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

http://creativecommons.org/licenses/by-nc/3.0/us/

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

© 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.

ii

I certify that I have read this dissertation and that, in my opinion, it is fully adequatein 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 adequatein 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 adequatein 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 inUniversity Archives.

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Abstract

iv

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.

Abstract

v

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

Abstract

vi

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.

Dedication

vii

DEDICATION

Esta dissertação é dedicada

à minha irmã Bárbara Barros.

Acknowledgments

viii

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.

Acknowledgments

ix

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.

Table of Contents

x

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

Table of Contents

xi

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

Table of Contents

xii

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

List of Tables

xiii

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

List of Figures

xiv

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 Fig. 2.10 ISI + ICI power penalty at PS = 10−4 for L = 80 km, D = 17 ps/(nm.km), Nc = 64, Nu = 52 and R = 26.7 GHz. ................................................................................................... 19 Fig. 2.11 Overall power penalty at PS = 10−4 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

List of Figures

xv

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 Fig. 2.29 Receiver sensitivity penalty at PS = 10−4, 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

List of Figures

xvi

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 achieve Pb = 10−3. 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 achieve a Pb = 10−3 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 Fig. 3.16 Normalized optical SNR required for DC-OFDM to achieve Pb = 10−3 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 Fig. 3.18 OSNR values (over 0.1 nm) required to obtain Pb = 10−3 at 10.7 Gbit/s for OFDM and for OOK with MLSD [3]. .......................................................................................... 79 Fig. 3.19 OSNR values (over 0.1 nm) required to obtain Pb = 10−3 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 Fig. 4.8 Receiver electrical SNR required to obtain 10 Gbit/s at Pb = 10−4 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 symbol rate as OOK ( OFDM

sR = Rs = 10 GHz). ................................................. 96 Fig. 4.9 Optical spectra of M-PAM and OFDM. The symbol rate for OFDM is twice that for M-PAM, OFDM

sR = 2Rs. ...................................................................................................... 97 Fig. 4.10 Receiver electrical SNR required to achieve Pb = 10−4 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 formats is twice that for OOK, OFDM

sR = 2Rs = 20 GHz. ............... 98

List of Figures

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Fig. 4.11 Receiver electrical SNR required for DC-OFDM to achieve Pb = 10−4 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 discrete bit allocations to achieve Pb = 10−4 at 10 Gbit/s for fibers of 1-km length. The symbol rate for all OFDM formats is OFDM

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

sR = 20 GHz. ....................................................................................... 101 Fig. 4.14 Cumulative distribution function (CDF) of the required receiver electrical SNR to obtain Pb = 10−4 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 Fig. 4.15 Receiver electrical SNR required to obtain 20 Gbit/s at Pb = 10−4 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 obtain 20 Gbit/s at Pb = 10−4 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 Fig. 5.4 Electrical SNR required to achieve Pb = 10−4 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 Fig. 5.5 Electrical SNR required to achieve Pb = 10−4 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 Fig. 5.7 Electrical SNR required to achieve Pb = 10−4 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

List of Figures

xviii

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

Introduction

1

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

Introduction

2

f

Bandwidth

OFDM

fBandwidth

Single-Carrier

Time Time

t

TOFDM

t

1

-1Ts

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.

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.

Applications WAN (>1000 km)

MAN (50 – 500 km)

LAN (0.3 – 5 km)

PAN (10 m)

Goal 100-1000 Gbit/s 10-100 Gbit/s 10-100 Gbit/s 300 Mbit/s

Medium Single-Mode Fiber

Single-Mode Fiber

Multimode Fiber Free Space

Modulation Electric Field (Complex)

Electric Field or Intensity Intensity Intensity

Detection Method

Coherent (Electric field)

Direct Detection (Intensity)

Direct Detection (Intensity)

Direct Detection (Intensity)

Dominant Dispersion

Chromatic/Polar. (Linear)

Chromatic (Nonlinear)

Multimode (Linear)

Multipath (Linear)

Introduction

3

Dominant Noise

Amplifier (AWGN)

Amplifier (Chi-squared χ2)

Thermal (AWGN)

Ambient Light

(Shot noise)

Noise Type Signal-Indep. Signal-Dep. Signal-Indep. Signal-Indep.

OFDM Formats OFDM

DC-OFDM ACO-OFDM SSB-OFDM

DC-OFDM ACO-OFDM

DC-OFDM ACO-OFDM

Other Effects

Fiber Nonlinearity Clipping Noise Clipping

Noise Clipping

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

o Modal dispersion: different modes travel with different propagation constants

o Large delay spreads

o Refractive index changes with frequency (GVD)

o Fiber asymmetry can cause polarization-mode dispersion (PMD)

o Small delay spreads

Low bandwidth-distance product (5 Gbit/s·km)

High bandwidth-distance product (200 Tbit/s·km)

Introduction

4

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.

Introduction

5

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.

Introduction

6

In Chapter 6, we give the concluding remarks of this thesis and discuss

directions for future work.

Wide Area Networks

7

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

Wide Area Networks

8

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

Wide Area Networks

9

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.

Wide Area Networks

10

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.

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.

λ

…

1λ

50 GHz

2λ 1−Nλ Nλ

AWG

…

1λTX 1

TX 2

TX N−1

TX N…

RX 1

RX 2

RX N−1

RX N

AWG

2λ

1−Nλ

Nλ

SMF

TX RX

Span 1 Span NA

SMF DCF SMF DCF

Wide Area Networks

11

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

Laser

Encoder

Bits0 1 0 1 0 1 1 0

0 1 0 1 0 1 1 0RX

ElectricalLPF

Decision

0 1 0 1 0 1 1 0

Fiber

TX

Wide Area Networks

12

( ) ( )∑ ∑−

=

−=k

N

q

tqfjsymkqofdm

cdekTtbXtx

1

0

2,

π , (2.1)

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.

Fig. 2.4 Digital implementation of OFDM transmitter and receiver.

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

( ) ( )∑ ∑−

=

−=k

N

q

Nqnj

symckqcofdm

ccekTnTbXnTx

1

0

2

,

π

. (2.2)

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,

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

Hchannel(ω) is given by

)()()()( ωωωω VHPH fiberchannel = , (2.3)

Add Prefix(Npre)

+Windowing

(Nwin)

.

.

.

X0,k

X1,k

XN-2,k

XN-1,k

.

.

.

.

.

.

P/S Fiber

IDFT

Remove Prefix(Npre)

+Windowing

(Nwin)

.

.

.

.

.

. DFT

Coherent O/E Down-conversion

LO

S/P

.

.

.

Y0,k

Y1,k

YN-1,k

PulseShaping

x

x

x

Heq(N-1)

Heq(1)

Heq(0)

X0,k

X1,k

XN-1,k

p(t) v(t)

xofdm(t) = I(t)+jQ(t)

hfiber(t)

Anti-Aliasing

D/A

D/A PulseShaping

I(t)

Q(t)p(t)

hchannel(t) = p(t) × hfiber(t) × v(t)

Anti-Aliasing

I'(t)

Q'(t) v(t)

A/D

A/D

MZlaser

Wide Area Networks

13

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).

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

( ) ( )∑−

=⎟⎟⎠

⎞⎜⎜⎝

⎛−=

1

0

22 2

2sinc

cN

qd

symq

c

symxx qf

TP

NT

S πωπ

ω , (2.4)

where Pq = E[|Xq|2] 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.

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

…… …○ ○● ●□□■ ■

cN negNposNwinN

winpreofdmsym TTTT ++=

Effective Tx time (Tofdm)

…

symTπ2

( ) ⎟⎟⎠

⎞⎜⎜⎝

⎛+−π≈ω

symdcBW TfN 212

dfπ2

ω

( )ωxxS

Wide Area Networks

14

dcsym

dcBW fNT

fN )1(22)1(2 +≤⎟⎟⎠

⎞⎜⎜⎝

⎛+−≈ ππω , (2.5)

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

RN

NNN

N

c

prec

c

cBW

)()1(2++

≈ πω . (2.6)

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 Rsc 22 ×≈ πω [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

Wide Area Networks

15

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.

Fig. 2.7 Pulse spreading due to fiber GVD.

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

Lj

fiber eH 222

)(βω

ω−

= , (2.7)

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.

t t

SMF

Input

Output

Wide Area Networks

16

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

)()(

)()()(

1

11

1

11

uhuNNnx

rhrNNNnxnISI

channel

N

Lunegck

channel

L

Nrpospreckk

neg

n

p

pos

∑

∑−−

+−=+

−

+=−

−−−+

−+++=. (2.8)

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

⎟⎟⎠

⎞⎜⎜⎝

⎛+= ∑∑

−−

+−=

−

+=

1

1

21

1

222 )()()(neg

npos

N

Lnn

Lp

NppsISI qHqHq σσ , (2.9)

where σs2 = E[|x(n)|2] 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

c

c

nc

c

p

NvqjnN

LNvchanneln

NvqjL

pvchannelp

evhqH

evhqH

π

π

21

21

)()(

)()(

−−−

−=

−−

=

∑

∑

=

=. (2.10)

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

Npos… …

Symbol k+1Symbol k

hchannel(t)

t

NnegNpos Nneg

… …

Wide Area Networks

17

variance as the ISI. Thus, the variance of the total interference on subcarrier q is given

by

⎟⎟⎠

⎞⎜⎜⎝

⎛+= ∑∑

−−

+−=

−

+=+

1

1

21

1

222 )()(2)(neg

n

p

pos

N

Lnn

L

NppsICIISI qHqHq σσ . (2.11)

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

∑−

= + ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟

⎟⎠

⎞⎜⎜⎝

⎛

+=

1

022

0 )(211 uN

q ICIISI

q

upre

u

us q

PNN

NerfcN

Pσσ

, (2.12)

where Nu/(Nu+Npre) is the extra energy wasted on the cyclic prefix samples, Pq is the

average power per symbol of subcarrier q, σ02 is the variance of sampled additive

white Gaussian noise (σ02 = N0R), and σ2

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

Wide Area Networks

18

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.

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.

Since the majority of the FEC codes have a threshold around Ps = 10−3, we will

measure the power penalty at Ps = 10−4 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.

6 7 8 9 10 11 12 13 14 15 1610

-5

10-4

10-3

10-2

10-1

Es/No (dB)

P s

IdealCyclic PrefixCyclic Prefix + GVDSimulation

Wide Area Networks

19

Fig. 2.10 ISI + ICI power penalty at PS = 10−4 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

Fig. 2.11 Overall power penalty at PS = 10−4 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

1 3 5 7 9 11 13 15 17 19 21 23 25 27

1

2

3

4

5

6

7

Prefix Length (Npre)

ISI +

ICI P

ower

Pen

alty

(dB

)

L = 80 km

AnalyticalSimulation

1 3 5 7 9 11 13 15 17 19 21 23 25 270

1

2

3

4

5

6

7

8

Prefix Length (Npre)

Ove

rall

Pow

er P

enal

ty (d

B)

L = 80 km

AnalyticalSimulation

ISI dominated

Prefix dominated

Wide Area Networks

20

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.

1 2 3 4 5 6

0

0.5

1

1.5

2

2.5

3

Prefix Length (Npre)

Ove

rall

Pow

er P

enal

ty (d

B)

(a) L = 1000 km, 2% Uncompensated

Nu= 12

Nu= 26

Nu= 52

Nu=104

Nu=208

1 2 3 4 5 6 7 80

1

2

3

4

5

6

Prefix Length (Npre)

Ove

rall

Pow

er P

enal

ty (d

B)

(b) L = 2000 km, 2% Uncompensated

Nu= 26

Nu= 52

Nu=104

Nu=208

1 2 3 4 5 6 7 8 9 10 11 120

1

2

3

4

5

6

7

Prefix Length (Npre)

Ove

rall

Pow

er P

enal

ty (d

B)

(c) L = 3000 km, 2% Uncompensated

Nu= 26

Nu= 52

Nu=104

Nu=208

1 2 3 4 5 6 7 8 9 10 11 120

1

2

3

4

5

6

7

8

Prefix Length (Npre)

Ove

rall

Pow

er P

enal

ty (d

B)

(d) L = 4000 km, 2% Uncompensated

Nu= 52

Nu=104

Nu=208

Wide Area Networks

21

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.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 160

1

2

3

4

5

6

7

8

9

10

Prefix Length (Npre)

Ove

rall

Pow

er P

enal

ty (d

B)

(e) L = 5000 km, 2% Uncompensated

Nu= 52

Nu=104

Nu=208

Wide Area Networks

22

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 .

Total Length (km)

Residual Dispersion

D·L.(ps/nm)

DFT Size Nc

Modulated Subcarriers

Nu

Cyclic Prefix Npre

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.

Wide Area Networks

23

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.

200 600 1000 1400 1800 2200 2600 3000

50

100

150

200

250

300

350

400

450

Length (km)

Pref

ix L

engt

h (N

pre)

Nu = 208

Nu = 416

Nu = 832

Nu = 1664

X Pol.

Y Pol.

ElectricalLPF

t

t

t

τ

τX

Y

Z

Wide Area Networks

24

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

output time-domain signals are denoted by x(t)=[x1(t) x2(t)]T and y(t)=[y1(t) y2(t)]T,

respectively. Their Fourier transforms can be related as

⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡)()(

)()()(

2

1

2

1

ωω

ωωω

XX

UYY

, (2.13)

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

112 )()( −Λ= RRU ωω , (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

⎥⎦

⎤⎢⎣

⎡=⎥

⎦

⎤⎢⎣

⎡ −=

− 2

2

12

21

00jωωτ

jωωτ

*ii

*ii

i ee

Λ rrrr

R , (2.15)

where

iiiii

iiiii

jrjr

εθεθεθεθ

sincoscossinsinsincoscos

2

1

+=−= . (2.16)

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

Lβjω

fiber eU(ωH(ωH

(H 2

1

222

)())

)−

=⎥⎦

⎤⎢⎣

⎡= ωω , (2.17)

Wide Area Networks

25

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.

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

Fiber

DualPolarization

CoherentReceiver

OFDMTx 1

OFDMTx 2

LO

OFDMRx 1

OFDMRx 2

I1(t)

Q1(t)

I2(t)

Q2(t)

xofdm,1(t)

xofdm,2(t)

I'1(t)

Q'1(t)

I'2(t)

Q'2(t)D/A

D/A

D/A

D/A

A/D

A/DPBS

A/D

A/D

MZ

MZ

Wide Area Networks

26

22

2

22

12

11

1 2)(Lqβ

Nπfj

c

sfibereq

c

s

eRΛRN

qπfHqH⎟⎟⎠

⎞⎜⎜⎝

⎛+

−−− =⎟⎟⎠

⎞⎜⎜⎝

⎛= . (2.18)

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.

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.

A typical PMD parameter is Dp = 0.1 ps/km1/2 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

222

22

22 Lq

Nfjq

Nfj ss

eβπτπ

⎟⎠⎞

⎜⎝⎛++

222

22

22 Lq

Nfjq

Nfj ss

eβπ

τπ

⎟⎠⎞

⎜⎝⎛+−

Wide Area Networks

27

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

2/2/)( ωτωτω jjPMD beaeH −+= , (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

∑−

= ⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎟⎠

⎞⎜⎜⎝

⎛

+=

1

022

0 )/2(1

211 uN

q ccPMD

q

upre

u

us

NqfH

PNN

NerfcN

Pπσ

, (2.20)

where Nu/(Nu+Npre) is the extra energy wasted on the cyclic prefix samples, Pq is the

average power per symbol of each subcarrier, σ02 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., ( ) ( )22)( τδτδ −++= tbtathPMD . 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

Wide Area Networks

28

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

6 7 8 9 10 11 12 13 14 15 1610

-5

10-4

10-3

10-2

10-1

Es/No (dB)

P s

IdealCyclic PrefixCyclic Prefix + PMDSimulation

Wide Area Networks

29

EEjEtEj

zE 2

2

22 ||

22γαβ

=+∂∂

+∂∂ , (2.21)

where ( )tzE , is the electric field, α is the attenuation coefficient, 2β 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.

Re

Ima)

ФNL

Re

Imb)

Wide Area Networks

30

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., ( )NLφvar [14].

When only linear effects are present in the fiber, the phase noise variance is

approximately equal to ( )SNR21var

⋅≈NLφ for high SNR. For multi-carrier systems, the

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

( ) ( )u

u

NLNL

NN

n

nOFDM/1

1

varvar ⎥⎦

⎤⎢⎣

⎡= ∏

=

φφ , (2.22)

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

to a phase noise variance of 0.08 rad2 for QPSK systems. For OFDM transmission, we

Wide Area Networks

31

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,

γSMF = 1.2 W−1/km, αDCF = 0.6 dB/km, DDCF = −85 ps/nm·km and γDCF = 5.3 W−1/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.

Wide Area Networks

32

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.

-10 -8 -6 -4 -2 0 2 410-2

10-1

100

Power (dBm)

Pha

se n

oise

var

ianc

e (ra

d)2

FEC limitOFDM SC (4-QAM)

Linear

Nonlinear

(a)

0 10 20 30 40 50 6010-4

10-3

10-2

10-1

100

Number of Spans

FEC limitOFDM

SC (4-QAM)

Pha

se n

oise

var

ianc

e (ra

d )2 (b)

Wide Area Networks

33

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.

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.

(a)

OFDM

SC (4-QAM)

-10 -8 -6 -4 -2 0 2 410-2

10-1

100

Power (dBm)

FEC limit

Pha

se n

oise

var

ianc

e (ra

d )2

-10 -8 -6 -4 -2 0 2 410-2

10-1

100

Power (dBm)

FEC limit

OFDM

SC (4-QAM)Pha

se n

oise

var

ianc

e (ra

d )2 (b)

Wide Area Networks

34

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

s

OFDMsbs

OFDMbTxOFDM

MNRTMNN

TTNNNO

⋅+=

=

ν)(log4

/)(log4)(

2

2

.

[ ][ ]

s

OFDMsbsu

OFDMbuRxOFDM

MNRTMNNN

TTNNNNO

⋅++

=

+=

ν6)(log4

/6)(log4)(

2

2

(2.23)

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

[ ]

MN

TTN

/TTNNNO

taps

sbtaps

sbtapstapsDQAM

2log/8

/8

44)(

=

=

+=

. (2.24)

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

Wide Area Networks

35

MBLNLN

NOf

BtapsBtapsFFTQAM

2

2

log6))1(log(8)1(

)(+−+⋅⋅−+

= . (2.25)

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.

Transmission Distance

(km)

Single-Carrier OFDM

Direct FFT

Transmitter Receiver Block Size (LB) Complexity

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

Wide Area Networks

36

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

Wide Area Networks

37

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]

[ ]2

2

)(

|)(|maxPAR

txE

txt= , (2.26)

where 2|)(|max txt

is the peak value squared of the signal x(t) and E[x(t)2] is the

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

2||maxPAR

k

kku XE

XN= , (2.27)

where 2||max kkX is the largest symbol magnitude squared on the kth subcarrier, E[Xk

2]

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

uN=PAR . (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]

⎟⎟⎠

⎞⎜⎜⎝

⎛=

π

πV

tVtEtE m

in

out

2)(sin

)()( , (2.29)

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

Wide Area Networks

38

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]

int12int

int12int

2)()2)(exp(1

)()(

LfdjLfLfdjLf

tVtV

d

m

παπα

−+−−

= , (2.30)

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.

Fig. 2.22 Quadrature MZ modulator.

Laser

I

Q90

Quadrature MZ

Fiber

MZ

MZ

vd,i(t)

vd,r(t)

Wide Area Networks

39

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.

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

PowerLaser CW

Power Modulated AverageOPE = , (2.31)

Laser

X1

IDFT

Parallel-to-

Serial+

CyclicPrefix

N

XN D/AIm{ }

D/ARe{ }

I

Q90

Quadrature MZ

Fiber

… …

MZ

MZ

vd,i(t)

vd,r(t)

Wide Area Networks

40

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

2)(

2)(

2)(sin

)()(

πππ

πππV

tVV

tVV

tVtEtE mmm

in

out (2.32)

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

⎪⎩

⎪⎨

⎧

><<−

−<−=

AnxAAnxAnx

AnxAnVin

)( if,)( if),(

)( if,)( (2.33)

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

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛=

|)(|max)(arcsin2)(nV

nVVnVinn

inout π

π , (2.34)

where Vin(n) and Vout(n) are the digital input and output voltages of the pre-distortion

device. Note that AnVinn=|)(|max because of the clipping operation and such that the

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.

Wide Area Networks

41

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.

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.

-1 -0.75 -0.5 -0.25 0 0.25 0.5 0.75 1

0

Vin(t)

Vou

t(t)-V

π

Vπ

Hard ClipperCL

−CL CL−CL

sin−1( )Vπ

−CL CL−Vπ

Hard ClipperCL

−CL CL−CL

sin−1( )Vπ

−CL CL−Vπ

D/A

X1

OFDM

TX

…XN

Re{ }

Im{ }

Helec.( )−1

fB

Helec.( )−1

fB

D/A

QuadratureMZLaser

Clipping Pre-Distortion Equalization

Vd,i(t)

Vd,r(t)Vd,r(n)

Vout,r(n)Vin,r(n)

Vout,i(n)Vin,i(n)

Vd,i(n)

Wide Area Networks

42

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

σA

=CR , (2.35)

where A is the clipping level and σ is the rms power of the OFDM signal, i.e.,

[ ]2)(CR txE= . 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

Wide Area Networks

43

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.

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.

0 10 20 30 40 50-9

-8

-7

-6

-5

-4

-3

-2

-1

0

Frequency (GHz)

20 lo

g 10(|V

m(f)

/Vd(f)

|) (d

B)

Wide Area Networks

44

Fig. 2.28 Optical power efficiency for Nc = 64, Nu = 52 and R = 29.66 GHz.

Since typical current FEC codes have a threshold around PS = 10−3, we

measure the receiver sensitivity penalty at PS = 10−4 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.

Fig. 2.29 Receiver sensitivity penalty at PS = 10−4, Nc = 64, Nu = 52 and R = 29.66 GHz.

1 2 3 4 5 6 7 8 9 10 110

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

Opt

ical

Pow

er E

ffici

ency

(W/W

)

Clipping Ratio (CR)

Full Comp.Pre-Comp.No Comp.

1 2 3 4 5 6 7 8 9 10 110

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Rec

eive

r Sen

sitiv

ity P

enal

ty (d

B)

Clipping Ratio (CR)

Full Comp.Pre-Comp.No Comp.

Wide Area Networks

45

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.

Fig. 2.30 Optical power efficiency for different number of subcarriers.

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Opt

ical

Pow

er E

ffici

ency

(W/W

)

Clipping Ratio (CR)

Nu = 26

Nu = 52

Nu = 104

Wide Area Networks

46

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.

1 2 3 4 5 6 7 8 9 10 11 12 13 14 150

0.5

1

1.5

2

2.5

3

3.5

4

Rec

eive

r Sen

sitiv

ity P

enal

ty (d

B)

Clipping Ratio (CR)

Nu = 26

Nu = 52

Nu = 104

Wide Area Networks

47

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

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

1 2 3 4 5 6 7 8 9 10 110

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Opt

ical

Pow

er E

ffici

ency

(W/W

)

Clipping Ratio (CR)

n = 4 bitsn = 5 bitsn = 6 bitsn = 7 bitsn = 8 bits

1 2 3 4 5 6 7 8 9 10 110

0.5

1

1.5

2

2.5

3

3.5

4

Rec

eive

r Sen

sitiv

ity P

enal

ty (d

B)

Clipping Ratio (CR)

n = 4 bitsn = 5 bitsn = 6 bitsn = 7 bitsn = 8 bits

Wide Area Networks

48

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

⎥⎦

⎤⎢⎣

⎡⎟⎟⎠

⎞⎜⎜⎝

⎛+⎟⎟

⎠

⎞⎜⎜⎝

⎛=

ππ

ππV

tVjV

tVjtEtE mm

in

out )(exp)(exp21

)()( 21 , (2.36)

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.

⎥⎥⎦

⎤

⎢⎢⎣

⎡+=

(t)j(t)j m2VπVπ

m1VπVπ

ee21

(t)E(t)E

in

out

Wide Area Networks

49

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.

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

( )( )

( )( )qnAnVnV

qnAnVnV

out

out

πθπ

πθπ

π

π

2)(arccos)()(

2)(arccos)()(

2

1

+−=

++= (2.37)

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.

1

x(n)⎩⎨⎧

⎭⎬⎫

(t)E(t)EIm

in

out

)(2

21 nV

Vπj mπe

⎩⎨⎧

⎭⎬⎫

(t)E(t)ERe

in

out

)(1

21 nV

Vπj mπe

Wide Area Networks

50

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.

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.

D/A

X1

IDFT

Parallel-to-

Serial+

CyclicPrefix

N

…

XN

Re{ }

Im{ }

…

Helec.( )−1

fB

Helec.( )−1

fB

D/A

Dual-DriveMZLaser

Hard-Clipping+

Dual-Drive Pre-Distortion

Vd1(t)

Vd2(t)

Vout1(n)

Vout2(n)

Vd1(n)

Vd2(n)

1

⎩⎨⎧

⎭⎬⎫

(t)E(t)EIm

in

out

⎩⎨⎧

⎭⎬⎫

(t)E(t)ERe

in

outQuadrature

MZ

Dual-DriveMZ

CR

Wide Area Networks

51

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

1 , ),()(subject to)()(minimize

,1

21

+==Δ+Δ

rrnnVnVrVrV

ioutm

mm (2.38)

where )1()()( +−=Δ rVrVrV mimimi 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.

0 20 40 60 80 100-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

Time (ps)

Rea

l{Opt

ical

Fie

ld}

Dual-DriveDesired

Wide Area Networks

52

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

0 200 400 600 800 1000 1200-4

-2

0

2

4a) No Trajectory Optimization (-vπ ≤ vm,i, ≤ vπ)

Re{

Opt

ical

Fie

ld}

0 200 400 600 800 1000 1200

0 200 400 600 800 1000 1200Time (ps)

-4

-2

0

2

4R

e{O

ptic

al F

ield

}

-4

-2

0

2

4

Re{

Opt

ical

Fie

ld}

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

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

Wide Area Networks

53

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.

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.

1 2 3 4 5 6 7 80

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45O

ptic

al P

ower

Effi

cien

cy (W

/W)

Clipping Ratio (CR)

Wide Area Networks

54

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.

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

1 2 3 4 5 6 7 80

0.5

1

1.5

2

2.5

3

Rec

eive

r Sen

sitiv

ity P

enal

ty (d

B)

Clipping Ratio (CR)

1 2 3 4 5 60

0.5

1

1.5

2

2.5

3

Oversampling Ratio (M)

Rec

eive

r Sen

sitiv

ity P

enal

ty (d

B)

Quadrature MZDual-drive MZ

Wide Area Networks

55

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.

Metropolitan Networks

56

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

Metropolitan Networks

57

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.

Metropolitan Networks

58

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.

TX RX

SMF Amplifier + Filter

Metropolitan Networks

59

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.

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

∫∫ −→∞−→∞==

T

TT

T

T optT

t dttxT

KdttPT

P )(21

lim)(21

lim , (3.1)

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

Current (mA)

Laser Power (mW)

Ith

x(t)

Popt (t)

Metropolitan Networks

60

( )SNRBC

+= 1log2 , (3.2)

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

⎟⎠⎞

⎜⎝⎛

Γ+=

SNRBR 1log2 , (3.3)

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.

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

3

2α=Γ , (3.4)

where )(1ePQ−=α , Pe is the symbol-error probability and Q-1 is the inverse Q

function. As an example, the gap is 8.8 dB at Pe = 10−6 and is 9.5 dB at Pe = 10−7 for

uncoded QAM. The use of forward error-correction (FEC) codes reduces the gap. A

2 4 6 8 10 12 14 160

1

2

3

4

5

6

R/B

(bit/

s/H

z)

SNR (dB)

Capacity(Γ = 0 dB)

3 dB

6 dB

9 dB

Metropolitan Networks

61

well-coded system may have a gap as low as 0.5 dB at Pe < 10−6. 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

∑

∑−

=

−

=

=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+=

1

0

1

02

2

2P

subject to

1logb maxn

N

nnt

N

n

nn

PP

ΓσHP

n , (3.5)

where |Hn|2, 2nσ 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.

aH

ΓσP

nn

n =+ 2

2

, (3.6)

where a is a constant chosen such that ∑=n

nt PP . The optimum power allocation is

then

⎪⎪

⎩

⎪⎪

⎨

⎧

<

≥−

=

2

2

2

2

2

2

,0

,

n

nnn

H

Γσa

H

Γσ a

H

Γσa

Pn

nn

. (3.7)

The optimum power allocation is illustrated in Fig. 3.5.

Metropolitan Networks

62

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

∑

∑−

=

−

=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛+=

=

1

02

2

2

1

0tP

1logb subject to

P minn

N

n

nn

N

nn

nΓσHP

P

. (3.8)

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.

[ ][ ] 2

2

22

HNσΓ

[ ][ ] 2

2

33

HNσΓ

[ ][ ] 2

2

44

HNσΓ

[ ][ ] 2

2

NHNNσΓ

n1 2 3 4 N

water level a

total power P

[ ][ ] 2

2

11

HNσΓ

[ ] 01 =P[ ]2P

[ ]3P

[ ]4P

[ ]5P

n1 2 3 4 N

Metropolitan Networks

63

3.3 OFDM System Model for Metro Links

The OFDM system model is shown in Fig. 3.6.

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

( ) Lj

fiber eH 222 β

ωω = , (3.9)

where β2 is the fiber GVD parameter and L is the fiber length. The overall optical

transfer function is given by

( ) ( ) ( )ωωω filterfiber HHH ⋅= , (3.10)

.

.

.

X0

X1

XN−2

XN−1

OpticalOFDM

Tx

Fiber OpticalFilter

Anti-Aliasing Filter

OFDMRx

Amplifier ...

Y0

Y1

YN−2

YN−1

Metropolitan Networks

64

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.

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

Optical Spectrum

XkXk*f

XN/2−1

X1

X0

IDFT

X1

*

Parallelto

Serial+

Cyclic Prefix

.

.

.

D/AXN/2−1

*

.

.

.

Clip

Real Signal

Fiber

XN/2

Laser

DC bias

QAM

QAM*

Metropolitan Networks

65

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

( ) ( ) ( ) 22det )( 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 )( thtxAty ac ⊗+= . (3.12)

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

detected signal becomes

( )

( )*

2/3

2

2/3

2

det

8)(

2)(

8)(

2)()(

⎥⎦

⎤⎢⎣

⎡⊗⎟⎟

⎠

⎞⎜⎜⎝

⎛+−+×

⎥⎦

⎤⎢⎣

⎡⊗⎟⎟

⎠

⎞⎜⎜⎝

⎛+−+=

thA

txAtxA

thA

txAtxAty

acac

acac …

(3.13)

After expanding the terms in Eq. (3.13), the detected signal simplifies to

( ) ( ) ( ) ( )

( ) ( ) ( ) ( ) …+⊗

−⊗

−

⊗+

⊗+=

Athtx

Athtx

thtxthtxAty

acac

acac

88

22)(

**22

**

det

(3.14)

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

Metropolitan Networks

66

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., ( ) ( )txtx acac*= . Thus, we can write the equivalent linear channel for

DC-OFDM as

( ) ( ) ( )2

* thththeq+

= . (3.15)

The transfer function is then

( ) ⎟⎠⎞

⎜⎝⎛=

+=

−

LeeHLjLj

eq 2cos

222

222222

βωω

βωβω

. (3.16)

The equivalent transfer function is plotted on Fig. 3.8.

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

0 2 4 6 8 10 12 14 16 18 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

|Heq

(f)|

Frequency (GHz)

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67

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.

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

QAM

QAM

XkXk*f

X1

XN/2−1

IDFT

Parallelto

Serial+

Cyclic Prefix

D/A

Fiber

QAM

Real Signal

0QAM

.

.

.

0

0

0

0

.

.

.

Clip

X0X1

*

...XN/2−1

*

.

.

.

XN/2

Laser

*

*

Optical Spectrum

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68

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.

IDFT

Parallelto

Serial+

Cyclic Prefix

.

.

.

D/A

DC bias

.

.

.

Re0

0D/A

QuadMZ

Im

Optical Spectrum

Fiber

X0X1

XN/2

XN−1

QAM

BBf

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69

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

written as

( ) ( )( ) ( ) 222det )( thetsAtEty tfj b ⊗+== π , (3.17)

where ( ) tfj bets π2 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

( ) ( )[ ] ( ) ( )[ ]( ) ( ) ( ) ( )

( ) ( ) 2

*2*22

*22det

)()()(

thts

thetsAthetsAA

thetsAthetsAtytfjtfj

tfjtfj

bb

bb

⊗+

⊗⋅+⊗⋅+=

⊗+⋅⊗+=− ππ

ππ

. (3.18)

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.

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

( ) ( )ωω HAHSSB ⋅= , (3.19)

where A is the DC bias.

OFDM* OFDMf

BB

0 fb-fb

Intermodulation Products

(f)Sdety

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70

3.4.4 Effects of Amplifier Noise

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

( ) ( ) 2det )( tntEty += , (3.20)

where E(t) is the output electric field and n(t) is the filtered ASE, i.e.,

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

( )( ) ( )( )( ) ( ) ( ) 2**2

*det

)()()(

)()()(

tntntEtntEtE

tntEtntEty

+⋅+⋅+=

+⋅+=. (3.21)

From Eq. (3.21), we observe that the signal is corrupted by signal-spontaneous

and spontaneous-spontaneous beat noises. We define

( ) ( ) 2** )()()()( tntntEtntEtw +⋅+⋅= 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)0(

)(||)(

)(||)()(

22

2*

*2

''

''

τRR

RmR

RmRR

IIII nnnn

nnEEE

nnEEEww

++

⋅++

⋅+=

ττ

τττ

, (3.22)

where ( ) EmtEtE −= )(' , mE is the average value of the received electric field, )('' τEE

R

is the autocorrelation of ( )tE ' , Rnn(τ) is the autocorrelation of the noise n(t) and

( )τIInnR is the autocorrelation of the real component of n(t). In Eq. (3.22), there are

three main noise components:

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

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

• Spontaneous-spontaneous noise: )(2)0( 22 τRRIIII nnnn +

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71

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

LR22βγ = , (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

RN

PSNR opt

opt0

= , (3.24)

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

R

OSNRSNRoptGHz 5.12

= , (3.25)

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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72

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.

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.

R2

OFDM*

( )fOFDMDCS −

fOFDM OFDM

( )fOFDMSSBS −

fR2

R2

OFDM*

( )fOFDMACOS −

fOFDM

R2f

( )fOOKS

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73

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

σ

biasR

DCB = , (3.26)

where electP=σ 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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74

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.

Power allocation(water-f illing)

Update DC subcarrier using

the bias ratio

Measure received subcarrier

SNRs/gains

Power allocation changed?

End

No

Yes

Initialize the subcarrier

SNRs/gains

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Typical high-performance FEC codes for optical systems have a threshold of

the order of Pb = 10−3, so we compute the minimum optical power required to achieve

Pb = 10−3 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.

Fig. 3.14 Normalized optical SNRs required for the different OFDM formats for γ = 0.25 to achieve Pb = 10−3. 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

0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.411

12

13

14

15

16

17

18

19

20

21

DC-OFDMNu =208416832

ACO-OFDMNu = 104 208 416

SSB-OFDMNu =52104208

Req

uire

d S

NR

opt(

dB)

Bias Ratio, BR

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76

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.

Fig. 3.15 Subcarrier power distribution for DC-OFDM, ACO-OFDM, SSB-OFDM and to achieve a Pb = 10−3 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

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Normalized Subcarrier Frequency (fn / R)

Sub

carri

er P

ower

(Pn/ P

n,m

ax) DC-OFDM

ACO-OFDM

SSB-OFDM

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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.

Fig. 3.16 Normalized optical SNR required for DC-OFDM to achieve Pb = 10−3 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.

0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 2.2 2.412

13

14

15

16

17

18

19

20

21

Bias Ratio, BR

Req

uire

d S

NR op

t(dB

)

Lower Bound

Simulation

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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

achieve Pb = 10−3. 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

0.25 N = 2048 Nu = 832 ν = 18

N = 2048 Nu = 416 ν = 18

N = 1024 Nu = 208 ν = 11

0.5 N = 2048 Nu = 832 ν = 31

N = 2048 Nu = 416 ν = 31

N = 1024 Nu = 208 ν = 21

0.75 N = 2048 Nu = 832 ν = 44

N = 2048 Nu = 416 ν = 44

N = 2048 Nu = 416 ν = 32

1 N = 2048 Nu = 832 ν = 59

N = 2048 Nu = 416 ν = 59

N = 2048 Nu = 416 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

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 110

12

14

16

18

20

22

24

Req

uire

d S

NR

opt(

dB)

DC-OFDM

ACO-OFDM

SSB-OFDM

γ

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79

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

measured over a 12.5-GHz band (OSNR) to achieve Pb = 10−3.

Fig. 3.18 OSNR values (over 0.1 nm) required to obtain Pb = 10−3 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

0 50 100 150 200 250 300 350 4008

10

12

14

16

18

20

22

24

L (km)

Req

uire

d O

SN

R (d

B)

DC-OFDM

ACO-OFDM

SSB-OFDM OOK with MLSD (Bosco)

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80

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

GHz).

Fig. 3.19 OSNR values (over 0.1 nm) required to obtain Pb = 10−3 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

0 50 100 150 200 250 300 350 4008

9

10

11

12

13

14

15

16

17

18

L (km)

Req

uire

d O

SN

R (d

B) DC-OFDM

ACO-OFDM

SSB-OFDM

OOK with MLSD (Bosco)

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KMO MTMLSD b

⋅⋅= 22)( , , (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

OFDMb

TxTOFDM TTNNNO

b/)(log)( 2 , =

[ ] OFDMbuRx

TOFDM TTNNNNOb

/)(log)( 2 , += , (3.28)

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.

γ MLSD SSB-OFDM 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.

Local Area Networks

82

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)

Local Area Networks

83

[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.

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84

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.

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

TX RX

MMF

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85

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.

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

.

.

.

OFDMTx

MultimodeFiber Anti-Aliasing

FilterOFDM

Rx

.

.

.

X1

X0

XN−1

XN−2 YN−2

YN−1

Y0

Y1

Laser

Local Area Networks

86

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]

( ) ( ) )()()(, thzntnzth pulsen

MMF ⊗⎟⎠

⎞⎜⎝

⎛−= ∑ τδα , (4.1)

where the set α(n) is known as the mode power distribution, τ(n) is the group delay per

unit length of the nth 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.

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87

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.

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.

0 5 10 15 200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

Principal Mode Number

Mod

e G

roup

Del

ay (n

s/km

)

0 5 10 15 20 25-60

-50

-40

-30

-20

-10

0

Frequency (GHz)

20 lo

g 10(|H

(f)|)

(dB

)

0 5 10 15 200

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

Principal Mode Number

Mod

e P

ower

Dis

tribu

tion

b)a)

c)

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88

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

)()()()( tnthtxRty MMF +⊗⋅= , (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 ( 0)( ≥tx ) and

the average transmitted optical power Popt is given by

[ ])()(21

lim txEdttxT

PT

TTopt == ∫−∞→

, (4.3)

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

optMMF PHP )0(= , (4.4)

where HMMF(0) is the DC gain of the channel, i.e., ∫∞

∞−= dtthH MMFMMF )()0( .

0 1 2 3 4 5 6 7 8 9 10 11 12 130

5

10

15

20

25

3 dB Bandwidth (GHz)

Fibe

r Dis

tribu

tion

(%)

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89

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]

b

optMMF

b RNPHR

RNPRSNR

0

222

0

22 )0(== , (4.5)

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

2

22

)(

)()(

⎥⎥⎦

⎤

⎢⎢⎣

⎡ −=

∫∫

dtth

dtthtD

MMF

MMFμ, (4.6)

where the mean delay μ is given by

∫

∫ ⋅=

dtth

dttht

MMF

MMF

)(

)(2

2

μ . (4.7)

Fig. 4.6 shows the channel root-mean-square (rms) delay spread D distribution

of all the fibers used in our analysis.

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90

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):

b

T TDD = . (4.8)

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 ]|)([| 22 txE=σ . 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

[ ] dxNxtxEP clipopt ∫+∞

) ⋅==0

2,()( σγ , (4.9)

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.160

2

4

6

8

10

12

Rms Delay Spread, D (ns)

Fibe

r Dis

tribu

tion

(%)

Local Area Networks

91

where N(γ, σ2) is the Gaussian pdf with mean γ and variance σ2. Doing a variable

change, z = x − γ, we obtain

[ ]

2

2

2

22

2

21

,0(,0(

,0()()(

σγ

γγ

γ

πσ

σγγ

σσγ

σγ

−

∞+

−

∞+

−

+∞

−

+⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛−=

) ⋅+) =

) ⋅+=

∫∫

∫

eQ

dzNzdzN

dzNztxE clip

, (4.10)

where Q is the Gaussian Q function [4]. The average optical power for DC-OFDM is

then

[ ] ⎟⎟⎠

⎞⎜⎜⎝

⎛⎟⎠⎞

⎜⎝⎛−+==

−

− σγγ

πσ σ

γ

QetxEP clipOFDMDC 12

)( 2

2

2 . (4.11)

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 πσ 2/ and 2/2σ , respectively

[61]. Hence, the average optical power for ACO-OFDM is [61]

π

σ2

=−OFDMACOP . (4.12)

Local Area Networks

92

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.

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.

Optical Spectrum

XkXk*f

XN/2−1

X1

X0

IDFT

X1

*

Parallelto

Serial+

Cyclic Prefix

.

.

.

D/AXN/2−1

*

.

.

.

Clip

Real Signal

Fiber

XN/2

0

j·PAM

−j·PAM

Local Area Networks

93

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]

π

σ2

=−DMTPAMP , (4.13)

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

⎟⎟⎠

⎞⎜⎜⎝

⎛Γ

+= 2

21

2||1log

σHPb t

ACO , (4.14)

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

( )

aH

σΓP

nn

n =+ 2

2 2. (4.15)

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

Local Area Networks

94

δ

δ

−=

+=

2

2

2

1

topt

topt

PP

PP, (4.16)

where δ is given by

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛−Γ= 2

12

2

2 1141

HHσδ . (4.17)

The maximum normalized bit rate for PAM-DMT is

( ) ( )

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

Γ−

+⋅Γ

++=

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

Γ++

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

Γ+=

22

12

21log

21log

21

21log

21

2

22

2

21

2

2

222

22

211

2

σδ

σδ

σσ

HPHP

HPHPb

tt

optopt

PAM

. (4.18)

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

( ) ( )

( )

ACO

t

t

ttPAM

b

HP

HHP

HPHPb

<

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

Γ+<

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

Γ−

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛

Γ+≤

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

Γ−

+⋅Γ

++≤

2

21

2

2

2

21

2

2

21

2

2

21

2

21

2

1log

222

1log

22

12

21log

σ

σδ

σ

σδ

σδ

. (4.19)

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

Local Area Networks

95

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-

error ratio (BER) of the order of Pb = 10−3. In order to provide a small margin, we

compute the minimum required SNR to achieve Pb = 10−4 for the different modulation

formats. A BER Pb = 10−4 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-

PAM or OOK, Rs = Rb). We let OFDMsR denote the equivalent symbol rate for OFDM

[25]. In an attempt to provide a fair comparison between unipolar M-PAM and

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

. 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

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96

sampling frequency of OFDMssRM . 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

= 10 Gbit/s and sOFDMs RR = , 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.

Fig. 4.8 Receiver electrical SNR required to obtain 10 Gbit/s at Pb = 10−4 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 symbol rate as OOK ( OFDM

sR = 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

same for both modulation schemes, i.e., OFDMsR = 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

0.1 0.2 0.3 0.4 0.5 1 28

10

12

14

16

18

20

22

24

26

Normalized Delay Spread (DT)

Rec

eive

r Ele

ctric

al S

NR

(dB

)

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

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97

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

approximately OFDMsR [25]. Hence, for the same symbol rate as M-PAM ( OFDM

sR = 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.

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

sR2f

( )fsRPAMS )(

( )fsROFDMS )2(

OFDM*

fOFDM

sR2

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98

symbol rate of OOK, i.e., OFDMsR = 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.

Fig. 4.10 Receiver electrical SNR required to achieve Pb = 10−4 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 formats is twice that for OOK, OFDM

sR = 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.

0.1 0.2 0.3 0.4 0.5 1 25

10

15

20

25

30

Normalized Delay Spread (DT)

Rec

eive

r Ele

ctric

al S

NR

(dB

)

ACO-OFDM (2Rs)

MMSE-DFE OOK (Rs)

DC-OFDM (2Rs)

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99

Fig. 4.11 Receiver electrical SNR required for DC-OFDM to achieve Pb = 10−4 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

SNR to achieve 10 Gbit/s at Pb = 10−4 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

1 1.5 2 2.5 3 3.524.5

25

25.5

26

26.5

27

27.5

Bias Ratio

Rec

eive

r Ele

ctric

al S

NR

(dB

)

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100

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.

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

sR = 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.

0.1 0.2 0.3 0.4 0.5 1 26

8

10

12

14

16

18

20

22

24

26

Rec

eive

r Ele

ctric

al S

NR

(dB

)

Normalized Delay Spread (DT)

ACO-OFDM (Discrete, β = 0.25)

PAM-DMT (Discrete, β = 0.25)

ACO-OFDM (Continuous)

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Fig. 4.13 Subcarrier power distribution for ACO-OFDM with continuous bit allocation and with discrete bit loading (with granularity β = 0.25) for 10 Gbit/s at Pb = 10−4 in fiber 10. The symbol rate is

OFDMsR = 20 GHz.

Fig. 4.14 shows the cumulative distribution function (CDF) of the required

electrical SNR, which is defined as

( )SNRxSNR ≤= Prob)(CDF , (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,

the target Pb ≤ 10−4 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.

0 100 200 300 400 5000

0.2

0.4

0.6

0.8

1

1.2

Data Subcarrier Number

Sub

carri

er P

ower

(Pn

/ Pn,

max

)

ContinuousDiscrete (β = 0.25)

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Fig. 4.14 Cumulative distribution function (CDF) of the required receiver electrical SNR to obtain Pb = 10−4 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.

Fig. 4.15 Receiver electrical SNR required to obtain 20 Gbit/s at Pb = 10−4 for the different

5 10 15 20 25 300

0.2

0.4

0.6

0.8

1

1.2

Receiver Electrical SNR (dB)

CD

F(S

NR

)

ACO-OFDM and PAM-DMT (2Rs)

DC-OFDM (2Rs)

MMSE-DFE OOK (Rs)

ACO-OFDM andPAM-DMR (Rs)

0.2 0.3 0.4 0.5 1 2 3 45

10

15

20

25

30

35

Rec

eive

r Ele

ctric

al S

NR

(dB

)

ACO-OFDM (2Rs)

ACO-OFDM (Rs)

MMSE-DFE 4-PAM (Rs)

Normalized Delay Spread (DT)

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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

Gbit/s at Pb = 10−4 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.

Fig. 4.16 Cumulative distribution function (CDF) of the required receiver electrical SNR to obtain 20 Gbit/s at Pb = 10−4 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

5 10 15 20 25 30 35 400

0.2

0.4

0.6

0.8

1

1.2

Receiver Electrical SNR (dB)

CD

F(S

NR

)

MMSE-DFE 4-PAM (Rs)

ACO-OFDM and PAM-DMT (2Rs)

ACO-OFDM andPAM-DMR (Rs)

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sampling frequency required for 4-PAM using an oversampling ratio of two, i.e., 20

GHz.

Personal Area Networks

105

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

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106

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

Personal Area Networks

107

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.

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108

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 ∫∞

∞−

−= dtethfH ftj π2)()( . 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

.

.

.

OpticalOFDM

TxIndoor Wireless

ChannelAnti-Aliasing

Filter

OFDMRx

.

.

.

X1

X0 Y0

Y1

h(t)

Noise

+

XN−2

XN−1

YN−2

YN−1

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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 ( 0)( ≥tx ) and the average transmitted optical power

Pt is given by

[ ])()(21

lim txEdttxT

PT

TTopt == ∫−∞→

, (5.2)

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

tPHP )0(= , (5.3)

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

∞−= dtthH )()0( .

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, ( ) …,2,1,0),( =kth k . 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

Personal Area Networks

110

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.

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]

b

t

b RNPHR

RNPRSNR

0

222

0

22 )0(== , (5.4)

10 20 30 40 50 60 70 80 90 10002468

101214161820

Time (ns)

Impu

lse

resp

onse

(s-1)

h(1)(t)

h(2)(t)

h(3)(t)h(4)(t) h(5)(t)

h(t)

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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

2

22

)(

)()(

⎥⎥⎥

⎦

⎤

⎢⎢⎢

⎣

⎡ −=

∫∫

∞

∞−

∞

∞−

dtth

dtthtthD

μ, (5.5)

where the mean delay μ is given by

∫∫

∞

∞−

∞

∞−=dtth

dttth

)(

)(2

2

μ . (5.6)

We also compute the normalized delay spread DT, which is the rms delay

spread D divided by the bit period Tb (Tb = 1/Rb).

b

T TDD = . (5.7)

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

Personal Area Networks

112

directed toward an infinite diffuse reflector. Considering a single bounce, the impulse

response can be obtained in closed form as

( ))(6)( 7

6

tuat

athc += , (5.8)

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 1)0( =H . The rms delay

spread is related to a by

( )1113

12)( athD c = . (5.9)

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

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113

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

systems have a bit-error ratio (BER) thresholds of the order of Pb = 10−3 [71]. In order

for our system to be compatible with such FEC codes while providing a small margin,

we compute the minimum required SNR to achieve Pb = 10−4 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).

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114

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

= Rb). We let OFDMsR 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, we initially set the two symbol rates to be equal: sOFDMs RR = [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

are the same for both modulation schemes, i.e., OFDMsR = 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.

Fig. 5.4 Electrical SNR required to achieve Pb = 10−4 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.

0.001 0.01 0.1 1 105

10

15

20

25

30

ACO-OFDM (Rs)

OOK MMSE-DFE (Rs)

Duration BitSpreadDelay RMS

=TD

Req

uire

d E

lect

rical

SN

R (d

B)

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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

electrical bandwidth (Be)2 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

electrical bandwidth of approximately sPAMe RB ≈ . On the other hand, OFDM at

symbol rate OFDMsR only requires an electrical bandwidth of only 2/OFDM

sOFDMe RB ≈

[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.

Personal Area Networks

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PAM ( OFDMsR = Rs), OFDM requires about half the bandwidth of M-PAM with

rectangular NRZ pulses. If we double the symbol rate for OFDM, both modulations

schemes use approximately the same electrical bandwidth: sPAMe

OFDMe RBB == . The

electrical bandwidth required for all modulation formats is summarized in Table 5.1.

Bit Rate (Mbit/s)

DC-OFDM ACO-OFDM PAM-DMT OOK RS 2·RS RS 2·RS RS 2·RS RS

50 N = 512 Nu = 208 ν = 10 Be = 25

N = 512 Nu = 208 ν = 12 Be = 50

N = 1024 Nu = 208 ν = 10 Bo = 25

N = 1024 Nu = 208 ν = 12 Be = 50

N = 1024 Nu = 416 ν = 10 Be = 25

N = 1024 Nu = 416 ν = 12 Be = 50

Nf = 28 Nb = 6 Be = 50

100 N = 512 Nu = 208 ν = 12 Be = 50

N = 512 Nu = 208 ν = 15

Be = 100

N = 1024 Nu = 208 ν = 12 Bo = 50

N = 1024 Nu = 208 ν = 15

Be = 100

N = 1024 Nu = 416 ν = 12 Be = 50

N = 1024 Nu = 416 ν = 15

Be = 100

Nf = 44 Nb = 10 Be = 100

300 N = 512 Nu = 208 ν = 18

Be = 150

N = 1024 Nu = 416 ν = 26

Be = 300

N = 1024 Nu = 208 ν = 18

Bo = 150

N = 2048 Nu = 416 ν = 26

Be = 300

N = 1024 Nu = 416 ν = 18

Be = 150

N = 2048 Nu = 832 ν = 26

Be = 300

Nf = 98 Nb = 28 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

ssOFDMss RMRM 2⋅= , 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

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

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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.

Fig. 5.5 Electrical SNR required to achieve Pb = 10−4 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

requires the lowest SNR to achieve a Pb = 10−4 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.

ACO-OFDM (2Rs)

OOK MMSE-DFE (Rs)

CB for ACO-OFDM (2Rs)

CB for OOK (Rs)

0.001 0.01 0.1 1 105

10

15

20

25

30

Duration BitSpreadDelay RMS

=TD

Req

uire

d E

lect

rical

SN

R (d

B)

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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:

( )SNRxSNR ≤= Prob)(CDF , (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

electrical SNR is at least 30 dB, the target Pb ≤ 10−4 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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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.

We observe in Fig. 5.6, as in Fig. 5.5, that ACO-OFDM using twice the

symbol rate of OOK requires the lowest SNR to achieve a Pb = 10−4. 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.

5 10 15 20 25 30 350

0.2

0.4

0.6

0.8

1

1.2

Required Electrical SNR (dB)

CD

F(S

NR

) DC-OFDM (2Rs)

ACO-OFDM and PAM-DMT (Rs)

OOK MMSE-DFE (Rs)

ACO-OFDM and PAM-DMT (2Rs)

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Fig. 5.7 Electrical SNR required to achieve Pb = 10−4 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.

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.

ACO-OFDM (2Rs)

ACO-OFDM (Rs)

4-PAM MMSE-DFE (Rs)

0.01 0.1 1 1010

15

20

25

30

35

40

45

Req

uire

d E

lect

rical

SN

R (d

B)

Duration BitSpreadDelay RMS

=TD

DC-OFDM (Rs)

10 15 20 25 30 35 40 450

0.2

0.4

0.6

0.8

1

1.2

Required Electrical SNR (dB)

CD

F(S

NR

)

DC-OFDM (2Rs)

ACO-OFDM and PAM-DMT (Rs)

4-PAM MMSE-DFE (Rs)

ACO-OFDM and PAM-DMT (2Rs)

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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-

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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.

We consider a channel realization to be in outage if Pb > 10−4 for a desired bit

rate. We define outage probability Poutage as the fraction of channels with Pb > 10−4

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

( ) ( )∑=

=realizN

kk

realizmean fH

NfH

1

1 , (5.11)

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

channel realizations are in outage, i.e., have Pb > 10−4.

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).

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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.

5 10 15 20 25 30 35 400

20

40

60

80

100

120

Required Electrical SNR (dB)

Pou

tage

(%)

ACO-OFDM (Mean Channel Loading)

ACO-OFDM (Equal Loading)

OOK MMSE-DFE

ACO-OFDM (CB Loading)

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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 (Mbit/s) 50 100 300

ACO-OFDM (RS)

Tx 48.5 48.5 48.1 Rx 50.0 50.0 49.6

ACO-OFDM (2·RS)

Tx 97.0 96.7 106.5 Rx 100.0 99.6 109.4

PAM-DMT (RS)

Tx 48.5 48.5 48.1 Rx 51.5 51.5 51.1

PAM-DMT (2·RS)

Tx 97.0 96.7 106.5 Rx 102.9 102.6 112.4

DC-OFDM (RS)

Tx 43.0 43.0 42.4 Rx 45.9 45.9 45.3

DC-OFDM (2·RS)

Tx 86.1 85.4 95.3 Rx 91.9 91.2 101.1

OOK (RS)

Lf 101 213 415 Lb 59 119 229

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

Personal Area Networks

126

complexity among OFDM formats, since it uses smaller FFT sizes than ACO-OFDM

or PAM-DMT.

Conclusions

127

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

Conclusions

128

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-

Conclusions

129

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.

Conclusions

130

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).

Appendix A

131

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

)()(

)()()(

1

11

1

11

uhuNNnx

rhrNNNnxnISI

channel

N

Lunegck

channel

L

Nrpospreckk

neg

n

p

pos

∑

∑−−

+−=+

−

+=−

−−−+

−+++=

A.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

c

neg

n

p

pos

c

Nnqj

channel

N

Lunegck

channel

L

Nrpospreck

N

nk

euhuNNnx

rhrNNNnxqISI

π21

11

1

11

1

0

)()(

)()()(

−−−

+−=+

−

+=−

−

=

⎟⎟⎠

⎞−−−+

⎜⎜⎝

⎛−+++=

∑

∑∑ A.2

Performing a variable change on the different sums, we get

c

negc

nc

c

c

p

pos

c

Nnqj

cchannel

nNN

nLNznegk

N

n

Nnqj

channel

nL

nNvpospreck

N

nk

eNznhzNx

evnhvNNNxqISI

π

π

22

1

1

0

21

11

1

0

)1()1(

)()()(

−−−−

−−=+

−

=

−−−

−+=−

−

=

+−+−−−+

+−++=

∑∑

∑∑ A.3

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

Appendix A

132

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

∑∑

∑∑−

−−=

−

+

−−

+−=

−−

=

−

−

−

+=

−−−+−++

−+++=

1 2

1

1

1

1

0

2

1

1

1

)1()1(

)()()(

c

nc

c

neg

n

p

c

p

pos

N

zLNn

Nnqj

negkcchannel

N

Lz

vL

n

Nnqj

pospreckchannel

L

Nvk

ezNxNznh

evNNNxvnhqISI

π

π

A.4

Substituting n + v= p and n+ z = m in the first and second double sums,

respectively, we get

)()1(

)()(

)1()1(

)()(

)1()1(

)()()(

1

1

2

1

21

11

1 21

1

2

1

1 221

11

1 )(21

11

1 )(21

11

qHezNx

qHevNNNx

eNmhezNx

ephevNNNx

eNmhzNx

ephvNNNxqISI

z

N

Lz

Nzqj

negk

vN

vqjL

Nvpospreck

zN

LNm

Nmqj

cchannel

N

Lz

Nzqj

negk

L

vp

Npqj

channelN

vqjL

Nvpospreck

zN

LNm

Nqzmj

cchannel

N

Lznegk

L

vp

Nqvpj

channel

L

Nvpospreckk

neg

n

c

c

p

pos

c

nc

c

neg

n

c

p

cc

p

pos

c

nc

c

neg

n

p

c

p

pos

∑

∑

∑∑

∑∑

∑∑

∑∑

−−

+−=+

−

+=−

−−

−=

−−−

+−=+

−

=

−−

+=−

−−

−=

−−−−

+−=+

−

=

−−−

+=−

−−−+

−++=

+−−−−+

−++=

+−−−−+

−++=

π

π

ππ

ππ

π

π

A.5

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,

the variance σ2ISI(q) on each subcarrier q is

Appendix A

133

[ ]

[ ]

[ ]

[ ]

[ ] c

neg

n

p

pos

c

p

pos

neg

n

c

neg

n

neg

n

c

p

pos

p

pos

Nqvz

j

posprecknegk

N

Lz

L

Nvvz

Nqzv

j

negkpospreck

L

Nv

N

Lzzv

Nqzz

j

negknegk

N

Lz

N

Lzzz

Nqvv

j

pospreckpospreck

L

Nv

L

Nvvv

ISI

evNNNxzNxE

qHqH

ezNxvNNNxE

qHqH

ezNxzNxE

qHqH

evNNNxvNNNxE

qHqH

qISIqISIEq

)(2*

2121

1

12

1

12

*22

)(2*

1111

1

11

1

11

*11

)(2*

2111

1

11

1

12

*21

)(2*

2111

1

11

1

12

*21

*2

22

11

21

21

)()1(

)()(

)1()(

)()(

)1()1(

)()(

)()(

)()(

)()()(

−

−+

−−

+−=

−

+=

−

+−

−

+=

−−

+−=

−

++

−−

+−=

−−

+−=

−

−−

−

+=

−

+=

−++−−−

+

−−−−++

+

−−−−−−

+

−++−++

=

=

∑ ∑

∑ ∑

∑ ∑

∑ ∑

π

π

π

π

σ

A.6

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

⎟⎟⎠

⎞⎜⎜⎝

⎛+= ∑∑

−−

+−=

−

+=

1

1

21

1

222 )()()(neg

n

p

pos

N

Lzz

L

NvvsISI qHqHq σσ A.7

where σs2 = E[|x(n)|2] 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.

Appendix A

134

Fig. A-1 DWDM system with 50 GHz channel spacing.

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

)()(

)()()(

1

1

1

1

uhuNNnx

rhrNNNnxnICI

channel

N

Lunegck

channel

L

Nrpospreckk

neg

n

p

pos

∑

∑−−

+−=

−

+=

−−−−

−+++−=

A.8

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.

Hence, the ISI and ICI have the same variance, σ2ISI(q) = σ2

ICI(q). Thus, the variance

of the total interference on subcarrier q can be written as

⎟⎟⎠

⎞⎜⎜⎝

⎛+= ∑∑

−−

+−=

−

+=+

1

1

21

1

222 )()(2)(neg

n

p

pos

N

Lzz

L

NvvsICIISI qHqHq σσ A.9

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.

Npos Nneg

Npos NnegExtp Extn

Extp Extn

-

Symbol k with ICI

Symbol k without ICI

ICI

0 0 0

References

135

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