Performance Analysis ofMultipulse PPM on MIMO
Free-Space Optical Channels
A Thesis
Presented to
the Faculty of the School of Engineering and Applied Science
University of Virginia
In Partial Fulfillment
of the Requirements for the Degree
Master of Science
Electrical Engineering
by
Michael L. Baedke
August 2004
Approval Sheet
This thesis is submitted in partial fulfillment of therequirements for the degree of
Master of Science Electrical Engineering
Author
This thesis has been read and approved by the examining Committee:
Dissertation advisor
Accepted for the School of Engineering and Applied Science:
Dean, School of Engineering andApplied Science
August 2004
Acknowledgements
Over the course of the past few years, there have been many people who helped to make
this point in my education possible. In particular, I would like to thank my advisor, Professor
Wilson, whose dedication to learning and contagious passion for the field has been an inspiration
to me. Thank you for taking a personal interest in sharing your knowledge and providing me
with the best education imaginable.
I would also like to thank my wife, Alison for her unfailing love and support. Thank you
for putting your dreams on hold this past year so I could follow mine, and thank you for always
believing in me and encouraging me to be my best.
Thanks also goes to my parents for always offering their financial and emotional support
over the past few years. Being able to concentrate fully on our educational goals would have
been impossible without a ‘safety net’ to rely on whenever we needed it.
Finally, I would like to thank Dr. Guess and Dr. Brandt-Pearce, who also happen to be on
my committee, for their time and assistance this past year. You each made time for me whenever
I needed your help, and I’ve really enjoyed getting to know both of you.
i
Abstract
Free-space-optics (FSO) 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. In
FSO systems, information is transmitted between two points by modulating a light source, much
like with traditional fiber optic communication. However, FSO is a wireless technology in that
it operates via line-of-sight, transmitting the data through the air, potentially over distances on
the order of 1 km.
Its main advantages over other competing technologies are that it can provide high speed
access into the Gbps range, it operates in an unregulated frequency domain, and unlike fiber
optic systems or other wired services, it avoids the need for trenching, which is much slower
and more costly.
The main challenge to FSO systems is the atmosphere itself. The systems must be designed
such that the potentially harsh atmospheric effects of turbulence and aerosol scattering can be
mitigated. Traditional solutions include increasing the link margin (supplying excess power to
overcome potentially deep fades), and keeping the link distances small. However, these solu-
tions are limited: Link margins become prohibitively difficult (often impossible) to maintain
during deep fades, and keeping link distances small is often impractical in real-world imple-
mentations.
In this thesis, we study a method to combat link fading based on the multiple-input, multiple-
output (MIMO) approach that has seen much success in the RF domain. By using multiple
transmitters and receivers spaced sufficiently far from one another, we are able to create mul-
tiple, uncorrelated paths over which to send the data. The probability that all of the paths are
simultaneously faded is much lower than when only relying on a single path from transmitter
to receiver, as with a single-input, single-output (SISO) system.
For our analysis of the MIMO FSO system, we explore multiple pulse position modulation
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(MPPM). This is a modulation technique where the duration of a signal is divided intoQ slots,
and the laser array is pulsed simultaneously duringw of them, creating(
Qw
)possible patterns.
Traditional PPM is a special case of MPPM, wherew = 1. We show that MPPM is superior to
PPM with respect to bandwidth efficiency (and maximized whenw = bQ2c), and exhibits supe-
rior symbol error performance when the system is peak-power-limited. PPM exhibits superior
symbol error performance when the system is average-power-limited.
In this thesis, we develop the maximum likelihood detectors for the system operating in the
perfect photon counting (Poisson) and thermal-noise-limited (Gaussian) regimes. We demon-
strate that for non-faded channels, having multiple receivers improves symbol error perfor-
mance due to the increase in receiver aperture size. We also demonstrate that for faded chan-
nels, performance gains are seen for increases in the number of transmittersand receivers. Full
transmitter and receiver diversity is observable by analyzing the Rayleigh fading case.
We also analyze the information-theoretic channel capacity of the Poisson regime by look-
ing at the ergodic (average) capacity and the outage probability (probability of the instantaneous
capacity dropping below some set threshold). We see from these results that the maximum ca-
pacity is achieved at lower power levels for MIMO and SIMO systems over non-faded channels,
and for MIMO, SIMO, and MISO systems over faded channels. The outage probability curves
are steeper and show the beneficial effect of transmitter and/or receiver diversity. Full diversity
is once again observable by looking at the outage probability curves in the Rayleigh fading case.
We conclude that MIMO system design is a technique that improves MPPM FSO sys-
tem performance under various fading cases, and that full transmitter and receiver diversity
is achievable in this system.
Contents
1 Introduction 1
1.1 What is free-space optics? . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 The history of free-space optical communication . . . . . . . . . . . . . . . . 1
1.3 Uses of free-space optics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3.1 Military communication . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3.2 Satellite and deep-space communication . . . . . . . . . . . . . . . . . 3
1.3.3 The last-mile solution . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2 System overview 6
2.1 Information . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.2 Source and channel encoders . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
2.3 Multiple pulse position modulator . . . . . . . . . . . . . . . . . . . . . . . . 7
2.4 The transmitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
2.5 The channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.5.1 Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.5.2 Aerosol scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.5.3 Fading models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.6 The receiver . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.6.1 p-i-n photodiodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.6.2 Avalanche photodiodes . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.6.3 Bandwidth and noise considerations in p-i-n and APD receivers . . . . 17
2.6.3.1 Bandwidth . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.6.3.2 Shot noise . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.6.3.3 Background noise . . . . . . . . . . . . . . . . . . . . . . . 18
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2.6.3.4 Thermal noise . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.6.3.5 Excess APD noise . . . . . . . . . . . . . . . . . . . . . . . 18
2.7 MPPM demodulator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
2.8 Source decoder, channel decoder, and retrieved information . . . . . . . . . . 19
3 MIMO applied to FSO systems using MPPM - background, system model and
definitions 20
3.1 Research on MIMO and MPPM FSO communications systems . . . . . . . . . 20
3.1.1 MIMO in wireless systems . . . . . . . . . . . . . . . . . . . . . . . . 20
3.1.1.1 Application of MIMO concepts to free-space optics . . . . . 22
3.1.2 Multiple pulse position modulation (MPPM) . . . . . . . . . . . . . . 22
3.1.3 Error probability - Gaussian vs. Poisson . . . . . . . . . . . . . . . . . 23
3.2 Power comparisons between PPM and MPPM . . . . . . . . . . . . . . . . . . 24
3.3 System model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.3.1 MPPM signaling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3.3.2 Transmitter array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3.3 Receiver array . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3.4 Channel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
3.3.5 Detector and observable . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3.5.1 Poisson Regime . . . . . . . . . . . . . . . . . . . . . . . . 29
3.3.5.2 Gaussian Regime . . . . . . . . . . . . . . . . . . . . . . . 29
4 Maximum likelihood (ML) detection 33
4.1 ML Detection for the poisson regime . . . . . . . . . . . . . . . . . . . . . . 33
4.1.1 Case 1: ML detection with no background and no fading . . . . . . . . 34
4.1.2 Case 2: ML detection with no background and fading . . . . . . . . . . 36
4.1.3 Case 3: ML detection with background radiation and no fading . . . . . 36
4.1.4 Case 4: Background radiation and fading . . . . . . . . . . . . . . . . 37
4.2 General ML detection in the Gaussian regime . . . . . . . . . . . . . . . . . . 38
4.2.1 Thermal noise dominates over shot noise . . . . . . . . . . . . . . . . 40
4.2.1.1 No fading present . . . . . . . . . . . . . . . . . . . . . . . 41
4.2.1.2 Fading present . . . . . . . . . . . . . . . . . . . . . . . . . 41
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4.2.2 Shot noise dominates over thermal noise . . . . . . . . . . . . . . . . . 41
4.2.2.1 Fading present . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.2.2.2 No fading present . . . . . . . . . . . . . . . . . . . . . . . 42
4.2.3 Shot and thermal noise are not dominant . . . . . . . . . . . . . . . . . 42
4.2.3.1 Fading Present . . . . . . . . . . . . . . . . . . . . . . . . . 43
4.2.3.2 No fading present . . . . . . . . . . . . . . . . . . . . . . . 43
5 Error analysis of MIMO FSO system using MPPM 44
5.1 Error analysis in the Poisson regime . . . . . . . . . . . . . . . . . . . . . . . 44
5.1.1 No background radiation . . . . . . . . . . . . . . . . . . . . . . . . . 44
5.1.1.1 No background, no fading . . . . . . . . . . . . . . . . . . . 46
5.1.1.2 No background, Rayleigh fading . . . . . . . . . . . . . . . 47
5.1.1.3 No background, log-normal fading . . . . . . . . . . . . . . 49
5.1.2 Error probability in the presence of background radiation . . . . . . . . 52
5.1.2.1 Background radiation, no fading . . . . . . . . . . . . . . . 52
5.1.2.2 Background radiation, Rayleigh fading . . . . . . . . . . . . 57
5.1.2.3 Background radiation, log-normal fading . . . . . . . . . . . 58
5.2 Error probability in the Gaussian regime . . . . . . . . . . . . . . . . . . . . . 58
5.2.0.4 No fading . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
5.2.0.5 Rayleigh fading . . . . . . . . . . . . . . . . . . . . . . . . 63
5.2.0.6 Log-normal fading . . . . . . . . . . . . . . . . . . . . . . . 65
6 Capacity of the MIMO FSO system using MPPM in the Poisson regime 68
6.1 No background radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
6.1.1 No background, no fading . . . . . . . . . . . . . . . . . . . . . . . . 70
6.1.2 No background, Rayleigh fading . . . . . . . . . . . . . . . . . . . . . 72
6.1.3 No background, log-normal fading . . . . . . . . . . . . . . . . . . . . 74
6.2 Background radiation and no fading . . . . . . . . . . . . . . . . . . . . . . . 77
6.2.1 Background radiation, no fading . . . . . . . . . . . . . . . . . . . . . 80
6.2.2 Background radiation, Rayleigh fading . . . . . . . . . . . . . . . . . 82
6.2.3 Background radiation, log-normal fading . . . . . . . . . . . . . . . . 85
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7 Conclusions 88
List of Figures
2.1 FSO system block diagram. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Output light power vs. input drive current for all three most common light
sources [1]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Probability density functions for Rayleigh and log-normal distributions. . . . . 14
2.4 A typical FSO transceiver [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.5 Simplified detector circuit employing a p-i-n photodiode. . . . . . . . . . . . . 16
2.6 Simplified model of an APD and integrator. . . . . . . . . . . . . . . . . . . . 17
3.1 The MIMO concept. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2 Relative energy efficiencies vs.Q for w ∈ 1, bQ/2c. . . . . . . . . . . . . . 26
3.3 Norton equivalent noise model. . . . . . . . . . . . . . . . . . . . . . . . . . . 29
3.4 Poisson and Gaussian p.d.f.’s with equal means and variances ofλ = 200. . . . 31
3.5 Poisson and Gaussian p.d.f.’s with equal means and variances ofλ = 2. . . . . 32
5.1 Symbol error probability vs. average power with no fading and no background
radiation, andQ = 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
5.2 Symbol error probability vs. peak power with no fading and no background
radiation, andQ = 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
5.3 Symbol error probability vs. average power with Rayleigh fading, no back-
ground radiation, andQ = 8. . . . . . . . . . . . . . . . . . . . . . . . . . . 49
5.4 Symbol error probability vs. peak power with Rayleigh fading, no background
radiation, andQ = 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
5.5 Symbol error probability vs. average power with log-normal fading (S.I. = 1.0),
no background radiation, andQ = 8. . . . . . . . . . . . . . . . . . . . . . . 51
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5.6 Symbol error probability vs. peak power with log-normal fading (S.I. = 1.0), no
background radiation, andQ = 8. . . . . . . . . . . . . . . . . . . . . . . . . 51
5.7 An example of a definite error. . . . . . . . . . . . . . . . . . . . . . . . . . . 53
5.8 An example of an indefinite error (where the receiver chooses incorrectly from
the 3 possible modulator symbols). . . . . . . . . . . . . . . . . . . . . . . . 54
5.9 Symbol error probability vs. average power with no fading,PbTb = −170 dbJ,
andQ = 8 – dashed and solid lines overlap. . . . . . . . . . . . . . . . . . . . 56
5.10 Symbol error probability vs. peak power with no fading,PbTb = −170 dbJ, and
Q = 8 – dashed and solid lines overlap. . . . . . . . . . . . . . . . . . . . . . 56
5.11 Symbol error probability vs. average power with Rayleigh fading,PbTb =
−170 dbJ, andQ = 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
5.12 Symbol error probability vs. peak power with Rayleigh fading,PbTb = −170
dbJ, andQ = 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.13 Symbol error probability vs. average power with Log-normal fading, S.I. = 1.0,
PbTb = −170 dbJ, andQ = 8. . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.14 Symbol error probability vs. peak power with Log-normal fading, S.I. = 1.0,
PbTb = −170 dbJ, andQ = 8. . . . . . . . . . . . . . . . . . . . . . . . . . . 59
5.15 Symbol error probability vs. average power with no fading,Q = 8, R = 100
Ω, T0 = 290 K, PbTb = −170 dbJ, andRb = 100 Mbps. . . . . . . . . . . . . 63
5.16 Symbol error probability vs. peak power with no fading,Q = 8, R = 100 Ω,
T0 = 290 K, Pb = −90 dBW, andRb = 100 Mbps. . . . . . . . . . . . . . . . 64
5.17 Symbol error probability vs. average power with Rayleigh fading,Q = 8,
R = 100 Ω, T0 = 290 K, Pb = −90 dBW, andRb = 100 Mbps. . . . . . . . . 64
5.18 Symbol error probability vs. peak power with Rayleigh fading,Q = 8, R = 100
Ω, T0 = 290 K, Pb = −90 dBW, andRb = 100 Mbps. . . . . . . . . . . . . . 65
5.19 Symbol error probability vs. signal power with Rayleigh fading using equal
gain combining (EGC) or optimal gain combining (OGC).Q = 8, R = 100 Ω,
T0 = 290 K, Pb = −90 dBW, andRb = 100 Mbps. . . . . . . . . . . . . . . . 66
5.20 Symbol error probability vs. average power with log-normal fading,S.I. = 1.0,
Q = 8, R = 100 Ω, T0 = 290 K, Pb = −90 dBW, andRb = 100 Mbps. . . . . 66
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5.21 Symbol error probability vs. peak power with log-normal fading,S.I. = 1.0,
Q = 8, R = 100 Ω, T0 = 290 K, Pb = −90 dBW, andRb = 100 Mbps. . . . . 67
5.22 Symbol error probability vs. signal power with log-normal fading using equal
gain combining (EGC) or optimal gain combining (OGC).Q = 8, R = 100 Ω,
T0 = 290 K, Pb = −90 dBW, andRb = 100 Mbps. . . . . . . . . . . . . . . . 67
6.1 Ergodic capacity vs. average power with no fading, no background radiation,
andQ = 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
6.2 Ergodic capacity vs. peak power with no fading, no background radiation, and
Q = 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
6.3 50% outage probability vs. average power with no fading, no background radi-
ation, andQ = 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72
6.4 50% outage probability vs. peak power with Rayleigh fading, no background
radiation, andQ = 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6.5 Ergodic capacity vs. average power with Rayleigh fading, no background radi-
ation, andQ = 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
6.6 Ergodic capacity vs. peak power with Rayleigh fading, no background radia-
tion, andQ = 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6.7 50% outage probability vs. average power with Rayleigh fading, no background
radiation, andQ = 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
6.8 50% outage probability vs. peak power with Rayleigh fading, no background
radiation, andQ = 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6.9 Ergodic capacity vs. average power with log-normal fading, no background
radiation,Q = 8, andS.I. = 1.0. . . . . . . . . . . . . . . . . . . . . . . . . 76
6.10 Ergodic capacity vs. peak power with log-normal fading, no background radia-
tion, Q = 8, andS.I. = 1.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.11 50% outage probability vs. average power with log-normal fading, no back-
ground radiation, andQ = 8. . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6.12 50% outage probability vs. peak power with log-normal fading, no background
radiation, andQ = 8. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6.13 Ergodic capacity vs. average power with no fading,Q = 8, andPbT = −170
dBJ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
xi
6.14 Ergodic capacity vs. peak power with no fading,Q = 8, andPbT = −170 dBJ. 81
6.15 50% outage probability vs. average power with no fading,Q = 8, andPbT =
−170 dbJ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
6.16 50% outage probability vs. peak power with no fading,Q = 8, andPbT =
−170 dBJ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.17 Ergodic capacity vs. average power with Rayleigh fading,Q = 8, andPbT =
−170 dBJ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
6.18 Ergodic capacity vs. peak power with Rayleigh fading,Q = 8, andPbT =
−170 dBJ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.19 50% outage probability vs. average power with Rayleigh fading,Q = 8, and
PbT = −170 dbJ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
6.20 50% outage probability vs. peak power with Rayleigh fading,Q = 8, and
PbT = −170 dBJ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
6.21 Ergodic capacity vs. average power with log-normal fading,Q = 8, andPbT =
−170 dBJ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.22 Ergodic capacity vs. peak power with log-normal fading,Q = 8, andPbT =
−170 dBJ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.23 50% outage probability vs. average power with log-normal fading,Q = 8, and
PbT = −170 dbJ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
6.24 50% outage probability vs. peak power with log-normal fading,Q = 8, and
PbT = −170 dBJ. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
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List of Symbols
a Substitution foranm in integration.
A M ×N matrix representing the path gains from each receiver to each
transmitter.
anm One realization of the individual path gain from transmitterm to receiver
n; this quantity is squared to find power gain.
b, g, i, j, k, l Generic variables used to index summations and for other tasks.
C Information theoretic capacity of a channel, measured in bits per channel
use.
Cd Junction capacitance of a photodetector.
d0 Correlation distance, measured in meters.
Es Energy per modulator symbol contributed by the entire transmit array to
one receiver element.
f Frequency of transmitted signal. For a 1550 nm laser diode, this would be
approximately2× 1014 Hz.
F (M) Excess noise factor in an avalanche photodiode.
G Average gain of an avalanche photodiode.
h Planck’s constant, approximately equal to6.6× 10−34.
H(X) Entropy ofX, measured in bits.
L Link length, measured in meters.
m Denotes a single transmitter.m ∈ 1, 2, ..., MM Number of transmitters in the system.
n Denotes a single receiver.n ∈ 1, 2, ..., NN Number of receivers in the system.
P Peak ‘on’ power observed at the receiver.
Pave ‘On’ power observed at the receiver, averaged over the duration of the
modulator symbol.
Pb Background power observed at the receiver.
Pdef Probability of a ‘definite’ error; occurs when one or more ‘off’ slots have a
higher count rate than ‘on’ slots.
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Pindef Probability of an ‘indefinite’ error; occurs when one or more ‘off’ slots
have a count rate equal to one or more ‘on’ slots, with no ‘off’ slots having
a count rate greater than any ‘on’ slot.
Ppeak Also used to denote peak ‘on’ power at the receiver.
Q Number of slots per MPPM symbol.
Qion The set ofw ‘on’ slots for a MPPM symbol.
Qioff The set ofQ− w ‘off’ slots for a MPPM symbol.
Rb Bit rate of system.
S.I. Scintillation index. A measure of strength of fading in the log-normal
fading model.
T Duration of one slot of a pulse position modulation symbol, measured in
seconds.
Tb Duration of a single bit. Inverse is the bit rate.
Ts Duration of the entire pulse position modulation symbol, measured in
seconds.
w Number of ‘on’ slots per MPPM symbol.
X The estimate of the receiver; the modulator symbol chosen by the receiver
as the most probable symbol received.
Z A N ×Q matrix representing the received observations by allN detectors
over allQ slots.
Znq Observable at output of integrator at receivern at slotq.
β Probability of a deep fade detrimentally affecting a link path from
transmitterm to detectorn.
ζn Defined as(λon,n + λoff )/λoff
η Quantum efficiency of detector; the average number of electron-hole pairs
generated per incident photon
λ Wavelength of the transmitted signal; we have used 1550 nm for our
analysis.
λon Poisson parameter denoting average number of photoelectrons observed
during an ‘on’ slot, summed over allN photodetectors.
λon,n Poisson parameter denoting average number of photoelectrons observed
during an ‘on’ slot at a single detectorn.
xiv
λoff Poisson parameter denoting average number of photoelectrons
observed during an ‘off’ slot at a single detectorn.
µon Used for the Gaussian regime, as the average number of photoelectrons
observed during an ‘on’ slot, summed over allN photodetectors.
µon,n Used for the Gaussian regime, as the average number of photoelectrons
observed during an ‘on’ slot at detectorn.
µoff Used for the Gaussian regime, as the average number of photoelectrons
observed during an ‘off’ slot at detectorn.
µ2X Denotes the mean of the log-normal fading variable.
σ2 Used for the Gaussian regime, when thermal noise dominates, as the
variance of the number of photoelectrons observed during an ‘on’ or
‘off’ slot.
σ2on Used for the Gaussian regime, when thermal noise is not dominant, as
the variance of the number of photoelectrons observed during an
‘on’ slot.
σ2off Used for the Gaussian regime, when thermal noise is not dominant, as
the variance of the number of photoelectrons observed during an
‘off’ slot.
σ2X Denotes the variance of the log-normal fading variable.
τ0 Correlation time, measured in seconds.
φpdf (k, α) Probability that a Poisson random variable with parameterα equalsk.
φcdf (k, α) Probability that a Poisson random variable with parameterα is less than or
equal tok.
Chapter 1
Introduction
1.1 What is free-space optics?
Free-space optics is a method of communication that involves using a light source, usually
a laser, to transmit information through space or the atmosphere to a receiver. It is similar to
traditional fiber-optics communication, in that it uses light to communicate information, but
the difference is the medium through which the information travels. Fiber-optics, as the name
implies, uses a fiber to carry the light wave from transmitter to receiver. Free-space optics,
however, relies on a line-of-sight approach. As a result, the medium could be a near-vacuum,
as it is for satellite-to-satellite communication, or it could be the atmosphere, which includes
atmospheric and other natural obstructions that come into the path of the light.
1.2 The history of free-space optical communication
Although free-space optical (FSO) communication as we know it originated in the 1970s
[3], the history of optical communication through free-space using light really began when
signal fires were used to send messages across long distances. Paul Revere’s lanterns, as well
as manually operated lanterns on ships are examples of early FSO communication systems [4].
The first major technological advancement in FSO communication, that moved us past the
ages of signal fires, happened in 1880, when Alexander Graham Bell invented the photophone.
The device could send intensity-modulated sunlight over a distance of a few hundred feet. How-
1
2
ever, there were no major advancements in FSO communication for nearly a century, until the
laser was invented in the 1960s [4].
Within a few months of the invention of the laser, there was interest in using the technology
to communicate through the atmosphere. Bell Labs engineers brought an early ruby laser to the
top of a microwave tower at Murray Hill, NJ, and pointed it at a large screen 40 km away. A
colleague watched the screen for signs of the red pulses coming through, but not many pulses
made it. Although the results were poor, the engineers were able to make a spot as large as a
dining table glow like a fireplace at a distance of 2-6 km [5].
FSO communication, which also sometimes goes by the names “optical wireless communi-
cation,” “lasercom” (patented by McDonnell Douglas in the 1980’s), “wireless optical commu-
nication,” and “fiber-free optical communication,” really started to take off in the 1980s. Fund-
ing was increased in the United States and Europe as governments tried to plan for next gen-
eration communication technologies for air-to-air, satellite-to-submarine, air-to-satellite, and
satellite-to-satellite links [6].
There was also an interest for non-military communication, as well, but after the first few
trials of FSO communication, interest began to decline. One reason is that communicating
through optical fibers seemed so far superior to communicating through the atmosphere [5].
A lack of reliable components was also always an issue with early optical systems. Almost
every single component had to be developed, including laser sources, detectors, high-speed
electronics, and high accuracy pointing components to name a few. Because of these technical
difficulties, as well as financial and political issues as well, one-by-one, almost all early FSO
communications programs met with an early demise [6].
However, in the past few years, mainly due to a need to solve the “last-mile” problem,
an increased interest has been seen in non-military uses of FSO communication [5]. In fact,
Acampora cites a study that predicts that the FSO communications industry could grow from
an annual 120 million dollars in 2000 to more than 2 billion dollars annually by 2006 [3].
3
1.3 Uses of free-space optics
1.3.1 Military communication
Free-space optics is an attractive method of communication for military applications. The
reason is security. Using traditional radio frequency (RF) communications methods, eaves-
dropping on a conversation is much easier than with free-space optics, since the RF waves are
transmitted over a large area. This makes it possible to receive the signal while in the vicinity
of the system, although it is still necessary to demodulate and decode it. A FSO link, on the
other hand, has a very narrow beam divergence [4], typically milliradians, so the only way to
intercept the signal is to be in the path of transmission.
1.3.2 Satellite and deep-space communication
Free-space optics has several advantages that make it well suited to satellite communication.
First, it can provide high data-rate communication links between satellites at geosynchronous
distances and beyond. Second, it has several attributes that are superior to traditional RF com-
munication methods.
Traditionally, satellite communication has been accomplished using microwaves, but these
systems are bulky and expensive [4]. Free-space optics has the advantage of being much smaller
and far more inexpensive, which is an enormous asset to any space vehicle. For this reason,
NASA is developing a deep-space optical communication transceiver in its X2000 program,
also known as the Advanced Deep-Space Systems Development Program [7]. Early in the
X2000 program, NASA plans to support tens of kilobits per second of data from the Mars range
[8]. It also plans on building the first of two 10 m class ground receiving telescopes by 2008
[7].
The International Space Station (ISS) Engineering Research and Technology Development
program is sponsoring the development of a high data rate FSO transmitter from the low-earth-
orbit range (on board the ISS) [7], that is projected to be able to support a data rate of 2.5 Gbps.
FSO communications appears to be the technology that will meet the needs of future space
ventures, including near-earth, solar-system, and interstellar missions [8].
4
1.3.3 The last-mile solution
Probably one of the most compelling and timely uses of free-space optics is to provide a
solution to the “last-mile” problem (or “first-mile” problem, [9] depending on your perspective).
The multibillion-dollar optical fiber backbone that was built to provide high-speed broadband
access to offices and homes has come up less than one mile short for 9 of 10 US businesses
with 100 or fewer workers. As a result, only 2-5% of the fiber network is actually being used
today [3]. Most businesses and homes are currently connected to the fiber backbone using
traditional copper wires, which do not possess the gigabit-per-second capacity required to carry
bandwidth-intensive applications [3].
Laying fiber optic cable to each home and business that needs broadband access would be
the ideal solution to this problem, but it is slow and expensive. The process can take 6-12
months, and can cost anywhere from $100,000 to $500,000 per mile[3], with up to 85% of the
cost due to trenching and installation [2]. Free-space optics, on the other hand, can be up and
running in a few days, and costs 1/3 to 1/10 of the cost of a fiber installation [3]. Trenching
also causes traffic jams, displaces trees, and can destroy historical areas. For these reasons,
Washington D.C. is considering a moratorium on fiber trenching [2].
FSO communication is thought by many experts to have the best chance at succeeding over
other fiber-free technologies (like DSL, microwave radio, etc.) at bridging the last-mile gap
[3]. In addition to the cost and speed benefits, it has a greater potential because it operates
in an unlicensed band (which is an enormous cost benefit as well), it’s scalable (unlike RF
networks [9]), and it can be set up in a mesh configuration to carry full duplex gigabit-per-
second communications around a city town or region [3].
One challenge that faces FSO communications systems is building sway. Because of the
very narrow beamwidths possible from the lasers, very small changes in the position of either
the transmitter or receiver can cause the laser beam to miss its target. The two possible solutions
to this problem are to increase the beam divergence at the receiver (which also reduces the power
density) or, for especially tall buildings or narrow beamwidths, to use an active tracking system;
mirrors continually adjust to keep the beam centered on the target.
The biggest challenge facing free-space optics in terrestrial applications, which is also a
major focus of this thesis, is overcoming limitations caused by the atmosphere. Bad weather,
especially thick fog, can severely attenuate the signal before it reaches the receiver [3]. In fact,
5
weather is the reason links in non-desert regions are often kept to 200-500m to ensure carrier-
class availability (99.999% availability) [2].
In this thesis we analyze an approach called spatial diversity which attempts to overcome
these atmospheric difficulties. This approach employs multiple transmitters and/or receivers
simultaneously sending and/or receiving the information. The idea is to keep the transmitters
and receivers sufficiently far from one another (which is a surprisingly small distance, as we
will see), such that all of the individual paths from transmitters to receivers would have to be
simultaneously faded (a much lower probability event) in order to degrade system performance.
Chapter 2
System overview
In this section, we will describe the free-space optical link from beginning to end. The
physical devices, channel, and noise and other disturbances will be considered, however in this
section we are most concerned with the hardware used to construct such a system, and a better
understanding of the physical phenomena that affect both the hardware and the channel to make
them non-ideal.
An overall block diagram depicting the FSO system is shown below in Figure 2.1.
Figure 2.1: FSO system block diagram.
6
7
2.1 Information
For the purpose of this paper, we are going to assume that the information to be transmitted
is already in binary format. This could be any kind of data including, but not limited to, Internet
and intranet traffic, multimedia applications (including streaming audio and video), and file
transfers or data exchanges of any kind.
2.2 Source and channel encoders
From an information-theoretic point of view, the raw information described in Section 2.1
contains natural redundancies that can be removed to make the system more efficient. This is
known as compression or source coding. The channel encoder then adds “intelligent” redun-
dancies back into the stream of data that make the system more resistant to errors. More detail
on these topics can be found in texts by Wilson [10] and Cover and Thomas [11]. Upper limits
on the performance of channel coding will be investigated in Chapter 6.
2.3 Multiple pulse position modulator
For this system, we are focusing on multiple pulse position modulation (MPPM), which is
a intensity modulation technique. This is, of course, not the only possible way to build this
system. Since a laser is, in effect, an optical oscillator [5], any modulation that is possible
with RF communication is also possible with optical communication (including coherent mod-
ulation/demodulation techniques). The drawback, however, comes from the fact that coherent
techniques like phase or frequency modulation are far more complex and expensive to build
[12].
The analysis and a detailed description of MPPM will be completed in Section 3.1.2. Until
then, it suffices to say that the modulator can take a certain number of bits (from the channel
encoder), and map them to a single MPPM symbol that will be sent across the channel by the
transmitter.
8
2.4 The transmitter
In Figure 2.1, the transmitter and modulator are depicted as being separate entities, but there
are actually different ways to construct this. It is possible for the transmitter to be constantly on,
and then be modulated as it is passed on to the channel, or the laser can be directly modulated
in one step.
Here we consider the transmitter to be the light source, which simply has the task of sending
light over the channel.
There are three different types of light sources that are commonly used in free-space optics:
• Light Emitting Diode (LED): LED’s can produce light in the 800-900 nm band, they are
cheap, and they can produce radiation with low current drive levels. However, they have
limited output powers (1-10 mW), there is more frequency spreading than the other light
sources, and the light tends to be incoherent and unfocused [1].
• Laser: Lasers have power outputs of 0.1-1 W, but are much bulkier than LED’s. The
laser is an optical cavity filled with light amplification material and mirrored facets at
each end. When the cavity “lases,” an initiated optical field crosses back and forth in a
self-sustaining reaction. A small aperture in one of the mirrored facets allows some of the
energy to escape as radiated light. In the linear range of operation (see Figure 2.2), lasers
are unstable, so they are usually operated as continuous-wave devices at peak power [1].
• Laser Diode: Like LED’s, laser diodes are semiconductor junction devices [1], but they
operate more like lasers with reflecting etched substrates which act like small reflectors
(like the reflectors in the laser). Laser diodes are small, rugged, and very power-efficient.
They require more drive current than LED’s, but also generate more power. A laser diode
produces about a hundred milliwatts of useable optical power [4] with a more focused
beam than with LED’s [13].
All of the three light sources have the same output power characteristics, shown in Figure
2.2. From this, one can see a distinct linear region of operation, where an increase in input
current would result in a proportional increase in output light power.
The wavelength chosen for FSO systems usually falls near one of two wavelengths, 850 nm
or 1550 nm. The shorter of the two wavelengths is cheaper and is favored for shorter distances.
9
Figure 2.2: Output light power vs. input drive current for all three most common light sources
[1].
The 1550 nm light source is favored for longer distances since it has an allowed power that
is two orders of magnitude higher than at 850 nm [2]. These power limits are determined by
the American National Standards Institute (ANSI) Z136.1 Safety Standard [9]. The reason for
the higher allowed power is that laser-tissue interaction is very dependent on wavelength. The
cornea and lens are transparent to visible wavelengths (such as 850 nm) so the power can reach
the retina at the back of the eye. At 1550 nm retinal absorption is much lower, since the power
is absorbed mostly by the lens and cornea before it can reach the retina. The power at 1550 nm
is not unlimited, however, since it can still cause photo-keratitis and cataracts at higher levels
[14].
The 1550 nm wavelength is also preferred since more photons per watt of power arrive for
longer wavelengths, and therefore more photocurrent is produced per watt of incident power for
equal efficiency devices[4].
For the remainder of this thesis, we will assume the laser to be an ideal, infinite bandwidth
light source. For the simulations in Chapters 5, 6, and 5.2 we will make use of the 1550 nm
wavelength.
2.5 The channel
In a terrestrial free-space optical link, the channel is simply the atmosphere plus any other
disturbances through which the optical signal will pass. This is a very important component of
10
our system, since the channel is often the limiting factor for how long the link can be.
The atmospheric channel is uncontrolled in that the designers have no way of preventing
obstructions and other disturbances from coming between the transmitter and receiver. The
engineer will attempt put the system in a location where it is unlikely for obstructions to occur,
but it is always possible for a bird, for example, to temporarily pass through the beam. However,
in a packet-switched network, short duration interruptions are easily handled by retransmitting
the data [9].
A more serious threat is the atmosphere itself. Zhu and Kahn classify atmospheric effects on
the FSO channel into two categories, atmospheric turbulence and aerosol scattering [12]. These
are discussed further below.
2.5.1 Turbulence
Atmospheric turbulence is also known as scintillation. Even on a clear day, there are con-
tinual variations in the intensity of the light at the receiver due to inhomogeneities in the tem-
perature and pressure of the atmosphere. The Kolmogorov turbulence model is often used to
describe atmospheric turbulence [12, 15] and predicts that changes in the air temperature as
small as 1 degree Kelvin can cause refractive index changes as large as several parts per mil-
lion [15]. These pockets of air with different refractive indices, or eddies, act like time-varying
prisms [2] whose size ranges from a few millimeters to a few meters [12], and whose time scale
is related to wind speed [16] among other things.
These eddies cause the light to diffract along the path to the receiver in a time-varying
manner, affecting the intensity of the light. This phenomenon is visible to the naked eye by
watching the stars twinkle at night, or by watching the horizon shimmer on a hot day [2].
The effect of scintillation on a FSO communications link can be a wandering beam when
the eddies are bigger than the beam diameter and move the beam completely off target [2],
fluctuating power at the receiver [9], and changes in the phase of the received light wave [12].
For weak turbulence, the intensity of the received signal is a random variable best approximated
by a log-normal distribution [13, 16]. This model is described further in Section 2.5.3.
To describe turbulence-induced fading, we do so using parameters in the spatial and tem-
poral domains. The first useful parameter is the correlation length, which we calld0. This is
simply the distance for which the intensity of a light wave at two points in the atmosphere is
11
essentially uncorrelated. This distance can be approximated byd0 ≈√
λL, whereλ is the wave-
length of the transmitted wave, andL is the length of the FSO link [13]. This approximation
is valid for most FSO communication systems using visible or infrared lasers for link lengths
ranging from a few hundred meters to a few kilometers [12], and is approximately 1-10 cm for
most terrestrial links [13]. The importance of correlation distance will become evident as we
talk about spatial diversity as a method of mitigating the effect of turbulence on FSO links.
The second useful parameter is the correlation time, which we callτ0. When observing a
single point in the atmosphere at two different times,τ0 represents the amount of time between
observations for which the atmospheric parameters are uncorrelated. The time scale for scintil-
lation is about the time it takes a volume of air the size of the beam to move across the path, and
is therefore related to wind speed [16]. Typical values for terrestrial links are 1-10 ms [13].
Correlation time is important to our discussion in order to justify spatial diversity as a
method of mitigating block fading. At the transmission rates desirable for a FSO system
(2.5 Gbps for example), a deep fade that could last 1-10 ms could potentially affect 2.5 to
25 megabits of data. The normal approach to counteract block fades is to interleave the data
before coding, but this is an unattractive solution due to the enormous size of the interleaver that
would be necessary to be effective [15]. Spatial diversity, which will be discussed in Section
3.1.1 is a method that avoids the need for such large interleavers.
2.5.2 Aerosol scattering
The most detrimental atmospheric phenomenon that affects FSO links is fog, which is clas-
sified as aerosol scattering. According to Acampora, susceptibility to fog has slowed the com-
mercial development of free-space optics, since it so severely limits the range of a FSO link
[3].
The exact amount of signal attenuation caused by fog varies with its density. Acampora
states that the link might lose 90% of its power for every 50 meters in moderately dense fog
[3]. This translates into a loss of 200 dB/km. Other sources give ranges in attenuation from 16
dB/km in light fog [9] to 300 dB/km in dense fog [17].
There are various ways to combat link fade due to fog. One such method is to simply
increase the power, also known as increasing the link margin. The link margin is simply extra
transmit power that is in excess of what is normally needed to communicate. The only problem
12
with increasing the link margin is that power levels are limited for any system, both because
of eye safety as well as practical limitations in the system itself. For moderately dense fog,
increasing the link power by a large amount, 60 dB (a factor of one million) for example, would
still only allow for an extra 300 m in link length.
Fading can also be mitigated by making the link length as small as possible. Longer links
can be accommodated by arranging the transmitters and receivers in a mesh-topology. In an
urban setting, the mesh could jump from building-to-building or house-to-house, so that the
signal propagates only over shorter distances and has multiple paths to reach any point in the
network [3].
The wavelength of the link also affects the link’s susceptibility to fog. Future FSO systems
will most likely take advantage of the long wavelength infrared range (LWIR) spectrum (8µm <
λ < 14µm), also known as the night-vision spectrum. LWIR systems are called all-weather
systems because they are 10-20 times less sensitive to fog, rain, smog, and other atmospheric
disturbances. These wavelengths are also far less dangerous to eye safety so allowable power
levels are higher than those for the 0.7-1.55µm range. [18].
Point-to-point microwave radio is an alternative to free-space optics that is immune to fog.
However, this technology requires spectrum licensing, which is a major disadvantage when
compared to FSO systems [3].
Effectively overcoming challenges imposed by foggy weather for any particular FSO link
would most likely involve a combination of the aforementioned solutions. Using spatial diver-
sity to combat atmospheric effects, which is the focus of this thesis, can be incorporated into
almost any well-designed system that also uses link margin, a mesh topology, and LWIR lasers.
The effect of also using spatial diversity is to introduce yet one more weapon in the arsenal of
the communications engineer to combat link fading.
2.5.3 Fading models
There are two widely used models for fading. One is the log-normal distribution, the other
is the Rayleigh distribution. In all of the fading cases, we keep the expected path gainE[A2]
equal to one, in order to make fair comparisons between them and the non-fading case.
The log-normal distribution is very often used in the literature to describe atmospheric tur-
bulence as experienced in FSO systems. In the log-normal distribution, the amplitude of the
13
path gain is a random variableA whereA = eX andX is normal with meanµX and vari-
anceσ2X . By definition, the logarithm ofA follows a normal distribution. The optical intensity
I = A2 is also log-normally distributed.
The p.d.f. forA is
fA(a) =1
(2πσ2X)1/2a
exp(−(loge a− µX)2/2σ2X), a > 0 (2.1)
In order to keep the mean path intensity unity, i.e.E[A2] = 1, it can be shown thatµX =
−σ2X . For the log-normal distribution, we make use of a parameter called the scintillation index,
defined as
S.I. =E[A4]
E2[A2]− 1 (2.2)
This quantity is proportional to the degree of fading, as seen in Figure 2.3, and can be related
to the varianceσ2X by S.I. = e4σ2
X − 1. Typical values appearing in the literature for S.I. are in
the range 0.4-1.0.
They Rayleigh distribution is used less often in the literature than log-normal fading to
analyze FSO systems, but has some nice properties that make it an attractive model to use. First
of all, the Rayleigh fading case exhibits deeper fading than log-normal fading because of the
higher concentration of low-amplitude path amplitudes (see Figure 2.3). Second, with Rayleigh
fading, the diversity order of the MIMO system becomes apparent when analyzing the slopes
of the curves for symbol error probability.
In Rayleigh fading, the amplitude of the path gain follows a Rayleigh distribution. The
wavelength of the light is modeled to be large compared with the size of the scatterer, such that
the composite field is produced by a large number of non-dominating scatterers, each contribut-
ing random optical phase upon arrival at the detector. The central limit theorem then gives a
complex Gaussian field, whose amplitude is Rayleigh:
fA(a) = 2ae−a2
, a > 0 (2.3)
where we have normalized so thatE[A2] = 1. The random intensityI = A2 is a one-sided
exponential random variable, whose density function is heavily concentrated at low (deeply
faded) values. 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, as shown in
Figure 2.3 [19].
14
0 0.5 1 1.5 2 2.5 3 3.50
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
a
f A(a
)
Log−normal fading, S.I. = 0.4Log−normal fading, S.I. = 0.6Log−normal fading, S.I. = 1.0Rayleigh fading
Figure 2.3: Probability density functions for Rayleigh and log-normal distributions.
We will make use of both of these fading models in Chapters 5 and 6 when we discuss the
effect of fading on error probability and channel capacity.
2.6 The receiver
Once the transmitted signal passes through the atmosphere, it must be collected and mea-
sured by the receiver. As we mentioned before, both coherent and noncoherent detection
schemes are possible, but for complexity and cost reasons noncoherent (or direct detection)
is preferred. For this thesis, we will only consider noncoherent systems.
In noncoherent optical signal detection, the detectors rely on the photoelectric effect – inci-
dent photons are absorbed by the detector and free-carriers are generated and can be measured.
This is a probabilistic phenomenon, since it is possible for a photon to pass through the pho-
todetector without generating any free-carriers. However, in a well-designed photodetector, the
probability of an incident photon causing a free-carrier is high [4].
There are two models that we will use in our analysis of the system. In the ideal photon
counting model, we assume no thermal noise is present, and the system is capable of counting
current ‘blips’ that occur as each photoelectron is produced. Integrating the photocurrent over
15
a certain period of time (called a ‘slot’) is equivalent to counting the current ‘blips.
In the Gaussian model, we assume that zero mean, additive white Gaussian noise (AWGN)
is added to the generated photocurrent. We still integrate over a slot, and the integration process
should, on average, remove the noise power from the observable.
Figure 2.4 shows what a FSO transceiver (receiver and transmitter in one unit) might look
like, showing other components also present in many FSO systems.
Figure 2.4: A typical FSO transceiver [2].
According to Alexander, photodetectors fall into one of four categories: photomultipliers,
photoconductors, photodiodes, and avalanche photodiodes [4]. There are numerous configura-
tions and variations in these four categories, so we will concentrate on the two most popular
detectors in optical receivers for communication, p-i-n photodiodes and avalanche photodiodes.
2.6.1 p-i-n photodiodes
A p-i-n photodiode is made up of a p-type, and an n-type layer of semiconductor, separated
by an intrinsic layer (hence the name p-i-n photodiode). The p-type layer is made to be very thin,
so incident photons can pass directly through to the intrinsic region where they can generate
electron-hole pairs. Any pairs that are generated are quickly swept into the p- and n-type layers
where they contribute to the photocurrent.
16
The p-i-n photodiode has a quantum efficiency associated with it that depends on the re-
flectivity of the p-type layer, the absorption length of the intrinsic region, and the length of
the depletion region [4]. The quantum efficiency is often denoted byη and is a measure of
the average number of electron-hole pairs generated per incident photon. In a practical p-i-n
photodiode,η ranges from 0.3 to 0.95 [20].
A simplified model of the p-i-n photodiode with its biasing voltage and integrator is shown
in Figure 2.5 below.
Figure 2.5: Simplified detector circuit employing a p-i-n photodiode.
2.6.2 Avalanche photodiodes
An avalanche photodiode (APD) is constructed similarly to a p-i-n photodiode. In some
models, there is a second p-type layer between the intrinsic layer and the n-type layer (the layers
are p-i-p-n). Incident photons still generate electron-hole pairs, but now there is an “avalanche”
effect – each free electron and/or hole has the potential to create more free electrons and/or
holes as it traverses the gain region (the extra p-type layer and part of the n-type layer). Each
newly created electron or hole can then repeat the process until all carriers have exited the gain
region.
This avalanche process creates multiple carriers for every incident photon. This increase in
the number of carriers is known as the APD gain. It is the ratio of observable photocurrent at the
APD terminals to the internal photocurrent before multiplication [4], and is a random variable
with meanG. A simplified model of an APD is shown in Figure 2.6.
17
Figure 2.6: Simplified model of an APD and integrator.
2.6.3 Bandwidth and noise considerations in p-i-n and APD receivers
2.6.3.1 Bandwidth
An ideal photodetector would be noiseless and have an infinite bandwidth. An actual pho-
todetector, however, has neither of these attributes due to the physical qualities of the photode-
tector itself and the accompanying electronics that take part in the detection process.
A photodetector has a junction capacitanceCd associated with it that is proportional to the
aperture size. Increasing the capacitance of the detector also increases the time-constant which
lowers the bandwidth. Therefore there is a tradeoff between increasing the field-of-view (FOV)
for a detector and the detector’s bandwidth.
The time-constant, and consequently the bandwidth is also dependent on the load resistor
(R in Figures 2.5 and 2.6). Kedar and Arnon give an estimate for the data rate based on these
parameters [21]
Rb ≈ 1
2πRCd
(2.4)
which is simply the inverse of the time constant converted to frequency in Hz. Clearly, increas-
ing the capacitance (by increasing the aperture size) or increasing the resistance decreases the
data rate.
2.6.3.2 Shot noise
There are also many sources of noise that must be considered when doing analysis on pho-
todetection circuits. The first source we consider is optical shot noise, which occurs because
18
of the randomness of the creation of photoelectrons. We adopt the semi-classical view of pho-
todetection, in which light arrives as a wave, and produces a stream of photoelectrons from
the detector. The number of photoelectrons produced during a slot time can be described by
a Poisson random variable with a mean and varianceλ. This variance in the generation of
photoelectrons can be seen as noise, which can affect the probability of error.
2.6.3.3 Background noise
In a terrestrial free-space optical system, there is also a very good chance that background
noise will enter the receiver along with the signal. Sources of background noise include the sun
and artificial lighting, and can enter the receiver’s aperture directly or by reflecting off of other
surfaces. One way of minimizing background noise is by blocking other sources of radiation so
that light can enter the receiver only from approximately the direction of the transmitter.
Another method for minimizing background noise is to use a frequency selective filter in
front of the receiver to only allow a narrow band around the center frequency of the laser. The
potential drawback to this method is that it reduces the strength of the received signal.
2.6.3.4 Thermal noise
Thermal noise is also called Johnson noise, and is a result of thermally induced random
fluctuations in the charge carriers in a resistive element. Thermal noise is technically present in
any semiconductor where thermally induced charge carriers can be present, which even includes
the photodetector itself, but it is only significant in the load resistor (see Figures 2.5 and 2.6)
whose resistance is higher than the other sources.
Both the p-i-n photodiode and APD are affected by thermal noise, but it is more detrimental
for the p-i-n photodiode. The APD has internal amplification that can be seen as a low-noise
amplifier, whereas the p-i-n photodiode relies completely on the circuitry for amplification.
2.6.3.5 Excess APD noise
The APD has the advantage of internal multiplication to raise the overall SNR, but it does so
in spite of the excess noise factorF (M), caused by the random nature of the gain mechanism.
F (M) depends on the semiconductor material, the average gain of the APD, the ratio of the
ionization coefficients for electrons and holes, and is largest in devices where both holes and
19
electrons produce ionizing collisions [4]. The SNR after the avalanche process for the APD is
reduced by multiplying the SNR before amplification byF (M)−1.
2.7 MPPM demodulator
The MPPM demodulator has the task of taking the electronic signals delivered by the re-
ceiver and deciding which of the MPPM symbols was sent. This is a non-trivial task, and the
maximum likelihood decision metric is investigated further in Chapter 4.
2.8 Source decoder, channel decoder, and retrieved informa-
tion
Once the MPPM demodulator has decided which MPPM symbol was sent, the source and
channel decoders perform the inverse operations of the source and channel encoders. If the
symbol was chosen correctly, the retrieved information matches the input information to the
system. If the symbol was chosen incorrectly, the error may be detected and possibly even
corrected (depending on the error and the coding scheme). It is also possible that the error could
remain undetected and the retrieved information would not exactly match input information, i.e.
bit errors would occur. These aspects are studied in detail in Section 5.
Chapter 3
MIMO applied to FSO systems using
MPPM - background, system model and
definitions
In this chapter we look at the application of multiple-input, multiple-output (MIMO) tech-
niques to the FSO system using multiple pulse position modulation (MPPM).
3.1 Research on MIMO and MPPM FSO communications
systems
3.1.1 MIMO in wireless systems
Multiple Input Multiple Output (MIMO) systems have recently emerged as one of the most
significant breakthroughs in modern communications. The idea behind MIMO systems can be
explained quite simply. At both the transmitter and receiver end, the system employs multiple
antennas. The effect of MIMO approaches is that the signals at the transmitter and receiver can
be combined such that the bit error rate or the data rate (in bits/sec) is improved. In the wireless
RF domain this is done at no extra cost of spectrum – only added hardware and complexity [22].
The MIMO concept is depicted in Figure 3.1.
The advantage from the MIMO setup can be utilized through two different concepts, spatial
multiplexing and spatial diversity.
20
21
Figure 3.1: The MIMO concept.
In spatial multiplexing, such as with the BLAST technique, the incoming high-rate data is
decomposed intoM independent data streams, and sent to allM antennas to be transmitted
simultaneously over the channel. The receiver array, having learned the mixing channel matrix
through training sequences, can identify each of the individual data streams and recombine
them to retrieve the original message. The result is that the spectral efficiency improves; the
transmitter array can send at a new data rateM times faster than with a single antenna [22].
The second beneficial concept is spatial diversity. In MIMO systems, theMN path gains
from each transmitter to each receiver can be described in aM × N matrix form (see Figure
3.1). If the antennas are situated far enough apart, the paths can be considered decorrelated
and the effects of random fading caused by multipath or other phenomena can be mitigated.
The improvement of a MIMO system is directly related to the number of decorrelated antenna
elements, also known as thediversity order, whose maximum isMN [22].
There has been much research into the performance advantages of MIMO systems. The
consensus is that MIMO design can increase the capacity and decrease the bit error rate over a
single input, single output (SISO), MISO, or SIMO system with a given power and bandwidth.
The results of this research have focused heavily on RF systems, and the reader is directed
to [11] or [22] (among many possibilites) for more detail.
22
3.1.1.1 Application of MIMO concepts to free-space optics
Free-space optical communications systems can also benefit from spatial diversity, as has
been shown in [15, 23–25] . Although a well-designed FSO link will not suffer from traditional
multipath effects (except for diffuse FSO systems), atmospheric fading (the most serious prob-
lem facing FSO systems) can be mitigated through MIMO design. The key is to place the lasers
and photodetectors sufficiently far from one another to ensure with a high degree of probability
that each of theMN path gains are independent.
The distance that each laser or photodetector should be placed away from one another can
be calculated from the correlation distanced0 (see Section 2.5.1). This is the distance for which
two points in the atmosphere are uncorrelated and can be approximated byd0 ≈√
λL, whereλ
is the wavelength of the transmitted wave, andL is the length of the FSO link [13].
As an example, if the length of the FSO link is 1 km and the wavelength is 1550 nm, the
correlation distance would be approximately 4 cm. Therefore, keeping the lasers and photode-
tectors separated by at least 4 cm would ensure with a high degree of probability that theMN
path gains are independent. This small of a separation distance is perhaps surprising – but
it is small enough for MIMO to be considered for terrestrial FSO links, where the laser and
photodetector arrays would be placed on rooftops or even behind windows.
The result of keeping theMN path gains independent is seen by considering the probability
that a path gainanm is sufficiently small, such that the signal falls behind the background level.
If we call the probability of this eventβ, then the probability of a deep fade detrimentally
affecting all of the paths for one realization of the path gain matrixA is βMN . Therefore, the
system has the potential to achieve the diversity orderMN .
3.1.2 Multiple pulse position modulation (MPPM)
The capacity and performance of PPM and MPPM has been studied in detail in [26–33].
We will first investigate the attributes of PPM signaling, and then transition to MPPM.
In pulse position modulation (PPM), an observable time period (the duration of a modulator
symbol,T ) is divided up into slots each having a duration ofTs. A symbol is represented by
sending a pulse in only one of the time slots. If there areQ slots in a symbol, then there are
consequentlyQ possible symbols, each representing up tolog2 Q bits of information. PPM is a
23
Q-ary orthogonal signaling scheme.
To increase the throughput of a PPM system, it is necessary to increaseQ, which decreases
the pulsewidth [28]. This is an attractive solution for a number of reasons. First, decreasing the
width of the slot also decreases the number of background photons that will be received [29],
since we are assuming the background count rate remains the same. Second, the probability of
symbol error for noncoherent detection of M-ary orthogonal signals decreases for an increasing
number of symbols for a fixed energy per bit,Eb [10].
Decreasing the slot width, however, has its limitations. Namely, an increase in the required
bandwidth, implying more thermal noise. For high data rate applications, MPPM is a more
attractive alternative [34]. Multipulse transmission works with the same concept as PPM, but
instead of having only one ‘on’ slot, there arew of Q slots that can be on for each modulator
symbol, giving(
Qw
)possible symbols. The bandwidth efficiency, defined as the number of bits
that can be transmitted per is superior in MPPM, and as we will see later, the MPPM system
has an improved performance when the system is peak-power-limited.
This was studied in detail by Atkin and Fung in [33]. In their analysis, they compared
different schemes with similar bandwidths, and found that MPPM can outperform standard
PPM in coded and uncoded systems. Our analysis differs from theirs in a few ways. We allow
all(
Qw
)symbols to remain in the set, whereas they would limit the symbol set size to a power
of two, which is a logical limitation to place on the modulation scheme. The result is that their
error analysis was limited to an upper bound on error probability. By allowing all(
Qw
)symbols
to remain in the set, we preserve symmetry in the problem, and can often obtain closed form
expressions for error probability. We also consider MIMO techniques overlayed with MPPM,
whereas one of their focuses was on Reed-Solomon coding of a MPPM system.
3.1.3 Error probability - Gaussian vs. Poisson
In optical detection the Gaussian approximation is often used to analyze systems that em-
ploy APD’s as detectors [35, 36]. The properties of APD’s and the accuracy of the Gaussian
approximation were established in the early 1970’s in work by McIntyre, Conradi, and Webb
[37–39], and the approximation is used to take many factors specific to APD’s into consider-
ation. However, we are concerned with detection using p-i-n photodiodes, and details of the
operation of APD’s is beyond the scope of this thesis.
24
Instead, when we speak of the Gaussian approximation, we are either interested in approx-
imating the Poisson point process at the output of the photodetector when the signal and back-
ground power levels are both large, or when additive white Gaussian noise (thermal noise) is
introduced by the receiver.
3.2 Power comparisons between PPM and MPPM
Care must be taken to fairly compare system performance asw varies. The reason for this is
that increasingw with a constant peak signaling power (the ‘on’ power during a slot) increases
the average power consumption for the system. Also, increasingw while keeping the symbol
durationTs constant would increase the bit rate.
To address the first concern, we will need to make a distinction between peak-power-limited
systems, and average-power-limited systems. In a peak-power-limited system, the signal power
in an ‘on’ slot is limited toPpeak, regardless of how many ‘on’ slots there are. This means the
total received optical energy per symbol is equal to the peak power multiplied by the duration
of the ‘on’ slots.
Es = PpeakTw (3.1)
In an average-power-limited system, the average power in allQ of the slots (both ‘on’ and
‘off’) observed over the duration of the symbol must remain constant, and the optical energy
per symbol is equal to
Es = PaveTQ (3.2)
Therefore, combining (3.1) and (3.2), we can state that the relationship between average
power and peak power is
Pave = Ppeakw
Q(3.3)
Pave is easily recognized asPpeak times the duty cycle of a symbol.
To consider the second concern, we observe that the bit rate of the system is related tow.
The bit timeTb multiplied by the number of bits per modulator symbol is equal to the symbol
time,
Ts = Tb log2
(Q
w
)(3.4)
25
and the slot time is therefore
T =Tb log2
(Qw
)
Q(3.5)
We can address both of these concerns by defining a general optical energy parameterPT
with which to compare systems (used in plots in Chapters 5 and 6), whereP is the signal-
ing power andT is the slot duration. Peak and average power are handled by the following
conversions:
PT = Pave
Tb log2
(Qw
)
wor PT = Ppeak
Tb log2
(Qw
)
Q(3.6)
For a better understanding of the effect of average or peak power limitations on system
performance, we can define a relative energy efficiency for equal asymptotic performance, based
on (3.6) as the multiplier onPTb:
ρave =log2
(Qw
)
wbits/slot (3.7)
ρpeak =log2
(Qw
)
Qbits/slot (3.8)
which shows that for a given(Q,w) pair, multipulse is more efficient than single pulse in the
peak-power-limited system. In fact, for the peak-power-limited system, the relative energy
efficiency is at a maximum whenw = bQ/2c. The efficiencies as a function ofQ are shown
below in Figure 3.2.
It is important to be careful in interpreting this plot. The first thing that the plots reveal is
that for a givenPTb, the average-power-limited system will always outperform the peak-power-
limited system. The interpretation of this goes back to (3.3). Notice that if we are givenP , and
interpret it as average power, the corresponding peak would beQ/w times greater than ifP
were interpreted as peak power.
Also interesting to note is that for the peak-power-limited system, the MPPM system has
a superior relative energy efficiency, and in an average-power-limited system, standard PPM is
superior.
It is also interesting to note from this plot that the efficiency is unbounded for the average-
power-limited system wherew = 1 (unlike all of the other curves). From a logical standpoint,
this makes sense; asQ increases, the efficiency increases at a rate oflog2 Q. However, what
26
0 5 10 15 20 25 300
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
Q
Rel
ativ
e E
nerg
y E
ffici
ency
ρave
, w=1ρ
ave, w=Q/2
ρpeak
, w=Q/2ρ
peak, w=1
(Q,w) = (8,1)
(Q,w) = (8,4)
3.00
1.53
0.766
0.375
Figure 3.2: Relative energy efficiencies vs.Q for w ∈ 1, bQ/2c.
is not seen from this plot is that for the average power to remain constant asQ becomes large,
the peak power must also become large. Another way to think of this is that all of the power is
concentrated in a single slot – the ‘on’ slot is constantly becoming narrower asQ grows, which
squeezes the peak power (as well as the bandwidth) up toward some large value. Because of
power and bandwidth limitations in any real system, the advantages of this phenomenon become
impractical to implement for large values of Q. The zig-zag trajectories are due to the fact that
w = bQ/2c only changes at even values ofQ.
Another important advantage of the MPPM system is its spectral efficiency. We can define
this to beψ, which is measured in bps per Hertz. If we state that the bandwidth is inversely
proportional to the slot durationT , ψ is calculated as
ψ =Rb
1/T(3.9)
= Rb
Tb log2
(Qw
)
Q(3.10)
= Rb
log2
(Qw
)
RbQ(3.11)
=log2
(Qw
)
Q(3.12)
27
whereRb is the bit rate of the system.
Since(
Qw
)is maximized whenw = bQ/2c, we can state that the spectral efficiency of
MPPM is also maximized forw = bQ/2c. As Q becomes large, the spectral efficiency of
MPPM approaches one for thew = bQ/2c system, and approaches zero for thew = 1 system.
This can also be seen in Figure 3.2, since the spectral efficiencyψ is equivalent to the relative
energy efficiency for the peak-power-limited case,ρpeak. Although bandwidth conservation is
not of particular interest in FSO communications, since it operates on unregulated spectrum, the
spectral efficiency is more of a measure of the necessary electronics speed to operate at a given
bit rate. Alternatively, the spectral efficiency shows us that for a given bandwidth (or minimum
slot duration allowed by the electronics), the MPPM system will allow the system designer to
transmit at a higher bit rate. This is one of the most attractive features of MPPM for high data
rate communications.
3.3 System model
3.3.1 MPPM signaling
We send binary information using multipulse pulse-position modulation, where each mod-
ulator symbol is of durationTs seconds and is comprised ofQ slots, eachT seconds long. The
symbols are created by turning the laser on forw of theQ slots. This results in(
Qw
)different
symbols which we assume are equiprobable. Each symbol can representlog2
(Qw
)bits of infor-
mation. For simplification of the analysis, we ignore the fact that(
Qw
)may not be a power of
two.
We will often revert to the notation that an ‘on’ slot is denoted by a one and an ‘off’ slot
by a zero. For example, aQ = 8, w = 4 symbol, where the firstw slots are ‘on’ can be
denoted by[1 1 1 1 0 0 0 0]. We also make use of the notation thatQion denotes the set of
‘on’ slots for modulator symboli, andQioff is the set of ‘off’ slots. In our example above, for
q ∈ Qion, q = 0, 1, 2, 3 and forq ∈ Qi
off , q = 4, 5, 6, 7.
28
3.3.2 Transmitter array
For our system, the transmitters are modeled to have infinite bandwidth, so it is assumed
possible to have the laser on at some constant power for the duration of a slot. Each laser is also
completely off (full extinction) for the entire duration of any ‘off’ slot.
The number of lasers is denoted byM . In order to compare systems differing in the number
of lasers, we divide the transmitter power at each laser byM so the power delivered by the entire
transmitter array is constant as we varyM . We also assume that the lasers are noncoherent,
without any special precautions. The wavelength we have chosen in simulations for the system
is 1550 nm, which corresponds to a frequency of2× 1014 Hz.
3.3.3 Receiver array
Each of theN receivers is assumed to be perfect-photon-counting devices, also with infinite
bandwidth. The arrival of the signal and background photons at the receiver is modeled as a
Poisson process, where the number of photons arriving at detectorn during a single slot time
has a mean and variance ofλon,n for ‘on’ slots andλoff for ‘off’ slots.
We also assume perfect synchronization, so signal photons are only received during ‘on’
slots, and ‘off’ slots can only contain background photons.
3.3.4 Channel
The channel was described in detail in Section 2.5. We assume that theM transmitters and
N receivers are placed sufficiently far from one another such that each of the individual paths
from transmitter to receiver is independent.
The intensity gain along each path from transmitterm to receivern is denoted bya2nm, and
it is a random variable following the distributions described in Section 2.5.3. We denote a single
realization of all of the amplitude path gains as aM ×N matrix,A.
The signal received at detectorn is a composite of the signals received from all of the
M transmitters simultaneously. Therefore, at each detectorn, the signal power received is
proportional to∑
m a2nm, assuming all transmitters send with the same power.
29
3.3.5 Detector and observable
3.3.5.1 Poisson Regime
We first focus on the Poisson regime, which is applicable when the background count rate
is low and the variance in the observable at detectorn for slot q is Znq due to thermal noise in
the amplifier is small. Each of theN receivers is modeled according to Figure 2.5, and consists
of a photodetector and integrator. The photodetector is analyzed using the semi-classical view
of photodetection. The optical wave is received, and the output is a flow of photoelectrons that
obey Poisson statistics over any slot interval. The observable at detectorn and slotq is depicted
asZnq, and has a mean ‘on’ slot countλon,n, and a mean ‘off’ slot countλoff . These are given
by
λon,n =ηPT
hfM
∑m
a2nm +
ηPbT
hf(3.13)
and
λoff =ηPbT
hf, n = 1, ..., N (3.14)
respectively, wherePb is the power received due to background radiation, andη is an efficiency
factor for the detector, defined as the ratio of generated photoelectrons to incident photons.
3.3.5.2 Gaussian Regime
For this case, it is convenient to model the noise using a Norton equivalent circuit, as in
Figure 3.3.
Figure 3.3: Norton equivalent noise model.
30
In the figure,I is the mean current, equal to the average number of photoelectrons created
by the photodetector times the charge associated with each photoelectron,q, giving
I =
(ηP
hfM
∑m a2
nm + ηPb
hf
)q, signal present;(
ηPbhf
)q, no signal present.
(3.15)
The voltage across the (noiseless) resistorR is equal toIR. Since the mean values of the
optical shot noiseis(t) and thermal noiseit(t) are equal to zero, and the mean of an ‘on’ or ‘off’
slot is assumed to be constant over the slot timeT , the output of the integratorZ has a mean
equal to
E[Z] =
(ηPThfM
∑m a2
nm + ηPbThf
)Rq, signal present;(
ηPbThf
)Rq, no signal present.
(3.16)
The generation of photoelectrons is a Poisson point process, so the variance in the photo-
electron count during an ‘on’ or ‘off’ slot is equal to (3.13) and (3.14). The voltage at the input
to the integrator caused by this optical shot noise current is this value multiplied by a constant
qR. Since a random variableX with a variance ofσ2x times a constantk has a new variance of
k2σ2x, we can state that the variance of the output due to optical shot noise is equal to
V ar[Z]shot =
(ηPThfM
∑m a2
nm + ηPbThf
)R2q2, signal present;(
ηPbThf
)R2q2, no signal present.
(3.17)
The variance in the thermal noise currentit(t) can be described by its spectral noise density
St(f) = 2kT0/R, whereT0 is the absolute temperature of the resistorR [40]. Multiplying this
current by the resistance again means multiplying its variance byR2. Taking the integral over
T seconds gives
V ar[Z]thermal = 2kT0TR (3.18)
Since the thermal and shot noise processes are independent, we can add them together to get
the variance at the output of the integrator. Putting everything together, we have that the output
of the integrator is a random variableZ described by
E[Z] =
(ηPThfM
∑m a2
nm + ηPbThf
)Rq, signal present;(
ηPbThf
)Rq, no signal present.
(3.19)
31
and
V ar[Z] =
(ηPThfM
∑m a2
nm + ηPbThf
)R2q2 + 2kT0TR, signal present;(
ηPbThf
)R2q2 + 2kT0TR, no signal present.
(3.20)
At this point, we may choose to use (3.19) and (3.20) in a Gaussian approximation to model
Z. However, we must be careful to determine when this approximation is valid.
The thermal noise is modeled as a Gaussian random variable. Therefore, if its variance is
much larger than the optical shot noise variance, the first term may be neglected in (3.20), and
Z may be accurately approximated by a Gaussian random variable.
Also, if the level of background powerPb is high enough, the p.d.f. for the count of photo-
electrons, which is Poisson in nature, becomes increasingly Gaussian in shape (see Figure 3.4).
When this is added to the Gaussian noise, the resulting p.d.f. may also be modeled as Gaus-
sian. This fact is also true of high signal powerP , but high background powerPb is a sufficient
condition for the Gaussian approximation to hold, and necessary when thermal noise is low.
Figure 3.4: Poisson and Gaussian p.d.f.’s with equal means and variances ofλ = 200.
In the case where the background powerPb and thermal noise power levels are both low,
the Gaussian approximation ceases to yield accurate results. The Poisson distribution would
32
be poorly approximated by a Gaussian distribution (see Figure 3.5). As a result, we must let
the model for the distribution of the ‘off’ slots remain as Poisson. Thus, the convolution (due
to adding independent random variables) of the Gaussian and Poisson p.d.f.’s would no longer
have a Gaussian shape. Instead, for low mean count rates, the convolution of the two would
create a p.d.f. having many non-overlapping Gaussian shapes, each centered and weighted at
the different integer values given by the Poisson p.d.f., which would be poorly approximated by
a Gaussian distribution.
Figure 3.5: Poisson and Gaussian p.d.f.’s with equal means and variances ofλ = 2.
Chapter 4
Maximum likelihood (ML) detection
In this section, the general ML detection rules are derived for MIMO FSO detection in both
the Poisson and Gaussian regimes.
4.1 ML Detection for the poisson regime
LetZnq be the photoelectron counts for detectorn and slotq, andZ = Znq, n = 1, ..., N, q =
0, ..., Q − 1 represent the set of received observations. If we send one of(
Qw
)binary patterns
represented byXi, then the ML decision is
X = arg maxXi
f(Z|Xi) (4.1)
Using the definitions ofλon,n andλoff from Section 3.3.5, and recognizing thatZnq at each of
theN detectors andQ slots is independent, the conditional distribution of theN × Q random
matrixZ can be written as aN ×Q-fold product over all of the individual elements inZ. These
elements, which are represented asZnq, are conditioned on whether they are in an ‘on’ slot, or
an ‘off’ slot.
X = arg maxXi
∏n
∏
q∈Q(i)on
exp(−λon,n)(λon,n)Znq
Znq!
∏
q∈Q(i)off
exp(−λoff )(λoff )Znq
Znq!(4.2)
Since theZnq! terms in the denominator are invariant toXi, they can be removed without
affecting the outcome of the ML decision. Thus
33
34
X = arg maxXi
∏n
∏
q∈Q(i)on
exp(−λon,n)(λon,n)Znq∏
q∈Q(i)off
exp(−λoff )(λoff )Znq (4.3)
If we assume that the number of ‘on’ slots for all modulator symbols is equal tow, the
exponential terms are equal toexp (−wλon,n) andexp (−(Q− w)λb) for all Xi and can also be
eliminated, leaving
X = arg maxXi
∏n
∏
q∈Q(i)on
(λon,n)Znq∏
q∈Q(i)off
(λoff )Znq (4.4)
Next, we take the logarithm of the entire quantity to find the log-likelihood function.
X = arg maxXi
∑n
∑
q∈Q(i)on
ln((λon,n)Znq) +∑
q∈Q(i)off
ln((λoff )Znq)
(4.5)
which can be rewritten as
X = arg maxXi
∑n
∑
q∈Q(i)on
Znq ln(λon,n) +∑
q∈Q(i)off
Znqln(λoff )
(4.6)
= arg maxXi
∑n
∑
q∈Q(i)on
Znq ln(λon,n) +∑
all q
Znq ln(λoff )−∑
q∈Q(i)on
Znqln(λoff )
(4.7)
= arg maxXi
∑n
∑
q∈Q(i)on
Znq ln
(λon,n
λoff
) (4.8)
Therefore, the ML detector would make a decision based on a weighted sum over the ‘on’
slots.
What will be interesting to note in the following subsections is that of the four cases we
will consider in the Poisson regime, the optimal detector simplifies to finding thew largest
column sums with equal gain combining for three of them. The exception to this rule is the case
where background radiation and fading are both present, and the optimal detector searches for
a weighted sum over the ‘on’ and ‘off’ slots.
4.1.1 Case 1: ML detection with no background and no fading
For this case, we start with (4.6), and expressλon,n andλoff explicitly:
35
X = arg maxXi
∑n
[ ∑
q∈Q(i)on
Znq ln(λon,n) +∑
q∈Q(i)off
Znqln(λoff )
](4.9)
= arg maxXi
∑n
[ ∑
q∈Q(i)on
Znq ln
(ηPT
hfM
∑m
a2nm +
ηPbT
hf
)+
∑
q∈Q(i)off
Znqln
(ηPbT
hf
)](4.10)
For the cases where no background radiation is present,Pb = 0 andZnq = 0 for all ‘off’
slots. Since
limx→0+
x× log(x) = 0 (4.11)
(4.10) can be restated as
X = arg maxXi
∑n
∑
q∈Q(i)on
Znq ln
(ηPT
hfM
∑m
a2nm
)(4.12)
In the no fading case, all of the fading variablesA are equal to one, meaning that the
ln(
ηPThfM
∑m a2
nm
)term is constant for allXi leading to
X = arg maxXi
∑n
∑
q∈Q(i)on
Znq (4.13)
= arg maxXi
∑
q∈Q(i)on
∑n
Znq (4.14)
If we define∑
n Znq = Sq, which we will refer to as a ‘column sum’,
X = arg maxXi
∑
q∈Q(i)on
Sq (4.15)
which shows that the optimal detector simply searches for thew largest column sums. In the
case where ties occur and there are more thanw column sums that could provide an optimal
solution, the detector must resort to making a random choice among all of the equally likely
codewords.
36
4.1.2 Case 2: ML detection with no background and fading
With no background and fading, we know that any slot that has a non-zero countmustbe
an ‘on’ slot, and any slot that has a zero countmaybe an ‘off’ slot. Intuitively, we can easily
state that ifw slots have non-zero counts at one or more detectors, there is no ambiguity in the
transmitted symbol and the system will err with zero probability.
Therefore, no outside information is required to make the maximum likelihood decision.
Any scheme involving weighting each column sum differently based on the current realization
of A cannot assist in the outcome of the decision, so the optimal detector simply searches for
thew largest (or simply non-zero) column sums.
4.1.3 Case 3: ML detection with background radiation and no fading
In this case, we again start with
X = arg maxXi
∑n
[ ∑
q∈Q(i)on
Znq ln
(ηPT
hfM
∑m
a2nm +
ηPbT
hf
)+
∑
q∈Q(i)off
Znqln
(ηPbT
hf
)](4.16)
Here, the∑
m a2nm term is equal toM , and bias terms can be eliminated, yielding
X = arg maxXi
∑n
[ ∑
q∈Q(i)on
Znq ln (P + Pb) +∑
q∈Q(i)off
Znqln Pb
](4.17)
Here, we can make the observation that logarithms are monotonic increasing functions, and
P andPb are both positive numbers or zero. Therefore we can state thatln(P + Pb) ≥ ln Pb for
all P andPb. If we make a substitution and letτ = ln(Pb) andτ + ∆ = ln(P + Pb) (so that
∆ ≥ 0 is the difference between the two logarithmic terms), we can rewrite (4.17) as
X = arg maxXi
∑n
[(τ + ∆)
∑
q∈Q(i)on
Znq + τ∑
q∈Q(i)off
Znq
](4.18)
= arg maxXi
∑n
[∆
∑
q∈Q(i)on
Znq + τ∑
all q
Znq
](4.19)
37
Since theτ∑
all q Znq term is independent ofXi, it can be eliminated.∆ is also a scale
factor which is independent ofXi, leaving the following ML decision rule:
X = arg maxXi
∑n
∑
q∈Q(i)on
Znq (4.20)
= arg maxXi
∑
q∈Q(i)on
∑n
Znq (4.21)
= arg maxXi
∑
q∈Q(i)on
Sq (4.22)
This once again shows that the optimal detector simply searches for thew largest column
sums.
4.1.4 Case 4: Background radiation and fading
When background radiation and fading are both present, we again start with
X = arg maxXi
∑n
[ ∑
q∈Q(i)on
Znq ln
(ηPT
hfM
∑m
a2nm +
ηPbT
hf
)+
∑
q∈Q(i)off
Znqln
(ηPbT
hf
)](4.23)
Eliminating bias terms leaves
X = arg maxXi
∑n
[ ∑
q∈Q(i)on
Znq ln
(P
M
∑m
a2nm + Pb
)+
∑
q∈Q(i)off
Znqln Pb
](4.24)
If we again employ the same method applied to the previous case, we can state that
ln [(P/M)∑
m a2nm + Pb] ≥ ln Pb for all P , Pb, M , andanm. Therefore, we can make the
following substitutions: Letτ = ln Pb and τ + δn = ln [(P/M)∑
m a2nm + Pb], such that
δn ≥ 0 is the difference between the two logarithmic terms, and is dependent onn through the
fading distributionA.
We can then rewrite the ML decision rule as
38
X = arg maxXi
∑n
[(τ + δn)
∑
q∈Q(i)on
Znq + τ∑
q∈Q(i)off
Znq
](4.25)
= arg maxXi
∑n
[(τ + δn)
∑
q∈Q(i)on
Znq + τ∑
q∈Q(i)off
Znq
](4.26)
= arg maxXi
∑n
[δn
∑
q∈Q(i)on
Znq + τ∑
all q
Znq
](4.27)
The last summation term is independent ofXi and can be eliminated, which leaves
X = arg maxXi
∑n
[δn
∑
q∈Q(i)on
Znq
](4.28)
whereδn = ln [(P/M)∑
m a2nm + Pb]−ln Pb. This shows that the optimal detector will perform
a weighted sum of the ‘on’ slots.
A reasonable approximation to this detector is to perform equal gain combining and search
for the w largest column sums, treating this case like the other 3 cases. This is only slightly
suboptimal, as was shown in [19], and is the method that was chosen for the analysis in Chapters
5 and 6.
4.2 General ML detection in the Gaussian regime
Assuming the necessary conditions are met for the Gaussian approximation to be justified
(see Section 3.3.5.2), we can start to develop the general ML detector in the Gaussian regime.
For now, we will not assume that thermal noise or shot noise is dominant, and we will
assume the background and signal power levels are such that the Gaussian approximation is
justified. This will allow us to develop a more general ML detector. More specific cases will be
considered in the following subsections.
Based on the analysis in Section 3.3.5.2 we can state that the observableZ is a Gaussian
random variable with parameters
µon,n =
(ηPT
hfM
∑m
a2nm +
ηPbT
hf
)Rq (4.29)
39
µoff =
(ηPbT
hf
)Rq (4.30)
σ2on,n = (µon,n) Rq + 2kT0TR (4.31)
σ2off = (µoff ) Rq + 2kT0TR (4.32)
Similar to the development from the Poisson regime, we can claim thatN ×Q elements in
Z are independent, and therefore the distribution ofZ is simply aN × Q-fold product over all
detectors and slots of the (Gaussian) distribution of eachZnq, which are conditioned onq ∈ Qon
or q ∈ Qoff :
X = arg maxXi
N∏n=1
∏
q∈Q(i)On
(1√
2πσ2on,n
e− (Znq−µon,n)2
2σ2on,n
)×
∏
q∈Q(i)Off
1√
2πσ2off
e− (Znq−µoff )2
2σ2off
(4.33)
Eliminating the1/√
2π scale factors and taking the logarithm gives
X = arg maxXi
N∑n=1
[ ∑
q∈Q(i)On
(−(Znq − µon,n)2
2σ2on,n
− ln(σon,n)
)+
∑
q∈Q(i)Off
(−(Znq − µoff )
2
2σ2off
− ln(σoff )
)](4.34)
= arg maxXi
N∑n=1
[ ∑
q∈Q(i)On
((−Z2
nq + 2µon,nZnq − µ2on,n)
2σ2on,n
− ln(σon,n)
)+
∑
q∈Q(i)Off
((−Z2
nq + 2µoffZnq − µ2off )
2σ2off
− ln(σoff )
)](4.35)
Theµ2on,n, µ2
off , ln(σon,n), andln(σoff ) terms are constant for allXi, so
40
X = arg maxXi
N∑n=1
∑
q∈Q(i)On
(2µon,nZnq − Z2
nq
2σ2on,n
)+
∑
q∈Q(i)Off
(2µoffZnq − Z2
nq
2σ2off
) (4.36)
= arg maxXi
N∑n=1
[ ∑
q∈Q(i)On
(2µon,nZnq − Z2
nq
2σ2on,n
)+
∑
q∈Q(i)All
(2µoffZnq − Z2
nq
2σ2off
)−
∑
q∈Q(i)On
(2µoffZnq − Z2
nq
2σ2off
)](4.37)
= arg maxXi
N∑n=1
[ ∑
q∈Q(i)On
(2µon,nZnq − Z2
nq
2σ2on,n
− 2µoffZnq − Z2nq
2σ2off
) ](4.38)
= arg maxXi
N∑n=1
[ ∑
q∈Q(i)On
(µon,nσ
2off − µoffσ
2on,n
)Znq +
(σ2
on,n − σ2off
2
)Z2
nq
](4.39)
We will use this as the starting point for the different cases in the Gaussian regime.
4.2.1 Thermal noise dominates over shot noise
When the thermal noise is the dominant noise in the system, we can neglect the variance
due to the shot noise, and state that the slot count at each receiver,Znq follows a Gaussian
distribution with mean and variance of
µon,n =
(ηPT
hfM
∑m
a2nm +
ηPbT
hf
)Rq, signal present (4.40)
µoff =
(ηPbT
hf
)Rq, no signal present (4.41)
σ2 = 2kT0TR, either case (4.42)
Therefore, to determine the ML decision in the Gaussian regime, we start with (4.39), but
let the variance terms be dominated by2kT0TR. This results in
X = arg maxXi
N∑n=1
(µon,n − µoff ) Znq (4.43)
41
After making the appropriate substitutions, eliminating the scale factors gives the ML de-
tector as
X = arg maxXi
N∑n=1
∑
q∈Q(i)On
(∑m
a2nm
)Znq
(4.44)
4.2.1.1 No fading present
When no fading is present (and regardless of the presence of background radiation),∑
m a2nm =
M . This becomes a scale factor that can be eliminated, giving
X = arg maxXi
∑n
∑
q∈Q(i)on
Znq (4.45)
= arg maxXi
∑
q∈Q(i)on
∑n
Znq (4.46)
= arg maxXi
∑
q∈Q(i)on
Sq (4.47)
which shows that the optimal detector searches for the largest column sums. This is an iden-
tical result to the Poisson regime when no fading was present, where the optimal detector also
searched for thew largest column sums.
4.2.1.2 Fading present
With fading present (and regardless of the presence of background radiation), (4.44) is irre-
ducible and gives the optimal detector, which searches for the largest weighted column sums,
dependent on the fading distributionA.
4.2.2 Shot noise dominates over thermal noise
For this case, we will start with (4.39), and simply plug in the mean and variance expressions
given by
µon,n =
(ηPT
hfM
∑m
a2nm +
ηPbT
hf
)Rq (4.48)
42
µoff =
(ηPbT
hf
)Rq (4.49)
σ2on,n = (µon,n) Rq (4.50)
σ2off = (µoff ) Rq (4.51)
4.2.2.1 Fading present
After plugging in the appropriate means and variances into (4.39) and eliminating the scale
factors, the ML detector is
X = arg maxXi
N∑n=1
∑
q∈Q(i)On
(∑m
a2nm
)Z2
nq
(4.52)
This differs from the Poisson regime, where the optimal detector looks for weighted column
sums. Here, the optimal detector will square the observable at each detector before multiplying
by the weighting coefficient and summing over detectors and ‘on’ slots.
4.2.2.2 No fading present
With no fading present,∑
m a2nm = M , and just as when thermal noise was dominant, the
optimal detector becomes
X = arg maxXi
∑n
∑
q∈Q(i)on
Z2nq (4.53)
= arg maxXi
∑
q∈Q(i)on
∑n
Z2nq (4.54)
This also differs from the Poisson regime since the optimal detector will square the observ-
able at each detector, and then look for the ‘w’ largest column sums (as opposed to simply
looking for the ‘w’ largest column sums in the Poisson regime).
4.2.3 Shot and thermal noise are not dominant
In this case, we use the means and variances defined earlier as
43
µon,n =
(ηPT
hfM
∑m
a2nm +
ηPbT
hf
)Rq (4.55)
µoff =
(ηPbT
hf
)Rq (4.56)
σ2on,n = (µon,n) Rq + 2kT0TR (4.57)
σ2off = (µoff ) Rq + 2kT0TR (4.58)
4.2.3.1 Fading Present
Plugging these values into (4.39) and eliminating scale factors terms gives us
X = arg maxXi
∑n
∑
q∈Q(i)on
∑m
a2nm
(2kT0T ) Znq +
(q
2
)Z2
nq
(4.59)
This result is similar to other fading results in the Gaussian and Poisson regimes, since it
shows the optimal detector to be a weighted sum based on the fading distribution,A.
In most instances, one would expect(q/2)Z2nq to negligible, in which case2kT0T becomes
a scale factor. In these instances, the optimal detector is identical to the optimal detector for the
other fading cases in the Gaussian regime.
4.2.3.2 No fading present
When no fading is present, (4.60) becomes
X = arg maxXi
∑n
∑
q∈Q(i)on
(2kT0T ) Znq +
(q
2
)Z2
nq
(4.60)
Again, when(q/2)Z2nq is negligible,2kT0T becomes a scale factor and the optimal detector
is simply searching for thew largest column sums.
Chapter 5
Error analysis of MIMO FSO system
using MPPM
In this chapter, we will perform symbol error probability analysis of the MIMO MPPM
FSO system in both the Poisson and Gaussian regimes. For simplicity of the analysis, we will
assume equal gain combining for all cases. This is optimal in three of the four cases under
consideration, and was shown to be a very good approximation for the Poisson regime in the
case where background radiation and atmospheric fading are both present [19]. We also show
this to be a very good approximation in the Gaussian regime in Section 5.2.
5.1 Error analysis in the Poisson regime
For the Poisson regime, we consider a number of possible scenarios that are combinations
of the following: no fading, Rayleigh fading, log-normal fading, background radiation, and no
background radiation.
5.1.1 No background radiation
For the case of no background radiation, thew column sums with the largest counts will be
thew ‘on’ slots in the transmitted symbolXj unlessi > 0 of them are zero (due to quantum
effects in the photodetector).
With i of the column sums equaling zero, the detector may have partial information (or no
44
45
information if i = w), with which it can make a guess at the transmitted symbol. The receiver
must decide among(
Q−w+ii
)equally probable symbols. This quantity can be interpreted by
noting that this is the number of ways the receiver can place thei non-received signal pulses in
Q− w + i vacant slots. The detector errs with probability
t(Q,w, i) =
(Q−w+i
i
)− 1(Q−w+i
i
) (5.1)
As described in Section 3.3.5, for ‘on’ slots, we can model the photoelectron slot countZnq
at each of theN receivers as a Poisson process with a Poisson parameter ofλon,n andλoff for
‘on’ and ‘off’ slots. With equal gain combining, the sum over all receivers for each ‘on’ slot is
also a Poisson random variable (conditional onA) with a parameter of
λon =ηPT
hfM
N∑n=1
M∑m=1
a2nm (5.2)
If we defineSq =∑N
n=1 Znq, an ‘on’ slot will have zero photoelectrons with probability
P (Sq = 0) = p = e−λon (5.3)
Treating each column count as an independent random variable, we get
P [i of w columns = 0] =w∑
i=1
(w
i
)pi(1− p)w−i (5.4)
Therefore, we can state for the no background case, that the error probability conditioned
on the fading variablesA is
Ps|A =w∑
i=1
t(Q,w, i)
(w
i
)pi(1− p)w−i (5.5)
We can take the(1− p)w−i term and rewrite it using the binomial expansion:
(1− p)w−i =w−i∑
l=0
(−1)l
(w − i
l
)pl (5.6)
This gives the probability expression
Ps|A =w∑
i=1
t(Q,w, i)
(w
i
)pi
w−i∑
l=0
(−1)l
(w − i
l
)pl (5.7)
46
which is easily rewritten
Ps|A =w∑
i=1
w−i∑
l=0
(−1)l
(Q−w+i
i
)− 1(Q−w+i
i
)(
w
i
)(w − i
l
)p(i+l) (5.8)
This is the probability of symbol error, conditioned on no background radiation and the
fading variablesA. In each of the next three subsections we will use different distributions of
A, and show how the system performs.
5.1.1.1 No background, no fading
In the non-fading case,a2nm = 1 for all n andm. Plugging this value into (5.2), (5.3), and
(5.8) gives
λon =ηPTN
hf(5.9)
P (Sq = 0) = p = e−ηPTN
hf (5.10)
and
Ps|no fading =w∑
i=1
w−i∑
l=0
(−1)l
(Q−w+i
i
)− 1(Q−w+i
i
)(
w
i
)(w − i
l
)e−
ηPTN(i+l)hf (5.11)
respectively.
Observe that the average number of photoelectron counts and the probability of error are
independent of the number of lasersM in the non-fading case. This is because theM path
gains from each laser to each detector are unity, and we kept the total transmit power constant
by dividing the transmit power at each laser byM . However, performance does depend on the
number of receivers because the total aperture size increases by a factor ofN .
For consistency, we could have removed that dependency on receivers by also decreasing the
total aperture size byN (as done by Shin and Chan in [15]), but in a realistic setting increasing
the number of receivers will also increase the aperture size, so our choice is justified.
As can be seen by Figures 5.1 and 5.2, no change in performance is observable for varying
M . However, a 6 dB gain in performance is seen in the cases whereN is equal to four. This
can be attributed to the exponent ofp being increased by a factor of four.
47
It is also important to note that multipulse PPM outperforms standard PPM only in a peak-
power-limited system. This makes intuitive sense, since average-power-limited systems would
require that the total signal (‘on’) power be distributed among thew on slots – in the average-
power-limited system, asw increases, each ‘on’ slot has a lower peak power, which increases
the probability that a zero is observed at the detector during an ‘on’ slot.
−210 −200 −190 −180 −170 −160 −15010
−12
10−10
10−8
10−6
10−4
10−2
100
PTb,dBJ
Ps
M=1,N=1,w=1M=4,N=1,w=1M=1,N=4,w=1M=4,N=4,w=1M=1,N=1,w=4M=4,N=1,w=4M=1,N=4,w=4M=4,N=4,w=4
Figure 5.1: Symbol error probability vs. average power with no fading and no background
radiation, andQ = 8.
5.1.1.2 No background, Rayleigh fading
For the Rayleigh fading case, we refer back to Section 2.5.3, which discusses different
fading models. (2.3) gives the p.d.f. for Rayleigh fading, and is repeated here for convenience:
fA(anm) = 2anme−a2nm , anm > 0 (5.12)
We assume that each realization of a channel fading gainanm follows this distribution, and
therefore the expected error probability can be found by averaging (5.8) with respect to the
Rayleigh distribution for allMN channel paths.
48
−210 −200 −190 −180 −170 −160 −15010
−12
10−10
10−8
10−6
10−4
10−2
100
PTb,dBJ
Ps
M=1,N=1,w=1M=4,N=1,w=1M=1,N=4,w=1M=4,N=4,w=1M=1,N=1,w=4M=4,N=1,w=4M=1,N=4,w=4M=4,N=4,w=4
Figure 5.2: Symbol error probability vs. peak power with no fading and no background radia-
tion, andQ = 8.
Ps|Rayleigh fading =N∏
n=1
M∏m=1
∫ ∞
0
w∑i=1
w−i∑
l=0
(Q−w+i
i
)− 1(Q−w+i
i
)(
w
i
)(w − i
l
)×
(−1)l2anme−[ηPTa2nm(i+l)]/hfMQdanm (5.13)
=w∑
i=1
w−i∑
l=0
(Q−w+i
i
)− 1(Q−w+i
i
)(
w
i
)(w − i
l
)(−1)l ×
M∏m=1
N∏n=1
∫ ∞
0
2anme−[ηPTa2nm(i+l)]/hfMQdanm (5.14)
This is aNM -fold integration, but each term is identical so it can be rewritten
Ps|Rayleigh fading =w∑
i=1
w−i∑
l=0
(Q−w+i
i
)− 1(Q−w+i
i
)(
w
i
)(w − i
l
)(−1)l ×
[∫ ∞
0
2ae−[ηPTa2(i+l)]/hfMQda
]MN
(5.15)
and applying the solution to this integral gives the final closed-form result
Ps|Rayleigh fading =w∑
i=1
w−i∑
l=0
(Q−w+i
i
)− 1(Q−w+i
i
)(
w
i
)(w − i
l
)(−1)l
[1
1 + ηPT (i+l)hfMQ
]MN
(5.16)
49
This is plotted in Figures 5.3 and 5.4. Since the error probability drops by a factor of10MN
for every 10 dB increase in signal power, we claim that the system achieves a diversity equal
to MN . This shows a performance advantage to increasing the number of lasers as well as the
number of receivers. However, comparing theM = 1, N = 4 to theM = 4, N = 1 curves
for constantw shows that for a given diversity, increasing the number of receivers has more
of a beneficial effect than increasing the number of transmitters, due to the increase in overall
receiver aperture size.
−210 −200 −190 −180 −170 −160 −150 −14010
−12
10−10
10−8
10−6
10−4
10−2
100
PTb, dBJ
Ps
M=1 N=1, W=1M=4 N=1, W=1M=1 N=4, W=1M=4 N=4, W=1M=1 N=1, W=4M=4 N=1, W=4M=1 N=4, W=4M=4 N=4, W=4
Figure 5.3: Symbol error probability vs. average power with Rayleigh fading, no background
radiation, andQ = 8.
5.1.1.3 No background, log-normal fading
With log-normal fading, the channel path gainsanm follow the distribution in (2.1), repeated
here for convenience
fA(anm) =1
(2πσ2X)1/2anm
exp(−(loge anm − µX)2/2σ2X), anm > 0 (5.17)
whereµX = −σ2X andS.I. = E[A4]/E2[A2]− 1 = e4σ2
X − 1 ∈ [0.4, 1.0].
The symbol error probability could be found by averaging over the log-normal distribution:
50
−210 −200 −190 −180 −170 −160 −150 −14010
−12
10−10
10−8
10−6
10−4
10−2
100
PTb, dBJ
Ps
M=1 N=1, W=1M=4 N=1, W=1M=1 N=4, W=1M=4 N=4, W=1M=1 N=1, W=4M=4 N=1, W=4M=1 N=4, W=4M=4 N=4, W=4
Figure 5.4: Symbol error probability vs. peak power with Rayleigh fading, no background
radiation, andQ = 8.
Ps|Log−normal fading =N∏
n=1
N∏m=1
∫ ∞
0
w∑i=1
w−i∑
l=0
(Q−w+i
i
)− 1(Q−w+i
i
)(
w
i
)(w − i
l
)(−1)l ×
e−[ηPT (i+l)]/hfMQ 1
(2πσ2X)1/2anm
e(−(ln anm−µX)2/2σ2X)danm (5.18)
=w∑
i=1
w−i∑
l=0
(Q−w+i
i
)− 1(Q−w+i
i
)(
w
i
)(w − i
l
)(−1)l ×
[∫ ∞
0
e−[ηPT (i+l)]/hfMQ 1
(2πσ2X)1/2a
e(−(ln a−µX)2/2σ2X)da
]MN
(5.19)
However, unlike the Rayleigh fading case, the resulting integral is not easily solved in closed
form. Instead, we can choose from a number of methods that still give reasonable accuracy,
including numerical integration, importance sampling, and Monte Carlo simulation.
The method chosen for this thesis is numerical integration. The log-normal density is sam-
pled for the values0 to 10.0 in step sizes of10−5, and using a S.I. of 1.0. The resulting plots
can be seen in Figures 5.5 and 5.6.
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. Most notable are the
51
−200 −195 −190 −185 −180 −175 −170 −165 −160 −155 −150 −14510
−12
10−10
10−8
10−6
10−4
10−2
100
PTb,dBJ
Ps
M=1,N=1,w=1M=4,N=1,w=1M=1,N=4,w=1M=4,N=4,w=1M=1,N=1,w=4M=4,N=1,w=4M=1,N=4,w=4M=4,N=4,w=4
Figure 5.5: Symbol error probability vs. average power with log-normal fading (S.I. = 1.0), no
background radiation, andQ = 8.
−200 −195 −190 −185 −180 −175 −170 −165 −160 −155 −150 −14510
−12
10−10
10−8
10−6
10−4
10−2
100
PTb,dBJ
Ps
M=1,N=1,w=1M=4,N=1,w=1M=1,N=4,w=1M=4,N=4,w=1M=1,N=1,w=4M=4,N=1,w=4M=1,N=4,w=4M=4,N=4,w=4
Figure 5.6: Symbol error probability vs. peak power with log-normal fading (S.I. = 1.0), no
background radiation, andQ = 8.
52
effects of transmitter and receiver diversity. Just as in the Rayleigh fading case, a considerable
performance gain is also achievable by increasing only the number of lasers. However, the
increased aperture size resulting from adding receivers is also visible, showing that receiver
diversity is even more effective than transmitter diversity.
5.1.2 Error probability in the presence of background radiation
The analysis of error probability for the case of background radiation present is more com-
plicated than for no background radiation. In the no background case, errors only occurred
when an ‘on’ slot was received as a zero at allN receivers. In the background case, the ‘off’
slots have the potential to have an equal or higher count than any of the ‘on’ slots, in which case
the receiver could pick the wrong modulator symbol.
For this reason, a closed-form solution, even for the non-fading case is an intractable prob-
lem. For the non-fading case, an infinite-summation expression is possible. For both fading
cases, however, some sort of Monte Carlo simulation is necessary.
5.1.2.1 Background radiation, no fading
To obtain the infinite-summation solution we first classify the possible symbol error types
into two categories. The first are errors caused when one or more of the ‘off’ slots has a higher
count than one or more of the ‘on’ slots. In this case, the receiver will certainly make an error.
We will refer to this as adefinite errorand use the symbolPdef . This scenario is depicted in
Figure 5.7.
We will start by analyzing definite errors. When the receiver receives a vectorZ representing
theQ slots summed over allN photodetectors,w of them were originally sent as ‘on’ slots, and
Q − w as ‘off’ slots. Of the ‘on’ slots, there will bei ∈ 1, . . . , w of them that will have the
lowest slot countu. If j ∈ 1, . . . , Q − w of the ‘off’ slots have a value (labeledv in Figure
5.7) which is greater thanu, a definite error has occurred.
LettingSon,l, l = 1, ..., w denote the column sums for ‘on’ slots, andSoff,l, l = 1, ..., Q−w denote the column sums for ‘off’ slots, we can state this for all possible values ofu as
53
Figure 5.7: An example of a definite error.
Pdef =∞∑
u=0
P
[(|l : Son,l = u| = i) ∩ (|l : Son,l > u| = w − i) (5.20)
∩ (|l : Soff,l > u| = j) ∩ (|l : Soff,l ≤ u| = Q− w − j)
](5.21)
=∞∑
u=0
w∑i=1
(w
i
)P (Son = u)iP (Son > u)w−i ×
Q−w∑j=1
(Q− w
j
)P (Soff > u)jP (Soff ≤ u)Q−w−j (5.22)
For simplicity, we will make the following symbolic substitutions regarding Poisson random
variables with parameterα:
φpdf (k, α) = P (slot count= k|α) =αke−α
k!, k = 0, 1, 2... (5.23)
φcdf (k, α) = P (slot count≤ k|α) =k∑
b=0
αbe−α
b!, k = 0, 1, 2... (5.24)
(being careful to note thatφcdf andφpdf are both equal to zero fork < 0). If we let λon =
ηPT∑
n
∑m a2
nm/hfM + NηPbT/hf andλoff = NηPbT/hf ,
54
Pdef =∞∑
u=0
w∑i=1
(w
i
)φpdf (u, λon)i(1− φcdf (u, λon))w−i ×
Q−w∑j=1
(Q− w
j
)(1− φcdf (u, λoff ))
jφcdf (u, λoff )Q−w−j (5.25)
The second error type, which we will call anindefinite error(Pindef ), happens when one or
more of the ‘off’ slots has the same count as one or more of the lowest-valued ‘on’ slots. We
make the distinction of lowest-valued, because an ‘off’ slot tying with an ‘on’ slot whose count
is higher than any of the other ‘on’ slots will cause a definite error. Indefinite errors (depicted in
Figure 5.8) cause symbol errors a fraction of the time, since the receiver makes a correct guess
at the modulator symbol at least some of the time. The probability of symbol error is the sum
of definite and indefinite error probabilities.
Figure 5.8: An example of an indefinite error (where the receiver chooses incorrectly from the
3 possible modulator symbols).
Similar to the development for definite errors, we can define an expression for indefinite
errors. Here, we are concerned aboutg ’off’ slots with countu when thei lowest-valued ‘on’
slots also have a count ofu. This is expressed as
55
Pindef =∞∑
u=0
(g+i
i
)− 1(g+i
i
) × P
[(|l : Son,l = u| = i) ∩ (|l : Son,l > u| = w − i)
∩ (|l : Soff,l = u| = g) ∩ (|l : Son,l < u| = Q− w − g)
](5.26)
=∞∑
u=0
w∑i=1
(w
i
)φpdf (u, λon)i(1− φcdf (u, λon))w−i ×
Q−w∑g=1
(g+i
i
)− 1(g+i
i
)(
Q− w
g
)φpdf (u, λoff )
gφcdf (u− 1, λoff )Q−w−g (5.27)
where the((
g+ii
)− 1)/(
g+ii
)term represents the probability of the receiver choosing the wrong
symbol if i ‘on’ slots have the same count asg ‘off’ slots, and the remainingw − i slots have
the highest counts. The full expression is then
Ps = Pdef + Pindef (5.28)
A reasonable numerical approximation to this can be obtained from any mathematical soft-
ware package. We take advantage of the built-in Matlab functionspoisspdfandpoisscdfwhich
correspond directly to what we defined asφpdf andφcdf respectively. To avoid an infinite sum-
mation, feedback can be used in the summation-loop to determine when the probability of error
has asymptotically approached the final value to within a certain threshold (1% of the final value
in plots generated for this thesis).
This advantage of this method over the Monte Carlo approach is reasonable accuracy for
error probabilities as low as10−12 without the need for large numbers of trials. The result of
this simulation is shown in Figures 5.9 and 5.10.
Looking at the plots, we again notice that there is no advantage to increasingM , since this
is a non-fading situation. The advantage to increasing the number of receivers is caused by the
increased aperture size at the receiver.
It is interesting to note that with−170 dBJ of background radiation present, the advantage
for havingN = 4 overN = 1 drops from 6 dB in the no background case to 4 dB, showing that
background radiation cuts into the advantage provided by increasing the receiver aperture size.
Also, comparing Figures 5.9 and 5.10 to Figures 5.1 and 5.2 respectively, background radiation
shifts the curves to the right by amounts ranging from 4 to 7 dB, and causes them to drop off
56
−210 −200 −190 −180 −170 −160 −15010
−12
10−10
10−8
10−6
10−4
10−2
100
PTb,dBJ
Ps
M=1,N=1,w=1M=4,N=1,w=1M=1,N=4,w=1M=4,N=4,w=1M=1,N=1,w=4M=4,N=1,w=4M=1,N=4,w=4M=4,N=4,w=4
Figure 5.9: Symbol error probability vs. average power with no fading,PbTb = −170 dbJ, and
Q = 8 – dashed and solid lines overlap.
−210 −200 −190 −180 −170 −160 −15010
−12
10−10
10−8
10−6
10−4
10−2
100
PTb,dBJ
Ps
M=1,N=1,w=1M=4,N=1,w=1M=1,N=4,w=1M=4,N=4,w=1M=1,N=1,w=4M=4,N=1,w=4M=1,N=4,w=4M=4,N=4,w=4
Figure 5.10: Symbol error probability vs. peak power with no fading,PbTb = −170 dbJ, and
Q = 8 – dashed and solid lines overlap.
57
more steeply than in the no background radiation case. This suggests a signal power threshold
for reliable communications caused by the presence of background radiation.
5.1.2.2 Background radiation, Rayleigh fading
When background radiation and fading are present, the infinite summation expression from
the previous subsection could be combined with a Monte Carlo approach to obtain the desired
error probability. A large number of fading variables could be drawn according to a fading
distribution, plugged-in to (5.28), and averaged to achieve accurate results for values of the
error probability much lower than possible with a pure Monte Carlo approach. However, this
is a prohibitively slow process due to the large number of fading variables combined with the
‘infinite’ summation.
For results in the two fading cases, we resort to simple Monte Carlo simulation, generating a
large number of Poisson and Rayleigh (or log-normal) fading variables, and dividing the number
of errors that occur by the number of trials.
As Figures 5.11 and 5.12 show, the system achieves diversity equal toM ×N (measurable
for M ×N ≤ 4) in the presence of Rayleigh fading, even with background present.
−210 −200 −190 −180 −170 −160 −15010
−6
10−5
10−4
10−3
10−2
10−1
100
PTb, dBJ
P s
M=1,N=1,w=1M=4,N=1,w=1M=1,N=4,w=1M=4,N=4,w=1M=1,N=1,w=4M=4,N=1,w=4M=1,N=4,w=4M=4,N=4,w=4
Figure 5.11: Symbol error probability vs. average power with Rayleigh fading,PbTb = −170
dbJ, andQ = 8.
58
−210 −200 −190 −180 −170 −160 −15010
−6
10−5
10−4
10−3
10−2
10−1
100
PTb, dBJ
P s
M=1,N=1,w=1M=4,N=1,w=1M=1,N=4,w=1M=4,N=4,w=1M=1,N=1,w=4M=4,N=1,w=4M=1,N=4,w=4M=4,N=4,w=4
Figure 5.12: Symbol error probability vs. peak power with Rayleigh fading,PbTb = −170 dbJ,
andQ = 8.
5.1.2.3 Background radiation, log-normal fading
Results for the log-normal fading case were also obtained using Monte Carlo simulation.
Unfortunately, due to the large number of trials that would be necessary, the results are only
accurate to symbol probabilities of10−5.
Once again, we notice the advantage to transmitter and receiver diversity, with the greatest
advantage being gained at the receiver due to the increased aperture size.
5.2 Error probability in the Gaussian regime
We will now investigate the probability of error in the Gaussian regime. In this regime, we
are dealing with continuous random variables, which changes our analysis to a certain degree
from the Poisson regime.
Again, we will assume equal gain combining, such that our detector will look for thew slots
with the highest observable,∑
n Znq = Sq. An error occurs if the highest-valued ‘off’ slot has
a higher count rate than the lowest-valued ‘on’ slot. We will ignore tie-break situations, since
we are dealing with continuous random variables where ties are zero-probability events.
Assuming that modulator symbolXi is sent, we define
59
−210 −200 −190 −180 −170 −160 −15010
−6
10−5
10−4
10−3
10−2
10−1
100
PTb,dBJ
P s
M=1,N=1,w=1M=4,N=1,w=1M=1,N=4,w=1M=4,N=4,w=1M=1,N=1,w=4M=4,N=1,w=4M=1,N=4,w=4M=4,N=4,w=4
Figure 5.13: Symbol error probability vs. average power with Log-normal fading, S.I. = 1.0,
PbTb = −170 dbJ, andQ = 8.
−210 −200 −190 −180 −170 −160 −15010
−6
10−5
10−4
10−3
10−2
10−1
100
PTb,dBJ
P s
M=1,N=1,w=1M=4,N=1,w=1M=1,N=4,w=1M=4,N=4,w=1M=1,N=1,w=4M=4,N=1,w=4M=1,N=4,w=4M=4,N=4,w=4
Figure 5.14: Symbol error probability vs. peak power with Log-normal fading, S.I. = 1.0,
PbTb = −170 dbJ, andQ = 8.
60
U = minq∈Qi
on
(Sq) (5.29)
V = maxq∈Qi
off
(Sq) (5.30)
whereSq =∑N
n=1 Znq. Therefore, the conditional probability of error is
Ps|A = P (V > U |A) (5.31)
If we write the joint density function ofU andV asf(u, v), we can state that
P (V > U) =
∫ ∞
−∞
[∫ ∞
u
fUV (u, v)dv
]du (5.32)
SinceU andV are independent, this simplifies to
P (V > U) =
∫ ∞
−∞
∫ ∞
u
fU(u)fV (v)dvdu
=
∫ ∞
−∞fU(u)
[∫ ∞
u
fV (v)dv
]du (5.33)
If we state the c.d.f. ofV asFV (u) = P (V ≤ u) =∫ u
−∞ fV (u)du,
then
[1− FV (u)] =
∫ ∞
u
fV (v)dv (5.34)
and
P (V > U) =
∫ ∞
−∞fU(u) [1− FV (u)] du (5.35)
We are first interested in determining the densityfU(u). We do so by first looking at the
c.d.f., FU(u), whose derivative will give us the desired p.d.f. Similar to the development in
Papoulis [40], we can writeFU(u) = P (S1 = s1 ≤ u⋃
S2 = s2 ≤ u⋃
...⋃
Sw = sw ≤ u),
whereS1...Sw are the column sums in thew ‘on’ slots.
Whenw = 2, for example,FU(u) can be expressed as
FU(u) = FS1(u) + FS2(u)− FS1S2(u, u) (5.36)
which is recognized as the sum of the individual c.d.f.’s minus the joint c.d.f.
61
To convert this to a p.d.f. as required in (5.35), we simply take the derivative of the respective
c.d.f.’s. For the last term, the chain rule is required, giving
fU(u) =dFU(u)
dv= fS1(u) + fS2(u)− [FS1(u)fS2(u) + FS2(u)fS1(u)] (5.37)
= fS1(u) [1− FS1(u)] + fS2(u) [1− FS2(u)] (5.38)
SinceS1 andS2 are i.i.d., both being in the set of ‘on’ slots, we can write them asSon
instead, and simplify (5.38):
fU(u) = 2fSon(u) [1− FSon(u)] (5.39)
Extending this to arbitraryw results in
fU(u) = wfSon(u)[1− FSon(u)w−1
](5.40)
Since we are dealing with Gaussian density functions, theFSon(u)w−1 term can be expressed
in terms of aQ−function, and then rewritten using the binomial expansion as
FSon(u)w−1 = P (Son ≤ u)w−1 (5.41)
= [1−QSon(u)]w−1 (5.42)
=w−1∑i=0
(w − 1
i
)(−1)iQSon(u)i (5.43)
By isolating thei = 0 term in the summation, we can pull out a 1, and eliminate the ‘1-’
from the expression forfU(u):
fU(u) = wfSon(u)[1− FSon(u)w−1
](5.44)
= wfSon(u)
[1−
w−1∑i=0
(−1)i
(w − 1
i
)QSon(u)i
](5.45)
= wfSon(u)
[1− 1−
w−1∑i=1
(−1)i
(w − 1
i
)QSon(u)i
](5.46)
= wfSon(u)
[−
w−1∑i=1
(−1)i
(w − 1
i
)QSon(u)i
](5.47)
62
Now focusing our attention on the1− FV (u) term in (5.35), it can be rewritten as
1− FV (u) = 1− FSb(u)Q−w
= 1− [1−QSb(u)]Q−w (5.48)
whereSb is the column sum at an ‘off’ slot receiving only background radiation, if present. The
Q−w in the exponent is due to each of theQ−w ‘off’ slots being independent. The right-most
term can be rewritten by once again using the binomial expansion formula:
1− [1−QSb(u)]Q−w = 1−
Q−w∑
l=0
(−1)l
(Q− w
l
)QSb
(u)l (5.49)
To eliminate the ’1-’ term thel = 0 term is expressed explicitly:
1− [1−QSb(u)]Q−w = 1−
Q−w∑
l=0
(−1)l
(Q− w
l
)QSb
(u)l (5.50)
= 1− 1−Q−w∑
l=1
(−1)l
(Q− w
l
)QSb
(u)l (5.51)
= −Q−w∑
l=1
(−1)l
(Q− w
l
)QSb
(u)l (5.52)
Putting everything together, we get
Ps =
∫ ∞
−∞wfSon(u)
[−
w−1∑i=1
(−1)i
(w − 1
i
)QSon(u)i
]×
[−
Q−w∑
l=1
(−1)l
(Q− w
l
)QSb
(u)l
]du (5.53)
which is a general expression for the probability of symbol error for all of the cases in the
Gaussian regime. Implicit in this expression is that the mean, variance, background noise,
number of lasers, number of receivers, number of ‘on’ and ‘off’ slots, and fading will all affect
the probability of symbol error through their respectiveQ−functions.
5.2.0.4 No fading
For the non-fading case, we can simply take (5.53) and perform a numerical integration over
a large range of values foru – large enough to thoroughly encompass the tail regions of the ‘on’
and ‘off’ Gaussian p.d.f.’s.
63
Values were chosen for the load resistor,R, the system temperature,T0, and the bit rateRb
and the integral was performed, yielding Figures 5.15 and 5.16. The curves exhibit the same
general shape as in the Poisson regime1 . Again, the multipulse system outperforms the single
pulse system only in the peak-power-limited system. The effect ofN = 4 is also observed as
approximately 6 dB of improved performance.
−90 −85 −80 −75 −70 −65 −60 −55 −5010
−12
10−10
10−8
10−6
10−4
10−2
100
Pave
,dBW
Ps
M=1,N=1,w=1M=4,N=1,w=1M=1,N=4,w=1M=4,N=4,w=1M=1,N=1,w=4M=4,N=1,w=4M=1,N=4,w=4M=4,N=4,w=4
Figure 5.15: Symbol error probability vs. average power with no fading,Q = 8, R = 100 Ω,
T0 = 290 K, PbTb = −170 dbJ, andRb = 100 Mbps.
5.2.0.5 Rayleigh fading
With Rayleigh fading, the integral would need to be averagedM ×N times for all possible
fading paths from transmitter to receiver. This is prohibitively difficult, and instead, a simple
Monte Carlo approach using equal gain combining was chosen. As can be seen from the slopes
of the curves in Figures 5.17 and 5.18, full transmitter and receiver diversity is achieved in the
Gaussian regime as well.
Optimal gain combining, using the ML detector developed in Chapter 4 gives only slightly
improved results for faded channels whereN > 1. This is shown for selected cases with
1Note here, however, that the error probability is plotted versus received powerP in dBW, as opposed toPTb
in dBJ for the Poisson regime.
64
−90 −85 −80 −75 −70 −65 −60 −55 −5010
−12
10−10
10−8
10−6
10−4
10−2
100
Ppeak
,dBW
Ps
M=1,N=1,w=1M=4,N=1,w=1M=1,N=4,w=1M=4,N=4,w=1M=1,N=1,w=4M=4,N=1,w=4M=1,N=4,w=4M=4,N=4,w=4
Figure 5.16: Symbol error probability vs. peak power with no fading,Q = 8, R = 100 Ω,
T0 = 290 K, Pb = −90 dBW, andRb = 100 Mbps.
−90 −85 −80 −75 −70 −65 −60 −55 −50 −45 −4010
−6
10−5
10−4
10−3
10−2
10−1
100
Pave
,dBW,ECG
Ps
M=1,N=1,w=1M=4,N=1,w=1M=1,N=4,w=1M=4,N=4,w=1M=1,N=1,w=4M=4,N=1,w=4M=1,N=4,w=4M=4,N=4,w=4
Figure 5.17: Symbol error probability vs. average power with Rayleigh fading,Q = 8, R = 100
Ω, T0 = 290 K, Pb = −90 dBW, andRb = 100 Mbps.
65
−90 −85 −80 −75 −70 −65 −60 −55 −50 −45 −4010
−6
10−5
10−4
10−3
10−2
10−1
100
Ppeak
,dBW,EGC
Ps
M=1,N=1,w=1M=4,N=1,w=1M=1,N=4,w=1M=4,N=4,w=1M=1,N=1,w=4M=4,N=1,w=4M=1,N=4,w=4M=4,N=4,w=4
Figure 5.18: Symbol error probability vs. peak power with Rayleigh fading,Q = 8, R = 100
Ω, T0 = 290 K, Pb = −90 dBW, andRb = 100 Mbps.
Rayleigh fading in Figure 5.22.
5.2.0.6 Log-normal fading
Again, with log-normal fading, Monte Carlo simulation was chosen, and the results are
shown below in Figures 5.20 and 5.21. The MIMO system clearly exhibits superior performance
to the SISO system, as we have come to see.
Figure 5.22 shows that the optimal detector only slightly outperforms equal gain combining
in the log-normal fading case.
66
−90 −85 −80 −75 −70 −65 −60 −55 −50 −45 −4010
−6
10−5
10−4
10−3
10−2
10−1
100
P,dBW
Ps
M=1,N=1,w=1, Pave
, EGCM=1,N=1,w=1, P
ave, OGC
M=1,N=4,w=1, Ppeak
, EGCM=1,N=4,w=1, P
peak, OGC
M=4,N=1,w=4, Pave
, EGCM=4,N=1,w=4, P
ave, OGC
M=4,N=4,w=4, Ppeak
, EGCM=4,N=4,w=4, P
peak, OGC
Figure 5.19: Symbol error probability vs. signal power with Rayleigh fading using equal gain
combining (EGC) or optimal gain combining (OGC).Q = 8, R = 100 Ω, T0 = 290 K,
Pb = −90 dBW, andRb = 100 Mbps.
−90 −85 −80 −75 −70 −65 −60 −55 −50 −45 −4010
−6
10−5
10−4
10−3
10−2
10−1
100
Pave
,dBW
Ps
M=1,N=1,w=1M=4,N=1,w=1M=1,N=4,w=1M=4,N=4,w=1M=1,N=1,w=4M=4,N=1,w=4M=1,N=4,w=4M=4,N=4,w=4
Figure 5.20: Symbol error probability vs. average power with log-normal fading,S.I. = 1.0,
Q = 8, R = 100 Ω, T0 = 290 K, Pb = −90 dBW, andRb = 100 Mbps.
67
−90 −85 −80 −75 −70 −65 −60 −55 −50 −45 −4010
−6
10−5
10−4
10−3
10−2
10−1
100
Ppeak
,dBW
Ps
M=1,N=1,w=1M=4,N=1,w=1M=1,N=4,w=1M=4,N=4,w=1M=1,N=1,w=4M=4,N=1,w=4M=1,N=4,w=4M=4,N=4,w=4
Figure 5.21: Symbol error probability vs. peak power with log-normal fading,S.I. = 1.0,
Q = 8, R = 100 Ω, T0 = 290 K, Pb = −90 dBW, andRb = 100 Mbps.
−90 −85 −80 −75 −70 −65 −60 −55 −50 −45 −4010
−6
10−5
10−4
10−3
10−2
10−1
100
P,dBW
Ps
M=1,N=1,w=1, Pave
, EGCM=1,N=1,w=1, P
ave, OGC
M=1,N=4,w=1, Ppeak
, EGCM=1,N=4,w=1, P
peak, OGC
M=4,N=1,w=4, Pave
, EGCM=4,N=1,w=4, P
ave, OGC
M=4,N=4,w=4, Ppeak
, EGCM=4,N=4,w=4, P
peak, OGC
Figure 5.22: Symbol error probability vs. signal power with log-normal fading using equal
gain combining (EGC) or optimal gain combining (OGC).Q = 8, R = 100 Ω, T0 = 290 K,
Pb = −90 dBW, andRb = 100 Mbps.
Chapter 6
Capacity of the MIMO FSO system using
MPPM in the Poisson regime
In this chapter, we investigate the information-theoretic capacity for the MIMO MPPM FSO
channel. This is broken down into different cases using the same assumptions about the system
in the Poisson regime as before in Chapter 5.
6.1 No background radiation
For the channel capacity of the multipulse case, we first start by considering the mutual
information betweenZ, the random array of counts for theQ slots, and the input symbolX:
I(X; Z) = H(X)−H(X|Z) (6.1)
For each modulator symbolXj, w slots will be ‘on’ slots, andQ − w will be off. There is
a probability that allN of the detectors will generate zero photoelectrons fori of thew slots,
in which case the detector must choose from(
Q−w+ii
)equally probable symbols. Since we are
assuming no background radiation, uncertainty exists only wheni > 0 of thew column sums
for ‘on’ slots equal zero at the detector.
Since this is a symmetric channel, mutual information is maximized when the input symbols
are equiprobable, and the capacity is given by
68
69
C = H(X)−w∑
i=1
H(X|i out of w column counts = 0)×
P (i out of w column counts = 0) (6.2)
= log2
(Q
w
)−
w∑i=1
log2
(Q− w + i
i
)P (i out of w column counts = 0) (6.3)
We repeat the argument developed in Chapter 5, that with equal gain combining, the sum
over all receivers for each ‘on’ slot is a Poisson random variable with a parameter ofλon =
ηPThfM
∑Nn=1
∑Mm=1 a2
nm, and an ‘on’ slot will have zero photoelectrons with probabilityP (Sq =
0) = p = e−λon.
Treating each slot count as an independent random variable, we get
P (i out of w column counts = 0) =w∑
i=1
(w
i
)pi(1− p)w−i (6.4)
and thus
H(X|Z) =w∑
i=1
log2
(Q− w + i
i
)(w
i
)pi(1− p)w−i (6.5)
The last term can be expanded using the binomial expansion formula, which after simplifi-
cation gives
H(X|Z) =w∑
i=1
w−i∑
l=0
(−1)l log2
(Q− w + i
i
)(w
i
)(w − i
l
)pi+l (6.6)
Replacingp gives the general formula for capacity. Observe that the capacity,C depends on
fading,A:
CA = log2
(Q
w
)−
w∑i=1
w−i∑
l=0
(−1)l log2
(Q− w + i
i
)×
(w
i
)(w − i
l
)e−ηPT
∑Mm=1
∑Nn=1 a2
nm(i+l)/hfM (6.7)
There are two different metrics we will use when analyzing the capacity of this system: er-
godic capacity and outage probability. The ergodic capacity is simply the time average capacity
of a channel, whereas the outage probability is simply the probability that a single realization
of the channel will cause the instantaneous capacity to drop below some threshold (one-half of
70
the maximum attainable capacity in our analysis). If a system designer is coding at a rate equal
to this threshold, and the instantaneous capacity is below the threshold, then it is not possible to
drive the error probability arbitrarily low during that instant.
The ergodic capacity, averaged over the fading distributionfA(anm) (which we will do in
the following sections), can be expressed as such:
EA[C] = log2
(Q
w
)−
w∑i=1
w−i∑
l=0
(−1)l log2
(Q− w + i
i
)(w
i
)(w − i
l
)×
M∏m=1
N∏n=1
∫ ∞
0
fA(anm)e−ηPTa2nm(i+l)/hfMdanm (6.8)
Outage capacity is computed differently for the various cases, and will be discussed in each
individual section.
6.1.1 No background, no fading
In the non-fading case,∑M
m=1
∑Nn=1 a2
nm = MN , and ergodic capacity can be expressed as
EA[C] = log2
(Q
w
)−
w∑i=1
w−i∑
l=0
(−1)l log2
(Q− w + i
i
)(w
i
)(w − i
l
)e−ηPTN(i+l)/hf (6.9)
Here, just as with the error analysis, there is no advantage to increasing the number of
lasers in the absence of fading. However, performance improves with increasingN , due to the
increase in the overall aperture size. The ergodic capacity is shown in Figures 6.1 and 6.2. For
consistency with the analysis and plots developed in Chapter 5, this is plotted for aQ = 8
system with different values ofM , N , andw all in 1,4.In the plots for ergodic capacity, we can notice that at low power, the capacity asymptotically
approaches zero, and as power is increased, the expected capacity for the channel asymptotically
approacheslog2
(Qw
), which is 3.00 and approximately 6.13 bits per channel use for the(Q,w) =
(8, 1) and(8, 4) systems, respectively.
For the system to experience an outage as we are defining it, its instantaneous capacity would
have to drop below the coding rate, which we are allowing to be0.5 log2
(Qw
). For the non-fading
case,∑M
m=1
∑Nn=1 a2
nm = MN which removes the conditioning from (6.8). In fact, capacity is
deterministic in this case, and is equivalent to (6.9). Therefore, we can look at Figures 6.1 and
71
−220 −215 −210 −205 −200 −195 −190 −185 −180 −175 −1700
1
2
3
4
5
6
7
PTb,dBJ
E[C
], bi
ts/c
hann
el u
se
M=1,N=1,w=1M=4,N=1,w=1M=1,N=4,w=1M=4,N=4,w=1M=1,N=1,w=4M=4,N=1,w=4M=1,N=4,w=4M=4,N=4,w=4
Figure 6.1: Ergodic capacity vs. average power with no fading, no background radiation, and
Q = 8.
−220 −215 −210 −205 −200 −195 −190 −185 −180 −175 −1700
1
2
3
4
5
6
7
PTb,dBJ
E[C
], bi
ts/c
hann
el u
se
M=1,N=1,w=1M=4,N=1,w=1M=1,N=4,w=1M=4,N=4,w=1M=1,N=1,w=4M=4,N=1,w=4M=1,N=4,w=4M=4,N=4,w=4
Figure 6.2: Ergodic capacity vs. peak power with no fading, no background radiation, and
Q = 8.
72
6.2 and state that the outage probability curve is simply an inverted unit step function, equal to
one for power levels when the capacity is below0.5 log2
(Qw
)and equal to zero for power levels
when capacity is above0.5 log2
(Qw
). This is shown in Figures 6.3 and 6.4. One can note the 6
dB advantage to havingN = 4 versusN = 1, due to increased aperture size.
−220 −215 −210 −205 −200 −195 −190 −185 −180 −175 −1700
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
PTb,dBJ
Pou
t
M=1,N=1,w=1M=4,N=1,w=1M=1,N=4,w=1M=4,N=4,w=1M=1,N=1,w=4M=4,N=1,w=4M=1,N=4,w=4M=4,N=4,w=4
Figure 6.3: 50% outage probability vs. average power with no fading, no background radiation,
andQ = 8.
6.1.2 No background, Rayleigh fading
To find the ergodic capacity with Rayleigh fading, we letfA(anm) be the Rayleigh p.d.f. and
perform theM ×N integrations over all values ofanm:
EA[C] = log2
(Q
w
)−
w∑i=1
w−i∑
l=0
log2
(Q− w + i
i
)(w
i
)(w − i
l
)(−1)l ×
N∏n=1
M∏m=1
∫ ∞
−∞e−ηPrT
∑Mm=1
∑Nn=1 a2
nm(i+l)/hfMQ2anme−a2nmdanm (6.10)
Just as in the development for Chapter 5, we assume theanm terms are all independent, so
we can replace them with a generic fading variablea, and rewrite in the following form:
73
−220 −215 −210 −205 −200 −195 −190 −185 −180 −175 −1700
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
PTb,dBJ
Pou
t
M=1,N=1,w=1M=4,N=1,w=1M=1,N=4,w=1M=4,N=4,w=1M=1,N=1,w=4M=4,N=1,w=4M=1,N=4,w=4M=4,N=4,w=4
Figure 6.4: 50% outage probability vs. peak power with Rayleigh fading, no background radi-
ation, andQ = 8.
EA[C] = log2
(Q
w
)−
w∑i=1
w−i∑
l=0
log2
(Q− w + i
i
)(w
i
)(w − i
l
)×
(−1)l
[∫ ∞
0
2ae−[ηPTa2(i+l)+1]/hfMda
]MN
(6.11)
This then integrates to give the following closed-form expression for the ergodic capacity
with respect to Rayleigh fading:
EA[C] = log2
(Q
w
)−
w∑i=1
w−i∑
l=0
log2
(Q− w + i
i
)(w
i
)(w − i
l
)×
(−1)l
[1
1 + ηPT (i + l)/hfM
]MN
(6.12)
This is plotted in Figures 6.5 and 6.6. As can be seen from the plots, transmitterandreceiver
diversity are beneficial in a fading environment. It is also interesting to note that while the
(M, N) = (4, 1) system shows a superior performance to the(1, 1) system, the(4, 4) system
is only slightly better than the(1, 4) system. This suggests that diversity is very helpful in a
fading environment to achieve a high capacity, but that adding diversity at both ends of the link
74
yields a diminishing return in terms of channel capacity. Just as in the performance analysis, the
plots also indicate that if one were choosing between multiple transmitters or multiple receivers,
it would be more advantageous to use the extra resources at the receiver, where the increased
aperture size can be exploited (assuming the total power at the transmitter array is fixed with
respect to the number of transmitters).
−220 −215 −210 −205 −200 −195 −190 −185 −180 −175 −1700
1
2
3
4
5
6
7
PTb,dBJ
E[C
],bits
/cha
nnel
use
M=1,N=1,w=1M=4,N=1,w=1M=1,N=4,w=1M=4,N=4,w=1M=1,N=1,w=4M=4,N=1,w=4M=1,N=4,w=4M=4,N=4,w=4
Figure 6.5: Ergodic capacity vs. average power with Rayleigh fading, no background radiation,
andQ = 8.
The outage probability curves for the Rayleigh fading case, shown in Figures 6.7 and 6.8, are
important in that they once again show full transmitter and receiver diversity in their respective
slopes.
6.1.3 No background, log-normal fading
For log-normal fading, the ergodic capacity analysis is remarkably similar to the Rayleigh
fading case. We replace the log-normal fading distribution, (2.1) forfA(anm) in (6.8), and
perform theM × N -fold integration. Unfortunately, when assessing the ergodic capacity, the
closed-form solution is not as easily found as in the Rayleigh case. However, just as with
Section 5.1.1.3, we can solve the integral numerically to obtain the desired results. The result
of this procedure is shown in Figures 6.9 and 6.10.
75
−220 −215 −210 −205 −200 −195 −190 −185 −180 −175 −1700
1
2
3
4
5
6
7
PTb,dBJ
E[C
],bits
/cha
nnel
use
M=1,N=1,w=1M=4,N=1,w=1M=1,N=4,w=1M=4,N=4,w=1M=1,N=1,w=4M=4,N=1,w=4M=1,N=4,w=4M=4,N=4,w=4
Figure 6.6: Ergodic capacity vs. peak power with Rayleigh fading, no background radiation,
andQ = 8.
−195 −190 −185 −180 −175 −170 −165 −160 −15510
−4
10−3
10−2
10−1
100
PTb,dBJ
Pou
t
M=1,N=1,w=1M=4,N=1,w=1M=1,N=4,w=1M=4,N=4,w=1M=1,N=1,w=4M=4,N=1,w=4M=1,N=4,w=4M=4,N=4,w=4
Figure 6.7: 50% outage probability vs. average power with Rayleigh fading, no background
radiation, andQ = 8.
76
−190 −185 −180 −175 −170 −165 −160 −15510
−4
10−3
10−2
10−1
100
PTb,dBJ
Pou
t
M=1,N=1,w=1M=4,N=1,w=1M=1,N=4,w=1M=4,N=4,w=1M=1,N=1,w=4M=4,N=1,w=4M=1,N=4,w=4M=4,N=4,w=4
Figure 6.8: 50% outage probability vs. peak power with Rayleigh fading, no background radi-
ation, andQ = 8.
−220 −215 −210 −205 −200 −195 −190 −185 −180 −175 −1700
1
2
3
4
5
6
7
PTb,dBJ
E[C
],bits
/cha
nnel
use
M=1,N=1,w=1M=4,N=1,w=1M=1,N=4,w=1M=4,N=4,w=1M=1,N=1,w=4M=4,N=1,w=4M=1,N=4,w=4M=4,N=4,w=4
Figure 6.9: Ergodic capacity vs. average power with log-normal fading, no background radia-
tion, Q = 8, andS.I. = 1.0.
77
−220 −215 −210 −205 −200 −195 −190 −185 −180 −175 −1700
1
2
3
4
5
6
7
PTb,dBJ
E[C
],bits
/cha
nnel
use
M=1,N=1,w=1M=4,N=1,w=1M=1,N=4,w=1M=4,N=4,w=1M=1,N=1,w=4M=4,N=1,w=4M=1,N=4,w=4M=4,N=4,w=4
Figure 6.10: Ergodic capacity vs. peak power with log-normal fading, no background radiation,
Q = 8, andS.I. = 1.0.
Once again, we can see that transmitter and receiver diversity is advantageous with respect
to ergodic capacity, with the same general result as with Rayleigh fading: Diversity can be
achieved on both ends of the FSO link, but increasing the number of transmitters alone provides
more improvement in ergodic capacity than increasing the number of receivers alone, due to the
increased aperture size.
The outage probability plots show the very important result that both transmitter and receiver
diversity is achievable in the presence of log-normal fading. A system designer using a MIMO
FSO system, and employing a rateR = 0.5 log2
(Qw
)code could reduce the probability of outage
to a desired level with far less power than with a comparable SISO system.
6.2 Background radiation and no fading
When doing an analysis on the channel capacity with background radiation present, the
problem is complicated even further. In the situation with no background, uncertainty inX
could only occur ifi of the ”on” slots at the transmitter are received as ”off” slots at the receiver.
In the background radiation case, uncertainty exists to a certain extent ineveryreceived symbol,
78
−195 −190 −185 −180 −175 −170 −165 −16010
−5
10−4
10−3
10−2
10−1
100
PTb,dBJ
Pou
t
M=1,N=1,w=1M=4,N=1,w=1M=1,N=4,w=1M=4,N=4,w=1M=1,N=1,w=4M=4,N=1,w=4M=1,N=4,w=4M=4,N=4,w=4
Figure 6.11: 50% outage probability vs. average power with log-normal fading, no background
radiation, andQ = 8.
−190 −185 −180 −175 −170 −165 −160 −15510
−4
10−3
10−2
10−1
100
PTb,dBJ
Pou
t
M=1,N=1,w=1M=4,N=1,w=1M=1,N=4,w=1M=4,N=4,w=1M=1,N=1,w=4M=4,N=1,w=4M=1,N=4,w=4M=4,N=4,w=4
Figure 6.12: 50% outage probability vs. peak power with log-normal fading, no background
radiation, andQ = 8.
79
since each of the column sums has the potential of being non-zero, regardless of which symbol
was sent.
Again, the channel is symmetric, so we can make all symbol probabilities equal to achieve
capacity.
We first use an alternative, but equivalent, definition for mutual information:
I(X; Z) =∑
x
∑z
p(x)p(z|x) log2
(p(z|x)
p(z)
)(6.13)
By symmetry of the channel, we can rewrite the conditional probability ofz for any of
the modulator symbols,X = x0 for example. Therefore, we can eliminate the summation
(averaging) over all of the symbols, giving
I(X; Z) =∑
z
p(z|x0) log2
(p(z|x0)
p(z)
)(6.14)
Upon inspection, this is simply the formula for
EZ|X=x0
[log2
(p(z|x0)
p(z)
)](6.15)
Therefore, to evaluate the capacity of the channel with background radiation, we can per-
form a Monte-Carlo simulation forlog2
(p(z|x0)
p(z)
)and use the following equation:
E[C] = EZ|X=x0
[log2
(p(z|x0)
p(z)
)](6.16)
To perform the simulation, it is necessary to first look at the probability distribution forZ
givenx0. If we defineQ0 to be the set of ”on” slots in modulator symbolx0, andQC0 to be the
complement of that set signifying the ”off” slots, we can state that for symbolx0, the probability
distribution is
p(Z|X = x0) =N∏
n=1
∏
q∈QO
(λon,n + λoff )Znqe−(λon,n+λoff )
Znq!
∏
q∈QCO
(λoff )Znqe−(λoff )
Znq!
(6.17)
Similarly, the distribution forZ given any equiprobable symbolxi with weightw is
p(Z) =∑
allxi
1(Qw
)P (Z|X = xi) (6.18)
80
p(Z) =1(Qw
)∑
allxi
∏Nn=1
[∏q∈Qi
(λon,n + λoff )Znqe−Wλon,n
∏q∈QC
iλ
Znq
off e−Qλoff
]∏
allq Znq!(6.19)
Therefore,
p(Z)
p(Z|X = x0)=
1(Qw
)∑
allxi
∏Nn=1
[∏q∈Qi
(λon,n + λoff )Znq
∏q∈QC
iλ
Znq
off
]
∏Nn=1
[∏q∈QO
(λon,n + λoff )Znq∏
q∈QCO
λZnq
off
] (6.20)
or equivalently,
p(Z)
p(Z|X = x0)=
1(Qw
)∑
allxi
∏Nn=1
[(λon,n + λoff )
∑q∈Qi
Znq(λoff )(∑
all q Znq−∑
q∈QiZnq)
]
∏Nn=1
[(λon,n + λoff )
∑q∈Q0
Znq(λoff )(∑
all q Znq−∑
q∈Q0Znq)
]
(6.21)
We can define a new parameterζn = (λon,n + λoff )/λoff and cancel like terms, to give
p(Z)
p(Z|X = x0)=
1(Qw
)∑
allxi
N∏n=1
[(ζn)(
∑q∈Qi
Znq−∑
q∈Q0Znq)
](6.22)
Plugging this into the ergodic capacity expression gives
E[C] = EZ|X=x0
[log2
(p(z|X = x0)
p(z)
)](6.23)
= EZ|X=x0
[log2
(Q
w
)− log2
∑
allxi
N∏n=1
[(ζn)(
∑q∈Qi
Znq−∑
q∈Q0Znq)
]](6.24)
= log2
(Q
w
)− EZ|X=x0
[log2
∑
allxi
N∏n=1
[(ζn)(
∑q∈Qi
Znq−∑
q∈Q0Znq)
]](6.25)
From this, it becomes easy to devise a programming strategy for simulating the terms in
(6.22) for each trial.
6.2.1 Background radiation, no fading
Figures 6.13 and 6.14 show the ergodic capacity for the non-fading case when background
radiation is present. As expected, increasingM shows no improvement in ergodic capacity, but
increasingN does, just as with the no background case.
81
−200 −195 −190 −185 −180 −175 −170 −165 −1600
1
2
3
4
5
6
7
PTb,dBJ
E[C
],bits
/cha
nnel
use
M=1,N=1,w=1M=4,N=1,w=1M=1,N=4,w=1M=4,N=4,w=1M=1,N=1,w=4M=4,N=1,w=4M=1,N=4,w=4M=4,N=4,w=4
Figure 6.13: Ergodic capacity vs. average power with no fading,Q = 8, andPbT = −170 dBJ.
−200 −195 −190 −185 −180 −175 −170 −165 −1600
1
2
3
4
5
6
7
PTb,dBJ
E[C
],bits
/cha
nnel
use
M=1,N=1,w=1M=4,N=1,w=1M=1,N=4,w=1M=4,N=4,w=1M=1,N=1,w=4M=4,N=1,w=4M=1,N=4,w=4M=4,N=4,w=4
Figure 6.14: Ergodic capacity vs. peak power with no fading,Q = 8, andPbT = −170 dBJ.
82
Unlike the no background case, the outage probability of the background and no fading case
is no longer a deterministic function. Instantaneous capacity can drop below the threshold due
to the probability of not receiving an ‘on’ pulse in an ‘on’ slot, and also due to ‘off’ slots having
non-zero counts. This gives the recognizable ‘waterfall’ curves in Figures 6.15 and 6.16. Again,
the advantage is clearly in theN = 4 system, independent ofM .
−195 −190 −185 −180 −175 −170 −165 −160 −15510
−3
10−2
10−1
100
PTb,dBJ
Pou
t
M=1,N=1,w=1M=4,N=1,w=1M=1,N=4,w=1M=4,N=4,w=1M=1,N=1,w=4M=4,N=1,w=4M=1,N=4,w=4M=4,N=4,w=4
Figure 6.15: 50% outage probability vs. average power with no fading,Q = 8, andPbT =
−170 dbJ.
6.2.2 Background radiation, Rayleigh fading
Figures 6.17 and 6.18 show the ergodic capacity when Rayleigh fading and background
radiation are both present. Again, we notice the gains attributable to transmitter and receiver
diversity, as well as to the increase in aperture size at the receiver, showing the superiority of
MIMO systems.
When Rayleigh fading and background radiation are present, the outage probability plots
are similar in appearance to the corresponding symbol error probability plots from Chapter 5.
Here, we again see evidence of full transmitter and receiver diversity when looking at the slopes
of the lines for the different systems.
83
−195 −190 −185 −180 −175 −170 −165 −160 −15510
−3
10−2
10−1
100
PTb,dBJ
Pou
t
M=1,N=1,w=1M=4,N=1,w=1M=1,N=4,w=1M=4,N=4,w=1M=1,N=1,w=4M=4,N=1,w=4M=1,N=4,w=4M=4,N=4,w=4
Figure 6.16: 50% outage probability vs. peak power with no fading,Q = 8, andPbT = −170
dBJ.
−200 −195 −190 −185 −180 −175 −170 −165 −1600
1
2
3
4
5
6
7
PTb,dBJ
E[C
],bits
/cha
nnel
use
M=1,N=1,w=1M=4,N=1,w=1M=1,N=4,w=1M=4,N=4,w=1M=1,N=1,w=4M=4,N=1,w=4M=1,N=4,w=4M=4,N=4,w=4
Figure 6.17: Ergodic capacity vs. average power with Rayleigh fading,Q = 8, andPbT =
−170 dBJ.
84
−200 −195 −190 −185 −180 −175 −170 −165 −1600
1
2
3
4
5
6
7
PTb,dBJ
E[C
],bits
/cha
nnel
use
M=1,N=1,w=1M=4,N=1,w=1M=1,N=4,w=1M=4,N=4,w=1M=1,N=1,w=4M=4,N=1,w=4M=1,N=4,w=4M=4,N=4,w=4
Figure 6.18: Ergodic capacity vs. peak power with Rayleigh fading,Q = 8, andPbT = −170
dBJ.
−195 −190 −185 −180 −175 −170 −165 −160 −15510
−6
10−5
10−4
10−3
10−2
10−1
100
PTb,dBJ
Pou
t
M=1,N=1,w=1M=4,N=1,w=1M=1,N=4,w=1M=4,N=4,w=1M=1,N=1,w=4M=4,N=1,w=4M=1,N=4,w=4M=4,N=4,w=4
Figure 6.19: 50% outage probability vs. average power with Rayleigh fading,Q = 8, and
PbT = −170 dbJ.
85
−195 −190 −185 −180 −175 −170 −165 −160 −15510
−6
10−5
10−4
10−3
10−2
10−1
100
PTb,dBJ
Pou
t
M=1,N=1,w=1M=4,N=1,w=1M=1,N=4,w=1M=4,N=4,w=1M=1,N=1,w=4M=4,N=1,w=4M=1,N=4,w=4M=4,N=4,w=4
Figure 6.20: 50% outage probability vs. peak power with Rayleigh fading,Q = 8, andPbT =
−170 dBJ.
6.2.3 Background radiation, log-normal fading
For log-normal fading with background radiation, the ergodic capacity and outage probabil-
ity curves are once again indicative of the strength of the MIMO system over its SISO counter-
part. We see a diversity advantage of increasing the number of transmitters and receivers.
86
−200 −195 −190 −185 −180 −175 −170 −165 −1600
1
2
3
4
5
6
7
PTb,dBJ
E[C
],bits
/cha
nnel
use
M=1,N=1,w=1M=4,N=1,w=1M=1,N=4,w=1M=4,N=4,w=1M=1,N=1,w=4M=4,N=1,w=4M=1,N=4,w=4M=4,N=4,w=4
Figure 6.21: Ergodic capacity vs. average power with log-normal fading,Q = 8, andPbT =
−170 dBJ.
−200 −195 −190 −185 −180 −175 −170 −165 −1600
1
2
3
4
5
6
7
PTb,dBJ
E[C
],bits
/cha
nnel
use
M=1,N=1,w=1M=4,N=1,w=1M=1,N=4,w=1M=4,N=4,w=1M=1,N=1,w=4M=4,N=1,w=4M=1,N=4,w=4M=4,N=4,w=4
Figure 6.22: Ergodic capacity vs. peak power with log-normal fading,Q = 8, andPbT = −170
dBJ.
87
−195 −190 −185 −180 −175 −170 −165 −160 −15510
−6
10−5
10−4
10−3
10−2
10−1
100
PTb,dBJ
Pou
t
M=1,N=1,w=1M=4,N=1,w=1M=1,N=4,w=1M=4,N=4,w=1M=1,N=1,w=4M=4,N=1,w=4M=1,N=4,w=4M=4,N=4,w=4
Figure 6.23: 50% outage probability vs. average power with log-normal fading,Q = 8, and
PbT = −170 dbJ.
−195 −190 −185 −180 −175 −170 −165 −160 −15510
−6
10−5
10−4
10−3
10−2
10−1
100
PTb,dBJ
Pou
t
M=1,N=1,w=1M=4,N=1,w=1M=1,N=4,w=1M=4,N=4,w=1M=1,N=1,w=4M=4,N=1,w=4M=1,N=4,w=4M=4,N=4,w=4
Figure 6.24: 50% outage probability vs. peak power with log-normal fading,Q = 8, and
PbT = −170 dBJ.
Chapter 7
Conclusions
In this thesis, we have analyzed MIMO MPPM FSO systems and developed the ML de-
tection schemes for the Poisson and Gaussian regimes. For the Poisson regime, we see that
the optimal detector in three of the four cases simply combines the received signal from allN
receivers equally, and chooses the symbol with thew highest slot counts. In the case where
background radiation and fading are both present, the optimal detector maximizes a weighted
sum over the ‘on’ and ‘off’ slots, but as we saw from previous research, equal gain combining
achieves essentially equal performance. For the Gaussian regime, the ML detector is also equal
gain combining for all but one scenario: when the system is thermal noise limited and fading is
present.
We have shown that MPPM used with a MIMO FSO system is a robust and effective way
to mitigate the negative effects of atmospheric turbulence. In all but the case where there is no
fading, increasing the number of lasers (MISO) resulted in better error performance, with the
ergodic capacity approaching its maximum value at lower power levels, and a steeper outage
probability, when compared to a SISO system.
In all cases, including non-fading, we demonstrated that increasing the number of receivers
(SIMO) resulted in similar improvements in system performance compared with the MISO sys-
tem. When compared to a MISO system, the SIMO system displayed an overall improvement
in all cases due to the increased aperture size at the receiver.
The MIMO system performed the best in all of the fading cases, and equivalent to SIMO
in the non-fading case. In the case of Rayleigh fading, we were able to ascertain from the
symbol error probability and outage probability plots, that the system achieves full transmitter
88
89
and receiver diversity. The improvements in symbol error probability seen in the Poisson regime
were echoed in the Gaussian regime, with full transmitter diversity being observable in the
presence of Rayleigh fading.
The use of MPPM was a good choice for this system. In the case where average power is
limited, traditional PPM (w = 1) exhibited better error performance than MPPM (w > 1) due
to the distribution of signal power across thew on slots. However, in the case where peak power
is limited, MPPM is the clear winner. MPPM also has the advantage of being more bandwidth
efficient for increasingw until w = Q2
, consuming less unit bandwidth per given bit rate.
There are many possibilities for future work on this topic. Implementation of a decoder
for modulator symbol sets where(
Qw
)is not a power of 2 is a particularly interesting design
challenge which should be addressed. Also, study of different channel coding schemes applied
to the MPPM system, with the goal of achieving coding gain on the channel, is of interest and
should be investigated further.
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