Diversity Optical Communication over the Turbulent Atmospheric Channel
by
Etty J. Shin
B.A.Sc., Computer Engineering
University of Waterloo, 1999
Submitted to the Department of Electrical Engineering and Computer Science
In Partial Fulfillment of the Requirements for the Degree of
Master of Science in Electrical Engineering
at the
Massachusetts Institute of Technology
BARKE R
June 2002
OF TECEHNOLOGY
©2002 Massachusetts Institute of Technology
All rights reserved
Signature of Author.....
JUL 3 1 2002
LIBRARIES
............................................
Deprtment of Electrical Engineering and Computer Science
May 10, 2002
C ertified by...................,..............
...............................
.......................
Vincent W.S. Chan
Science and
Computer
&
Engineering
of
Electrical
Joan and Irwin M. Jacobs Professor
Aeronautics & Astronautics, Director of Laboratory for Information and Decision
Systems
Thesis Supervisor
Accepted
by ....................................
Arthur C. Smith
Chairman, Department Committee on Graduate Students
Diversity Optical Communication over the Turbulent Atmospheric
Channel
by
Etty J. Shin
Submitted to the Department of Electrical Engineering and Computer Science
on May 10, 2002 in Partial Fulfillment of the
Requirements for the Degree of Master of Science in
Electrical Engineering
ABSTRACT
Optical communication through the atmosphere provides a means for high data rate
communication (gigabits per second) over relatively short distances (kilometers).
However, the turbulence in the atmosphere leads to fades of varying depths, some of
which may lead to heavy loss of data. For example, at a data rate of 2.5 gigabits per
second, as many as 250 x 10 consecutive bits can be lost in a single 100 millisecond
deep fade. It is feasible to recover the data loss in these fades via error correcting codes
but only via substantial hardware complexities and processing delays. Thus, it would be
of great benefit if we could reduce the probability of a fade.
In this thesis, we examine spatial diversity at the transmitter and receiver as well as time
diversity as a means to mitigate the short-term loss of signal strength. Using direct
detection receivers and binary pulse position modulation as an example, we derive the
outage probability of several diversity systems: receiver diversity systems that use Equal
Gain Combining, Optimal Combining, or Selection Combining, transmitter diversity
systems, combined transmitter and receiver diversity systems, and time diversity systems.
The outage probabilities for the various diversity systems are compared and the power
gain of using these diversity systems is established. It is found that the power gain of
diversity systems over non-diversity systems is substantial and that Equal Gain
Combining has performance almost equivalent to Optimal Combining.
Thesis Supervisor: Vincent W.S. Chan
Title: Joan and Irwin M. Jacobs Professor of Electrical Engineering & Computer Science
and Aeronautics & Astronautics, Director of Laboratory for Information and Decision
Systems
2
Acknowledgments
I would like to thank my parents, brother, and Dennis Lee for their support and
encouragement. They have been great friends and were understanding in all respects. I
would also like to thank my thesis advisor Vincent W.S. Chan for his guidance and
support. This research was possible through the financial support of the Defense
Advanced Research Projects Agency (DARPA) under the Steered Agile Beam (STAB)
Program.
3
Table of Contents
1 INTRODUCTION AND BACKGROUND ............................................................
9
1.1 INTRO D U CTION ..........................................................................................................
9
1.2 KOLMOGOROV TURBULENCE MODEL....................................................................10
1.3 HUYGENS-FRESNEL PRINCIPLE AND EXTENDED HUYGENS-FRESNEL PRINCIPLE..... 11
1.4 INTRODUCTION TO FOLLOWING CHAPTERS .........................................................
13
2 DIVERSITY SYSTEMS ........................................................................................
15
2.1 GENERAL SPATIAL DIVERSITY SYSTEM................................................................15
2.2 TIM E D IVERSITY SYSTEM ........................................................................................
16
2.3 FADING, RECEIVER TYPE, AND MODULATION SCHEME.......................................16
2.4 C OM BINING M ETHODS ..........................................................................................
18
3 ANALYSIS OF DIVERSITY SYSTEMS.............................................................
3.1 PERFORMANCE METRICS ......................................................................................
3.2 MAXIMUM LIKELIHOOD DECISION RULE..............................................................22
19
19
3.3 No DIVERSITY (ONE TRANSMITTER, ONE RECEIVER).........................................23
3.4 R ECEIVER DIVERSITY ..........................................................................................
26
3.4.1 Receiver Diversity with Equal Gain Combining........................................26
3.4.2 Receiver Diversity with Optimal Combining ............................................
41
3.4.3 ComparingPerformanceGain of Receiver Diversity with Optimal
Combining and Receiver Diversity with Equal Gain Combining.......................56
3.4.4 Receiver Diversity with Selection Combining...........................................69
3.5 COMBINED TRANSMITTER AND RECEIVER DIVERSITY WITH EQUAL GAIN
C OM B IN IN G ....................................................................................................................
74
3.6 TIME DIVERSITY AT RECEIVER .............................................................................
75
4 CONCLUSIONS......................................................................................................
76
A INTUITIVE UNDERSTANDING OF FADING MODEL .................................
79
B LOG NORMAL APPROXIMATIONS...............................................................
B.1 MEAN AND VARIANCE OF U WHERE Z=EU
4
XI+. . .+XN AND
81
Z IS LOG NORMAL .... 81
B.2 MEAN AND VARIANCE OF U WHERE Z'=EU' =
2
+.
..
+0N2 WHERE Z IS LOG NORMAL
.......................................................................................................................................
83
C R EC EIV ED POW ER .................................................................................................
85
C. I RECEIVED SIGNAL POWER ......................................................................................
C.2 RECEIVED BACKGROUND NOISE POW ER.............................................................
BIBLIO G R A PH Y .......................................................................................................
5
85
86
90
List of Figures
Figure 1.1 Physical Setup for Huygens-Fresnel Principle.............................................12
Figure 2.1 Spatial D iversity System Setup....................................................................
Figure 2.2 Tim e Diversity System Setup......................................................................
15
16
Figure 3.1 General Probability of Error Curve for AWGN Channels..........................20
Figure 3.2. System w ith No D iversity ..........................................................................
23
Figure 3.3 System with Receiver Diversity and Equal Gain Combining.....................26
Figure 3.4 Probability Density Function of the Sum of 9 Log Normal Random Variables
ai where ln(xi) c< N(-20x 2,4Ax 2 ) and ax=0. 3 ............................. ................... ...... . . 35
Figure 3.5 Probability Density Function of the Sum of 9 Log Normal Random Variables
oc where ln(a ) c N(-2 yx 2,4 yx 2) and Tx=0. 5 ............................. ................... ...... . . 36
Figure 3.6 Outage Probability For Two Receiver Diversity and Equal Gain Combining,
KEGC= I ........--..---.....-..--..--..-....
--------------. -- -- -- ---...............................................
38
Figure 3.7 Outage Probability For Four Receiver Diversity and Equal Gain Combining,
KEGC=
-............-..--..-..........-....------------............... ................................................
38
Figure 3.8 Outage Probability For Nine Receiver Diversity and Equal Gain Combining,
KEGC= I ...........--
------
...........-...--------------.
..................................................
39
Figure 3.9 Outage Probability, KEGC= 1, X=O. 1, Log Normal Approximation............40
Figure 3.10 Outage Probability, KEGC=1, aX=0.3, Log Normal Approximation..........40
Figure 3.11 Outage Probability, KEGC=1, =X=0.5, Log Normal Approximation..........41
Figure 3.12 System with Receiver Diversity and Optimal Combining.......................42
Figure 3.13 Probability Density Function of the Sum of 9 Log Normal Random
Variables
2
uj
where ln(i 2 ) oc N(-4yx2, 16(yx 2) and FX=0.3 ....................................
51
Figure 3.14 Probability Density Function of the Sum of 9 Log Normal Random
Variables oQ where lnai2 ) cc N(-4Gx 2 , l6o2) and x=O.I ...................................
52
Figure 3.15 Outage Probability For Two Receiver Diversity and Optimal Combining,
KEGC= I ......---------------------------------------------------------------...............................................
53
Figure 3.16 Outage Probability For Four Receiver Diversity and Optimal Combining,
KEGC= 1 ...........--.....................
-...----------.---------...-................................................
54
Figure 3.17 Outage Probability For Nine Receiver Diversity and Optimal Combining,
KEGC= I ...........----.-------------
.....-.-.....-..-------
.
.
54
Approximation..........55
Approximation..........56
Approximation..........60
Approximation..........61
------------................................................
Figure 3.18 Outage Probability, KEGC= 1, TX=0.1, Log Normal
Figure 3.19 Outage Probability, KEGC= 1, yX=0.3, Log Normal
Figure 3.20 Outage Probability, KEGC= 1, X=0. 1, Log Normal
Figure 3.21 Outage Probability, KEGC= 1, X=0.3, Log Normal
Figure 3.22 Outage Probability, KEC= , X=0.5 .............................................................
Figure 3.23 Power Gain, KEGC= , TX=O.1, Log Normal Approximation .....................
Figure 3.24 Power Gain, KEGC=1, oX=0.3, Log Normal Approximation .....................
6
61
63
63
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
Figure
3.25
3.26
3.27
3.28
3.29
3.30
3.31
3.32
3.33
64
Pow er G ain, KEGC= , 6%=0.5 ........................................................................
Power Gain as the Number of Receivers Approaches Infinity ................. 65
Power Gain, KEGC= , TX 1, Log Normal Approximation ..................... 66
Power Gain, KEGC=I, GX=0.3, Log Normal Approximation ..................... 66
Power Gain, KEGC=I, yT=0.5, Log Normal Approximation ..................... 67
Power Gain for EGC as the Number of Receivers Approaches Infinity.......68
Power Gain for OC as the Number of Receivers Approaches Infinity ......... 68
System with Receiver Diversity and Selection Combining ...................... 69
Outage Probability of a Two Receiver Diversity with Selection Combining
System and a No Diversity System, Klbranch= I----------------......................................71
Figure 3.34 Outage Probability of a Four Receiver Diversity with Selection Combining
System and a No Diversity System, Klbranch= I-...................................72
Figure 3.35 Outage Probability of a Nine Receiver Diversity with Selection Combining
System and a No Diversity System, Klbranch= --------.............
.............. .....
72
Figure 3.36 System with Transmitter and Receiver Diversity and Equal Gain Combining
...................................................................................................................................
74
Figure A. I Visualization of Wave Propagation from Transmitter to Receiver............79
Figure C. 1 Geometry of Single Transmitter, Single Receiver Setup ...........................
85
Figure C.2 Geom etry of Single Receiver ....................................................................
86
Figure C.3 Field of View and Diffraction Limited Angle of a No Diversity Receiver.... 88
Figure C.4 Field of View and Diffraction Limited Angle of a Multiple Receiver System
...................................................................................................................................
89
7
List of Tables
Table
Table
Table
Table
Table
3.1
3.2
3.3
3.4
3.5
Outage Probability of No Diversity System.................................................
Outage Probability of Receiver Diversity with Equal Gain Combining ..........
Outage Probability of Receiver Diversity with Optimal Combining .......
Power Gain of Receiver Diversity with Equal Gain Combining .................
Power Gain of Receiver Diversity with Optimal Combining .....................
8
57
57
58
58
59
Chapter 1
Introduction and Background
1.1 Introduction
Optical communication through the Earth's atmosphere can support high data rate
communication (on the order of gigabits per second) over short distances (on the order of
a kilometer). However, fades due to air turbulence can span several milliseconds to one
tenth of a second, which in turn can lead to loss of a large number of consecutive bits.
For example, at a data rate of 2.5 gigabits per second, as much as 250 x 106 bits can be
lost in a single 100 millisecond deep fade. The durations of the fades roughly equal the
time it takes for crosswinds or thermally induced air moments to move the turbules across
the laser beam. Error correcting codes can be used to correct for errors due to fades but
they will require an impractically large interleaver. Thus, it would be of great benefit if
we could reduce the probability of a fade and thus of loss of data.
The goal of this thesis is to analyze several system methods to mitigate these fades and to
quantify the possible gains of these methods. The methods we will explore are spatial
diversity (at transmitter and receiver) and time diversity coupled with various signal
combining methods.
The optical communication system that will be discussed in this thesis involves a
modulated optical wave that propagates through the Earth's atmosphere. If we consider
9
the atmosphere to be a vacuum, the transmitted signal undergoes no random attenuation
or phase modification and thus the performance analysis is relatively simple. However,
the Earth's atmosphere is quite a different medium from free space.
conditions such as fog, rain, snow and hail cause absorption.
Bad weather
Even in clear weather
conditions, the mixing of eddies of air with slightly different temperatures (on the order
of I degree Kelvin) leads to slight variations in refractive index (on the order of 10-6).
Although this fluctuation may seem small in absolute value, it has a great impact on
optical communications.
Only optical communication systems through air turbulence,
without fog, rain, snow or hail, are analyzed in this thesis.
1.2 Kolmogorov Turbulence Model
The Kolmogorov model describes the statistics of temperature and refractive index
fluctuations (atmospheric turbulence). Turbulent eddies beginning with outer scale size
Lo (typically approximately 10-100m) transfer their energy to smaller eddies which in
turn transfer their energy to even smaller eddies. When the eddy eventually reduces to
inner scale size Io (typically 10-3M), viscous damping takes over and dissipation occurs.
Within the inertial sub-range 1o to Lo, Kolmogorov showed that the temperature structure
function follows
D, (T)=< (T(T-0 + T) - T(=CT|
))2>
, <<
",3
<< L
where < . > denotes expectation and that the spectral distribution of the structure function
follows
7T(
2)=
1
CT(r
exp(- jk -F T(
(2x)
2
1111/3
0.033 K
CT ,
ff2
-<<
Lo
10
(1.2)
-
2IT
K <<
10
K- 2
where CFT(AT(-rO+ r-)T(2 )) for AT =T- <T>
structure constant.
CT2
CT2
n
r
is the temperature
is approximately 10-4 for very weak turbulence and I for very
strong turbulence.
When the inertial sub-range contributes significantly to the propagation fluctuation, then
the refractive index can be modeled as
(1.3)
n(F)=1 + An(r)
where An(r)
10-6 AT(F).
Thus a change in temperature of I Kelvin results in a change
Thus, the refractive index has similar statistics to the
of 10-6 in refractive index.
temperature statistics. Specifically,
Dn,
(r
=C,2
(Dn, (K) = 0.0331K
where C,2
=10-" CT
2
/ 3,
1/3
Cn 2,
(1.4)
10 << r << L
IT
L0
-
2r
(1.5)
In
is typically 10-6 for very weak turbulence and 10-12 for very strong
turbulence.
1.3 Huygens-Fresnel Principle and Extended Huygens-Fresnel
Principle
The Huygens-Fresnel Principle allows us to represent an optical wave after it has traveled
in free space from one plane to another parallel plane a distance L away. It is based on
the scalar wave equation and takes into account diffraction. The principle is stated as
11
follows: given a quasi-monochromatic optical field Ui(p,t) in the plane z=O, the field
after it has propagated to the plane z=L is given by
)l dp
1fJUj(,t-)exp[jk(L+
jAL RC2
UO',)=
(4.6)
where p and p' are the coordinate vectors in the z=O and z=L planes respectively, and
R, is the transmitting pupil area as shown in Figure 1.1.
L
RI
Figure 1.1 Physical Setup for Huygens-Fresnel Principle
The Extended Huygens-Fresnel Principle extends the Huygens-Fresnel Principle to take
into account atmospheric turbulence. The field at z=L is given by
UO(',t)=
ja
where X and
#
I
Ui(,t L)exp[jk(L+
iAL
2L
)]exp[x(P',;)+jO(P',P)] d#
(1.7)
are random variables that model amplitude and phase fluctuation as the
field travels through atmospheric turbulence.
We can assume that X is a Gaussian
process with known (measurable) mean and variance. Kolmogorov turbulence leads to
the following statistics for the log-amplitude fluctuation X
(otherwise known as
scintillation) in the case of horizontal propagation where the turbulence strength is
uniform over the path:
<>= -o
12
and
(1.8)
var(X) = q. = 0.1 24k7 1 6 C L""'6
The variance is always less than 0.5; larger values do not occur.
(1.9)
This is due to a
phenomenon called saturation of scintillation. Typical values of q2 lie between 0.01 and
0.25. The phase
one. Thus,
#
# is
a Gaussian random variable whose variance is much greater than
is modeled as uniformly distributed on [0, 27c].
If we consider a point source or source that is much smaller than the atmospheric
coherence length, then the turbulence factor exp(X+j#) can be factored out of the integral
in (1.7). Thus, the output field can be written as
U
"P"=jAL
exp[(-')+j#(')] fRU(y,t-)exp[Jk(L+
P
epj(+2L
)(J dt
(1.10)
The fading reduces to a multiplicative amplitude and phase factor where the amplitude
factor is log normal and the phase factor is uniformly distributed. Due to the log normal
fading statistics of the atmosphere, when o 2 > 0.18, fade depths of 10 dB or deeper occur
with probability 1% or more. See Appendix A for an intuitive understanding of why the
amplitude factor is modeled as log normal.
1.4 Introduction to Following Chapters
In this chapter, we discussed the Kolmogorov model for turbulence in the atmosphere and
how the Huygens-Fresnel Principle is extended to account for the turbulence. We also
established that optical communication through air turbulence may lead to the loss of a
large number of contiguous bits due to the fades. Thus, there is a need to mitigate the
effects of this fading by somehow reducing the probability of a fade. In this thesis, will
explore spatial and time diversity in order to do exactly this.
13
In Chapter 2, we describe the general setup for spatial diversity and time diversity
systems along with assumptions that are made in the rest of the thesis. In Chapter 3, we
first analyze a system with no diversity so that it can be used as a basis of comparison for
all diversity systems.
Next, we analyze receiver diversity systems and consider the
possible gain of using multiple receivers under Equal Gain Combining, Optimal
Combining, and Selection Combining.
We then proceed to analyze a combined
transmitter and receiver diversity system when received Airy patterns can be resolved.
Finally, we will analyze a time diversity system. In Chapter 4, conclusions are made
regarding the performance gain of diversity systems for optical communication through
atmospheric turbulence.
14
Chapter 2
Diversity Systems
2.1 General Spatial Diversity System
As shown in Figure 2.1, a general spatial diversity system uses multiple transmitters to
transmit the signal and multiple receivers to receive the faded signals. In general, there
can be M transmitters and N receivers where M, N >1 and at least one of M or N are
greater than one. The outputs of the N receivers may be combined in any desired way to
produce the final observation(s) that is used to make a decision on whether a 0 bit or 1 bit
was sent. Spatial diversity systems make use of the fact that as M and N are increased, it
becomes less likely for all of the paths to be severely faded simultaneously.
By
appropriate selection of the type of combining, for example, as a function of the amount
of fading on each of the paths, the diversity system can maximize the performance.
Transmitter
1
Transmitter
7.
Receiver I
Receiver 2
Com-bining
Decision
Transmitter M
Receiver N
Figure 2.1 Spatial Diversity System Setup
15
2.2 Time Diversity System
Time diversity may involve the transmitter sending the signal multiple times, say N
times, separated by fixed time periods T that are much larger than a typical deep fade
duration. Figure 2.2 shows time diversity graphically with N=3. The receiver receives N
faded copies of the signal, combines them in some appropriate way, and determines
whether a '0' or '1' was sent. The hope of time diversity, just as in spatial diversity, is
that at least one of the N received versions of the signal will not be deeply faded.
Repeat Interval T
|]]I|I
E
Transmitter
bits
bits
Receiver
bits
Detect
Combine
-
Detect
Delay
T
Detect
Figure 2.2 Time Diversity System Setup
Another form of time diversity is using an error correcting code with an interleaver
several times longer than a fade. If enough unfaded symbols are received, the message
can be successfully recovered. This form of time diversity will not be considered in this
thesis.
2.3 Fading, Receiver Type, and Modulation Scheme
In general, if each transmitter and receiver pupil size is less than the coherence length,
then the fading experienced by a wave as it propagates through atmospheric turbulence
from a transmitter to a receiver is modeled by a random multiplicative factor e+JO. As
16
described in Chapter 1, the amplitude fading portion er is log normal distributed where g
is Gaussian, N(-oai, oqj), and the random phase 0 is uniformly distributed over [0, 27T].
The power fading factor, is then given by the log normal random variable a=e2 X where
ln(a) has a Gaussian distribution, N(-2ao, 4orY). In the spatial diversity systems we will
be considering, we define "j to be the time-varying power fading factor from transmitterj
to receiver i. In the time diversity system, we define "- to be the time-varying power
fading factor in time slot i.
The reason that we focus only on the power fading statistics and not the phase statistics is
that this thesis will be concerned with incoherent receivers.
Incoherent receivers,
otherwise known as direct detection receivers, detect only the energy of the incident field,
ignoring the phase portion of the field. Coherent receivers on the other hand utilize both
the amplitude and phase of the received field. However, they are significantly more
complex and difficult to implement. We assume that the receiving detector area sizes are
fixed and that they do not change when diversity is used or not used.
In this thesis, we consider binary pulse position modulation (BPPM) as the modulation
scheme for simplicity of the optimum receiver. On-off keying would require a threshold
that needs to be estimated from measured parameters at the receiver. In BPPM, the
transmitting laser is turned on for the first half of the bit period Tbi, if we send a '0' and
for the second half if we send a ''.
We assume that Ho (sending a '0') and H, (sending a
'1') are equally likely, and that each transmitter and receiver has a diameter of less that a
coherence length. Moreover, we assume that each receiver is separated by more than an
amplitude coherence length so that the fading seen by each receiver is independent of one
another. This is a realistic and plausible assumption as the coherence length is on the
order of centimeters. We further consider the transmitters as being separated by more
than a spatial mode of the receiver aperture so that at each of the N detectors, the M Airy
patterns can be resolved. The fading coherence time, which is generally on the order of
milliseconds to one tenth of a second, is much larger than the bit interval time, which is
on the order of femto to picoseconds. Thus, the factors "j and "-I are modeled accurately
as independent random variables with constant value over each bit interval time Tbi,.
17
2.4 Combining methods
Given that we have N received versions of a signal in a diversity system, we would like to
take into account various methods of combining the N received signals. The following
are three possible combining methods:
1) simply add the N received signals after detection (Equal Gain Combining)
2) combine the N received signals optimally (Optimal Combining)
3)
select the signal with the least amount of fading and discard the other N-1 signals
(Selection Combining)
In order to minimize the probability of decision error, the maximum likelihood (ML)
decision rule is used after combining the N received signals, to determine if a '0' or '1'
was sent. When analyzing diversity systems with particular combining schemes, their
performance will compared against the performance of a no-diversity system that uses the
ML decision rule. Thus, the no-diversity system will serve as a benchmark.
18
Chapter 3
Analysis of Diversity Systems
In this chapter, we analyze the performance of various spatial diversity configurations:
no diversity, receiver diversity with Equal Gain Combining, receiver diversity with
Optimal Combining, receiver diversity with Selection combining, and combined
transmitter and receiver diversity with Equal Gain Combining. We also analyze a time
diversity system. We define the outputs of receiver i in the first half and second half bit
periods as Ri,0 and Rij respectively. Ri,0 and Rij will consist of a signal component and
noise component due to thermal, shot, background and dark current noise. Since each of
the noise components is additive and independent of one another, we model them as one
lumped additive white Gaussian noise variable. We further define m as the link margin,
or increase in transmitted power over that necessary for a no turbulence link, provided by
the optical communication system to minimize the outage probability. The performance
gain of the diversity systems will be analyzed with respect to outage probability, where
outage probability is defined in Section 3.1.
3.1 Performance Metrics
The usual performance metric in analyzing communication systems is the probability of
bit error.
However, when analyzing systems for optical communication through
atmospheric turbulence, the average probability of error is not the best metric. This is
19
because errors resulting from signal fades are no longer independent and large strings of
data (duration from 1-100 milliseconds) can be lost. Moreover, the average bit error rate
is not available in closed form.
Using the performance metric of outage probability allows closed form evaluation of
diversity systems.
More importantly, it indicates how often the system is below
performance threshold.
Outage probability is defined as the probability that the short
term bit error rate Pe (over a duration of less than the channel coherence time) is above a
determined required value Pe*. As shown in Figure 3.1, for an AWGN channel with onesided power spectral density No, the bit error rate P, of a single transmitter, single
receiver system is a function of y--E/No=P/(NR) where P and E are the receiver output
power and energy respectively (includes fading) due to the signal, and R is the symbol
rate.
Pr(error)
(dB)
Pe
mEb*
Eb*INo
O
Figure 3.1 General Probability of Error Curve for AWGN Channels
Let )*=Eb*/No=P*/(NoR) be the required y value to achieve Pe* and let the transmitter
transmit just enough power to let Eb* be the received signal energy per bit (or
equivalently, P* be the receiver output power per bit) under no fading. In Figure 3.1, the
outage probability can be seen as the probability that the operating point moves along the
curve left of the threshold Eb *No. The outage probability expressed as
Pr(outage) = Pr(Pe> Pe*) = Pr(r<*)
20
(3.1)
In the case of direct detection receivers, if we let S denote the receiver output current due
to the detected optical signal, then
E, = (S)2
2
oisson
(
=
bit
(3.2)
arrival2
P 2a r i
q
where q is the charge of an electron, q7 is the efficiency of the photodetector, h is Planck's
constant, v is the optical frequency of the transmitted wave, and Pr is signal power
impinging on the receiving pupil. Appendix C derives an expression of Pr as a function
of the transmitted signal power and system geometries.
Using this expression, and
incorporating a fading factor a,
,i
= (q 77 ajp2
Eb
where
hv
(3.3)
2
is the fraction of transmitted power detected by the receiver when there is no
turbulence. If the transmitter emits just enough power P,* to result in Eb=Eb* under no
fading, then the energy per bit at the output of the receiver is
E,* = q
I
hv
P,*
2
(3.4)
Tbit
2
If the transmitter provides link margin m beyond the original transmitted power P,*, then
E,
=
=
q
=qh
v am[2
am
(Cn)2 E*
21
J
Tbit
2
(3.5)
For use in later sections of this chapter, we define S* to be the receiver output current
under no fading that results in bit error probability Pe'.
S= q 7 {P
(3.6)
Eb = (cmS *)2 Tbj,
2
(3.7)
hv
So,
If the optical communication system functions without loss of data for any probability of
error less than the threshold Pe* (which will happen if coding is used), then it would be
useful to compare the outage probability of an N receiver system with that of a single
receiver system.
We define power gain of a spatial diversity system to be the fractional decrease in
required transmitted power in a spatial diversity system compared to a non-diversity
system to achieve the same specified outage probability. This definition of power gain
provides us with a useful means to evaluate the gain of using spatial diversity systems.
3.2 Maximum Likelihood Decision Rule
When the transmitter sends a '0' (hypothesis HO) or '1'
(hypothesis HI) with equal
probability, the decision rule that minimizes the probability of error is the Maximum
Likelihood (ML) decision rule. Denoting the received observation as the vector r, the
ML decision rule is to choose HO if
p(rI HO)
p(rI HI)
fp(a)p(rI H 0 ,a4Ia f p(a)p(rI H,,apla
0
0
22
8
and H, otherwise. We will now prove that choosing Hi that maximizes p(rHi,a)is an
equivalent optimum decision rule. Denoting the range of r in which the receiver decides
Ho or H1 was sent as Q0 and Q, respectively, the average probability of making a correct
decision is
P(C) = p(HO)J
=
p(r I H
dr + p(Hj)J
p(r IH,)dr
p(H0 )L f
= )p(a)p(r I H0 ,a adr + p(H
= p(HO)
=
f p(a)p(|IH0 ,aladr+p(H,
)f
p(a p(r | H,a)dadr
-
1
p(ap(r|H,,adadr]
+(p(a)(p(.r|H0,a)-p(rjH,,a) adrl
The expression is maximized if
p(r|Ho,a) _p(r|Hi,a)
(3.10)
for all r in Q0 or equivalently if p(r|Ho,a) p(rHj, a)for all r in Q1. Thus, the optimal
decision rule that minimizes the probability of error is to choose Hi that maximizes
p(rIHi, a). We use this decision rule for all the spatial diversity systems analyzed in the
remainder of this chapter. However, we note that the observations r seen by the decision
stage will differ between spatial diversity setups dependent on the type of combining
performed on the received signals.
3.3 No Diversity (One Transmitter, One Receiver)
Figure 3.2 shows an optical communication system that does not use spatial diversity.
Transmitter
Transm ittPA I
upil area At
..................................
Receiver
Pupil area1 Ar
Detector area Ad
R0 , R1
Figure 3.2. System with No Diversity
23
e cision
The system has transmitting pupil area A, receiving pupil area Ar, and detector area Ad.
The outputs of the receiver, under hypothesis HO and Hi are given by
H 0 : Ro =aIImIS* +noIrc
R = n',1rec
(3.11)
H,: RO=nO,Irec
R,
=am]S*+ nO,Irec
where Ro is the receiver output in the first half bit interval (0, Tbi,/2)
R, is the receiver output in the second half bit interval (Tbi/2, Tbit)
no,Irec is a Gaussian noise random variable distributed as N(O, Jrec2
S* is the receiver output current in the single receiver setup that corresponds to the
bit error probability threshold Pe* when the white Gaussian noise level
No re/2=Yrec2,
link margin m=J, and fading factor all=] (no fading)
m is the power link margin provided by the transmitter above the required power
to achieve Pe* under no fading (the subscript 1 refers to the 1 receiver system)
all is the log normal fading factor from transmitter 1 to receiver 1
The noise variance
oJi
rec2=NoIrec/2
is due to the sum of variances of the thermal,
background, shot and dark noise of the receiver.
We assume that the receiver is
background noise limited. i.e. that background noise dominates.
Under equiprobable hypotheses, the decision rule that minimizes the probability of
decision error, found by the Likelihood Ratio Test, is to choose Hi that maximizes
p(Ro,RIHi). As shown in Section 3.2, an equivalent decision rule is to choose Hi that
maximizes p(Ro,RI|Hi, all). Thus, the optimum decision rule chooses HO if
24
I HO,a,1) p(RO,R,| H,,a, )
p(RO,R
p(RO IHO,a1 )p(R| HO,a 1) p(RO IH,,a)p(R, IH,a,,
1)
e
2fo -
x
1
2
-(RO -a 1 m1 S*)
e
20 -2e
-
2 O -2
20 2
x (Irec
(R - am
S *)2
-R
>-R 2-(R,
-aimS*)2
2
202
Irec
Irec
2
-(R,
-R
1
-
-anmS*
(3.12)
,
O
Ie
2
RO !RI
and H1 otherwise. The outage probability is given by
Pr(outageof no diversity system)= Pr(amS.
=Pr
J
<(S* )2
aH <rIj
Q - a-x +
exp -
(3.13)
In(m,)
- -+
x
n(mi)j.
where the last equality is the Chernoff Bound. Using this Chernoff Bound, the link
margin required by the system to obtain an outage probability of Poua* is
m,= 2u- (Q-(2PO*.,)+ a,)
~exp(29, (- 2]n(2 Po* )+ a-)
= exp(2o-
(3.14)
-2ln2P)*,,)exp(2o-)
Notice that for a given turbulence strength, the link margin m, required by the system to
achieve outage probability Pout* can be calculated using (3.14) i.e. m, and Pout* are
algebraically related to each other by (3.14). The outage probability (3.13) and link
margin (3.14) of the no diversity system of this section will be the basis of comparison
25
for all the spatial diversity systems in the remainder of the chapter. They are plotted in
Section 3.4.3 along with the outage probability and link margin of receiver diversity
systems that are investigated in Section 3.4.1 and 3.4.2.
3.4 Receiver Diversity
3.4.1 Receiver Diversity with Equal Gain Combining
Figure 3.3 shows an optical communication system that uses spatial diversity at the
receiving end with N_>2 receivers and Equal Gain Combining (EGC).
seceivern
Rpi areaA/N
Ddectr ea
Poo
AP~il area
-N0
te 1
F'-
aeaA/N
Idectcr amaN
Figure 3.3 System with Receiver Diversity and Equal Gain Combining
Each receiving pupil area is A,/N so that the sum of the N pupil areas is the same as the
pupil area of the no diversity system.
This scaling of areas is done so that under no
fading, all the described systems have the same received signal powers. This allows the
systems to be compared fairly.
The outputs at the receiving end after Equal Gain
Combining are
26
H :
RO
=
aimNEGCS IN +nEGC
ZR I
N
RI =
R
-n0,EGC
(3.15)
N
HI :
RO =
R,=
R,
ZRi1
-n0,EGC
4
aimN.EGCS
N +n0,EGC
where Ri,O is receiver i's output in the first half bit interval (0, Tbil2)
Rij, is receiver i's output in the second half bit interval (Tbi 1/2 , Tbi,)
nO,EGC is a Gaussian noise random variable distributed as N(O, JEGC)
S* is the same as defined in Section 3.3
mN,EGC
is the power link margin provided by the transmitter above the required
power to achieve Pe* under no fading (the subscript NEGC refers to the N
receiver system with EGC)
oi; is the log normal fading factor from transmitter 1 to receiver i
The noise variance JEGC2=NoEGC/2 is due to the sum of the variances of the thermal,
background, shot and dark noise of the N receivers. We again assume that the receivers
are background limited.
Background noise variances for same field of view receivers are proportional to the
receiving pupil area (see Appendix C). Since each of the N receivers in Figure 3.3 has
pupil area A,/N, they are each subject to 1/N times the amount of noise seen by the single
receiver in Section 3.3. The combined noise of the N receivers contributes the same
absolute amount to
7EGC
2
2
as the single receiver with pupil area Ar does to arrec2
if
diffraction limited receivers are used instead of fixed field of view receivers, each
receiver, in either the no diversity or receiver diversity system, sees the same background
noise variance. So the total background noise variance for the system in Figure 3.3 in this
case would be N times that of the no diversity setup. We denote the fractional change in
total noise variance of the receiver diversity with EGC setup from the no diversity setup
27
as KEGC=NO
EGC
ENo
IJre
rec.
If fixed field of view receivers are used, KEGC=J and if diffraction
limited receivers are used, KEGC=N. KEGC=] is the more realistic assumption for low cost
medium range links without active spatial tracking.
As shown in Section 3.2, the optimal decision rule is to choose Hi that maximizes
p(Ro,Ri Hi,_g).
Thus, the optimal decision rule, given Equal Gain Combining, is to
choose Ho if
p(RO, R, |Ho,a)
p(RO IHo, )p(R| Ho,cq)
N
p(RO, R, |H ,a)
p(Ro IHIa,)p(R,| H
mN.EGC
S
)
N
i=1
,a)
exp
- R
exp
2U72
1 rec
2
N
mNEGC
N
exp
2f&
-K
N
-
a
2O
rec
Irec
R0
exp
0
S,
mN
N,EGC
N)
2
-R
mNEGC
R1
>-R2 -
(3.16)
N
and H1 otherwise. The resulting outage probability is given by
a 'mN GS*
(N
Pr(OutageofReceiverDiversitywith EGC)= Pr
I
N.EGC
(=1 N
2
*
N 0EGC
No
N
=Pr(
Sai
<
N
MN.EGC
28
)KEGC
(3.17)
There is no closed form expression for the exact probability density function (pdf) of the
sum of N log normal random variables. Thus, it would be useful for analysis purposes to
make an approximation to this pdf.
3.4.1.1 Log normal Approximation
If we take the sum of log normal random variables co to be well approximated by a log
normal random variable Z=e", i.e.
N
ai = eu
zZ
(3.18)
i=1
then assuming that the fading seen by each receiver is independent, we find that
UocN(puu, aU2) where pu and ov2 are given by
I exp(4U2)-1
N
pUU = In(N) -0.5 In 1+
()72
=ln I+
U
p(N)).
N
(3.19)
(3.20)
See Appendix B for the derivation of (3.19) and (3.20). Thus, using (3.17) and the log
normal approximation, the outage probability of the receiver diversity with Equal Gain
Combining is
29
Pr(Outageof Re ceiver Diversity with EGC)
ai,<
=Pr
=Pre
N
MNEGC
C
<
EGC
MN.EGC
=Pr U <In
N
G
MN,EGC
Q
in
NEGCNEG
K
EGC
2I
exp(4'
2K1 ex(4c~)-
LnNEGC
mNEGC
2
1+ -
x
+-exp(4oj)-
21n
)
J
2
x
+
where the last line in (3.21) is the Chernoff Bound. Recall that we defined mi and mN,EGC
to be the link margin required by the no diversity and receiver diversity with EGC
systems respectively to achieve outage probability P 0 )r*at any bit error probability Pe*. If
we set the outage probability of the no diversity system and receiver diversity with Equal
Gain Combining systems to be the same (by equating (3.13) and (3.21)), we can find
mN,EGC
in terms of mi.
n(m
- '~~,r
+
-
a
+
ln(m)= In mN
2
Knexp(4U
KG
=Q 2u-
,EG C
KEGC
I)
1
IJN
+
In
-
U 2__
_________
mN+EGC =IK EGCK1
exp(4mE)GC
II
NN
N
30
2,
(3.22)~)
The power gain of the receiver spatial diversity system over the no diversity system is
Power Gain of Re ceiver Diverisity with EGC =m
I mN
/GC
m1
In~m,
)
exp(4o2) -1
N
KEGC
exp
-
exp
2In(2P, ,)r 2a, -
0 X+
n
exp(42)NX
KEGC
F+exp(407
( 2az
n
N
exp(2-
N
-
)
(3.23)
1
where the last equality comes from expressing m, in terms of Po,' as given by (3.14).
The last line in (3.23) expresses the power gain of receiver diversity with Equal Gain
Combining as a function of Po,,
As N approaches infinity,
Power Gain of Receiver Diversity with EGC as N
-
=m
M
KEGC
(3.24)
exp(2c A[-2ln(2P*,,)exp(2K)
KEGC
The outage probability (3.21), and power gains (3.23) and (3.24) of receiver diversity
with Equal Gain Combining are listed in Tables 3.2 and 3.4 in Section 3.4.3 for ease of
comparison.
3.4.1.2 Gaussian Approximation
If N is large, by the Central Limit Theorem, we can take the sum of log normal random
variables c;j to be well approximated by a Gaussian random variable YocN(py, oy).
31
Assuming that the fading seen by each receiver is independent, we find that py and cY
are given by
(3.25)
pY = N
2
(3.26)
= N(exp(4j) -1)
since
p
a
as shown in Appendix B.
=1I and
= exp(4
2)
(3.27)
(3.28)
-1.
Using (3.17) and the Gaussian approximation, the outage
probability of the receiver diversity with EGC is
Pr(Outageof
N
Re ceiver Diversity with EGC) = Pr(Y <
KEGC
MNEGC
N
__NEGC
I(1
-
EGC
MN,EGC
eXP(4ji)2
j
We will show in Section 3.4.1.4 that for moderate N, the Gaussian approximation is poor
compared to the log normal approximation.
So the outage probability using the log
normal approximation, (3.21), is more accurate than the outage probability using the
Gaussian approximation, (3.29). Thus, the Chernoff Bound of (3.29) will not be used for
analysis. By the Central Limit Theorem, for infinite N, the Gaussian approximation is a
good one so we will use (3.29) to find the power gain for infinite N. If we set the outage
probability of the no diversity system and receiver diversity with Equal Gain Combining
systems to be the same (by equating (3.13) and (3.29)), we can find mN,EGC in terms of
MI1 .
32
K- GC
(
Q
ln(mi)
-07r + --
2a-
MNEGC
=Q
)
exp(4j)-1
j
(3.30)
C()-
1
ln(m)=
+
--
MNEGC
2ax
exp(4U-I
+
exp(4a. )I-1
Ii)
I
m
-- ,fN21X
1
K
MnNEGC
m
exp(4o)-1
-I
K
MNEGC
o
+
I ln(m))
zT
The power gain of the receiver spatial diversity system with EGC over the no diversity
system is
PowerGain of
M
KEGC
Re ceiver Diversity with EGC
exp(4o) - I
(3.31)
ln(mi)
L
N
= m, imN,EGC
2ax
As N approaches infinity,
Power Gain of Receiver Diversity with EGC as N -
oo =
_
KEGC
exp(2-a
(3.32)
-21n(2P*,
exp(2o0
KEGC
where the last equality comes from expressing mi in terms of Put as given by (3.14).
Notice that the power gain of receiver diversity with EGC as N approaches infinity gives
the same expression whether we use the log normal approximation or the Gaussian
approximation.
This provides confirmation that our expression for power gain of
receiver diversity with EGC for infinite N is correct.
33
3.4.1.3 Exact
In Sections 3.4.1.1 and 3.4.1.2, we described approximate expressions for the outage
probability and power gain of a receiver diversity system with Equal Gain Combining. In
this section, we describe how the exact outage probability is calculated.
The probability density function (pdf) of the sum of N independent random variables is
the convolution of the N pdfs. Letting
S
=
(3.33)
a,
the pdf of S is the convolution of the log normal pdf of each o .
PS (S) = PI (S)
P
2 (S)
®...
0
P'N(S)
(3.34)
Since the outage probability of receiver diversity with Equal Gain Combining is given by
(3.17), it can be calculated by the following integration
N
Pr(Outage of Re ceiver Diversity with FCC)
~NG
N
ps (s)ds
(3.35)
KG(
Pa (S)
P,(S)@...@®pN (s)ds
Since there is no closed form expression for the convolution of N log normal random
variables, we can resort to calculating (3.35) numerically.
3.4.1.4 Comparison of Log Normal and Gaussian Approximations
Figure 3.4 plots the pdf of
S = a, +...+
34
a9
(3.36)
where ln(a) cc N(-2cTJ,4o 2 ) and oq=0.3 (moderate turbulence).
The pdf ps(s) is
calculated using 3 methods: using the log normal approximation, using the Gaussian
approximation, and by numerically convolving the pdfs of ca and integrating.
---
0.2
-
0
Convolution
Gaussian Approximation
Log-Normal Approximation
0.18 0.16
n
.4
-
0.12
-
/
0.1
0.08
2
0
!b
0.06
0.04
Des
b
As
0.02
07
0
1
2
3
4
5
6
7
8
9
s
Figure 3.4 Probability Density Function of the Sum of 9 Log Normal Random Variables x
where ln(%) -, N(-2yx 2 4y 2)and yx=0. 3
We see in Figure 3.4 that the log normal approximation is, percentage-wise, very accurate
in the main hump of the distribution but becomes larger than the actual distribution in the
tail. The calculation of the outage probability of the receiver diversity system involves
integrating the left tail of the distribution (integrating below Z=sqrt(KEGc)N/mNEGC)Thus, the log normal approximation is accurate when this threshold is significantly large
such that the upper portion of the integration dominates. At low outage probability, the
log normal approximation yields an upper bound to the outage probability.
35
The log
normal approximation becomes less accurate as the turbulence parameter O-r is increased
and more accurate as a. is decreased (compare Figures 3.4 and 3.5).
- ---- -
I
Convolution
Gaussian Approximation
Log-Normal Approximation
/
/
0.12
-
-
/
1~
0.1
//
a,
5 0.08
-0.06
0
CL
0.04-
/
0.02 /
/
/
/
/
/
I
0
1
2
3
5
4
I
I
6
7
8
9
Figure 3.5 Probability Density Function of the Sum of 9 Log Normal Random Variables ji
2
,4y 2 ) and yx=0.5
where ln(%) oc N(-2 X
The outage probability expression that uses the Gaussian approximation is clearly not
very accurate for moderate sized N such as N=9. This is because the left tail of the
Gaussian distribution is the region used to calculate Pr(outage), and when using the
Gaussian approximation, the tail becomes less accurate for smaller values of N.
Moreover, the actual distribution of S is zero for s<O whereas the Gaussian
approximation has a non-zero pdf for s<O.
As N---oo, the Gaussian approximation is accurate by the Central Limit Theorem and the
power gain expression (3.32) is accurate. The power gain as N--oo of receiver diversity
36
with Equal Gain Combining is plotted in Section 3.4.3 along with the power gain of
receiver diversity with Optimal Combining, which is derived in the next section.
Figures 3.6, 3.7, and 3.8 plot the outage probability of receiver diversity with Equal Gain
Combining for N=2, 4 and 9 receivers respectively.
probability
using
the
log normal
approximation,
Each figure plots the outage
Gaussian
approximation
and
convolution. We see that in all cases, the log normal approximation is more accurate
than the Gaussian one. Also, for N taking a value up to 9, we see from the figures that
the accuracy of the log normal approximation decreases as the outage probability
decreases. Also, the accuracy becomes worse as turbulence increases. For example, the
log normal approximation for outage probability of receiver diversity with Equal Gain
Combining has the following accuracy for the following outage probability ranges:
a) for low turbulence o-=O. I : 0.1 dB link margin accuracy for any fixed probability of
outage above 10-12
b)
for moderate turbulence o-7=0.3 : 0.5 dB link margin accuracy for any fixed
probability of outage above 10-12
c) for high turbulence o--=O.5 : within I dB link margin difference for fixed probability
of outage above 10-.
37
10 a
.. ....
I
..........
.......
.....
..
...
..
....
....
10-1 . ........... a .*.:.. '
.
..
..........
.........
...
. ....
..
''...
....
.......
..
...
...
..
..
...
...
...
..
..
..
......
.....
...
...
........
...
...................................... :.................. -----------.... ... .
..
..
....
..........
.............
................................. ............
....... ........
.......
..............
........
........
.......
......... .................
... ........
. . ......
-;.'I . ..
..
.. .........
.........
........
.........
........ - ..... .
....
..
.......
.
.............
MW
.......... ..................
..........
..... ..
....
.........
.. ...
.
........
. .
..
......
...
...
...
. ...
....
...
...
--.....
W...
.................
......... ...
I...
.........
.....
...........
........ .......
*...
........
...
...
...
.....
...
.........:..
...
..
. ........
...
..
.....
..
.
.......
......
..
....
.. ....
..
...
..
..
....
...
...
..
...
....
...
...
....
...
..
...
.
..
....
........ .... ...
......
..
.
10-2
-3
----- -------------..............
..... ....
. ....
...
......
.....
........
....
........ ...........
.............
.............. .
- - - - .....
....
. ...... ....... - ...
.......... ..... ..
:
.........
......... ....
...
..
..
.... ....
.... ....
..
..
.
...
..
......
....
......
...
...
..
..
....
..
....
..
..
......
...........
...........
..... .......
+ a -0.1, N.2, EUC, Ino
.......
rM
..
....
.....
........... ................
.......
a x 0.31 N=2, EGQ Ino rMl
...............
....... ...
....
:-.............
- - - ...... .........
........
.. ......
. ........ ..
..................... ......... ............. ..............
aX'0.5, N-2, EGC, Inorm
..........
........ .....
...... .; .
X
10
41
0
CL
Q
0
............
. ...........
.
10-6
......
ax .01
-0.3, N=2
N=2,
or.-O-5, N-2,
-o.1, N-2,
M0.3, N=2,
x
ax'0.5, N-2,
- ...............
.. ............
...... .
......
...................
...............................
- ..'......
I
.......... .......
................
... .... ......
........... .....
........
...... WI: 6
........
EGC CLT
EGC,
EGC,
EGQ
EGQ
EGQ
CLT
CLT
conv
conv
conv
...... ... .... .. .. .....
.............
......
............
......
.. .
A .7-
-..-
----77-'.'.
.7
-7--
-7-1-7-
7 ---
......
........
.....
*............
..........
--------------
......
-7-77.
............. . .. ...........
.. . .. .........
-
..........
..... .
------7.'.'
.......... ...........
......
... . .......
... ...........
0
1
2
................. - .............
..........
3
. .....
4
.
...
....
... ....... .... .... .. ...............
5
6
link margin rn (dB)
... ... ....................
7
8
9
10
Figure 3.6 Outage Probability For Two Receiver Diversity and Equal Gain Combining,
KECC= I
0
10
EB
. .......... .
...
........
10-2
EB
...
...
....
......
...
...
.....
.. ....
. ..
....
.. ..
....
..
....
. ................ .............
10-4
M
a z0.1, N=4, F(3(;,
X, 0.3, N=4, EGQ
a MO.5, N-4, EGQ
x
a '0 *11 N=4, EGQ
a -0.3, N-4, EGC,
aX'0.5' Ns4, EGQ
x
.0.1, N-4, EGC,
x
'0A N-4, EGQ
a '0.5, N-4, EGQ
0
x
......
00 10-6
W
10-8
............
10- 10k
0
Inorm 1-:1..........
Inorm
Inorm
CLT
CLT
CLT
conv
conv
conv
1
2
3
4
5
6
link margin rn (dB)
7
8
9
10
Figure 3.7 Outage Probability For Four Receiver Diversity and Equal Gain Combining,
38
KEGC= I
-
10
10
.
-..
...... ....
.......
-
..
- .
...
...
-4
10
-ED
a
0
10
-r0.
... ...
10 1.
1,No9
a .0.3, N-9,
* 05, N-9,
0.1, N-9,
10.3, N-9,
1
-G , nr
EGC, Inorm
EGC, Inorm
EGC, CLT
EGC, CLT
0.5, N-9, EGC, CLT
-00.1, N-9, EGC, conv
0.3, N.9, EGC, conv
0.5, N-9, EGC, conv
0
1
2
3
4
5
6
link margin m (dB)
7
8
9
10
Figure 3.8 Outage Probability For Nine Receiver Diversity and Equal Gain Combining,
Figures 3.9-3.11
show the outage probability
for EGC using the
KEGC=l
log normal
approximation (3.21) and the Chernoff Bound to this approximation (3.21) for low to
high turbulence.
We see that although the Chernoff Bound for the outage probability
becomes slightly less accurate for higher turbulence or lower N, it quite accurate (at worst
1.5 dB link margin accuracy for fixed outage probabilities greater than 10-4).
39
0
10
+
10
2
10 -4
co
-0
0
.......... X
+
X
+
i( 0
+
0
0
.........................
X .0 .....
0
X 0 .
I -Z -"'
+
X U
X
+
+
0
M
.........
............
10
CD
0)
C13
Z5
0
N=1
N=11,
N=2,
N=4,
N=9,
N=2,
N=4,
N=9,
x
........
.
.......
Chernoff
EGC
EGC
EGC
EGC, Chernoff
EGC, Chernoff
EGC, Chernoff
.......
+
:
-8
10
X
............ ........... ..............
.....
....... ...........
.....
+
10
XQ
........... X .() ......... *...........it ........... A6
X0
............ ............
XO
0
1
2
3
4
link margin m (dB)
Figure 3.9 Outage Probability,
5
6
7
(Tx=O. 1, Log Normal Approximation
KECC= 1,
1 0 0 ..
..
......
............
...........
.........
4 ..
........... ........ ....... .......... ....
...
...
......
....
.4 ..
..
...
....
..
....
..
...
...
...
...
...
....
...
...
..
....
.....
....
...
....
.....
....
...
...
.....
...
....
....
.....
.....
....
....
...
N=1
............
N=11, Chernoff
..........
X
X N=2, EGC
+
+ N=4, EGC
---------------- +
10 ..............
......... ........ ......... 11
+
+ N=9, EG C
...............
..........
........ 0
...
0. X.. .. .........
0 N=2, EGC, Chernoff
.............i- ......... [D .X ..... ....
>-.
,
.
'
4
D N=4, EGC, Chernoff
..............
...
....... X
............
..
....
K
.....
.....
N=9, EGC, Chernoff
co
X
+
X
0
L- 10 -2
+ V'.
CIL
.............
..............
a)
........
...............
......... X ..... ......
...... ........ ...... ...........
...... ....
...... T--
X ...
.X
X
.....
. . .. . .. .. .. . ..
C13
.. . ... ... .. .
....
. .. . .. .
.. . .. .. .. .. .. .. ..
..........
... .....
. . .. .. . ... ... . ..
.. .. .. . .
+
?c ....
X
:X
.
...............
..........
......
..............
X
10 -3
.. . . . .. . ..
.. . .. .. .
..
. .....
.. . . .. . .. ...
. .. . .. .. .. ..
..
X
W -: ..
.. .. . . .
.
. . ..
X
. . . . ....
..............
..........
.............
...
..
.........
.....
.
......
+
.
..
'
Q
0
0
2
4
Figure 3.10 Outage Probability,
.. .. ..
. . . . . . . . . . . ..
. . . . . . . . ..
..........
0
....
. . . .. .
. . . ... . . . . .
.. . .. .. . .
...
. .
+
......
...............
.
....
:
X
X ....
X ....
X
X
.:
50 ...
........
X
X
......
.......
Q
Y
6
8
link margin m (dB)
KEGC=1,
40
:! % ......
............
.......
.....
.....
10
. .....
12
cyx=0.3, Log Normal Approximation
1
10 0
.....
...N
. . ..
.......
~
~..
~....
.. ..
:+
10 1
........
~
+
-0
-
-
= 1SN=1, Chernoff
X
x
4+
++
+
N=2, EGC
N=4,
N=9,
N=2,
G N=4,
0.* N=9,
EGC
E GC
E GC, Chernoff
E GC, Chernoff
EGC, Chernof
& i" 2
a)
0
10-3
2
4
6
8
10
12
link margin m (dB)
Figure 3.11 Outage Probability,
KEGC=l,
14
16
18
20
cx=O.5, Log Normal Approximation
3.4.2 Receiver Diversity with Optimal Combining
Figure 3.12 shows an optical communication system that uses spatial diversity at the
receiving end with Optimal Combining (OC). The number of receivers N 2 and each
pupil area is A/N so that the sum of the N pupil areas is the same as the pupil area of the
no diversity system.
The outputs of the receivers are given by
I
H 0:
R0 =a mN.OCS* I N+noi
R, =noi
HI :
(3.37)
R, 0 =n0 ,
RI =aImNOCS*
for i between 1 and N and where
41
IN+noi
Ri,o, Ri, 1, S*, and og are the same as defined in Section 3.4.1
noi is a Gaussian random variable distributed as N(O, o)
mN,oc is the power link margin provided by the transmitter above the required
power to achieve Pe* under no fading (the subscript N, OC refers to the N
receiver system with OC)
The noise variance o2 is due to the variances of the thermal, shot, background, and dark
noise of receiver i. We assume again that we are using background limited receivers and
that fixed field of view receivers are used in both the diversity and no diversity
configurations.
1
Pipil araA/N
Receiver
RI0, RI
etctor ara Ad
Estinate u
Transnitter I
Ppilaa A
.
.r
Receivcr 2
Ripil am A/N
ara Ad
Rzo, RI
Optinr
Con-bining
Estinute ct
Receiver N
A/N
Pupil
Itecr
aiva Ad
Estiniate M
Ro, RN I
P
Figure 3.12 System with Receiver Diversity and Optimal Combining
The optimal decision rule is to choose Ho if
42
Decision
p(R1 0 ... RNo R1
.
p(R,0 ,...RNi,o R1,1,.. .RN, I Hl, a)
RN] HO, a)
jp(R,0 I H 0 ,a)p(R i
JJ p(Ri) I H ,ca)p(R, I H, a)
HO,ca)
i=1
/.=I
a
N
1
N
<N
mN,OCS
(R i'
KO
N
N
exp
exp
K 2rc3
RiI R"
2;To7,2
KNiN ep
N
Ri,, 0 -
Z ai
2a
Na
K2l
MN,OC
exp
No~
N
N
MN,OC S
+ R2
R 2+
>
N
R
-
I ilRi' 2
i=1
S
42]
N
(3.38)
N
i=1
Sai
mNOC
aiRi
and H1 otherwise. The fading factors ai; can be estimated accurately since the coherence
time is much larger than the bit interval time. Thus, we will replace the actual oi's with
the estimated ot;'s in (3.38). We define
N
R0 Za',
1
(3.39)
Ro
N
(3.40)
i=1
The noise variance of both RO and of R, given oaj is
N
N
72 =
1
2
EGC
U 2=a
1=1
The resulting outage probability is given by
43
N
NoC
0
2
(3.41)
Pr(Outageof Re ceiverDiversitywith OC) = Pr
*
ai,2mN,OC
lec
N0
N
N
2
N
=P
*
i MN,OC
<(*Y
N
a
i=1
N
2
(3.42)
NEGC
N01 e
No
NN
= Pr
EGC
, <
i=1
N2
KEGC ,
where KEGC .
MN.OC
2 is
Since oa~ is a log normal random variable,
.0lrec
No
also a log normal random variable.
There is no closed form expression for the exact probability density function (pdf) of the
sum of N log normal random variables. Thus, it would be useful for analysis purposes to
make an approximation to this pdf.
3.4.2.1 Log Normal Approximation
Let us take the sum of log normal random variables ei2 to be well approximated by a log
normal random variable Z' =e
,
i.e.
Z'=
a2 =
eU'
(3.43)
By assuming that the fading seen by each receiver is independent, we find that
U'cocN(pu',au) where
plu. =
In(N)+ 4U2
U2 =
z2
exp(16U2) -1
N
In
In I+ exp(
(3.45)
N
Thus, using the log normal approximation, the outage probability is
44
(3.44)
Pr(Outageof Re ceiver Diversity with OC)
=Pr
2 KEGC
MN.OC
= Pr eU.
N 2
KEGC
MNOC
=Pr U'<ln
N
2 KEGC
MN.OC
N
-I
/U
2 KEGC
N.OC
CU'
=Q
QIn
1n1N,0C
Imvoc6X2_
KEGC
+4X
2e
/
inri+
-
N1aX
N
22
-
In
MO C
KEGC
I
2lnrI+
+40-2
N
16X
2
N
(3.46)
Because the ratio of
-z/z-
is larger than cz/uz of the previous section, the log normal
approximation is valid over a smaller range of N, a- and Po
0 ,,*
for the Optimal
Combining scenario than the Equal Gain Combining scenario.
In order to find the link margin mNoc required by the receiver diversity with Optimal
Combining system in terms of the link margin m, required by the no diversity system for
the same outage probability, we equate (3.13) with (3.46).
45
Q
+--n(m,>I
±
2a
=
)
Q in
4uj
-±""
-
ep(6
n1
pN
1je
+ eXp(16c)
KEGC.
(3.47)
-oX+
I- n(m)=
In
2ax
m N,C
= j
in
+41
KEGC
r+exp(I6o)
+ exp
(
1+
-1
N
6u2)-11/exprI
)
207X
2uxJ
exp(6
N )
Nn
The power gain of the receiver diversity system with Optimal Combining is
PowerGain = m,
/
mN.OC
K+ exp(16Q ) -1
I
exp
exp(2u
KGc rI
exp!
+
-
In(mi)
2ax
-
-2n(2P 0 ,)
N
2a- -
-21n(2P ,,)intj
exp(4-I
)
+
exp(
607) )
-2U2
)
+exp(16
exp 2
KEGC1+
In
2In(2P,4 )exp(2o-
exp(1 6)-1
N
X)-I
I+exp(16 U2
)Nx~~j-
2UJ
(2
11/
XN
(3.48)
where the second last equality is found by expressing m, in terms of Pout using (3.14).
The last line in (3.48) expresses the power gain of receiver diversity with Optimal
Combining as a function of Pout.
As N approaches infinity,
46
Power Gain of Re ceiver Diversity with OC as N
->oo =
m1
K
exp(2o7)
(3.49)
EGC
exp(2az -2in2(P., ))exp(4o4)
KEGC
The outage probability (3.46), and power gains (3.48) and (3.49) of receiver diversity
with Optimal Combining are listed in Tables 3.3 and 3.5 in Section 3.4.3 for ease of
comparison.
3.4.2.2 Gaussian Approximation
Just as in Section 3.4.1.2, if N is large, by the Central Limit Theorem, we can take the
sum of log normal random variables ci1 to be well approximated by a Gaussian random
variable Y'OcN(py,, oy ). Assuming that the fading seen by each receiver is independent,
we find that uy, and Jy2 are given by
p, =Nexp(4j)
02
= Nexp(8U 4(exp(16
(3.50)
) -1)
(3.51)
since
pa
=exp(4o2) and
oa 2 =exp(8J,)exp(I16aj)-1)
(3.52)
(3.53)
as shown in Appendix B. Using this Gaussian approximation, the outage probability of a
receiver diversity system with Optimal Combining is then
47
Pr(Outageof Re ceiver Diversity with OC)= Pr Y'<
N
KEGC
N.OC
N
N
'r
~
EGC
KEGC
2
MNOC
= Pr
N exp(4a )-
Q
N
KEGC
MN,OC
N exp(8X(exp(16
TI exp(449
)-
KEGC
mN.OC
Q
exp(8o-
) -6U)
)(exp(1 6U)-l
(3.54)
Again, just as in Section 3 4.1.2, the Chernoff Bound of (3.54) is not useful for analysis
of outage probability for moderate N since (3.54) is only a good approximation for
However, we will use this equation to find the power gain of receiver
infinite N.
diversity with Optimal Combining as N approaches infinity.
If we set the outage
probability of the no diversity system and receiver diversity with Optimal Combining
systems to be the same (by equating (3.13) and (3.54)), we can find mNoc in terms of ml.
KEGC
exp(4u2-r)_KmN
C
____ex
Q -Qr
+
ln(mi)
I
=Q
N,C
V exp(8a
2UX
exp(1 6u)
-1)
(3.55)
(355
::I
Ne(4,)
n(m,) =N,OC
29X
Vexp(8a,'Xexp(1 6a,) -1
1
0o-
--
1/12
exp(8a)(exp(n6)-- 1
+1
mN.OC
N
KEGC
1e+I
2a
)
So, the power gain of the receiver spatial diversity system over the no diversity system is
48
PowerGain of Re ceiver Diversity with OC = m I mN.OC
______exp(8Qa
=+
1
exp(4N
)exp(1
e-1
a
N
KEGC
(3.56)
/2
U21 -1
XP(802)
6u
+
-3n(m.5
X2ax
As N approaches infinity,
Power Gain of Re ceiver Diversity with OC as N
-+
-
m1
K EGC
exp(2
exp(2U2)
(3.57)
- 21n 2(P.,,))exp(4o )
K EGC
where the last equality comes from expressing mi in terms of P0 ut as given by (3.14). Just
as for power gain of receiver diversity with EGC, notice that the power gain of receiver
diversity with OC as N approaches infinity gives the same expression whether we use the
log normal approximation or the Gaussian approximation. This provides confirmation
that our expression for power gain of receiver diversity with OC for infinite N is correct.
3.4.2.3 Exact
In Sections 3.4.2.1 and 3.4.2.2, we described approximate expressions for the outage
probability and power gain of a receiver diversity system with Optimal Combining. In
this section, we describe how the exact outage probability is calculated.
Similar to Section 3.4.1.3, the pdf of the sum of N log normal random variables
2
Xj
is the
convolution of N log normal pdfs. Letting
N
S'=
(3.58)
ai,
the pdf of S' is
Px,(S') = P (S')
P
2
49
(S')®...0 p
(s').
(3.59)
Since the outage probability of receiver diversity with Optimal Combining is given by
(3.42), it can be calculated by the following integration
N
Pr(Outageof Re ceiver Diversity with
OC
"IC
K
p
5
(s)ds'
(3.60)
N
- -"cNGC Paj(s')()Pa
2
(S')®~
PC
(s)ds'
Since there is no closed form expression for the convolution of N log normal random
variables, we can resort to calculating (3.60) numerically.
3.4.2.4 Comparison of Gaussian and Log Normal Approximations
Figure 3.13 plots the pdf of
s'=
where ln(o)
cc
a2 +... + a,
(3.61)
N(-4J,16ui) and u,=0.3 (moderate turbulence).
The pdf ps(s) is
calculated using 3 methods: using the log normal approximation, using the Gaussian
approximation, and by numerically convolving the pdfs of a0
.
It is seen in this figure
that the log normal approximation is more accurate than the Gaussian approximation.
However, just as in Section 3.4.1.4, the log normal approximation leads to a pdf that is
larger than the actual distribution in the tail. The calculation of the outage probability of
the receiver diversity system involves integrating the left tail of the distribution
2
Thus, the outage probability calculated with the
(integrating below Z'=KEGcN/mN0c).
log normal approximation yields an upper bound to the outage probability.
The log
normal approximation becomes less accurate as the turbulence parameter GX is increased
and more accurate as o is decreased (compare Figures 3.13 and 3.14). We note that the
log normal approximation is valid over a smaller range of N, a. and Po,
1 ,* for Optimal
Combining than for Equal Gain Combining.
50
The Gaussian approximation is clearly not accurate for moderate sized N such as N=9.
However, as N--oo, the Gaussian approximation is accurate by the Central Limit
Theorem and the power gain expression (3.57) is accurate. The power gain as N--+a of
receiver diversity with Optimal Combining is plotted in Section 3.4.3 along with the
power gain of receiver diversity with Equal Gain Combining.
- - Convolution
0.08 - Gaussian Approximation
- -
Log-Normal Approximation
0.07
-247
0.06
b
0.05
.
0
-
0. 04
-
0.03
0.02
---
0.01
0
)
1
2
3
4
5
6
7
8
9
Figure 3.13 Probability Density Function of the Sum of 9 Log Normal Random Variables
where ln(c; 2 )
oc
N(-4Gx 2 ,16(y 2 ) and sYX=0. 3
51
.;2
1
I
SI
--
0.3
-
Con volution
Gau ssian Approximation
Log -Normal Approximation
0.25
1
"
x
0.2 -
0.15
0
/1
V. I r-
-I-
0.05
0
V
0
1
2
3
4
5
6
7
8
9
Figure 3.14 Probability Density Function of the Sum of 9 Log Normal Random Variables Ce; 2
where ln(ot;2 ) oc N(-4yx2 , 16a 2) and sTx=0. I
Figures 3.15-3.17 plot the outage probability of receiver diversity with Optimal
Combining for N=2, 4 and 9 receivers respectively.
probability using
the log normal
approximation,
Each figure plots the outage
Gaussian
approximation
and
convolution. Again, just as in Section 3.4.1.4, we see that in all cases, the log normal
approximation is more accurate than the Gaussian one. Also, for N taking a value up to
9, we see from the figures the same trends as we saw in Section 3.4.1.4: the accuracy of
the log normal approximation decreases as the outage probability decreases and as
turbulence increases. For example, the log normal approximation for outage probability
of receiver diversity with Optimal Combining has the following accuracy for the
following outage probability ranges:
a) for low turbulence qX=0.1 : 0.1 dB link margin accuracy for any fixed probability of
outage above 1012
52
b) for moderate turbulence or=0.3 : 1 dB link margin accuracy for fixed probability of
outage above 10-4
c) for high turbulence a=0.5: 1 dB link margin accuracy for fixed probability of outage
above 0.2
We note that for high turbulence, the outage probability of receiver diversity with
Optimal Combining found using either the Gaussian or log normal approximations are
not very accurate for reasonable outage probabilities. (The accuracy is worse than several
dB for outage probabilities below 10-2).
100
.. .. ......
- ... ....111.1- ........
........ - ......... ....
.......................
...........
4
101
...........
..................
-4
4 "-A
......... ........
Vf
.................
.. .
. ..........
........
.. .........
..........
...............
..
..................
.............
......
..
...
.........
...
10-2
.... .. ......
.. ...... .....
....
.........
. ....
10-3
..........
.. . ....
.. .....
........
..... ....
.
..... ....
.............................
.. .... ... ..
............
......
.. ..............
...........
........ ...............
. . .. .........
........ ....... ........
..............
.....
.......
........ ........
.....
........
....
.......
...........
---------..........
.................
.....
......
..........
..........
.........................
c0
=0. 1, N nZ,
0.3, N .2,
X
a M0.5, N-2,
0.1, N -2,
q'-0.3,
N -2,
X
-0.5, N-2,
a '0. 1, N m2,
a 0.3, N-2,
z z0.5, N -2,
d,
......... ........
..
C
10
-5
........ ......
.......
........
0
............. ...............
10-6 .....
.......
........... ...
. .......
...
...
...
..
....
.....
..
...
10-
-.
77777--77
.......
...
...
........
.:
... . ... ....
. .....................
.....
- . -.. ....
:-- .......
. ...
......
. .... ...
............
......
...
.
......
. .. ......
........
..... .... .... .
.........
....... ....
... .. ..
................
0
..............
M , InoFff-,,,,
O C, Inorm ..........
........
........ .......
.......
OC, Inorm
X".
OQ CLT
.............
OC , CLT ...........
..........
..................
...
..........
.
......... ....
....
OC, CLT
......
O C conv
....
...
.... ...... ...........
.................
............
................
OQ conv
. ......... ......................
.....
OC , conv
...
..
...
...
...
:.....
...
....
...
..
.....
..
.
..
..
..
...
..
..
.......
..
....
...
...
..
....
.....
...
...
..
.
1
...... ...
.. .........
......
...........
2
3
.... .. ... .....I .... ...
...........1:......... ........
. ... ..
4
......
........
.. .... ...
5
6
link margin m (dB)
...-
........
7
-------------------
.......... . .... .... ....
.. .....
... ...... ..
.............
8
9
10
Figure 3.15 Outage Probability For Two Receiver Diversity and Optimal Combining,
53
KEGc=l
10 0
j
I
-C'
10
-2
-
10
K '
-4
-6
.10
0>
N=4, OC, Inorm
* -0.5, N-4, OC, Inorm
N-4, CC,
-a =0.1,
=0.1, N=4,
CC, CLT
CLT
a .,
10
10
a.0.3, N=4, CC,
a .0.5, N-4, CC,
a0.1,
=
N-4, CC,
- 0.3, N-4, CC,
a 0.5, N-4, CC,
-
-8
-10
-
CLT
CLT
conv
conv
conv
+
U
0
1
3
2
4
5
6
7
8
9
10
link margin m (dB)
Figure 3.16 Outage Probability For Four Receiver Diversity and Optimal Combining,
KEGc=l
10
*
2
>*
C
*
+
k
10
..
....
....
..
.o
10
10
.
-
1
-
-
+
-0.5,
-
-
-
0.1,N9,C, inbrm
a0.3, N-9, CC, Inorm
.5, N-9, CC, Inorm
S
ax0.1, N=9, CC, CLT
M0.3, N.9, OC, CLT
N-9, OC, CLT
.0.1, N-9, CC, conv
0.3, N-9, CC, conv
a0.5, N=9, CC, conv
10
-
-
I
0
1
--.
-
.-
...
2
3
4
5
6
link margin m (dB)
7
4..
--
8
9
10
Figure 3.17 Outage Probability For Nine Receiver Diversity and Optimal Combining,
54
KEGc=l
Figures 3.18 and 3.19 show the outage probability for OC using the log normal
approximation (3.46) and the Chernoff Bound to this approximation (3.46) for low to
We see that although the Chernoff Bound for the outage
moderate turbulence.
probability becomes slightly less accurate for higher turbulence or lower N, it is quite
accurate (at worst 1.5 dB link margin accuracy for fixed outage probabilities greater than
104).
100
-
+
x
-------- * -
10
-:-
+
10
+
-----
-x X-
x
N=2, OC-
+
0
+
+
0
i
N=4,
N=9,
N=2,
N=4,
x+
4
. ...
+
+
. ..
LftN=9,
+ ....
* ...............
zMx
+N=1
N=1, Chernoff
xO
......
OC
OC
OC, Chernoff
OC, Chernoff
OC, Chernoff
.......
+
-
.x.
10-10
10 ...........................
........
t. ...............
. .....
x0
0
1
2
Figure 3.18 Outage Probability,
3
4
link margin m (dB)
KEGC=l,
55
5
6
7
ay=O. 1, Log Normal Approximation
10............ ................
4.........I....
-.
.
....
X
±
+
0
10
.±.......
+..
1o 2--
2
re 3.0 9
A-
OC
OC
OC, Chernof:
OC, ChernofV
OC, Chernoft
4
±
mg
m (dB)
tage
...P. ba..i..ty KEGC...... 3.....a.pr
....
x
+
0
0
+
+
0
1 1
Chernoff
OC
--
0
Ig
x
N=1
N=1,
N=2,
N=4,
N=9,
N=2,
N=4,
N=9,
2.
.t
+
10
4.
12.
3.4.3 Comparing Performance Gain of Receiver Diversity with
Optimal Combining and Receiver Diversity with Equal Gain
Combining
In this section, we will compare the performance of receiver diversity with Optimal
Combining and Equal Gain Combining. Tables 1 to 3 summarize the outage probability
expressions for systems with no diversity, receiver diversity with EGC, and receiver
diversity with OC respectively.
Tables 4 and 5 summarize the power gain of receiver
diversity with EGC and OC.
We found in the previous sections that the Gaussian
approximation expressions are not as accurate as the log normal approximation, thus we
are concerned only with the expressions obtained using the log normal approximation.
We will first compare the outage probability and power gain of receiver diversity when
OC and EGC are used. This analysis will be performed using the expressions derived
56
with the log normal approximation.
Next, we will show that using the approximate
expression (3.13) for m1 , we still obtain accurate algebraic expressions for power gain.
Pr(Outage of no Diversity)
Pr
al <I
=Q(ux + I In(m)
11
xp- -
+
in(m )
Chernoff Bound
Table 3.1 Outage Probability of No Diversity System
Pr(Outage of Receiver Diversity with EGC)
Exact Expression
<N
N
M=1N.EGC(
Log Normal
Approximation
(
Q1 In
exp
NEGC
21+exp(4a)N
Gaussian
Approximation
TN
)
N
+
KEGC
iN,EGc
exp(4cr)-1
-
I
exp(4u3)-1
n+
N,EGCf
N
Chernoff Bound
21n +
K---
(1_exp(472)
NL(C)
-J
Table 3.2 Outage Probability of Receiver Diversity with Equal Gain Combining
57
Pr(Outage for Receiver Diversity with OC)
Exact
Expression
N
N
N
Pr
a
.
<
2
E
where K.;
KEGC
EGC
No
'~0C
M N1
Log Normal
Approximation
Q In
1+
+4.
+n
e
KEGC
N
2
- exp
-
(
M2
In
+40. 2
2
e16,
21n 1±
, 216
(
N
Chernoff Bound
KEG(N
Gaussian
Approximation
' -N
exp(4 2)- Kj(;(N
2
exp(89
exp(16U) -1J
Table 3.3 Outage Probability of Receiver Diversity with Optimal Combining
I Power Gain of Receiver Diversity with EGC
Exact
Expression
Log Normal
Approximation
MI /
MN,EGC
MI
KEGC. + exp(44
exp
') eXp
-2n(2P,,,)
2a
+ l(m,
n
+exp(4c )-1
1+exp(290
n
Using approximate m, expression
K EG
Gaussian
Approximation
As N- oo
(Log Normal or
Gaussian
Approximation)
MI
r
exp(4a,2) -
exp(4C- )_I
+
a
+n(m,
)
+EGC 2a-m
e
M,
-KEGC
exp (2o
-2 In1,))exp(2u2)
'E'
Using approximate m, expression
Table 3.4 Power Gain of Receiver Diversity with Equal Gain Combining
58
Power Gain of Receiver Diversity with OC
Exact Expression
mI
/mN.OC
Log Normal
Approximation
MI
KC-
+ exp(16)
exp6Cj)
)-
exp! - 21n(2P,)[2az
-±
1
+ exp(16u,)
inri+ exp(16a ) -1jjep
4
-2
)
Using approximate m1 expression
Gaussian
1
Approximation
AsN-oo
K
+(1(
2 1n(2 P, )exp(8
exp 2ex-2ln(2r,2)exp(4/4
)
)(exp(16
)-
N
m exp(2oi,)
(Log Normal or
Gaussian
(Using
approximate m1 expression
Approximation)
Table 3.5 Power Gain of Receiver Diversity with Optimal Combining
Figures 3.20 and 3.21 compare the outage probability of receiver diversity with EGC and
OC (both calculated using the log normal approximation) for
=0. 1 and 0.3 respectively.
We see that for low to moderate turbulence, EGC and OC have approximately the same
outage probability. (The reason that the outage probability curve for EGC is below that
of OC in Figures 3.20 and 3.21 is the accuracy of the log normal approximation for OC;
the OC curve with the log normal approximation is only accurate to 1 dB as the outage
probability approaches i0-4 from above).
Figure 3.22 shows the outage probability of receiver diversity with EGC and OC for
Cb=0.5. The EGC curves are plotted using the log normal approximation whereas the OC
curve is calculated using convolution of pdfs. (The reason that OC curve is plotted using
the convolution rather than log normal approximation is that we found in Section 3.4.2.4
that for
,= 0 .5,
the log normal approximation for OC has accuracy that gets
59
progressively worse than I dB as outage probability decreases.) We see that that for high
turbulence, the outage probability of OC slightly lower than for EGC.
100
-- - - -
102
N=1
N=2, EGC
N=4, EGC
+
N=9, EGC
+0
N=2, 0C
A N=4, 0C
N=9, 0C
x
x
+
+
-to--
0
LI
0-4
-.-.-.....
-0--
-
o
10
)
0)
Co
...
. .
0108
.
...
...
10- 10
±i+
0
1
2
4
3
5
6
7
link margin m (dB)
Figure 3.20 Outage Probability,
KEGC=l,
60
yC=O- 1, Log Normal Approximation
. . . . . . . ... . .
.. .. . . ... . .. .
.. . .. .. .... . .. .
........ ........
.........................................
....... ........ ........ ...............
........
...
........ ........
:+
......... ..
10-
....................
.......
........
t
......
co
-0
0
+0
+
..........
..
..........
-2
CIL 10
a)
cm
-I-.-ca
X
.........
..X
...................
........
......... ..
.....
...........
......
..........
.. .. . ..
. .. . . . .. . .. .
4
1
.. .. . .. .
X .
44
.............
. . .. . .. .. .
.. .. . .. ..
. . . . . . . . . . . . . . . ..
..
X
0
...
..
..
..
....
.........
.
EJ. ....... X
...........
+ EX
..............
+
..
.....
....
..
....
...
.
...
...........
.......
...
...
...
...
..
. .........XX.......
......
......
...
....
...
......
....
. .....
...................
................
......
..... IX
0
X
:,X*
XX
:
+
10-3
1
0
2
3
10
7
4
5
6
link margin m (dB)
Figure 3.21 Outage Probability,
KEGC=I,
.. .
................. .......... I........................ .................. ........ . ........
.......
.............
.. ......... ........ ......
...............
......... .......
...
.....
..
.......... .................. ............
.............. ....
......
...
.. ...
...
.......
.....
...
r........
... .. ......
8
9
10
(yx=0.3, Log Normal Approximation
.... ....
....
....
..
......
....
..
.. .....
.I....
...
..
. ..
..
.....
..
. ..... ....
N= 1
N=2, O C , conv
N=4, OC, conv
N=9, OC, conv
N=2, EGC, Inorm
N=4, EGC, Inorm
N=9, EGC, Inorm
....
. .....
...
..
...
......
...
...
..
....
...
...
...
.....
.....
..
....
.
.. .. ....
...
...
..... ..
...
....
......
......
......... ..
.. ..
...
...
....
.....
..
..
....... ...
......
....
...
....
......
..
...
...
....
..
......
...........
....... -...........
.......... .
......
10-
...
. ....
..
..
.
.....
..
..
........... .......
X .......
........
1.........
..........
10-2
..
X
........ ............... + .*.........
0
. .. ..
. . .. . .. .
. . . . . ....
. . ..
... . . . . . . .
X
=3
CL
N=1
N=2 , E GC
N=4, EG C
N=91, EGC
N=2. OC
N=4 0 C
N=9, OC
X
+
+
0
0
xt
X
+
+
0
. .....
..... ...
....
.. .
...... ............... ..
........... ...
......
......... ........
......
........
...
...
..
....
..
...
....
...
.
..
.....
. ........
....
....
..
.. ..... ....
I...........
......
..
...
....
. ....
..
.....
..
.....
..
..
........
....
...
...
..
..
..
....
...
...
...
...
..
..
....
.... .....
.
..
....
....
..
......
...
..
...
...
...
....
....
..
..
0
...
..
..
...
.....
......
..
....
....
.....
....
...
...
....
..
..
..
...
...
..
..
+
+
X
................
..... ............ ....................
.......
-----------------........
..........----------................
.......... ............
....
.....
I......................
........
........... .....
....................
.......
........ ....
.....
....
..
.....
..........
....
".*.,.,.,:
+......
..........
...........
...
....
......
..
...
...
..
..
..
... ..
...
....
...
..
..
...
..
...
..
.....
. .. ..
..
.......
....... .......
........
.....
.....
.. ....
..........
WX
.....
.......
.................
..... ......
........... . .....
.
.....
. . ........... ........... .....
0
2
4
6
8
10
12
link margin m (dB)
14
...........
...
16
Figure 3.22 Outage Probability, KEGc=l, (yx=0.5
61
..... .......
.....
18
20
Figures 3.23 and 3.24 show the power gain of receiver diversity with EGC and OC for
o7=0.1 and 0.3 respectively. The plots are calculated using the (3.23), (3.48), (using the
log normal approximation) and the first line of (3.14).
systems is seen to be approximately the same.
The power gain of the two
Again, due to the accuracy of the
expressions as described in Sections 3.4.1.4 and 3.4.2.4, the curve for EGC is slightly
above that of OC in Figure 3.24. Figure 3.25 shows the power gain of receiver diversity
with EGC and OC for u,=0.5 (the EGC plot is calculated (3.23) and the OC curve is
plotted by comparing the no diversity outage probability curve with the outage
probability for receiver diversity with OC curve using convolution of pdfs). Recall that
for a,=0.5, the accuracy of the EGC power gain is within 1 dB for outage probabilities
above 104. We see that that for high turbulence, the power gain of OC is just slightly
higher than for EGC.
Thus, for all turbulence levels, the outage probability of receiver diversity using Equal
Gain Combining and receiver diversity using Optimal Combining are approximately
equal.
Therefore, we do not achieve significant additional power gain by using OC
compared to using ECC. This is because deep fades occur only occasionally. When a
received signal is deeply faded, it adds little or no performance with or without optimum
combining. For the detectors with signal power close to nominal, straight addition is
close to Optimal Combining.
The power gain from using receiver diversity with either EGC or OC comes from the fact
that it is less likely that all N receivers in the receiver diversity system see deep fades
than it is for the receiver in the no diversity system to see a deep fade. We see from
Figures 3.23-3.25 that power gain increases for larger N but that the incremental gain
decreases for larger N.
This makes sense because we would expect that the more
receivers the system uses, the better the performance will be and that there is a limit to
the performance gain that can be achieved. From Figures 3.23-3.25, we also see that the
power gain is larger for more turbulent air conditions or for lower outage probabilities.
Intuitively this makes sense because we would expect that diversity will improve system
performance under more stringent conditions.
62
2.5
......................
.......................
......................
......................
.....................
2 ...................... ....................... .......................
....................
1.5
. ..................... .......................
....................... ....................... ....................... ? ......................
....................
......................
V
0
.S
CO
C)
....................... ...................... ....................... ........................ ....................
1 ..................... ............................................ .......................
0
CL
2
EGC, P
Out"" 10
0
..........
.
.......................
0C,
P
10 -2
...............................................
0. 5 ..............................................
out
10-4
EGC, P
X
X
out
4
100
OC, P
0
out
UW
1
2
3
7
6
5
Number of Receivers N
4
0
9
8
Figure 3.23 Power Gain, KECC=I, (yx=0.1, Log Normal Approximation
8
.....................
......................
.......................
.............
..................
.......................
............
.......................
7 ......................
....................
.................
......................
..............
.......................
.......................
...............................................
6 ......................
................
.................... ...................... ....................... ....................... ....................
.......
....................
1...............
.......................
a:S
C)
.S
cis 4
0
.....................
.......................
.......................
......................
.......................
.....................
......................
.....................
-t...................
6 3
....................
......................
.......................
.......................
...............
........................
......................
.......................
.....................
......
CL
2
....................
....................... ....................... ....................... .........
1
1
F-
......................
2
3
Figure 3.24 Power Gain,
4
E C I P out =10
0
0
OC, P*
X
X
EGC, P *
0
0
7
6
5
Number of Receivers N
KEGC=I,
4-
-2
= 10-2
out
= 10-4
out
OC, P* = 10-4
out
8
9
(yx=0.3, Log Normal Approximation
63
10
1~
1
(D
2......................
1
............
4 .........
.....
8.
+
+
(0
CL
+
+
-.......... -.......
x
............-........................
2 .
x
out 10-2, Inorm
0 OC, Pou
-2, conv
out
E G C , P out 10-4 , Ino rm
EGC, P
0 :C,
n.
_
1
_
2
__L
_
3
4
P
out
-
10~, conv
__________
5
6
7
Number of Receivers N
8
9
10
Figure 3.25 Power Gain, KEGC=l, c=X=0.5
Now let us consider the power gain of using receiver diversity when the number of
receivers approaches infinity. This will provide a limit on the possible power gain that a
receiver diversity system can obtain by increasing the number of receivers. Figure 3.26
shows a plot of power gain for receiver diversity under low to high turbulence levels
when the number of receivers is infinite (calculated using (3.31) and (3.57) which used
the log normal approximation). Link margin m, was calculated in terms of P,,,, by using
the first line of (3.14). We see that the power gain is higher for higher turbulence levels
and also for lower outage probabilities.
We also observe that receiver diversity with
Equal Gain Combining provides significant power gain over a single receiver system.
Using Optimal Combining does not provide much additional power gain over using
Equal Gain Combining. The implementation of Optimal Combining is more complex
than Equal Gain Combining since it requires estimation of the fading factors a and also
requires N additional
multiplication blocks.
A
system that uses
the simpler
implementation of Equal Gain Combining will benefit by almost the performance gain of
the Optimal Combining.
64
0 0EGC: = 0.1
'V 'AJ''":
'''iii"'
''''i
X XIt' II II m
I illll
II'Of
3 1IIIl
III
EGC cy., O 3
CY=0.3
x~xEGC,
SI I s I I
Nvlklt I
iI
gilil
I
~+OCa=,0.3
C , CY =0.5
XX
o
*
, . .
I Bi Ii IiI
Ii iII
iIII
1losti
i
I
, I Im
i
i - & I aiO
T -- T - r I-Ii- r T' r i i - XI -
I ' T -1 :
IiB I
4--
. : , , ,',
" ,',
1 I 1 B i EiiIBl
i
I
i X .
I iii1E
I I II
M
I
'L L J --Ii
C3 I I i I IEl
1 0 is -:-.L BII I
ii
B i iI iA
I I I I
a)
I
>'AiI
'V
IL
E
L I I.JI LE
t
i fiIl
I I Bl i p
al s
0
I
E
l l
1 i i
allI
10
10
I
I I 1 11 1I
1
10
lr
i1 i i i i
1 i
10
10
I
10
10
Outage Probability
Figure 3.26 Power Gain as the Number of Receivers Approaches Infinity
Recall that when we substituted the approximate expression for mj (last line of 3.14) into
the power gain expressions, we obtained algebraic expressions for power gain as a
function of outage probability. We now consider how close these algebraic expressions
are to the power gain calculated using the exact mj value (first line of 3.14). Figures
3.27-3.29 plot the power gain using the log normal approximation Ojust as in Figures
3.23-3.25) and also the power gain using the log normal approximation and the
approximate expression for mj in terms of outage probability (the last line of (3.14)).
These figures show that the approximate expressions for power gain (3.23) and (3.48),
which are functions of outage probability, are fairly accurate (to within
turbulence and within 1 dB for moderate to high turbulence).
expressions for power gain are accurate.
65
dB for low
Thus the algebraic
3
2.5
. ..................... .......................
.......................
.......................
.......................
.......................
.......................
...................
t
.
.
..................
.
......................
.......................
.......................
.....................
...................
-f.................
2
...................... ...................... 4
.................... -
CO
.......................
.......................
......................
......
.................
.......................
.......................
.......................
......................
1.5
....................
0
0
a-
.......................
.......................
.....................
....................
1
+
+
0
0
10 -4
out=
EGC, P =10- 4 Chernoff
out
EGC, P
.......................
....................
X
0. E ...................... .......................
X
0 C, P
0
2
1
3
Figure 3.27 Power Gain,
4
0
OC, P
out
out
8
5
6
7
Number of Receivers N
KEGC= 1,
1 0-4
=10- 4 Chernoff
9
10
(y.=O. 1, Log Normal Approximation
...................
...........
. ..................... ............................................... ............................................... ........
....................
.................
......
......................
....................
.......................
.......................
.......................
.......................
6 ......................
....................
......................
......................
........................
......................
. ....................
.....................
.......................
5 ......................
0
C:
.......................:....................... ....................
......................
.......................
.......................
...............................................
4 ......................
.....................
...................... .......................
.......................
........................
.......................
........
0 3 ....................
a-4
10
+
+
EGC, P
o ut
.......................
.......................
.......................
2 .....................
-4
0
X
1 ...................... ....................... ....................... ....................... ......
1
2
3
4
0
X
EGC, P out =10
0C, P
0C,
5
6
7
Number of Receivers N
P
Ou t
Chernoff
=10 4
4
0ut 10- Chernoff
8
9
Figure 3.28 Power Gain, KEcc=l, (TX=0.3, Log Normal Approximation
66
10
1
12 ---
10 .......
-
-.
0
.
...
..... ..+...........+
. +.
+--
G+.C P
=
_
mO
0
++
0-
+
..+
2 ........-...
S
OE3
1
2
3
4
5
EGOP
10
put
EGC, Pout 10
7
6
8
4
Chernoff
9
10
Number of Receivers N
Figure 3.29 Power Gain, KEGC=l,
OX =0.5,
Log Normal Approximation
The power gain limit (3.32) and (3.57) using the exact value of mj and the approximate
expression for mj are plotted Figures 3.30 to 3.31. These figures show that the algebraic
expressions for power gain as N approaches infinity ((3.32) and (3.57)) are accurate to
within 1 dB for low turbulence and 2 dB for high turbulence.
67
25
x IX
<-I
.
20
x x
I tite
i
EGC
=0.1 ,Chernoff
EC. ,rO=.3, Chernoff
EGCc=0.3, Chernoff
EGC, a=0.1
I--I ,
...
i
|,||
|
,
-
,.
|
|
z
.
4
.1
T "-:-:
-v-
-
-
xil
T
x
a=0.3
EGC,
S.x,
a)
-
EGO
a=0.5
r
IT'
-I
T
--
1I
r
15
a
i
i
i
i
--o
a i llt. I I
i*mmi
eil
I
j ------
C
CD
a)
iIi
t4 4ii
&
xl
1 11
11X1'
11111
I 41
IIIII
L -P-1-Ij LL:
--
L I L UJ t
10
3*
r T T r:l-
C"
- -
1"E'",TI
- - -
1"" T
B ~ ~~ ~ ~
5
r rIT
-
i- 1ITj
I
r
T -
T
~
I
-r~m$1~j
-- -
r"lrI-
p4
~
*
*
T -
"
-rm
r
T
"l
i a a i ll
-
~ s ~ i ~ i isiiiI
~ ~
I
'I
I
Tm
" "
4 4
yr
m IIm
-.' e I L I
~
1 4.4
I I
1T
0
10
6
10Outage Probability
10"6
10-1
10-2
10~4
Figure 3.30 Power Gain for EGC as the Number of Receivers Approaches Infinity
30
''
*
I
I
I
I
II
T
,Ir
I
,
IIII
-- +1"-r'--r - 'err r
1i-il
-
-
,
,
z
.
i
,-
"|
E 20
i
z;-
i
i
-I
i
i
ill,:
i I 11:iI
,,I
i
I Ii eI
iIII
00, a =0.3, Ohernoff
=4 0.6, Chernoff
OC,
-
IIIII
vw--r.
It
,1-
0Cy0 1 Chermoff
-
0..C
.,,a
,
±OC, a =0.3
u
I
111.1I.VC>
.
i
III -
It
I
N'm.2%-...
C
M
-
15'
-r---T-- i I -:-- ' - .- -. ------ L . - --J _ A -A i- i- 1 -rO
i I I I -- 1= 2. : ---2 -.
+4
I
fI.I
C
I I
I~~~~~~~~~~~~~~~~~~
:I i I I
:I : I : I : I :
II
al
4 c
1111
-r-
11 1111I
I
- T II
- I
-- -T
I
I-l
'
"
II
SI
II
I
11
1
1
1
I,
I
II
I -I
I
iIIII
, II
I
?'I.i
imu
.. l4J
I
I
I
I
4
IIi
I
i t :
1
Ir I i i
4 i -Iit- - -
I
1a
1
Vj)
T
I .
I
T
n-a-, EI
T I._j j j
it
it
L'' .LIi i i I l
1
1 1
1II I
1
1
1
I
n
10
1-5
1-4
10-3
1-2
10-1
Outage Probability
Figure 3.31 Power Gain for OC as the Number of Receivers Approaches Infinity
68
3.4.4 Receiver Diversity with Selection Combining
Figure 3.32 shows an optical communication system that uses spatial diversity at the
receiving end with Selection Combining (i.e. selection of kth branch with largest fading
factor ak1). It is assumed that the fading factors ctj for i=1 to N can be tracked. This is a
realistic assumption since the bit period is much larger than the coherence time. The
number of receivers N22 and each pupil area is A/N so that the sum of the N pupil areas
is the same as the pupil area of the no diversity system.
Peceiver 1
IlareaiAN
&m Al/
DeteoraaAd
EstinAte o
Transnitter I
Pupil aea A
.
Receiver 2
.uiRijil .....
area AVN
ILetor armad
ppI
Select rrh
k Wth largest
IDwision
Oki
Estinte ot
Figure 3.32 System with Receiver Diversity and Selection Combining
The outputs of the receivers are given by
H0 : R,o =ai mNSELS* /N+noi
Ri'l = noi
Hl :
(3.62)
Ri.0 = n0 i
Ria =ailmN.SELS*
for i between I and N and where
69
I N + no,
iz, and noi are the same as defined in Section 3.4.1
Ri,o, Ri,1 , S*,
mN,SEL
is the power link margin provided by the transmitter above the required
power to achieve Pe* under no fading (the subscript NSEL refers to the N
receiver system with Selection Combining)
Say that the decision rule is to choose Ho if
RkO
(3.63)
> R.I
and H, otherwise. The outage probability of receiver diversity with Selection Combining
is
Pr(outage of receiver diversity system with best branch selection)
= Prrakl
mN,Sel Sj
= Pr ail
MN,Sel S
<
(S
)2 Nobtaflc
Nj
No
lbranch
for i from Ito N
Ilrec
N
N(
I
2
N
MNSel S
=
Pr
al
N
=Prf
=
Pr
a
LY
where KIijranici
<S* )2 Klbranch
N"'ranc
N0
N( e
-N
N, eN K-l
<
N,SeI
ln(ail)
]
J
< In
N
Kbr, e
MN,Sel
-N
-2a
N
2-in
mN,Sel
(3.64)
2a
=4
Q -ax
+
N
I n
2ax
+ 2N
mNSel
N Klbraf
2
exp -
70
MNSel
+ In
2ax
NVK,,,cl
For fixed field of view receivers, and background limited noise,
KIb,,,h=I1N.
Figures
3.33 to 3.35 compare the outage probability (3.64) against the outage probability of a no
diversity system (3.13).
10
....I...
. .. .. .. .. ..
. . . . . . . . . . .. . . . . . . . . . ..
. .. . ... .. .. .
. . . . . . . . . . . . . . . . . . . . . . . ... . . . . . .
. . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
.........................
............
.........................
............
............ ..........................
.. .. .. .. .. ... .. .. .. .. .. . . .. .. .. .. ...... . .. .. .. ..
10-
Ug
1
.
.......
...........
................
.
..............
..........
................
. . .. .. . . . . ... . 0
10-2
. . . .. . . .
....
.. .. ..
.............................
........
..... ...
.
. ......
............
.....
.....
.......
. .....
..........
......
. ......
.5t+*.'*'.*.'.*.*.'.'.*.'
......
. .. .. .. .
............
........
. .
. . .. .. ... ... . .. .. ..
...
. ... .. .. .. .
.. . . . . . . . . . . ..
....
...
...... .............
. .........
..........
.. ......I . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
. . . . . . . . . . . ....
. . .. .. .. .. .
. . . . . . . . . . . . . . . . . . . . . . . ..
........
10-3
......
...
..............
.............
........... .......
..............
...........: * * * * , * ' ' * ' ' *., * * * , * ' '
. . . . . . . . . . . : . .. . . I . . . .. . . .
. . ... . .. . . .......
.. ..
D .. : .
C2= 0 . 1
........... .............. .
10-4 ---------------------
.....
.......
X
........
..
..................
.....
............ ............
........... ...........
4
=0.62
6
: * ' ' * * * , * * * , *..
. . . . . . . . . . . ....
... . .. ..
. ... . .
..
..
...........
......
...
....
..
..
. ...
.
...
.......
..............
............
............
............
JBI
.......
..........
.............. .........
N = 2 , S el8
...........
........
................ ..........
........
=1
-=O .3
2
=0.5
=O. 12, N=2,
=0.3 2 N=2.
10-5
0
..... .........
2
.....%.
.......
....
..
.
-''*
...
..
.....
..
..........
... (i
..
...
...
....
...
...
....
....
...
...
.....
....
......
...
.X
..
.............
...........
.....
.....
...
......
10
.....
12
14
Figure 3.33 Outage Probability of a Two Receiver Diversity with Selection Combining System
and a No Diversity System, KI branch= I
71
a
n
"
n
10"
0
10
:
:%
S
10
x
x
)=o0.32, N=1
=0 .52 , N = 1
o
:l3:2
+
..
-.
-..--..--..-..
- . - .-.
-.
-..- ...- .
..
- -
.12,
1 N=4, Sel
0 () 1$
0
:
1
0.32 N=4, Sol
4
.52N.46
4
2 ,3N=47S9l1
2
3
4
5
5
6
7
8
9
10
11
Figure 3.34 Outage Probability of a Four Receiver Diversity with Selection Combining System
and a No Diversity System, KIbrach=l
100
10-2
10
6
+=0.1
+
-;
10
(
4
ci
+
<>
0
i
+
K>
1
,N=1
+
~
0.32,N=1
052,N=1
xx
8
2
A 0.12, N=9, Sel
0.32 N=9, Sel
II2N=9SeI
.
2
3
4
5
6
7
8
9
10
11
Figure 3.35 Outage Probability of a Nine Receiver Diversity with Selection Combining System
and a No Diversity System, Klbranch=l
72
When a=O.1 (low turbulence), the receiver diversity system with Selection Combining
has worse performance than the no diversity system for small link margin m<<N. This
is because using the signal from only one branch means that under no fading, the receiver
output signal is, on average, 1/N times that of the no diversity system (since the pupil area
of each receiver in the diversity scheme is Ap/N). If low turbulence and low link margin
(1<m<<N) are added to the system, the output signal will almost always be below
threshold.
However, as m is increased, the a,=O.] diversity curve intersects the no
diversity curve. This is because link margin m is multiplied by fading factor
Xkl
as the
wave propagates from the transmitter to receiver. As m is increased, there is a better
chance that the received signal on the best branch will be greater than the threshold (due
to the increase in average power and magnified instantaneous power by
ai).
When there is more turbulence, at say a,=0.3 or a,=0.5, for low link margins, the
Selection Combining scheme will have slightly lower outage probability than at low
turbulence. In other words, for low link margins, turbulent channels provide better
performance than less turbulent channels. Though this may not seem intuitive at first, the
reason for this behavior is that the average received power is much lower than the
required threshold so the additional turbulence is more likely to push the signal above the
threshold. However, as the link margin is increased even more, the diversity curve for
higher turbulence intersects the diversity curve for lower turbulence. For link margin
values above this point, the higher turbulence performance is worse than for low
turbulence. This can be explained by the fact that after the link margin is increased to a
certain value, without fading, the received signal is above the required threshold. So if
the link margin is fixed at that value, and turbulence is added, as turbulence is increased,
there is more chance that the signal will drop below the threshold required value. Thus,
for low link margins (m<<N), turbulence helps us and for high link margins (m>N),
turbulence adversely affects us. It is clear from the outage probability plots that Selection
Diversity does not have as good performance as Equal Gain Combining or Optimal
Combining.
73
3.5 Combined Transmitter and Receiver Diversity with Equal
Gain Combining
Figure 3.36 shows an optical communication system that uses spatial diversity at the
transmitting and receiving ends and uses Equal Gain Combining at the receiving end.
The number of transmitters M 2 and number of receivers N_>2 with M not necessarily
equal to N. Each transmitter pupil area has area A/M so that the sum of the M pupil areas
is the same as the transmitter pupil area of the no diversity system. Thus, the power
transmitted by each of the M transmitters is IM times that transmitted in the no diversity
system. Each receiving pupil has area A/N so that the sum of the N pupil areas is the
same as the pupil area of the no diversity system. Thus, under no fading, this system has
the same transmitted and received powers as all the systems analyzed thus far.
...
Transmitter I
Pupil area '/M
Transmitter 2
Pupil area /M
Transmitter M
Pupil area A/M
Receiver I
Pupil area AIN
Ri, ,R1,1
Detector area Ad
'
Receiver 2
Pupil area A/N
Detector area Ad
.. -.....
.RN,o,
.
....................
'
Sum
Ro, R,
- >
eiso
RN, I
Receiver N
Pupil area A/N
Detector area Ad
Figure 3.36 System with Transmitter and Receiver Diversity and Equal Gain Combining
The outputs at the receiving end after Equal Gain Combining are
74
H:R
N
R j = O , GEG
,MEG
R = ~R 1 -nEGC
(3.65)
N
HI: R 0 = I3Ri 0 -nEGC
R=
NRi.[1
a
N
iMEGCS*
+]nOEGC
where Ri,o, Ri 1 , S*, and nO,EGC is the same as defined in Section 3.4.1
mM,N,EGC is the power link margin provided by the transmitters above the required
power to achieve Pe* under no fading (the subscript M,N,EGC refers to the M
transmitter, N receiver system with EGC)
aij is the log normal fading factor from transmitterj to receiver i
The system shown in Figure 3.34 is isomorphic to the receiver spatial diversity system
with EGC of Section 3.4.1 with the number of receivers being NM.
3.6 Time Diversity at Receiver
Let the transmitter send the signal N times, separated by fixed time periods T that are
much larger than a typical deep fade
duration.
Then the receiver receives N
independently faded copies of the signal, combines them in some appropriate way, and
determines whether a 0 or I was sent. If the receiver aperture area is Ar/N, then the time
diversity system is isomorphic to the spatial diversity with one transmitter and N
receivers.
75
Chapter 4
Conclusions
Optical communication through atmospheric turbulence is a desirable means of
communication because it provides high transmission speeds (gigabits per second) over
short distances (kilometers).
However, optical communication through the turbulent
atmosphere poses difficulty due to eddies of air that mix and cause deep fades that last as
long as 100 milliseconds. Rather than using the "brute force" method of increasing the
transmit power by tens of decibels to overcome fades, one can use a sensible system
technique. Coding is not a practical alternative because the long coherence times, which,
at high data rates (gigabits per second) are on the order of billions of times longer than bit
periods, would require enormous interleavers. In this thesis, we studied diversity as the
system technique to mitigate fades in atmospheric optical systems.
We found that
diversity can significantly improve the system outage probability and thereby provide
substantial power gain.
The diversity schemes considered in this thesis are receiver diversity with various
combining schemes, combined transmitter and receiver diversity with Equal Gain
Combining, and time diversity.
We identified the second diversity scheme to be
isomorphic to receiver diversity with Equal Gain Combining and the third scheme to be
isomorphic to receiver diversity. Thus, the results of receiver diversity performance may
be applied directly to combined transmitter-receiver
Combining and to time diversity.
76
diversity with Equal Gain
In analyzing the performance of receiver diversity systems, the combining techniques
considered were Equal Gain Combining, Optimal Combining, and Selection Combining.
We found Selection Combining to have inferior performance compared to Equal Gain
and to Optimal Combining.
Moreover, Equal Gain Combining provides substantial
power gain. For example, at an outage probability of 10-4, using just four receivers with
Equal Gain Combining can provide approximately 8dB of power gain.
We further
established that there is relatively little additional gain in combining optimally rather than
equally. This result, though initially surprising, is a valuable one. Optimal Combining is
considerably more complex to implement than Equal Gain Combining because it requires
channel estimators. Thus, Equal Gain Combining is the combining method of choice.
When designing a practical diversity system, in order to make maximum use of
resources, one should note the general performance trends.
For all of the receiver
diversity systems we analyzed, the power gain is larger when atmospheric turbulence is
high and when the required outage probability is low. This is an intuitively pleasing
result since we expect diversity to provide performance improvement under more
stringent conditions. We also found that the performance of receiver diversity increases
as the number of receivers increases.
However, the marginal gain decreases as the
number of receivers increases. Substantial gains can already be realized with a modest
number of receivers. Using these trends, a cost analysis may be considered to design a
diversity system with the optimal number of receivers.
A major benefit of diversity in optical communication is that it is not susceptible to
drastic performance changes due to inexact models. The analysis performed using the
Kolmogorov turbulence model and Extended Huygens-Fresnel Principle modeled the
fading as log normal. However, in reality, the actual tail of the fading probability density
function, where occurrences are very unlikely, will be different from that predicted by the
idealized log normal model. It would be a mistake to design a system that relies on the
exact statistics of these low probability events. Diversity results in the operation in the
high probability region of the probability density function thereby eliminating the need to
77
rely on inexact statistics of the tail. Diversity is a sensible technique to mitigate fades
due to atmospheric turbulence.
78
APPENDIX A
Intuitive Understanding of Fading Model
This appendix describes an intuitive explanation for the validity of modeling the
amplitude fluctuation as log normal distributed.
Consider an optical signal that is
transmitted at one point and received at a distance away. As shown in Figure A.1, the
path that the laser beam takes can be thought of as being comprised of a large number of
smaller paths, each contributing an amplitude path loss
i
and phase change eil to the
signal passing through it.
transmitter
s(t)
Kiexp(joo)
r(t)
receiver
/
Kiexpoo1 )
K2exp($ 2)
Figure A. I Visualization of Wave Propagation from Transmitter to Receiver
If we denote the original transmitted signal as s(t), then the signal impinging on the
receiver is
79
r(t) = s(t)Kl K 2 -- KN exp(j(
+ 01 + .. + ON))
=s(t) exp(ln(K K 2 ...KN))exp(j(bl +
=
+
.+
ON))
s(t) exp IIn K exp(j(#A +0 +...+ON))
N
N
By the Central Limit Theorem, as N becomes large, both
In K and
Oi are Gaussian
random variables. Thus, the amplitude change and phase change as a laser beam travels
from the transmitter to receiver can be modeled and log normal and Gaussian
respectively.
80
APPENDIX B
Log Normal Approximations
B.1 Mean and Variance of U where Z=eU =-C1+...+CN and Z is
Log Normal
Consider the log normal random variables a,, ..., aN where ln(ai) for i=1 to N are
N
Gaussian, N(-2q,2 ,4q,2). If the random variable Z = Lai
=
e
is log normal, then the
i=I
mean and variance of U are given by
pt =ln(N) --0.5 In 1+ ex
(B.1)
N
o~
=n~iexp(4c')
ae- I
t
in
(B.2)
Euto=
Equations (B3.1) and (B.2) are developed in this appendix.
In general, a log normal
random variable X, where ln(X) is Gaussian N(, 02), has mean and variance given by
E[X ]= exp(p + 0.5.2)
(B.3)
- = (E[X ])2(eXp(.)1)
(B.4)
Since o is log normal distributed where ln(aj) is Gaussian N(-20$j,43/'),
81
and
(B.5)
= exp(4a ) -
(B.6)
E[a]
a
1
If we define a new random variable Z as
N
(B.7)
a,,
Z=
i=I
and know that Z is a log normal random variable, then it must have mean and variance
given by
E[Z]= NE[a,]
- =
=
2
expJu
+0.50-2)
(E[Z]) 2 ((2
1)
(B.8)
(B.9)
where Z=eJ where U is Gaussian N(pu, a2). Using (B.8) and (B.9), we can solve for pu
2
and au2.
Dividing (B.9) by (B.8) squared,
N(72 "
N 2E[a]
1+
=exp(a )-2
2
=exp(Cr
O
2
2
NE[a]
(B. 10)
a2 =In 1+±"
U
NE[ a] 2
=lnl+
p(
2
)-
N
Substituting this into (B.8) gives
plu =ln(NE[a])-0.5n( 1+ NE [ "E
a]a2(B
=
ln(N) - 0.5 In I +) exp(41
N
82
I
1
B.2 Mean and Variance of U where Z'=eU
a 2+
.+aN2 where
Z is log normal
Consider the log normal random variables a.
where ln(u ) for i=1 to N are
a
Gaussian, N(-4ou7 2 16o,2 ). If the random variable Z=
N
ai
=u
is log normal, then the
mean and variance of U' are given by
plu, = In(N) +4a' -0.5 In I+ ex(
a'
N
exp(I 6a') - 1
UU,=In
(B.12)
(B.13)
N
The expressions (B.12) and (B.13) are developed in this portion of the appendix. The
development is similar to the development of (B.1) and (B.2). Since a;2 is log normal
distributed where ln(oi) is Gaussian N(-4af,16a2), we can use the general equations for
the mean and variance of a log normal random variable, (B.3) and (B.4), to express the
mean and variance of a;2 as
E[a] = exp(4j)
a
= exp(8o, )(exp(16o-)
(B.14)
-1)
(B.15)
If we define a new random variable Z' as
N
Zl= Lai2
(B.16)
1=1
and know that Z' is a log normal random variable, then it must have mean and variance
given by
83
E[Z']= NE[a2]=exp(,u +0.5U2,)
(B.17)
= Na 2 = (E[Z'])2(exp(U,) -1)
(B.18)
2,
where Z'=eU' where U' is Gaussian N(,uu,a
2).
Using (B.17) and (B.18), we can solve
for pu, and au, 2 . Dividing (B. 17) by (B. 17) squared,
N.a
N 2 E~t2 ]2
+
a]2
NE[a2i 2
(B.19)
2
YU =In I
+
",
NE[ari2 12
=ln
K+
exp(6
N
1
)
Substituting this into B. 17 gives
pu, =
In(NE[a2])
62_
)-1J
~O.1nl±exp(16cY
0.5 In 1+~
N
(B.20)
=
ln(N)+ 4u2 --0.51nI + exp(160j)- 1
N
84
APPENDIX C
Received Power
This appendix derives an expression for received signal and background noise power as a
function of the transmitted signal power and system geometries.
C.1 Received Signal Power
Figure C. 1 shows the geometry of a transmitter and receiver separated by a distance L in
an optical communication setup. The receiving pupil is assumed to be in the far field of
the transmitter. The transmitting pupil has diameter D, and area Ar, the diffraction limited
angle is O~A/D, where A is the wavelength of the wave, the main footprint of the
propagated wave's intensity at the receiving plane has diameter Dfoo,.tpit and area Aoorint,
and the receiving pupil has area Ap.
Signal
Footprint,
Dfootprint,
Afootprint
...........
...............
Detector
...
O=X/Dt
Transmitting
Pupil, Dt, A
Receiving
Pupil, AP
L
Figure C. 1 Geometry of Single Transmitter, Single Receiver Setup
85
The transmitted wave's footprint area at the receiving plane is given by
Af
=
""
KD; fi-
,,,
)2
2
=;2
_ Z AL)2
- -t- =~-
=(L)2
2
4 D,
(L2
4A,
2
ZL2
--
=
A,
(C.1)
4
Hence, under no fading, the fraction of the transmitted power that the receiving pupil
harnesses, assuming uniform intensity across the main hump of the intensity pattern at the
receiving plane, is
A~
AA,
.r
rAL
Afootr t
(C.2)
C.2 Received Background Noise Power
Figure C.2 shows a circular receiving pupil, circular detector and their corresponding
field of view and diffraction limited angle. The receiving pupil has diameter dp and area
A 1 = zdp2/4 while the detector has diameter dd and area Ad=zrdd2 /4.
QFV
pupil
detector
QDL
dd, Ad
dp, Ap
Figure C.2 Geometry of Single Receiver
The diffraction limited angle of the receiver is a function of the pupil size and is
approximately given by
ODL
dp
So,
86
(C.3)
DL
r
4)T dp
4)
(C .4)
/=-
2
A,
The field of view of the receiver is a function of the detector size and is given by
OFV
Ffc
(C.5)
dd
So,
QFV
FV 4
fJ ,
Ad
(C.6)
f'2
The received number of modes by the receiver is a function of the diffraction limited
angle and field of view and is given by
received number of modes -
FV
Q DL
-
(C.7)
dP
(f-4)2
2
2
c
We note that the received number of modes is proportional to both the pupil area and the
detector area.
The accepted model for background noise is a uniformly radiating source. A uniform
extended background source is assumed to be infinite in extent and is present in the field
of view of any receiver. The collected background light power by the receiver in Figure
C.2 is,
Pb = Pb0
FV
-
(C.8)
O
= Po - (received number of modes)
where
87
Po =No- Af
hf
exp
(C.9)
-;
Since the number of received modes is proportional to the pupil area and detector area,
the collected background light power is also proportional to these areas.
In practice, the detector size is fixed for a desired data rate. So we assume that each
receiver, whether it is part of a system with or without diversity, has the same size
detector. When the receiving pupil size changes, the receiving telescope is modified to
focus the light received by the pupil onto the detector. Consider the no diversity receiver
of Figure C.3 and the receiver diversity system of Figure C.4.
QFV
pupil
detector
fDL
Adl
Ap'
Figure C.3 Field of View and Diffraction Limited Angle of a No Diversity Receiver
88
QFV
pupil I
detector I
QDL 2=NK2DL1
AP2=AP /N
AI
pupil N
p
detector N
QDL=Nt DL
AP2=AP /N
Ad I
Figure C.4 Field of View and Diffraction Limited Angle of a Multiple Receiver System
Each detector area in the two figures has the same area. Thus the field of view of each of
the receivers is the same. However, the pupil area in each of the receivers in Figure C.
is 1/N times the pupil area in Figure C.3. Thus the diffraction limited angle of the second
system is N times that of the first system. Since each receiver in the second system has
the same field of view but N times the diffraction limited angle of the first system, each
of these receivers collects, on average, ]/N times the background noise as the single
receiver in the first system. Thus, if the outputs of the N detectors in the second system
are summed, then the sum will have the same noise power as the first system.
89
Bibliography
[1] V. W. S. Chan, "Coding for the Turbulent Atmospheric Optical Channel," IEEE
Transactionson Communications,vol. COM-30, no. 1, January, 1982.
[2] R. M. Gagliardi and S. Karp, Optical Communications, New York: Wiley, 1976.
[3] J. W. Goodman, Introduction to FourierOptics, New York: McGraw-Hill, 1968.
[4] E. V. Hoversten, R. 0. Harger and S. J. Halme, "Communication Theory for the
Turbulent Atmosphere,", Proceedings of the IEEE, vol. 58, pp. 1626-1650, October
1970.
[5] S. M. Haas, J. H. Shapiro, V. Tarokh, "Space-Time Codes for Wireless Optical
Channels," Eurasip Journalon Applied Signal Processing, vol. 2002, no. 3, March,
2002.
[6] R. L. Mitchell, "Permanence of the Log-Normal Distribution," Journalof the Optical
Society of America, vol. 58, pp. 1267-1272, September 1968.
[7] R. Ramaswami and K.N. Sivarajan, Optical Networks: A PracticalPerspective,
Boston: Morgan Kaufmann Publishers, 1998.
[8] S. Rosenberg and M. C. Teich, "Photocounting Array Receivers for Optical
Communication Through the Lognormal Atmospheric Channel. 2: Optimum and
Suboptimum Receiver Performance for Binary Signaling," Applied Optics, vol. 12,
pp. 2625-2634, November 1973.
[9] J.H. Shapiro, Optics of the Turbulent Atmosphere. Cambridge: MIT, 1980.
[10] J.H. Shapiro, J.W. Strohbehn (Ed.), Laser Beam Propagation in the Atmosphere,
New York: Springer-Verlag, 1978, pp. 172-183.
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