This article describes a statistical model to analyze the signal intensity received at the solid-state imaging (SSI) camera of the Galileo optical communications system from an Earth-based transmitter (GOPEX) demonstration. The analytical model assumes that the optical beam possesses a Gaussian profile and the communication channel has a log-normal scattering characteristic. The atmospheric-induced jitter is modelled as two independent zero mean Gaussian random variables. By modelling the system parameters as a set of independent and identically distributed (iid) random variables, the combined impact of uncertainties due to system parameters and the turbulent atmosphere can be approximated by a log-normal distributed signal intensity at the spacecraft. A Monte-Carlo software simulation package was also developed to compute the confidence interval probabilities for general optical beam profiles. Numerical results show that the approximation is valid for a wide range of operation scenarios

Kiesaleh, K.

Yan, T.-Y.

English

NASA Technical Reports Server

TDA ProgressReport42-111 November15, 1992
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z
A Statistical Model for Evaluating GOPEX Uplink
Performance
K. Kiesaleh and T.-Y, Yan
Communications Systems Research Section
This article describes a statistical model to analyze the signal intensity received at
the solid-state imaging (SSI) camera of the Galileo optical communications system
from an Earth-based transmitter (GOPEX) demonstration. The analytical model
assumes that the optical beam possesses a Gaussian profile and the communication
channel has a log-normal scattering characteristic. The atmospheric-induced jitter is
modelled as two independent zero mean Gaussian random variables. By modelling
the system parameters as a set of independent and identically distributed Oid)
random variables, the combined impact of uncertainties due to system parameters
and the turbulent atmosphere can be approximated by a log-normM distributed
signal intensity at the spacecraft. A Monte-Carlo software simulation package has
a/so been developed to compute the confdence interval probabilities for genera/
optical beam profiles. Numerical results show that the approximation is valid for a
wide range of operation scenarios.
I. Introduction
In this article, analytical expressions for the probabil-
ity density functions (pdf's) of the optical field intensity
and the number of observed photoelectrons per pixel of
Galileo's solid-state camera for a laser pulse interval in the
Galileo optical communication from an Earth-based trans-
mitter (GOPEX) experiment are derived. These pdf's play
a critical role in assessing the performance of this optical
uplink and provide a means of acquiring meaningful design
predicts. Furthermore, the aforementioned pdf's shed light
on the characteristics of the underlying processes involved
in this experiment and the impacts of various parameters
on the detection probability.
It is assumed that the main sources of disturbance
are the atmosphere-induced jitter, for the short-exposure
model at hand, and the log-normal large-particle channel
scattering, which proves to be significant for the operating
characteristics of this experiment. In general, there are a
number of parameters, described below, that directly im-
pact the observed signal intensity at the spacecraft. These
parameters are assumed to be independent and identically
325
https://ntrs.nasa.gov/search.jsp?R=19930009735 2020-03-17T07:20:34+00:00Z
distributed (iid) random variables. It is imperative to note
the importance of this observation, since as a result of
this assumption the signal intensity in the absence of at-
mospheric effects may be approximated by a log-normal
random variable as well. In this article, the log-normal
channel scattering is shown to be the dominant effect for
the operating characteristics of this experiment. Thus,
it is concluded that the combined impact of random sys-
tem parameters and turbulent atmosphere would yield a
log-normal distributed signal intensity at the spacecraft.
Before going any further, a description of the system pa-
rameters that directly impact the observed signal intensity
and which are needed to conduct the ensuing analytical
study of the GOPEX uplink is presented.
The optical field intensity at the spacecraft is a func-
tion of transmitter optics efficiency, r/t, receiver optics ef-
ficiency, r/v, atmospheric transmittance, r/at,n, transmitter
laser energy (in joules), Wt, distance from the spacecraft,
Z, and the predefined angular beam diameter, 0,. Since
the number of photoelectrons per pixel, observed over a
laser pulse duration, is of importance here, one must also
include the effective detector area (pixel area), Arec, and
the quantum efficiency of the photodetector, rk, in deter-
mining the overall statistical modelling. In the event of a
uniform beam pattern, and for Z >> 1, one can describe
the optical field intensity in joules/m 2 at the spacecraft as
4_
< I(x,y,t) >t -- _2.,ir/tqrrl_tm (1)
7rz_ Ua
where < I(x, y, t) >t describes the intensity in the plane
of observation at coordinates (x,y), and < . >t signifies
short-term intensity averaging over time, since the obser-
vation interval is limited to only 6 to 12 nsec, the laser
pulse duration. The number of observed photoelectrons in
the laser pulse duration is then
= < >, A..r/c (2)
These expressions will be modified in the next section to in-
clude the Gaussian beam profile and the log-normal large-
particle scattering effect.
II. Uplink Analysis: Gaussian Beam Profile
In this section, a transmitted beam with a Gaussian
profile is assumed. Therefore, for a short exposure model,
one may assume that the Gaussian beam profile maintains
its integrity in the plane of observation. In this case, the
optical field intensity at (x, y) in the plane of observation
may be approximated by
4 Wt secfe)
< I(x,y,t) >t _-, _r/tr/,'r/o " "
Iz [- 4( 2 + (3)
where now It accounts for the log-normal atmospheric
scattering effect, 7/o is the atmospheric loss at zenith,
and a is the spacecraft's zenith angle. This implies that
Bee(e)
r/atrn = r/O , which agrees with most experimental and
theoretical studies of atmospheric absorption. In this anal-
ysis, assume a negligible pointing error, and thus the field
intensity must be evaluated for (x,y) = (0,0). Further-
more, since the receiving area of the charged-coupled de-
vice (CCD) camera is significantly smaller than tile beam
footprint at the spacecraft location, the CCD camera may
be considered a point detector. Therefore, Eq. (3) at
(x,y) = (0, 0) accurately describes the field intensity for
all the detectors of the CCD camera.
The pdf of It is given by the well-known log-normal
density
1
f1,(i,) = V/_t it
; it > 0 (4)
where ¢rI is, in turn, given by [1]
o-? see
_o Hx C2n(h)h _ dh (5)
In the above equation, A is the wavelength of the laser
in meters, H is the height of the atmosphere, and Cn(h)
is the medium index of the refraction structure constant.
Major stumbling blocks in determining an accurate esti-
mate of cr_ are the dependency of Cn on various random
channel parameters and the unavailability of an accurate
model for Cn. However, in the literature a number of ap-
proximate models for this structure constant are available.
326
For a complete list, refer to [2]. Among those references,
the following expression, originally proposed by Hufnagel,
approximates this structure constant:
C_(h) = 8.2 x lO-S6V2hl° e -^/1°°°
+ 2.7 x 10-16e -h/is°° (6)
where h is the altitude in meters above sea level and V is
the rms wind velocity, averaged over 5 to 20 km altitude,
and is considered to be Gaussian distributed about a mean
of 27 m/sec with a standard deviation of 9 m/sec. This
model, however, is valid only for altitudes in excess of 5
km. For altitudes below 5 kin, an approximate expression
from Hufnagel and Stanely for the structure constant is
given by
C2(h) _-, 1.5 x lO-la/h (7)
For the parameters of the GOPEX project, and based
upon the above approximations, _q _< 0.9.
In a turbulent medium, a Gaussian beam, as described
above, experiences deflection in various directions due
to gaseous blobs and other turbulent particles flowing
through the path of the beam. For a short exposure model
(6- to 12-nsee observation interval), the atmospherically
induced deflections may accurately be modelled as atmo-
spherically induced pointing "jitter." This implies that x
and y in Eq. (3) may now be viewed as two independent
and zero mean Gaussian random variables with standard
deviation _Z. Thus, a represents the rms angular (half-
beam) jitter due to turbulence. When pointing error is
present, its standard deviation may be added directly to
(r. However, caution must be exercised in applying the
ensuing results to a model with a constant pointing error.
In this event, x and y are nonzero mean Gaussian random
variables. For this analysis, however, the constant point-
ing error is considered to be negligible, and thus one may
consider x and y as zero mean Gaussian random variables.
The standard deviation _r can be estimated for a given
index of refraction profile by using the following expression
[3]:
2[a = _ 1.46k 2see (8)
fo°°°° s/s]a/sx <c.'(0(12o000'j (s)
where k is the wave number. For the parameters of this
experiment, _ is estimated to be 9/_rad for near zenith and
14 prad for a zenith angle of 60 deg. Define the random
variable Ig as follows:
v) = exp .j (9)
It can readily be shown that for jointly Gaussian x and
y, 19 is a special case of beta distributed random variables
with pdf
11,(/9) -- fli(aZ-U; 0 < i9 -< 1 (10)
where 3 = 0_/8 az. In the following, fl ranges from 5 to 19
(0, = ll0prad and a m 9 to 14_urad) for all practical pur-
poses. It is interesting to note that for 3 = 1 this density
reduces to a uniform density, signifying the detrimental
impact of a turbulent medium.
The pdf of It =Iah can now be found. It represents
the combined impact of log-normal medium scattering and
atmospherically induced pointing jitter. This pdf is ex-
pressed as
e
,11,
where Q(x) = 1/(v/_) f.oo exp (-s2/2) ds. For a reason-
able range of system parameters, this density is depicted
in Figs. 1 through 4. It is important to note that at plays
a critical role in determining the behavior of this random
variable. Unfortunately, for zenith angles in the range of
40 to 55 deg, A = 0.532/_m, and arms wind velocity of
27 m/s, al proves to be in excess of 0.2, resulting in a pdf
that can accurately be approximated by a log-normal den-
sity function. This implies that for all practical purposes,
1
fl'(iO _ _i,
On(i,) - }xexp ; i,>_O (12)
where
327
xQ 1,2,],.-.  13/
 1'1
These expressions can be numerically evaluated for a de-
sired set of system parameters. The validity of approxi-
mating the characteristics of It with that of a log-normal
random variable is examined in a number of plots (see
Figs. 5 through 8). As noted, for a wide range of system
parameters the penalty for this approximation is comfort-
ably small.
One can now express the number of observed photo-
electrons in a pulse interval as
4W,
Ne,e,tron = --"_5-_ t/,o, It (15)
.z,-t%
5
where t/tot = I-L=x t/i, with t/1 : t/r, 172 = t/t, 1']3 = t/_ec(0),
t/4 = tk, and t/s = Aree. If one considers t/i's as lid random
variables, it is possible to approximate t/tot with a log-
normal random variable. Due to a lack of sufficient data
for accurate characterization of these parameters or for
considering the worst-case scenario, one may assume that
t/i is uniform over (t/_i., t/_.=) for all i. In this event,
Y.,o,(.o,)
where rn_
mT i m
[in(t/,o,) - m_] 2 }x exp _-a_ ;t/,o, > 0 (16)
_i_1 a2 with= )-']_=1 mr, and a, = 0,
I f _7.=
t/3a _ -- t/mi n dT_i" In(s) ds
1
t/p._ t/m,n
x [t/_a= ln(r/_n.=) _ t/,_in ln(t/_ni,)] _ 1 (17)
aT, - t/?._ - t/'di',
x [lnZ(t/_"=)-21n(_)]
t/_i"
x [ln2(t/_i'_)-21n(_'o)]-mo 2, (18)
Because the product of two log-normal random vari-
ables yields log-normal statistics
1
fN, o,(ntot) ._ _ntot
[in(n,o,) - r.u] _}x exp _ ; n,o, >_0 (19)
= _ +a_.where Ntot t/toth, mN = m T + mt, and a_ = a T
Finally, the total number of photoelectrons observed over
a pulse interval is given by
4W_ N
Net,ctron - rZ2----_] tot (20)
Note that an alternate means of computing o"N and m N
is to first compute the mean and mean-square of O,otI,,
which are given by
/3 _ (t/?"'_ + t/7'')
E{N,o,} - 1 +/3 2 (21)
i=1
E{NL, } = exp(at 2) /3
2+/_
(t/?_.)2 t/y°._t/?,.]
x II [(t/i"=Y+ + (221
3
i=1
and use the following expressions:
328
(r_v -- log [E{Nt_ot}/E2{N, ot} + 1] (23)
l 2
mN = log [E{N,a}] - -_o" N (24)
It is imperative to note that the accuracy of the above
model is highly sensitive to at and other system parame-
ters. In particular, caution must be exercised in assuming
a log-normal scattering model for values of at > 0.75, since
for such values of o'_ the variance of the normalized in-
tensity fluctuation due to scattering (i.e., e _,_ - 1) exceeds
0.75. In this event, log-normal statistics are no longer valid
in describing the characteristics of the scattering channel.
Instead, an exponential pdf must be employed to char-
acterize the scattering channel [3,4]. For the experiment
at hand, crl for V _< 27 m/sec and 0 _< 55 deg is less
than 0.9. This, therefore, makes the log-normal assump-
tion rather suspect for some portion of this experiment
where the spacecraft takes on large zenith angles, tIow-
ever, since the rms wind velocity is a random variable, and
al is highly sensitive to this parameter, one may approx-
imate the channel scattering effect with the log-normal
statistic.
III. Uplink Analysis: General Beam Profile
This section provides a brief description of a software
simulation package that was developed for the analysis of
the GOPEX uplink when the optical beam possesses a gen-
eral profile. To compute confidence interval probabilities,
one has to resort to a Monte Carlo simulation to include
the impacts of atmospherically induced jitter, log-normal
channel scattering, and uncertainties in other system pa-
rameters that are modelled as uniformly distributed ran-
dora variables (see above).
This package reads the beam matrix profile, recorded
at a known distance, and evaluates the beam profile at
the location of the spacecraft. The turbulent medium is,
once again, modelled as a zero mean Gaussian pointing
jitter in both x and y coordinates. This program provides
the mean, standard deviation, and the pdf of the num-
ber of observed photoelectrons in a pulse duration for a
specified interval. The confidence probability for a given
confidence interval is also computed. Since no theoreti-
cal results were available to test the validity of the results
of this program for a general beam profile, a Gaussian
beam was specified as the input beam profile. The system
parameters were set at the following values: Wt = 0.25
joules, ,_ = 0.532pm, 0s = ll0/Jrad, r/t = 0.705, r/_ = 0.24,
qc = 0.37, r/0 = 0.715 (atmospheric attenuation at zenith
and station altitude of 2.286 km [5]), A_,c = 0.01825 m 2,
Z = 0.6 × 106 km, V = 27 m/sec, and 0 = 55 deg. From
the above analytical results for the Gaussian beam profile,
the mean and standard deviation of the observed photo-
electrons were estimated at 111.991 × 103 and 126.08 × 103,
respectively. The large standard deviation is a clear indi-
cation of log-normal statistics due to dominant channel
scattering. The simulation for 10,000 samples yielded the
following mean and standard deviation: 110.834 x 103 and
119.535 × 103. These numbers improved to 111.950 × 103
and 124.361 × 103 for 50,000 samples. It is quite clear
that a reasonable accuracy may be achieved with a rela-
tively small number of samples. However, with the aid of
stratified or importance sampling, the required number of
samples may be drastically reduced. The above calculation
was repeated for a 60-deg elevation, i.e., 0 = 30 deg, with
the remaining parameters fixed at the above values. From
theoretical results, the mean and standard deviation were
found to be 142.025 × 10 3 and 97.166 × 103, respectively.
For 10,000 samples, simulation yielded 141.27 × 103 and
94.877 × 103 as the mean and standard deviation of the
observed photoelectrons, respectively. Once again, these
numbers improved to 142.095 × 103 and 96.536 × 103 for
50,000 samples, clearly indicating the consistency of the
simulation.
IV. Conclusion
This article described a statistical model to evaluate the
signal intensity received at the solid-state imaging camera
for the GOPEX demonstration. The model includes the
effect of log-normal channel scattering, atmospherically
induced jitter, and uncertain system parameters. It has
been shown that the resulting probability density function
can be well approximated by a log-normal distribution. A
Monte Carlo simulation software package has been devel-
oped to analyze the uplink performance when the opti-
cal beam profile is non-Gaussian. The confidence-interval
probability can be computed to analyze the GOPEX ex-
perimental data during demonstration.
329
References
[1]
[;]
[3]
[4]
[5]
V. I. Tatarskii, Wave Propagation in a Turbulent Medium, New York: McGraw
Hill, 1961.
W. L. Wolf and G. J. Zissis, Chapter 6 in The Infrared Handbook, Detroit: The
Infrared Information Analysis (IRIA) Center, Environmental Research Institute
of Michigan, 1989.
S. Karp, R. Gagliardi, S. E. Moran, and L. B. Stotts, Optical Channels, New
York: Plenum Press, pp. 178-180, 1988.
R. Fante, Electromagnetic Beam Propagation in Turbulent Medium, an Update,
Proc. IEEE, vol. 68, pp. 1424-1443, 1980.
K. S. Shaik, "Atmospheric Transmission Calculations for Optical Frequencies,"
Proceedings of the Thirteenth NASA Propagation Experiments Meeting (NAPEX
XIII), pp. 158-162, June 29-30, 1989.
330
3.0
Z
O
W- 2.5
o
z
D
u.. 2.0
z 1.5
iEl
-i 1.0
en
<
en
O 0.5
n.-
0-
0
0
I 1 I I I l I
_ =o.ol
7 -
¢ -
0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00
NORMALIZED INTENSITY
Fig. 1. Probability density function of the normalized intensity,
/_=3.
6
F-
0/
0
I I I
GO= 0,01
0.25 0.50 0.75 1.00 1.25 1.50 1.75
NORMALIZED INTENSITY
m
m
i
i
2.00
Fig. 3. Probability density function of the normalized Intensity,
_--7.
z
o
1
n-
Q_
I t I I I I I
(_s = 110 p.rad
- G = 18 wad -
i
'_1 = 0.0_
0.3
0.25 0.50 0.75 1,00 1.25 1.50 1.75 2.00
NORMALIZED INTENSITY
Fig. 2. Probability density function of the normalized Intensity,
/_ = 4.6.
z
o
z
DIJ_
O3
Z
ul
a
_J
o
Q_
10
0
0
I I f I I 1 I
0.53 0.
-- 0.3
0.25 0.50 0.75 .00 1.25
NORMALIZED INTENSITY
o2 = 0.01
q I
1.50 1.75 2.00
Fig. 4. Probability density function of the normalized Intensity,
_=10.
331
1.2 r I I I l
O 1.0 LOG-NORMAL
O
z
E)
u. 0.8
_ 0.6
_ 0,4 -
i 0.2
0,
0 0.5 1.0 1.5 2.0 2.5 3.0
NORMALIZED INTENSITY
Fig. 5. Probability density function of the normalized Intensity,
= 5 and o"l = 0.5, and Its approximated log-normal counter-
part.
z
0
I-
o
z
:3
LL
z
a
_J
<
0
if,
2.5 I t I I I
LOG-NORMAL
PART
2.0-
1.5
1.0
0.5
0 JJ I t _ I I
0 0.5 1.0 1.5 2.0 2,5 3.0
NORMALIZED INTENSITY
Fig. 7. Probability density function of the normalized Intensity,
/_ = 5 and <Tt = 0.2, and Its approximated log-normal counter-
part.
2.5
2.0
I I r r I
z
O
(.3
Z
'-, LOG-NORMAL
u_ _ _ COUNTERPART
1.5-
n
z
wQ
_ 1.0-
0.5-
o
n-
O _ i _ I /
0.5 1.0 1.5 2.0 2.5
NORMALIZED INTENSITY
Fig. 6. Probability density function of the normalized Intensity,
_ = 10 and o"l = 0.2, and its approximated log-normal counter-
part.
-- Z
o
3,0 i
1,4 I f I I 1
1.2 _
/ X -"- LOG-NORMAL
1.0
0.8
0.6
0.4
0.2
01 I
0 0.5 1.0 1.5 2.0 2.5
NORMALIZED INTENSITY
3.0
Fig. 8. Probability density function of the normalized Intensity,
= 5 and crl = 0.5, and its approximated log-normal counter-
part.
332


A statistical model for evaluating GOPEX uplink performance

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