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Released to SC-159 WG 6

1 August 2007

Aerospace Electronic Systems

Navigation, Communications and Control

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

© Honeywell,International, Inc. 2007 All rights reserved

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

Navigation, Communications and Control

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

Circle radius R

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

© Honeywell,International, Inc. 2007 All rights reserved

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

Navigation, Communications and Control

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

© Honeywell,International, Inc. 2007 All rights reserved

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

Navigation, Communications and Control

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

© Honeywell,International, Inc. 2007 All rights reserved

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

Navigation, Communications and Control

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

© Honeywell,International, Inc. 2007 All rights reserved

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

Navigation, Communications and Control

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

© Honeywell,International, Inc. 2007 All rights reserved

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

Navigation, Communications and Control

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

© Honeywell,International, Inc. 2007 All rights reserved

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 dont 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.

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