# fermi problem

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Released to SC-159 WG 6
1 August 2007
Aerospace Electronic Systems
Research and Technology Center
7000 Columbia Gateway Drive
Columbia, MD 21046
MEMO Number NCCRTC-EFCL-0276
DATE: 1-August 2007
TO: R. J. Kelly, Kelly Systems Engineering
FROM: EFC LaBerge
SUBJECT: Random Emitters on a Flat Plate
1
INTRODUCTION
This note performs some statistical computations on the total RF interference at a single point
(the aircraft) from a finite number of discrete sources placed independently and uniformly on a
circle centered on the sub-aircraft point. With certain general assumptions, the form of the
expected value of the total interference is exactly the same as the result from a uniform power
flux density across the circle, under the assumption that the power flux is spatially white.
2
ANALYSIS
The geometry of the problem is shown in Figure 1. The aircraft is located at a position (0,0, h) in
Cartesian or cylindrical coordinates. The emitters are constrained to lie within a circle of radius
R centered at the sub-aircraft point. The k-th emitter is located at ( xk , yk ,0) in Cartesian
coordinates, or, equivalently, at (rk ,fk ,0) in cylindrical coordinates. The range from the k-th
emitter to the aircraft is
d k = xk2 + yk2 + h 2 = rk2 + h 2
(1.1)
The problem is to assess the total interference present at the aircraft antenna output. To start
with, assume that the antenna is an isotropic antenna with isotropic antenna gain
GI =
?2
4p
The incremental power from the k-th emitter measured at the aircraft is then
p ?2
pak = sk 2 2
(4p ) d k
(1.2)
(1.3)
where pak is the contribution at the aircraft of the k-th emitter and psk is the emitter effective
isotropic radiated power. For the remainder of this analysis, we will assume that all of the
emitters have the same emission power, that is, psk = ps , k = 1, 2,…N . We will assume that there
are N emitters.
Page: 1
EFCL0276 Random emitters on a flat plate for SC-159.doc saved 2/10/2010 8:10:00 AM printed 1/28/2012
11:36:00 AM
Released to SC-159 WG 6
1 August 2007
Aerospace Electronic Systems
Research and Technology Center
7000 Columbia Gateway Drive
Columbia, MD 21046
Aircraft (0,0, h)
y
Distance to a/c d k
fk
k -th emitter (xk , yk , h), or (rk ,fk ,0)
rk
x
Figure 1 Geometry of Random Emitter Problem
The random element of the problem is the position of the k-th emitter. Assume that the
N emitters are randomly and independently positioned across the surface of the circle, and the
spatial probability density function of the emitter position is uniform across the circle. That is,
the probability that an emitter will be in a small square of area dx × dy is constant for all x, y
within the circle, and zero for all x, y outside the circle
? K for x 2 + y 2 = R 2
p( x, y,)dxdy = ?
? 0 otherwise
(1.4)
By the fundamental law of probability
??
x2 + y 2 = R2
p ( x, y )dxdy =
??
Kdxdy = 1
(1.5)
x2 + y 2 = R2
Page: 2
EFCL0276 Random emitters on a flat plate for SC-159.doc saved 2/10/2010 8:10:00 AM printed 1/28/2012
11:36:00 AM
Released to SC-159 WG 6
1 August 2007
Aerospace Electronic Systems
Research and Technology Center
7000 Columbia Gateway Drive
Columbia, MD 21046
so that K =
1
.
p R2
Consider the probability that an emitter lies within a small circle of radius 0 = r = R. Then
Pr{rk = r} =
dxdy r 2
?? p R 2 = R 2
x2 + y 2 = r 2
(1.6)
Now, (1.6) is the cumulative distribution function of the random variable rk . The value is
positive and monotonically non-decreasing (in fact increasing) for all values of rk = xk2 + yk2 ,
and Pr{rk = R} = 1. So the probability density function of the r-coordinate in a cylindrical
coordinate system is
pr ( r ) =
d ? r 2 ? 2r
? ?=
dr ? R 2 ? R 2
(1.7)
The range to the aircraft d k , given by (1.1) is not a function of f , so pr (r ) provides sufficient
information to perform the necessary statistical analyses.
The (ensemble) expected value of the contribution of the k-th emitter at the aircraft is computed
by taking the expected value of (1.3) as a function of r, with respect to r.
ps ? 2
R
2
0
=
ps ? 2
( 4p ) ? r
2
ps ? 2
2
k
R 2 + h2
2
( 4p R ) h?
2
=
The dimensions are
ps ?
2
k
pr ( r )dr
1 ? 2r ?
?
? dr
+ h2 ? R 2 ?
R
0
=
1
( 4p ) ? d
Epak =
(1.8)
1
du , where u = r 2 + h 2 , and du = 2rdr
u
? R2 ?
log
?1 + 2 ? , where log(*) is the natural logarithm
2
h ?
( 4p R )
?
2
W-m 2
= W , as required.
m2
Page: 3
EFCL0276 Random emitters on a flat plate for SC-159.doc saved 2/10/2010 8:10:00 AM printed 1/28/2012
11:36:00 AM
Released to SC-159 WG 6
1 August 2007
Aerospace Electronic Systems
Research and Technology Center
7000 Columbia Gateway Drive
Columbia, MD 21046
The (ensemble) mean square value is
? p ?2 ?
Ep = ? s 2 ?
? ( 4p ) ?
?
?
2
2
ak
? p ?2 ?
=? s 2 ?
? ( 4p ) ?
?
?
2
2
? 1 ?
?0 ?? d k2 ?? pr (r )dr
R
2
? 1 ? ? 2r ?
?0 ?? rk2 + h2 ?? ?? R 2 ?? dr
R
? p ?2 ?
=? s 2 ?
? ( 4p ) R ?
?
?
2
R 2 + h2
?
h2
2
?1?
2
2
? ? du , where u = r + h , and du = 2rdr
?u?
(1.9)
2
? p ?2 ? ? 1
1 ?
=? s 2 ? ? 2 – 2
? ( 4p ) R ? ? h
R + h 2 ??
?
?
2
2
? p ? 2 ? ? R 2 + h2 – h2 ? ? p ? 2 ? ?
1
?
=? s 2 ? ? 2 2
=? s 2 ? ? 2 2
4 ?
? ( 4p ) R ? ? h R + h ? ? ( 4p ) ? ? h R + h 4 ??
?
?
?
?
The dimensions are
W 2 -m 4
= W 2 , as required
m4
The (ensemble) variance of the power contributed by a single emitter is thus
Var ( pak ) = Epak2 – ( Epak )
2
2
2
? p ?2 ? ?
? R2 ? ?
1
? ? ps ?
?
=? s 2 ? ? 2 2

log
? 1 + 2 ? ??
? ( 4p ) ? ? h R + h 4 ?? ? ( 4p R )2
h ??
?
?
?
?
? p ?2 ?
=? s 2 ?
?
?
? ( 4p ) ?
2
2
??
1
1 ? ? R2 ? ? ?
?
?? 2 2
– 4 ? log ?1 + 2 ? ?? ?
4 ?
h ?? ?
?? h R + h ? R ?? ?
?
?
2
(1.10)
2
? p ?2 ?
= ? s 2 ? f ( h, R )
? ( 4p ) ?
?
?
Page: 4
EFCL0276 Random emitters on a flat plate for SC-159.doc saved 2/10/2010 8:10:00 AM printed 1/28/2012
11:36:00 AM
Released to SC-159 WG 6
1 August 2007
Aerospace Electronic Systems
Research and Technology Center
7000 Columbia Gateway Drive
Columbia, MD 21046
W 2 -m 4
= W 2 , as required. The function f ( h, R) is a constant for fixed
m4
geometry, with units m -4 .
The dimensions are
The aggregate power at the aircraft from N identical emitters randomly placed is
ps ? 2
ps ? 2 N 1
=
?
2 2
(4p )2 k =1 d k2
k =1 (4p ) d k
N
N
pa = ? pak = ?
k =1
(1.11)
The positions are independent by assumption, and (ensemble) expected value is
N
Epa = ? Epak =
k =1
Nps ? 2
? R2 ?
log
?1 + 2 ?
2
h ?
( 4p R )
?
(1.12)
The (ensemble) variance of the aggregate power is
2
1 ? Np ? 2 ?
? N
?
Var ( pa ) = Var ? ? pak ? = NVar ( pak ) = × ? s 2 ? f (h, R )
N ?? ( 4p ) ??
? k =1
?
where the final equality uses the form N =
(1.13)
N2
.
N
It is important to note that the total power of the (discrete) emissions contained in the circle is
Nps .
Equations (1.12) and (1.13) state that the ensemble expected value of the power at the output of
an isotropic antenna at the aircraft position has some variation, and that this variation is caused by
the random positions of the N emitters.
Now consider the case where the emission power is uniformly distributed across the circle with a
power flux density of ? watts/m 2 . The total power emitted by a patch at cylindrical coordinates
(r ,f ,0) is ? rdrdf , and the differential contribution of this emission at the aircraft is
dpa =
? 2 ? rdrdf
(4p )2 ( r 2 + h 2 )
(1.14)
Page: 5
EFCL0276 Random emitters on a flat plate for SC-159.doc saved 2/10/2010 8:10:00 AM printed 1/28/2012
11:36:00 AM
Released to SC-159 WG 6
1 August 2007
Aerospace Electronic Systems
Research and Technology Center
7000 Columbia Gateway Drive
Columbia, MD 21046
2p R
The total power from the whole circle is pS =
? ? ? rdrdf = p R ? watts.
2
0 0
The total power received at the output of an isotropic antenna at the aircraft is obtained by
integrating (1.14) over the circle
2p R
pa =
? 2 ? rdrdf
? 2 ? 2p R rdrdf
=
?0 ?0 (4p )2 ( r 2 + h2 ) (4p )2 ?0 ?0 ( r 2 + h2 )
2p? 2 ?
rdr
2 ?
2
(4p ) 0 ( r + h 2 )
R
=
2p? 2 ?
=
2 × (4p )2
=
R 2 + h2
?
h2
(1.15)
du
u
? R2 ?
p? 2 R 2 ?
log
?1 + 2 ?
h ?
(4p R )2
?
If we constrain the total power emitted within the disk to a constant value in both the continuous
and discrete emitter cases, then Nps = p R 2 ? , and the results of (1.12) and (1.15) are identical.
This means that the (ensemble) average value of the discrete emitter case is equal to the
deterministic result of the continuous power flux density case.
The discrete case, however, has a finite variance given by (1.13), whereas the continuous case is a
constant dependent on the geometry. Furthermore, the Central Limit Theorem (CLT) assures that
the probability density function of the aggregate power received at the aircraft will approach a
Gaussian distribution with parameters µ = Epa and s 2 = Var ( pa ) given by (1.12) and (1.13),
respectively.
Under the constraint that the total power of the emissions within the circle is fixed at pS = Nps ,
we can rewrite the ensemble variance of the aggregate power given (1.13) as
2
1
Var ( pa ) =
N
? P ?2 ?
× ? T 2 ? f (h, R ), PT = Nps
? ( 4p ) ?
?
?
(1.16)
This variance is both positive and monotonically decreasing as N increases, and therefore
approaches zero in the limit, thus confirming a Weak Law of Large Numbers for this particular
analysis.
Page: 6
EFCL0276 Random emitters on a flat plate for SC-159.doc saved 2/10/2010 8:10:00 AM printed 1/28/2012
11:36:00 AM
Released to SC-159 WG 6
1 August 2007
Aerospace Electronic Systems
Research and Technology Center
7000 Columbia Gateway Drive
Columbia, MD 21046
Under the power constraint, we can then use the CLT result we can estimate the probability that
the ratio of the true aggregate power to the ensemble mean (i.e., the continuous result) is no more
than
? p
?
Pr ? a = 1 + e = C | e = 0 ?
? Epa
?
{
= Pr pa = Epa + d Var(pa )
}
(1.17)
? e Epa ?
= 1 – Q (d ) = 1 – Q ?
?
? Var(p ) ?
a ?
?
where Q( x) =
1
8
?e
2p
-a 2 / 2
da .
We can use (1.17) and (1.16) to solve for an N sufficient to
x
meet a desired C or e value. In such situations, if the number of discrete emitters N > Ne , then
the continuous approximation is valid with an error limited to 10log10 (C ) = 10log10 (1 + e )
decibels.
Page: 7
EFCL0276 Random emitters on a flat plate for SC-159.doc saved 2/10/2010 8:10:00 AM printed 1/28/2012
11:36:00 AM
MEMO Number
DATE:
TO:
FROM:
SUBJECT:
1
INTRODUCTION
[Why are your writing this tech note? What will be the final answer you want the reader to
remember? What is the context of the note (background, etc.)?
2
BACKGROUND
[Are there things the readers need to know before they can understand the document that they might
not know? If you dont need this part, you can delete the section and header.?
3
DISCUSSION
[Make the points you want to make. Provide support for your arguments. References to other
work(s) are usually shown here. There may be other sections for analysis or other mathy kind of
things.]
4
SUMMARY AND CONCLUSIONS
[Briefly summarize the key points of what you said in the discussion. If you are presenting new
data or an analysis, provide the reader with a conclusion.
References [put the references at the end. Here are examples]
[1]
W. C. Lindsey and M. Simon, Telecommunication Systems Engineering. Englewood Cliffs,
NJ: Prentice Hall, 1973.
[2]
M. K. Simon, S. M. Hinedi, and W. C. Lindsey, Digital Communication Techniques Signal
Design and Detection. Englewood Cliffs, NJ: Prentice-Hall, Inc., 1995.
[3]
J. M. Wozencraft and I. M. Jacobs, Principles of Communication Engineering. New York:
John Wiley & Sons, 1965.
[4]
J. G. Proakis and M. Salehi, Digital Communications 5th ed. New York: McGraw-Hill,
2008.
[5]
E. F. C. LaBerge, “Extension of North Atlantic Traffic Model to Determine Peak
Instantaneous Communication Load for Oceanic Airspace,” RTCA, Inc, Washington, DC,
SC-222/WP-026, July 8, 2009.
Page: 1
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