficient cyclic prefix.

In Fig. 2.9, it can be observed that the power penalty increases at small Ps. −3 Since the majority of the FEC codes have a threshold around Ps = 10 , we will −4 measure the power penalty at Ps = 10 in order to have some margin.

Fig. 2.10 and 2.11 illustrate the ISI + ICI penalty and the overall penalty, respectively, as a function of the cyclic prefix length, holding the data rate constant.

18 Wide Area Networks

L = 80 km 7 Analytical 6 Simulation

5

4

3

2 ISI +ISI ICI PowerPenalty (dB) 1

1 3 5 7 9 11 13 15 17 19 21 23 25 27 Prefix Length (N ) pre −4 Fig. 2.10 ISI + ICI power penalty at PS = 10 for L = 80 km, D = 17 ps/(nm.km), Nc = 64, Nu = 52 and R = 26.7 GHz.

In Fig. 2.11, we observe that the overall power penalty has two different regions: ISI + ICI-dominated and cyclic prefix- dominated. In the former, the cyclic prefix is much shorter than the fiber impulse response and therefore severe ISI + ICI occur, impairing the system performance

L = 80 km 8 Analytical 7 Simulation ISI dominated 6

5

4

3 Prefix dominated 2 Overall Power Penalty (dB) Penalty Power Overall

1

0 1 3 5 7 9 11 13 15 17 19 21 23 25 27 Prefix Length (N ) pre −4 Fig. 2.11 Overall power penalty at PS = 10 for L = 80 km, D =17 ps/(nm.km), Nc = 64, Nu = 52 and R = 26.7 GHz.

In the latter region, the cyclic prefix is sufficiently long that the ISI + ICI are negligible, but a large fraction of the transmitted energy is wasted on the cyclic prefix

19 Wide Area Networks samples, leading to a power penalty. For example, for the same scenario as in Fig. 2.11, if one chose a cyclic prefix length such that 98% or 95% of the fiber’s impulse response energy was contained in that same duration, one would be required to use a cyclic prefix length of 14 and 12 samples, respectively, while the optimum cyclic prefix is 9 samples. The optimum cyclic prefix length results in a penalty from ISI+ICI equal to the penalty from energy wasted in the cyclic prefix samples.

As explained above, it may be desirable to minimize the number of subcarriers employed. Hence, we would like to calculate the minimum combination of number of subcarriers and cyclic prefix that generate a specified power penalty. For concreteness, we consider a system transmitting at a symbol rate R = 26.7 GHz through fiber spans having dispersion D = 17 ps/(nm⋅km), with 98% inline optical dispersion compensation (i.e., the residual dispersion is 0.34 ps/(nm⋅km)). Fig. 2.12 shows the overall power penalties for fiber lengths between 1000 km and 5000 km.

(a) L = 1000 km, 2% Uncompensated (b) L = 2000 km, 2% Uncompensated 3 6 N = 12 N = 26 u u N = 26 N = 52 2.5 u 5 u N = 52 N =104 u u N =104 N =208 2 u 4 u N =208 u 1.5 3

1 2 Overall Power Penalty (dB) Penalty Power Overall Overall Power Penalty (dB) Penalty Power Overall 0.5 1

0 0 1 2 3 4 5 6 1 2 3 4 5 6 7 8 Prefix Length (N ) Prefix Length (N ) pre pre (c) L = 3000 km, 2% Uncompensated (d) L = 4000 km, 2% Uncompensated 7 8 N = 26 N = 52 u u 7 6 N = 52 N =104 u u N =104 N =208 u 6 u 5 N =208 u 5 4 4 3 3 2 2 Overall Power Penalty (dB) Penalty Power Overall Overall Power Penalty (dB) Penalty Power Overall

1 1

0 0 1 2 3 4 5 6 7 8 9 10 11 12 1 2 3 4 5 6 7 8 9 10 11 12 Prefix Length (N ) Prefix Length (Npre) pre

20 Wide Area Networks

(e) L = 5000 km, 2% Uncompensated 10 N = 52 9 u N =104 u 8 N =208 u 7

6

5

4

3

Overall Power Penalty (dB) Penalty Power Overall 2

1

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 Prefix Length (Npre) Fig. 2.12 Overall power penalties at R = 26.7 GHz for 2% undercompensated GVD lengths of a) 1000 km b) 2000 km c) 3000 km d) 4000 km and e) 5000 km. The dashed lines are the analytical values and the solid lines are the simulations results with insufficient cyclic prefix.

In Figs. 2.10, 2.11 and 2.12, we observe some discrepancies between the simulation and theoretical results. Eqs. (2.11) and (2.12) are exact given that there is no correlation between the OFDM samples. This condition is satisfied only if the transmitted symbols are uncorrelated and if the oversampling ratio Ms is equal to 1, i.e., no oversampling. Since an oversampling ratio Ms = 1.2 is required to avoid aliasing (with the Butterworth filters we have used), Eqs. (2.11) and (2.12) are only an approximation of the ICI + ICI variance, as explained in Appendix A. The error is maximum when the cyclic prefix length is short since it corresponds to the situation of highest ISI+ICI. However, we verified that for high ISI/ICI, Eq. (2.11) predicts the ISI

+ ICI variance with a maximum error on the order of 10-15% when Ms = 1.2.

In Fig. 2.12, we can also observe that in some cases increasing the cyclic prefix length also increases or maintains the power penalty. This is because increasing the cyclic prefix length while keeping the data rate constant requires increasing the sample rate which, in turn, increases the OFDM bandwidth, increasing the temporal spread caused by GVD. On the other hand, a longer cyclic prefix can compensate longer tails of the impulse response. It is not obvious which effect will dominate. When the additional pulse spreading is longer than the increased cyclic prefix length, the overall power penalty increases.

21 Wide Area Networks

From Fig. 2.12, we can extract the minimum number of subcarriers and cyclic prefix required to achieve a desired power penalty. The values required to achieve a penalty of 1 dB are given in Table 2.1 .

Residual Modulated Total Length DFT Size Cyclic Prefix Dispersion Subcarriers (km) Nc Npre D·L.(ps/nm) Nu 1000 340 32 26 3 2000 680 64 52 4 3000 1020 64 52 6 4000 1360 128 104 6 5000 1700 128 104 9 Table 2.1 Number of subcarriers and cyclic prefix required to achieve an overall power penalty less than 1 dB. We consider 4-QAM subcarriers, R = 26.7 GHz, D = 17 ps/(nm·km) with 98% inline optical dispersion compensation.

Note that by making other choices of these parameters, it is possible to make the total penalty arbitrarily small. As the uncompensated dispersion becomes larger, this may involve choosing a very large number of subcarriers, which may be undesirable, for reasons cited above. In choosing these parameters, one should note that some synchronization schemes used in practice require that the cyclic prefix length exceeds a certain fraction of the total number of subcarriers [17], [18]. The results presented in Table 2.1 are compatible with those synchronization schemes.

Finally, in Fig. 2.13 we plot the optimum number subcarriers and cyclic prefix length as a function of the fiber length when no optical dispersion compensation is used.

22 Wide Area Networks

450 N = 208 400 u N = 416 u 350 N = 832 u )

pre 300 N = 1664 u 250

200

150 Prefix Length (N Length Prefix 100

50

200 600 1000 1400 1800 2200 2600 3000 Length (km) Fig. 2.13 Optimum number of used subcarriers and cyclic prefix length as a function of the fiber length for R = 26.7 GHz when no optical dispersion compensation is used.

2.4 Polarization-Mode Dispersion

2.4.1 Theory

Fiber asymmetry (birefringence) introduces coupling between the two degenerate modes in the SMF. Furthermore, due to random nature of the fiber birefringence, the two degenerate modes arrive at the receiver with different delays [19]. The delay τ between the degenerate modes is random and is Maxwellian distributed [19]. This impairment is called polarization-mode dispersion (PMD) and is illustrated in Fig. 2.14.

X Pol.

t Y Pol. τ

X τ t

Z t Electrical Y LPF

23 Wide Area Networks

Fig. 2.14 Polarization-mode dispersion (PMD.

The average delay τ between the degenerate modes is given by √, where Dp is the PMD parameter and L is the fiber length. A single-mode fiber with PMD can be described by a frequency-dependent Jones matrix [19,20,21,22]. Input and output signals can be described by two-component Jones vectors. The input and T T output time-domain signals are denoted by x(t)=[x1(t) x2(t)] and y(t)=[y1(t) y2(t)] , respectively. Their Fourier transforms can be related as

⎡Y1 ω)( ⎤ ⎡X1 ω)( ⎤ ⎢ ⎥ = U ω)( ⎢ ⎥ , (2.13) ⎣Y2 ω)( ⎦ ⎣X 2 ω)( ⎦ where U(ω) is a frequency-dependent matrix. There is an analytical expression for U(ω) only for first-order PMD, i.e., the delay τ is independent of frequency. For first- order PMD, the matrix U(ω) is given by

−1 2Λ= ωω )()( RRU 1 , (2.14) where R1 and R2 are frequency-independent rotation matrices representing a change of basis into the input and output principal states of polarization (PSPs), respectively, and Λ is a frequency-dependent matrix representing the delay between the fast and slow

PSPs [19]. Explicit expressions for Ri and Λ [21], [22] are

⎡ − rr * ⎤ ⎡e jωωτ2 0 ⎤ 1i 2i , Ri = ⎢ * ⎥ Λ = ⎢ − jωωτ2 ⎥ (2.15) ⎣ rr 12 ii ⎦ ⎣ 0 e ⎦ where

r = coscos − j sinsin εθεθ 1i ii ii . (2.16) r2i = cossin ii + j sincos εθεθ ii

The θi, εi are independent random variables representing the fast PSP azimuth and ellipticity angles, respectively. τ is a random variable representing differential group delay (DGD). Considering GVD and PMD, the fiber frequency response is then

β H (ω) − jω2 2 L ⎡ 2 ⎤ 2 fiber(H ω) = ⎢ ⎥ = ω)( eU , (2.17) ⎣H1(ω) ⎦

24 Wide Area Networks

We will now discuss two approaches to reception in the presence of PMD. The first approach uses a dual-polarization receiver and the second approach uses a single- polarization receiver with polarization control.

2.4.2 Dual-Polarization Receiver

A dual-polarization receiver enables electronic polarization control and the use of polarization-multiplexed signals which can double the bit rate. An OFDM polarization-multiplexed system is shown in Fig. 2.15.

I1(t) I'1(t) D/A PBS A/D OFDM MZ Fiber OFDM Tx 1 A/D Rx 1 D/A xofdm,1(t) Q' (t) Q1(t) Dual 1 Polarization Coherent I2(t) Receiver I'2(t) D/A OFDM A/D OFDM MZ Tx 2 A/D Rx 2 D/A xofdm,2(t) LO Q (t) Q'2(t) 2 Fig. 2.15 OFDM polarization-multiplexed system.

The transmitter consists of two independent OFDM modulators. The two modulated optical signals are combined in orthogonal polarizations using a polarization beam splitter (PBS). After passing through single-mode fiber, the received signal is split into two copies, which are mixed with the LO in two orthogonal polarizations. Each polarization is detected using a 90° hybrid and two balanced photodetectors. The electrical outputs are the in-phase and quadrature components associated with the two orthogonal polarizations. These signals are then low-pass filtered and sampled [16].

The dual-polarization receiver can also be used with a single-polarization transmitter. In this case, one of the transmitters of Fig. 2.15 would not be used.

In order to compensate for PMD and GVD, the receiver must invert the fiber frequency response. The equalizer corresponds then to a 2×2 matrix multiplication between the signals in the two polarizations on each subcarrier q after the DFT operation. For first-order PMD, it can be written as

25 Wide Area Networks

2 ⎛ 2πfs ⎞ β2 2 ⎛ 2πf q ⎞ + j⎜ ⎟ Lq −1 ⎜ s ⎟ 1 −− 1 ⎝ Nc ⎠ 2 . (2.18) eq )( = HqH fiber ⎜ ⎟ = R1 Λ 2 eR ⎝ Nc ⎠ In writing down Eq. (2.18), for simplicity, we have not included the pulse- shaping and anti-aliasing filters. Fig. 2.16 shows the combined equalizer for PMD and GVD.

2 2πf ⎛ 2πf ⎞ β j s τ +− jq ⎜ s ⎟ 2 Lq2 e 2N ⎝ N ⎠ 2

2 2πf ⎛ 2 f ⎞ βπ j s τ ++ jq ⎜ s ⎟ 2 Lq2 e 2N ⎝ N ⎠ 2 Fig. 2.16 Combined PMD and GVD equalizer for an OFDM polarization-multiplexed system.

In the absence of polarization-dependent losses, using a dual-polarization receiver, PMD causes no loss of information, since PMD is a unitary transformation. In order to avoid ISI, the cyclic prefix length should be no smaller than the duration of the channel impulse response. For example, to compensate first-order PMD only (in the absence of GVD), the cyclic prefix duration should be no shorter than the DGD τ. 1/2 A typical PMD parameter is Dp = 0.1 ps/km and therefore, for a fiber length of 5000 km, the mean DGD is E[τ] ≈ 7 ps. Assuming the symbol rate per polarization is R =

26.7 GHz and assuming 4-QAM-modulated subcarriers, the sample period is about Tc ≈ 35 ps for 128 subcarriers (Table 2.1). Assuming it is necessary to compensate a DGD of five times the mean in order to achieve low outage probability, the maximum

DGD is τmax = 5 E[τ] ≈ 35 ps. A cyclic prefix length of only one sample would be sufficient. In practice, the overall power penalty would be dominated by GVD, as it requires a cyclic prefix length of several samples.

An exact analytical form for the Jones matrices describing higher-order PMD has not yet been developed [21], [22]. However, without loss of generality, the configuration in Fig. 2.16 could be used to mitigate any order of PMD, provided that the cyclic prefix is chosen to be sufficiently long. If the complete statistics were

26 Wide Area Networks known for higher-order PMD, Eq. (2.11) could be used to estimate the ISI and ICI from the tails of the impulse response not covered by the cyclic prefix, and therefore a design choice could be done for a desired outage probability. However, we believe that using a cyclic prefix somewhat longer than five times the mean DGD should be sufficient to combat almost the entire duration of the impulse response of higher-order PMD [21].

2.4.3 Single-Polarization Receiver

A single-polarization receiver was already shown in Fig. 2.4. We assume that polarization control is used such that the LO polarization is locked to the polarization at the carrier frequency. Since only one polarization is detected, the frequency response corresponds to one of the rows of the Jones matrix given by Eq. (2.14). It can be written as

jωτ 2/ − jωτ 2/ H PMD ω)( += beae , (2.19) where a and b are complex-value constants from the matrices R1 and R2. As one can observe, Eq. (2.19) is very similar to multi-path propagation in wireless system. The probability of symbol error for 4-QAM-modulated subcarriers can be written as

⎛ ⎞ Nu −1 ⎛ ⎞ 1 ⎜ 1 ⎛ N ⎞⎛ Pq ⎞ 1 ⎟ P = erfc ⎜ u ⎟⎜ ⎟⎜ ⎟ , s ∑ ⎜ ⎜ ⎟⎜ 2 ⎟⎜ 2 ⎟ ⎟ (2.20) N u q=0 ⎜ 2 pre + NN u ⎝ σ 0 ⎠ π NqfH )/2( ⎟ ⎝ ⎝ ⎠ ⎝ PMD cc ⎠ ⎠ where Nu/(Nu+Npre) is the extra energy wasted on the cyclic prefix samples, Pq is the 2 average power per symbol of each subcarrier, σ0 is the AWGN variance and HPMD(ω) is the frequency response given by Eq. (2.19). We note again that the cyclic prefix length should be equal to the DGD to avoid ISI as can be seen from the impulse response corresponding to Eq. (2.19), i.e., PMD = δ( +τ 2)( ) + δ()tbtath −τ 2 . However, we point out that while the cyclic prefix can be made long enough to avoid ISI up to a desired outage probability, there is SNR degradation in the single-polarization receiver due to the frequency-dependent attenuation of the channel.

Fig. 2.17 shows the probability of symbol error for transmission at a symbol rate R = 26.7 GHz using 4-QAM-modulated subcarriers

27 Wide Area Networks

-1 10 Ideal Cyclic Prefix Cyclic Prefix + PMD -2 10 Simulation

-3 s

P 10

-4 10

-5 10 6 7 8 9 10 11 12 13 14 15 16 E /N (dB) s o Fig. 2.17 Probability of symbol error in presence of first-order PMD using a single-polarization receiver, assuming τ = 21 ps, a| = |b| = 0.5, Nu = 52, Nc = 64, Npre = 1 and R = 26.7 GHz.

The DGD is τ = 21 ps, i.e., 3 times the mean of a link with 5000 km with Dp = 0.1 ps/km1/2. As a worst case, equal power splitting between the PSPs (|a| = |b| = 0.5) is assumed. The error probability computed using Eq. (2.20) is in good agreement with the simulation results.

2.5 Fiber Nonlinearity

2.5.1 Theory

Nonlinear impairments in fiber arise from the Kerr effect, where the refractive index of the transmission medium changes with intensity of signal (i.e., the square magnitude of the applied electric field). Although the Kerr effect is extremely small in silica, the confinement of light in the core of single mode fiber is such that even at realistic transmit powers, nonlinear effects can be problematic in long haul systems. Nonlinearity is ultimately a limiting factor to the theoretical capacity of optical fiber.

In single-polarization transmission where PMD is negligible, signal propagation is described by a scalar nonlinear Schrödinger equation (NLSE) along the reference polarization of the fiber

28 Wide Area Networks

∂E ∂ 2 Ej αβ + 2 =+ γ || 2 EEjE , (2.21) ∂z ∂t 2 22

where (),tzE is the electric field, α is the attenuation coefficient, β 2 is the dispersion parameter, γ is the nonlinear coefficient, and z and t are the propagation direction and time, respectively.

Nonlinear effects have deterministic and statistical components. The nonlinearity experienced by a signal due to its own intensity is known as self-phase modulation (SPM). In WDM systems, a signal also suffers from nonlinear effects due to neighboring channels. These are known as cross-phase modulation (XPM) and four- wave-mixing (FWM) [14], and their impact can be reduced by using non-zero dispersion fiber which causes “pulse walkoff” [23]. In the absence of ASE noise, all nonlinear effects are deterministic, so it is possible in principle to pre-compensate them at the transmitter [14]. It is also possible to perform multi-channel detection at the receiver but the system complexity can be prohibitive.

In amplified long-haul systems, interaction between amplifier noise (i.e., ASE noise) and signal through the Kerr nonlinearity leads to nonlinear phase noise (NLPN) [24]. The phase noise arising from the interaction of ASE noise with the channel of interest is called SPM-induced NLPN, while the phase noise arising from the interaction of ASE noise with the signals of neighboring channels is called XPM- induced NLPN [14]. Since the nonlinear interactions between signal and noise are non-deterministic, they cannot be perfectly compensated. Fig. 2.18 shows the interaction between ASE noise and signal through the fiber nonlinearity when there is no fiber dispersion.

Im Im a) b)

ФNL Re Re

29 Wide Area Networks

Fig. 2.18 Impact of nonlinearity on the transmission of BPSK: (a) without nonlinearity, (b) with nonlinearity.

In Fig. 2.18, we observe that the received signal in the presence of nonlinearity has a spiral-shaped constellation. The spiraling arises because points that are further away from the origin (i.e., have a higher instantaneous intensity) experience greater nonlinear phase shift as a result of the Kerr effect.

2.5.2 Simulation Results

Since ASE noise and signal interact through the fiber nonlinearity, the received nonlinear noise is no longer Gaussian distributed and therefore the conventional SNR metric cannot be used to evaluate the system performance. A good performance metric for single-carrier systems when fiber nonlinear effects are present is the phase noise variance, i.e., var()φ NL [14].

When only linear effects are present in the fiber, the phase noise variance is 1 approximately equal to var()φ ≈ for high SNR. For multi-carrier systems, the NL 2⋅SNR geometric SNR was shown to be good predictor of the performance as a whole [6]. Having these figures in mind, we propose the geometric phase noise variance to quantify the OFDM performance in the presence of nonlinearity. The geometric phase noise variance is given by

/1 Nu ⎡ Nu ⎤ var()φ OFDM = var ()φ n , (2.22) NL ⎢∏ NL ⎥ ⎣ n=1 ⎦ where Nu is the number of used subcarriers.

In this study, we consider a single-wavelength system transmitting over multiple spans with inline optical amplification and dispersion compensation. We consider a polarization-multiplexed QPSK system transmitting at 100 Gbit/s; after inclusion of forward error-correction (FEC) coding with 7% overhead, the symbol rate is R =26.7 Gsym/s. The FEC limit is at a bit-error ratio BER = 2×10-3, corresponding 2 to a phase noise variance of 0.08 rad for QPSK systems. For OFDM transmission, we

30 Wide Area Networks optimize the number of subcarriers and cyclic prefix for each dispersion map according to section 2.3.

The total amplifier gain is equally distributed between the two amplifiers and is set to exactly compensate the loss incurred in the SMF and DCF. Each amplifier has a spontaneous emission factor nsp = 1.41. The parameter values for the SMF and DCF are typical of terrestrial links: LSMF = 80 km, αSMF = 0.25 dB/km, DSMF = 17 ps/nm·km, −1 −1 γSMF = 1.2 W /km, αDCF = 0.6 dB/km, DDCF = −85 ps/nm·km and γDCF = 5.3 W /km. The amount of optical dispersion compensation for each dispersion map is determined by the length of the DCF. At the receiver, DSP-based equalization is implemented for residual GVD compensation after passing through a 5th order Butterworth lowpass filter. In single-carrier transmission, the received signal is oversampled by a factor of 2, and is processed by a MMSE linear equalizer. In OFDM transmission, the received signal is oversampled by a factor close to Ms ≈ 1.2., and is processed by a single-tap equalizer on each subcarrier.

We evaluate system performance by numerical solution of the nonlinear Schrödinger equation with random noise sources, and we quantify system performance by phase noise variance. We ignore laser phase noise and PMD. For simplicity, we perform simulations using single-polarization transmitters and receivers, modeling propagation in only one polarization. While we neglect the nonlinearity induced by the orthogonal polarization, it should affect single-carrier and OFDM in a similar fashion, so including it should not change the conclusions. Considering only one polarization, in single-carrier systems, nonlinear impairments include intra-channel four-wave mixing (IFWM) or nonlinear phase noise or a combination of the two, depending on the dispersion map. In OFDM systems, in the absence of GVD, the impact of nonlinearity can be understood as four-wave mixing (FWM) between the different subcarriers, i.e., intermodulation between the subcarriers. In order to compare the performance of single-carrier and OFDM, we use for OFDM the geometric phase noise variance given by Eq. (2.22).

Fig. 2.19 a) shows the phase variance comparison for the case of 100% dispersion compensation per span for 20 spans of propagation.

31 Wide Area Networks

0 0 10 10 2

(a) 2 (b) OFDM -1 10 FEC limit

-1 OFDM SC (4-QAM) -2 10 FEC limit 10 SC (4-QAM)

-3 Linear 10 Phase noise variance (rad )

Nonlinear Phase noise variance (rad ) -2 -4 10 10 -10 -8 -6 -4 -2 0 2 4 0 10 20 30 40 50 60 Power (dBm) Number of Spans Fig. 2.19 Phase noise variance for OFDM and SC transmissions with 100% dispersion compensation per span: a) for 20 spans, as a function of launched power, and b) as function of the number of spans.

In the linear regime where signal powers are low, system performance is ASE noise-limited for both modulation formats and OFDM performs slightly worse than single-carrier. This is due to the power penalty incurred in the cyclic prefix [25]. However, this penalty can be made arbitrarily small, so the performance becomes equal to that of single-carrier, by increasing the number of subcarriers. In the high- signal-power regime, where system performance is nonlinearity-limited, we observe that single-carrier has significant lower phase noise variance than OFDM. A lower phase noise variance means that the system can achieve longer distances. Fig. 2.19 b) shows the minimum phase variance comparison for the case of 100% dispersion compensation per span as a function of the number of spans. We observe that single- carrier has always a smaller phase variance than OFDM, and single-carrier can achieve 52 spans before reaching the FEC threshold, while OFDM is only able to achieve 33 spans.

The low performance of OFDM for the case of 100% dispersion compensation is because of the high number of FWM products. For 100% dispersion compensation, the accumulate phase mismatch between the subcarriers is small and therefore the FWM efficiency is high [14]. In order to reduce the FWM efficiency, we can deliberately introduce phase mismatch between the subcarriers by, for example, not fully compensating the SMF dispersion after each span.

32 Wide Area Networks

Fig. 2.20 a) and b) compare OFDM and single-carrier for 20 spans of propagation for the cases of 90% and 0% dispersion compensation per span, respectively.

0 0 10 10 2 (a) 2 (b)

OFDM

-1 -1 10 FEC limit 10 FEC limit SC (4-QAM) OFDM Phase noise variance (rad ) Phase noise) variance (rad SC (4-QAM) -2 -2 10 10 -10 -8 -6 -4 -2 0 2 4 -10 -8 -6 -4 -2 0 2 4 Power (dBm) Power (dBm) Fig. 2.20 Phase noise variance for OFDM and SC transmissions for 20 spans, as a function of launched power: a) 90% dispersion compensation per span and b) 0% dispersion compensation per span.

We note that for the case of 0% compensation, there is no DCF and there is only one amplifier per span. The small performance difference between OFDM and SC in the linear regime is again attributed to the cyclic prefix penalty, as described above. However, in the nonlinearity-limited regime, the performance gap between OFDM and SC for 90% and 0% is smaller than that of 100% dispersion compensation per span. This can be understood by the incoherent additions and partial cancelations of the FWM products from one span to another in the presence of residual dispersion in each span. Furthermore, we observe that for 0% compensation per span, OFDM and single-carrier have very similar performance in the nonlinear regime.

2.6 Computational Complexity

For OFDM, the IDFT and DFT operations are performed efficiently using a fast Fourier transform (FFT) algorithm. An FFT of size N requires 4·N·log2(N) real operations (multiplications plus additions) [26]. Specifically, the number of additions is 8/3·N·log2(N) and the number of multiplications is 4/3·N·log2(N). Thus, the number of real operations required per second for the transmitter is 4·N·log2(N)/TOFDM, where

TOFDM is the OFDM symbol period, which is given in [25]. For the OFDM receiver, we need to take into account the complex single-tap equalizer on each used subcarrier.

33 Wide Area Networks

Assuming that complex multiplications are implemented with the usual 4 real multiplications and 2 real additions) [26], the number of real operations required per second for the receiver is (4·N·log2(N) + 6·Nu)/TOFDM, where Nu is the number of used subcarriers. The overall complexity order in real operations per bit for OFDM is

Tx OFDM = 2 b /)(log4)( TTNNNO OFDM )(log4 RTMNN OFDM . = 2 sbs ν ⋅+ MN s (2.23) Rx OFDM = [ 2 + ] bu /6)(log4)( TTNNNNO OFDM []+ 6)(log4 RTMNNN OFDM = 2 sbsu ν ⋅+ MN s For single-carrier (M-QAM), the complexity of directly implementing the convolution operation for the feedforward MMSE equalizer is 2Ntaps /Ts complex operations (multiplications plus additions) per second, where Ntaps is the number of taps of the feedforward filter and Ts is the symbol period (Ts = 1/Rs = Tb·log2M). The equalizer computes an output once per symbol interval. The overall complexity order in real operations per bit for single-carrier (M-QAM) in a direct implementation is

D QAM [ taps += 44)( ] /TTNNNO sbtaps

= /8 TTN sbtaps . (2.24)

= Ntaps log/8 2 M For a large number of taps, linear equalization is more efficiently implemented using an FFT-based block processing such as overlap-add or overlap-save [4].

Suppose a block length of LB is chosen. An FFT-based implementation has a complexity of (Ntaps + LB −1)·(4×2·log2(Ntaps + LB −1) + 6) real operations per block

LB for an equalizer with Ntaps taps. For a given Ntaps, there exists an optimum block length LB that minimizes the number of operations required. We assumed that the equalizer taps come from a lookup table since the SMF impulse response typically changes slowly and therefore the same equalizer taps can be used for many symbols. The overall complexity order in real operations per bit for M-QAM with a MMSE feedforward equalizer in a FFT implementation is

34 Wide Area Networks

FFT ( + LN Btaps − ⋅ ⋅ 2 (log(8)1 + LN Btaps − + 6))1 QAM NO )( = . (2.25) f log2 MB We compute the number of operations required per bit for single-carrier and OFDM in Table 2.2 for same system parameters as in section 2.3, i.e., 4-QAM subcarriers, R = 26.7 GHz, D = 17 ps/(nm·km) with 98% inline optical dispersion compensation. For single-carrier, we calculate the complexity of direct implementation and FFT based implementation using an optimized block size LB subject to having a FFT size that is an integer power of two. In Table 2.2, we observe that an OFDM system requires two digital signal processors (DSPs) with approximately 40% less computational complexity than the single DSP required for single-carrier. On the other hand, an OFDM system requires an A/D and D/A while single-carrier requires only one A/D.

Single-Carrier OFDM Transmission Distance FFT Direct Block Size Transmitter Receiver (km) Complexity (LB) 1,000 48.0 6 53.3 24.6 30.6 2,000 96.0 27 66.4 29.5 35.5 3,000 128.0 25 71.6 29.5 35.5 5,000 208.0 52 78.8 34.5 40.4 Table 2.2 Computational complexity in real operations per bit for the various modulation formats. We consider 4-QAM subcarriers, R = 26.7 GHz, D = 17 ps/(nm·km) with 98% inline optical dispersion compensation.

2.7 Optical Modulator

Multi-carrier signals like OFDM have a high peak-to-average ratio (PAR), which is proportional to the number of used subcarriers [5]. Optical OFDM is modulated using Mach-Zehnder (MZ) modulators having nonlinear, peak-limited transfer characteristics. Experiments have been performed demonstrating the potential of OFDM with coherent detection in optical systems [27,9,10]. Some previous work has investigated the possibility of pre-distortion and clipping to mitigate the effects of MZ nonlinearity on OFDM system performance [10], [28]. Our study extends that previous work by considering the frequency-dependent MZ electrode losses, by

35 Wide Area Networks quantifying optical power efficiency, and by introducing the dual-drive MZ as an alternative option for the OFDM modulator.

There are several options to generate an optical OFDM signal. A popular technique shifts the electrical OFDM signal to an intermediate frequency (IF) and then uses it to drive a single MZ modulator [27,9,10]. Another option is to use a quadrature MZ. A quadrature MZ modulates directly the baseband electrical OFDM signal to the optical domain without the need of an IF. Hence, this technique, sometimes called direct conversion [10], reduces the required electrical bandwidth by a factor of two, at the expense of a more complicated modulator design.

A conventional solution to the PAR problem is to reduce the operating range in the MZ to accommodate the OFDM peak. However, this solution results in a significant power efficiency penalty, which may require the use of optical amplification at the transmitter to boost the signal level. An alternative solution would be using peak-reduction algorithms studied for wireless systems [5]. All of these peak- reduction algorithms, however, lead to undesired effects, such as increased coding overhead and/or increased average transmitted power [29,30,31,13]. An increased coding overhead requires an increased sampling rate in order to maintain the desired bit rate, exacerbating the impact of GVD and PMD, and requiring faster D/A converters. An increased average transmitted power can lead to increased nonlinear impairment; in [11], it was shown that the variance of phase noise caused by four- wave mixing is proportional to the power in each subcarrier, but is not simply related to the instantaneous peak power. Furthermore, all of the peak-reduction algorithms present a computational burden to the transmitter [13], which might be prohibitive at the speeds of optical systems.

In this section, we present hard clipping with pre-distortion as a simple and effective approach to combat the nonlinearity in the quadrature MZ and increase the optical power efficiency. Since the OFDM peaks occur with a very low probability, clipping can be an effective technique. In addition, we study the combined effects of having a finite number of bits in the D/A and the MZ nonlinearity. We then extend our

36 Wide Area Networks study to the dual-drive MZ, which proves to require a higher oversampling ratio to achieve the same performance as the quadrature MZ.

2.7.1 PAR and MZ Review

2.7.1.1 Peak-to-Average Power Ratio

The PAR is defined as [5]

tx |)(|max 2 PAR = t , (2.26) []txE )( 2 where tx |)(|max 2 is the peak value squared of the signal x(t) and E[x(t)2] is the t average signal power. The peak of the signal determines the dynamic range required of the D/As and modulators in the circuit. Thus, it is desirable to have signals with low PAR. In the case of OFDM, i.e., identical constellations on all of the subcarriers, the PAR can be written as [5], [13]

2 X k ||max k (2.27) PAR = Nu 2 , []XE k

2 th 2 where X k ||max is the largest symbol magnitude squared on the k subcarrier, E[Xk ] k is the average power per symbol on subcarrier k and Nu is the number of used subcarriers. In particular, if all the subcarriers are modulated using QPSK, Eq. (2.27) becomes

PAR = Nu . (2.28)

2.7.1.2 Mach-Zehnder Modulator

If complementary drive signals are used, the transfer characteristic of a single- drive MZ modulator is [14]

out tE )( ⎛ π m tV )( ⎞ = sin⎜ ⎟ , (2.29) in tE )( ⎝ 2Vπ ⎠ where Eout(t) and Ein(t) are the output and input electric fields, respectively, Vm(t) is the electrical modulator signal and Vπ is the voltage that must be applied to the single

37 Wide Area Networks electrode to produce a differential phase shift of π between the two waveguides. The modulator electrode has frequency-dependent loss, so that high-frequency components of the drive signal are attenuated. The electrode frequency response can be modeled by [14]

tV )( − −α + π LfdjLf )2)(exp(1 m = int int12 , (2.30) d tV )( α int − 2)( π LfdjLf int12 where Vd is the input drive voltage, Vm is the modulating voltage of the MZ transfer characteristic, α(f) is the frequency-dependent loss, d12 is the velocity mismatch difference between the optical and electrical waveguides and Lint is the interaction length. The model for single-drive MZ modulator is shown in Fig. 2.21.

Fig. 2.21 Model for single-drive MZ modulator, corresponding to one phase of a quadrature MZ modulator.

Two single-drive MZ modulators can be combined to create a quadrature MZ. The quadrature MZ comprises two single-drive MZs, whose output optical fields are added in quadrature. The quadrature MZ is useful to modulate a complex baseband signal directly to the optical domain without the need of shifting the signal to an IF. Fig. 2.22 shows the block diagram of a quadrature MZ.

Quadrature MZ

vd,r(t) I Fiber MZ Laser MZ 90 Q

vd,i(t) Fig. 2.22 Quadrature MZ modulator.

38 Wide Area Networks

For the remainder of the chapter, in block diagrams, we use single lines to represent electrical signals and double lines to represent optical signals.

2.7.2 Quadrature MZ Optimization

2.7.2.1 Quadrature MZ Optimization

Fig. 2.23 shows an optical OFDM modulator using a quadrature MZ.

Quadrature MZ Re{ } Parallel D/A v (t) X1 d,r -to- I Fiber Serial MZ … … Laser IDFT + MZ 90 N Cyclic Q Prefix vd,i(t) XN D/A Im{ } Fig. 2.23 OFDM transmitter using quadrature MZ modulator.

In Fig. 2.23, the transmitted symbols are modulated using the inverse discrete Fourier transform (IDFT) and the cyclic prefix is added to the signal. After D/A conversion, the real and imaginary components are used to drive the MZ modulators. Then, one of the MZ outputs is delayed by 90o in order to generate the optical in-phase (I) and quadrature (Q) components. These are then summed and injected into the fiber.

We note that in Fig. 2.23 not all the Nc subcarriers are used to transmit data. Some of the subcarriers are used for oversampling [25]. The oversampling ratio is defined as Ms = Nc/Nu, where Nc is the IDFT size and Nu is the number of used subcarriers [25]. Furthermore, the used subcarriers are properly positioned within the DFT block such that the resulting OFDM spectrum is centered and, typically, the D.C. subcarrier is not used for data transmission.

The MZ modulators in Fig. 2.23 can introduce two impairments: low optical power efficiency and intermodulation products between subcarriers. In order to quantify the first impairment, we define OPE as

Power Modulated Average Modulated Power OPE = , (2.31) Pow erLaser CW PowerLaser

39 Wide Area Networks

A backoff in the MZ operating range is required to accommodate the peak of any drive signal. Since the OFDM signals have a high PAR, the OPE can be very low.

For example, if Nc and Ms = 1.2, i.e., 52 used subcarriers, the OPE is less than 1% (Fig. 2.28).

The second impairment results from the MZ nonlinear transfer characteristic. Expanding Eq. (2.29) into a Taylor series, we get

3 out tE )( ⎛ m tV )( ⎞ m tV )( ⎛ πππ m tV )( ⎞ = sin⎜ ⎟ −≈ ⎜ ⎟ +… (2.32) in tE )( ⎝ 2Vπ ⎠ 2Vπ ⎝ 2Vπ ⎠ The cubic term in Eq. (2.32) will generate in-band intermodulation products between the subcarriers, which cannot be removed by filtering.

In order to mitigate the previous impairments, we will use pre-distortion to compensate the MZ nonlinearity and hard clipping to increase the OPE. The clipping operation is described as

⎧− if, )( −< AnxA ⎪ in nV )( = ⎨ if),( )( <<− AnxAnx (2.33) ⎪ ⎩ if, )( > AnxA where Vin(n) is the clipped signal at sample time n, and A is the clipping level. The clipping operation is done separately on each of the real and imaginary components of the OFDM signal. Furthermore, we will use a super-Gaussian filter (SGF) at the MZ output in order to eliminate any out of band leakage generated by hard clipping (in a practical WDM system, this filtering function would be performed by the multiplexer). The pre-distortion function is given by

2V ⎛ nV )( ⎞ nV )( = π arcsin⎜ in ⎟ , out ⎜ ⎟ (2.34) π in nV |)(|max ⎝ n ⎠ where Vin(n) and Vout(n) are the digital input and output voltages of the pre-distortion device. Note that in |)(|max = AnV because of the clipping operation and such that the n output signal Vout(n) swing is always between −Vπ and +Vπ due to the mapping of the pre-distortion curve. The pre-distortion transfer characteristic is shown in Fig. 2.24.

40 Wide Area Networks

V π

(t) 0 out V

-V π -1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1 V (t) in Fig. 2.24 Pre-distortion transfer characteristic for quadrature MZ modulator.

In addition to the pre-distortion curve, we need to compensate the electrode frequency response of each MZ. We note that if the electrode frequency response is compensated, then Vm(n) = Vout(n). The optimized OFDM transmitter is shown in Fig. 2.25 and the corresponding waveforms are shown in Fig. 2.26.

Clipping Pre-Distortion Equalization V (n) −1 −1 d,r X1 Hard Clipper sin ( ) Helec.( ) Vd,r(t) Re{ } Vin,r(n) Vout,r(n) CL Vπ D/A

−CL CL −CL CL −CL −Vπ f … OFDM B Quadrature Laser −1 −1 MZ TX Hard Clipper sin ( ) Helec.( ) V (n Im{ } CL in,i ) Vπ Vout,i(n)

−CL CL −CL CL D/A f XN −Vπ Vd,i(t) −CL B Vd,i(n) Fig. 2.25 OFDM transmitter including hard clipping, pre-distortion and electrode frequency response compensation.

In Fig. 2.26, we note that the peak after the D/A can exceed the value limited by the digital hard clipping device because of interpolation effects [13], [32]. In order to avoid an analog clipping device, we will allow the driving signal to exceed Vπ. Later on, we will show that this effect has negligible impact on system performance.

41 Wide Area Networks

Fig. 2.26 Electrical and optical MZ waveforms. The solid and dotted lines represent the waveforms with and without hard clipping, respectively.

2.7.2.2 Clipping Simulation Results

In order to make the simulation results independent of the number of subcarriers, we will define a normalized clipping level called the clipping ratio (CR) [33], which is defined as

A CR = , (2.35) σ where A is the clipping level and σ is the rms power of the OFDM signal, i.e., = []txE )(CR 2 . For example, if there is no clipping, then = PARCR . A value CR = 2 means that the signal is clipped at twice the rms power level. We note that the PAR of the real and imaginary components is twice the PAR of the complex baseband OFDM signal, since the real and imaginary parts have half the power but the same peak value as the complex baseband OFDM signal.

In our simulations, we consider a polarization-multiplexed system with a total bit rate of 118 Gbit/s (103 Gbit/s with 15% FEC overhead). The subcarriers are modulated using QPSK, so the symbol rate on each polarization is 29.66 GHz. The

FFT size is 64 and the oversampling ratio is Ms = 1.2, i.e. only 52 subcarriers are used. The cyclic prefix is chosen accordingly to [25], and is approximately 1/10 of the total

42 Wide Area Networks number of subcarriers. We assume for now that the D/A has an infinite number of bits. We neglect all transmission impairments, such as fiber nonlinearity, GVD and PMD. We use a homodyne receiver, followed by an anti-aliasing filter that is a 5th-order Butterworth lowpass filter having a 3-dB bandwidth equal to 17.2 GHz, the first null in the baseband OFDM spectrum [25]. The 3-dB bandwidth of the SGF is set equal to the OFDM bandwidth [25]. The MZ 3-dB bandwidth is 30 GHz, which is representative of currently available devices [34]. The MZ frequency response is shown in Fig. 2.27.

0

-1

-2

-3 (f)|) (dB)(f)|) d -4 (f)/V m -5 (|V 10 -6

20 log -7

-8

-9 0 10 20 30 40 50 Frequency (GHz) Fig. 2.27 MZ electrode frequency response.

We note that the compensation of the MZ frequency response shown in Fig. 2.25 requires one extra DFT and IDFT operation to scale the subcarriers by the inverse of the MZ electrode frequency response. We consider three scenarios for the modulator: Full Compensation where we compensate the MZ electrode frequency response as shown in Fig. 2.25, Pre-Compensation where we move the electrode frequency response compensation to the OFDM transmitter as a pre-distortion operation, i.e., we are swapping the order of the nonlinear and linear effects and No Compensation where we do not compensate the MZ electrode frequency response. The simulation results are shown in Fig. 2.28 and Fig. 2.29.

43 Wide Area Networks

0.45 Full Comp. 0.4 Pre-Comp. No Comp. 0.35

0.3

0.25

0.2

0.15

0.1 Optical Power Efficiency (W/W) 0.05

0 1 2 3 4 5 6 7 8 9 10 11 Clipping Ratio (CR) Fig. 2.28 Optical power efficiency for Nc = 64, Nu = 52 and R = 29.66 GHz.

−3 Since typical current FEC codes have a threshold around PS = 10 , we −4 measure the receiver sensitivity penalty at PS = 10 in order to have some margin.

In Fig. 2.28, we observe that the OPE is around 1% if there is no clipping CR = 10.2). As we start clipping, the power efficiency increases significantly. Moreover, we can also observe that the three compensation options have approximately the same OPE. For the case of no compensation, the OPE is slightly inferior, due to the frequency dependence of the MZ electrode frequency response.

5 Full Comp. 4.5 Pre-Comp. 4 No Comp.

3.5

3

2.5

2

1.5

1 Receiver Sensitivity Penalty (dB) Penalty Sensitivity Receiver 0.5

0 1 2 3 4 5 6 7 8 9 10 11 Clipping Ratio (CR) −4 Fig. 2.29 Receiver sensitivity penalty at PS = 10 , Nc = 64, Nu = 52 and R = 29.66 GHz.

44 Wide Area Networks

In Fig. 2.29, we observe that the power penalty is approximately constant down to about CR = 2.5, below which, the signal becomes severely distorted and the receiver sensitivity penalty increases rapidly. Furthermore, we observe that the three compensation scenarios have approximately the same performance. For the case of no compensation, we note that there is a residual receiver sensitivity penalty even if there is no clipping. The residual receiver sensitivity penalty is due to the power lost at some frequencies in the MZ. By going from CR = 10.2 (no clipping) down to CR = 2.5, the OPE increases from less than 1% to 14%, a 12-dB power gain at the cost of a 0.5-dB sensitivity penalty.

Finally, in order to confirm that the CR makes the optimized clipping level independent of the number of subcarriers, we repeated the simulations with a different number of subcarriers. Fig. 2.30 and Fig. 2.31 show the OPE and system performance curves for different numbers of subcarriers for the case of no compensation of the MZ electrode frequency response.

0.5 N = 26 0.45 u N = 52 u 0.4 N = 104 u 0.35

0.3

0.25

0.2

0.15

OpticalPower Efficiency (W/W) 0.1

0.05

0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Clipping Ratio (CR) Fig. 2.30 Optical power efficiency for different number of subcarriers.

45 Wide Area Networks

4 N = 26 u 3.5 N = 52 u N = 104 3 u

2.5

2

1.5

1 Receiver Sensitivity Penalty Receiver(dB) Penalty Sensitivity 0.5

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Clipping Ratio (CR) Fig. 2.31 Receiver sensitivity penalty for different number of subcarriers.

2.7.2.3 Quantization Effects

For the remainder of the chapter, we consider only the situation where the electrode frequency response is not compensated, since we verified that this compensation has little effect on OPE and system performance. In addition, not compensating the electrode frequency response minimizes hardware complexity and power consumption.

In a practical OFDM transmitter, the D/A necessarily will have a finite number of bits. Fig. 2.32 and Fig. 2.33 show the OPE and system performance curves for different number of bits in the D/A.

46 Wide Area Networks

0.4 n = 4 bits 0.35 n = 5 bits n = 6 bits 0.3 n = 7 bits n = 8 bits 0.25

0.2

0.15

0.1 OpticalPower Efficiency (W/W) 0.05

0 1 2 3 4 5 6 7 8 9 10 11 Clipping Ratio (CR) Fig. 2.32 Optical power efficiency for different number of bits in the D/A, Nc = 64, Nu = 52 and R = 29.66 GHz.

4 n = 4 bits 3.5 n = 5 bits n = 6 bits 3 n = 7 bits n = 8 bits 2.5

2

1.5

1 Receiver Sensitivity Penalty (dB) Penalty Receiver Sensitivity 0.5

0 1 2 3 4 5 6 7 8 9 10 11 Clipping Ratio (CR) Fig. 2.33 Receiver sensitivity penalty for different number of bits in the D/A, Nc = 64, Nu = 52 and R = 29.66 GHz.

We note that the D/A reference voltage is always equal to Vπ regardless of the

CR, since the pre-distortion device maps the clipped peak value to ±Vπ.

In Fig. 2.32, we observe that the OPE is independent of the number of bits in the D/A. On the other hand, in Fig. 2.33 we observe that for a fixed number of bits, the receiver sensitivity penalty decreases as we further clip the signal until around CR = 2.5. This is because the D/A with a limited number of quantization levels can better represent the OFDM signal as the peak is further clipped. We also observe that as we

47 Wide Area Networks increase the number of bits, the receiver sensitivity penalty decreases and converges to the case of infinite-resolution value, as expected. If a value CR = 2.5 is used, 6 bits would be sufficient to obtain good performance. Finally, we note that although we considered Nu = 52 in Fig. 2.32 and Fig. 2.33, the receiver sensitivity penalty due to clipping and quantization is independent of the number of subcarriers for a given value of CR.

2.7.3 Dual-Drive MZ Optimization

2.7.3.1 Dual-Drive MZ Modulator

We have shown that the quadrature MZ can be used as an efficient OFDM transmitter. In an attempt to reduce complexity, we consider using a dual-drive MZ as an OFDM transmitter. The dual-drive MZ has two independent drive electrodes, and has transfer characteristic given by

tE )( 1 ⎡ ⎛ π tV )( ⎞ ⎛ π tV )( ⎞⎤ out m1 m2 , = ⎢exp⎜ j ⎟ + exp⎜ j ⎟⎥ (2.36) in tE )( 2 ⎣ ⎝ Vπ ⎠ ⎝ Vπ ⎠⎦ where Eout(t) and Ein(t) are the output and input electric fields, respectively, Vm1(t) and

Vm2(t) are independent electrical modulator signals and Vπ is the drive voltage that must be applied differentially between the two electrodes to produce a differential phase shift of π between the two waveguides. The model for the dual-drive MZ is shown in Fig. 2.34.

π π E (t) 1⎡ j Vm1(t) j Vm2(t)⎤ out = ⎢ Vπ +ee Vπ ⎥ E (t) 2 in ⎣⎢ ⎦⎥

48 Wide Area Networks

Fig. 2.34 Model for dual-drive MZ modulator.

The frequency response of each electrode in Fig. 2.34 is given by Eq. (2.30). In the dual-drive MZ, by adding together two independent signals whose phases are proportional to the two independent drive voltages, we can generate an output electric field having arbitrary magnitude and phase. In other words, we can generate any point inside the unit circle of the complex plane Eou(t)/Ein(t), as shown in Fig. 2.35.

⎧ (t)E ⎫ Im⎨ out ⎬ ⎩ in (t)E ⎭ x(n)

π j nV )( π 1 m 2 j nV )( Vπ 1 m1 e Vπ e 1 2 ⎧ (t)E ⎫ 2 Re⎨ out ⎬ ⎩ in (t)E ⎭

Fig. 2.35 Dual-drive MZ plane.

The drive voltages required to generate a point x(n) = A(n)·ejθ(n) (0 ≤ A(n) ≤ 1) are given by

V nV π ()θ n += ()+ 2)(arccos)()( πqnA out1 π (2.37) V nV π ()θ n −= ()+ 2)(arccos)()( πqnA out2 π

We note that due to the symmetry of the problem, the Eqs. for Vout1(n) and

Vout2(n) are periodic with period 2Vπ and that they could be interchanged and the point x(n) would still be obtained. We note that if the frequency response of each electrode is compensated, then Vm1(n) = Vout1(n) and Vm2(n) = Vout2(n).

As for the quadrature MZ modulator, we use hard clipping in the dual-drive MZ to increase the OPE. The optimized OFDM transmitter using the dual-drive MZ is shown in Fig. 2.36.

49 Wide Area Networks

V (n) −1 d1 Helec.( ) Vd1(t) Parallel Re{ } Vout1(n) D/A X1 -to- Hard-Clipping Serial f + B Dual-Drive … … IDFT + Laser −1 MZ N Dual-Drive Helec.( ) Cyclic Im{ } Pre-Distortion Vout2(n) Prefix D/A f XN B Vd2(t) Vd2(n) Fig. 2.36 OFDM transmitter using the dual-drive MZ with hard-clipping, pre-distortion and electrode frequency response compensation.

Besides reduced hardware complexity, another advantage of the dual-drive MZ over the quadrature MZ is a reduced clipping probability for the same clipping level. As shown in Fig. 2.37, the dual-drive MZ can span the indicated circle in the complex plane Eou(t)/Ein(t) for a given clipping level, while the quadrature MZ can only span the indicated square.

⎧ (t)E ⎫ Im⎨ out ⎬ (t)E ⎩ in ⎭ Dual-Drive MZ

⎧ (t)E ⎫ CR 1 out Quadrature Re⎨ ⎬ ⎩ in (t)E ⎭ MZ

Fig. 2.37 OFDM span regions for a fixed clipping level. The dotted and solid regions correspond to the quadrature MZ and the dual-drive MZ, respectively.

If the drive signal samples Vm1(n) and Vm2(n) are digitally pre-distorted, the MZ will generate the correct optical electric field at the sampling instants. However, the drive voltages Vm1(t) and Vm2(t) between samples are obtained by interpolation in the D/A converter, and they are not generally the values required to generate the correct optical electric field in between the sampling instants. As shown in Fig. 2.38, the optical waveform will differ from the correct OFDM waveform, which can degrade system performance. This problem can be controlled by increasing the oversampling ratio.

50 Wide Area Networks

2 Dual-Drive 1.5 Desired

1

0.5

0

-0.5 Real{Optical Field}Real{Optical -1

-1.5

-2 0 20 40 60 80 100 Time (ps) Fig. 2.38 Real component of the dual-drive MZ output electric field.

However, for a fixed oversampling ratio, there are some degrees of freedom in selecting the drive voltage values at the sampling instants since the dual-drive nonlinear transfer characteristic is periodic with period 2Vπ and drive voltage values

Vm1(n) and Vm2(n) can be interchanged. The correct OFDM values in between the sampling instants are obtained by smooth interpolation (ideally sinc() interpolation). So, in order to minimize the error in between the sampling instances, the trajectory spanned by the drive voltage vectors in the complex plane Eou(t)/Ein(t) should also be smooth, which is analogous to

minimize Δ+Δ rVrV )()( m1 m2 (2.38) subject to m1 = ,iout ),()( , rrnnVnV += 1 where mi mi mi rVrVrV +−=Δ )1()()( and i = 1, 2. In Eq. (2.38), we note that once the pre-distortion equation for Vm1(n) is chosen, the equation for Vm2(n) is constrained such that the optical electrical field at the sampling instant is equal to the OFDM sample value.

Fig. 2.39 illustrates the output optical electrical field generated by the dual- drive MZ for the cases of no trajectory optimization and with trajectory optimization for an oversampling ratio of Ms = 2.5 before the SGF.

51 Wide Area Networks

a) No Trajectory Optimization (-v ≤ v ≤ v ) 4 π m,i, π

2

0

-2 Re{Optical Field} -4 0 200 400 600 800 1000 1200

b) Trajectory Optimization (-vπ ≤ vm,i, ≤ vπ) 4

2

0

-2 Re{Optical Field} -4 0 200 400 600 800 1000 1200

c) Trajectory Optimization (-2vπ ≤ vm,i, ≤ 2vπ) 4

2

0

-2 Re{Optical Field} Re{Optical -4 0 200 400 600 800 1000 1200 Tim e (ps) Fig. 2.39 Output electric field of the dual-drive MZ for the cases of no trajectory optimization and with trajectory optimization before the super-Gaussian filter. The green and blue lines represent the output electric field and the correct OFDM waveform, respectively.

In Fig. 2.39 a), no trajectory optimization was used and the drive voltages were set to Vm1(n) = Vout1(n) and Vm2(n) = Vout2(n) for all discrete time n and they were limited to the range between −Vπ to +Vπ. We observe that the optical field in between the sampling instants is not equal to the correct OFDM waveform for the majority of the time. However, in Fig. 2.39 b), we observe that the error in between the sampling instants is significantly minimized when trajectory optimization is used. The receiver sensitivity penalty is around 1-dB for an oversampling ratio Ms = 2.5 when trajectory optimization is used. On the other hand, if no trajectory optimization is used, the oversampling ratio to achieve 1-dB receiver sensitivity penalty is Ms ≥ 5. Finally, in Fig. 2.39 c), we have expanded the drive voltages range to ±2Vπ and the error in between samples was further reduced since there are now more degrees of freedom for the drive voltages values. The receiver sensitivity penalty for this case is around 0.7 dB. We note that increasing the drive voltage range by a factor of two reduces the effective vertical resolution of the DAC by the same factor which, in turn, means that

52 Wide Area Networks a larger number of bits is required to maintain the same performance. However, we believe that in a practical MZ modulator, the drive voltages values will be constrained to ±Vπ so we will only consider this case for the remainder of the chapter.

Fig. 2.40 and Fig. 2.41 show the dual-drive MZ optical power efficiency and receiver sensitivity penalty, respectively, for different clipping levels.

0.45

0.4

0.35

0.3

0.25

0.2

0.15

0.1 Optical Power Efficiency (W/W) 0.05

0 1 2 3 4 5 6 7 8 Clipping Ratio (CR) Fig. 2.40 Dual-drive MZ receiver sensitivity penalty with trajectory optimization for Ms = 2.5, Nu = 52 and R = 29.66 GHz.

We observe in Fig. 2.40 that the OPE increases significantly as we further clip the signal, like in the quadrature MZ. In Fig. 2.41, we notice that the power penalty is approximately constant down to about CR = 2.5, below which, the signal becomes severely distorted and the receiver sensitivity penalty increases rapidly. Finally, from Fig. 2.40 and Fig. 2.41, we conclude that the optimum clipping level is CR = 2.5, as for the quadrature MZ.

53 Wide Area Networks

3

2.5

2

1.5

1

Receiver Sensitivity Penalty (dB) Penalty Sensitivity Receiver 0.5

0 1 2 3 4 5 6 7 8 Clipping Ratio (CR) Fig. 2.41 Dual-drive MZ optical power efficiency with trajectory optimization for Ms = 2.5, Nu = 52 and R = 29.66 GHz.

In Fig. 2.42, we compare the receiver sensitivity penalty for the two types of modulators as a function of the oversampling ratio at the optimum clipping level, CR = 2.5.

3 Quadrature MZ Dual-drive MZ 2.5

2

1.5

1

Receiver Sensitivity Penalty (dB) Penalty Receiver Sensitivity 0.5

0 1 2 3 4 5 6 Oversampling Ratio (M) Fig. 2.42 Dual-drive and quadrature MZ receiver sensitivity penalties as a function of the oversampling ratio with no frequency loss compensation and with trajectory optimization for CR = 2.5, drive voltages range between ±Vπ and R = 29.66 GHz.

We observe in Fig. 2.42 that the quadrature MZ penalty remains constant as we increase the oversampling ratio. The 0.4 dB residual penalty can be decomposed into 0.15 dB from the MZ frequency-dependent electrode losses and 0.25 dB from the

54 Wide Area Networks clipping noise. On the other hand, the dual-drive MZ penalty decreases as the oversampling ratio increases. This is expected, as increasing the number of sampling points reduces the error between the correct and attainable waveform trajectories in the complex plane. Finally, we note that the dual-drive receiver sensitivity penalty converges to the same value as that for the quadrature MZ as the oversampling ratio increases.

55 Metropolitan Networks

3 METROPOLITAN NETWORKS

3.1 Introduction

Receiver-based electronic signal processing in metropolitan networks has also been the subject of many recent studies. In long-haul systems using coherent detection, linear equalizers have been shown to fully compensate linear fiber impairments in single-mode fiber (SMF), such as group-velocity dispersion (GVD) and polarization-mode dispersion (PMD) [16]. On the other hand, in metropolitan systems using direct-detection (DD), linear equalizers offer little performance improvement, because the nonlinear photodetection process destroys information on the phase of the received electric field. Recently, maximum-likelihood sequence detection (MLSD) was shown to be effective in mitigating GVD and PMD impairments in DD links [35], [36]. Although the computational complexity of MLSD increases exponentially with the channel memory, using low-complexity branch metrics, near-optimal performance can be achieved with manageable computational complexity, provided the effective channel memory does not exceed a few symbol intervals [3].

An alternate approach to combating fiber impairments in amplified DD systems is to use multicarrier modulation, such as orthogonal frequency-division multiplexing (OFDM). There are three major approaches for combining OFDM with DD. The first two techniques are based on intensity modulation (IM), and so require some means to make the OFDM signal nonnegative. The first technique adds a DC bias to reduce the negative signal excursions and then clips the remainder of the negative excursions. This method is called DC-OFDM. The second technique clips the entire negative excursion of the waveform, avoiding the need for a DC bias [37], [38]. Clipping noise is avoided by appropriate choice of the subcarrier frequencies. This technique is called asymmetrically clipped optical OFDM (ACO-OFDM) [38]. The third technique is based on single-sideband modulation of the complex-valued optical

56 Metropolitan Networks electric field by an OFDM signal. A DC bias (carrier component) is added, leaving a guard band between the carrier and the OFDM signal in order to avoid intermodulation products caused by photodetection [39], [40]. This method is called single-sideband OFDM (SSB-OFDM).

There have been several studies of the different OFDM techniques (e.g., [38]), but there has been no comparison of power efficiencies among the various direct- detection OFDM methods in SMF, and to conventional baseband methods, such as on- off keying (OOK). Furthermore, in previous work, the DC bias and the powers of the subcarriers were not jointly optimized based on the channel response and the nonlinear beat noises, since there is no closed-form solution for this optimization. We present an iterative procedure based on known bit-loading algorithms with a new modification, the bias ratio (BR), in order to obtain the optimum power allocation. We compare the performance of the three OFDM techniques using optimized power allocations to the performance of OOK with MLSD.

This chapter is organized as follows. In section 3.2, we present the metro networks and review methods for power and bit allocation for multicarrier systems and describe the optimal water-filling solution. We present our system model in section 3.3. In section 3.4, we review the different OFDM formats and derive equivalent linear channel models for each one, assuming only GVD is present in the SMF. Furthermore, we discuss the effects of amplifier noise in direct-detection systems and compute an expression for the autocorrelation function of the photodetected noise. In section 3.5, we compare the optical power required to transmit at a given bit rate for the different optimized OFDM formats and for OOK with MLSD, making use of previously published results for the latter format. In that section, we also compare the computational complexities of the various formats.

57 Metropolitan Networks

3.2 Metropolitan Networks and Power Allocation Review

3.2.1 Metropolitan Networks

Metropolitan networks (or metro) typically use single-mode fiber (SMF) and also operate around 1.55 μm which is the region with the lowest fiber attenuation. Metro networks normally use light sources with small linewidths like lasers. Metro systems demutliplex the incoming high-speed traffic from long-haul systems into several lower bit rate streams, typically between 1 Gbit/s and 40 Gbit/s. The demultiplexed traffic is transmitted in the metro network (e.g. core network of New York city) and the distances vary between 50 km to 500 km. Fig. 3.1 shows a block diagram of a metro network.

Fig. 3.1 Metro network diagram.

Metro systems typically use a single span between terminal equipment and do not employ any dispersion-compensating fiber (DCF). At the end of the link, an optical amplifier is used to compensate the SMF attenuation. An optical filter is also normally placed after the amplifier in order to reduce out-of-band amplifier noise (i.e., ASE noise). Fig. 3.2 shows a block diagram of a metro link.

SMF Amplifier + Filter

TX RX

58 Metropolitan Networks

Fig. 3.2 Metro link with an optical filter.

Metro systems use direct-detection (i.e., detect the instantaneous optical power) for its simplicity and reduced hardware cost. Furthermore, metro links normally employ intensity modulation (IM), i.e., the transmitter modulates the instantaneous optical power (intensity) but electric field modulation can also be used. In IM, the modulating waveform has to be nonnegative in order to avoid loss of information. IM can be achieved, for example, by direct modulation of the laser current, as shown in Fig. 3.3.

Laser Power (mW)

Popt (t)

Ith Current (mA)

x(t) Fig. 3.3 Intensity modulation (IM) by direct modulation of the laser current.

We note that in IM, the average transmitted optical power Pt is proportional to the average of the modulating waveform E[x(t)], and is given by

1 T 1 T Pt = lim opt )( = KdttP lim )( dttx , (3.1) T→∞ 2T ∫−T T→∞ 2T ∫−T rather than the usual time-average of |x(t)|2 (i.e., E[|x(t)|2]), which is appropriate when x(t) represents amplitude.

3.2.2 Power and Bit Allocation Review

3.2.2.1 Gap Approximation

On an AWGN channel, the maximum achievable bit rate is given by the Shannon capacity

59 Metropolitan Networks

C ()1log += SNR , (3.2) B 2 where C is the capacity, B is the channel bandwidth and SNR is the signal-to-noise ratio. Any real system must transmit at a bit rate less than capacity. For QAM modulation, the achievable bit rate can be expressed approximately as

R ⎛ SNR ⎞ 2 ⎜1log += ⎟ , (3.3) B ⎝ Γ ⎠ where R is the bit rate and Г is called the gap constant. The gap constant, introduced by Cioffi et al [41] and Forney et al [42], represents a loss with respect to the Shannon capacity.

6

5

4 Capacity (Γ = 0 dB) 3

R/B (bit/s/Hz) 3 dB 2 6 dB 9 dB 1

0 2 4 6 8 10 12 14 16 SNR (dB) Fig. 3.4 Achievable bit rates as a function of SNR for various values of the gap Γ.

The gap analysis is widely used in the bit loading of OFDM systems, since it separates coding gain from power-allocation gain [43]. For uncoded QAM, the gap constant is given by

α 2 =Γ , (3.4) 3

−1 -1 where α = PQ e )( , Pe is the symbol-error probability and Q is the inverse Q −6 −7 function. As an example, the gap is 8.8 dB at Pe = 10 and is 9.5 dB at Pe = 10 for uncoded QAM. The use of forward error-correction (FEC) codes reduces the gap. A

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−6 well-coded system may have a gap as low as 0.5 dB at Pe < 10 . A gap of 0 dB means the maximum bit rate has been achieved and therefore R = C. Fig. 3.4 shows the attainable bit rates for various gap values. In the case of real-valued PAM, the gap approximation is valid, but the bit rate is reduced by a factor of two as compared to Eq. (3.3). For the remainder of the thesis, we define the normalized bit rate = / BRb , which has units of bits/s/Hz.

3.2.2.2 Optimum Power Allocation

The OFDM signal splits the transmission channel into N parallel channels. When the total average transmitted power is constrained, the maximum obtainable bit rate can be written as

2 N−1 ⎛ HP ⎞ ⎜ nn ⎟ 1logb max = 2 1logb + P ∑ 2 n n=0 ⎜ Γσ ⎟ ⎝ n ⎠ , (3.5) N−1 subject to t = ∑ PP n n=0

2 2 where |Hn| ,σn and Pn are the channel gain, noise variance and transmitted power at subcarrier n, respectively, and Pt is total average transmitted power. The solution of Eq. (3.5) is the known water-filling solution.

Γσ 2 n Pn 2 =+ a , (3.6) H n where a is a constant chosen such that . The optimum power allocation is t = ∑PP n n then

⎧ Γσ 2 Γσ 2 n n ⎪a − 2 , a ≥ 2 ⎪ H n H n Pn = ⎨ . (3.7) Γσ 2 ⎪ ,0 a < n ⎪ 2 ⎩ H n The optimum power allocation is illustrated in Fig. 3.5.

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water level a 2 Γσ N [1] H[]1 2 2 total power P Γσ 2 [N] Γσ N [2] N 2 []NH 2 H[]2 2 Γσ N [4] 2 2 H[]4 Γσ N [3] H []3 2 n 123 4 N

P[3]

P[4]

P[2] P[5] P[]= 01 n 123 4 N Fig. 3.5 Optimal power allocation for OFDM.

After the optimum power allocation is determined, the number of bits to be transmitted on each subcarrier is computed using Eq. (3.3). While the ideal value of b = R/B is an arbitrary nonnegative real number, in practice, the constellation size and FEC code rate are adjusted to obtain a close rational approximation. In our analysis, we will neglect the difference between the ideal value of b and its rational approximation.

We note that when the total average transmitted power is constrained, the optimal power allocation yields a variable data rate. For some applications, a fixed data rate is required. In this case, the optimal design minimizes the average power required to transmit at a given fixed bit rate. The power minimization can be written as

N−1

P min P t = Pn P ∑ n n=0 2 . (3.8) N−1 ⎛ HP ⎞ ⎜ nn ⎟ 1logb subject to = ∑ 2 1logb + 2 n=0 ⎜ Γσ ⎟ ⎝ n ⎠ The solution of Eq. (3.8) is also the water-filling solution given by Eq. (3.6). However, in this case the constant a is chosen such that bit rate is equal to the desired value. We can interpret this solution as the water/power being poured until the required bit rate is achieved.

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3.3 OFDM System Model for Metro Links

The OFDM system model is shown in Fig. 3.6.

X0 Y0

X1 Y1 Optical Fiber Anti-Aliasing . Amplifier Filter Filter . Optical OFDM . OFDM . Rx . Tx .

XN−2 YN−2

XN−1 YN−1

Fig. 3.6 OFDM system model for metro links.

An optical OFDM modulator encodes transmitted symbols onto an electrical OFDM waveform and modulates this onto the intensity (instantaneous power) of an optical carrier. The modulator can generate one of DC-OFDM, SSB-OFDM or ACO- OFDM. Details of modulators for particular OFDM schemes are described in section 3.4. After propagating through the SMF, the optical signal is optically amplified and bandpass-filtered. We assume that the optical amplifier has a high gain so that its amplified spontaneous emission (ASE) is dominant over thermal and shot noises. The ASE noise is modeled as complex additive white Gaussian noise (AWGN) with zero mean. The ASE can be expressed as nASE(t) = nI(t) + j·nQ(t), where nI(t) and nQ(t) are uncorrelated Gaussian random processes, each having half the variance of nASE(t).

In our analysis, PMD and fiber nonlinearity are neglected, and GVD is the only fiber impairment considered. In the optical electric field domain, the fiber is modeled as a linear system with transfer function given by

β ω 2 2 Lj 2 (3.9) fiber ()ω = eH , where β2 is the fiber GVD parameter and L is the fiber length. The overall optical transfer function is given by

(ω) = fiber (ω)⋅ HHH filter (ω), (3.10)

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where Hfilter(ω) is the transfer function of the optical bandpass filter.

After bandpass filtering, the optical signal intensity is detected, and the electrical current is lowpass filtered. The OFDM signal is demodulated and equalized with a single-tap equalizer on each subcarrier to compensate for the channel distortion [38], [39].

3.4 Analysis of Direct-Detection OFDM Schemes

3.4.1 DC-Clipped OFDM

The main disadvantage of using OFDM with IM is the required DC bias to make the OFDM signal nonnegative. Since OFDM signals have a high peak-to- average power ratio, the required DC bias can be excessively high. A simple approach to reduce the DC offset is to perform hard-clipping on the negative signal peaks [37], [38]. This technique is usually called DC-OFDM and the transmitter is shown in Fig. 3.7.

DC bias X0

X1 . QAM . Parallel Real Laser Fiber . to XN/2−1 Signal Serial XN/2 IDFT D/A Clip + * XN/2−1 Cyclic . Prefix Optical Spectrum * QAM . . X* 1 f * Xk Xk Fig. 3.7 Block diagram of a DC-OFDM transmitter.

In Fig. 3.7, the transmitted symbols are modulated such that the time-domain waveform is real. This is achieved by enforcing Hermitian symmetry in the symbols input to the inverse discrete Fourier transform (IDFT). We note that the 0th or the DC subcarrier is not modulated and it is equal to the DC offset. After D/A conversion, the

64 Metropolitan Networks electrical OFDM signal is hard-clipped such that the waveform is nonnegative and then the signal is intensity modulated onto the optical carrier.

After propagating through the optical system, the detected signal can be written as

2 2 det )( () ()⊗== ()thtxtEty , (3.11) where E(t) is the output electric field, x(t) is the DC-OFDM signal and h(t) is the impulse response of the overall optical system, given by the inverse Fourier transform of Eq. (3.10). For now, we have neglected amplifier noise in Eq. (3.11).

We can expand the OFDM signal as x(t) = A + xac(t) + nclip(t), where A is the average of the OFDM signal after clipping and xac(t) is the OFDM signal with the DC subcarrier equal to zero and nclip(t) is the clipping noise. We note that if the DC bias is chosen such that there is no clipping, then A is equal to the DC bias and nclip(t) = 0. For now, we neglect the clipping noise. The detected signal is then

2 det )( ac ()⊗+= ()thtxAty . (3.12)

We can now use the Taylor series expansion on the square-root term. The detected signal becomes

⎡⎛ tx )( 2 tx )( ⎞ ⎤ )( ⎜ Aty ac ac +−+= …⎟ ⊗ th det ⎢⎜ 2/3 ⎟ ()⎥ ⎣⎝ 2 A 8A ⎠ ⎦ * (3.13) ⎡⎛ tx )( 2 tx )( ⎞ ⎤ ⎜ A ac ac +−+× ⎟ ⊗ th ⎢⎜ 2/3 ⎟ ()⎥ ⎣⎝ 2 A 8A ⎠ ⎦ After expanding the terms in Eq. (3.13), the detected signal simplifies to

( )⊗ ( ) * ( )⊗ * (thtxthtx ) )( Aty += ac + ac det 2 2 (3.14) 2 ()⊗ ()thtx *2 ()⊗ * ()thtx − ac − ac +… 8A 8A The first two convolution terms in Eq. (3.14) correspond to an equivalent linear channel. The remaining terms in Eq. (3.14) correspond to the interaction between the detector nonlinearity and the fiber GVD. They can be interpreted as an

65 Metropolitan Networks equivalent nonlinear detection noise that will degrade the receiver SNR. We note that these terms decrease as the bias level increases.

The linear terms in Eq. (3.14) can be further simplified since the OFDM signal

* is real valued, i.e., ac ()= ac (txtx ). Thus, we can write the equivalent linear channel for DC-OFDM as

( ) + * (thth ) ()th = . (3.15) eq 2 The transfer function is then

β β ω2 2 − ω 2 2 LjLj 2 2 + ee ⎛ 2 β2 ⎞ (3.16) H eq ()ω = = cos⎜ω L⎟ . 2 ⎝ 2 ⎠ The equivalent transfer function is plotted on Fig. 3.8.

1

0.9

0.8

0.7

0.6 (f)|

eq 0.5 |H 0.4

0.3

0.2

0.1

0 0 2 4 6 8 10 12 14 16 18 20 Frequency (GHz) Fig. 3.8 Equivalent transfer function for DC-OFDM for R = 20 Gbit/s, L = 100 km and D = 17 ps/nm/km, corresponding to γ = 0.87.

In Fig. 3.8, we observe that the equivalent channel transfer function is not unitary and is frequency-selective. Thus, in order to maximize the bit rate, we need to use variable bit loading on each subcarrier.

3.4.2 Asymmetrically Clipped Optical OFDM

Armstrong and Lowery [37], [38] proposed adding no DC bias and clipping the entire negative excursion of an electrical OFDM signal before modulating it onto the

66 Metropolitan Networks intensity of an optical carrier. They showed that clipping noise is avoided (at least in the absence of dispersion) by encoding information symbols on only the odd subcarriers. This technique is called ACO-OFDM. Fig. 3.9 shows a block diagram of an ACO-OFDM transmitter.

0 X0 QAM X1 0 . QAM . Parallel Real 0 Laser Fiber . . to Signal . XN/2−1 Serial . XN/2 IDFT D/A Clip + . * . XN/2−1 Cyclic . . Prefix Optical Spectrum 0 . * QAM . 0 * QAM* X1 f * Xk Xk Fig. 3.9 Block diagram of an ACO-OFDM transmitter.

As in DC-OFDM, the OFDM subcarriers are assumed to have Hermitian symmetry, so that the time-domain waveform is real. Because only the odd subcarriers are used to transmit data, for a given choice of signal constellation, ACO-OFDM has only half the spectral efficiency of DC-OFDM. After D/A conversion, the electrical OFDM signal is hard-clipped at zero and intensity modulated onto an optical carrier.

The equivalent transfer function of ACO-OFDM is the same as DC-OFDM, and is given by Eq. (3.16). This can be shown by using the same approach as in section 3.4.1, setting A equal to the average of the ACO-OFDM waveform.

3.4.3 Single-Sideband OFDM

Another approach to reducing the required DC bias is to use SSB-OFDM. In SSB-OFDM, only one of the DC-OFDM sidebands is transmitted and there is a frequency guard band between the sideband and the optical carrier. The bandwidth of the guard band must be no smaller than that of the OFDM sideband. Fig. 3.10 shows a block diagram of a digital implementation of a SSB-OFDM transmitter

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DC bias X0 Re X1 0 D/A . Parallel . to Fiber QAM . Serial Quad XN/2 IDFT + MZ Cyclic . Prefix Optical Spectrum 0 . . D/A

XN−1 Im f B B Fig. 3.10 Block diagram of an SSB-OFDM transmitter.

SSB-OFDM can be obtained by applying a Hilbert transform to a DC-OFDM signal. A Hilbert transform can be generated at the OFDM transmitter by setting the negative subcarriers to zero, as shown in Fig. 3.10. The frequency guard band is obtained by modulating the high-frequency subcarriers and setting to zero the low- frequency subcarriers. The IDFT output is an analytical signal, and its real and imaginary components are used for the drive voltages of the inphase and quadrature inputs of a quadrature Mach-Zehnder modulator. Hence, contrary to the previous techniques, in SSB-OFDM the signal is actually modulated onto the optical electric field, and not simply onto its intensity, and therefore no hard clipping is required. The DC bias in SSB-OFDM sets the amplitude of the optical carrier, affecting system performance, as we show in the following.

SSB-OFDM has the same optical spectral efficiency as DC-OFDM. However, the transmitter considered here requires twice the bandwidth of DC-OFDM for a digital implementation of SSB-OFDM. Other transmitter implementations that require less bandwidth are discussed in [40]. There have been some studies on the optimization of SSB-OFDM, particularly in the trade-off between spectral efficiency and performance as a function of the optical carrier power (i.e. DC bias) [44], [45]. However, in the previous work, the DC bias and the powers of the subcarriers were not jointly optimized based on the channel response and the nonlinear beat noises.

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After propagating through the optical system, the detected signal can be written as

2 2 2π btfj det )( () ( +== () )⊗ ()thetsAtEty , (3.17)

2π btfj where ()ets is the OFDM sideband centered at frequency fb, A is the DC bias and h(t) is the optical channel impulse response, given by the inverse Fourier transform of Eq. (3.10). We can expand the detected signal as

* 2π btfj 2π btfj det []()( += ( ) ( ) [() +⋅⊗ ( ) ) ⊗ (thetsAthetsAty )] 2 ()2π btfj ()⋅+⊗⋅+= * ()− 2π btfj ⊗ * ()thetsAthetsAA . (3.18) ()⊗+ ()thts 2

Assuming that the OFDM sideband has a bandwidth B centered at fb, the quadratic term in Eq. (3.18) occupies a bandwidth from –B to +B. Thus, if the frequency guard band is equal or greater than B, the quadratic intermodulation products are avoided. This concept is illustrated in Fig. 3.11.

(f)S ydet BB

OFDM* OFDM f -fb 0 fb Intermodulation Products

Fig. 3.11 PSD of the detected signal in SSB-OFDM.

Finally, since the quadratic intermodulation products are avoided by using the guard band and the desired signal is the OFDM sideband centered at frequency fb, we conclude from Eq. (3.18) that the transfer function for SSB-OFDM is a scaled version of the optical transfer function given by Eq. (3.10), and can be written as

SSB (ω) = ⋅ HAH (ω), (3.19) where A is the DC bias.

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3.4.4 Effects of Amplifier Noise

If we include amplifier noise, the detected signal can be written as

2 det )( ( ) += (tntEty ) , (3.20) where E(t) is the output electric field and n(t) is the filtered ASE, i.e.,

()ASE ()⊗= filter ()thtntn . We can further expand the detected signal as

* det )( ( ( ) ) ( ( )+⋅+= tntEtntEty )()( ) . (3.21) ()2 ()* * () +⋅+⋅+= tntntEtntEtE )()()( 2 From Eq. (3.21), we observe that the signal is corrupted by signal-spontaneous and spontaneous-spontaneous beat noises. We define ()* * () +⋅+⋅= tntntEtntEtw )()()()( 2 as the total noise. By using the moment generating function for n(t) or Gaussian moments theorems [46] and assuming that the transmitted signal is uncorrelated with the amplifier noise, we can write the autocorrelation of the total noise w(t) as

*2 '' RmRR )(||)()( ww ( EE ττ E )⋅+= nn τ * 2 ()EE '' τ E ⋅++ RmR nn τ )(||)( , (3.22) )(2)0( 2 ++ RR 2 τ)(2)0( nn II nn II

' R '' τ )( where () )( −= mtEtE E , mE is the average value of the received electric field, EE

' is the autocorrelation of (tE ) , Rnn(τ) is the autocorrelation of the noise n(t) and R ()τ is the autocorrelation of the real component of n(t). In Eq. (3.22), there are nn II three main noise components:

*2 • DC-spontaneous noise: E nn +⋅⋅ RRm nn ττ ))()((||

* * • Signal-spontaneous noise: EE '' nn EE '' ⋅+⋅ RRRR nn ττττ )()()()(

• Spontaneous-spontaneous noise: R2 + R2 τ)(2)0( nn II nn II

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From Eq. (3.22), we observe that the DC-spontaneous beat noise power increases proportionally with the DC bias. In DC-OFDM, if the DC bias is excessively high, the DC-spontaneous beat noise dominates, and the receiver SNR is low. On the other hand, if the DC bias is small, the clipping noise and the nonlinear detection noise from Eq. (3.14) dominate, and the receiver SNR is also low. Thus, the DC bias needs to be optimized for best performance.

In SSB-OFDM, if the DC bias is high, the receiver SNR is unaffected since both the OFDM signal and the dominant noise are amplified by the DC offset. However, the optical power efficiency is very low in this case. On the other hand, if the DC bias is small, the other noise terms dominate and the receiver SNR is low.

3.5 Comparison of Direct-Detection Modulation Formats

In order to make our results independent of bit rate, we use the dimensionless dispersion index γ, defined as

2 = βγ 2 LR , (3.23) where β2 is the fiber GVD parameter, L is the fiber length and R is the bit rate. Furthermore, in order for the required optical SNR to be independent of the bit rate, the noise bandwidth should be matched to the signal bandwidth. Thus, we use the normalized optical SNR defined as

Popt SNRopt = , (3.24) 0 RN where Popt is the optical power, N0 is the ASE power spectral density and R is the bit rate. The normalized optical SNR is related to the conventional optical SNR (OSNR) measured in a 0.1-nm (12.5-GHz) bandwidth as

GHz 5.12 GHz SNR = OSNR , (3.25) opt R The system model is shown in Fig. 3.6. As mentioned before, we neglect all transmission impairments except for GVD. We consider a fiber with a dispersion D =

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17 ps/(nm⋅km). The optical filter is modeled as a second-order super-Gaussian filter with a 3-dB bandwidth B0 = 35 GHz, which is realistic for commercial systems with 50-GHz channel spacing. We also assume that the optical amplifier has a high gain so that the ASE is dominant over thermal and shot noises from the receiver. At the receiver, the anti-aliasing filter is a fifth-order Butterworth lowpass filter having a 3- dB cutoff frequency equal to the first null of the OFDM spectrum [25].

In order to perform a fair comparison with OOK, we maintain OFDM optical bandwidth constant and equal to the optical bandwidth required to transmit at a bit rate R using OOK. Table 3.1 summarizes the required electrical and optical bandwidths for the different modulation techniques.

SOOK()f SACO−OFDM( f )

OFDM* OFDM f f 2R 2R

S −OFDMDC ()f S −OFDMSSB ( f )

OFDM* OFDM OFDM f f 2R 2R Fig. 3.12 PSD of the different modulation formats.

Modulation Receiver BW Optical BW OOK R 2R DC-OFDM R 2R ACO-OFDM R 2R SSB-OFDM 2R 2R Table 3.1 Electrical and optical bandwidths required to transmit at a bit rate R.

For all the OFDM formats, we use an oversampling ratio of Ms = 64 /52 ≈ 1.2 to avoid aliasing. As an initial estimate, we choose the cyclic prefix and the number of subcarriers using the equivalent linear channel described in section 3.4 for each OFDM format [25]. Then, we increase the cyclic prefix until the interference penalty is completely eliminated. Typically, the final result is very close to the initial estimate.

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After selecting the OFDM parameters, we proceed to optimally allocate power among the subcarriers. We note that we cannot use Eqs. (3.6) and (3.7) directly, since the nonlinear detection noise in each subcarrier depends on the power of all the subcarriers. Thus, we perform the power allocation iteratively. For a given power allocation, the SNR measured at each subcarrier is used to compute an updated water- filling solution. We repeat this process until the power allocation no longer changes. In our simulations, we do not actually implement the coding required to transmit the number of bits given by the water-filling solution. In practice, the constellation used on each subcarrier will be M-ary QAM (M-QAM) with coding (e.g., trellis coded modulation) in order to achieve the number of bits given by the water-filling algorithm. As seen before, the number of bits that can be allocated to a subcarrier depends only on the received SNR of that subcarrier. For simplicity, we used QPSK on each subcarrier with the same power given by the water-filling solution in order to obtain the same SNR that would be required to transmit the specific number of bits given by the bit allocation.

In addition to optimizing the power/bit allocation at each subcarrier, we need to optimize the DC bias for DC-OFDM and SSB-OFDM. In order to minimize the DC bias, we use a bias level proportional to the square-root of the electrical power. We define the proportionality constant as the bias ratio (BR), which is given by

DC B = bias , (3.26) R σ where σ = Pelect is the standard deviation of the electrical OFDM waveform. Using the BR insures that the water-filling solution minimizes the DC bias, and thus the optical power required. Fig. 3.13 shows the flow chart for power minimization of DC- OFDM and SSB-OFDM using the bias ratio.

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Initialize the subcarrier SNRs/gains

Measure received Power allocation subcarrier (water-f illing) SNRs /gains

Power Update DC allocation Yes subcarrier using changed? the bias ratio

No

End

Fig. 3.13 Flow chart of DC-OFDM and SSB-OFDM optimization using the bias ratio (BR).

For example, in DC-OFDM, if, at iteration k, the electrical power is high after subcarrier power allocation, the DC bias will also be high (since it is proportional to the square-root of electrical power), and the optical power will be high. A high DC bias reduces both the clipping noise and the intermodulation terms, and therefore the received SNR on each subcarrier is high. At the next iteration k +1, the water-filling algorithm removes some of the excess electric power in order to lower the SNR to the desired value, thereby reducing the DC bias and the optical power. On the other hand, if, at iteration k, the DC bias is low, the clipping noise and the intermodulation terms will be high, and therefore the received SNR on each subcarrier is low. At the next iteration k +1, the water-filling algorithm adds more electrical power to increase the electrical SNR, and consequently the DC bias and optical power will increase. For a given value of the bias ratio BR, we repeat this process until the power allocation and DC bias no longer change. The minimum required optical SNR is obtained by performing an exhaustive search over the value of the bias ratio BR, as shown in Fig. 3.14.

Finally, we note that the water-filling solution is optimum for Gaussian noise but the noises in an optically amplified direct-detection receiver are not Gaussian. However, if the number of subcarriers is large, we expect the noise to be approximately Gaussian after the DFT, an assumption confirmed in our simulations.

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Typical high-performance FEC codes for optical systems have a threshold of −3 the order of Pb = 10 , so we compute the minimum optical power required to achieve −3 Pb = 10 for the different OFDM formats. Fig. 3.14 shows the normalized optical SNR required for a dispersion index of γ = 0.25. As an example, γ = 0.25 corresponds to 100 km of standard single-mode fiber (SSMF) without optical dispersion compensation at 10 Gbit/s or 70 km of SSMF with 90% inline dispersion compensation at 40 Gbit/s.

21 ACO-OFDM 20 Nu = DC-OFDM 104 19 Nu = 208 208 416 (dB) 18 416

opt 17 832 16 15 14 SSB-OFDM Required SNR N = 13 u 52 12 104 208 11 0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4

Bias Ratio, BR

Fig. 3.14 Normalized optical SNRs required for the different OFDM formats for γ = 0.25 to achieve Pb −3 = 10 . The number of used subcarriers, Nu, is indicated for each curve.

In Fig. 3.14, we notice that for all OFDM formats the required optical SNR decreases slightly with the number of subcarriers, since the prefix penalty decreases as the number of subcarriers increases. In addition, OFDM uses the available channel bandwidth more efficiently as the number of subcarriers increases, which contributes to the reduction of the required optical SNR. In Fig. 3.14, we observe that the performance achieved with the chosen number of subcarriers is very close to the best performance achievable for each OFDM technique.

We also verify that SSB-OFDM requires the lowest optical SNR to achieve Pb = 10−3. Furthermore, for SSB-OFDM, the required optical SNR increases linearly with the DC bias for BR values greater than 1.4. This is expected since from Eq. (3.19) the electrical SSB-OFDM signal is scaled by the DC bias, and the dominant noise for a

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high bias level is the DC-spontaneous beat noise. As we decrease the BR, the signal- spontaneous beat noise is no longer negligible, and the minimum required optical SNR is achieved at a BR = 1. We note that the optimum BR is independent of the number of used subcarriers. If we further decrease the BR, more power has to be allocated to each subcarrier to compensate for the signal-spontaneous beat and therefore the required optical SNR increases.

For DC-OFDM, the required optical SNR also increases linearly with the DC bias for BR values greater than 1.6, since the dominant noise is the DC-spontaneous beat noise. As we decrease the BR, the clipping and the nonlinear detection noises are no longer negligible and the minimum required optical SNR is achieved at a BR = 1.1.

We note again that the optimum BR is independent of the number of used subcarriers.

If we further decrease the BR, more power must be allocated to each subcarrier to compensate for the clipping and nonlinear noise, and therefore the average value of the OFDM waveform increases. Thus, the required optical SNR also increases.

Fig. 3.15 shows an example of the subcarrier power allocation for the different OFDM formats.

1 DC-OFDM ) 0.9 ACO-OFDM 0.8 n,max

/ P 0.7 SSB-OFDM n 0.6 0.5 0.4 0.3 0.2 Subcarrier Power (P 0.1 0 0 0.2 0.4 0.6 0.8 1 Normalized Subcarrier Frequency (fn / R) Fig. 3.15 Subcarrier power distribution for DC-OFDM, ACO-OFDM, SSB-OFDM and to achieve a Pb −3 = 10 for γ = 0.25. The number of used subcarriers is 416, 416 and 208 for DC-OFDM (BR = 1.1), ACO-OFDM and SSB-OFDM (BR = 1.0), respectively.

We observe that the power allocation is consistent with the equivalent linear channel described in section 3.4. For DC-OFDM and ACO-OFDM, we observe a

76 Metropolitan Networks frequency-selective channel with notches, as expected. For SSB-OFDM, there are no notches, and the power allocation follows the shape of the signal-spontaneous beat noise.

For DC-OFDM, we can obtain an analytical lower bound on the required optical SNR by theoretically assuming that the only noises sources are the DC- spontaneous and the spontaneous-spontaneous beat noises from Eq. (3.22) and the by using the equivalent linear channel from Eq. (3.16). If we fix the DC bias, the optimum power allocation is given directly by Eqs. (3.6) and (3.7), since the noises do not depend on the subcarrier powers. Fig. 3.16 compares the analytical lower bound with the results obtained by simulation.

21

20

19 Simulation 18 (dB)

opt 17

16

15 Required SNR Required Lower Bound 14

13

12 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.4 Bias Ratio, BR −3 Fig. 3.16 Normalized optical SNR required for DC-OFDM to achieve Pb = 10 for γ = 0.25. The number of used subcarriers is Nu = 832.

From Fig. 3.16, we verify that for high BR (BR > 2), the lower bound is equal to the simulation results. Thus, we conclude that using water-filling iteratively with the

BR normalization for the DC level converges to the optimum power allocation in the linear regime. Furthermore, from Fig. 3.14 and Fig. 3.16, we observe that DC-OFDM can never be more power efficient than SSB-OFDM.

Fig. 3.17 shows the minimum required optical SNR for various dispersion indexes γ. The OFDM parameters are summarized in Table 3.2.

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24

22

ACO-OFDM 20

(dB) DC-OFDM

opt 18

16

Required SNR 14

SSB-OFDM 12

10 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 γ Fig. 3.17 Minimum normalized optical SNR required for various dispersion indexes γ for the different OFDM formats.

In Fig. 3.17, we observe that SSB-OFDM requires the lowest optical SNR to −3 achieve Pb = 10 . The high performance of SSB-OFDM comes at the price of the extra electrical bandwidth required to avoid the quadratic intermodulation products. Furthermore, unlike OOK with MLSD, SSB-OFDM requires two D/As and a quadrature modulator at the transmitter.

γ DC-OFDM ACO-OFDM SSB-OFDM N = 2048 N = 2048 N = 1024 0.25 Nu = 832 Nu = 416 Nu = 208 ν = 18 ν = 18 ν = 11 N = 2048 N = 2048 N = 1024 0.5 Nu = 832 Nu = 416 Nu = 208 ν = 31 ν = 31 ν = 21 N = 2048 N = 2048 N = 2048 0.75 Nu = 832 Nu = 416 Nu = 416 ν = 44 ν = 44 ν = 32 N = 2048 N = 2048 N = 2048 1 Nu = 832 Nu = 416 Nu = 416 ν = 59 ν = 59 ν = 42

Table 3.2 OFDM parameters for the various dispersion indexes γ. N is the DFT size, Nu is the number of used subcarriers and ν is the cyclic prefix.

From Fig. 3.17, we can also verify that the required optical SNR for ACO- OFDM is higher than DC-OFDM for γ greater than 0.25. This happens because the

78 Metropolitan Networks fiber dispersion destroys the orthogonality between the subcarriers and therefore there is significant nonlinear inter-carrier interference (ICI). Furthermore, since half the subcarriers are zero, the used subcarriers need to transmit at twice the data rate, which exacerbates the ICI. We note that the power allocation for ACO-OFDM does not converge for γ greater than 0.5. On the other hand, for low dispersion the subcarriers remain orthogonal and the ICI is avoided due to the zero subcarriers. In this regime, ACO-OFDM performs better than DC-OFDM.

In order to compare OFDM with single-carrier, we use results on OOK with MLSD from [3], since OOK with MLSD achieves the best optical power efficiency among known IM/DD techniques. For ease of comparison with [3], we present our results for 10.7 Gbit/s in Fig. 3.18 using the conventional definition of optical SNR −3 measured over a 12.5-GHz band (OSNR) to achieve Pb = 10 .

24

22 ACO-OFDM 20 DC-OFDM 18

16

14 Required OSNR (dB) 12 SSB-OFDM OOK with MLSD (Bosco)

10

8 0 50 100 150 200 250 300 350 400 L (km) −3 Fig. 3.18 OSNR values (over 0.1 nm) required to obtain Pb = 10 at 10.7 Gbit/s for OFDM and for OOK with MLSD [3].

We see that OOK with MLSD performs better than DC-OFDM or ACO- OFDM. SSB-OFDM performs as well as OOK with MLSD but requires twice the electrical bandwidth. For 10 Gbit/s systems, we can further optimize the OFDM signals by using the full available optical bandwidth (B0 = 35 GHz). In this case, we are trading optical bandwidth for optical power. Fig. 3.19 shows the OSNR required to

79 Metropolitan Networks

−3 obtain Pb = 10 when the OFDM signal occupies the full channel bandwidth (B0 = 35 GHz).

18

17 ACO-OFDM

16

15 DC-OFDM

14

13

12 OOK with MLSD (Bosco)

Required OSNR (dB) 11

10 SSB-OFDM 9

8 0 50 100 150 200 250 300 350 400 L (km) −3 Fig. 3.19 OSNR values (over 0.1 nm) required to obtain Pb = 10 at 10.7 Gbit/s for OFDM and for OOK with MLSD [3]. In this case, the OFDM signal occupies the full channel bandwidth (B0 = 35 GHz).

As we can observe, SSB-OFDM performs better than MLSD if we use the extra available optical bandwidth. We also notice that the power allocation for ACO- OFDM converges because of the extra degrees of freedom as compared to the case in Fig. 3.17. However, OOK with MLSD still performs better than ACO-OFDM and DC-OFDM.

3.5.1 Computational Complexity

Another important criterion is the computational complexity of a modulation/detection technique. For OOK with MLSD, the complexity of the Viterbi algorithm depends on the number of trellis states, given by N = 2M, where M is the channel memory measured in bit intervals. In the Viterbi algorithm, complexity is dominated by computation of the branch metrics. In each bit interval, 2·N·K branch metrics must be evaluated by the receiver, where K is the number of samples per bit interval [3]. Thus, the complexity per second for OOK with MLSD is at least

2·N·K/Tb, where Tb is the bit interval. The complexity order per bit for OOK with MLSD is then at least

80 Metropolitan Networks

MO 22)( M ⋅⋅= K , , TMLSD b (3.27) In our case, K = 2 and the memory M required for different values of dispersion index γ are listed in Table 3.3 [3].

γ 0.25 0.5 0.75 1 M 3 5 6 8 Table 3.3 Memory required for various values of the dispersion index γ for OOK with MLSD [3].

For OFDM, the IDFT and DFT operations are performed efficiently using a fast Fourier transform (FFT) algorithm. As shown in Chapter 2, an FFT of size N requires 4·N·log2(N) real operations (multiplications plus additions) [26]. The overall complexity order per bit for OFDM is then

Tx , TOFDM = 2 b /)(log)( TTNNNO OFDM b (3.28) Rx = [ + ] /)(log)( TTNNNNO , , TOFDM b 2 bu OFDM where TOFDM is the OFDM symbol period. Using data from Table 3.2 and Table 3.3 and Eqs. (3.27) and (3.28), in Table 3.4 we compare the computational complexity per bit for SSB-OFDM and OOK with MLSD.

MLSD SSB-OFDM γ Receiver Transmitter Receiver Total 0.25 32 47.9 48.8 96.7 0.5 128 46.8 47.8 94.6 0.75 256 52.1 53.1 105.2 1 1024 51.5 52.4 103.9 Table 3.4 Number of real operations required per bit for SSB-OFDM and OOK with MLSD for the various dispersion indexes γ.

We observe that SSB-OFDM requires fewer operations per bit than OOK with MLSD. The relatively low complexity of OFDM is due to the efficiency of the FFT algorithm. The high complexity of OOK with MLSD is caused by the exponential dependence of the number of trellis states on the channel memory length.

81 Local Area Networks

4 LOCAL AREA NETWORKS

4.1 Introduction

Multicarrier modulation has been proposed to combat inter-symbol interference (ISI) in multimode fibers since the symbol period of each subcarrier can be made long compared to the delay spread caused by modal dispersion [47,48,49].

The main drawback of multicarrier modulation in systems using intensity modulation (IM) is the high DC bias required to make the OFDM waveform nonnegative, as shown in Chapter 3. There are several approaches for reducing the DC power such as DC-clipped OFDM (DC-OFDM) and asymmetrically clipped optical OFDM (ACO-OFDM) [37] as discussed in Chapter 3. Recently, a new technique called pulse-amplitude modulated discrete multitone (PAM-DMT) has been proposed to reduce the DC power for systems using intensity modulation with direct-detection (IM/DD). This technique clips the entire negative excursion of the waveform similarly to ACO-OFDM, but clipping noise is avoided by modulating only the imaginary components of the subcarriers [50].

There have been several studies of the different OFDM techniques (e.g., [51], [52]) in multimode fibers, but to our knowledge, there has been no comparison of power efficiencies among the various OFDM methods, and to conventional baseband methods, such as on-off keying (OOK). Furthermore, in previous work, the powers of the subcarriers were not optimized for finite bit allocation based on the channel frequency response. We present an iterative procedure for DC-OFDM based on known bit-loading algorithms with a new modification, the bias ratio (BR), in order to obtain the optimum power allocation as presented in Chapter 3.

The optimum detection technique for unipolar pulse-amplitude modulation (PAM) in the presence of ISI is maximum-likelihood sequence detection (MLSD), but its computational complexity increases exponentially with the channel memory. ISI in multimode fiber is well-approximated as linear in the intensity (instantaneous power)

82 Local Area Networks

[53], and for typical fibers, PAM with minimum mean-square error decision-feedback equalization (MMSE-DFE) achieves nearly the same performance as MLSD and requires far less computational complexity. Hence, we compare the performance of the three aforementioned OFDM techniques using optimized power allocations to the performance of PAM with MMSE-DFE at different spectral efficiencies.

This chapter is organized as follows. In section 4.2, we review local area networks (LANs) and discrete bit allocation for multicarrier systems, known as the Levin-Campello algorithm [54]. We present our system and fiber models in section 4.3. In that section, we also discuss the performance measures used to compare different modulations formats. In section 4.4, we review the different OFDM formats and study analytically the performance differences between ACO-OFDM and PAM- DMT. In section 4.5, we compare the receiver electrical SNR required to transmit at a given bit rate for the different OFDM formats and for unipolar PAM with MMSE- DFE equalization at different spectral efficiencies.

4.2 Local Area Networks and Discrete Bit Allocation Review

4.2.1 Local Area Networks

Local area networks (LANs) use multimode fiber (MMF) for its low cost and ease of connection since MMF is more tolerant to misalignments. LANs typically operate at 850 nm or 1330 nm and use low cost lasers like vertical-cavity surface- emitting lasers (VCSEL). LANs can support bit rates between 1 Gbit/s and 10 Gbit/s and typical distances are between 300 m to 5 km. Fig. 4.1 shows a block diagram of a LAN.

83 Local Area Networks

Fig. 4.1 LAN block diagram.

LANs use neither optical amplification nor optical dispersion compensation in the link. Fig. 4.2 shows a block diagram of a LAN link.

MMF

TX RX Fig. 4.2 LAN link.

LAN systems use intensity modulation with direct-detection (IM/DD) for its simplicity and reduced hardware cost. IM can be achieved, for example, by direct modulation of the laser current, as shown in Chapter 3.

4.2.2 Discrete Bit Allocation Review

As shown in Chapter 3, on a bipolar channel with AWGN, the optimum power and bit allocations for an OFDM system are given by the water-filling solution. While the optimal value for the number of bits (b = Rb/B) for each subcarrier is an arbitrary nonnegative real number, in practice, the constellation size and FEC code rate need to be adjusted to obtain a rational number of bits. The optimal discrete bit allocation method is known as the Levin-Campello algorithm [54]. We first choose the desired bit granularity β for each subcarrier, i.e., the bit allocation on each subcarrier will be an integer multiple of β. Next, we choose an initial bit allocation for all the subcarriers (not necessarily optimal). We make the initial bit allocation optimum by using the “efficientizing” (EF) algorithm [54]. Then, depending on the system design, we can either use the “E-tightening” algorithm to obtain the maximum achievable bit rate for a given total power Pt or the “B-tightening” algorithm to obtain the minimum total

84 Local Area Networks

power required to transmit at a constant bit rate Rb. We can summarize the Levin- Campello algorithm [54] as

• Choose an initial bit distribution according to β.

• Make the initial bit distribution optimal using the EF algorithm.

• Either use “E-tightening” to obtain the maximum achievable bit rate for a given total power or “B-tightening” to obtain the minimum total power required to transmit at a constant bit rate.

4.3 System Model for LANs

4.3.1 Overall System Model

The OFDM system model is shown in Fig. 4.3.

X0 Y0 X Y 1 Multimode 1 Anti-Aliasing Laser Fiber . Filter . . OFDM OFDM . Tx Rx . .

XN−2 YN−2

XN−1 YN−1

Fig. 4.3 OFDM system model for LANs.

An optical OFDM transmitter encodes transmitted symbols onto an electrical OFDM waveform and modulates this onto the intensity (instantaneous power) of an optical carrier. The modulator can generate one of DC-OFDM, ACO-OFDM or PAM- DMT. Details of modulators for particular OFDM schemes are described in Chapter 3 and in section 4.4.

We assume that there is no optical amplification in the system. After propagating through the multimode fiber, the optical signal intensity is detected, and the electrical current is lowpass filtered. Since we are trying to minimize the optical power required to transmit at given bit rate Rb, we assume the receiver operates in a

85 Local Area Networks regime where signal shot noise is negligible, and the dominant noise is thermal noise arising from the preamplifier following the photodetector. We model the thermal noise as real baseband AWGN with zero mean and double-sided power spectral density

N0/2.

After lowpass filtering, the electrical OFDM signal is demodulated and equalized with a single-tap equalizer on each subcarrier to compensate for channel distortion [51], [52].

4.3.2 Multimode Fiber Model

There have been several studies on accurately modeling modal dispersion in multimode fibers [55,56,57,58]. We can express a fiber’s intensity impulse response as [55], [57]

⎛ ⎞ MMF ()= ⎜∑ (−τδα )⎟ ⊗ pulse thzntnzth )()()(, , (4.1) ⎝ n ⎠ where the set α(n) is known as the mode power distribution, τ(n) is the group delay per th unit length of the n principal mode and hpulse(t) is the pulse shape of a principal mode group. The parameters α(n) and τ(n) depend not only on the fiber index profile but also on the fiber input coupling, the input excitation, connectors offsets, and fibers bends. Hence, these parameters are usually modeled as random variables.

The group delay parameter τ(n) is characterized by the differential-mode delay (DMD), which is the difference between the fasted and slowest principal mode groups. In our model, we choose τ(n) as an exponential random variable, α(n) as a Gaussian power profile and hpulse(t) as a second-order super-Gaussian pulse. We choose the full- width at half-maximum (FWHM) of the pulse shape of a principal mode group

(hpulse(t)) to vary inversely with the mean group delay (E[τ(n)]) such that the fiber 3- dB bandwidth would scale inversely proportional to the fiber length, as presented in [57]. We adjust the parameters such that our set of fibers is very similar to the fibers in [55], [56] at a wavelength of 850 nm. Specifically, we have created 1728 fibers with DMDs between 0.2 and 0.7 ns/km and with 19 principal mode groups propagating.

86 Local Area Networks

Fig. 4.4 shows the mode power distribution, mode delays and frequency response for an exemplary fiber from our set. In our study, we choose a fiber length of 1 km.

0.18 0.7 0.16 0.6 0.14 0.5 0.12 0.1 0.4

0.08 0.3 0.06 0.2

Mode Power Distribution 0.04 Mode Group Delay (ns/km) 0.1 0.02 a) b) 0 0 0 5 10 15 20 0 5 10 15 20 Principal Mode Number Principal Mode Number 0

-10

-20

(|H(f)|) (dB) -30 10

-40 20 log

-50 c)

-60 0 5 10 15 20 25 Frequency (GHz) Fig. 4.4 (a) Mode power distribution, (b) mode delays and (c) frequency response for fiber 1183. We consider a 1-km length in computing the delays and frequency response.

Fig. 4.5 shows the 3-dB bandwidth distribution of all the fibers used in our analysis.

87 Local Area Networks

25

20

15

10 Fiber Distribution (%)

5

0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 3 dB Bandwidth (GHz) Fig. 4.5 3-dB bandwidth distribution of the multimode fibers simulated. All fibers have 1-km length.

4.3.3 Performance Measures

Our baseband channel model is given by

= ⋅ ⊗ MMF + tnthtxRty )()()()( , (4.2) where y(t) is the electrical detected signal, x(t) is the input intensity waveform, hMMF(t) is the fiber intensity impulse response, n(t) is the thermal noise from the photodetector preamplifier, and R is the photodetector responsivity (A/W). This model differs from conventional electrical systems because the channel input represents instantaneous optical power (i.e. intensity). Hence, the channel input is nonnegative ( tx ≥ 0)( ) and the average transmitted optical power Popt is given by

1 T Popt = lim = []txEdttx )()( , (4.3) T ∞→ 2T ∫−T rather than the usual time-average of |x(t)|2 (i.e., E[|x(t)|2]), which is appropriate when x(t) represents amplitude. The average received optical power can be written as

= MMF )0( PHP opt , (4.4)

∞ where HMMF(0) is the DC gain of the channel, i.e., H MMF )0( = MMF )( dtth . ∫ ∞−

88 Local Area Networks

In order to facilitate comparison of the average optical power requirements of different modulations techniques at a fixed bit rate, we define electrical SNR as [59]

PR 22 22 )0( PHR 2 SNR == MMF opt , (4.5) 0 RN b 0 RN b where Rb is the bit rate and N0 is the (single-sided) noise power spectral density. We see that the SNR given by Eq. (4.5) is proportional to the square of the received optical signal power P, in contrast to conventional electrical systems, where it is proportional to the received electrical signal power. Hence, a 2-dB change in the electrical SNR (Eq. (4.5)) corresponds to a 1-dB change in average optical power.

A useful measure of the ISI introduced by a multimode fiber is the temporal dispersion of the impulse response expressed by the channel root-mean-square (rms) delay spread D [60]. The rms delay spread D can be calculated as [60]

2/1 ⎡ 22 ⎤ − μ MMF )()( dttht D = ⎢∫ ⎥ , (4.6) ⎢ 2 )( dtth ⎥ ⎣ ∫ MMF ⎦ where the mean delay μ is given by

2 ⋅ MMF )( dttht μ = ∫ . (4.7) 2 )( dtth ∫ MMF Fig. 4.6 shows the channel root-mean-square (rms) delay spread D distribution of all the fibers used in our analysis.

89 Local Area Networks

12

10

8

6

4 Fiber DistributionFiber (%)

2

0 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 Rms Delay Spread, D (ns) Fig. 4.6 Rms delay spread (D) distribution of the multimode fibers simulated. All fibers have 1-km length.

We will also use the normalized delay spread DT, which is a dimensionless parameter defined as the rms delay spread D divided by the bit period Tb (Tb = 1/Rb):

D DT = . (4.8) Tb

4.4 Analysis of IM/DD OFDM Schemes

4.4.1 DC-Clipped OFDM

We use the same DC-OFDM format with hard-clipping and bias ratio (BR) optimization described in Chapter 3.4.1.

For a high number of subcarriers, we can model the electrical OFDM signal x(t) as a Gaussian random variable with mean equal to the DC bias and variance equal to the electrical power σ 2 = txE 2 ]|)([| . After hard-clipping at zero, we obtain only the positive side of the Gaussian distribution. The average optical power is equal to the average of the clipped waveform and can be written as

+∞ = ⋅= ,()( σγ 2 ) dxNxtxEP opt []clip ∫ , (4.9) 0

90 Local Area Networks where N(γ, σ2) is the Gaussian pdf with mean γ and variance σ2. Doing a variable change, z = x − γ, we obtain

+∞ ) ⋅+= ,0()()( σγ 2 ) dzNztxE []clip ∫ −γ ∞+ ∞+ 2 2 = ,0( σγ ∫∫ ⋅+) ,0( σ ) dzNzdzN , (4.10) −γ −γ

γ 2 − ⎛ ⎛ γ ⎞⎞ σ 2σ 2 γ ⎜1−= Q⎜ ⎟⎟ + e ⎝ ⎝σ ⎠⎠ 2π where Q is the Gaussian Q function [4]. The average optical power for DC-OFDM is then

γ 2 − σ 2 ⎛ ⎛ γ ⎞⎞ 2σ (4.11) P −OFDMDC = []clip )( = etxE γ ⎜1−+ Q⎜ ⎟⎟ . 2π ⎝ ⎝σ ⎠⎠ We note that if the DC bias is chosen such that there is no clipping (γ → ∞), then the optical power is equal to the DC bias γ, as expected. If we choose the DC bias to be proportional to the square root of the electrical power, i.e., γ = BR·σ where BR is the bias ratio discussed in Chapter 3.4.1, then minimizing the electrical power also minimizes the optical power required.

4.4.2 Asymmetrically Clipped Optical OFDM

In our study, we use the same ACO-OFDM format described in Chapter 3.4.2. For a high number of subcarriers, we can model again the electrical OFDM signal x(t) as a Gaussian random variable but for ACO-OFDM the electrical signal has a zero mean. After hard-clipping at zero, we obtain again only the positive side of the Gaussian distribution. Armstrong and Lowery showed that for ACO-OFDM, the average and variance of the electrical waveform are 2/ πσ and σ2 2/ , respectively [61]. Hence, the average optical power for ACO-OFDM is [61]

σ P = . (4.12) ACO−OFDM 2π

91 Local Area Networks

As we can observe in Eq. (4.12), the average optical power is proportional to the square root of the electrical power. We can obtain the same result as Armstrong for ACO-OFDM [61] by setting the DC bias to γ = 0 and Eq. (4.11) simplifies to Eq. (4.12).

4.4.3 PAM-Modulated Discrete Multitone

Similar to ACO-OFDM, PAM-DMT [50] clips the entire negative excursion of the electrical waveform to minimize average optical power. Lee and Koonen [50] showed that if data is modulated using PAM only on the imaginary components of the subcarriers, clipping noise does not affect system performance since that noise is real valued, and is thus orthogonal to the modulation. Fig. 4.7 shows a block diagram of a PAM-DMT transmitter.

0 X0 X1 . . j·PAM Parallel Real Fiber . to XN/2−1 Signal Serial XN/2 IDFT D/A Clip + * XN/2−1 Cyclic . Prefix Optical Spectrum −j·PAM . . X* 1 f * Xk Xk Fig. 4.7 Block diagram of a PAM-OFDM transmitter.

As in DC-OFDM and ACO-OFDM, the OFDM subcarriers for PAM-DMT are assumed to have Hermitian symmetry, so that the time-domain waveform is real. Contrary to ACO-OFDM, in PAM-DMT all of the subcarriers are used, but the modulation is restricted to just one dimension. Hence, PAM-DMT has the same spectral efficiency as ACO-OFDM. After D/A conversion, the electrical OFDM signal is hard-clipped at zero and intensity modulated onto an optical carrier.

92 Local Area Networks

Using an analysis similar to that for ACO-OFDM, Lee and Koonen showed that the average transmitted optical power for PAM-DMT is given by [50]

σ P = , (4.13) PAM −DMT 2π where σ is the square root of the electrical power of the unclipped OFDM waveform.

Up to our best knowledge, no comparison has been done between the performance of ACO-OFDM and PAM-DMT in frequency selective channels. We first compare the performance in the limit of a small number of subcarriers. The minimum number of subcarriers is five, such that two are used for data, the other two are Hermitian conjugates of the data subcarriers and the fifth is the DC subcarrier, which is set to zero. For ACO-OFDM, only one of the two possible subcarriers is used for data transmission. Thus, for a given total power Pt, the normalized bit rate is given by

⎛ HP || 2 ⎞ b ⎜1log += t 1 ⎟ , ACO 2 ⎜ 2 ⎟ (4.14) ⎝ Γσ ⎠ where H1 is the channel gain at the first subcarrier.

In PAM-DMT, all the subcarriers are used for data transmission but the modulation is restricted to one dimension, i.e., PAM. The optimum power allocation for PAM-DMT is given by the waterfiling algorithm in Chapter 3.2.2.2 and can be written as

(σΓ 2 2) n Pn 2 =+ a . (4.15) H n We note that the noise variance in Eq. (4.15) is half than in ACO-OFDM because in PAM-DMT only one dimension is being modulated. The constant a can be found by knowing that Pt = P1 + P2. The optimum power allocation for the two subcarriers is then

93 Local Area Networks

opt Pt P1 += δ 2 , P (4.16) Popt t −= δ 2 2 where δ is given by

1 ⎛ 11 ⎞ σδ 2 ⎜ −Γ= ⎟ . (4.17) 4 ⎜ 2 2 ⎟ ⎝ 2 HH 1 ⎠ The maximum normalized bit rate for PAM-DMT is

2 2 1 ⎛ opt HP ⎞ 1 ⎛ opt HP ⎞ b ⎜1log += 11 ⎟ ⎜1log ++ 22 ⎟ PAM 2 2 ⎜ σ 2 2 ⎟ 2 2 ⎜ σ 2 2 ⎟ ⎝ Γ ⎠ ⎝ Γ ⎠ . (4.18) ⎛ 2 2 ⎞ ⎜ ()t 2 + δ HP 1 ()t 2 −δ HP 2 ⎟ 1log += 1+⋅ 2 ⎜ Γσ 2 2 Γσ 2 2 ⎟ ⎝ ⎠ We now assume that the channel has a lowpass frequency response, such that

|H2| < |H1|. This assumption is valid for the majority of multimode fibers. We obtain an upper bound on Eq. (4.18) as

⎛ ()2 + δ HP 2 ()2 − δ HP 2 ⎞ b ⎜ 1log +≤ t 1 1+⋅ t 1 ⎟ PAM 2 ⎜ Γσ 2 2 Γσ 2 2 ⎟ ⎝ ⎠ ⎛ 2 2 ⎞ ⎜ ⎛ 2 HP 2 ⎞ ⎛ δ H 2 ⎞ ⎟ ⎜ ()t 1 ⎟ ⎜ 1 ⎟ ≤ 2 ⎜ 1log + 2 − 2 ⎟ . ⎜ ⎜ Γσ 2 ⎟ ⎜ Γσ 2 ⎟ ⎟ (4.19) ⎝ ⎝ ⎠ ⎝ ⎠ ⎠ ⎛ HP 2 ⎞ ⎜1log +< t 1 ⎟ 2 ⎜ Γσ 2 ⎟ ⎝ ⎠

< bACO From Eq. (4.19), we observe that ACO-OFDM is more power efficient than PAM-DMT for a small number of subcarriers on lowpass channels. Furthermore, from

Eq. (4.19), we see that if the channel is flat, i.e., |H1| = |H2|, then δ = 0 and both techniques have exactly the same power efficiency.

For a high number of subcarriers, we can apply a similar analysis for each pair of used subcarriers. Since |Hn+1| ≈ |Hn| for a high number of subcarriers, the power

94 Local Area Networks efficiency of PAM-DMT converges asymptotically to that of ACO-OFDM for lowpass channels in the limit of a high number of subcarriers.

4.5 Comparison of IM/DD Modulation Formats

The system model is shown in Fig. 4.3. As mentioned previously, we neglect all transmission impairments except for modal dispersion of the multimode fiber. We employ the fiber model discussed in section 4.3.2, assuming a fiber length of 1 km and transmission at 850 nm.

We assume the dominant noise is thermal noise, modeled as real baseband

AWGN with zero mean and double-sided power spectral density N0/2. We choose N0 = 10−22 A2/Hz, which is a typical value for commercial optical receivers. We assume a photodetector quantum efficiency of 90%, corresponding to responsivity R = 0.6 A/W at 850 nm. At the receiver, the anti-aliasing filter is a fifth-order Butterworth lowpass filter. For OFDM, we set the 3-dB cutoff frequency of the anti-aliasing filter equal to the first null of the OFDM spectrum [25]. For M-PAM, we set the 3-dB cutoff frequency to 0.8Rs, where Rs is the symbol rate.

Typical high-performance FEC codes for optical systems have a threshold bit- −3 error ratio (BER) of the order of Pb = 10 . In order to provide a small margin, we −4 compute the minimum required SNR to achieve Pb = 10 for the different modulation −4 formats. A BER Pb = 10 corresponds to a gap of Г = 6.6 dB. We choose a granularity β = 0.25 for the discrete bit loading algorithms, since it is straightforward to design practical codes whose rates are multiples of 0.25.

We let Rs denote the symbol rate for unipolar M-PAM (in the special case of 2-

OFDM PAM or OOK, Rs = Rb). We let Rs denote the equivalent symbol rate for OFDM [25]. In an attempt to provide a fair comparison between unipolar M-PAM and

OFDM OFDM, unless stated otherwise, we will let the two symbol rates be equal, s = RR s

. For all OFDM formats, we use an oversampling ratio of Ms = 64 /52 ≈ 1.23 to avoid noise aliasing. In this case, OFDM requires an analog-to-digital (A/D) converter

95 Local Area Networks

OFDM sampling frequency of RM ss . While an equalizer for M-PAM can employ an arbitrary rational sampling frequency such as 3/2 to achieve good performance, an oversampling ratio of 2 is often chosen because it yields slightly better performance than 3/2, while greatly simplifying the equalizer structure [16]. Assuming a bit rate Rb

OFDM = 10 Gbit/s and s = RR s , the required A/D sampling frequency is 12.3 GHz for OFDM and 20 GHz for OOK.

Fig. 4.8 shows the receiver electrical SNR required to achieve a bit rate of 10 Gbit/s for several fibers of 1-km length using ACO-OFDM and OOK.

26

24

22

20

18

16 MMSE-DFE OOK (Rs) 14 ACO-OFDM (Rs)

Receiver Electrical SNR (dB) Electrical Receiver 12

10

8 0.1 0.2 0.3 0.4 0.5 1 2

Normalized Delay Spread (DT) −4 Fig. 4.8 Receiver electrical SNR required to obtain 10 Gbit/s at Pb = 10 for ACO-OFDM and OOK for fibers with 1-km length. The bit allocation granularity is β = 0.25 and ACO-OFDM has the same OFDM symbol rate as OOK ( Rs = Rs = 10 GHz).

In order to make our results independent of the bit rate, we present our results in Fig. 4.8 as a function of the normalized delay spread DT. The symbol rate is the

OFDM same for both modulation schemes, i.e., Rs = Rs = 10 GHz. For ACO-OFDM, the

FFT size is N = 1024 and the number of used subcarriers is Nu = 208. We set the cyclic prefix equal to the duration (in samples) of the worst fiber impulse response, corresponding to Npre = 14 samples measured at the OFDM symbol rate. For OOK, we use a fractionally spaced MMSE-DFE at an oversampling ratio of two. We use the

96 Local Area Networks same number of taps for all fibers, which is chosen based on the worst fiber: 41 taps for the feedforward filter and 15 taps for the feedback filter.

In Fig. 4.8, we observe that ACO-OFDM requires a higher SNR than OOK for all fibers in our set when both modulations use the same symbol rate. In this case, which corresponds to a spectral efficiency of 1 bit/s/Hz, the performance difference is about 1 dB. ACO-OFDM requires a higher SNR than OOK because it requires an average of 4 bits (16-QAM) on the used subcarriers to compensate the information rate loss when half of the subcarriers are set to zero. However, we can improve the performance of OFDM by using more optical bandwidth (which also requires more electrical bandwidth). At a symbol rate (or baud rate) Rs, M-PAM requires an optical bandwidth of approximately 2Rs, which corresponds to the interval between spectral nulls on either side of the carrier. On the other hand, OFDM requires a bandwidth of

OFDM OFDM approximately Rs [25]. Hence, for the same symbol rate as M-PAM ( Rs = Rs), OFDM requires half the bandwidth of M-PAM. If we double the symbol rate for OFDM, both modulations schemes use approximately the same optical bandwidth1

2Rs, as shown in Fig. 4.9.

S f RPAM s )( ( )

f 2Rs

S f ROFDM s )2( ( )

OFDM* OFDM f 2Rs Fig. 4.9 Optical spectra of M-PAM and OFDM. The symbol rate for OFDM is twice that for M-PAM, OFDM Rs = 2Rs.

Fig. 4.10 shows the receiver electrical SNR required to achieve a bit rate of 10 Gbit/s for several fibers of length 1 km when all the OFDM formats use twice the

1 The optical bandwidths stated here assume that the source linewidth is small compared to the modulation bandwidth. If the source linewidth is large, changing the symbol rate may have little effect on the optical bandwidth, although it does affect the electrical bandwidth.

97 Local Area Networks

OFDM symbol rate of OOK, i.e., Rs = 2Rs = 20 GHz. The sampling frequency required for OFDM in this case is 1.23×20 = 24.6 GHz. We set again the cyclic prefix equal to the duration of the worst fiber impulse response, obtaining Npre = 24 samples measured at the OFDM symbol rate. The prefix penalty is computed as in [25]. We choose the number of used subcarriers Nu equal to 416 for DC-OFDM, such that the prefix penalty is negligible. The FFT size is N = 1024 for DC-OFDM. For ACO- OFDM, the FFT size is N = 2048, such that the number of used subcarriers is the same as DC-OFDM. For PAM-DMT, we set the FFT size to N = 2048 (Nu = 832), in order to make a fair comparison with ACO-OFDM.

30

DC-OFDM (2Rs) 25

20

15 MMSE-DFE OOK (Rs)

Receiver Electrical SNR (dB) 10

ACO-OFDM (2Rs) 5 0.1 0.2 0.3 0.4 0.5 1 2 Normalized Delay Spread (DT) −4 Fig. 4.10 Receiver electrical SNR required to achieve Pb = 10 at 10 Gbit/s for different modulations formats in fibers with 1-km length. The bit allocation granularity is β = 0.25. The symbol rate for all OFDM OFDM formats is twice that for OOK, Rs = 2Rs = 20 GHz.

For DC-OFDM, we perform the power allocation iteratively using the bias ratio (BR) as described in Chapter 3.5. The minimum required optical SNR is obtained by performing an exhaustive search on the BR value, as shown in Fig. 4.11.

98 Local Area Networks

27.5

27

26.5

26

25.5 Receiver Electrical SNR (dB) 25

24.5 1 1.5 2 2.5 3 3.5 Bias Ratio −4 Fig. 4.11 Receiver electrical SNR required for DC-OFDM to achieve Pb = 10 at 10 Gbit/s for different values of the bias ratio in fiber 295.

In Fig. 4.10, we observe again that OOK with MMSE-DFE requires the lowest −4 SNR to achieve 10 Gbit/s at Pb = 10 for most fibers, even when all OFDM formats use twice the symbol rate of OOK. We note that when OFDM uses twice the symbol rate of OOK, OFDM is allowed to use twice the bandwidth it would normally require to transmit the same bit rate as OOK. Even with this advantage, all OFDM formats still perform worse than OOK for the majority of the fibers, as shown in Fig. 4.10. However, for fibers with a large bandwidth (> 7 GHz), ACO-OFDM outperforms OOK. We note that in computing Fig. 4.10 for OOK with MMSE-DFE, we have used correct decisions at the input of the feedback filter. If we use detected symbols for the feedback filter input, the required SNR increases by 0.4 dB for fiber bandwidths less than 4 GHz, due to error propagation. For fiber bandwidths beyond 4 GHz, the increase in required SNR is less than 0.2 dB. The difference between ACO-OFDM and OOK is within 0.8 and 1 dB, so OOK with error propagation is still more power efficient than ACO-OFDM.

DC-OFDM requires the highest SNR because of the DC bias required to make the OFDM waveform non-negative. Fig. 4.11 shows the receiver electrical SNR required for DC-OFDM for different BR values. We can see that if the BR is too high, then the required SNR is also high, because of the power wasted in the DC bias. On

99 Local Area Networks

the other hand, if the BR is too low, more power has to be allocated to each subcarrier to compensate for the high clipping noise, and therefore the required SNR increases.

Fig. 4.12 compares ACO-OFDM and PAM-DMT using β = 0.25 with ACO- OFDM using continuous bit allocation. In Fig. 4.12, we observe that there is no significant performance difference between PAM-DMT and ACO-OFDM. The difference in SNR between ACO-OFDM and PAM-DMT is less than 0.1 dB. This is to be expected, since for a high number of subcarriers, PAM-DMT converges asymptotically to the same performance as ACO-OFDM. Since PAM-DMT and ACO- OFDM have the same performance, we choose ACO-OFDM as the OFDM format for comparison for the remainder of the chapter.

26

24 PAM-DMT (Discrete, β = 0.25) 22 ACO-OFDM (Discrete, β = 0.25) 20 ACO-OFDM (Continuous)

18

16

14

12

Receiver Electrical SNR (dB) Electrical Receiver 10

8

6 0.1 0.2 0.3 0.4 0.5 1 2

Normalized Delay Spread (DT) Fig. 4.12 Receiver electrical SNR required for various OFDM formats with continuous and discrete bit −4 allocations to achieve Pb = 10 at 10 Gbit/s for fibers of 1-km length. The symbol rate for all OFDM OFDM formats is Rs = 20 GHz.

We also see that there is no significant difference between ACO-OFDM with discrete loading (β = 0.25) and ACO-OFDM with continuous bit allocation. This means that using OFDM with coding rates that are multiples of 0.25 is sufficient to achieve optimal power performance. The subcarrier power distribution for discrete and continuous ACO-OFDM is shown in Fig. 4.13.

100 Local Area Networks

1.2

Continuous 1 ) Discrete (β = 0.25) n,max 0.8 / P n

0.6

0.4 Subcarrier Power (P 0.2

0 0 100 200 300 400 500 Data Subcarrier Number Fig. 4.13 Subcarrier power distribution for ACO-OFDM with continuous bit allocation and with −4 discrete bit loading (with granularity β = 0.25) for 10 Gbit/s at Pb = 10 in fiber 10. The symbol rate is OFDM Rs = 20 GHz.

Fig. 4.14 shows the cumulative distribution function (CDF) of the required electrical SNR, which is defined as

SNR = Prob)(CDF ( ≤ SNRx ), (4.20) where SNR takes all possible values for the required SNR for a given modulation format. We assume that all 1728 fiber realizations occur with equal probability. The

CDF corresponds to the fraction of channels on which the target Pb is reached at a given SNR. For example, Fig. 4.14 shows that if the electrical SNR is at least 25 dB, −4 the target Pb ≤ 10 is met for all the channels when using MMSE-DFE OOK at 10 Gbit/s.

In Fig. 4.14, we observe that OOK with MMSE-DFE requires less SNR for the majority of the fibers, as expected. Furthermore, we also notice that doubling the symbol rate for ACO-OFDM only improves the performance for fibers with low RMS delay spreads. For high RMS delay spreads, doubling the symbol rate gives practically no performance increase.

101 Local Area Networks

1.2

1

0.8

MMSE-DFE OOK (Rs) 0.6 DC-OFDM (2R ) CDF(SNR) ACO-OFDM and s 0.4 PAM-DMT (2Rs)

0.2 ACO-OFDM and PAM-DMR (Rs) 0 5 10 15 20 25 30 Receiver Electrical SNR (dB) Fig. 4.14 Cumulative distribution function (CDF) of the required receiver electrical SNR to obtain Pb = −4 10 at 10 Gbit/s for different modulation formats. The symbol rate for OOK is Rs = 10 GHz and the symbol rate for OFDM is the same or twice that for OOK, as indicated in the figure.

It is also interesting to check if the same performance difference is obtained at higher spectral efficiencies, i.e., for unipolar M-ary PAM (M-PAM). Fig. 4.15 and Fig. 4.16 show the receiver electrical SNR required to achieve a bit rate of 20 Gbit/s for ACO-OFDM and unipolar 4-PAM. The symbol rate for 4-PAM is kept constant at 10 GHz.

35

30

25 ACO-OFDM (Rs)

20

MMSE-DFE 4-PAM (Rs) 15

Receiver Electrical SNR (dB) Electrical Receiver ACO-OFDM (2Rs) 10

5 0.2 0.3 0.4 0.5 1 2 3 4

Normalized Delay Spread (DT) −4 Fig. 4.15 Receiver electrical SNR required to obtain 20 Gbit/s at Pb = 10 for the different

102 Local Area Networks

modulations formats for fibers of 1-km length. The bit allocation granularity is β = 0.25 and the symbol rate for 4-PAM is Rs = 10 GHz. The symbol rate for ACO-OFDM is the same or twice that for 4-PAM, as indicated in the figure.

In Fig. 4.15, we observe that 4-PAM requires the lowest SNR to achieve 20 −4 Gbit/s at Pb = 10 for most fibers. This can also be easily seen in Fig. 4.16. We observe that the difference in SNR between ACO-OFDM and unipolar 4-PAM is about 4 dB when both modulations use the same symbol rate. We conclude that increasing the spectral efficiency from 1 bit/s/Hz (Fig. 4.8) to 2 bit/s/Hz (Fig. 4.15), increases difference in SNR requirement between ACO-OFDM and M-PAM from 1 dB to 4 dB. This is to be expected, since increasing the spectral efficiency from 1 bit/s/Hz to 2 bit/s/Hz for ACO-OFDM requires doubling the average number of bits on each subcarrier from 4 bits (16-QAM) to 8 bits (256-QAM) when the symbol rate for ACO-OFDM is the same as that for M-PAM.

1.2

1

0.8

0.6 ACO-OFDM and

CDF(SNR) PAM-DMR (Rs) 0.4 ACO-OFDM and PAM-DMT (2Rs)

0.2

MMSE-DFE 4-PAM (Rs) 0 5 10 15 20 25 30 35 40 Receiver Electrical SNR (dB) Fig. 4.16 Cumulative distribution function (CDF) of the required receiver electrical SNR to obtain 20 −4 Gbit/s at Pb = 10 for different modulation formats. The symbol rate for 4-PAM is Rs = 10 GHz and the symbol rate for ACO-OFDM is the same or twice that for 4-PAM, as indicated in the figure.

We also observe that ACO-OFDM with twice the symbol rate of 4-PAM has practically the same performance as 4-PAM. Furthermore, ACO-OFDM now outperforms M-PAM for a larger fraction of fibers. However, this performance increase for ACO-OFDM requires doubling the electrical bandwidth and increasing the sampling frequency from 12.3 GHz to 24.6 GHz, which is 23% higher than the

103 Local Area Networks sampling frequency required for 4-PAM using an oversampling ratio of two, i.e., 20 GHz.

104 Personal Area Networks

5 PERSONAL AREA NETWORKS

5.1 Introduction

Indoor optical wireless transmission has been studied extensively in recent decades [62,59,63,64]. The visible and infrared spectral regions offer virtually unlimited bandwidth that is unregulated worldwide. Light in the infrared or visible range penetrates through glass, but not through walls or other opaque barriers, so that optical wireless transmissions are confined to the room in which they originate. Furthermore, in a visible or infrared wireless link employing intensity modulation with direct detection (IM/DD), the short carrier wavelength and large-area photodetector lead to efficient spatial diversity that prevents fading. Nevertheless, the existence of multiple paths between the transmitter and receiver causes multipath distortion, particularly in links using non-directional transmitters and receivers, or in links relying upon non-line-of-sight propagation [62], [59]. This multipath distortion can lead to inter-symbol interference (ISI) at high bit rates.

Multicarrier modulation has been proposed to combat ISI in optical wireless links, since the symbol period of each subcarrier can be made long compared to the delay spread caused by multipath distortion [65]. Multicarrier modulation is usually implemented by orthogonal frequency-division multiplexing (OFDM) [63,66,38]. The main drawback of multicarrier modulation in systems using intensity modulation (IM) is the high DC bias required to make the multicarrier waveform nonnegative. There have been several approaches for reducing the DC bias in IM OFDM systems, such as DC-clipped OFDM (DC-OFDM) [63], [66], asymmetrically clipped optical OFDM (ACO-OFDM) [38] and PAM-modulated discrete multitone (PAM-DMT) [50].

There have been several studies comparing the performance of different OFDM techniques (e.g., [38]) but these comparisons have been made for ideal additive white Gaussian noise (AWGN) channels. To our knowledge, previous studies have not compared the OFDM methods to conventional baseband methods, such as

105 Personal Area Networks on-off keying (OOK) or unipolar pulse-amplitude modulation (PAM), nor have they considered the dispersive nature of optical wireless channels. Furthermore, in previous work, the powers of the subcarriers and the DC bias for DC-OFDM were not jointly optimized according to the channel frequency response in order to obtain the lowest required optical power. We present an iterative procedure for DC-OFDM based on known bit-loading algorithms with a new modification, the bias ratio, in order to obtain the optimum power allocation.

The optimum detection technique for unipolar PAM in the presence of ISI is maximum-likelihood sequence detection (MLSD), but its computational complexity increases exponentially with the channel memory. ISI in optical wireless links is well- approximated as linear in the instantaneous power [59], and for typical wireless links, PAM with minimum mean-square error decision-feedback equalization (MMSE-DFE) achieves nearly the same performance as MLSD and requires far less computational complexity. Hence, we compare the performance of the three aforementioned OFDM techniques using optimized power allocations to the performance of PAM with MMSE-DFE at different spectral efficiencies.

This chapter is organized as follows. We introduce the optical wireless networks in section 5.2. We present our system and indoor optical wireless models in section 5.3. In section 5.4, we compare the receiver electrical SNR required to transmit at several bit rates for the different OFDM formats and for unipolar M-PAM with MMSE-DFE equalization at different spectral efficiencies. Furthermore, we compare the receiver electrical SNR required for the different modulation formats when there is no channel state information (CSI) available at the transmitter. Finally, we also compare the computational complexity required for OFDM and M-PAM at different bit rates in section 5.5.

5.2 Personal Area Networks

Personal area networks (PANs) typically use optical wireless for its low cost. Indoor optical wireless networks operate at infrared or visible light since these spectral

106 Personal Area Networks regions offer virtually unlimited bandwidth that is unregulated worldwide. Optical wireless networks typically use noncoherent light sources such as light-emitting diodes (LED). LEDs emit an average power of several tens of mW that is concentrated within a semiangle of 15° –30°. LED emission wavelength typically lies between 850 and 950 nm and are cheaper than lasers. PANs can support bit rates between 10 Mbit/s and 300 Mbit/s and the transmission is normally confined within a room. Fig. 5.1 shows a block diagram of an indoor optical wireless system.

Fig. 5.1 Indoor optical wireless transmission.

Indoor optical wireless use neither optical amplification nor optical dispersion compensation in the link. Typically, the transmitter is placed in the ceiling and the receiver is located at desk floor (~ 1 m), as shown in Fig. 5.1. Optical wireless systems use intensity modulation with direct-detection (IM/DD) for its simplicity and reduced hardware cost.

5.3 System Model and Performance Measures

5.3.1 Overall System Model

The OFDM system model is shown in Fig. 5.2.

107 Personal Area Networks

X0 Y0

X1 Y1 Noise . . Optical . OFDM . OFDM h(t) + Rx . Tx . Indoor Wireless Anti-Aliasing XN−2 Channel Filter YN−2

XN−1 YN−1

Fig. 5.2 OFDM system model for LANs.

An OFDM modulator encodes transmitted symbols onto an electrical OFDM waveform and modulates this onto the instantaneous power of an optical carrier at infrared or visible frequencies. The modulator can generate one of DC-OFDM, ACO- OFDM or PAM-DMT. Details of modulators for particular OFDM schemes are described in Chapter 4.4. After propagating through the indoor wireless link, the optical signal intensity is detected and the electrical photocurrent is low-pass filtered. Since we are trying to minimize the optical power required to transmit at given bit rate

Rb, we assume the receiver operates in a regime where signal shot noise is negligible, and the dominant noise is the shot noise from detected background light or thermal noise from the preamplifier following the photodetector. After low-pass filtering, the electrical OFDM signal is demodulated and equalized with a single-tap equalizer on each subcarrier to compensate for channel distortion [25].

5.3.2 Optical Wireless Channel

Multipath propagation in an indoor optical wireless channel [67], [68] can be described by an impulse response h(t) or by the corresponding baseband frequency

∞ response = )()( − 2πftj dtethfH . Including noise, the baseband channel model is ∫ ∞− [59]:

= ⋅ ⊗ + tnthtxRty )()()()( , (5.1) where y(t) is the detected photocurrent, x(t) is the transmitted intensity waveform, R is the photodetector responsivity, and n(t) represents ambient light shot noise and

108 Personal Area Networks thermal noise. Optical wireless channels differ from electrical or radio frequency channels because the channel input x(t) represents instantaneous optical power. Hence, the channel input is nonnegative ( tx ≥ 0)( ) and the average transmitted optical power

Pt is given by

1 T Popt = lim = []txEdttx )()( , (5.2) T ∞→ 2T ∫−T rather than the usual time-average of |x(t)|2 (i.e., E[|x(t)|2]), which is appropriate when x(t) represents amplitude. The average received optical power can be written as

= )0( PHP t , (5.3)

∞ where H(0) is the DC gain of the channel, i.e., = )()0( dtthH . ∫ ∞−

We use the methodology developed by Barry et al [67] to simulate the impulse responses of indoor optical wireless channels, taking account of multiple bounces. A similar model can be found in [69]. The algorithm in [67] partitions a room into many elementary reflectors and sums up the impulse response contributions from kth-order bounces, ()k kth = ,2,1,0),( … . More recently, Carruthers developed an iterative version of the multi-bounce impulse response algorithm which greatly reduces the computational time required to accurately model optical wireless channels [70]. In this study, we use a toolbox developed by Carruthers to implement the algorithm in [70].

We place the transmitter and receiver in a room with dimensions 8 m × 6 m × 3 m (length, width, height). Room surfaces are discretized with a spatial resolution of 0.2 m, and are assumed to have diffuse reflectivities as in configuration A in [67]. All the source and receiver parameters are the same as in configuration A in [67]. We assume the transmitter has a Lambertian radiation pattern, with intensity per unit solid angle proportional to the cosine of the angle with respect to the transmitter normal [67]. We also assume that the receiver area is equal to 1 cm2 and it only detects light whose angle of incidence is less than 90º with respect to the receiver normal [67]. According to [70], considering three to five bounces in calculating the impulse

109 Personal Area Networks response should be sufficient to accurately model most indoor environments with typical reflectivities and geometries. Hence, we consider five bounces.

We generate 170 different channels by placing the transmitter in different locations within the room. In all channel realizations, the receiver is placed in the middle of the room 1 m above the floor and pointed upward. Our channel ensemble includes line-of-sight (LOS) configurations (transmitter placed at the ceiling and pointed down) and diffuse configurations (transmitter placed 1 m above the floor and pointed up). In order to simulate shadowing by a person or object next to the receiver, we block the LOS path (i.e., h(0)(t)) of some of the impulse responses. Fig. 5.3 shows the impulse response for an exemplary diffuse channel from our set.

20 18 h(t) )

-1 16 14 12 10 h(1)(t) 8 h(2)(t) 6

Impulse response (s (3) 4 h (t) h(4)(t) 2 h(5)(t) 0 10 20 30 40 50 60 70 80 90 100 Ti m e (ns) Fig. 5.3 Impulse response of an exemplary non-directional, non-LOS (diffuse) channel. This channel has no LOS component h(0)(t). The contributions of the first five reflections, h(1)(t),…, h(5)(t), are shown.

5.3.3 Performance Measures

In order to compare the average optical power requirements of different modulation techniques at a fixed bit rate, we define electrical SNR as [59]

PR 22 )0( PHR 222 SNR == t , (5.4) 0 RN b 0 RN b

110 Personal Area Networks

where Rb is the bit rate and N0 is the (single-sided) noise power spectral density. We see that the SNR given by Eq. (5.4) is proportional to the square of the received optical signal power P, in contrast to conventional electrical systems, where it is proportional to the received electrical signal power. Hence, a 2-dB change in the required electrical SNR Eq. (5.4) corresponds to a 1-dB change in the required average optical power.

Given a channel’s impulse response h(t), we compute the channel’s root-mean- square (rms) delay spread D using [68], [60]

∞ 2/1 ⎡ − μ 22 )()( dttht ⎤ ⎢∫ ∞− ⎥ , ()()thD = ∞ (5.5) ⎢ 2 )( dtth ⎥ ⎣⎢ ∫ ∞− ⎦⎥ where the mean delay μ is given by

∞ 2 )( dttth ∫ ∞− . μ = ∞ (5.6) 2 )( dtth ∫ ∞−

We also compute the normalized delay spread DT, which is the rms delay spread D divided by the bit period Tb (Tb = 1/Rb).

D DT = . (5.7) Tb

The normalized delay spread DT is known to be a useful measure of multipath- induced ISI, at least when using OOK or pulse-position modulation (PPM). In [68], [60], it was found that over a wide ensemble of experimentally measured channels, for a given modulation and equalization method, the ISI power penalty on a given channel can be predicted with reasonable accuracy based solely on the normalized delay spread DT, independent of the particular time dependence of the impulse response h(t).

5.3.4 Ceiling-Bounce Model

Carruthers and Kahn [60] derived a channel model based on a diffuse link comprising a Lambertian transmitter co-located with a non-directional receiver, both

111 Personal Area Networks directed toward an infinite diffuse reflector. Considering a single bounce, the impulse response can be obtained in closed form as

6a6 th )( = tu )( , (5.8) c ()+ at 7 where a is the round-trip propagation time and u(t) is the unit step function. The impulse response given by Eq. (5.8) is normalized such that H = 1)0( . The rms delay spread is related to a by

a 13 ()thD )( = . (5.9) c 12 11 Carruthers and Kahn referred to the impulse response in Eq. (5.8) as the ceiling-bounce (CB) model. They found that the simple functional form of the impulse response given by Eq. (5.8) exhibits approximately the same relationship between ISI penalty and normalized delay spread DT as a wide ensemble of experimentally measured channels, at least for OOK and PPM [60].

5.4 Comparison of IM/DD Modulation Formats

In the following analysis, we use the system model shown in Fig. 5.2. We employ the wireless channel model discussed in section 5.3.2. In our analysis, we consider the IM/DD OFDM formats described in Chapter 4, such as DC-OFDM, ACO-OFDM and PAM-DMT.

We model the dominant noise as real baseband AWGN with zero mean and double-sided power spectral density N0/2. We assume a photodetector quantum efficiency of 90%, which corresponds, for example, to a responsivity of R = 0.6 A/Hz at 850 nm. At the receiver, the anti-aliasing filter is a fifth-order Butterworth low-pass filter. For OFDM, we set the 3-dB cutoff frequency of the anti-aliasing filter equal to the first null of the OFDM spectrum [25]. For M-PAM, we set the 3-dB cutoff frequency to 0.8Rs, where Rs is the symbol rate. We note that the anti-aliasing filter cannot cause noise enhancement for any of the modulation formats, since the noise is added before the anti-aliasing filter. In other words, the performance comparison

112 Personal Area Networks between the modulation formats is independent of the anti-aliasing filter type, given an adequate receiver oversampling ratio and an adequate cyclic prefix length or number of equalizer taps.

We would like to minimize the oversampling ratios for OFDM and M-PAM in order to minimize the A/D sampling frequency while still obtaining optimal performance. In OFDM, oversampling is performed by inserting zero subcarriers, and hence, it is possible to employ arbitrary rational oversampling ratios. For OFDM, we find that an oversampling ratio Ms = 64 /52 ≈ 1.23 is sufficient to obtain optimal performance. For M-PAM, arbitrary rational oversampling ratios are possible, but require complex equalizer structures [16]. While good performance can be achieved at an oversampling ratio of 3/2, we choose an oversampling ratio of 2, since it achieves slightly better performance than 3/2 while minimizing equalizer complexity. Note that in this case, OOK requires an A/D sampling frequency 63% higher than OFDM.

Typical high-performance forward error-correction (FEC) codes for optical −3 systems have a bit-error ratio (BER) thresholds of the order of Pb = 10 [71]. In order for our system to be compatible with such FEC codes while providing a small margin, −4 we compute the minimum required SNR to achieve Pb = 10 for the different modulation formats.

On a bipolar channel with AWGN, the optimum power and bit allocations for an OFDM system are given by the water-filling solution [4]. While the optimal value for the number of bits for each subcarrier is an arbitrary nonnegative real number, in practice, the constellation size and FEC code rate need to be adjusted to obtain a rational number of bits. The optimal discrete bit allocation method on bipolar channels is known as the Levin-Campello algorithm [54]. In this algorithm, we first set the desired bit granularity β for each subcarrier, i.e., the bit allocation on each subcarrier is an integer multiple of β (Chapter 4). In our study, we choose a granularity β = 0.25, since it is straightforward to design codes whose rates are multiples of 0.25 [72]. Furthermore, we observe that there is no significant performance difference between OFDM with discrete loading (β = 0.25) and with continuous loading (Chapter 4).

113 Personal Area Networks

5.4.1 OOK and OFDM Performance

In this section, we analyze the effect of delay spread on the required SNR for different modulation formats at a spectral efficiency of 1 bit/symbol. We present our results using the normalized delay spread DT, given by Eq. (5.7).

We let Rs denote the symbol rate for unipolar M-PAM (for 2-PAM or OOK, Rs

OFDM = Rb). We let Rs denote the equivalent symbol rate for OFDM [25]. In order to perform a fair comparison at a given fixed bit rate Rb between unipolar M-PAM and

OFDM OFDM, we initially set the two symbol rates to be equal: s = RR s [4].

Fig. 5.4 shows the electrical SNR required to achieve bit rates of 50, 100 and 300 Mbit/s for all channels using ACO-OFDM and OOK. In Fig. 5.4, the symbol rates

OFDM are the same for both modulation schemes, i.e., Rs = Rs, and they are chosen to be 50, 100 and 300 MHz, respectively. The receiver and transmitter electrical bandwidths are scaled accordingly to the symbol rate in use.

30

25

ACO-OFDM (Rs)

20

15

OOK MMSE-DFE (Rs) Required Electrical SNR (dB) SNR Electrical Required 10

5 0.001 0.01 0.1 1 10 Delay RMS Delay Spread DT = Duration Bit Duration −4 Fig. 5.4 Electrical SNR required to achieve Pb = 10 vs. normalized delay spread DT at bit rates of 50, 100 and 300 Mbit/s (spectral efficiency of 1 bit/symbol) for ACO-OFDM and OOK. The bit allocation granularity is β = 0.25 and the symbol rates for ACO-OFDM are the same as those for OOK, as indicated.

114 Personal Area Networks

For ACO-OFDM with a symbol rate of 300 MHz, the FFT size is N = 1024 and the number of used subcarriers is Nu = 208. We set the cyclic prefix equal to three times the rms delay spread (D) of the worst impulse response, corresponding to ν = 18 samples measured at the OFDM symbol rate. For OOK, we use a fractionally spaced MMSE-DFE at an oversampling ratio of 2. We use the same number of taps for all wireless channels, which is chosen based on the worst channel: 98 taps for the feedforward filter and 28 taps for the feedback filter when the symbol rate is 300 MHz. The parameters for the various modulation formats and bit rates are listed in Table 5.1.

In Fig. 5.4, we notice that as the normalized delay spread increases, the available electrical bandwidth decreases and hence more power is required to maintain the bit rate. We also observe that ACO-OFDM requires a higher SNR than OOK for all channels in our set when both modulations use the same symbol rates to transmit the same bit rates. In this case, which corresponds to a spectral efficiency of 1 bit/symbol, the performance difference is about 1.3 dB. ACO-OFDM requires a higher SNR than OOK because it requires an average of 4 bits (16-QAM constellation) on the used subcarriers to compensate the information rate loss of setting half of the subcarriers to zero.

One option to improve the performance of OFDM is to use more electrical bandwidth in order to reduce the constellation size on each subcarrier. We define 2 electrical bandwidth (Be) as the span from DC to the location of the first spectral null of the power spectral density (PSD) of the transmitted waveform x(t) [59]. At a symbol rate Rs, M-PAM with rectangular non-return-to-zero (NRZ) pulses requires an

PAM electrical bandwidth of approximately e ≈ RB s . On the other hand, OFDM at

OFDM OFDM OFDM symbol rate Rs only requires an electrical bandwidth of only e ≈ RB s 2/ [25], since it has a more confined spectrum. Hence, for the same symbol rate as M-

2 We note that the electrical bandwidth of a modulation technique has little influence on the optical bandwidth Bo occupied by an IM signal when using typical broadband light sources (e.g., a light emitting diodes or multimode laser diodes) [59], since the optical bandwidth Bo is dominated by the source linewidth. For example, a 1-nm linewidth corresponds to 469 GHz, assuming a wavelength of 800 nm.

115 Personal Area Networks

OFDM PAM ( Rs = Rs), OFDM requires about half the bandwidth of M-PAM with rectangular NRZ pulses. If we double the symbol rate for OFDM, both modulations

OFDM PAM schemes use approximately the same electrical bandwidth: e e == RBB s . The electrical bandwidth required for all modulation formats is summarized in Table 5.1.

Bit Rate DC-OFDM ACO-OFDM PAM-DMT OOK (Mbit/s) RS 2·RS RS 2·RS RS 2·RS RS N = 512 N = 512 N = 1024 N = 1024 N = 1024 N = 1024 N = 28 N = 208 N = 208 N = 208 N = 208 N = 416 N = 416 f u u u u u u N = 6 50 ν = 10 ν = 12 ν = 10 ν = 12 ν = 10 ν = 12 b Be = 50 Be = 25 Be = 50 Bo = 25 Be = 50 Be = 25 Be = 50 N = 512 N = 512 N = 1024 N = 1024 N = 1024 N = 1024 N = 44 N = 208 N = 208 N = 208 N = 208 N = 416 N = 416 f u u u u u u N = 10 100 ν = 12 ν = 15 ν = 12 ν = 15 ν = 12 ν = 15 b Be = 100 Be = 50 Be = 100 Bo = 50 Be = 100 Be = 50 Be = 100 N = 512 N = 1024 N = 1024 N = 2048 N = 1024 N = 2048 N = 98 N = 208 N = 416 N = 208 N = 416 N = 416 N = 832 f u u u u u u N = 28 300 ν = 18 ν = 26 ν = 18 ν = 26 ν = 18 ν = 26 b Be = 300 Be = 150 Be = 300 Bo = 150 Be = 300 Be = 150 Be = 300 Table 5.1 System parameters for the various modulation formats for different bit rates and symbol rates. N is the DFT size, Nu is the number of used subcarriers, ν is the cyclic prefix, Nf is the number of taps of the feedforward filter, Nb is the number of taps of the feedback filter and Be is the required electrical bandwidth in MHz.

However, we note that when OFDM uses twice the symbol rate of M-PAM, OFDM uses twice the electrical bandwidth it would normally require to transmit the same bit rate as M-PAM. We also note that when the OFDM symbol rate is doubled, OFDM requires an analog-to-digital (A/D) converter sampling frequency of

OFDM ss ⋅= 2RMRM ss , which is 23% higher than the sampling frequency required for M-PAM using an oversampling ratio of 2. We also want to point out that the usable electrical bandwidth becomes eventually limited by the channel multipath distortion. Hence, for channels with high delay spreads, there is very little benefit in increasing the transmitter and receiver electrical bandwidths.

Having the previous considerations in mind, we study the trade-off between electrical bandwidth and SNR performance for OFDM in Fig. 5.5 and Fig. 5.6. Fig. 5.5 shows the receiver electrical SNR required to achieve the bit rates of 50, 100 and

OFDM 300 Mbit/s when ACO-OFDM uses twice the symbol rate of OOK, i.e., Rs = 2Rs.

116 Personal Area Networks

The symbol rates for OOK are 50, 100 and 300 MHz, respectively and the symbol rates for ACO-OFDM are 100, 200 and 600 MHz, respectively. The receiver and transmitter electrical bandwidths are scaled according to the symbol rate in use.

30

25

20

15 OOK MMSE-DFE (Rs)

ACO-OFDM (2Rs)

Required SNR Electrical (dB) 10 CB for OOK (Rs) CB for ACO-OFDM (2Rs) 5 0.001 0.01 0.1 1 10 Delay RMS Delay Spread DT = Duration Bit Duration −4 Fig. 5.5 Electrical SNR required to achieve Pb = 10 vs. normalized delay spread DT at bit rates of 50, 100 and 300 Mbit/s (spectral efficiency of 1 bit/symbol) for different modulations formats. The dashed lines correspond to the SNR requirement predicted using the ceiling-bounce (CB) model. The bit allocation granularity is β = 0.25 and the symbol rates for ACO-OFDM are twice those for OOK, as indicated.

In Fig. 5.5, we observe that ACO-OFDM using twice the symbol rate of OOK −4 requires the lowest SNR to achieve a Pb = 10 for all channels at the bit rates of 50,

100 and 300 Mbit/s. For low delay spreads (DT < 0.1), corresponding to LOS channels, ACO-OFDM significantly outperforms OOK. For high delay spreads, however, OOK with MMSE-DFE performs very close to ACO-OFDM. We again note that when OFDM uses twice the symbol rate of OOK, OFDM uses twice the bandwidth it would normally require to transmit the same bit rate as OOK. Even with this advantage, for a given bit rate, ACO-OFDM achieves approximately the same performance as OOK for high delay spreads, as shown in Fig. 5.5. ACO-OFDM performs better for low delay spreads because it can use more bandwidth and transmit lower constellation sizes on each subcarrier. For high delay spreads, the wireless channel becomes more bandwidth-constrained, and there is very little gain by using more bandwidth for ACO-OFDM.

117 Personal Area Networks

In Fig. 5.5, we also show the SNR requirements for OOK and ACO-OFDM estimated using the ceiling-bounce (CB) model [60]. Carruthers and Kahn [60] found a simple functional approximation of the impulse response that exhibits approximately the same relationship between ISI penalty and normalized delay spread DT as a wide ensemble of experimentally measured channels, at least for OOK and PPM [60]. We observe that the CB model provides a reasonable is estimate of the SNR requirement. We note that in computing Fig. 5.5 for OOK with MMSE-DFE, we have assumed correct decisions at the input of the feedback filter.

In order to make a fair comparison with ACO-OFDM, the number of used subcarriers in DC-OFDM should be the same as for ACO-OFDM. For PAM-DMT, since only one dimension is used to transmit data, the number of used subcarriers should be twice that for ACO-OFDM. In computing the optimized subcarrier power allocation for DC-OFDM, we use an iteratively water-filling solution with the bias ratio BR, as described in Chapter 3.5, performing an exhaustive search over the value of BR as in [73]. We have plotted the SNR requirements for DC-OFDM and PAM- DMT separately in Fig. 5.6, in order to make Fig. 5.5 more legible.

Fig. 5.6 shows the cumulative distribution function (CDF) of the required electrical SNR, which is defined as:

SNR = Prob)(CDF ( ≤ SNRx ), (5.10) where SNR takes all possible values for the required SNR for a given modulation format. We assume that all 170 channels realizations occur with equal probability. The

CDF corresponds to the fraction of channels on which the target Pb is reached at a given SNR. For example, Fig. 5.6 shows that if the transmitter has CSI and the −4 electrical SNR is at least 30 dB, the target Pb ≤ 10 is met for all the channels when using ACO-OFDM (or PAM-DMT) at any of the three bit rates, 50, 100 or 300 Mbit/s.

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1.2

1

0.8 ACO-OFDM and DC-OFDM (2Rs) PAM-DMT (2Rs) 0.6

CDF(SNR) ACO-OFDM and 0.4 PAM-DMT (Rs)

0.2

OOK MMSE-DFE (Rs) 0 5 10 15 20 25 30 35 Required Electrical SNR (dB) −4 Fig. 5.6 Cumulative distribution function (CDF) of the electrical SNR required to achieve Pb = 10 at bit rates of 50, 100 and 300 Mbit/s (spectral efficiency of 1 bit/symbol) for different modulation formats. The symbol rates for OFDM are the same as or twice those for OOK, as indicated.

We observe in Fig. 5.6, as in Fig. 5.5, that ACO-OFDM using twice the −4 symbol rate of OOK requires the lowest SNR to achieve a Pb = 10 . On the other hand, if ACO-OFDM uses the same symbol rate as OOK, OOK is more power efficient. Furthermore, we verify that there is no significant performance difference between ACO-OFDM and PAM-DMT. This is expected, since it was proven in [74] that ACO-OFDM and PAM-DMT achieve very similar performance on low-pass channels. Finally, we observe that DC-OFDM requires the highest SNR because of the optical power used in the DC bias to reduce clipping noise.

5.4.2 Unipolar 4-PAM and OFDM Performance

In this section, we compare the performance of the different modulation formats at higher spectral efficiencies to check if there are any significant differences from the previous section. Fig. 5.7 and Fig. 5.8 show the electrical SNR required to achieve bit rates of 100, 200 and 600 Mbit/s for different modulation formats at a spectral efficiency of 2 bit/symbol. The symbol rates for unipolar 4-PAM are 50, 100 and 300 MHz, respectively. The receiver and transmitter electrical bandwidths are scaled according to the symbol rate in use.

119 Personal Area Networks

45

40

35 ACO-OFDM (Rs)

30

4-PAM MMSE-DFE (R ) 25 s

20 Required Electrical SNR (dB)

15 ACO-OFDM (2Rs)

10 0.01 0.1 1 10 Delay RMS Delay Spread DT = Duration Bit Duration −4 Fig. 5.7 Electrical SNR required to achieve Pb = 10 vs. normalized delay spread DT at bit rates of 100, 200 and 600 Mbit/s (spectral efficiency of 2 bit/symbol) for different modulations formats. The bit allocation granularity is β = 0.25 and the symbol rates for ACO-OFDM are the same as or twice those for 4-PAM, as indicated.

In Fig. 5.7 and Fig. 5.8, we observe that ACO-OFDM and PAM-DMT using twice the symbol rate of 4-PAM again requires the lowest SNR for all channels.

1.2

1

ACO-OFDM and DC-OFDM (2Rs) 0.8 PAM-DMT (2Rs)

0.6 DC-OFDM (Rs) CDF(SNR) 0.4 ACO-OFDM and PAM-DMT (Rs)

0.2 4-PAM MMSE-DFE (Rs)

0 10 15 20 25 30 35 40 45 Required Electrical SNR (dB) −4 Fig. 5.8 Cumulative distribution function (CDF) of the electrical SNR required to achieve Pb = 10 at bit rates of 100, 200 and 600 Mbit/s (spectral efficiency of 2 bit/symbol) for different modulation formats. The symbol rates for OFDM are the same as or twice those for 4-PAM, as indicated.

120 Personal Area Networks

In Fig. 5.8, we observe again that PAM-DMT has the same performance as ACO-OFDM for any symbol rate. Since both modulations have the same performance, we choose ACO-OFDM as our basis of comparison for the remainder of the chapter. The performance improvement for ACO-OFDM (and PAM-DMT) in Fig. 5.7 and Fig. 5.8 requires doubling the bandwidth and increasing the sampling frequency, which becomes 23% higher than the sampling frequency required for 4-PAM with an oversampling ratio of 2. Comparing Fig. 5.6 and Fig. 5.8, we see that the performance difference between ACO-OFDM with twice the symbol rate and 4-PAM increases with the spectral efficiency. We also observe in Fig. 5.8 that DC-OFDM with twice the symbol rate performs close to 4-PAM.

When all the modulations use equal symbol rates, 4-PAM requires the lowest SNR of all the modulation formats, and the difference in SNR between ACO-OFDM (or PAM-DMT) and unipolar 4-PAM is about 4 dB. We conclude that increasing the spectral efficiency from 1 bit/symbol (Fig. 5.6) to 2 bit/symbol (Fig. 5.8) increases differences in SNR requirements between ACO-OFDM and M-PAM from 1 dB to 4 dB. This is to be expected, since increasing the spectral efficiency from 1 bit/symbol to 2 bit/symbol for ACO-OFDM requires doubling the average transmitted number of bits on each subcarrier from 4 bits (16-QAM) to 8 bits (256-QAM). For higher spectral efficiencies, ACO-OFDM require constellations that are increasingly large and therefore M-PAM has increasingly better performance than ACO-OFDM with equal symbol rates.

Finally, we also note in Fig. 5.8 that DC-OFDM performs very close to ACO- OFDM (or PAM-DMT) when the OFDM symbol rate is the same as that for 4-PAM and the spectral efficiency is 2 bit/symbol. This is because the clipping noise penalty in DC-OFDM becomes less significant when compared to the additional SNR required to transmit higher constellation sizes in each ACO-OFDM subcarrier.

Although DC-OFDM uses twice as many subcarriers as ACO-OFDM (or PAM-DMT) to transmit data, the clipping noise and DC bias limit the power efficiency of DC-OFDM for a spectral efficiency of 1 bit/symbol, as shown in Fig. 5.6. However, as spectral efficiency increases, the clipping noise penalty in DC-

121 Personal Area Networks

OFDM becomes less significant when compared to the additional SNR required to transmit higher constellation sizes in each ACO-OFDM subcarrier, and eventually DC-OFDM performs better than ACO-OFDM.

5.4.3 Outage Probability

It is also useful to analyze the receiver performance when the transmitter does not have CSI. This happens, for example, in a broadcast configuration where the transmitter sends the same information to several receivers or when there is no return path from the receiver (i.e., uplink). When CSI is unavailable at the transmitter, some data-bearing subcarriers might be lost due to multipath distortion. Hence, we use coding across the subcarriers such that the information can be recovered from the good subcarriers. We choose a shortened Reed-Solomon (RS) code with (n, k) = (127, 107) over GF(8) for simplicity3. In order for the comparison between OFDM and OOK to be fair, OOK also employs this code.

−4 We consider a channel realization to be in outage if Pb > 10 for a desired bit −4 rate. We define outage probability Poutage as the fraction of channels with Pb > 10 over the entire ensemble of channels realizations. In practice, a system might not be expected to work on the most severe channel realizations (e.g., shadowed channels), but in our study we include all channels as a worst-case scenario.

We assume a desired bit rate of 300 Mbit/s. For OOK, we increase the symbol rate from 300 MHz to 356 MHz to compensate for coding overhead. For OFDM, we use ACO-OFDM with twice the data symbol rate of OOK, i.e., 600 MHz. We compensate for the information loss due to coding by increasing the constellation size on each subcarrier. The same performance could be achieved with PAM-DMT.

3 In an optimized implementation, this code (used to recover subcarriers lost due to multipath distortion) might be integrated with the outer FEC code (used to combat additive noise) into a single concatenated coding scheme. Likewise, more powerful coding schemes (such as turbo or low-density parity-check codes) could be used to recover the lost subcarriers. Given the simple nature of indoor optical wireless channels, we would expect our conclusions to remain valid even if more optimized coding schemes were employed.

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We assume that the channel changes slowly enough such that many OFDM blocks are affected by a channel realization. In this scenario, there is no benefit to using time diversity (e.g., interleaving) in the bit allocation on the subcarriers. We also assume that the transmitter has knowledge of the mean channel, which has a frequency response defined by

1 Nrealiz mean ()fH = ∑ k ()fH , (5.11) Nrealiz k=1 which is an average over all 170 channel realizations. For ACO-OFDM, we consider three different bit allocation schemes: mean channel loading, where the transmitter does integer bit loading using the frequency response of the mean channel; ceiling-bounce loading, where the transmitter performs integer bit allocation on a ceiling-bounce channel having the same rms delay spread D as the mean channel,4 and equal loading, where all the subcarriers carry the same number of bits. After having performed the bit allocation, we sweep the transmitter power and observe which −4 channel realizations are in outage, i.e., have Pb > 10 .

Since each RS block in OFDM has a different uncoded Pb, we simulate RS decoding with error counting in order to obtain the correct Pb after RS decoding. For

OOK, we use a binomial expansion to obtain Pb after RS decoding, since each bit has the same uncoded Pb.

Fig. 5.9 shows the outage probability for coded ACO-OFDM and coded OOK with MMSE-DFE for a desired bit rate of 300 Mbit/s.

4 For the purpose of computing the rms delay spread D of the mean channel, the impulse response of the mean channel is computed as the inverse Fourier transform of Hmean(f).

123 Personal Area Networks

120

100 ACO-OFDM (CB Loading)

80 (%) 60 ACO-OFDM

outage (Equal Loading) P 40 ACO-OFDM (Mean Channel Loading) OOK 20 MMSE-DFE

0 5 10 15 20 25 30 35 40 Required Electrical SNR (dB) Fig. 5.9 Outage probability for OOK and ACO-OFDM with coding and various bit allocations averaged over all channels. All modulation formats use the code RS(127,107) over GF(8). The information bit rate is 300 Mbit/s (spectral efficiency of 1 bit/symbol) and the symbol rates for OOK and OFDM are 356 MHz and 600 MHz, respectively. The bit allocation granularity is β = 1, i.e., integer bit allocation.

We observe that OOK has a significantly lower outage probability than OFDM for the same SNR. In other words, when the transmitter does not have CSI, OOK with MMSE-DFE performs better than ACO-OFDM with twice the symbol rate, in contrast to Fig. 5.6 and Fig. 5.8. Results for 50 and 100 Mbit/s are qualitatively similar to those for 300 Mbit/s, particularly at low outage probabilities. These results are not shown in Fig. 5.9 to maximize legibility. At low outage probabilities (Pb < 10%), CB loading performs slightly better than the other loading schemes. For outage probabilities between 25% and 70%, CB loading is significantly better than the other schemes. For high outage probabilities, equal loading performs the best. Finally, we observe that over a wide range of outage probabilities, mean channel loading generally outperforms equal loading. We note that when the outage probability is high, the system can only operate on channels having low delay spreads, which are generally those without shadowing. In practice, a user could improve performance by moving his receiver to avoid shadowing.

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5.5 Computational Complexity

For OFDM, the IDFT and DFT operations are performed efficiently using a fast Fourier transform (FFT) algorithm. For M-PAM, the DFE equalization can be done by direct implementation or by FFT-based block processing, as discussed in Chapter 2. In our comparison, we use the same analysis and equations from Chapter 2.6.

Using the parameters in Table 5.1, we compute the number of operations required per bit for OOK and OFDM in Table 5.2.

Bit Rate 50 100 300 (Mbit/s) ACO-OFDM Tx 48.5 48.5 48.1 (RS) Rx 50.0 50.0 49.6 ACO-OFDM Tx 97.0 96.7 106.5 (2·RS) Rx 100.0 99.6 109.4 PAM-DMT Tx 48.5 48.5 48.1 (RS) Rx 51.5 51.5 51.1 PAM-DMT Tx 97.0 96.7 106.5 (2·RS) Rx 102.9 102.6 112.4 DC-OFDM Tx 43.0 43.0 42.4 (RS) Rx 45.9 45.9 45.3 DC-OFDM Tx 86.1 85.4 95.3 (2·RS) Rx 91.9 91.2 101.1

Lf 101 213 415 OOK Lb 59 119 229 (RS) FFT 137.2 150.8 174.5 Direct 272 432 1008 Table 5.2 Computational complexity in real operations per bit for the various modulation formats.

For OOK, we calculate the complexity of direct implementation and FFT based implementation using optimized block sizes Lf and Lb subject to having a FFT size that is an integer power of two. In Table 5.2, we observe that an OFDM system with twice the symbol rate of OOK requires two digital signal processors (DSPs) with approximately 40% less computational complexity than the single DSP required for OOK. On the other hand, an OFDM system requires an A/D and D/A while OOK requires only one A/D. We also note that DC-OFDM has the lowest computational

125 Personal Area Networks complexity among OFDM formats, since it uses smaller FFT sizes than ACO-OFDM or PAM-DMT.

126 Conclusions

6 CONCLUSIONS

6.1 Conclusions

We have shown that in long-haul systems using coherent detection, OFDM with a one-tap equalizer is able to compensate for linear distortions such as GVD and PMD, provided that an adequate cyclic prefix length is chosen. Compensation of PMD requires a dual-polarization receiver. We have derived analytical penalties for the ISI and ICI occurring when an insufficient cyclic prefix is used. Using these penalties, we have computed the minimum number of subcarriers and cyclic prefix length required to achieve a specified power penalty for GVD and first-order PMD. We observed that GVD is the dominant impairment, since it requires several cyclic prefix samples, whereas first-order PMD typically requires only one cyclic prefix sample. We verified that an oversampling ratio of 1.2 is sufficient to minimize noise aliasing. By contrast, single-carrier systems typically require an oversampling ratio of 1.5 or 2 to avoid noise aliasing. We have also shown that when nonlinear effects are present in the fiber, single-carrier with digital equalization outperforms OFDM for various dispersion maps. However, when no inline dispersion compensation is used, OFDM has a similar performance as single-carrier because of incoherent additions and partial cancelations of the FWM products. We have also shown that OFDM requires less computational complexity per DSP than single-carrier but requires two DSPs. In terms of the optical modulator, we have shown that the quadrature MZ with pre- distortion and hard clipping is able to achieve good performance without additional oversampling, while achieving high optical power efficiency. We observed that the MZ electrode frequency dependent losses can be neglected, since they have very little impact on system performance. We have also shown that the optimum clipping level is approximately CR = 2.5, since this value yields a 12-dB gain in optical power efficiency with minimal receiver sensitivity degradation. Furthermore, we verified that the D/A requires at least 6 bits, and that the optimum clipping level is also approximately CR = 2.5 when quantization noise is present. We have also shown that

127 Conclusions the dual-drive MZ can also be used as an efficient OFDM modulator. It requires less hardware than the quadrature MZ and minimizes the clipping probability. On the other hand, the dual-drive MZ requires a higher oversampling ratio which, in turn, means that faster D/As and wider electrode frequency responses are necessary.

Considering metropolitan area networks, we have evaluated the performance of three different OFDM formats in amplified DD optical systems: SSB-OFDM, DC- OFDM and ACO-OFDM. We have derived equivalent linear channel models for each format in the presence of GVD. We showed how to minimize the average optical power required to achieve a specified error probability by iteratively adjusting the subcarrier power allocation and the bias ratio (BR). We found that for a given dispersion, SSB-OFDM requires the smallest optical power. We presented an analytical lower bound for the minimum required optical SNR for DC-OFDM and concluded that DC-OFDM can never achieve the same optical power efficiency as SSB-OFDM. We showed that at low dispersion, ACO-OFDM performs close to SSB- OFDM but at high dispersion ACO-OFDM performs worse than DC-OFDM, because of nonlinear ICI. Using published results on OOK with MLSD, we showed that SSB- OFDM can achieve the same optical power efficiency as OOK with MLSD, at the expense of requiring twice the electrical bandwidth and therefore requiring also a higher A/D sampling rate. Furthermore, SSB-OFDM requires a quadrature modulator, which also increases the hardware complexity. On the other hand, we showed that SSB-OFDM requires significantly lower computational complexity than OOK with MLSD.

For application to local area networks, we have evaluated the performance of three different IM/DD OFDM formats in multimode fibers: DC-OFDM, ACO-OFDM and PAM-DMT. We have derived the optimal power allocation for PAM-DMT and have shown that the performance of PAM-DMT converges asymptotically to that of ACO-OFDM as the number of subcarriers increases. We have also shown how to minimize the average optical power required for DC-OFDM to achieve a specified error probability by iteratively adjusting the subcarrier power allocation and the bias ratio (BR). For a given symbol rate, we have found that unipolar M-PAM with MMSE-

128 Conclusions

DFE has a better optical power performance than all OFDM formats. Furthermore, we have found that the performance difference between M-PAM and OFDM increases as the spectral efficiency increases. We have shown that at a spectral efficiency of 1 bit/symbol, OOK performs better than ACO-OFDM with twice the symbol rate of OOK. For higher spectral efficiencies, ACO-OFDM at twice the symbol rate of M- PAM performs close to M-PAM, but requires more electrical bandwidth and 23% faster A/D converters than those required for M-PAM at 2 samples per symbol.

Considering indoor optical wireless systems, we have evaluated the performance of the three IM/DD OFDM formats, DC-OFDM, ACO-OFDM and PAM-DMT, and have compared them to unipolar M-PAM with MMSE-DFE. When using the same symbol rate for all modulation methods, we have found that unipolar M-PAM with MMSE-DFE has better optical power efficiency than all OFDM formats over a range of spectral efficiencies. Furthermore, we have found that as spectral efficiency increases, the performance advantage of M-PAM increases, since the OFDM formats require increasingly large signal constellations. We have also found that ACO-OFDM and PAM-DMT have virtually identical performance at any spectral efficiency. They are the best OFDM formats at low spectral efficiency, but as spectral efficiency increases, DC-OFDM performs closer to ACO-OFDM, since the clipping noise penalty for DC-OFDM becomes less significant than the penalty for the larger constellations required for ACO-OFDM. When ACO-OFDM or PAM-DMT are allowed to use twice the symbol rate of M-PAM, these OFDM formats have better performance than M-PAM. However, at a spectral efficiency of 1 bit/symbol, OOK with MMSE-DFE has performance similar to ACO-OFDM or PAM-DMT for high delay spreads. When CSI is unavailable at the transmitter, M-PAM significantly outperforms all OFDM formats even when they use twice the symbol rate of M-PAM. When using the same symbol rate for all modulation methods, M-PAM requires approximately three times more computational complexity per DSP than all OFDM formats and 63% faster A/D converters. When OFDM uses twice the symbol rate of M-PAM, OFDM requires 23% faster A/D converters than M-PAM but OFDM requires 40% less computational complexity than M-PAM per DSP.

129 Conclusions

6.2 Future Work

In long-haul systems, future work could focus on nonlinearity compensation algorithms in order to improve the OFDM performance, and possibility, determine the maximum performance achievable.

Future work in metro systems is needed to determine new techniques or algorithms that can achieve good performance while requiring much less computational complexity.

Finally, future work in IM/DD systems, such as LANs and indoor optical wireless, is still needed to compute the information theoretic channel capacity of non- negative channels with bipolar Gaussian noise. The main difficulties are that the transmitted waveform x(t) is constrained to be non-negative and the parameter to be minimized is the optical power, which corresponds to the first moment of x(t).

130 Appendix A

A APPENDIX

Derivation of the ISI and ICI variance on the different subcarriers is crucial for determining the probability of symbol error. A similar derivation for unilateral channels can be found in [75]. Here, we generalize to bilateral channels and include the effect of correlation when oversampling is used.

If Npre = Npos + Nneg samples are used for the cyclic prefix, and if the channel impulse response has positive and negative lengths Lp−1 and Ln−1, respectively, the residual ISI on the nth time-domain sample in the kth OFDM symbol can be written as

Lp −1 k )( = ∑ k −1 ( c pre pos −+++ channel rhrNNNnxnISI )() Nr pos += 1 A.1 Nneg −− 1 + ∑ k +1 ( c neg −−− channel uhuNNnx )() Lu n +−= 1 where n varies from 0 to Nc−1. Eq. A.1 represents the ISI as a linear function of the OFDM samples. As the number of subcarriers increases, by the Central Limit Theorem [5], the pdf of the samples approaches a Gaussian. Since the ISI is a linear function of the samples, its pdf is also Gaussian. The signal after the DFT becomes

L −1 Nc −1 ⎛ p qISI )( = ⎜ ( −+++ rhrNNNnx )() k ⎜ ∑∑ k−1 c pre pos channel n=0 ⎝ Nr pos += 1 2πnq A.2 Nneg −− 1 ⎞ − j + ( −−− )() ⎟euhuNNnx Nc ∑ k+1 c neg channel ⎟ Lu n +−= 1 ⎠ Performing a variable change on the different sums, we get

1−− nL 2πnq Nc −1 p − j Nc k qISI )( = ∑∑ −1( ck pre pos −++ channel + )() evnhvNNNx n=0 1−+= nNv pos A.3 2−−− nNN 2πnq Nc −1 negc − j Nc + ∑∑ k+1 neg −−− channel ()1( c +−+ )1 eNznhzNx n=0 nc −−= nLNz Next we interchange the inner and outer sums. In both of the double summations, we require that Nc > max(Lp−Npos−1, Ln−Nneg−1). This means that the OFDM block must be longer than the impulse response duration or interference from symbols k+2 or k−2 will take place. In the first double sum, as n increases, the upper

131 Appendix A

limit of the inner sum v decreases accordingly. The lower limit Npos+1 is reached when n = Lp−Npos−1. If we now take the sum over v for the outer sum of the first double sum, we obtain the upper limit for the inner sum n as Lp−1−v. The same idea can be applied to the second double sum, and the lower limit of the inner sum z is reached when n = Ln−Nneg−1. If we now take the sum over z for the outer sum of the second double sum, we obtain the lower limit for the inner sum n as Nc−Ln−z. Thus, Eq. A.3 becomes

2πnq Lp −1 p 1−− vL − j Nc k qISI )( = ∑∑ channel −1()( preck pos −+++ )evNNNxvnh Nv pos += 1 n=0 A.4 N −− 1 2πnq neg Nc −1 − j Nc + ∑∑ channel( kc +1 neg −−−+−+ )1()1 ezNxNznh Lz n +−= 1 nc −−= zLNn Substituting n + v= p and n+ z = m in the first and second double sums, respectively, we get

π − )(2 qvp Lp −1 Lp −1 − j Nc k )( = ∑ −1 ( ck pre pos −++ ∑ channel )() ephvNNNxqISI Nv pos += 1 =vp

N −− 1 π − )(2 qzm neg c 1−− zN − j Nc ∑ k+1 neg −−−+ )1( ∑ channel c +− )1( eNmhzNx Lz n +−= 1 −= LNm nc 2πvq 2πpq Lp −1 j Lp −1 − j Nc Nc = ∑ −1 ( ck pre pos −++ ) ∑ channel )( ephevNNNx Nv += 1 =vp pos A.5 N −− 1 2πzq 2πmq neg j c 1−− zN − j Nc Nc + ∑ k+1 neg −−− )1( ezNx ∑ channel c +− )1( eNmh Lz n +−= 1 −= LNm nc 2πvq Lp −1 j Nc = ∑ −1 ( ck pre pos −++ v qHevNNNx )() Nv pos += 1 2πzq Nneg −− 1 j Nc + ∑ k+1 neg −−− z qHezNx )()1( Lz n +−= 1

Note that the expressions for Hv(q) and Hz(q) correspond to the DFTs of the positive and negative tails of the channel impulse response hchannel(n). After the DFT, 2 the variance σ ISI(q) on each subcarrier q is

132 Appendix A

2 * σ ISI = [ qISIqISIEq )()()( ]

Lp −1 Lp −1 * = ∑∑ vv 21 qHqH )()( Nv pos += 11 Nv pos += 12 π − )(2 qvv j 21 * Nc ( []−1 ( preck pos −++ −11 () preck pos −++ 2 ) evNNNxvNNNxE

Nneg −− 1 Nneg −− 1 * + ∑∑1 zz 2 qHqH )()( Lz n +−= 11 Lz n +−= 12 π − )(2 qzz j 21 * Nc []k+1 neg k+11 neg −−−−−− 2 )1()1( ezNxzNxE A.6

Lp −1 Nneg −− 1 * + ∑∑ zv 11 qHqH )()( Nv pos += 11 Lz n +−= 11 π − )(2 qzv j 11 * Nc ( []−1 ( preck pos k+11 neg −−−−++ 1)1() ezNxvNNNxE

Nneg −− 1 Lp −1 * + ∑∑2 vz 2 qHqH )()( Lz n +−= 12 Nv pos += 12 π − )(2 qvz j 22 * Nc ()1( []k+1 neg −−− −12 ()1( preck pos −++ 2 ) evNNNxzNxE

* If we assume that the symbols are uncorrelated, i.e., E[x(n)k x(l)m ] = 0 for n ≠ l, Eq. A.6 simplifies to

⎛ Lp −1 Nneg −− 1 ⎞ 2 q)( = σσ 2 ⎜ qH )( 2 + qH )( 2 ⎟ ISI s ⎜ v ∑∑ z ⎟ A.7 ⎝ Nv pos += 1 Lz n +−= 1 ⎠ 2 2 where σs = E[|x(n)| ] is the mean power per sample of the time-domain waveform. Eq. A.7 gives the variance of the interference induced on subcarrier q in the OFDM symbol k by the previous and subsequent symbols, k−1 and k+1. It remains now to calculate the interference caused by the current symbol k on itself, i.e., the ICI. ICI occurs because when the cyclic prefix is not sufficiently long, linear convolution between the channel and symbol k does not correspond to one period of a circular convolution. In order to compute the ICI, we assume an extension within the received symbol k so that it is equivalent to one period of a circular convolution between the channel and the transmitted symbol k. The removal of this extension would then be the ICI. The extension concept is illustrated in Fig. A-1.

133 Appendix A

Symbol k without ICI

Npos Extp Extn Nneg

Npos Nneg -

0 Extp 00Extn Symbol k with ICI ICI Fig. A-1 DWDM system with 50 GHz channel spacing.

The ICI is then similar to Eq. A.1 and can be written as

Lp −1 k )( −= ∑ k ( c pre pos −+++ channel rhrNNNnxnICI )() Nr pos += 1 A.8 Nneg −− 1 ( − ∑ k ( c neg −−− channel uhuNNnx )() Lu n +−= 1 If we follow the same steps as in derivation of the ISI, we see that the ICI and ISI differ only by a sign, which disappears when squaring to compute the variance. 2 2 Hence, the ISI and ICI have the same variance, σ ISI(q) = σ ICI(q). Thus, the variance of the total interference on subcarrier q can be written as

⎛ Lp −1 Nneg −− 1 ⎞ σ 2 q = 2)( σ 2 ⎜ qH )( 2 + qH )( 2 ⎟ +ICIISI s ⎜ v ∑∑ z ⎟ A.9 ⎝ Nv pos += 1 Lz n +−= 1 ⎠ Eqs. A.7 and A.9 are exact when the OFDM samples are uncorrelated. However, if oversampling is used, the samples are no longer uncorrelated and therefore Eqs. A.7 and A.9 represent an approximation.

134 References

REFERENCES

[1] S. J. Savory, G. Gavioli, R. I. Killey, and P. Bayvel, "Transmission of 42.8 Gbit/s polarization multiplexed NRZ-QPSK over 6400 km of standard fiber with no optical dispersion compensation," in Proc. Optical Fiber Commun. Conf. (OFC 2007), OTuA1, 2007.

[2] J. B. Carruthers and J. M. Kahn, "Modeling of Nondirected Wireless Infrared Channels," IEEE Trans. on Commun., vol. 45, no. 10, pp. 1260–1268, 1997.

[3] G. Bosco, P. Poggiolini, and M. Visintin, "Performance Analysis of MLSE Receivers Based on the Square-Root Metric," J. Lightw. Technol., vol. 26, no. 14, pp. 2098–2109, 2008.

[4] J. G. Proakis, Digital Communications, 4th ed.: McGraw-Hill, 2002.

[5] G. L. Stuber, Principles of Mobile Communication, 2nd ed.: Kluwer Academic Publishers, 2001.

[6] K. Sistanizadeh, P. Chow, and J. M. Cioffi, "MultiTone Transmission for Asymmetric Digital Subscriber Lines (ADSL)," in ICC'91, Switzerland, 1993, pp. 756–760.

[7] W. Shieh, "PMD-Supported Coherent Optical OFDM Systems," IEEE Photon. Technol. Lett., vol. 19, no. 3, pp. 134–136, 2007.

[8] S.L. Jansen, I. Morita, N. Takeda, and H. Tanaka, "20-Gb/sOFDM Transmission over 4160 km SSMF Enabled by RF-Pilot Tone Phase Noise Compensation," in Proc. Optical Fiber Commun. Conf. (OFC 2007), PDP15, 2007.

[9] B. Schmidt, A. Lowery, and J. Armstrong, "Experimental Demonstration of 20 Gbit/s Direct-Detection Optical OFDM and 12 Gbit/s with a colorless transmitter," in Proc. Optical Fiber Commun. Conf. (OFC 2007), PDP18, 2007.

135 References

[10] Y. Tang, W. Shieh, X. Yi, and R. Evans, "Optimum Design for RF-to-Optical Up-Converter in Coherent Optical OFDM Systems," IEEE Photon. Technol. Lett., vol. 19, no. 7, pp. 483–485, 2007.

[11] A. Lowery, S. Wang, and M. Premaratne, "Calculation of power limit due to fiber nonlinearity in optical OFDM systems," Optics Express, vol. 15, no. 20, pp. 13282–13287, 2007.

[12] S. Wu and Y. Bar-Ness, "OFDM Systems in the Presence of Phase Noise: Consequences and Solutions," IEEE Trans. on Commun., vol. 52, no. 11, pp. 1988–1996, 2004.

[13] J. Tellado, Multicarrier Modulation with Low PAR: Applications to DSL and Wireless.: Kluwer Academic Publishers, 2000.

[14] K.-P. Ho, Phase-modulated optical communication systems, Springer, Ed., 2005.

[15] K. Ho and J. M. Kahn, "Spectrum of Externally Modulated Optical Signals," J. Lightw. Technol., vol. 22, no. 2, pp. 658–663, 2004.

[16] E. Ip and J. M. Kahn, "Digital Equalization of Chromatic Dispersion and Polarization Mode Dispersion," J. Lightw. Technol., vol. 25, no. 8, pp. 2033– 2043, 2007.

[17] T. M. Schmidl and D. C. Cox, "Robust frequency and timing synchronization for OFDM," IEEE Trans. on Commun., vol. 45, no. 12, pp. 1613–1621, 1997.

[18] J. J. van de Beek, M. Sandell, and P. O. Borjesson, "ML estimation of time and frequency offset in OFDM systems," IEEE Trans. Signal Process., vol. 45, no. 12, pp. 1800–1805, 1997.

[19] G. J. Foschini and C. D. Poole, "Statistical theory of polarization dispersion in single mode fibers," J. Lightw. Technol., vol. 9, no. 11, pp. 1439–1456, 1991.

[20] E. Forestieri and G. Prati, "Exact analytical Evaluation of Second-order PMD Impact on the Outage Probability for a Compensated System," J. Lightw.

136 References

Technol., vol. 22, no. 4, pp. 988–996, 2004.

[21] E. Forestieri and L. Vincetti, "Exact evaluation of the Jones matrix of a fiber in the presence of polarization mode dispersion of any order," J. Lightw. Technol., vol. 19, no. 12, pp. 1898–1909, 2001.

[22] H. Kogelnik, L. E. Nelson, J. P. Gordon, and R. M. Jopson, "Jones Matrix for second-order polarization mode dispersion," Optics Letters, vol. 25, no. 1, pp. 19–21, 2000.

[23] A. P. T. Lau and J. M. Kahn, "Signal design and detection in presence of nonlinear phase noise," J. Lightw. Technol., vol. 25, no. 10, p. 3008−3016, 2007.

[24] J. P. Gordon and L. F. Mollenauer, "Phase noise in photonic communications systems using linear amplifiers," Optics Letters, vol. 15, no. 23, p. 1351−1353, 1990.

[25] D. J. F. Barros and J. M. Kahn, "Optimized Dispersion Compensation Using Orthogonal Frequency-Division Multiplexing," J. Lightw. Technol., vol. 26, no. 16, pp. 2889–2898, 2008.

[26] S.G. Johnson and M. Frigo, "A modified split-radix FFT with fewer arithmetic operations," IEEE Trans. Signal Process., vol. 55, no. 1, pp. 111–119, 2007.

[27] S. Jansen, I. Morita, T. Schenk, N. Takeda, and H. Tanaka, "Coherent optical 25.8-Gb/s OFDM transmission over 4160-km SSMF," J. Lightw. Technol., vol. 26, no. 1, pp. 6–15, 2008.

[28] Y. Tang, K. Ho, and W. Shieh, "Coherent Optical OFDM Transmitter Design Employing Predistortion," IEEE Photon. Technol. Lett., vol. 20, no. 11, pp. 954– 956, 2008.

[29] B. Krongold and D. Jones, "PAR reduction in OFDM via active constellation extension," IEEE Trans. Broadcast., vol. 49, no. 3, pp. 258–268, 2003.

137 References

[30] S. Sezginer and H. Sari, "OFDM peak power reduction with simple amplitude predistortion," IEEE Commun. Lett., vol. 10, no. 2, pp. 65–67, 2006.

[31] C. Ciochina, F. Buda, and H. Sari, "An analysis of OFDM peak power reduction techniques for WiMAX systems," in Proc. IEEE ICC’06, 2006, pp. 4676–4681.

[32] J. Armstrong, "Peak-to-average power reduction for OFDM by repeated clipping and frequency domain filtering," Electronics Letters, vol. 38, no. 5, pp. 246–247, 2002.

[33] X. Li and J. L. Cimini, "Effects of clipping and filtering on the performance of OFDM," IEEE Commun. Lett., vol. 2, no. 5, pp. 131–133, 1998.

[34] 40 Gbps LiNbO3 External Modu. (2004) Fujitsu. [Online]. http://www.fujitsu.com/global/services/telecom/optcompo/lineup/40gln/

[35] O. E. Agazzi, M. R. Hueda, H. S. Carrer, and D. E. Crivelli, "Maximum- likelihood sequence estimation in dispersive optical channels," J. Lightw. Technol., vol. 23, no. 2, pp. 749–763, 2005.

[36] T. Foggi, E. Forestieri, G. Colavolpe, and G. Prati, "Maximum-likelihood sequence detection with closed-form metrics in OOK optical systems impaired by GVD and PMD," J. Lightw. Technol. , vol. 24, no. 8, pp. 3073–3087, 2006.

[37] J. Armstrong, B. J. C. Schmidt, D. Kalra, H. A. Suraweera, and A. J. Lowery, "Performance of asymmetrically clipped optical OFDM in AWGN for an intensity modulated direct detection system," in IEEE GLOBECOM, 2006.

[38] J. Armstrong and B. J. C. Schmidt, "Comparison of Asymmetrically Clipped Optical OFDM and DC-Biased Optical OFDM in AWGN," IEEE Comm. Letters, vol. 12, no. 8, pp. 343–345.

[39] A. J. Lowery, "Improving Sensitivity and Spectral Efficiency in Direct-Detection Optical OFDM Systems," in Proc. Optical Fiber Commun. Conf. (OFC 2008), OMM4, 2008.

138 References

[40] B. J. C. Schmidt, A. J. Lowery, and J. Armstrong, "Experimental demonstrations of electronic dispersion compensation for long-haul transmission using direct- detection optical OFDM," J. Lightw. Technol., vol. 26, no. 1, pp. 196–203, 2008.

[41] J. M. Cioffi, G. P. Dudevoir, M. V. Eyuboglu, and G.D. Forney Jr, "MMSE decision-feedback equalizers and coding – Part II: Coding results," IEEE Trans. on Commun., vol. 43, no. 10, pp. 2595–2604, 1995.

[42] G. D. Forney and G. Ungerboeck, "Modulation and coding for linear Gaussian channels," IEEE Trans. on Info. Theory, vol. 44, no. 6, pp. 2384–2415, 1998.

[43] W. Yu, G. Ginis, and J. M. Cioffi, "An adaptive multiuser power control algorithm for VDSL," in IEEE GLOBECOM, 2001.

[44] A. Ali, J. Leibrich, and W. Rosenkranz, "Spectral Efficiency and Receiver Sensitivity in Direct Detection Optical-OFDM," in Proc. Optical Fiber Commun. Conf. (OFC 2009), OMT7, 2009.

[45] A. Ali, H. Paul, J. Leibrich, W. Rosenkranz, and K. D. Kammeyer, "Optical Biasing in Direct Detection Optical-OFDM for Improving Receiver Sensitivity," in Proc. Optical Fiber Commun. Conf. (OFC 2010), JThA12, 2010.

[46] J. W. Goodman, Statistical Optics.: Wiley-Interscience, 1985.

[47] J. M. Tang, P. M. Lane, and K. A. Shore, "Transmission performance of adaptively modulated optical OFDM signals in multimode fiber links," IEEE Photon. Technol. Lett., vol. 18, no. 1, pp. 205–207, 2006.

[48] S. Randel, F. Breyer, and S. C. J. Lee, "High-Speed Transmission over Multimode Optical Fibers," in Proc. Optical Fiber Commun. Conf. (OFC 2008), OWR2, 2008.

[49] J. M. Tang, P. M. Lane, and K. A. Shore, "High-speed transmission of adaptively modulated optical OFDM signals over multimode fibers using directly modulated DFBs," J. Lightw. Technol., vol. 24, no. 1, pp. 429–441, 2006.

139 References

[50] S. C. J. Lee, S. Randel, F. Breyer, and A. M. J. Koonen, "PAM-DMT for Intensity-Modulated and Direct-Detection Optical Communication Systems," IEEE Photon. Technol. Lett., vol. 21, no. 23, pp. 1749–1751, 2009.

[51] X. Q. Jin, J. M. Tang, P. S. Spencer, and K. A. Shore, "Optimization of adaptively modulated optical OFDM modems for multimode fiber-based local area networks," J. Optical Networking, vol. 7, no. 3, pp. 198–214, 2008.

[52] S. C. J. Lee, F. Breyer, S. Randel, H. P. A. van den Boom, and A. M. J. Koonen, "High-speed transmission over multimode fiber using discrete multitone modulation," J. Optical Networking, vol. 7, no. 2, pp. 183–196, 2008.

[53] M. B. Shemirani and J. M. Kahn, "Higher-Order Modal Dispersion in Graded- Index Multimode Fiber," J. Lightw. Technol., vol. 27, no. 23, pp. 5461–5468, 2009.

[54] J. Campello, "Optimal discrete bit loading for multicarrier modulation systems," in IEEE Symp. Info. Theory, 1998.

[55] P. Pepeljugoski, S. E. Golowich, A. J. Ritger, P. Kolesar, and A. Risteski, "Modeling and simulation of next-generation multimode fiber links," J. Lightw. Technol., vol. 21, no. 5, pp. 1242–1255, 2003.

[56] P. Pepeljugoski et al., "Development of system specification for laser-optimized 50-μm multimode fiber for multigigabit short-wavelength LANs," J. Lightw. Technol., vol. 21, no. 5, pp. 1256–1275, 2003.

[57] X. Q. Jin, J. M. Tang, K. Qiu, and P. S. Spencer, "Statistical Investigations of the Transmission Performance of Adaptively Modulated Optical OFDM Signals in Multimode Fiber Links," J. Lightw. Technol., vol. 26, no. 18, pp. 3216–3224, 2008.

[58] X. Zheng, J. M. Tang, and P. S. Spencer, "Transmission Performance of Adaptively Modulated Optical OFDM Modems Using Subcarrier Modulation

140 References

over Worst-Case Multimode Fibre Links," IEEE Commun. Lett., vol. 12, no. 10, pp. 788–790, 2008.

[59] J. M. Kahn and J. R. Barry, "Wireless Infrared Communications," Proc. of the IEEE, vol. 85, no. 2, pp. 265–298, 1997.

[60] J. B. Carruthers and J. M. Kahn, "Modeling of Nondirected Wireless Infrared Channels," IEEE Trans. on Commun., vol. 45, no. 10, pp. 1260–1268, 1997.

[61] J. Armstrong and A. J. Lowery B, "Power efficient optical OFDM," Electronics Letters, vol. 42, no. 6, pp. 370–372, 2006.

[62] F. R. Gfelle and U. Bapst, "Wireless In-House Data Communication via Diffuse Infrared Radiation," Proc. of the IEEE, vol. 67, no. 11, pp. 1474–1486, 1979.

[63] O. Gonzalez, R. Perez-Jimenez, S. Rodriguez, J. Rabadan, and A. Ayala, "Adaptive OFDM system for communications over the indoor wireless optical channel," IEE Proc. – Optoelectronics, vol. 153, no. 4, pp. 139–144, 2006.

[64] H. Joshi, R. J. Green, and M. S. Leeson, "Multiple Sub-carrier Optical Wireless Systems," in ICTON 2008, vol. 4, 2008, pp. 184–188.

[65] J. B. Carruthers and J. M. Kahn, "Multiple-Subcarrier Modulation for Non- Directed Wireless Infrared Communication," IEEE J. Sel. Areas in Commun., vol. 14, no. 3, pp. 538–546, 1996.

[66] S. K. Wilson and J. Armstrong, "Transmitter and Receiver methods for improving Asymmetrically-Clipped Optical OFDM," IEEE Trans. on Wireless Commun., vol. 8, no. 9, pp. 4561–4567, 2009.

[67] J. R. Barry, J. M. Kahn, W. J. Krause, E. A. Lee, and D. G. Messerschmitt, "Simulation of Multipath Impulse Response for Wireless Optical Channels," IEEE J. Sel. Areas in Commun., vol. 11, no. 3, pp. 367–379, 1993.

[68] J. M. Kahn, W. J. Krause, and J. B. Carruthers, "Experimental Characterization of Non-Directed Indoor Infrared Channels," IEEE Trans. on Commun., vol. 43, no.

141 References

234, pp. 1613–1623, 1995.

[69] J. Grubor, O. C. Gaete Jamett, J. W. Walewski, S. Randel, and K.-D. Langer, "High-Speed Wireless Indoor Communication via Visible Light," in ITG FACHBERICHT, 2007.

[70] J. B. Carruthers and P. Kannan, "Iterative site-based modeling for wireless infrared channels," IEEE Trans. on Antennas and Propag., vol. 50, no. 5, pp. 759–765, 2002.

[71] E. Forestieri, Optical Communication Theory and Techniques.: Springer, 2005.

[72] T. J. Richardson and R. L. Urbanke, Modern Coding Theory.: Cambridge Univ. Pr., 2008.

[73] D. J. F. Barros and J. M. Kahn, "Comparison of Orthogonal Frequency-Division Multiplexing and On-Off Keying in Amplified Direct-Detection Single-Mode Fiber Systems," J. Lightw. Technol., vol. 28, no. 12, pp. 1811–1820, 2010.

[74] D. J. F. Barros and J. M. Kahn, "Comparison of Orthogonal Frequency-Division Multiplexing and On-Off Keying in Direct-Detection Multimode Fiber Links," in press for J. Lightw. Technol., 2011.

[75] W. Henkel, G. Taubock, P. Odling, P.O. Borjesson, and N. Petersson, "The Cyclic Prefix of OFDM/DMT – An Analysis," in Proc. International Seminar on Broadband Communications, 2002.

142