fermi problem

please read the question and tell me if you can help. i will send a template that explain how your answer should be and the word document is the template
was.png

solved_example.pdf

Don't use plagiarized sources. Get Your Custom Essay on
fermi problem
Just from $13/Page
Order Essay

template.docx

Unformatted Attachment Preview

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 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
20180407022415template saved 5/9/2019 7:44:00 PM printed 5/9/2019 7:44:00 PM

Purchase answer to see full
attachment

GradeAcers
Calculate your paper price
Pages (550 words)
Approximate price: -

Why Work with Us

Top Quality and Well-Researched Papers

We always make sure that writers follow all your instructions precisely. You can choose your academic level: high school, college/university or professional, and we will assign a writer who has a respective degree.

Professional and Experienced Academic Writers

We have a team of professional writers with experience in academic and business writing. Many are native speakers and able to perform any task for which you need help.

Free Unlimited Revisions

If you think we missed something, send your order for a free revision. You have 10 days to submit the order for review after you have received the final document. You can do this yourself after logging into your personal account or by contacting our support.

Prompt Delivery and 100% Money-Back-Guarantee

All papers are always delivered on time. In case we need more time to master your paper, we may contact you regarding the deadline extension. In case you cannot provide us with more time, a 100% refund is guaranteed.

Original & Confidential

We use several writing tools checks to ensure that all documents you receive are free from plagiarism. Our editors carefully review all quotations in the text. We also promise maximum confidentiality in all of our services.

24/7 Customer Support

Our support agents are available 24 hours a day 7 days a week and committed to providing you with the best customer experience. Get in touch whenever you need any assistance.

Try it now!

Calculate the price of your order

Total price:
$0.00

How it works?

Follow these simple steps to get your paper done

Place your order

Fill in the order form and provide all details of your assignment.

Proceed with the payment

Choose the payment system that suits you most.

Receive the final file

Once your paper is ready, we will email it to you.

Our Services

No need to work on your paper at night. Sleep tight, we will cover your back. We offer all kinds of writing services.

Essays

Essay Writing Service

No matter what kind of academic paper you need and how urgent you need it, you are welcome to choose your academic level and the type of your paper at an affordable price. We take care of all your paper needs and give a 24/7 customer care support system.

Admissions

Admission Essays & Business Writing Help

An admission essay is an essay or other written statement by a candidate, often a potential student enrolling in a college, university, or graduate school. You can be rest assurred that through our service we will write the best admission essay for you.

Reviews

Editing Support

Our academic writers and editors make the necessary changes to your paper so that it is polished. We also format your document by correctly quoting the sources and creating reference lists in the formats APA, Harvard, MLA, Chicago / Turabian.

Reviews

Revision Support

If you think your paper could be improved, you can request a review. In this case, your paper will be checked by the writer or assigned to an editor. You can use this option as many times as you see fit. This is free because we want you to be completely satisfied with the service offered.

Order your essay today and save 15% with the discount code DISCOUNT15