Electrical and Computer Engineering - ULisboa · Artur Rafael Gama Tavares Duarte Thesis to obtain...

100 Gb/s MB-OFDM metropolitan networksemploying SSBI mitigation in presence of fiber PMD

effect

Artur Rafael Gama Tavares Duarte

Thesis to obtain the Master of Science Degree in

Electrical and Computer Engineering

Supervisors: Prof. Dr. Adolfo da Visitacao Tregeira Cartaxo

Dr. Tiago Manuel Ferreira Alves

Examination Committee

Chairperson: Prof. Dr. Jose Eduardo Charters Ribeiro da Cunha Sanguino

Supervisor: Prof. Dr. Adolfo da Visitacao Tregeira Cartaxo

Member of the Committee: Prof. Dr. Armando Humberto Moreira Nolasco Pinto

May 2015

This dissertation was performed under the project “Metro networks based on multi-band

orthogonal frequency-division multiplexing signals”

(MORFEUS-PTDC/EEI-TEL/2573/2012) funded by Fundacao para a Ciencia e

Tecnologia from Portugal, under the supervision of

Prof. Dr. Adolfo da Visitacao Tregeira Cartaxo

Dr. Tiago Manuel Ferreira Alves

i

Acknowledgments

I would like to thank all the people who contributed in some way to the work described in this

dissertation. First, I offer my sincerest gratitude to my supervisor Prof. Dr. Adolfo Cartaxo,

and co-supervisor Dr. Tiago Alves who have supported me throughout my dissertation with

their patience, knowledge and high quality academic guidance.

I would also like to thank my parents and elder brother for the unequivocal support they

always gave me. In addition, I thank Instituto de Telecomunicacoes (IT) which has provided

me the support and equipment I needed to produce and complete my dissertation.

I am also grateful to my labmates, Ph.D. student Pedro Cruz and Eng. Joao Rosario, for

their encouragement and helpful advice, and for providing a friendly daily work environment.

Finally, I would like to thank all my friends for all the support and good moments which

gave me the will to continue this journey.

iii

Abstract

In this dissertation, the impact of polarization mode dispersion (PMD) on 100 Gb/s multi-

band (MB) orthogonal frequency division multiplexing (OFDM) metropolitan networks em-

ploying signal-signal beat interference (SSBI) mitigation is evaluated through numerical

simulation. The numerical simulations are performed using software mostly developed by

the author in MATLAB®.

The principles of the PMD are described. The study of PMD can be broken into two

different approaches in what concerns the PMD modeling: the first-order and second-order

PMD. The SSBI mitigation technique considered consists of a digital signal processing (DSP)

based iterative algorithm that estimates the SSBI component and then subtract it from the

received signal corrupted by the SSBI.

In this work, the first- and second-order PMD effect is emulated using a coarse-step method

where the optical fiber is considered as a concatenation of short fiber segments with a

given mean birefringence and random coupling angles. As this study is performed using

computational simulation, the long periods of time needed to perform such simulations are

a concern. Therefore, the trade-off between the number of fiber segments considered and

the quality of the PMD emulation is studied.

The results show that the quality of the PMD emulation remains almost constant for a

number of fiber segments equal to 50 or higher. Also, the results show that the impact of

first- and second-order PMD on the performance of the considered system is neglectable for

standard single mode fibers link lengths up to 400 km. The number of iterations needed for

the DSP-based iterative SSBI mitigation algorithm to converge is the same either considering

or neglecting the PMD effect.

Keywords: polarization mode dispersion (PMD), orthogonal frequency division multiplex-

ing, signal-signal beat interference, PMD emulation, multi-band, direct detection

v

Resumo

Nesta dissertacao e avaliado, a partir de simulacao numerica, o impacto da disperssao por

modos de polarizacao (PMD) em redes metropolitanas a operar em multi-banda (MB)

com multiplexagem por divisao ortogonal na frequencia (OFDM) a 100 Gb/s usando mit-

igacao da interferencia por batimento sinal-sinal (SSBI). As simulacoes numericas foram

realizadas com recurso a software em MATLAB® maioritariamente desenvolvido pelo au-

tor. Os princıpios da PMD sao descritos em profundidade. O estudo da PMD pode ser

dividido em duas abordagens diferentes no que se refere a sua representacao. Sao elas a

PMD de primeira e segunda ordem. A tecnica de mitigacao de SSBI considerada consiste

num algoritmo de processamento digital de sinal que estima a componente SSBI e a subtrai

ao sinal recebido, sinal este affectado pelo SSBI.

Neste trabalho, os efeitos da PMD de segunda ordem sao emulados considerando a fibra

optica como uma concatenacao de segmentos de fibra mais curtos, cada com uma deter-

minada birrefringencia media e angulo de acoplamento aleatorio. Dado que este estudo e

realizado usando simulacao computacional, os longos tempos de simulacao necessarios para

realizar tais simulacoes representam um problema. Por esse motivo, foi feito um estudo do

compromisso entre o numero de segmentos de fibra considerado e a qualidade da emulacao

da PMD.

Mostrou-se que a qualidade da emulacao da PMD, no que se refere a sua natureza estatistica

e oscilacoes ao longo do comprimento de onda, mantem-se quase constante e boa para um

numero de segmentos igual ou superior a 50.

Mostrou-se tambm que o impacto da PMD de primeira e segunda ordem na performance do

sistema negligencivel em fibras mono-modo padro para comprimentos de fibra at 400 km.

O nmero de iteraes necessrias para que o algoritmo de processamento digital de sinal que

mitiga o SSBI convirja igual caso se considere ou no efeito de PMD.

vii

Contents

i

Acknowledgments iii

Abstract v

Resumo vii

List of Acronyms xvii

List of Symbols xxi

1 Introduction 1

1.1 Scope of the work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.1 Optical networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.1.2 OFDM based networks . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.3 Objectives and structure of the dissertation . . . . . . . . . . . . . . . . . . 4

1.4 Main original contributions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

2 General principles of OFDM and MB-OFDM systems 7

2.1 OFDM basic concepts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2.1.1 Mathematical formulation of an OFDM signal . . . . . . . . . . . . . 7

2.1.2 Discrete Fourier transform implementation of OFDM . . . . . . . . . 8

2.1.3 Cyclic prefix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.1.4 Spectral efficiency of OFDM . . . . . . . . . . . . . . . . . . . . . . . 11

2.1.5 OFDM system description . . . . . . . . . . . . . . . . . . . . . . . . 13

2.1.6 Equalizer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.2 Multiband OFDM basic concepts . . . . . . . . . . . . . . . . . . . . . . . . 16

2.2.1 Singleband OFDM vs. multiband OFDM . . . . . . . . . . . . . . . . 16

2.2.2 Relation between MB-OFDM system parameters . . . . . . . . . . . 17

2.3 Optical OFDM systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.3.1 MB-OFDM system description and operation . . . . . . . . . . . . . 21

ix

2.3.2 Coherent optical OFDM and direct-detection optical OFDM . . . . . 22

2.3.3 E-O conversion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3.4 O-E conversion and thermal noise . . . . . . . . . . . . . . . . . . . . 25

2.3.5 Signal-signal beat interference and virtual carrier motivation . . . . . 26

2.3.6 Optical noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

2.4 Performance evaluation of the system . . . . . . . . . . . . . . . . . . . . . . 31

2.4.1 Error vector magnitude . . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.4.2 Exhaustive Gaussian approach . . . . . . . . . . . . . . . . . . . . . . 32

2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

3 SSBI mitigation techniques and optical fiber dispersion effects 33

3.1 SSBI mitigation techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.1.1 Beat interference cancellation receiver . . . . . . . . . . . . . . . . . . 33

3.1.2 Signal-phase-switching . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.1.3 DSP-based iterative SSBI mitigation algorithm . . . . . . . . . . . . 36

3.2 VBG restriction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.3 Optical fiber dispersion effects . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.3.1 Chromatic dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.3.2 First-order polarization mode dispersion . . . . . . . . . . . . . . . . 45

3.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4 Study of the impact of first- and second-order PMD on direct detection

MB-OFDM systems 53

4.1 Theoretical modeling and statistical analysis of first- and second-order PMD 53

4.1.1 Theoretical modeling of first- and second-order PMD . . . . . . . . . 54

4.1.2 Statistical properties of first- and second-order PMD . . . . . . . . . 55

4.1.3 Optimization of the parameters of the PMD numerical simulation . . 57

4.2 Impact of PMD fluctuations along time . . . . . . . . . . . . . . . . . . . . . 60

4.3 Performance evaluation of SSBI mitigation algorithm in presence of first- and

second-order PMD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67

5 Conclusion and future work 69

5.1 Final conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

5.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

A MB-OFDM signal 73

B Super Gaussian band selector 77

List of Figures

1.1 Optical network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

2.1 Output values sm of the IDFT after the parallel-to-serial module. . . . . . . 9

2.2 Motivation for the use of CP in OFDM systems. Illustrative representation

of an OFDM signal with two subcarriers in the presence of a dispersive channel. 10

2.3 Spectrum of Nsc subcarriers . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 Block diagram of an OFDM system. . . . . . . . . . . . . . . . . . . . . . . 13

2.5 OFDM signal replicas produced at DAC and zero padding motivation. Tc -

chip time. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.6 Illustrative spectrum of a MB-OFDM signal. Bb,n - bandwidth of the nth

band; BG - guard band; ∆B - OFDM band slot. . . . . . . . . . . . . . . . . 17

2.7 Optical link employing external modulation. Single line arrow - electrical

domain; Double line arrow - optical domain. . . . . . . . . . . . . . . . . . . 21

2.8 Conceptual diagram of a MORFEUS node. MEB - MORFEUS extraction

block; MIB - MORFEUS insertion block; ROADM - reconfigurable optical

add-drop multiplexer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.9 Possible scheme of a MZM. Ein - input static light wave; va(t), vb(t) - refrac-

tive index voltage control of the upper and lower arms respectively; eMZM(t)

- output light signal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.10 Normalized electrical field at the output of the MZM as a function of the ap-

plied electrical voltage normalized to the switching voltage. MBP - minimum

bias point; QBP - quadrature bias point. . . . . . . . . . . . . . . . . . . . . 24

2.11 DPMZM structure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.12 Photodetector scheme of a direct detection system. RL - load resistor; Vbias

- bias voltage; hν - photon energy; iPIN(t) - current generated by the PIN;

iout(t) - current at photodetector output. . . . . . . . . . . . . . . . . . . . . 25

2.13 Illustration of the SSBI positioning. fgap - frequency gap; fλ - frequency

of the optical carrier; Bs - bandwidth of the OFDM band signal; BSSBI -

bandwidth of the SSBI. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.14 Illustration of a SSB MB-OFDM signal spectrum and the frequency gap. fgap

- frequency gap; fλ - frequency of the optical carrier; Bb - bandwidth of the

OFDM band signal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

xi

2.15 Main frequency parameters of MB-OFDM signals employing virtual carriers.

BG - band gap; VBG - virtual carrier-to-band gap; fc,1, fc,2 - central frequency

of the first and second OFDM band respectively; ∆B - band spacing. . . . . 29

2.16 Constellation diagram and EVM . . . . . . . . . . . . . . . . . . . . . . . . . 31

3.1 BICR structure. PD1, PD2 - photodiode 1 and 2, respectively. . . . . . . . . 34

3.2 SPS conceptual diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3.3 Schematic diagram of the SSBI mitigation algorithm in training mode for a

SSB MB-OFDM system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.4 Schematic diagram of the SSBI mitigation algorithm in data mode for a SSB

MB-OFDM system. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.5 EVM as a function of VBG/∆fsc for VBPR=10 dB and modulation index of

5%. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.6 EVM as a function of the SSBI iteration number for the four cases under

study. Cases detailed in table 3.1. . . . . . . . . . . . . . . . . . . . . . . . . 41

3.7 In black - normalized PSD of the photodetected OFDM signal before the

SSBI mitigation algorithm; in grey - normalized PSD of the OFDM signal

after the SSBI mitigation algorithm. . . . . . . . . . . . . . . . . . . . . . . 42

3.8 Normalized PSD of the estimated SSBI component. . . . . . . . . . . . . . . 42

3.9 EVM as a function of VBG/∆fsc for OSNR=30 dB and VBPR=7 dB. . . . 43

3.10 Impact of the CD on the constellation at the receiver before the equalizer. . 44

3.11 EVM after the SSBI mitigation algorithm as a function of the fiber length in

presence of CD. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.12 Effect of PMD on a OFDM signal. . . . . . . . . . . . . . . . . . . . . . . . 46

3.13 Impact of PMD on an OFDM signal after photodetection when condition

3.25 is not verified. System without noise. . . . . . . . . . . . . . . . . . . . 47

3.14 Impact of PMD on the transmitted signal. ∆τ = 70 ps and VBG = 150∆fsc.

CD as been neglected. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.15 EVM of the system as a function of the DGD. System without optical noise. 50

3.16 BER as function of the fiber length in presence of CD, first-order PMD and

optical noise simultaneously. . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.1 Illustration of the concatenation of Nseg fiber segments. The angle αn is the

coupling angle between the (n− 1)th and nth segments, and hn is the length

of the nth segment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

4.2 DGD as a function of the equivalent baseband frequency for a 400 km fiber

(DPMD = 0.5 ps/√

km) where the length of the segments is constant and

equal to 500 meters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.3 DGD as a function of the equivalent baseband frequency for a 400 km fiber

(DPMD = 0.5 ps/√

km) where the length of the segments are randomly gener-

ated from a Gaussian distribution around the mean length per segment equal

to 500 meters with standard deviation equal to 30% of the mean length per

segment. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.4 Statistical distribution of DGD for a 400 km optical fiber (〈∆τ〉 = 10 ps),

composed by 800 concatenated unequal segments (mean length per segment

equal to 500 meters). Both figures represent the histogram of the DGD val-

ues obtained from simulation superimposed with the theoretical Maxwellian

distribution with mean value equal to 10 ps. . . . . . . . . . . . . . . . . . . 57

4.5 Similarity measure as a function of the fiber length with Nseg = 100. The

results presented were obtained from 1000 fiber realizations. . . . . . . . . . 58

4.6 Similarity measure as a function of the number of segments considered for a

400 km fiber. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.7 Statistical distribution of DGD for a 400 km optical fiber (〈∆τ〉 = 10 ps).

Both figures represent the histogram of the DGD values obtained from sim-

ulation, superimposed with the theoretical Maxwellian distribution. . . . . . 59

4.8 DGD of the fiber affected by the PMD effect as a function of the equivalent

baseband frequency with Nseg equal to 5 and 800. Lf = 400 km. . . . . . . . 60

4.9 Absolute value of the slope of the DGD as a function of the equivalent base-

band frequency for a given fiber realization. Lf = 400 km. . . . . . . . . . . 61

4.10 BER of the system and occurrence probability of each DGD category fiber

realization, for a 100 km fiber link. . . . . . . . . . . . . . . . . . . . . . . . 65

4.11 Weighted BER of each DGD category fiber realization for a 100 km fiber link. 65










4.18 Weighted BER, 〈BER〉, as a function of the fiber length in presence of first-

and second-order PMD, CD and optical noise. . . . . . . . . . . . . . . . . . 67

A.1 16 QAM constellation at the transmitter symbol mapper output. . . . . . . . 73

A.2 Signal waveform at the output of the DAC and LPF used at the transmitter

side. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

A.3 Normalized PSD at DAC output. . . . . . . . . . . . . . . . . . . . . . . . . 74

A.4 Normalized PSD at LPF output. . . . . . . . . . . . . . . . . . . . . . . . . 74

A.5 Normalized PSD at IQM output. . . . . . . . . . . . . . . . . . . . . . . . . 74

A.6 Normalized PSD of the MB-OFDM signal. . . . . . . . . . . . . . . . . . . . 74

A.7 16 QAM constellation at the receiver side. . . . . . . . . . . . . . . . . . . . 75

B.1 Spectra of an optical MB-OFDM signal, in equivalent baseband frequency,

in a system with a 2nd order super Gaussian BS. In black - signal before BS;

in grey - signal after BS. The BS has a -3 dB bandwidth of 2.2 GHz and a

detuning (relatively to the central frequency of the OFDM band) of 300 MHz. 77

List of Tables

2.1 Values of β corresponding to each M = 2n, n ∈ N. . . . . . . . . . . . . . . . 19

2.2 Values of BER corresponding to each M = 22n, with integer n. . . . . . . . . 20

3.1 System parameters defined for each of the four cases under study. The un-

derlined values highlights the parameters that differs from the reference test

(case A). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.2 Fiber parameters considered in this work. . . . . . . . . . . . . . . . . . . . . 44

3.3 DGD phase shift and resulting attenuation of the first and last subcarriers. . 49

4.1 List of non-empty DGD categories for fiber lengths equal to 100, 200, 300

and 400 km. A→ B stands for interval between A and B. . . . . . . . . . . 64

xv

List of Acronyms

ADC Analog-to-digital converter

ADM Add/drop multiplexers

ADSL Asymmetric digital subscriber line

BER Bit error ratio

BS Band selector

CD Chromatic dispersion

CO-OFDM Coherent detection orthogonal frequency division multiplexing

CP Cyclic prefix

DAC Digital-to-analog converter

DD-OFDM Direct detection orthogonal frequency division multiplexing

DEC Direct error counting

DFT Discrete Fourier transform

DGD Differential group delay

DSP Digital signal processing

DPMZM Dual-parallel Mach-Zehnder modulator

DVB-C Digital video broadcasting-cable

DVB-T Digital video broadcasting-terrestrial

E-O Electro-optical

EGA Exhaustive Gaussian approach

EVM Error vector magnitude

FEC Forward error correction

xvii

FFT Fast Fourier transform

HSSG High speed study group

ICI Intercarrier interference

IDFT Inverse discrete Fourier transform

IEEE Institute of electrical and electronics engineers

IFFT Inverse fast Fourier transform

IQDM In-phase and quadrature demodulator

IQM In-phase and quadrature modulator

ISI Intersymbol interference

ITU-T International Telecommunication Union - Telecommunication

Standardization Sector

LASER Light amplification by stimulated emission of radiation

LED Light emitting diode

LPF Low-pass filter

LTE Long-term evolution

MB-OFDM Multiband orthogonal frequency division multiplexing

MBP Minimum bias point

MEB MORFEUS extraction block

MIB MORFEUS insertion block

MORFEUS Metro networks based on multiband orthogonal frequency-division

multiplexing signals

MZM Mach-Zehnder modulator

O-E Opto-electric

OADM Optical add-drop multiplexers

OFDM Orthogonal frequency division multiplexing

OSNR Optical signal-to-noise ratio

PSD Power spectrum density

PSP Principal states polarization

PMD Polarization mode dispersion

QAM Quadrature amplitude modulation

QBP Quadrature bias point

RF Radio frequency

RMS Root mean square

ROADM Reconfigurable optical add-drop multiplexer

SB-OFDM Singleband orthogonal frequency division multiplexing

SNR Signal-to-noise ratio

SSB Single sideband

SSBI Signal-signal beat interference

WDM Wavelength-division multiplexing

Wi-Max Wireless metropolitan area networks

Wifi Wireless local area networks

List of Symbols

Symbol Designation

aPMD,n attenuation caused by the PMD on the nth subcarrier

b PMD coefficient of the fiber

B−3dB bandwidth of the filter at -3 dB

Bb,n bandwidth of the nth band

Bch,avai channel bandwidth available due to non-ideal filters

Bch channel bandwidth

BERn BER resulting from the nth DGD category

〈BER〉 weighted BER

BG guard band

BMB−OFDM overall bandwidth of the MB-OFDM system

BOFDM approximation of OFDM signal bandwidth

Bn(ω) birefringence matrix of the nth segment

BSSBI bandwidth of the SSBI

Bs OFDM signal bandwidth

c speed of light in vacuum

C optical carrier

cki ith OFDM symbol at kth subcarrier

DPMD PMD parameter of the optical fiber

Dλ0 dispersion parameter of the optical fiber at the wavelength λ0

E0 optical carrier~Ea(t) input optical field projected onto the two PSPs at the input of the fiber~Eb(t) input optical field projected onto the two PSPs at the output of the fiber

Ein static light wave

eMZM(t) optical signal at MZM output

eout optical signal at the output of the DPMZM

ePIN optical signal incident on the PIN

xxi

Es OFDM signal

Es,(k) OFDM signal resulting from the kth iteration

EVM [k] EVM of the kth subcarrier

fc central frequency

fgap frequency gap

fk frequency of the kth subcarrier

fn noise figure of the pre-amplifier

fsc,max frequency of the subcarrier with the higthest frequency

f∆τ (∆τ) probability density function of the Maxewellian distribution

fλ frequency of the optical carrier

g gain of the electrical pre-amplifier

Gsist transmission gain from the DPMZM output to the photodiode input

Ha attenuation of the optical fiber

HCD transfer function of the CD

Hch,load(k) channel loading response

Hchannel,n(k) channel transfer function resultimg from the nth OFDM training symbol

Heq(k) equalizer transfer function

hn length of the nth segment

HPMD,1st first-order PMD transfer function

HSG(f) transfer function of a super Gaussian filter

hν photon energy

IA original photocurrent of the OFDM signal

IB phase shifted photocurrent of the OFDM signal

in(t) thermal noise current resulting from RL

iout current at the output of the electrical pre-amplifier

iPIN(t) current generated by the PIN

kB Boltzmann constant

Lf fiber length

M order of the modulation scheme

NB number of OFDM bands

Nclass number of classes which the histogram is divided

nIFFT (N) number of mathematical computations performed by an IFFT with N inputs

Ninfo total number of information OFDM symbols

Nrealiz number of fiber realizations

Ns total number of OFDM symbols

Nsc number of subcarriers

Nseg number of segments of the optical fiber

Nslots number of frequency slots available in a optical channel

ntotal,MB−OFDM number of mathematical computations performed by a MB-OFDM system

ntotal,SB−OFDM number of mathematical computations performed by a SB-OFDM system

Ntrain total number of training symbols

pb power of the OFDM band

pn thermal noise power

Pteo Theoretical Maxwellian probability density function

Psim DGD histogram generated by simulation

pvc power of the virtual carrier

Rb bit-rate a an OFDM system

Rb,noGI bit-rate a an OFDM system without guard interval

Rb,withGI bit-rate a an OFDM system with guard interval

RL load resistor

rm received OFDM signal sampled at every time interval TsNsc

R(αn) rotation matrix between the (n− 1)th and nth segments

Rλ PIN responsivity

s+fn

(t) signal corresponding to the nth subcarrier that propagates on PSP+

s−fn(t) signal corresponding to the nth subcarrier that propagates on PSP-

si(ω) PSP representation vector of the optical signal at the input of the fiber

sli[k] signal corresponding to the kth subcarrier of the lth OFDM symbol

of the ideal constalation

sk(t) waveform of the kth subcarrier

sm mth sample of a OFDM symbol

Sn noise power spectral density

so(ω) PSP representation vector of the optical signal at the output of the fiber

slo[k] signal corresponding to the kth subcarrier of the lth OFDM symbol

s(t) OFDM signal

Sλ0 dispersion slope parameter at the wavelength λ0

T temperature in kelvin of the resistor

Tc chip time duration

td time delay between two subcarriers

tG guard interval

Tn period of the nth subcarrier

ts time duration of the DFT window

Ts OFDM symbol period

tsim,BER time spend to calculate the BER of the system resulting

from one fiber realization

tsim,seg time spend to generate one fiber segment

T (ω) Jones matrix

u DGD in ps

v(t) RF signal

va(t) applied voltage to the upper arm of the MZM

Vb bias voltage applied to the MZM

vb(t) applied voltage to the lower arm of the MZM

Vb,3 applied voltage used to control the phase difference

Vbias bias voltage

vsig signal to be transmitted

Vsv switching voltage

x arbitrary signal

Xeq(k) symbols at the equalizer output

xH Hilbert transform of x

XR(k) receiver symbol at the input of the equalizer

Xsc,k(f) spectrum of the kth OFDM subcarrier

xSSB SSB signal of x

αn coupling angle between the (n− 1)th and nth segments

β filling factor

β(Ω) propagation constant as a function of the baseband equivalent frequency

β0 propagation constant at υ0

β1 propagation constant adjustment to υ

β2 group velocity dispersion

β3 second order group velocity dispersion

γ PMD power-splitting ratio

Γadj adjustment transmission coefficient

Γc transmission coefficient

Γini initial value of the transmission coefficient

∆B OFDM band slot

∆DGD DGD variation

∆inc incremental factor in decibel

∆fsc frequency spacing between two adjacent subcarriers

∆τ DGD between the two PSPs

∆τmax maximum DGD

〈∆τ〉 mean value of DGD

∆ω frequency interval in radians

εa+ principal state of polarization at fiber input

εa− principal state of polarization at fiber input

εb+ principal state of polarization at fiber output

εb− principal state of polarization at fiber output

η spectral efficiency

ηinfo percentage of OFDM symbols that carries information

λ0 wavelength of the optical carrier

µ standard deviation

ξ Euclidean distance

Π(t) OFDM symbol shaping function

ρkl correlation coefficient between kth and lth subcarrier

τ0 polarization independent group delay

υ0 optical frequency

ϕI phase shift on the in-phase component

φn phase shift due to temperature variations along the fiber

φPMD phase shift caused by the PMD

ϕQ phase shift on the quadrature component

Ω baseband equivalent frequency

Chapter 1

Introduction

1.1 Scope of the work

This work is performed in the scope of the optical telecommunication systems, more pre-

cisely, the multiband (MB) orthogonal frequency division multiplexing (OFDM) metropoli-

tan networks using direct-detection. The traffic growth experienced in the last years en-

courage the development of next generation systems with 100 Gb/s per wavelength to be

employed in metropolitan networks. Therefore, a system operating at such bit rate is con-

sidered is this work.

1.1.1 Optical networks

Recent optical networks provide a common infrastructure over which a variety of services

can be delivered [1]. Triple play services and the consequent increase of bandwidth demand,

requires the network to be capable of delivering bandwidth in a flexible manner where and

when needed. Optical fiber offers much higher bandwidth than copper cables and is less

susceptible to electromagnetic interference and other undesirable effects [2]. As depicted in

figure 1.1, the optical network can be broken up into: access network, metropolitan (metro)

network and long-haul network [2]. The long-haul network interconnects different cities,

regions or even countries reaching hundreds to thousands of kilometers between nodes. The

access network extends from a node of the metro network out to end users which reach is

typically a few kilometers. Metropolitan networks are responsible by aggregating traffic from

the access networks, transferring traffic between different access networks and, if required,

by delivering the traffic to the long-haul network. The metro network is the part of the

network that lies within a large city or a region. Typical extension of these networks is

usually about 200 km to 300 km [1].

The most common metro network architecture is the ring topology due to its reliability [3].

The nodes of a metro network might be responsible for traffic aggregation (add function)

or extraction (drop function) from the metro to the access, long-haul or even other metro

network. Such process is performed by devices called add/drop multiplexers (ADM). The

1

...

Metro ring

Metropolitan network Access networkLong-haul network

Node A

Node B

Node C

Users

Figure 1.1: Optical network

ADM nodes were employed in the first generation of optical networks. Networks using ADM

are commonly referred as opaque architectures because the optical signal is converted to the

electrical domain at each node. This causes the network to be inefficient since all the traffic

must be converted to electrical domain even if it is not going to be extracted. In the second

generation, transparent nodes were deployed. In this case, optical to electric conversions are

no longer needed. This kind of optical network uses optical ADM (OADM) in the nodes.

Recently, reconfigurable OADM (ROADM) has been employ on the network nodes due to

its ability to remotely switch traffic without affecting traffic already passing thought it.

Despite current optical networks consider the metro and the access network as different

and independent networks, the integration into a single hybrid metro-access (HMA) optical

network has been appointed as a good solution from network operators viewpoint to reduce

cost and increase the energy efficiency [1]. It has been shown that the convergence of

multiple services over fiber is possible if the modulation formats of the signals used to

transmit those services are similar [4]. Recent high capacity wireless signal standards such

as ultrawideband (UWB), long-term evolution (LTE), and worldwide interoperability for

microwave access (WiMAX), use OFDM and its variants [5]. Therefore, a fully convergence

quintuple-play (5th-play) service is a strong candidate for future networks. However, the

increase of the bit-rate per user leads to high costs to interconnect the access and metro

networks. One solution to avoid the costs of this HMA is to use a long-reach passive optical

network (LR-PON) architecture [4]. However, traditional 20 km span of the PONs are not

viable for HMA networks, therefore, studies have been performed with the goal of extend

the coverage span of PONs up to 100 km [6].

2

1.1.2 OFDM based networks

The OFDM has been widely used as a modulation technology to achieve high data rate

transmission in telecommunication networks. Many wireless standards such as WiMAX,

wireless local area networks (WiFi, IEEE 802.11a/g), fourth generation mobile communica-

tions technology LTE and digital video broadcasting-terrestrial (DVB-T), as well as, cable

standards such as DVB-cable (DVB-C) and asymmetric digital subscriber line (ADSL), have

adopted the OFDM modulation technology as a mean to offer great capacities to end users

[7]. OFDM is a particular form of multi-carrier transmission and is suited for frequency

selective channels. OFDM allows to precisely adapt the transmitted signal to the frequency

characteristics of the channel, that is, by avoiding frequencies with low SNR, and increasing

the constellation size on subcarriers with better performance [8]. Although radio frequency

(RF) OFDM has been extensively studied during approximately the past 20 years [9], the

usage of OFDM in optical systems is far from being an easy task. Issues such as chromatic

dispersion, polarization mode dispersion (PMD) and non-linearity of the optical fiber rises

new challenges comparatively to wireless applications [9]. Even though wavelength division

multiplexing (WDM) technique allows the accommodation of huge quantities of traffic, the

access to high granularity (sub-wavelength) levels within the optical channel in WDM-based

systems is only possible in the electric domain.

OFDM is a powerful solution to provide the much needed finner granularity, switching capa-

bilities and high spectral efficiency. Such finner granularity is accomplished by transmitting

several OFDM bands within a wavelength with reduced guard band between adjacent OFDM

bands. This kind of signal is denominated MB-OFDM. The MB-OFDM signals allows fine

tuning the information rate of each OFDM band, which is accomplished by adjusting the

bandwidth occupied by the OFDM band and the modulation used in the different subcar-

riers [1]. In this work, the system under study is the metro networks based on multi-band

orthogonal frequency-division multiplexing signals (MORFEUS) network proposed in [1],

which consists of a metro network based on MB-OFDM signals and employing virtual-

carrier assisted direct-detection. The main difference between the MORFEUS network and

previous MB-OFDM direct-detection networks lies on the fact that the virtual carriers of

MORFEUS network are generated in the electrical domain together with each OFDM band

and one virtual carrier per OFDM band is employed enabling the use of a low-bandwidth

and low-cost receiver. The performance of a 42.8 Gb/s MORFEUS network employing a

2-band, 3-band and 4-band MB-OFDM signal has been evaluated in [1] for a 240 km optical

metro link, where obtained a bit error rate (BER) equal to 10−3 was obtained for a required

optical signal to noise ratio (OSNR) equal to 24 dB.

The traffic growth experienced in the last years encourage the development of next gener-

ation systems with 100 Gb/s per wavelength to be employed in metro networks. Whereas

direct-detection OFDM is more suitable for cost-effective short-reach applications, the supe-

rior performance of coherent-detection OFDM makes it an excellent candidate for long-haul

3

transmission systems. Yang and Shieh in [10] demonstrated the first 107 Gb/s coherent-

detection OFDM reception using multiple orthogonal bands by employing 2x2 multiple-

input and multiple-output (MIMO) OFDM signal processing. It has been experimentally

demonstrated in [11] that dual-polarization MB-OFDM is a credible solution for modern

dispersion compensating fiber (DCF) free 100 Gb/s transmission. Also in [11] the results

revealed in the current paper are very encouraging when considering the next generation

400 Gb/s wavelength division multiplexing (WDM) systems based on OFDM technology. In

[12], an experimental proof of concept of optical band switching over a 100 Gb/s MB-OFDM

signal. An optical add-drop of an OFDM band as narrow as 8 GHz inside a 100 Gb/s dual-

polarization MB-OFDM signal constituted of four bands spaced by 4 GHz guard-bands, has

been successfully performed.

1.2 Motivation

The OFDM technology employed in optical systems provides higher spectral efficiency and

switching capabilities than conventional modulation schemes. The traffic growth experi-

enced in the last years encourage the development of next generation systems with 100

Gb/s per wavelength to be employed in metro networks. However, problems that arise on

such demanding system must be overcome. One of those problems is the fact that virtual-

carrier assisted direct-detection systems (the one employed in MORFEUS network) generate

signal-signal beat interference (SSBI) due to the photodetection process. Although other

photodetection techniques, such as coherent detection, have a superior performance when

compared with direct-detection, the choice of direct-detection is motivated by its reduced

cost. The SSBI degrades the quality of the signal, and so, it must be mitigated. One way

to mitigate the SSBI is by using a digital signal processing (DSP) based iterative algorithm.

It has been shown that such algorithm removes the SSBI successfully [1]. However, the per-

formance of this SSBI mitigation algorithm in presence of fiber PMD is yet to be assessed.

Therefore, this work focuses on the study and characterization of the impact of the PMD

on a direct detection OFDM system at 100 Gb/s. Particularly, the impact of the PMD on

the SSBI mitigation algorithm is assessed using numerical simulation.

1.3 Objectives and structure of the dissertation

The main objective of this work is to evaluate the performance of 100 Gb/s MB-OFDM

metropolitan networks employing SSBI mitigation in presence of fiber PMD effect.

This dissertation is composed by 5 chapters and 2 appendix.

In Chapter 2, the general principles of orthogonal frequency division multiplexing (OFDM)

are introduced. First, a detailed mathematical formulation of the OFDM signal, cyclic prefix

concept and spectral efficiency is presented. Then, a description of the OFDM system is

4

presented. Basic concepts of multiband OFDM are also introduced, such as the signal

characteristics as well as the system operation. Also, optical telecommunication systems

based on OFDM technology is introduced.

In Chapter 3, three different SSBI mitigation techniques are presented. One of those tech-

niques is employed in this work, thus, a rigorously and detailed explanation of that technique

is presented. Also, chromatic dispersion (CD) and first-order polarization mode dispersion

(PMD) of the optical fibers are introduced. The SSBI mitigation technique employed in the

presence of such dispersion effects is evaluated.

In Chapter 4, the performance of SSBI mitigation algorithm in presence of second order

PMD is evaluated. To do so, a coarse-step method is used to simulate an optical fiber, in

which, the fiber is considered as being composed by several concatenated segments with

different birefringence and coupling angles. Also, the trade-off between the number of fiber

segments considered and the quality of the PMD emulation is studied.

In Chapter 5, the final conclusions of this dissertation are presented, and proposals for future

work on this subject are presented.

In Appendix A, the MB-OFDM system operation is explained detail. The signal character-

istics at the different points of the system is shown as well.

In Appendix B, the super Gaussian band selector to be employed on this work is described.

1.4 Main original contributions

In the author’s opinion, the main original contributions of the work developed in this dis-

sertation are:

Behavior assessment of the DSP-based iterative SSBI mitigation algorithm in presence

of first-order PMD for links with high PMD levels,

Performance evaluation of the DSP-based iterative SSBI mitigation algorithm in pres-

ence of first and second-order PMD for links up to 400 km.

Study of the trade-off between the number of segments considered in the second-order

PMD model and the quality of the second-order PMD emulation in order to optimize

the computational simulation time.

5

6

Chapter 2

General principles of OFDM and

MB-OFDM systems

In chapter 2, the general principles of orthogonal frequency division multiplexing (OFDM)

are introduced. First, a detailed mathematical formulation of the OFDM signal, cyclic prefix

concept and spectral efficiency is presented. Then, a description of the OFDM system is

presented. Basic concepts of multiband OFDM are also introduced, such as the signal

characteristics as well as the system operation. Also, optical telecommunication systems

based on OFDM technology are introduced. Section 2.1 follows closely the theoretical

analysis performed in [9].

2.1 OFDM basic concepts

In single-carrier systems, the bit stream is transmitted over one carrier resulting in high

symbol rates for high capacity systems. Those symbols, due their short period, become

very sensible to the dispersive characteristics of the channel, resulting in symbol spreading.

This spreading extends symbols beyond their designated time slot to adjacent symbol slots,

resulting in intersymbol interference (ISI). This effect leads to an increase of complexity of

the receiver to mitigate those effects.

The basic principal of OFDM is to split a high-rate bit stream into a number of lower-

rate bit streams that are transmitted simultaneously over the same number of subcarriers.

After the OFDM process, the symbol period has longer duration, making the symbol less

susceptible to the dispersive characteristics of the channel. The main feature of OFDM

resides in the orthogonality between the subcarriers leading to null intercarrier interference

(ICI) and allowing spectral overlapping providing high spectral efficiency.

2.1.1 Mathematical formulation of an OFDM signal

An OFDM signal is represented analytically by [9]

7

s(t) =+∞∑i=−∞

Nsc∑k=1

ckisk(t− iTs) (2.1)

where cki is the ith OFDM symbol at the kth subcarrier, Nsc is the number of subcarriers,

sk(t) is the waveform of the kth subcarrier which is described as

sk(t) = Π(t)ej2πfkt (2.2)

Π(t) =

1, 0 < t ≤ Ts

0, t ≤ 0, t > Ts(2.3)

where fk is the frequency of the kth subcarrier, Ts is the OFDM symbol period, and Π(t) is

the OFDM symbol shaping function. Using the concept of orthogonality, it is possible for

the signal power spectrum of the subcarriers to overlap without interference. Such concept

can be verified using the correlation coefficient between any two subcarriers as shown by

ρkl =1

Ts

Ts∫0

sk(t)sl∗(t)dt =

1

Ts

Ts∫0

exp(j2π(fk−fl)t)dt = exp(j2π(fk−fl)Ts)sin(π(fk − fl)Ts)π(fk − fl)Ts

(2.4)

Notice that ρkl = 0 for

fk − fl = n1

Ts, n ∈ Z \ 0 (2.5)

resulting in no correlation between the subcarriers k and l, that is, no ICI. Assuming ∆fsc

as the frequency spacing between two adjacent subcarriers

∆fsc = n1

Ts, n ∈ Z \ 0 (2.6)

This means that two subcarriers spaced in frequency by ∆fsc are orthogonal and, therefore,

recoverable although the spectral overlapping. Notice that, to achieve maximum spectral

efficiency, n must be as lower as possible.

2.1.2 Discrete Fourier transform implementation of OFDM

It was first shown by Weinsten and Ebert that OFDM modulation/demodulation can be

performed by inverse discrete Fourier transform (IDFT)/discrete Fourier transform (DFT)

[9]. Considering one OFDM symbol, i = 0, we obtain from equation 2.1 and 2.2

si=0(t) =Nsc∑k=1

ck,0 ej2πfkt, 0 < t ≤ Ts (2.7)

8

Assuming si=0(t) is sampled at every interval of TsNsc

, t is defined as

t =(m− 1)Ts

Nsc

(2.8)

where m ∈ 1, 2, ..., Nsc. The mth sample of si=0(t) becomes

sm =Nsc∑k=1

ck,0 ej2πfk

(m−1)TsNsc (2.9)

Using equation 2.6, we obtain the frequency of the kth subcarrier

fk = (k − 1)∆fsc = (k − 1)n

Ts, k ∈ 1, 2, ..., Nsc (2.10)

In order to maximize the spectral efficiency it is imposed n = 1, and so, we arrive at

fk =k − 1

Ts, k ∈ 1, 2, ..., Nsc (2.11)

By combining equations 2.9 and 2.11, we obtain

sm =Nsc∑k=1

ckej2π

(k−1)(m−1)Nsc = IDFTck (2.12)

In a similar way, at the receiver, we have

c′k = DFTrm (2.13)

where rm is the received signal sampled at every time interval TsNsc

. Afterwards, the values

of sm go through a parallel-to-serial converter to become the values of the OFDM signal

as illustrated in figure 2.1 for a sequence of OFDM symbols. Note that the stream of sm

time

sNsc

s1

s2

s3

...... ... ...sNsc

s1

s2

s3

sNsc

s1

s2

s3

i = -1 i = 0 i = 1 i = 2

Figure 2.1: Output values sm of the IDFT after the parallel-to-serial module.

values in figure 2.1 are in the digital domain. Afterwards, the stream passes through a

digital-to-analog converter to obtain an analog signal. A more detailed explanation of such

digital-to-analog conversion is given in section 2.1.5. For the transmission channel to affect

each subcarrier as a flat channel, there is the need to use a large number of subcarriers in

the OFDM signal.

9

2.1.3 Cyclic prefix

Ts

time

fslow

ffast

(a) At the transmitter side and without CP.

timeDFT window, t

s

fslow

ffast

Ts

(b) At the receiver side without CP.

time

CP CP

CPCP

tG

fslow

ffast

Ts

ts

(c) At the transmitter side and with CP.

time

CP CP

CPCP

td DFT window

fslow

ffast

tG

Ts

ts

(d) At the receiver side with CP.

Figure 2.2: Motivation for the use of CP in OFDM systems. Illustrative representation ofan OFDM signal with two subcarriers in the presence of a dispersive channel.

The cyclic prefix (CP) for OFDM emerges as a solution to avoid ISI and ICI caused by the

fact that OFDM subcarriers travel at different speeds in the dispersive channel. Suppose an

10

OFDM signal with two subcarriers. The slowest subcarrier reaches the receiver with a delay

relatively to the faster one. This brings a severe problem because, when the DFT window

is applied at the receiver, the slow subcarrier has crossed the symbol boundary leading to

interference between neighboring OFDM symbols leading to ISI [9]. Furthermore, because

the OFDM waveform in the DFT window for the slow subcarrier is incomplete, the critical

orthogonality condition for the subcarriers established in equation 2.4 is lost, resulting in

ICI [9]. The solution is to extend the OFDM symbols by copying a slice of the end of each

symbol, and placing it at the beginning of that same symbol. This way, even if delays occurs

between subcarriers, the DFT window does not conflict with the adjacent symbols. This

process is illustrated in figure 2.2 where Ts is the OFDM symbol period, ts is the duration

the DFT window, td is the time delay between the subcarriers and tG is the guard interval.

Notice that

Ts = ts + tG (2.14)

The important condition for ISI-free and ICI-free OFDM transmission is given by [9]

td < tG (2.15)

2.1.4 Spectral efficiency of OFDM

For an OFDM system, the signal bit-rate is given by [9]

Rb =Nsc

Tslog2M (2.16)

where M is the order of the modulation scheme used in each subcarrier and log2M is the

number of bits carried by each subcarrier. Thus, Nsc log2M is the number of bits per OFDM

sysmbol. Dividing the number of bits per OFDM sysmbol by the OFDM symbol period in

which the bits are sent, we obtain the bit-rate of the system. In the specific scenario of an

OFDM symbol without guard interval, i.e., Ts = ts, thus, the bit-rate is given by

Rb,noGI =Nsc

tslog2M (2.17)

For an OFDM signal with guard interval we have to account that some of the OFDM symbol

period does not carry information.

As introduced in subsection 2.1.1, since OFDM subcarriers are orthogonal, their spectrum

is overlapped. The spectrum of an individual OFDM subcarrier has a

Xsc,k(f) =

∣∣∣∣sin(πTs(f − fk))πTs(f − fk)

∣∣∣∣2 (2.18)

shape, centered in the frequency of the kth subcarrier fk, due to the rectangular baseband

shaping function [13]. The spectrum of the OFDM signal at the OFDM transmitter output

11

BOFDM

1 2 3 Nsc

1/ts 1/Ts

frequency

. . . .

1/ts1/ts1/Ts

Figure 2.3: Spectrum of Nsc subcarriers

is illustrated in figure 2.3. Adjacent subcarriers are spaced in frequency by ∆fsc = 1ts

(equation 2.6 where n = 1 and Ts is replaced by ts). This replacement is due to the fact

that the DFT window is applied over a period of time ts. Regarding the frequency of the

nulls of the individual subcarrier spectrum, it is concluded from equation 2.18 that the

frequency spacing between the peak of the main lobe and the 1st null is equal to 1Ts

due

to the presence of the guard interval which enlarge the OFDM symbol period. The overall

OFDM signal bandwidth is then, given by

BOFDM =2

Ts+Nsc − 1

ts(2.19)

Since typical OFDM systems have a large number of subcarriers, we have that

Nsc − 1

ts 2

Ts(2.20)

Nsc − 1 ≈ Nsc (2.21)

Thus, an approximation of the OFDM signal bandwidth is given by

BOFDM ≈Nsc

ts(2.22)

From equations 2.16 and 2.22, the bit-rate of an OFDM band is given by

Rb = BOFDM log2M (2.23)

The general expression for the spectral efficiency, η, is given by

η =Rb

BOFDM

(2.24)

12

By replacing equations 2.16 and 2.22 in 2.24, we obtain

η =ts

ts + tGlog2M (2.25)

2.1.5 OFDM system description

Figure 2.4 shows a block diagram of a singleband OFDM system. An OFDM system converts

a bit stream into quadrature amplitude modulation (QAM) symbols, and then, generates

an OFDM signal. Figure 2.4 represents a block diagram of an OFDM system. The OFDM

S/P Symb. Map.

IFFT CP P/S

Symb. Demap.

FFTS/P

RemoveCP

P/S Equalizer

DAC

DAC

LPF

LPF

ADC

ADC

LPF

LPF

channel

Input bit stream

Outputbit stream

...... ... ...

... ... ... ...

IQM

IQDM

cos(2πfct)

-sin(2πfct)

cos(2πfct + φ

I)

-sin(2πfct + φ

Q)

Figure 2.4: Block diagram of an OFDM system.

modulation and demodulation process comprise several steps. Those steps are:

S/P (Serial-to-parallel): first, the data bit stream is converted from a single stream

to Nsc parallel streams.

Symb. Map. (Symbol mapper): each of the Nsc bit streams are mapped using a

M -ary symbol mapper. At the symbol mapper output, each symbol is represented by

a complex number with real and imaginary parts.

IFFT (Inverse fast Fourier transform): the IFFT is an algorithm used to compute the

IDFT. The Nsc symbols generated in the symbol mapper enter the IFFT where they

are transformed from frequency to time domain.

CP (Cyclic prefix): the cyclic prefix is added to each OFDM symbol to prevent ISI

and ICI.

13

P/S (Parallel-to-series): at this point, the sm values are converted from parallel to

series. The real and imaginary parts of sm are separated into two streams.

DAC (Digital-to-analog converter): at this stage, the digital signals, composed by

a series of discrete values, are converted to analog. A typical DAC converts those

discrete values into a sequence of impulses that are processed by a reconstruction

filter using some form of interpolation method to fill in data between the impulses,

creating a smoothly varying signal. A simple method uses sample and hold technique.

LPF (Low-pass filter): the DAC process creates replicas of the OFDM signal spectrum

along the frequency spectrum, each centered at a integer multiple of 1Tc

where Tc,

designated chip time, is the time duration of each time sample of the output of the

IFFT. A more detailed explanation of the chip time is done in appendix A. Those

replicas need to be removed otherwise aliasing occurs. Such removal is performed by

a LPF.

IQM (In-phase quadrature modulator): in the up-conversion process, the signals car-

rying the real and imaginary parts are combined in an IQ modulator where fc is

the frequency of the electrical carrier. The in-phase signal is multiplied by a cosine,

while the in-quadrature signal is multiplied by a sine. This process generates two

independent signals to be transmitted. They are simply added one to the other and

transmitted through the channel. This kind of modulation allows each one of the two

streams to be perfectly recovered at the receiver.

IQDM (In-phase quadrature demodulator): in the down-conversion, the up-converted

OFDM signal is splitted. One of them is multiplied by a cosine with phase shift ϕI

and the other one by a sine with phase shift ϕQ, both with frequency fc. The purpose

of the phases ϕI and ϕI is ensure synchronization. The in-phase and in-quadrature

signal spectrum is, at this point, at baseband.

LPF (Low-pass filter): the IQDM process generates replicas at ±2fc which are filtered

out by a LPF.

ADC (Analog-to-digital converter): the analog signal is sampled at every Tc time

instant.

S/P and remove CP: the time samples are converted from serial to Nsc parallel

streams and the CP is removed.

FFT (Fast Fourier transform): in this block, the inverse process of IFFT is performed.

The Nsc time values corresponding to one OFDM symbol period enter the FFT block,

obtaining at the output the amplitude values of each subcarrier.

14

Equalizer: the equalizer uses training symbols to estimate the channel characteristic.

This information is then used to compensate for the amplitude and phase distortion

caused by the transmission channel. Ideally, the equalizer transfer function must

match the inverse of the channel transfer function.

Symb. Demap. (Symbol demapper): in this block, the hard decision is preformed

and the mapping symbols, are converted back to bits.

P/S (Parallel-to-series): the Nsc parallel bit streams are joint into a single bit stream.

The DAC process, at the transmitter, produces replicas of the OFDM signal spectrum which

are spaced in frequency by 1Tc

where Tc is the chip time which is the time duration that each

sample is held. This effect brings a problem because, since the frequency spacing between

the left- and rightmost subcarriers is 1Tc

, the replicas are stick side by side. As a consequence,

when a non-ideal LPF is to be used to select the baseband spectrum, some power of the

adjacent replicas passes through the filter. This process is illustrated in figure 2.5a. In order

to solve this problem, the size of the IFFT is duplicated, 2Nsc, and zeros are inserted at

the inputs corresponding to the highest frequencies. This procedure is called zero padding.

Such procedure results in twice the frequency spacing between the replicas, therefore, the

baseband spectrum can easily be selected by a non-ideal LPF. The zero padding process is

illustrated in figure 2.5b.

1

N

...

IFFT

...

0 1T c

−1T c

f [Hz]

OFDM signal spectrum

Signal replicas

Non-ideal LPF

freq

uenc

y

tim

e

(a) Without zero padding.

1

IFFT

...

0 1T c

−1T c

f [Hz]

OFDM signal spectrum

Signal replicas

Non-ideal LPF

freq

uenc

y

tim

e

N/2N/2+1

3N/23N/2+1

2N

0

0

0

0

......

...

(b) With zero padding.

Figure 2.5: OFDM signal replicas produced at DAC and zero padding motivation. Tc - chiptime.

2.1.6 Equalizer

The equalizing process consists in adjusting the amplitude and shifting the phase of spe-

cific frequency components in order to compensate for the amplitude and phase distortion

caused by the transmission channel. The equalizer uses training symbols generated by the

transmitter to estimate the channel transfer function. Those training symbols are known

by the receiver, and so, the channel transfer function can be estimated at the receiver by

15

dividing each received training symbol by the corresponding transmitted training symbol.

The equalizer transfer function is then given by

Heq(k) =

⟨1

Hchannel,n(k)

⟩, n ∈ 1, 2, ..., Ntrain (2.26)

where k is the subcarrier index, Ntrain is the total number of OFDM training symbols and

Hchannel,n(k) is the channel transfer function resulting from the nth OFDM training symbol

given by

Hchannel,n(k) =sQAM,Rx,n(k)

sQAM,Tx,n(k), n ∈ 1, 2, ..., Ntrain (2.27)

where sQAM,Tx,n(k) and sQAM,Rx,n(k) are the mapping symbols at the transmitter and re-

ceiver respectively. Notice that one OFDM training symbol corresponds to Nsc mapping

symbols. The symbols at equalizer output are then given by

sQAM,Eq,n(k) = Heq(k)sQAM,Rx,n(k), n ∈ 1, 2, ..., Ntrain (2.28)

where sQAM,Rx,n(k) are the received symbols at the input of the equalizer. The number of

training symbols, Ntrain, must be sufficiently large in order to compensate the fluctuations

caused by the noise. The equalizer transfer function Heq(k) results from the channel char-

acteristic at the time interval when the training symbols where transmitted. Therefore, due

to fluctuations of the channel characteristic along time, the equalizer transfer function may

not perfectly describe the channel. However, if the time interval between the transmission

of the training symbols sequences is short enough, the channel characteristic fluctuations

do not cause much impact.

2.2 Multiband OFDM basic concepts

2.2.1 Singleband OFDM vs. multiband OFDM

To better understand the concept behind multiband OFDM (MB-OFDM), it is useful to

compare it with the singleband OFDM (SB-OFDM). Suppose that we want to transmit

the same bit rate using both SB-OFDM and MB-OFDM systems. Also, consider the same

parameters for both systems, that is, modulation order M , OFDM symbol period Ts, and

total number of subcarriers Nsc. In SB-OFDM the bit stream is converted in an IFFT

module with a large number of inputs, while MB-OFDM is implemented using several IFFT

modules with reduced number of inputs, i.e., lower number of subcarriers per module. It is

known that for an IFFT with N inputs there is the need for [9]

nIFFT (N) =N

2log2N (2.29)

mathematical computations. Considering nIFFT (x) being the number of mathematical com-

16

putations done by one IFFT module x-sized and N the total number of subcarriers of the

system, the total number of mathematical computations made during one period of OFDM

symbol for the SB-OFDM is given by

ntotal,SB−OFDM = nIFFT (N) =N

2log2(N) (2.30)

In MB-OFDM, the computational efforts are divided by α IFFT modules resulting in

ntotal,MB−OFDM = α nIFFT

(N

α

)= α

(Nα

)2

log2

(N

α

)=N

2log2

(N

α

)(2.31)

As observed in equation 2.31, the greater the value of α, the lower the ntotal. Special

attention for the fact that, since the logarithm is base 2, α can only assume values of the

power of 2, otherwise the result would not be integer. In terms of spectrum, MB-OFDM

for optical applications, is the partition of a WDM channel into several independent bands.

Suppose that multiple bands have the same destiny or, at a certain point, the same path.

MB-OFDM allows these bands to be joint into a single wavelength. One or more bands can

be reserved for a single service depending on the needs, as well as, several bands reserved

to several services in which the resources are shared depending on each necessity at the

moment. MB-OFDM offers a better management of the spectral resources.

2.2.2 Relation between MB-OFDM system parameters

MB-OFDM is composed by multiple OFDM bands. This means that each band carries a

certain bit-rate, and so, the total bit-rate of a MB-OFDM signal is given by

Rb,MB−OFDM =

NB∑n=1

Rb,n (2.32)

where Rb,n is the bit-rate of each individual band and NB the number of bands.

frequency

BMB-OFDM

Bb,1

BG

Bb,2

BG

Bb,3

Bb,NB

....

ΔB ΔB

Figure 2.6: Illustrative spectrum of a MB-OFDM signal. Bb,n - bandwidth of the nth band;BG - guard band; ∆B - OFDM band slot.

The overall bandwidth is given by

17

BMB−OFDM =

NB∑n=1

Bb,n + (NB − 1)BG (2.33)

where Bb,n is the bandwidth of the nth band and BG is the guard band. Figure 2.6 illustrates

the spectrum of a MB-OFDM signal. In order to better understand the relation between

parameters of a MB-OFDM system, suppose a system in which the bandwidth of the bands

are the same, that is, Bb,n = Bb for all n. Lets consider that we want to achieve 100 Gb/s per

channel since it is the required bit-rate of this work. However, considering the existence of

12% overheads, the overall bit-rate required is 112 Gb/s. This overhead takes into account

the FECs and the 64b/66b coding of the information rate of 100 Gb/s specified by the

IEEE-high speed study group (HSSG) [14]. For this work purpose, lets consider an optical

channel bandwidth of Bch = 50 GHz, as defined by ITU-T Recommendation. Due to the

non-ideal characteristics of the filters only about 80% of the channel bandwidth is available:

Bch,avai = 0.8Bch = 40 GHz (2.34)

Since 3.125 GHz is the most likely slot bandwidth for future optical grid standardized by

ITU, lets assume ∆B = 3.125 GHz. The number of spectrum slots available in a channel

can be computed:

Nslots =

⌊Bch,avai

∆B

⌋= b12.8c = 12 (2.35)

From equation 2.32 and 2.23 we get

Rb,MB−OFDM =

NB∑n=1

Rb,n = NBRb = NBBb log2 M = NB β ∆B log2 M (2.36)

where β is the filling factor and it is given by

β =Bb

∆B(2.37)

We have two parameters influencing the bit-rate, M and β. Solving the equation 2.36 in

order to β we obtain

β =Rb,MB−OFDM

NB ∆B log2 M(2.38)

Table 2.1 shows the values of β for several values of M assuming Rb,MB−OFDM = 112 Gb/s

and NB = Nslots = 12. Notice that, as shown in table 2.1, β > 1 for M = 2 and M = 4.

This means that those values of M cannot be used. Also, M = 8 cannot be used since it

implies that the spectrum of the MB-OFDM signal does not have guard band. This fact

establishes 16 as the lower limit for M .

As the modulation order increases, the decision regions of the constellation shrinks making

18

M β2 2.9874 1.4938 0.99616 0.74732 0.59764 0.498128 0.427256 0.373

Table 2.1: Values of β corresponding to each M = 2n, n ∈ N.

the system more susceptible to errors. Thus, an upper limit for M must be found. The

error probability of a system can be evaluated using the bit error rate (BER). The BER of

each OFDM subcarrier at the receiver can be computed from the error vector magnitude

(EVM) of each OFDM subcarrier. A more detailed explanation of the BER and the EVM

is done in section 2.4. By using standard expressions used to evaluate the performance of

M -ary quadrature amplitude modulation (QAM) formats with M = 22n, where n ∈ N, the

BER of the kth subcarrier can be computed by [15]:

BER[k] = 21− 1√

M

log2Merfc

√3 log2

√M

M − 1

1

EVM2RMS[k] log2M

(2.39)

where EVMRMS[k] is the EVM root mean square of the kth sub-carrier and erfc(x) is the

complementary error function given by

erfc(x) =2√π

∞∫x

e−t2

dt (2.40)

The overall BER of a OFDM band with Nsc subcarriers can be computed by averaging the

BER of each subcarrier over all the subcarriers:

BER =Nsc∑k=1

BER[k]

Nsc

(2.41)

For simplicity lets assume that all the subcarriers have the same BER. We then get

BER = BER[k] (2.42)

By establishing a typical -25 dB as the minimum value for the EVM, that is, best case

scenario for optical networks, the BER for several values of M can be evaluated. Those

values can be seen in table 2.2. Note that, unlike table 2.1 where M = 2n, in table 2.2 the

equation 2.39 is restricted to values of M = 22n, that is, constellations with square shape.

Assuming a maximum BER requirement of 10−3 for the system [1], table 2.2 shows that 64

19

M BER2 10−12 ≈ 04 10−12 ≈ 016 10−12 ≈ 064 3.04× 10−5

256 1.26× 10−2

1024 6.50× 10−2

Table 2.2: Values of BER corresponding to each M = 22n, with integer n.

is the maximum modulation order allowed. By crossing the information from table 2.1 and

table 2.2, we can conclude that constellation sizes between 16 and 64 are the only candidates

for the desired system.

A more detailed discussion of the MB-OFDM system operation, as well as the signal char-

acteristics at different points of the system, are presented in appendix A.

2.3 Optical OFDM systems

In sections 2.1 and 2.2, the electrical OFDM system is described. OFDM in optical com-

munication systems has been widely used due to their high spectral efficiency, resilience to

linear fiber effects and a much welcome finer granularity. An optical communication link is

summarily composed by a radio frequency (RF) signal transmitter, an electro-optical (E-O)

converter, an optical channel, an opto-electrical (O-E) converter and a RF signal receiver.

At the RF signal transmitter and receiver, the electrical OFDM signal is modulated and

demodulated respectively. The E-O conversion of the RF signal can be performed using

two different methods: direct modulation or external modulation. In direct modulation, the

RF signal drives directly the optical source, which can be a LED (light emitting diode) or

a LASER (light amplification by stimulated emission of radiation). Due to the frequency

chirp of the optical source combined with the chromatic dispersion of the fiber, this type of

modulation is limited in distance for high bit rates [16]. In the external modulation method,

the RF signal modulates a continuous optical wave generated by an optical source in order

to mitigate the limitations imposed by the optical source chirp. For that same reason, ex-

ternal modulation is a better choice compared with direct modulation, and the one used in

MORFEUS network [1]. In what concerns to the optical source, poor chromatic purity and

low modulation bandwidth imposes great limitations on the LED employment, therefore,

the LASER is preferable over the LED [17]. Figure 2.7 illustrates an OFDM optical link

scheme employing external modulation.

20

RF-OFDMtransmitter

OpticalModulator

RF-OFDMreceiver

LASER

Fiber link

Photodetector

Figure 2.7: Optical link employing external modulation. Single line arrow - electrical do-main; Double line arrow - optical domain.

2.3.1 MB-OFDM system description and operation

Since this work is done in the scope of the MORFEUS project, the MORFEUS network is

considered. Figure 2.8 illustrates a conceptual diagram of a MORFEUS node.

DE

MU

X MU

X

Band blocker

Tunable OF

MIBMEB

ROADM

…

Optical OFDM band

From fiber

To fiber

To other metro ring

To access network

Figure 2.8: Conceptual diagram of a MORFEUS node. MEB - MORFEUS extraction block;MIB - MORFEUS insertion block; ROADM - reconfigurable optical add-drop multiplexer

The MORFEUS node purpose is to extract and insert optical OFDM bands from or to a

metro ring. This kind offers much more granularity in terms of spectral management than

typical reconfigurable optical add-drop multiplexers (ROADM) which are limited to wave-

length granularity. First, the WDM MB-OFDM signal coming form the optical fiber enters

a ROADM where wavelength-demultiplexing is preformed. Each output of the DEMUX

is connected to an optical switch that may or not redirect the signal to the next stage of

the process. Then, the signal enters the MORFEUS extraction block (MEB) where two

different situations may occur. If a specific OFDM band is to be extracted, then the signal

passes through an tunable optical filter (OF) where the desired band is filtered. This band

can either be routed, in optical domain, to other metro ring or to the access network. If the

21

wish is to insert a new OFDM band, then signal that had entered the MEB passes through

a band blocker in order to free the frequency slot where the new band will be inserted. The

insertion is performed in the MORFEUS insertion block (MIB). Finally, the signal with the

new band in it, enters the ROADM where it is multiplexed and transmitted back to the

fiber.

2.3.2 Coherent optical OFDM and direct-detection optical OFDM

There are mainly two kinds of detection techniques used in optical OFDM systems, direct

detection OFDM (DD-OFDM) and coherent detection OFDM (CO-OFDM). In DD-OFDM

the photodetection is performed by a single PIN photodiode, whereas, in CO-OFDM, the

photodetection in some architectures, involve four PIN photodiodes and a local oscillator

making CO-OFDM much more complex than DD-OFDM. Therefore, DD-OFDM has much

lower cost than CO-OFDM making it more suitable for cost-effective short reach applica-

tions. CO-OFDM has a superior performance compared with DD-OFDM since it allows

the system to have better spectral efficiency, higher sensitivity and robustness against po-

larization dispersion. However the high cost of the coherent-detection systems remains a

critical factor, leaving direct-detection systems as the preferred detection method for metro

applications. Thus, section 2.3.4 focus on DD-OFDM since it is the method used in the

MORFEUS network.

2.3.3 E-O conversion

As mentioned in section 2.3, external modulation is a better choice compared to direct mod-

ulation due to the lower frequency chirp and better performance of the external modulation.

The type of external modulator to be used for this work is the Mach-Zehnder modulator

(MZM). A possible scheme of a MZM is illustrated in figure 2.9. At the input of the MZM,

a static light wave Ein is spited by a 50/50 directional coupler [18]. Each light wave enters

a waveguide which refractive index can be controlled by an applied electrical field induced

by the voltages va(t) and vb(t) for the upper and lower arms respectively. By changing the

refractive index of the material, the phase of the wave propagating through that arm is

shifted [19]. The two light waves are then combined into another directional coupler. Due

to the difference in phase between the two waves, they can constructively or destructively

interfere. By manipulating the applied voltages, the amount of light at the output of the

MZM can be controlled, that is, amplitude modulation. The optical signal at MZM output,

eMZM(t), is given by [20]

eMZM(t) =Ein2

exp

[jπva(t)

2Vsv

]+Ein2

exp

[jπvb(t)

2Vsv

](2.43)

where Vsv is the switching voltage. In order to make an optical modulator with a single RF

signal input, the applied voltages va(t) and vb(t) are imposed as the following

22

va(t) = −vb(t) = v(t) (2.44)

where v(t) RF signal input. This way, the phase shifts obtained in each phase modulator

have push-pull symmetry between each other. Notice that, in equation 2.43, Ein is multiplied

by 1/2 due to the fact that the signal at the input of MZM goes through two 50/50 directional

couplers. Remember that each 50/50 directional coupler causes a 1/√

2 attenuation in terms

of amplitude since the signal power at the input is split in half. Therefore, the overall

attenuation due to the two directional couplers is given by 1√2

1√2

= 1/2. By applying

equation 2.44 in equation 2.43, and after some trigonometric simplifications, the optical

signal at MZM output is then given by

eMZM(t) = Ein cos

(π

2

v(t)

Vsv

)(2.45)

Figure 2.10 represents the normalized electrical field at the output of the MZM as a function

of the applied electrical voltage normalized to the switching voltage. Notice that eMZM(t) is

a periodic function with period 4Vsv and it is null when v(t) = ±Vsv. In order to establish an

operating point in the MZM characteristic, a bias voltage is added to the applied electrical

voltage. Therefore, v(t) is defined as

v(t) = −Vb + vsig(t) (2.46)

where Vb is the bias voltage and vsig(t) is the signal to be transmitted. Two possibilities

are commonly considered for the bias point: minimum bias point (MBP) and quadrature

bias point (QBP). In MBP, the operating point is set at the point where the electrical field

eMZM(t) is null, that is Vb = Vsv. The main advantage of this operating point is the fact that

we are at the linear region of MZM, which reduces distortion effects. The main disadvantage

is the fact that no optical carrier is generated. In QBP, the operating point is set at the

point where the electrical field at output is eMZM(t) = 1√2Ein, that is Vb = Vsv/2. The

advantage of this operating point is the fact that an optical carrier is generated. However,

the signal is much more affected by distortion due to the non-linear characteristic of the

MZM at the QBP, and some of the signal power is used on the optical carrier which does

not carry any information. In order to manage the levels of distortion, the amplitude of

the signal vsig(t) must be controlled. This kind of E-O converter architecture generates

a dual-sideband signal. However, as it will be introduced in section 2.3.5, a MORFEUS

network requires that the optical signal must be a single sideband (SSB) signal. One way

to generate such signal is using a MZM structure with four phase modulators in parallel

called dual-parallel MZM (DPMZM). In this particular structure, depicted in figure 2.11, a

MZM is inserted in each arm of an MZM. The input-output characteristic of the DPMZM

is given by [20],

23

Ein eMZM(t)va(t)

vb(t)

Directional coupler

Figure 2.9: Possible scheme of a MZM. Ein -input static light wave; va(t), vb(t) - refractiveindex voltage control of the upper and lowerarms respectively; eMZM(t) - output light sig-nal

−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−1

−0.75

−0.5

−0.25

0

0.25

0.5

0.75

1

v(t)/Vsv

e MZM(t)/Ein

QBP

MBP

Figure 2.10: Normalized electrical field atthe output of the MZM as a function of theapplied electrical voltage normalized to theswitching voltage. MBP - minimum biaspoint; QBP - quadrature bias point.

eout(t) =Ein2

[exp

(jπ

2VsvVb,3

)e1(t)

Ein,1+ exp

(−j π

2VsvVb,3

)e2(t)

Ein,2

](2.47)

e1,2(t) =Ein,1,2

2

[exp

(jπ

2Vsvv1,2(t)

)+ exp

(−j π

2Vsvv1,2(t)

)](2.48)

v1(t) = −Vb,1 + vsig(t) (2.49)

v2(t) = −Vb,2 + TH vsig(t) (2.50)

where TH vsig(t) is the Hilbert transform of the RF signal and Vb,3 is the voltage used to

control the phase difference between the upper and lower arms signal.

By definition, a SSB signal is obtain doing the following operation

xSSB = x+ jxH (2.51)

where x is the original dual sideband signal, xH is the Hilbert transform of x and xSSB is the

resulting SSB signal. When xH is applied on equation 2.51, it produces a signal which has

no negative-frequency components. These negative frequency components can be discarded

with no loss of information, but with the cost of leading to a complex-valued signal instead.

Notice that, in equation 2.51 xH comes multiplied by j which in fact translates in a phase

delay of π/2 applied to xH . In order to obtain a π/2 phase delay between the upper and

lower arms of the outer MZM, Vb,3 must be set at Vsv/2.

24

Ein eout(t)

v1(t)

v2(t)

Ein,1

Ein,2

e1(t)

e2(t)

Vb,3

Figure 2.11: DPMZM structure.

2.3.4 O-E conversion and thermal noise

Thermal noise is the electronic noise generated by the thermal agitation of the charge car-

riers inside an electrical conductor. The thermal noise is independent of the applied voltage

and, in an ideal resistor, it is approximately white [2]. This means that the power spectral

density is nearly constant along the frequency spectrum. The power of the thermal noise

can be statistically described by a Gaussian distribution. As mentioned in section 2.3.2, the

opto-electric (O-E) conversion of a MORFEUS link is performed by a single photodetector

which scheme is illustrated in figure 2.12 [17]. The photodetection is performed by a PIN

h υPIN

iout (t )

iPIN (t )

Pre-amplifier

V bias

RL

Figure 2.12: Photodetector scheme of a direct detection system. RL - load resistor; Vbias -bias voltage; hν - photon energy; iPIN(t) - current generated by the PIN; iout(t) - currentat photodetector output.

photodiode which is polarized by the resistor RL and the bias voltage Vbias. The photodetec-

tion generates a current iPIN(t), which drives the voltage at the input of the pre-amplifier

due to the presence of the resistor RL. Notice that the pre-amplifier is assumed to have

infinite input impedance, and so, all the current generated by the PIN goes thought the

resistor. At the output of the pre-amplifier the resulting current is given by

iout(t) = g iPIN(t) + in(t) (2.52)

where g is the gain of the pre-amplifier and in(t) is the thermal noise current resulting from

25

RL. In order to evaluate in(t), there is the need to describe quantitatively the thermal noise.

For this specific scheme of figure 2.12, the noise power spectral density of the noise at the

output of the photodetector is given by [17]

Sn =2kBTfng

2

RL

[W/Hz] (2.53)

where kB is the Boltzmann constant, T is the temperature in kelvin of the resistor, fn is the

noise figure of the pre-amplifier and g is the gain of the pre-amplifier. The thermal noise

power contained in a certain bandwidth B can be computed as

pn =

∫B

Sn df = Sn B (2.54)

As mentioned earlier in the present section, the thermal noise can be statistically described

by a Gaussian distribution when limited to a finite bandwidth. Hereupon, the thermal noise

current, in(t), is given by a Gaussian distribution characterized by: mean value equal to

zero and standard deviation equal to√pn.

2.3.5 Signal-signal beat interference and virtual carrier motiva-

tion

Typical metro fiber networks can reach hundreds of kilometers leading to important values

of chromatic dispersion. If the signal to be sent is double sideband, the two signal side-

bands suffer different phase shift due to chromatic dispersion dependence on wavelength.

The square law characteristic of the photodetector causes a beat between the two sidebands

leading to destructive interference due to the difference in phase between the sidebands [1].

This destructive effect is called chromatic dispersion induced power fading. In order to

suppress such power fading, two methods can be applied. One method is to compensate the

chromatic dispersion of the fiber link. Other method, and the one employed on MORFEUS

network, is the single sideband (SSB) transmission which consists in transmitting only one

optical signal sideband. The choice of SSB transmission over chromatic dispersion compen-

sation is justified by the desire of network operators to move all the network complexity to

the transmitter and receiver sides of the network in order to be able to up-grade the network

with no need to access or modify equipment deployed in intermediate points of the network

[1].

As mentioned in section 2.3.4, photodetection is performed by a PIN photodiode. The

relation between the incident optical field and the PIN current follows a square law charac-

teristic. The PIN current is given by

iPIN(t) = Rλ |ePIN(t)|2 (2.55)

where Rλ is the PIN responsivity and ePIN(t) is the optical field incident on the PIN.

26

Assuming that the optical field at the input of the PIN is composed by an optical SSB

signal s(t) and an optical carrier C, we have that ePIN(t) is given by

ePIN(t) = C + s(t) (2.56)

The spectrum of ePIN(t) is illustrated in figure 2.13a. Notice that the optical carrier and

the SSB signal are spaced in frequency by fgap. By using equation 2.56 in equation 2.55 the

PIN current is obtained,

iPIN(t) = Rλ

[C2 + 2CRes(t)+ |s(t)|2

](2.57)

Notice that equation 2.57 is composed by a DC component C2 which can be easily filtered

out, a fundamental term 2CRes(t) containing linear relation with the OFDM signal, and

a non-linear second-order term |s(t)|2 that needs to be removed. The non-liner term |s(t)|2

is usually called signal-signal beat interference (SSBI). Several methods have been proposed

to deal with the SSBI [9]. One way is to make the frequency gap fgap larger than the

bandwidth occupied by the SSBI, BSSBI . Usually, BSSBI is similar to the bandwidth of the

OFDM signal Bs [1], and so, by establishing

fgap > Bs (2.58)

it is guarantee that the SSBI does not interfere with the linear term of the signal. Figure

2.13b illustrates the signal spectrum at PIN output and the frequency gap used to accom-

modate the SSBI. However, this method brings a severe disadvantage since the existence

f λ

f gaps(t )

Bs

C

f [Hz]

(a) Signal spectrum at PIN input

f gapBSSBI

C 2

0

|s (t)|22C Res(t)

f [Hz]

(b) Signal spectrum at PIN output

Figure 2.13: Illustration of the SSBI positioning. fgap - frequency gap; fλ - frequency of theoptical carrier; Bs - bandwidth of the OFDM band signal; BSSBI - bandwidth of the SSBI.

of a frequency gap reduces the spectral efficiency of the system, especially if the OFDM

signal is composed by multiple bands leading to a large Bs, and so, a large fgap. A solution

to improve the spectral efficiency of a multiband OFDM system is to make the frequency

gap just large enough to accommodate the SSBI corresponding to one OFDM band. Such

spectral arrangement is illustrated in figure 2.14a. However, this solution implies that the

photodetection can only be performed over one OFDM band at a time. Also, there is an-

other drawback. Since both the optical carrier and the desired OFDM band need to be

27

filtered, this method requires a dual band filter with a bandwidth of a few GHz for each

band filter and both bands spaced in frequency by a few GHz. As well as, the fact that the

filter must have a high selectivity in order to filter out adjacent OFDM bands producing low

linear crosstalk. Nowadays, such optical filters are very difficult to develop and their cost

are very high [1]. One way to overcome the need for such demanding optical filters is to use

virtual carriers close to each OFDM band to assist the photodetection of that same band.

Similarly to the scenario illustrated in figure 2.13a, each OFDM band and the corresponding

virtual carrier should be spaced in frequency by the same bandwidth of the OFDM band in

order to accommodate the SSBI, as illustrated in figure 2.14b. By doing so, each band can

be photodetected individually relieving the demand on the optical filter. However, the fact

that each OFDM band needs its own frequency gap, causes the spectral efficiency of the

system to be drastically reduced. In order to increase the spectral efficiency, the frequency

gap must be reduced, but with the consequence of additional distortion of the OFDM signal

caused by the SSBI. One way to reduce the distortion caused by the SSBI is to increase the

power of the virtual carrier comparatively to the OFDM band. By doing so, the SSBI is

delectable when compared to the fundamental term of equation 2.57, that is,

2CRes(t) |s(t)|2 (2.59)

Other possibility, is to reconstruct the SSBI term at the receiver, and remove it from the pho-

todetected signal using digital signal processing (DSP) algorithms. The main disadvantage

of DSP techniques is the system complexity increase. Both the control of the power ratio

between the virtual carriers and the OFDM band, and the employment of DSP algorithms,

are studied in MORFEUS project.

f gap

Bb

MB-OFDM signal

f λ

…

f [Hz]

(a) Without virtual carriers

f gap

Bb

f λ

...

virtual carrier

f [Hz]

(b) With virtual carriers

Figure 2.14: Illustration of a SSB MB-OFDM signal spectrum and the frequency gap. fgap- frequency gap; fλ - frequency of the optical carrier; Bb - bandwidth of the OFDM bandsignal.

Regarding the choice of VBG, it is important to take into account the fact that the virtual

carrier must not interfere with the OFDM signal. This issue is very important specially if

the VBG is small. Therefore, the best positioning for the virtual carrier in the spectrum is

where the nulls of the sinc functions of the subcarries occur. So, the VBG can be given by

28

the following equation

VBG = k ∆fsc, k ∈ N0 (2.60)

where ∆fsc is the frequency gap between adjacent subcarriers. Also, ∆fsc can be given by

∆fsc =Rb,MB−OFDM

NB

1

Nsc log2(M)(2.61)

The main frequency parameters that characterize MB-OFDM signals employing virtual

carriers are represented in figure 2.15 [21], where, BG is the band gap between adjacent

OFDM bands, VBG is the virtual carrier-to-band gap, fc,1 is the central frequency of the

first OFDM band and ∆B is the band spacing between adjacent OFDM bands. It is also

important to define the ratio between the power of the virtual carrier and the corresponding

band, therefore, we define the virtual carrier-to-band power ratio (VBPR) as

VBPR = 10 log

(pvcpb

)[dB] (2.62)

where pvc and pb are the power of the virtual carrier and the corresponding band, respectively.

VBG

BG

ΔB f [Hz]fc,1

...

fc,2

Figure 2.15: Main frequency parameters of MB-OFDM signals employing virtual carriers.BG - band gap; VBG - virtual carrier-to-band gap; fc,1, fc,2 - central frequency of the firstand second OFDM band respectively; ∆B - band spacing.

2.3.6 Optical noise

In optical communication systems, optical amplifiers are used to compensate for fiber losses

when the signal is too weak to be managed. The most commune optical amplifier, and

the one used in MORFEUS networks are the erbium-doped fibre-based amplifiers (ED-

FAs). These EDFAs, if needed, are installed at the nodes of a metro network to work as a

preamplifier prior to the reconfigurable optical add-drop multiplexer (ROADM). However,

the amplification process generates noise known as amplified spontaneous emission (ASE)

noise. The ASE noise can be decomposed into two orthogonal polarizations, a parallel (‖)and a perpendicular (⊥). Also, the ASE noise in each polarization has an in-phase (I) and

a quadrature component (Q). Therefore, the ASE noise power is equally divided by those

four components. Thus, the spectral density of the ASE noise for each component is given

by

29

SASE,I,‖(υ0) = SASE,I,⊥(υ0) = SASE,Q,‖(υ0) = SASE,Q,⊥(υ0) =SASE(υ0)

4(2.63)

and the ASE noise power is given by

pASE =∣∣eout,EDFA,‖(t)∣∣2 + |eout,EDFA,⊥(t)|2 = SASE(υ0)B0 (2.64)

where eout,EDFA,‖(t) and eout,EDFA,⊥(t) are the parallel and perpendicular components of the

optical signal at the output of the EDFA, respectively, and B0 is the reference optical noise

bandwidth of the optical receiver which is equal to 0.1 nm. The optical signal at the output

of the EDFA is given by

eout,EDFA(t) = [ein,‖,I(t) + jein,‖,Q(t) + nASE,‖,I(t) + jnASE,‖,Q(t)]e‖

+ [ein,⊥,I(t) + jein,⊥,Q(t) + nASE,⊥,I(t) + jnASE,⊥,Q(t)]e⊥ (2.65)

where ein(t) is the signal at the input of the EDFA and nASE(t) is the ASE noise. Considering

that the optical signal at the input of the PIN is equal to the optical signal at the output

of the EDFA, the photocurrent after the PIN is given by

iPIN = Rλ

∣∣eout,EDFA,‖(t)∣∣2 +Rλ |eout,EDFA,⊥(t)|2 (2.66)

where Rλ is the responsivity of the PIN. Notice that, as shown in equation 2.66, the pho-

todetection process of the parallel and perpendicular components of the optical signal is

independent, being summed together afterwards in term of current. Given this properties,

the analyses of the parallel component photodetection process is equivalent to the perpendic-

ular. Therefore, the photocurrent correspondent to the parallel component can be expanded

as follows:

iPIN,‖(t) = Rλ

∣∣eout,EDFA,‖(t)∣∣2= Rλ

∣∣ein,‖,I(t) + jein,‖,Q(t) + nASE,‖,I(t) + jnASE,‖,Q(t)∣∣2

= Rλ[(ein,‖,I(t))2 + (ein,‖,Q(t))2 + 2Re

ein,‖,I(t)je

∗in,‖,I(t)

+ 2Re

ein,‖,I(t)jn

∗ASE,‖,I(t)

+ 2Re

ein,‖,I(t)jn

∗ASE,‖,Q(t)

+ 2Re

ein,‖,Q(t)jn∗ASE,‖,I(t)

+ 2Re

ein,‖,Q(t)jn∗ASE,‖,Q(t)

+ 2Re

nASE,‖,I(t)jn

∗ASE,⊥,Q(t)

+ (nASE,‖,I(t))

2 + (nASE,⊥,Q(t))2] (2.67)

From inspection of equation 2.67 one sees that the photocurrent is composed by signal-signal

beat terms, signal-noise beat terms and noise-noise beat terms. The signal-noise beat term

is seen as noise since there is no way to extract the signal information. If the signal is much

larger than the noise, an usual scenario in optical systems, then the signal-noise beat terms

are much larger than the noise-noise beat terms. Therefore, noise-noise beat terms can be

30

neglected.

As mentioned in section 2.3.4, the photodetector circuit of a DD-OFDM system generates

thermal noise which is added to the photocurrent. The question now is the importance of

the thermal noise when compared to the ASE noise. In fact, if the optical power at the

input of the PIN is high, in the order of 0 dBm, than the thermal noise can be neglected.

In this work, such approach has been taken.

2.4 Performance evaluation of the system

In a digital communication system, the transmitted signal may be affected by undesirable

effects such as noise, interference, distortion, synchronization problems and attenuation,

leading to information errors at the receiver. Therefore, the performance evaluation of the

system is a much important matter to be considered. There are several ways to evaluated the

performance of a system. The simplest approach is the bit error ratio (BER) by direct error

counting (DEC) which is computed by dividing the number of bit errors at the receiver by the

total number of bits that have been transmitted. Since this method is done by DEC it gives

an accurate estimate for the system performance, however it requires long computation times

to achieve low BER levels [22]. In order to do a fast and efficient performance evaluation

other methods must be used.

2.4.1 Error vector magnitude

The error vector magnitude (EVM) measures the difference between the complex value

of the demodulated symbol and the ideal symbol, which is illustrated in figure 2.16. This

so

siEVM

I

Q

Figure 2.16: Constellation diagram and EVM

method has the advantage of allowing the analytical evaluation of the BER from EVM using

standard expressions for M-ary QAM formats such as the one in equation 2.39 without the

need for the long computation times required by the direct error counting method. The

EVM of each subcarrier can be individually evaluated using the following expression

EVM[k] =〈|so(l)[k]− si(l)[k]|2〉〈|si(l)[k]|2〉

(2.68)

31

where so(l)[k] is the signal corresponding to the kth subcarrier of the lth OFDM symbol of

the constellation obtained at the receiver, and si(l)[k] is the signal corresponding to the kth

subcarrier of the lth OFDM symbol of the ideal constellation. The overall EVM is then

given by

EVM = 〈EVM[k]〉 (2.69)

2.4.2 Exhaustive Gaussian approach

The exhaustive Gaussian approach (EGA) is a method to evaluate the BER through numer-

ical simulation. The EGA proposed in [23] is a method to be employed in DD-ODFM using

square and cross QAM. This method provides a fast and accurate estimates for the BER

independently of the BER levels, and assumes that the received in-phase and quadrature

components of each OFDM subcarrier are well described by a Gaussian distribution. Since

EGA is performed on a statistical approach basis, a sufficient number of noise runs must

the done in order to achieve good BER estimates. For this work purpose, it is considered

100 noise runs as a good number of runs in terms of accuracy and simulation time trade-off

[1]. It is shown that the EGA requires about three orders of magnitude less in computation

time than the DEC method for BER levels around 10−6 [23].

2.5 Conclusion

In this chapter, the basic concepts of OFDM were given. First, a detailed mathematical

formulation of the OFDM signal, cyclic prefix concept and spectral efficiency is presented.

Then, a description of the OFDM system is presented. Basic concepts of MB-OFDM are

also introduced, such as the signal characteristics as well as the system operation. Is as

been shown that constellation sizes between 16 and 64 are the only candidates for a 12-

bands MB-OFDM system with 40 GHz available bandwidth per channel operating at 112

Gb/s. Also, optical telecommunication systems based on OFDM technology is introduced.

A brief description of the components that comprise an optical system is explained, such as

detection methods and E-O/O-E conversion. It has been concluded that the thermal noise

generated at the photodetector circuit can be neglected when compared to the ASE noise

of the system under study. The issue concerning the SSBI caused by the direct detection

is also introduced. Lastly, two methods of performance evaluation are introduced where,

for the EGA method, 100 noise runs was established as a good number of runs in terms of

accuracy and simulation time trade-off.

32

Chapter 3

SSBI mitigation techniques and

optical fiber dispersion effects

In this chapter, three different SSBI mitigation techniques are presented. The technique

employed in this work is analyzed and explained in detail. Also, an introduction of chromatic

dispersion (CD) and first-order polarization mode dispersion (PMD) of the optical fibers is

performed. The performance evaluation of the SSBI mitigation technique employed in the

presence of such dispersion effects is performed.

3.1 SSBI mitigation techniques

As mentioned in section 2.3.5, the square-law characteristic of the photodiode generates

SSBI which needs to be removed. Two techniques to reduce the impact of the SSBI were

presented in section 2.3.5. One technique is to insert a sufficiently large frequency gap,

fgap, between the OFDM band and the virtual carrier in order to accommodate the SSBI

spectrum. Since the SSBI bandwidth, BSSBI , is approximately equal to the bandwidth of the

photodetected OFDM band, fgap must be greater or equal to BSSBI . The other technique

is to increase the amplitude of the virtual carrier comparatively to the OFDM band causing

the photocurrent to be dominated by the virtual carrier-signal beat when compared with

the signal-signal beat. In the following sections, three different SSBI mitigation techniques

are presented.

3.1.1 Beat interference cancellation receiver

A beat interference cancellation receiver (BICR) has been proposed in [24]. It represents

a relatively simple method to remove SSBI as it only requires one optical filter and one

balanced receiver in the optical receiver. Figure 3.1 depicts the structure of a BICR [25].

The received optical signal is split by an optical coupler into two parallel branches. The

signal in the upper branch is directly transmitted to the photodiode. The signal in the lower

branch passes through an optical filter to remove the virtual carrier. After photodetection,

33

Optical filter

+-

RF-OFDM receiver

PD1

PD2

B

0 Hz

0 Hz B

B

Virtual carrier

Suppressedvirtual carrier

Residual virtual carrier-band beat

From band

selector

Figure 3.1: BICR structure. PD1, PD2 - photodiode 1 and 2, respectively.

the photocurrent in the upper branch contains the SSBI term, the fundamental term and

a DC component, whereas in the lower branch, only the SSBI term exists. By subtracting

the signal at the output of the upper branch from the one in the lower branch, the SSBI

is removed. However, the main problem of this technique is the the difficult to remove the

virtual carrier in the lower branch which frequency between the virtual carrier and the band

is reduced. That situation requires an optical filter with very high selectivity which may

not be feasible. Therefore, the virtual carrier in the lower branch may not be completely

removed leading to a residual beat between the virtual carrier and the OFDM signal. Also

the right- and leftmost OFDM subcarriers suffer more attenuation due to the non-ideal

optical filter causing the SSBI in the upper and lower branches to be slightly different,

leading to unperfect SSBI removal. Furthermore, the filtering process causes a time shift

on the signal that must be compensated before the photocurrents from the upper and lower

branches being subtracted. The spectral efficiency of a system employing this technique

depends largely on the selectivity of the optical filter due to the fact that the frequency gap

between the OFDM band and the virtual carrier must be large enough for the optical filter to

be capable of suppressing the virtual carrier without excessively attenuate the OFDM band.

In systems like MORFEUS, BICR cannot be implemented because the required frequency

gap between the OFDM band and the virtual carrier, in the order of dozens of MHz, is very

small.

3.1.2 Signal-phase-switching

The principle of the signal-phase-switching (SPS) is to transmit the same OFDM symbol

two consecutive times where the symbol replica has a phase shift of 90 degrees relatively to

the first one [26]. The optical carrier is continuously transmitted without any shift. Such

process is illustrated in figure 3.2.

After photodetection, the resultant photocurrent from the two copies of the OFDM symbol

are given by

34

Es,1 jEs,1 Es,2 jEs,2

E0

Optical carrier

Signal ...

...

A B A B

Figure 3.2: SPS conceptual diagram.

IA = |E0 + Es|2 (3.1)

IB = |E0 + jEs|2 (3.2)

where IA and IB are the original and the phase shifted photocurrents of the OFDM signal, E0

is the optical carrier and Es is the OFDM signal. The complex signal Es can be reconstruct

from equations 3.1 and 3.2 as follows

I1 = IA − jIB = (1− j)|E0|2 + 2E∗0Es + (1− j)|Es|2 (3.3)

By re-arranging equation 3.3 in order to Es, we obtain

Es =I1 − (1− j)|E0|2 − (1− j)|Es|2

2E∗0(3.4)

Notice that Es is present in both sides of equation 3.4. The main goal of this SPS technique

is to remove SSBI, and so, obtain a good representation of the OFDM signal. Therefore,

the next stage of this technique is to obtain an approximation of Es. That is accomplished

using an iterative process applied to equation 3.4, which leads us to:

Es,(k+1) =I1 − (1− j)|E0|2 − (1− j)|Es,(k)|2

2E∗0(3.5)

where Es,(k) is the desired signal resulting from the kth iteration. In [26], two methods to

derive the initial iteration are purposed, which are

|Es,(0)|2 =I2

2 (IA + IB)(3.6)

|Es,(0)|2 =(IA + IB)±

√(IA + IB)2 − 2I2

2(3.7)

where I2 is given by

I2 =(IA − |E0|2

)2 −(IB − |E0|2

)2(3.8)

35

In practice, the carrier power |E0|2 can be estimated using pilot subcarriers or symbols.

The main disadvantage of SPS technique is the fact that, since the OFDM symbols are

transmitted twice, the overall capacity of the system is reduced to half.

3.1.3 DSP-based iterative SSBI mitigation algorithm

In this section, the DSP-based iterative SSBI mitigation algorithm used in [1] is described

and its operation is shown.

Description of the DSP-based iterative SSBI mitigation algorithm

The goal of the SSBI mitigation algorithm is to estimate the SSBI term generated due to the

photodiode characteristic and subtract it from the photodetected signal in order to obtain

an virtually SSBI-free signal. The SSBI mitigation algorithm is composed by two stages:

the training mode and the data mode. Firstly, in the training mode, the SSBI mitigation

algorithm uses the training OFDM symbols to estimate the number of iterations required

to obtain an accurate estimation of the SSBI term and the adjusting coefficient Γc used to

fit the amplitude of the estimated SSBI to the SSBI contained in the photodetected signal.

After the training mode is completed, the data mode takes place. In the data mode, the

information OFDM symbols are processed using the parameters previously computed in the

training mode. In this way, the SSBI term can be efficiently mitigated from the signal with

less time and computational effort as the required number of iterations and the values of Γc

are already identified. Figure 3.3 represents the schematic diagram of the SSBI mitigation

algorithm in training mode for a MB-OFDM receiver. The training mode in composed by

two different types of iterations: the iterations used to estimate the SSBI (designated as

SSBI iteration), and the iterations used to adjust the Γc (designated as Γc iteration).

The SSBI mitigation algorithm in training mode comprises the following steps [27]:

1) The DC component resultant from the beat between the virtual carrier with itself of

the photodetected signal (the C2 term of equation 2.57) is removed and the resultant signal

is stored (in store block A).

2) Each OFDM training symbol passes through a sequence of blocks similar to the OFDM

receiver in figure 2.4, down-conversion, FFT and equalization. The received QAM training

symbols are then obtained. These symbols are stored (in store block B).

3) The received QAM training symbols corrupted by the SSBI are sent to the SSBI es-

timation block. In this block, several processes take place. First, the stored QAM training

symbols are loaded and hard decision of these symbols is accomplished. Notice that, some

of the symbols may be wrongly decided due to the distortion caused by SSBI. Then the

channel loading recreates the channel effects on the symbols. The channel loading response

36

BS PIN StoreA

DC block

Up conversion

FFTEqualizerHeq(k)

StoreB

EVM calculation

ADC -

|EVMj – EVMj-1| < 0.1 dB ?>

j= Γs, iter

Switch 1

j=1

j=1

i = i + 1j = 1

i = i + 1

i = SSBIiter?

Load(store B)

Ideal Hilbert

transform

Hard decision

IFFT Hch(k)| . |2

| . |2+

Down conversion

StoreC

Γc adjustment

Store

Γc(i)=Γadj(j)Γadj(j)Γc(i)=Γinic

SSBI estimation block

Switch 2

j=1

j=1

j = j + 1

j = 1i = 0

Data mode

Y

N

N

Y

Load(store C)

Training Mode

Figure 3.3: Schematic diagram of the SSBI mitigation algorithm in training mode for a SSBMB-OFDM system.

is obtained by normalizing and inverting the channel equalizer previously computed which

can be written as:

Hch(k) =

(Heq(k)

min(Heq(k))

)−1

(3.9)

where Heq(k) is the equalizer transfer function and k is the subcarrier index. Then the signal

passes through the IFFT and up-conversion blocks in order to obtain a replica of the original

electrical OFDM signal, but, no virtual carrier is added. The rebuilt signal is split into two

branches, in which, to the bottom one is applied an ideal Hilbert transform. A square-law

operator is applied to both branches to emulate the PIN characteristic. Then, both signals

are summed together. Notice that, since the signal at the input of the square-law operator

does not have a virtual carrier, the resultant signal contains only the reconstructed SSBI.

Then, the reconstructed SSBI is stored in store block C.

4) After the SSBI estimation block, the reconstructed SSBI stored in store block C, is

multiplied by the transmission coefficient (Γc) to fit the amplitude of the estimated SSBI

term. Since the reconstructed SSBI is an estimation of the real SSBI term, Γc needs to be

adjusted to obtain the best fitting. This adjustment is preformed by a series of iterations

37

where an initial value of the transmission coefficient is defined as Γini, which is incremented

by a factor ∆inc in decibel at every iteration as Γadj(j) = Γini + (j − 1)∆inc. The decision

of the most accurate Γc is obtained when the difference of the EVM of two consecutive Γc

iterations are less than 0.1 dB.

Since each SSBI iteration estimates a newly SSBI term, Γc must be re-calculated for each

SSBI iteration. The optimal Γc values are stored to be used later in the data mode, thus,

reducing the computational effort during the processing of the information symbols.

In order to the Γc adjustment to be time efficient, an adequate value for Γini must be defined.

In [27] is proposed a mean to calculate Γini where E-O conversion is considered to be a linear

process. Thus, Γini is given by

Γini = Rλ|Ein|2π2

16V 2sv

Gsist (3.10)

where Gsist represents the transmission gain from the DPMZM output to the photodiode

input.

5) The estimated SSBI term, is then subtracted from the stored OFDM symbol. The

resulting symbol from the subtraction is a partially SSBI-free symbol.

6) The partially SSBI-free OFDM symbol is then demodulated again. Since the SSBI of the

signal has been partially removed, the hard decision process of the next iteration may have

less errors. Therefore, the new SSBI term is estimated with better accuracy comparing to

the previous iteration and, consequently, leads to a better SSBI removal at each iteration.

Steps 2 to 5 are repeated until the EVM performance no longer improves. Notice that, it is

possible to calculate the EVM of the received training symbols since the training symbols

are known by the receiver. Similarly to the Γc iterations, the number of SSBI iterations

required is obtained when the difference of the EVM of two consecutive SSBI iterations are

less than 0.1 dB. The number of required SSBI iterations is stored to be used as a stopping

criteria in the data mode. However, if there are too many wrong symbols at the hard deci-

sion in the initial iteration, the reconstructed SSBI, can be an inaccurate estimation of the

true SSBI causing the iterative process difficult to converge [28].

Figure 3.4 represents the schematic diagram of the DSP-based iterative SSBI mitigation

algorithm in the data mode. The data mode is similar to the training mode except for

the fact that the Γc coefficients are already known, and so, no Γc iterations are performed.

When the algorithm reaches the last SSBI iteration, established by the training mode, the

algorithm stops and the virtually SSBI-free QAM information symbols are demapped.

38

BS PIN StoreA

DC block

Up conversion

FFT StoreB

ADC -

i = SSBIiter?

Load(store B)

Ideal Hilbert

transform

Hard decision

IFFT Hch(k)| . |2

| . |2+

Down conversion

SSBI estimation block

j = 1i = 0

N

Y

Load(store B)

Demapping

LoadΓc(i) from data

mode

Data Mode

i = i + 1

EqualizerHeq(k)

Figure 3.4: Schematic diagram of the SSBI mitigation algorithm in data mode for a SSBMB-OFDM system.

Performance of the DSP-based iterative SSBI mitigation algorithm

In this section, the performance of the DSP-based iterative SSBI mitigation algorithm is

evaluated. The following tests are focused on assessing the behavior of the algorithm when

the main parameters of the system such as, VBG, VBPR, OSNR and modulation index of

the DPMZM change, and how they affect the EVM of the system. Also, the optimal value

for VBG is defined based on the obtained results.

The system considered is a 12-band MB-OFDM system at 112 Gb/s with 2.333 GHz per

band and a 16 QAM modulation scheme. Figure 3.5 represents the EVM as a function of

VBG/∆fsc, where ∆fsc is the frequency spacing between two adjacent subcarriers. Concern-

ing the Γc iterations, the increment is established as ∆inc = 0.5 dB and the stopping criteria

was defined as the EVM difference between two consecutive iterations being less than 0.1

dB. Figures 3.5a and 3.5b represent the EVM of the system as a function of VBG/∆fsc

without and with optical noise, respectively. In both cases consider VBPR= 10 dB and

modulation index of 5%. Notice that, the lower the VBG, the greater the impact of the

SSBI term on the signal. However, both figures shows that the EVM of the system, when

SSBI mitigation is employed, remains almost constant. Thus, the SSBI mitigation algorithm

succeed in removing the SSBI.

In order to assess the behavior of the algorithm as a function of the VBPR, OSNR and

modulation index parameters, four study cases were established. Case A is defined as a

control in which there is no optical noise, VBPR has a high value so that the amplitude of

the SSBI term is relatively small compared to the OFDM signal, and the modulation index

39

10 30 50 70 90 110 130−35

−30

−25

−20

−15

−10

VBG/∆f sc

EV

M [dB

]

without SSBI mitigationwith SSBI mitigation

(a) Without optical noise.

10 30 50 70 90 110 130−17

−16

−15

−14

−13

−12

−11

VBG/∆fsc

EV

M [dB

]


(b) With optical noise, OSNR=23 dB.

Figure 3.5: EVM as a function of VBG/∆fsc for VBPR=10 dB and modulation index of5%.

is small to minimize the distortion caused by the non-linearities of the DPMZM. In cases B,

C and D, each of the three system parameters is modified. Table 3.1 summarizes the four

cases under study.

Case A Case B Case C Case DOSNR [dB] ∞ 23 ∞ ∞VBPR [dB] 10 10 5 10mod. index [%] 5 5 5 10

Table 3.1: System parameters defined for each of the four cases under study. The underlinedvalues highlights the parameters that differs from the reference test (case A).

Notice that, case B represents a system strongly affected by optical noise, case C represents

a system strongly affected by SSBI and case D represents a system affected by the non-

linearity of the DPMZM. Figure 3.6 represents the EVM of the system as a function of the

SSBI iteration number for the four cases under study. Notice that, the EVM of case A and

case D is overlap, thus, the modulation index does not affect the SSBI mitigation algorithm

performance for values comprised between 5% and 10%. From the inspection of the case B,

we conclude that the OSNR affects greatly the final value of EVM but does not remarkably

affects the number of SSBI iterations required. For the case C, it can be observed that,

although the reduction of VBPR does not affect the final value of EVM, the number of

iterations required increase to more than twice the ones needed in the reference case (case

A) leading to a remarkable increase of the computation time required.

Figure 3.7 represents the normalized power spectral density of the OFDM signal before

and after the SSBI mitigation algorithm in color black and grey, respectively. Figure 3.8

40

0 1 2 3 4 5 6 7−30

−26

−22

−18

−14

−10

−6

SSBI iteration number

EV

M [dB

]

Case ACase BCase CCase D

Figure 3.6: EVM as a function of the SSBI iteration number for the four cases under study.Cases detailed in table 3.1.

represents the normalized power spectral density of the estimated SSBI component. The

parameters of the system are the same as in case A and VBG is equal to 50∆fsc for a better

viewing of the SSBI component. Notice that, the spectral components corresponding to the

SSBI term are almost completely removed from the OFDM signal.

3.2 VBG restriction

It is obvious that to obtain the best spectral efficiency, VBG must be as small as possible.

For that reason, figure 3.9 shows the EVM as a function of VBG/∆fsc for small values

of VBG so that a suitable value for VBG could be chosen. Notice that EVM has huge

fluctuations with minimums when VBG is an integer product of ∆fsc. This happens due

to the overlapping between the virtual carrier and the sinc shape function of the OFDM

subcarrier spectrum as explained in section 2.3.5. Therefore, the most suitable value for

VBG is ∆fsc. From equation 2.61, we get

VBG = ∆fsc =Rb,MB−OFDM

NB

1

Nsc log2(M)≈ 18.23 MHz (3.11)

3.3 Optical fiber dispersion effects

In this section, the dispersion effects of the optical fiber, namely the CD and the PMD, are

introduced. Optical fibers are the transmission medium used in optical wired communica-

tions. Standard single-mode fiber (SSMF) is the most common fiber in metro applications.

Therefore, in order to keep the results of this work comparable to real systems, SSMF was

41

-4 -3 -2 -1 0 1 2 3 4-50

-40

-30

-20

-10

0

10

20

frequency [GHz]

norm

. P

SD [dB

]

Figure 3.7: In black - normalized PSD ofthe photodetected OFDM signal before theSSBI mitigation algorithm; in grey - normal-ized PSD of the OFDM signal after the SSBImitigation algorithm.

-4 -3 -2 -1 0 1 2 3 4-50

-40

-30

-20

-10

0

10

20

frequency [GHz]

norm

. P

SD [dB

]

Figure 3.8: Normalized PSD of the estimatedSSBI component.

used as the optical transmission medium. Although an optical fiber is a nonlinear trans-

mission medium, for the sake of simplicity, it is assumed that the optical fiber has a linear

behavior. Such assumption is made with the premise that the power launched into the fiber

is low enough for the non-linearities to be neglected. In the third window of the optical

frequency grid (the one to be considered in this work) the SSMF has an attenuation of

about 0.2 dB/km. The attenuation of the optical fiber is given by

Ha = exp

(−αLf

2

)(3.12)

where α is the attenuation coefficient in Np/m and Lf is the length of the fiber in m.

3.3.1 Chromatic dispersion

Due to the chromatic dispersion (CD), different frequency components of the optical signal

presents different propagation velocities. The CD of an optical fiber can be described by

a transfer function that performs a frequency dependent phase shift on the optical signal.

Such transfer function is given by

HCD(Ω) = exp (−jβ(Ω)Lf ) (3.13)

where Lf [m] is the fiber length and β(Ω) [rad/m] is the propagation constant as a function

of the baseband equivalent angular frequency Ω [rad/s]. This baseband equivalent angular

42

0.1 1 2 3 4−24

−22

−20

−18

−16

−14

−12

−10

−8

VBG/∆f sc

EV

M [dB

]


Figure 3.9: EVM as a function of VBG/∆fsc for OSNR=30 dB and VBPR=7 dB.

frequency represents the deviation from the optical frequency υ0 [Hz] and is given by

Ω = 2π(υ − υ0) (3.14)

Since β(Ω) is described by complex equations, in order to simplify the calculations, the

propagation constant can be expressed in terms of a Taylor series [29]

β(Ω) ≈ β0 + β1Ω +β2

2Ω2 +

β3

6Ω3 (3.15)

where β0 = β(υ0) and β1 = 1/vg where vg is the group velocity. These terms are neglected

in this work, since they do not impose temporal broadening to the signal. β2 corresponds

to the group velocity dispersion (GVD) and is given by [9]

β2 = −λ20Dλ0

2πc(3.16)

β3 corresponds to the second-order GVD parameter and is given by [9]

β3 =

(λ2

0

2πc

)2

Sλ0 +λ3

0Dλ0

2π2c2(3.17)

where λ0 is the wavelength corresponding to the optical carrier frequency given by λ0 = c/υ0,

Dλ0 is the dispersion parameter of the optical fiber at wavelength λ0, Sλ0 is the dispersion

slope parameter at wavelength λ0, and c is the speed of light in vacuum (c = 299792458 m/s).

By using equation 3.15 in equation 3.13, we obtain

HCD(Ω) = exp

[(−j β2

2Ω2 − j β3

6Ω3

)Lf

](3.18)

43

Note that, as mentioned before, β0 and β1 were neglected. The fiber parameters considered

in this work are summarized in table 3.2.

υ0 [THz] 193.1λ0 [nm] 1552.52Dλ0 [ps/nm/km] 17Sλ0 [fs/nm2/km] 70

Table 3.2: Fiber parameters considered in this work.

−1 0 1 2 3 40

5

10

15

20

25

30

35

40

45

50

frequency [GHz]

phas

e [d

egre

es]

21.5º

OFDM band

(a) Phase response of the CD transfer function for a120 km fiber, superimposed on an illustrative repre-sentation of the transmitted OFDM band.

−9 −6 −3 0 3 6 9

x 10−7

−9

−6

−3

0

3

6

9x 10

−7

I

Q

21.5º

(b) 16 QAM constellation at the receiver before theequalizer.

Figure 3.10: Impact of the CD on the constellation at the receiver before the equalizer.

In order to understand the impact of the CD on the constellation at the receiver before the

equalizer, the following simulation was conducted. One OFDM band with bandwidth equal

to 2.333 GHz and positioned at the first OFDM band slot is transmitted. Figure 3.10a

represents the phase response of the CD transfer function, HCD(Ω), for a 120 km fiber. In

addition, figure 3.10a shows also an illustrative representation of the spectral occupation of

the transmitted OFDM band. Notice that, the phase shift between the left- and rightmost

subcarrier of the OFDM band is equal to 21.5. This phase shift results in an azimuthal

spreading of the QAM symbols of the received constellation. The azimuthal spreading is

approximately equal to the phase shift between the left- and rightmost subcarrier as seen in

figure 3.10b which represents the constellation at the receiver before the equalizer in presence

of CD. Figures 3.11a and 3.11b represent the EVM of the system after SSBI mitigation,

without and with optical noise, respectively, as a function of the fiber length, Lf . The

fiber length varies from 0 km up to 1000 km with 50 km steps. Notice that, in both cases,

the EVM remains almost constant. Therefore, we conclude that the equalizer perfectly

44

0 200 400 600 800 1000−30

−28

−26

−24

−22

−20

Lf [km]

EV

M [dB

]

(a) Without optical noise.

0 200 400 600 800 1000−30

−28

−26

−24

−22

−20

EV

M [dB

]

Lf [km]

(b) With optical noise, OSNR=23 dB.

Figure 3.11: EVM after the SSBI mitigation algorithm as a function of the fiber length inpresence of CD.

compensates the CD effects. Remember that, as explained in section 2.1.3, the cyclic prefix

has the important role of avoiding ISI and ICI. Thus, the constellation in figure 3.10b is the

result of an ISI- and ICI-free OFDM signal.

3.3.2 First-order polarization mode dispersion

When fiber asymmetry occurs as a result of mechanical stress or geometrical distortions, the

transmission properties of the fiber become polarization dependent [30]. As a consequence,

different polarizations propagate at different velocities causing a delay between polarizations

at the receiver. This effect is called polarization mode dispersion (PMD). The propagation

of an optical pulse through a long length of fiber is very complicated due to the random

variation of PMD orientation along the fiber. However, in absence of polarization-dependent

loss, the first-order PMD can be characterized by two orthogonal-polarization states called

principal states of polarization (PSP) [31]. The first-order approximation of PMD is a simple

case where it is assumed that the differential group delay (DGD) between the two PSPs

is constant along wavelength. However, when DGD has a non-negligible dependence on

wavelength, high order distortions take place [32]. Such effect will be addressed in chapter

4.

For each pair of input PSPs, εa+ and εa−, there is a corresponding pair of orthogonal PSPs

at fiber output, εb+ and εb−, where all PSPs are expressed as Jones vectors. Suppose a

polarized field ~Ea(t) = Ea(t)εa as the input optical signal to the fiber. This input field can

be projected onto the two input PSPs as [33]

~Ea(t) =√γEa(t)εa+ +

√1− γEa(t)εa− (3.19)

45

where γ is the PMD power-splitting ratio. The output field of the fiber is given by

~Eb(t) =√γEa

(t− τ0 −

∆τ

2

)εb+ +

√1− γEa

(t− τ0 +

∆τ

2

)εb− (3.20)

where τ0 is the polarization-independent group delay, and ∆τ is the DGD between the two

PSPs. By inspection of equation 3.20 one can notice that the PMD effect can be seen as a

differential phase shift on both PSPs polarization. Therefore, the first-order PMD transfer

function can be written as [33]

HPMD,1st(f) =√γ exp

[j2πf

(−∆τ

2

)]εb+ +

√1− γ exp

[j2πf

(∆τ

2

)]εb− (3.21)

where the polarization-independent group delay was neglected since it does not cause tem-

poral broadening to the signal. The two PSPs propagate independently through the entire

system from the modulator to the photodetector. The current resulting from the photode-

tection is give by

iPIN(t) = Rλ

∣∣∣∣√γEa(t− τ0 −∆τ

2

)∣∣∣∣2 +Rλ

∣∣∣∣√1− γEa(t− τ0 +

∆τ

2

)∣∣∣∣2 (3.22)

PSP +

PSP -

......

Δ time

s+(t)f1

s+(t)f2

s+(t)fN

s-(t)f1s-(t)f2

s-(t)fN

τ

OFDM symbol

OFDM symbol

Figure 3.12: Effect of PMD on a OFDM signal.

Figure 3.12 illustrates the PMD effect where CD is neglected. s+fn

(t) and s−fn(t) are the

time waveforms corresponding to the nth subcarrier that propagate on each orthogonal

polarization (PSP + and PSP - respectively). The resulting signal at the photodetector can

be viewed as the sum of the nth subcarrier of one polarization with the corresponding nth

subcarrier of the other polarization. Considering the DGD (∆τ) caused by the PMD as a

phase shift (φPMD), the sum of both nth subcarriers can be performed. Taking into account

equation 2.57 which gives us the signal at PIN output, the fundamental term corresponding

46

to the nth subcarrier can be calculated as

2C Res+fn

(t) + s−fn(t) ∝ cos(2πfnt) + cos(2πfnt+ φPMD)

∝ 2 cos

(4πfnt+ φPMD

2

)cos

(2πfnt− 2πfnt− φPMD

2

)∝ 2 cos

(2πfnt+

φPMD

2

)cos

(−φPMD

2

)(3.23)

where fn is the frequency of the nth subcarrier after photodetection. Notice that,

cos

(−φPMD

2

)= 0⇔ φPMD = π ± 2πk, k ∈ Z (3.24)

Thus, this cosine term causes the attenuation of the subcarrier resulting from the sum of

both PSPs, which can even lead to a total cancellation of that subcarrier for a specific value

of PMD-induced phase shift. In order to prevent such cancellation, |φPMD| < π must always

be verified. Translating this condition in terms of time delay, and taking into account that

π radians corresponds to half period of a sinusoid, we obtain the following condition

∆τ T

2⇔ ∆τ 1

2

1

fsc,max(3.25)

where fsc,max is the frequency of the subcarrier of the OFDM signal with the highest fre-

quency. The highest frequency subcarrier is the one that has the shortest period, and so,

the most affected.

-3 -2 -1 0 1 2 3-40

-30

-20

-10

0

10

20

30

40

frequency [GHz]

norm

. P

SD [dB

]

(a) In grey - OFDM band spectrum without PMD; inblack - OFDM band spectrum with PMD.

1 17 33 49 65 81 97 113 128−30

−20

−10

0

10

20

subcarrier index

EV

M [dB

]

without PMDwith PMD

(b) EVM as a function of the subcarrier with and with-out PMD.

Figure 3.13: Impact of PMD on an OFDM signal after photodetection when condition 3.25is not verified. System without noise.

47

Figure 3.13a shows the spectrum of an OFDM band after photodetection for VBG = ∆fsc =

18.23 MHz. In grey color is represented the OFDM band spectrum without PMD. In black

color is represented the OFDM band spectrum where condition 3.25 is not verified. Notice

that cancellation occurs at fn ≈ ±1.46 GHz which the corresponding subcarrier can be

calculated as follows:

fn = VBG + (n− 1)∆fsc ⇔ n =fn − VBG

∆fsc+ 1 ≈ 80 (3.26)

In fact, this cancellation effect is confirmed in figure 3.13b where the EVM as a function

of the subcarrier is represented. As expected, a peak occurs around the 80th subcarrier.

Notice that, the behavior shown in figure 3.13b is only true for this hypothetical case where

system does not have noise. If we were to consider noise in the system, then the peak would

not be so much concentrated around the 80th subcarrier, but would enlarge. This is due

to the signal amplitude of the subcarriers being smaller in frequencies closer to the deep

caused by PMD, leading to a degradation of the signal-to-noise ratio.

−6 −4 −2 0 2 4 6−6

−4

−2

0

2

4

6

I

Q

a

bc

c

(a) 16 QAM constellation at the receiver before theequalizer. a/b ≈ 0.38. c is the margin that accountsfor the DAC effect (see figure A.7).

−4 −3 −2 −1 0 1 2 3 4−4

−3

−2

−1

0

1

2

3

4

I

Q

(b) 16 QAM constellation at the receiver after theequalizer.

Figure 3.14: Impact of PMD on the transmitted signal. ∆τ = 70 ps and VBG = 150∆fsc.CD has been neglected.

Figure 3.14a represents a constellation at the receiver before equalization when DGD is

equal to 70 ps. VBG is equal to 150∆fsc in order to visualize the QAM symbols not affect

by SSBI. CD has been neglected for a better understanding of the PMD impact on the

transmitted signal. Notice that, radial spreading occurs on the QAM symbols of the received

constellation as a result of the PMD-induced attenuation effect. The higher the frequency

of the subcarrier, the greater the attenuation it suffers. Lets say that the attenuation of the

nth subcarrier is given by

48

aPMD,n =

∣∣∣∣cos

(−φPMD,n

2

)∣∣∣∣ (3.27)

where

φPMD = 2π∆τ

Tn⇔ φPMD = 2πfn∆τ (3.28)

where Tn and fn are the period and frequency of the nth subcarrier respectively after

photodetection. Using equations 3.27 and 3.28 we can evaluate the attenuation suffered by

the first and last subcarrier. The results are summarized in table 3.3.

fn [GHz] φPMD [degrees] aPMD,n

n = 1 2.735 68.92 0.825 (−0.835 dB)n = 128 5.050 127.3 0.444 (−3.53 dB)

Table 3.3: DGD phase shift and resulting attenuation of the first and last subcarriers.

From the results obtained in table 3.3, the attenuation difference caused by the PMD be-

tween the first and last subcarrier can be calculated as follows:

aPMD,n=1 − aPMD,n=128 = 0.381 (3.29)

This is an important result because it can be related with the radial spread effect observed

in figure 3.14a. Figure 3.14b represents the constellation after the equalizer. Notice that,

the PMD effect seems to be perfectly compensated by the equalizer. However, this is a case

were the SSBI term does not interfere with the OFDM band, and so, no conclusions about

the performance of the SSBI mitigation algorithm in presence of first-order PMD can be

made. To provide insight about this point, two simulations have been performed to evaluate

the performance of the SSBI mitigation algorithm when the PMD increases. In the first

case, the SSBI overlaps the OFDM band and, in the other case, the VBG is large enough

so that the SSBI term does not interfere with the OFDM band. The following study is

performed on a system similar to the one described in section 3.1.3 (12-band MB-OFDM

system at 112 Gb/s with 2.333 GHz per band and a 16 QAM modulation scheme). For the

sake of a correct analyses of the impact of the PMD, we have to ensure that, in both cases,

the PMD causes the same amount of distortion. Lets consider the maximum DGD for this

analyses as 90% of the DGD value that causes a total cancellation on the highest frequency

subcarrier after photodetection. By using equation 3.25, the maximum DGD is given by

∆τmax = 0.91

2

1

fsc,max= 0.9

1

2

1

VBG +BOFDM

(3.30)

Therefore, for the first case we have

VBG = ∆fsc → ∆τmax = 19.14× 10−11 ≈ 190 ps (3.31)

49

and for the second case

VBG = 150∆fsc → ∆τmax = 8.880× 10−11 ≈ 90 ps (3.32)

where ∆fsc = 18.23 MHz and BOFDM = 2.333 GHz.

0 10 20 30 40 50 60 70 80 90−35

−30

−25

−20

−15

−10

−5

EV

M [dB

]

without equalizerwith equalizer

Δ τ

(a) For VBG = 150∆fsc.

0 20 40 60 80 100 120 140 160 190−30

−25

−20

−15

−10

−5

EV

M [dB

]

without equalizerwith equalizer

Δ τ

(b) For VBG = ∆fsc.

Figure 3.15: EVM of the system as a function of the DGD. System without optical noise.

Figures 3.15a and 3.15b represent the EVM of the system at the receiver for VBG = 150∆fsc

and VBG = ∆fsc, respectively, as a function of the DGD. Notice that the values of DGD

are comprised between 0 ps and the respectively ∆τmax as defined in equations 3.31 and

3.32. In each case, the simulation is performed with and without equalization. As expected,

in figure 3.15a, the equalizer compensates the PMD effect for a wide range of PMD values.

One can question the fact that the value of EVM should be the same at ∆τ = 0 with and

without equalizer since there is no PMD affecting the signal. This discrepancy is justified by

the fact that, in the case where there is no equalization, the distortion caused by the DAC

process is not compensated. A more detailed explanation concerning this DAC distortion is

done in appendix A. By inspection of figure 3.15b, one can observe that, for values of DGD

higher than 70 ps, the performance of the system is strongly affected by the PMD even with

equalization. However, a DGD value of 70 ps only occurs for a very large fiber length. The

mean value of DGD for high random-mode coupling fibers can be calculated as

〈∆τ〉 = DPMD

√Lf (3.33)

where DPMD is the PMD parameter of the optical fiber in ps/√

km and Lf is the length of

the fiber in km. By using equation 3.33, and considering a typical DPMD = 0.5 ps/√

km for

modern fibers, one can notice that DGD values higher than 70 ps occurs for lengths of fiber

in the order of tens of thousands of kilometers. This leads to the important conclusion that,

despite the SSBI mitigation algorithm fails to remove the SSBI in presence of strong first

order PMD, in the scope of the metro networks where the longest links have typically 400

50

km, apparently, the PMD is so small that does not affect the system performance. In order

to validate this statement, a study of the system performance in presence of CD, first-order

PMD and optical noise simultaneously must be done. The band selector (BS) employed is

a second-order super Gaussian filter. A detailed explanation concerning this filter in done

in appendix B. The VBPR value is established as 7 dB, and held in the reaming sections of

this dissertation, since it is the value that optimizes the photodetection as explained in [1].

Before making such study, we need to establish an OSNR value to be used in the following

simulations. The task is to find the required OSNR to achieve acceptable values of BER. To

establish such value, the BER as a function of the OSNR for a back-to-back system config-

uration is obtained. The EGA method considering 100 noise runs [34] is used to calculate

the BER. Simulations were performed for each of the 12 OFDM bands that comprise the

MB-OFDM spectrum. Results have shown that the 11th band has the worst performance,

with a required OSNR = 31.3 dB to achieve BER = 10−3.

0 100 200 300 400−3.4

−3.2

−3

−2.8

−2.6

−2.4

Lf

log 10 BER

Figure 3.16: BER as function of the fiber length in presence of CD, first-order PMD andoptical noise simultaneously.

Figure 3.16 represents the BER as function of the fiber length in presence of CD, first-

order PMD (DPMD = 0.5 ps/√

km) and optical noise (OSNR = 31.3 dB). The fiber CD

parameters considered are shown in table 3.2. The number SSBI iterations required for the

algorithm to stabilize, there is, the difference between the EVM of two consecutive SSBI

iterations being less than 0.1 dB, is 5. Notice that the BER of the system remains almost

constant, with slight non-relevant fluctuations, for fiber lengths between 0 and 400 km.

Therefore, it is possible to conclude that the performance of the DSP-based iterative SSBI

mitigation algorithm is not affected by first-order PMD for fiber links up to 400 km.

51

3.4 Conclusion

In this chapter, three different SSBI mitigation techniques have presented. The first two

techniques, BICR and SPS, have been presented in a very brief way where the principle

of operation, main advantages and disadvantages have been explained. The other SSBI

mitigation technique is the DSP-based iterative SSBI mitigation algorithm which has been

rigorously detailed. The behavior of the SSBI mitigation algorithm when different system

parameters change has been analyzed. It as been shown that, if the optical noise of the

system increases, than the number of SSBI iterations required remains constant but the

EVM resulting from the last SSBI iteration increases. Also, if the VBPR of the system

decreases, than the EVM resulting from the last SSBI iteration remains constant but the

number of SSBI iterations required increases. Lastly, it as been shown that neither the

number of SSBI iterations or the EVM resulting from the last SSBI iteration is affected by

the modulation index for values between 5% and 10%.

The optimal VBG has been defined as 18.23 MHz, which takes into account the overlapping

between the virtual carrier and the sinc shape spectrum of the left- and rigthmost subcarriers

of the OFDM band that extends beyond the band limits.

Also, an introduction of CD and first-order PMD of the optical fiber has been made such as

its impact on the constellation of the received OFDM signal. The performance evaluation

of the SSBI mitigation algorithm technique employed in the presence of such dispersion

effects has been preformed. It has been shown that the CD alone does not affect the SSBI

mitigation algorithm for a length of fiber up to 1000 km. As for the first order PMD, it has

been shown that, in the case of VBG being large enough to accommodate the SSBI term

after photodetection, the equalizer is able to perfectly compensate the distortion caused by

first order PMD up to 90 ps of DGD. However, if VBG is small, the performance of SSBI

mitigation algorithm degrades for values of DGD higher than 70 ps. However, such values

of DGD fall outside the scope of the metro networks. Finally, a simulation where CD, first-

order PMD and optical noise are considered has been performed. The results conclude that

first-order PMD does not affect the DSP-based iterative SSBI mitigation algorithm for fiber

links up to 400 km.

52

Chapter 4

Study of the impact of first- and

second-order PMD on direct

detection MB-OFDM systems

In this chapter, the theoretical modeling and statistical analyses of first- and second-order

PMD is introduced and the performance of the DSP-based iterative SSBI mitigation algo-

rithm in presence of first- and second-order PMD is evaluated. As introduced in section

3.3.2, the first-order approximation of PMD is a simple case where it is assumed that the

DGD between the PSPs is constant along wavelength. However, when DGD has a non-

negligible dependence on wavelength, high order distortions take place [32]. The second-

order PMD accounts the DGD variations along wavelength which causes additional signal

distortion [35]. In this section, the second-order PMD effect is emulated using a coarse-step

method where the optical fiber is considered as a concatenation of shorter fiber segments

with a given mean birefringence and random coupling angles between consecutive segments.

Experimental results have shown that such method provides good descriptions either for

the first-order PMD statistics or second-order PMD [35]. As this study is performed using

computational simulation, the long periods of time needed to perform such simulations is a

concern. Hence, a study of the trade-off between the number of fiber segments considered

and the quality of the PMD emulation has been done.

4.1 Theoretical modeling and statistical analysis of first-

and second-order PMD

In this section, the theoretical modeling of second-order PMD is introduced. Then the

statistical properties of second-order PMD are introduced and verified using numerical sim-

ulation. Also, a study of the trade-off between the number of fiber segments considered and

the quality of the PMD emulation is performed in order to optimize the simulation times.

53

4.1.1 Theoretical modeling of first- and second-order PMD

In order to study the first- and second-order PMD, an optical fiber can be viewed as a con-

catenation of shorter fiber segments with a given mean birefringence and random coupling

angles between consecutive segments. Figure 4.1 illustrates the concatenation of Nseg fiber

segments fiber segments. The angle αn is the coupling angle between the (n− 1)th and nth

segments, and hn is the length of the nth segment.

(1) (2) (3) (Nseg)

α1 α2 α3 αN seg

Nseg

...h1 h2 h3 h

Figure 4.1: Illustration of the concatenation of Nseg fiber segments. The angle αn is thecoupling angle between the (n − 1)th and nth segments, and hn is the length of the nthsegment.

It is possible to calculate the optical field at the output of the fiber for a given optical field

at the input as follows [36]

~Eb(ω) = T (ω) ~Ea(ω) (4.1)

with

~Ea(ω) =

Ea+(ω)

Ea−(ω)

~Eb(ω) =

Eb+(ω)

Eb−(ω)

(4.2)

where T (ω) is the Jones matrix that describes a concatenation of Nseg segments, ~Ea(ω)

and ~Eb(ω) are the PSP representations of the signal at the input and output of the fiber,

respectively. For simplicity, the effect of CD is only applied after the PMD emulation by

individually multiplying each orthogonal PSP representation of the signal at the output of

the fiber by the frequency response of the CD as follows:

~Eout(Ω) =

Eout+(Ω)

Eout−(Ω)

=

HCD(Ω)Eb+(Ω)

HCD(Ω)Eb−(Ω)

(4.3)

where ~Eout(Ω) is the equivalent baseband frequency representation of the signal affected by

both PMD and CD, and Eb(Ω) is the the optical field after PMD emulation represented on

the equivalent baseband frequency. At photodetection, the current at the PIN output can

be calculated using equation 2.55 as follows

iPIN(t) = Rλ |Eout+(t)|2 +Rλ |Eout−(t)|2 (4.4)

54

where Eout+(t) and Eout−(t) are the inverse Fourier transform of Eout+(Ω) and Eout−(Ω),

respectively.

The Jones matrix T (ω) can be calculated as [35]

T (ω) =

Nseg∏n=1

Bn(ω)R(αn)

=

Nseg∏n=1

ej(√

3π/8 bω√hn/2+φn

)0

0 e−j

(√3π/8 bω

√hn/2+φn

) cos(αn) sin(αn)

− sin(αn) cos(αn)

(4.5)

where Bn(ω) is the birefringence matrix of the nth segment, R(αn) is the rotation matrix

that emulates the coupling between the (n−1)th and nth segments, b is the PMD coefficient

of the fiber in ps/√

km, ω is the optical frequency in rad/s and hn is the length in km of the

nth segment. The phase φn accounts for small temperature fluctuations along the fiber which

can be described by an uniform distribution between 0 and 2π. The angle αn is the coupling

angle between the (n − 1)th and nth segments also described by an uniform distribution

between 0 and 2π. It is known that the matrix T (ω) has the following mathematical property

[29]

T (ω) =

T11 T12

T21 T22

=

T11 T12

−T ∗12 T ∗11

(4.6)

The DGD for a single wavelength can be calculated from the matrix T (ω) as [36]

∆τ(ω) = 2

√(dT11

dω

)2

+

(dT12

dω

)2

(4.7)

This can be approximated by

∆τ(ω) ≈ 2

√√√√(T11

(ω + ∆ω

2

)− T11

(ω − ∆ω

2

)∆ω

)2

+

(T12

(ω + ∆ω

2

)− T12

(ω − ∆ω

2

)∆ω

)2

(4.8)

where ∆ω must be as small as possible in order to obtain a good approximation to the real

value of the DGD.

4.1.2 Statistical properties of first- and second-order PMD

In this chapter, it is considered a SSMF with PMD parameter of DPMD = 0.5 ps/√

km.

The choice of such DPMD is justified by the fact that modern fibres are designed to have

low PMD, typically less than 0.5 ps/√

km as recommended in [37]. Figure 4.2 represents

the DGD as a function of the equivalent baseband frequency for a 400 km long fiber with

55

1 2 3 4 5 6 70

10

20

30

40

frequency [THz]

DG

D [ps

]

0

DGD period DGD period

Figure 4.2: DGD as a function of the equivalent baseband frequency for a 400 km fiber(DPMD = 0.5 ps/

√km) where the length of the segments is constant and equal to 500

meters.

1 2 3 4 5 6 70

10

20

30

40

DG

D [ps

]

frequency [THz]

0

Figure 4.3: DGD as a function of the equivalent baseband frequency for a 400 km fiber(DPMD = 0.5 ps/

√km) where the length of the segments are randomly generated from

a Gaussian distribution around the mean length per segment equal to 500 meters withstandard deviation equal to 30% of the mean length per segment.

constant length fiber segments obtained through numerical simulation. This simulation

is performed using a method based on equation 4.5. Steps with length of 500 meters are

considered. A criterion for the length of the fiber segments is established in the section 4.1.3.

A periodic behavior is clearly seen indicating that this model does not correctly describe

the second-order PMD [38]. Such periodic behavior is explained by the fact that the length

of all segments is the same [38]. This statement can be verified by figure 4.3 where the

length of the segments are randomly generated from a Gaussian distribution around the

mean length per segment equal to 500 meters with standard deviation equal to 30% of the

mean length per segment. Notice that, by using the non-constant approach for the segments

length as suggested in [35], the periodic behavior disappears.

In order to perform a statistical analysis of the DGD, two sets of fiber realizations (one

with 100 realizations and other with 10000 realizations) have been simulated. The value of

56

0 5 10 15 20 25 300

2

4

6

8

10

12

14

DGD [ps]

norm

. fr

eque

ncy

[x10

−2]

(a) Obtained from 100 fiber realizations.

0 5 10 15 20 25 300

2

4

6

8

10

12

14

DGD [ps]

norm

. fr

eque

ncy

[x10

−2 ]

(b) Obtained from 10000 fiber realizations.

Figure 4.4: Statistical distribution of DGD for a 400 km optical fiber (〈∆τ〉 = 10 ps),composed by 800 concatenated unequal segments (mean length per segment equal to 500meters). Both figures represent the histogram of the DGD values obtained from simulationsuperimposed with the theoretical Maxwellian distribution with mean value equal to 10 ps.

DGD at a fixed and arbitrary frequency of each fiber realization is taken and organized into

a histogram. The resulting histograms for the equivalent baseband frequency of 0 GHz are

represented in figure 4.4, superimposed with the theoretical Maxwellian distribution given

by [39]

f∆τ (∆τ) =

√2

π

∆τ 2

µ3exp

(−∆τ 2

2µ2

), ∆τ ∈ [0,+∞[ (4.9)

where µ is the standard deviation of DGD given by [39]

µ =

√π

8〈∆τ〉 (4.10)

Figure 4.4 shows that the developed numerical simulator describes very well DGD random

nature as the numerical simulated realizations of DGD follow closely the Maxwellian dis-

tribution [35]. Similar accordance between the theoretical distribution and the numerical

simulation estimates is observed for other equivalent baseband frequencies.

4.1.3 Optimization of the parameters of the PMD numerical sim-

ulation

The result shown in figure 4.4 is for the specific case where the segments have approximately

500 meters in length. So, the following question arises: can the length of the fiber segments

be larger than 500 meters without excessively affect the quality of the PMD emulation? In

order to answer this question, the dependence of the quality of the PMD emulation when the

length and the number of the fiber segments considered change must be evaluated. There-

57

fore, a similarity measure parameter must be defined to assess the similarity between the

histograms resulting from the fiber realizations and the theoretical Maxwellian distribution.

The Euclidean distance is used as a similarity measure, which is given by

ξ =

√√√√Nclass∑n=1

|Pteo[un−1 ≤ u < un]− Psim[un−1 ≤ u < un]|2 (4.11)

where u is the DGD in ps and Nclass is the number of classes in which the histogram is

divided. The probabilities Pteo[un−1 ≤ u < un] and Psim[un−1 ≤ u < un] are the probability

of the DGD of a given fiber realization being comprised between un−1 and un estimated from

the theoretical Maxwellian distribution and from the numerical simulation, respectively.

50 100 150 200 250 300 350 4000

0.04

0.08

0.12

0.16

0.2

Lf [km]

ξ

Figure 4.5: Similarity measure as a function of the fiber length with Nseg = 100. The resultspresented were obtained from 1000 fiber realizations.

Figure 4.5 represents the similarity measure as a function of the fiber length where 100 fiber

segments are considered. This figure only consider fiber lengths up to 400 km since, in the

scope of metro network, the longest links have typically 400 km. This figure is obtained

from 1000 fiber realizations. Notice that the fiber length being equal to 50 km and 400 km

corresponds to having the length of each segment approximately equal to 500 meters and 4

km, respectively. By inspection of figure 4.5, we can notice that the similarity between the

resulting histograms and the theoretical Maxwellian distribution remains almost constant

for lengths of fiber comprised between 50 km and 400 km. Hereupon, the most important

conclusion that we can extract from this result is the fact that the statistical nature of

PMD does not depend on the length of the fiber segments if they are comprised between

500 meters and 4 km. Next, we need to understand how that same statistical nature of the

PMD depends on the number of fiber segments.

Figures 4.6a and 4.6b represent the similarity measure as a function of the number of

segments for the Nseg intervals 1 up to 100, and 100 up to 800, respectively. Notice that the

similarity remains almost constant for a number of segments higher than 5. In fact, figure 4.7

58

1 20 40 60 80 1000

0.04

0.08

0.12

0.16

0.2

N seg

ξ

(a) Nseg interval between 1 and 100.

100 200 300 400 500 600 700 8000

0.04

0.08

0.12

0.16

0.2

Nseg

ξ

(b) Nseg interval between 100 and 800.

Figure 4.6: Similarity measure as a function of the number of segments considered for a 400km fiber.

shows the histograms for Nseg = 5 and Nseg = 200 where it can be seen that the probability

distribution is similar for both cases. Such result may wrongly lead to the conclusion that

the quality of the PMD emulation does not depend on the number of segments for Nseg

higher than 5. However, that might not be true since the analysis based on the histograms

does not take into account the fluctuations of the DGD along the frequency.

0 10 20 30 400

2

4

6

8

10

12

DGD [ps]

norm

. fr

eque

ncy

[x10

−2]

(a) For Nseg = 5

0 10 20 30 400

2

4

6

8

10

12

DGD [ps]

norm

. fr

eque

ncy

[x10

−2

]

(b) For Nseg = 200

Figure 4.7: Statistical distribution of DGD for a 400 km optical fiber (〈∆τ〉 = 10 ps). Bothfigures represent the histogram of the DGD values obtained from simulation, superimposedwith the theoretical Maxwellian distribution.

In order to extend the study to consider such fluctuations, it is important to make a compar-

ison between the characteristic of the DGD along frequency for different fiber realizations

where different number of segments are considered. Figure 4.8 represents the DGD of the

59

0 100 200 300 400 500 6000

5

10

15

20

25

f [GHz]

DG

D [ps

]

Nseg

= 5

Nseg

= 800

Figure 4.8: DGD of the fiber affected by the PMD effect as a function of the equivalentbaseband frequency with Nseg equal to 5 and 800. Lf = 400 km.

fiber affected by PMD effect as a function of the equivalent baseband frequency with Nseg

equal to 5 and 800. The DGD is calculated using equation 4.8. It can be seen in figure

4.8 that the lower the number of segments, the smaller the fluctuations of the frequency

response. Thus, the number of segments is an important parameter in order to obtain a

correct PMD emulation. So, which is the most suitable number of segments that should

be considered? To answer this question the absolute value of the slope of the DGD as a

function of the equivalent baseband frequency is presented in figure 4.9 for Nseg equal to

5, 20, 50, 100, 300 and 800. Notice that, if the number of segments considered is 5 or

10, the slope values of the DGD are small, which means that the fluctuations of the DGD

along the frequency are smooth. On the other hand, if the number of segments considered

is higher than 50, such fluctuations are stronger and seems to remain similar for values up

to Nseg = 800. Assuming that the higher the number of segments, the more realistic is

the PMD model, then the best choice would be 800 segments. However, due to long sim-

ulation times required to perform the PMD emulation (this subject is discussed in section

4.3), which are unacceptable for this work purpose, the choice of Nseg = 100 offers a good

trade-off between simulation time and PMD emulation quality.

4.2 Impact of PMD fluctuations along time

Long term studies have concluded that the DGD variations along time for a given wavelength

also follows a Maxwellian distribution over long periods of time [31]. Such PMD variations

can be caused by mechanical and temperature fluctuations and its time interval during while

the PMD properties of the fiber remain almost unchanged is of the order of few milliseconds

[9]. However, these fluctuations along time may not affect the system performance if it

is ensured that the refreshing rate of the channel estimation is sufficiently high. Since

60

0 0.5 1 1.5 20

2

4

6

8x 10

−10

f [THz]

|dD

GD

/df | [p

s/H

z]

(a) For Nseg = 5

0 0.5 1 1.5 20

2

4

6

8x 10

−10

|dD

GD

/df | [p

s/H

z]

f [THz]

(b) For Nseg = 10

0 0.5 1 1.5 20

2

4

6

8x 10

−10

|dD

GD

/df | [p

s/H

z]

f [THz]

(c) For Nseg = 50

0 0.5 1 1.5 20

2

4

6

8x 10

−10

|dD

GD

/df | [p

s/H

z]

f [THz]

(d) For Nseg = 100

0 0.5 1 1.5 20

2

4

6

8x 10

−10

|dD

GD

/df | [p

s/H

z]

f [THz]

(e) For Nseg = 300

0 0.5 1 1.5 20

2

4

6

8x 10

−10

|dD

GD

/df | [p

s/H

z]

f [THz]

(f) For Nseg = 800

Figure 4.9: Absolute value of the slope of the DGD as a function of the equivalent basebandfrequency for a given fiber realization. Lf = 400 km.

the channel estimation is performed using the OFDM training symbols, it is important to

find the maximum number of OFDM symbols that should be transmitted before the next

61

sequence of OFDM training symbols. Such calculation can be performed using the following

equation

Ns = Ntrain +Ninfo =δtDGDts + tG

(4.12)

where Ntrain and Ninfo are the number of OFDM training symbols and OFDM information

symbols, respectively, δtDGD is the interval during while the PMD properties of the fiber

remain almost unchanged, ts is the OFDM symbol duration and tG is the guard time.

From equation 2.22, it is possible to calculate the OFDM symbol duration for the system

considered in this work:

ts ≈Nsc

BOFDM

=128

2.333× 109= 54.9 ns (4.13)

The guard time, as mentioned in section 2.1.3, must be larger than the inter-subcarrier

delay caused by the CD. The CD delay between the fastest and the slowest subcarriers of

an OFDM band is given by

td =angle(HCD(f +BOFDM + ∆f

2))− angle(HCD(f +BOFDM − ∆f

2))

∆f

−angle(HCD(f + ∆f

2))− angle(HCD(f − ∆f

2))

∆f(4.14)

where f is an arbitrary frequency on the equivalent baseband frequency, HCD(f) (given by

equation 3.18) is the frequency response of the SSMF that takes the CD into account for a

400 km optical fiber and ∆f is a given frequency interval that should be as small as possible

in order to obtain an accurate result. The parameters of the CD considered are written in

table 3.2. Since BOFDM = 2.333 GHz, then we obtain td ≈ 0.803 ns. Using equation 4.12

and considering δtDGD = 1 ms [29], the maximum number of OFDM symbols that may be

transmitted before the next set of training OFDM symbol can be calculated as

Ns =1 ms

55.7 ns≈ 18000 (4.15)

Taking into account the number of OFDM training symbols considered on the simulations

performed in this work (Ntrain = 75), the fraction of OFDM symbols that carries information

is given by

ηinfo =Ninfo

Ns

=18000− 75

18000≈ 0.9958 (4.16)

Such result leads us to the conclusion that the fluctuations of the PMD along time do not

constitute a major problem in terms of impact on the efficiency of the system. Notice that

this result is obtained for the specific case where the timescale of change of the PMD is of

the order of a few milliseconds. Since the PMD is highly dependent on factors such as the

62

environment and mechanical stresses, such conclusion may not be valid for other timescales.

4.3 Performance evaluation of SSBI mitigation algo-

rithm in presence of first- and second-order PMD

Since the PMD has a random nature, for a specific length of optical fiber, different fiber

realizations may lead to different performance results. The simplest approach to perform a

performance evaluation of the system in presence of first- and second-order PMD would be

to generate a huge number of fiber realizations and calculate the mean over all the BERs

corresponding to each one of those realizations. However, such procedure would lead to

huge simulation times. The total simulation time required to estimate the mean BER for a

given link length is given by

Tsim,tot = NrealizNsegtsim,seg +Nrealiztsim,BER (4.17)

where Nrealiz is the number of fiber realizations, tsim,seg is the time needed to generate one

fiber segment and tsim,BER is the time needed to calculate the BER of the system resulting

from one fiber realization. Notice that the total simulation time increases linearly with the

number of fiber realizations. A 3.5 GHz Intel Core i7-3770K PC with 32 GB of RAM was

used to perform the simulations leading to tsim,seg = 1.8 s and tsim,BER = 1500 s. Using the

number of fiber segments equal to 100, defined in subsection 4.1.3, and considering 10000

as a suitable number of fiber realizations to obtain good results (as shown in figure 4.4b),

the total simulation time can be calculated as follows

Tsim,tot = 10000× 100× 1.8 + 10000× 1500 = 1.68× 107 s ≈ 194 days (4.18)

It is obvious that such simulation time is completely unacceptable for this work purpose.

Therefore, a more suitable method to evaluate the performance of the DD MB-OFDM

system in presence of first- and second-order PMD must be adopted.

In order to evaluate the performance which accounts for all possible DGD values within a

reasonable time period, the following method was used. For a given fiber length, the DGD

range is divided in 0.4 ps width intervals from 0 ps up to 40 ps (100 intervals). For this

work purpose, these intervals are denominated as DGD categories. The system considered

for the performance evaluation is a 12-band MB-OFDM system at 112 Gb/s with 2.333 GHz

of bandwidth per band and a 16 QAM modulation scheme. A 2nd order super Gaussian

filter is used as a BS. From a set of approximately 2500 fiber realizations (for a given

fiber length), the DGD value at the frequency where the center of the OFDM band to be

selected is positioned, is calculated. Then, each fiber realization is organized into the DGD

categories. As mentioned in section 3.3.2, the 11th OFDM band is the band that leads to

the worse performance, therefore, the one to be selected by the BS. Then, one realization

63

from each DGD category is chosen to serve as an example, and it is used as a channel for

the system under study. In these simulations, both CD and optical noise are considered,

with OSNR = 31.3 dB. The BER resulting from each fiber realization is calculated from

100 noise runs using EGA method. Then, a weighted mean over the resulting BERs is

performed. The BER resulting from each one of the DGD categories is multiplied by the

probability of that same category. Then, all those values are summed in order to obtain the

overall weighted mean BER. The overall weighted mean BER calculation is summarized in

the following equation

〈BER〉 =100∑n=1

BERnPteo[0.4(n− 1) ≤ DGD < 0.4n] (4.19)

where n is the DGD category and BERn is the BER resulting from the nth category.

Notice that the theoretical Maxwellian distribution, Pteo, differs according to the fiber length

considered. Although a set of 2500 fiber realizations for a given fiber length seems to be

a considerable statistical sample to describe the PMD, some DGD categories may end up

empty. This is due to the occurrence probability of certain DGD values being extremely

low. In table 4.1, it is listed the non-empty DGD categories for the fiber length equal to

100, 200, 300 and 400 km. Notice that the smaller the fiber length, the fewer the filled DGD

categories.

Lf [km] DGD categories100 1→ 34200 2→ 45, 47, 53300 3→ 51, 53, 54, 56, 57, 61, 62, 64, 72400 3→ 62, 66, 67, 70, 75

Table 4.1: List of non-empty DGD categories for fiber lengths equal to 100, 200, 300 and400 km. A→ B stands for interval between A and B.

By employing this method to evaluate the performance of the system in presence of second-

order PMD, the total simulation time is greatly reduced when compered with the method

described at the beginning of the present section. Using equation 4.17, the total simulation

time can be calculated as follows

Tsim,tot ≤ 2500× 100× 1.8 + 100× 1500 = 6× 105 s ≈ 7 days (4.20)

which is a reasonable time for this work purpose. The less or equal sign (≤) is used in

equation 4.20 because, as mentioned earlier in the present section, although the maximum

number of filled categories is 100, same of those categories may end up empty.

64

1 26 51 76 1000

0.02

0.04

0.06

0.08

DGD category

Pro

b [D

GD

cat

egor

y]

1 26 51 76 100−4

−3.6

−3.2

−2.8

−2.4

−2

log

B

ERn

10

Figure 4.10: BER of the system and occur-rence probability of each DGD category fiberrealization, for a 100 km fiber link.

1 26 51 76 100−9

−8

−7

−6

−5

−4

−3

DGD category

log

10 (

wei

gthe

d B

ER

)n

Figure 4.11: Weighted BER of each DGD cat-egory fiber realization for a 100 km fiber link.

1 26 51 76 1000

0.02

0.04

0.06

0.08

1 26 51 76 100−4

−3.6

−3.2

−2.8

−2.4

−2

Pro

b [D

GD

cat

egor

y]

DGD category

log

B

ERn

10


1 26 51 76 100−9

−8

−7

−6

−5

−4

−3

DGD category

log

10 (

wei

gthe

d B

ER

)n


65

1 26 51 76 1000

0.02

0.04

0.06

0.08

1 26 51 76 100−4

−3.6

−3.2

−2.8

−2.4

−2P

rob

[DG

D c

ateg

ory]

DGD category

log

B

ERn

10


1 26 51 76 100−9

−8

−7

−6

−5

−4

−3

DGD category

log

10 (

wei

gthe

d B

ER

)n


1 26 51 76 1000

0.02

0.04

0.06

0.08

1 26 51 76 100−4

−3.6

−3.2

−2.8

−2.4

−2

Pro

b [D

GD

cat

egor

y]

DGD category

log

B

ER

n10


1 26 51 76 100−9

−8

−7

−6

−5

−4

−3

DGD category

log

10 (

wei

gthe

d B

ER

)n


Figure 4.10 represents the BER of the system resulting from each DGD category fiber

realization, superimposed with the occurrence probability of each DGD category for a 100

km fiber link. Figure 4.11 shows the weighted BER of each DGD category fiber realization

which is calculated using the argument of the summation from equation 4.19. The same

information is presented in figures 4.12 up to 4.17 but for fiber link lengths equal to 200

km, 300 km and 400 km. Notice that, despite the fluctuations of the BER value (figures

4.10, 4.12, 4.14 and 4.16) along the DGD categories, the BER values do not present an

evident increase when the DGD increase. This is the first indication that, apparently, the

second-order PMD may not affect the performance of the system. Also, in figures 4.11,

66

4.13, 4.15 and 4.17, we can see that the contribution of each DGD category for the weighted

mean BER is progressively lower (exponential decaying) for higher values of DGD. This is

an important conclusion because, despite the existence of empty DGD categories for high

DGD values, the statistical sample of the PMD remains almost unaffected.

0 100 200 300 400−4

−3.5

−3

−2.5

−2

Lf [km]

log 10 <BER>

Figure 4.18: Weighted BER, 〈BER〉, as a function of the fiber length in presence of first-and second-order PMD, CD and optical noise.

Figure 4.18 represents the weighted BER (〈BER〉) as a function of the fiber length in

presence of CD, optical noise, first- and second-order PMD. Similarly to the link where

only first-order PMD is considered, the number of SSBI iterations required for the DSP-

based algorithm to stabilize, is also 5. Notice that the BER of the system remains almost

constant, with slight non-relevant fluctuations, for fiber lengths between 0 and 400 km.

Therefore, it is possible to conclude that the performance of the DSP-based iterative SSBI

mitigation algorithm is not affected by second-order PMD for fiber link lengths up to 400 km.

Notice that such result assumes that the time interval between two consecutive sequences of

training OFDM symbols is very short when compared with the PMD variations along time.

4.4 Conclusion

In this chapter, the performance of the DSP-based iterative SSBI mitigation algorithm in

presence of optical noise, CD, first- and second-order PMD has been evaluated. It has

been shown that the performance of the DSP-based iterative SSBI mitigation algorithm

is not affected by the second-order PMD for optical link lengths up to 400 km. Such

result assumes that the time interval between two consecutive sequences of training OFDM

symbols is shorter than the PMD variations along time.

Also, the trade-off between the number of fiber segments considered and the quality of the

PMD emulation has been studied in order to optimize the computational simulation times.

It has been shown that the quality of the PMD emulation remains almost constant for a

67

number of fiber segments equal to 50 or higher. Therefore, 100 fiber segments was chosen

as a good trade-off taking into account the time limitations of the simulation.

The time dependence of PMD has been discussed. The maximum number of OFDM sym-

bols, that may be transmitted before the next sequence of training OFDM symbols is trans-

mitted, has been identified as 18000, for a timescale of the PMD of 1 ms. This number is

much larger than the number of OFDM training symbols considered (Ntrain = 75).

68

Chapter 5

Conclusion and future work

In this chapter, the final conclusions of the work developed in this dissertation are presented,

as well as suggestions for future work.

5.1 Final conclusions

In this dissertation, the impact of first and second-order PMD on 100 Gb/s MB-OFDM

metropolitan networks employing DSP-based iterative SSBI mitigation technique is eval-

uated through numerical simulation. Also, a study of the trade-off between the number

of fiber segments considered in the second-order PMD model and the quality of the PMD

emulation has been done.

In chapter 2, the basic concepts of OFDM and MB-OFDM have been presented. It has been

shown that constellation sizes between 16 and 64 are the only candidates for a 12-bands MB-

OFDM system with 40 GHz available bandwidth per channel operating at 112 Gb/s. Also,

optical telecommunication systems based on OFDM technology have been introduced. The

components that comprise an optical system have been described briefly, such as detection

methods and E-O/O-E conversion. It has been concluded that the thermal noise generated

at the photodetector circuit can be neglected when compared to the ASE noise of the system

under study. Also, the issue concerning the SSBI caused by the direct detection has been

introduced. Lastly, two methods of performance evaluation have been introduced where, for

the EGA method, 100 noise runs have been established as a good number of runs in terms

of accuracy and simulation time trade-off.

In chapter 3, three different SSBI mitigation techniques have been presented. The first two

techniques, BICR and SPS, have been presented in a very brief way where the principle

of operation, main advantages and disadvantages have been explained. The other SSBI

mitigation technique is the DSP-based iterative SSBI mitigation algorithm which has been

rigorously detailed. The behavior of the SSBI mitigation algorithm when different system

parameters change has been analyzed. It has been shown that, if the optical noise of the

system increases, then the number of SSBI iterations required remains constant but the

69

EVM resulting from the last SSBI iteration increases. Also, if the VBPR of the system

decreases, then the EVM resulting from the last SSBI iteration remains constant but the

number of SSBI iterations required increases. Lastly, it has been shown that neither the

number of SSBI iterations nor the EVM resulting from the last SSBI iteration is affected

by the modulation index for values between 5% and 10%.

The optimal VBG has been defined as 18.23 MHz, which takes into account the overlapping

between the virtual carrier and the sinc shape spectrum of the left- and rigthmost subcarriers

of the OFDM band that extends beyond the band limits.

Also, the effects of CD and first-order PMD of the optical fiber have been introduced such as

its impact on the constellation of the received OFDM signal. The performance evaluation

of the SSBI mitigation algorithm technique employed in the presence of such dispersion

effects has been preformed. It has been shown that the CD alone does not affect the SSBI

mitigation algorithm for a length of fiber up to 1000 km. As for the first order PMD, it

has been shown that, in the case of VBG being large enough to accommodate the SSBI

term after photodetection, the equalizer is able to perfectly compensate for the distortion

caused by first order PMD up to 90 ps of DGD. However, if VBG is small, the performance

of SSBI mitigation algorithm degrades for values of DGD higher than 70 ps. However, such

values of DGD fall outside the scope of the metro networks. Finally, a simulation where

CD, first-order PMD and optical noise are considered has been performed. The results show

that first-order PMD does not affect the DSP-based iterative SSBI mitigation algorithm for

fiber link lenghts up to 400 km.

In chapter 4, the performance of the DSP-based iterative SSBI mitigation algorithm in

presence of optical noise, CD, first- and second-order PMD has been evaluated. It has been

shown that the performance of the DSP-based iterative SSBI mitigation algorithm is not

affected by the first- and second-order PMD for optical link lengths up to 400 km. Such

result assumes that the time interval between two consecutive sequences of training OFDM

symbols is shorter than the PMD variations along time.

Also, the trade-off between the number of fiber segments considered and the quality of the

PMD emulation has been studied in order to optimize the computational simulation times.

It has been shown that the quality of the PMD emulation remains almost constant for a

number of fiber segments equal to 50 or higher. Therefore, 100 fiber segments was chosen

as a good trade-off taking into account the time limitations of the simulation.

A simple discussion concerning the time dependence of PMD has been made, where the

maximum number of OFDM symbols that may be transmitted before the next sequence of

training OFDM symbols is transmitted was identified as 18000, considering a timescale of

the PMD equal to 1 ms.

70

5.2 Future work

Some work topics for future investigation are suggested in order to complement or further

develop the work accomplished in this dissertation:

To perform a detailed study on the PMD variations along time and its impact on DD

MB-OFDM systems at 100 Gb/s.

Experimental demonstration of the results obtained in this dissertation concerning the

impact of PMD on the system.

To perform a study on the trade-off between the number of OFDM training symbols

to obtain a good channel estimation, and the resulting net bit-rate on DD MB-OFDM

systems.

Analysis of the impact of PMD on long-haul fiber links and assess the employment of

adaptive modulation for links highly affected by PMD.

To perform a performance evaluation of a DD MB-OFDM system in presence of PMD

effect where the bandwidth of the bands are larger than the ones considered in this

dissertation.

71

72

Appendix A

MB-OFDM signal

The purpose of this appendix is to give a more detailed explanation of the MB-OFDM

system operation, as well as, to show the signal characteristics at the different points of

the system. The following analysis takes into account the system parameters discussed in

section 2.2.2, that is: Bch = 50 GHz, ∆B = 3.125 GHz and NB = 12. For instance, lets

consider the number of subcarriers as Nsc = 128 and a modulation order of M = 16 which

corresponds to β = 0.747 as seen in table 2.1. As defined in equation 2.37, β = 0.747

means that approximately 75% of the frequency slot is occupied by the signal spectrum,

while the remaining corresponds to the guard band. CP is not added since no channel was

implemented for this simple approach. For simplicity reasons, the DAC is performed by a

sample and hold, that is, each sample is held for a certain period of time. The low-pass

filter (LPF) placed at DAC output is a rectangular filter with bandwidth Bb. The main

goal is to achieve the 112 Gbit/s per MB-OFDM signal.

−4 −2 0 2 4−4

−2

0

2

4

I

Q

Figure A.1: 16 QAM constellation at thetransmitter symbol mapper output.

1.8 2.2 2.6 3 3.4 3.8−0.3

−0.2

−0.1

0

0.1

0.2

0.3

0.4

time [ns]

ampl

itud

e [V

]

DAC outputLPF output

Figure A.2: Signal waveform at the outputof the DAC and LPF used at the transmitterside.

Figure A.1 represents the 16-sized constellation originated by the symbol mapper at the

transmitter. Figure A.1 shows that all the symbols are perfectly placed at their designated

73

−12 −8 −4 0 4 8 120

20

40

60

80

frequency [GHz]

norm

. P

SD [dB

]

Figure A.3: Normalized PSD at DAC output.

−12 −8 −4 0 4 8 120

20

40

60

80

frequency [GHz]

norm

. P

SD [dB

]

Figure A.4: Normalized PSD at LPF output.

−12 −8 −4 0 4 8 120

20

40

60

80

frequency [GHz]

norm

. P

SD [dB

]

Figure A.5: Normalized PSD at IQM output.

−40 −30 −20 −10 0 10 20 30 4040

60

80

100

120

frequency [GHz]

norm

. P

SD [dB

]130

Figure A.6: Normalized PSD of the MB-OFDM signal.

spot. The symbols then enter the IFFT block. From equations 2.22 and 2.37, the OFDM

symbol duration ts can be computed as

ts =Nsc

β ∆B= 54.9 ns (A.1)

Since CP was assumed as non-mandatory, tG is null. Figure A.2 illustrates the evolution

of the OFDM signal waveform in a given time window at the output of the DAC and LPF

of the transmitter. Note that, after the LPF, the signal has a smooth behavior. This is

a consequence of the higher frequencies of the signal generated by the DAC being filtered

out by the LPF. The normalized power spectrum density (PSD) of the OFDM signal at

DAC output can be seen in figure A.3. Notice that the replicas generated by the DAC are

gradually lower in power as they apart from the central frequency. This is a consequence

of the DAC transfer function which has a sinc shape characteristic with nulls at ±k 2/Tc,

74

with k ∈ N as explained in section 2.1.5, where Tc is the chip time. The periodic nulls of

the DAC transfer function justifies the nulls in the normalized PSD observed in the DAC

output signal spectrum. Figure A.4 shows the normalized PSD at the LPF output. Then,

the OFDM signal is up-converted by a IQ modulator. The PSD of the signal at IQ modulator

output is shown in figure A.5. The spectrum is now centered at fIQ,1 = ±3.125/2 GHz at

both positive and negative sides. The positioning of the bands that comprise the multiband

signal, as seen in figure A.6, follows the equation

fIQ,n = ±(

∆B

2+ (n− 1) ∆B

)(A.2)

where ∆B = 3.125 GHz. For simplicity, the channel is not considered, and so, equalizing

is not mandatory since no relevant distortion occurs. The constellation of the symbols

resulting from the FFT process at the receiver can be seen in figure A.7. The symbols of

the constellation are not perfectly positioned due to the DAC sample and hold process. As

mentioned earlier, the DAC process is equivalent to multiplying the OFDM spectrum by

a sinc shaped transfer function. Since the main lobe of the sinc is not flat, the OFDM

spectrum is not flat either. The higher the subcarrier frequency, more attenuation it suffers.

This causes that some symbols of the constellation are attenuated in magnitude.

−1 −0.6 −0.2 0.2 0.6 1

x 10−6

−1

−0.6

−0.2

0.2

0.6

1x 10

−6

I

Q

Figure A.7: 16 QAM constellation at the receiver side.

75

76

Appendix B

Super Gaussian band selector

The band selectors (BS) used in real systems are far from having ideal shapes. In fact,

currently, optical filters with high selectivity and bandwidth in the order of 2 GHz are not

commercially available. Therefore, for this work purpose, we have to consider a type of BS

that meets the requirements even if it is still not commercially available. A good candidate

for such filter is the super Gaussian filter of order 2. The transfer function of a super

Gaussian filter is described by the following function

HSG(f) = exp

[−22n log(

√2)

(f − fcB−3dB

)2n]

(B.1)

where fc is the central frequency of the filter, B−3dB is the -3 dB bandwidth and n is the

filter order.

0 1 2 3 4 5 6 7 8 90

20

40

60

80

frequency [GHz]

norm

. P

SD [dB

]

Figure B.1: Spectra of an optical MB-OFDM signal, in equivalent baseband frequency, in asystem with a 2nd order super Gaussian BS. In black - signal before BS; in grey - signal afterBS. The BS has a -3 dB bandwidth of 2.2 GHz and a detuning (relatively to the centralfrequency of the OFDM band) of 300 MHz.

Due to the small guard band between adjacent bands, the optimal bandwidth and central

frequency of the filter must be defined. In [40], the optimal parameters for a super Gaussian

77

BS used in an optical MB-OFDM system with frequency slot of 3.125 GHz and OFDM

bandwidth of 2.333 GHz are B−3dB = 2.2 GHz and a detuning of 300 MHz. The motivation

for the choice of this detuning is to ensure that the virtual carrier of the left adjacent band is

sufficiently attenuated so that the beat between that virtual carrier of the left adjacent band

and the desired OFDM band is small. Figure B.1 represents the spectrum of an MB-OFDM

signal where the second band is selected using a super Gaussian BS with the earlier defined

parameters.

78

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