The probability of exceeding interference power levels and the duration of interference at the Deep Space Network (DSN) antenna is calculated parametrically when the state vector of an Earth-orbiting satellite over the DSN station view area is not known. A conditional probability distribution function is derived, transformed, and then convolved with the interference signal uncertainties to yield the probability distribution of interference at any given instant during the orbiter's mission period. The analysis is applicable to orbiting satellites having circular orbits with known altitude and inclination angle

Ruggier, C. J.

English

NASA Technical Reports Server

TDA Progress Report 42-109 May 15, 19'.
N92-29386
Generalized Probability Model for Calculation of
Interference to the Deep Space Network Due to
Circularly Earth-Orbiting Satellites
C. J. Ruggier
TelecommunicationsSystems Section
The probability of exceeding interference power levels and the duration of in-
terference at the Deep Space Network (DSN) antenna is calculated parametrically
when the state vector of an Earth-orbiting satellite over the DSN station view area is
not known. A conditional probability distribution function is derived, transformed,
and then convolved with the interference signal uncertainties to yield the proba-
bility distribution of interference at any given instant during the orbiter's mission
period. The analysis is applicable to orbiting satellites having circular orbits with
known altitude and inclination angle.
I. Introduction
Knowing the orbital parameters and transmitting
power level of a potentially interfering satellite is prereq-
uisite to estimating tile received interference power level
incident at any DSN tracking antenna. However, there
are occasions when the satellite launch and orbit injection
data are uncertain or unavailable at the time of the analy-
sis. In this case, the satellite orbital position relative to the
ground station antenna beam cannot be predicted using
the usual deterministic analytical methods: A probabilis-
tic approach to the analysis is more practicable. Reason-
able probabilities of interference can be calculated if the
geometrical and signal parameter uncertainties are defined
for the orbiting satellite. The calculated probabilities can
then be useful in determining whether any significant in-
terference threats to the DSN will exist during the orbiting
satellite mission period.
The probability of interference is sought from the prob-
ability of the orbiting satellite being incident at some
point in the antenna beana, for some DSN tracking an-
tenna pointing angle, during the period when the satellite
is in view. This conditional probability fimction is de-
rived fi'om the geometrical relationships of the satellite's
orbit. From the conditional probability, the off-beam an-
tenna gain probability can be determined and then con-
volved with the interference signal transmission uncertain-
ties to yield the probability of received interference power.
Finally, the probability of received interference power is
scaled by the ratio of view area contained within the satel-
356
https://ntrs.nasa.gov/search.jsp?R=19920020143 2020-03-24T07:15:21+00:00Z
lite's ground track coverage area and the probability of
tracking time of the DSN antenna during a 24-hour period.
The probability of the DSN station exceeding a given inter-
ference power level (density) at any given time can then be
determined. In addition to the probability of interference,
the period of the interference signal and its probability of
occurrence is also considered, given that the interference
event has occurred.
III. Statistical Interference Model
The probability density of the satellite interference sig-
nal power Pr at tile DSN antenna is defined as
P(Pr) = [P(P,) * POo) * P(-L)] Po Pv (1)
where * denotes convolution and
II. Rationale for the Interference Model
P(p )t
Figure A-1 (Appendix A) shows the geometrical re-
lationships involved in the derivation of the probability P(ga) ,x
model. It is assumed that the interfering satellite trajec-
tory over the DSN station view area is constrained on the
surface of a spherical sector with radius (Re + h) and swept P(-L) a__
out by a given antenna off-beam angle, a. Re is tile mean
Earth radius, and h is the altitude of tile orbiting satellite Po_
above ground. Furthermore, the uncertainty of the satel-
lite's orbiting position is equiva]ent to tile satellite entering
tile view area randomly with uniformly distributed prob- Pv "
ability.
probability density function of the satellite
transmit power (dBm), where the satellite a.n-
tenna is assumed to have isotropic gain
probability density function of the DSN tracking
off-beam antenna gain (dBi)
probability density function of free space loss
(-dB)
probability of DSN antenna tracking tbne over
a 24-hour period
ratio of the view area contained within the total
satellite ground track area
h/2R(sin _)
satellite altitude, km
Re (Earth radius) +h, km
__asatellite inclination angle, 0 deg < ( < 90 deg
h&
R_
The probability model is derived from both the ran-
dora and periodic variables which interplay during tile en-
counter between the DSN station antenna beam and the
orbiting satellite. There exist two fundamental random
variables: the receiving antenna pointing angle and the
satellite's position relative to the antenna off-beam angle.
Tile antenna pointing angle variable can be further re-
solved into the elevation and azimuth components. Tile
azimuth angle is primarily a function of the Earth's diur- integral of P (p,):
nal rate motion, and hens been assumed to be nonrandom.
Ilowever, the elevation angle is considered to be random
and to have some probability density, relative to the space-
craft trajectory being tracked. Deterministic and periodic
variations are associated with the azimuth angle, the an-
gle of inclination of the Earth's axis with respect to the
ecliptic (23.44 deg), and the rapid changes in tile elevation
angle at the rise and set times. Thus, if averaged over long
periods of time, the periodic variations can be neglected
without affecting the overall derived probabilities.
The circular orbiting system is also assumed to be er-
godic, by virtue of the satellite's constant angular velocity
in a circular orbit. This property enables one to transform
the sate/rite's time-averaged periods within the view area
into probabilities; these probabilities can then be used to
scale the derived probability distribution function. Simi-
larly, DSN spacecraft tracking period statistics can be uti-
lized as additional probability scaling factors.
The probability of exceeding the satellite interDrence
signal power at the DSN antenna can be defined as the
T(p_) =
¢YD
f P(Pr) dp,
P_
The probabilities arise from the uncertainty of the satel-
lite's orbital position and transmission parameters. P(p_)
can be represented by a triangular probability density
function defined from the transmitter power minimum,
maximum, and most probable va.lues; P(g_,) is the off-
beam antenna gain probability density which is derived
in this analysis; and P(-L) is the uncertainty associated
with the free space loss and can be represented by a uni-
form probability density function, with space loss values
corresponding to line-of-sight range between the antenna
and the interfering satellite. The aggregate effect of the
system uncertainties can then be determined from the con-
volution of these probability densities. Note that when the
357
satellite transmit power variability is negligible, the prob-
ability density function can be replaced with a Dirac Delta
function, which sums the mean value of Pt with the ran-
dom variable values of P(9_).
The probability density P(ga), is tile derivative of the
probability distribution P (g_). P (g_) is transformed
using the International Radio Consultative Committee
(CCIR) antenna gain pattern Eq. (2); P(a) represents the
probability of the satellite being inside the antenna beam
cone, bounded by the off-axis angle c_ and is derived in
Appendix A.
The CCIR generalized antenna gain pattern is given as 1
g = Gmax, dBi (maximum antenna gain) )
0 < a < 1 deg
g =32-25log(a), dBi, 1 deg < _ < 48deg (2)
g = -10dBi, 48 deg _< c_ _< 180deg
The probability distribution of a is
90
P(_) = f P(_lc) P(c)d_ (3)
0
where P (a [ e) is the conditional probability distribution
of a, given c, which is the DSN station antenna elevation
angle.
P(c) is the probability density function of the DSN an-
tenna elevation angle. If this probability density function
is not known, it can be approximated with a triangular
probability density function, similar to that shown in Ap-
pendix B.
Figure A-1 in Appendix A shows the geometrical re-
lationships used to determine P (a I e). It is required
to know the probability of finding the satellite anywhere
inside the antenna beam bounded by the off-beam angle
and the portion of the antenna beam which is in view of
the satellite. The antenna beam cuts the satellite's or-
bital sphere at r1 and r2, for any given elevation angle c;
this sphere contains orbital surfaces bounded by an ellipse
projection whose eccentricity is a function of the antenna
pointing angle. Surface asymmetry is also affected by the
a The equations apply to systems with antenna gains >48 dBi. Due
to the very small probabilities associated with the &ngle a values
between 0 deg and 1 deg, only values equal to or greater than 1 deg
considered in the model.
variable off-beam angle _ (representative of the satellite
probable position inside the antenna beam cone). For tl
ease of calculation, first-order ellipses are assumed.
Furthermore, the portion of the antenna beam that
blocked by the ground and out of sight of the satellit
is a major consideration. Instead of directly calculatill
the antenna beam cone surface area blockage, the ratio c
the projected beam blockage to the total projected area
calculated and used as a scaling factor.
IV. Interference Period Probability
Given that the interference occurs, it becomes useful t,
determine the interference period and its associated prob
ability. The probability of t seconds of interference for
given interference threshold level, corresponding to som,
fixed off-beam angle ao, can be written as
where
t_o _ longest interference period at c_o
A
c,,w = most probable antenna elevation angle
ema_ = maximum antenna elevation angle
the period t is computed piecewise to also account for the
Earth-blocked segment of the antenna beam. For the un-
blocked view area,
s ( 2rlr2 cos(2ao) - ra2 - r22)t_o=_=cos -1 l+ 2R 2 .
R
x 77-,, ao < e _< 90 deg (5)
va
where s is the longest possible path length (km) traversed
by the satellite over the antenna beam cone bounded by
c_o, and V8 is the mean orbital velocity of the satellite
(km/sec). For the blocked view area,
s ( 2rlpcos(e+_o)-rl2 -- p 2)t_o V, cos -1 1 += -- = 2R_
R
xG, p = x/m- (6)
358
where the inverse cosine functions are in radians, and the
space loss is averaged over the region bounded by a fixed
antenna off-axis angle.
V. Application of Model
An example is now given of the model's application to
determine the probability of interference when the satel-
lite orbiting mission period and frequency band coincide
with those of a DSN mission. Consider a circularly Earth-
orbiting satellite and a DSN station with the following
parameters:
Earth radius, F% 6378 km
Satellite inclination angle, +75 deg
Satellite altitude above Earth, h 637.8 km
Satellite transmit power spectral
density 2 -26.9 dBm/tIz
Satellite orbital velocity 7.6 km/sec
Satellite/DSN station frequency
band 8.0 GHz
DSN antenna diameter 70 m
Po, the probability of tracking time over a 24-hour period,
is assumed to be 1.0; Pv, the ratio of the view area, is
calculated as 0.06. It is also assumed that the satellite
transmitter is switched on throughout the satellite's Earth-
orbiting mission period.
2Assume that satellite equivalent isotropic radiated power is at 1 W
(0 dBW) and the transmit data bandwidth is 500 kHz.
By applying these specified parameters to the model,
tile derived P(ga) can be convolved with P(p_) and P(-L)
to yield tile probability of exceeding a given power spectral
density level. In this example P(Pt) is treated _s a constant
and can be convolved as a Dirac Delta function. P(-L) is
derived in Appendix B.
Figures 1 and 2 show tile probability of exceeding the
total received power and power spectral density, respec-
tively, for the above orbital and transmission parameters.
From Fig. 2, it can be noted that tile probability of
exceeding the CCIR recommended interference threshold
level of -190 dBm/Hz is 4.5 x l0 -6 or 0.00045 percent.
The interference periods and the probabilities of oc-
currence of those periods for the most probable antenna
elevation angles are given in Table 1.
VI. Conclusion
It has been shown that when the instantaneous posi-
tion of an Earth-orbiting satellite is unknown, one can de-
termine the probability of interference to a DSN tracking
station from the conditional probability of the satellite's
signM being within the antenna beam cone, given any el-
evation angle. The analytical derivation of the interfer-
ence probability function is made possible entirely from
the known orbital geometrical properties and transmission
parameters of the orbiting satellite.
The probability model provides an early first-order as-
sessment of the potential interference to the DSN, and en-
ables one to determine whether more comprehensive anal-
ysis will be necessary.
359
Table 1. Interference periods and probabilities.
Most probable elevation Inted'erence period, Probability,
angle, deg sec percent
15 102.8 0.16
25 65.2 0.42
35 44.1 0.88
45 32.3 1.50
55 25.5 2.40
65 21.5 3.30
75 19.3 4.30
36O
<_
o3
co
O
co
[]
<
(.9
Z
w
LLI
C.)
<
[]
O
rr"
O-
t 0 -3
10 -4
10--5
10 -6
f ii i i i f I
-t 70 -160 -150 -140 -130 -t 20
RECEIVED TOTAL POWER, dBm
Fig. 1. Probabillty of exceeding the total received power.
I0 -3
-220
LLLLLLLLLI]_J_I ,L
--210 --200 --190 --180
RECEIVED POWER SPECTRAL DENSITY, dBm/Hz
-170
Fig. 2. Probability of exceeding the power spectral density.
36t
Appendix A
Calculation of P (ga)
The projected surface area is given by [1]
1
A = _ [s'/- c(/- h')]
where
' - h"_
s' = 2/cos -1 \ r' /
arc length of projected segment (not shown)
/ = ,1 sin (_)
radius of projected segment
c = [4(2h'/- h?)] 1/2
chord of projected segment (not shown)
h' -- rl sin (e + a)
co_ (_)
for 0 < _ <a,
1 _< a _< 48 deg
(A-l)
(A-2)
(A-3)
(A-4)
(A-5)
Case 1.
blockage).
In this case, ¢ < a (occurrence of view area
The conditional probability distribution of the satellite
being in the antenna beam cone bounded by the off-beam
angle a for a given elevation angle e can be approximated
by expressions for the blocked and unblocked view area.
For the blocked view area, the conditional probability dis-
tribution P (c_ ] e) is given as
S(a, e)Av (a, e) I sec 6 1
P (_ I ¢) _ _R 2[1 - I_/R] AT
fore _< a, 0 < e _< a,
1 < a < 48deg (A-6)
362
where AT = tab is tile total area of tile ellipse projecte
from the orbital surface area bounded by the angle c_; a an
b are computed from Eqs. (A-12) and (A-13), respective],
R_P_ + h
and the visible projected area Ap is determined as
[ tan(0]Ap = rx2 sin 2 (a) cos-' tan (o)J
[2,._ sin (_.) sin (_+ _)t cos (_)
,_ sin 2 (c + o)]
cos2(_) l
r, [ sin (_ +__)]x T sin (a) cos' (¢) J'
1/2
for 0 __<¢ < a, 1 < a < 48deg (A-7)
Note: The inverse sine function is in radians.
I 2xy I (A-8)Isec (_)1= v2 +;_- z2
where
and
• = (d +d- 2,1,2cos 'n
y= [2r12(1 -- cos 2or)] 112
z = r2 -- "1
"1 =- R_sin (e+a)
+ [n__ R_cos _(,+_)],n (A-9)
,_ =- Re sin (e-a)
+ (R _ - Re 2cos _(_ - a)) 11_ (A-IO)
Thesurfaceareacoveredby tile orbitingsatellitefor a
givena and e is the surface area of the sphere (satellite
orbital space) cut by the ellipse (antenna beam).
The surface integral S(a,e) is given as
S(a, ¢) = 27rR2
_,R/JR2_
o
where
and
M w
b2(_ ' _) ] 1/2
M(a,e) sin 2 0 + 1J
b2 _ a2
a 2
dO
(A-11)
r_ _ (_,_)
ra (a, e) r2(a, e) cos 2a (A-12)
2
b2(a,e) = -_ (a,e) (1 - cos 2a) (A-13)
By expanding S(a, e) as a power series [2],
j b2/R 2S(a,e) = 2R 2 (Msin 2 0+1)
0
dO
2
R 2 (b2/R2) 2
+ -2- (M sin 2 0 + 1) 2
dO
2
R _ (b_/R2) 3
+ T (M sin 2 0 + 1) 3
0
dO + ... (A-14)
Due to tile fast convergence of the series, two terms will
adequately approximate S(a, e) and can be further simpli-
fied as
S(a,Q _ _ l+ _ 8(l+M)
(A-15)
Case 2. In this case, e > a.
For the unblocked view area, the conditional probability
distribution is given as
P(_ l0 = S(_,e) , for e > o,,
o_ < e _.< 90deg, 1 _< ct _< 48deg
(A-16)
where S(a,Q is given in Eq. (A-15). Then, from Eq. (3),
9O
P(o) = f P(_ 10 v(0 dc
O
the probability density P(a) is obtained by differentiating
P(o0; hence,
d
P(c_) = _ [P(_)] (A-17)
and P(a) is tranformed into P(ga) by using Eq. (2).
363
SATELLITE r ..._'_ _
ORBITAL _- 1 / I __ _UNBLOCKED
SPHERE -_ ___i__iro _,! t _ SEGMENT
\ /_ hl__ _ _ -4
EARTH'S _ P ----'_ ' I |
X/ / Is" " / .J_--_---_'_4x_J _, BLOCKED
_ SEGMENT
rl, r2 A= INTERFERENCE CONE Re &__RADIUS OF EARTH
BOUNDS
c_ A__ANTENNA OFF-BEAM ANGLE
r 0 =_ ANTENNA BORESIGHT
¢ &_. ANTENNA ELEVATION ANGLE
p tl SATELLITE ALTITUDE
ABOVE GROUND S &__SURFACE AREA
(MAXIMUM DISTANCE) BOUNDED BY (x
h &= SATELLITE ALTITUDE 5 A PROJECTION ANGLE
ABOVE GROUND
(MINIMUM DISTANCE)
Fig. A-1. Orbital geometry of an Earth-orblting satellite over a DSN station.
364
Appendix B
Consideration of P(Pt) and P(L)
In the cases where the uncertainty of the satellite trans-
mit power becomes a significant part of the analysis, a
triangular probability density function can be used. The
satellite transmit power probability density can be repre-
sented as [3]
2(p_- a)
(b- a)(c- a) if a_<pt <_ c
P(P') = 2(b - p,)
(b - a)(b- c) if c<pt <_ b
where
A
a = minimum power
b _ maximum power
A
c = most probable power
The variability of free space loss due to the continu-
ous changes in the satellite's range to the DSN tracking
antenna can introduce an additional uncertainty in the re-
ceived interference power. This uncertainty becomes very
significant for lower Earth-orbiting satellites where fluctu-
ations in space loss can be several orders of magnitude.
The probability density function of the free space loss
uncertainty P(L) can be modeled as a uniform probability
density:
where
P(L) = 1
_-a ifa<_L <b
0 otherwise
a _ minimum loss = 20 loglo (47r/,k)+20 loglo
h, dB
b a__maximum loss = 20 log_0 (4r/A)
+ 10 logl0(R 2 - R_), dB
)_ _ wavelength of interference signal, meters
h a_ satellite altitude above ground, meters
P_ _ Earth's radius, meters
R _a_ Re + h, meters
365
References
[1] C. E. Jordan, Reference Data for Engineers: Radio Electronics, Computer, and
Communications, 7th ed., 3rd printing, Indianapolis, Indiana: IIoward W. Sams
and Co., 1986.
[2] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, New
York: Academic Press, 1980.
[3] A. M. Law and D. Kelton, Simulation Modeling and Analysis, New York:
McGraw-ttill, 1982.
366


Generalized probability model for calculation of interference to the Deep Space Network due to circularly Earth-orbiting satellites

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