PERFORMANCE ANALYSIS

OF COMMUNICATION SYSTEMS

OVER MIMO FREE SPACE OPTICAL CHANNELS

By

Qianling Cao

SUBMITTED IN PARTIAL FULFILLMENT OF THE

REQUIREMENTS FOR THE DEGREE OF

MASTER OF SCIENCE

AT

UNIVERSITY OF VIRGINIA

CHARLOTTESVILLE, VIRGINIA

JAN 2005

c© Copyright by Qianling Cao, 2005

UNIVERSITY OF VIRGINIA

DEPARTMENT OF

ELECTRICAL AND COMPUTER ENGINEERING

The undersigned hereby certify that they have read and recommend

to the Faculty of Graduate Studies for acceptance a thesis entitled

“Performance Analysis of Communication Systems over

MIMO Free space Optical Channels” by Qianling Cao in partial

fulfillment of the requirements for the degree of Master of Science.

Dated: Jan 2005

Supervisor:Maıte Brandt-Pearce

Readers:Stephen G. Wilson

Yibin Zheng

ii

UNIVERSITY OF VIRGINIA

Date: Jan 2005

Author: Qianling Cao

Title: Performance Analysis of Communication Systems

over MIMO Free space Optical Channels

Department: Electrical and Computer Engineering

Degree: M.Sc. Convocation: Jan Year: 2005

Permission is herewith granted to University of Virginia to circulate andto have copied for non-commercial purposes, at its discretion, the above titleupon the request of individuals or institutions.

Signature of Author

THE AUTHOR RESERVES OTHER PUBLICATION RIGHTS, ANDNEITHER THE THESIS NOR EXTENSIVE EXTRACTS FROM IT MAYBE PRINTED OR OTHERWISE REPRODUCED WITHOUT THE AUTHOR’SWRITTEN PERMISSION.

THE AUTHOR ATTESTS THAT PERMISSION HAS BEEN OBTAINEDFOR THE USE OF ANY COPYRIGHTED MATERIAL APPEARING IN THISTHESIS (OTHER THAN BRIEF EXCERPTS REQUIRING ONLY PROPERACKNOWLEDGEMENT IN SCHOLARLY WRITING) AND THAT ALL SUCH USEIS CLEARLY ACKNOWLEDGED.

iii

To Xiao Qun and my parents.

iv

Table of Contents

Table of Contents v

List of Figures vii

Acknowledgements ix

Abstract x

1 Introduction 1

2 System Model 6

2.1 Q-ary Pulse Position Modulation . . . . . . . . . . . . . . . . . . . . 8

2.2 Free Space Channel Model . . . . . . . . . . . . . . . . . . . . . . . . 9

2.3 Transmitter and Receiver . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4 Link Budget . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17

3 Performance Analysis 19

3.1 Case I: No Channel Fading, No Background Radiation . . . . . . . . 19

3.1.1 Error Probability Analysis . . . . . . . . . . . . . . . . . . . . 19

3.1.2 Peak Power Constraint and Average Power Constraint . . . . 21

3.1.3 Modulation Efficiency . . . . . . . . . . . . . . . . . . . . . . 23

3.2 Case II: Fading Channel, No Background Radiation . . . . . . . . . . 25

3.3 Case III: No Channel Fading, with Background Radiation . . . . . . . 29

3.4 Case IV: Fading Channel, with Background Radiation . . . . . . . . . 34

4 Conclusion and Summary 41

4.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

4.2 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

v

Bibliography 44

vi

List of Figures

2.1 System model of free space optical communications. . . . . . . . . . . 7

2.2 System block diagram of free space optical communications. . . . . . 7

2.3 Probability density functions of channel gain A under Rayleigh and

log-normal distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4 Optical detection model of free-space communication system . . . . . 12

3.1 System error probability for non-fading, no background radiation case. 22

3.2 System error probability for fading, no background radiation case with

M = 1 and N = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.3 Symbol error probability for Rayleigh and log-normal fading, no back-

ground radiation, Q = 8, w = 4,M ∈ (1, 2, 4), N = 1, and peak power

definition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.4 Symbol error probability for log-normal fading (S.I = 1.0), no back-

ground radiation, Q = 2, w = 1,M ∈ (1, 2, 4), N ∈ (1, 2), and peak

power definition. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.5 Symbol error probability in different background radiation levels with-

out channel fading, for binary PPM and 8-ary PPM with w = 1,

M = 1, N = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

3.6 Symbol error probability under background radiation without channel

fading, for binary PPM and 8-ary PPM with w ∈ (1, 4), M = 1, N = 1,

PbTb = −170 dBJ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

vii

3.7 Simulation result of symbol error probability for optimal combining and

equal-gain combining, Rayleigh and log-normal fading, and background

radiation, M = 1, N = 4, PbTb = −170 dBJ . . . . . . . . . . . . . . 36

3.8 Simulation of symbol error probability for Rayleigh and log-normal

(S.I.=1.0) fading, Q = 8, w = 4, background energy = -170 dBJ . . . 40

viii

Acknowledgements

I would like to thank Professor Maıte Brandt-Pearce, my supervisor, for her many

suggestions and constant support during this research. I am also thankful to Professor

Stephen G. Wilson for his guidance through the years of my study in this university.

Dr. Bo Xu expressed his interest in my work and supplied me with the preprints

of some of his recent work, which gave me a better perspective on my own results.

Michael Baedke shared with me his knowledge and provided many useful references

and friendly encouragement.

Of course, I am grateful to my husband for his patience and love. Without him

this work would never have come into existence (literally). I also give thanks to my

parents, their support is my confidence in my study.

Finally, The author gratefully acknowledges the support of the National Science

Foundation for funding the research under the project Space-Time Coding for Optical

MIMO Channels.

Qianling Cao,

January, 2005

ix

Abstract

The global telecommunications network has seen massive expansion over the last few

years. Optical communication through clear atmosphere provides a means for high

data rate communication over relatively short distances (e.g. 2km). However, the

turbulence in the atmosphere leads to fades of varying depths, some of which may

lead to heavy loss of data. Free space optical communication system using multi-input

multi-output with QPPM is described in this thesis. We use spatial diversity at both

the transmitter and receiver as a means to mitigate the channel fading. Using direct

detection receivers and QPPM modulation, we derive the symbol error probability of

MIMO systems that use ML detection and equal gain combining with and without

background radiation. We demonstrate that for faded channels, performance gains

are seen as the number of transmitters and receivers increases. Full transmitter and

receiver diversity is obtained and observed by analyzing the Rayleigh fading case.

We also show that multipulse QPPM is superior to PPM with respect to bandwidth

efficiency, and exhibits superior symbol error performance when the system is peak-

power-limited. Hence, we conclude that MIMO systems can be used effectively as a

technique for atmospheric optical channels.

x

Chapter 1

Introduction

The global telecommunications network has seen dramatic expansion over the last

decade, catalyzed by the telecommunications deregulation of 1996. First came the

tremendous growth of long-haul, wide-area networks (WANs), then followed by a

more recent emphasis on metropolitan area networks (MANs). Meanwhile, local area

networks (LANs) and gigabit Ethernet ports are being deployed with a comparable

growth rate. In order for this tremendous capacity to be exploited, and for the

users to be able to utilize the broad array of services becoming available, network

designers must provide some flexible and cost-effective means for the users to access

the telecommunications network. As a consequence, there is a strong need for a

high-bandwidth bridge (the ‘last mile’) between the LANs and MANs or WANs.

Free-space optical communication systems represent one of the most promising

approaches for addressing the emerging broadband access market and its ‘last mile’

bottleneck. Since light travels through air less expensive than through fibre and it

is more convenient to build a line-of-sight link without fiber, it provides a natural,

reliable approach for broadband access.

Mention optical communications and most people think of fiber optics. But light

1

2

travels through air in a simpler way. So it is hardly a surprise that smart entrepreneurs

and technologists are borrowing many of the devices and techniques developed for

fiber-optic systems and applying them to what some call fiber-free optical communi-

cations. Free-space optical systems, which establish communication links by trans-

mitting laser beams directly through the atmosphere, have been engineered to provide

robust performance that is highly competitive with other access approaches, offering

high capacity, excellent availability, fiber-like bandwidth, low operational cost per bit

per second, and rapid deployment (for example, 2 hours). Available systems offer

capacities in the range of 100 Mbps to 10 Gbps. These systems are compatible with

a wide range of applications and markets and they are sufficiently flexible as to be

easily implemented using a variety of different architectures.

As we know, modern free-space optical communication originated in the 1970’s

[11]. It really started to develop in the 1980’s. But as communications through optical

fibers boomed, the interest in free-space optical communication began to decline.

However, as the requirement for solving the ‘last-mile’ problem arise, the market for

free-space optical communications began to grow.

Free-space optical interconnects can provide high bandwidth with no physical

contact, but are hampered by signal fading effects due to particulate scattering in the

line-of-sight path caused by atmospheric turbulence. In particular the atmospheric

turbulence causes fluctuations in both the intensity and the phase of the received light

signal, producing additional space losses as well as possible beam distortion. Even

in clear weather, channels may suffer fading due to inhomogeneities in the index

of refraction of the optical path. Related research results can be found in several

references [2], [3] and [11]. Incorporating large link margins to combat the optical

3

propagation effects is not efficient.

Furthermore, the narrow optical beam-width also implies critical pointing from

the transmitter to the receiver, especially when we establish the free-space optical

system between two high buildings since the sway can make it impossible to keep both

transmitter and receiver pointing towards each other without any active tracking.

To address these challenges, especially the intensity fluctuations, we consider the

use of optical arrays instead of a single transmitter laser and a single photodetector to

reduce fading effect. This creates a multiple-input multiple-output channel. Specif-

ically, we use M separate lasers, assumed to be intensity modulated (IM), together

with N photodetectors, assumed to be ideal direct detection (DD) receivers. The

sources and detectors are situated separately enough so that we can assume the fad-

ing experienced between every laser-photodetector pair is statistically independent.

Thus diversity benefits can accrue from the multiple-input multiple-output (MIMO)

channel and the pointing issue can also be improved. However, the assumption of

independence is based on the fading character and it may not merely depend on the

spacing of the devices. For example, in foggy weather, all the laser-detector pair links

will induce large fades. Related work on MIMO optical communication can be found

in [17] by Haas et al by analyzing pairwise code error probability. Shin et al [18] also

treat the problem of receiver diversity.

The multi-source, multi-detector configuration also helps the pointing issue. Hor-

izontal roof-top transmit/receiver arrays will experience fading instead of total point-

ing loss in the presence of substantial sway.

MIMO processing has been used with great success to combat fading in RF wire-

less communication systems [44][45]. The free-space optical intensity modulation,

4

direct detection communication system that we propose has several aspects different

from traditional RF wireless technology. First, the MIMO channel input symbols are

non-negative real intensities. Second, the channel gains are real and non-negative.

This is unlike the RF wireless system where both input symbols and channel gains

are typically described as complex numbers. Third, the noise mechanisms are quite

different - in this thesis, we analyze the circumstance that the signal-dependent shot

noise in optical communications limits the system performance. On the other hand,

RF wireless communications are often thermal-noise-limited. Our study focuses on

the performance analysis of optical MIMO systems in various fading environments

with and without background radiation.

For this system, we are focusing on multiple pulse position modulation (MPPM),

which is an intensity modulation technique. Research in pulse position modulation

(PPM) in optical free-space communication can be dated as early as 1960’s, [1] and

[14]. These works showed that PPM becomes more average-energy efficient as the

number of time slots per symbol increases. Further work on coded PPM includes

[32] and [33] in the 1980’s and some recent work [12]. In the 1990’s, MPPM has

been studied and combined with traditional block codes to improve performance [19].

Some recent studies related to MIMO channels with direct detection are found in

[18]. In this thesis, we analyze the system performance by an approach called spatial

diversity which attempts to overcome the atmospheric difficulties.

The rest of the thesis is organized as follows. A theoretical system model con-

structed based on the MIMO channel is described in Chapter 2 including a detailed

link budget. In Chapter 3 we analyze the performance of the system and also offer

simulation results. Chapter 4 then summarizes the results and also provides a brief

5

discussion of the advantages and the limitations of the proposed method.

Chapter 2

System Model

A free-space optical communication system is composed of three basic parts: a trans-

mitter, the propagation channel and a receiver. A typical simple diagram illustrating

the system is displayed in Figure 2.1.

In our system, M lasers, intensity-modulated by input symbols, all point toward

a distant array of N photodetectors. Every laser beamwidth is sufficiently wide to

illuminate the entire photodetector array. The MN laser-photodetector path pairs

may experience fading and the amplitude of the path gain from laser m to detector

n is designated as anm.

Figure 2.2 shows the block diagram of the proposed free space optical MIMO

system. In the transmitter, binary data bits are converted into a stream of pulses

corresponding to QPPM symbol described below, and sent to the M lasers. All

lasers send the same symbol towards every photodetector (repetition coding). Every

photodetector counts the photoelectrons it receives in every QPPM symbol slot. The

received symbol of the nth photodetector, is a vector of Q photoelectron counts

Znq, q = 1, · · · , Q and it is passed to the processor and finally decoded to binary

data bits.

6

7

Photodetector ArrayLaser Array

500 - 5000 meters

Figure 2.1: System model of free space optical communications.

MPPM Modulation

bits in Turbulent Atmosphere

Medium

X

MQ M M x x x ... 2 1

Q x x x 1 22 21 ...

Q x x x 1 12 11 ...

PD2

PD1

PDN

Processor

output bits out

Q z z z 1 12 11 ...

NQ N N z z z ... 2 1

Q z z z 2 22 21 ...

Laser M

Laser 1

Laser 2

Figure 2.2: System block diagram of free space optical communications.

8

The wavelength chosen for free space optical systems usually falls near one of two

wavelengths, 0.85µm or 1.55µm. The shorter of the two wavelengths is cheaper and is

favored for shorter distances. The 1.55µm light source is favored for longer distances

since it has an allowed power that is two orders of magnitude higher than at 0.85µm

[16]. The reason for the higher allowed power is that laser-tissue interaction is very

dependent on wavelength. The eye hazard at 1.55µm is much lower than at 0.85µm.

2.1 Q-ary Pulse Position Modulation

Pulse-position modulation, or PPM, is a powerful and widely used technique for

transmitting information over an optical direct-detection channel [14]. PPM is a

modulation technique that uses pulses that are of uniform amplitude and width but

displaced in time by an amount depending on the data to be transmitted. It is

also sometimes known as pulse-phase modulation. It has the advantage of requiring

constant transmitter power since the pulses are of constant amplitude and duration.

PPM also has the advantage of good noise immunity since all the receiver needs to

do is detect the presence of a pulse at the correct time; the duration and amplitude

of the pulse are not important [13].

Q-ary Pulse Position Modulation, or QPPM, is an energy-efficient and well de-

veloped modulation method. At the transmitter, the encoder maps blocks of L con-

secutive binary data bits into a single PPM channel symbol by placing a laser pulse

into one of several time slots. In this method, every symbol interval of duration Ts

is subdivided into Q slots, each of duration TQ = Ts/Q. If each bit is Tb seconds in

duration, then L bits take Ts = L× Tb seconds to transmit.

9

We use multi-pulse QPPM (M-QPPM) in our system, which means that in ev-

ery symbol the lasers turn on for w time slots out of a possible Q time slots. Of

course, w = 1 represents conventional QPPM described above. Instead of sending

a single pulse as in traditional QPPM, w pulses are sent in certain symbol slots

to transfer a digital message. Every symbol represents L = log2

(Q

w

)bits. So

Ts = Tb log2

(Q

w

)where Tb is the bit duration. After establishing slot and symbol

synchronization, the receiver detects the un-coded M-QPPM symbols by determining

which w out of the Q slots contains the laser pulses, and performs the inverse map-

ping operation to recover the bit stream. If Eb is the energy per bit, then the symbol

energy is Es = Eb log2

(Q

w

).

In this thesis, we assume repetition coding across all lasers, that is, each of the

M lasers transmits the same w-pulse symbol at the same time. While this constraint

restricts the permissible bit rate, relative to an unconstrained set of patterns, the

receiver processing is simple, performance analysis is more direct and, as we shall see,

performance is remarkably good on the MIMO channel.

2.2 Free Space Channel Model

Atmospheric turbulence can degrade the performance of free-space optical commu-

nication systems, particularly over ranges longer than 1 km. Inhomogeneities in the

temperature and pressure of the atmosphere lead to variations in the refractive index

along the transmission path. These index inhomogeneities cause fluctuations in both

the intensity and the phase of the received signal. As a consequence, these fluctua-

tions lead to an increase in the system error probability, limiting the performance of

10

the communication system.

Atmospheric turbulence has been studied by many scientists, and various theoret-

ical models have been proposed to describe the intensity fluctuations (i.e, the signal

fading). None is universally accepted due to the difference of atmospheric conditions.

Among these models, log-normal and Rayleigh models are widely used.

Due to the turbulence of the atmosphere, the field strength received at the detector

becomes a random field. We adopt both log-normal and Rayleigh models - which are

the most accurate among them. In the log-normal model the path gain is A = eX

where X is Gaussian distributed with mean µX and variance σ2X . By definition, the

logarithm of A follows a normal distribution. Strohbehn first showed that for optical

atmospheric channels the gain can be assumed to be log-normal distributed and the

phase has a uniform distribution [2]. The p.d.f. of A is

fA(a) =1

(2πσ2X)

12 a

exp(−(loge a− µX)2

2σ2X

), a > 0 (2.2.1)

We restrict the mean path intensity to unity, i.e. E[A2] = 1. This requires

µX = −σ2X .

The scintillation index, used to measure the strength of fading, is defined as

S.I. =

√E[A4]

E2[A2]− 1 (2.2.2)

Typical values of S.I. are in the range of 0.4-1.0 for log-normal distributions.

The Rayleigh model is also widely used to describe the channel gain. It is used less

often in the literature than log-normal fading to analyze free-space optical systems,

but has some nice mathematical properties that make it an attractive model to use.

First of all, the Rayleigh fading case exhibits deeper fading than log-normal fading

11

because of the higher concentration of low-amplitude path amplitudes and it can be

considered the worst case. Furthermore, with Rayleigh fading, the diversity order -

which means the number of independently fading propagation paths - of the MIMO

system becomes apparent by analyzing the slopes of the symbol error probability

curves . The p.d.f of A under the Rayleigh distribution is

fA(a) = 2ae−a2

, a > 0 (2.2.3)

The scintillation index for the Rayleigh situation is 1, though the distribution is

quite different from the log-normal case, especially in the small-amplitude tail. Figure

2.3 shows the probability density functions for Rayleigh and log-normal distributions.

The density function of Rayleigh is more concentrated at low (deeply faded) values.

We also assume that the spatial coherence distance of the field at the detector

is large relative to the size of one photodetector. Spatial coherence distance in the

turbulent atmosphere are reported to range between 10 centimeters and 1 meter [1]

for a wavelength of 1.55µm as we use in our system.

2.3 Transmitter and Receiver

Our study is based on a semi-classical treatment of photodetection, where the incident

field is treated as a wave, and this wave produces a modulated Poisson point process

of photoelectrons at the receiver end that contributes to the detector current at the

output of the photodetector. An ideal photon-counting model with a typical quantum

efficiency is assumed. Figure 2.3 illustrates the receiver model for our system.

The aggregate optical field from all the lasers is detected by each photodetector

12

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 50

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

f A(a

)

a

log−nomral fading, S.I.=0.4log−nomral fading, S.I.=0.6log−nomral fading, S.I.=1.0Rayleigh fading

Figure 2.3: Probability density functions of channel gain A under Rayleigh and log-normal distribution

Integral Photodetector Optical

field

Receiver Aperture

Z 1, Z 2 ,....,Z q

Figure 2.4: Optical detection model of free-space communication system

13

and we denote the total incident signal power at one photodetector for a non-fading

channel from all the lasers as Pr when a pulse is transmitted. Znq is the number of

photoelectrons at slot q collected by photodetector n. Znq is a Poisson random vari-

able with mean value depending on the receiving power, background power, receiver

efficiency and receiver time slot duration. The average number of signal photoelec-

trons generated in a QPPM slot in which a pulse is transmitted is denoted as

λs =ηPrTQ

hf(2.3.1)

where η is the detector’s quantum efficiency factor, defined as the ratio of generated

photoelectrons to incident photons, assumed to be 0.5 here. TQ is the slot duration

equal to Ts/Q. h is Planck’s constant, and f is the optical center frequency.

In addition to the signal, background radiation is received. The average number

of photoelectrons due to the background field is denoted as

λb =ηPbTQ

hf(2.3.2)

where Pb is the incident background power on one photodetector. At the receiver end,

in a slot a pulse is sent (what we call an ‘on’ slot ), the photodetector receives both

incident power and background noise. The probability mass function for the number

of counts in an ‘on’ slot is

Pr(Znq = k) =(λs + λb)

k exp (− (λs + λb))

k!, k = 0, 1, 2, · · · (2.3.3)

In the slot where no pulse is sent ( what we call an ‘off’ slot ), the photodetector

receives only background noise. The probability mass function for the number of

counts in an ‘off’ slot is

14

Pr(Znq = k) =(λb)

k exp (− (λb))

k!, k = 0, 1, 2, · · · (2.3.4)

In the fading case, we denote the path gain from the mth laser to the nth pho-

todetector as anm. The mean number of the photoelectrons at the nth detector in

the signal ‘on’ slot is derived from the sum of incident powers from all M lasers plus

background noise; the mean number becomes λon = 1M

∑Mm=1 a2

nmλs + λb.

To be fair in our comparisons, we keep the total laser power constrained by assum-

ing the transmit power is equally shared among M lasers, so Pr is the power received

form all M lasers to one photodetector. This parallels the standard assumption for

the microwave MIMO system.

We designate the collection of slot-by-slot photoelectron counts for the nth pho-

todetector at the qth slot in a symbol as [Znq, n = 1, · · · , N, q = 1, · · · , Q], where n

describes the photodetector number and q describes the slot number. Then Z = [Znq]

is the received observation matrix.

Suppose X is the whole set of possible symbols, i.e, every x ∈ X is a Q slot symbol

with w slots ‘on’ and (Q−w) slots ‘off’. For every fading matrix A with entries anm,

the maximum likelihood detector is given by

x = arg maxx∈X

f (Z|x,A) (2.3.5)

Since the Znq are all independent of each other, the conditional distribution of

the N ×Q random matrix Z can be written as a N ×Q-fold product over all of the

individual elements Znq.

15

x = arg maxx∈X

N∏n=1

Q∏q=1

exp(−λs

Mxq

∑Mm=1 a2

nm + λb

)(λs

Mxq

∑Mm=1 a2

nm + λb

)Znq

Znq!

= arg maxx∈X

N∏n=1

Q∏q=1

exp

(−λs

Mxq

M∑m=1

a2nm + λb

)(λs

Mxq

M∑m=1

a2nm + λb

)Znq

(2.3.6)

We define the set of the ‘on’ slots as Qon and the set of all the ‘off’ slot as Qoff .

Their sizes are w and Q − w respectively. The sets Qon and Qoff depend on the

symbol x. These elements are conditioned on whether they are in Qon, or in Qoff .

x = arg maxx∈X

N∏n=1

∏q∈Qon

exp

(−λs

M

M∑m=1

a2nm + λb

)(λs

M

M∑m=1

a2nm + λb

)Znq

∏q∈Qoff

exp (−λb) (λb)Znq (2.3.7)

where x ∈ X is possible sending symbol.

We take the logarithm of the entire quantity to find the log-likelihood function.

The ML detector becomes

x = arg maxx∈X

N∑n=1

∑q∈Qon

(−λs

M

M∑m=1

a2nm + λb

)+ Znq log

(λs

M

M∑m=1

a2nm + λb

)

+N∑

n=1

∑q∈Qoff

(−λb + Znq log (λb)) (2.3.8)

which can be rewritten as

16

x = arg maxx∈X

N∑n=1

∑

q∈Qon

Znq log

(λs

M

M∑m=1

a2nm + λb

)+

∑q∈Qoff

Znq log (λb)

= arg maxx∈X

N∑n=1

∑q∈Qon

Znq log

(λs

M

M∑m=1

a2nm + λb

)

+N∑

n=1

(∑q∈Q

Znq log (λb)−∑

q∈Qon

Znq log (λb)

)

= arg maxx∈X

N∑n=1

∑q∈Qon

Znq log

(λs

M

∑Mm=1 a2

nm + λb

λb

)(2.3.9)

Therefore, the ML detector would make a decision based on a weighted sum over

the ‘on’ slots.

If background noise can be ignored, the ML detector chooses the slots that photons

are received and makes a free guess if there are some slots pulses were sent but nothing

received. In case there is no channel fading, the ML detector does not need to weight

Znq. The ML detector just compares the sum of the photoelectrons counts of all N

photodetector in every slots and chooses the largest w slots. When channel fading and

background noise are both present, we propose to use an equal-gain combiner instead

of the ML detector since monitoring the channel gains increases the complexity of

the receiver and has only a slight benefit, as we show in Section 3.4. An equal-

gain combiner simply adds the output of every detector without weighting them. In

summary, in all cases we form the sum over all detector counts slot by slot and choose

the slots with the largest counts.

x = arg maxx∈X

∑q∈Qon

N∑n=1

Znq (2.3.10)

17

2.4 Link Budget

Assume we want to establish a free space optical communication system between

two buildings separated by 2 kilometers to transmit a data stream of 100 Mbps. To

minimize the eye hazard, we use a wavelength of 1.55 µm. The transmitter laser power

Pt is 100 mW . The full laser transmit beam angle θ is 10 mrad (about 0.6 degrees).

The receiver aperture diameter DR is 1 cm and we assume all photodetectors have

the same size. We also assume that the whole path experiences no link fading and

perfect alignment.

At the receiver end, the power flux density is

P (d) =Pt

2πd2(1− cos( θ2))≈ 4Pt

πd2θ2mW/m2 (2.4.1)

where d is the distance between the laser and detector. The power intercepted by the

receive aperture is

Pr = P (d)Arec =PtD

2R

d2θ2= 1.25× 10−8 W (2.4.2)

Assuming binary PPM without considering any background radiation, the slot

time is 5 nanoseconds. In every signal ‘on’ slot the average number of received signal

photons is

λs = (ηPr/hf)TQ ≈ 244 photons (2.4.3)

where the quantum efficiency factor η is 0.5.

In addition to the desired source power, a receiver also collects undesirable strong

background radiation falling within the spatial and frequency ranges of the detector.

The background power levels can be calculated as

Pb = W · Arec ·∆λΩfv = 1.84× 10−9 watts (2.4.4)

18

where W is the spectral radiance function defined as the power radiated at the wave-

length of interest per unit of bandwidth into a unit solid angle per unit of source area

[1], which in our system is the background noise from the sky. ∆λ is the received

wavelength bandwidth assume to be 10−9 m. Ωfv is the receiver field of view. The

field of view angle is 100 mrad so Ωfv = π4× 10−2 sr here. For wavelength of 1.55

µm , W = 3× 10−4W/(cm2 − µm− sr) [1].

For a binary PPM system, in every signal ‘off’ slot the average number of received

photons is

λb = (ηPb/hf)TQ ≈ 36 photons (2.4.5)

Chapter 3

Performance Analysis

We consider four cases: without or with background radiation, and non-fading or

fading links. We discuss a general theory, and illustrate with specific results for the

most interesting cases. The situation without background radiation and non-fading

links is the easiest and is treated first.

3.1 Case I: No Channel Fading, No Background

Radiation

3.1.1 Error Probability Analysis

With no loss of generality, we assume the symbol with the first w of total Q slots

‘on’ is sent. At the receiver end, we receive a matrix Z with elements [Znq, n =

1, · · · , N, q = 1, · · · , Q] where n indicates the receiver number and q indicates the

slot number. Since there is no background radiation, then λb = 0. If slot q ∈ Qoff

Znq will be zero. The channel gain is the same for all paths with anm = 1. The

19

20

maximum likelihood detector becomes

x = arg maxx∈X

N∑n=1

∑q∈Qon

Znq log

(λs

M

∑Mm=1 a2

nm + λb

λb

)

= arg maxx∈X

N∑n=1

∑q∈Qon

Znq log

(λs + λb

λb

)

= arg maxq

N∑n=1

∑q∈Qon

Znq (3.1.1)

In this case, an error will only occur when one or more of the w ‘on’ slots register zero

counts at all N detector outputs and likelihood ties represents the only mechanism

for decision error. When Q = 2 and w = 1 this is equivalent to a binary erasure

channel.

Specifically, suppose i of the w ‘on’ slots (i ≤ w) produce a column of zeros in the

Z matrix where non-zero counts are expected. Then, a likelihood tie occurs among(

Q−w+ii

)candidates and tie-breaking errors have probability

P [making an error] =

(Q−w+i

i

)− 1(Q−w+i

i

) = t(Q,w, i) (3.1.2)

By the Poisson property and independence we have that the probability of exactly i

of w columns registering zero counts is

P [i of w columns = 0] =

(w

i

)pi(1− p)w−i (3.1.3)

where p = e−λs from (2.3.3), and λs = ηPrTQ/hf = ηPrTs/hfQ from (2.3.1). Putting

this altogether we can derive the symbol error probability in no background radiation,

for a non-fading channel

21

Ps =w∑

i=1

(w

i

)t(Q,w, i)pi(1− p)w−i (3.1.4)

By expanding the last term using a binomial expansion, i.e.

(1− p)w−i =w−i∑

l=0

(−1)l

(w − i

l

)pl,

we can combine terms to get a finite series expansion for symbol error probability:

Ps =w∑

i=1

w−i∑

l=0

(−1)l

(w

i

)(w − i

l

)t(Q,w, i)pi(1− p)w−ie−λsN(i+l) (3.1.5)

This says that for a fixed total transmitter energy, the probability of symbol er-

ror is independent of M , i.e., there is no phased-array gain attached to the multiple

sources, since these are non-coherent sources. The effective received power does in-

crease linearly with N , the effect of increasing receiving aperture size.

3.1.2 Peak Power Constraint and Average Power Constraint

To cast this symbol error probability in terms of peak power and a common informa-

tion rate from the bit-symbol relation (TQ = Ts/Q and Ts = Tb log2

(Qw

)) we get

PrTQ = PaveTb

log2

(Qw

)

w(3.1.6)

or

PrTQ = PpeakTb

log2

(Qw

)

Q(3.1.7)

The symbol error probability Ps is shown in Figure 3.1 for non-fading links without

background radiation versus PaveTb and PpeakTb. P represents either Pave for average

power or Ppeak for peak power in this and all subsequent plots.

22

−200 −195 −190 −185 −180 −175 −170 −165 −16010

−10

10−8

10−6

10−4

10−2

100

PTb in dBJ

Ps

Ppeak

, Q = 2P

peak, Q = 8, w = 4

Ppeak

, Q = 8, w = 1P

ave, Q = 2

Pave

, Q = 8, w = 4P

ave, Q = 8, w = 1

Figure 3.1: System error probability for non-fading, no background radiation case.

23

In this figure, we compare binary PPM and 8-ary PPM, showing both classic

QPPM and M-QPPM with w = Q/2 = 4 for Q = 8. In the non-fading regime,

the choice of M has no impact since we fix the total laser array power. We choose

M = N = 1.

Notice that the multipulse case with w = Q/2 exhibits a superior energy efficiency

under a peak power constraint. On the other hand, if average energy per bit is

the criterion, classic PPM with w = 1 is much superior. In optical system using

semiconductor laser, peak power is the most logical comparison.

3.1.3 Modulation Efficiency

Modulation efficiency of various system designs depends on whether one considers the

peak or average power; efficiencies provide different result. By studying (3.1.5) for

the non-fading case above, the dominant term in Ps at large values of PTb (or small

Ps) is found to be the term that one signal slot receives zero photoelectron count, i.e,

i = 1 and l = 0 in (3.1.5). Thus, we may derive an approximation of efficiency as the

multiplier of PT in (3.1.6) and (3.1.7).

This approximation gives the asymptotic relative efficiency. The results are

γave =log2

(Qw

)

w(3.1.8)

and

γpeak =log2

(Qw

)

Q(3.1.9)

For purposes of comparison, we have γave = 1 in the binary PPM case, (Q,w) = (2, 1).

For any fixed (Q,w) pair, the peak energy efficiency is always w/Q times the average

24

energy efficiency. We may also measure the effect of w > 1 using these expressions.

For example, comparing Q = 8 with w = 1, 4 we obtain that γpeak = γave/2 when

w = 4 and γpeak = γave/8 when w = 1. We can draw the following conclusion: in

case of average power constraint, large Q with w = 1 can get best performance and

in case of peak power constraint, large Q with w = bQ/2c is the best choice.

In general, if we keep the energy per bit fixed, the average power efficiency im-

proves with increasing Q, and decreasing w. On the other hand, the peak power

efficiency is best when w = bQ/2c, and in this case, as Q grows, the efficiency ap-

proaches 1 from below. Since a large Q may increase the difficulty in QPPM decoding

and receiver synchronization, from an engineering perspective, Q = 8, w = 4 is an

attractive choice for peak-power-limited design, and we emphasize this choice in the

rest of the thesis.

Spectral efficiency is of lesser concern in free-space optical systems, but due to

the required clock speed and the relative difficulty of receiver synchronization, spec-

tral efficiency also affects the relative difficulty of implementation. If we measure

bandwidth as proportional to bit rate 1/Tb, then the spectral efficiency in bps/unit

bandwidth is proportional to

β =log2

(Qw

)

Q(3.1.10)

For a given Q, β is maximized for w = bQ/2c, and when we choose w = bQ/2c, as Q

grows large, β monotonic approaches 1. In the binary PPM case, β = 0.5. For Q = 8

and w = 4, β = 0.766.

25

3.2 Case II: Fading Channel, No Background Ra-

diation

First, we assume the channel gain of every laser-detector pair is fixed over a symbol

duration. Letting amn denote the amplitude fading on the path from laser m to

photodetector n, we define the channel gain matrix as A with element [anm, n =

1, · · · , N, m = 1, · · · ,M ]. The probability of symbol error conditioned on the fading

variables is

Ps|A =w∑

i=1

w−i∑

l=0

(−1)l

(w

i

)(w − i

l

)t(Q, w, i)e−

λsM

∑n

∑m a2

mn(i+l) , (3.2.1)

where again λs = ηPTQ/hf from (2.3.1).

To extend the analysis of non-fading link and no background radiation case to the

case of link fading, we can simply average the (conditional) symbol error probability

of (3.2.1), with respect to the joint fading distribution of the Anm variables. We

emphasize that this produces the symbol error probability averaged over fades.

Formally, we find Ps by evaluating

Ps =

∫Ps|AfA(a)da (3.2.2)

where the integral is interpreted as an MN -dimensional integral. Since the Anm

variables are assumed independent, the above averaging leads to

Ps =w∑

i=1

w−i∑

l=0

(−1)l

(w

i

)(w − i

l

)t(Q,w, i)

(∫ ∞

0

e−λsM

(i+l)a2

fA(a)da

)MN

(3.2.3)

26

which is a function of received energy per slot, number of lasers and photodetectors,

and fading distribution.

If the channel is under Rayleigh fading, the averaging in (3.2.3) may be done

analytically, and produces a simple form

PsRayleigh=

w∑i=1

w−i∑

l=0

t(Q,w, i)

(w

i

)(w − i

l

)(−1)l

[1

1 + (λs/M)(i + l)

]MN

(3.2.4)

In case of log-normal fading, we can at least evaluate (3.2.3) numerically.

Figure 3.2 presents results for Rayleigh fading and log-normal fading with varying

values of scintillation index by using a single laser and photodetector. Clearly, the

log-normal fading case causes a degradation in system performance compared to the

non-fading case, although not as severe as with Rayleigh fading.

A study of (3.2.4) as a function of PTb reveals that Ps is an inverse-MN-power

function of the signal energy in the large signal regime, which we take as the definition

of the system achieving full diversity, MN . In contrast to a similar microwave sys-

tem, notice that attainment of full transmit diversity, M , is obtained without resort

to exotic space-time constructions here. The diversity gains are quite large in the

Rayleigh case, though smaller for the log-normal model. However, there is a power

penalty due to power sharing in the denominator of (3.2.4), but the diversity order is

nonetheless MN .

As in the non-fading case, the symbol error probability expressions can be plotted

versus either PaveTb or PpeakTb. Figure 3.3 illustrates the performance versus peak

power for Q = 8,M ∈ 1, 2, 4, N = 1, showing a diversity order M is attained in

the Rayleigh case. Similar conclusions pertain to average power as were made for the

non-fading case.

27

−195 −190 −185 −180 −175 −170 −165 −160 −155 −150 −14510

−10

10−8

10−6

10−4

10−2

100

PTb in dBJ

Ps

No fadingS.I. = 0.4S.I.= 0.6S.I. = 1.0Rayleighfading

Figure 3.2: System error probability for fading, no background radiation case withM = 1 and N = 1

28

−200 −190 −180 −170 −160 −150 −14010

−10

10−8

10−6

10−4

10−2

100

PTb in dBJ

Ps

no fadingM = 1M = 2M = 4

Rayleigh fading

log-normal fading

Figure 3.3: Symbol error probability for Rayleigh and log-normal fading, no back-ground radiation, Q = 8, w = 4,M ∈ (1, 2, 4), N = 1, and peak power definition.

29

From the Rayleigh fading curves in this figure we can see the error probability

drops by a factor of 10MN for every 10 dB increase in signal power, we claim that

the system achieves a diversity equal to MN. Just as in the Rayleigh fading case, a

considerable performance gain is also achievable by increasing only the number of

lasers in the log-normal cases.

It may also be noted that the interchange of M and N is not symmetric, due to

the power division by M at the transmitter (note the c/M factor in (3.2.4)) Thus,

though they have the same diversity order, (M,N) = (2, 1) and (1, 2) cases are 3 dB

different in favor of the latter. Actually, if we fix the total receive aperture as in [18],

then M and N are interchangeable.

Figure 3.4 shows the advantage of using multiple photodetector. Notice that be-

sides the diversity effect that we are seeking, adding detectors also improves efficiency

due to large total aperture .

3.3 Case III: No Channel Fading, with Background

Radiation

For the case of background radiation, the evaluation of error probability is more

complicated.

At the receiver end, we receive a matrix Z with elements [Znq, n = 1 · · ·N, q =

1 · · ·Q]. We assume λb is the Poisson count random variable parameter due to the

background radiation, and if slot q is an ‘off’ slot, Znq will be also a Poisson distributed

random variable with parameter λb. For signal ‘on’ slot, the Poisson count random

variable parameter is λs+λb. The channel gain is the same for all paths with anm = 1.

30

−200 −195 −190 −185 −180 −175 −170 −165 −160 −15510

−10

10−8

10−6

10−4

10−2

100

PTb in dBJ

Ps

no fadingM = 1, N = 1M = 2, N = 1M = 4, N = 1M = 1, N = 2M = 2, N = 2M = 4, N = 2

Figure 3.4: Symbol error probability for log-normal fading (S.I = 1.0), no backgroundradiation, Q = 2, w = 1,M ∈ (1, 2, 4), N ∈ (1, 2), and peak power definition.

31

Again, we assume without lose of generality that the symbol with the first w slot ‘on’

is send. The maximum likelihood detector becomes

x = arg maxx∈X

N∑n=1

∑q∈Qon

Znq log

(λs

M

M∑m=1

a2 + λb

)

= arg maxx∈X

∑q∈Qon

N∑n=1

Znq (3.3.1)

where Qon = [1, · · · , w] is the set of all ‘on’ slot that pulses are sent and Qoff =

[w + 1, · · · , Q] is the set of all ‘off’ slot that only background noise is received. De-

tection is correct only if all of the noise slot counts Zq are less than all the signal slot

counts. Thus, we can upper bound the symbol error probability.

Ps ≤ 1− P (all signal slot counts greater than noise slot counts) (3.3.2)

Adding the tie-break part, we can get the exact error probability.

Letting Zon be the set of the∑N

n=1 Znq for all slots q ∈ Qon and Zoff be∑N

n=1 Znq

in all slots q ∈ Qoff , for any (Q,w) pair, the symbol error probability will be

Ps ≤ P [min(Zon) < max(Zoff )]

= 1−∞∑i=1

(Poisson pmf (N (λs + λb) , i + 1))

×(1− (Poisson cmf (N (λs + λb) , i)))w−1

×(Poisson cmf (N (λb) , i))Q−w−1 (3.3.3)

where Poissonpmf(x, y) represent the Poisson probability density function at value

y using the corresponding parameter x and Poissoncmf(x, y) represent the Poisson

probability cumulative function at value y using the corresponding parameter x.

32

For the exact error probability we can get a precise form

Ps = P [min(Zon) < max(Zoff )]

+w∑

k=1

Q−w∑s=1

(k + s

s

)P [min Zon = max Zoff ] (3.3.4)

where s minimal signal slots have the same counts as k maximum noise slots. Then

the exact error probability can be written as

Ps = 1−∞∑i=1

(Poisson pmf (N (λs + λb) , i))

×(Poisson cmf (N (λs + λb) , i))w−1 × (Poisson cmf (Nλb, i−)Q−w)

−∞∑i=1

w∑

k=1

Q−w∑s=1

(k + s

s

)(Poisson pmf (N (λs + λb) , i))k

×(Poisson cmf (N (λs + λb) , i))w−k ×Q−w∑s=1

(Poisson pmf (Nλb, i)s)

×(Poisson cmf (Nλb, i− 1)Q−w−s)

(3.3.5)

where k is the number of ‘on’ slot which have count i and all the other ‘on’ slots have

counts greater that i while, at the same time, there are s ‘off’ slots that have count i

and all other ‘off’ slot have smaller counts. In this case, the probability of error is also

independent of the number of transmitters. An increase in the number of receivers

provides gain due to the increase in receiving aperture size.

Figure 3.5 shows the symbol error probability under different background radiation

levels. Comparing Figure 3.5 to Figure 3.1, background radiation shifts the curves

to the right by amounts ranging from 4 to 7 dB without changing the shape of curve

heavily.

33

−190 −185 −180 −175 −170 −165 −160 −15510

−12

10−10

10−8

10−6

10−4

10−2

100

Eb in dBJ

Ps

Binary PPM, noise −170dBJBinary PPM, noise −160dBJ8−ary PPM, noise −170dBJ8−ary PPM, noise −160dBJ

Binary PPM

8−ary PPM

Figure 3.5: Symbol error probability in different background radiation levels withoutchannel fading, for binary PPM and 8-ary PPM with w = 1, M = 1, N = 1

34

Figure 3.6 shows the symbol error probability under a background radiation with

energy PbTb = 10−17 joules - which is close to the background level we calculate in

Chapter 2 - fixed for binary PPM and QPPM. We notice that the shape of the curve is

slightly different from Case I, the new results being steeper – essentially some minimal

level of signal power is required to overcome the background noise, and once this level

is exceeded, performance improves sharply.

3.4 Case IV: Fading Channel, with Background

Radiation

This case is the most general in practice, and there is no simple expression for the

symbol error probability. Here some incorrect symbol can have higher likelihood (an

incorrect set of w slots have larger weighted column sums) same as in Case III. One

can formally sum conditional probabilities over the error region and correctly handle

ties, but we have resorted to Monte Carlo simulation using importance sampling

instead. Our simulation uses an equal-weight combiner instead of the ML detector,

which is slightly suboptimal only in this case of fading and background radiation, and

only when N > 1. In cases where it is an issue, Figure 3.7 shows that the loss of the

equal-gain combiner compared to the ML detector is very small, prompting us to use

it in simulations. The result shows that the curves of equal-weight combining has the

same shape as the curves of ML detection with less than 1 dB shift.

By using equal-gain-combiner, the upper bound on Ps conditioned on fading path

gain matrix A is

35

−200 −195 −190 −185 −180 −175 −170 −165 −16010

−10

10−8

10−6

10−4

10−2

100

Ps

PTb in dBJ

Pave

, Q = 2P

ave, Q = 8, w = 4

Pave

, Q = 8, w = 1P

peak, Q = 2

Ppeak

, Q = 8, w = 4P

peak, Q = 8, w = 1

Figure 3.6: Symbol error probability under background radiation without channelfading, for binary PPM and 8-ary PPM with w ∈ (1, 4), M = 1, N = 1, PbTb = −170dBJ

36

−190 −185 −180 −175 −170 −165 −160 −155 −150 −145 −14010

−10

10−8

10−6

10−4

10−2

100

PTb in dBJ, noise = −170 dBJ

Ps

log−normal, equal gainRayleigh, equal gainlog−normal, ML detectionRayleigh, ML detection

Figure 3.7: Simulation result of symbol error probability for optimal combining andequal-gain combining, Rayleigh and log-normal fading, and background radiation,M = 1, N = 4, PbTb = −170 dBJ

37

Ps|A ≤ 1−∞∑i=1

(Poisson pmf

(N∑

n=1

M∑m=1

a2nm

λs

M+ Nλb, i + 1

))

×(

1− (Poisson cmfN∑

n=1

M∑m=1

a2nm

λs

M+ Nλb)

)w−1

×(Poisson cmf (N (λb) , i− 1))Q−w−1 (3.4.1)

We can get the overall symbol error probability by averaging the conditional sym-

bol error probability. Numerical integration is a prohibitively slow process due to

the large number of fading variables combined with the infinite summation. In this

case, using numerical integration is not an efficient way to calculate the symbol er-

ror probability. We use simulation instead to analyze the performance of the MIMO

system.

Monte Carlo methods are a way of using random numbers to perform numerical

integrations. By way of example consider the one variable integral

Ps =

∫Ps|af(a)da (3.4.2)

For a given background and signal power, a normal Monte Carlo simulation does

not select sampling points but instead it chooses points at random, then perform the

detection and count errors. In our system, the symbol error probability could be

evaluated as

Ps =1

N

N∑i=1

I(ai) (3.4.3)

where I(ai) is the indicator function and in our system it is defined as an error event

38

for fading sample ai.

I(ai) =

1 error occur

0 detection correct(3.4.4)

In simple, we count the number of errors and divide it by the total number of symbol

we send.

This procedure takes a very long simulation time if low error probabilities are

sought. Importance sampling, also called biased sampling, is one of the variance-

reducing techniques in Monte Carlo methods. Monte Carlo calculations can be carried

out using sets of random points picked from a different channel gain probability

distribution in our system. The choice of distribution obviously makes a difference

to the efficiency of the method. In most cases, Monte Carlo calculations carried out

using uniform probability distributions give very poor estimates of high-dimensional

integrals and are not a useful method of approximation. In 1953, however, Metropolis

introduced a new algorithm for sampling points from a different probability function.

This algorithm enables the incorporation of ‘importance sampling’ into Monte Carlo

integration. Instead of choosing random variables from a uniform distribution, they

are now chosen from a distribution which concentrates the points where the function

being integrated is large. The equation (3.4.2) can be written as

Ps =

∫Ps|af(a)da =

∫Ps|a

f(a)

g(a)g(a)da (3.4.5)

where the function g(a) is chosen to be a distribution different from f(a). The integral

can be estimated numerically by choosing the random points from the probability

distribution g(a) and evaluating f(ai)/g(ai) at these points. The average of these

evaluations gives an estimate of I. The Monte Carlo estimate by using importance

sampling of the integral is then,

39

Ps =1

N

N∑i=1

I(ai)f(ai)

g(ai)(3.4.6)

The method of importance sampling applied to our system produces a much higher

frequency of errors than normal Monte Carlo simulation; weighting the error counts

appropriately we obtain an unbiased estimate of Ps. If the biased distributed ran-

dom variable is correctly chosen, the variance of the estimate can be greatly reduced

relative to that of the Monte Carlo procedure with the same number of trials. Since

biasing the Poisson parameter give us non-monotonic curves due to the discontinuity

of the Poisson distribution, in our procedure we bias the fading distribution. We used

a one-sided exponential for the amplitude variable, which has the effect of decreasing

the mean signal counts, i.e, exaggerating the fades.

A sample result is shown in Figure 3.8, for both Rayleigh and log-normal fading.

Background power remains fixed at PbTb = −170 dBJ. In our simulation, normal

Monte Carlo simulation was used for Ps greater than 10−3, that means, a large number

of Poisson and Rayleigh (or log-normal) fading variables are generated. For Ps smaller

than 10−3, we use an exponential distribution to generated random variables of path

gain instead of the Rayleigh or log-normal distribution, to achieve more errors. The

symbol error probability is the sum of weighted errors divided by number of trails.

Again full diversity is observed. As seen from Figure 3.8 and Figure 3.2, the

MIMO system clearly exhibits superior performance to the single-input single-output

system over fading channels, for environment with or without background radiation.

40

−190 −185 −180 −175 −170 −165 −160 −155 −150 −145 −14010

−10

10−8

10−6

10−4

10−2

100

PTb in dBJ, noise = −170 dBJ

Ps

M = 1M = 2M = 4no fading

Rayleigh fading

log-normal fading

Figure 3.8: Simulation of symbol error probability for Rayleigh and log-normal(S.I.=1.0) fading, Q = 8, w = 4, background energy = -170 dBJ

Chapter 4

Conclusion and Summary

4.1 Conclusion

Free space optics communication has emerged as a technology that has the potential to

bridge the last-mile gap that separates homes and businesses from high speed access to

the Internet [11]. Atmospheric turbulence causes significant transmission impairment

for an open air optical communication system and free space optical communication

through atmosphere turbulence is now under active research. We have found that

MIMO systems can significantly reduce the symbol error probability and provide di-

versity gain over single-input single-output systems. This thesis has presented an

optical MIMO system employing QPPM with direct detection and developed opti-

mal and sub-optimal detection schemes. The optimal detector in Case I - non-fading

channel without background radiation, Case II - fading channel without background

radiation and Case III - non-fading channel with background radiation, simply com-

bines the received signal from all N receivers equally, and chooses the symbol with

w slots with the largest counts. In the case where background radiation and fading

41

42

are both present, the optimal detector maximizes a weighted sum of observation. We

found that equal gain combining achieves essentially equal performance to simply

combining the received signal as in other three cases. Both log-normal and Rayleigh

fading models are studied, assuming independent fading on laser/photodetector path

pairs. The analysis shows that diversity gain is obtained from using multiple sources

and detectors without spatial coding. In the case of Rayleigh fading, we were able to

ascertain this from the symbol error probability plots.

Some aspects of the optical MIMO system resemble those of the microwave MIMO

system. Full transmit and receive diversity can be shown analytically for the no-

background and Rayleigh fading case. Even for the cases with background radiation

in fading channel, QPPM also can approach the performance of non-fading cases by

increasing the number of lasers. Increasing the number of receivers is able to achieve

better performance both from diversity gain and from aperture gain.

Furthermore, multipulse transmission exhibits clear superiority over classic QPPM

when peak laser power is constrained. For fixed Q, using of w = bQ/2c can maximize

the energy and spectral efficiencies in this case. A particularly attractive choice

appears to be Q = 8, w = 4, providing 70 patterns per symbol interval.

4.2 Future work

Many extensions to this work can be made. A more accurate analysis of the proposed

system would include thermal noise which has been ignored in this thesis. As in fiber

systems, avalanche photodetectors provide one way to make the optical (shot) noise

exceed the electronic/thermal noise. This comes at the expense of an excess noise

penalty however that will degrade the performance from the ideal photon-counting

43

analysis presented here [43].

Besides the repetition constant weight QPPM considered here, more general space-

time patterns could be considered, in order to increase the throughput even further.

We have concluded in other research, however, that attainment of full transmit di-

versity is precluded when one moves out of the repetition coding regime.

Channel coding overlayed with block, e.g. Reed-Solomon, or trellis codes is cer-

tainly possible, but will not further increase the diversity order in quasi-static chan-

nels. Coding gain is possible however. A study of M-QPPM MIMO capacity would

help in the choice of optimal coding rates.

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