Channel Equalization and Phase Estimation for Reduced...

Channel Equalization and Phase Estimation for

Reduced-Guard-Interval CO-OFDM Systems

Qunbi Zhuge

Department of Electrical & Computer Engineering

McGill University

Montreal, Canada

December 2011

A thesis submitted to McGill University in partial fulfillment of the requirements for the

degree of Master of Engineering.

2011 Qunbi Zhuge

i

Abstract

Reduced-guard-interval (RGI) coherent optical (CO) orthogonal frequency-division

multiplexing (OFDM) is a potential candidate for next generation 100G beyond optical

transports, attributed to its advantages such as high spectral efficiency and high tolerance

to optical channel impairments. First of all, we review the coherent optical systems with

an emphasis on CO-OFDM systems as well as the optical channel impairments and the

general digital signal processing techniques to combat them. This work focuses on the

channel equalization and phase estimation of RGI CO-OFDM systems. We first propose

a novel equalization scheme based on the equalization structure of RGI CO-OFDM to

reduce the cyclic prefix overhead to zero. Then we show that intra-channel nonlinearities

should be considered when designing the training symbols for channel estimation.

Afterwards, we propose and analyze the phenomenon of dispersion-enhanced phase noise

(DEPN) caused by the interaction between the laser phase noise and the chromatic

dispersion in RGI CO-OFDM transmissions. DEPN induces a non-negligible

performance degradation and limits the tolerant laser linewidth. However, it can be

compensated by the grouped maximum-likelihood phase estimation proposed in this

work.

ii

Sommaire

Le multiplexage rpartition en frquence orthogonale (MRFO) optique la dtection

cohrente avec intervalle protection rduite (IPR) est un candidat potentiel pour la

prochaine gnration des systmes de transport optique au-del de 100G. Cette mthode

dmontre un rendement spectral lev et une grande tolrance aux dgradations du canal

optique. En premier lieu, nous prsentons un bilan sur les systmes optiques la

dtection cohrente avec lemphase sur MRFO, les dgradations du canal optique, et

ainsi les techniques gnrales de traitement numrique du signal pour corriger ces

dgradations. Ce travail se concentre sur lgalisation du canal et lestimation de phase

des systmes MRFO optique la dtection cohrente avec IPR. Nous commenons par

proposer une nouvelle faon dgalisation base sur MRFO optique la dtection

cohrente avec IPR pour rduire la marge de la prfixe cyclique zro. Ensuite, nous

prsentons que les non-linarits intra-canal devrait tre considres pendant la

conception des symboles de rfrence pour lestimation du canal. Prochainement, nous

proposons et analysons le phnomne du bruit de phase la dispersion amliore (BPDA)

qui est cause par linteraction entre le bruit de phase du laser and la dispersion

chromatique dans les transmissions MRFO optique la dtection cohrente avec IPR. Le

BPDA entrane une dgradation de performance non-ngligeable et limite la tolrances de

la largeur spectrale du laser. Cependant, le BPDA peut tre compens par lestimation de

phase groupe la vraisemblance maximum propose dans ce travail.

iii

Acknowledgement

This thesis would not have been accomplished without the help and support of many

people.

Above all, I would like to thank my supervisor Prof. David V. Plant for his generosity

and encouragement in supporting my studies and research works. Ive really enjoyed

working with him throughout my Masters degree. I appreciate everything he has done

for me, and Im also very thankful for the example he has provided as a successful person.

I would like to thank my colleagues Dr. Chen Chen and Mohamed Osman. It is my

privilege to collaborate with both of them. They provided me with many useful

suggestions on the projects involved in this thesis and spent a lot of time editing my

papers. Ive learned a lot from both of them on how to do good research and how to

write a good paper.

I am also grateful to all people around me at McGill for their help and supports in both

research and life.

I would like to thank my girlfriend Huanhuan Shui for being with me during the past

two years. She makes me happy and motivates me to become a true man. Without her,

my life in Montreal would not have been so wonderful and so peaceful.

Finally, I would like to thank my parents. They selflessly support me in pursuing my

dream. They provided me a great environment to grow up in and become a happy and

positive person. Making them proud of me is my biggest motivation in life.

iv

Contribution of Authors

The work presented in this thesis has been published in the following journals and

conference proceedings:

[1] C. Chen, Q. Zhuge and D. V. Plant, Zero-Guard-Interval Coherent Optical OFDM with Overlapped Frequency-Domain CD and PMD Equalization, Optics Express,

vol. 19, pp. 7451-7467, 2011.

[2] C. Chen, Q. Zhuge and D. V. Plant, Coherent Optical OFDM with Zero Cyclic Prefix Using Overlapped Frequency-Domain CD and PMD Equalization, Optical

Fiber Communications (OFC) Conference, paper OWE7, 2011.

[3] Q. Zhuge, C. Chen and D. V. Plant, Impact of Intra-Channel Fiber Nonlinearity on Reduced-Guard-Interval CO-OFDM Transmission, Optical Fiber

Communications (OFC) Conference, paper OWO3, 2011.

[4] Q. Zhuge, C. Chen and D. V. Plant, Dispersion-enhanced phase noise effect on reduced-guard-interval CO-OFDM transmission, Optics Express, vol.19, pp.

4472-4484, 2011.

[5] Q. Zhuge, and D. V. Plant, Compensation for Dispersion-Enhanced Phase Noise in Reduced-Guard-Interval CO-OFDM Transmissions, Signal Processing in

Photonic Communications (SPPCom) Conference, paper SPMB4, 2011.

[6] Q. Zhuge, M. Osman, and D. V. Plant, Analysis of dispersion-enhanced phase noise in CO-OFDM systems with RF-pilot phase compensation, Optics Express,

vol. 19, pp. 24030-24036, 2011.

I co-authored the papers [1] and [2] with my colleague Dr. Chen Chen. Chen proposed

the novel equalization scheme and wrote the papers. I provided my code of reduced-

guard-interval CO-OFDM in both Matlab and Optisystem to help him build the

simulation setup. I also provided suggestions to improve the scheme, analyze the

performance and calculate the complexity. In addition, I edited the papers and

commented on them.

I co-authored the papers [3] and [4] with my colleague Chen. I developed the ideas,

did the simulations and wrote the papers. Chen provided a lot of useful suggestions and

edited the papers very carefully. He also taught me how to write a good paper in this

process.

I co-authored the paper [6] with my colleague Mohamed Osman. I developed the idea,

did the simulation and wrote the paper. Mohamed gave me many helpful suggestions on

how to make this idea more convincing. He also spent a lot of time editing and

commenting on this paper.

Other publications not directly related to this thesis:

[7] Q. Zhuge, B. Chtelain, and D. V. Plant, Comparison of Intra-Channel Nonlinearity Tolerance between Reduced-Guard-Interval CO-OFDM Systems and

Nyquist Single Carrier Systems, accepted to Optical Fiber Communications

(OFC) Conference, 2012.

v

[8] Q. Zhuge, M. E. Pasandi, X. Xu, B. Chtelain, Z. Pan, M. Osman and D. V. Plant, Linewidth-Tolerant Low Complexity Pilot-Aided Carrier Phase Recovery for M-

QAM using Superscalar Parallelization, accepted to Optical Fiber

Communications (OFC) Conference, 2012.

[9] M. Osman, Q. Zhuge, L. R. Chen, and D. V. Plant, Joint Mitigation of Laser Phase Noise and Fiber Nonlinearity for Polarization-Multiplexed QPSK and 16-

QAM Coherent Transmission Systems, Optics Express, vol. 19, pp. B329-B336,

2011.

[10] M. Osman, Q. Zhuge, L. R. Chen, and D. V. Plant, Feedforward carrier recovery via pilot-aided transmission for single-carrier systems with arbitrary M-QAM

constellations, Optics Express, vol. 19, pp. 24331-24343, 2011.

[11] Q. Zhuge, B. Chtelain, C. Chen and D. V. Plant, Mitigation of Equalization-Enhanced Phase Noise Using Reduced-Guard-Interval CO-OFDM, European

Conference on Optical Communication (ECOC), paper Th.11.B.5, 2011.

[12] Q. Zhuge, C. Chen, and D. V. Plant, Low Computation Complexity Two-Stage Feedforward Carrier Recovery Algorithm for M-QAM, Optical Fiber

Communications (OFC) Conference, paper OMJ5, 2011.

[13] J. D. Schwartz, Q. Zhuge, Y. Zhu, J. Azaa, and D. V. Plant, Broadband Microwave and MM-Wave Dispersion Using Periodic Structures, IEEE

Microwave Photonics Conference, paper WE4-7, 2010.

vi

Contents

Chapter 1 Introduction ........................................................................................................ 1

1.1 Motivation ................................................................................................................. 1

1.2 Thesis Problem Statement ......................................................................................... 2

1.3 Thesis Contribution and Organization ...................................................................... 3

Chapter 2 Literature Review ............................................................................................... 5

2.1 Coherent Optical Communication Systems .............................................................. 5

2.2 Coherent Optical OFDM Systems ............................................................................ 7

2.2.1 Conventional CO-OFDM Systems ..................................................................... 7

2.2.2 Reduced-Guard-Interval (RGI) CO-OFDM Systems ......................................... 9

2.3 Fiber Channel Impairments and Digital Signal Processing .................................... 11

2.3.1 Dispersion in Single Mode Fiber ...................................................................... 11

2.3.2 Laser Phase Noise ............................................................................................. 16

2.3.3 Fiber Nonlinearities .......................................................................................... 20

2.4 Conclusions ............................................................................................................. 23

Chapter 3 Channel Equalization for RGI CO-OFDM ...................................................... 24

3.1 Zero-Guard-Interval CO-OFDM ............................................................................. 25

3.1.1 Channel Equalization Algorithm Description .................................................. 26

3.1.2 Numerical Results and Discussions .................................................................. 31

3.1.3 Complexity Comparison ................................................................................... 36

3.2 Impact of Intra-Channel Fiber Nonlinearities ......................................................... 40

3.2.1 Impact of Intra-Channel Fiber Nonlinearities on Training Symbols ................ 40

3.2.2 Simulation Results ............................................................................................ 42

3.2.3 Improvement in Tolerance to Intra-Channel Fiber Nonlinearity...................... 44

3.3 Conclusions ............................................................................................................. 45

Chapter 4 Dispersion-Enhanced Phase Noise (DEPN) ..................................................... 47

4.1 DEPN with Pilot Subcarrier Phase Estimation ....................................................... 47

4.1.1. System Model .................................................................................................. 48

4.1.2. Performance Analysis ...................................................................................... 51

4.1.3. Numerical Results and Discussion .................................................................. 54

vii

4.2 DEPN with RF-Pilot Phase Compensation ............................................................. 58

4.2.1 Analysis of DEPN with RF-pilot Phase Compensation ................................... 59

4.3.1 Simulation Results and Discussions ................................................................. 62

4.3 DEPN Compensation .............................................................................................. 64

4.3.1 Grouped Maximum-Likelihood (GML) DEPN Compensation ........................ 64

4.3.2 Improved GML Algorithm with PS-based Phase Estimation........................... 68

4.4 Conclusions ............................................................................................................. 70

Chapter 5 Conclusions and Future Work .......................................................................... 71

5.1 Conclusions ............................................................................................................. 73

5.2 Future Works ........................................................................................................... 74

Appendix ........................................................................................................................... 75

References ......................................................................................................................... 77

viii

List of Acronyms

ADC: Analog-to-Digital Converter

ASE: Amplified Spontaneous Emission

AWGN: Additive White Gaussian Noise

BER: Bit Error Ratio

CD: Chromatic Dispersion

CDP: Correlated Dual-Polarization

CO: Coherent Optical

CP: Cyclic Prefix

CPE: Common Phase Error

DAC: Digital-to-Analog Converter

DCF: Dispersion-Compensating Fiber

DEPN: Dispersion-Enhanced Phase Noise

DFT: Discrete Fourier Transform

DGD: Differential Group Delay

DP: Dual Polarization

DSP: Digital Signal Processing

EDFA: Erbium-Doped Fiber Amplifier

FDE: Frequency Domain Equalizer

FDI: Frequency Domain Interpolation

FFT: Fast Fourier Transform

FWM: Four-Wave Mixing

GML: Grouped Maximum-Likelihood

ICI: Inter-Carrier Interference

IFFT: Inverse Fast Fourier Transform

ISFA: Intra-Symbol Frequency-Domain Averaging

LO: Local Oscillator

LPF: Low-Pass Filter

MC: Monte Carlo

ix

MIMO: Multiple-Input and Multiple-Output

NLS: Nonlinear Schrdinger

OFDE: Overlapped Frequency Domain Equalizer

OFDM: Orthogonal Frequency Division Multiplexing

OSNR: Optical Signal-to-Noise Ratio

OOK: On-Off Keying

PAPR: Peak-to-Average Power Ratio

PDF: Probability Density Function

PMD: Polarization Mode Dispersion

PDM: Polarization-Division Multiplexing

PRT: Phase Rotation Term

PS: Pilot Subcarrier

QAM: Quadrature Amplitude Modulation

QPSK: Quadrature Phase-Shift Keying

RGI: Reduced-Guard-Interval

RF: Radio Frequency

RMS: Root Mean Square

RPS: Residual Phase Shift

SINR: Signal-to-Interference-Plus-Noise Ratio

SP: Single-Polarization

SPM: Self-Phase Modulation

SSMF: Standard Single Mode Fiber

SSFM: Split-Step Fourier Method

TS: Training Symbol

WDM: Wavelength-Division Multiplexing

XPM: Cross-Phase Modulation

ZGI: Zero-Guard-Interval

x

List of Figures

Fig. 1. Block diagram of a coherent optical communication system. PBC: polarization

beam combiner. PBS: polarization beam splitter. LO: local oscillator............................... 6

Fig. 2. OFDM subcarriers in the (a) frequency domain and (b) time domain. ................... 8

Fig. 3. Illustration of the OFDM channel equalization with CP. ........................................ 8

Fig. 4. Block diagram of CO-OFDM system. ..................................................................... 9

Fig. 5. (a) Block diagram of RGI CO-OFDM; (b), (c), and (d): illustration the CD

compensation for RGI CO-OFDM signals. ...................................................................... 10

Fig. 6. The effect of CD on the transmitted signal of (a) single carrier systems for one

pulse and (b) OFDM systems for one symbol including many subcarriers. ..................... 12

Fig. 7. Illustration of a PMD model in conventional fibers. ............................................. 13

Fig. 8. Block diagram for DSP-based dispersion compensation. ..................................... 14

Fig. 9. (a) single carrier systems and (b) OFDM systems with two-stage dispersion

compensation. TDE: time domain equalizer. .................................................................... 15

Fig. 10. The illustration of single-polarization TS ............................................................ 16

Fig. 11. (a) Model of the laser phase noise applied to the signal. (b) Illustration of the

evolution of laser phase noise. .......................................................................................... 17

Fig. 12. Constellation of (a) transmitted QPSK symbols, (b) single carrier symbols

applied with laser phase noise, and (c) OFDM subcarriers applied with laser phase noise.

........................................................................................................................................... 18

Fig. 13. Block diagram of the Mth

power scheme ............................................................. 18

Fig. 14. Illustration of the PSs allocation. ....................................................................... 19

Fig. 15. (a) The spectrum of the OFDM signal with a RF pilot tone. (b) The estimated

phase from the RF-pilot based scheme and the CPE based scheme. ................................ 20

Fig. 16. Illustration of using the SSFM to model the signal propagation in a fiber link. . 23

Fig. 17. (a) CO-OFDM receiver structure and (b) OFDM frame. S/P: serial to parallel .. 27

Fig. 18. (a) Schematic of FDI. (b) OFDM spectrum before and after applying HFDI- 1

.The

top curve shows the real part of a in the channel matrix H (open dot) and the interpolated

channel matrix HFDI (thin line). Phase variations across modulated subcarriers before and

xi

after PMD compensation in Step (4) for a (c) deterministic DGD=320ps and (d)

stochastic PMD with =100ps. .............................................................................. 30

Fig. 19. Q vs. deterministic DGD for three different CO-OFDM systems. ...................... 32

Fig. 20. Contour plot of the estimated phase variations across modulated subcarriers on x-

polarization before (a) and after (b) Step 4. =5 ps is assumed............................. 33


polarization before (a) and after (b) Step 4. =10 ps is assumed........................... 34


polarization before (a) and after (b) Step 4. =25 ps is assumed........................... 34

Fig. 23. Q factor distribution after transmission over a fiber link with 500 different PMD

for (a) ZGI-CO-OFDM (0% CP) (b) ZGI-CO-OFDM (0.8% CP) and (c) RGI-CO-OFDM

(3.13% CP). We assume =10 ps. ......................................................................... 35



(3.13% CP). We assume =25 ps. ......................................................................... 35



(3.13% CP). We assume =50 ps. ......................................................................... 36

Fig. 26. (a) Number of complex multiplications per useful bit as a function of NFFT for the

conventional, RGI- and ZGI-CO-OFDM. (b) Percentage of extra computation complexity

of ZGI- over RGI- CO-OFDM, as a function of NOFDE and oversampling factor. ........... 39

Fig. 27. Illustration of the non-uniform power for SP-TSs induced by the large CD

relative to symbol duration in RGI CO-OFDM systems. ................................................. 41

Fig. 28. The constellation of payload data symbols estimated by SP-TSs. Left: total

subcarriers; Middle: first subcarrier; Right: last subcarrier. ............................................. 41

Fig. 29. Illustration of the uniform power for CDP-TSs in RGI CO-OFDM systems

during the transmission. .................................................................................................... 42

Fig. 30. Block diagram of the RGI CO-OFDM system. ................................................... 43

Fig. 31. Simulated Q-factor of RGI CO-OFDM and conventional CO-OFDM (Con) with

different channel estimation methods. .............................................................................. 43

Fig. 32. Mean PAPR versus the number of subcarriers for OFDM signal. ...................... 44

xii

Fig. 33. Comparison of the nonlinear tolerance for RGI CO-OFDM and conventional

CO-OFDM (Con) with different clipping ratios. .............................................................. 45

Fig. 34. Block diagram of RGI CO-OFDM systems. ....................................................... 48

Fig. 35. The communication channel model. .................................................................... 49

Fig. 36. (a) The applied laser phase noise at the transmitter side; (b) the applied laser

phase noise at the receiver side. ........................................................................................ 50

Fig. 37. The normalized variance of PRT for each subcarrier obtained from both theory

and simulation using either UD-PSs or C-PSs with no ASE noise (left) and with a SNR

of 11 dB (right). L = 3200 km, = 1 MHz and Nc = 80. ................................................. 54

Fig. 38. The constellation of the received symbols after phase estimation with UD-PSs. L

= 3200 km, = 1 MHz and Nc = 80. (a): total subcarriers; (b): center subcarriers; (c):

edge subcarriers. ............................................................................................................... 55

Fig. 39. The normalized average variance of PRT and ICI versus the number of

subcarriers Nc. L = 3200 km, = 1 MHz and SNR = 11 dB. ............................................ 56

Fig. 40. BER versus (a) SNR with L = 3200 km and (b) transmission distance with SNR =

11 dB. = 1 MHz for both systems. ................................................................................. 57

Fig. 41. Required SNR at BER = 10-3

versus transmission distance L with (a) different

numbers of subcarriers Nc with = 1 MHz and (b) varying linewidths and Nc = 80. ... 57

Fig. 42. The illustration of DEPN with RF-pilot phase compensation. (a) The dispersion-

induced walk-off between OFDM subcarriers within one symbol and the RF-pilot tone.

The constellations of (b), the middle subcarriers, and (c), the edge subcarriers for systems

with 320 subcarriers, 2 MHz linewidth and 3200 km transmission. ................................ 59

Fig. 43. The illustration of the relationship between RPS & ICI and the number of

subcarriers. (a) RGI OFDM with 80 subcarriers. (b) RGI OFDM with 320 subcarriers. (c)

Conventional OFDM with 1280 subcarriers. For each figure, the left constellation is for

back-to-back case, and the right constellation is for L = 3200 km. The curves correspond

to the right constellation with = 2 MHz and no ASE noise. For RPS, the unit is rad2. . 61

Fig. 44. OSNR penalty at BER = 10-3

versus the transmission distance L. (a) 28 Gbaud.

(b) 56 Gbaud. Conv denotes conventional CO-OFDM. ................................................... 62

Fig. 45. OSNR penalty at BER = 10-3

versus the laser linewidth . (a) 28 Gbaud. (b) 56

Gbaud. ............................................................................................................................... 63

xiii


versus (a) the number of subcarriers in each group

Ng with Np = 4 and (b) number of PSs Np with Ng = 20 for PS-based phase estimation. 65

Fig. 47. Illustration of the effectiveness of GML phase estimation with (a) = 1 MHz and

(b) = 2 MHz. .................................................................................................................. 67

Fig. 48. BER versus OSNR for DP-QPSK transmissions. Solid: RGI CO-OFDM. Dashed:

Conventional CO-OFDM. (a) 28 Gbaud. (b) 56 Gbaud. .................................................. 68


versus (a) the transmission distance L with a

linewidth of 2 MHz and (b) the laser linewidth for both Tx and Rx lasers at L = 4800 km.

........................................................................................................................................... 70

1

Chapter 1: Introduction

Chapter 1

Introduction

1.1 Motivation

High capacity optical communication technologies are being actively investigated and

researched in order to satisfy the unabated exponential growth of data network traffic for

local area networks, wide area networks, and especially data-centric users [1, 2]. High

spectral efficiency and adequate transmission distance along with cost-effective

underlying techniques are the key attributes for next generation optical transports with a

target of 20 b/s/Hz capacity and over 1500 km links in about 5 years [3]. Since it enables

an increase in channel capacity and suppresses the channel impairments [4], coherent

detection with digital signal processing (DSP) has been adopted as the underlying

technology for 100G products [1, 2].

Coherent optical (CO) orthogonal frequency division multiplexing (OFDM) systems

have been considered a potential candidate for high speed optical transports [5], with

advantages such as high spectral efficiencies, low required sampling rates, and flexible

bandwidth scalability and allocation [5]. More recently, a reduced-guard-interval (RGI)

CO-OFDM system, which compensates the chromatic dispersion (CD) prior to the

OFDM demodulation in order to reduce the cyclic prefix (CP) overhead, was proposed in

[6]. Several 100G beyond transmission experiments using RGI CO-OFDM have been

conducted [6-9], with impressively high spectral efficiencies (up to 7.76 b/s/Hz) and long

transmission distances (up to 4800 km), demonstrating its potential for next generation

100G beyond transports. However, for this novel system, DSP algorithms developed for

conventional CO-OFDM are not necessarily the most appropriate approaches to

2


compensate for the channel impairments and recover the data, considering the special

system structure, the shortened symbol duration and the impact of dispersion and fiber

nonlinearities. Therefore, there is motivation to optimize the conventional DSP

algorithms, or design new algorithms, in order to fully take advantage of RGI CO-OFDM

systems.

1.2 Thesis Problem Statement

Channel equalization (including the polarization demultiplexing) and phase estimation (or

carrier phase recovery) are mandatory for coherent optical systems [4, 5]. OFDM

provides the computational efficiency and ease of channel and phase estimation for the

two aforementioned DSP procedures, at the cost of increased overhead for training

symbols (TSs) and pilot subcarriers (PSs). However, the largely required overhead is

considered to be a limiting factor for the deployment of OFDM systems when compared

to the corresponding single-carrier format [10].

Contrary to conventional CO-OFDM, RGI CO-OFDM implements the dispersion

compensation in two stages: 1) CD compensation using a frequency domain equalizer

(FDE) and 2) polarization mode dispersion (PMD) compensation and polarization

demultiplexing using the conventional OFDM channel estimation. Since CP length and

OFDM symbol length is no longer determined by the considerably long channel memory

caused by CD, an OFDM signal with much shorter symbol duration can be used with an

even smaller CP overhead (around 3%) [6]. However, it is still important to further

reduce the CP overhead in order to make OFDM competitive with single carrier systems,

which do not require CP for channel equalization at all.

When symbol duration is shortened, the CD-induced walk-off between subcarriers

within each OFDM symbol becomes relatively larger, and it will not only influence the

design of TSs for channel estimation, but also enhance the laser phase noise. Therefore,

it is also important to characterize the impact of these effects and to propose new DSP

techniques that make the system more robust in various scenarios.

3


1.3 Thesis Contribution and Organization

Chapter 2 reviews the background of optical communication systems. Conventional CO-

OFDM and RGI CO-OFDM are described separately, focusing on the structure and

merits of each. The main fiber channel impairments, including the dispersion, laser phase

noise and fiber nonlinearities, are briefly introduced and the corresponding DSP

algorithms are discussed.

Chapter 3 focuses on the issue of channel equalization for RGI CO-OFDM. Firstly, a

novel cost-effective equalization approach based on the structure of RGI CO-OFDM is

proposed, which enables the removal of the CP and thus reduces the CP overhead to zero.

Secondly, the impact of intra-channel fiber nonlinearities on the design of TSs for

channel estimation is investigated. In addition, we demonstrate that RGI CO-OFDM is

more tolerant to intra-channel fiber nonlinearities than conventional CO-OFDM.

Chapter 4 investigates the interaction between the accumulated CD and the laser phase

noise for CO-OFDM, denoted as dispersion-enhanced phase noise (DEPN). The origin of

this phenomenon is studied analytically for PS based phase estimation and RF-pilot phase

compensation, respectively. Numerical results show that DEPN limits the laser linewidth

tolerance to hundreds of kHz for 112 Gb/s quadrature phase shift keying (QPSK) systems,

which implies that high-cost external cavity lasers might be required. Fortunately, DEPN

mainly induces phase shifts for RGI CO-OFDM, which can be easily compensated. We

propose grouped maximum-likelihood (GML) algorithms to compensate the DEPN

induced phase shifts, and the laser linewidth is increased to the level of MHz, enabling

the use of low-cost distributed feedback lasers.

Chapter 5 summarizes this work, and discusses potential future research on this topic.

4


5

Chapter 2: Literature Review

Equation Section 2

Chapter 2

Literature Review

2.1 Coherent Optical Communication Systems

In the 1980s and 1990s, coherent detection attracted a great deal of research interest, and

was considered a promising technology with potential to improve the receiver sensitivity

[11]. However, following the invention and commercial availability of the erbium-doped

fiber amplifier (EDFA), a device which improves the noise tolerance in a more cost-

effective way, the interest in coherent detection diminished until a decade later. Yet

coherent detection has another even more important capability, which is to linearly map

the optical field to the electrical field. Therefore the receiver obtains access to both the

amplitude and phase information of the optical signal. This capability enables channel

impairment compensation, polarization demultiplexing, and carrier recovery using DSP.

Also, attributed to the rapid development of high-speed silicon-based DSP technologies,

great improvements in both spectral efficiency and transmission distance are achieved

based on the DSP-assisted coherent detection. Therefore, coherent technology dominates

not only research work, but also commercial products, nowadays [2].

Fig. 1 depicts a general coherent optical communication system. At the transmitter, the

bit sequence is first mapped to modulation symbols, such as QPSK symbols. Then the

DSP can be used to do pulse shaping for single carrier systems, inverse fast Fourier

transform (IFFT) for OFDM systems, or channel pre-compensation. It should be noted

that for single carrier systems, DSP is not a must at the transmitter since the symbol

modulation can be done using analog components [12]. In contrast, DSP and digital-to-

analog converters (DACs) are required for OFDM systems at the transmitter for the IFFT

6


operation, unless all-optical OFDM systems are employed [13]; a subject which is outside

the scope of this work. Two optical IQ modulators are used to convert the electrical

signals to the optical domain and a polarization beam combiner combines the two optical

signals, forming one polarization-division multiplexing (PDM) signal.

Demodulation

and DSP

Modulation

and DSP

DAC

DAC

DAC

DAC

Data X

Data Y

/2

/2

90o

Optical

Hybrid

90o

Optical

Hybrid

ADC

ADC

ADC

ADC

Data X'

Data Y'

EDFA

Span N

PBC

PBS

Laser

Laser

LO LaserPBS

Ix

Qx

Iy

Qy

I'x

Q'x

I'y

Q'y

Transmitter

Receiver

Link

Fig. 1. Block diagram of a coherent optical communication system. PBC: polarization beam combiner. PBS:

polarization beam splitter. LO: local oscillator.

For this link, unlike non-coherent systems where the dispersion-compensating fiber

(DCF) is inserted at the end of each span in order to compensate CD, there are only fibers

and EDFAs for coherent systems; this is referred to as dispersion-unmanaged

transmissions, as all the dispersion is compensated at the receiver. In addition, Fig. 1

shows only a single channel model, and in wavelength-division multiplexing (WDM)

systems more components are required in the link such as arrayed waveguide gratings

and wavelength-selective filters. Moreover, in optical networks, a reconfigurable optical

add-drop multiplexer is used to add or remove channels from WDM transmissions.

The received optical signal is passed through a 90o optical hybrid along with the

output light of the local oscillator (LO) laser, followed by four balanced photodetectors

7


converting the optical signal to an electrical signal. By doing so, the optical field is

linearly mapped to the electrical field, which is then sampled at or above the Nyquist

sampling rate. Having the DSP include channel equalization and phase estimation at the

receiver is a key part of coherent systems, and it will be discussed throughout this work.

After symbols are recovered using DSP, decisions are made to obtain the received bit

sequence.

2.2 Coherent Optical OFDM Systems

2.2.1 Conventional Coherent Optical (CO)-OFDM Systems

As a special form of the multicarrier modulation family, OFDM has been widely

deployed in wireless communications thanks to its ability to combat channel impairments

such as multipath-induced frequency-selective fading. When DSP-assisted coherent

detection emerged as a promising technology for pushing the capacity of the optical

channel, OFDM was brought to optical communications as a competitive candidate

[14, 15]. There are several advantages of OFDM: 1) ease of channel and phase estimation,

2) high spectral efficiency and low required sampling rate, 3) modulation format (e.g.

quadrature amplitude modulation (QAM)) scalability, and, 4) flexible bandwidth

scalability and allocation [5]. Although OFDM suffers from fiber nonlinearities due to its

high peak-to-average power ratio (PAPR), it has been shown that in dispersion-

unmanaged transmissions, the difference of the tolerance in terms of fiber nonlinearities

between OFDM and single carrier systems is actually very small since the dispersion also

induces a large PAPR for the latter [16].

Instead of one wideband carrier, OFDM systems transmit a large number of

narrowband subcarriers, which are orthogonal to each other and closely spaced in the

frequency domain as shown in Fig. 2. The most efficient way to generate an OFDM

signal is to use the IFFT, which avoids a large number of modulators and filters at the

transmitter. Fig. 2 shows the OFDM subcarriers in the frequency domain and time

domain. In the frequency domain, in spite of the significant overlap between them,

subcarriers can be perfectly recovered at the receiver due to the orthogonality between

them. As a consequence, the spectral efficiency of OFDM signals is inherently high. In

8


the time domain, data is transmitted by a number of carriers with different frequencies at

the same time, as shown in Fig. 2 (b).

... ...

...

...

( )a ( )b

Fig. 2. OFDM subcarriers in the (a) frequency domain and (b) time domain.

Cyclic Prefix (CP) is required to prevent inter-symbol interference caused by linear

impairments of the fiber channel such as CD and PMD. In particular, CP is a copy of part

of the signal. For example, in Fig. 3 (a), the CP highlighted in blue is a copy of the red

part of the signal. After transmission in the fiber, there is a walk-off between subcarriers

as shown in Fig. 3 (b). The guard interval provided by CP ensures that there is no leaking

signal from adjacent symbols. At the receiver, after the CP removal and FFT, the

subcarriers will be recovered by channel equalization without any inter-symbol

interference, as plotted in Fig. 3 (c) and (d). However, CP contains no useful information

and therefore induces an overhead. Since the CP length should be greater than the CD

length, which is generally very long in long-haul optical communications, the OFDM

signal has to be generated with a long symbol duration in order to limit the CP overhead.

Fig. 3. Illustration of the OFDM channel equalization with CP.

9


Fig. 4 depicts the block diagram of a general CO-OFDM system. At the transmitter,

the input data bit sequence is first mapped to modulated symbols. Then training symbols

(TSs) and pilot subcarriers (PSs) are inserted before the IFFT. CP is inserted for each

symbol in order to combat the inter-symbol interference. At the receiver, after sampling

by the analog-to-digital converters (ADCs) with an oversampling factor typically from

1.2 to 1.6, CP is removed, and the data symbols on each subcarrier are obtained by the

FFT operation. The channel estimation is then implemented using a 22 multiple-input

and multiple-output (MIMO) processing to compensate for the inter-symbol interference

caused by CD and PMD. The phase information acquired from the PSs is used to

compensate for the common phase error (CPE). Finally, the recovered symbols are

decided on and decoded to obtain the bit sequence. Note that there are other required DSP

procedures such as the time synchronization, frequency offset compensation, and clock

recovery, which are out of the scope of this work and will not be discussed further.

Fig. 4. Block diagram of CO-OFDM system.

2.2.2 Reduced-Guard-Interval (RGI) CO-OFDM Systems

CD is one of the major fiber channel effects. The CD parameter of a standard single mode

fiber (SSMF) is typically 17 ps/nmkm. For a 28 Gbaud signal with a 1500 km

transmission distance, the CD length is 5.7 ns. As introduced in the previous subsection,

the CP length should be larger than the CD length in order to prevent the inter-symbol

interference. Thus the OFDM symbol duration has to be larger than 57 ns to make the CP

overhead less than 10%. Such a symbol duration corresponds to 1596 subcarriers. With

the increasing of either the baud rate or the transmission distance, the symbol duration

has to be longer for a fixed CP overhead. There are several issues with large symbol

durations: 1) FFT/IFFT with such a large size is complicated to implement in real-time

systems, 2) long symbol duration suffers from a more severe laser phase noise induced

Receiver Transmitter

Optical Channel Data

X-pol

Y-pol

Data

X-pol

Ser

ial

to P

aral

lel

TS

/PS

in

sert

ion

IFF

T

Sy

mb

ol

Map

pin

g

CP

In

sert

ion

DA

C

DA

C

DA

C

DA

C

IQ Mod

IQ Mod

Co

her

ent

Det

ecti

on

AD

C

AD

C

AD

C

AD

C

FF

T

Ch

ann

el E

stim

atio

n

Ser

ial

to P

aral

lel

Ph

ase

Est

imat

ion

Par

alle

l to

Ser

ial

CP

Rem

ov

al

Par

alle

l to

Ser

ial

Sy

mb

ol

Map

pin

g

Y-pol

10


inter-carrier interference [17], and, 3) the PAPR increases as the number of subcarrier

increases, leading to a reduced fiber nonlinearity tolerance [18].

Transmitted OFDM symbol with CP

Time

Fre

qu

en

cy

Time

Fre

qu

en

cy After transmission

Time

Fre

qu

en

cy After CD compensation( )c ( )d

( )a( )b

Receiver

Data

X-pol

Y-pol

Co

her

ent

Det

ect

ion

AD

C

AD

C

AD

C

AD

C

FF

T

Ch

ann

el E

stim

atio

n

Ser

ial

to P

aral

lel

Ph

ase

Est

imat

ion

Pre

fix

Rem

ov

al

Par

alle

l to

Ser

ial

Sy

mb

ol

Map

pin

g

CD

Eq

ual

izer

Fig. 5. (a) Block diagram of RGI CO-OFDM; (b), (c), and (d): illustration the CD compensation for RGI

CO-OFDM signals.

RGI CO-OFDM was proposed to decouple the CD compensation from the

conventional OFDM channel estimation, in order to reduce the CP overhead as well as

the symbol duration [6]. RGI CO-OFDM has the same transmitter setup as conventional

OFDM, but the DSP procedures at the receiver are different; the RGI CO-OFDM receiver

is shown in Fig. 5 (a). In particular, CD is compensated by a FDE prior to the OFDM

demodulation, and the OFDM channel estimation is used to only compensate for the

residual inter-symbol interference such as PMD and the filtering effect. Consequently, the

CP length is independent of the CD length and thus can be much shorter. As illustrated in

Fig. 5 (b), (c), and (d), the CD induced walk-off is compensated by the CD equalizer, and

therefore the symbol duration can be much shorter while still being able to reduce the CP

overhead. Although the computational complexity of RGI CO-OFDM becomes larger

than conventional CO-OFDM due to the two-stage dispersion compensation, it is almost

the same as that of single carrier systems employing a similar two-stage dispersion

compensation [19]. More importantly, RGI CO-OFDM is more tolerant to fiber

nonlinearities and laser phase noise compared to conventional CO-OFDM because of the

reduced PAPR and symbol duration, while preserving its advantages such as high spectral

efficiency.

11


2.3 Fiber Channel Impairments and Digital Signal Processing

2.3.1 Dispersion in Single Mode Fiber

CD and PMD are the major linear impairments in single mode fibers, and should be

carefully handled when designing transmission systems. CD induces wavelength

(frequency) dependent group velocities, i.e. different spectral components transmit at

different group velocities. Therefore, it is also referred to as group-velocity dispersion.

PMD results from the randomly varying fiber birefringence, which leads to a pulse

broadening and the varying state of polarization. For the previous on-off keying (OOK)

system, CD and PMD both broaden the pulse width and increase the crosstalk between

adjacent pulses, therefore limiting the achievable data rate. Moreover, bulky and high loss

DCFs are deployed in each span in order to compensate for CD. However, in coherent

systems, CD and PMD can be easily compensated using DSP, so DCF can be removed

from the link to save cost and simplify the system design.

2.3.1.1 Chromatic Dispersion

The group velocity for a specific spectral component is defined as

1

g

dv

d

(2.1)

where is the propagation constant, which is related to the index n, frequency , and

light speed in vacuum c, as n c . After travelling through a single-mode fiber of

length L, the group delay g can be thus obtained by

g

g

L dL

v d

(2.2)

Then the group delay difference (or the pulse broadening) g can be calculated as

2

22

g

g

d dL L LD

d

(2.3)

where 2 22 d d is known as the group-velocity dispersion parameter. The transfer

function of the CD effect is given by

22

1

2( )j L

CDH e

(2.4)

12


It is more customary in optical communication systems to use the dispersion

parameter D [ps/nmkm] instead of 2 because it is related to the wavelength difference

. D is defined as:

2

22 2 2

2 2c d cD

d

(2.5)

It can also be expressed as the sum of the material dispersion DM and the waveguide

dispersion DW:

2 2

2 2 2 2 2 2

2 1 2 2 2, M W M W

g

c d dn d n dn d nD D D D D

d v d d d d

(2.6)

The material dispersion occurs because the refractive index of the fiber core material n, is

a function of the wavelength , while the waveguide dispersion arises due to the fact that

the propagation constant is dependent on fiber parameters such as the core radius and

the refractive index difference between the fiber core and cladding materials. Normally,

D ranges from 15 to 18 ps/(kmnm) for the single-mode fiber near 1.55 um, which is of

considerable interest for optical communications due to the benefit of low loss.

Time

Am

plit

ud

e

Time

Fre

qu

en

cy

( )a ( )b

transmittertransmitter

receiver

receiver

Fig. 6. The effect of CD on the transmitted signal of (a) single carrier systems for one pulse and (b) OFDM

systems for one symbol including many subcarriers.

CD influences single carrier and OFDM systems in different ways, as is illustrated in

Fig. 6. For the single carrier signal, since each symbol (or pulse) contains all the spectral

components, CD will broaden the pulse as different spectral components travels at

different speeds. For the OFDM signal, because the spectrum of each subcarrier is very

narrow the pulse broadening for each subcarrier is negligible. But CD will cause a

frequency-dependent walk-off between different subcarriers within each symbol, leading

to an inter-symbol interference if CP is not inserted.

13


2.3.1.2 Polarization Mode Dispersion

Due to the departure from the perfect cylindrical core with a uniform diameter, real

single-mode fibers acquire birefringence, the degree of which is characterized by the

index difference of orthogonally polarized modes m x yB n n , where nx and ny denote

the indices of the orthogonally polarized modes. The birefringence results in a periodic

power exchange between the two modes. The period is known as beat length and is given

by

B

m

LB

(2.7)

Unless a linearly polarized light is polarized along one of the principle axes, in which

case it will remain linearly polarized during the transmission, state of polarization of the

light will periodically change from linear to elliptical and then back to linear over the

length LB.

Another important phenomenon associated with PMD is the differential group delay

(DGD). In particular, for a constant birefringence, the components of different

polarizations travel at different speeds due to the different group velocities. The

difference of the arrival time at the output of the fiber is referred to as DGD, and can be

calculated by

1 2 1x y

gx gy

L LL L

v v (2.8)

where x and y represent the two principle states of polarization and 1 is related to the

group velocity difference between the two states.

Different Fiber Sections

Birefringence Axes

Fig. 7. Illustration of a PMD model in conventional fibers.

For conventional fibers, however, because the birefringence randomly varies along the

fiber as well as with time, PMD is much more complex to analyze and deal with. One

14


way to model the PMD in conventional fibers is to divide the fiber into a large number of

sections. The birefringence changes from section to section and the birefringence within

each section remains the same as shown in Fig. 7.

For non-coherent systems, such as OOK and differential phase shift keying (DPSK)

systems, the state of polarization is not of concern since the photodiode used at the

receiver to detect the light power is insensitive to the polarization state. However, the

PMD-induced pulse broadening is a limiting factor for such systems, especially for long

distances and high bit rates. Since PMD is a random process, its effect on pulse

broadening is characterized by the root-mean-square (RMS) value of the DGD, which is

typically in the range of 0.01-10 ps/(km)1/2

and

15


temperature and vibrations, the coefficients of the equalizer should be adaptively updated

over time using adaptive algorithms such as the constant modulus algorithm. On the other

hand, in dispersion-unmanaged (without DCF) systems, the accumulated CD is normally

very large, requiring a large number of coefficients. Therefore, it would not be cost-

effective if all the coefficients were updated every time, since most of them remain the

same because CD is static. Consequently, it is customary to implement the equalizer with

two stages for single carrier systems as plotted in Fig. 9(a). The first stage employs an

overlapped frequency domain equalizer (OFDE) to compensate the static CD using the

inverse transfer function CDH , while the second stage uses a butterfly time domain

equalizer applied to adaptively compensate PMD. For OFDM, two-stage equalization as

shown in Fig. 9(b), is also preferred for different reasons, e.g. reduced overhead and

higher tolerance to fiber nonlinearities and laser phase noise as already introduced in

Section 2.2.

IFF

T

Fiber

Channel

FF

T

IFF

T

FF

T

C DH

FF

T

IFF

T

FF

T

C DH

B

C

A

IFF

T

OFDE

Fil

ter

Fiber

Channel

FF

T

IFF

T

Fil

ter

C DH

FF

T

IFF

T

C DH

b

c

a

dFil

ter

OFDE

Fil

ter

FDE

TDE

( )a

( )b

D

Fig. 9. (a) single carrier systems and (b) OFDM systems with two-stage dispersion compensation. TDE:

time domain equalizer.

In order to equalize the channel, it is crucial to estimate the channel transfer function.

Single carrier systems use blind adaptive algorithms, such as the constant modulus

algorithm and the least mean square algorithm, to converge to the inverse channel

transfer function. OFDM systems normally use the TS-based channel estimation. For

16


PDM optical transmission, single-polarization (SP) TS can be transmitted and processed

at the receiver to estimate the channel [5]. In particular, as shown in Fig. 10, we first

transmit only one TS in one polarization and in the next symbol time slot we transmit

only one TS in the other polarization, so that the TSs pair occupies two symbol time

slots and can be expressed as

1 2[ ,0] , [0, ]T T

x yt t t t (2.9)

Then, with the received TSs given by

1 1 1 2 2 2[ , ] , [ , ]T T

x y x yt t t t t t (2.10)

the channel transfer function can be estimated by

1 2

1 2

' ( ) / ( ) ' ( ) / ( )( ) ( )

' ( ) / ( ) ' ( ) / ( )( ) ( )

x x x y

y x y y

t k t k t k t ka k b k

t k t k t k t kc k d k

(2.11)

where k is the index of the subcarrier.

yt

xt

freq

uen

cy

y-pol

x-pol

time

Fig. 10. The illustration of single-polarization TS

2.3.2 Laser Phase Noise

2.3.2.1 Impact of Laser Phase Noise

Laser phase noise is caused by spontaneous emission, which broadens the spectral

linewidth of the laser output. For coherent signals, the broadened laser linewidth denoted

as applies a phase shift from both the transmit laser denoted as ( )t t , and the LO laser

denoted as ( )r t , to the optical signal as plotted in Fig. 11(a). The evolution of the laser

phase noise ( )t is shown in Fig. 11(b) and is modeled as a Wiener process:

0( ) 2 ( )

t

t n d (2.12)

where n(v) is a Gaussian variable with a zero mean and a variance of /(2).

17


t

()t

( )rj te( )tj te

( )s t ( )r t

( )a ( )b

Fig. 11. (a) Model of the laser phase noise applied to the signal. (b) Illustration of the evolution of laser

phase noise.

Laser phase noise is not an issue for non-coherent systems such as OOK and DPSK,

because they either utilize no phase information or detect the phase difference between

adjacent symbols where the laser phase noise is quasi-static. However, for coherent

detections, since the optical signal is linearly mapped to the electrical signal, the detected

signal r(t) becomes

( ) ( )( ) ( ) t r

j t tr t s t e

(2.13)

The impact of such phase noise is different for single carrier and OFDM systems. Since

the single carrier symbol is very short, the phase noise over the symbol duration remains

the same. However, for different symbols there will be a phase shift, thus we will observe

constellations with random phase shifts as shown in Fig. 12(b) with respect to the original

QPSK constellation as shown in Fig. 12(a). It is impossible to decide the received symbol

unless the carrier phase is recovered. Therefore, carrier phase recovery is an

indispensable procedure for single carrier systems. As for OFDM, over the much longer

symbol duration the non-negligible phase fluctuations partially destroy the orthogonality

of the subcarriers. Consequently, each subcarrier experiences the interference from the

other subcarriers; a phenomenon known as inter-carrier interference (ICI). As expected,

we observe not only the phase shift but also the amplitude noise induced by ICI in

Fig. 12(c). Therefore, OFDM is considered to be more sensitive to laser phase noise than

single carrier systems. In Section IV, we will show that employing RGI CO-OFDM

systems can significantly reduce ICI, but another phenomenon called DEPN arises and

should be carefully approached in order to increase the linewidth tolerance.

18


( )a ( )b ( )c

Fig. 12. Constellation of (a) transmitted QPSK symbols, (b) single carrier symbols applied with laser

phase noise, and (c) OFDM subcarriers applied with laser phase noise.

2.3.2.2 Carrier Phase Recovery and Phase Estimation

For coherent single carrier systems, the procedure to approach laser phase noise is called

carrier phase recovery. Traditionally, either an analog or digital phase-locked loop is used

to synchronize the phase of the LO laser with the transmit laser for the purpose of phase

recovery. However, its performance in high data rate optical transmission systems is

limited by the propagation or implementation delay, which is usually very large due to

the massive parallelization and pipelining in such high-speed, real-time circuits at the

receiver. Therefore, feedforward algorithms are preferred, such as the Mth

power scheme

[20], also known as Viterbi and Viterbi algorithm, and blind phase search [21]. Next, we

will briefly introduce the Mth

power scheme as an example of carrier phase recovery

algorithms.

[ ]r n

( )M LPF

1arg( )

M

Phase

Unwrap

[ ]j ne

[ ]r n

[ ]S n

Fig. 13. Block diagram of the Mth

power scheme

The Mth

power scheme applies to M-ary PSK modulations including QPSK. The block

diagram is plotted in Fig. 13. By omitting any noise interference, the received M-ary PSK

symbol r[n] can be expressed by

2[ ]

[ ] , 0,1,..., 1

mj n

Mr n e m M

(2.14)

19


where [ ]n is the combined phase shift of both the transmit laser and LO laser on each

symbol. By raising the symbol to the Mth

power, the modulation can be removed as

2

[ ]2 [ ] [ ][ ]

mj n M

M j m jM n jM nMr n e e e e

(2.15)

Then, by taking the argument and dividing the result by 1/M, the phase shift [ ]n is

obtained. Since the noise interference is unavoidable, a low-pass filter (LPF) is required

to remove the noise in order to estimate a more accurate phase. In addition, the argument

function only provides the wrapped phase angle in the range of to , so the estimated

phase is limited from /M to /M. Therefore, phase unwrap is needed to extend the

range of the estimated phase. After that, the obtained phase is applied to the received

symbol to complete the process of the carrier phase recovery.

The situation for OFDM systems is much simpler in terms of phase estimation.

Traditionally, it is considered that subcarriers within each OFDM symbol experience

exactly the same phase drift because they are overlapped and travel simultaneously in the

time domain. This common phase drift is called CPE, and can be estimated by sending

PSs. Basically, we allocate several PSs that are known to the receiver in the transmitted

symbols as shown in Fig. 14. Then, at the receiver, the phase of data subcarriers can be

easily estimated from the phase of PSs because they have been acted on by the same

phase noise. Averaging over the phase of PSs is required in order to remove the noise

interference.

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

Frequency

pilot subcarriers

...

data subcarriers

Po

we

r

Fig. 14. Illustration of the PSs allocation.

Although the computational complexity of the PS based CPE estimation is very low, it

induces extra overhead, as these subcarriers are not transmitting data. More importantly,

this method is unable to compensate for ICI, which limits the laser linewidth tolerance for

conventional CO-OFDM systems. Another phase estimation scheme employing the RF-

pilot tone was proposed to combat ICI [22]. Fig. 15 plots the spectrum of an OFDM

20


signal with a RF-pilot inserted in the middle. The subcarriers in the middle should be

turned off to avoid interference with RF-pilot. Therefore the RF-pilot based scheme also

consumes some spectrum space, similar as that of the PS based scheme. The RF-pilot

tone will collect the same phase noise as the OFDM symbols; therefore, at the receiver,

the phase noise can be compensated for by inverting the phase of the RF-pilot and

multiplying it with the signal in the time domain. As shown in Fig. 15(b), since the phase

is compensated sample by sample instead of symbol by symbol which is the case in the

PS based CPE compensation, the ICI can be significantly reduced [23]. The details of the

RF-pilot phase compensation will be discussed in Chapter 4.

f

RF-pilotOFDM spectrum

( )a ( )b

Fig. 15. (a) The spectrum of the OFDM signal with a RF pilot tone. (b) The estimated phase from the

RF-pilot based scheme and the CPE based scheme.

2.3.3 Fiber Nonlinearities

2.3.3.1 Impact of Fiber Nonlinearities

Fiber nonlinearities are considered to be the main factor limiting the achievable fiber

channel capacity [24]. There are two major groups of fiber nonlinearities. The first group

is related to the Kerr effect, which is attributed to the dependence of the refractive index

on the light intensity. The second group includes the stimulated Raman scattering and

stimulated Brillouin scattering. In this section, we will focus on the nonlinearities in the

first group including self-phase modulation (SPM), cross-phase modulation (XPM) and

four-wave mixing (FWM), since these nonlinearities dominate over the long-haul

transmission.

We first introduce the well-known nonlinear Schrdinger (NLS) equation, which

describes the propagation of the optical signal in fiber and has the following form:

21


222

22 2

iA AA i A A

z T

(2.16)

where A is the complex amplitude of the field envelope, which is a function of the

distance z and the time T. Also, 2 is the group velocity dispersion parameter, is the

attenuation coefficient, and is the nonlinear coefficient given by

22

eff

n

A

(2.17)

where n2 denotes the nonlinear Kerr coefficient and Aeff denotes the effective fiber core

area. Note that the third-order dispersion is neglected in Eq. (2.16), since it is negligible

in practice.

SPM occurs in one WDM channel as a result of its own power. To show the SPM

effect, we first neglect the fiber dispersion and attenuation, and the NLS equation (2.16)

has a solution as follows

2exp( )A A jz A (2.18)

Basically, SPM applies a phase shift 2

NL z A to the signal in the time domain, which

is proportional to the power of the signal itself. However, in practice it is not as simple as

this because the dispersion, as well as the attenuation, will change the envelope of the

signal and subsequently change the nonlinear phase shift.

XPM occurs in one WDM channel as a result of other channels power. Thus in

multichannel systems, one specific channel experiences not only the nonlinear phase shift

induced by its own power, but also by the power of the other channels. This can be

expressed as

2 2

,

1,

2N

NL n n i

i i n

z A A

(2.19)

where n denotes the specific channel, and N is the total number of channels. It is seen that

the XPM is much more severe than SPM, as shown in Eq. (2.19). Fortunately, the

dispersion-induced walk-off limits the efficiency of XPM due to the field averaging.

Therefore, only close channels interact with each other by means of XPM.

FWM is another nonlinear phenomenon in multichannel transmissions. In particular,

FWM generates a new optical wave at the frequency ijk i j kf f f f , when three

22


signals with frequencies fi, fj and fk co-propagate inside the fiber, provided that the phase

matching condition ijk i j k is satisfied. In quantum-mechanical terms, FWM

occurs while photons are destroyed and new photons are created at different frequencies

such that net energy and momentum conservation is satisfied. FWM will have significant

impacts as the newly generated frequencies will coincide with existing channels and

distort the transmitting signals. Also, the dispersion-induced walk-off will prevent the

phase-matching condition and thus reduce the efficiency of FWM.

2.3.3.2 Compensation and Mitigation of Fiber Nonlinearities

Fiber nonlinearities compensation in optical long-haul transmissions has been a hot topic

for decades, especially for coherent systems where both the amplitude and phase

information is accessible at the receiver. One well-known and effective method is

backpropagation. Before we talk about this algorithm, we first introduce the split-step

Fourier method (SSFM), which is used to numerically solve the NLS equation. In order

to do so, we rewrite the NLS equation as

A

D N Az

(2.20)

where D accounts for the CD and attenuation and N accounts for the fiber

nonlinearities. If we neglect high order dispersions, they are given by

2

2

2

2 2

iD

T

(2.21)

2N i A (2.22)

The dispersion and nonlinearity interact with each other along the fiber, and thus the

NLS equation does not generally lend itself to analytical solutions. The principle of the

SSFM is to assume that over a small distance L the dispersive and nonlinear effects act

independently as

( ) exp( )exp( ) ( , )A z L hD hN A z T (2.23)

As introduced in previous sections, the dispersive effect can easily be calculated in the

frequency domain while the nonlinear effect can easily be calculated in the time domain.

Therefore, in the first step we calculate only the dispersion in the frequency domain,

23


assuming there is no nonlinearity. After that, we calculate the nonlinearity in the time

domain neglecting the dispersion. Fig. 16 illustrates the process of SSMF.

...D N

L

D N D N D N

Fiber Link

Fig. 16. Illustration of using the SSFM to model the signal propagation in a fiber link.

The principle of backpropagation is to use the SSFM to build an inverse fiber

channel, which has the opposite effect of the dispersion and nonlinearity. Ideally, fiber

nonlinearities and dispersions will be fully compensated for, provided that the parameters

of the fiber link such as the fiber length, the dispersion parameter and the nonlinear

coefficient, are known [25]. The effectiveness of the backpropagation has been

demonstrated through many works, e.g. [25-28]. However, it is quite difficult to

implement this method in real systems due to its extremely large computational

complexities. Still, efforts have been made to reduce the complexity of this algorithm

[29]. Another drawback of the backpropagation algorithm is that it is not able to

compensate for inter-channel nonlinearities including XPM and FWM, since in WDM

transmissions the information of the transmitted symbols of other channels are not

available to a certain given channel.

In addition to the backpropagation, other approaches have also been investigated to

compensate or mitigate fiber nonlinearities. For example, it has been demonstrated that

the pulse shape of single carrier signals can be optimized to improve the nonlinear

tolerance [30]. The RF-pilot tone has also been employed to partially compensate for

both intra- and inter-channel nonlinearities for both OFDM and single carrier systems

[31-33]. Furthermore, fiber nonlinearities can be compensated using Volterra filters [34].

2.4 Conclusions

In this chapter, the relevant background and literature have been introduced and reviewed.

We first discussed coherent optical communication systems, which currently dominate in

both academic research and industrial products. RGI CO-OFDM is one of the potential

candidates for next generation 100G beyond transports. We reviewed the principle of

OFDM systems, and showed that compared to conventional CO-OFDM systems,

24


RGI CO-OFDM benefits from lower CP overhead and higher tolerance to optical channel

impairment. Afterwards, we discussed the main optical channel impairments including

fiber dispersion, laser phase noise and fiber nonlinearities. The general DSP procedures

in coherent transmissions such as dispersion compensation, carrier phase recovery and

nonlinearity compensation were also briefly presented.

25

Chapter 3: Channel Equalization for RGI CO-OFDM

Equation Section (Next)

Chapter 3

Channel Equalization for RGI CO-OFDM

In this chapter, we focus on the channel estimation and equalization of RGI CO-OFDM

systems. In Section 3.1, we propose a novel equalization scheme, which uses the OFDE

to compensate all inter-symbol interference, including CD and PMD [35, 36]. By doing

so, the CP can be removed from the data symbols and the CP overhead is therefore

reduced to zero. The details of the proposed equalization scheme are described and its

performance demonstrated through simulations. Afterwards, its complexity in terms of

operations is discussed. In Section 3.2, we analyze the impact of intra-channel

nonlinearities on the design of TSs [18]. We show that the correlated dual-polarization

(CDP) TSs should be used for RGI CO-OFDM rather than conventional SP TSs when

considering the CD-induced walk-off and the intra-channel nonlinearities. Moreover, we

demonstrate the improvement of RGI CO-OFDM systems in the tolerance to intra-

channel nonlinearities compared to conventional CO-OFDM systems when CDP TSs are

employed.

3.1 Zero-Guard-Interval CO-OFDM

As already introduced in Chapter 2, using RGI CO-OFDM can significantly reduce the

required CP length, enabling to reduce the CP overhead as well as the FFT size. However,

CP is still required to accommodate the residual linear effects such as PMD and filtering

effects. In this section, we propose a novel equalization scheme to completely remove CP

from data symbols [35, 36]. We first describe the operation principle and the algorithm of

the new equalization scheme which is based on the structure of RGI CO-OFDM systems,

namely an OFDE for dispersion compensation prior to OFDM demodulation. In the new

26


scheme, instead of compensating CD only, the first-stage OFDE simultaneously

compensates CD and the residual dispersion (for the sake of simplicity, we assume the

residual dispersion includes PMD only in this work). It offers the key to the zero CP

transmission, and this system is therefore referred to as ZGI CO-OFDM. Basically, the

channel transfer function estimated from the OFDM channel estimation is applied at the

OFDE stage, which then becomes able to compensate both CD and PMD. But a

frequency domain interpolation (FDI) is required, because the FFT size used in the OFDE,

NOFDE, is usually larger than that used for OFDM symbols, NFFT. We also show the

principle of the FDI implementation. In Subsection 3.1.2, we demonstrate the system

performance of a 112 Gb/s PDM ZGI CO-OFDM. Besides removing CP from data

symbols, the new scheme provides additional system benefits. We show that it achieves a

larger PMD tolerance compared to RGI CO-OFDM. Moreover, an even smaller FFT size

NFFT (e.g. 16 and 32) can be used while CP overhead is still zero. However, this

improvement in reducing the overhead comes with a tradeoff between the CP overhead

and the computation complexity of the algorithm when compared to the conventional and

RGI CO-OFDM. So in Subsection 3.1.3, we provide an analytical comparison of the

computational complexity between the conventional, RGI and ZGI CO-OFDM. We show

that ZGI CO-OFDM requires reasonably small additional computation effort compared to

RGI CO-OFDM, while providing several system benefits.

3.1.1 Channel Equalization Algorithm Description

Fig. 17 plots the CO-OFDM receiver structure (based on the structure of RGI CO-OFDM)

with the proposed equalization scheme. The OFDM transmitter and fiber link are the

same as those in the conventional OFDM. Besides CD compensation, one key feature of

the OFDE in our new scheme is to acquire the channel transfer function (in form of a 2-

by-2 matrix H[k] for each kth

modulated subcarrier) from the OFDM channel estimation

based on TSs, and then to compensate PMD by applying the inverse of the channel

matrix. Therefore, with both CD and PMD compensated at the OFDE, subsequent data

symbols dont require CP for equalization at the receiver. Fig. 17(b) shows the OFDM

frame used with the new equalization scheme. Different from the typical ones used in the

conventional or RGI CO-OFDM, in which the same CP is encapsulated in both TSs and

data symbols, ZGI CO-OFDM requires a small CP for TSs in order to obtain accurate

27


channel estimation while no CP is allocated for data symbols. When compared to

RGI CO-OFDM in [6], where CP with length NCP=4 was used for all TSs and data

symbols, in our system NCP=4 is only used for each TS. It is worth noting that a very

short CP length (for example, NCP=1) can be still inserted to each data symbol, so our

system would become more robust against any residual inter-symbol interference from

the imperfect channel equalization at the OFDE. However, a small CP overhead will be

required as a tradeoff.

Fig. 17. (a) CO-OFDM receiver structure and (b) OFDM frame. S/P: serial to parallel

A. Two-stage equalization algorithm

The algorithm is described step-by-step as follows:

Step 1: The OFDE compensates CD for the incoming TSs (t1 and t2) and

produces the output t1 and t2. Note that t1 and t2 are received from the x and y

polarizations, respectively. The CD compensation is done by simply multiplying the

incoming signals with the inverse of the CD transfer function HCD,

2 ( /2) /[ ] , 1, ,OFDE OFDE nOFDE OFDE OFDECD

jDL k N f cH k e k N (3.1)

where D is the dispersion parameter, L is the fiber transmission distance, is the

optical carrier wavelength in vacuum, nf is the frequency spacing in the OFDE, and

(b)

...

300 data symbols4 training symbols (TSs)

NCP = 4 NCP = 0

Dual-

polarization

optical hybrid

ADC

Synchro

niz

ation

S/P

OF

DE

FF

T &

IF

FT

=2048

FF

T=

128

Channel estim

ation

with I

SF

A

Channel com

pe

nsa

tion

Sym

bol

mappin

g

ADC

ADC

ADC

Sym

bol

mappin

g

P/S

P

/S

56Gb/s

(x-pol)

56Gb/s

(y-pol)

OFDM demodulator

(a)

Ix

Qx

Qy

Iy

)()(

)()()(

kdkc

kbkakH

Optical

signal

28


c is the speed of light in vacuum. The FFT size used in the OFDE, NOFDE, is often

larger than that used for OFDM symbols, NFFT, thus nf is smaller than the subcarrier

spacing. kOFDE is the frequency index at the OFDE.

Step 2: At the OFDM demodulator, estimate the channel matrix H[k], which only

includes PMD, for each of the Nsc modulated subcarriers. The intra-symbol

frequency-domain averaging (ISFA) can be applied, but the averaging length m is

limited by the amount of the uncompensated PMD [37]. For the kth

modulated

subcarrier, the estimated channel matrix is

[ ] [ ]

[ ] , 1, ,[ ] [ ]

a k b kH k k Nsc

c k c k

(3.2)

Step 3: The OFDE acquires H from the OFDM demodulator, and performs the

frequency-domain interpolation (FDI) to map H to HFDI. H represents Nc 2-by-2

matrices, while the number of 2-by-2 matrices in HFDI is equal to ( / )SC OFDE FFTN N N ,

and NOFDE is usually larger than NFFT.

[ '] [ ']

[ '] , ' 1, , ( / )[ '] [ ']

FDI FDI

FDI SC OFDE FFT

FDI FDI

a k b kH k k N N N

c k c k

(3.3)

Step 4: At the OFDE, multiply t1 and t2 by HFDI-1

to produce t1 and t2. In

essence, this is a multiple-input-multiple-output (MIMO) demodulation process,

which compensates DGD and de-multiplexes the two polarizations. In the

conventional and RGI CO-OFDM, however, this MIMO demodulation occurs at the

OFDM demodulator instead of at the OFDE stage. This step is the key to enable our

system to operate with zero CP overhead.

Step 5: At the OFDM demodulator, estimate the new channel matrix H[k] for

each of the Nsc subcarriers now using t1 and t2. Same as H, H represents Nsc

2-by-2 channel matrices. Now a large ISFA length m can be used to improve the

estimation accuracy of H, because both CD and PMD have been compensated. H

will be saved and used to demodulate the subsequent data symbols. Subsequently, we

can perform phase noise estimation and compensation after applying H to the data

symbols.

Unlike the channel estimation/compensation used for the conventional and RGI CO-

OFDM, our new scheme relies on a collaborative effort between the OFDE and OFDM

29


channel estimation, and it requires processing the same TS and performing channel

estimation twice. To understand our algorithm from a mathematical standpoint, we

express each of the Nsc demodulated OFDM subcarriers rout obtained after Step 5 as,

1 1 1

[ ] '[ ] [ ]out cd FDI ink H H H k kr r (3.4)

In Eq. (3.4) rin[k] is the kth

received OFDM subcarrier at the OFDE input. Both rout and

rin are the 2-by-1 column vectors, representing symbols on both x- and y-polarization.

1 1( )cd FDIH H

is applied to the received signal at the OFDE, while

1'[ ]H k

is multiplied at

the OFDM demodulator. Also1 1

( )cd FDIH H is a function of NOFDE frequency points, and

we dont express it as function of k, because there is a lack of the explicit one-to-one

correspondence between n and k. Note that the term for phase noise compensation is not

included in Eq. (3.4), since it is the same process as that for the conventional and RGI

CO-OFDM.

In comparison with ZGI CO-OFDM, RGI CO-OFDM uses a relatively straightforward

two-stage equalization process but it has to incorporate a larger CP. In RGI CO-OFDM

the OFDE only compensates CD, while the channel compensation for PMD, polarization

de-multiplexing and phase noise estimation are performed at the OFDM demodulator.

Mathematically, we can express the equalization process of RGI CO-OFM as,

1 1

[ ] [ ] [ ]out cd RGI ink H H k kr r (3.5)

All the channel equalization operations at the OFDM demodulator are lumped in

1[ ]RGIH k

, which is obtained using the TS based channel estimation. Finally, the channel

equalization for the conventional CO-OFDM is even more straightforward, which can be

simply expressed as

1

[ ] [ ] [ ]out CONV ink H k kr r (3.6)

But this system must have a very large CP to avoid the inter-symbol interference

mainly caused by CD. Comparing Eq. (3.4), (3.5) and (3.6), a tradeoff can be observed

between the equalization algorithm complexity and CP overhead. We will compare the

computation complexity of the three CO-OFDM algorithms analytically in

Subsection 3.1.3.

30


B. Frequency domain interpolation (FDI)

As mentioned earlier ZGI CO-OFDM requires FDI to map H to HFDI at Step 3, due to the

mismatch between NOFDE and NFFT. To illustrate FDI implementation, we use a

ZGI CO-OFDM system with NOFDE=2048 and NFFT=128, so the interpolation factor is

NOFDE/NFFT=16. In Fig. 18(a) the open circles are used to indicate the channel matrix H,

and the curve connecting the open circles is the result from FDI. In this particular

example, we assume each OFDM symbol contains 64 modulated subcarriers occupying

the center half of the NFFT=128 window, so that the oversampling factor at the DAC/ADC

is 2. Fig. 18(b) illustrates the spectrum of the real part of the x-pol OFDM signal before

and after applying HFDI- 1

at Step 4, where we assume that the fiber link has a

deterministic DGD of 320 ps. We show only the real part of the signal in order to

highlight the difference in spectrum before and after applying HFDI- 1

, because HFDI- 1

only

imparts a phase rotation given our assumption of a deterministic DGD.

Fig. 18. (a) Schematic of FDI. (b) OFDM spectrum before and after applying HFDI- 1

.The top curve shows

the real part of a in the channel matrix H (open dot) and the interpolated channel matrix HFDI (thin line).

Phase variations across modulated subcarriers before and after PMD compensation in Step (4) for a (c)

deterministic DGD=320ps and (d) stochastic PMD with =100ps.

)1(H )2(H )62(H )63(H)3(H )64(H

Interpolation factor = FFTOFDE / FFTOFDM (a)

...

0 256 512 768 1024 1280 1536 1792 2048-150

-100

-50

0

50

100

FFT index at OFDE

Am

plit

ud

e (

a.u

.)

real(t1')

real(t1'')

real(a)

(b)

'

'

''

''

2

11

2

1

t

tH

t

tFDI

0 8 16 24 32 40 48 56 64-15

-10

-5

0

5

10

15

Subcarrier Index

Phase (

Radia

n)

0 8 16 24 32 40 48 56 64-40

-20

0

20

40

Subcarrier Index

Phase (

radia

n)

x-pol w/o PMD EQ

y-pol w/o PMD EQ

x-pol w/ PMD EQ

y-pol w/ PMD EQ

(c) (d)DGD=320ps =100ps

( )c ( )d

31


Moreover, the OFDM signal is confined in the center half of the OFDE FFT window

as expected from only modulating 64 middle subcarriers, and HFDI- 1

is only multiplied by

the OFDM signal within this center half of the spectrum. On the top of the signal

spectrum, the curve shows the real part of aFDI (see Eq. (3.3)), and the open circles

connected by the curve indicate the real part of a from 64 channel matrices H (see

Eq. (3.2)). It is worth noting that the periodicity of the curve is inversely proportional to

DGD, and it would become irregular when considering higher-order PMD [38].

Therefore, in order to compensate a large DGD for which the periodicity will become

small, one must increase TSs NFFT to ensure a fine frequency resolution before FDI.

Otherwise, a coarse frequenc

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