Copyright

by

Kumar Appaiah

2013

e Dissertation Commiee for Kumar Appaiahcertifies that this is the approved version of the following dissertation:

Signal Processing and Incoherent-MIMO for

Multimode Optical Fibers

Commiee:

Sriram Vishwanath, Supervisor

Seth R. Bank, Supervisor

Ray T. Chen

Julian Cheng

Robert W. Heath Jr.

Todd E. Humphreys

Signal Processing and Incoherent-MIMO for

Multimode Optical Fibers

by

Kumar Appaiah, B. Te., M. Te.

DISSERTATION

Presented to the Faculty of the Graduate School of

e University of Texas at Austin

in Partial Fulfillment

of the Requirements

for the Degree of

DOCTOR OF PHILOSOPHY

THE UNIVERSITY OF TEXAS AT AUSTIN

May 2013

Dedicated to my father, sister and family.

Anowledgments

I owe several people gratitude for helping me get through graduate school.

First and foremost, I shall be perennially indebted to my advisors, Dr. Vishwanath

and Dr. Bank. Most people know Dr. Vishwanath as a theorist, his intuition and

insight on building practical communication links has been invaluable for helping

me get across the many roadblocks I met on the way. From Dr. Bank, I have learnt

how to design experiments, and on the importance of discipline, not only during

the conduct of research, but while presenting it to an audience. anks are also

due to the faculty who were on my dissertation commiee, for helping me gain a

beer perspective on my research.

In the LINC group, I am grateful to several students who have helped me

in various aspects of research. Among the senior students, Jubin Jose and Rajiv

Soundararajan were immensely helpful during the initial stages of graduate school

by guiding me on effective presentation of research. Abhik Das has always been

a great friend and collaborator who has helped me throughout grad school. Ioan-

nis Mitliagkas, Joyce Ho, Ankit Rawat, Avhishek Chaerjee, Deepjyoti Deka and

Ethan Elenberg have always been happy to hear me whenever I needed someone

to discuss some issues. I would also like to thank, Ozan Koyluoglu and Hongbo Si

for helping me with research in information theory. anks also to everyone else

who has made my time in the group worthwhile and enjoyable.

v

e LASE group members have been very accommodating in helping me

get work done in the lab, even though my research interest may not align directly

with theirs. Vaishno Dasika, Adam Crook and Hari Nair have always offered me

useful insights on understanding optics and optoelectronic devices beer. Rodolfo

Salas has helped me several times with circuit related doubts, and has gone out of

his way to help me get my experiments working. Sco Maddox's insights have

oen helped me view or present experimental data beer. I would also like to

conveymy gratitude to Erica Krivoy, not only for offering technical help whenever

I needed it, but also for patiently helping me learn official procedures, such as

equipment procurement. Most importantly, she has always been ready to listen to

my long winding rants about the ups and downs of graduate school, and helping

me lighten up and helping me get back on my feet. I also wish to thank the new

group members, namely Sco Sifferman, Kyle McNicholas, Daniel Ironside and

Nathanial Sheehan, for the numerous useful discussions I've had with them. My

collaborator Sagi Zisman has spent many an hour helping me write code as well

as debug experiments in the lab, for which I owe him gratitude.

Being a member of the Wireless Networking and Communications Group

(WNCG) at UT Austin has offered me an excellent opportunity to interact with

people working in various layers of the communication system, and has broadened

my horizon significantly. Several friends in WNCG have also been very helpful

to me throughout graduate school. I'd like to thank Omar El Ayach, Zrinka Puljiz,

SharayuMoharir, PraneethNetrapalli, Srinadh Bhojanapalli, Aneesh Reddy, Aditya

Gopalan, Shreeshankar Bodas and several others whom I have not named here.

vi

Special thanks to Siddhartha Banerjee, whom I have now known well for a decade,

for always being friendly and helpful.

My stay in Austin has been made an enjoyable one, thanks to several good

friends. In particular, I'd like to thank Aswin, Shravan, Harsha, Apurva, Ganesh,

Siva and Akarsh for the several fun times we've had.

vii

Signal Processing and Incoherent-MIMO for

Multimode Optical Fibers

Publication No.

Kumar Appaiah, Ph.D.

e University of Texas at Austin, 2013

Supervisors: Sriram VishwanathSeth R. Bank

Multimode fibers (MMF) are generally used in short and medium haul opti-

cal networks owing to the availability of low cost devices and inexpensive packag-

ing solutions. However, the performance of conventional multimode fibers is lim-

ited primarily by the presence of high modal dispersion owing to large core diame-

ters. While electronic dispersion compensation methods improve the bandwidth-

distance product of MMFs, they do not utilize the fundamental diversity present

in the different modes of the multimode fiber. is thesis draws from develop-

ments in wireless communication theory and signal processing to motivate the

use of multiple-input multiple-output (MIMO) and signal processing techniques

in MMF links. MIMO techniques that utilize the diversity of modes present in the

fiber increase data rates and link reliability. eoretical models for propagation

effects in MMF systems are used to analyze and design the geometry of laser and

detector arrays for MIMO-MMF links, and study how the design of these arrays

viii

impacts link performance. esemodels are also used to develop and evaluate low-

complexity algorithms that efficiently utilize dense detector arrays, with “greedy

subset selection” based on submodular optimization. Experimental evaluation of

1×1, 2×2, 3×3 and 4×4MIMO systems have been conducted over various MMF

media, including 100 m - 3 km silica MMF with externally modulated distributed

feedback lasers and directly modulated vertical cavity surface emiing lasers (VC-

SELs), as well as with Fabry Perot lasers over 10m - 100m plastic MMF.e use of

off-the-shelf components as well as the role of axial offset coupling in enhancing

modal diversity has been experimentally quantified. e experimental techniques

discussed in this thesis have enabled an increase of over 25× in the bandwidth-

distance product of the MMF link, when compared to currently deployed MMF

systems, such as 10GBASE-SR.

ix

Table of Contents

Anowledgments v

Abstract viii

List of Tables xiv

List of Figures xv

Chapter 1. Introduction 11.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 e Case for MIMO in Multimode Fibers . . . . . . . . . . . . . . . 31.3 Comparison with Wavelength Division Multiplexing . . . . . . . . 51.4 State-of-the-art in DSP based MMF links . . . . . . . . . . . . . . . 71.5 Problems addressed in this thesis . . . . . . . . . . . . . . . . . . . 91.6 Contributions of this thesis . . . . . . . . . . . . . . . . . . . . . . 111.7 Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

Chapter 2. Signal Processing and MIMO-MMF: Models and Abstractions 142.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.2 DSP-based MIMO Communication Systems for Optical Fibers . . . 15

2.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . 152.2.2 Benefits of MIMO . . . . . . . . . . . . . . . . . . . . . . . . 152.2.3 Digital Signal Processing for MMF Links . . . . . . . . . . . 17

2.3 Channel Model for an Optical MIMO Link . . . . . . . . . . . . . . 192.3.1 Pulse-spreading in a Multimode Fiber . . . . . . . . . . . . 202.3.2 Spatial Effects of Multiple Modes . . . . . . . . . . . . . . . 23

2.3.2.1 Parallels between Wireless and Optical MIMO . . . 242.3.2.2 Main differences . . . . . . . . . . . . . . . . . . . 28

x

2.4 Signal Processing Algorithms and Paradigms for Optical-MIMO links 292.4.1 Pilot-based estimation . . . . . . . . . . . . . . . . . . . . . 292.4.2 Reliability vs. HigherData Rates: eDiversity-Multiplexing

Tradeoff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312.4.3 Diversity and Multiplexing Schemes without channel state

feedback . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322.4.3.1 Transmit Diversity: Alamouti Code . . . . . . . . . 332.4.3.2 Receive Diversity: Maximum Ratio Combining . . 342.4.3.3 Spatial multiplexing without feedback . . . . . . . 35

2.4.4 MIMO with feedback: Beamforming and Spatial Multiplexing 362.4.4.1 Beamforming . . . . . . . . . . . . . . . . . . . . . 382.4.4.2 Feedback based spatial Multiplexing . . . . . . . . 39

2.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

Chapter 3. Experimental evaluation: 2 × 2 MIMO-MMF link with off-the-shelf components 42

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.2 MIMO-MMF Experimental Setup . . . . . . . . . . . . . . . . . . . 423.3 Modulation and Coding . . . . . . . . . . . . . . . . . . . . . . . . 453.4 Experiments: Characterization of Impulse Response and Noise . . 47

3.4.1 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.4.2 Description of the experiments . . . . . . . . . . . . . . . . 473.4.3 Observations and results . . . . . . . . . . . . . . . . . . . . 48

3.5 Experiments: Modal Diversity and Advanced Modulation . . . . . 493.5.1 Purpose . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493.5.2 Description of the Experiments . . . . . . . . . . . . . . . . 503.5.3 Observations and Results . . . . . . . . . . . . . . . . . . . 52

3.6 Experiments: Feedback and Precoding . . . . . . . . . . . . . . . . 543.6.1 Feedback and Precoding . . . . . . . . . . . . . . . . . . . . 54

3.6.1.1 Beamforming . . . . . . . . . . . . . . . . . . . . . 553.6.1.2 Spatial Multiplexing . . . . . . . . . . . . . . . . . 57

3.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 593.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

xi

Chapter 4. Analysis and Design of Laser and Detector Array Geometry 644.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 644.2 Multimode fiber model . . . . . . . . . . . . . . . . . . . . . . . . . 67

4.2.1 Propagation matrix . . . . . . . . . . . . . . . . . . . . . . . 674.3 Analysis of MIMO system matrix . . . . . . . . . . . . . . . . . . . 72

4.3.1 System transfer matrix . . . . . . . . . . . . . . . . . . . . . 724.3.2 Metrics for optimization of device configurations . . . . . . 75

4.4 Optimizing placement of devices . . . . . . . . . . . . . . . . . . . 764.4.1 Exhaustive search . . . . . . . . . . . . . . . . . . . . . . . 784.4.2 Submodular search . . . . . . . . . . . . . . . . . . . . . . . 78

4.5 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . 824.5.1 Rate benefits due to MIMO . . . . . . . . . . . . . . . . . . 844.5.2 Effect of device positions on achievable rate using exhaus-

tive search . . . . . . . . . . . . . . . . . . . . . . . . . . . . 844.5.3 Comparison of exhaustive and greedy search: coarse grids . 874.5.4 Detector arrays using fine grids and greedy search . . . . . 894.5.5 From device arrays to segmented detectors . . . . . . . . . 92

4.6 Using dense detector arrays: dynamic detector selection . . . . . . 954.6.1 Complexity of MIMO Decoding . . . . . . . . . . . . . . . . 964.6.2 Submodularity: Greedy Detector Selection . . . . . . . . . . 984.6.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . 99

4.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101

Chapter 5. Offset coupling, feedba and spatial multiplexing in a 4 × 4incoherent-MIMO multimode fiber link 103

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1035.2 Metrics for evaluating MIMO-MMF systems . . . . . . . . . . . . . 1055.3 System description . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.3.1 Optical System . . . . . . . . . . . . . . . . . . . . . . . . . 1095.3.1.1 Silica MMF with DFB lasers . . . . . . . . . . . . . 1095.3.1.2 Silica MMF with VCSELs . . . . . . . . . . . . . . . 1125.3.1.3 Plastic MMF with Fabry-Perot laser . . . . . . . . . 1135.3.1.4 Mode scramblers and fiber offsets . . . . . . . . . . 114

xii

5.3.2 Modulation and coding . . . . . . . . . . . . . . . . . . . . 1165.4 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . 120

5.4.1 4 × 4 link over Silica MMF with DFB lasers . . . . . . . . . 1245.4.2 2 × 2 link over Silica MMF with VCSELs . . . . . . . . . . . 1365.4.3 2 × 2 link with plastic MMF . . . . . . . . . . . . . . . . . . 139

5.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1415.6 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143

Chapter 6. Summary and Conclusions 145

Appendices 148

Appendix A. Submodularity of Channel Capacity 149

Bibliography 153

Index 170

Vita 172

xiii

List of Tables

3.1 Results for feeedback based schemes. . . . . . . . . . . . . . . . . 59

5.1 OFDM system parameters . . . . . . . . . . . . . . . . . . . . . . 1165.2 Optimal axial offsets for 4× 4 system . . . . . . . . . . . . . . . . 132

xiv

List of Figures

1.1 An example showing the different paths taken by an incident rayin a multimode waveguide . . . . . . . . . . . . . . . . . . . . . . 4

2.1 A schematic of the implementation of a digital signal processingfor dispersion compensation in a multimode fiber link. . . . . . . 17

2.2 A representation showing the resolved components while sam-pling the impulse response of a system . . . . . . . . . . . . . . . 22

2.3 An example of delayed copies of signals sent and received by mul-tiple antennas in wireless multipath. . . . . . . . . . . . . . . . . . 23

2.4 e correspondence between multiple transmiers and multipledetectors in the wireless and MMF cases . . . . . . . . . . . . . . 25

2.5 A schematic showing feedback based beamforming . . . . . . . . 38

3.1 Schematic of the system setup used for the experiments. . . . . . 433.2 Pilot placement and channel estimates obtained via interpolation. 473.3 Impulse response for an impulse (approximated as a 100 ps signal. 483.4 Computed and observed step response. . . . . . . . . . . . . . . . 493.5 Distribution of the observed noise. . . . . . . . . . . . . . . . . . . 503.6 Measured BER vs. SNR for various MIMO configurations: Im-

provement is observed when more transmiers or detectors areemployed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

3.7 A comparison of the BER vs. received power for the 1× 1 systemand the 2× 1 beamforming system. . . . . . . . . . . . . . . . . . 56

3.8 A comparison of average SNR of the 1×1 and beamforming (2×1)are compared for each subcarrier. . . . . . . . . . . . . . . . . . . 57

3.9 Optional caption for list of figures . . . . . . . . . . . . . . . . . . 573.10 BER vs. received power for the two spatial multiplexing streams

compared with standard 1× 1. . . . . . . . . . . . . . . . . . . . . 58

xv

4.1 Uiprop is a random matrix that describes intermodal coupling with

in a section. In particular, it transforms a vector containing theweights of each guided mode to provide a vector which has thenew weights aer the signal has undergone intermodal couplingwithin the fiber section. . . . . . . . . . . . . . . . . . . . . . . . . 70

4.2 Ri is a randommatrix that describes rotation of the polarization ofthe electric field at section junctions. It rotates the polarization ofeach mode within the fiber section based upon propagation effectsof the fiber. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.3 Mi is a random matrix that describes rotation of the electric fieldprofile due to fiber twists within the ith fiber section. . . . . . . . 70

4.4 Binning approach to to evaluate the sum rate of the fiber acrossall frequency ranges. e frequency response of the fiber channelwas split into several bins of 100MHz each, and the achievable ratewas evaluated within each bin assuming a frequency-flat channelresponse, and added up to get the net achievable rate. is can bethought of as a frequency division multiplexing approach to rateevaluation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4.5 e beam evolution over 1 km of graded-index multimode fiber.e physical fiber measurement is performed using a beam pro-filer, while the simulated profile uses one channel realization ob-tained using the fiber model. . . . . . . . . . . . . . . . . . . . . . 83

4.6 Multiple lasers and detectors. e devices were assumed to fill 90%of the fiber core area. . . . . . . . . . . . . . . . . . . . . . . . . . 85

4.7 Achievable rate versus SNR for 1 × 1, 2 × 2, and 3 × 3 MIMOsystems for the best device configuration . . . . . . . . . . . . . . 85

4.8 (a) Achievable rate versus SNR for a 2 × 2 MIMO system for the“best” device configuration that achieves the highest rate and theaverage over all possible configurations. (b) e configuration oflasers and detectors in the best configuration. e four circles atthe boom represent the fiber cross sections and the laser and de-tector placements on the cross sections that yielded both the bestand suboptimal configurations. . . . . . . . . . . . . . . . . . . . . 86

4.9 (a) Achievable rate versus SNR for a 3 × 3 MIMO system for the“best” device configuration that achieves the highest rate and theaverage over all possible configurations. (b) e configuration oflasers and detectors in the best configuration. . . . . . . . . . . . 86

4.10 Comparing the configurations obtained by the exhaustive searchand the greedy search. It can be observed that about 92% of therate of the optimal exhaustive search can be obtained by the greedysearch in this case. . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

xvi

4.11 e laser array utilized with the 50 µm fiber. e lasers have amode field diameter of 5 µm and a pitch of 8 µm . . . . . . . . . . 89

4.12 e detector configurations obtained by the greedy algorithm fordetectors of diameter 4 µm for various grid structures. Interest-ingly, there is a significant preference towards detectors closer tothe fiber core, indicating the fact that much of the received powerin graded-index MMFs propagates close to the axis. . . . . . . . . 90

4.13 e rate trends obtained with the detector configurations shownin Figure 4.12. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.14 (a) A fiber was analyzed to obtain the best 23 locations to placesmall circular detectors on a 11 × 11 grid as discussed in Sec-tion 4.5.3. (b) Clustering these detectors to obtain larger squaresegments to improve the fill factor. (c) A regular four-element de-tector array without using the design from the algorithm. . . . . . 92

4.15 Comparison of achievable rate trends for the detector paerns shownin Figure 4.14. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94

4.16 Operations required by a MIMO-MMSE equalizer . . . . . . . . . 974.17 (a) e detector array paern. e 29 detectors have a pitch of 8

µm a mode-field diameter of 5µm. (b) Progression of greedy se-lection: selecting eight detectors for a channel realization obtainedwith 3 lasers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

4.18 Comparison of the greedy and optimal searches with (a) 20 laserand (b) 29 lasers at the transmier. e plots show the fraction ofdata rate obtained using a subset of detectors vs. using all detectorsfor decoding, with a target BER of 10−9. . . . . . . . . . . . . . . . 100

5.1 Schematic of the 4×4MIMO experimental setup. e offset launchand detection components were realized using nanoprecision fiberalignment stages, while the MMF couplers were all 2 × 1 cou-plers. e inset shows the offset positions for the launch and detectstages, which were placed in a square grid with offset intervals of2 µm. Fibers and couplers colored blue represent SMF, while redrepresents MMF. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5.2 e transmier and receiver arrangement for the VCSEL case. . . 1135.3 e transmier and receiver arrangement for the evaluation of

plastic fibers. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1145.4 e impact of amode scrambler comparedwith a fiber offset launch

aer propagating through a 1mMMF patch cord. While the modescrambler causes an expansion of the beam and signal spread intoneighboring modes, the alignment stage allows control for excita-tion of higher order modes. . . . . . . . . . . . . . . . . . . . . . . 121

xvii

5.5 Data rate versus length for a 1× 1 system with and without signalprocessing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

5.6 Data rate versus length for various lengths of fiber with V-BLAST.ese data rates were observed with optimized offset launch anddetection with V-BLAST. e solid curve indicates the rated fiberbandwidth-length product, while the doed line indicates the satu-ration speed for the detector beyond which the eye diagram showsa mostly closed eye. . . . . . . . . . . . . . . . . . . . . . . . . . . 124

5.7 Average SNR for each stream in the 3 km 4×4 case, correspondingto the data streams labeled x1, x2, x3 and x4 in Fig. 5.1.. e energyof each subcarrier was averaged over 100 OFDM symbols for anSNR estimate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125

5.8 e radial variation of the capacity for various detector alignmentstage offsets for a fixed position of the transmier side alignmentstage, (a) in 3-D, as well as (b) along one cross-section along a planeto the right. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

5.9 Constellations received in the 35th subcarrier over a 3 km 4 × 4link with offset launch and detection that correspond to the datastreams labeled x1, x2, x3 and x4 in Fig. 5.1. While streams 1 and 3have a very low BER (∼ 10−5), 2 and 4 suffer from a much higherBER (∼ 10−2). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 128

5.10 e radial variation of the capacity for various detector alignmentstage offsets for a fixed position of the transmier side alignmentstage, in 3-dimensions above as well as the cross-section along aplane below. As can be seen, an increase in length reduces theachievable data rate, but increases the tolerance with which thereceiver stage needs to be aligned. . . . . . . . . . . . . . . . . . . 130

5.11 Constellation for the two virtual channels which were used formodulation with spatial multiplexing for a 1 kmMMF. (a) Stream 1(highest singular value): a QAM-64 constellationwas supported bythis virtual channel. (b) Stream 2 (second highest singular value):Only a QAM-4 constellation was supported in this virtual channel.e remaining two channels were ignored, since their SNRs werevery low. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132

5.12 Data rate versus length for various lengths of fiber with spatialmultiplexing. ese data rateswere observedwith optimized offsetlaunch and detection with spatial multiplexing with feedback ofchannel coefficients. . . . . . . . . . . . . . . . . . . . . . . . . . . 135

5.13 Initial VCSEL beampaern and aer being shaped by amode scram-bler . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

xviii

5.14 Data rate performance of the 2× 2 VCSEL link. e benefits fromspatial multiplexing are not very significant, although there is no-table improvement at longer fiber lengths. . . . . . . . . . . . . . 138

5.15 Data rate performance of the 2× 2 plastic fiber link. . . . . . . . . 1395.16 Beam propagation in plastic fiber sections of various lengths. e

heavy intermodal coupling causes a spatial spread of the signaleven within tens of meters. . . . . . . . . . . . . . . . . . . . . . . 141

xix

Chapter 1

Introduction

1.1 Motivation

Fiber optic communication technology has seen several advances in the

past few decades, wherein technological advances have permied communication

of several terabits per second through a single optical fiber [1]. e availability

of optical fiber links that can span several hundreds of kilometers has enabled

large amounts of information to be transmied across long distances with great

efficiency. While long-haul optical links generally use single-mode fibers (SMFs),

short-haul links oen utilize multimode fibers (MMFs). e main reason for this

is that the overall cost of building a network for a small area is minimized with

multimode fiber components, as opposed to single-mode fibers, while stringent

performance requirements make single-mode fibers more suitable for long-haul

networks.

e predominant problem that needs to be addressed in building MMF

based optical links is to limit or compensate for the spreading of the optical pulse

as it traverses through the fibermedium, due to several physical effects collectively

known as dispersion. Conventional optical networks are oen operated within the

dispersion imposed limits, which is the prime motivation for using single-mode

1

fibers [2]. Single-mode fibers, by design, eliminate modal dispersion. Compensa-

tion of dispersion has also been proposed, using physical methods, such as with

dispersion compensated fibers and photonic crystals [1, 3]. However, these meth-

ods are generally used in conjunction with single-mode fibers. Multimode fibers,

which are generally more severely restricted due to larger dispersion, offer only a

low data rate when operated within dispersion limits, as is generally done today.

Digital signal processing (DSP) technologies have advanced significantly

over the past two decades, both in terms of algorithms as well as in terms of the

availability of speedier and inexpensive hardware. With demands driven by the

ever growing need for faster signal processing for bothwired andwireless systems,

the performance of signal processing implementations has now progressed to be-

ing suitable for much higher speeds of communication, such as those required for

optical transmiers and receivers, and the ever reducing costs of CMOS hardware

provides us motivation to explore the use of signal processing for optical commu-

nication links [4]. e adoption of signal processing for optical system provides

a rich array of tools, ranging from electronic dispersion compensation techniques

to enhancements such as multiple-input multiple-output (MIMO) techniques to

increase the available signaling dimensions to increase data rates.

Multimode fibers currently account for over 90% of all deployed short-

haul optical links, such as in the context of local-area and metropolitan-area net-

works [5, 6]. With the ever increasing demand for bandwidth, the utilization of

already existing multimode fiber is essential to obtain a cost-effective means for

upgrading network speeds. Signal processing offers an efficient solution to achieve

2

this. However, to effectively facilitate the use of signal processing techniques, spe-

cific algorithms and hardware that cater to the unique demands imposed by optical

communication systems have to be developed. is thesis considers a theoretical

and experimental approach to developing and evaluating such techniques.

1.2 e Case for MIMO in Multimode Fibers

e performance limits of an optical link are determined predominantly

by the limitations imposed by the nature of noise, pulse spreading, and dispersion.

Noise considerations are accounted for in the power budget, but pulse spreading

is either a hard limitation, or requires additional correction in the form of disper-

sion compensating fibers, photonic crystals, or electronic dispersion compensation

techniques [7]. Since the pulse spreading due to dispersion increases with length,

use of longer fibers result in lower data rate limits.

e common metric for comparing the performance of optical links is the

bandwidth-length product. is metric is appropriate for our comparisons be-

cause it accounts for the fact that longer fibers, which are affected more due to

dispersion, have smaller data rate limits. Due to the high group velocity disper-

sion caused by modal dispersion, multimode fibers have a very low bandwidth-

length product, and thus, are less preferred for deployment in high-speed links.

However, the large core diameter of multimode fibers also provides them the abil-

ity to carry significantly larger amounts of power without greatly distorting the

signal [8, 9]. is can be explained by the fact that fiber nonlinearities, such as

stimulated Brillouin scaering and Raman scaering, are less pronounced at lower

3

cross-sectional power densities, which is the case with multimode fibers that have

large core diameters.

Figure 1.1: An example showing the different paths taken by an incident ray in amultimode waveguide

In addition, the larger core significantly simplifies alignment and pack-

aging [10]. However, the large core diameter also translates to a larger number

of "modes" in which the electromagnetic wave propagates. is propagation in

a larger number of modes, where several modes have different group velocities,

causes the data pulses to spread in time; this effect is called modal dispersion.

Over a sufficient length of fiber, the transmied symbols start overlapping with

each other. is overlap of successive data symbols is known as "inter-symbol

interference". us, to augment the useful data rate through multimode fibers,

this inter-symbol interference has to be compensated for. In our approach, we use

signal processing to correct for this undesirable effect.

To gain benefits fromMIMO techniques in multimode fibers, the large core

diameter and the availability of several orthogonal propagating modes of the mul-

timode fiber can be used for multiplexing several data streams. e success of

MIMO techniques in the wireless communication realm allows us to draw paral-

lels in the wayMIMO benefits can be realized in multimode fibers. However, there

are several differences in the kinds of systems used, properties of the channel and

4

limitations of the link for multimode fiber links that make it more challenging than

a direct reapplication of MIMO concepts developed for wireless communication.

1.3 Comparison with Wavelength Division Multiplexing

Wavelength divisionmultiplexing (WDM) is the current standardmeans to

increase the capacity of optical links, wherein multiple wavelengths carry distinct

information streams and are multiplexed and demultiplexed through the fiber [11].

e MIMO techniques described here are all for deployment on a per-wavelength

basis. As such, all techniques described here are applicable for WDM-MMF net-

works and provide similar rate enhancements, i.e., optical MIMO techniques can

be laid over existing standard MMF-WDM networks.

From a comparative perspective, the primary tradeoff between choosing

WDM and MIMO-MMF is summarized as follows:

• WDM links, specifically with DWDM, provide immense capacity increases

over fiber links. However, each wavelength requires an additional laser,

modulator, filter and detector chain. In the MIMO case, the use of a sin-

gle wavelength could result in component reduction if specifically designed

laser and detector arrays are utilized.

• e use of a single wavelength in MIMO obviates the need to use powerful

temperature stabilization needed in DWDM.

• While the capacity with MIMO theoretically grows linearly with number of

modulators/detectors, this has not been experimentally verified with a large

5

number of devices. us, it remains to be seen if benefits from a pure MIMO

based approach are comparable to the improvements that can be obtained

with WDM in a similar situation.

However, we emphasize that, since the demands placed by the MIMO im-

plementation are very simple, overlaying MIMO on existing MMF-WDM systems

could result in a much improved performance, and could be the right system trade-

off to work towards. e following table lists a brief comparison:

Parameter SMF-WDM MMF-WDM MIMO-MMFTemperature control Required Required Not requiredData rate/channel 100+ Gb/s 10 Gb/s 50 Gb/s1

Distances Long Short Short-MediumComplexity2 High High (Potentially) Low

Temp. control3 Needed Needed (Potentially) UnneededLatency Low Low Higher4

Channels 40 40 401: ough the actual capacity of MIMO-MMF links is not known, our re-

sults indicate that much higher speeds are achievable.

2: e complexity is in the number of components interconnected, leading

to various losses, as well as the cost of components.

3: Assuming that wavelengths used in forWDM inMIMO-MMF are spaced

far apart for them to not need temperature stabilization.

4: e additional signal processing involves some additional overhead, but

it can be minimized using parallel processing.

6

1.4 State-of-the-art in DSP based MMF links

Increasing the data rates supported in optical fibers has been an active re-

search area for several decades, although the focus has been largely on long-haul

related applications. e primary technique employed there to boost data rates is

the through the use of WDM over single-mode fibers. e current developments

in multimode fiber technology can roughly be separated into two parts: the first

one is the design and development of new media with the aim of improving data

rates, and the second one is the development of signal processing techniques to

increase the reliability of links using conventional MMF as well as new media.

Recently, it has been demonstrated that significant increases in data rate

can be obtained using multiple modes as degrees of freedom. MIMO techniques

have been applied in single mode fibers, utilizing polarization modal diversity [12,

13] and orthogonal band multiplexing [14, 15]. ere is also a significant new

body of work emerging on the application of similar MIMO techniques to mod-

ern optical media, specifically multi-core and few-mode fibers [16–20]. Multi-core

fibers are optical fibers which possess multiple cores within the same cladding, to

support several parallel signals within a single fiber. Current multicore fibers gen-

erally have seven or more cores, and modern design techniques have enabled sig-

nal propagation through these cores with lile or no inter-core interactions [21].

Since each core possess SMF-like low dispersion characteristics, multicore fibers

have been demonstrated to possess terabits-per-second speeds [17, 18, 22]. Few-

mode fibers, on the other hand, are fibers that possess a core radius very slightly

larger than single-mode fibers to enable the propagation of two or three modes. In

7

the strict sense of the definition, they are multimode fibers manufactured with a

controlled core radius to allow a few modes explicitly used for multiplexing. e

advantage of few-mode fibers over both single-mode and multimode fibers is that

they possess the low-dispersion characteristics of single-mode fibers while also al-

lowing multiplexing of multiple signals in a controlled fashion. e complication

with using few-mode fibers is that launching into the modes of the fiber require

accuratemode filtering of signals alongwith coherent detection to separate signals

at the receiver. Recent developments have enabled long-haul links with few-mode

fibers and signal processing to function at terabits-per-second [23–25]. e fun-

damental distinction between these approaches and the topics considered in this

thesis are that the prior approaches involve MIMO techniques using a coherent

communication framework, on new media specifically designed for multiplexing

many data streams in order to improve data rate, with a focus on long-haul and

ultra long-haul reach. e techniques utilized in these approaches do not directly

apply to conventional multimode fiber communication, since these media exhibit

different propagation properties and intermodal interactions. In addition, these

media generally require more sophisticated alignment and packaging solutions

for the laser and detector components, owing to their specific geometries, and

utilize coherent detection for the received information. e requirement of co-

herent communication and advanced media makes these suitable for high-speed

long-haul links, but prohibitively expensive for short range deployments due to

the increased complexity.

e use of multiplexing in conventional MMF based links was first consid-

8

ered in [26], where a nominal speed of 50 Mb/s was achieved in a proof-of-concept

2 × 2 MIMO-MMF link. Subsequently, coherent approaches to MIMO in MMFs

have been demonstrated [27, 28], where advanced modulation and detection en-

abled a data rate of 800 Mb/s . However. this method requires the recovery of the

laser carrier and phase at the receiver, making the deployment an expensive and

complex proposition for short, inexpensive links. e use of incoherent techniques

for multiplexing include mode group diversity multiplexing [29–31], square law

detection approaches [32], although these approaches are spectrally limited due

to their restriction to binary modulation. eoretical considerations concerning

device properties for MIMO on MMFs have been also been studied [33].

1.5 Problems addressed in this thesis

is thesis aims to provide a deeper understanding of the problems and

limitations imposed by multimode fiber links, and the extent to which these prob-

lems can be tackled within the purview of signal processing. We begin this by

studying the propagation characteristics of pulses in multimode optical fibers, and

creating abstractionswhichwould facilitate the development of specific signal pro-

cessing techniques. We follow this by developing appropriate modulation, coding

and pulse shaping which would allow us to maximize the date rate through the

fiber, and evaluate this with an MMF system built using off-the-shelf components

for quick prototyping. Based on the inferences drawn from these results, we de-

velop a propagation model to study the impact of launch conditions and offset

coupling at the fiber launch and detect stages to get beer reliability and multi-

9

plexing. Finally, we experimentally evaluate the performance boosts obtainable

with offset launch and detection, and the impact of these on both silica and plastic

MMF sections of various lengths. We also discuss the impact of these results on

the development of arrays of lasers and detectors designed to facilitate effective

multiplexing in conjunction with MIMO and signal processing.

We focus on the use of advanced modulation and MIMO techniques such

as spatial multiplexing to harness the diversity benefit offered by the MMF. In

particular, we connect several physical parameters of the fiber that are measurable

using signal processing, and use these to optimize the data rates on both plastic and

silica MIMO-MMF links with fiber sections of various lengths. Moreover, we also

study the impact of using different optical components, such as nanopositioning

stages, mode scramblers and lasers, and gauge the impact these components have

on the performance of the optical link.

e metric of interest considered in this thesis for evaluating the perfor-

mance of high speed links is the bandwidth-length product. Since there is no

accepted standard for DSP based high speed links, related work in this field has

not standardized the list of electro-optic and rf components used in various exper-

imental evaluations. To obtain a baseline standard for current MMF communica-

tion strategies we consider the following data rate values:

• No DSP/MIMO: For the case where the link is used with conventional on-off

keying with no additional dispersion compensation, the bandwidth-length

product for the rated fiber is used as the baseline. is is generally between 1

10

Gb/s-km and 2Gb/s-km for 62.5µm core diameter silicaMMF, and 200Mb/s-

km for 62.5 µm core diameter perfluorinated plastic MMF. e 10GBASE-

SR and 10GBASE-LRM standards for silica fiber are the most widely used

formats currently [34], and they support data rates of 10 Gb/s over links of

up to 200 m, thus corresponding to a bandwidth-length product of 2 Gb/s-

km.

• With DSP/MIMO: For the case with DSP and MIMO, we refer to recent lit-

erature focused on DSP based communication in conventional multimode

fibers [35–39]. Based on these and other recent results, we assume that

the state-of-the-art bandwidth-length products achieved on DSP basedMMF

links are generally between 5 - 10 Gb/s-km.

For want of a standard that defines the data rates achievable with signal

processing in MMF links, the experimental evaluations described in this thesis

refer to the above metrics for evaluating the benefits obtainable with incoherent

MIMO techniques. Moreover, the focus of this thesis is on describing how various

parameters, such as fiber offsets, coupling, modulation and coding can affect the

achievable data rates through the fiber, using the 1 × 1 system deployment as

baseline. is enables a fair comparison of these techniques in a manner agnostic

to the exact equipment used for the deployment.

1.6 Contributions of this thesis

e contributions of this thesis can be summarized as follows:

11

• We develop a signal processing framework for optical fiber systems, using

which dispersion compensation and data rate enhancement can be achieved

from a signal processing perspective. Chapter 2 covers this part.

• We develop techniques to analyze and design devices (laser and photode-

tector arrays) which specifically take advantage of the MIMO structure to

improve reliability of communication and increase data rates. is work is

primarily discussed in Chapter 4.

• We experimentally determine and characterize the gains obtainable by us-

ing the developed signal processing methods in a MIMO multimode fiber

system, and identify the causes for gaps in performance from theoretical

predictions. is is covered in Chapters 3 and 5.

1.7 Organization

is document is organized into five chapters in addition to this introduc-

tory chapter. Chapter 2 introduces the signal processing abstractions for the op-

tical MIMO channel, and describes the algorithms and techniques useful for en-

hancing performance. Chapter 3 justifies the channel model used, and then pro-

ceeds to provide several experimental evaluations of the techniques using off-the-

shelf components. Chapter 4 considers several problems connected to theoreti-

cally modeling MIMO-MMF links, and proposes a model to study the propagation

properties of MMFs, and how the use of laser/detector arrays and efficient sig-

nal processing techniques can enhance data rates while keeping signal processing

12

complexity in check. Chapter 5 describes the experimental evaluation of a MIMO-

MMF link with plastic and silica fibers of various lengths, and how different ap-

proaches to signal processing and the use of different optical components can im-

pact the data rates in such links. Finally, Chapter 6 summarizes the contributions

of this thesis and describes future problems connected to the topics covered here.

13

Chapter 2

Signal Processing and MIMO-MMF: Models andAbstractions

2.1 Introduction

Developments in signal processing processing technology have revolution-

ized communication systems over the past decades. Motivated primarily by devel-

opments in wireless communications, several signal processing algorithms have

been developed to combat the problem of dispersion-limited data rates. e use

of multiple antennas is now well-established technology in modern wireless sys-

tems. e use of multiple antennas in wireless systems results in a significant

increase in data-rates for a fixed power budget, which can drastically improve the

performance of an otherwise highly constrained wireless medium. us, MIMO

has become the key enabling technology for reliable high-speed wireless commu-

nication. Prior work, discussed in Section 1.4, has shown that similar concepts are

applicable in multimode fibers, owing to the availability ofmodal diversity. In this

chapter, we focus on developing an abstraction of the MMF that allows the devel-

opment of an incoherent MIMO link. Such an abstraction that encapsulates the

physical transformation effected by the fiber in a mathematical structure allows

for evaluating several diversity multiplexing schemes in MIMO-MMF links.

14

is chapter will present the fundamental concepts and assumptions made

for both the theoretical and experimental evaluation of signal processing parame-

ters of the MMF optical links considered in this thesis.

2.2 DSP-basedMIMOCommunication Systems for Optical Fibers2.2.1 Introduction

Multiple-input multiple-output for optical fibers (referred to in short as op-

tical MIMO) is a means to utilize the multiple degrees of freedom naturally avail-

able in multimode fibers in their modes to transmit more data reliably. e modes

in optical fibers offer a spatially diverse set of propagation paths, which can be ac-

tivated selectively with appropriate signaling, and signal processing at the receiver

can be used to appropriately combine the signals received from different modes

or mode groups to obtain a higher effective SNR [26]. e realization of the opti-

cal MIMO system by means of a larger number of transmiers (modulators) and

receivers (detectors), and appropriate use of signal processing to superpose and

correct for dispersion and coupling between different modes enables the multiple

modes of an optical fiber to be used as additional degrees of freedom for signaling.

2.2.2 Benefits of MIMO

As is well known, the aim of digital communication is to transport bits of

information across a channel as accurately and efficiently as possible. e key

challenge in the implementation of such a system is in understanding the vagaries

of the channel, and designing techniques that efficiently combat the limitations

15

and restrictions imposed by these effects. e fundamental limit to communication

is determined by the channel characteristics and noise, and the common tradeoffs

are those between bandwidth and bits per symbol. Due to the constant variations

which affect the signals naturally, the channel is assumed to be probabilistic in

nature.

e simplest ways to scale up data rates are by increasing the transmission

power, or by increasing the bandwidth over which the data is transmied. How-

ever, once the limits of power and bandwidth allowance have been reached, the

technique employed in wireless communication for faster, more robust communi-

cation links is by introducing "space" as an additional degree of freedom [40]. e

key benefit of having channels available by spatial separation is that they provide

the "diversity" of spreading the data over multiple paths, some of which may be

of higher quality than the other. e way to realize the benefits of multiple paths

in practice is by appropriate placement of multiple antennas at both the transmit-

ter and/or the receiver, so that they capture independent signals which are later

weighted appropriately and combined with signal processing algorithms.

e additional dimension provided by MIMO can be utilized in different

ways [40]. One way to utilize this benefit is to increas robustness, and enable

transmission at a lower bit-error rate for the same power budget (this is enabled by

"diversity"). Alternately, the newly available paths can be used to signal multiple

streams of data with the additional degrees of freedom (referred to as "multiplex-

ing"). ese advantages provide a flexible interface for advanced signaling, where

a tradeoff can be had between robustness and higher data rates, and opens up a

16

wide variety of MIMO modulation and coding techniques for MMF optical links.

2.2.3 Digital Signal Processing for MMF Links

Digital signal processing has not been considered a viable data processing

means for optical systems, primarily because the computation requirements for

processing data at data optical fiber link speeds were not tractable till recently.

However, the latest developments in circuit technology have pushed down the

costs while increasing speeds of digital signal processors. For instance, there have

been Gigabit Ethernet solutions for optics based completely on DSPs for several

years [41]. is improvement in speed and cost-effectiveness motivates the devel-

opment of more advanced signal processing solutions for optical links.

Data pulse

Dispersive (multimode) fiber

Received pulse

Digitize

Inverse filter

Photodetector

Output data

Reconstructed pulse

Opticalmodulator

Digital signal processor

Figure 2.1: A schematic of the implementation of a digital signal processing fordispersion compensation in a multimode fiber link.

Signal processing allows for techniques such as waveform shaping and ad-

vanced modulation which allow for several benefits, including simpler algorithms

17

for dispersion compensation and beer modulation for more spectrally efficient

signaling [42]. e main reason for the effectiveness of digital signal process-

ing in combating physical phenomena is because signal processing abstracts the

effects produced by various components of the communication system, and con-

verts the problem of recovering the transmied information into one of computa-

tion [43, 44]. With the ever increasing efficiency and reducing costs for computa-

tion, DSP implementations are becoming more powerful while also being afford-

able, thus making them ideally suited to solve these kinds of problems. is is the

primary reason for the proliferation of DSP technology in most of today's wireless

and wired systems, and this motivates us to use the power afforded to us by DSPs

to solve the problems posed by the optical fiber medium.

A possible implementation of a digital signal processing based electronic

dispersion compensation system is shown in Figure 2.1. As is conventional in such

dispersion compensation systems, the digital signal processing circuitry aempts

to first "learn" the nature of the variations caused in the received waveform by the

channel (which, in this case, consists of the dispersion caused due to propagation

through the optical fiber). is process is called channel estimation. Using the

estimated channel, the processor then computes the optimal digital filter which

would best "undo" or "compensate" for the changes effected by the fiber, and pass

received data through such a filter. is step is called equalization. e realization

of such a circuit has the following requirements [45]:

• e sampling rate should bemore than twice the symbol rate. While this seems

like a severe limitation in high-speed optical systems, this can easily be over-

18

come by parallelizing data streams using techniques such as Time/Frequency

Division Multiplexing (TDM/FDM), much like TDM is the norm in separat-

ing data streams in / systems.

• e system should be operated in the linear or near linear regime. Signal pro-

cessing algorithms are oen available and are effective in coping with sev-

eral non-linearities in systems. However, most of the optimal algorithms in

signal processing offer performance guarantees only when the system op-

erates in the linear regime. In addition, the interaction between different

components of the transmied symbol is also assumed to be linear, in that

non-linear transformations of transmied signals, such as that which occurs

in Four-Wave Mixing, is not considered in our implementation [11]. is is

a justified assumption, owing to the fact that power densities are lower in

multimode fibers due to their larger core diameters.

• e system parameters should be quasi-static. is is essential for being able

to estimate the system parameters electronically and compensate for them.

In other words, the assumption of time-invariance is necessary for electronic

dispersion compensation algorithms to perform as expected [46].

2.3 Channel Model for an Optical MIMO Link

To effectively develop and implement DSP algorithms for a MIMO-Optical

links, it is essential to arrive at an abstraction of the multimode-fiber in the current

context, to model all the fiber effects which need to be compensated for in order

19

to communicate successfully. In this section, we first develop a linear and time-

invariant model to describe and encapsulate the pulse spreading caused by the

multimode fiber, and extend it to the case where there are multiple transmiers

and detectors. We refer to existing literature on developing such channel models

and adapt it in our case. We also assume that the length of the fiber is sufficiently

long, so that phases of the pulse arriving at different times are uncorrelated.

2.3.1 Pulse-spreading in a Multimode Fiber

We consider the pulse spreading in a multimode fiber due to the varying

velocities with which different modes propagate. In maintaining consistency with

conventional digital communication literature, we assume that our signal is a com-

plex waveform in baseband, and modulating with a carrier to obtain a real signal

is mathematically understood using the Hilbert Transform [47].

Let m(t) represent the baseband signal, which is a shaped waveform con-

taining the coded information bits to be conveyed to the receiver. is is modu-

lated to a higher radio frequency by means of a carrier radio signal (a sinusoid)

whose frequency is fc. is signal is used to drive the electro-optic modulator

to optically modulate the intensity of the laser. We remark that linear intensity

modulation in the context of optical communication is very similar to amplitude

modulation in the radio frequency (r) case. We represent the modulation of the

signal to the carrier frequency to obtain the transmit signal as

x(t) = Re(ej2πfctm(t)

). (2.1)

In order to model the effects of the fiber, we assume that the effect of each mode

20

produces a different amplitude transformation as well as phase transformation on

the message signal. We assume that the fiber possesses Q effective modes, and

that the k-th mode manifests at the detector with transformed amplitude of Ak

and an effective phase of ϕk, while producing a delay of τk. e additive noise is

represented as w(t). us, the effective received signal is given as

y(t) = Re

Q∑q=1

Aqejϕqej2πfc(t−τq)x(t− τq)

. (2.2)

is equation is very similar to that in the wireless case, wherein the cause for

multiple delayed copies of a signal appearing is the multiple reflections a signal

undergoes before arriving at the detector [48]. In order to obtain a baseband equiv-

alent signal, the reverse operation to equation 2.1 is performed, to obtain:

yb(t) = Re(e−j2πfcty(t)

). (2.3)

It can be shown that the signal obtained from equation 2.3 permits us to represent

the effective channel as a simple baseband impulse response filter. e details

of this are omied, but can be found in [46, 47]. e effective baseband impulse

response is given by

h(t) =

Q∑q=1

Aqejϕqδ(t− τq). (2.4)

Finally, we discretize the impulse response to obtain a discrete-time filter repre-

senting the transformation of the discrete symbols effected by the fiber medium.

In this step, we remark that, owing to the sampling rate at the receiver not being

sufficiently fast, the different received signal components corresponding to each

21

Time

Rec

eive

dA

mpl

itud

e (a

.u.)

{ { { {Effective combined discrete components

Figure 2.2: A representation showing the resolved components while sampling theimpulse response of a system

of the Q modes do not appear as separate samples. In other words, due to limita-

tions in "resolvability" of different closely arriving components [40], some of the

modes' contributions are averaged out in the receiver, as shown in Figure 2.2.

e selective resolvability of the signal can be explained by the fact that

the separation between any two copies of the signal arriving in close proximity of

times is much smaller than the inverse of the signal bandwidth [46]. e compo-

nents which are inseparable, i.e. for those components whose τ1 ≈ τ2, are merged

into a single value which averages the amplitude and phase transformations of

each of these. us, simplifying all these effects into one, we get an effective

discrete-time channel as

y[n] =N∑

n=0

h[N − n]x[n] + w[n] (2.5)

where x[n] is the discrete transmied signal, y[n] is the received signal, w[n] is

22

Figure 2.3: An example of delayed copies of signals sent and received by multipleantennas in wireless multipath.

the effective noise and h[n] is the effective impulse response obtained aer the

discretization process. Since the system is operation conditions considered here

are circuit noise limited, as opposed to optical noise limited, the distribution of the

additive noise is white and Gaussian distributed.

With the effective channel model in place, we now extend the model to

account for the MIMO case.

2.3.2 Spatial Effects of Multiple Modes

In this section, we extend the above described channel model to account

for the spatial effects caused by transmission through a multimode fiber. As in the

earlier case, our model for signal processing is motivated by the channel model

for the wireless MIMO case, as is shown in Figure 2.3, which we shall adapt. We

shall also describe the key differences between these models and how that affects

the design of the system.

23

2.3.2.1 Parallels between Wireless and Optical MIMO

e similarity of the wireless and MMF-optical channels allows us to use

signal processing techniques similar to those used in wireless communication. e

primary similarity between the channels is both have a concept of signals being

delayed in time, thus causing the pulse to spread. It is this similarity between

multipath and dispersion which allows us to employ signal processing techniques

such as estimation and equalization to compensate for this dispersion, and over-

come the natural dispersion limit of the fiber. In addition, the orthogonality of

the multiple modes of the multimode optical fiber closely resembles the notion of

independent paths provided by wireless channels. e fact that intermodal cou-

pling causes signals to shi their energies across different modes is similar to the

reception of signals from various paths superposed on receiver antennas in the

wireless case [2]. We can thus utilize the rich techniques afforded to us by MIMO

signal processing in order to fully benefit from the additional signaling capability

provided to us by the diversity of modes in multimode fibers. Figure 2.4 shows a

possible model for the MIMO-MMF Optical system.

Modes in a multimode optical fiber not only possess different group veloc-

ities, but also varying spatial geometries across the cross-section of the fiber. is

occurs naturally due to the dielectric boundary conditions imposed by the weakly

guiding interface between the fiber core and cladding. e difference in geometry

results in slight variation in the channel properties as observed between different

transmiers and detectors. It is this key property which allows us to exploit the

modal diversity offered by modes in an optical fiber, and thus, communicate with

24

Figure 2.4: e correspondence between multiple transmiers and multiple detec-tors in the wireless and MMF cases

greater reliability.

To arrive at a model for this system, we assume a system with M transmit

components (modulators) which can modulate independent signals which can be

coupled into the fiber, andN detectors at the receiver. Since we operate the system

in the linear regime, we characterize the system, much like in the earlier section,

except that we employ a matrix transfer function to describe the system. In order

to not depend on modulating different modes independently, we account for the

fact that the modes excited by each transmier may be a superposition of various

modes. In addition, due to intermodal coupling, the energy in different modes

fluctuate among each other for a sufficiently long distance of propagation. In a

linear superposition regime, this process can be accounted for in estimating the

matrix impulse response of the system.

25

We assume that the transfer function can be described in components. Let

hij(t) represent the impulse response from the j-th transmier to the i-th receiver.

We represent the transmit signal on modulator j as xi(t), and that received on

detector i as yi(t). Finally, we represent the receiver noise on detector i as wi(t).

en we have the receiver component-wise received signal can be represented as

yi(t) =M∑j=1

∫ ∞

−∞hij(t− τ)xj(τ)dτ + wi(t), i = 1, 2, . . . , N. (2.6)

Representing this in a convenient matrix form, with the ∗ operator denoting con-

volution, equation 2.6 can be reformulated asy1(t)y2(t)

...yN(t)

=

h11(t) h12(t) · · · h1M(t)h21(t) h22(t) · · · h2M(t)

......

. . ....

hN1(t) hN2(t) · · · hNM(t)

∗

x1(t)x2(t)

...xM(t)

+

w1(t)w2(t)

...wN(t)

.

(2.7)

Representing the above equation in matrix form, using boldface notation for cor-

responding vectors, we get

y(t) = H(t) ∗ x(t) +w(t) (2.8)

where y(t) is the N × 1 receive vector, x(t) is the M × 1 transmit vector, w(t) is

the N × 1 noise vector and H(t) is the N ×M channel matrix.

Finally, to obtain a discrete input-output relationship, we adopt a frequency

division based approach, in which the useful frequency range is split into several

subsections, such as in the case of OFDM. is allows us to individually treat the

channel for each frequency bin as being a single-tap (flat-fading) channel. We omit

26

the details of this here, since these are discussed in detail in Section 4.3 of this

dissertation, as well as in greater detail in Chapter 3 of [40]. Using this technique,

we arrive at the effective discrete-time input-output representation for a single

frequency bin. Suppressing the frequency bin number, the system input/output

relationship for each discrete-time index n can be represented as:

y[n] = H[n]x[n] +w[n]. (2.9)

With this model, we develop signal processing techniques to perform the follow-

ing:

• eoretically analyze and predict themaximumdata rates achievable through

such a channel medium, and determine the best modulation and coding ap-

propriate for such a medium.

• Utilize the linear system model to develop and implement algorithms to

learn the transfer functions which occur due to the transmission medium,

and generate dynamic digital compensation filters to achieve the same in an

scenario without feedback, where the transmier has no prior knowledge

of the transfer function.

• Develop a feedback based approach to digital dispersion compensation, where

some information about the channel transfer function is made available by

the receiver to the transmier in order to perform some precompensation.

27

2.3.2.2 Main differences

e primary difference between the wireless channel and the MMF-optical

channel is the fact that the former is an unguided medium, while the optical fibers

are a guided medium. is leads to certain other interesting variations in the way

signals propagate through these media.

e guided nature of the optical fiber channel leads to propagation of the

electromagnetic wave only in certain quantized “modes” that satisfy the boundary

conditions imposed by the radially varying refractive index. esemodes have dif-

ferent group velocities, and thus, travel at different velocities with respect to one

another, causing pulse spreading. is is unlike the wireless case, where trans-

mission is generally isotropic, and the pulse spreading is due multiple paths taken

by the transmied signal.

Finally, the guided nature of the optical medium also causes another key

difference in the MIMO paradigm in the optical case. Traditional wireless MIMO

receivers receive roughly the same amount of power irrespective of the number of

receive antennas. However, in the conventional optical MIMO receiver case, since

the received power has to be split among all the receivers, arbitrarily increasing

the photodetectors as part of an optical MIMO receiver is not feasible without the

use of an amplifier at the receiver. In this thesis, our focus is on simple imple-

mentations that do not consider the use of amplifiers at the receiver (such as the

system described in [49]).

28

2.4 Signal ProcessingAlgorithms and Paradigms forOptical-MIMOlinks

is section details some of the common signal processing algorithms that

are useful in MMF optical links, particularly from a MIMO perspective. e focus

is on how to learn the channel properties, how to adapt the transmier and re-

ceiver dynamically to changes in operating conditions and how to precompensate

the signal to minimize distortion aer propagation through the medium. While

the concepts discussed are generic, the focus is on orthogonal frequency division

multiplexing (OFDM) and discrete multitone (DMT) systems.

2.4.1 Pilot-based estimation

e concept of pilot-based estimation rests on the assumption that the

channel conditions that affect the transmied signal do not vary significantly over

a sufficiently long duration of time, in comparison to the symbol duration. is al-

lows the transmier to periodically send some "probe" signals (called pilot signals),

which are known a priori to the receiver. e receiver makes use of this informa-

tion to "estimate" the channel parameters, and thus, can learn how to compensate,

or "equalize" for the changes which the medium causes. is allows the medium

to transmit data successfully by periodically devoting some symbols as overhead

for the receiver to estimate the channel.

e stability of optical links has been established in the context of mode

group diversity multiplexing [50]. Since the Optical-MIMO link involves using

a very similar channel medium, we can expect the channel properties to remain

29

stable in this context as well.

To demonstrate a conventional pilot based estimation routing, consider an

M ×N MIMO link with M modulators and N receivers. We consider the single-

tap channel case (known as flat-fading in wireless parlance), since most estimation

problems can be converted to this simplified case. e input-output relation for

the link is given in 2.9. In order to estimate the channel, we can use a pilot signal

denoted by X, which is an M × K link, where K ≥ N for an adequate number

of estimation samples. We interpret the rows of X as the symbols sent on each

transmit antenna simultaneously at a certain time index, while we take the differ-

ent columns to represent different times. Suppressing the time-index, 2.9 can be

stated in this context as

Y = HX+W. (2.10)

Here, Y is the N × K receive matrix, whose columns represent the time-index,

while the rows represent the symbols received at each of the detectors on the

receive side. H is theN ×M channel matrix, whileW is theN ×K noise matrix.

SinceX is a predetermined vector known to the receiver in advance, the design of

X is made in a way which facilitates the estimation process well.

Conventionally, the metric chosen to estimate the channel is the mean

squared error, and it uses the statistics of the noise in order to determine the best

estimate. With this metric, it can be shown that the best estimate for the channel

matrix H can be obtained by matrix manipulation and inversion [43]. e de-

tails are omied here. Once the channel estimate H is obtained, it can be used to

harness the benefits of MIMO as converted from the modal diversity to a signal

30

processing problem. We now delve into the various algorithms and techniques

which facilitate this procedure.

2.4.2 Reliability vs. Higher Data Rates: e Diversity-Multiplexing Tradeoff

We briefly discuss a fundamental tradeoff that occurs naturally in MIMO

systems, which is the one between the diversity andmultiplexing. We shall briefly

describe these here, but we omit the details, which are described in [40, 51].

e primary reason why MIMO techniques offer benefits is because the

MIMO channel matrix (considered probabilistic in nature) allows transmission of

multiple symbols of information simultaneously. An alternate means of viewing

this is that there are multiple virtual "paths" available to transmit the same piece

of information. with a greater reliability. ere is a continuous range of choices

within a regime where the MIMO channel can operate, between the highest data

rate and the highest reliability. e fact that multiple streams of data can be sent

along the MIMO channel refers to "multiplexing", while the fact that multiple ver-

sion of the same information can be sent along the same paths refers to "diver-

sity". Naturally, multiplexing multiple streams results in a lower reliability, while

using full diversity results in a lower data rate (while maintaining other param-

eters, such as modulation, the same). is fundamental tradeoff is known as the

diversity-multiplexing tradeoff.

To demonstrate this concept, we use an example. Consider the input-

output equation 2.9, restated here with the time-index suppressed and for the flat-

31

fading case.

y = Hx+w. (2.11)

Here, if we view the transformation of x by H as a matrix multiplication, we can

easily see that the maximum dimension of the vector space in which y can lie is

min(M,N). us, the maximum number of parallel streams that can be multi-

plexed simultaneously is min(M,N). However, if the same data is sent through

the various parallel paths available through the channel, the diversity gain corre-

sponds to the maximum number of independent entries of the matrix H, which,

ideally, is MN . us, communication schemes (such as space-time codes) can

be developed which optimally operate with the optimal amount of diversity and

multiplexing. It must be noted that, for multiplexing, since the rank of the channel

matrix must be more than 1, more than one modulator and more than one detector

are needed.

In our work, we work on implementations which mainly focus on operat-

ing on the extremes, which involve full diversity or full multiplexing, since this

sufficiently demonstrates the efficacy of these MIMO techniques in MMF links.

2.4.3 Diversity and Multiplexing Semes without annel state feedba

is section details several algorithmswhich allow the receiver to compen-

sate for the effects caused by the channel without any channel state information

being present at the transmier. is is particularly useful, since it allows opera-

tion of the systemwithout a reverse data link between the receiver and transmier,

and thus significantly simplifies the implementation of MIMO systems. is sec-

32

tion discusses transmit diversity, receive diversity as well as a spatial multiplexing

scheme.

2.4.3.1 Transmit Diversity: Alamouti Code

Space-time codes are a class of codes which allow for diversity benefits to

be obtained from MIMO channels. In particular, space-time block codes are codes

which operate on data block-by-block to harness the multiple independent paths

provided by the channel. ey offer a solution to get maximum diversity ben-

efit, wherein linear precoding and post-processing of the transmit data provides

reliability benefits.

A particularly powerful space-time block code is the Alamouti code , which

has several favourable properties. We briefly describe the Alamouti code for the

2×1 channel, which has only one detector and two modulators. We can represent

the channel equation, suppressing the time-indices, as

y =[h1 h2

] [ x1

x2

]+ w (2.12)

where the form is similar to equation 2.9. Now, to demonstrate the Alamouti code,

we group complex data symbols into groups of two each, say x1, x2, and transmit,

in successive time intervals, the symbols (x1, x2) and (−x∗2, x

∗1) on modulators

(1, 2) respectively (here, ∗ represents complex conjugation). We can represent the

effective transmission as[y1 y2

]=[h1 h2

] [ x1 −x∗2

x2 x∗1

]+[w1 w2

](2.13)

33

which can be restated as[y1y∗2

]=

[h1 h2

h∗2 −h1∗

][x1

x2

]+

[w1

w∗2

]. (2.14)

With channel knowledge at the receiver, the original data can be recovered by

projecting the received signal onto the columns of the effective channel matrix,

we obtain the following sufficient statistics ri:

ri = ||h||xi + wi, i = 1, 2. (2.15)

is effective SNR is ||h||2 = |h1|2+ |h2|2, which is greater than that which would

have been obtained in the case of separate decoding, i.e. |hi|2, i = 1, 2. is

concept can be extended for the case where the number of detectors is greater

than 1. For the case where more than two modulators exist, a different family of

space-time codes exist. ese details can be found in [52].

2.4.3.2 Receive Diversity: Maximum Ratio Combining

Possessingmultiple detectors also results in a significant benefit in aMIMO

context. In the case where multiple detectors are present, the received signals are

combined in an appropriate manner to obtain a benefit in effective SNR, which

translates into a lower BER. Like space-time coding, receive diversity methods

such as maximum-ratio combining and equal gain combining also provide a useful

method for geing a performance gain. Here, we briefly describe maximum ratio

combining (MRC) for the 1× 2 case.

e input-output equation for the 1× 2 case can be represented as[y1y2

]=

[h1

h2

]x+

[w1

w2

](2.16)

34

where the symbols have their usual meanings. In this situation, the optimal re-

ceiver strategy is to obtain a weighted combination of the received signals on the

two detectors. e weights are the normalized versions of the channel estimates.

Assuming accurate estimates, the sufficient statistic is

r = ||h||x+ w′ (2.17)

which resembles the equation 2.15 (e noise is w′ due to the matrix transforma-

tion, but the statistics remain the same). us, receive diversity with MRC and

transmit diversity with Alamouti coding offer similar diversity benefits.

2.4.3.3 Spatial multiplexing without feedba

We now consider a spatial multiplexing scheme, where multiple transmit

modulators and detectors are employed to harness the multiplicity of paths in

MIMO systems with no feedback of channel state information to the transmit-

ter. e spatial multiplexing scheme is a coding and detection scheme which al-

lows the recovery ofmultiple streams of data by transmiing independent symbols

through different transmit antennas, and using appropriate detection algorithms

at the receiver to isolate the transmied symbols.

As in the earlier examples, we consider a simple 2× 2 example. We repre-

sent the input-output relation as[y1y2

]=

[h11 h12

h21 h22

][x1

x2

]+

[w1

w2

](2.18)

35

or, more succinctly as

y = Hx+w. (2.19)

Now, the receiver first identifies which of the streams has the higher effective

SNR, and isolates that signal. Once the strongest signal is isolated, it is used to

jointly decode the remaining streams recursively, by progressing to the next most

powerful stream. e details can be found in [40]. emaximal likelihood decoder

can be specified as

[x1, x2] = arg minx1,x2

||y −H

[x1

x2

]||. (2.20)

2.4.4 MIMO with feedba: Beamforming and Spatial Multiplexing

Feedback based MIMO methods are useful, in that they operate by sharing

knowledge on the channel state with the transmier. is is an aractive solution

when a reverse link is available between the receiver and transmier. is offers

several advantages:

• e transmier can precompensate the data, which results in improved per-

formance using more appropriate coding suitable to the current channel

state. For instance, the available power can be distributed appropriately in

the modulators to obtain the best performance benefits offered by the optical

link.

• Performing some of the compensation computations in advance at the trans-

mier significantly simplifies the algorithms to be used at the receiver. is

36

distribution of computation load could prove useful in designing signal pro-

cessing algorithms whose speeds need to scale to the high speeds offered by

optical fiber links.

In many communication systems, particularly those in which the chan-

nel is not benign, and causes distortion of the transmied signal, an open loop

approach to transmission and reception, where no exchange of channel state in-

formation occurs between the transmier and receiver, requires that the receiver

learn the channel details. However, this approach adds significant complexity

to the receiver structure, and if reliable channel state information cannot be in-

ferred by the receiver, the link performance is adversely affected. An alternate

approach, wherein the receiver conveys some information about the state of the

channel to the receiver, significantly improves the performance of the system in

many cases [53]. Even in the case of channels which vary with time, as is the case

withwireless communication, feedback of channel state at regular intervals within

which the channel is assumed to be stationary proves to be useful for transmier

preprocessing. e temporal duration over which the link is considered stationary

is termed the "coherence time".

e most significant issue with feedback of channel coefficients is to ac-

curately transmit the channel state to the transmier. In general, this is achieved

by obtaining an accurate channel estimate, and then feeding it back with much

protection in the form of redundancy, since inaccurate channel estimates under-

mine the utility of this method. While perfect channel knowledge at the trans-

mier would completely do away with the requirement of channel estimation and

37

equalization at the receiver, the complexity in this method rests on providing a

sufficiently accurate channel estimate to the transmier.

In this section, we briefly describe the a diversity scheme that uses channel

state information at the transmier, viz. beamforming, as well as spatial multi-

plexing approach that uses channel state feedback to enhance data rates.

Transmitter Receiver

h1

h2

×h∗

1

|h1|

×h∗

2

|h2|

x1

x2

s

Feed back h1, h2

Figure 2.5: A schematic showing feedback based beamforming

Figure 2.5 shows an example of feedback used for precoding information

to accomplish beamforming, which is a feedback based transmit diversity scheme,

described in the next section.

2.4.4.1 Beamforming

Beamforming is a simple scheme by which the transmier performs a pre-

multiplication on the transmied symbols in order to align them along the same

direction in the complex plane, so as to get the maximum received SNR. To put it

in perspective, beamforming corresponds to precoding to obtain an equivalent of

the maximum ratio combining by using pure transmier preprocessing.

38

To demonstrate this transmit diversity scheme in the 2 × 1 case, we refer

to the simplified form of equation 2.9 for this case, which has been presented in

equation 2.12. In order to align the symbols, we use the equal gain combining

approach [54]. is gives us the following input output relationship:

y =[h1 h2

] [ h∗1

|h1|xh∗2

|h2|x

]+ w. (2.21)

us, the effective sufficient statistic to obtain x is given by

r = (|h1|+ |h2|)x+ w (2.22)

which differs slightly from the MRC sufficient statistic derived in equation 2.17

owing to the lack of a joint power constraint. is is similar to the equal gain

transmission technique used in MIMO wireless systems [54]. us, conventional

beamforming allows benefits similar to maximum-ratio combining using simple

transmier pre-processing, thereby obviating the need for complicated channel

estimation and equalization (though these might still be necessary if the channel

estimates are not accurate). One observation in this technique is that knowledge

of the phase of the channel parameters h1, h2 is more important than complete

knowledge with amplitude, since it cophasing the transmit signals is the main

source of benefit. is gives rise to simple and efficient quantization schemes that

can be utilized for conveying channel state information [53, 55].

2.4.4.2 Feedba based spatial Multiplexing

Feedback based spatial multiplexing enables multi-stream communication,

allowing transmission of min(M,N) streams across the MIMO channel. is is

39

facilitated by possessing the channel state information at both the transmier and

receiver. As in the earlier cases, we demonstrate the method using an example for

the M ×N case.

Once more, we use the input-output equation 2.11 for the MIMO channel,

and using the singular-value decomposition, we can expand H as H = UΣV∗,

where U is a N × N unitary matrix, V is a M ×M unitary matrix, Σ is a N ×

M diagonal matrix and ∗ represents the Hermitian transpose of a matrix. Σ is a

diagonal matrix which has r non-zero diagonal elements, where r is the rank of

H. ese non-zero (non-negative) entries are known as singular-values, and are

represented as σi, i = 1, 2, . . . r.

Since the singular-value decomposition is unique, possessing the channel

information is sufficient to determine U, V and Σ uniquely.

us, to take advantage of this method, the channel matrix H must be

available at the receiver. en, the transmit symbol vector, say x0 (M × 1) is

premultiplied by the matrix V , to obtain the precoded transmit matrix, to obtain

x = Vx0. e decoder performs a premultiplication with U∗ to obtain y0. e

effective transmission matrix is

y0 = U∗y = U∗UΣV∗Vx0 +U∗w (2.23)

which can be simplified to

y0 = Σx0 +w′ (2.24)

where w′ is the transformed noise, which has the same statistics as w owing to U

is a unitary matrix.

40

Since Σ is a diagonal matrix, we can write the effective parallel transmit

equations as

y0(i) = σix0(i) +w′(i), i = 1, 2, . . . r. (2.25)

us, in effect, the channel is parallelized into r parallel channels, with the i-

th channel having an effective SNR of σ2i /N0. us, r independent data streams

can be transmied. Based on the SNR of each parallel channel, the constellation

chosen for each stream is optimized based on the effective SNR of the channel.

While spatial multiplexing is the optimal strategy to transmit multiple data

streams, an imperfect channel estimate would result in suboptimal performance.

However, an additional channel estimation and equalization step at the receiver is

sufficient to recover the data with sufficient integrity.

2.5 Conclusion

e framework described in this chapter provides a simplified encapsula-

tion of the MMF system from a DSP perspective that allows the implementation

of MIMO techniques on MIMO-MMF systems. We build upon the abstractions de-

veloped here in the experimental evaluations described in chapters 3 and 5. e

modeling section in Chapter 4 provides a more detailed description of the spa-

tial characteristics of the MMF channel and how the propagation properties of the

MMF channel can impact the performance of a MIMO-MMF link. Wherever ap-

plicable, we shall refer to the assumptions and models described in this chapter,

and specify how they apply in the various theoretical and experimental models

considered in this thesis.

41

Chapter 3

Experimental evaluation: 2 × 2 MIMO-MMF link withoff-the-shelf components

3.1 Introduction

is chapter utilizes the framework developed in Chapter 2, and discusses

the implementation of an off-the-shelfMIMO-MMF optical link. e diversity ben-

efits with the use ofMIMO are enabled by the spatial diversity inherently produced

by the MMF couplers and offsets in interconnections. While this does not guar-

antee good multiplexing performance, it still provides a quick means to evaluate

the techniques outlined in the previous experiment, and serves as motivation for

further refinement of the multiplexing techniques, as is discussed in the following

chapters. e work in this chapter has been extensively covered in [56–58].

3.2 MIMO-MMF Experimental Setup

In this section, we describe the components of the opticalMIMO evaluation

setup. A schematic is presented in Figure 3.1.

• Laser: e laser used for the experiments was a 1517 nm distributed feed-

back laser.

42

Vectorpredistortion

Complexmodulation

Vectorpredistortion

DAC

DAC

Complexreceived symbols

3 kmMultimode

Fiber

Intensity modulatedoptical signal

Electricalsignal

ADC

ADC

Vectorchannel

(impulse response)estimation

Receivedelectrical

signal(distorted)

Data bits

VectorSignal

processing(distortion

compensation)

Demodulation

Received bits

Transmitter

Digital Signal Processor

Digital Signal Processor

Receiver

Complex modulated(QAM/PSK) symbols

OpticalModulator

Laser

2 × 1Multimode

coupler

2 × 1Multimode

splitter

OpticalModulator

I

t

I

t

Feedback links

Photodetector

Photodetector

Figure 3.1: Schematic of the system setup used for the experiments.

• Modulator: eelectro-opticmodulators usedwere JDSUX5Mach-Zehnder

modulators designed to operate in the C-band.

• Optical Couplers: Multimode optical 2 × 1 (3 dB) couplers for the C-band

were used to split and combine the laser signals at the transmier and re-

ceiver.

• Optical Fiber: e optical fiber used was a silica multi-mode fiber with core

radius 62.5 µm.

• Function Generator: e arbitrary waveform generator was a Tektronix

dual channel waveform generator which has a capability of 10 GS/s. ese

are used to generate the electrical data signal which drives the electro-optic

modulators.

43

• Detectors: e optical detectors are wideband high-speed Discovery Semi-

conductor PIN detectors. ese are connected to a high-speed oscilloscope

to sample the received data.

• Oscilloscope: AHP high speed sampling oscilloscope connected to a PCwas

used to sample the signal for further processing.

We used an optical link, consisting of the pigtailed diode laser connected

to two Mach-Zehnder modulators. e signals for the multiple transmit arms

were generated as baseband signals, and were fed to the modulators from an arbi-

trary waveform generator which modulated the intensity of the laser signal. e

modulated optical signals were then combined by means of the 3 dB coupler and

launched into the 3 km section 62.5 µm diameter multimode optical fiber, whose

bandwidth-length product was rated to be 1 GHz-km. e receiver subsystem

consisted of a 1 × 2 splier, with each output arm connected to a photodetector.

e oscilloscope was used to store the received signals, and signal processing and

detection was performed offline. e transmit and receive systems were appro-

priately adapted to support different MIMO configurations (1 × 1, 1 × 2, 2 × 1

and 2× 2; where, in each pair, the former number indicates the number of active

modulators, the laer the number of active photodetectors).

A standard personal computer was used for signal processing. Some parts

of the signal processing were performed real time, but the data decoding was per-

formed offline.

44

3.3 Modulation and Coding

e concepts described in the previous chapter are sufficiently generic to

be applied to any modulation and coding methods. However, for the experiments

performed here, we adopt orthogonal frequency division multiplexing (OFDM).

OFDM [59] and DMT are the most widely used modulation techniques in both

wired and wireless standards today, wherein the data to be transmied is modu-

lated onto finely spaced frequency bands, called subcarriers. e receiver extracts

the data from these frequency bands and performs signal processing to compen-

sate for changes which the channel has caused and detects the data. OFDM is the

standard modulation technique employed by several communications standards

operating at various speeds in different media; ranging from phone-line ADSL

(ITU G.992.1) to high-speed wireless standards such as WiMAX and 3GPP LTE.

is is because of its flexible and robust operation, which simplifies tasks such

as estimation and equalization, as well as ease of implementation using the Fast

Fourier Transform (FFT). Its widespread deployment, especially in wireless en-

vironments where it is useful as a means to combat multipath pulse spreading,

suggests that it could also be useful as a tool to combat dispersion in optical fibers.

e modulation format we chose for our experimental evaluations was

OFDMwith quadrature amplitude modulation (QAM), while constraining the out-

put to real signals, owing to the availability of fast and efficient implementations

of the Fast Fourier Transforms [46]. Use of OFDM has already been suggested

for optical links [60–63], which motivates us to choose it for our implementation,

although the concepts discussed herein do not explicitly require a particular mod-

45

ulation format.

We used a MIMO-OFDM based modulation approach with signal process-

ing at the receiver to compensate for dispersion and to coherently combine copies

of data obtained at different receivers. Coded OFDM with a cyclic-prefix (CP)

is a robust and widely used technique, which, along with signal processing, sig-

nificantly simplifies the task of coherently combining delayed copies of signals

obtained in wireless systems due to the multipath propagation effect. In addition,

the same signal processing techniques extend themselves in a very simple way for

the MIMO case. In our system we used a pilot-based estimation and equalization

approach, which is used in most wireless standards. e transmier fills certain

subcarriers with data known to the receiver, and the receiver uses this data in or-

der to form an estimate of the system channel response. e advantage of this ap-

proach is that changes in the channel with time are easily captured and the signal

processing algorithms adapt themselves to these changes. We use a simple pilot

allocation scheme with equally spaced pilots in fixed locations in an OFDM sym-

bol; and linearly interpolating to obtain the complete channel frequency response

estimate, as shown in Figure 3.2. Periodically placing pilots in OFDM symbols is

necessary to compensate for the dispersion caused by propagation over 3 km of

multimode fiber. In addition, simple signal processing algorithms are meaningful,

since they lend themselves to fast implementations, which is essential for optical

systems operating at high speeds.

Finally, while the standard operation may not ensure achievement of a suf-

ficiently low BER, forward error correction was employed to reduce it to 10−9. To

46

Figure 3.2: Pilot placement and channel estimates obtained via interpolation.

achieve this, an appropriate error-control code was chosen from a family of Reed

Solomon codes.

3.4 Experiments: Characterization of Impulse Response andNoise3.4.1 Purpose

An important assumption for implementing signal processing algorithms

is that the channel is sufficiently linear and the noise statistics satisfy the standard

assumptions. Specifically, we verify that

• e system satisfies linearity assumptions in the mode of operation under

consideration.

• e noise is Gaussian and white.

3.4.2 Description of the experiments

An impulse is transmied of the modulators, and one detector is used to

characterize and store the impulse response. is is then followed by transmission

of a step signal, and the step response and the expected step response computed

47

from the impulse response are verified and compared. Finally, the statistics of the

noise are evaluated.

3.4.3 Observations and results

e observed impulse response is shown in Figure 3.3. Due to the presence

of a 10 GS/s waveform generator, the impulse is approximated by a 100 ps unit

signal. e plot is in arbitrary units.

Figure 3.3: Impulse response for an impulse (approximated as a 100 ps signal.

Figure 3.4 shows the calculated and measured step responses. Since the

predicted and measured step responses are fairly close, we can make the conclu-

sion that for the power ranges in which the system is being operated, the assump-

tion of linearity is valid.

Finally, the noise histogram is shown in Figure 3.5. In addition, the cor-

48

Figure 3.4: Computed and observed step response.

relation of the noise was evaluated and the noise sampled found to be mutually

uncorrelated. It is clear from the observation that the channel is white, thus justi-

fying the assumption that the channel is an additive white Gaussian noise.

3.5 Experiments: Modal Diversity and Advanced Modulation3.5.1 Purpose

ese experiments establish the utility of advanced modulation and MIMO

techniques in optical fibers. ey aim to demonstrate the availability of modal di-

versity in a multimode fiber, and provide means to utilize signal processing tech-

niques to enhance the reliability of signals and enhance data rates through the

fiber.

49

Figure 3.5: Distribution of the observed noise.

3.5.2 Description of the Experiments

Using the aforementioned setup, we evaluated the performance of MIMO-

OFDM in the intensity-modulation and direct-detection based optical link by trans-

miing and receiving several OFDM symbols and evaluating the bit-error rate

(BER). We used a 128-point Discrete Fourier Transform (DFT) with a 5 symbol

cyclic prefix, and utilized four equally spaced symbols as pilot symbols dedicated

to channel estimation. In order to have real baseband transmit signals, we dupli-

cated and conjugated the data on half of the subcarriers onto the other half, so

that the Inverse-DFT operation produced a real output. Bits were generated and

modulated to a adrature Amplitude (QAM) constellation. ese symbols were

split into two copies for the purpose of multiple transmiers for MIMO. e two

copies were each individually preprocessed and sent out to one of the channels of

50

the arbitrary waveform generator, operating at 10 GS/s. e waveform was gen-

erated with an upsampling factor of 2, to facilitate easy recovery at the receiver,

with tolerance for timing errors. e collected symbols were then post processed.

Pilots were used to estimate all independent channel responses, and this estimate

was used to perform equalization to compensate for dispersive effects. Finally, the

symbols were decoded to reveal the transmied data, and the BER was evaluated.

We performed the experiment in the following configurations to compare

performance over a baseline 1× 1 system:

• A single transmier (modulator) was activated and two detectors were used,

to have a 1×2 system. In the two detectors, the pilots were used to estimate

the two channel responses, and maximum ratio combining (MRC) [40] was

used to combine the two received signals to obtain the diversity benefit.

• In the second configuration, both transmiers were activated but only one

detector was used, to have a 2× 1 system. Here, twice the number of pilots

were used: one set for estimating the channel from transmier 1, and the

other set for transmier 2. We also had another layer of complex coding of

the symbols with the Alamouti code [40].

• Finally, both transmiers and receivers were activated, to get a 2×2 system.

Signaling was done in two different ways: (1) an Alamouti code was used,

much like a 2× 1 system, and (2) V-BLAST [40] was employed to send two

streams of data simultaneously.

51

For each of the above cases, the modulation schemewas characterized with

optimal performance for that configuration. For all experiments, of the available

64 subcarriers, 58 are modulated with QAM symbols, the rest being nulled or dedi-

cated to pilots. Reed-Solomon codes of different rates for forward error correction

(FEC) were employed to achieve a BER of 10−9.

3.5.3 Observations and Results

We first verified that using multiple transmit and/or receive devices pro-

vides a lower bit-error rate than the SISO case. For this, we employ a QAM-16

constellation and vary the input power and observe the BER. e observations in

Figure 3.6, shown for the 1× 2 and 2× 1 cases, show that modal diversity indeed

improves performance. e summary of the performance results for each MIMO

6 8 10 12 14 16 18

SNR10−4

10−3

10−2

10−1

100

BE

R

1× 1 (SISO)1× 2 (SIMO)2× 1 (MISO)

Figure 3.6: Measured BER vs. SNR for various MIMO configurations: Improve-ment is observed when more transmiers or detectors are employed.

52

configuration, when the laser is operated at rated power, are presented as follows:

• For the 1× 1 case, an uncoded data rate of 8.72 Gb/s is obtained with QAM-

16 with BER 1.2× 10−6. A coding overhead of 7% yielded an effective data

rate of 10−9 at 8.11 Gb/s.

• For the 1 × 2 case, a QAM-64 constellation yielded an uncoded data rate

of 13.082 Gb/s BER of 10−5. 13% FEC was needed to reduce the BER to

acceptable levels of 10−9 and achieve an effective data rate of 11.38 Gb/s.

• e 2 × 1 case with Alamouti coding shows a similar BER performance as

the 1 × 2 situation, though the requirement of additional pilots along with

FEC overheads allowed a lower effective data rate of 10.98 Gb/s for a BER of

10−9.

• For the 2 × 2 case, with an Alamouti code and QAM-64, we were able get

an effective data rate of 12 Gb/s at a BER of 10−9, since the coding overhead

reduces to 8%. With V-BLAST, we were able to send two data streams, one

at 8.1 Gb/s, and the other at 2 Gb/s at a BER of 10−9 each. e incremental

improvement over the previous cases indicates that modal coupling does not

provide four completely independent paths, and the correlation among the

paths could limit the improvements obtained by increasing transmiers and

receivers.

We were able to operate at an effective bandwidth-length product of 22.5 GHz-

km; well in excess of the rated fiber characteristic bandwidth-length product of 1

53

GHz-km [64].

us, we have demonstrated the operation of a MIMO-OFDM based op-

tical communication system, which uses simple direct detection along with sig-

nal processing techniques to compensate for modal dispersion. Accounting for

(small) overheads incurred due to forward error correction, the effective data rate

achieved is 12 Gb/s. us, MIMO and signal processing can facilitate reliable com-

munication over multimode fibers, with the operating point well exceeding the

rated bandwidth-length product of the fiber by over 20-fold.

3.6 Experiments: Feedba and Precoding3.6.1 Feedba and Precoding

In this section, we describe experiments to evaluate the performance of

feeding back channel state information to the transmier, so that precoding can

be performed to predistort the transmied waveform for dispersion and diversity

benefits. Channel state information is obtained at the receiver using a pilot OFDM

symbol consisting of known values, and the receiver's estimate of the channel is

provided to the transmier for preprocessing further transmied OFDM symbols.

For simplicity of implementation, we feed back the channel estimates directly to

the transmier without quantization. Despite this simplification, the results are

indicative of what is possible with the use of feedback based precoding. A practical

system could potentially perform beer when using more sophisticated limited

feedback techniques, where much less information feedback is required to achieve

a similar performance [53].

54

3.6.1.1 Beamforming

With the above setup, a QAM-16 constellation was used to transmit a data

stream for various input powers, and compare it with the performance with a

single transmier. Operating the laser at rated power, we employed a QAM-16

constellation and observe a bit-error rate (BER) of 9.1 × 10−7. By using a Reed-

Solomon code for error correction, we incurred an overhead of 7% and achieved an

effective data rate of 8.67 Gb/s. e constellations of received symbols are shown

in Figure 3.9. We observe from Figure 3.9b that precoding alone was insufficient to

perform complete equalization. As has been discussed in Section 2.2.2, this can be

aributed to the quality of the channel estimate fed back to the transmier, since

the estimate could be noisy. However, a simplified estimation and equalization

step at the receiver allowed us to recover the data symbols reliably. We emphasize

that the utility of the feedback based precoding is in simplifying the estimation and

equalization algorithms used at the receiver, as well as in reducing the amount of

estimation overhead, which is much larger in pilot based methods for acquiring

an accurate channel estimate.

In order to verify the presence of a diversity benefit, we measured the bit-

error rate obtained when the optical system is operated in the 2× 1 configuration

with feedback and beamforming and compared it to the performance of the 1× 1

case, and the data is shown in Figure 3.7. We observe that the slope for the 2× 1

system is beer than that for the 1×1 system, due to the more effective use of the

modal properties of the multimode fiber.

Finally, we repeated the transmission experiment with a QAM-64 constel-

55

-67 -60 -53 -47 -40 -33 -26 -20 -13Received power (dBm)

10−3

10−2

10−1

100

BE

R

1× 1

2× 1 with beamforming

Figure 3.7: A comparison of the BER vs. received power for the 1× 1 system andthe 2× 1 beamforming system.

lation, and observed an uncoded BER of 4.3 × 10−6 for a transmission rate of

13.98 Gb/s. With an overhead of 9% for a Reed-Solomon error correcting code,

the effective data rate observed was 12.60 Gb/s at a BER of 10−9. e operating

bandwidth-length product was 15 GHz-km. In order to visualize the utility of the

processing technique, the average SNR of each subcarrier is shown in Figure 3.8

for both the 1 × 1 case as well as beamforming case. is was obtained by av-

eraging the signal properties over 100 OFDM symbols which were during which

channel conditions did not change appreciably. We can infer that the signal pro-

cessing, indeed, is able to utilize the diversity present in the system. Moreover, the

benefits are more pronounced at the higher subcarriers, owing to their presence

at higher frequency bands, where the modal dispersion is higher.

56

0 10 20 30 40 50 60Subcarrier index

4

5

6

7

8

9

10

11

12

13

Ave

rage

SN

R(d

B)

1× 1

2× 1 with beamforming

Figure 3.8: A comparison of average SNR of the 1 × 1 and beamforming (2 × 1)are compared for each subcarrier.

(a) (b) (c)

Figure 3.9: Constellation diagrams at various stages at the receiver: (a) Unequal-ized: No precoding, (b) Unequalized: precoded, (c) Aer equalizer.

3.6.1.2 Spatial Multiplexing

Weconducted a spatialmultiplexing experiment as described in Section 2.4.4.

In this case, the channel information is utilized by both the transmier and receiver

to perform precoding and post-compensation using the singular-value decompo-

57

Figure 3.10: BER vs. received power for the two spatial multiplexing streams com-pared with standard 1× 1.

sition. Two parallel data streams are transmied over the 2× 2 channel using the

singular value decomposition (SVD) technique described in [40]. e amount of

overhead required for the feedback is about 100 bits for every 10 Mb of data for

the beamforming case, and 200 bits for the spatial multiplexing case, which is less

than 0.001%.

eBER vs. received power for each of the two spatialmultiplexing streams

is shown in Figure 3.10. e power represented on the x-axis of the figure is the

net mean power launched into the multimode fiber. e fact that the 1× 1 curve

and the first (higher SNR) spatial stream follow a similar trend indicates that we

can expect a performance and data rates from the first stream that is similar to a

58

Table 3.1: Results for feeedback based schemes.

MIMO seme Data Rate (Gb/s) @ BER 10−9

1× 1 8.112× 1 (Beamforming) 12.6

2× 2 (Spatial multiplexing) 12.2

conventional 1× 1 link. However, since the second stream offers a higher BER, it

is likely to support a lower data rate than the first stream.

We observed that the two streams which could be sent at a BER of 10−9

were 8.1 Gb/s and 4.1 Gb/s respectively. While this is an improvement over the

V-BLAST data rates obtained in [56] under a similar experimental setup, it is not

a significant improvement over beamforming. is can be aributed to multiple

causes. First, the modal diversity present in the system is likely insufficient to

produce a significant improvement in the 2 × 2 case, owing to correlation of the

paths (this possibility has been alluded to in [65,66]). is effect is compounded by

the fact that the channel estimate passed back from the receiver to the transmier

is not sufficiently accurate to achieve the best possible transmission rate owing to

implementation constraints. With a more sophisticated channel estimation and

feedback mechanism, this performance is likely to improve. e effective data

rates are summarized in Table 3.1.

3.7 Discussion

In this section, we analyze some of the observations and compare them to

theoretical predictions. In particular, we comment on the diversity order observed.

59

It was shown in [56] that the the measured diversity order for the experimental

cases is about 1.5, which is less than the ideal diversity order 2. Similarly, the ex-

perimentally evaluated performance of themultiplexingmethods discussed in [56]

is found to be less than that which could be expected in a conventional MIMO sys-

tem with appropriately similar parameters. is can be aributed to the fact that

the system, as is, does not fully leverage all available independent paths, and there

is a significant correlation among the modes utilized during transmission [65,66].

Nevertheless, the performance benefits was obtainedwithout explicit optimization

of launch conditions, which indicates the simplicity in realizing such systems. e

presence of modal diversity can be aributed to the modal dynamics of the MMF

which cause variations in fiber characteristics that can be exploited by signal pro-

cessing techniques such as those discussed in this thesis. A detailed theoretical

consideration of the mode characteristics can be found in [67, 68]. In addition,

we remark that the diversity achieved in this scheme is comparable, or even an

improvement, over the diversity achieved in wireless systems, primarily owing to

the presence of correlated signals on wireless transceiver antennas.

is experiment was conducted using off the shelf equipment, including

conventionalMMF couplers, similar to the experiment described in [28], and this is

sufficient to exploit the the multiplexing capabilities of the multimode fiber. In or-

der to fully utilize the spatial diversity offered by the system, several methods such

as offset launch and multisegmented detectors [30] can be considered to increase

diversity. is, along with dispersion compensation by means of channel estima-

tion and equalization could close the gap between the observed performance and

60

theoretical best performance. Implementing a more efficient system using these

approaches is a topic for future research.

e feedback mechanisms we use in this thesis represent simplistic as-

sumptions on the reverse channel. However, based on the actual coherence time

of the optical link, an improved feedback quantization method can be developed,

which is tailored appropriately to the requirements of the link in question. By

using the best quantization scheme for the optical link in question, the maximum

benefit from feedback can be obtained for a given constraint in the rate available

from the feedback channel.

e direct detection based approach results in a significant simplification

of system implementation in terms of transceiver complexity. e significant ad-

vantage of this approach is that it obviates the need to have a laser at the receiver

that is matched in frequency and phase to the transmit laser to aid carrier recov-

ery, while also not needing inteferometers and matched detectors, thus making

it much more suitable for inexpensive links. Coherent detection is also suscepti-

ble to phase noise of the laser, since the phase of the laser carries the modulated

data [65]. Since phase shi modulation encodes the phase onto the phase of the

laser signal, sudden variations in the laser phase owing to phase noise would cause

erroneous detection at the receiver, both in case of PSK modulation [69] as well

as advanced modulation such as OFDM [70]. However, in the case of intensity

modulation, the signal is carried on an rf carrier or baseband signal modulated

onto the laser intensity. Due to the fact that the phase of signals with in the rf

range can be maintained with sufficient fidelity at the frequencies of interest to us

61

in this experiment (10 GHz), intensity modulated communication does not suffer

from the phase noise limitation.

Finally, it remains to be seen how the performance would scale in prac-

tice with more than two devices at the transmier and receiver from a MIMO and

signal processing perspective. Specifically, the modeling and design of optical

components with large numbers of lasers/modulators and detectors, and improv-

ing signal processing techniques to scale the performance in MIMO-MMF systems

is a topic for future research.

3.8 Conclusion

In this chapter, we have described a framework for building a MIMO-MMF

system using off-the-shelf components to enhance data rates using signal process-

ing and the inherent diversity present in the system. Such a system uses MIMO

techniques similar to those developed for wireless and some optical systems, with

a key difference being the use of incoherent intensity modulation in this imple-

mentation, compared to coherent techniques used in conventional MIMO systems.

Modal dispersion is traditionally seen as the bane of fiber optic communication,

and considerable efforts are devoted in nulling, canceling and avoiding it through

fiber design and other optical means. We illustrate that signal processing tech-

niques can naturally be used to compensate for modal dispersion and, more im-

portantly, that themultimode nature of the fiber is not a necessarily an impairment

for fiber optic communication. e presence of multiple modes results in multiple

coupled but distinct paths from source to destination, which can be leveraged using

62

vector signal processing techniques to get significant performance improvements

in optical systems. In addition, we used feedback and preprocessing effectively

to combat the distortion introduced by the system, and thus allowed for flexible

implementation of signal processing algorithms. Our experiments revealed that

the efficient use of multiple modulators and detectors and signal processing with

feedback in MMF links enable data rates in excess of 12 Gb/s over a multimode

fiber; exceeding the bandwidth-length product by a factor of 15. Further exper-

iments would involve refining the feedback methods and studying the effect on

optimizing coupling conditions via offset coupling and detection of the signals

to improve diversity gains in the MMF link. e forthcoming chapters aempt

to address some of these concerns, both using theoretical studies as well as with

experimental validation.

63

Chapter 4

Analysis and Design of Laser and Detector ArrayGeometry

4.1 Introduction

e previous chapters have introduced the basic use of MIMO techniques

in incoherent MMF links, along with a comparison of how these techniques differ

from the accepted use of MIMO in wireless communication systems. An experi-

mental evaluation with off-the-shelf components has been described in Chapter 3,

although its performance was limited due to the unavailability of control over

which modes or mode groups were excited. Further improvements would require

greater access to a wider array of modes of the MMF to realize the benefits that

modal diversity has to offer. In this chapter, we consider a theoretical and simu-

lation based formation on how arrays of lasers and detectors, along with MIMO

techniques, can enhance MMF links.

e primary motivation for this work stems from antenna placement con-

cepts in wireless MIMO systems. In wireless MIMO systems, carefully designed

antenna placement ensures the independence of channel coefficients, which en-

hances the channel's reliability and multiplexing capabilities [40]. e conditions

for obtaining the maximum diversity in a wireless link are met by separating an-

64

tennas of a wireless system with appropriate spacing; a condition that is easily

satisfied, given the unguided nature of the wireless propagation medium. e

analogous problem in the MIMO-MMF context is to determine optimal placement

strategies for lasers and detectors, since this would enable effective separation

of signals across various orthogonal modes of the MMF. is problem is compli-

cated by the fact that the modal diversity in MMFs stems from orthogonal modes

that occupy the same spatial region of the guiding medium (the MMF core). is

makes it difficult to draw conclusions regarding the optimal signal launch and de-

tection geometries that maximize data rates in MIMO-MMF links. To this end,

we adopt the field propagation analysis technique developed in [71] and apply it

to the MIMO-MMF channel to study the role of laser and detector placement on

link performance. Our simulations indicate that optimized placement strategies

yield 2-3 times the achivable rate of arbitrary placements, due to improved modal

diversity.

While earlier work has shown the usefulness of MIMO techniques for con-

ventional MMFs [26, 29, 30, 32, 33, 56], a fundamental analytical framework for

MIMO communication over MMFs is yet absent. Offset coupling, namely the

launch of signals into multimode fibers with a radial offset to the fiber axis, to

improve the bandwidth-length product, has been shown to be a useful technique

to improve MMF data rates [31, 35, 72]. e feasibility of using a higher numbers

of lasers and detectors to improve data rate performance was analyzed in ref. [33],

where a power diffusion approach was employed to quantify the effectiveness of

sending multiple parallel streams of data through the fiber using mode groups.

65

However, the impact of device placement geometry on the achievable data rates

over MIMO-MMF has not been studied.

In this chapter, we utilize a field propagation analysis technique [71], and

develop a framework to optimize laser and detector placement by maximizing the

(information theoretic) achievable rate of MIMO-MMF links. Simulations using

thismodel revealed that laser and detector geometry on the input and output facets

of the fiber affects the achievable rate, and this was used to determine configura-

tions of lasers and detectors that achieve the largest information theoretic data

rate using a grid based search. For fine grids, exhaustively searching for these

configurations becomes computationally expensive, and we introduce a submod-

ular, “greedy” search that produces detector array geometries for fixed laser arrays,

that aain in excess of 90% of the achievable rate of the optimal configurations,

while requiring far fewer computations (over 99% reduction). e greedy search

approach aggressively seeks to place detectors where the dominant modes' power

is received. While the efficient greedy search technique is useful for developing

static detector arrays, the small number of computations can also be used to enable

dynamically reconfigurable detector arrays for reduced complexity signal process-

ing, as described along with some examples in this chapter.

e chapter is organized as follows: Section 4.2 describes the physical

model of the fiber channel and its statistical nature. Section 4.3 outlines the formu-

lation of a MIMO system matrix for each channel realization, develops the input-

output model, and discusses the metrics for determining the quality of the channel

for given device configurations. Section 4.4 discusses the techniques for efficient

66

optimization of source and detector array geometries. Section 4.5 describes the

simulation results for select configurations and shows that a “good” configuration

can be found under certain constraints. Clustering the arrays to consolidate them

into segmented detectors having beer fill factors is also discussed. Section 4.6

describes a low complexity algorithm based on greedy selection that considerably

simplifies the efficient use of dense detector arrays while not compromising the

MIMO benefits that can be obtained. Finally, Section 4.7 concludes the chapter.

e work in this chapter has also been covered in [73–78].

4.2 Multimode fiber model4.2.1 Propagation matrix

To model signal propagation through a MMF, we build upon tools devel-

oped by Shemirani et al. [71] to arrive at MIMO and signal processing metrics that

can be optimized. Other models that can potentially be used to model propagation

in MMFs is the diffusion power flow approach which treats modal coupling as a

continuous power diffusion equation along the length of the fiber [79]. is dif-

fusion model assumes that coupling only occurs between nearest neighbor modes

and accounts for a power loss mechanism using a mode-dependent parameter that

can be measured experimentally [80]. While this approach is suitable for model-

ing modal coupling, it does not account for other factors, including the polariza-

tion of the electric field and changes to the polarization by fiber non-idealities

and birefringence. In contrast to the diffusion-based model, our approach treats

modal coupling in a perturbation framework where the perturbing effect is due

67

to bending and twisting of the fiber, thus making it suitable for analyzing MIMO

transmission over real MMFs.

e modes of an optical fiber form a spanning set for the solution of the

Helmholtz wave equation. ese eigenmodes are the well-established Hermite-

Gauss functions [81]. us, we can decompose an electric field distribution guided

by the fiber into a linear combination of these eigenmodes with complex coeffi-

cients. Using this basis, a vector can that fully describes the electric field profile

in the cross-section of the fiber can be constructed. is entries of this vector are

the projections of an incoming electric field profile at the input facet of the optical

fiber onto each of the fiber eigenmodes. We refer to this as the “mode-vector”.

To find the mode vector for a particular laser/detector, we assumed that the

device electric field is a circularly symmetric Gaussian beam (TEM00 mode) polar-

ized randomly along the x-y plane (assuming that z is the propagation direction),

and that the size of the beam reflects the effective area of the device. is assump-

tion is motivated by the fact that small devices, such as VCSELs, produce near-

Gaussian beams and can be readily fabricated into two dimensional arrays [82].

Let EL(x, y) be the incoming electric field due to a laser source polarized in the

x-y plane, where the subscript L denotes a laser. e divergence of the laser beam

is considered small, as the devices are assumed to be bu-coupled to the fiber face.

Let E(pq)F (x, y) be the electric field distribution of the fiber eigenmode indexed by

mode numbers (p, q) and polarized in the x-y plane. en the complex entries of

68

the vector corresponding to the incoming electric field are given by [81]

apq =⟨EL, E(pq)F ⟩√

⟨EL, EL⟩⟨E(pq)F , E(pq)F ⟩(4.1)

where ⟨A,B⟩ =∫∫R R

A(x, y)B∗(x, y) dx dy.

Using the overlap integral, the input vector can be found as follows:

aL =

⟨EL, EF1⟩⟨EL, EF2⟩

...⟨EL, EFM

⟩

(4.2)

where aL is an M × 1 vector of complex numbers, with M being the number of

modes propagating in the fiber. e number of modes supported by an optical

fiber depends on the core diameter, wavelength of light, and the differences in

refractive indices of the core and cladding [2]. In particular, the number of modes

increases with increasing core diameter. For instance, a weakly-guided multimode

fiber with a diameter of 50 µm, nominal refractive index of 1.45 and core-cladding

refractive index difference of 0.01 supports 55 guided modes at 1550 nm. However,

the electric field is polarized in the cross-section of the fiber and contains x and y

components for each of the guided modes of the fiber. In other words, accounting

for the two states of polarization of each spatial mode, the fiber possesses a total of

55× 2 = 110 modes. us, M defined here is twice the number of guided modes

in the fiber and our model takes into account polarization variations of each mode.

e simulations performed in this chapter assumed a source wavelength 1550 nm

sources, since this wavelength lies in the lowest loss window of the fiber.

69

+50% 50%

Figure 4.1: Uiprop is a random matrix that describes intermodal coupling with in a

section. In particular, it transforms a vector containing the weights of each guidedmode to provide a vectorwhich has the newweights aer the signal has undergoneintermodal coupling within the fiber section.

Figure 4.2: Ri is a random matrix that describes rotation of the polarization of theelectric field at section junctions. It rotates the polarization of each mode withinthe fiber section based upon propagation effects of the fiber.

Once an input vector is determined, an M ×M propagation matrix Utotal

transforms the input electric field vector into the output electric field vector. e

propagation matrix includes information about the power transfer between the

eigenmodes and losses incurred during propagation from fiber perturbations. De-

composing the fiber intoN infinitesimally short, cascaded sections, a propagation

matrix can be found for each of the i sections, Uisection. Multiplying these together

yields the total propagation matrix Utotal =∏N

i=1Uisection.

Figure 4.3: Mi is a randommatrix that describes rotation of the electric field profiledue to fiber twists within the ith fiber section.

70

e propagation matrix for the ith section is a product of three matrices

that account for three different effects. e first, Uiprop, encapsulates the effects of

intermodal coupling due to the curvature of the fiber. As shown in Figure 4.1, a

small curvature causes coupling of a mode into its nearest neighbor modes. is

assumption is valid in situations where the bend radius ri is large in comparison

to the fiber core diameter. In Figure 4.2, a second matrix, Ri, accounts for the

rotation of the electric field vector at section junctions due to fiber twists. Fi-

nally, Figure 4.3 shows matrix Mi that accounts for mode profile rotations also

due to fiber twists. e propagation matrix for the ith section is then given by

Uisection =

∏Ni=1M

iRiUiprop, which combines the effect of fiber variations on the

modes within a section. For each section, these matrices are generated based on

statistics representative of physical fiber parameters [71].

To determine the response at the photodetector, an output vector aD de-

scribing the coupling from the output facet to the photodetector is found using a

technique similar to Equation (4.2). e inner product of this vector with the out-

put field vector at the output facet of the optical fiber, along with additive noise

from the photodetector, can be wrien conveniently in matrix notation as

y = aHDUtotalaLx+ n (4.3)

where x represents the scalar input signal, y represents the output signal at the de-

tector, and n represents the additive noise. Note that aHD represents the conjugate

transpose of aD.

71

4.3 Analysis of MIMO system matrix

In this section, we derive a MIMO system matrix model from the propaga-

tion framework for the MMF discussed in the previous section. Subsequently, we

use this model to analyze how device placement can affect the performance of the

link from a signal processing purview, and examine and design optimal laser and

detector placement strategies to improve achievable data rates through the MMF.

4.3.1 System transfer matrix

To derive aMIMO channel matrix, for each laser and detector in the config-

uration, an input vector describing its electric field profile can be formed using the

techniques of Section 4.2.1. ese can be combined to obtain an effective channel

matrix that describes the MIMO input-output channel state from the transmier

to the receiver. Performance metrics, such as capacity, can be calculated with this

MIMO matrix. Such models can be used to evaluate the effectiveness of a partic-

ular laser/detector configuration in terms of information theoretic rate.

We assumed that the time-varyingmatrixUtotal, describing themodal trans-

formation induced by theMMF, changes slowly with time. is enables estimation

of the fiber transformation and compensation for its effects [28, 56, 83]. e im-

plication of this assumption on the optimization metric considered is discussed in

Section 4.4. e effect that these changes cause on the data rate are factored into

our model.

Using the overlap integral technique described in Equation (4.2), the input

coupling vectors of each of the NL lasers, represented as aL1, aL2

, . . . aLNLcan be

72

found. e output coupling vectors describing the overlap of different modes with

each of the ND detectors, represented by vectors aD1, aD2

, . . . aDNDcan also be

calculated similarly. Since the overlap integral satisfies the properties of the inner

product, the projection of the initial vector induced by the array of lasers as well as

the projection of the modes onto the array of detectors can be described as follows:

H(t) = AHDUtotal(t)AL (4.4)

AL =

· · ·aL1

aL2· · · aLNL

· · ·

AD =

· · ·aD1

aD2· · · aDND

· · ·

Here, H is the ND ×NL matrix describing the channel state of the fiber channel,

andAL andAD are, respectively,M×NL andM×ND matrices that describe the

modal interaction of the lasers and detectors with the optical fiber. is framework

allows the input-output relationship to be described in a compact fashion with in-

put signals x1, x2, . . . xNL, outputs y1, y2, . . . yND

and detector noise n1, n2, . . . nND

as y1(t)y2(t)

...yND

(t)

= H(t)

x1(t)x2(t)

...xNL

(t)

+

n1(t)n2(t)

...nND

(t)

. (4.5)

In the equations above, the signals xi(t), 1 ≤ i ≤ NL modulate the ith laser, while

the signals yj(t), 1 ≤ j ≤ ND are received at the jth photodetector. To analyze

73

100 MHz bin

Figure 4.4: Binning approach to to evaluate the sum rate of the fiber across all fre-quency ranges. e frequency response of the fiber channel was split into severalbins of 100 MHz each, and the achievable rate was evaluated within each bin as-suming a frequency-flat channel response, and added up to get the net achievablerate. is can be thought of as a frequency division multiplexing approach to rateevaluation.

the performance of the MIMO system, we assumed that the system is in the ther-

mal noise limited regime, as opposed to being shot noise limited. erefore, the

additive noise was modeled as white and Gaussian [28]. To factor in the effect of

dispersion, the achivable ergodic rate was evaluated over various frequency bins,

assuming that the channel remained flatwithin each bin, and thesewere integrated

to get a net rate, as shown in Figure 4.4. e frequency bin for each flat subchannel

was chosen to be 100 MHz, and rate values were evaluated up to 10 GHz, beyond

which dispersion effects made the frequency response insignificant from a data

rate perspective. We determined that using a frequency domain bin size of 100

MHz provides a sufficiently accurate evaluation of the frequency-domain integral.

In the following subsection, we briefly describe the metrics considered and

the techniques adopted to optimize device geometries for increasing data rates.

74

4.3.2 Metrics for optimization of device configurations

e primary metric we use for optimizing the configurations of laser and

detector arrays is the maximum achievable data rate through the fiber. For char-

acterizing the potential data rate through the fiber, we considered the achievable

ergodic rate obtainable in the fiber channel without channel state knowledge at

transmier, given by [40]:

C = E

[log2 det

(IND

+ρ

NL

HHH

)][b/s/Hz] (4.6)

where INDis the ND × ND identity matrix, ρ is the signal-to-noise ratio (SNR),

and det represents the determinant of a matrix. Our analysis assumed a narrow

frequency band signal within each subband. us, the rate could be obtained in

bits-per-second by integrating it over the frequency response of the channel, as

described in the previous subsection. e utility of ergodic rate as the metric for

optimization is that it averages over the variations that occur in the fiber channel

with time, thereby ensuring that the optimal array performs well under varying

fiber channel conditions in the absence of channel state information at the trans-

mier.

Ergodic capacity is defined as the maximum rate that is obtainable through

the fiber channel averaged over all realizations of the channel state H, optimized

over all input distributions. erefore, it is based solely on the statistics of the

channel, as opposed to specific realizations of channel states. In the case of a Gaus-

sian MIMO channel whose channel matrix coefficients are i.i.d. Rayleigh entries,

it has been shown that using i.i.d. Gaussian signaling on each transmit antenna

75

is optimal [84]. However, for the MIMO-MMF channel considered in this chapter,

the structure of the input/output relationship obtained by placing lasers and de-

tectors in an array complicates the characterization of the input distribution that

aains the ergodic capacity. In general, the obtaining optimal input distribution

that achieves the ergodic capacity would require numerical computation of the in-

put covariance. us, we restrict ourselves to using achievable ergodic rate using

Gaussian signaling with an identity covariance matrix as our optimization metric.

In practice, since the fibers under consideration possess weak mode coupling and

small mode-dependent losses, the achievable ergodic rate can be expected to be a

close approximation to the ergodic capacity of the fiber [71].

It must be noted that operating close to the rate as characterized by Equa-

tion (4.6) would require amplitude modulation and coherent detection. Intensity

modulation allows modulation of only half of the laser waveform swing, and no

ability to use the laser phase. e model that we considered accommodates the

use of intensity modulation by restricting the data signals to be positive and real.

Since it is known that the capacity trends with increasing modulators and detec-

tors for incoherent detection shows an increasing trend, similar to the coherent

case, at high SNR [40], the ergodic rate trends with various device configurations

are also representative of rate trends in systems that employ direct detection.

4.4 Optimizing placement of devices

In this section, we discuss two different grid-based techniques for obtain-

ing the optimal laser and detector configurations that maximize the ergodic rate

76

achievable over the channel. e first, an exhaustive search over all possible de-

vice placements, yields the highest achievable rate for the grid; however, this

method required significant computation time and became prohibitive for fine

grids. e second method, a submodular optimization by “greedy” selection, re-

duced the number of computations required for evaluating the detector array for

a pre-selected laser configuration. With greedy selection, the rate obtainable with

the resulting configuration is only guaranteed to be within a constant factor of

the maximal (ergodic) rate for the grid under consideration. While this may ap-

pear to be a limitation, it enabled the design of detector configurations for finer

grids where an exhaustive search was computationally not tractable with stan-

dard computational resources. Even in coarse grids where an exhaustive search

can be used, the greedy algorithm resulted in a configuration that aained a rate

fairly close (over 90%) to the maximal rate, with fewer computations. is also

motivates the development of dense arrays of detectors from which a subset can

be selected dynamically to enable reduced complexity signal processing. For our

analysis and for ease of implementation, we restricted ourselves to circular lasers

and detectors which formed arrays. However, to improve fill factors and adapt to

other geometric constraints, we suggest an alternate mechanism in Section 4.5.5

to suitably adapt the resulting array structures. e techniques presented here are

applicable for designing device arrays with grids of arbitrary geometries, but in

this section, we have restricted the discussion to square grids to allow for easy

comparison between the various computational approaches.

77

4.4.1 Exhaustive sear

An exhaustive search for the optimal device configurations on an N ×

N square grid involves calculating the achievable rate of the channel for each

possible combination of lasers and detectors on the grid positions and selecting

the configurations that aain the highest rate. Leing the number of lasers be

NL and the number of detectors be ND, this amounts to an exhaustive search

over(N2

ND

)×(N2

NL

)possible device configurations. While this method always yields

the maximum achievable rate for the channel, it requires significant computation

for a fine grid or large number of devices. For instance, using a 10 × 10 grid to

determine the placement of 20 lasers and detectors would require(10020

)×(10020

)≈

1041 comparisons. us, we restrict ourselves to applying exhaustive searches

only on coarse grids.

4.4.2 Submodular sear

To design arrays with a larger number of devices, a finer grid structure of

the fiber core region is necessary. However, as discussed in the previous section,

determining optimal device placements was computationally prohibitive in such

structures. We thus considered an alternative optimization technique, based on

the observation that the rate function satisfies the property of “submodularity.”

Submodularity enables the use of a “greedy selection” algorithm that performs this

optimization to within a constant factor of the global maximum, while requiring

a significantly lower number of computations. In wireless MIMO systems, it is

known that the problem of selecting a subset of antennas at the receiver to utilize

78

for signal detection is submodular [85]. In the MMF case, the link rate is observed

to be submodular with respect to the grid of detectors chosen, while keeping the

laser configurations fixed. Let the set of potential detector locations on theN×N

grid be indexed by the set U = {1, 2, . . . N2}. e definitions and performance of

submodular optimization are clarified in the theorems below:

eorem 4.4.1. e ergodic rate achievable over MIMO link with no channel state in-

formation (information about the channel matrixH) at the transmier given in (4.6)

is submodular in the choice of the subsets of receive devices (detectors) chosen. In

other words, if the rate achievable at SNR ρ for a subset of receive devices S of size

ND, denoted by RS with is given by:

RS = log2 det

(IND

+ρ

ND

HSH†S

), (4.7)

where H†S represents the Hermitian transpost of HS , then RS is submodular in S ⊆

U . In addition, the function is monotone in the subsets; i.e., for S ⊆ T ⊆ U , we have

CT ≥ RS .

e proof for this theorem is provided in Appendix A.e fact that the rate

is submodular in the subset of detectors used leads to a performance guarantee

with the use of Algorithm 1, that is based on greedy selection.

eorem 4.4.2. Let R∗ be the achievable rate when the ND detectors placed in the

best ofN2 grid locations, andRS be the data rate achievable with detectors placed at

the locations given by the set S of sizeND, the output of the greedy selection approach

described by Algorithm 1. en we have:

79

Algorithm 1 Greedy selection algorithm

initialize S = , U = {1, 2, . . . N2}for i = 1 to ND do

select d = argmaxx∈U\S

(CS∪{x} −RS

)set S = S ∪ {d}

end forreturn S

C∗(1− e−1) ≤ RS ≤ C∗. (4.8)

e proof for the general case of eorem 4.4.2 is provided in several texts,

including [86].

eorem 4.4.3. e achievable ergodic rate of the multi-device link with no channel

knowledge at the transmier, which is given by:

E[RS] = E

[log2 det

(IND

+P

NL

HSH†S

)]

(where the expectation is over the probability distribution of H), is submodular and

monotonic in the choice of the subsets of receive devices (detectors) S ⊆ U .

Proof. Sinceeorem 4.4.1 holds for eachH, and a linear combination of submod-

ular functions is submodular, the ergodic rate is submodular. Since monotonicity

is also proved pointwise, ergodic rate is also monotone in the choice of the receive

devices.

80

It must be noted that the submodular optimization technique is useful only

for optimizing detector configurations for a pre-determined laser array configura-

tion. is is because the submodularity ofMMF link rate does not extend to subsets

of lasers selected from a grid. us, when utilizing submodular optimization, we

restricted the analysis to pre-determined laser configurations and focused on op-

timizing the detector configurations. e restriction of structure imposed due to

submodularity did not allow laser positions to be optimized using this technique.

Although this appears to be a limitation, significant performance benefits can be

obtained even with regular geometries for laser arrays [76]. For finer grids, an

exhaustive search was prohibitive, and submodularity provided an alternate and

more viable solution. In fact, the rate obtained using a greedy search guaranteed

to be within a factor of (1− e−1) of that obtained with the optimal detector array

searched exhaustively. is provides a computationally feasible way to search for

device configurations that are reasonably close to optimal for fine grids. Without

additional structure in the problem, a performance guarantee beer than (1−e−1)

cannot be obtained with a polynomial-time algorithm. It can be proved that with

just submodularity and no additional structure, obtaining any guarantee on op-

timality beer than a factor of (1 − e−1), i.e. a guarantee of (1 − e−1) + ϵ for

any ϵ > 0, would require exponentially large computation resources for the most

general optimization problems [87].

81

4.5 Simulation results

We performed simulations on Matlab to optimize laser and detector con-

figurations that yielded the greatest ergodic rate over the ensemble of channel

realizations under various scenarios. From Section 4.3, for a given configuration,

the system matricesH are derived fromUtotal matrices. An ensemble of randomly

generated Utotal matrices corresponds to the ensemble of system realizations and

thus, an ensemble of system matrices H. We generated 700 realizations of Utotal

matrices. To generate eachUtotal, themodel split the graded-index fiber into 10,000

sections, each 10 cm in length. e fiber had a diameter of 50 µm, core index of

refraction of 1.444 and a numerical aperture of 0.19. e lasers were assumed

to operate at a wavelength of 1.55 µm, and the spatial electric field paern they

produced was assumed to be circularly symmetric. e choice of 1550 nm was

made since this wavelength band has a low propagation loss through the fiber.

In addition, the field propagation approach for modeling MMF behavior has been

experimentally shown to be accurate [88]. e detectors' response was evaluated

based on the overlap of the received signal on each detector. e statistical nature

of the fiber is a result of the curvature and twists in each section which are mod-

eled by parameters κi and θi respectively. Both are modeled as Gaussian random

variables with κi having a standard deviation of 0.95 m−1 and θi having a stan-

dard deviation of 0.6 radians. ese parameters were determined by correlating

experimentally observed beam profiles obtained aer propagation through a 1 km

graded-index multimode fiber for tuning the model to match physical parameters.

An example of simulated and physical realizations of the fiber beams is provided

82

1 km MMF(measured)

1 km MMF(simulated)

Figure 4.5: e beam evolution over 1 km of graded-index multimode fiber. ephysical fiber measurement is performed using a beam profiler, while the simu-lated profile uses one channel realization obtained using the fiber model.

in Figure 4.5. e 10 cm fiber sections that the model considers are sufficiently

small in comparison to the radius of curvature of the fiber, thus being effective

in characterizing modal propagation effects [71]. To simplify the simulation com-

plexity in this study, the device configurations were restricted to 2-D rectangular

arrays centered about the fiber axis. e motivation for this assumption comes

the fact that the fabrication of devices with this geometry has been demonstrated

in practice [82, 89].

is section demonstrates two ways of enhancing the rate achievable over

the channel. e first approach involves increasing the total number of lasers and

detectors, and the second approach involves fixing the laser and detector positions

of anNL×ND MIMO system selectively on the grid and showing the existence of

optimal device configuration on the grid. is optimal device configuration was

found through exhaustive search over all possible the device positions on the grid.

When the grid becomes fine, an exhaustive search became computationally ex-

pensive. Instead, a suboptimal “greedy” search was performed for detector arrays,

83

for a fixed laser array. We compared the values of ergodic rate obtained using the

exhaustive search and greedy search based algorithms for a coarse grid to establish

its utility.

4.5.1 Rate benefits due to MIMO

To demonstrate the effect of increasing the number of devices on the rate

achievable over the MMF link, simple segmented lasers and detectors were consid-

ered, as shown in Figure 4.6. e mode field diameter of the lasers and detectors

was assumed to be 45 µm, and the ergodic rate was evaluated for 1× 1, 2× 2 and

3× 3 links with these laser and detector geometries. In each case, the achievable

rate was averaged over 700 randomly generated system matrices that represented

the channel conditions to obtain the ergodic rate, as stated in Equation (4.6). e

results of this simulation are shown in Figure 4.7. Such simple segmented lasers

and detectors may not necessarily be the optimal geometries to extract the max-

imum data rate from the MIMO link; nevertheless, the results of this simulation

indicate that a significant increase in data rates can be obtained with MIMO tech-

niques. Compared with a 1× 1 system at 10 dB SNR, the optimal 2× 2 and 3× 3

implementations yielded gains of 40% and 80% respectively, supporting the claim

that when an increasing number of devices is used, the rate improves significantly.

4.5.2 Effect of device positions on aievable rate using exhaustive sear

To examine how the achievable rate over a NL × ND MIMO system can

be increased with appropriate choice of laser and detector positions, a 3 × 3 grid

84

Laser/Detector

Fiber core

(a) 1× 1

Laser/Detector

segment

(b) 2× 2 (c) 3× 3

Figure 4.6: Multiple lasers and detectors. e devices were assumed to fill 90% ofthe fiber core area.

0 2 4 6 8 10SNR (dB)

0

1

2

3

4

5

6

7

8

9

Ach

ieva

ble

rate

(Gb

s−1)

1× 1

2× 2

3× 3

Figure 4.7: Achievable rate versus SNR for 1× 1, 2× 2, and 3× 3 MIMO systemsfor the best device configuration

was used as an example for optimizing the geometries of 2 × 2 and 3 × 3 MIMO

systems. e average rate was found for each combination of 2 (or 3) lasers and

2 (or 3) detectors on the grid. Figure 4.8(a) shows the rate versus SNR of a 2 × 2

MIMO system for the best combination, and compares it with the average rate

over all possible positions of lasers and detectors. e rate obtained using the

85

0 2 4 6 8 10SNR (dB)

0

1

2

3

4

5

6

Ach

ieva

ble

rate

(Gb

s−1)

Best configurationAverage over all configurations

(a)

Lasers Detectors

(b) Best configuration

Figure 4.8: (a) Achievable rate versus SNR for a 2× 2 MIMO system for the “best”device configuration that achieves the highest rate and the average over all possi-ble configurations. (b) e configuration of lasers and detectors in the best config-uration. e four circles at the boom represent the fiber cross sections and thelaser and detector placements on the cross sections that yielded both the best andsuboptimal configurations.

0 2 4 6 8 10SNR (dB)

1

2

3

4

5

6

7

8

Ach

ieva

ble

rate

(Gb

s−1)

Best configurationAverage over all configurations

(a)

Lasers Detectors

(b) Best configuration

Figure 4.9: (a) Achievable rate versus SNR for a 3× 3 MIMO system for the “best”device configuration that achieves the highest rate and the average over all pos-sible configurations. (b) e configuration of lasers and detectors in the best con-figuration.

best arrangement of devices far exceeds that obtained when the device placement

is done randomly, as is clear from the fact that the rate averaged over all posi-

86

tions is significantly smaller. In this case, the data rate of the best configuration

at an SNR of 10 dB is more than twice that of the average over all configurations.

e location of the lasers and detectors that obtains the best rate is shown in Fig-

ure 4.8(b). A similar analysis was performed for a 3 × 3 MIMO system with the

results, shown in Figure 4.9(a). From the figures, it can be seen that if the best

configuration is chosen at an SNR of 10 dB, the data rate can be increased to over

twice the data rate obtained by averaging over all configurations for the 3 × 3

case. ese simulation results indicate that the maximum achievable data rate can

be significantly increased with appropriately chosen configurations of laser and

detector arrays. As most configurations of laser and detector positions on the grid

result in suboptimal performance, it is essential to use an suitable configuration of

device geometries to ensure good MIMO performance.

4.5.3 Comparison of exhaustive and greedy sear: coarse grids

As mentioned in Section 4.4.2, a “`greedy” selection can be used to find the

best configuration of detectors in finer grids, where an exhaustive search is com-

putationally expensive. To demonstrate the usefulness of the search technique and

to provide an example of the data rates that result from it, an illustrative case with

small numbers, a 5×5MIMO system used on a 5×5 grid, was evaluated. For such

a system, an exhaustive search is computationally reasonable. However, even for

this small grid and MIMO system, an exhaustive search requires(255

)= 53, 130

comparison operations, while a greedy search requires only ∼ 115 comparisons.

e positions of the five lasers were chosen as in Figure 4.10(a). Given this place-

87

0 2 4 6 8 10SNR (dB)

1.5

2.0

2.5

3.0

3.5

4.0

4.5

5.0

5.5

6.0

Ach

ieva

ble

rate

(Gb

s−1)

Configuration from exhaustive searchConfiguration from greedy searchAverage over all configurations

(a) Laser positions(fixed)

(b) Detector posi-tions: exhaustivesearch

(c) Detectorpositions: greedysearch

Figure 4.10: Comparing the configurations obtained by the exhaustive search andthe greedy search. It can be observed that about 92% of the rate of the optimalexhaustive search can be obtained by the greedy search in this case.

ment of lasers, an exhaustive search selected a detector configuration shown in

Figure 4.10(b) and a submodular search selected a detector configuration as shown

in Figure 4.10(c). From the plot in Figure 4.10, at an SNR of 10 dB, the rate of the

channel using the greedy configuration yielded roughly 92% of the optimal rate

obtained by an exhaustive search, while requiring only∼0.2% of the comparisons.

For reference, the average of the rate over all positions is also presented, and it

88

is significantly lower than the rate achievable with the configurations obtained

by the exhaustive and greedy selection procedures. As the grids are made finer,

with a larger number of potential locations to place detectors, a greedy search re-

quires a significantly lower number of comparisons while guaranteeing at least a

(1 − e−1) factor of the rate obtained with the optimal configuration, making the

submodular technique useful in such situations. While this discussion focuses on

developing efficient static arrays of detectors using the greedy selection technique,

one could also employ this technique on a dense grid of detectors to dynamically

select a subset of detectors for reduced complexity MIMO signal processing while

retaining all the diversity benefits of a fine-grained array of detectors.

4.5.4 Detector arrays using fine grids and greedy sear

5 μm

8 μm

Figure 4.11: e laser array utilized with the 50 µm fiber. e lasers have a modefield diameter of 5 µm and a pitch of 8 µm

Performing an exhaustive search to optimize detector geometries for a 6×6

MIMO system becomes computationally prohibitive with finer grids, such as a

7× 7, 9× 9 or 11× 11 grids. For example,(496

)≈ 107,

(816

)≈ 108 and

(1216

)≈ 109

comparison operations are required for the 7×7, 9×9, 11×11 grids respectively;

in other words, the complexity of this task is O(NND) in the Big-O notation [90],

89

where N is the grid size and ND is the number of detectors, making an exhaus-

tive search a computationally expensive proposition. erefore, we resorted to

fixing a laser configuration and performing a submodular search on the detector

configurations, which reduced the number of computations significantly. For in-

stance, for placing 6 detectors in the 11 × 11 grid, only ∼ 711 comparisons are

needed, which represents massive computation savings when compared to the

optimal search requiring 109 comparisons. In other words, the greedy approach

imposes a complexity of only O(NND). As an example, we considered the place-

ment of detectors for a 6 × 6 MIMO system with a fixed laser array as shown in

Figure 4.11.

(a) 7× 7 grid (b) 9× 9 grid (c) 11× 11 grid

Figure 4.12: e detector configurations obtained by the greedy algorithm for de-tectors of diameter 4 µm for various grid structures. Interestingly, there is a sig-nificant preference towards detectors closer to the fiber core, indicating the factthat much of the received power in graded-index MMFs propagates close to theaxis.

With this fixed laser configuration, we evaluated the rate obtainable over

the link with six photodetectors, with each photodetector having a mode-field di-

ameter of 4 µm, and obtained the best configuration by submodular optimization.

e resulting detector array configurations obtained by simulation with these grid

configurations is shown in Figure 4.12. e data rate achieved by each of these

90

0 2 4 6 8 10SNR (dB)

2

4

6

8

10

12

14

16

18

Ach

ieva

ble

rate

(Gb

s−1)

6× 6 system with 7× 7 grid6× 6 system with 9× 9 grid6× 6 system with 11× 11 grid

Figure 4.13: e rate trends obtained with the detector configurations shown inFigure 4.12.

configurations is shown in Figure 4.13. It can be observed that the achievable rate

improves for the 9× 9 grid when compared to the 7× 7 grid, due to the improved

flexibility in placement of devices. However, the further improvement is marginal

for the 11 × 11 case, indicating a trend of diminishing returns with the increas-

ing granularity of the grid structure. is trend can be aributed to the fact that

the rate is dependent on choosing detector placements that overlap with high-

intensity points at the output facet of the fiber. A 4 µm mode-field diameter for

the detectors, chosen in accordance with the pitch of the grid structure, provides

structures that cover the high-intensity points sufficiently for a 9× 9 grid and the

finer detector positions offered by a 11× 11 grid do not significantly improve the

overlap, thus indicative of diminishing returns with finer grid structures. So, while

the use of finer grids for obtaining detector configurations is made possible using

91

this approach, the diminishing returns obtained as the grids become finer indicate

that extreme granularity of the grid structure is not necessary for obtaining good

detector structures that provide MIMO benefits.

4.5.5 From device arrays to segmented detectors

(a) (b) (c)

Figure 4.14: (a) A fiber was analyzed to obtain the best 23 locations to place smallcircular detectors on a 11 × 11 grid as discussed in Section 4.5.3. (b) Clusteringthese detectors to obtain larger square segments to improve the fill factor. (c) Aregular four-element detector array without using the design from the algorithm.

While the simple device array models presented above confirm the utility

of choosing an appropriate geometry for laser and detector arrays for MIMO-MMF

systems, it is oen desirable to employ detectors that possess a larger profile in

order to improve the fill factor, and thereby increase the received SNR and reduce

their susceptibility to speckle noise [33]. To this end, we used the fine grid simula-

tion result as a template to build a consolidated laser/detector array that provided

a much more robust solution to capturing a large fraction of the received signal.

We describe this technique using an example.

92

Figure 4.14(a) shows the result of the algorithm described in Section 4.5.3

for a 11×11 grid, with the circles indicating ideal locations for 23 detectors. Now,

the algorithm obtained a detector array that possessed a poor fill factor of about

1%. While the diversity benefit offered by 23 detectors is high, the SNR required

to reap these benefits is also prohibitively large. us, we assumed that neighbor-

ing detectors receive fairly correlated signals, and by combining these detectors

which receive correlated signals into one block, the diversity benefit is not signif-

icantly diminished, but the SNR is greatly improved. In the current example, we

clustered the detector locations in Figure 4.14(a) into four sectors, and obtained a

new configuration shown in Figure 4.14(b), significantly improving the fill factor.

We restricted the choice to square segments for implementation ease.

e predicted performance of these systems is shown in Figure 4.15, where

we considered a system with one laser at the transmier and various configu-

rations at the receiver; here, Figure 4.14(a) shows the detector configuration for

the “Small circular detectors” array, Figure 4.14(b) represents the detector system

shown as “Consolidated square segmented detectors,” and finally, Figure 4.14(c)

shows a “Regular square segmented detector” array constructed without the opti-

mization procedure. We observed that the fill factor improvement is significant for

both of the square array structures, where the rate is about 3.5× the rate achiev-

able with the small circular detector array at an SNR of 0 dB. However, the slope

in the rate curve is diminished, and at very high SNRs (in excess of 50 dB), the

performance of the square arrays falls due to the loss in diversity. Despite this

limitation, over a practical range of SNRs, the consolidated detector array sig-

93

nificantly outperforms both the small circular detectors and the regular detector

system, that does not possess as much of the diversity advantage as the circular

and consolidated square arrays.

0 10 20 30 40 50 60 70SNR (dB)

0

10

20

30

40

50

Ach

ieva

ble

rate

(Gb

s−1)

Small circular detectorsConsolidated square segmented detectorsRegular square segmented detector

Figure 4.15: Comparison of achievable rate trends for the detector paerns shownin Figure 4.14.

While the device geometry design algorithm presented in Section 4.3.2 gen-

erates array structures that are efficient from a diversity perspective, the small fill

factors of resulting arrays could make them unsuitable for practical use owing to

their susceptibility to speckle noise. By effectively consolidating the structures

obtained from the algorithm, useful detector arrays that possess a high fill factor,

while still providing diversity benefits over regular detector structures, could be

obtained. An alternate approach to avoiding the problem of small detectors would

be to use a lens to couple to an array of photodetectors that is larger in size, such

as the arrays described in reference [91]. While such an approach would allow for

94

the development large, dense arrays of photodetectors, it could also impose more

strict alignment tolerances.

4.6 Using dense detector arrays: dynamic detector selection

e previous sections have established that the use of multiple lasers and

detectors along with effective signal processing techniques can provide signifi-

cant increases in data rates. However, with more devices in MIMO-MMF links,

the complexity of and computational overhead signal processing techniques in-

creases significantly, thus increasing the energy-per-bit cost. e following sec-

tions discuss the algorithmic complexity of decoding data on a signal processing

based multiple-laser multiple-detector link, and the proposition of selecting a lim-

ited subset of the available detectors at the receiver to efficiently decode informa-

tion without significantly degrading the achievable rate. Efficient implementation

of dynamic selection, to compensate for temporal variations, can allow reliable

decoding while significantly reducing processing complexity. In many cases, the

complexity of signal processing algorithms can be reduced by 98%, while retaining

over 90% of the achievable rate.

e combinatorial nature of the problem of selecting the optimal subset of

detectors makes it prohibitive for practical real-time implementation when a large

number of detectors is available. e previous sections have used simulation to

indicate that the achievable rate of a multiple-input multiple-output (MIMO) link

satisfies themathematical property of submodularity. Submodularity allows for an

efficient “greedy” search algorithm that provides performance guarantees and, in

95

practice, provides gains fairly close to optimal subset selection. Greedy algorithms

are well-studied and are optimal for certain optimization problems [92]; in other

cases, they are suboptimal, but have performance guarantees for a class of prob-

lems [93]. Submodularity and greedy selection is useful in designing geometries

for arrays of detectors to optimize the achievable rate. In the following sections,

we consider a fixed detector array, and utilize the fact that the selection of detec-

tors in the MIMO-MMF link can be viewed as a submodular problem. is enables

the use of techniques from submodular optimization to efficiently select the best

detectors to use for a reduction of signal processing complexity, while preserving

rate guarantees. Using the propagation model described above for MIMO-MMF

links, we then describe a technique to analyze the performance of detector arrays

in MIMO-MMF links and provide simulation results that predict the performance

of the algorithm and compare it to the optimal selection procedure.

4.6.1 Complexity of MIMO Decoding

To compute the number of digital signal processing operations needed for

decoding and detection in MIMO links, we note that the receiver needs to possess

knowledge of the channel state H. us, the transmier sends symbols known to

the receiver, whereby the receiver initially estimates H. Since this operation only

occurs once in every coherence interval, the major contributor to the complexity

at the receiver is in decoding symbols. To study the decoding complexity, we con-

sider a generic matrix inversion equalizer: minimum mean square error (MMSE)

equalization. e MMSE equalizer for the system in Equation (2.9) for a signal to

96

Figure 4.16: Operations required by a MIMO-MMSE equalizer

noise ratio (SNR) of ρ is given by [40]:

X =

(I+

ρ

NL

HH†)−1

H†Y (4.9)

where the A† operation represents the Hermitian transpose of A. e vector X

is then used to detect the transmit symbols based on the modulation and coding

scheme chosen by the transmier. e boleneck in the decoding process is the

matrix inverse operation described in Equation (4.9), whose complexity is gener-

allyO(n3) operations in the number of rows ofH (O(n2.8) for large matrices) [90].

As an example, the number of processing operations performed by aMIMO-MMSE

equalizer in decoding a single symbol is given in Figure 4.16. We observe from

Figure 4.16 that the complexity scales polynomially. For example, for 5 active de-

tectors, the number of mathematical operations performed is 1.5× 103, while for

6 detectors, the number rises to 2.6×103, which is a 73% increase. However, as is

discussed in Section 4.6.2 and the simulations in Section 4.6.3, the benefit in per-

97

formance with additional detectors generally diminishes, indicating diminishing

returns with more detectors even as the computation requirement grows.

4.6.2 Submodularity: Greedy Detector Selection

A larger number of detectors at the receiver generally provides greater re-

liability at the receiver, as this provides for higher diversity, and consequently,

a higher data rate. However, using all detectors simultaneously in the decoding

process requires a larger number of mathematical operations, thus making the

problem intractable. Consequently, a simpler solution would be to utilize a subset

of the “best” nD < ND detectors that work in conjunction to provide the lowest

BER in the decoding process, making this choice once during each coherence inter-

val. However, this task is complicated by the combinatorial nature of the problem,

where choosing nD of ND requires(ND

nD

)operations. For instance, to choose 10

detectors out of 30 requires(3010

)≈ 107 comparisons, which imposes prohibitive

complexity for real-time implementation. us, we utilize a mathematical tech-

nique called submodular optimization to obtain a selection efficiently [86,94]. e

details of how this technique can prove useful in detector design along with math-

ematical proofs are provided in the sections above; however, in this section, we

utilize this concept for constructing a subset of detectors by choosing one detector

at a time, in a way that optimally augments link rate. is technique is guaranteed

to achieve at least a (1− e−1) fraction of link rate obtained when detector subsets

are selected optimally. In other words, greedy selection ensures that the obtained

detector set of size nD would achieve at least (1−e−1) ∼ 63% of the rate obtained

98

with optimal subset selection. For particular links, simulations indicate that the

rate obtainable with a subset of detectors can even be as high as 90% of the rate ob-

tainable with optimal selection. in Section 4.6.3. Additionally, greedy selection of

nD detectors requires O(n2D) operations as opposed to the combinatorial solution

for the optimal subset. For instance, the choice of the 10 best detectors among 30

needs only around a 100 comparisons, as opposed to∼ 107 for the optimal search.

4.6.3 Simulation Results

Fiber coreDetectors

(a) (b)

Figure 4.17: (a) e detector array paern. e 29 detectors have a pitch of 8 µma mode-field diameter of 5µm. (b) Progression of greedy selection: selecting eightdetectors for a channel realization obtained with 3 lasers.

To simulate an optical link with laser and detector arrays as discussed in

Section 4.5, we utilized the MMF channel model developed above. We assumed a

50 µm core diameter graded-index MMF of 1 km length, and generated 700 chan-

99

1 2 3 4 5 6 7 8Number of detectors used

60

65

70

75

80

85

90

95

100

Frac

tion

ofda

tara

tew

ithal

ldet

ecto

rs(%

)

Optimal selectionGreedy selection

(a)

1 2 3 4 5 6 7 8Number of detectors used

60

65

70

75

80

85

90

95

100

Frac

tion

ofda

tara

tew

ithal

ldet

ecto

rs(%

)

Optimal selectionGreedy selection

(b)

Figure 4.18: Comparison of the greedy and optimal searches with (a) 20 laser and(b) 29 lasers at the transmier. e plots show the fraction of data rate obtainedusing a subset of detectors vs. using all detectors for decoding, with a target BERof 10−9.

nel matricesHwhich describe the transformation induced by the fiber, along with

a 29 detector array at the receiver, shown in Figure 4.17a. e SNR at the receiver

was assumed to be 10 dB. First, for each channel realization, the subset of detec-

tors used was selected using greedy selection as well as optimal selection based on

the selection criteria discussed in Section 4.6.2. We used an OFDM implementa-

tion and simulated the transmission of bits, with a target BER of 10−9, calculated

the maximum achievable data rate for each configuration, and averaged the data

rates over several channel realizations. e results of the simulation are shown in

Figures 4.18a with 20 lasers at the transmier and 4.18bwith 29 lasers. Figure 4.17b

shows the progression of the algorithm for a particular channel realization. We ob-

serve that there are diminishing benefits in data ratewhenmore detectors are used,

hence needlessly increasing the number of operations. In addition, the greedy al-

100

gorithm performs almost as well as the optimal selection. is can be aributed to

the nature of MMF propagation, which causes successive detectors to produce di-

minishing results even when utilized simultaneously, thereby making the selected

subset perform almost as well as the optimal set. Using 8 detectors instead of 29

for decoding requires just (8/29)3 ∼ 2% of the computational cost, thus providing

a significant saving in computation while still achieving a significant part of the

optimal capacity.

4.7 Conclusion

In this chapter, we have developed a statistical channel model for a MIMO

based multimode fiber link, that is capable of quantifying the impact of the geom-

etry of laser and detector arrays on the achievable data rate. Such an input-output

signal processing model enables us to analyze the rate achievable overa multi-

mode fiber link containing multiple lasers and detectors. In particular, we have

provided a method to analyze the link performance with particular geometries of

laser and detector arrays, and emphasize the importance in designing appropriate

arrays of devices for maximum performance. With ergodic rate as the design crite-

rion, we numerically determined the optimal device arrays for coarse grids under

appropriate feasibility constraints using an exhaustive search. ese simulations

revealed that systems with optimal device configurations could outperform arbi-

trarily chosen device arrays by over 200%. For finer grids, an optimal exhaustive

search becomes computationally demanding due to a large number of compar-

isons. On the other hand, a submodular “greedy” algorithm, which is guaranteed

101

to yield an achievable rate of at least (1 − e−1) factor of the rate obainable with

optimal selection, is used to alleviate complexity. Optimal exhaustive searches for

device geometries was compared to greedy searches, and this revealed that device

configurations that aain over 90% of the rate of an optimal exhaustive search

with could be obtained with less than 0.2% of the comparisons for a fine grids,

illustrating the potential utility of the greedy search. e high performance and

low complexity of the greedy search makes it an effective tool for implementing

dynamically reconfigurable detector arrays with reduced DSP complexity. To ad-

dress the issue of low fill factors with the detector structures obtained from the

algorithm, we provided a method to consolidate several detectors to obtain seg-

mented detectors with improved fill factors while retaining diversity benefits. In

addition, decoding symbols in MIMO-MMF links is complicated in the presence of

many detectors, due to the large number of DSP operations that have to be per-

formed. Selecting a subset of detectors for decoding provides much of the benefits;

however, conventional methods of selecting the optimal subset involve an expen-

sive combinatorial search. A greedy selection approach ensures an efficient, yet

effective, means to achieve performance near the rate obtained with optimal selec-

tion over the link. Simulations reveal that the method can provide in excess of 90%

of the rate with only 2% of the computation, making these low-complexity of these

algorithms ideal for implementing practical MIMO-MMF systems with several de-

tectors. Future work will involve experimental verification of these concepts as

well as an extension to other guided media, such as plastic fibers.

102

Chapter 5

Offset coupling, feedba and spatial multiplexing in a4 × 4 incoherent-MIMO multimode fiber link

5.1 Introduction

e previous chapters have discussed several aspects of incoherent MIMO-

MMF links. In particular, Chapter 3 discussed the implementation of a link with

offset coupling that established the utility of MIMO over MMF, while the theoret-

ical and simulation based approach described in Chapter 4 predicted the benefits

that appropriate shaping of the input and modal filtering of the output can have

on the data rate capacity of the MIMO-MMF link. e use of offset coupling in

MMF links has been considered in the past. Mode group diversity multiplexing is

a related technique, where mode groups are excited by means of offset launch for

multiplexing data streams within dispersion limits [29]. Launching into the MMF

with a radial offset to the fiber axis, with the aim of improving the bandwidth-

length product while using on-off modulation has been considered [31, 35, 72].

e applicability of using a higher number of lasers and detectors for data rate im-

provement in MMF links was analyzed in [33]. It has also been demonstrated that

data throughput through plastic MMF can be significantly improved with the use

of signal processing [95, 96], and the use of MIMO techniques to provide further

improvements has also been proposed [97]. However, a comprehensive evalua-

103

tion of channel quality and the impact of changing fiber length, launch offset and

the use of advanced modulation and signal processing techniques over different

MMF media is yet absent. Characterization of the impact of link parameters such

as wavelength, fiber material, modulation and coding and axial offsets on fiber

data rates is essential in understanding the capabilities and limitations of incoher-

ent MIMO-MMF links. In this chapter, we have described a detailed experimental

evaluation of the impact of varying these parameters and how this data leads to

conclusions on developing more effective multiplexing and dispersion compensa-

tion solutions for incoherent MIMO-MMF links. To introduce axial offsets, two

bu-coupled fibers were aligned using a fiber alignment stage, and the axis of one

fiber was displaced perpendicular to the axis of the other. Controlling the offset

distance allowed the launched modal distribution to be altered. Due to the avail-

ability of only one fiber alignment stage each at the transmit and receive sides,

mode scramblers were also employed to further increase modal diversity. ese

axial offsets and mode scramblers allowed signals to propagate in the MMF with

different modal footprints, resulting in spatial separation among data streams for

improved multiplexing. We have investigated the impact these techniques have

on the multiplexing performance that MIMO and signal processing can provide

when used with 2×2, 3×3 and 4×4 configurations over MMF sections of various

lengths, with and without channel state information feedback to the transmier.

e term “spatial multiplexing” is oen used in the context of MMF links

to refer to directly multiplexing using the modes of the fiber. However, in this

chapter, we use the definition of spatial multiplexing from the wireless MIMO con-

104

text, as defined in [40], where the physical channel is mathematically parallelized

into several virtual parallel channels in order to facilitate transmission of multiple

streams in conjunction with signal processing. is definition is appropriate for

characterizing the multiplexing capabilities of the fiber channel in the case where

information about the fiber channel state is made available to the transmier.

e rest of this chapter is organized as follows: In Section 5.2, we describe

the metrics evaluated to characterize the properties of the MMF link. Section 5.3

describes the components of the optical system as well as the signal processing

and modulation/coding parameters used in the system. Section 5.4 describes the

actual experimental results. Section 5.5 provides a discussion of the experimental

results and puts them in the context of developing improved datacom solutions

for MMF links. Finally, Section 5.6 summarizes results and discusses some future

directions. e work described in this chapter has also been covered in [98, 99].

5.2 Metrics for evaluating MIMO-MMF systems

In order to evaluate and compare various systemparameters, we first present

the abstraction used as part of the signal processing structure during the encod-

ing and decoding process. ese parameters provide a means to directly view

how the channel states of the fiber and the various transformations that it effects

translate to changes in data rate and multiplexing capabilities of the system. A key

difference in the approach described here is that we do not transmit and receive

directly on the physical MIMO channels, but we transform them to virtual parallel

channels and transmit and receive information on these processed channels. us,

105

Mode scramblerModulator

Modulator

Modulator

Modulator

Mode scrambler

MMF

Waveform generator 1

Waveform generator 2

Offset detect

SMFMMFElectrical

Offset launch1541 nm

1517 nm

Oscilloscope

and PC

(ADC, DSP)

Channel statefeedback

Figure 5.1: Schematic of the 4×4MIMO experimental setup. e offset launch anddetection components were realized using nanoprecision fiber alignment stages,while the MMF couplers were all 2 × 1 couplers. e inset shows the offset po-sitions for the launch and detect stages, which were placed in a square grid withoffset intervals of 2 µm. Fibers and couplers colored blue represent SMF, while redrepresents MMF.

the resources of the physical channels are efficiently utilized to facilitate efficient

transmission and reception signals over these processed virtual channels.

To simplify the implementation and description, we utilize a subcarrier

based approach, in this case, orthogonal frequency division multiplexing (OFDM),

since it has been demonstrated that OFDM is effective in combating dispersion

in MMF links [61, 100]. is allows each subcarrier to possess a single-tap (flat-

fading) channel parameter, thus making per-subcarrier equalization just a process

of dividing by the estimated channel value, for zero-forcing equalization. In our

implementation, we used minimummean-square equalization to correct for chan-

nel impairments.

For anM -transmierN -detectorMIMO system, let hij represent the chan-

nel parameter from the j-th transmier to the i-th receiver. We represent the

transmit signal on modulator j as xj , and that received on detector i as yi. Finally,

106

we represent the receiver noise on detector i aswi. enwe have the detector-wise

received signal for i = 1, 2, . . . N can be represented in matrix form as

e effective channel for this subcarrier can be represented as follows:

y = Hx+w

where y =[y1 y2 · · · yN

]T

H =

h11 h12 · · · h1M

h21 h22 · · · h2M...

.... . .

...hN1 hN2 · · · hNM

w =

[w1 w2 · · · wN

]T.

(5.1)

With this formulation, the following data rate can be achieved over this subcarrier:

R = log2 det (I+ ρHH∗) b/s/Hz (5.2)

where ρ is the signal to noise ratio of the system, andH∗ represents the Hermitian

transpose of the matrix H. us, to compare various configurations of the fiber

linkwith tunable parameters, such as offset launch, zeroing in on the configuration

that maximizes the achievable rate provides for an effective method of tuning the

system to operate effectively.

To further understand and characterize the system from a spatial multi-

plexing perspective, we use the singular-value decomposition (SVD) of the chan-

nel matrix. e SVD of the channel matrix H can be represented as:

H = UΣV∗ (5.3)

107

where U and V are unitary matrices, and Σ = diag{σ1, σ2, . . . σmin(M,N)} is a di-

agonal matrix containing non-negative elements on its diagonals, called singular-

values. To understand the significance of the SVD in interpreting multiplexing

through a MIMO link, we transform equation (5.1) using equation (5.3) as follows

U∗y = U∗UΣV∗x+U∗w

⇒ y = Σx+ w(5.4)

where y = U∗y, x = V∗x and w = U∗w, with the noise w and w being statis-

tically similar. By effecting this transformation, we obtain what are essentially a

pair of parallel “effective” channels of the form:

yi = σixi + wi, i = 1, 2, . . .min(M,N).

With this new formulation, the achievable rate from equation (5.2) becomes

R = log2 det(I+ ρΣ2

)=

min(M,N)∑i=1

log2(1 + ρσ2

i

)b/s/Hz.

(5.5)

us, we can view the MIMO channel from a multiplexing perspective as sev-

eral parallel virtual channels whose SNRs are determined by the singular-values.

While this formulation makes the assumption of a linear system, this assump-

tion has been found to hold for intensity modulation systems under mild condi-

tions [26, 66, 101]. us, with an experimental setup of a MIMO-MMF system, the

above formulation provides us with metrics that allow for a detailed study on how

changing various system parameters, such as launching and detector stage offsets,

mode scramblers, modulation choices etc. affect the achievable data rate. In the

108

following sections, we describe the design of such a link and describe the tech-

niques we use to optimize the link to achieve the for maximum data rate using

this formulation.

5.3 System description

In this section, we describe the design and implementation of a 4×4MIMO-

MMF system, describing the various components as well as themodulation, coding

and receiver aspects.

5.3.1 Optical System

Since the system is operated with various different fiber media and config-

urations, each of these is described separately.

5.3.1.1 Silica MMF with DFB lasers

e optical system schematic for a 4 × 4 link with silica fibers is shown

in Fig. 5.1. e transmier consists of two (non-identical) C-band fiber coupled

distributed feedback lasers. e specific lasers used were a 1541 nm JDSU telecom

laser and an NEL 1517 nm telecom laser, each having an optical power output of

13 dBm (20 mW). Two lasers were employed because the power requirements

for links above 1 km could not be met by spliing a single laser for a 4 × 4 sys-

tem. e optical signal from each of these lasers was split into two parts using

SMF spliers. e four resulting signals were modulated using external intensity

modulators, whose rf input signals were produced using two Tektronix 7102 ar-

109

bitrary waveform generators from four quadrature amplitude modulated (QAM)

data streams. e modulators used in this setup were four JDSU Mach-Zehnder

modulators rated for 10 Gb/s operation. e two pairs of signals were then coupled

together, with one of them connected to the coupler using a conventional patch

cord and the other connected through amode scrambler to induce a different mode

profile [102]. e mode scramblers consist of tightly wound multimode fiber (into

about 100 turns of radii less than 1 cm) that are designed to facilitate heavy in-

termodal coupling in the transmied signal, resulting in the randomization of the

modal content of the signal. Inducing a different mode paern among each of

the signals through a modal filtering or scrambling process serves to differentiate

the spatial mode content of each data stream [26]. Distributing the signal across

different modes, maintaining mechanical stability of the fiber, and the ability to

perform digital signal processing at the receiver make the system robust against

coherent modal interference [39]. Aer combining the two pairs of signals orig-

inating from each laser, these were further combined by means of an MMF cou-

pler; one of them was bu-coupled, while the other was launched with an specific

offset using a 3-axis nanoprecision fiber alignment stage. e optical signal thus

obtained was then transmied and received over a conventional 62.5 µm diameter

graded-index multimode fiber (OM1), whose rated bandwidth-length product was

2 Gb/s-km at 1550 nm and 1 Gb/s-km at 850 nm. is class of fibers is commonly

deployed in 10GBASE-SR systems [103].

On the receiver side, the signals were split using aMMF coupler, and one of

these arms was bu-coupled into another MMF coupler directly, and the other was

110

connected to an MMF coupler via a nanoprecision fiber alignment stage, which

acted as a modal filtering system. e resulting four signals were directly detected

by four 10 Gb/s fiber coupled InGaAs photodetectors. Each photodetector's electri-

cal output was connected to a Tektronix DPO70604 oscilloscope, which acted as an

analog to digital converter (ADC).e detected signals were processed offline with

signal processing algorithms. To obtain an accurate bit-error rate (BER) statistic,

the transmission characteristics were evaluated over several OFDM symbols, and

the errors from these were averaged to obtain a BER estimate. Feedback of chan-

nel state information, obtained using training symbols (pilots) at the receiver, was

returned to the transmier for the experiments where a closed-loop modulation

setup was required. e percentage overhead incurred for the feedback was lim-

ited to 0.1% of the data rate, since the channel conditions were not fast-varying in

comparison to the data rates. e transmission of each stream by the waveform

generator was at 10 gigasamples/s, while the receiver sampling of the electrical

output from the photodetectors at the oscilloscope was at 20 gigasamples/s. e

fiber offset locations at the launch and detect fiber alignment stages were spread

over a 40 µm × 40 µm grid, as shown in the inset of Fig. 5.1. e data rate was

evaluated by launching and detecting at 2 µm offsets at the transmit and receive

alignment stages, so as to obtain the best data rate.

e DFB laser based system's performance metrics and data rate capabili-

ties were investigated over a wide array of fiber lengths; specifically 10 m, 100 m,

500 m, 1 km and 3 km. e various MIMO configurations ranging from 1 × 1 to

4× 4 were obtained with slight modifications to the system in Fig. 5.1 as follows:

111

• 1× 1: e couplers were removed from transmit and receive ends, and only

the first laser was used at 10 dBm output power.

• 2× 2: Streams x1 and x3 were used, along with the first and third photode-

tector directly connected (without couplers). e two lasers each operated

at 10 dBm output power and the modulators and photodetectors were con-

nected directly without couplers.

• 3 × 3: Streams x1, x2 and x3 were used, along with the first three detec-

tors. e first laser operated at 13 dBm, and was split by a 2 × 1 coupler,

while the second laser operated at 10 dBm and was directly connected to the

modulator corresponding to stream x3.

• 4×4: e full system as, shown in Fig. 5.1, was active, with each laser output

set to 13 dBm.

e power levels of the laserswere adjusted to ensure that the same amount

of power was launched into the fiber, for a fair comparison. e combined and

scrambler insertion losses were less than 1 dB, and thus, did not appreciably alter

our data rate measurements.

5.3.1.2 Silica MMF with VCSELs

For the VCSEL case, the system design was fairly similar to the DFB based

system, except that a 2× 2 system was employed. e optical transmit subsystem

consisted of two Finisar 850 nm VCSELs, each of which was biased and directly

112

Waveform generator 1

Waveform generator 2

Bias-tee

Bias-tee

MMF

Oscilloscope

and PC

(ADC, DSP)

Offset launch

Offset detect

SMFMMFElectrical

Channel statefeedback

850 nm

850 nm

Figure 5.2: e transmier and receiver arrangement for the VCSEL case.

modulated by means of a 100 Hz-12 GHz bias-tee. e VCSELs were capable of

emiing at -3 dBm (0.5 mW). e system is shown in Fig. 5.2. For the 1× 1 case,

the couplers were removed from both the transmier and receiver, and only one

VCSEL and detector were used.

e VCSEL based system's performance metrics and data rate capabilities

were investigated over only 10 m, 100 m, 500 m and 1 km MMF fiber sections.

Longer lengths were not considered since the optical power of the VCSELs was

insufficient to support a sufficiently large SNR with longer fiber sections to ensure

reliable communication at high speeds.

5.3.1.3 Plastic MMF with Fabry-Perot laser

In addition to evaluating the system parameters for conventional silica

MMF, a similar experiment was performed for a 2 × 2 optical link with a plas-

tic MMF. e schematic for this setup is shown in Fig. 5.3. Like in the VCSEL

113

system, for the 1 × 1 case, the couplers were removed from both the transmier

and receiver, and only one VCSEL and detector were used.

Waveform generator 1

Waveform generator 2

Offset detect

Oscilloscope

and PC

(ADC, DSP)

Modulator

Modulator

PlasticMMF

SMFMMFElectrical

Offset launch

Channel statefeedback

1310 nm

Figure 5.3: e transmier and receiver arrangement for the evaluation of plasticfibers.

For these evaluations, perfluorinated graded-index plastic optical fiberwith

62.5 µm diameter was used. is choice was made to allow for consistency in the

use of components across the silica and plastic fiber setups, and to facilitiate com-

parison with the 62.5 µm diameter silica GI-MMF used in the other experiments.

e rated bandwidth-length product of this fiber was 200 Mb/s-km, and the ex-

periment was conducted on sections of length 1 m, 10 m and 100 m. e laser

used was a single-mode fiber 13 dBm (20 mW) coupled Fabry Perot laser whose

wavelength was 1310 nm, and the modulator was a JDSU lithium niobate external

modulator designed for use in the O-band (1260-1360 nm).

5.3.1.4 Mode scramblers and fiber offsets

e primary benefit of MIMO for data transmission through MMFs is ob-

tained by separating multiple streams through different modes or mode groups of

114

the fiber. To facilitate this, we took two approaches: mode-scramblers and offset-

launch and detecting.

Passing an optical signal through a mode-scrambler causes the signal to

gain a randomized modal footprint [102, 104]. e aim of using multiple mode

scramblers, each producing a random mode footprint, was to allow signals to pos-

sess spatial diversity by virtue of traveling through different modes. e signifi-

cant advantage of using mode scramblers is that they are easy to interface with an

existing fiber-coupled setup. However, they are limited in the variation of modal

footprint they provide, since they offer no means to alter or control the modal

transformation they effect.

e use of fiber alignment stages at the transmier and receiver provides

a convenient means to evaluate the complete spatial characteristics of the fiber

channel. By traversing all lateral offsets during signal launch and detection into

the fiber, they can provide fine-grained access to the mode-groups of the fiber,

along with simultaneous control at the transmit and receive stages to maximize

the data rate. However, they also significantly complicate the deployment of the

optical link. Our motivation in the use of the optical stage was to determine how

offset coupling enhances the fiber data rates, and to gain intuition on geometries

for the design of device structures (such as arrays of lasers and detectors) which

would possess the combined benefits of enhanced data rates as well as deployment

ease. It must be noted that the use of fiber alignment stages by themselves does

not allow for access to individual modes or mode groups of the fiber. Modulat-

ing individual modes would require sophisticated techniques, such as spatial light

115

Table 5.1: OFDM system parameters

Sampling rate 10 GS/sFFT Size 128

Cyclic Prefix 5Occupied subcarriers 58 (including 4 pilot subcarriers)

Constellation QAM-2, 4, 8, 16, 64Reed-Solomon code (255, 239), (255, 231) and (255, 223) codes

modulation or accurate spatial filters, and would make the system highly sensitive

to fiber parameters and require tight alignment tolerances. Since the focus of this

work is to utilize a diversity of modes as opposed to specific modes, we utilize

offset launch and detection and do not focus on matching accurately to individual

fiber modes or mode groups.

5.3.2 Modulation and coding

In this section, we discuss the modulation and coding utilized in the system

implementation, as well as the MIMO techniques used for system evaluation and

data rate measurements.

To evaluate the system from a MIMO perspective, we utilize the spatial

multiplexing technique described in Section 5.2. First, channelmeasurementswere

obtained at the receiver by utilizing the pilot symbols. Using this channel coeffi-

cient data, the achievable data rate through each subcarrier was calculated and

added up to obtain a net achievable data rate estimate. In addition, by studying

the singular values of the system, an inference can be drawn on how effective vari-

ous virtual streams are in their ability to transmit data. esemetrics are evaluated

116

with various configurations of the optical system described above, including mode

scramblers as well as various offset launching and detecting, as shown in the inset

of Fig. 5.1. Once we obtain a position where the parameters are optimized, we

evaluate the data rate obtained by optimizing the modulation and error control

coding parameters.

For the data rate experiments, the techniques evaluated included:

• e vertical Bell Labs layered space-time (V-BLAST) code, which is an open-

loop multiplexing technique that does not require feedback of channel coef-

ficients.

• e spatial multiplexing technique, which uses feedback of channel coeffi-

cients to parallelize the channel, as described in Section 5.2. e key advan-

tage of this technique over V-BLAST is that it results in focused allocation

of resources along the virtual parallel channels which offer the highest SNR,

although the complexity of transmiing channel coefficients accurately to

the transmier makes it more difficult to realize massive gains in practice.

To evaluate the data transmission performance of the system for the V-

BLAST case, we transmied four streams of data (labeled x1, x2, x3 and x4 in

Fig. 5.1) using various modulation schemes, and evaluated the BER performance.

In order to obtain reliable data rate measurements, we utilized Reed-Solomon

codes of different rates for forward error correction (FEC) to restrict the BER to

10−9 or lower. To encode the bits on each stream and process them for transmis-

sion, we employed QAM and orthogonal frequency division multiplexing (OFDM).

117

is choice was made for convenient receiver implementation and because QAM-

OFDM has been shown to be an effective modulation technique in optical fiber

links [105]. e QAM modulation used was chosen based on the subcarrier SNR

and waterfilling; QAM-2 was used for SNRs below 1 dB, QAM-4 when the SNR

was between 1 and 6 dB, and QAM-16 was used when the SNR was above 6 dB to

maximize the data rate. Once these parameters were set, they were kept constant

for each experimental run, and were not altered based on feedback information.

eOFDM implementation in this experiment utilized a 128-point Discrete Fourier

Transform with a 5 symbol cyclic prefix. e dc subcarrier and the adjacent 3 low-

est frequency subcarriers were not used, ensuring that data was transmied at

a sufficiently high frequency to allow for MIMO benefits [66]. To obtain a real

baseband signal during the OFDM modulation operation, data symbols were al-

located to half of the frequency subbands, and the conjugate of these symbols

was allocated onto the other half. ese data streams drove the external intensity

modulators. e vertical Bell Labs layered space-time (V-BLAST) [40] signaling

technique was adopted in this for decoding and separating data streams at the re-

ceiver. Here different data symbols sent on each transmit arm were decoded using

a successive interference cancellation style decoding at the receiver on each sub-

carrier. During transmission, the symbols meant for each subcarrier were trans-

mied simultaneously on each modulator without any knowledge of the channel

state. Since symbols sent on different transmit arms interfere with each other at

the receiver, possessing an accurate knowledge of the channel transformation for

each subcarrier is essential for the receiver to compensate for intermodal coupling

118

in the V-BLAST case. To estimate the channel properties, an OFDM symbol con-

sisting of a pilot sequence was sent for every 100 OFDM data symbols, to estimate

and compensate for dispersive effects and coupling among the signals at the re-

ceiver. Finally, the symbols were detected and the resulting BER was measured.

e experiment was repeated for all MIMO configurations (1× 1, 2× 2, 3× 3 and

4× 4) and fiber lengths (100 m, 500 m, 1 km and 3 km).

For the case of feedback based spatial multiplexing, channel state infor-

mation was found at the receiver by sending an OFDM symbol with pilots on all

subcarriers. e channel estimates thus obtained at the transmier were immedi-

ately used for modulating further data for the spatial multiplexing case. e delay

in transmission of the feedback information was within a few hundred millisec-

onds, to minimize the impact temporal channel variations, although the stability

of the MMF channel is well known [28, 83].

To transmit feedback information to the transmier, a simple approach of

quantizing the channel estimates and relaying them to the transmier was used.

For obtaining an accurate estimate of channel coefficients, an OFDM symbol with

pilots on all subcarriers was sent, so that the receiver could infer information about

the channel state of every subchannel. is was followed by the actual data trans-

mission, with modulation parameters optimized based on the channel state in-

formation. e accuracy of quantized feedback has a significant impact on the

performance of a feedback based MIMO link [106], although a simple quantiza-

tion approach is sufficient for us to demonstrate the value added by feedback in

MIMO-MMF links, while retaining ease of practical implementation. Another sig-

119

nificant fact utilized in this experiment was that the link coherence time, during

which the channel coefficients are assumed to not change appreciably, is suffi-

ciently large for the feedback of channel coefficients to be useful [28, 83]. Since

the data occupied by the feedback channel generally amounts to a small fraction of

the data rate, the burden of additional overheads incurred due to the transmission

of feedback information is very small. e simplistic quantization approach de-

scribed here results in noisy channel estimates, so an additional equalization step

was performed at the receiver to compensate for this before the data was decoded.

For each of the above described transmission scenarios, the data rate was

evaluated for the best combination of QAM modulation and Reed-Solomon code

to obtain the maximum possible data rate, for various MIMO configurations and

fiber lengths.

5.4 Experimental results

is section discusses in detail the results of each experiment conducted

within the purview of this chapter. Initially, we describe the common procedure

for all experiments, and then discuss the specific performance results for various

hardware and fiber media.

In our experiments, we also compare the performance results obtained by

employing only mode scramblers and no fiber alignment stages, in order to char-

acterize the mode diversity benefits provided by these solutions. In general, it

was observed that while mode scramblers and the inherent asymmetry in mode

couplers introduces some mode changes, they fall short of the diversity benefits

120

obtained by the flexible control of offsets with fiber alignment stages. e example

shown in Fig. 5.4 shows the output of a laser beam aer propagation through an

MMF for two cases: one aer passing through a mode scrambler, and the other

aer launching into a patch cord with an axial offset of 20 µm before connecting

to the MMF coupler. While a mode scrambler allows access to the modal diversity

of an MMF without advanced components, the use of offset launch provides finer

control, and is likely to result in beer data rates.

Mode

scrambler

20 µm offsetlaunch

Figure 5.4: e impact of a mode scrambler compared with a fiber offset launchaer propagating through a 1 m MMF patch cord. While the mode scramblercauses an expansion of the beam and signal spread into neighboring modes, thealignment stage allows control for excitation of higher order modes.

efiber systemwas set up for each experiment as indicated in Figures 5.1, 5.2

and 5.3. e fiber alignment stages were computer controlled, and measurements

of channel parameters and data signals were obtained by programmatically mov-

ing the fiber alignment stages over a 40 µm × 40 µm grid, with an increment of 2

µm between grid locations, as shown in the inset of Fig. 5.1. Before we delve into a

detailed discussion of the experimental results, we discuss some of the limitations

in measurements caused by speed restrictions imposed by the rf components of

121

the system.

100 500 1000 3000Length (m)

0

5

10

15

20

25

30D

ata

rate

(Gb/

s)

Detector speedlimit

Rated BW-Lengthproduct (2 Gb/s-km)

6.5 Gb/s-km

1× 1 (equalizer off)1× 1 (equalizer on)

No DSP

Figure 5.5: Data rate versus length for a 1 × 1 system with and without signalprocessing.

Fig. 5.5 compares the performance of a 1× 1 MMF link of various lengths,

with “No DSP” referring to conventional on-off keying without dispersion com-

pensation (as in the case of 10GBASE-SR systems [103]), “equalizer o” referring

to the case where OFDMwith QAMmodulation was used without performing per-

forming an equalization of the channel effects on the subcarriers at the receiver,

and “equalizer on” referring to a complete minimum mean-squared error equal-

ization on a per-subcarrier basis. ere are some important observations that can

be made from these observations:

• e achieved data rate does not exceed the rated bandwidth-length product

122

of the fiber for short lengths (around 200 m or less). is can be aributed

to the fact that at short lengths, the data rate limitation is primarily due

to restrictions in the maximum bandwidth of the signals generated by the

waveform generator and the speeds of the modulators and detectors (a com-

bined limitation of 7 GHz). For longer lengths of fiber, dispersion impacts

the link more significantly, and becomes the dominant effect contributing

to signal impairment for lengths of over 200 m.

• e benefits of using signal processing for dispersion compensation risewith

increasing distance. From the figure, it is clear that the pulse shaping and

spectrally efficient signaling with OFDM raises the bandwidth-length prod-

uct to about 6.5 Gb/s-km. However, with dispersion compensation, the per-

formance benefits over the “no equalization” case rise with distance, indicat-

ing that the benefits of using signal processing increase as fiber dispersion

affects the system more significantly.

erefore, in all our experimental results, we make a distinction between

these two data transmission regimes: onewhere the system is limited by the equip-

ment speed, and the other, where it is limited by fiber dispersion.

We now discuss data rates measured on these systems and their implica-

tions on the design of MIMO-MMF links.

123

5.4.1 4 × 4 link over Silica MMF with DFB lasers

e link shown in Fig. 5.1 was evaluated with and without offset launch,

and the performance of V-BLAST as well as spatial multiplexing was considered.

Each of the DFB lasers used was fiber coupled to a single-mode fiber, with a core

diameter of 8 µm and a numerical aperture of 0.14, thus resulting in a beam which

was very small in comparison to the core diameter of the MMF, which is 62.5 µm.

e small beam size allowed fiber alignment during the launch stage to be effective

and offered a significant control over the mode profile that could be obtained.

100 500 1000 3000Length (m)

0

5

10

15

20

25

30

Dat

ara

te(G

b/s)

Detector speedlimit

Fiberlimited

Equipmentlimited

Rated fiber BW-Length product

1× 1

2× 2

3× 3

4× 4

No DSP

Figure 5.6: Data rate versus length for various lengths of fiber with V-BLAST.ese data rates were observed with optimized offset launch and detection with V-BLAST. e solid curve indicates the rated fiber bandwidth-length product, whilethe doed line indicates the saturation speed for the detector beyond which theeye diagram shows a mostly closed eye.

e data rates achieved using system shown in Fig. 5.1 was evaluated under

124

various configurations, as discussed in Section 5.3.1. e performance with on-off

keying and no additional dispersion compensation, as used in standards such as

10GBASE-LRM, is labeled “No DSP” in Fig. 5.6. e bandwidth of the generated

signals was smaller than the full bandwidth supported by the fiber at lengths of

200m or less, since the speed of the modulators, photodetectors and the waveform

generators was limited to 7 GHz [2]. e impact of this limitation is indicated by

the shaded region in the le part of Fig. 5.6. For longer fiber lengths, fiber disper-

sive effects became more pronounced, and signal processing techniques enabled

data rates to significantly exceed the rated bandwidth-length product of the fiber.

e fiber channel responses were measured for each data stream using the pilot

0 10 20 30 40 50Subcarrier index

0

2

4

6

8

10

12

14

Ave

rage

SN

R(d

B)

Stream 3

Stream 1

Stream 4

Stream 2

Figure 5.7: Average SNR for each stream in the 3 km 4× 4 case, corresponding tothe data streams labeled x1, x2, x3 and x4 in Fig. 5.1.. e energy of each subcarrierwas averaged over 100 OFDM symbols for an SNR estimate.

sequences, and the best offsets for the two fiber-alignment stages were those that

125

(a) (b)

Figure 5.8: e radial variation of the capacity for various detector alignment stageoffsets for a fixed position of the transmier side alignment stage, (a) in 3-D, as wellas (b) along one cross-section along a plane to the right.

maximized the total data rate. A sample-averaged SNR for the 4× 4 system with

a 3 km fiber section is shown in Fig. 5.7. For most situations, an axial offset of

∼10 µm from the fiber center at both the launch and detection stages was found

to be effective for increasing the data rate for all fiber lengths considered. e

temporal variation of the channel was sufficiently slow and the change in the best

offset location was not significant (as in references [28, 83]), allowing for estima-

tion and equalization digital compensation of dispersion and intermodal coupling

while fixing fiber offsets. Fig. 5.8 shows the variation of data rate as the detec-

tor offset is varied, for the 3 km case. We observe that that perturbations of up to

±3 µm do not reduce data rate by more than 5%, indicating that MIMO-MMF links

very tolerant to misalignments while still benefiting significantly from modal di-

versity. Once the best fiber alignment parameters were established, random bits

were generated for each data stream and transmied over the optical link. Upon

reception, the signals were processed offline to evaluate the BER. A sample re-

ceived constellation on subcarrier 35 for a 4 × 4 MIMO link over a 3 km MMF

126

section is presented in Fig. 5.9. Here, it can be observed that V-BLAST decoding

results in two streams (in this case, streams 1 and 3) that possess a much lower

BER than the other two. Even though the average power received at each pho-

todetector was roughly the same (around -5 dBm), streams 1 and 3 outperform

streams 2 and 4. is is because x1 and x3 possess a higher effective SNR when

decoding them from the signals received on all of the photodetectors combined.

Aer decoding these, the remaining interference due to intermodal coupling was

too great for V-BLAST decoding to recover x2 and x4. e performance observed

in streams 1 and 3 was similar to that which can be obtained in wireless MIMO sys-

tems, where two channels seen by a pair of appropriately spaced antennas possess

significant spatial diversity [40]. In the optical case, the spatial separation between

these streams is provided using alignment stages. Conversely, for streams 2 and

4, the incremental benefits mirror the wireless MIMO case for antennas that are

not appropriately spaced, resulting in heavily correlated channels and diminished

diversity gains [40]. is suggests that the benefits offered by controlled spatial

offsets of fiber axes significantly exceed those from the inherent spatial diversity

offered by couplers and mode scramblers.

Next, the data rates supported by the system for various MIMO config-

urations and various fiber lengths with V-BLAST was observed. To verify the

efficacy of the system in compensating for differential model delay for higher or-

der modes, the 1 × 1 system was evaluated with several offsets, and dispersion

compensation was found to be effective up to offsets of 22 µm. Subsequently,

the performance evaluation was conducted for all relevant MIMO configurations,

127

Figure 5.9: Constellations received in the 35th subcarrier over a 3 km 4×4 link withoffset launch and detection that correspond to the data streams labeled x1, x2, x3

and x4 in Fig. 5.1. While streams 1 and 3 have a very low BER (∼ 10−5), 2 and 4suffer from a much higher BER (∼ 10−2).

as described in 5.3.1. e modulation and coding rates were adjusted to obtain

a coded BER of 10−9 and the measured BER was then verified to be within this

limit by averaging bit-error counts over several transmissions. e data rate im-

provement from the 1 × 1 to 2 × 2 case was significant, but diminished benefits

were obtained with further increase in the numbers of transmiers/receivers, due

to the lack of additional modal diversity. From observing the performance versus

fiber length, it is evident that effective signal processing with MIMO was able to

increase data rates through the MMF link, ranging from 26 Gb/s to 16.5 Gb/s for

4 × 4 systems with lengths increasing from 100 m through 3 km. Significantly,

MIMO signaling and signal processing was able to increase data rates to 16.5 Gb/s

over 3 km, exceeding the bandwidth-length product by 25-fold in comparison to

10GBASE-SR data rates. e same improvement was not observed at shorter fiber

lengths due to the bandwidth limitations of the waveform generators and photode-

tectors. e use of faster components that raise the bandwidth limitation from the

current 7GHz to 10GHz is expected to provide up to 30% increase in data rates for

128

lengths of 200 m or less [2]. Equipment limitations notwithstanding, the trends

observed at all lengths confirm the benefits of using MIMO-based multiplexing to

effectively boost the data rate in MMF links. Offset launch was able to provide

a consistent 60-70% increase in data rates at all lengths, thus underscoring the

benefits of the higher modal diversity harnessed from offset launch and detection.

Following the V-BLAST experiment, we conducted a detailedmeasurement

and analysis of how offset launch and detection improves data rate performance.

Initially, the launch conditions were configured such that the fiber alignment stage

at the transmier, shown in Fig. 5.1, was fixed to an offset of 10 µm from the fiber

axis, and the channel parameters weremeasured bymoving the receiver alignment

stage over the set of positions shown in the inset of Fig. 5.1. Using the channel ma-

trix, the achievable data rate was calculated with the formula from equation (5.2),

and ploed in Fig. 5.10 for 10 m, 100 m, 500 m, 1 km and 3 km fiber sections. A

two-dimensional cross-section is also displayed for beer comprehension of the

plot. Although 10 µmmay not be the optimal offset at the transmit stage to maxi-

mize the achieved data rate, the study of how the achievable data rate varied at the

receiver serves to illustrate the spatial sensitivity of the detector offset, and how

this sensitivity changes with the length of the fiber section.

e results in Fig. 5.10 allow us to draw some interesting inferences. It

is easy to see that the maximum achievable data rate falls with increasing fiber

length, owing to fiber losses and additional dispersion. Having no offset causes a

drop in the data rate since there is insufficient modal spread among the transmied

data in order to support multiple streams. In addition, the alignment tolerances

129

x-offset (µm)

2010

010

20 y-offse

t (µm)

2010

010

20 Data

rate

(Gb/

s)

0510152025

0 5 10 15 20 25Data rate (Gb/s)

20 15 10 5 0 5 10 15 20Offset (µm)

0

5

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25

Data

rate

(Gb/

s)

95% of max.data rate

(a) 10 m

x-offset (µm)

2010

010

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t (µm)

2010

010

20 Data

rate

(Gb/

s)

0510152025

0 5 10 15 20 25Data rate (Gb/s)

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0

5

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25

Data

rate

(Gb/

s)

95% of max.data rate

(b) 100 m

x-offset (µm)

2010

010

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t (µm)

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rate

(Gb/

s)

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25

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rate

(Gb/

s)

95% of max.data rate

(c) 500 m

x-offset (µm)

2010

010

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010

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rate

(Gb/

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rate

(Gb/

s)

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(d) 1 km

x-offset (µm)

2010

010

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t (µm)

2010

010

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rate

(Gb/

s)

0510152025

0 5 10 15 20 25Data rate (Gb/s)

20 15 10 5 0 5 10 15 20Offset (µm)

0

5

10

15

20

25

Data

rate

(Gb/

s)

95% of max.data rate

(e) 3 km

Figure 5.10: e radial variation of the capacity for various detector alignmentstage offsets for a fixed position of the transmier side alignment stage, in 3-dimensions above as well as the cross-section along a plane below. As can be seen,an increase in length reduces the achievable data rate, but increases the tolerancewith which the receiver stage needs to be aligned.

130

for achieving the maximum data rate are more stringent for shorter fiber lengths,

and are more relaxed for longer fiber lengths. For instance, to achieve 95% or

more of the maximum achievable 25 Gb/s for the 10 m case shown in Fig. 5.10a,

the tolerance turns out to be ±2 µm, while for the 18.7 Gb/s in the 3 km case,

this number rises to ±6 µm. is can be aributed to spreading of the modal

content into more neighboring modes (that have similar spatial characteristics) as

the fiber section lengths increase progressively [71]. Such mode mixing did not

significantly affect the performance of signal processing using linear techniques,

since the group velocities of these neighboring modes are very similar.

Overall, the results from Fig. 5.10 indicate that a significant improvement

in data rate can be obtained with appropriate signal processing and offset coupling

with a fair amount of tolerance to misalignments. In the spatial multiplexing case,

by seing the transmit and receive alignment stages to a 10 µm of offset, we were

able to aain within 5% of the maximum data rate obtained with optimized offsets

of the receiver alignment stage for each fiber section length. In addition, the power

received did not vary much once the alignment stages were set to the optimal

position, as has been observed in related experiments [39, 83]. us, even MMF

systems with fixed offsets that would be simpler to deploy would be able to realize

significant gains with the use of MIMO techniques like the ones described here,

while retaining relaxed alignment tolerances.

Finally, for each fiber section, the positions of both of the optical align-

ment stages at the transmit and receive ends were moved across the 40 µm × 40

µm square grid with points separated by 2 µm, as shown in Fig. 5.1, and, for the

131

Stream 1

(a)

Stream 2

0.00.20.40.60.81.0 0.0

0.2

0.4

0.6

0.8

1.0

(b)

Figure 5.11: Constellation for the two virtual channels which were used for mod-ulation with spatial multiplexing for a 1 km MMF. (a) Stream 1 (highest singularvalue): a QAM-64 constellation was supported by this virtual channel. (b) Stream2 (second highest singular value): Only a QAM-4 constellation was supported inthis virtual channel. e remaining two channels were ignored, since their SNRswere very low.

Table 5.2: Optimal axial offsets for 4× 4 system

MMF Length Transmit stage Receive stage Tolerance10 m 14 µm 12 µm ±2 µm100 m 12 µm 10 µm ±2 µm500 m 10 µm 8 µm ±4 µm1 km 10 µm 10 µm ±4 µm3 km 12 µm 12 µm ±6 µm

132

offset seings in which the best channel quality was obtained, the data rate was

evaluated. To evaluate all possible configurations of the offset stages, the transmit

stage was initially set to one of the 441 grid locations indicated by the dots, and

the receiver parameters were measured for each of the 441 points. is was then

repeated for each of the transmit stage positions. In this case, since the channel is

parallelized into four parallel channels, more appropriate modulation formats can

be chosen based on the SNR of the channel. Table 5.2 lists the optimal offset posi-

tions of the fiber alignment stages at the transmier and receiverwhere the highest

data rate was obtained. e tolerance is defined as the minimum offset that can be

tolerated at the transmier or receiver that causes the data rate to fall below 95%

of the maximum. e best axial offsets for each length were in between 8 µm to

12 µm at both the transmier and receiver alignment stages, and the data rate did

not vary appreciably with small perturbations, as discussed previously. For each

of the cases, it was observed that, upon parallelizing the channels to obtain vir-

tual channels and ordering these based on singular values, the first two channels

were usable, and the remaining two were very noisy, and did not support a high

data rate. is is indicative of the fact that the predominant benefit of the fiber

offsets is effectively captured in the first two virtual channels. e highest possi-

ble QAM constellation was used in these channels to maximize the data rate, and

error control coding was used to keep the BER below 10−9. Although only two vir-

tual channels were used, the data rates supported on them were higher than those

obtained in the case of V-BLAST. As an example, Fig. 5.11 shows the two con-

stellations obtained aer transmission over the best virtual channels for the 1 km

133

case, where the ability to transmit a rich constellation (QAM-64) over the primary

virtual channel compensates for the inaccessibility of the remaining two channels.

e singular values realized for the four virtual channels, with the largest singular

value normalized to unity, were [1.00, 0.33, 0.04, 0.001], thus making the effective

SNRs of the second, third and fourth channels respectively 10 dB, 28 dB and 60 dB

lower than the first. e observation that first two channels are usable, while the

last two have very low in SNR, can be understood by realizing that we have the

flexibility of only one offset each at the transmier and receiver, thus possessing

only one spatial degree of freedom with full control in addition to direct center

launch. e flexibility to introduce more axial offset streams to effectively utilize

different modes of the fiber is likely to increase the third and fourth singular values

as well, making them useful by endowing them with a higher effective SNR.

Optimal axial offsets of the alignment stages at the transmier and receiver

were ∼8 µm and did not vary much with length, possibly due to the inherently

large beam size of the VCSEL output. e alignment tolerance, measured as the

maximum offset introduced at the transmier or receiver for the data rate to re-

duce by 5%, was about 2 µm up to 100 m and 4 to 6 µm for lengths of 500 m and

above. e results of the data rate experiments for the spatial multiplexing case are

shown in Fig. 5.12. For 100 m and 500 m, the data rate increase is not significant

in comparison to the V-BLAST data rates, as shown in Fig. 5.6. is is likely be-

cause the benefits obtained with spatial multiplexing with a simple (noisy) channel

quantization might prove insufficient to harness the full benefits of parallelizing

the channels with channel state feedback. However, for the 1 km and 3 km cases,

134

100 500 1000 3000Length (m)

0

5

10

15

20

25

30

Dat

ara

te(G

b/s)

Detector speedlimit

Fiberlimited

Equipmentlimited

Rated fiber BW-Length product

1× 1

2× 2

3× 3

4× 4

No DSP

Figure 5.12: Data rate versus length for various lengths of fiber with spatial mul-tiplexing. ese data rates were observed with optimized offset launch and detec-tion with spatial multiplexing with feedback of channel coefficients.

135

the data rate improves noticeably, by 5% and 10% respectively. While these data

rate improvements are modest, the value of these techniques is indicative of fur-

ther benefits which can be exploited with more sophisticated feedback techniques

in optical links. Using V-BLAST and spatial multiplexing with offset coupling, a

bandwidth-length product of about 57 Gb/s-km (for the 3 km case) was obtained,

representing an improvement of 28-fold over the rated fiber bandwidth-length pa-

rameter of the fiber.

5.4.2 2 × 2 link over Silica MMF with VCSELs

An experiment similar to the one conducted with the DFB laser setup was

conducted with directly modulated VCSELs for the 2×2 case, as shown in Fig. 5.2.

However, the fiber lengthswere restricted to 1 km, since the power emied by each

laser (-3 dBm) was insufficient to support a 3 km MIMO link with an acceptable

SNR at 3 km.

e VCSELs had an cross-section diameter of 45 µm, which caused them to

emit beams that were several times larger than the beams obtained in the DFB case.

Moreover, the resulting beam is shaped according to the inherent spatial mode

paern of the emiing laser. As shown in Fig. 5.13, passing the signal through a

mode scrambler was effective in shaping the beam, although the impact on data

rate due to this transformation was not appreciable. us, the ability to control

which modes get activated was significantly diminished in this case.

A procedure similar to that of the DFB link was performed for the evalu-

ation of channel parameters, and by varying the modulation and coding for the

136

Modescrambler

VCSEL beam After mode scrambler

Figure 5.13: Initial VCSEL beam paern and aer being shaped by a mode scram-bler

position with highest rate, evaluated by substituting channel parameters in Equa-

tion (5.2) and optimizing the modulation and coding parameters, the best data rate

wasmeasured. e results are shown in Fig. 5.14, where the rated fiber bandwidth-

length product is 1 Gb/s, and “Sp. mux.” refers to the performance using spatial

multiplexing with feedback. e signal processing and multiplexing benefits are

able to exceed the bandwidth length product significantly for links of 100 m or

more, but the benefits are significantly diminished in comparison to the DFB case.

In addition, while feedback and spatial multiplexing improved data rate perfor-

mance for the 500 m and 1 km cases, the benefits are small. For example, exam-

ining the 35th subcarrier reveals that the ratio of the second singular value to the

highest one diminishes from 0.55 to 0.18 as the length ranges from 10 m to 1 km.

As a result, 10 Gb/s could not be reached for the 1 km case, and the bandwidth-

length product increase over the rated fiber parameter of 1 Gb/s was restricted to

9×; much less than the 28× improvement obtained in the DFB 4× 4 case.

137

10 100 500 1000Length (m)

0

5

10

15

20

Dat

ara

te(G

b/s)

Detector speedlimit

Rated fiberBW-Length product

Fiberlimited

Equipmentlimited

Fiberlimited

Equipmentlimited

1× 1

2× 2 - V-BLAST2× 2 - Sp. mux.No DSP

Figure 5.14: Data rate performance of the 2×2 VCSEL link. e benefits from spa-tial multiplexing are not very significant, although there is notable improvementat longer fiber lengths.

138

5.4.3 2 × 2 link with plastic MMF

e channel parameters and data rate performance was evaluated with

plastic fibers in a manner similar to the measurements conducted with silica fibers,

as described in the earlier subsections. However, due to the lossy, dispersive na-

ture of the plastic fibers, the lengths of fiber sections for which the experiments

were conducted were restricted to 1 m, 10 m and 100 m. In addition, for these

experiments, the waveform generators were operated in their “interleaved” mode,

wherein each waveform generator generated a 20 gigasamples/s signal.

1 10 100Length (m)

0

10

20

30

40

50

Dat

ara

te(G

b/s)

Detector speedlimit

Rated fiberBW-Length product

Fiberlimited

Equipmentlimited 1× 1

2× 2 - V-BLAST2× 2 - Sp. mux.No DSP

Figure 5.15: Data rate performance of the 2× 2 plastic fiber link.

e optimal fiber alignment stage offsets at the transmier and receiver

were ∼8 µm and did not change much with the plastic fiber length. e align-

ment tolerance, measured as the maximum offset introduced at the transmier or

139

receiver for the data rate to reduce by 5%, was about 4 µm for all lengths. e

data rates measured in this experiment are shown in Fig. 5.15, where the rated

fiber bandwidth-length product is 200 Mb/s-km, and “Sp. mux.” refers to the per-

formance using spatial multiplexing with feedback. An interesting observation is

that spatial multiplexing does not improve the performance significantly in com-

parison to V-BLAST. More significantly, the benefits from multiplexing display a

diminishing trend at longer lengths of fiber. For instance, the 35th subcarrier sees

the ratio of the second singular value to the highest one diminish as 0.65, 0.53 and

0.2 as the length ranges as 1 m, 10 m and 100 m respectively. is is indicative

of the fact that the multiplexing capabilities of the channel are poorer than in the

silica case, likely due to significant intermodal coupling in 100 m of plastic fiber.

is is consistent with earlier observations of significant mode coupling over short

distances of perfluorinated plastic fiber [107, 108]. For a visual manifestation for

this effect, Fig. 5.16 shows the impact on a beam launched from a glass fiber patch

cord (referred to as “Input”) into the plastic fiber sections used in this experiment.

It is clear that even as the distance rises to about 10 m, the beam spreads signifi-

cantly across the cross section of the fiber, thus indicative of the signal spreading

across several modes. In terms of the singular values of the MIMO channel, the

second singular value was significantly diminished at 10 m, and almost negligible

at 100 m. us, the bandwidth-length product increase is restricted to about 12×.

In view of these observations, multiplexing through plastic fibers seems

to offer significant benefits only at short distances. At lengths longer than tens

of meters, while signal processing without MIMO (the 1 × 1 case in Fig. 5.15) re-

140

Input beam 1 m 10 m 100 m

Figure 5.16: Beam propagation in plastic fiber sections of various lengths. eheavy intermodal coupling causes a spatial spread of the signal even within tensof meters.

mains effective, the benefits of multiplexing seem to diminish significantly. us,

the bandwidth-length product improvement can be aributed largely to advanced

modulation and signal processing, as opposed to multiplexing, for long sections of

plastic MMF.

5.5 Discussion

In this section, we discuss the observations and results presented in the

previous section, and comment on what the implications of these results are for

some of the practical aspects of designing components and signal processing for

realizing efficient and effective MIMO-MMF systems.

e first realization based on the experimental results is that, in conjunc-

tion with mode scramblers and offset launch and detection, linear processing tech-

niques such as channel estimation and equalization for V-BLAST and spatial multi-

plexing are effective in intensity modulated MIMO links. While the use of square-

law based intensity detection could reduce their effectiveness, within the power

ranges of the transmier considered in this chapter, these techniques were use-

141

ful, especially with offset launch and detection. While more advanced techniques

could involve the use of coherent detection or nonlinear processing, these add a

significant amount of cost and complexity to the implementation, thus making

such systems a less aractive proposition for short, inexpensive fiber optic links.

Futurework should involve a detailed comparison of cost, complexity and data rate

with coherent and incoherent detection, and determine the system limits where

the utility of each technique lies.

e use of feedback in optical links is relevant, especially sincemost optical

links involve a bidirectional setup, thus providing very high bandwidth paths from

both transmiers to receivers. Moreover, since channel conditions vary slowly in

comparison to the data rates, the amount of feedback can be made very small (of

the order of 0.1%). e feedback methods used in the experiments described in

this chapter to enable channel parallelization using spatial multiplexing are sim-

plistic due to constraints in implementation, yet they offer evidence of the utility

of providing channel state information to the transmier. A further refinement

of techniques to estimate and effectively quantize and transmit channel state in-

formation can be expected to provide further increases in performance and the

flexibility to enable a larger number of streams of data through the MMF with

high effective SNRs.

Finally, the effectiveness of offset launch and detection in conjunction with

signal processing facilitates a significant increase the data rates in MMF links. Due

to equipment limitations, only one alignment stage was employed at the transmit-

ter, and one at the receiver. e observations from these experiments strongly in-

142

dicate that offset coupling makes multiplexing effective, and the ability to launch

multiple data streams, with the flexibility of axial offsets and greater modal di-

versity, promises to further increase data rates. While the use of complicated

alignment solutions, such as nanoprecision fiber alignment stages provides sig-

nificant control and accuracy, it is not practical to use these tools when deploying

such links. However, the results from Section 5.4 indicate that there is a signif-

icant amount of flexibility and tolerance in positioning the launch and detection

stages for optimal data rates, motivating the design of devices such as arrays of

lasers and detectors with appropriate geometries to harness modal diversity bene-

fits. e use of such devices would provide a convenient and and easily deployable

solution that would enable multiplexing through MMF links.

5.6 Conclusion

e use of conventional multimode fibers has been limited, owing to the

data rate restrictions imposed by modal dispersion. While electronic dispersion

compensation enables some improvement in data rates, a further boost requires

techniques such as advanced modulation, signal processing and MIMO. To gain

a beer understanding of how various parameters of the system, such as offset

coupling, fiber channel parameters and different optical components, affect sys-

tem performance, we have performed an extensive experimental characterization

of the fiber channel for various fiber media over several fiber sections of various

lengths. In addition to a thorough evaluation of the fiber channel parameters, we

have also presented an evaluation of various modulation and feedback methods

143

that are able to provide bandwidth-length product increases ranging from 10×

to 28×. e observed data rates ranged from 16 - 25 Gb/s with DFB lasers and

VCSELs on silica fibers with lengths of 100 m to 3 km. While plastic fibers ex-

hibited significant intermodal coupling at longer lengths, MIMO techniques still

enabled a 12× improvement in data rate, achieving 22 Gb/s over 100 m. Future

work should investigate how multiple axially offset streams can be used to over-

come diversity limits, and motivate the design of arrays of lasers and detectors

that can take advantage of modal diversity. In addition, investigations are needed

to study how improved feedback quantization can improve the performance of

spatial multiplexing for beer preprocessing of data symbols at the transmier.

144

Chapter 6

Summary and Conclusions

Although single-mode fibers have eclipsed conventional large core multi-

mode fibers in terms of data rate capacity, the ability to multiplex multiple sig-

nals through multimode fibers offers them a new lease of life. e wide existing

deployment of multimode fibers along with the advantage of relaxed alignment

tolerances makes them aractive media for high data rates. In this thesis, we have

considered the use of MIMO signal processing techniques motivated by develop-

ments in MIMO wireless systems. By adapting these techniques to incoherent

MIMO links, we realized significant benefits over state-of-the-art MIMO systems.

Our experimental evaluations using off-the-shelf components, described in Chap-

ter 3, established the basic utility of MIMO techniques in MMF links, although the

lack of control over mode excitation limits the improvement in bandwidth-length

product of the fiber to around 15× over current multimode fiber based standards,

such as 10GBASE-SR. Subsequently, considering a statistical model for the fiber,

the data rate limits of the fiber were evaluated in Chapter 4. Simulations revealed

that positioning of lasers and detectors at the fiber face can have a significant

impact on the data rate. Using ergodic capacity as the metric, the geometry of

laser and detector arrays was optimized for various situations and efficient signal

processing algorithms for large arrays of detectors were also presented. Finally,

145

Chapter 5 describes a complete experimental characterization of a 4 × 4 MIMO-

MMF link with axial offset coupling for various offsets for plastic and silica fibers

of various lengths. e improved strategies discussed in these experiments in-

creased the bandwidth-length product by up to 25×. In addition, the impact of

using directly modulated VCSELs was compared with the use of externally mod-

ulated distributed feedback lasers.

While this thesis has described the development of a MIMO-MMF testbed

and analyzed the theoretical aspects of MIMO-MMF links, there are several fur-

ther developments that require due consideration to make a MIMO-MMF system

practical and easily deployable:

• is thesis has taken a purely signal processing based approach to analyze

the impact of variousmodes andmode groups on the impulse response of the

fiber. Particularly in the experimental sections, the solution to the optimal

launch and detection situation was found by exhaustively searching over

several offset and detection positions using information theoretic capacity

as themetric. Studying the impact of offset launch andmodal filtering on the

delay and dispersion characteristics can allow direct design of the impulse

response of the fiber, like the description in [71].

• is thesis has restricted considerations to incoherent approaches to MIMO-

MMF links. is restriction was imposed since the aim was to consider a

low-cost implementation for short to medium haul optical fiber links. How-

ever, for longer links, a cost evaluation should be conducted with coherent

146

detection, since the use of coherent detection improves receiver sensitivity,

while also ensuring that linear detection and signal processing techniques

function effectively.

• e theoretical fiber model is currently based on a finite element based im-

plementation. Recent developments in MMF modeling have produced sta-

tistical encapsulations that can effectively capture the impact of propagation

in simplified probability distributions, such as in [67, 109]. Such models can

be adapted to suit MIMO-MMF links, thereby significantly reducing com-

putation overheads and simplify the analysis, design and implementation of

MIMO-MMF links.

• A real-time implementation of a MIMO-MMF link would establish the prac-

tical utility of several of the techniques discussed in this thesis, and would

significantly upgrade the capabilities of the MIMO-MMF testbed.

147

Appendices

148

Appendix A

Submodularity of Channel Capacity

To allow the results presented in Chapter 4 to be self-sufficient, we provide

the proof for the submodularity results used in this chapter. e MIMO informa-

tion theory related notation closely follows the notation used in [40]. e proof is

similar to the proof outline presented in [85].

eoremA.1.1. e entropy of a random vectorX = [X1, X2, . . . Xn] is submodular

in the choice of subsets of {X1, X2, . . . Xn}.

Proof. Let h denote the entropy of a random vector. We note that a sufficient

condition for the entropy to be submodular in the subsets of the elements of X is

h(XA∪B) + h(XA∩B) ≥ h(XA) + h(XB)∀A,B ⊆ {X1, X2, . . . Xn} .

We prove this below.

To denote subsets of the random vector, we use the notation XA = {Xi :

i ∈ A}, where A ⊆ {1, 2, . . . n}. Let A and B be subsets of {1, 2, . . . n}. With

149

this, we have

h(XA∪B) + h(XA∩B)− h(XA)− h(XB)

=[h(XA) + h(XB\A|XA)

]+ h(XA∩B)− h(XA)− h(XB)

= h(XB\A|XA) + h(XA∩B)−[h(XA∩B)− h(XB\A|XA∩B)

]= h(XB\A|XA)− h(XB\A|XA∩B)

≥ 0

us, this function is submodular.

Proof of eorem 4.4.1

Proof. We prove this by using eorem A.1.1 and making an observation on aug-

mented matrices. Let P be the SNR of the received signal and ND be the number

of detectors. First, consider anND × 1 complex Gaussian random vector Z whose

covariance matrix is RZ = IND+ P

NDHH†. We denote the subvector of Z con-

sisting of the elements in the set S by ZS . en entropy of this random vector is

given by

h(ZS) = log((πe)|S| det

(RZS

))= log

((πe)|S| det

(I|S| +

P

ND

HSH†S

))= |S| log(πe) + CS.

us, for any set of receive devices S, we have

CS = h(ZS)− |S| log (πe) .

150

We also note that the addition of receive device (detector) to the system

would transform the system matrix as follows, assuming the row x of H, repre-

sented by hx is added:

HS∪{x} =

[HS

hx

]which results in the new capacity

CS∪{x} = log det

I|S|+1 +P

ND

[HS

hx

] [H†

S h†x

]= log det

(RZS∪{x}

)= h(ZS∪{x})− (|S|+ 1) log (πe)

where RZS∪{x} is the covariance matrix a subvector of the random vector Z con-

sisting of elements with indices given in S ∪ {x}.

Consider now a set of receive devices T , such that S ⊆ T ⊆ {1, 2, . . . ND}.

We have

CS − CS∪{x} −(CT − CT∪{x}

)= h(ZS∪{x})− (|S|) log (πe)−

(h(ZS)− |S| log (πe)

)−h(ZT∪{x}) + (|T |) log (πe) +

(h(ZT )− |T | log (πe)

)=

[h(ZS∪{x})− h(ZS)−

(h(ZT∪{x})− h(ZT )

)]≥ 0 (due to the submodularity of entropy)

e last step follows from eorem A.1.1.

151

To prove monotonicity, we repeatedly utilize the fact that det(I+AB) =

det(I+BA). We begin with:

CS∪{x} = log det

I|S|+1 +P

ND

[HS

hx

] [H†

S h†x

] .

Proceeding thus, we have

CS∪{x} = log det

I|S|+1 +P

ND

[HS

hx

][HS

hx

]†= log det

IND+

P

ND

[HS

hx

]† [HS

hx

]= log det

(IND

+P

ND

[H†

SHS + h†xhx

])= log det

(IND

+P

ND

H†SHS

)+ log det

(IND

+

(IND

+P

ND

H†SHS

)−1

h†xhx

)

= log det

(IND

+P

ND

H†SHS

)+ log

(1 + hx

(IND

+P

ND

H†SHS

)−1

h†x

)= CS + log(1 + b)

where b = hx

(IND

+P

ND

H†SHS

)−1

h†x

Now, by observing that b is of the form hxMh†x, whereM is a positive semidefinite

matrix, we have that b ≥ 0, and thus, log(1+ b) ≥ 0. us, we may conclude that:

CS∪{x} = CS + log(1 + b) ≥ CS

thereby establishing monotonicity.

152

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169

Index

Abstract, viiiAcknowledgments, vAlamouti code, 33antenna placement, 64Appendices, 148

Beamforming, 55beamforming, 36, 38Bibliography, 169

channel estimation, 29coherence time, 37Complexity, 96

Dedication, ivDFT, 50digital signal processing, 2dispersion, 1dispersion compensated fibers, 2dispersion compensation, 3diversity, 16DSP, see digital signal processing

electronic dispersion compensation, 2ergodic capacity, 75ergodic rate, 75

FDM, 19Feedback, 54few-mode fibers, 7forward error correction, 52

greedy search, 66greedy selection, 77

intermodal coupling, 24

Mach-Zehnder, 44maximum ratio combining, 34MIMO-OFDM, 46modal diversity, 14, 24mode group diversity multiplexing, 9multicore fibers, 7

OFDM, 45offset coupling, 103overlap integral, 69

photonic crystals, 2pilot, 29plastic MMF, 113power diffusion, 65Precoding, 54

QAM, 45

receive diversity, 34

SDH, 19singular value decomposition, 58SONET, 19space-time codes, 33Spatial Multiplexing, 57Spatial multiplexing, 35spatial multiplexing, 36, 39Submodular search, 78submodularity, 66

TDM, 19transmit diversity, 33

170

VCSEL, 112

wavelength division multiplexing, 5WDM, see wavelength division mul-

tiplexing

171

Vita

Kumar Appaiah received the B.Tech. degree in Electrical Engineering and

the M.Tech. degree in Communication Engineering from the Indian Institute of

Technology Madras, India, in 2008. He is currently pursuing the Ph.D. degree in

Electrical and Computer Engineering at the University of Texas at Austin. His

research interests include signal processing for wireless and optical communica-

tion.

Permanent address: 1, Balaji Apartments,32, ird Street, East Abiramapuram,Mylapore, Chennai - 600004India

is dissertation was typeset with LATEX† by the author.

†LATEX is a document preparation system developed by Leslie Lamport as a special version ofDonald Knuth's TEX Program.

172