ATMOSPHERIC TURBULENCE AND MULTIPLE SCATTERING EFFECTS
ON THE BIT ERROR RATE OF THE OPTICAL RECEIVER FOR PPM
MULTIGIGABIT LASER COMMUNICATIONS
by
Fotios Panagiotis Kourouniotis
B.S. E.E., University of Colorado at Denver, 1985
A thesis submitted to the
Faculty of the Graduate School of the
University of Colorado in partial fulfillment
of the requirements for the degree of
Master of Science
Department of Electrical and Computer Engineering
This thesis for the Master of Science degree by
Fotios Panagiotis Kourouniotis
has been approved for the
Department of
Electrical and Computer Engineering
by
A. K. Majumdar
E. T. Wall
J. Thomas
Date: August 7, 1991
Kourouniotis, Fotios Panagiotis (M.S. Electrical Engineering)
Atmospheric Turbulence and Multiple Scattering Effects on the Bit Error Rate
of the Optical Receiver for PPM MultiGigabit Laser Communications
Thesis directed by Professor Arun K. Majumdar
This thesis discusses the turbulence and multiple scattering effects on
the impulse response functions of the atmospheric system, as well as the Bit
Error Rate (BER) of the optical receiver, in the case of Pulse Position
Modulation (PPM) laser communications.
The mathematical techniques required to obtain expression for the
impulse response functions, Gj (t), G2(t), and G(t), for both turbulence and
multiple scattering, are presented. Then, these impulse response functions
are graphed for both strong and weak turbulence, and in the case of multiple
scattering, for a variety of atmospheric conditions, such as rain, cloud, haze,
and fog. Subsequently, these graphs are used to obtain appropriate curves
for the Bit Error Rate of the optical receiver.
The impulse response functions of the nonturbulent free space are
also developed, based on the results obtained for the turbulence and
multiple scattering cases.
In the case of multiple scattering, the Bit Error Rate (BER) for the
optical receiver is graphed for dense rain, heavy rain, and light fog. BER
curves are also plotted for both weak and strong atmospheric turbulence.
Comparisons of the graphs lead to the following conclusion: Given an input
pulse power of approximately 0.2W2W, we may be able to significantly
decrease the adverse effects of multiple scattering in the three cases, so that
the probability of the optical receiver detecting the correct slot position in a
given word, will increase significantly. In other words, the probability of bit
error, or Bit Error Rate (BER) will be greatly reduced to acceptable levels. In
fact, with an input pulse power higher than 2W, the effects of multiple
scattering on the Bit Error Rate can reach negligible levels, especially in the
cases of rain and light fog. In cloud, haze, and dense fog, however, the pulse
i
spreading will be very large, leading to unacceptable BER levels. Also, in
the case of strong turbulence, the BER of the receiver will again reach
unacceptable levels, even for very large amounts of input pulse power. To
lower these BER levels, we could use enormous amounts of input pulse
power, perhaps greater than 100MW, which in any case, would be extremely
impractical and expensive.
Computer programs in the C programming language were written to
numerically evaluate the expressions of the impulse response functions of
the atmosphere, and the Bit Error Rate of the optical receiver, for both
turbulence and multiple scattering, and are included in this thesis.
Dedication
This thesis is dedicated to my parents, who suffered a great deal after
their immigration to the United States, in order to provide me with the best
emotional, and financial help. Their love and patience has been exceptional,
and their support has been immeasurable. This dedication is a small token
of my great appreciation for them. Thank you very much.
Acknowledgements
I would like to express my gratitude to Professor Arun K. Majumdar,
for the many hours that he spent in discussions, comments, and
explanations of complex material. His encouragements, guidance, and
support have been exceptional, and without them, this work would never
have been finished.
I would also like to thank Mr. William Brown for his corrections and
explanations on the derivations of certain mathematical expressions. His
contributions were quite significant and greatly appreciated.
Further acknowledgements and thanks are extended to: Professor Ed
Wall for his helpful comments on this work; the Committee for their extra
effort at impromptu scheduling; Christy for her patience to withstand and
fulfill my numerous, urgent requests; Nick, George, and Ulysses for their
great help with programming and the computer systems; and Kathy for
proofreading and correcting the manuscript.
Contents
Chapter 1
Atmospheric Effects on Laser Optical Signal Propagation
1. Introduction.............................................1
2. Turbulence versus Multiple Scattering....................2
3. Evaluation of the Bit Error Rate (BER) in PPM Format.....4
Chapter 2
The Impulse Response Function of the Atmosphere in the
Case of Turbulence
1. The Transfer Function G(tz/c)..........................12
2. Bit Error Rate (BER) and Slot Count Statistics..........24
Chapter 3
The Impulse Response Function of the Atmosphere in the
Case of Multiple Scattering
1. The Discrete Random Medium..............................33
2. Slot Count Statistics for Multiple Scattering...........36
Chapter 4
Pulse Propagation in Free Space
1. Introduction............................................40
2. The Impulse Response Function of Free Space.............41
Contents
(continued)
Chapter 5
Results and Conclusion
1. The G, G2, and G, Impulse Response Functions......46
2. The Bit Error Rate (BER) Function...................50
a. Turbulence.....................................50
b. Multiple Scattering............................52
3. Conclusion..........................................53
Bibliography
.......................................................56
Appendix A
Figures and Graphs.....................................57
Appendix B
The Programs Used for the Evaluation of G, G2, G, and
the BER Functions......................................89
List of Figures
Figure #1: The Trapezoidal Approximation...............58
Figure #2: G1 Turbulence Function...................59
Figure #3: G2 Turbulence Function...................60
Figure #4: G Turbulence Function.......................61
Figure #5: G1 Multiple Scattering; Rain.............62
Figure #6: G1 Multiple Scattering; Rain.............63
Figure #7: G1 Multiple Scattering; Dense Rain..........64
Figure #8: G1 Multiple Scattering; Heavy Rain..........65
Figure #9: G1 Multiple Scattering; Light Fog...........66
Figure #10: G1 Multiple Scattering; Fog................67
Figure #11: G1 Multiple Scattering; Fog................68
Figure #12: G1 Multiple Scattering; Cloud..............69
Figure #13: G1 Multiple Scattering; Cloud..............70
Figure #14: G1 Multiple Scattering; Cloud..............71
Figure #15: G1 Multiple Scattering; Haze...............72
Figure #16: G1 Multiple Scattering; Haze...............73
Figure #17: BER; Strong Turbulence; 10ps...............74
List of Figures
(continued)
Figure #18: BER; Strong Turbulence; 20ps..............75
Figure #19: BER; Strong Turbulence; 50ps..............76
Figure #20: BER; Weak Turbulence; 10ps................77
Figure #21: BER; Weak Turbulence; 20ps................78
Figure #22: BER; Weak Turbulence; 50ps................79
Figure #23: BER; Multiple Scattering;
Dense Rain; 10ps.......................80
Figure #24: BER; Multiple Scattering;
Dense Rain; 20ps.......................81
Figure #25: BER; Multiple Scattering;
Dense Rain; 50ps.......................82
Figure #26: BER; Multiple Scattering;
Heavy Rain; 10ps.......................83
Figure #27: BER; Multiple Scattering;
Heavy Rain; 20ps.......................84
Figure #28: BER; Multiple Scattering;
Heavy Rain; 50ps.......................85
Figure #29: BER; Multiple Scattering;
Light Fog; 10ps........................86
Figure #30: BER; Multiple Scattering;
Light Fog; 20ps........................87
List of Figures
(continued)
Figure #31: BER; Multiple Scattering;
Light Fog; 50ps........
List of Programs
Program #1: The Program for the G1 Turbulence
Function..........................90
Program #2: The Program for the G2 Turbulence
Function......................... 91
Program #3: The Program for the G Turbulence
Function..........................92
Program #4: The Program for the G1 Dense Rain
Multiple Scattering Function....93
Program #5: The Program for the G1 Heavy Rain
and Light Fog Multiple Scattering
Function........................94
Program #6: The Program for the BER Function
Strong Turbulence, 10ps.........95
Program #7: The Program for the BER Function
Strong Turbulence, 20ps.........96
Program #8: The Program for the BER Function
Strong Turbulence, 50ps.........97
Program #9: The Program for the BER Function
Weak Turbulence, 10ps...........98
Program #10: The Program for the BER Function
Weak Turbulence, 20ps.............99
Program #11: The Program for the BER Function
Weak Turbulence, 50ps..........
100
List of Programs
(continued)
Program #12: The Program for the BER Function
Multiple Scattering, Dense,
Heavy Rain, Light Fog,
10ps, 20ps, 50ps...........
1
Chapter 1
Atmospheric Effects on Laser Optical
Signal Propagation
1. Introduction
With the ever increasing demands for more advanced, sophisticated
and reliable communication systems, scientists and engineers are trying to
discover new ways of transmitting information faster, more efficiently and
with as little corruption of the propagating signal as possible.
The traditional way of communicating information so far has been
through conventional electronic signal transmission in the forms of radio
signals, short waves, microwaves, etc. However, a potentially revolutionary
way of transmitting and receiving information today can be achieved through
the use of Optical Lasers. Lightwave communications offer a number of
2
advantages over the conventional methods. These advantages include, but
are not limited to, transmission of information at the speed of light, no short
circuit problems, lighter and more cost effective system components, and
more reliable overall system performance. These reasons make laser light
transmission much more desirable than the conventional ways.
When a specific sequence of pulses is transmitted at the optical
frequencies through a turbulent medium, many effects occur, such as pulse
broadening, amplitude fluctuations, time arrival delays, etc. These effects are
usually due to the random nature of the transmission medium (channel)
which corrupts the signal by adding different kinds of noise. The kinds of
noise we are going to deal with are turbulence and multiple scattering as
they occur when an optical signal travels through the Earths turbulent
atmosphere.
2. Turbulence versus Multiple Scattering
Now, we will try to define these two kinds of noise.
a) Turbulence:
By turbulence, we mean the kind of noise added to an
optical signal due to the large variations of the atmospheric
variables. The atmosphere, being a prime example of a
turbulent medium, is constantly changing. These dynamics
affect all kinds of communications. There are too many
variables to account for; nevertheless, we can still come up
with an accurate model of the atmosphere by incorporating
only the most prevalent of these variables. The sudden
changes in temperature, the shifts in wind direction and
velocity, the changes of air density from place to place and
according to the elevation from Earths surface, the random
variations of solar activities, the random distribution of the
moisture content of the atmosphere, the clouds, fog, rain, snow,
and lightning, all contribute to the corruption of a transmitted
signal. These atmospheric conditions usually account for
random amplitude fluctuations, pulse broadening and time
arrival delays. These effects are discussed in Chapter 2, where
the impulse response of the atmospheric system is evaluated
for the case of turbulence.
b) Multiple Scattering:
This is an even more fascinating subject than
turbulence. Lets briefly outline what we mean by multiple
scattering. In conventional electronic communications, the
information signal modulates a carrier which leaves the source
having a particular wavelength, X. Now, it is well known that the
atmosphere contains millions upon millions of particles on the
order of picometers. The wavelength of the propagating carrier
is normally many orders of magnitude larger than the
dimensions of these picoparticles. Therefore, when the signal
encounters these particles, it just ignores them and passes by
to continue on. The resulting effect of these particles on the
propagating signal is quite negligible. On the other hand, the
same picoparticles can have a very significant effect on the
4
propagation of optical signals since an optical carriers
wavelength is also in the order of picometers. When a laser
beam encounters a particle with dimensions of the same order
of magnitude as its wavelength, some of the signal will go
through the particle, while some will be reflected back and
some will scatter in many different directions. The scattered
beams will then encounter other particles with dimensions
similar to their wavelengths and scatter again. The same
procedure can go on for millions (!) of times before the signal
finally reaches its destination. So, the light beam will go
through multiple scattering. As in the case of turbulence,
multiple scattering causes similar distortions on the
propagating pulse. However, in the multiple scattering case,
the effects are even more amplified. These effects are
discussed in Chapter 3.
For both of these cases, the mathematics involved for obtaining the Bit Error
Rate (BER) and the impulse response functions for the atmospheric system
is similar. However, one should remember that turbulence refers to a
continuous random medium, whereas multiple scattering refers to a discrete
random medium. In the next section, we will derive the general expression
for the Bit Error Rate (BER) of the receiver. The following derivation will be
valid for both turbulence and multiple scattering.
3. Evaluation of the Bit Error Rate (BER) in PPM Format
5
In this section, we are going to derive the general expression for the
Bit Error Rate (BER) at the receiver side of a communication system after a
laser signal has travelled through the turbulent atmosphere for some
distance. The modulation format assumed is Pulse Position Modulation
(PPM). The primary advantage of Pulse Position Modulation (PPM), is that it
enables us to transmit very narrow, highly peaked optical pulses, at a high
rate, over noisy channels. ^ ^
Suppose we transmit a series of words, each of length M. Assume
that each word consists of only one slot, q, that contains the information
(logic 1), and an M1 number of slots, j, that contain no information at all
(logic 0). As these pulses propagate through the atmosphere, they become
noisy as the photon counts change in each slot. For example, the j slots that
contain no information could pick up some photons along the way, while the
q slot could lose some of its photons. In some cases, the loss of photons in
the q slot could be large enough so that the total number of photoelectrons
counted by the receiver for this slot is less than the total number of
photoelectrons counted for some particular j slot. In this case, the
photodetector will assign a logic 1 in the j slot, and a logic 0 in the q slot.
Therefore, the receiver will make an error in detecting the correct slot for the
transmitted information, which is called a symbol error. This section deals
with the probability of symbol error (PSE) which will lead us to the probability
of bit error, or Bit Error Rate (BER).
The receiver reads the power received in each individual slot in a
frame consisting of M slots. If the information is contained in the q slot, then
the remaining j slots contain no signal.
If rj is the random output count for the jth slot, then the probability of
symbol error will be ^
6
(1.3.1)
M
PSE = Â£ Prob(rj>rq).
Equation (1.3.1) evaluates the probability that the count of
photoelectrons in the j slots will exceed the count in the q slot. Therefore, the
Bit Error Rate (BER) of the system will be ^
First of all, the probability that the count in a slot j will exceed the
count in the q slot needs to be evaluated. That is, we need to determine
Prob(rj>rq).
Now, if we assume that the expected counts in all slots are fixed, then
we need to find Prob(rj>rq  mj, mq), where mj, mq are the expected counts
in slots j and q respectively. Using the formulas for conditional probability,
we can write
j=1
j*q
(1.3.3) Prob(rj>rq  mj, mq) =
Prob(rj>rq n mj, mq)
Prob(mj, mq)
or
(1.3.4) Prob(rj>rq  mj, mq) =
Prob(mj, mq  rj>rq) Prob(rj>rq)
Prob(mj, mq)
The term Prob(mj, mq  rj>rq) means: What is the probability of the
expected values mj, mq being constant, given that the count in a j slot
exceeds the count in the q slot? For all practical purposes, the two events
may be considered to be independent so that Prob(mj, mq  rj>rq) = Prob(mj,
mq). Therefore, equation (1.3.4) becomes
(1.3.5) Prob(rj>rq  mj, mq) = Prob(rj>rq).
Now, we only need to evaluate Prob(rj>rq). Lets attempt to write
Prob(rj>rq) in terms of probability mass functions (PMFs). Since the
measured counts, rj and rq, are random variables, then Prob(rj>rq) really
consists of two events: (a) the event where the random variables rj and rq
take on some value rj and rq, and (b) at the same time, rj > rq.
So, Prob(rj>rq) = Prob(rj = rj, rq = rq n rj>rq) may also be written as
(1.3.6)
Prob(rj = rj n rq = rq n rj>rq).
However, expression (1.3.6) is a joint probability mass function
(1.3.7) prjrq(rj rq> Provided that rj>rq
Also note that the counts at slots j and q are independent since we
assume statistical independence between the time slots. This means that
expression (1.3.7) becomes
prjrJrj rq> = pr.(rj) prJrq> Provided that rj>rq
I H J H
(1.3.8)
Since the expected values mj and mq are also independent of the
measured values, then equation (1.3.8) can be written as
(1.3.9)
prjCj)
prq prjmj
rqmq
(rqlmq). provided that rj>rq.
If we add up all the probabilities that the counts in all j slots exceed
the count in the q slot, then
(1.3.10)
Prob(r]>rq)=Â£Prj
rj=1
j*q
mjOW Â£ prqmq
rq=0
rÂ£
Note that the second term, ^ prqmq^rqlmq^ takes 'nt0 account
rq=0
only those cases where rj>rq.
So far we have assumed that mj and mq are constant. If we assume
that they are variable and continuous on (0, ), then the random variables
mj and mq will have a probability density function (pdf) description of fm.(mj)
and fm (mq) respectively. If we multiply equation (1.3.10) by fm.(mj)
q j
fm (mq) and integrate over mj and mq, we obtain
(1.3.11) Prob(rj>rq) =
9
oo oo
oo
prjmj(rJ'mP Â£ prqmq dmidmq'
fq=0
j*q
Substituting equation (1.3.11) into equation (1.3.1), we obtain the
probability of symbol error (PSE)
(1.3.12) PSE =
M
OO
Â£
1=1
j*q
Prjn.j Â£ Prqm dmjdmq
fq=0
Finally, by substituting equation (1.3.12) into equation (1.3.2), we
obtain the the desired expression for the Bit Error Rate (BER)
M
M log2MX7
<1313>BERn>i>nr2^ J j) V"^*
j=1
j*q
oo
rj=1
prjmj Â£ PrqmqCqlnlq) dmidmq
rq=0
j*q
Equation (1.3.13) evaluates the Bit Error Rate (BER) of the receiver,
given the probability density functions (pdfs) of the expected values at slots j
and q respectively are known. These pdfs typically have exponential or log
10
normal distributions. In the case of an exponential distribution, the
exponential term in fm.(mj) goes to zero much more quickly than the
exponential term in fm (niq). This situation is expected since fm(mj)
represents the pdf of the expected counts in the j slots which contain no
signal.
The probability mass functions (PMFs) of the random counting
processes, Pr.m.(rjlmj) and pr mij(rqlmq) are ta^en t0 be conditional
Poisson discrete counting processes ^
(1.3.14)
mfj
J
Prjln,/rih>=_ijremiwhereri = 12...
The word length M is a 2n number, where n is an integer.
Equation (1.3.13) can be evaluated by numerical methods only. To
this end, programs were written in the C programming language which
calculated the BER for different word lengths M, different input pulses, and
different pdfs fm.(mj) and fmq(mq) (See Appendix B for a sample of these
programs.) At this point, it is very important to notice that these pdfs depend
on the received expected pulse shape. The pdfs of mj and mq in the case of
turbulence differ from those in the case of multiple scattering. Also, In
multiple scattering, the transmitted pulse is expected to spread out more
than in turbulence. These are indeed the results which were obtained.
These results are discussed in Chapter 5, and graphed in Appendix A. The
expected pulse shape, which depends on the impulse response function of
the atmospheric system, is discussed in Chapter 2 for the case of turbulence,
and Chapter 3 for multiple scattering.
12
Chapter 2
The Impulse Response Function of the Atmosphere
in the Case of Turbulence
1. The Transfer Function G(tz/c)
As mentioned in the previous chapter, the transmitted pulse picks up
noise as it travels through the turbulent atmosphere. This causes a random
fluctuation of its amplitude, as well as spreading. The amount of spread can
be approximately determined from the transfer function characteristics of the
atmospheric model. In the case of turbulence, the transfer function of the
atmosphere is G(tz/c), where c is the speed of light. This transfer function
can actually be obtained from the convolution of G(tz/c) with G2(tz/c),
where G(tz/c) corresponds to the highfrequency component of the transfer
function G(tz/c) and G2(tz/c) corresponds to the lowfrequency one. The
reason for the separation of the G function into these two components comes
from the fact that the output pulse shape can be expressed in terms of the
twofrequency mutual coherence function (MCF) ^
(2.1.1) r=r(ti,t2. to! ,2),
which can be considered as the correlation between output fields.
The correlation function, Ru, of the output pulse, is given by
oo oo
(2.1.2) Ru(ti,t2) = J J Uj(o>i)Uj(co2)r(t,t2,coi,co2)exp(icoit1 +ico2t2) dc^dco^
where Uj (co) is the spectrum of the complex envelope of the input pulse, Uj(t),
and the asterisk^*) denotes the complex conjugate of. The spectrum of Uj(t)
is given by
(2.1.3)
OO
The correlation function, equation (2.1.2), may be changed to the
intensity function, l(t), of the output pulse when tj = t2 = t.
OO OO
(2.1.4) l(t) = J J Uj (co!) Uj(co2) ToCt.co!,co2) exp[i(co1 co2)t] dci^do^,
OO OO
where r0 is T evaluated at = t2.
Equation (2.1.4) gives us the output response of the atmospheric
system for any input signal, Uj(t). The desired impulse response function,
G(t), can be obtained when Uj(t) = 8(t). Substituting this into equation (2.1.3),
we obtain
(2.1.5)
Substitution of equation (2.1.5) into equation (2.1.4) yields
(2.1.6)
oo oo
J J To(t, 1, 2) exp[i(i 2)t] d!d2.
50 OO
In many practical applications, the twofrequency mutual coherence
function is a slowly varying function of COh = ^ (1 + 2), and t = ^ (tj +12);
therefore, we can assume that T is a function of d only. This is called the
widesense stationary uncorrelated scattering (WSSUS) channel. With this
in mind, the double integral in equation (2.1.6) can be transformed into
another double integral according to the formula
(2.1.7) J J f(x,y)dxdy= J J f(x(u,v), y(u,v)) J dudv,
R R
where J, is the Jacobian
15
(2.1.8)
Bx Bx
Bu Bv
By By
5u Bv
'Bx By^ B_x^ By_
JBu Bv Bv Bu
In our case, we have co^h = (1 + 2), and cod = 1 co2. Solving for
oo1 and co2, we obtain
(2.1.9)
1
1 COcoh 2 d>
and
1
 ^coh 2 ^d
Since the Jacobian is 1, equation (2.1.6) can be written as
oo oo
(2.1.10) G(t) = ~ J J r0(cocoh, cod) e'lcod1 dcocohdcod.
^ oo oo
However, because r0 is a slowly varying function of coCOh, we can
assume that there is no variation with respect to coCOh. Therefore, the double
integral in equation (2.1.10) becomes a single integral with respect to cod.
(2.1.11)
OO
G(t) =" j r(cod) d
oo
where, for convenience, we dropped the subscript o from Tq.
16
Equation (2.1.11) will give us the impulse response function, G(t), as
the Fourier transform of the MCF function r(cDd).
Now, we need to determine r(cod). Lets write r(cod) as p 427^
(2.1.12) rccad) = r^toc) exp( k/kfoh2).
or
(2.1.13) r(cod) = Ti (kd / kcohi, p/po) exp( / k^oh2).
So, our task is to determine IY Note that the following derivation is
good for both turbulence and multiple scattering.
The governing equation for T is actually a function of an integral
equation which will not be derived here, but is as stated below. ([51> P315)
(2.1.14) r = +
J P dfg {[exp(iKRi iKgR^J/RiFy x J dQ f f2 I exp(iKrÂ§ Vtd).
The first term on the righthand side of the above equation represents
the coherent part, Tc, and the second term represents the incoherent part, Tj.
Note that in most practical applications, the laser beam will have to
propagate through relatively large distances to reach its destination, so that
the incoherent part of equation (2.1.14) will be the dominant one. All
derivations in this chapter are based on that fact. After all, the coherent
17
impulse response G(t), is just that: an impulse, or mathematically speaking, a
delta function.
The general solution of equation (2.1.14) is not available. However,
we can obtain a much simpler approximate differential equation when the
signal propagates in a random medium with particle sizes comparable to or
greater than a wavelength. In this case, the waves are scattered mostly in
the forward direction, and if we also assume that the waves are plane waves
propagating in the z direction, equation (2.1.14) can take the following
approximate form ([5], p 316)
where Vd is the twodimensional Laplacian operator with respect to pd.
By expanding the exponential and keeping the first two terms, we
obtain
Now, we let z/L = z and p where L is the propagation
distance. p0 is chosen such that equation (2.1.16) takes the form
(2.1.15) {a/az + iaVd + pnas[1 exp(k2pd/4ap)]} r^cod) = 0,
(2.1.16)
p/dz + iaVd + bpd ] n (d) = 0,
where a = kd / 2k2, and b = pnos k2 / 4ap.
(2.1.17)
[d/dz + i(cod/coCoh)Vd + Pd'2] rl(d) = 0.
The following technique is used to obtain the exact solution for F1 (00^).
Lets assume Ti (cod) has the following form PP321325)
(2.1.18) r^Wd) = [f(z')]i exp[g(z)pd'2],
where f(z) and g(z) are functions of z only. Substituting this into equation
(2.1.17), we obtain
(2.1.19)
1 3f
f dz
7 + i4ag +
777 + i4ag2 + 1
dz
= 0.
Now, since the first two terms of equation (2.1.19) are functions of z
only, regardless of pd2, we should require that
(2.1.20) 1 T^ + '4ag =
and
(2.1.21) 777 + i4ag2 + 1 dz
with g(z) = 0, and f(z) = 1 at z = 0.
Equation (2.1.21) is a Riccati equation and can be solved to give
(2.1.22)
, tan[(i4a)1/2z]
9(2) = ' (i4x)i/a
19
Substituting equation (2.1.22) into equation (2.1.20), f can be
obtained.
(2.1.23)
f(z) = cos[(i4a)1/2z].
Since we are interested in what happens at the receiver side, we
should, therefore, evaluate equations (2.1.22) and (2.1.23) at z=L, or z=1.
Hence, the final solution for r^coj) will be given by
(2.1.24)
___]___ exJ'2
cos[(i4a)1/2] (i4a)1/2 Pd
where a = co
Then, equation (2.1.11) can be written as
(2.1.25) G(t) =
CO
1 f 1 tan[(i4a)1/2] ,2"
2k J cos[(i4a)1/2] (i4a)2 Pd
oo
exp( k^/kcoh2) e'jCd(t) dd
Evaluating equation (2.1.25) at pd =0, we obtain
(2.1.26)
G(t) =
w 271
exp( k^/kfoh2) e'i^dW dcod.
cos[(i4a)1/2]
CO
20
However, since we are considering light propagating in the z
direction, equation (2.1.26) can be written as
oo
(2.1.27) <***Â£ J exp(^h2)ei^c)d0)d.
OO
Equation (2.1.27) can be written as the convolution of G(tz/c) and
G2(tz/c). The result is as follows:
(2.1.28)
OO
G(t) = Jgi (ttz/c) G2(tz/c) dt,
OO
where
(2.1.29)
G (tz/c) =
2ji
OO
c 1
cos[(i4a)1/2]
/
eiooa(iz/c) dci)
and
OO
(2.1.30) G2(tz/o) = ~ J exp( k^k^) dcod.
^^OO
The integration in equation (2.1.30) can be carried out easily to yield
21
(2.1.31)
G2(tz/c) =
1
expJ
(tz/c)
r2
21
for continuous random media, or turbulence. T2 is given by
(2.1.32)
T2=^ 1.2056 Cn L/6L1/2.
The integral in equation (2.1.29) is a complex integral, and requires
the use of the method of residues. After a change of variables through the
relationships a = o)d/cocoh, and T = coCOh (tz/c), equation (2.1.29) may be
written as
(2.1.33)
G (tz/c) =
Q^coh
2k
1
cos[(i4a)1/2]
eiaT da.
OO
The poles a = an of the integrand are given by
(2.1.34)
(i4ttn)1/2 = (2n + 1) (rc/2), n = 0,1,2,
Therefore, we have a series of double poles a = an
[(2n + 1) (tc/2)]2
an =4], n = 0,1,2......
(2.1.35)
22
since the poles at n = 1, 2,.coincide with the poles at n = 0, 1, 2.
These poles are all in the lower half of the a plane along the imaginary axis.
Therefore, G\ (tz/c) = 0 for tz/c<0. For tz/oO, we close the contour in the
lower half plane and obtain a series of residues at the poles. Then, Gj (tz/c)
becomes
(2.1.36) Gj (tz/c) = 27ci Â£ residues.
n=0
The evaluation of the residues is easy, and we obtain
oo
(2.1.37) Gi(tz/c) = 4^2^ (1 )n(2n+1 )expj(2n+1 )2 ^ }, 1z/oO,
n=0
= 0, * , tz/c<0,
where Ti = 1/cocoh. The coherence bandwidth of the random medium, cocoh,
is given by
(2.1.38) 0)coh = 1.28c Cn12/5 k^5 C11'5
p
If k0 = 27i/X is the wavenumber of the optical signal, Cn is the strength
of turbulence parameter, and L is the propagation path length through the
turbulent region, then
T (_J______\ n12/5.2/5 ,11/5
Ti = \1.28c/ ^ L
(2.1.39)
23
The convolution of equations (2.1.37) and (2.1.31) will give us the
simplified expression for the G function in the case of turbulence. (
represents convolution.)
G(tz/c) = G (tz/c) G2(tz/c) =
oo oo
Jgj [(tt)z/c] G2(tZ/C) dt = Jg1 (t'z/c) G2[(tt)z/c)] dt =
^X('i)n (2n+i> exp[ (2n+i>2
^ n=0
exp
f(tt')z/c12
dt,
which, with a little simplification, becomes
(2.1.40) G(tz/c) =
Vrc"
4T!T2
OO
* exp [(tt)Z/c]2' o
^2
oo
I
n=0
(1)n(2n+1)exp
;(2n+1,a^
dt.
J
OO
Equation (2.1.40) is the impulse response function of the atmosphere
in the case of turbulence. The only way to evaluate this expression for
different values of T and T2 is by numerical methods. The C programming
language was used again. The results are discussed in Chapter 5, and a
sample of the C program is given in Appendix B.
In the next section, we will establish the relationship between the Bit
Error Rate (BER), expression (1.3.13), and the impulse response function
G(tz/c), equation (2.1.40).
2. Bit Error Rate (BER) and Slot Count Statistics
In this section, we will determine the pdf of the expected count in a
slot j as a function of the transfer function, G(tz/c). Since ultimately we are
interested in a plot of G versus time, we can safely use G(t) instead of G(t
z/c). The factor z/c will just shift G along the time axis by an amount
dependent on the numerical value of z/c.
Generally speaking, the received pulse power over the effective
aperture area of the telescope is a random function which depends upon the
impulse response G(t) and the background noise of the receiver. Let the
received pulse power per unit area be ^
(2.2.1) P(t) = IS(t),
where I is a random intensity function representing amplitude fluctuations
due to atmospheric turbulence, and S(t) is a unitless quantity representing
the output pulse shape without the noise. S(t) is defined as follows:
25
OO
(2.2.2) S(t)=^ J Pi(t) G(W) df,
OO
where Pj(t) is the input pulse power, and y is a parameter dependent on the
full width transmit beam angle, 0, and on the range, R, and is defined as y =
zz. , is the average, or expected mean value, of the intensity random
process I.
Since the receiver reads power and not amplitude, the expected
instantaneous count for the slot q, which contains the information, will then
be
qTs
(2.2.3) mq = a Arec J P(t)
(qDTs
or
qTs
mq =aArec JI S(t) dt + nq,
(qi)Ts
71
where a = is a constant which depends on the system parameters, t is
the detector quantum efficiency, h is Plancks constant, and v is the
operating optical frequency. Arec is the effective aperture area, and nq is the
combined background and receiver noise count at the q slot. If we assume
that I is constant overtime intervals as short as Ts, then equation (2.2.3) may
be written as
26
qTs
(2.2.4) mq = a Arec I J S(t) dt + nq = a Arec I Aq + nq,
(ql)Ts
where Aq is the expected pulse shape area in the q slot,
qTs
(2.2.5) Aq = f S(t) dt.
(qi)Ts
Lets examine equation (2.2.4). Since mq = a Arec I Aq + nq, and I and
nq are independent random processes, then
(2.2.6)
fmq fqArecIAq(a Arec 1 ^ fnq(nq>
Since a, Arec, I, and Aq are all independent events, equation (2.2.6)
may be written as
(2.2.7)
fmq W) 'Arec
Since Arec and Aq are deterministic, and a is a constant, fa(a),
fA (Arec), and fA (Aa) are just delta functions having a strength of unity;
Mrec Mq ^
therefore, they cannot affect fmq(q). Hence, equation (2.2.7) becomes
(2.2.8)
tmqfl,nq
If we assume that the pdf of I is known and use equation (2.2.4)
without including the receiver and background noise, we obtain
27
mq = a Arec I Aq.
Solving for I yields
(2.2.9)
nv
ot Arec Aq
Substituting equation (2.2.9) into equation (2.2.8) and adding the
noise process of the receiver, we finally obtain
(2.2.10)
f_ (ma) = fi(~ ^~A ~) fn (na).
mqV q IVa Arec Aq7 "qv W
Equation (2.2.10) is the one that shows us the relationship between
the probability density functions (pdfs) of the expected slot counts and the
impulse response function of the turbulent atmosphere. This is because
m
q
(mq) depends on the expected pulse shape area Aq for each slot, which
in turn depends on the impulse response function G(t). Since the Bit Error
Rate, (BER), equation (1.3.13), is a function of fm (mq), it is now obvious
q
what the relationship between G(t) and the BER is. Note that we have
assumed that the expected noise count in each slot is the same.
Something that we still need to determine is an expression for the
expected pulse shape area, Aq. Equation (2.2.5) can be also written as
qTs
/*
C oo
Aq = ^ J P(f) G(tf) df dt,
OO
(q1)Ts
qTs
/*
oo
(2.2:11)
Y
J Pj(t) G(tt) dt dt.
"
OO
(qi)Ts
Y
So, Aq, can be expressed as a product of the factor ^ and the area
under the convolution of Pj(t) with G(t). According to one of the theorems
about the convolution integral, we know that the area under a convolution is
equal to the product of the area under the factors. Therefore, the value of
the integral on the righthand side of equation (2.2.11) is equal to the
product of the area under the functions Pj(t) and G(t). Since Pj(t) is the input
pulse power function of constant amplitude Pt, and duration tp seconds,
then, the area under it over that amount of time, will be equal to the energy
contained in the transmitted pulse, and will be equal to Pttp.
Now, since G(t) is a convolution between the functions Gj (t) and
G2(t), its area will be equal to the product of the areas under Gi (t) and G2(t).
First, let us evaluate the area under Gi (t). Integrating both sides of equation
(2.1.37) with respect to time t from z/c to we obtain the following
expression.
OO
29
OO
oo
16 T
n=0
(2n+1 )2k2
z/c
Since the exponential term within the brackets, ({}), is equal to 1 and after
some simplification, equation (2.2.12) becomes
under the G(tz/c) function is unity. Note that this result is good for both
turbulence and multiple scattering, and is independent of T. The area
under the G2(tz/c) function can be evaluated as follows. Integrating both
sides of equation (2.1.31) with respect to time t from z/c to we obtain the
following.
OO
OO
(2.2.13)
n
The infinite sum in equation (2.2.13) converges to ^ so that the area
OO
OO
(2.2.14)
z/c
Changing tz/c to x, we obtain
OO
OO
(2.2.15)
0
30
Now, since int =
oo
exp
o
dx, then,
(2.2.16)
int"
OO OO
' x_i .dx * x_i
exp. 0 exp.
J T2 T2
J y *
0 0
which may be also written as
(2.2.17)
OO OO
Changing from cartesian to polar coordinates, we obtain
(2.2.18)
dr d6.
The evaluation of the integral is very easy, and we finally obtain
which implies that
(2.2.20)
Substituting equation (2.2.20) into (2.2.15), we obtain
(2.2.21)
J G2(x) dx
1 Vet i
t2^ 2 2"2'
which is the area under the G2(tz/c) function. So, the area under the G(tz/c)
function is 1* \ = \ ; therefore, the area under the convolution integral on the
Pttn
righthand side of equation (2.2.11) is
Finally, the desired area, Aq, is
(2.2.22)
An^k
^ 2'
This result can be used together with equation (2.2.10) to obtain the
final form of fmq(mq) Also, since we have assumed that the expected noise
count, nq, in each slot is constant, then its pdf, fn (nq), must be a delta
function of the form 8(mqnq), where mq is just a variable representing any
real number on the mq axis, and nq is a constant. So, the convolution in
equation (2.2.10) may be written as
32
OO
(2.2.23)
Wmq) =
mqmg
\
^rec Aq^/
S(mqnq) dmq,
which implies that
(2.2.24)
(mq) = f
r rngrig >
Arec Aq>'
Equation (2.2.24) will give us the final form of the pdf of mq, once the
pdf for I, fj(l), is known.
The next chapter discusses the impulse response function of the
atmosphere and the slot count statistics for the case of multiple scattering.
33
Chapter 3
The Impulse Response Function of the Atmosphere
in the Case of Multiple Scattering
1. The discrete random medium
As shown in the previous chapter, the impulse response function, G(t
z/c), can be expressed as a convolution
(3.1.1) G(tz/c) = G (tz/c) G2(tz/c).
The same mathematical techniques which were used to obtain G(t
z/c) for turbulence can be applied here to obtain G(tz/c) for multiple
scattering. Thus, the expressions for G) (tz/c) and G2(tz/c) here will be very
similar to those obtained in Chapter 2. The only difference is that a turbulent
34
medium is considered a continuous random medium, whereas a medium
consisting of scatterers of random sizes, shapes, orientations, velocities and
densities, is considered a discrete random medium. These two different
kinds of random media will, primarily, affect the mutual coherence function
(MCF), r, which in the case of multiple scattering becomes
where W0 is the albedo of a scatterer, x = pnosz, which is the optical
distance, and os is the scattering cross section. Note that the albedo is
and is equal to os+oa, where oa is the absorption cross section of some part
of the signal by the randomly distributed particles.
Equation (3.1.2) is equivalent to equation (2.1.12). Since G(tz/c) is
dependent on r(cod) only and not on the exponential term in equation
(3.1.2), and Fi (cod) remains the same for multiple scattering, then Gj(tz/c)
will be exactly the same as well. For convenience, equation (2.1.37) is
repeated here.
(3.1.2)
CTe
defined as W0 = , where ot is the total cross section of the random medium,
OO
n=0
= 0
, tz/c<0,
35
G2(tz/c), however, depends on the exponential term of equation
(3.1.2). So, equation (2.1.30) may be rewritten as
(3.1.4)
OO
G2(tz/c) = ^ J expj
OO
1 W0
Wo
!id(tz/c) dcQd
The integration can be carried out to yield
(3.1.5)
G2(tz/c) = 5(tf) expj
9
for discrete random media, or for multiple scattering. ^
The impulse response function, G(tz/c), is obtained by substitution of
equations (3.1.3) and (3.1.5) into equation (3.1.1),
oo
r
(3.1.6) G(tz/c) =
j
4?^ (1)" (2n+1) exp[(2n+1)2 <^]
n=0
*
js(t t 1) exp[ (1^) x]Jdt.
Using the sifting property of the delta function, we obtain the following
form for G(tz/c)
(3.1.7) G(tz/c) =
36
OO
7Z
4T7 exP1
'1 Wp
Wn
(_1)n (2n+1) exp{ ~(2n+1)2 T6 (t~t/C)
n=0
Equation (3.1.7) is the impulse response function of the atmosphere
f n W1 1
for the case of multiple scattering. Note that expj ^ xj i
is the
attenuation factor for a pulse wave propagating in a discrete random
medium. In fact, equation (3.1.7) can be used to describe the impulse
response function for multiple scattering in any medium, as long as we know
the albedo, W0, and the optical distance, x. Once again, the C programming
language was used to evaluate equation (3.1.7) and obtain results for
different atmospheric conditions.
Another very interesting question is what happens to Gi and G2
when the laser beam propagates in free space? This case may arise when
an optical signal originates at a satellite far above the earths atmosphere,
and is sent either downwards towards the earths surface, or towards the
empty space, perhaps in an attempt to communicate with other civilizations
in the Universe! This, will be discussed in Chapter 4.
2. Slot count statistics for multiple scattering
As in the case of turbulence, the mathematical techniques used in
order to find the pdf of the expected count in the q slot, fm (mq), are
identical. The only difference will be the slightly different expression for the
expected pulse area, Aq. This is because the area under the G2(tz/c)
37
function is different in the case of multiple scattering. To find the area under
G2, we integrate both sides of equation (3.1.5), with respect to t, from z/c to
(3.2.1) J G2(tz/c) dt = j 8 expj
1 W0
W0
x > dt.
z/c
The integration can be easily carried out to yield,
OO
(3.2.2)
J G2(tz/c) dt = expj
1 W0'
W0
The area under the G1 (tz/c) function is still unity, so that the area
1 w0
under the impulse response function, G(tz/c), will be expj
and therefore, the expected pulse area will be
W0
n.
(3.2.3)
expf
1 w0
W0
Equation (2.2.24) remains the same, so that the final form of fm (mq)
will still be
(3.2.4)
fmq = fl
f mgng ^
Arec Aq^
however, Aq, is now given by equation (3.2.3).
38
calculated. Unlike the turbulence case, the multiple scattering case is not
affected by the random amplitude variations of the intensity random process,
I. In multiple scattering, the amplitude of the process, I, may be considered to
be a constant over the entire word length, Mts, so that its pdf will be a delta
function. With this in mind, equation (3.2.4) becomes
where lq is the constant power per unit area amplitude of the intensity
random process, I. The pdf for the amplitude of the process, I, in the j slots
will have the same form as the pdf in equation (3.2.5). The only difference
will be the change of the subscript q to the subscript j. To find the Bit Error
Rate, we need to substitute equation (3.2.5) into equation (1.3.13). After the
substitution we obtain the following result,
or
(3.2.5)
Vnq(mÂ£^ = S(mq_nq"'qa ^rec Aq)>
M
j=1
j*q
which, by using the properties of the delta functions, can be simplified to give
the following expression.
where lave = lq = lj is the average constant power per unit area amplitude of
the intensity random process, I, and is approximately equal to 10nW. Also,
note that we have used the conditional Poisson discrete counting process
for the PMFs of rj and rq. Equation (3.2.7) can be evaluated only numerically,
and the results are discussed in Chapter 5. Graphs of the BER functions are
included in Appendix A, and a sample of the programs used for the
evaluation of equation (3.2.7) is included in Appendix B.
The next chapter deals with the interesting case of propagation of an
optical signal in free space. The development of the impulse response
function, G(t), as well as a short discussion about the nature of the slot count
statistics of free space are included.
(3.2.7) BER =
j=1
j*q
40
Chapter 4
Pulse propagation in free space
1. Introduction
So, the question is: What happens to a light beam when it propagates
in vacuum? Well, for one thing, free space is not a turbulent medium. There
is no way that a signal can be corrupted by noise when travelling through it.
In addititon, free space, ideally, does not contain any particles that would
cause scattering of the signal. Therefore, a laser beam will not experience
any turbulence or multiple scattering effects when propagating in free space.
In the following section, we will examine the form of the impulse
response function, G(tz/c), which corresponds to the free space case.
41
2. The impulse response function of free space
The same equations for the impulse response function, G(tz/c), that
were derived for turbulence and multiple scattering may also be applied in
the case of a pulse propagating in free space.
Since there is no turbulence present, the strength of the turbulence
o
parameter, Cn, must be zero. Then, equation (2.1.39) becomes
Taking the limit of equation (2.1.37) as T approaches 0, we obtain
(4.2.2) lim G1(tz/c) =
T*0
In Chapter 2, section 2, we proved that the area under the Gi curve is
unity, independent of the value of Tj. As Ti gets smaller and smaller, the Gj
function gets narrower and narrower, with a corresponding increase in
amplitude in order to keep the area under the curve equal to 1. At the limit,
when T = 0, the curve becomes an impulse of unit area, so that
(4.2.1)
r
oo
lim Gi(tz/c) = 8(tz/c).
T^O
(4.2.3)
Now, since the turbulence parameter, Cn is zero, then from equation
(2.1.32), T2 must be zero as well. Taking the limit of equation (2.1.31) as T2
approaches zero, we obtain
(4.2.4) lim G2(tz/c) = lim ^=exp.
t2>o t2>0 T 2"\ it
which implies that
(4.2.5) lim G2(tz/c) = 5(tz/c).
t2>o
The convolution of G(tz/c) with G2(tz/c) will give us the impulse
response function of free space, and is equal to
G(tz/c) = G(tz/c) G2(tz/c) = J 8(ttz/c) S(tz/c) dt,
which implies that
(4.2.6) G(tz/c) = 5ft 2f\
This is the impulse response function of free space in the absence of
turbulence. Therefore, the output pulse shape, S(t), will be
4 3
(4.2.7)
OO
S(t) = ^ f Pi(tt') sfr yW
2z/c V J
which, as was expected, is a shifted version of the input pulse, Pj(t), with a
different amplitude.
Now, lets see what happens in the absence of multiple scattering. In
this case, G (tz/c) is identical to Gi (tz/c) in the absence of turbulence. That
is, Gi (tz/c) = 8(tz/c). Also, since there is no attenuation of the propagating
signal in vacuum, the exponential factor in equation (3.1.5) must be equal to
1, so that G2(tz/c) = 8(tz/c). Of course, the convolution of these two functions
is identical to equation (4.2.6), and therefore, the final pulse shape in the
absence of multiple scattering is identical to the one given by equation
(4.2.7), as expected. Thus, when a signal propagates in free space, it
preserves its initial shape, and the length of time required to reach its
destination is given by 2z/c. So, as expected, there is no energy loss when a
signal propagates in free space, and the pulse received is an exact replica
of the transmitted pulse.
Additionally, this analysis, verifies the expressions obtained for the
impulse response functions of the atmosphere, as they, give us the
appropriate results when they are taken to the limiting cases. Both,
turbulence parameters and multiple scattering attenuation factors are zero at
the limiting case, which would represent signal propagation in a perfectly
nonturbulent medium.
Furthemore, it is noteworthy to observe that the Bit Error Rate is zero
when a pulse propagates in free space. This can be verified mathematically
with the following procedure.
44
Since there are no disturbances in free space, there will be no loss or
gain of photons in either the j slots, or the q slot. So, the photon count in all
slots in a frame will remain the same, no matter how long the signal
propagates before it arrives at its destination. Since there is no information
in the j slots, the photon counts in these slots will always be zero. On the
other hand, the photon count in the q slot will be a fixed number, say, n.
Therefore, the probability density functions (pdfs) for the counts in all slots
will just be delta functions, with the pdfs for the counts in the j slots being
(4.2.8) fm (mj) = 8(mj),
which is a delta function centered at mj=0 since the counts in the j slots must
be 0.
The pdf for the count in the q slot will be
(4.2.9) fmq(mq) = S(mqn),
which is a delta function centered at mq=n since the count in the q slot is
constant and equal to n.
Inserting equations (4.2.8) and (4.2.9) into equation (1.3.13), we obtain
M logpM
BER = 2(M1)
*
M
I
j=1
j*q
00 00 oo n1
1 8 Â£ Prqm dmidmq
rj=1 rq=
rj= 1
j*q
45
which gives us,
(4.2.10) BER = 0,
as expected. So, we should not expect any Bit Error Rate when an optical
signal propagates in free space.
As we can see, since there cannot be any turbulence or multiple
scattering effects, optical signal propagation in free space must be a very
efficient way to transmit information. Indeed, the results obtained are very
encouraging, and much more research should be devoted to the fascinating
subject of laser optical signal propagation in free space.
However, a word of caution must be given here: Free space may not
always be free of turbulence. In fact, the universe contains an infinite number
of radiation emmitting stars, black holes, intergalactic and interstellar
material, galaxies, comets, and many other objects, that are capable of
corrupting, and even preventing the propagation of a signal. Nevertheless,
there is great excitement about pulse propagation in random media,
something that started as an idea around the beginning of this century, or
earlier, but only recently began to be materialized, and has a very promising
future ahead.
Who knows? That may just be the beginning for a different kind of
adventure into the infinite mystery and beauty of the universe.
46
Chapter 5
Results and conclusion
1. The G, G2, and G, impulse response functions
In this chapter, we will discuss the results obtained for the impulse
response functions, G, G2, and G, and the Bit Error Rate (BER) of the
receiver, for both turbulence and multiple scattering. We will start with the
impulse response functions first.
As mentioned in previous chapters, the programming language C
was used as the vehicle to run the numerical evaluation of the impulse
response functions. The trapezoidal rule was used in order to evaluate the
integrals in expressions (2.1.31), (2.1.37), and (2.1.40). The following
approximation was used for the evaluation of the area under any function
f(t),
47
(5.1.1) Jf(t)dtAt^ f[(n+l)Al],
where At is the time increment. The approximation is very good since tens of
thousands of increments were taken for the evaluation of the area under
each particular function. A typical case is shown in Figure #1.
By numerically evaluating expressions (2.1.31), (2.1.37), and (2.1.40),
based on equation (5.1.1) (with an appropriate change of variables, of
course), a set of curves was obtained for the impulse response functions.
In the case of turbulence, these functions did not exhibit any
appreciable amount of spreading to cause intersymbol interference between
the time slots j and q, as a typical set of curves for G, G2, and G indicates
for both strong and weak turbulence. Also, graphs were obtained for the Bit
Error Rate for both cases, and for different input pulse widths. The results are
discussed in the following section.
In the case of multiple scattering, the impulse response functions, G,
and G, were graphed for a variety of atmospheric conditions, including
different kinds of clouds, haze, fog, and rain (The G2 functions are just delta
functions of strength less or equal to unity, depending on the albedo W0,
located at t=z/c, and therefore they were not graphed). Examination of these
curves show that in the case of cloud cover, haze, and fog, the pulse
spreading is extremely large, ranging from 5ns in fog, to 300ns in haze. This
implies that not only does a pulse overlap into adjacent slots to slot q,
causing intersymbol interference, but it will also spread over to adjacent
words in a particular frame, causing interframe interference as well. Both
conditions are undesirable because they will result in a very large Bit Error
Rate by greatly increasing the probabilities that the photoelectron counts in
the j slots are greater than the photoelectron counts in the q slots. On the
other hand, the resulting pulse spreading due to rain is far smaller than the
one in the other cases (between 6ps and 15ps). In this case, there will be
virtually no overlap in the j slots, so that the Bit Error Rate will be far smaller.
The very interesting case, however, arises when the pulses will have to
propagate through dense rain, heavy rain, or light fog. Here, the pulse
spreading will be sufficiently large, between 300ps and 600ps in dense and
heavy rain, to 1ns in light fog, to cause intersymbol interference in the slots
adjacent to it, but sufficiently small to not spread over to frames adjacent to
its own frame causing interframe interference. Still, however, most of the
pulse energy will be contained in the q slot, where the originally transmitted
pulse was located. Some very interesting results for the Bit Error Rate were
obtained in this case, which are discussed in the following section.
It is worthwhile to note that in turbulence, the shape of the G impulse
response function resembles the function Gj close to the origin, but is
almost entirely dictated by the shape of G2 thereafter. This is because G is
almost an impulse compared to G2, and their convolution will very closely
resemble the G2 function. On the other hand, since the G2 function is indeed
an impulse in multiple scattering, the shape of the G function will be identical
f p w0
to the G1 function, only attenuated by the factor, exa ^
x >. This
can be explained as follows: In a continuous random medium, the higher the
carrier frequency, the more efficient is the transmission. Lower frequencies
tend to have much more difficulty to efficiently propagate through the
medium, and the same rule applies to optical signals as well. Thats why G2
49
(corresponding to the lowfrequency component of the impulse response
function) is the dominant function in the case of turbulence. On the other
hand, in a discrete random medium, a greater the wavelength of the carrier
frequency will cause less scattering. For large wavelengths, the relative size
of the atmospheric picoparticles is negligibly small, so that the signal will
just ignore them, and continue through. When the wavelength, however,
gets very small, the signal will go through a lot of scattering, and thats why
G (corresponding to the highfrequency component of the impulse
response function) is the dominant function in the case of multiple scattering.
Also, note that the amplitude of the G function, in the case of
turbulence, is a few orders of magnitude greater than the amplitude of G2.
This is expected, since the area under the curves is 1, and 1/2, respectively.
The area under the impulse response function, G, is 1/2 in the case of
1 W0 ,
x fin the case of multiple scattering.
turbulence, and exps
W0
Because the area under G is less than 1, the fact that there is energy loss
when a pulse propagates in a turbulent medium is verified. Note, however,
that this is entirely due to the contribution from G2.
In addition, the contribution from G2, in the case of multiple scattering,
is variable; it could be any number between 0 and 1. In the case of a
lossless medium, there will be no absorption of any part of the signal due to
the random distribution of the atmospheric particles. In this case, oa = 0, and
subsequently, W0 =1, so that the factor, ex
4
'1 w
w0
will be equal to
1. On the other hand, when the signal is totally absorbed, the absorption
cross section oa will be infinitely large, and therefore, the exponential factor,
j ri^o
1L w
ex
x \ will be equal to 0. This would be the case of an ideal
lossy medium, where nothing can go through. In reality, this may very well
be the case where a light beam encounters a black hole somewhere in the
universe!
2. The Bit Error Rate (BER) function
a. Turbulence:
As mentioned before, equation (1.3.13) was evaluated numerically to
obtain a set of graphs for the Bit Error Rate (BER) of the receiver. The C
programming language was used. The BER expression was evaluated for
both strong and weak turbulence, and for three different input pulse
durations of 10, 20, and 50 picoseconds (ps). The probability density
functions for the expected counts in the j and q slots have the exponential
form in the case of strong turbulence, and the lognormal form in the case of
weak turbulence, as indicated below.
(5.2.1)
fn,q =
1
aArecAq
1
mgng
.& Arec Aq
where, , is the mean value of the intensity random process I. Other
system parameters, and their typical values, are listed below.
a= 1.0668* 1018 Ws'1
T) = Detector Quantum Efficiency = 0.2,
X = Optical Wavelength = 1.06pm,
Arec = Effective Aperture Area = 0.7854m2
Ts = 167ps, for 2 Gbps Data Rate,
Pt = Transmitted Optical Power = 1W,
x = Optical Path Transmissivity = 0.8,
xT = Transmitter Optical Efficiency = 0.9,
xQ = Receiver Optical Efficiency = 0.9,
tl = Pointing Losses = 1,
h = Plancks Constant = 6.624 10'34Js,
0 = Transmit Beam Angle = 15pRad,
R = Range = 3.6 107m.
In weak turbulence, the expression for the pdf of mq becomes,
In f I niqnq j + 0.2028 > 2
_tx Arec Aq >
0.811 >
which is a lognormal distribution.
In both cases, the pdf of mj takes the following exponential form,
(5.2.4)
1.53 106
exp{50.5 mj}, mj > 0.
These expressions were substituted into equation (1.3.13), and a set
of curves were obtained for the Bit Error Rate (BER) (See Figures #17 #22).
The important element about these curves is that as the input pulse
decreases in duration, the Bit Error Rate increases. This is expected, since
the narrower the pulse, the more relative spreading it will experience,
52
according to the impulse response curves. Intuitively, this makes sense
because a narrow pulse contains less energy; therefore, it will experience
more relative losses as it propagates through the turbulent atmosphere.
Additionally, as indicated by the BER curves, the Bit Error Rate drops in an
almost linear fashion as the transmit power increases in magnitude.
We also notice that in strong turbulence, we would need a
tremendous amount of source power in order to achieve acceptable BER
levels (about 10'8). An input pulse power of 100MW, or higher, would be
required to do so. Of course, this is not a realistic amount. In contrast, in the
case of weak turbulence, we would be able to achieve acceptable BER
levels with only 2W to 10 W of input pulse power.
b. Multiple Scattering
In this case, equation (3.2.7) was evaluated numerically when the
pulse spreading was large enough to produce overlapping in adjacent time
slots. The results obtained here for the Bit Error Rate are very interesting,
especially when compared to the ones for turbulence (see Figures #23 
#31). As we can see, the Bit Error Rate curves for the multiple scattering
cases will be very similar to the ones for weak turbulence, as long as the
transmit power is fairly small (up to almost 0.1W). But, notice what happens
once the transmit power exceeds 0.1 W: All the sudden, the Bit Error Rate
curves (in multiple scattering), experience a very steep exponential decline,
and by the time the transmit power has reached almost 1W, the Bit Error
Rate of the receiver is far smaller in the multiple scattering case than in the
weak turbulence case! This phenomenon may be explained from the fact
that the capability of the atmospheric particles to scatter an optical signal
53
drops exponentially as the input pulse power increases beyond a certain
point. On the other hand, the curves for the Bit Error Rate in turbulence show
a much smoother decline as the input pulse power increases (For an input
transmit power of lOWatts, the log of the Bit Error Rate, even in the case of
weak turbulence, did not drop below 18). This may be due to the fact that no
matter how high the transmit power (even for a 1,000W input transmit power
in the strong turbulence case, the logarithm of the BER did not drop below 
3.5), a pulse propagating in a continuous medium will still experience
severe random amplitude fluctuations, which are capable of corrupting it
much more than in the case of multiple scattering. This produces a much
more linear decline in the BER curves, as the input transmit power
increases. Because the amplitude fluctuations of the intensity random
process, I, can be severe in the turbulence case, their contribution to the Bit
Error Rate curves can be large. This may explain the very large BER of the
optical receiver, especially in the case of strong turbulence. On the other
hand, there are not any amplitude fluctuations of the intensity random
process, I, in the multiple scattering case, so that there cannot be any really
significant contribution to the BER curves. This may explain the lower BER
obtained here in the three cases of multiple scattering that were graphed,
despite the fact that the pulse spreading was much higher, as indicated by
the G multiple scattering impulse response function curves.
3. Conclusion
The physics, as well as the mathematics, involved in laser optical
pulse propagation through a random medium were investigated. The
54
expressions for the impulse response functions of the atmosphere, G, G2.
and G, for both turbulence, and multiple scattering were numerically
evaluated. From these results, we can conclude that a pulse, in general, will
experience very large amounts of spreading in the case of multiple
scattering, leading to unacceptable Bit Error Rates for the receiver. Only
when a pulse propagates through rain, the scattering effects are minimized
enough to obtain acceptable levels for the receiver Bit Error Rate. In the case
of turbulence, the pulse spreading is far smaller than in multiple scattering.
The exception to this is in the case of rain, where the resultant impulse
response functions have somewhat comparable amounts of spreading.
However, as the Bit Error Rate curves indicate, the optical receiver will
most likely perform far better in determining the correct slot position for a
received pulse in the case of multiple scattering, even when only a small
amount of source power is used, (about 0.5W), than in turbulence. This is a
very encouraging result because with a relatively small amount of power, we
may be able to transmit information much more effectively over larger
distances by efficiently minimizing the multiple scattering effects in the cases
of heavy rain or light fog.
However, we must keep in mind that turbulence and multiple
scattering are not two separate theories that occur one independently of the
other, but both act together at the same time to corrupt a propagating pulse
with noise. We may be able to minimize the multiple scattering effects by
using more input power, but we may not be able to bring the corresponding
turbulence effects (especially of the strong one) down as much. On the other
hand, even a small increase from 1W to 2W of input power will probably
create more financial trouble than expected! In other words, more input
power usually means more money must be spent for the design and
55
implementation of costeffective components, which are required for
accurate detection and extraction of an optical signal from noise. But, we
may always be able to find the way to build very lowcost optical
components, and still be able to use a higher input power.
To put all these results together, it seems that as long as we avoid
transmitting a signal through strong turbulence, dense cloud cover, haze, or
fog, we should be able to obtain a reasonably good quality of PPM multi
Gigabit laser communications.
In addition, the very interesting case of pulse propagation in free
space was discussed in Chapter 4, which shows us that, because of free
spaces nonturbulent nature, optical communications can have a bright
future in helping us explore some other distant worlds.
However, much more research will needs to take place to fully
understand and utilize the big advantages of using laser optical pulses in
transmitting information over the conventional electronic ways of
transmission.
56
Bibliography
[1] Gagliardi, Robert M., and Karp, Sherman: Optical Communications,
John Wiley & Sons, New York, 1976, pp 261277.
[2] Prati, Giancarlo, and Gagliardi, Robert M.: Decoding with Stretched
Pulses in Laser PPM Communications, IEEE Transactions on
Communications, Vol. COM31, No. 9, September 1983.
[3] Majumdar, Arun K., and Brown, William C.: Atmospheric Turbulence
Effects on the Performance of MultiGigabit Downlink PPM Laser
Communications, SPIE, vol. 1218, FreeSpace Laser Communication
Technologies II, 1990.
[4] Abshire, James B.: Performance of OOK and LowOrder PPM
Modulations in Optical Communications when Using APDBased
Receivers, IEEE Transactions on Communications, Vol. COM32, No.
10, October 1984.
[5] Ishimaru, Akira: Wave Propagation and Scattering in Random Media,
Vol. 2, Academic Press, 1978, pp 313314.
[6] Hong, Shin Tsy; Sreenivasiah, I.; Ishimaru, Akira: Plane Wave Pulse
Propagation Through Random Media, IEEE transactions on Antennas
and Propagation, Vol. AP25, No. 6, November 1977.
57
Appendix A
Figures and graphs
This section includes the graphs for the impulse response functions,
G, G2, and G, and the Bit Error Rate (BER) curves, for both turbulence and
multiple scattering.
The impulse response functions for turbulence and multiple scattering
are shown first, followed by the Bit Error Rate curves.
58
Figure #1
The Trapezoidal Approximation
o
59
Figure #2
G1 Turbulence Function
AMPLITUDE
xlO15
G1 TURBULENCE FUNCTION
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
TIME (sec)
xlO15
60
Figure #3
G2 Turbulence Function
AiMPLITUDE
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
TIME (sec)
xlO
Figure #4
G Turbulence Function
AMPLITUDE
0 1 2 3 4 5 6
TIME (sec)
xlO
Figure #5
G1 Multiple Scattering; Rain
AMPLITUDE
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
TIME (sec)
xlO
Figure #6
G1 Multiple Scattering; Rain
AMPLITUDE
xlO11
G1 MULTIPLE SCATTERING; RAIN; T1 = 1.8373 E12
TIME (sec)
xlO
Figure #7
G1 Multiple Scattering; Dense Rain
AMPLITUDE
xlO9 G1 MULTIPLE SCATTERING; DENSE RAIN; T1 = 3.8738 Ell sec
TIME ( sec)
xlO10
Figure #8
G1 Multiple Scattering; Heavy Rain
AMPLITUDE
xlO9 G1 MULTIPLE SCATTERING; HEAVY RAIN; T1 = 7.6440 Ell sec
TIME (sec)
xlO"9
Figure #9
G1 Multiple Scattering; Light Fog
AMPLITUDE
xlO9
G1 MULTIPLE SCATTERING; LIGHT FOG; T1 = 1.489 E10 sec
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
TIME (sec)
xlO9
Figure #10
G1 Multiple Scattering; Fog
AMPLITUDE
xlOB
G1 MULTIPLE SCATTERING; FOG; T1 = 5.9069 E10 sec
68
Figure #11
G1 Multiple Scattering; Fog
AMPLITUDE
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
TIME (sec)
xlO
69
Figure #12
G1 Multiple Scattering; Cloud
AMPLITUDE
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
TIME ( sec)
xlO
70
Figure #13
G1 Multiple Scattering; Cloud
AMPLITUDE
xlO7 G1 MULTIPLE SCATTERING; CLOUD; T1 = 2.0305 E8 sec
TIME (sec)
xlO
Figure #14
G1 Multiple Scattering; Cloud
AMPLITUDE
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
TIME (sec)
xlO
Figure #15
G1 Multiple Scattering; Haze
AMPLITUDE
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6
TIME (sec)
xlO6

PAGE 1
ATMOSPHERIC TURBULENCE AND MULTIPLE SCATTERING EFFECTS ON THE BIT ERROR RATE OF THE OPTICAL RECEIVER FOR PPM MULTIGIGABIT LASER COMMUNICATIONS by Fotios Panagiotis Kourouniotis B.S. E.E., University of Colorado at Denver, 1985 A thesis submitted to the Faculty of the Graduate School of the University of Colorado in partial fulfillment of the requirements for the degree of Master of Science Department of Electrical and Computer Engineering 1991
PAGE 2
This thesis for the Master of Science degree by Fotios Panagiotis Kourouniotis has been approved for the Department of Electrical and Computer Engineering by A. K. Majumdar E. T. Wall J. Thomas Date: August 7, 1991
PAGE 3
Kourouniotis, Fotios Panagiotis (M.S. Electrical Engineering) Atmospheric Turbulence and Multiple Scattering Effects on the Bit Error Rate of the Optical Receiver for PPM MultiGigabit Laser Communications Thesis directed by Professor Arun K Majumdar This thesis discusses the turbulence and multiple scattering effects on the impulse response functions of the atmospheric system, as well as the Bit Error Rate (BER) of the optical receiver, in the case of Pulse Position Modulation (PPM) laser communications. The mathematical techniques required to obtain expression for the impulse response functions, G1 (t), G2(t), and G(t), for both turbulence and multiple scattering, are presented. Then, these impulse response functions are graphed for both strong and weak turbulence, and in the case of multiple scattering, for a variety of atmospheric conditions, such as rain, cloud, haze, and fog. Subsequently, these graphs are used to obtain appropriate curves for the Bit Error Rate of the optical receiver. The impulse response functions of the nonturbulent free space are also developed, based on the results obtained for the turbulence and multiple scattering cases. In the case of multiple scattering, the Bit Error Rate (BER) for the optical receiver is graphed for dense rain, heavy rain, and light fog. BER curves are also plotted for both weak and strong atmospheric Comparisons of the graphs lead to the following conclusion: Given an input pulse power of approximately 0.2W2W, we may be able to significantly decrease the adverse effects of multiple scattering in the three cases, so that the probability of the optical receiver detecting the correct slot position in a given word, will increase significantly. In other words, the probability of bit
PAGE 4
error, or Bit Error Rate (BER) will be greatly reduced to acceptable levels. In fact, with an input pulse power higher than 2W, the effects of multiple scattering on the Bit Error Rate can reach negligible levels, especially in the cases of rain and light fog. In cloud, haze, and dense fog, however, the pulse spreading will be very large, leading to unacceptable BER levels. Also, in the case of strong turbulence, the BER of the receiver will again reach unacceptable levels, even for very large amounts of input pulse power. To lower these BER levels, we could use enormous amounts of input pulse power, perhaps greater than 1 OOMW, which in any case, would be extremely impractical and expensive. Computer programs in the C programming language were written to numerically evaluate the expressions of the impulse response functions of the atmosphere, and the Bit Error Rate of the optical receiver, for both turbulence and multiple scattering, and are included in this thesis.
PAGE 5
Dedication This thesis is dedicated to my parents, who suffered a great deal after their immigration to the United States, in order to provide me with the best emotional, and financial help. Their love and patience has been exceptional, and their support has been immeasurable. This dedication is a small token of my great appreciation for them. Thank you very much.
PAGE 6
Acknowledgements I would like to express my gratitude to Professor Arun K. Majumdar, for the many hours that he spent in discussions, comments, and explanations of complex material. His encouragements, guidance, and support have been exceptional, and without them, this work would never have been finished. I would also like to thank Mr. William Brown for his corrections and explanations on the derivations of certain mathematical expressions. His contributions were quite significant and greatly appreciated. Further acknowledgements and thanks are extended to: Professor Ed Wall for his helpful comments on this work; the Committee for their extra effort at impromptu scheduling; Christy for her patience to withstand and fulfill my numerous, urgent requests; Nick, George, and Ulysses for their great help with programming and the computer systems; and Kathy for proofreading and correcting the manuscript.
PAGE 7
Contents Chapter 1 Atmospheric Effects on Laser Optical Signal Propagation 1. lntroduction ............................................................................................................ 1 2. Turbulence versus Multiple Scattering ............................................................. 2 3. Evaluation of the Bit Error Rate (BER) in PPM Format.. ................................ .4 Chapter 2 The Impulse Response Function of the Atmosphere in the Case of Turbulence 1. The Transfer Function G(tz/c) .......................................................................... 12 2. Bit Error Rate (BER) and Slot Count Statistics .............................................. 24 Chapter 3 The Impulse Response Function of the Atmosphere in the Case of Multiple Scattering 1. The Discrete Random Medium ......................................................................... 33 2. Slot Count Statistics for Multiple Scattering ................................................... 36 Chapter 4 Pulse Propagation in Free Space 1. lntroduction .......................................................................................................... 40 2. The Impulse Response Function of Free Space ........................................... 41
PAGE 8
Contents (continued) Chapter 5 Results and Conclusion 1. The G1, G2. and G, Impulse Response Functions ....................................... 46 2. The Bit Error Rate (BER) Function ................................................................... 50 a. Turbulence ............................................................................................... 50 b. Multiple Scattering .................................................................................. 52 3. Conclusion ........................................................................................................... 53 Bibliography .................................................................................................................................. 55 Appendix A Figures and Graphs ..................................................................................... 57 Appendix B The Programs Used for the Evaluation of G1, G2, G, and tl1e BER Functions ...................................................................................... 8 9
PAGE 9
List of Figures Figure #1: The Trapezoidal Approximation ......................... sa Figure #2: G1 Turbulence Function .......................................... 59 Figure #3: G2 Turbulence Function .......................................... so Figure #4: G Turbulence Function ............................................. 61 Figure #5: G1 Multiple Scattering; Rain ................................. 62 Figure #6: G1 Multiple Scattering; Rain .................................. 63 Figure #7: G1 Multiple Scattering; Dense Rain ................ 64 Figure #8: G1 Multiple Scattering; Heavy Rain ................ ss Figure #9: G1 Multiple Scattering; Light Fog .................... ss Figure #1 0: G1 Multiple Scattering; Fog ............................. 67 Figure #11 : G1 Multiple Scattering; Fog ............................... sa Figure #12: G1 Multiple Scattering; Cloud ......................... 69 Figure #13: G1 Multiple Scattering; Cloud ......................... 7o Figure #14: G1 Multiple Scattering; Cloud ......................... 71 Figure #15: G1 Multiple Scattering; Haze .......................... 72 Figure #16.: G1 Multiple Scattering; Haze .......................... 73 Figure #17: BER; Strong Turbulence; 10ps ........................ 74
PAGE 10
List of Figures (continued) Figure #18: BER; Strong Turbulence; 20ps ........................ 75 Figure #19: BER; Strong Turbulence; 50ps ........................ 76 Figure #20: BER; Weak Turbulence; 10ps ........................... 77 Figure #21 : BER; Weak Turbulence; 20ps .......................... 78 Figure #22: BER; Weak Turbulence; 50ps ........................... 79 Figure #23: BER; Multiple Scattering; Dense Rain; 1 Ops ..................................................... so Figure #24: BER; Multiple Scattering; Dense Rain; 20ps ..................................................... 81 Figure #25: BER; Multiple Scattering; Dense Rain; 50ps ....................................................... 82 Figure #26: BER; Multiple Scattering; Heavy Rain; 1 Ops ....................................................... 83 Figure #27: BER; Multiple Scattering; Heavy Rain; 20ps ....................................................... 84 Figure #28: BER; Multiple Scattering; Heavy Rain; 50ps ....................................................... 85 Figure #29: BER; Multiple Scattering; Light Fog; 10ps ............................................................ ss Figure #30: BER; Multiple Scattering; Light Fog; 20ps ............................................................ 87
PAGE 11
List of Figures (continued) Figure #31 : BER; Multiple Scattering; Light Fog; 50ps ......... ................................................... aa
PAGE 12
list of Programs Program #1 : The Program for the G1 Turbulence Function .............................................................. 90 Program #2: The Program for the G2 Turbulence Function ..................... ; ........................................ 91 Program #3: The Program for the G Turbulence Function .................. ............................................ 92 Program #4: The Program for the G1 Dense Rain Multiple Scattering Function ................ 93 Program #5: The Program for the G1 Heavy Rain and Light Fog Multiple Scattering Function .............................................................. 94 Program #6: The Program for the BER Function Strong Turbulence, 1 Ops ........................ 95 Program #7: The Program for the BER Function Strong Turbulence, 20ps ........................ 96 Program #8: The Program for the BER Function Strong Turbulence, 50ps ........................ 97 Program #9: The Program for the BER Function Weak Turbulence, 1 Ops ........................... 98 Program #1 0: The Program for the BER Function Weak Turbulence, 20ps ........................... 99 Program #11 : The Program for the BER Function Weak Turbulence, 50ps ........................ 1 oo
PAGE 13
List of Programs (continued) Program #12: The Program for the BER Function Multiple Scattering, Dense, Heavy Rain, Light Fog, 1 Ops, 20ps, 50ps ........................................ 1 01
PAGE 14
Chapter 1 Atmospheric Effects on Laser Optical Signal Propagation 1. Introduction With the ever increasing demands for more advanced, sophisticated and reliable communication systems, scientists and engineers are trying to discover new ways of transmitting information faster, more efficiently and with as little corruption of the propagating signal as possible. The traditional way of communicating information so far has been through conventional electronic signal transmission in the forms of radio signals, short waves, microwaves, etc. However, a potentially revolutionary way of transmitting and receiving information today can be achieved through the use of Optical Lasers. Lightwave communications offer a number of 1
PAGE 15
advantages over the conventional methods. These advantages include, but are not limited to, transmission of information at the speed of light, no short circuit problems, lighter and more cost effective system components, and more reliable overall system performance. These reasons make laser light transmission much more desirabie than the conventional ways. When a specific sequence of pulses is transmitted at the optical frequencies through a turbulent medium, many effects occur, such as pulse broadening, amplitude fluctuations, time arrival delays, etc. These effects are usually due to the random nature of the transmission medium (channel) which corrupts the signal by adding different kinds of noise. The kinds of noise we are going to deal with are turbulence and multiple scattering as they occur when an optical signal travels through the Earth's turbulent atmosphere. 2. Turbulence versus Multiple Scattering Now, we will try to define these two kinds of noise. a) Turbulence: By turbulence, we mean the kind of noise added to an optical signal due to the large variations of the atmospheric variables. The atmosphere, being a prime example of a turbulent medium, is constantly changing. These dynamics affect all kinds of communications. There are too many variables to account for; nevertheless, we can still come up with an accurate model of the atmosphere by incorporating 2
PAGE 16
only the most prevalent of these variables. The sudden changes in temperature, the shifts in wind direction and velocity, the changes of air density from place to place and according to the elevation from Earth's surface, the random variations of solar activities, the random distribution of the moisture content of the atmosphere, the clouds, fog, rain, snow, and lightning, all contribute to the corruption of a transmitted signal. These atmospheric conditions usually account for random amplitude fluctuations, pulse broadening and time arrival delays. These effects are discussed in Chapter 2, where the impulse response of the atmospheric system is evaluated for the case of turbulence. b) Multiple Scattering: This is an even more fascinating subject than turbulence. Let's briefly outline what we mean by multiple scattering. In conventional electronic communications, the information signal modulates a carrier which leaves the source having a particular wavelength, A.. Now, it is well known that the atmosphere contains millions upon millions of particles on the order of picometers. The wavelength of the propagating carrier is normally many orders of magnitude larger than the dimensions of these picaparticles. Therefore, when the signal encounters these particles, it just ignores them and passes by to continue on. The resulting effect of these particles on the propagating signal is quite negligible. On the other hand, the same picaparticles can have a very significant effect on the 3
PAGE 17
propagation of optical signals since an optical carrier's wavelength is also in the order of picometers. When a laser beam encounters a particle with dimensions of the same order of magnitude as its wavelength, some of the signal will go through the particle, while some will be reflected back and some will scatter in many different directions. The scattered beams will then encounter other particles with dimensions similar to their wavelengths and scatter again. The same procedure can go on for millions (!) of times before the signal finally reaches its destination. So, the light beam will go through multiple scattering. As in the case of turbulence, multiple scattering causes similar distortions on the propagating pulse. However, in the multiple scattering case, the effects are even more amplified. These effects are discussed in Chapter 3. For both of these cases, the mathematics involved for obtaining the Bit Error Rate (BER) and the impulse response functions for the atmospheric system is similar. However, one should remember that turbulence refers to a continuous random medium, whereas multiple scattering refers to a discrete random medium. In the next section, we will derive the general expression for the Bit Error Rate (BER) of the receiver. The following derivation will be valid for both turbulence and multiple scattering. 3. Evaluation of the Bit Error Rate (BER) in PPM Format
PAGE 18
In this section, we are going to derive the general expression for the Bit Error Rate (BER) at the receiver side of a communication system after a laser signal has travelled through the turbulent atmosphere for some distance. The modulation format assumed is Pulse Position Modulation (PPM). The primary advantage of Pulse Position Modulation (PPM), is that it enables us to transmit very narrow, highly peaked optical pulses, at a high rate, over noisy channels. [1 ]; [ 21 Suppose we transmit a series of words, each of length M. Assume that each word consists of only one slot, q, that contains the information (logic 1 ), and an M1 number of slots, j, that contain no information at all (logic 0). As these pulses propagate through the atmosphere, they become noisy as the photon counts change in each slot. For example, the j slots that contain no information could pick up some photons along the way, while the q slot could lose some of its photons. In some cases, the Joss of photons in the q slot could be large enough so that the total number of photoelectrons counted by the receiver tor this slot is less than the total number of photoelectrons counted for some particular j slot. In this. case, the photodetector will assign. a logic 1 in the j slot, and a logic 0 in the q slot. Therefore, the receiver will make an error in detecting the correct slot for the transmitted information, which is called a symbol error. This section deals with the probability of symbol error (PSE) which will lead us to the probability of bit error, or Bit Error Rate (BER). The receiver reads the power received in each individual slot in a frame consisting of M slots. If the information is contained in the q slot, then the remaining j. slots contain no signal. If rj is the random output count for the jth slot, then the probability of symbol error will be [31 '5
PAGE 19
M (1.3.1) PSE = L, Prob(rj>rq). j=1 j:t:q Equation (1.3.1) evaluates the probability that the count of photoelectrons in the j slots will exceed the count in the q slot. Therefore, the Bit Error Rate (BER) of the system will be 31: 41 (1.3.2) BER M log2M M log2M M 2 (M1 ) PSE = 2 (M1 ) L, Prob(rj>rq). j=1 j:t:q First of all, the probability that the count in a slot j will exceed the count in the q slot needs to be evaluated. That is, we need to determine Prob(rj>rq). Now, if we assume that the expected counts in all slots are fixed, then we need to find Prob(rj>rq I mj. mq). where mj. mq are the expected counts in slots j and q respectively. Using the formulas for conditional probability, we can write (1.3.3) or (1.3.4) Prob(rj>rg n mj. mg) Prob(mj, mq) 6
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The term Prob(mj. mq 1 rj>rq) means: What is the probability of the expected values mj. mq being constant, given that the count in a j slot exceeds the count in the q slot? For all practical purposes, the two events may be considered to be independent so that Prob(mj, mq I rj>rq) = Prob(mj, mq). Therefore, equation (1.3.4) becomes (1.3.5) Now, we only need to evaluate Prob(rj>rq) Let's attempt to write Prob(rj>rq) in terms of probability mass functions (PMF's). Since the measured counts, rj and rq. are random variables, then Prob{rj>rq) really consists of two events: {a) the event where the random variables rj and rq take on some value rj and rq. an? (b) at the same time, rj > rq. So, Prob(rj>rq) = Prob(rj = rj. rq = rq n rj>rq) may also be written as (1.3.6) However, expression (1.3.6) is a joint probability mass function (1.3.7) Also note that the counts at slots j and q are independent since we assume statistical independence between the time slots. This means that expression (1.3.7) becomes (1.3.8) 7
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Since the expected values mj and mq are also independent of the measured values, then equation (1.3.8) can be written as (1.3.9) P r(rj) P r (rq) = P rlm(rjlmj) P r lm (rqlmq). provided that rj>rq. J q J J q q If we add up all the probabilities that the counts in all j slots exceed the count in the q slot, then oo r1 (1.3.1 0) Prob(rj>rq) = L p rjlmj(rjlmj) r p rqlmq(rqlmq). rj=1 rq=O j:;tq r1 Note that the second term, r P r q I mq (r ql mq). takes into account rq=O only those cases where rj>r q So far we have assumed that mj and mq are constant. If we assume that they are variable and continuous on (0, oo), then the random variables mj and mq will have a probability density function (pdf) description of fmj(mj) and fmq(mq) respectively. If we multiply equation (1.3.10) by fmj(mj) fm (mq) and integrate over mj and mq. we obtain q (1.3.11) Prob(rj>rq) = 8
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Substituting equation (1.3.11) into equation {1.3.1 ), we obtain the probability of symbol error (PSE) {1.3.12) PSE = M I l fmq(mq) I, Pr;lmtilmj) P,qlmq(rqlmq) dmjdmq. j=1 rj=1 rq=O j:tq j:tq Finally, by substituting equation {1.3.12) into equation {1.3.2), we obtain the the desired expression for the Bit Error Rate {BER) 9
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Equation (1.3.13) evaluates the Bit Error Rate (BER) of the receiver, given the probability density functions (pdfs) of the expected values at slots j and q respectively are known. These pdfs typically have exponential or log normal distributions. In the case 'of an exponential distribution, the exponential term in fm. ( m j) goes to zero much more quickly than the J exponential term in fm (mq). This situation is expected since fm.(mj) q J represents the pdf of the expected counts in the j slots which contain no signal. The probability mass functions (PMFs) of the random counting processes, P 1 .(rjlmJ) and P 1 (rqlmq). are taken to be conditional r 1 m 1 rq mq Poisson discrete counting processes [31 (1.3.14) r m.l P 1 .(rJImJ) = r.: emj, where r1 = 1, 2, ... oo. r1 m1 1 The word length M is a 2n number, where n is an integer. Equation (1.3.13) can be evaluated by numerical methods only. To this end, programs were written in the C programming language which calculated the BER for different word lengths M, different input pulses, and different pdfs fm.(mj) and fm (mq). (See Appendix B for a sample of these J q programs.) At this point, it is very important to notice that these pdfs depend on the received expected pulse shape. The pdfs of mj and mq in the case of turbulence differ from those in the case of multiple scattering. Also, In multiple scattering, the transmitted pulse is expected to spread out more than in turbulence. These are indeed the results which were obtained. These results are discussed in Chapter 5, and graphed in Appendix A. The expected pulse shape, which depends on the impulse response function of 1 0
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the atmospheric system, is discussed in Chapter 2 for the case of turbulence, and Chapter 3 for multiple scattering. 1 1
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Chapter 2 The Impulse Response Function of the Atmosphere in the Case of Turbulence 1. The Transfer Function G(tz/c) As mentioned in the previous chapter, the transmitted pulse picks up noise as it travels through the turbulent atmosphere. This causes a random fluctuation of its amplitude, as well as spreading. The amount of spread can be approximately determined from the transfer function characteristics of the atmospheric model. In the case of turbulence, the transfer function of the atmosphere is G(tz/c), where c is the speed of light. This transfer function can actually be obtained from the convolution of G1 (tz/c) with G2(tz/c), where G1 (tz/c) corresponds to the highfrequency component of the transfer function G(tz/c) and G2(tz/c) corresponds to the lowfrequency one. The 12
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reason for the separation of the G function into these two components comes from the fact that the output pulse shape can be expressed in terms of the twofrequency mutual coherence function (MCF) [S] (2.1.1) which can be considered as the correlation between output fields. The correlation function, Ru, of the output pulse, is given by 00 00 (2.1.2) RuCt1.t2) = J J Ui(ro1)U;(ro2)t(t1,t2,ro1,ro2)exp(iro1t1+iro2t2) dro1dffi2, 00 00 where Ui (ro) is the spectrum of the complex envelope of the input pulse, ui(t), and the asterisk (*) denotes ''the complex conjugate of." The spectrum of ui(t) is given by 00 (2.1.3) 1 J irot Ui (ro) = ui(t) e dt. 'J21t oo The correlation function, equation (2.1.2), may be changed to the intensity function, l(t), of the output pulse when t 1 = t 2 = t. 00 00 (2.1.4) l(t) = J J Ui (ro1 ) u; (ro2 ) r0(t,ro1 ,ro2 ) exp[i(ro 1 ro2 )t] dro1dro2 oo ;oo where r0 is revaluated at t 1 = t2 13
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Equation (2.1 .4) gives us the output response of the system for any input signal, u;(t). The desired impulse response function, G(t), can be obtained when u;(t) = o(t). Substituting this into equation (2.1.3), we obtain 00 00 (2.1.5) U; (ro) = J u;(t) e irot dt = J o(t) e irot dt = v 21t oo v 21t oo v 21t Substitution of equation (2.1.5) into equation (2.1.4) yields 00 00 (2.1.6) G(t) = ;1t J J ro(t. ro1 ro2) exp[i(ro 1 ro2 )t] dro1dro2. oo oo In many practical applications, the twofrequency mutual coherence function is a slowly varying function of IDcoh = (ro1 + ro2), and t = (t1 + t2); therefore, we can assume that r is a function of rod only. This is called the widesense stationary uncorrelated scattering (WSSUS) channel. With this in mind, the double integral in equation (2.1.6) can be transformed into another double integral according to the formula (2.1.7) J J f(x,y) dxdy = J J f(x(u,v), y(u,v)) IJI dudv, R R where J, is the Jacobian 1 '4
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ax ax (2.1.8) au av =(ax ay ay au av av au au av In our case, we have rocoh = (co1 + co2), and cod= co1 co2. Solving for co1 and co2, we obtain (2.1.9) and Since the Jacobian is 1, equation (2.1.6) can be written as 00 00 (2.1.10) G(t) = _1 J J fo(rocoh cod) eicodt drocohdcod. 21t oo oo However, because ro is a slowly varying function of COcoh we can assume that there is no variation with respect to COcoh Therefore, the double integral in equation (2.1.1 0) becomes a single integral with respect to cod. (2.1.11) 00 G(t) = 1 J f(rod) eirodt drod, 21t oo where, for convenience, we dropped the subscript o from r 0 15
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Equation (2.1.11) will give us the impulse response function, G(t), as the Fourier transform of the MCF function r(rod). Now. we need to determine r(rod). Let's write r(rod) as U51 P 427 ) (2.1.12) or (2.1.13) So, our task is to determine r 1 Note that the following derivation is good for both turbulence and multiple scattering. The governing equation for r is actually a function of an integral equation which will not be derived here, but is as stated below. ([51 P 315 ) The first term on the righthand side of the above equation represents the coherent part, r c and the second term represents the incoherent part, r;. Note that in most practical applications, the laser beam will have to propagate through relatively large distances to reach its destination, so that the incoherent part of equation (2.1.14) will be the dominant one. All derivations in this chapter are based on that fact. After all, the coherent 1 6
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impulse response G(t), is just that: an impulse, or mathematically speaking, a delta function. The general solution of equation (2.1.14) is not available. However, we can obtain a much simpler approximate differential equation when the signal propagates in a random medium with particle sizes comparable to or greater than a wavelength. In this case, the waves are scattered mostly in the forward direction, and if we also assume that the waves are plane waves propagating in the z direction, equation (2.1.14) can take the following approximate form ([SJ, P 316) (2.1.15) where V d is the twodimensional Laplacian operator with respect to Pd By expanding the exponential and keeping the first two terms, we obtain (2.1.16) 2 2 where a= 2k and b = PnO"s k I 4ap. Now, we let z/L = z' and PdiPo = pd' where L is the propagation distance. Po is chosen such that equation (2.1.16) takes the form (2.1.17) 1 7
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The following technique is used to obtain the exact solution for r1 (rod). Let's assume r 1 (rod) has the following form ([5], PP 321325) (2.1.18) where f(z) and g(z) are functions of z only. Substituting this into equation (2.1.17), we obtain (2.1.19) .! [ ag 2 J f az' + 14
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Substituting equation (2.1.22) into equation (2.1.20), f can be obtained. (2.1.23} f(z') = cos[(i4a)112z']. Since we are interested in what happens at the receiver side, we should, therefore, evaluate equations (2.1.22} and (2.1.23} at Z=L, or z'=1. Hence, the final solution for r 1 (cod) will be given by (2.1.24) 1 J tan[(i4a)112] 2 ] r1(rod) = cos[(i4a)112] ex1(i4a)112 Pd where a= coc:Ycocoh Then, equation (2.1.11) can be written as (2.1.25) G(t) = 00 _1 J 1 [ tan[(i4a)112] ] 2 2 jrod(t) 21t cos[(i4a)112] exp (i4a)112 Pd exp(kd1kcoh2) e dcod. oo Evaluating equation (2.1.25) at pd'=O, we obtain 00 (2.1.26) G(t) = ;, J cos[(i:a)112] exp(?l{.h2) ejO>,J(t) d
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However, since we are considering light propagating in the zdirection, equation (2.1.26) can be written as 00 (2.1.27) G(tz/c): ;11 J cos[(i:..)1'2] exp(ei"'d(tz/c) d"'d. oo Equation (2.1.27) can be written as the convolution of G1 (tz/c) and G2(tz/c). The result is as follows: (2.1.28) where (2.1.29) and (2.1.30) 00 G(t) = J G1 (tt'z/c) G2(t'z!c) dt', oo 00 G1 (tz/c) = _1 J 1 eirod(tz/c) drod, 21t cos[(i4a)112] 00 00 G2(tz/c) = 1 J exp(e irod(tzlc) drod. 21t_oo The integration in equation (2.1.30) can be carried out easily to yield 20
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(2.1.31) 1 { (tz/c)2} G2(tz/c) = 1 exp 2 T2 for continuous random media, or turbulence. T 2 is given by (2.1.32) The integral in equation (2.1.29) is a complex integral, and requires the use of the method of residues. After a change of variables through the relationships a = rod/rocoh and T = rocoh (tz/c), equation (2.1.29) may be written as 00 (2.1.33) G1.(tz/c) = O>coh J 1 eiaT da 21t cos[(i4a)112] oo The poles a= an of the integrand are given by (2.1.34) (i4an) 112 = (2n + 1) (1t/2), n = 0, , .... Therefore, we have a series of double poles a = an (2.1.35) [(2n + 1) (1t/2)]2 4i n = 0, 1, 2, ... 21
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since the poles at n = 1, 2, .... coincide with the poles at n = 0, 1, 2, .... These poles are all in the lower half of the a plane along the imaginary axis. Therefore, G1 (tz/c) = 0 for tzlcO, we close the contour in the lower half plane and obtain a series of residues at the poles. Then, G1 (tz/c) becomes (2.1.36) 00 G1 (tz/c) = 21ti L residues. n=O The evaluation of the residues is easy, and we obtain 00 1t { 1t2 (tz/c) } (2.1.37) G1(tz/c) = 4T1L..J (1)"(2n+1)exp (2n+1)216 T1 tz/c>O, n=O =0, tz/c:::;Q, where T1 = 1/rocoh The coherence bandwidth of the random medium, rocoh is given by (2.1.38) 1 28 ct5 lc 215 L11/5 Olcoh C n ''0 If k0 = 21t/A. is the wavenumber of the optical signal, is the strength of turbulence parameter, and L is the propagation path length through the turbulent region, then (2.1.39) T ( 1 ) r. 12/5 L 11/5 1 1.28c '"'t1 "O 22
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The convolution of equations (2.1.37) and (2.1.31) will give us the simplified expression for the G function in the case of turbulence. ( represents convolution.) G(tz/c) = G1 (tz/c) G2(tz/c) = 00 00 J G1 [(tt')zlc] G2(t'z!c) dt' = J G1 (t'z!c) G2[(tt')z/c)] dt' = oo 00 00 f { 1 ( 1 )" (2n+ 1) exp[ (2n+ 1 )2 } 00 {_1exp[ } dt', T2 which, with a little simplification, becomes (2.1.40) G(tz/c) = 00 00 4T T exp 2 n+ exp n+ 16 T v; I [ [(tt')z/c]2]L ( 1 )"(2 1) [ (2 1. )2 1t2 (t'z/c)J dt' 1 2 T2 n=O 1 oo 23
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Equation (2.1.40) is the impulse response function of the atmosphere in the case of turbulence. The only way to evaluate this expression for different values of T1 and T 2 is by numerical methods. The C programming language was used again. The results are discussed in Chapter 5, and a sample of the C program is given in Appendix B. In the next section, we will establish the relationship between the Bit Error Rate (BER), expression (1.3.13), and the impulse response function G(tz/c), equation (2.1.40). 2. Bit Error Rate (BER) and Slot Count Statistics In this section, we will determine the pdf of the expected count in a slot j as a function of the transfer function, G(tz/c). Since ultimately we are interested in a plot of G versus time, we can safely use G(t) instead of G(t z/c). The factor z/c will just shift G along the time axis by an amount dependent on the numerical value of z/c. Generally speaking, the received pulse power over the effective aperture area of the telescope is a random function which depends upon the impulse response G(t) and the background noise of the receiver. Let the received pulse power per unit area be [31 (2.2.1) P(t) = I S(t), where I is a random intensity function representing amplitude fluctuations due to atmospheric turbulence, and S(t) is a unitless quantity representing the output pulse shape without the noise. S(t) is defined as follows: 24
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00 (2.2.2) S(t) = f Pi(t') G(tt') dt' I oo where Pi(t') is the input pulse power, and 'Y is a parameter dependent on the full width transmit beam angle, e, and on the range, R, and is defined as 'Y = <1>, is the average, or expected mean value, of the intensity random e R process I. Since the receiver reads power and not amplitude, the expected instantaneous count for the slot q, which contains the information, will then be (2.2.3) or qTs mq =a Arec J P(t) dt + nq. (q1)Ts qTs mq = aArec J I S(t) dt + nq, (q1)Ts where is a constant which depends on the system parameters, 11 is the detector quantum efficiency, h is Planck's constant, and v is the operating optical frequency. Arec is the effective aperture area, and nq is the combined background and receiver noise count at the q slot. If we assume that I is constant over time intervals as short as T S then equation (2.2.3) may be written as 25
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qTs (2.2.4) mq =a Arec.l J S{t) dt + nq =a Arec I Aq + nq. {q1)T 5 where Aq is the expected pulse shape area in the q slot, qTs (2.2.5) Aq = J S{t) dt. (q1)T5 Let's examine equation (2.2.4). Since mq =a Arec I Aq + nq. and I and nq are independent random processes, then (2.2.6) fm {mq) = faA lA (a Arec I Aq) fn (nq). q .recq q Since a, Arec I, and Aq are all independent events, equation (2.2.6) may be written as (2.2.7) fm (mq) =fa( a) fA (Ared f 1(1) fA (Aq) fn (nq). q q q Since Arec Aq are deterministic, and a is a constant, fa(a), fA (Ared and fA (Aq) are just delta functions having a strength of unity; q . therefore, they cannot affect fmq(mq) Hence, equation (2.2.7) becomes (2.2.8) If we assume that the pdf of I is known and use equation (2.2.4) without including the receiver and background noise, we obtain
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Solving for I yields (2.2.9) mq = aArec I Aq. I = __ m_g...___ a. Arec Aq Substituting equation (2.2.9) into equation (2.2.8) and adding the noise process of the receiver, we finally obtain (2.2.1 0) Equation (2.2.1 0) is the one that shows us the relationship between the probability density functions (pdfs) of the expected slot counts and the impulse response function of the turbulent atmosphere. This is because fmq(mq) depends on the expected pulse shape area Aq for each slot, which in turn depends on the impulse response function G(t). Since the Bit Error Rate, (BER), equation (1.3.13), is a function of fmq (mq). it is now obvious what the relationship between G(t) and the BER is. Note that we have assumed that the expected noise count in each slot is the same. Something that we still need to determine is an expression for the expected pulse shape area, Aq. Equation (2.2.5) can be also written as 27
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(2.2:11) qTs Aq = I "( Joo Pi(t') G(tt') dt' dt, oo (q1)T5 qTs = I 1Pj(l') G(t1') dt' dt. (q1)T s So, Aq. can be expressed as a product of the factor and the area under the convolution of Pi(t) with G(t). According to one of the theorems about the convolution integral, we know that the area under a convolution is equal to the product of the area under the "factors". Therefore, the value of the integral on the righthand side of equation (2.2.11) is equal to the product of the area under the functions Pi(t) and G(t). Since Pi(t) is the input pulse power function of constant amplitude P1 and duration tp seconds, then, the area under it over that amount of time, will be equal to the energy contained in the transmitted pulse, and will be equal to P1tp. Now, since G(t) is a convolution between the functions G1 (t) and G2(t), its area will be equal to the product of the areas under G1 (t) and G2(t). First, let us evaluate the area under G1 (t). Integrating both sides of equation (2.1.37) with respect to time t from z/c to oo, we obtain the following expression. 00 (2.2.12) j G 1 (tz/c) dt = 28
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00 1t L 16 T 1 { [ 1t2 (tz/c)J }oo 4 T ( 1 ) n ( 2 n + 1 ) 2 2 exp ( 2 n + 1 ) 2 16 T 1 n=O (2n+ 1) 1t 1 zJc Since the exponential term within the brackets, ({}), is equal to 1 and after some simplification, equation (2.2.12) becomes 00 00 (2.2.13) 4 (1)" f G1 (tz/c) dt1t ( 2 n+ 1 ) zlc n=O The infinite sum in equation (2.2.13) converges to: so that the area under the G1 (tz/c) function is unity. Note that this result is good for both turbulence and multiple scattering, and is independent of T 1 The area under the G2(tz/c) function can be evaluated as follows. Integrating both sides of equation (2.1.31) with respect to time t from z/c to oo, we obtain the following. 00 00 (2.2.14) j G2(tz/c) dt = f 1 { (tz/c) 2 } dt 1 exp 2 T2 z/c Changing tz/c to x, we obtain 00 00 (2.2.15) J G2(x} dx = f exp{dx, 0 29
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00 = 1 exp 2 x = ,.....mt. 1 I { x2}d 1 .. T2v1t T2 T2v1t 0 00 Now, since in!= I exp{dx, then, 0 00 00 (2.2.16) I exp{dx I exp{dx, 0 0 which may be also written as (2.2.17) Changing from cartesian to polar coordinates, we obtain (2.2.18) The evaluation of the integral is very easy, and we finally obtain 30
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(2.2.19) which implies that (2.2.20) 2 1tT2 Jnt = 4 2 ,{; mt=yT2. Substituting equation (2.2.20) into (2.2.15), we obtain (2.2.21) which is the area under the G2(tz/c) function. So, the area under the G(tz/c) function is 1 = ; therefore, the area under the convolution integral on the righthand side of equation (2.2.11) is Finally, the desired area, Aq, is (2.2.22) rPttp Aq = 2<1>. This result can be used together with equation (2.2.1 0) to obtain the final form of fmq (mq). Also, since we have assumed that the expected noise count, nq. in each slot is constant, then its pdf, fnq (nq). must be a delta function of the form B(mqnq). where mq is just a variable representing any real number on the mq axis, and nq is a constant. So, the convolution in equation (2.2.1 0) may be written as 3 1
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00 (2.2.23) J ( mgmg' ) fm (mq) = f 1 A A B(mq'nq) dmq'. q a rec q 0 which implies that (2.2.24) Equation (2.2.24) will give us the final form of the pdf of mq. once the pdf for I, f 1(1), is known. The next chapter discusses the impulse response function of the atmosphere and the slot count statistics for the case of multiple scattering. 32
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Chapter 3 The Impulse Response Function of the Atmosphere in the Case of Multiple Scattering 1. The discrete random medium As shown in the previous chapter, the impulse response function, G(t z/c), can be expressed as a convolution (3.1.1) G(tz/c) = G1 (tz/c) G2(tz/c). The same mathematical techniques which were used to obtain G(t z/c) for turbulence can be applied here to obtain G(tz/c) for multiple scattering. Thus, the expressions for G1 (tz/c) and G2(tz/c) here will be very similar to those obtained in Chapter 2. The only difference is that a turbulent 33
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medium is considered a continuous random medium, whereas a medium consisting of scatterers of random sizes, shapes, orientations, velocities and densities, is considered a discrete random medium. These two different kinds of random media will, primarily, affect the mutual coherence function (MCF), r, which in the case of multiple scattering becomes (3.1.2) where W0 is the albedo of a scatterer, t = Pna5z, which is the optical distance, and a5 is the scattering cross section. Note that the albedo is a defined as W0 = where at is the total cross section of the random medium, Gt and is equal to a5+aa, where aa is the absorption cross section of some part of the signal by the randomly distributed particles. Equation (3.1.2) is equivalent to equation (2.1.12). Since G1 (tz/c) is dependent on r 1 (rod) only and not on the exponential term in equation (3.1.2), and r 1 (rod) remains the same for multiple scattering, then G1 (tz/c) will be exactly the same as well. For convenience, equation (2.1.37) is repeated here. 00 1t { 1t2 (tz/c) } (3.1.3) G1 (tz/c) = 4 T1..L.,. (1 )"(2n+1 )exp (2n+1 )2 16 T1 tzlc>O n=O 34
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The integration can be carried out to yiel (3.1.5) for discrete random media, or for multiple scattering. [&] The impulse response function, G(tz/c), is obtained by substitution of equations (3.1.3) and (3.1.5) into equation (3.1.1 ), 00 (3.1.6} G(tz/c} = f { 1 ?/ 1 )" (2n+ 1) exp [ (2n+ 1 )2 oo {s(tt'exp[Using the sifting property of the delta function, we obtain the following form for G(tz/c) (3.1.7) G(tz/c) = 35
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00 1t { [1 Wo] { 1t2 (t2z/c) } 4 T1 exp Wo t ...J {1 )" {2n+1) exp {2n+1 )2 T6 T1 n=O Equation {3.1. 7) is the impulse response function of the atmosphere for the case of multiple scattering. Note that exp {[ 1 0 J t} is the attenuation factor for a pulse wave propagating in a discrete random medium. In fact, equation {3.1.7) can be used to describe the impulse response function for multiple scattering in any medium, as long as we know the albedo, w0 and the optical distance, t. Once again, the C programming language was used to evaluate equation {3.1.7) and obtain results for different atmospheric conditions. Another very interesting question is what happens to G1 and G2 when the laser beam propagates in free space? This case may arise when an optical signal originates at a satellite far above the earth's atmosphere, and is sent either downwards towards the earth's surface, or towards the empty space, perhaps in an attempt to communicate with other civilizations in the Universe! This, will be discussed in Chapter 4. 2. Slot count statistics for multiple scattering As in the case of turbulence, the mathematical techniques used in order to find the pdf of the expected count in the q slot, fm {mq). are q identical. The only difference will be the slightly different expression for the expected pulse area, Aq. This is because the area under the G2{tz/c) 36
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function is different in the case of multiple scattering. To find the area under G2, we integrate both sides of equation (3.1.5), with respect to t, from z/c to oo, 00 (3.2.1) The integration can be easily carried out to yield, (3.2.2) 00 j G2(tz/c) dt =ex{ [ 1 <} The area under the G1 (tz/c) function is still unity, so that the area under the impulse response function, G(tz/c), will be exp { [ 1 0 ] t}. and therefore, the expected pulse area will be (3.2.3) { [1 W0 ] } Aq = exp Wo t Equation (2.2.24) remains the same, so that the final form of fm (mq) q will still be (3.2.4) however, Aq. is now given by equation (3.2.3). 37
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So, once the pdf of I, f 1(1), is known, fm (mq) can always be q calculated. Unlike the turbulence case, the multiple scattering case is not affected by the random amplitude variations of the intensity random process, I. In multiple scattering, the amplitude of the process, I, may be considered to be a constant over the entire word length, Mts, so that its pdf will be a delta function. With this in mind, equation (3.2.4) becomes 1 mgng ) f (mq) = lq mq a Arec Aq a Arec Aq or (3.2.5) where lq is the constant power per unit area amplitude of the intensity random process, I. The pdf for the amplitude of the process, I, in the j slots will have the same form as the pdf in equation (3.2.5). The only difference will be the change of the subscript q to the subscript j. To find the Bit Error Rate, we need to substitute equation (3.2.5) into equation (1.3.13). After the substitution we obtain the following result, 38
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oo rt1 B(mqnqlqcx Arec Aq) '_L P rjlmj(rjlmj) '_L P rqlmq(rqlmq) dmjdmq. rq=O j*q which, by using the properties of the delta functions, can be simplified to give the following expression. M log 2 M M (3.2.7) BER2(M1 ) exp[(cxAreclave(Aj+Aq))] j=1 j*q where lave = lq = lj is the average constant power per unit area amplitude of the intensity random process, I, and is approximately equal to 1 OnW. Also, note that we have used the conditional Poisson discrete counting process for the PMFs of rj and rq. Equation (3.2.7) can be evaluated only numerically, and the results are discussed in Chapter 5. Graphs of the BER functions are included in Appendix A, and a sample of the programs used for the evaluation of equation (3.2.7) is included in Appendix B. The next chapter deals with the interesting case of propagation of an optical signal in free space. The development of the impulse response 'function, G(t), as well as a short dis.cussion about the nature of the slot count statistics of free space are included. "39
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Chapter 4 Pulse propagation in free space 1. Introduction So, the question is: What happens to a light beam when it propagates in vacuum? Well, for one thing, free space is not a turbulent medium. There is no way that a signal can be corrupted by noise when travelling through it. In addititon, free space, ideally, does not contain any particles that would cause scattering of the signal. Therefore, a laser beam will not experience any turbulence or multiple scattering effects when propagating in free space. In the following section, we will examine the form of the impulse response function, G(tz/c), which corresponds to the free space case. 40
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2. The impulse response function of free space The same equations for the impulse response function, G(tz/c), that were derived for turbulence and multiple scattering may also be applied in the case of a pulse propagating in free space. Since there is no turbulence present, the strength of the turbulence parameter, must be zero. Then, equation (2.1.39) becomes (4.2.1) T1 = ( L 11/5 = 0. Taking the limit of equation (2.1.37) as T1 approaches 0, we obtain (4.2.2) lim G1 (tz/c) = { } 1t 1 1t2 tz/c 4 lim T (1 )n (2n+1) exp[ (2n+1 )2 16 ( T )] 1ft 1 In Chapter 2, section 2, we proved that the area under the G1 curve is unity, independent of the value of T1 As T1 gets smaller and smaller, the G1 function gets narrower and narrower, with a corresponding increase in amplitude in order to keep the area under the curve equal to 1. At the limit, when T 1 = 0, the curve becomes an impulse of unit area, so that (4.2.3) lim G1 (tz/c) = B(tz/c). 41
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Now, since the turbulence parameter, Cn is zero, then from equation (2.1.32), T 2 must be zero as well. Taking the limit of equation (2.1.31) as T 2 approaches zero, we obtain (4.2.4) which implies that (4.2.5) lim G2(tz/c) = o(tz/c). The convolution of G1 (tz/c) with G2(tz/c) will give us the impulse response function of free space, and is equal to 00 G(tz/c) = G1(tz/c) G2(tz/c) =J o(tt'z/c) o(t'z/c) dt', which implies that (4.2.6) G(tz/c) = o(t 2 cz) This is the impulse response function of free space in the absence of turbulence. Therefore, the output pulse shape, S(t), will be 42
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(4.2.7) 00 S(t) = J Pi(tt') 2 cz) dt' = P{t 2 cz) 2z/c which, as was expected, is a shifted version of the input pulse, Pi(t), with a different amplitude. Now, let's see what happens in the absence of multiple scattering. In this case, G1 (tz/c) is identical to G1 (tz/c) in the absence of turbulence. That is, G1 (tz/c) = o(tz/c). Also, since there is no attenuation of the propagating signal in vacuum, the exponential factor in equation (3.1.5) must be equal to 1, so that G2(tz/c) = o(tz/c). Of course, the convolution of these.two functions is identical to equation (4.2.6), and therefore, the final pulse shape in the absence of multiple scattering is identical to the one given by equation (4.2.7), as expected. Thus, when a signal propagates in free space, it preserves its initial shape, and the length of time required to reach its destination is given by 2z/c. So, as expected, there is no energy loss when a signal propagates in free space, and the pulse received is an exact replica of the transmitted pulse. Additionally, this analysis, verifies the expressions obtained for the impulse response functions of the atmosphere, as they, give us the appropriate results when they are taken to the limiting cases. Both, turbulence parameters and multiple scattering attenuation factors are zero at the limiting case, which would represent signal propagation in a perfectly nonturbulent medium. Furthemore, it is noteworthy to observe that the Bit Error Rate is zero when a pulse propagates in free space. This can be verified mathematically with the following procedure. 43
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Since there are no disturbances in free space, there will be no loss or gain of photons in either the j slots, or the q slot. So, the photon count in all slots in a frame will remain the same, no matter how long the signal propagates before it arrives at its destination. Since there is no information in the j slots, the photon counts in these slots will always be zero. On the other hand, the photon count in the q slot will be a fixed number, say, n. Therefore, the probability density functions (pdfs) for the counts in all slots will just be delta functions, with the pdfs for the counts in the j slots being (4.2.8) which is a delta function centered at mj=O since the counts in the j slots must be 0. The pdf for the count in the q slot will be (4.2.9) which is a delta function centered at mq=n since the count in the q slot is constant and equal ton. Inserting equations (4.2.8) and (4.2.9) into equation (1.3.13), we obtain BER M log 2 M ::c:* 2(M1) M L j=1 oo oo oo r1 J J li(mj) li(mqn) I P,jlm/rilmj) r prqlmq(rqlmq) dmjdmq. rj=1 rq=O j:;tq j:;tq 44
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which gives us, (4.2.10) BER=O, as expected. So, we should not expect any Bit Error Rate when an optical signal propagates in free space. As we can see, since there cannot be any turbulence or multiple scattering effects, optical signal propagation in free space must be a very efficient way to transmit information. Indeed, the results obtained are very encouraging, and much more research should be devoted to the fascinating subject of laser optical signal propagation in free space. However, a word of caution must be given here: Free space may not always be free of turbulence. In fact, the universe contains an infinite number of radiation emmitting stars, black holes, intergalactic and interstellar material, galaxies, comets, and many other objects, that are capable of corrupting, and even preventing the propagation of a signal. Nevertheless, there is great excitement about pulse propagation in random media, something that started as an idea around the beginning of this century, or earlier, but only recently began to be materialized, and has a very promising future ahead. Who knows? That may just be the beginning for a different kind of adventure into the infinite mystery and beauty of the universe. 45
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Chapter 5 Results and conclusion 1. The G1, G2, and G, impulse response functions In this chapter, we will discuss the results obtained for the impulse response functions, G1, G2, and G, and the Bit Error Rate (BER) of the receiver, for both turbulence. and multiple scattering. We will start with the impulse response functions first. As mentioned in previous chapters, the programming language C r was used as the vehicle to run the numerical evaluation of the impulse response functions. The trapezoidal rule was used in order to evaluate the integrals in expressions (2.1.31 ), (2.1.37), and (2.1.40). The following approximation was used for the evaluation of the area under .any function f(t), 46
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00 00 (5.1.1) J f(t) dt = At {( t\1]. where .!\tis the time increment. The approximation is very good since tens of thousands of increments were taken for the evaluation of the area under each particular function. A typical case is shown in Figure #1. By numerically evaluating expressions (2.1.31 ), (2.1.37), and (2.1.40), based on equation (5.1 .1) (with an appropriate change of variables, of course), a set of curves was obtained for the impulse response functions. In the case of turbulence, these functions did not exhibit any appreciable amount of spreading to cause intersymbol interference between the time slots j and q, as a typical set of curves for G1, G2, and G indicates for both strong and weak turbulence. Also, graphs were obtained for the Bit Error Rate for both cases, and for different input pulse widths. The results are discussed in the following section. In the case of multiple scattering, the impulse response functions, G1, and G, were graphed for a variety of atmospheric conditions, including different kinds of clouds, haze, fog, and rain (The G2 functions are just delta functions of strength less or equal to unity, depending on the albedo w0 located at t=zlc, and therefore they were not graphed). Examination of these curves show that in the case of cloud cover, haze, and fog, the pulse spreading is extremely large, ranging from 5ns in fog, to 300ns in haze. This implies that not only does a pulse overlap into adjacent slots to slot q, causing intersymbol interference, but it will also spread over to adjacent words in a particular frame, causing interframe interference as well. Both 4'J
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conditions are undesirable because they will result in a very large Bit Error Rate by greatly increasing the probabilities that the photoelectron counts in the j slots are greater than the photoelectron counts in the q slots. On the other hand, the resulting pulse spreading due to rain is far smaller than the one in the other cases (between 6ps and 15ps). In this case, there will be virtually no overlap in the j slots, so that the Bit Error Rate will be far smaller. The very interesting case, however, arises when the pulses will have to propagate through dense rain, heavy rain, or light fog. Here, the pulse spreading will be sufficiently large, between 300ps and 600ps in dense and heavy rain, to 1 ns in light fog, to cause intersymbol interference in the slots adjacent to it, but sufficiently small to not spread over to frames adjacent to its own frame causing interframe interference. Still, however, most of the pulse energy will be contained in the q slot, where the originally transmitted pulse was located. Some very interesting results for the Bit Error Rate were obtained in this case, which are discussed in the following section. It is worthwhile to note that in turbulence, the shape of the G impulse response function resembles the function G1 close to the origin, but is almost entirely dictated by the shape of G2 thereafter. This is because G1 is almost an impulse compared to G2, and their convolution will very closely resemble the G2 function. On the other hand, since the G2 function is indeed an impulse in multiple scattering, the shape of the G function will be identical to the G1 function, only attenuated by the factor, [ 1 0 ] t }. This can be explained as follows: In a continuous random medium, the higher the carrier frequency, the more efficient is the transmission. Lower frequencies tend to have much more difficulty to efficiently propagate through the medium, and the same rule applies to optical signals as well. That's why G2 48
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(corresponding to the lowfrequency component of the impulse response function) is the dominant function in the case of turbulence. On the other hand, in a discrete random medium, a greater the wavelength of the carrier frequency will cause less scattering. For large wavelengths, the relative size of the atmospheric piceparticles is negligibly small, so that the signal will just "ignore" them, and continue through. When the wavelength, however, gets very small, the signal will go through a lot of scattering, and that's why G 1 (corresponding to the highfrequency component of the impulse response function) is the dominant function in the case of multiple scattering. Also, note that the amplitude of the G1 function, in the case of turbulence, is a few orders of magnitude greater than the amplitude of G2. This is expected, since the area under the curves is 1, and 1/2, respectively. The area under the impulse response function, G, is 1/2 in the case of turbulence, and exp {[ 1 <}in the case of multiple scattering. Because the area under G is less than 1, the fact that there is energy loss when a pulse propagates in a turbulent medium is verified. Note, however, that this is entirely due to the contribution from G2. In addition, the contribution from G2, in the case of multiple scattering, is variable; it could be any number between 0 and 1. In the case of a lossless medium, there will be no absorption of any part of the signal due to the random distribution of the atmospheric particles. In this case, cra = 0, and subsequently, w0 =1 so thatthe factor, [ 1 0 J <} will be equal to 1. On the other hand, when the signal is totally absorbed, the absorption cross section cra will be infinitely large, and therefore, the exponential factor, [' 0J <}will be to 0. This would be the case of an ideal 49
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lossy medium, where nothing can go through. In reality, this may very well be the case where a light beam encounters a black hole somewhere in the universe! 2. The Bit Error Rate (BER) function a. Turbulence: As mentioned before, equation (1.3.13) was evaluated numerically to obtain a set of graphs for the Bit Error Rate (BER) of the receiver. The C programming language was used. The BER expression was evaluated for both strong and weak turbulence, and for three different input pulse durations of 10, 20, and 50 picoseconds (ps). The probability density functions for the expected counts in the j and q slots have the exponential form in the case of strong turbulence, and the lognormal form in the case of weak turbulence, as indicated below. (5.2.1) 1 { 1 [ mgng ]} fm (mq) = exp , q aArecAq a Arec Aq where, , is the mean value of the intensity random process I. Other system parameters, and their typical values, are listed below. a= 1.0668 1018 ws1 11 = Detector Quantum Efficiency = 0.2, A. = Optical Wavelength = so
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Arec = Effective Aperture Area = 0. 7854m2 T s = 167ps, for 2 Gbps Data Rate, Pt = Transmitted Optical Power= 1 W, t3 = Optical Path Transmissivity = 0.8, tT = Transmitter Optical Efficiency = 0.9, tR =Receiver Optical Efficiency= 0.9, tL =Pointing Losses= 1, h = Planck's Constant= 6.624 1 o34Js, 9 = Transmit Beam Angle = 15J,J.Rad, A= Range= 3.6 *107m. In weak turbulence, the expression for the pdf of mq becomes, { [ ( 1 [ mgng ]) J2 } In a Arec Aq + 0.2028 ( 5 2 3 ) fmq(mq) = 1.596211 exp 0.811 which is a lognormal distribution. In both cases, the pdf of mj takes the following exponential form, (5.2.4) These expressions were substituted into equation (1.3.13), and a set of curves were obtained for the Bit Error Rate (BER) (See Figures #17 #22). The important element about these curves is that as the input pulse decreases in duration, the Bit Error Rate increases. This is expected, since the narrower the pulse, the more relative spreading it will experience, 51
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according to the impulse response curves. Intuitively, this makes sense because a narrow pulse less energy; therefore, it will experience more relative losses as it propagates through the turbulent atmosphere. Additionally, as indicated by the BER curves, the Bit Error Rate drops in an almost linear fashion as the transmit power increases in magnitude. We also notice that in strong turbulence, we would need a tremendous amount of source power in order to achieve acceptable BER levels (about 1 o8 ). An input pulse power of 1 OOMW, or higher, would be required to do so. Of course, this is not a realistic amount. In contrast, in the case of weak turbulence, we would be able to achieve BER levels with only 2W to 1 0 W of input pulse power. b. Multiple Scattering In this case, equation (3.2. 7) was evaluated numerically when the pulse spreading was large enough to produce overlapping in adjacent time slots. The results obtained here for the Bit Error Rate are very interesting, especially when compared to the ones for turbulence (see Figures #23 #31 ). As we can see, the Bit Error Rate curves for the multiple scattering cases will be very similar to the ones for weak turbulence, as long as the transmit power is fairly small (up to almost 0.1 W). But, notice what happens once the transmit power exceeds 0.1 W: All the sudden, the Bit Error Rate curves (in multiple scattering), experience a very steep exponential decline, and by the time the transmit power has reached almost 1 W, the Bit Error Rate of the receiver is far smaller in the multiple scattering case than in the weak turbulence case! This phenomenon may be explained from the fact that the capability of the atmospheric particles to scatter an optical signal 52
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drops exponentially as the input pulse power increases beyond a certain point. On the other hand, the curves for the Bit Error Rate in turbulence show a much smoother decline as the input pulse power increases (For an input transmit power of 1 OWatts, the log of the Bit Error Rate, even in the case of weak turbulence, did not drop below 18). This may be due to the fact that no matter how high the transmit power (even for a 1,000W input transmit power in the strong turbulence case, the logarithm of the BER did not drop below 3.5), a pulse propagating in a continuous medium will still experience severe random amplitude fluctuations, which are capable of corrupting it much more than in the case of multiple scattering. This produces a much more linear decline in the BER curves, as the input transmit power increases. Because the amplitude fluctuations of the intensity random process, I, can be severe in the turbulence case, their contribution to the Bit Error Rate curves can be large. This may explain the very large BER of the optical receiver, especially in the case of strong turbulence. On the other hand, there are not any amplitude fluctuations of the intensity random process, I, in the multiple scattering case, so that there cannot be any really significant contribution to ttie BER curves. This may explain the lower BER obtained here in the three cases of multiple scattering that were graphed, despite the fact that the pulse spreading was much higher, as indicated by the G1 multiple scattering impulse response function curves. 3. Conclusion The physics, as well as the mathematics, involved in laser optical pulse propagation through a random medium were investigated. The 53
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expressions for the impulse response functions of the atmosphere, G1, G2, and G, for both turbulence, and multiple scattering were numerically evaluated. From these results, we can conclude that a pulse, in general, will experience very large .amounts of spreading in the case of multiple scattering, leading to unacceptable Bit Error Rates for the receiver. Only when a pulse propagates through rain, the scattering effects are minimized enough to obtain acceptable levels for the receiver Bit Error Rate. In the case of turbulence, the pulse spreading is far smaller than in multiple scattering. The exception to this is in the case of rain, where the resultant impulse response functions have somewhat comparable amounts of spreading. However, as the Bit Error Rate curves indicate, the optical receiver will most likely perform far better in determining the correct slot position for a received pulse in the case of multiple scattering, even when only a small amount of source power is used, (about O.SW), than in turbulence. This is a very encouraging result because with a relatively small amount of power, we may be able to transmit information much more effectively over larger distances by efficiently minimizing the multiple scattering effects in the cases of heavy rain or light fog. However, we must keep in mind that turbulence and multiple scattering are not two separate theories that occur one independently of the other, but both act together at the same time to corrupt a propagating pulse with noise. We may be able to minimize the multiple scattering effects by using more input power, but we may not be able to bring the corresponding turbulence effects (especially of the strong one) down as much. On the other hand, even a small increase from 1 W to 2W of input power will probably create more financial trouble than expected! In other words, more input power usually means more money must be spent for the design and 54
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implementation of costeffective components, which are required for accurate detection and extraction of an optical signal from noise. But, we may always be able to find the way to build very lowcost optical components, and still be able to use a higher input power. To put all these results together, it seems that as long as we avoid transmitting a signal through strong turbulence, dense cloud cover, haze, or fog, we should be able to obtain a reasonably good quality of PPM multi Gigabit laser communications. In addition, the very interesting case of pulse propagation in free space was discussed in Chapter 4, which shows us that, because of free space's nonturbulent nature, optical communications can have a bright future in helping us explore some other distant worlds. However, much more research will needs to take place to fully understand and utilize the big advantages of using laser optical pulses in transmitting information over the conventional electronic ways of transmission. 55
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Bibliography [1] Gagliardi, Robert M., and Karp, Sherman: Optical Communications, John Wiley & Sons, New York, 1976, pp 261277. [2] Prati, Giancarlo, and Gagliardi, Robert M.: Decoding with Stretched Pulses in Laser PPM Communications, IEEE Transactions on Communications, Vol. COM31, No. 9, September 1983. [3] Majumdar, Arun K., and Brown, William C.: Atmospheric Turbulence Effects on the Performance of MultiGigabit Downlink PPM Laser Communications, SPIE, val. 1218, FreeSpace Laser Communication Technologies II, 1990. [4] Abshire, James B.: Performance of OOK and LowOrder PPM Modulations in Optical Communications when Using APDBased Receivers, IEEE Transactions on Communications, Vol. COM32, No. 10, October 1984. [5] lshimaru, Akira: Wave Propagation and Scattering in Random Media, Vol. 2, Academic Press, 1978, pp 313314. [6] Hong, Shin Tsy; Sreenivasiah, 1.; lshimaru, Akira: Plane Wave Pulse Propagation Through Random Media, IEEE transactions on Antennas and Propagation, Vol. AP25, No. 6, November 1977. 56
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Appendix A Figures and graphs This section includes the graphs for the impulse response functions, G1, G2, and G, and the Bit Error Rate (BER) curves, for both turbulence and multiple scattering. The impulse response functions for turbulence and multiple scattering are shown first, followed by the Bit Error Rate curves. 5'7
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58 Figure #1 The Trapezoidal Approximation
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+'
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59 Figure #2 G1 Turbulence Function
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t=l ::> f:4 0.. ::g < xlOI5 4 5 G 1 TURBULENCE FUNCTION 3.5 ; ...... ,. ___________ 3 r....................... ... ! : 2.5 2 , 1.5 1 0.5 \. i I i i i j I I rrr++t I I I __ 1 I I I 0 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 TIME (sep) xl o15
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60 Figure #3 G2 Turbulence Function
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G2 TURBULENCE FUNCTION x1012 1.704 1.704 ...1.704 ,..... .................... .... l ....................................... 11.704 f4 ....l 1.703 < 1 .... .... : .... _________ 1 ... L ... ... r ... i '!''" _____ ,_, ; .................... .. .. .. .......... ) .. _, __ ,,,., I l r ... ....... 1.703 1.703 1.703 I i I ; i I i ttrt1.703 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 TIME (sec) x1 o14
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61 Figure #4 G Turbulence Function
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til ;:::J Et ...... 0.. < x1012 2.8 2.6 2.4 2.2 2 1.8 t1.6 1.4 1.2 0 I I I i I I I I I I I I 1 G TURBULENCE FUNCTION ( = Gl G2) I l L I I I I I i I I I l I I _. I I ...__ L_ [ I I I I 1 I I I I I I I I I I I I 2 3 4 5 6 TIME (sec) xtQ14
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62 Figure #5 G1 Multiple Scattering; Rain
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....... J i z < Q;: i .... ! . i. 0 i ............ i ... ! i l''' i .. ...... .................. ++! !! i I l I I I r.. rtt! i : _______ j t; ........... ..1 i I I I j +I ... t ......... i i l 3U!1.LI1dWV ;i i i '"'1 t ........ + co 0 1:'0 co 0 1.0 0 .:!' 0 M 0 N 0 0 0 0 I 0 II< 0 Q) Cf.l _. Eo
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63 Figure #6 G1 Multiple Scattering; Rain
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0 :::: E< ....... ....:I 04 ::a < xfOll 3 2.5 Gl MULTIPLE SCATTERING; RAIN; T1 = 1.8373 E12 i. ... .:. ........ !. .... 'l'"........ r.;. .................. ___________ 21tl ;... l .. t ....... i ........ _,, ...... ......... ........ ; ......... .... ................... .................... !'", _____ ""' .......................... ........ i 1.5 ; 1 ... ..! ... .. .. .. L. _____ .. ___________ j ___________ ;1I f_____ .l .... I .... t.. .... ______ ;_ __ j __________ T ____ +""[f0.5 ' t.. _____ ................ . i 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 TIME (sec) x1oto
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64 Figure #7 G1 Multiple Scattering; Dense Rain
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::> E4 :::3 p.. < x109 G1 MULTIPLE SCATTERING; DENSE RAIN; Tl 3.8738 E11 sec 10"""\8 '" l.... ____________ .J L! ; 6 I ! I l rrT1.......... L .. 1L! ! 4 ; 2 11i : 1 t .. r1......0 1 3 5 2 4 6 TIME (sec) x1 o10
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65 Figure #8 G1 Multiple Scattering; Heavy Rain
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tz:l E4 0.. < xf09 G1 MULTIPLE SCATTERING; HEAVY RAIN; T1 7.6440 E11 sec 6 !i.... i.. .ii...ii j ....... ti.. 5 ... : ,._,. ...... ........ l .. ............ . _; ____ t.......... .. !....... I I ' ...... . 1. .4 i .... f! i J .... !... .. i +1+j+3 i I T... I ++i1[+t++1 I""' I ' I ' l r i 2 1 !+ ! 1r+ 0 ; ; 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 O.B 0.9 1 TIME (sec) x109
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66 Figure #9 G1 Multiple Scattering; Light Fog
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;::J Et ....:I IJ.. ::g < 3 2.5 2 1.5 1 0.5 x109 G1 MULTIPLE SCATTERING; LIGHT FOG; T1 = 1.489 E10 sec "!. ____ J........ + + ++"+. I j i i i L+i 1 i i I : I i I I i I I : i I I I i I I : i lj: 1 I I l I i I I i 1rj; i i I I i J I I I I ! ! i I I i I !!;j_ 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 TIME (sec) x109
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67 Figure #10 G1 Multiple Scattering; Fog
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:;:J E< .....:1 p.. ::g < x10B 8 G1 MULTIPLE SCATTERING; FOG; Tl 5.9069 E10 sec 7 f1\.. ....... . ++.. """'''' +.. .. .. .... ...... ,..;._, ........ ___ .. .. 6 .. + ! .... 5 !!.... ]!i 4 ... ...i ............... ................. ). L................................ !L...... 3 : I i rr ttjl:... 2 i = i j i r1lJ r,..1 h( ______ : ______ J ___ J_ +__ _i ______ l_ ____ _L ______ L_ ____ 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 t TIME (sec) xtos
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68 Figure #11 G1 Multiple Scattering; Fog
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Q ::::;, f1 p.. ::g < x10a 2 1.8 1.6 1.4 G 1 MULTIPLE SCATTERING; FOG; T1 2.362 E9 sec .... ........... i.. .. ______ ,. _________ .......................... . ......... : ............... ___ ......... t. . .. ............. L ... ____ _______ ... ___ ................ ; ................. _________ ..... l....... =.. . . 1.2 ........... 1 ..... ... . ......... ... .. ...... ... _L. _________ .. __ l_ _____ _______ 1 ; .. ...... ....... ........ +....... .... ... L .................... L. .. ...... .. ) . ............ ...... r.. ... i .............. L .. ...... . ____ _ 0.8 . ..... ........................ ...... .................................... i ................. ..;. ......... ............. ............ + ......... ..... L .......................... ; ........................... 0.6 0.4 0.2 0 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 TIME (sec) xl07
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69 Figure #12 G1 Multiple Scattering; Cloud
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0 ::::> E< ....:l c.. ::;s
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70 Figure #13 G1 Multiple Scattering; Cloud
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t:il ;:J Et ...... ...:I p... < x107 2.5 2 G1 MULTIPLE SCATTERING; CLOUD; T1 2.0305 E8 sec i . .,,,.,,,,.,,,.,,., '''''''''"'""''''"1"""'" '' L ............................... i 1. 5 H\ ....... +.. . t.... 1....... +......................... 4 .............................. +.. .. +t__ .. _, ______ ,'! ........................... ,_ 1 \_,__ r1:rt 0. 5 \.1 .. f;!j(it[.. 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 TIME (sec) xl06
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7 1 Figure #14 G1 Multiple Scattering; Cloud
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C) Q,) Cll co I co 0 l.i.l C C\l II co 0 i>< co C\l .... i .. _]_ "'''!' ................. ...... J ..... .... .! ... f.. .. 1t t 0 .. i ......... {... ... t I co 3GD.LI1dWV I i i ... i..1. ; ; ... .. .. r +: .... L .... co 0') 0 co 0 l'0 co 0 ID 0 N 0 0 0 0 co I 0 i>< 0 Q,) Cll E
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72 Figure #15 G1 Multiple Scattering; Haze
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CJ Q,) Cll c::J I 0 C'j II CD 0 I>< co .::!' C\l ........ j ....... +: i i I r .... ; .......... i [ "!".. ... 0 .............. .. I ......... !._ .. .. l l [ ........... ..!. co 3Gil.LI1dWV ; .. ; t .. .... r t' _J .. __ I ! i i + .. i I"' .... r .. .. r.::!' C\l 0 co c::J 0 co 0 N 0 0 1:0 I 0 CJ Q,) Cll ..._ Eo
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73 Figure #16 G1 Multiple Scattering; Haze
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E< ...:I p.. ::s < xtos 16 G 1 MULTIPLE SCATTERING; HAZE; Tl 3.01 E8 sec 14 !.. .... ... _j. .. +... ... L .... .).L__J:12 10 8 6 !.. .. .. ;.. .. ... ............ __ .,, ......... ............. .................... .... ....... __ ,_j __________ i,. ... _______ ,, ............. ,,, ______ .,_, ........ ... .......... I .j ................................ t""'.................... .;. .. I l +.. Jtr; I i i i i !.. .. ____ ... rr ........ ................. .... ; .,i .. L.. ff1 : 1... .... _____ .s _______ +... t.. .. }........ ___ .. ........... .!._ ...... L ....... ................. __ I I I I 1 4 H. . \ : ., . +. . +... ____ _______ j_ _______________ j _______________ ...... L.L. ; ; i 2 __ J ..... .L .. ..!. __ J_ +________ [ __________________ L 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 TIME (sec) x1os
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74 Figure #17 BER; Strong Turbulence; 10ps
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..1]:1 _.. C' 0 ...:I BER; STRONG TURBULENCE; INPUT PULSE 10psec : 1 I ; : : I i 2 4 6 8 10 1 1 I I i : i : i : i I I rtr11 I I I i 1 _l_ ___ .,_._., _____ .,_l" 1"jtl[_ ______ +1 I I : '1 I I I I I ! I ________ _j _______ [ ________ l _____ j _____ ___ ____ j_____ __ j ____ i 1 . i I I i l I I i I I I 1 I i r l 1 : I I I I Jl _______ !t11tI I l i I .t I I 1 I I I : I : i I I i I I I I I I i I I I i I I I .. L.. r.. ... .i.............. !.. JL4lI I I I I . I I __________ L ________ .. .L _____ j ______________ L ___________ J _______ __t _______ ___ l _________ _L _________ J ________ ! i i ; I i i J I i ! I I 101 102 103 10" 105 106 107 108 109 toto 1011 TRANSMIT POWER (Watts)
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Figure #18 BER; Strong Turbulence; 20ps
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. w o:l .._... C' 0 BER; STRONG TURBULENCE; INPUT PULSE 20psec I I 2 I I L: 1 4 : f i I I . _ ________ .. ___ J _ _ ________ i ,i. ________ L_. _ ________ !. : I i i I i I I i I I I I I I li11 i '. 'I. I : I I I f+t:rr" .,i I i : ; I i I : I i i f I 1 1 I +;r___ _j _____ 6 8 10 12 to 102 103 105 106 109 1010 1011 107 108 to TRANSMIT POWER (Watts)
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76 Figure #19 BER; Strong Turbulence; SOps
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. o:= t::.:l C' 0 .....:l BER; STRONG TURBULENCE; INPUT PULSE 50psec i I l I I i l I ... t 1.i! ' 2 +!+14fl. j i i I 4 ttrI I I I I' ' ; ! I i i i l l : i I i I ,. : l i j I I i i T, I I I i !lt1 i I I i I I I i = 1 1 I : i I I ; 6 8 10 l2 100 101 104 105 109 1010 106 107 108 102 103 TRANSMIT POWER (Watts)
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77 Figure #20 BER; Weak Turbulence; 10ps
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ll:: r:z::J a:l t!J 0 ....:I BER; WEAK TURBULENCE; INPUT PULSE 10psec 2 4 6 8 10 1 i ; , i i i I i I I I i I ! ; I i I I' I l I I I ; ; I i I. i I I I' i I I I I I ! I ' .. I I I I I I I I I I 1 I i I ! i I l ! i : I I i : i I i I i I I I I I i I I I I I .. i i I i I I i I 'I i I I 11' I I I j f i ' I I I I I I I I : I I I I I I I I I I I I I ; J I I I I I I : I I I ! ; ; : I I I i ! I i ; : i I i : I I i I I I ; I I I i ; i I I l i ..... ; .. + t1{+ltftr+t i I : I I i ; i : I i I i i i I i I I ; I tlli tt to 100 to 102 TRANSMIT POWER (Watts)
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78 Figure #21 BER; Weak Turbulence; 20ps
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BER; WEAK TURBULENCE; INPUT PULSE 20psec .. ........ .. ... . 2 1.:. .......... .:....... f. .... 1 ; .., .. ,_ .. ________ ...,! ... 1 0 i : i i ; ' rtr. . i I I i i I I I i I i ! I I I I I ' ; ; ; 2 p:: Ill 4 ._ t!l 0 .....:l 6 8 I I I i ! i i i I _L ___ LlJ .. ir _______ L ____ l_ ___________ i ________ i ________ t_ ______ l_ __ i f l 10 ............... L .............................. j ........................ j ................. [. __ .......... j ............. [ ......... f ...... [. l 1l...... 1J..... J ... 12 _L__j____jl_Lj_J_0n__ l___L_j__j_j_l__lij 1 ot 100 101 TRANSMIT POWER (Watts)
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79 Figure #22 BER; Weak Turbulence; 50ps
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................................ .,.. .......... ,., _____ _________________ ,.,, ............. + .............................. .. ..... l .............. 1 11"1... .. t.! .... _____ I i J.i 1 I .. t ....... ___ !t........... f+.. .. _. _____ 1 ______ L, r' i .. .. .. i i.. ! .. I rr; r' + ....... i"' : i l...1 I i "t .. ... . ..i.. . ... ..................... ___ + .. .. ... ______ j ................ _______ .. _j ______ ... ... __ ... ______ l _________________ j ___________________ L_. __ !..... .... l. ____ __ ________ ... _.t .... .. J .. ____ ., i ....... _1._ i i ....... ":j"... .... 1_ .. ________ .. _____ .. j i.1 ....... .... .. J. .. .. + .......................... .. i i ... .. :! i .. ....... t r .. .... .. .. +1 ...................... ... _,, _____ ..... ......... ... .. L .. .. .. +.. ! ............. ,. ..... _____ ....... r.. +.. .. r.. ' .......... ..;. ................ _, __________ I f : .. .. .. t.. N I .::!' I I (H3S:)D01 co I 0 I 0 rn ....,J ....,J ctl lit: Q:: 0 lit: 0 0 c.. E< trl z < Q:: E<
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Figure #23 BER; Multiple Scattering; Dense Rain; 10ps 80
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BER; MULTIPLE SCATTERING; DENSE RAIN; INPUT PULSE lOpsec .. 0 2 Cll 4 _.. c.!) 0 6 __ _L __ __ j_ __ l__L_l_Ll_D l02 8 I ; i ! I I I ++J _J_LJJ_J_i_ ____ i _j __ __ I I I ; I I I I I : I I i I I I I ' l ' I i l : I i I I I i ; I I I I I ' : i i ! ! I I 11 l I I ; ! l I i I : I : I : : I I I I I I l I I [ I i ; I [ ; I .. :I +t;t+__ : ___ I l II I I I l I I I [ I I I I I : i I I I i I i I I ; I I I I I I I I' I I I I I I I I : : i 1 J I : ; i l I I I I i I I I I .. ;Trt I I I i I I I I I I I l ; I I ' l l i 1 [ l I i ; I I i I ; i i i I I I I i I I I l i I I I I I I I I i ! I I to 100 TRANSMIT POWER (Watts)
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Figure #24 BER; Multiple Scattering; Dense Rain; 20ps 81
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o:= ll:l '"' 0 BER; MULTIPLE SCATTERING; DENSE RAIN; INPUT PULSE 20psec 0 2 4 6 8 :.. ..l .............. .. i_____ J ___________ i_ .......... 1 ......... 1 i L1+ ; i ......... _L .... !.. l .... J! i I , i i +ti.i.l___ ;._ _____ J ___ I I ______ ,_ fLit+ l rt.... ;. ......... ;...... .,i .... ! i I j ; l. 1o :;_ 10 2 rl1tiltr1t\tl+i + J i I ! l i I 101 100 TRANSMIT POWER (Watts)
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Figure #25 BER; Multiple Scattering; Dense Rain; SOps 82
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i .... ... .................... _. ___ ,_i"................ ... t.. ............................ .......... r i i .... ................ + !+ji I l I r, i } 1 I i I I i 1 I I u ..  ..  ; :rrri I I i I i j i i rr1 I I I I i ... .. __ !_, ______ ___ __ _J_ _________ ,,. ___ ,_,,_, ................ .. .. .......... l ................... _.......... I I .......... i,, _____ ,, __ ________ ,,, ......... 1.lt. t1 i .. t.. r.. ; .. ....... ' I j I .. _ .. ____ ____ ,_ ..... _. __________ .. ...... ... i i ..!.'!I 1 i tfi1! ! ..... ......... ..... __ ......... .......... i I ... .L _____ Lj __ J ____ l 1 i +r (C138)!)01 0 0 ... I 0 Cll I oo 1 U) ...,J (lj r== et:= r== 0 c.. E::a U'J z < et:= E
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Figure #26 BER; Multiple Scattering; Heavy Rain; 10ps 83
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p:; I::Q .._ t:.!l 0 ....::I BER; MULTIPLE SCATTERING; HEAVY RAIN; INPUT PULSE lOpsec 2F =' ll '!! I! I! I! l I l!!! 'I 0 2 4 6 8 ++ _l_ .J _j __ .............. ; .. .. r..;....... 1 j j I i l J ....... ___ J __ ..... i ___ ; _u__ i ; ! 1L ... j .. i j LL. ____ j ___ _j ..... L.L 11L ..... LL! l I I I I 1 I ++li.L .. +l!1J.: i .. i l +jl+l : i : I i i l++i+lii .; ....... i fL i f .... L ..... l ......... j. I I I I ......... .......... l.. ____ ...... .... .. ,.i ....... l ........ J ..... 1 ... i .. 1 I I I i '+!+!+ + + 10 10 2 to too to TRANSMIT POWER (Watts)
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Figure #27 BER; Multiple Scattering; Heavy Rain; 20ps 84
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..III ._... c.!:! 0 BER; MULTIPLE SCATTERING; HEAVY RAIN; INPUT PULSE 20psec l I 0 f........ .l .. ]"'.. 2 4 6 8 i !... ... ,l .. ____ .. l{1l+r! I l i I : l i I i i I rf1i I I I i i I I I I I l I I I l i I I I lI I i I I I i I ; ; I I I I I I I I I I i i i i i I i i i I i ! i i i : i : j f1 ___ ; + l .J. I ____ L__j __ __ _jl__l __ t02 to too TRANSMIT POWER (Watts)
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Figure #28 BER; Multiple Scattering; Heavy Rain; SOps 85
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o::= r:z:l p:j _.. t.!J 0 BER; MULTIPLE SCATTERING; HEAVY RAIN; INPUT PULSE 50psec 0 .j ..... J. _____ lLlL ...... ___ i _______ .. L .. ! i : 2 4 6 i i i I I I I I l I . I I I I I I I I I i i : I I I I : : i ! I : : I I I I i : i I I ! .. .. ..... 1.  _______ _;_ L .>' I l I ; i i ! I I l i i I I II I i i I. I I i ; I i I I i I I i : I : i I ; i : l I i i : i I i I i I I i I I _i_. ________________ J. ........ _____ l _____ .L ______ J __ .l ___ J _____ l ____ ___________________________ L _______ L ______ j __ J. ___ J. __ L ______ J.. .. j ! j j [ ! I i i i J ; I 8 I ++++HJ, I i 1 r : I I I I ..... ....... t .... . !I;.. !!tlI I i to __ _L __ tot too TRANSMIT POWER (Watts)
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Figure #29 BER; Multiple Scattering; Light Fog 10ps
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rz:l r:Il ._ c.!:J 0 BER; MULTIPLE SCATTERING; LIGHT FOG; INPUT PULSE tOpsec 2 +1lL i 2 4 ____ __ I ! i : i I I i : ; I I ; ! i : I I ; f' .... . lr ..... r; 1;!r,.. i i i I I i i I 1 I j l i I ; I I I I ; I I : ; I I I i I i I I i I I I I I : ; I i I i I i ! i I I I i i t ; l I l I i I : I I I I I ; I I : i I I I I ; ; I i I : I : : 1 1 I : : I : I 1 : I I : 1 I Crr!ttiftr nnn:! 6 8 to 12 to 2 tot too 101 TRANSMIT POWER (Watts)
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Figure #30 BER; Multiple Scattering; Light Fog 20ps 87
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p:; p.;J a:l C!l 0 BER; MULTIPLE SCATTERING; LIGHT FOG; INPUT PULSE 20psec i 2 4 6 8 10 12 10::;2 _ r:lrt I ! I I i i I' I ! I _______ j_ ___ l_J ____ I _l_l__Ll _________ l. ________ j ____ _l_ i i i i I j i i I I I i i l I I I I I I "! I : I i i I i i I I I I I I I I I [ I I I I I I .___ [_ ___ .. ] _____ t1___ __ L 1_ l ________ j___ __ _j ___ 4 _J ____ J.I I I ! ; I I I I I I l i i i : I i I i i J : 1 I 1 i i I i : l j i i I i I II i I ' ' I I ' I I ___________________ L _________ L_ ___ J ___ .. 11 ___ j ____ 11 ____ [ ____ _____ _L ___ I I I I I I i I : I I i I l ! ; ! i ! I I : I i : . l 1 I i I I ' i +!1fl01 100 TRANSMIT POWER {Watts)
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Figure #31 BER; Multiple Scattering; Light Fog 50ps 88
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c.!l 0 HER; MULTIPLE SCATTERING; LIGHT FOG; INPUT PULSE = 50psec I : I : : : r r 1 l : I : : I I I I ; 1 I { ; I i I ' ; I i i I' II i I ; II I i I I ; I I I I I I : I I : I I I I i I I I II i tj1,..; 1I I I I I j l {; 1 I I 1 I ; I I I I I I I t++ ! ' I I I ! i I I I i I I I ' I I I I I I i I I I I I I I i I i l : I ; I i I I : I j l +ttj ; I I : ' I : ; I I I I ! I I. I' I I I I I __ ,._ ____ ,,,. I I I I I i I I I I I I I I' I ' L' I I I i 111 !I l1__ ]_ _:_ 11 u ll ; I I I I I 1 I , I _.J.__ I I I I I I I" I I I I ; I I I I I ' I l II I I' I I ' I I I I t' ! i .__1 _________ t I I I I I ,i I I I ;i i I I I I i I ; I I ! I i I I I i I I i I I I i I ; 'I I i I I i I I I I 1 ..:I I II ri I' li 1 i I I ,I 1 i 1: I ,r+ I I ' I ' li I II I I \ I! ,. I I I I , , I i I' : i I I I I I l i I I I I I i I II : i I I 2 4 6 B 10 t2 to2 to too TRANSMIT POWER (Watts)
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Appendix B The programs used for the evaluation of G1, G2, G, and the BER functions This section includes a sample of the programs used for the numerical evaluation of the impulse response functions, G1, G2, and G, and the Bit Error Rate (BER), for both turbulence and multiple scattering. 89
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Program #1 The Program for the G1 Turbulence Function 90
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#include #include #define PI 3.1415926536 #define KO 12.6e6 #define L 1000 #define C 3.0e8 #define CN 1e7 #define INCRE 0.1 #define MAXIMUM LIMIT 10.0 main() ( FILE *out; double temporary, c,k,l, exp4, t1, t2, p1, t, final, total; double normal, temp en, temp_ko, temp_l, temp; double time_nick; int n, flip; normal 0; out fopen( "turbg1.mat", 11W11 ); c 6.309573445e16; k .. 0.001444946; 1 2.51189e07; printf( "c: %g'\n11, c); print f ( 11 k : % q'\n 11 k l ; printf(11l: \g'\n", 1); t2 6.906743694e15; printf("t2: \e'\n", t2); for(t INCRE; t < MAXIMUM_LIMIT; t + (double) INCRE) ( ) total 0.0; for(n 0, flip 1 ( flip ... 1; n < 20;++n) p1 c (double) ( (2.0 (double) n) + 1.0) flip; ) temp pow((((double) 2.0 (double) n) + 1.0), (double) 2.0); exp4 (1.0) *temp* ((double) PII4) (double) t; total + (double) p1 exp((double) exp4); final t2 total; time nick t I t2 l; fprintf( out, 11%9 %g'\n11, time_nick, final); if(final > normal) ( normal final; ) printf(11%g %g'\n11, t, final 1 4.06748el5); printf(11Normal: \g'\n", normal); fclose (out ) ;
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Program #2 The Program for the G2 Turbulence Function 91
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#include #include #define DENOMINATOR 1.2056 #define DENOMINATOR2 1.4535 #define PI 3.14159 #define C 3.0e8 #define C2 9.0e16 #define CN 1e7 #define KO 12.6e6 #define L 1000 #define LO SO #define CONST 140392232.10 #define MAXIMUM LIMIT 1e14 #define INCREMENT 1e17 main() { FILE *out1 double en, lo, 1, g2, t2, t1 double exponential, total_constant, temp1 out fopen( "turbg2.mat", "w" )I en E pow((double) CN, (double) 1.0)1 lo pow((double) LO, (double) 5.016.0)1 1 pow( (double) L (double) 1.012.0) 1 total_constant ((double) CONST *en* lo 1)1 for(t INCREMENT! t < MAXIMUM_LIMITI t + INCREMENT) { t2 E pow((double) t, (double) 2.0)1 en pow((double) CN, (double) 2.0)1 lo pow((double) LO, (double) 10.016.0)1 1 E pow((double) L (double) 1.0)1 exponential (double) (1.0) (t2 C2 en lo 1) 1 (double) DENOMINATOR2 I* printf("Exponential: \g'n", exponential)! *I temp exp((double) exponential)! g2 E total constant temp; printf( "%e \e,n", t, g2); fprintf( out, "\e \e,n", t, g2); } fclose(out ll }
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Program #3 The Program for the G Turbulence Function 92
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#include #include #define C 3e8 #define CN 1e7 #define KO 12.6e6 #define L 1e3 #define LO 50 #define DT 1e17 main() { FILE *out; int i1 n; long int m; double x11X21x31x41X51X61x71x81x91x101x111con11con21con31c11c21c3; double totl 0. 01 t'otal.. 0. 01 t; out fopen( "turbg.mat"1 "w" ); x1 17.0 1 5.0; x2 .. 2.0 1 5.0; x32.7; x4 .. 5.0 1 6.0; x5 12.0 1 5.0; x6 .. x2; x7 11.0 I 5.0; xB2.0; x9 .. 5.0 I 3.0; xlO 1. 0; xll .. 2. 0; con1a 4.234lel6 pow( (double) CN1 xl) pow( (double) K01 x2) pow( (double) L1 x3) pow( (double) L01 x4) DT; con2 2.3687e8 pow( (double) CN1 x5) pow( (double) K01 x6) *pow( (double) L1 x7); con3 6.192e16 pow( (double) CN1 xB) pow( (double) L01 x9) *pow( (double) L1 x10); for( ta 1e171 m 1; (t (a 1e12) && (m <); t +., DT1 ++m { } for( ia 1; i < m; ++i { for( na 0; n < 20; ++n { c1a pow( (double) 1.01 (double) n ); c2 2.0 n + 1.0; c3 pow( (double) (t((i1 ) DT + ( DT 1 2.0)))1 (double) 2.0); tot1 + c1 c2 exp( (double) (c2 con2 (( i1.0 ) DT + ( DT I 2.0 )))) exp( (double) (con3 c3 )); } total +.. totl; totl"' 0. 0; } fprintf(out1"\e \e"n"1 t1 con1 total); totala 0.0; } fclose(out );
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93 Program #4 The Program for the G1 Dense Rain Multiple Scattering Function
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#include #include #define PI 3.1415926536 #define INCRE 4e13 #define MAXIMUM LIMIT 6e10 main() ( FILE *out; double temporary, c,k,l, exp4, t1, t2, p1, t, final, total; double temp, time_nick; int n, flip; out fopen( "msg1.mat", "w" ); t2 20.2747409e9; printf("t2: %e\n", t2); for(t INCRE; t < MAXIMUM_LIMIT; t + (double) INCRE) ( total 0.0; for(n 0, flip 1 ( flip 1; n < 20; ++n) p1 (double) ( (2.0 (double) n) + 1.0) flip; } temp pow((((double) 2.0 (double) n) + 1.0), (double) 2.0); exp4 (1.0) temp* ((double) PI/4) (double) t2 (double) t; total + (double) p1 exp((double) exp4); final t2 total; fprintf( out, "%g %g\n", t, final); printf("%g %g\n", t, final); } fclose(out ); }
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Program #5 The Program for the G1 Heavy Rain and Light Fog Multiple Scattering Function 94
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#include #include #define PI 3.1415926536 #define INCRE 4e13 #define MAXIMUM LIMIT lOe10 main() { FILE *out; double temporary, c,k,l, exp4, tl, t2, pl, t, final, total; double temp, time_nick; int n, flip; out fopen( "msgll.mat", "w" ); t2 10.2747409e9; print ( "t2: \e\n", t2); for(t INCRE; t < MAXIMUM_LIMIT; t + (double) INCRE) { total 0.0; for(n 0, flip 1 { flip ... 1; n < 20; ++n) pl E (double) ( (2.0 (double) n) + 1.0) flip; } temp pow((((double) 2.0 (double) n) + 1.0), (double) 2.0); exp4 c (1.0) *temp* ((double) PI/4) (double) t2 (double) t; total + (double) pl exp((double) exp4); final E t2 total; fprintf( out, "%9 \g\n", t, final); printf("%g \g\n", t, final); } fclose(out ); }
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Program #6 The Program for the BER Function Strong Turbulence, 1 Ops 95
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#include #include #define R ZERO (long double) 0 #define RONE (long double) 1 #define RTWO (long double) 2 #define R396 (long double) 79.2 #define R=7point5 (long double) 7.5 long double term1 (long double, int); long double term2 (int); long double term3 (int, int); long double term4 (int, int); long double factorial (int); main() { int a, k, 1, Rj, Rq; long double total, sum1, sum2, sum3, sum4, Pk, Ql, RRj, Pt; long double initial; for (a; a<; a++) { printf ("Please enter Pt: "); scanf ("%lf", &Pt); putchar ( '\n'); initial (long double) 185455 1 Pt; sum1 R ZERO; sum2 RZERO; sum3 RZERO; sum4 RZERO; for (kc11 k<; k++) { Pk term1 (Pt, k); for (la1; l
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bottom (long double) sqrt (((long double) 1K ONE) (long double) 0.02 + (long double) O.Ol)J return top / bottom; long double term3 (int 1, int Rj) { long double base; base m ((long double) lK ONE) (long double) 0.02 + (long double) 0.01; return (long double) (pow (case, (long double) Rj) /factorial (Rj)); long double term4 (int k, int Rg) { long double base; base m ((long double) kK ONE) (long double) 15 + K 7point5; return (long double) (pow (case, (long double) Rg) /factorial (Rq)); long double factorial (int i) { int k; long double product c (long double) 1; for k
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Program #7 The Program for the BER Function Strong Turbulence, 20ps 96
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#include #include #define K ZERO (long double) 0 #define KONE (long double) 1 #define KTWO (long double) 2 #define K396 (long double) 158.4 #define K=7point5 (long double) 7.5 long double terml (long double, int); long double term2 (int); long double term3 (int, int); long double term4 (int, int); long double factorial (int); main() { int a, k, 1, Rj, Rq; long double total, suml, sum2, sum3, sum4, Pk, Ql, RRj, Pt; long double initial; for (aal; a
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bottom (long double) sqrt (((long doublei 1K ONE) (long double) 0.02 + (long double) 0.01); return top 1 bottom; long double term3 (int 1, int Rj) { long double base; base a ((long double) 1K ONE) (long double) 0.02 + (long double) 0.01; return (long double) (pow (Ease, (long double) Rj) 1 factorial (Rj)); long double term4 (int k, int Rq) { long double base; base ((long double) kK ONE) (long double) 15 + K ?pointS; return (long double) (pow (Ease, (long double) Rq) 1 factorial (Rq)); long double factorial (int i) { int k; long double product (long double) 1; for (k2; k
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Program #8 The Program for the BER Function Strong Turbulence, SOps 97
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#include #include #define K ZERO (long double) 0 #define KONE (long double) l #define KTWO (long double) 2 #define K 396 (long double) 396 #define K=7point5 (long double) 7.5 long double terml (long double, int); long double term2 (int); long double term3 (int, int); long double term4 (int, int); long double factorial (int); main() ( int a, k, 1, Rj, Rq; long double total, suml, sum2, sum3, sum4, Pk, Ql, RRj, Pt; long double initial; for (am1; a<; a++) ( printf ("Please enter Pt: "); scanf ("\lf", &Pt); putchar ( '\.n'); initial (long double) 4636 1 Pt; suml K ZERO; sum2 KZERO; sum3 ., KZERO; sum4 KZERO; for (k; k<; k++) { Pk term1 (Pt, k); for (1; 1<; 1++) Ql term2 (1); for (Rj; Rj<; Rj++) RRj term3 (1, Rj); for (RqO; Rq
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bottom (long double) sqrt (((long double) 1K ONE) (long double) 0.02 + (long double) 0.01)i return top / bottom; long double term3 (int 1, int Rj) { long double base; base ((long double) 1K ONE) (long double) 0.02 +(long double) 0.01; return (long double) (pow (case, (long double) Rj) /factorial (Rj)); long double term4 (int k, int Rq) { long double base; base a ((long double) kK ONE) (long double) 15 + K 7point5; return (long double) (pow (case, (long double) Rq) 1 factorial (Rq)); long double factorial (int i) { int k; long double product (long double) 1; for (k; k
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Program #9 The Program for the BER Function Weak Turbulence, 1 Ops 98
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#include #include #define K ZERO (long double) 0 #define KONE (long double) 1 #define KTWO (long double) 2 #define K 396 (long double) 79.2 #define K:7point5 (long double) 7.5 long double term1 (long double, int); long double term2 (int); long double term3 (int, int); long double term4 (int, int); long double factorial (int); main () { int a, k, 1, Rj, Rq; long double total, suml, sum2, sum3, sum4, Pk, Ql, RRj, Pt; long double initial; for (aal; a<; a++) { print ("Please enter Pt: "); scanf ("%lf", &Pt); putchar ('\n'); initial (long double) 9201797; suml K ZERO; sum2 KZERO; sum3 KZERO; sum4 KZERO; for (k=lT k<; k++) { Pk .. term1 (Pt, k); for (lal; 1++) Ol .. term2 (1); ) ) for (Rj; Rj<; Rj++) RRj term3 (1, Rj); } for (RqmO; Rq> l l; long double term2 (int 1) { long double top, bottom;
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top (long double) exp ((long double) 0.515 (K TWO (long double) 1 K ONE)); bottom (long double) sqrt (((long double) 1K ONE) (long double) 0.02 + (long double) 0.01)1 return top 1 bottom; long double term3 (int 1, int Rj) { long double base; base m ((long double) lK ONE) (long double) 0.02 + (long double) 0.01; return (long double) (pow (case, (long double) Rj) 1 factorial (Rj)); long double term4 (int k, int Rq) { long double base; base((long double) kK ONE) (long double) 15 + K 7point5; return (long double) (pow (case, (long double) Rq) 1 factorial (Rq)); long double factorial (int i) { int k; long double product (long double) 1; for (km2; k
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Program #10 The Program for the BER Function Weak Turbulence, 20ps 99
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#include #include #define K ZERO (long double) 0 #define KONE (long double) 1 #define KTWO (long double) 2 #define K396 (long double) 158.4 #define x:7point5 (long double) 7.5 long double terml (long double, int); long double term2 (int); long double term3 (int, int); long double term4 (int, int); long double factorial (int); main () { int a, k, 1, Rj, Rq; long double total, suml, sum2, sum3, sum4, Pk, Ql, RRj, Pt; long double initial; for (acl; a
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top (long double) exp ((long double) 0.515 (R TWO* (long double) 1R ONE)); bottom (long double) sqrt (((long double) 1K ONE) (long double) 0.02 + (long double) O.OllT return top 1 bottom; long double term3 (int 1, int Rj) { long double base; base ((long double) 1K ONE) (long double) 0.02 + (long double) 0.01; return (long double) (pow (oase, (long double) Rj) 1 factorial (Rj)); long double term4 (int k, int Rq) { long double base; base m ((long double) kK ONE) (long double) 15 + K ?pointS; return (long double) (pow (oase, (long double) Rq) 1 factorial (Rq)); long double factorial (int i) { int k; long double product (long double) 1; for k
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Program #11 The Program for the BER Function Weak Turbulence, 50ps IO'Q
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#include #include #define K ZERO (long double) 0 #define KONE (long double) 1 #define KTWO (long double) 2 #define K396 (long double) 396 #define K=7point5 (long double) 7.5 long double term1 (long double, int); long double term2 (int); long double term3 (int, int); long double term4 (int, int); long double factorial (int); main () { int a, k, 1, Rj, Rq; long double total, sum1, sum2, sum3, sum4, Pk, Ql, RRj, Pt; long double initial; for (ac1; a<; a++) { print ("Please enter Pt: "); scan ("%1", &Pt); putchar ( '\n'); initial (long double) 1150225; sum1 ., K ZERO; sum2 ICZERO; sum3 ., KZERO; sum4 KZERO; for (klT k<; k++) { Pk term1 (Pt, k); for (1; 1<; 1++) Ql c term2 (1); } } for (Rja1; Rj<; Rj++) RRj term3 (1, Rj); } for Rq
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top (long double) exp ((long double) 0.515 (K TWO* (long double) lK ONE))J bottom m(long double) sgrt (((long double) 1K ONE) (long double) 0.02 + (long double) O.OllT return top 1 bottomJ long double term3 (int 1, int Rj) { long double base, base E ((long double) 1K ONE) (long double) 0.02 + (long double) O.OlJ return (long double) (pow (case, (long double) Rj) 1 factorial (Rj))J long double term4 (int k, int Rg) { long double baseJ base ((long double) kK ONE) (long double) lS + K 7point5J return (long double) (pow (oase, (long double) Rg) 1 factorial (Rq))J long double factorial (int i) { int kJ long double product (long double) lJ for (kJ k
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Program #12 The Program for the BER Function Multiple Scattering, Dense, Heavy Rain, Light Fog, 10ps, 20ps, 50ps 101
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#include #include #define R ZERO (long double) 0 #define RONE (long double) 1 #define RTWO (long double) 2 #define K396 (long double) 396 #define K=7point5 (long double) 7.5 long double term3 (int); long double term4 ( int); long double factorial (int); main () { int a, b, k, 1, Rj, Rq; long double total, final, sum3, sum4, Pk, Ql, Aj, RRj, Pt; long double Aq, tp, exponent, group, Sj, Sq, initial, rate, BER; printf ("Please enter Aq: "); scanf ("%lf", &Aq); putchar ( '\n'); printf ("Please enter tp: "); scanf ("%lf", &tp); putchar ( '\n'); for (ba1; b
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return (long double) (pow (base, (long double) Rj) 1 factorial (Rj)); } long double term4 (int Rq) { long double base, Sq; base (long double) Sq; return (long double) (pow (base, (long double) Rq) 1 factorial (Rq)); long double factorial (int i) { int k; long double product (long double) 1; for (k; k
