Simulation of Optical Communication for Formation Flying ... · Simulation of Optical Communication...

F08014

Examensarbete 30 hpApril 2008

Simulation of Optical Communication for Formation Flying Spacecraft

Pär-Johan Oscarsson

In loving memory of my dear grandparents Bertil and Greta Ek

Simulation of Optical Communication for FormationFlying Spacecraft

Par-Johan Oscarsson

April 7, 2008

Teknisk- naturvetenskaplig fakultet UTH-enheten Besöksadress: Ångströmlaboratoriet Lägerhyddsvägen 1 Hus 4, Plan 0 Postadress: Box 536 751 21 Uppsala Telefon: 018 – 471 30 03 Telefax: 018 – 471 30 00 Hemsida: http://www.teknat.uu.se/student

Abstract

Simulation of Optical Communication for FormationFlying Spacecraft

Pär-Johan Oscarsson

Free space optical communication between miniaturized spacecraft networkedtogether in clusters allows for the possibility to replace large expensive spacecraftperforming scientific measurements. In addition, measurements with the largeeffective apertures available with formation flying spacecraft can significantly improveimaging resolution.

This master thesis investigates the bit error rate for free space optical links as afunction of the parameters, aperture sizes, transmitter power, communicationdistances, modulation formats, and different kinds of receivers and pointing jittercaused by noise in the tracking system. Simulations in Matlab and Simulink show that abit error rate of 10-9 can be achieved with aperture sizes of 1 cm, and a transmitteroutput power of 12 mW for a distance of 10 km using avalanche photodiodereceivers or positive intrinsic negative photodiode receivers with an optical amplifier.

Sponsor: Rymdstyrelsen, VINNOVAISSN: 1401-5757, F08014Examinator: Tomas NybergÄmnesgranskare: Greger ThornellHandledare: Henrik Kratz

Contents

1 Introduction 1

2 History of free space optical communication 3

2.1 Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

2.2 Terrestrial . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

3 Theory and models 9

3.1 Signal power budget . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.2 Transmitter model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3.3 Background radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.4 Receiver models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

3.4.1 PIN diode receiver model . . . . . . . . . . . . . . . . . . . . . . . . 12

3.4.2 PIN diode receiver model with an optical amplifier . . . . . . . . . 14

3.4.3 Avalanche photo diode receiver model . . . . . . . . . . . . . . . . . 16

3.5 Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.5.1 On-off keying . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

3.5.2 Pulse position modulation . . . . . . . . . . . . . . . . . . . . . . . 20

3.6 Vibrations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

3.7 Networks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

4 Modeling 25

4.1 Case studies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25

4.1.1 Link budget for a transmitter and a receiver 1 and 10 km apart . . 25

4.1.2 Link budget for a constant aperture size of 1 cm . . . . . . . . . . . 26

4.1.3 Link budgets for vibrating transmitters and receivers . . . . . . . . 27

4.1.4 Link budgets for messages relayed through several spacecraft . . . . 27

4.2 Simulation setup in Matlab . . . . . . . . . . . . . . . . . . . . . . . . . . 27

4.2.1 Signal power budget in Matlab . . . . . . . . . . . . . . . . . . . . 27

4.2.2 Receiver models in Matlab . . . . . . . . . . . . . . . . . . . . . . . 27

4.2.3 Modulation formats in Matlab . . . . . . . . . . . . . . . . . . . . . 28

4.2.4 Vibration model in Matlab . . . . . . . . . . . . . . . . . . . . . . . 28

4.3 Simulation setup in Simulink . . . . . . . . . . . . . . . . . . . . . . . . . . 29

4.3.1 Signal power budget in Simulink . . . . . . . . . . . . . . . . . . . . 29

4.3.2 Receiver models in Simulink . . . . . . . . . . . . . . . . . . . . . . 31

4.3.3 PPM modulation in Simulink . . . . . . . . . . . . . . . . . . . . . 33

4.3.4 Vibration models in Simulink . . . . . . . . . . . . . . . . . . . . . 40

i

CONTENTS

5 Results 455.1 Link budget for a transmitter and a receiver 1 and 10 km apart . . . . . . 455.2 Link budget for a constant aperture size of 1 cm . . . . . . . . . . . . . . . 45

5.2.1 Matlab results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 455.2.2 Simulink results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

5.3 Link budgets for vibrating transmitters and receivers . . . . . . . . . . . . 565.3.1 Matlab results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 565.3.2 Simulink results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

5.4 Link budgets for networks . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

6 Discussion 696.1 Matlab results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696.2 Simulink results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

7 Conclusions 73

8 Outlook 758.1 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75

Acknowledgements 77

Bibliography 79

Appendices 81

A Matlab functions 81A.1 Ap G.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81A.2 Spaceloss.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81A.3 Pointing.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81A.4 meanvar PIN.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81A.5 meanvar PIN OA.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82A.6 meanvar APD.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84A.7 OOK.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85A.8 PPM.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85A.9 OOK vib APD.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86A.10 OOK vib PIN.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87A.11 OOK vib PIN OA.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88A.12 PPM vib APD.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88A.13 PPM vib PIN.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89A.14 PPM vib PIN OA.m . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

B Embedded Matlab functions 91B.1 Pointing error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91B.2 APD noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91B.3 PIN noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92B.4 PIN with optical amplifier noise . . . . . . . . . . . . . . . . . . . . . . . . 93B.5 Decision circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94B.6 Brownian walk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94B.7 Periodic vibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94B.8 Angular drift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95

ii

List of Figures

2.1 Illustration of the photophone. . . . . . . . . . . . . . . . . . . . . . . . . . 42.2 ARTEMIS communicating with SPOT-4. . . . . . . . . . . . . . . . . . . . 52.3 FSO site using multiple transmitters and receivers. . . . . . . . . . . . . . 6

3.1 Block diagram of a link between a transmitter and a receiver. . . . . . . . 93.2 Power levels of the laser signal. . . . . . . . . . . . . . . . . . . . . . . . . 103.3 Cross-section of a PIN diode. . . . . . . . . . . . . . . . . . . . . . . . . . 123.4 PIN receiver model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123.5 PIN receiver with an optical amplifier model. . . . . . . . . . . . . . . . . . 143.6 Principle of operation of an APD. . . . . . . . . . . . . . . . . . . . . . . . 173.7 APD receiver model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183.8 OOK signal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.9 PPM signal. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

4.1 Link model for a stationary transmitter and receiver. . . . . . . . . . . . . 294.2 Transmitter block. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 294.3 Aperture gain block. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304.4 Pointing error block with vibration models. . . . . . . . . . . . . . . . . . . 304.5 Pointing error block. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.6 Space loss block. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.7 Avalanche receiver block. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314.8 PIN receiver block. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324.9 PIN receiver with optical amplifier block. . . . . . . . . . . . . . . . . . . . 324.10 PPM signal source block. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334.11 PPM modulator block. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344.12 8-PPM modulator block. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354.13 8-PPM shift register. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 364.14 Receiver with PPM-demodulator. . . . . . . . . . . . . . . . . . . . . . . . 374.15 PPM demodulator block. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.16 8-PPM demodulator block. . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.17 8-PPM demodulator decision circuit block. . . . . . . . . . . . . . . . . . . 404.18 Random pointing jitter block. . . . . . . . . . . . . . . . . . . . . . . . . . 414.19 Brownian walk block. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.20 Periodic vibration block. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.21 Angular drift block. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

5.1 BER as a function of aperture sizes and different receivers for d = 1 km. . 465.2 BER as a function of aperture sizes and different receivers for d = 10 km. . 475.3 BER as a function of different receivers and constant aperture sizes of 1 cm. 48

iii

LIST OF FIGURES

5.4 Link model for a stationary transmitter and receiver 10.2 km apart. . . . . 495.5 BER as a function of time. . . . . . . . . . . . . . . . . . . . . . . . . . . . 495.6 Original messages and transmitted PPM signal. . . . . . . . . . . . . . . . 515.7 Outputs from the receiver and demodulator. . . . . . . . . . . . . . . . . . 525.8 Link model for a stationary transmitter and receiver 15 km apart. . . . . . 535.9 BER as a function of time. . . . . . . . . . . . . . . . . . . . . . . . . . . . 535.10 Original messages and transmitted PPM signal. . . . . . . . . . . . . . . . 545.11 Outputs from the receiver and demodulator. . . . . . . . . . . . . . . . . . 555.12 BER as a function of different receivers and jitter amplitude for d = 1 km. 575.13 BER as a function of different receivers and jitter amplitude for d = 10 km. 585.14 Link model for a vibrating transmitter and receiver. . . . . . . . . . . . . . 585.15 BER as a function of time. . . . . . . . . . . . . . . . . . . . . . . . . . . . 595.16 Total pointing error track for the vibrating transmitter and receiver. . . . . 595.17 Histograms of the transmitter vibrations generated by Brownian motion. . 605.18 Link model for a vibrating transmitter and receiver. . . . . . . . . . . . . . 605.19 BER as a function of time. . . . . . . . . . . . . . . . . . . . . . . . . . . . 615.20 Total pointing error track for the vibrating transmitter and receiver. . . . . 615.21 Histograms of the transmitter vibrations generated by the random jitter

model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.22 Link model for a vibrating transmitter and receiver. . . . . . . . . . . . . . 625.23 BER as a function of time. . . . . . . . . . . . . . . . . . . . . . . . . . . . 625.24 Total pointing error track for the vibrating transmitter and receiver. . . . . 635.25 Histograms of the transmitter vibrations generated by Brownian motion. . 635.26 Link model for a vibrating transmitter and receiver. . . . . . . . . . . . . . 635.27 BER as a function of time. . . . . . . . . . . . . . . . . . . . . . . . . . . . 645.28 Total pointing error track for the vibrating transmitter and receiver. . . . . 645.29 Histograms of the transmitter vibrations generated by the random jitter

model. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 655.30 Link model for the first case of 4 spacecraft networked together. . . . . . . 665.31 Link model for the second case of 4 spacecraft networked together. . . . . . 67

6.1 Total pointing error track for the vibrating transmitter and receiver. . . . . 70

iv

List of Tables

4.1 PIN receiver parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.2 APD receiver parameters. . . . . . . . . . . . . . . . . . . . . . . . . . . . 254.3 PIN receiver with an optical amplifier parameters. . . . . . . . . . . . . . . 264.4 Summarized parameters and requirements. . . . . . . . . . . . . . . . . . . 26

v

LIST OF TABLES

vi

Chapter 1

Introduction

Throughout the history of spacecraft communication there has always been of high priorityto find more efficient ways of transmitting and receiving data. In order to minimize thesize, mass, and power consumption of the communication payload onboard spacecraft,the carrier frequency in which information is modulated has continuously increased.

Today optical communication through optical fibres is the backbone for most commu-nication networks on earth and free space optical (FSO) links are becoming a commonsolution where digging trenches for optical fibres is to expensive or impossible. FSO com-munication has several advantages compared to radio frequency (RF) communication.Among them is the larger bandwidth, in which information is modulated, smaller sizeand weight of the communication payload, less transmitter power and high immunity tointerference. The main technological challenge when developing FSO payloads for use inspace is the pointing and tracking system which has to point the very narrow laser beamat the receiver. This is typically done by using the known ephemeris data of the spacecraftto estimate its own and neighbors position for rough pointing. To fine tune the pointing,one method is to use a beacon signal on one spacecraft and a tracking system on theother. Unfortunately it will always be some noise present in the sensors in the trackingsystem, this cause a small random error component in the estimation of the neighborsposition. This in combination with mechanical vibrations from components onboard thespacecraft causes both the pointing of the transmitter and receiver apertures to vibrate,also known as pointing jitter.

Free space optical communication between miniaturized spacecraft networked togetherconstitutes a high speed data distribution channel. This makes it possible to replace largespacecraft collecting scientific data with cheap miniaturized spacecraft flying in closeformation where each spacecraft performs measurements. All measured data is distributedin the network and by interpolating the measured data the network can in real-time actas one giant unit that collects and process large amounts of data.

ASTC (Angstrom Space Technology Centre) at Uppsala University is developing aFSO system to be used by miniaturized spacecraft flying in close formations. In orderto investigate the requirements for such a system it is desirable to make simulationson a system level to derive the link performance under different conditions. That iswhat this master thesis is about. Simulations are done on a system level to investigatehow the link performance depends on different environmental and component parameterssuch as modulation formats, communication distances, pointing jitter, aperture sizes andtransmitter power. The modulation formats investigated in this work is OOK and differentorders of PPM and the receivers are based on PIN and avalanche photo diodes (APDs).

1

CHAPTER 1. INTRODUCTION

Simulations are made by calculating link budgets in Matlab along with simulations in thetime domain using Simulink. The case studies investigated in this work treats spacecraftflying in close formations where the communication distances are between 1 and 10 km.

2

Chapter 2

History of free space opticalcommunication

In 1947 during the International Radiotelegraphic Conference held at Atlantic City (USA)the word ”telecommunication” got its definition as [1]:

”Any transmission, emission or reception of signs, signals, writings, images, soundsor information of any nature by wire, radioelectricity, optics or other electromagnetic sys-tems.”

This definition is very wide since electromagnetic waves both include electricity and op-tics. One of the earliest optical communication systems was used by the Romans wholinked the entire Roman Empire by fire signals placed on watch towers. With this systemthe Roman General Aetius forwarded his news of his victory over Attila to Rome in 451AD. A similar system is also described in the Iliad from around 400 BC [1].

At the end of the 18th century, before the invention of the electrical telegraph, largeparts of Europe were connected by an optical system called the optical air telegraph. Thissystem was made of large towers with wooden arms that used semaphore like codes toform letters. With the help of telescopes at each relay, station the distance between themcould be as far as a few tens of kilometers without loosing sight of each other.

When the electrical telegraph came in the end of the 19th century, the optical airtelegraph was quickly replaced but different primitive optical communication systemswere still used by the military for a long time.

The first optical system that truly resembles our modern systems was made in 1880by Alexander Graham Bell. He made an invention which he called the ”photophone”seen in Fig. 2.1. The photophone was able to transmit a human voice a distance ofabout 200 m using the sunlight as a carrier. The basic principle of the photophone isto focus sunlight onto a flexible reflective membrane. This membrane begins to vibratewhen someone speaks into it and modulates the reflected sun rays. The modulated raysare collected by a photoconductive selenium cell connected to a pair of ear-phones [1].Bell considered this invention greater than the telephone but for practical applicationsthe transmission lengths were too small, the sun is an unreliable source of light and itsavailability is unpredictable even during the day.

Shortly after the invention of the laser in the 1960s, research on laser technology forcommunication purpose began. Since then, laser communication with help of opticalfibres has become the backbone of the global communication network. Today a number

3

CHAPTER 2. HISTORY OF FREE SPACE OPTICAL COMMUNICATION

Fig. 2.1. On the top, someone is speaking into the transmitter and modulates reflectedsun rays. At the bottom, the rays are collected and demodulated [2].

of test-platforms for free space optical links between satellites and earth exist, and freespace optical links also begin to appear in terrestrial communication systems.

2.1 Space

The first free space laser experiments were made in the early 1960s when a NeHe laserbeam was reflected from the ECHO 1 A satellite from one location on earth to another [3].In one experiment a technician chopped the beam with his hand to send a messagewith Morse code to a colleague at the receiver station. The first attempt with lasercommunication from space was 1967 during the manned Gemini 7 mission. AstronautJames Lovell placed a hand held laser transmitter in the window and tried to communicatewith a ground station [3].

In the late 1970s both ESA (European Space Agency) and JPL (Jet Propulsion Lab-oratory) began to consider optical communications. JPLs work started with theoreticalinvestigations on the limits of the sensitivity of optical receivers. Results showed thatmultiple bits per photon seamed possible and their first laboratory demonstration systemshowed that 3 bits/detected photon was possible for reasonable bit error rates (below10−6) [4]. Further investigations showed that even greater efficiencies were possible andthat the practical limit is around 10 bits/photon [4]. At the same time, ESA made its firststudies of using semiconductor diode laser for potential use in intersatellite links. Majoradvances in laser, telescope, tracking and receiver systems were made during the 1980sand 1990s. During that time tests for both deep space and satellite-to-ground commu-nication were made and development of test platforms for intersatellite communicationsstarted.

In 1992, JPL performed an experiment with the Galileo probe where laser beams from

4

2.1. SPACE

Fig. 2.2. ARTEMIS communicating with SPOT-4 [5].

Earth were to be picked up by its onboard camera. The objective was to demonstrate that,based on trajectory predictions and pointing-calibrations of the ground telescopes, an up-link laser signal (to simulate an uplink beacon) could be transmitted to the spacecraft.Over eight nights the demonstration was performed and successful uplink detections oc-curred on seven of the actual demonstration nights. The first night the distance to Galileowas 600 000 km and the last night it was 6 000 000 km from Earth. This showed thatthe pointing of the uplink beam to a deep space probe could be made accurate enoughbut that scintillation on the beam due to atmospheric turbulence was quite severe [4].

In 1995, another space to ground optical communication experiment was made. TheJapanese launched the ETS-VI spacecraft which had a small laser communication terminalinstalled. ETS-VI was planed to be parked at a geostationary orbit (GEO) over Tokyobut an orbit insertion failure resulted in a different orbit. An agreement between NASAand the Japanese space agency (NASDA) was made to do a cooperative satellite-to-ground communication demonstration between the Japanese spacecraft and the groundtelescopes at Table Mountain Facility (TMF) in USA. Every third night from November1995 to May of 1996, except for nights with bad weather, various tests were made. Due tothe satellite orbit, the fly-by passes occurred later and later until well into daytime. Onalmost all of the days, a two-way lockup of the links (at 1 Mbps) was achieved. On manydays good signal strength were observed but on other days the signal faded in and out.This was believed to be caused by altitude fluctuations coupled with imprecise spacecraftcompensations for these fluctuations [4]. Even when the signal strength was reliable,the bit error rate was at times unstable due to scintillation of the beam. Further testsshowed that a multiple beam uplink significantly reduced the scintillation. By breakingup the beam into several independent beams (noncoherent relative to one another), theprobability that an undisturbed beam reached the satellite increased.

5


ESAs laser studies resulted in the start of the SILEX (Semiconductor Laser Inter-Satellite Link Experiment) programme in 1985 [6]. Several tests verified the feasibilityof the project. In 1989 an agreement with the developing team of the Earth observationsatellite SPOT-4 was made to include an optical terminal for communication [6]. This setthe start for the main SILEX development effort. The objective for ESA was to build adata-relay satellite ARTEMIS (Advanced Relay and TEchnology MIssion Satellite). Itspartner, SPOT-4, would use ESAs satellite to relay images back to Earth. SPOT-4 waslaunched in 1998 [7], and ARTEMIS was launched in 2001 [5]. Shortly thereafter, theirfirst laser data link was made. In Fig. 2.2 an artist impression of the two satellites isshown. Today ARTEMIS is located in a geostationary orbit and relays data to Earthusing many RF communication bands. With the optical link to SPOT-4, images arerelayed back to Earth in near real time. The data rate between the two is 50 Mbps andthe distance between them is approximately 38 500 km with a pointing error of less than2 microradians [7, 8, 9]. This project has shown very good performance in both trackingand communication, and has been so successful that JAXA (Japan Aerospace ExplorationAgency) has joined the project. In 2005, JAXA launched a satellite named Kirari whichalso uses ARTEMIS to relay data via the laser link [10]. The SILEX project has showngreat potential for intersatellite optical communications, and more projects are sure tofollow.

2.2 Terrestrial

FSO communication on the ground has a few more challenges than FSO in space, namelythe atmosphere and weather conditions. FSO systems on the ground have to withstand fog(far worse than rain and snow), scattering, absorption, scintillation, physical obstructions,building sway and seismic activity, and it have to satisfy strict regulations concerning

Fig. 2.3. FSO site using multiple transmitters and receivers [11].

6

2.2. TERRESTRIAL

human safety [12]. However, communications technology has been under development for40 years, and many solutions have been found. Fog is the biggest challenge since it modifiesthe light characteristics and might even hinder the passage of light by a combination ofabsorption, scattering and reflection. In regions where dense fog is common, FSO systemshave to be built in a network design to shorten the link distances and add redundantroutes.

Scattering is caused by small particles in the path of the beam. It causes a directionalredistribution of the beam and may cause a significant reduction of intensity over longdistances. Absorption reduces the intensity of the beam and is caused by water moleculessuspended in the atmosphere. Scintillation is caused by air pockets with different tem-peratures. These pockets have different indexes of refraction and cause a passing beamto slightly divert from its course, the turbulent nature of these pockets causes the beamto fluctuate. Another effect caused by scintillation, is beam spreading which over longdistances reduces the intensity of the beam.

Using appropriate power levels and multiple beams (also called spatial diversity, anexample can be seen in Fig. 2.3) mitigates the effect of scattering, absorption and scintil-lation and increases the network availability.

Flying birds and other passing object can temporarily block a single beam FSO sys-tem. These interruptions are typically short and the link can be resumed automatically.Multiple beams increase the probability to always have at least one unobstructed beamreaching the receiver.

Both building sway and seismic activity can upset receiver and transmitter alignment.The effect can be reduced by a combination of using beam divergence and a large receiverfield of view. An alternative is to have the transmitter and receiver track each other.

Today all of this is possible with current technology, and thousands of FSO links existson ground today. These offer stable solutions for large bandwidth at low error rates forlinks between places that would otherwise be difficult to reach, e.g. between buildingsin densely populated cities where digging new trenches for optical fibers would be veryexpensive or impossible [13].

7


8

Chapter 3

Theory and models

This chapter describes the theory and the basic models that are used throughout thiswork. A schematic block diagram of an optical system consisting of a laser transmitterand an optical receiver is shown in Fig. 3.1. The laser transmitter modulates a laser at a

Pointing error

Lpr

Receiver

Pr

Pointing error

Lpt

Free space loss

Ls

Aperture

Gr

Aperture

Gt

Tranmitter

Pt

Fig. 3.1. Block diagram of a link between a transmitter and a receiver (Pt, Gt, Lpt, etc.denote power gains and losses further described in Sec. 3.1).

desired average power, the laser beam is focused by a telescope aperture decreasing thedivergence angle of the beam. The alignment of the transmitter and receiver apertures isnot perfect resulting in a pointing error loss at the transmitter and receiver. When thebeam reaches the receiver the laser is focused by a telescope aperture into the receiver.The receiver converts the optical radiation into an electrical signal. A decision circuitdemodulates the signal and retrieves the information within. If the decision circuit makesan erroneous decision a symbol error occurs and leads to bit errors. If the receiver makesto many errors the link gets a high bit error rate (BER). The BER is the ratio betweenthe bit errors and the total amount of received bits.

3.1 Signal power budget

A signal power budget estimates the received optical power at the receiver. The budgetconsiders all power gains and losses throughout the communication path. For free spacecommunication only the transmitter power, the aperture gains, space loss, and pointinglosses are present. The received power Pr is given by [14]

Pr = PtGtLptLsLprGrLi (3.1)

where Pt is the transmitted power (W ), Gt and Gr are the gains (scalars) of the transmit-ter and receiver apertures. Lpt and Lpr are the pointing losses (scalars) at the transmitter

9

CHAPTER 3. THEORY AND MODELS

and receiver respectively and Ls is the free space loss (scalar). There are always imper-fections introduced in a manufactured system, this is compensated by the implementationloss and is denoted by Li (scalar) (this parameter has to be assumed at the design phase).The aperture gains Gt and Gr are given by [14]

G = η(πD

λ

)2

(3.2)

where D is the telescope diameter (m), η is the optical efficiency (scalar) of the lens (Dt

and ηt or Dr and ηr if the transmitter or the receiver is considered), and λ is the opticalwavelength (m). Due to the beam divergence free space loss in introduced and is givenby [14]

Ls =( λ

4πr

)2

(3.3)

which depends on the range or distance r (m). The pointing loss factors Lpr and Lpt is[14]

Lp = e−Gθ2

(3.4)

where G is the gain of the aperture and θ (Gt and θt or Gr and θr if the transmitter orthe receiver is considered) is the pointing offset in radians.

3.2 Transmitter model

The transmitter model consists of a laser source whose main objective is to operate thelaser according to the modulation format and to maintain a certain average power, and atelescope aperture that increases the directivity of the laser transmitter.

In this work OOK and PPM modulation are studied and are explained in Secs. 3.5.1and 3.5.2. OOK and PPM modulation transmit data by operating the laser in differentmodes to form symbols. Between pulses the laser still emits some light, since shutting ofthe laser causes time lags and upsets the timing of the pulses. Therefore the laser is keptslightly above the laser threshold to maintain stability. In Fig. 3.2 the definitions of Pmax,Pavg and Pmin are shown. The ratio between Pmax and Pmin is called the modulation

Pmax

Pavg

Pmin

P (W)

time (s)

Fig. 3.2. Power levels of the laser signal.

extinction ratio Mer and is given by

Mer =Pmax

Pmin

. (3.5)

10

3.3. BACKGROUND RADIATION

PPM modulation transmit one pulse per symbol and the average power for PPM modu-lation is

Pavg =Pmax

M

(1 +

1

Mer

(M − 1))

(3.6)

where M (scalar) is the order of the PPM modulation format. For OOK the averagepower is

Pavg =Pmax

2

(1 +

1

Mer

). (3.7)

A telescope aperture typically consists of a diffraction limited lens that focuses thelaser beam. The beamwidth θ (radians) of the main lobe for such a laser beam is [15]

θ = 2.24λ

D(3.8)

where D is the diameter of the transmitter aperture (m).

3.3 Background radiation

Background radiation interferes with the received signal. The sun, moon, planets, starsand earthshine produces light that may hit the receiver and cause background noise.Received background power Pb collected at the receiver depends on the background ir-radiance, effective receive area, receiver field of view, optical filter bandwidth and theoptical efficiency as [16]

Pb = HbArecΩfov∆ληr (3.9)

where Hb is the background irradiation energy density (W/m2/sr/µm), Arec is the effectivereceive area (m2), Ωfov is the receiver field of view measured in steradians (sr), ∆λ is theoptical filter bandwidth (µm) and ηr is the optical efficiency of the receiver (scalar). Thereceiver field of view is given by [16]

Ωfov =π

4θ2

fov (3.10)

where θfov is the planar angular detector field of view (radians) and depends on thetelescope aperture.

If the background noise originates from a point source and fits entirely in the receiverfield of view, the received background power can be written as [16]

Pb = NbArec∆ληr (3.11)

where Nb is the background radiance measured in (W/m2/µm).

3.4 Receiver models

The receiver models evaluate the mean and variance of the output signal current fromthe receiver. Important parameters are the input signal power and the total noise fromthe different noise sources at the receiver. Here, three different direct detection receivermodels are studied. Two receiver models use a PIN-photodiode and the third an avalanchephotodiode.

11


3.4.1 PIN diode receiver model

This section describes the typical noise characteristics of an optical receiver based on a PINdiode. In Fig. 3.3 a cross-section of a typical PIN diode is shown. The P-layer material at

SiO2

n-Si

i-Si

SiO2

p-Si

ElectricalContact(Ring)

ElectricalContact

Anti-ReflectionCoating

(very lightly n doped)

Fig. 3.3. Cross-section of a PIN diode [17].

the active surface and the N material at the substrate form a PN junction that operates asa photoelectric converter. In a classical PN diode a neutral layer is formed in the junctionbetween the P- and N-layers known as the depletion layer. This is the layer where asmuch light as possible is supposed to be absorbed, but for a PN diode the depletion layeris extremely thin and most of the light is absorbed in the doped material on either sideof the junction. In PIN diodes this is solved by making the depletion layer thicker byadding a very lightly doped layer called the intrinsic layer between the P- and N-layers.This increases the probability that an entering photon absorbs in the intrinsic layer. Awider junction reduces the capacitance across the junction. The lower the capacitancethe faster the device response becomes. Current can be carried across the junction intwo ways, by diffusion and by drift. A wide depletion layer favours current carriage bythe drift process which is faster than the diffusion process. Adding the intrinsic layerincreases the responsivity and decreases the response time of the diode [17].

No current will pass through a reversed biased P-N junction if nothing happens toexcite electrons from the valence band to the conduction band within the depletion layer.The principle of operation of a PIN diode is that when photons of greater energy thanthe bandgap energy of the material enter, they get absorbed. If the photons are absorbedin the intrinsic layer electron-hole pairs are created there. The electric field across thedepletion layer attracts the free electrons and holes to their opposite charges on eitherside of the junction and current flows.

A schematic view of the PIN receiver model is shown in Fig. 3.4. The model includes

Decision

circuitElectronic

filter

TelescopeData

out

PINOptical

input

Fig. 3.4. PIN receiver model.

a telescope, a PIN diode, an electronic filter, and a decision circuit. The current gener-ated from a PIN diode is proportional to the incident optical power multiplied with the

12

3.4. RECEIVER MODELS

conversion constant, called the responsivity R (A/W ) of the PIN which is given by [14, 18]

R =qη

hf(3.12)

where q is the electron charge (C), η the quantum efficiency (scalar), h Planck’s constant(Js) and f is the optical frequency (Hz). Since the responsivity is the conversion constantbetween the incident optical power and the generated current, the current µ generatedfrom an ideal PIN (without dark, background, and shot noise currents), is given by [14, 18]

µ = RPr (3.13)

where Pr is the incident received optical power (W ).Whenever an electrical current with average value µ is generated by a series of inde-

pendent, random charge carrier transits (as in a PIN diode) a noise current called shotnoise σsn is added to the average current [18]. There are several sources that give riseto shot noise, the current from the received signal, dark current, and current generatedfrom incident background light. Shot noise is not generally white, the actual spectrumdepends on the internal physics of the device exhibiting the noise. But over the range offrequencies employed in optical links the shot noise can be assumed to be white [18]. Thesquared shot noise current, σ2

sn (A2) for a received signal is [14, 18]

σ2sn = 2qRPrBw (3.14)

where Bw is the electrical bandwidth (Hz ) of the signal. Even if the PIN diode is inthe dark it still produce a small amount of current called dark current σdc. Dark noisenormally depends on the operating temperature of the PIN and its physical volume, acooler and smaller PIN decreases the noise [16]. The shot noise (A2) from the dark currentis given by [14, 16, 18]

σ2dc = 2qIdBw (3.15)

where Id is the dark current (A). Incident background light also contributes with shotnoise (A2) and is given by [14, 16]

σ2bg = 2qRPbBw (3.16)

where Pb is the background optical power (W ).Thermal noise arises from random movements of electrical carriers in conductors. The

warmer the conductor is, the larger and more frequent the movements of the electricalcarriers get. Thermal noise is white, has a zero mean, and has a mean-square currentgiven by [16, 18]

σ2th =

4kBTrBw

RL

(3.17)

where RL is the load resistance (Ω), Tr the electronic system noise temperature (K), andkB the Boltzmann constant (J/K ). The temperature Tr is the equivalent temperature ofthe loss and is usually the physical temperature of the load resistor [18].

Now, when all of the noise sources and their currents are known, the mean and thevariance of the output current from the receiver can be derived. The modulation formatscode the information by pulsing the laser in different modes (see Sec. 3.5). For thereceiver, this means that there are two cases that have to be considered: when a laser

13


pulse is received, and when no pulse is received. The mean output current for a receivedpulse is [16, 19]

µ1 = RPr1 + RPb + Id (3.18)

where Pr1 is the received optical power when a pulse is present (the ”1” in µ and Pr

denotes the presence of a pulse, ”0” denotes the absence of a pulse), both backgroundcurrent and dark current is included since they do not have a zero mean value. Shotnoises and thermal noise give rise to the variance (A2) of the total output [16, 19]

σ21 = σ2

sn1+ σ2

dc + σ2bg + σ2

th (3.19)

where σsn1 is the shot noise for a received signal pulse. The noise sources can be addedtogether since they are assumed to be Gaussian and independent of each other. Similarlythe mean current for when no pulse is present is [16, 19]

µ0 = RPr0 + RPb + Id (3.20)

where Pr0 is the received optical power when no pulse is sent. The variance is [16, 19]

σ20 = σ2

sn0+ σ2

dc + σ2bg + σ2

th (3.21)

where σsn0 is the signal shot noise for the absence of a pulse.

3.4.2 PIN diode receiver model with an optical amplifier

In Fig. 3.5, a PIN receiver model with an optical amplifier is shown. The model is derivedfrom [14] and [16] and includes a telescope, an optical band pass filter, input insertionlosses, an optical amplifier, output insertion losses, another optical band pass filter, a PINphotodiode, an electrical filter and a decision-rule circuit. The receiver telescope focuses

Decision

circuitElectronic

filter

Optical filter Output

insertion loss

Optical

amplifierInput insertion

lossOptical filterTelescope

Data

out

PINOptical

input

Fig. 3.5. PIN receiver with an optical amplifier model.

the beam onto the optical filter that discards most of the background radiation. Afterthe filter the beam propagates to the optical amplifier that has losses in its input andoutput due to reflection i.e. optical mismatch. The output from the optical amplifier is anamplified version of the input radiation with added optical amplifier noise. Some of thenoise from the amplifier is suppressed by the optical filter after the amplifier. The PINphotodiode convert the radiation to an electrical signal that passes through an electronicfilter. At the end, a decision circuit decides the kind of information received dependingon the arrival time and amplitude of the signal. To determine the amplitude of the signal,a signal and noise model is used. The signal model includes the PIN photodiode, whichconverts the optical power to an electronic signal current according to its responsivity,Eqn. 3.12. The optical power received is amplified by the optical amplifier before it reachesthe photodiode.

14


In the ideal case where no shot noises, amplified spontaneous emission (ASE) noises,relative intensity noise (RIN), thermal noise, background and dark currents are presentthe resulting electronic current µr for a received optical power Pr is [14]

µr = RGPr (3.22)

where G is the optical amplifier gain (scalar).In the PIN diode receiver model it is included that for every current generated in a

PIN, a corresponding shot noise is superimposed on it. There are three different kindsof shot noises arising from different current sources, one of those shot noises is the signalcurrent shot noise (A2) and is given by [14]

σ2sn = 2qGRPrBw. (3.23)

Another is the background current shot noise (A2) and is given by [14]

σ2bg = 2qGRPbBw (3.24)

where Pb is the power of the optical background noise (W ). Dark current does also giverise to shot noise, for a PIN dark current shot noise σ2

dc (A2) is given by Eqn. 3.15.There are several types of optical amplifiers (OA). They can be divided into two

categories: Semiconductor Laser Amplifier (SLA) and Active-Fibre or Doped-Fibre [20].Although there are different types of OAs, they still work by the same principle. AnSLA can be regarded as a diode laser without mirrors and doped fibres are stripes offibres doped with rear-earth ions that absorbs pump light [21]. The basic principle ofoperation is that either the diode or the ions within the fibre gets excited to a higherenergy level by a pump laser signal at a certain wavelength. After a short period of timethe electrons reach a state with a relatively long lifetime called a metastable state [22].The operational principle is that an incoming signal photon stimulates the transitionfrom the metastable state to the ground state. The signal laser source is matched to havean identical wavelength as the photon that is produced by the stimulated transition. Anincoming signal photon stimulates the emission of another identical photon, those two giverise to more stimulated photons that give rise to more and so on. This is a very practicalway of amplifying optical signals since they do not have to be converted into electricalsignals, become amplified (which will add thermal noise) and then be converted back tooptical signals again. OAs does however have another source of noise called amplifiedspontaneous emissions (ASE). This noise originates from the event when a spontaneoustransition from the metastable state to ground state occur which give rise to an erroneoussignal photon which also becomes amplified during its journey through the OA [20]. Dueto different circumstances three different types of ASE noise can occur, those are called thespontaneous emissions σse, amplified signal-spontaneous emissions σsg−sp, and amplifiedspontaneous spontaneous beat noise σsp−sp. The noise current (A2) from the spontaneousemissions is given by [14]

σ2se = 4Rqhfnsp(G− 1)L2

outBwBop (3.25)

where nsp is the spontaneous emission coefficient (scalar), Lout is the output insertion loss(scalar) of the optical amplifier. Bop is the optical filter bandwidth (Hz ) in the frequencydomain and is given by [14]

Bop ≈ ∆λ · cλ2

(3.26)

15


where c is the speed of light in vacuum (m/s), ∆λ the optical filter bandwidth (m), andλ is the wavelength of the laser. Secondly, the amplified signal-spontaneous beat noisecurrent (A2) is given by [14]

σ2sg−sp = 4R2PrGLinL

2out(G− 1)nsphfBw (3.27)

where Lin is the optical amplifier input insertion loss (scalar). Thirdly, the spontaneous-spontaneous beat noise current (A2) is given by [14]

σ2sp−sp = 8

(RLout(G− 1)nsphf

)BopBw. (3.28)

The amplified source relative intensity noise current (A2) is given by [14]

σ2RIN = 10

RIN10

(RGPr

)2

Bw (3.29)

where RIN is the relative intensity noise (dB/Hz ).Thermal noise (A2) from the electronics is given by [14]

σ2th =

4kBTrFBw

RL

(3.30)

where F the noise figure (scalar).Finally, the mean and variance of the output current from the receiver can be derived.

As for the PIN diode receiver model there are two cases to consider: when a pulse isreceived and when there is no pulse received. For a received laser pulse the mean outputcurrent becomes [14, 19]

µ1 = RGPr1 + RGPb + Id. (3.31)

Id is not amplified since it originates directly from the PIN. The variance (A2) for a signalpulse becomes [14, 19]

σ21 = σ2

sn1+ σ2

bg + σ2dc + σ2

se + σ2sg−sp1

+ σ2sp−sp + σ2

RIN1+ σ2

th (3.32)

where the index ”1” in the expressions for the mean and variance denotes the case whena laser pulse is present, ”0” denotes the absence of a pulse. The mean output current forthe absence of a pulse is [14, 19]

µ0 = RGPr0 + RGPb + Id (3.33)

and the variance is given by [14, 19]

σ20 = σ2

sn0+ σ2

bg + σ2dc + σ2

se + σ2sg−sp0

+ σ2sp−sp + σ2

RIN0+ σ2

th. (3.34)

3.4.3 Avalanche photo diode receiver model

An avalanche photodiode (APD) is essentially a PIN with a high electric field across thePN junction. In Fig. 3.6(a) the cross section of an APD is shown. Structurally, themain difference between a PIN and an APD is that the intrinsic layer (i-layer) is slightlyp-doped in an APD and is renamed to the π-layer. Typically the π-layer in the APD isthicker than the i-layer in the PIN and is carefully designed to ensure a uniform electricfield across the whole layer (see Fig. 3.6(b)) [17].

16


SiO2

p+ Si

π-Si

SiO2n+ Si

ElectricalContact(Ring)

ElectricalContact

Anti-ReflectionCoating

(absorptionregion)

Guard Ringn doped

p Si

multiplicationregion

(a) Cross-section of an APD diode [17].

n+ π p+p

Avalancheregion

Depletion region

ElectricField

Strength

(b) Electric field strength in an APD, note the thinavalanche region [17].

n+region

π

region

Electrical contacts Electrical contact

ArrivingPhoton

+ -50-200Volts

-

-

-

-

-

-

-

-

p+region

EnteringLight

p region

(c) Illustration of the avalanche process, the p-region (avalanche-region) is enlarged [17].

Fig. 3.6. Principle of operation of an APD [17].

17


The principle of operation is that incoming photons pass through the first N-layerand the avalanche region (since they are very thin) and are absorbed in the π-layer. Thisproduces electron-hole pairs in the π-layer, the electric field across the π-layer attracts theelectrons toward the avalanche region, and the holes towards the other direction. In theavalanche region, the electric field is very intense (see Fig. 3.6(b)), and arriving electronsfrom the π-layer are strongly accelerated by the field. When these electrons collide withatoms in the avalanche region, they have such a high energy that they produce newelectron-hole pairs. The new charge carriers (electrons and holes) are also accelerated,may ionize more atoms producing even more electron-hole pairs, and so on. This is calledthe multiplication process and is the cause of the internal gain. A schematic diagram ofan avalanche process is shown in Fig. 3.6(c), the electrons moves to the left and the holesto the right. If a newly produced hole ionize an atom at the far right in the avalancheregion it can produce another electron-hole pair. A new electron will accelerate acrossthe entire region and might start the entire process all over again. This can lead to anuncontrolled avalanche that will never stop. But if the device is designed so that one ofthe carriers has a significantly higher tendency to ionize other atoms than the other, anuncontrollable avalanche can be avoided [17].

A schematic view of an APD receiver model is shown in Fig. 3.7, the model includesa telescope, an APD, an electronic filter and a decision circuit. Incident light at the APD

Decision

circuitElectronic

filter

TelescopeData

out

APDOptical

input

Fig. 3.7. APD receiver model.

is converted into electrical signals proportional to the responsivity (given by Eqn. 3.12)and the power (W ) of the light. In the ideal case where no shot noise currents, thermalnoise, background and dark currents are present the APD converts incident light into acurrent µ as

µ = RGPr (3.35)

where G is the internal gain (scalar) of the APD.As explained in the PIN diode receiver model shot noise originates from the diode and

depends on the average current from different sources. The signal shot noise (A2) is givenby [23, 24]

σ2sn = 2qRPrG

2FBw (3.36)

where F is the excess noise factor of the APD and is given by [23, 24]

F = keffG + (1− keff )(2− 1/G) (3.37)

where keff is the ratio of the hole and electron ionization coefficients (scalar). Backgroundshot noise (A2) is produced by the same process as signal shot noise and is given by [23, 24]

σ2bg = 2qRPbG

2FBw. (3.38)

There are two kinds of dark currents in an APD: one that passes through the avalancheregion and become amplified, and another that does not go through the avalanche region.

18

3.5. MODULATION

The dark current shot noise (A2) is given by [23, 24]

σ2dc = 2qFG2IbBw + 2qIsBw (3.39)

where Ib is bulk leakage current (A) which become amplified and Is is the surface leakagecurrent (A) which does not pass through the avalanche region. Thermal noise σ2

th (A2) isidentical as the case for the PIN and is given by Eqn. 3.17.

Adding all currents and noise current sources, the mean and variance of the totaloutput from the receiver can be derived. There are two cases to consider: when a signalpulse is received and the time between when there is no pulse received. The mean outputcurrent from the receiver when a pulse is present is [19, 23, 25]

µ1 = RGPr1 + RGPb + GIb + Is (3.40)

where the index ”1” denotes a received pulse as before and a ”0” denotes the absence ofa pulse. The variance (A2) of the output current is given by [19, 23, 25]

σ21 = σ2

sn1+ σ2

bg + σ2dc + σ2

th. (3.41)

Similarly, when a pulse is absent the mean output current is [19, 23, 25]

µ0 = RGPr0 + RGPb + GIb + Is (3.42)

and the variance is [19, 23, 25]

σ20 = σ2

sn0+ σ2

bg + σ2dc + σ2

th. (3.43)

3.5 Modulation

Two intensity modulation formats will be examined, on-off keying (OOK) and pulse po-sition modulation (PPM).

3.5.1 On-off keying

On-off keying (OOK) is the most basic form of the intensity modulation formats. It placeseach bit in one slot, a one is sent as a laser pulse while a zero is denoted by the absenceof a pulse. The receiver decides every Ts seconds if a one or a zero has arrived. Since onebit is sent in each symbol the symbol time is

Ts = 1/Br (3.44)

where Br is the bit rate (bps). This implies that the occupied electrical bandwidth (Hz)by OOK modulation equals the bit-rate

Bw = Br. (3.45)

To maintain a constant average power, one can assume that equal amount of zeroes andones will be sent over time. Therefore the required average photon counts per bit is halvedand the transmitted signal power for a ”one” signal becomes Pt = 2Pavg (W ) where Pavg

is the average power. To determine if a received signal is a ”one” or a ”zero” the receivercompares the received signal value to a threshold value (the received signal value is a

19


1 0 1 0 1 0

Ts

Fig. 3.8. OOK signal.

stochastic variable and is denoted by X). The receiver chooses a ”one” if X > Topt and a”zero” otherwise. The optimal threshold value is when there is equal probability that areceived signal is either decoded as a ”one” or a ”zero”, Topt (A) is [25]

Topt =σ0µ1 + σ1µ0

σ0 + σ1

(3.46)

and the normalized distance (scalar) between the threshold and the distribution mean is[25]

Dn =µ1 + µ0

σ1 + σ0

. (3.47)

The bit error probability then becomes [14]

PEOOK= Q(Dn) (3.48)

where Q(x) is the Q-function and is given by

Q(x) =1√2π

∫ ∞

x

e−t2

2 dt. (3.49)

3.5.2 Pulse position modulation

Pulse position modulation (PPM) uses symbols of fix length to send messages, thesesymbols have a fix length in time and is divided into M slots. The number of slots setsthe length of the symbol. To represent a certain symbol a pulse is sent within only oneof the slots as shown in Fig. 3.9. Since PPM uses one pulse in every symbol the number

0 1 2 ......................... M-1

Slot period T Symbol-period Tslot s

Fig. 3.9. PPM signal.

of slots depends on the number of bits sent per symbol as [25]

M = k2 (3.50)

20

3.6. VIBRATIONS

where k is the number of bits. The duration of a symbol (s) depends on the bit-rate as

Ts =k

Br

, (3.51)

and the duration of a slot depends on the M -number and the bit-rate as [25]

Tslot =k

MBr

. (3.52)

The electrical bandwidth (Hz) required for PPM depends on the order of the PPM (M)and the bit-rate as

Bw =MBr

k. (3.53)

PPM modulation becomes more energy efficient for higher values of M. Since k number ofbits are sent per symbol, the average number of signal counts required per bit is dividedby k, which makes it more energy efficient than OOK modulation for k > 2. The receiverdecides what information is received by choosing the slot that has the highest signal count.If the receiver produces an erroneous decision, the number of bit errors becomes ≤ k. Theaverage number of bit errors per decision errors is [14, 25, 26]

Nbe =M

2(M − 1). (3.54)

The probability for the receiver to make a correct slot choice is [14, 25, 26]

Pcsc =

∫ ∞

−∞

1√2πσ1

e− (x−µ1)2

2σ21

[ ∫ x

−∞

1√2πσ0

e− (y−µ0)2

2σ20 dy

]M−1

dx. (3.55)

Combining (3.54) and (3.55) gives the bit error probability for the PPM modulationformat

PEPPM= Nbe(1− Pcsc). (3.56)

3.6 Vibrations

Due to the very narrow beamwidth of a laserbeam, the line of sight pointing betweenthe optics of two communicating satellites has to be very accurate. One method oftracking between satellites is to use a beacon signal on one satellite and a tracking systemand a quadrant detector on the other [14]. Noise in the tracking system and vibrationsfrom mechanical components onboard the satellite causes the transmitter and receivertelescopes to vibrate [14, 15].

Mathematical expressions for the pointing losses can be obtained if the pointing errorangle θT (radians) from a centered line between the transmitter and receiver satellites isassumed. One can also assume that the error angle is composed of a steady state pointingerror and a random pointing component. The pointing error is also assumed to consistof two independent and normally distributed error angles along the azimuthal axis θaz

(radians), and along the elevation axis θel (radians). These assumptions are typicallyused in calculations where random processes are involved [15]. The probability functionsof θaz and θel are given by [14]

θaz =1√

2πσaz

e− (θaz−µaz)2

2σ2az , (3.57)

21


and

θel =1√

2πσel

e− (θel−µel)

2

2σ2el (3.58)

where σaz, σel, µaz and µel are the standard deviations and mean values of the randomcomponents along the azimuthal and elevation error angles respectively. Since θaz and θel

are assumed to be independent of each other, the total radial error angle becomes

θT =√

θ2az + θ2

el, (3.59)

and by symmetry it is assumed that

σT = σaz = σel (3.60)

where σT is the standard deviation of the pointing error. For the case of zero bias (µaz =µel = 0) the total pointing error becomes Rayleigh-distributed with probability densityfunction [14, 15]

f(θT ) =θT

σ2T

e− θ2

T2σ2

T . (3.61)

The same assumptions can be made for the pointing error of the receiver

f(θR) =θR

σ2R

e− θ2

R2σ2

R (3.62)

where θT and θR are the total pointing error angles of the transmitter and the receiverrespectively.

The amplitude of the pointing jitter is given by the standard deviations σT and σR

for the transmitter and receiver pointing error angles respectively. To determine how thepointing jitter affects the total bit error probability of the link, the bit error probabilityfunction can be integrated with the pointing error probability density functions for allerror angles. The total bit error probability of the link budget then becomes [15]

Pbe =

∫ ∞

0

[∫ ∞

0

PE(θT , θR)f(θT )dθT

]f(θR)dθR (3.63)

where PE is the bit error probability function for a certain modulation format, with thepointing errors θT and θR included in the signal power budget.

3.7 Networks

This section describes a model that evaluates the BER for a message that is relayedthrough several satellites before it reaches the intended receiver. It is assumed that themessage passes between several satellites and the result of the model is the BER of thenetwork for a message passing through n links. The BER for the network is given by [14]

BERnet = 1−n∏

i=1

(1− PE(i)

)(3.64)

22

3.7. NETWORKS

where PE(i) is the BER for satellite i. If PE(i) is very small for every satellite i in thenetwork, the BER can be approximated by [14]

BERnet =n∑

i=1

(PE(i)

). (3.65)

From this it can be seen that two extreme cases can occur. If all satellites are identicaland the BER between each satellite is the same, the network BER becomes [14]

BERnet = nPE. (3.66)

The other case is when the BER for communication with one of the satellites is very highwhile the BER for the other satellites are low, the BER for the network becomes [14]

BERnet = PE(i) (3.67)

where PE(i) is the satellite with the high BER. This shows that the overall networkperformance can be governed by a single satellite in this model.

23


24

Chapter 4

Modeling

In this chapter different case studies and the Matlab and Simulink models used to describethem is presented. The results from the simulations are presented in Chap. 5.

4.1 Case studies

In this section the case studies are presented to describe how parameters such as aperturesizes, communication distances, and choices of receivers and modulation formats affectthe BER performance of a link. Link budgets are made to analyze the effect of theseparameters. All cases are assumed to use the same set of receivers. The PIN and APDreceiver parameters are shown in Tables 4.1, and 4.2, and the parameters for the PINreceiver with an optical amplifier are shown in Table 4.3.

Table 4.1. PIN receiver parameters.Parameter Value Unit

η 0.8 scalarId 1 nARl 100 ΩTr 300 K

Table 4.2. APD receiver parameters.Parameter Value Unit

G 200 scalarη 0.59 scalar

Keff 0.01 scalarIb 0.5 nAIs 5 nARl 100 ΩTr 300 K

4.1.1 Link budget for a transmitter and a receiver 1 and 10 kmapart

Two case studies of interest are when the distance between the transmitter and receiveris fixed to 1 and 10 km. Those are both distances that are realistic for formation flying

25

CHAPTER 4. MODELING

Table 4.3. PIN receiver with an optical amplifier parameters.Parameter Value Unit

η 0.8 scalarId 1 nARl 100 ΩTr 300 K

RIN -145 dB/Hznsp 2 scalarG 27 dBF 6 dBLin -1.5 dBLout -1.5 dB∆λ 1 nm

spacecraft. The goal is to evaluate the required aperture sizes needed to get a desiredBER.

First, assume a case where the distance is 1 km, the bit rate is 1 Gbps, and themaximum aperture size allowed is 1 cm. A PIN receiver with and without an opticalamplifier and an APD receiver along with a comparison between OOK and different ordersof PPM modulation are studied. The transmitter laser is limited to a maximum outputpower of 12 mW and its minimum output when held above the laser threshold is 1 mW .For bit-rates above 100 Mbps a bit error probability of at least 10−9 is recommended [27].A summary of the requirements is shown in Table 4.4. The same conditions are used forthe distance of 10 km and for all other case studies.

Table 4.4. Summarized parameters and requirements.Parameter Value Unit

Ptmax 12 mWPtmin

1 mWBr 1 GbpsDt 0 1 cmηt 0.8 scalarDr 0 1 cmηr 0.8 scalarλ 850 nmd 1000 - 10 000 m

Pback 10−13 WLi −2 dB

BER 10−9 scalar

4.1.2 Link budget for a constant aperture size of 1 cm

Assuming the same requirements as in the previous section, a new case where the aperturesizes is held constant at 1 cm is studied. This will evaluate how large the theoretical limitfor the communication distance is for those requirements.

26

4.2. SIMULATION SETUP IN MATLAB

4.1.3 Link budget for a vibrating transmitter and receiver 1 and10 km apart

This case study is an extension of the previous case explained in Sec. 4.1.1. The samereceiver and transmitter parameters as in previous cases are assumed, and the aperturesizes are held constant at 1 cm. To investigate how different amplitudes of the pointingjitter decrease the BER performance, link budgets for both 1 and 10 km are calculatedwith respect to the amplitude of the pointing jitter. It is also assumed that the receiverand transmitter apertures are subjected to the same vibration amplitude.

4.1.4 Link budgets for messages relayed through several space-craft

Two cases are investigated where messages are transmitted from one transmitting space-craft, relayed via two other spacecraft and finally received by a fourth. The distancesbetween every spacecraft are 10 km, the aperture sizes are 1 cm and the modulation for-mat used is 4-PPM. All spacecraft use PIN receivers with optical amplifiers with the sameparameters as shown in Table 4.3. The rest of the parameters are shown in Table 4.4.In the first case, the jitter amplitude of the transmitter and receiver apertures for allspacecraft is σT = σR = 10−5 radians. For the other case the jitter amplitude for thetransmitter and receiver apertures for one of the links is σT = σR = 10−5 radians and forthe other links the jitter amplitude is σT = σR = 6 · 10−6 radians.

4.2 Simulation setup in Matlab

The simulations in Matlab directly calculate the link performance for a certain environ-ment setup and components.

4.2.1 Signal power budget in Matlab

Implementing the signal power budget into Matlab functions is straightforward. Thesefunctions follow the Eqns. 3.2 - 3.4 and use the parameters in the equations as inputparameters. The functions can be seen in the Appendix A.1 - A.3.

4.2.2 Receiver models in Matlab

The receiver models use all the equations found in Sec. 3.4 to calculate the total meanand variances of the output signals for the APD- and PIN-based receiver models. Thesecalculations are performed both for the case when there is an incoming pulse and whenthere is none. In addition, different scalings of the power levels for maintaining a desiredaverage power for the different modulation formats are considered. There are a totalof three functions for this, one for every receiver model. Each calculates the outputstatistics for an incoming pulse and for when there is none. The functions are found inAppendix A.4 - A.6.

27

CHAPTER 4. MODELING

4.2.3 Modulation formats in Matlab

OOK

The function that calculates the bit error rate (BER) for the OOK modulation formatuses the mean and variances for an incoming signal for the case when there is a pulseand when there is none as input parameters. It uses the input parameters to calculatethe normalized distance between the threshold and the distribution mean, Dn, accordingto Eqn. 3.47 in Sec. 3.5.1. Then Dn is used as the input parameter to the Q-function.Matlab has a built-in solver, the complementary error function erfc(x) which relates tothe Q-function as

Q(Dn) =1

2erfc

(Dn√2

). (4.1)

The function OOK.m is found in Appendix A.7.

PPM

The function for calculating the BER for the PPM modulation format has the same inputparameters as the OOK function, and the M -number that decides the number of slots inthe PPM symbol. Calculating the bit error rate for PPM is more difficult than for OOKsince the equation is more complex. The probability that the decision circuit selects thecorrect slot, is the conditional probability that the receiver selects the correct slot whilediscarding the incorrect slots. This leads to a double integral to solve. The fenced termin the equation for Pcsc (3.55)

Pcsc =

∫ ∞

−∞

1√2πσ1

e− (x−µ1)2

2σ21

[∫ x

−∞

1√2πσ0

e− (y−µ0)2

2σ20 dy

]M−1

dx

is the cumulative normal distribution function, and is built into Matlab as the functionnormcdf(x, µ, σ). This gives an easy way to calculate the fenced term.

To avoid that an unnecessarily large integration range is used, an approximation ofthe infinity boundaries has to be made. Since the probability densities are normal, onecan decide where a large enough area of the gaussian curves are located. This is donewith the built-in norminv function, it is used to pinpoint the areas so that only 10−16

percent of the area falls outside the integration region. A for loop is used to numericallyintegrate the entire function. The Matlab function is shown in Appendix A.8.

4.2.4 Vibration model in Matlab

The entire link budget is integrated for every pointing error. Therefore every part ofthe link budget is included in this model. The input parameters are all the parametersincluded in all of the models, and the output is the BER of the entire link. The integral isshown in Eqn. 3.63. To evaluate this integral numerically the integral can be approximatedby

Pbe =∞∑

θR=0

[ ∞∑

θT =0

PE(θT , θR)f(θT )∆θT

]f(θR)∆θR (4.2)

where ∆θT and ∆θR are the step sizes of the summations, and the infinity can be approx-imated by a small number, since only a pointing error of a few microradians is enoughto miss the receiver. There are six different functions for this summation, each for everycombination of receiver and modulation format and is shown in Appendix A.9 - A.14.

28

4.3. SIMULATION SETUP IN SIMULINK

u

G

out

thetaTransmitter

To File

theta.matTo File 2

APD.mat

Symbol error rate

Tx

Rx

Out1

APD-receiver with

PPM-demodulator

PPM signal source

PPM-sig

Org-sig

Spaceloss

Display

Aperture Gainu

G

out

thetaAperture Gain

pointing error pointing error

Fig. 4.1. Link model for a stationary transmitter and receiver.

4.3 Simulation setup in Simulink

Simulink is a built-in simulation tool in Matlab. Simulink models are simulated in the timedomain where every time step is solved as a static model with a differential equation. InFig. 4.1 a model of a complete link budget between a stationary transmitter and a receiveris shown. The model includes the signal power budget, receiver models, PPM modulationformat, and the pointing error jitter model. Every block in the model except the pointingerror blocks are masked which means that if you double click on a block a window appearsthat asks for input parameters, for example the diameter of an aperture.

4.3.1 Signal power budget in Simulink

The signal power budget is divided into several blocks where each block represents onepart of the budget. These parts are the transmitter, aperture gain, pointing error and thespace loss. The transmitter receives signals from the signal source and modulation blockand scales the signal pulses to maintain a desired average power. In the aperture gainblock the telescope gain is determined and is multiplied with the signal. The pointingerror block includes models for the pointing jitter and the loss due to the total pointingerror. Models for the pointing jitter will be explained later. Finally there is the block forthe space loss which multiplies the signal with the damping due to the spreading of thelaser beam.

The transmitter block, Fig. 4.2, is specialized for PPM modulation. It contains one

Out1

1

Switch

Gain

-K-

Constant1

1/Mer

Constant

1

In1

1

Fig. 4.2. Transmitter block.

input for the signal, two constant value blocks, a switch, a gain block which multipliesthe signal, and an output. The switch selects one of the constant values depending onthe input signal. If the signal is a ”one”, it selects the constant value 1 which is later

29

CHAPTER 4. MODELING

multiplied with the gain block. This is the case when a pulse is transmitted. When theinput is ”zero”, it selects the constant 1

Merbecause the laser is biased above the laser

threshold to maintain stability. If the laser would be completely shut off, time lags couldoccur since it takes some extra time to start up the laser. Mer is the modulation extinctionratio. The equation in the gain block scales the signal so that a desired average power ismaintained, and is given by

GT = Pavg

(M

1 + M−1Mer

)(4.3)

where Pavg is the average power and M is the order of the PPM modulation.

Inside the aperture gain block, Fig. 4.3, the input signal is multiplied by Eqn. 3.2.Its first output is the output signal, its second the calculated telescope gain. A constant

G

2

sig

1

Product

Constant

-C-

In1

1

Fig. 4.3. Aperture gain block.

block contains the telescope gain formula, the input signal is multiplied with the calculatedconstant by the product block which produces the output. The constant is also sent tothe second output.

There are several blocks inside the pointing error block, shown in Fig. 4.4. Only the

theta

2

out

1

Terminator

Angular drift

Periodic vibration

Random pointing jitter

Stochastic Brownian vibration

Pointing error calc

u

theta

G

Out1

Out2

AddG

2

u

1

Fig. 4.4. Pointing error block with vibration models.

pointing error calc block in Fig. 4.5 will be described here since the others are parts ofthe vibration model. It contains an embedded Matlab function where ordinary Matlabcode can be inserted, Appendix B.1. The code is then compiled by Simulink to runfaster before the simulation starts. The embedded function evaluates the elevation andazimuthal pointing errors and calculates the total pointing error with Eqn. 3.59. By usingthe total error angle and the aperture gain calculated in the aperture gain block the fadingdue to the pointing error is calculated by Eqn. 3.4. Multiplying this fading with the inputsignal produces the output, which is the optical power that is sent to the receiver.

30


Out2

2

Out1

1

Embedded

MATLAB Function

u

theta_H

theta_V

G

y

theta

fcn

G

3

theta

2

u

1

Fig. 4.5. Pointing error block.

In free space the laser beam is only affected by spreading due to the distance betweenthe transmitter and receiver, and the wavelength of the laser. The space loss block isshown in Fig. 4.6, it contains a gain that multiplies its input with Eqn. 3.3.

Out1

1

Gain

-K-

In1

1

Fig. 4.6. Space loss block.

Those are all the blocks that calculates the signal power budget. The receiver also hasan aperture and a pointing error block.

4.3.2 Receiver models in Simulink

In Figs. 4.7 - 4.9, the APD receiver and the PIN based receiver models are shown. The

Out1

1

Random

Number

Pulse

Generator

Integrator

1

s

M

-C-

n

Tr

Rl

Pb

Ib

Is

Br

G

Keff

Avalanche noise model

u

Br

M

Keff

G

Is

Ib

Pb

Rl

Tr

n

lambda

Xs

stdXs

fcn

1

In1

Fig. 4.7. Avalanche receiver block.

APD receiver model contains an embedded Matlab function that has all the parameters

31

CHAPTER 4. MODELING

of the receiver model explained in Sec. 3.4.3 as inputs. Inside the embedded Matlab func-tion, a code similar to the Matlab functions described in Sec. 4.2.2 is used, Appendix B.2.A difference is that the input signal in this case is already scaled by the transmitter, whichmeans that the scaling does not have to be considered by the function. The output isthe mean and standard deviation of the integrated input signal with the noise included.To achieve a correct output signal, the standard deviation from the function is multipliedwith a random number generated from a gaussian random number generator with a stan-dard deviation of ”1”, and a ”0” mean. Adding the mean output signal with the standarddeviation simulates the output from the receiver with all of the APD noise included. Both

Out1

1

Random

Number

Pulse

Generator

PIN noise model

u

M

Br

Id

Pb

Rl

Tr

n

lambda

Xs

stdXs

fcn

Integrator

1

s

M

-C-

n

Tr

Rl

Pb

Br

Id

In1

1

Fig. 4.8. PIN receiver block.

Out1

1

PIN noise model

u

delta_lambda

F

M

Br

Lout

Lin

RIN

G

nsp

Id

Nb

Rl

Tr

n

lambda

Dr

meanPPM

stdPPM

fcn

1

s

Nb

nsp

G

RIN

Lin

Br

M

Lout

Dr

-C-

Tr

n

Rl

Id

F

-C-

1

Fig. 4.9. PIN receiver with optical amplifier block.

PIN based receivers are designed by the same principle as the APD. They evaluate all

32


of the parameters explained in Secs. 3.4.1 and 3.4.2, and embedded Matlab functions,Appendix B.3 and B.4, calculates the mean and standard deviation of the output cur-rent from the PIN based receivers. After the summation of the standard deviation andthe mean of the signal, the sum is integrated by an integration block that is restartedevery PPM slot period given by Eqn. 3.52. The integration block is restarted by a pulsegenerator that produces a pulse every Tslot seconds.

4.3.3 PPM modulation in Simulink

Constructing the modulation models in Simulink is different than coding the probabilityintegrals in Matlab. In Simulink several modulator and demodulator blocks for differentmodulation formats are already available, but that is not the case for PPM. Thereforenew blocks have been developed from scratch.

The modulator

The modulator block is inside the PPM signal source block which can be seen in Fig. 4.10.There are two blocks inside, a random number generator that produces equally distributed

Org

2

Out1

1

Random Integer

Generator 4

Random

Integer

MPPM -Modulator

MPPM -Modulator

Fig. 4.10. PPM signal source block.

numbers, and the PPM modulator block. The numbers represents messages that will besent over the link. Those are both sent to the modulator and to a second output forcomparison with the received decoded message. The PPM modulator is restricted tomodulate 2, 4, and 8 PPM, where the symbol length is selected by an input parameter,M.

In Fig. 4.11 the inside of the PPM modulator block is seen. Inside there are threeswitches that enables one of the separate modulator blocks. The modulation blocks aremade of enabled subsystems, which means that an enabling signal has to be sent to themto be activated. This is done by coupling the input parameter M to the first input portof the three switches. The first input port of the switches selects which one of the otherports that will be open. If for example, M = 8, port number 8 will be open on all threeswitches. For the first two switches no signal is sent through port 8, but for the third thesignal ”1” will pass through and enable the 8-PPM modulator block shown in Fig. 4.12.

A sample and hold block samples the input signal from the random number generator.The block samples the input with a sampling frequency given by the pulse generator,which generates a pulse each PPM symbol period given by Eqn. 3.51. It has to be notedthat the random number generator is also set to produce numbers at the same rate. Fromthe sample and hold, the sampled signal is sent to a switch case block. This block checksthe value of its input and sends an enabling signal to the corresponding switch case actionsubsystems (as an example, case[0] is the case for an input value of ”0” etc.).

33

CHAPTER 4. MODELING

Out1

1

Multiport

Switch3

Multiport

Switch2

Multiport

Switch1

Ground9

Ground8

Ground7

Ground6

Ground5

Ground28

Ground27

Ground26

Ground25

Ground24

Ground23

Ground22

Ground20

Ground19

Ground18

Ground17

Ground16

Ground14

Ground13

Ground12

Ground10

Constant9

M

Constant8

M

Constant7

M

Constant6

1

Constant5

1

Constant4

1

8-PPM

Modulator

In1 Out 1

4 -PPM

Modulator 1

In1 Out 1

2-PPM

Modulator 1

In1 Out 1

In1

1

Fig. 4.11. PPM modulator block.

34


Out 1

1

Switch Case Action

Subsystem 8

case : Out1

Switch Case Action

Subsystem 7

case : In1 Out1

Switch Case Action

Subsystem 6

case : In1 Out1

Switch Case Action

Subsystem 5

case : In1 Out1

Switch Case Action

Subsystem 4

case : In1 Out1

Switch Case Action

Subsystem 3

case : In1 Out1

Switch Case Action

Subsystem 2

case : In1 Out1

Switch Case Action

Subsystem 1

case : In1 Out1

Switch Case Action

Subsystem

case : In1 Out1

Switch Case

u1

case [ 0 ]:

case [ 1 ]:

case [ 2 ]:

case [ 3 ]:

case [ 4 ]:

case [ 5 ]:

case [ 6 ]:

case [ 7 ]:

case [ 8 ]:

Sample

and Hold

In<Lo>

S/H

Pulse

Generator 2

Pulse

Generator 1

Enable

In 1

1

Fig. 4.12. 8-PPM modulator block.

35

CHAPTER 4. MODELING

A 8-PPM symbol can represent one of eight different messages, in this case theyare chosen to be either one of the integers ”0” to ”7”. The case number eight is for theinitiation when the system is called for the first time, at that time a zero output is desired.All other cases enables one of the other switch case action subsystems, which producesthe desired PPM symbol. The outputs from all of the switch case action subsystems aresummed together to produce a single output. Only one of the cases is enabled at a time,this means that only one of the subsystems will produce a symbol while the others are”quiet”.

Fig. 4.13 shows the inside of one of the switch case action subsystems. This block

Out1

1

Switch

Sample

and Hold

In<Lo>

S/H

Sample

and Hold 7

In<Lo>

S/H

Sample

and Hold 6

In<Lo>

S/H

Sample

and Hold 5

In<Lo>

S/H

Sample

and Hold 4

In<Lo>

S/H

Sample

and Hold 3

In<Lo>

S/H

Sample

and Hold 2

In<Lo>

S/H

Sample

and Hold 1

In<Lo>

S/H

Constant

0

Action Port

case:

In1

1

Fig. 4.13. 8-PPM shift register.

36


produces a 8-PPM symbol, eight sample and hold blocks are coupled together to producea shift register. Each sample and hold block has an initial value, where seven of them areinitiated with ”0”, and one with ”1”. Which one of the block that has ”1” as initial valuedepends on what symbol the case block represents. If for example the symbol ”0” is to besent, the sample and hold block at the top has the ”1”. For every slot time period, everysample and hold block sends their current value to the one below. The bottom block bothsends its output to the block at the top, and to the output port of the subsystem.

A pulse generator in the 8-PPM modulator block is coupled to the input of everyswitch case action subsystem. The pulse generator produces a pulse every slot periodgiven by Eqn. 3.52. These pulses are sent to every sample and hold block to make surethat they are triggered at the same time and at the correct rate.

The pulse generator starts at the same time as the pulse generator that controls thesample and hold block at the input of the 8-PPM modulator block. This means that thepulse generator that controls the shift registers sends eight pulses for every integer value.This makes the values in the shift registers to complete a full ”lap” and end up at thesame block as they started. The output from the switch case action subsystem is thesame as the output from the bottom sample and hold block. This is how the symbol isdivided into eight slots with a pulse placed in one of them.

The switch before the output of the switch case action subsystem ensures that theblock has a zero output when no signal is expected.

The demodulator

This block decodes the PPM pulse train produced by the modulator. The demodulatorblock is inside the receiver with the PPM demodulator block shown in Fig. 4.14. It takesthe integrated output signal with added noise from the receiver model and attempts todecode the messages in the signal. The inside of the PPM demodulator block is shownin Fig. 4.15. This block works precisely as the modulator block, an input parameter,M, selects the order of PPM to be demodulated. If M = 8 the block that demodulates8-PPM will be selected.

Out1

1

MPPM -Demodulator

MPPM -Demodulator

APD2

Avalanche photo -

detectorIn1

1

Fig. 4.14. Receiver with PPM-demodulator.

A shift register reads the input signal so that each integrated slot value from thereceiver of the PPM symbol ends up in one out of eight sample and hold blocks in the8-PPM demodulator block, Fig 4.16. Every output of the sample and holds are sent tothe block representing the decision circuit, Fig. 4.17, and to the input of the sample andhold block below, except for the bottom block.

The pulse generator controlling the speed of the shift register is perfectly synchronizedwith the pulse generator in the modulator. In a more realistic case, a synchronizationblock has to synchronize the receiver with the transmitter. The pulse generator is set toproduce pulses each slot period with a phase delay of half of the slot duration time. Thedelay ensures that the pulses are sampled at the top of the pulse and not to close to theedges.

37

CHAPTER 4. MODELING

Out1

1

Multiport

Switch3

Multiport

Switch2

Multiport

Switch1

Ground9

Ground8

Ground7

Ground6

Ground5

Ground28

Ground27

Ground26

Ground25

Ground24

Ground23

Ground22

Ground20

Ground19

Ground18

Ground17

Ground16

Ground14

Ground13

Ground12

Ground10

Constant9

M

Constant8

M

Constant7

M

Constant6

1

Constant5

1

Constant4

1

8 -PPM

Demodulator

In1 Out 1

4 -PPM

Demodulator

In1 Out 1

2-PPM

Demodulator

In1 Out 1

In1

1

Fig. 4.15. PPM demodulator block.

38


Out 1

1

Decision Circuit

In 1

In 2

In 3

In 4

In 5

In 6

In 7

In 8

Out 1

Sample

and Hold 1

In<Lo>

S/H

Sample

and Hold 8

In<Lo>

S/H

Sample

and Hold 7

In<Lo>

S/H

Sample

and Hold 6

In<Lo>

S/H

Sample

and Hold 5

In<Lo>

S/H

Sample

and Hold 4

In<Lo>

S/H

Sample

and Hold 3

In<Lo>

S/H

Sample

and Hold 2

In<Lo>

S/H

Pulse

Generator 1

Pulse

Generator 2

Enable

In 1

1

Fig. 4.16. 8-PPM demodulator block.

39

CHAPTER 4. MODELING

Each slot value is sent to the decision circuit block, Fig. 4.17, to be compared withothers. This block compares all slot values and sorts out the greatest pulse. The output

Out1

1

Terminator

Embedded

MATLAB Function

u0

u1

u2

u3

u4

u5

u6

u7

y 0

y 1

fcn

Trigger

In8

8

In7

7

In6

6

In5

5

In4

4

In3

3

In2

2

In1

1

Fig. 4.17. 8-PPM demodulator decision circuit block.

from the decision circuit is the number of the slot that holds the largest pulse. Thisnumber is the decoded message. An embedded Matlab function is responsible for this,Appendix B.5, that uses the Matlab function max, it produces two outputs: the maximumvalue and the identity i.e. number of order, of the largest input variable.

4.3.4 Vibration models in Simulink

The vibration models constructed in Simulink are divided into four parts, where eachpart simulates different vibration characteristics. Two of them simulates random pointingerrors: one by using random numbers, and the other performs a random Brownian walk.The third, simulates periodic vibrations, and the last simulates drifting motion.

Random pointing jitter

This model simulates random pointing jitter by jumping the pointing of the laser beambetween normally distributed random error angles with a zero mean. The model is seenin Fig. 4.18 and includes two random number generators and two constant blocks. Theconstant values are given as inputs to the user by double clicking on the random pointingjitter block, seen in Fig. 4.4. Normally distributed random error angles are generatedby the number generator blocks, and the standard deviation is scaled by multiplying therandom values with the desired standard deviation from the constant block. The totalerror angle is divided into azimuthal and elevation error angles and are muxed togetherbefore the output.

Brownian walk

In this model the laser steps in random directions with a random angle. The total rangeof the walking motion is bounded by the desired value of the standard deviation from

40


theta

1

Constant2

-C-

Constant1

-C-

Fig. 4.18. Random pointing jitter block.

zero (perfect pointing). In this case the Brownian motion is divided into azimuthal andelevation error angles. The total error angle and the pointing loss is then calculated bythe Pointing error calc block, Fig. 4.4. In Fig. 4.19 the inside of the Stochastic brownianvibration block is shown. Two memory storage blocks stores the calculated azimuthal

theta

1

Gain

14

Embedded

MATLAB Function

theta_V

theta_H

sigma_V

sigma_H

theta_H 1

theta_V1

fcn

theta_H

theta_V

theta_H

theta_V

Data Store

Memory 1

theta_V

Data Store

Memory

theta_H

-C-

-C-

Fig. 4.19. Brownian walk block.

and elevational error angles from the previous time step. Along with the memory storageblocks, there are corresponding read and write blocks to the memories. An embeddedMatlab function, Appendix B.6, uses the error angle values from the memory, and twoconstant values that contains the desired standard deviation of the range of the walkingmotion. Those values are given by the user by double clicking on the Stochastic brownianvibration block.

Each step is of random angular length and in random direction, it is desired that aftera large number of steps that the total walk is centered around zero (perfect pointing). Inorder to achieve this the motion is bounded within a Hooke-law potential well. A springforce (F = −kx) proportionally drags the walking motion back towards the center thefurther away the walking motion deviates from it.

Periodic vibrations

This model simulates periodic vibrations generated from, e.g., spinning parts in a space-craft, such as momentum wheels and gyroscopes. The periodic vibrations are dividedinto azimuthal and elevation error angles in order to simulate different kinds of circularmotions. Double clicking on the Periodic vibration block generates a window where the

41

CHAPTER 4. MODELING

amplitudes, angular speeds, and the phase shifts for both the azimuthal and elevationalperiodic vibrations can be written as inputs. The Periodic vibration block consisting an

theta

1

Embedded

MATLAB Function

t

amp_V

amp_H

omega_V

omega_H

phase_V

phase_H

theta_H

theta_V

fcn

12:34

-C-

-C-

-C-

-C-

-C-

-C-

Fig. 4.20. Periodic vibration block.

embedded Matlab function with the amplitudes, angular speeds, and phase shifts as in-puts, Fig. 4.20. Apart from the other inputs the input at the top is a built-in Matlabblock called the Digital Clock, which generates the simulation time in seconds as output.

In the embedded Matlab function, Appendix B.7, a sine and a cosine function calculatethe azimuthal and elevation error angles, respectively, as

θaz = αaz sin (ωazt

2π− Φaz) (4.4)

and

θel = αel sin (ωelt

2π− Φel) (4.5)

where αaz and αel (scalars) are the amplitudes, ωaz and ωel (rad/s) are the angular speeds,and Φaz and Φel (rad) are the phase shifts of the periodical motions of the azimuthal andelevation error angles respectively. The time t is the simulation time in seconds (s).

From the embedded Matlab function there are two outputs, the azimuthal and eleva-tion error angles. Those are muxed together to produce a single output for the Periodicvibration block.

Angular drift

This block simulates drifting motion of the laser that could arise from some failure or astationary error in the tracking system.

In Fig. 4.21 the inside of the Angular Drift block is shown. It consists of three memorystorage blocks that stores the angular errors, and the simulation time from the previoustime step (their initial values are zero). Those values along with the current simulationtime and desired azimuthal and elevational angular speeds are used as inputs to an em-bedded Matlab function, Appendix B.8. The function uses the input to calculate theerror angles for the current time step as

θaz = θazl+ ωaz(t− tlast) (4.6)

andθel = θell + ωel(t− tlast) (4.7)

where θazland θell are the error angles of the previous time step, ωaz along with ωel are the

speeds of the angular drifting motions for the azimuthal and elevational error angles, and

42


theta

1

Embedded

MATLAB Function

t

t_last

speed_V

speed_H

theta_o_V

theta_o_H

theta_H

theta_V

t_o

fcn

12:34

t_last

theta_H

theta_V

t_last

theta_H

theta_V

Data Store

Memory 2

t_last

Data Store

Memory 1

theta_V

Data Store

Memory

theta_H

-C-

-C-

Fig. 4.21. Angular drift block.

tlast and t are the simulation time from the previous time step and the current simulationtime respectively.

There are three outputs from the embedded Matlab function: the new error angles,and the current simulation time. The error angles are muxed together to produce theoutput for the Angular Drift block, before they are muxed they are also stored in thememory blocks to be used as inputs for the next time step. This is also done for thecurrent time step, it is stored in a memory block to be used as an input for the next timestep.

43

CHAPTER 4. MODELING

44

Chapter 5

Results

In this chapter the results from the case studies explained in Chap. 4 are presented.These results detail the communication performance with respect to different variables.All studies use the same set of receivers, the parameters of which are given in Tables 4.1 -4.3.

5.1 Link budget for a transmitter and a receiver 1

and 10 km apart

This section investigates a case where the distance between the transmitter and receiver is1 km, the goal is to examine the required aperture size at the receiver and transmitter. It isassumed that the transmitter and receiver apertures have the same size. The requirementsfor this case is described in Sec. 4.1.1 and is summarized in Table 4.4. In Fig. 5.1 the BERperformance of the different receivers and modulation formats is shown. It shows thatthe APD and the PIN receiver have similar performance and that their required aperturesizes is roughly 0.3 cm depending on the modulation format. The PIN receiver does alsomeet the requirement of having an aperture size smaller than 1 cm, it needs aperture sizesof roughly 0.7 cm.

Fig. 5.2 show the BER plots as a function of the aperture sizes for 10 km transmission.All parameters are the same as for the 1 km case except for the distance.

5.2 Link budget for a constant aperture size of 1 cm

5.2.1 Matlab results

With the same parameters and requirements as in Tables 4.1 - 4.4 and setting the aperturesize constant at 1 cm, the theoretical distances reached for different demands of BERperformance is plotted. Fig. 5.3 show the plots for BER as a function of distance for thedifferent receivers and for OOK and M = 2 to M = 8 PPM modulation.

5.2.2 Simulink results

Two additional cases are simulated in Simulink, one where the communication distanceis 10.2 km and the other where the distance is 15 km. Both simulations use 8-PPMmodulation and an APD receiver.

45

CHAPTER 5. RESULTS

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.810

−10

10−8

10−6

10−4

10−2

100

Aperture diameter (cm)

BE

R

OOK2−PPM4−PPM8−PPM

(a) PIN receiver.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.3510

−10

10−8

10−6

10−4

10−2

100


BE

R


(b) APD receiver.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.3510

−10

10−8

10−6

10−4

10−2

100


BE

R


(c) PIN receiver with optical amplifier.

Fig. 5.1. BER as a function of aperture sizes and different receivers for d = 1 km.

46

5.2. LINK BUDGET FOR A CONSTANT APERTURE SIZE OF 1 CM

0 0.5 1 1.5 2 2.510

−10

10−8

10−6

10−4

10−2

100


BE

R


(a) PIN receiver.

0 0.2 0.4 0.6 0.8 110

−10

10−8

10−6

10−4

10−2

100


BE

R


(b) APD receiver.

0 0.2 0.4 0.6 0.8 110

−10

10−8

10−6

10−4

10−2

100


BE

R



Fig. 5.2. BER as a function of aperture sizes and different receivers for d = 10 km.

47

CHAPTER 5. RESULTS

1 2 3 4 5 6 7 8 9 1010

−10

10−8

10−6

10−4

10−2

100

Distance (km)

BE

R


(a) PIN receiver.

0 5 10 15 20 25 3010

−10

10−8

10−6

10−4

10−2

100

Distance (km)

BE

R


(b) APD receiver.

0 5 10 15 20 25 3010

−10

10−8

10−6

10−4

10−2

100

Distance (km)

BE

R



Fig. 5.3. BER as a function of different receivers and constant aperture sizes of 1 cm.

48


Fig. 5.4 shows the link model for the case where the distance is 10.2 km. The display

Transmitter

To File

SER.mat

Symbol error rate

Tx

Rx

Out1

APD-receiver with

PPM -demodulatorPPM signal source

PPM -sig

Org-sig

Spaceloss

Scope 4

Scope 2Scope 1

Display

1.501 e-008

4

2.666 e+008

Aperture Gain Aperture Gain

Fig. 5.4. Link model for a stationary transmitter and receiver 10.2 km apart.

shows the result of the total symbol error rate (SER) on the top, the value in the middleis the amount of receiver errors, and the last value is the number of received symbols. Toderive the BER Eqn. 3.54 is multiplied with the SER value. During the simulation theBER value changes over time and when the simulation was stopped the BER value was

BER =8

14· 1.501 · 10−8 = 8.577 · 10−9 (5.1)

which is comparable with the theoretical value of 4.157 · 10−9 derived from the Matlabmodel. In Fig. 5.5 a plot of the BER as a function of time is shown. The plot shows that

0 0.2 0.4 0.6 0.8 10

1

2

3

4

5

6

7

8

9x 10

−9

X: 0.7995Y: 8.577e−009

Time(s)

BE

R

Fig. 5.5. BER as a function of time.

the BER value has not yet converged to a stable value.Figs. 5.6 and 5.7 shows the signals from scopes 1-4 in Fig. 5.4 where the third scope

is connected to the output of the APD inside the APD-receiver and PPM-demodulatorblock. Fig. 5.6(a) shows the original random messages generated from the random integer

49

CHAPTER 5. RESULTS

generator shown in Fig. 4.10, the messages are modulated into 8-PPM signals and trans-mitted with the specified power levels and are illustrated in Fig. 5.6(b). At the receiverthe 8-PPM signals are collected by the receiver aperture and converted into electricalsignals by the APD receiver. Fig. 5.7(a) shows the output from the APD receiver. Thefilled signal is the mean value with the background and dark current included, the otheris the total output signal including all mean and noise currents. The electrical signalsare sampled by the demodulator that converts the signals back into the original messageswithout any receiver errors made, Fig. 5.7(b). Note the time lag of two symbol durationsbetween the transmitted and received messages. This is due to the sample and holds inthe modulation and demodulation process.

Figure 5.8 shows the result from the simulation where the distance between the receiverand transmitter is 15 km. The SER value shown at the display is 1.545 · 10−4 whichcorresponds to a BER value of 8.829 · 10−5. This is almost identical to the BER value8.911 · 10−5 derived form the link budget in Matlab. The BER as a function of time,Fig. 5.9, converges to a stable value unlike the previous case. Figs. 5.10 and 5.11 showsthe signals from scopes 1-4. This time the occurrence of a receiver error is shown, inFig. 5.11(a) the receiver fails to detect a pulse (the pulse in the middle of the time interval7.205− 7.210 µs). The integrated output current over the pulse duration is smaller thanone of the integrated output current over a slot where only noise is present. This slotis interpreted by the demodulator as a signal pulse and translates it to its equivalentmessage value. Fig. 5.11(b) shows the decoded values and one of the decoded messagesdeviates from the corresponding original message.

50


30.95 30.955 30.96 30.965 30.97 30.9750

1

2

3

4

5

6

7

8

Time (µs)

Orig

inal

mes

sage

(0−

7)

(a) Original messages.

30.95 30.955 30.96 30.965 30.97 30.9750

2

4

6

8

10

12

Time (µs)

Out

put p

ower

(m

W)

(b) Transmitted 8-PPM messages.

Fig. 5.6. Original messages and transmitted PPM signal.

51

CHAPTER 5. RESULTS

30.95 30.955 30.96 30.965 30.97 30.975

0

10

20

30

Time (µs)

Out

put c

urre

nt (

µA)

MeanMean + noise

(a) Output signal from the receiver.

30.95 30.955 30.96 30.965 30.97 30.9750

1

2

3

4

5

6

7

8

Time (µs)

Dec

oded

mes

sage

(0−

7)

(b) Decoded messages from the demodulator.

Fig. 5.7. Outputs from the receiver and demodulator.

52


Transmitter

To File

SER.mat

Symbol error rate

Tx

Rx

Out1

APD-receiver with

PPM -demodulatorPPM signal source

PPM -sig

Org-sig

Spaceloss

Scope 4

Scope 2Scope 1

Display

Aperture Gain Aperture Gain

0.0001545

6192

4.007e+007

Fig. 5.8. Link model for a stationary transmitter and receiver 15 km apart.

0 0.02 0.04 0.06 0.08 0.1 0.12 0.14

5.5

6

6.5

7

7.5

8

8.5

9

9.5

10x 10

−5

X: 0.1202Y: 8.83e−005

Time(s)

BE

R


53

CHAPTER 5. RESULTS

7.195 7.2 7.205 7.21 7.215 7.220

1

2

3

4

5

6

7

8

Time (µs)

Orig

inal

mes

sage

(0−

7)

(a) Original messages.

7.195 7.2 7.205 7.21 7.215 7.220

2

4

6

8

10

12

Time (µs)

Out

put p

ower

(m

W)

(b) Transmitted 8-PPM messages.

Fig. 5.10. Original messages and transmitted PPM signal.

54


7.195 7.2 7.205 7.21 7.215 7.22

0

5

10

15

20

Time (µs)

Out

put c

urre

nt (

µA)

MeanMean + noise

(a) Output signal from the receiver.

7.195 7.2 7.205 7.21 7.215 7.220

1

2

3

4

5

6

7

8

Time (µs)

Dec

oded

mes

sage

(0−

7)

(b) Decoded messages from the demodulator.

Fig. 5.11. Outputs from the receiver and demodulator.

55

CHAPTER 5. RESULTS

5.3 Link budget for a vibrating transmitter and re-

ceiver 1 and 10 km apart

5.3.1 Matlab results

In Figs. 5.12 and 5.13 the results for the case explained in Sec. 4.1.3 is presented. Fig. 5.12shows the BER as a function of the pointing jitter for the three different receivers andOOK, and 2, 4 and, 8 PPM modulation, where the distance between the transmitter andreceiver is 1 km. Fig. 5.13 shows the BER as a function of the pointing jitter for theAPD and PIN receiver with an optical amplifier where the distance is 10 km. The resultsfor the PIN receiver is not included since it already has a high BER even without thepointing jitter as seen in Fig. 5.3(a).

5.3.2 Simulink results

Based on the Matlab results, two additional cases where the distance is 10 km, are sim-ulated in Simulink. Case 1 uses an APD receiver, 2-PPM modulation, and an averagepointing jitter amplitude of σT = σR = 6 · 10−6 rad for both the transmitter and receiver.Case 2 uses a PIN receiver with an optical amplifier, 4-PPM, and an average pointingjitter amplitude of σT = σR = 10−5 rad. In addition, two methods (Brownian walk andrandom number generators) of simulating the pointing jitter are compared.

Case 1

First, the results for the APD receiver is shown in Fig. 5.14, the jitter is simulated withBrownian motion. The display shows that the BER (for the case of 2-PPM, BER =SER) at the time when the simulation was terminated was 5.14 · 10−7, and the BER as afunction of time is shown in Fig. 5.15. Deriving the BER in Matlab shows that the BERshould converge to 8.26 · 10−9, which is far below the Simulink result.

Fig. 5.16 shows the track of the laser beam as it has moved around over time. Thepointing error track is caused by the vibrations at the transmitter and receiver. Two redcircles displays the half power beamwidth (the inner circle) and the angular beam diameter(the outer circle) of the laser, to be used as a reference. The half power beamwidth iscalculated by using the pointing loss equations for the transmitter and receiver aperturesshown in Eqn. 3.4. When the pointing loss factor is 0.5 the received power is halvedcompared to when the pointing is perfect. Setting the Eqn. 3.4 to 0.5 and solving theequation with respect to θ, the radius for the half power beamwidth is found. For anaperture diameter of 1 cm, the radius becomes 2.52 · 10−5 radians and is shown as theinner circle. The diameter of the outer ring is calculated by Eqn. 3.8, for the same aperturethe angular radius of the laser beam is 9.52 · 10−5 radians.

The track shows where the center of the beam has wandered. If the pointing errortrack wanders outside the inner circle, the received power is halved compered to perfectpointing and when the beam has wandered outside the outer circle, the beam completelymiss the receiver aperture.

To verify that the Brownian walk for the azimuthal and elevational pointing errorangles are gaussian and have correct standard deviations, histograms along with a zeromean gaussian curve, with standard deviation of σT = 6 · 10−6 rad normalized to thehistograms are plotted, Fig. 5.17. Note that only the histograms for the transmitter areshown since the same method is used for the receiver.

56

5.3. LINK BUDGETS FOR VIBRATING TRANSMITTERS AND RECEIVERS

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.4510

−10

10−8

10−6

10−4

10−2

100

sigma2*G

BE

R


(a) PIN receiver.

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.4510

−10

10−8

10−6

10−4

10−2

100

sigma2*G

BE

R


(b) APD receiver.

0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.4510

−10

10−8

10−6

10−4

10−2

100

sigma2*G

BE

R



Fig. 5.12. BER as a function of different receivers and jitter amplitude for d = 1 km.

57

CHAPTER 5. RESULTS

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.4510

−10

10−8

10−6

10−4

10−2

100

sigma2*G

BE

R


(a) APD receiver.

0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.4510

−10

10−8

10−6

10−4

10−2

100

sigma2*G

BE

R


(b) PIN receiver with optical amplifier.

Fig. 5.13. BER as a function of different receivers and jitter amplitude for d = 10 km.

u

G

out

thetaTransmitter

To File

theta.matTo File 2

APD.mat

Symbol error rate

Tx

Rx

Out1

APD-receiver with

PPM-demodulator

PPM signal source

PPM-sig

Org-sig

Spaceloss

Display

Aperture Gainu

G

out

thetaAperture Gain

Display

5.141e-007

418

8.131e+008

XY Graph


Fig. 5.14. Link model for a vibrating transmitter and receiver.

58


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.90

1

2

3

4

5

6

7x 10

−7

X: 0.8131Y: 5.141e−007

Time (s)

BE

R


−1 0 1

x 10−4

−1

0

1x 10

−4

Azimuthal error angle (radians)

Ele

vatio

n er

ror

angl

e (r

adia

ns)

Fig. 5.16. Total pointing error track for the vibrating transmitter and receiver.

59

CHAPTER 5. RESULTS

−4 −3 −2 −1 0 1 2 3 4

x 10−5

0

2

4

6

8

10

12x 10

4

Error angle (radians)

Fre

quen

cy

(a) Histogram of the azimuthal vibration errorangles.

−4 −3 −2 −1 0 1 2 3 4

x 10−5

0

2

4

6

8

10

12x 10

4


Fre

quen

cy

(b) Histogram of the elevational vibration errorangles.

Fig. 5.17. Histograms of the transmitter vibrations generated by Brownian motion.

u

G

out

thetaTransmitter

To File

theta.matTo File 2

SER.mat

Symbol error rate

Tx

Rx

Out1

APD-receiver with

PPM-demodulator

PPM signal source

PPM-sig

Org-sig

SpacelossAperture Gainu

G

out

thetaAperture Gain

XY Graph


Display

2.19e-008

9

4.109e+008


The same case but where the jitter is simulated with random number generators ispresented in Fig. 5.18. In this case the BER value when the simulation stopped was2.19 · 10−8 and is only about 2.7 times larger than the Matlab value. The BER as afunction of time is shown Fig. 5.19 and the total pointing error track is shown in Fig. 5.20.Histograms for the azimuthal and elevational pointing errors for the transmitter are shownin Fig. 5.21 and verifies that the pointing error jitter is normal distributed with a standarddeviation of 6 · 10−6.

Case 2

The second case, Fig. 5.22, gave a corresponding BER value of 1.01 · 10−3, which can becompared with the value of 9.64 · 10−4 derived from the Matlab models. It is seen thatthe BER is still varying over time, Fig. 5.23, but is close to converging to a more stablevalue.

The histograms in Fig. 5.25 show that the Brownian walk, Fig. 5.24, of the transmitterpointing error was normal distributed with a standard deviation of 10−5.

In Fig. 5.26, the result for the same case where the pointing jitter is simulated byrandom number generators is presented. It gave a corresponding BER value of 9.31 ·10−4,and the BER, Fig. 5.27, has converged to a stable value. The histograms in Fig. 5.29show that the pointing jitter distribution of the total pointing error track, Fig. 5.28, isnormal with a correct standard deviation of 10−5.

60


0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.450

1

2

3

4

5

6

7

8x 10

−8

X: 0.4109Y: 2.19e−008

Time (s)

BE

R


−1 0 1

x 10−4

−1

0

1x 10

−4


Ele

vatio

n er

ror

angl

e (r

adia

ns)


61

CHAPTER 5. RESULTS

−4 −3 −2 −1 0 1 2 3 4

x 10−5

0

1

2

3

4

5

6x 10

4


Fre

quen

cy


−4 −3 −2 −1 0 1 2 3 4

x 10−5

0

1

2

3

4

5

6x 10

4


Fre

quen

cy


Fig. 5.21. Histograms of the transmitter vibrations generated by the random jitter model.

Display

0.001516

2.802 e+004

1.848 e+007

u

G

out

thetaTransmitter

To File

theta.matTo File 2

SER.mat

Symbol error rate

Tx

Rx

Out1

PPM signal source

PPM-sig

Org-sig


G

out

thetaAperture Gain

XY Graph


PIN -receiver with optical amplifier

and PPM -demodulator


0 0.005 0.01 0.015 0.02 0.025 0.03 0.035 0.040

0.2

0.4

0.6

0.8

1

1.2

1.4x 10

−3

X: 0.03696Y: 0.001011

Time(s)

BE

R


62


−1 0 1

x 10−4

−1

0

1x 10

−4


Ele

vatio

n er

ror

angl

e (r

adia

ns)


−5 0 5

x 10−5

0

500

1000

1500

2000

2500

3000

3500

4000

4500


Fre

quen

cy


−5 0 5

x 10−5

0

500

1000

1500

2000

2500

3000

3500

4000


Fre

quen

cy


Fig. 5.25. Histograms of the transmitter vibrations generated by Brownian motion.

Display

0.001397

1.275 e+004

9.131 e+006

u

G

out

thetaTransmitter

To File

theta.matTo File 2

SER.mat

Symbol error rate

Tx

Rx

Out1

PPM signal source

PPM-sig

Org-sig


G

out

thetaAperture Gain

XY Graph


PIN -receiver with optical amplifier

and PPM -demodulator


63

CHAPTER 5. RESULTS

0 0.005 0.01 0.015 0.020

0.2

0.4

0.6

0.8

1

1.2x 10

−3

X: 0.01826Y: 0.0009312

Time (s)

BE

R


−1 0 1

x 10−4

−1

0

1x 10

−4


Ele

vatio

n er

ror

angl

e (r

adia

ns)


64

5.4. LINK BUDGETS FOR NETWORKS

−5 0 5

x 10−5

0

200

400

600

800

1000

1200

1400

1600

1800

2000


Fre

quen

cy


−5 0 5

x 10−5

0

500

1000

1500

2000

2500


Fre

quen

cy


Fig. 5.29. Histograms of the transmitter vibrations generated by the random jitter model.

5.4 Link budgets for messages relayed through sev-

eral spacecraft

The two cases explained in Sec. 4.1.4 is simulated in Simulink and the result is comparedwith the theory presented in Sec. 3.7.

Figure 5.30 shows the result for the first case. All apertures have a pointing error ofσT = σR = 10−5 radians. The satellite blocks in the figure contain the same transmitterand receiver models as in previous case studies. Display 1 shows the SER for the linkbetween satellite 1 and 2, display 2 the SER for the link between satellite 2 and 3 anddisplay 3 the SER for the link between satellite 3 and 4. Finally, the network SER of0.004456, corresponding to a BER value of 0.002971, is shown in the top right display.

The BER value for a single link is derived in Matlab to 9.64 · 10−4. To get thetotal network BER, this value is multiplied with the number of links in the network themessages are sent through. In this case, there are three links resulting in a network BERof 0.002892, which is close to the Simulink value.

In the second case, where two of the links has a pointing jitter amplitude of 6 · 10−6

radians, and the link between satellite 2 and 3 is 10−5 radians, Fig. 5.31, the networkBER is almost completely governed by the weakest link which is between satellite 2 and3.

65

CHAPTER 5. RESULTS

total Symbol error rate

Tx

Rx

Out1

Spaceloss2

Spaceloss

Spaceloss1

Spaceloss

Spaceloss

Spaceloss

Satellite 4

Recei v

er

i np

ut

Decod

ed

sig

Recei v

er

poi n

t ing

err

or

Satellite 3

Recei v

er

i np

ut

Tra

nsm

itte

dsig

Decod

ed

sig

Org

-si g

Recei v

er

poi n

t ing

err

or

Tra

nsm

itte

rpo

int in

gerr

or

Satellite 2

Recei v

er

i np

ut

Tra

nsm

itte

dsig

Decod

ed

sig

Org

-si g

Recei v

er

poi n

t ing

err

or

Tra

nsm

itte

rpo

int in

gerr

or

Satellite 1

Tra

nsm

itte

dsig

Org

-si g

Tra

nsm

itte

rpo

int in

gerr

or

SER34

Tx

Rx

Out1

SER23

Tx

Rx

Out1

SER12

Tx

Rx

Out1

Display3

0.001447

8547

5.908e+006

Display2

0.001443

8525

5.908e+006

Display1

0.001579

9330

5.908e+006

Display

0.004456

2.633e+004

5.908e+006

Fig. 5.30. Link model for the first case of 4 spacecraft networked together.

66

5.4. LINK BUDGETS FOR NETWORKS

total Symbol error rate

Tx

Rx

Out1

Spaceloss2

Spaceloss

Spaceloss1

Spaceloss

Spaceloss

Spaceloss

Satellite 4

Recei v

er

i np

ut

Decod

ed

sig

Recei v

er

poi n

t ing

err

or

Satellite 3

Recei v

er

i np

ut

Tra

nsm

itte

dsig

Decod

ed

sig

Org

-si g

Recei v

er

poi n

t ing

err

or

Tra

nsm

itte

rpo

int in

gerr

or

Satellite 2

Recei v

er

i np

ut

Tra

nsm

itte

dsig

Decod

ed

sig

Org

-si g

Recei v

er

poi n

t ing

err

or

Tra

nsm

itte

rpo

int in

gerr

or

Satellite 1

Tra

nsm

itte

dsig

Org

-si g

Tra

nsm

itte

rpo

int in

gerr

or

SER34

Tx

Rx

Out1

SER23

Tx

Rx

Out1

SER12

Tx

Rx

Out1

Display3

0

0

7.476e+006

Display2

0.001424

1.064e+004

7.476e+006

Display1

1.338e-007

1

7.476e+006

Display

0.001424

1.065e+004

7.476e+006

Fig. 5.31. Link model for the second case of 4 spacecraft networked together.

67

CHAPTER 5. RESULTS

68

Chapter 6

Discussion

6.1 Matlab results

An interesting observation is that for higher orders of PPM modulation, a larger apertureis needed. This seams to indicate that higher orders of PPM are less energy efficientthan the lower order ones. The comparison made here is however unfair since the averagepower that is possible to use differs between the modulation formats and is lower forhigher orders of PPM because of the restriction of the peak laser power (12 mW ). Higherorders of PPM send shorter pulses further apart according to Fig. 3.9 and Eqn. 3.52. Thisforces the required amplitude of the pulses to rise if the same average power should bemaintained compared with a lower order.

Another observation is that the performance of different orders of PPM differs forthe APD receiver in comparison with the PIN receivers. For the PIN receiver, the OOKand 2-PPM formats require the same aperture sizes while the higher orders require largerapertures because of their lower average power. For the APD receiver the 2-PPM needsa larger aperture despite that the same average power was used. This is explained bytheir bandwidth efficiencies and the noise characteristics of the APD receiver. Lookingback at Sec. 3.4.3, it is seen that the noise components that passes through the avalancheregion of the APD are amplified by the square of the avalanche gain, G. Since all ofthe noise components depend linearly on the bandwidth, the APD receiver becomes moresusceptible to modulation formats that are less bandwidth efficient.

It is seen that the PIN receiver does not meet the requirement as the PIN alone requiresaperture sizes of roughly 2.2 - 2.3 cm depending on the modulation format. However, withan optical amplifier it is seen that a PIN can reach a communication distance of 10 km,it needs an aperture size of roughly 0.91 - 0.95 cm to meet the BER requirement of 10−9

depending on the modulation format. The APD receiver is also able to detect signalstransmitted 10 km away with aperture sizes smaller than 1 cm except for the 8-PPMmodulated signal.

6.2 Simulink results

Simulating link budgets in Simulink to evaluate BER values is a very slow process, es-pecially for cases were the BER is very low. It takes even longer times to derive BERvalues for vibrating transmitters and receivers since extra random components are addedthat can produce large number of errors during a short amount of time. These eventsdrastically increase the time it takes for the BER value to converge to a stable value.

69

CHAPTER 6. DISCUSSION

Since a large amount of received bits is needed to suppress the effect of those kinds ofoccurrences given that they are rare otherwise, a high BER is expected. None of the sim-ulations where the BER was low (in the order of magnitude of 10−9) converged before thesimulations were stopped. Simulating a Simulink model without vibration models 0.23seconds (in simulation time) takes about 24 hours, and with vibration models included0.15 seconds (in simulation time) could be reached during 24 hours. Given that at least afew hundreds or even thousands of error are needed to really be sure that the BER valueshas converged, there was not enough time to let the simulations run long enough.

Two methods of simulating vibrations were made, Brownian motion and adding ran-dom numbers to the pointing error. The first case where Brownian motion was used gavea result far from the value derived in Matlab. Both the transmitter and receiver hadcorrect vibration amplitude given by the standard deviation of the distributions shown bythe histograms in Fig. 5.17, still the total pointing error track was spread out over a largerarea than for the the simulation where the random jitter block was used. This is causedby a mistake during the simulation setup. The random number generators generating therandom steps were set at the same seed, which means that both the transmitter and thereceiver vibrations become identical and not independent of each other. This means thata covariance term is added to the total jitter and influences the BER negatively. It wouldbe desirable to make a new simulation of this case, but there is not enough time to makeone before the deadline of this thesis expires. However a new Brownian pointing track issimulated and is presented in Fig. 6.1. Comparing this track with the track in Fig. 5.20

−1 0 1

x 10−4

−1

0

1x 10

−4


Ele

vatio

n er

ror

angl

e (r

adia

ns)


shows that it has a similar range and that a new simulation probably would result ina value close to the one derived in Matlab. This along with the results from the othercase where a PIN receiver with an optical amplifier, 4-PPM modulation and a vibrationamplitude of 10−5 radians was used, indicates that both methods give comparable results.

However, the Brownian motion model better describes larger movements of the laserbeam as it moves around in time as opposed to jumping between positions. If this workis extended to include tracking between satellites, the Brownian motion model couldbe used to simulate random pointing errors during the tracking and lock-on procedure,

70

6.2. SIMULINK RESULTS

whereas the model based on random numbers could be used to describe pointing jitterassociated with mechanical vibrations of the spacecraft body. It should also be notedthat both the random pointing error model and the Brownian motion model can befurther adjusted. One way is to change how often new positions are randomized. Forthe Brownian motion, frequency and the step length can be changed. For tuning theseparameters, more knowledge of the spacecraft vibrations and tracking errors due to noisein the tracking system has to be gathered.

71

CHAPTER 6. DISCUSSION

72

Chapter 7

Conclusions

Simulations have shown that transmission lengths of 10 km can be achieved withgood BER performance using apertures of 1 cm in diameter.

Link performance changes drastically for small variations in pointing jitter ampli-tude, setting high demands on the tracking and pointing system. When deciding thelink margin for the system, an important advice is to allow some variations in thepointing jitter since there will always be some noise present in the tracking system.

Simulations of satellite networks where a message is relayed by several satellites showthat the number of satellites have a small affect on BER if the BER performancebetween every satellite is low. If one of the links have a high BER the total linkperformance is governed by the weakest link.

Apart from the results, tools for further investigations have been developed. Thesecan be expanded to cover more parameters.

73

CHAPTER 7. CONCLUSIONS

74

Chapter 8

Outlook

8.1 Future work

The next step for expanding this work should be to speed up the Simulink models andthe Matlab code. It is also advisable to improve the PPM modulator and demodulatorin Simulink so that arbitrary orders of PPM can be simulated. How much faster theSimulink models can be made is unknown. But comparing the simulation speed withRF communication links built with pre-compiled communication tools in Simulink for RFcommunication, shows that they are roughly 5 times faster. It should be noted thoughthat the RF links did not include models for pointing.

More simulations can also be made, in the cases investigated in this work a lasertransmitter with only one laser diode was considered. Therefore a constant average powercould not be maintained when simulating different orders of PPM. More cases where thesame average power is maintained should be investigated.

It is evident that there are a lot of parameters and design choices to consider whendesigning a communication system. This work can be extended to cover more kinds ofdirect detection modulation formats, such as differential PPM (DPPM), overlapping PPM(OPPM), and wavelength shift keying (WSK) [4]. It can also be expanded to includecoherent detection and modulation using receivers where the received optical signal ismixed with a local laser source. Such receivers can not only detect the amplitude ofthe signal, but also changes in signal frequency and phase. Coding can also improvelink performance considerably and can be included in the simulations. Models describingthese kinds of receivers and modulation formats can be implemented in both Matlab andSimulink.

Simulink is a very powerful simulation tool for dynamical systems, it can be used tosimulate more parameters that influence the communication performance than covered inthis work. Listed below are a few suggestions for further use of Simulink:

Synchronization of the receiver clocks controlling the sampling of the signal.

Simulate the tracking and lock on procedure before communication can start.

Tracking a moving spacecraft and keep pointing at it during communication.

Advanced simulations of several spacecraft flying in formations where pointing com-pensations are made for internally moving spacecraft within the formation.

75

CHAPTER 8. OUTLOOK

Many of these suggestions require simulations that last several minutes or even hours.Simulating BER values in Simulink has proven to be a very slow process, therefore adifferent way of evaluating the link performance during tracking has to be done in adifferent manner. One suggestion is to use results from Matlab simulations by for example,import data for BER as a function of signal to noise ratio or received signal strength intolookup table blocks. Then let the transmitter send a laser beam of constant power (ataverage power based on the modulation), the receiver can then evaluate the signal to noiseratio and let the lookup table block interpolate the BER. This will give instantaneous BERresults and speed up simulations considerably, since no modulation and demodulation ofthe signal has to be performed.

When formation flying spacecraft clusters can be simulated, different kinds of networktopologies and communication protocols can be simulated and evaluated.

76

Acknowledgements

First I would like to give a big thanks to my supervisor Henrik Kratz for continuoussupport, valuable inputs, finding literature and proofreading. I would also like to thankGreger Thornell for valuable inputs of the layout of the report. A big thanks to the PhDstudents and fellow thesis workers at ASTC, and to the entire material science group formaking my time here so pleasant.

Finally thanks to the sponsors of this project, VINNOVA (Swedish GovernmentalAgency for Innovation Systems) and SNSB (Swedish National Space Board) for makingthis project possible.

77

CHAPTER 8. OUTLOOK

78

Bibliography

[1] O. Bouchet, H. Sizun, C. Boisrobert, F. de Fornel, and P.-N. Favennec, Free-SpaceOptics - Propagation and Communication. ISTE, 2006.

[2] “Illustration of the photophone,” Jan. 2008. [Online]. Available: http://homemadecamera.blogspot.com/

[3] “FSO history,” Jan. 2008. [Online]. Available: http://www.robins.ball.com/lasercomm history 1960.html

[4] H. Hemmati, Ed., Deep Space Optical Communications. John Wiley & Sons, Inc,2006.

[5] “ARTEMIS illustration,” Jan. 2008. [Online]. Available: http://www.esa.int/esaCP/ESASGBZ84UC index 1.html

[6] “Optical Communications in Space - Twenty Years of ESA Effort,” Jan. 2008.[Online]. Available: http://www.esa.int/esapub/bulletin/bullet91/b91lutz.htm

[7] “SPOT-4,” Jan. 2008. [Online]. Available: http://spot4.cnes.fr/spot4 gb/

[8] “First laser data link between ARTEMIS and SPOT-4,” Jan. 2008. [Online].Available: http://www.esa.int/esaCP/ESASGBZ84UC index 0.html

[9] “ARTEMIS starts its journey to final orbit,” Jan. 2008. [Online]. Available:http://www.esa.int/esaCP/ESA9BAVTYWC index 0.html

[10] “ARTEMIS successfully concludes tests with Japanese Space Agency,” Jan.2008. [Online]. Available: http://telecom.esa.int/telecom/www/object/index.cfm?fobjectid=27835

[11] “Laser transmitters picture,” Lightpointe, Jan. 2008. [Online]. Available:http://www.lightpointe.com/images/gallery/p6 L.jpg

[12] “FSO history,” Lightpointe, Jan. 2008. [Online]. Available: http://www.freespaceoptics.org/freespaceoptics/default.cfm#history

[13] “Lightpointe homepage,” Lightpointe, Jan. 2008. [Online]. Available: http://www.lightpointe.com/company/default.cfm

[14] S. Arnon, S. R. Rotman, and N. S. Kopeika, “Performance limitations of free-spaceoptical communication satellite networks due to vibrations: direct detection digitalmode,” OE - Optical Engineering, vol. 36, no. 11, pp. 3148–3157, Nov 1997.

[15] D. G. Aviv, Laser space communications. Artech House Boston • London, 2006.

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BIBLIOGRAPHY

[16] S. G. Lambert and W. L. Casey, Laser communications in space. Artech HouseBoston • London, 1995.

[17] H. J. Dutton, Understanding Optical Communications, first Edition ed. IBM Cor-poration, International Technical Support Organization, Sep 1998.

[18] I. Charles H. Cox, Analog Optical Links Thory and Practice. CAMBRIDGE UNI-VERSITY PRESS, 2006.

[19] J. Jazayerifar, M. Salehi, “Atmospheric Optical CDMA Communication Systems viaOptical Orthogonal Codes,” Communications, IEEE Transactions on, vol. 54, no. 9,pp. 1614–1623, Sept. 2006.

[20] Z. Ghassemlooy, “Optical Fibre Communication Systems, Lecture 5 OpticalAmplifier,” School of Computing, Engineering and Information Sciences, TheUniversity of Northumbria U.K. (http://soe.unn.ac.uk/ocr), Jan. 2008. [Online].Available: http://soe.unn.ac.uk/ocr/teaching/fibre/pp/Opticalamp-L6.pdf

[21] D. R. Paschotta, “Rare-earth-doped fibers,” RP Photonics, Jan. 2008. [Online].Available: http://www.rp-photonics.com/rare earth doped fibers.html

[22] “Metastable states,” RP Photonics, Jan. 2008. [Online]. Available: http://www.rp-photonics.com/metastable states.html

[23] Technical information SD-28 Characteristics and use of Si APD (Avalanchephotodiode), HAMAMATSU PHOTONICS K.K., Solid State Division, 1126-1Ichino-cho, Hamamatsu City, 435-8558 Japan, http://www.hamamatsu.com, May2004. [Online]. Available: http://sales.hamamatsu.com/assets/applications/SSD/Characteristics and use of SI APD.pdf

[24] S. Arnon, S. Rotman, and N. Kopeika, “Optimum transmitter optics aperture forsatellite optical communication,” Aerospace and Electronic Systems, IEEE Transac-tions on, vol. 34, no. 2, pp. 590–596, Apr 1998.

[25] J. B. Abshire, “Performance of OOK and Low-Order PPM Modulations in OpticalCommunications When Using APD-Based Receivers,” IEEE Transactions on Com-munications, vol. 32, no. 10, pp. 1140–1143, Oct 1984.

[26] C.-C. Chen and C. Gardner, “Impact of random pointing and tracking errors onthe design of coherent and incoherent optical intersatellite communication links,”Communications, IEEE Transactions on, vol. 37, no. 3, pp. 252–260, Mar 1989.

[27] J. R. Wertz and W. J. Larson, Eds., Space Mission Analysis and Design, 3rd ed.Microcosm Press and Kluwer Academic Publishers, 2003.

80

Appendix A

Matlab functions

A.1 Ap G.m

function G = Ap G(D, lambda, n)% D = aperture diameter (m)% lambda = wavelength (m)% n = optics efficiency

G=(pi*D/lambda)ˆ2*n;

A.2 Spaceloss.m

function Ls = Spaceloss(lambda, d)% lambda = wavelength (m)% d = distance (m)

Ls = (lambda/(4*pi*d))ˆ2;

A.3 Pointing.m

function Lp = Pointing(Gt, theta)%theta = pointing offset (radians)%Gt = Transmitt telescope gain (real)

Lp = exp(−Gt.*theta.ˆ2);

A.4 meanvar PIN.m

function [meanPIN1, varPIN1, meanPIN0, varPIN0, SNR] =...meanvar PIN(Pmax, Pmin, M, Br, Id, Pback, Rl, Tr, n, lambda)

% Pmax = received power for a "one" (W).% Pmin = received power for a "zero" (W).% M = number of slots in a PPM symbol, for the case of

% OOK modulation, M = 1 (Scalar).

81

APPENDIX A. MATLAB FUNCTIONS

% Br = Bit rate (bps).% Id = dark current (A)% Pback = received background optical power (W).% Rl = Load resistance (ohm).% Tr = thermal temperature (K).% n = quantum efficiency (Scalar).% lambda = optical wavelength (m).

q = 1.60217646*10ˆ−19; % electron charge (C).Kb = 1.38065*10ˆ−23; % Boltzmann (J/K).h = 6.6260688*10ˆ−34; % plank (Js).c = 299792458; % speed of light in vaccum (m/s).

f = c/lambda; % Optical frequency (Hz).

R = q*n/(h*f); % Responsivity of the PIN (A/W).

Bw = Br*M/log2(M); % Occupied bandwidth of the modulation (Hz).

if M == 1 % Makes sure that OOK gets correct average power and bandwidth.Bw = Br;M = 2;

end

Pavg = Pmax*(1 + Pmin/Pmax*(M−1))/M; % Average received power (W).

%The variances of different noise sources for a PIN receiver without%optical amplifier.

varSn0 = 2*q*R*Pmin*Bw; %signal shot noise for "0" (Aˆ2).varSn1 = 2*q*R*Pmax*Bw; %signal shot noise for "1" (Aˆ2).varSnavg= 2*q*R*Pavg*Bw; %signal shot noise for average power (Aˆ2).

varDc = 2*q*Id*Bw; %Dark current noise (Aˆ2).

varBg = 2*q*Pback*R*Bw; %background noise (Aˆ2).

varTh = 4*Kb*Tr*Bw/Rl; %Thermal noise (Aˆ2).

% Mean (A) and variance (Aˆ2) for a "zero" signal.meanPIN0 = R*Pmin + R*Pback + Id;varPIN0 = varSn0 + varBg + varDc + varTh;

% Mean (A) and variance (Aˆ2) for a "one" signal.meanPIN1 = R*Pmax + R*Pback + Id;varPIN1 = varSn1 + varBg + varDc + varTh;

% The signal to noise ratio depends on the average received signal power.varPINavg = varSnavg + varBg + varDc + varTh;SNR = 10*log((R*Pavg)ˆ2/(varPINavg)); %dB

A.5 meanvar PIN OA.m

function [meanPINOA1, varPINOA1, meanPINOA0, varPINOA0, SNR] =...meanvar PIN OA(Pmax, Pmin, M, ∆ lambda, F, Br, Lout, Lin, RIN,...G, Pback, nsp, Id, Rl, Tr, n, lambda)

82

A.5. MEANVAR PIN OA.M

% Pmax = received maximum power (W).% Pmin = received minimum power (W).% M = number of slots in a PPM word, M = 1 for OOK (Scalar).% ∆ lambda = optical filter bandwidth (m).% F = noise figure (dB).% Br = Bit rate (bps).% Lout = Output optical amplifier insertion loss (dB).% Lin = Input optical amplifier loss (dB).% RIN = Relative intensity noise (dB/Hz).% G = Optical amplifier gain (Scalar).% Pback = Received background power (W).% nsp = spontaneous emission coefficient (Scalar).% Id = dark current (A).% Rl = Load resistance (Ohm).% Tr = thermal temperature (K).% n = quantum efficiency (Scalar).% lambda = optical wavelength (m).

q = 1.60217646*10ˆ−19; % electron charge (C).Kb = 1.38065*10ˆ−23; % Boltzmann (J/K).h = 6.6260688*10ˆ−34; % plank (Js).c = 299792458; % speed of light in vaccum (m/s).



Bop = ∆ lambda*c/lambdaˆ2; % Bop = optical filter bandwidth in the% frequency domain (Hz).



end

Pavg = Pmax*(1 + Pmin/Pmax*(M−1))/M; % Average received optical power (W).

%The variances of different noise sources.

%Amplified signal shot noise (Aˆ2).varSn 0 = 2*q*G*R*Pmin*Bw;varSn 1 = 2*q*G*R*Pmax*Bw;varSn avg = 2*q*G*R*Pavg*Bw;

%Amplified source relative intensity noise (Aˆ2).varRIN 0 = 10ˆ(RIN/10)*(R*G*Pmin)ˆ2*Bw;varRIN 1 = 10ˆ(RIN/10)*(R*G*Pmax)ˆ2*Bw;varRIN avg = 10ˆ(RIN/10)*(R*G*Pavg)ˆ2*Bw;

%Amplified signal−spontaneous beat noise (Aˆ2).varSgSp 0 = 4*Rˆ2*Pmin*G*Lin*Loutˆ2*(G−1)*nsp*h*f*Bw;varSgSp 1 = 4*Rˆ2*Pmax*G*Lin*Loutˆ2*(G−1)*nsp*h*f*Bw;varSgSp avg = 4*Rˆ2*Pavg*G*Lin*Loutˆ2*(G−1)*nsp*h*f*Bw;

83


%Dark current noise (Aˆ2).varDc = 2*q*Id*Bw;

%Thermal noise (Aˆ2).varTh = 4*Kb*Tr*F*Bw/Rl;

%Amplified background noise (Aˆ2).varBg = 2*q*G*R*Pback*Bw;

%Amplified spontaneous emissions (Aˆ2).varSe = 4*R*q*h*f*nsp*(G−1)*Loutˆ2*Bw*Bop;

%Amplified spontaneous−spontaneous beat noise (Aˆ2).varSpSp = 8*(R*Lout*(G−1)*nsp*h*f)ˆ2*Bop*Bw;

%The mean (A) and variance (Aˆ2) for the "1" signal.meanPINOA1 = R*G*Pmax + R*Pback + Id;varPINOA1 = varSn 1 + varRIN 1 + varSgSp 1 + varDc + varTh + varBg...

+ varSe + varSpSp;

%The mean (A) and variance (Aˆ2) for the "0" signal.meanPINOA0 = R*G*Pmin + R*Pback + Id;varPINOA0 = varSn 0 + varRIN 0 + varSgSp 0 + varDc + varTh + varBg...

+ varSe + varSpSp;

%The signal to noise ratio depends on the average received signal power.varPINOA avg = varSn avg + varRIN avg + varSgSp avg + varDc + varTh...

+ varBg + varSe + varSpSp;SNR = 10*log((R*G*Pavg)ˆ2/(varPINOA avg)); %dB

A.6 meanvar APD.m

function [meanAPD1, varAPD1, meanAPD0, varAPD0, SNR] =...meanvar APD(Pmax, Pmin, M, Br, Keff, G, Is, Ib, Pback, Rl, Tr, n, lambda)

% Pmax = maximum received optical power (W).% Pmin = minimum received optical power (W).% M = Numder of slots in PPM, M = 1 for OOK (Scalar).% Br = Bit rate (bps).% Keff = Effective ratio of the APD hole and electron ionization

% coefficients (Scalar).% G = APD Gain (Scalar).% Is = APD surface leakage current (A).% Ib = APD bulk leakage current (A).% Pback = incident background power (W).% Rl = Receiver load resistor (Ohm).% Tr = Thermal noise temperature (K).% n = detector quantum efficiency (Scalar).% lambda = optical wavelength (m).

q = 1.60217646*10ˆ−19; % electron charge (C).Kb = 1.38065*10ˆ−23; % Boltzmann (J/K).h = 6.6260688*10ˆ−34; % plank (Js).

84

A.7. OOK.M

c = 299792458; % speed of light in vaccum (m/s).


R = G*q*n/(h*f); % Responsivity of the APD (A/W).



end

Pavg = Pmax*(1 + Pmin/Pmax*(M−1))/M; % Average received power (W).

F = Keff*G + (1 − Keff)*(2 − 1/G); % APD excess noise figure (Scalar).

%The variances of different noise sources for an APD receiver

varSn0 = 2*q*F*Pmin*R*G*Bw; %signal shot noise for "0" (Aˆ2).varSn1 = 2*q*F*Pmax*R*G*Bw; %signal shot noise for "1" (Aˆ2).varSnavg= 2*q*F*Pavg*R*G*Bw; %signal shot noise for average power (Aˆ2).

varDc = 2*q*F*Gˆ2*Ib*Bw + 2*q*Bw*Is; %Dark current noise (Aˆ2).

varBg = 2*q*F*Pback*R*G*Bw; %background noise (Aˆ2).


% Mean (A) and variance (Aˆ2) for a "zero" signal.meanAPD0 = R*Pmin + R*Pback + G*Ib + Is;varAPD0 = varSn0 + varBg + varDc + varTh;

% Mean (A) and variance (Aˆ2) for a "one" signal.meanAPD1 = R*Pmax + R*Pback + G*Ib + Is;varAPD1 = varSn1 + varBg + varDc + varTh;

% The signal to noise ratio depends on the average received signal power.varAPDavg = varSnavg + varBg + varDc + varTh;SNR = 10*log((R*Pavg)ˆ2/(varAPDavg)); %dB

A.7 OOK.m

function Pb = OOK(Xs, Xns, varXs, varXns)

% "normalized distance" between the threshold and the distribution meanDN = (Xs − Xns)/(sqrt(varXs) + sqrt(varXns));

% Bit error probabilityPb = 1/2*erfc(DN/sqrt(2));

A.8 PPM.m

function Pb = PPM(Xs, Xns, varXs, varXns, M)

85


Nbs = M/(2*(M−1)); % Average number of bit errors per symbol error

% Determines the integration interval so it includes 1 − 2*10ˆ−16% percent of the entire area.int1 = norminv([10ˆ−16 (1 − 10ˆ−16)], Xs, sqrt(varXs));int2 = norminv([10ˆ−16 (1 − 10ˆ−16)], Xns, sqrt(varXns));

if int1(1) < int2(1)int min = int1(1);

elseint min = int2(1);

end

if int1(2) > int2(2)int max = int1(2);

elseint max = int2(2);

end

% Sets the step−size for the integration solverstep s = (int max − int min)/100;

% Evaluates the symbol error probabilityPcsc = 0;for x = int min:step s:int max

Pcsc = Pcsc + normpdf(x, Xs, sqrt(varXs))...

*normcdf(x, Xns, sqrt(varXns))ˆ(M−1)*step s;end

% Bit error probabilityPb = Nbs*(1−Pcsc);

% Makes the output look nicer for values that are out of bounds.if Pb < 10ˆ−16 | | Pb > 1

Pb=0;end

A.9 OOK vib APD.m

function Pb = OOK vib APD(var theta, Pmax, Pmin, Br, Keff, G, Is, Ib,...Pback, Rl, Tr, n, lambda, Dt, Dr, nt, nr, d, Lil)

% Solves the double integral that evaluates the mean BER for a certain% standard deviation of the pointing jitter.Pb = 0;for theta r = 0:2*10ˆ−6:10ˆ−4

P = 0;for theta t = 0:2*10ˆ−6:10ˆ−4

P = P + P OOK theta(Pmax, Pmin, Br, Keff, G, Is, Ib, Pback, Rl,...Tr, n, lambda, theta t, theta r, Dt, Dr, nt, nr, d, Lil)*...pointing error dist(theta t, var theta)*(2*10ˆ−6);

endPb = Pb + P*raylpdf(theta r, var theta)*(2*10ˆ−6);

end

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A.10. OOK VIB PIN.M

end

%Gives the BER for a certain transmitter and receiver pointing error.function P OOK = P OOK theta(Pmax, Pmin, Br, Keff, G, Is, Ib, Pback, Rl,...

Tr, n, lambda, theta t, theta r, Dt, Dr, nt, nr, d, Lil)

Pr1 = Pmax*Pointing(Ap G(Dt, lambda, nt), theta t)*Ap G(Dt,lambda,nt)...

*Spaceloss(lambda,d)*Pointing(Ap G(Dr, lambda, nr), theta r)...

*Ap G(Dr,lambda,nr)*10ˆ(−Lil/10);

Pr0 = Pmin*Pointing(Ap G(Dt, lambda, nt), theta t)*Ap G(Dt,lambda,nt)...



[Xs, varXs, Xns, varXns, SNR] = ...meanvar APD(Pr1, Pr0, 1, Br, Keff, G, Is, Ib, Pback, Rl, Tr, n, lambda);

P OOK = OOK(Xs, Xns, varXs, varXns);end

A.10 OOK vib PIN.m

function Pb = OOK vib PIN(Pmax, Pmin, var theta, Br, Id, Pback, Rl, Tr,...n, lambda, Dt, Dr, nt, nr, d, Lil)


P = 0;for theta t = 0:2*10ˆ−6:10ˆ−4

P = P + P OOK theta(Pmax, Pmin, Br, Id, Pback, Rl, Tr, n,...lambda, Dt, Dr, nt, nr, d, theta t, theta r, Lil)*...pointing error dist(theta t, var theta)*(2*10ˆ−6);


endend

%Gives the BER for a certain transmitter and receiver pointing error.function P OOK = P OOK theta(Pmax, Pmin, Br, Id, Pback, Rl, Tr, n,...

lambda, Dt, Dr, nt, nr, d, theta t, theta r, Lil)

Pr1 = Pmax*Pointing(Ap G(Dt, lambda, nt), theta t)*Ap G(Dt, lambda, nt)...

*Spaceloss(lambda, d)*Pointing(Ap G(Dr, lambda, nr), theta r)...

*Ap G(Dr, lambda, nr)*10ˆ(−Lil/10);

Pr0 = Pmin*Pointing(Ap G(Dt, lambda, nt), theta t)*Ap G(Dt, lambda, nt)...



[Xs, varXs, Xns, varXns, SNR] =...meanvar PIN(Pr1, Pr0, 1, Br, Id, Pback, Rl, Tr, n, lambda);


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A.11 OOK vib PIN OA.m

function Pb = OOK vib PIN OA(Pmax, Pmin, var theta, ∆ lambda, F, Br,...Lout, Lin, RIN, G, Pback, nsp, Id, Rl, Tr, n, lambda, Dt,...Dr, nt, nr, d, Lil)


P = 0;for theta t = 0:2*10ˆ−6:10ˆ−4

P = P + P OOK theta(Pmax, Pmin, ∆ lambda, F, Br, Lout, Lin,...RIN, G, Pback, nsp, Id, Rl, Tr, n, lambda, Dt, Dr, nt, nr,...d, theta t, theta r, Lil)*pointing error dist(theta t,...var theta)*(2*10ˆ−6);


endend

%Gives the BER for a certain transmitter and receiver pointing error.function P OOK = P OOK theta(Pmax, Pmin, ∆ lambda, F, Br, Lout, Lin,...

RIN, G, Pback, nsp, Id, Rl, Tr, n, lambda, Dt, Dr, nt,...nr, d, theta t, theta r, Lil)







[Xs, varXs, Xns, varXns, SNR] = ...meanvar PIN OA(Pr1, Pr0, 1, ∆ lambda, F, Br, Lout, Lin, RIN,...G, Pback, nsp, Id, Rl, Tr, n, lambda);


A.12 PPM vib APD.m

function Pb = MPPM vib APD(var theta, Pmax, Pmin, M, Br, Keff, G, Is, Ib,...Pback, Rl, Tr, n, lambda, Dt, Dr, nt, nr, d, Lil)


P = 0;for theta t = 0:2*10ˆ−6:10ˆ−4

P = P + P MPPM theta(Pmax, Pmin, M, Br, Keff, G, Is, Ib,...Pback, Rl, Tr, n, lambda, theta t, theta r, Dt, Dr, nt, nr,...

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A.13. PPM VIB PIN.M

d, Lil)*pointing error dist(theta t,var theta)*(2*10ˆ−6);endPb = Pb + P*raylpdf(theta r, var theta)*(2*10ˆ−6);

endend

% Gives the BER for a certain transmitter and receiver pointing error.function P MPPM = P MPPM theta(Pmax, Pmin, M, Br, Keff, G, Is, Ib,...

Pback, Rl, Tr, n, lambda, theta t, theta r, Dt, Dr,...nt, nr, d, Lil)

Pr1 = Pmax*Pointing(Ap G(Dt, lambda, nt), theta t)*Ap G(Dt,lambda,nt)...



Pr0 = Pmin*Pointing(Ap G(Dt, lambda, nt), theta t)*Ap G(Dt,lambda,nt)...



[Xs, varXs, Xns, varXns, SNR] = ...meanvar APD(Pr1, Pr0, M, Br, Keff, G, Is, Ib, Pback, Rl, Tr, n,lambda);

P MPPM = PPM(Xs, Xns, varXs, varXns, M);end

A.13 PPM vib PIN.m

function Pb = MPPM vib PIN(Pmax, Pmin, var theta, Br, M, Id, Pback, Rl,...Tr, n, lambda, Dt, Dr, nt, nr, d, Lil)


P = 0;for theta t = 0:2*10ˆ−6:10ˆ−4

P = P + P MPPM theta(Pmax, Pmin, Br, M, Id, Pback, Rl, Tr, n,...lambda, Dt, Dr, nt, nr, d, theta t, theta r, Lil)*...pointing error dist(theta t,var theta)*(2*10ˆ−6);


endend

% Gives the BER for a certain transmitter and receiver pointing error.function P MPPM = P MPPM theta(Pmax, Pmin, Br, M, Id, Pback, Rl, Tr, n,...

lambda, Dt, Dr, nt, nr, d, theta t, theta r, Lil)


*Spaceloss(lambda, d)*Pointing(Ap G(Dr, lambda, nr), theta r)*...Ap G(Dr, lambda, nr)*10ˆ(−Lil/10);


*Spaceloss(lambda, d)*Pointing(Ap G(Dr, lambda, nr), theta r)*...Ap G(Dr, lambda, nr)*10ˆ(−Lil/10);

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[Xs, varXs, Xns, varXns, SNR] = meanvar PIN(Pr1, Pr0, M, Br, Id, Pback,...Rl, Tr, n, lambda);


A.14 PPM vib PIN OA.m

function Pb = MPPM vib PIN OA(Pmax, Pmin, var theta, M, ∆ lambda,...F, Br, Lout, Lin, RIN, G, Pback, nsp, Id, Rl, Tr, n,...lambda, Dt, Dr, nt, nr, d, Lil)


P = 0;for theta t = 0:2*10ˆ−6:10ˆ−4

P = P + P MPPM theta(Pmax, Pmin, M, ∆ lambda, F, Br, Lout,...Lin, RIN, G, Pback, nsp, Id, Rl, Tr, n, lambda, Dt, Dr, nt,...nr, d, theta t, theta r, Lil)*pointing error dist(theta t,...var theta)*(2*10ˆ−6);


endend

% Gives the BER for a certain transmitter and receiver pointing error.function P MPPM = P MPPM theta(Pmax, Pmin, M, ∆ lambda, F, Br, Lout,...

Lin, RIN, G, Pback, nsp, Id, Rl, Tr, n, lambda, Dt,...Dr,nt, nr, d, theta t, theta r, Lil)



*Ap G(Dr, lambda, nr)*10ˆ(−Lil/10);Pr0 = Pmin*Pointing(Ap G(Dt, lambda, nt), theta t)*Ap G(Dt, lambda, nt)...



[Xs, varXs, Xns, varXns, SNR] = ...meanvar PIN OA(Pr1, Pr0, M, ∆ lambda, F, Br, Lout, Lin, RIN,...G, Pback, nsp, Id, Rl, Tr, n, lambda);


90

Appendix B

Embedded Matlab functions

B.1 Pointing error

function [y,theta] = fcn(u, theta H, theta V, G)% Calculates the "line of sight damping" between the transmitter and% receiver due to the pointing error.

theta = sqrt(theta Vˆ2 + theta Hˆ2);

y = u*exp(−G*thetaˆ2);

B.2 APD noise

function [Xs, stdXs]=fcn(u, Br, M, Keff, G, Is, Ib, Pb, Rl, Tr, n, lambda)

%u = Received optical power (W).%Br = Bit rate (bps).%M = The M number of the PPM (Scalar).%Keff = Effective ratio of the APD hole and electron ionization

%coefficients (Scalar).%G = APD Gain (Scalar).%Is = APD surface leakage current (A).%Ib = APD bulk leakage current (A).%Pb = incident background power (W).%Rl = Receiver load resistor (Ohm).%Tr = Thermal noise temperature (K).%n = detector quantum efficiency (Scalar).%lambda = optical wavelength (m).

q = 1.60217646*10ˆ−19; % Electron charge (C).Kb = 1.38065*10ˆ−23; % Boltzmann (J/K).h = 6.6260688*10ˆ−34; % Plank (Js).c = 299792458; % speed of light in vaccum (m/s).


R = G*q*n/(h*f); % Responsivity of the APD (A/W).

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APPENDIX B. EMBEDDED MATLAB FUNCTIONS


F = Keff*G + (1 − Keff)*(2 − 1/G); % APD excess noise figure (Scalar).

%The variances of different noise sources for an APD receiver.

varSn = 2*q*F*u*R*G*Bw; %Signal shot noise (Aˆ2).

varDc = 2*q*F*Gˆ2*Ib*Bw + 2*q*Bw*Is; %Dark current noise (Aˆ2).

varBg = 2*q*F*Pb*R*G*Bw; %Background noise (Aˆ2).


% Mean (A) and variance (Aˆ2) for the input signal.Xs = (R*u + R*Pb + G*Ib + Is)*T;stdXs = sqrt((varSn + varBg + varDc + varTh)*T);

B.3 PIN noise

function [Xs, stdXs] = fcn(u, M, Br, Id, Pb, Rl, Tr, n, lambda)

% u = Received power (W).% M = Number of slots in a PPM word (Scalar).% Br = Bit rate (bps)% Id = Dark current (A).% Pb = Received background optical power (W).% Rl = Load resistance (Ohm).% Tr = Thermal temperature (K).% n = Quantum efficiency (Scalar).% lambda = Optical wavelength (m).

q = 1.60217646*10ˆ−19; % Electron charge (C).Kb = 1.38065*10ˆ−23; % Boltzmann (J/K).h = 6.6260688*10ˆ−34; % plank (Js).c = 299792458; % Speed of light in vacuum (m/s).




%The variances of different noise sources for a PIN receiver.

varSn = 2*q*R*u*Bw; %signal shot noise for (Aˆ2).

varDc = 2*q*Id*Bw; %Dark current noise (Aˆ2).

varBg = 2*q*Pb*R*Bw; %background noise (Aˆ2).


% Mean (A) and variance (Aˆ2) for the signal.Xs = R*u + R*Pb + Id;

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B.4. PIN WITH OPTICAL AMPLIFIER NOISE

stdXs = sqrt(varSn + varBg + varDc + varTh);

B.4 PIN with optical amplifier noise

function [meanPPM, stdPPM] = fcn(u, ∆ lambda, F, M, Br, Lout, Lin,...RIN, G, nsp, Id, Pb, Rl, Tr, n, lambda)

% u = received power (W).% ∆ lambda = optical filter bandwidth (m).% F = noise figure (dB).% M = Number of slots in a symbol, M−PPM (Scalar).% Br = Bit rate (bps).% Lout = Output optical amplifier insertion loss (dB).% Lin = Input optical amplifier loss (dB).% RIN = Relative intensity noise (dB/Hz)% G = Optical amplifier gain (Scalar).% nsp = spontaneous emission coefficient (Scalar).% Id = dark current (A).% Pb = Background power (W).% Rl = Load resistance (Ohm).% Tr = thermal temperature (K).% n = quantum efficiency (Scalar).% lambda = optical wavelength (m).

q = 1.60217646*10ˆ−19; % electron charge (C).Kb = 1.38065*10ˆ−23; % Boltzmann (J/K).h = 6.6260688*10ˆ−34; % plank (Js).c = 299792458; % Speed of light in vacuum (m/s).



% Optical filter bandwidth in the frequency domain (Hz).Bop = ∆ lambda*c/lambdaˆ2;

Bw = Br*M/k; % Occupied bandwidth of the modulation (Hz).

%The variances of different noise sources.

%Amplified signal shot noise (Aˆ2).varSn = 2*q*G*R*u*Bw;

%Amplified source relative intensity noise (Aˆ2).varRIN = 10ˆ(RIN/10)*(R*G*u)ˆ2*Bw;

%Amplified signal−spontaneous beat noise (Aˆ2).varSgSp = 4*Rˆ2*u*G*Lin*Loutˆ2*(G−1)*nsp*h*f*Bw;

%Dark current noise (Aˆ2).varDc = 2*q*Id*Bw;

%Thermal noise (Aˆ2).varTh = 4*Kb*Tr*F*Bw/Rl;

%Amplified background noise (Aˆ2).

93

APPENDIX B. EMBEDDED MATLAB FUNCTIONS

varBg = 2*q*G*Pb*Bw;

%Amplified spontaneous emissions (Aˆ2).varSe = 4*R*q*h*f*nsp*(G−1)*Loutˆ2*Bw*Bop;

%Amplified spontaneous−spontaneous beat noise (Aˆ2).varSpSp = 8*(R*Lout*(G−1)*nsp*h*f)ˆ2*Bop*Bw;

%The mean (A) and variance (Aˆ2) for the signal for MPPM modultaion.meanPPM = R*G*u*T + R*G*Pb*T + Id*T;stdPPM = sqrt(varSn + varRIN + varSgSp + varDc + varTh + varBg +...

varSe + varSpSp);

B.5 Decision circuit

function [y0,y1] = fcn(u0, u1, u2, u3, u4, u5, u6, u7)

% Finds the greatest input, and outputs the greatest value in y0 and the% array number of the gretest value in y1.[y0 y1] = max([u0 u1 u2 u3 u4 u5 u6 u7]);

% Makes the output go from 0−7 instead of 1−8.y1 = y1 − 1;

B.6 Brownian walk

function [theta H1, theta V1] = fcn(theta V, theta H, sigma V, sigma H)%Generates a stochastic random walking motion.

%Sets the standard deviation of the step size for the brownian walk.d sigma H = sigma H/100;

%Generates a step of random length and direction along the azimuthal axis.d theta H = randn()*d sigma H − theta H*d sigma H/sigma H;

%the new error angle is produced by adding the random step the the current%error angle.theta H1 = theta H + d theta H;

%The same principle for the elevation axis.d sigma V = sigma V/100;d theta V = randn()*d sigma V − theta V*d sigma V/sigma V;theta V1 = theta V + d theta V;

B.7 Periodic vibration

function [theta H, theta V] = fcn(t, amp V, amp H, omega V, omega H, ...phase V, phase H)

%generates periodical motion.theta H = amp H*sin(omega H*t/(2*pi) − phase H);

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B.8. ANGULAR DRIFT

theta V = amp V*cos(omega V*t/(2*pi) − phase V);

B.8 Angular drift

function [theta H, theta V, t o] = fcn(t, t last, speed V, speed H, ...theta o V, theta o H)

%Generates a drifting angular motion in both elevational and azimuthal%directions at a desired speed.theta H = theta o V + speed H*(t − t last);theta V = theta o H + speed V*(t − t last);t o = t;

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