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A statistical model of atmospheric noise ..

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Title:
A statistical model of atmospheric noise ..
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Barney, John Marshall, 1924-
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English
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86 leaves : ; 28 cm.

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Subjects / Keywords:
Atmospherics ( jstor )
Density distributions ( jstor )
Eigenfunctions ( jstor )
Fourier Bessel transformations ( jstor )
Fourier transformations ( jstor )
Mathematical variables ( jstor )
Mellin transforms ( jstor )
Probabilities ( jstor )
Statistical models ( jstor )
Time functions ( jstor )
Dissertations, Academic -- Electrical Engineering -- UF
Electrical Engineering thesis Ph. D
Radio -- Interference ( lcsh )
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bibliography ( marcgt )
non-fiction ( marcgt )

Notes

Thesis:
Dissertation (Ph. D.) - University of Florida, 1954.
Bibliography:
Bibliography: leaves 83-85.
General Note:
Manuscript copy.
General Note:
Vita.

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Copyright [name of dissertation author]. Permission granted to the University of Florida to digitize, archive and distribute this item for non-profit research and educational purposes. Any reuse of this item in excess of fair use or other copyright exemptions requires permission of the copyright holder.
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Full Text











A STATISTICAL MODEL

OF

ATMOSPHERIC NOISE











By
JOHN MARSHALL BARNEY










A DISSERTATION PRESENTED TO THE GRADUATE COUNCIL OF
THE UNIVERSITY OF FLORIDA
IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE
DEGREE OF DOCTOR OF PHILOSOPHY










UNIVERSITY OF FLORIDA
August, 1954


















ACtKNOWLEDG1AWJT


The author acknowledges with sincere gratitude the continuous

guidance and encouragement given him by Dr. T. S. George, eM office

member of the Graduate Committee, University of Florida, and also

the many valuable discussions and suggestions made available to him

by the other members of the Committee.

He acknowledges also his indebtedness to Professor A. W. Sullivan

and Messrs. S. P. Hersperger, R. F. Brown, and J. D. Wells of the

Electrical engineering Department of the College of Engineering whose

measurements of the actual characteristics of atmospheric noise made

it possible to check his theoretical results.

He also takes this opportunity to express to his wife his appre-

ciation and gratitude for her patient forbearance as a dissertation

widow during the long period spent in the preparation of this paper.


























LIST OF TABLES


Table Page

1. Phenomena to Which the Poisson Model Applies 29


























LIST OF ILLUSTRATIONS


Figure Page

1. Comparison between Actual Atmospheric
Noise Distribution and Discrete Model 41

2. Change of Iamits of Integration .... ... 70

















TABLE OF CONTENTS
Page

LIST OF TABLES ......... iii

LIST OF ILLUSTRATIONS . iv

Chapter

I. STATISTICAL NOISE MDELS INTRODUCTION 1

II. METHOD OF SOLUTION 5

III. DERIVATION OF GENERAL DENSITY FUNCTIONS 7

Probability Theory and Random Processes
Non-Stationary Random Processes General Model
Stationary Random Processes General Model
Discussion of Expected Results
Example

IV. ATMOSPHERIC NOISE DEL ......... 32

Exponential Time Pulse General Model
Exponential Time Pulse Particular Results
Exponential Time Pulse A Discrete Solution
Triangular Time Pulse General Model

V. POIER SERIES APPROXIMATION AS A SOLUTION
OF GENERAL DENSITY FUNCTION 46

General Expression
Solution for Triangular Time Pulse

VI. SOLUTION BY METHOD OF INTEGRAL TRANSFORMS 52

Theory of Integral Transforms
Relations Between Certain Transforms and
Fourier Transforms
Applications to Atmospheric Noise Model
Definitions of Transforms











TABLE OF CONTENTS Continued


Chapter

VII. CONCLUSIONS ...

APPENDIX I. EXAMPLE OF USE OF FORLJE .

APPENDIX II. POWER SERIES APPROXIMATION ..

BIBLIOGRAPHY ... .

BIOGRAPHY ................... .


Page

64

65

75

83

85
















CHAPTER I


STATISTICAL NOISE MODELS INTRODUCTION


Lightning discharges in the atmosphere produce the major por-

tion of radio interference upon most communications systems in use

over the frequency range from ten kilocycles per second to ten

megacycles per second at the present time. Most radio listeners

have noted "static* occurring in their radios while a thunderstorm

was in progress in the surrounding area. In the last few years it

has been recognized that this "static" is an important limitation

to reliable communications, and studies are being made to further

our understanding of this phenomenon. As the lightning flashes

which produce the sstatic" are random in nature, statistical methods

are used studying this phenomenon.

This paper deals with the development of a mathematical model

of certain of these statistical measures. The model can be used as

a guide to ascertain the effects of atmospheric noise upon various

types of ooamnications systems. The present discussion is limited

to the output envelope of a normal amplitude-modulated receiver when

the input waveform consists of pulses of much shorter duration than

the reciprocal of the intermediate-frequency bandwidth of the

receiver.


-1-







-2-


A review of the work done in the field of statistical studies

of noise voltages and currents will prove helpful before considering

the immdeiate problem in more detail. One of the earliest works was

by Schottky() in 1918, in which he resented the results of studies

of spontaneous current fluctuations in conductors. Shortly thereafter

Hull and Williams discussed the dependence of noise voltage upon

current in certain nonmetallic resistors. Later Nyquist3 published

a paper concerning the thermal agitation of electrical charge in

conductors. Several other writers during this period noted the so

called "noise* currents in various electrical circuits and work was

begun to identify the types of variations being observed. (The

currents were called "noise% currents as they were random in nature

and could not be described explicitly by a particular type of time

function).

These studies led to the identification of the current or voltage

variations with the "Gaussian" or "Nomal" probability law of sta-

tistics. A more common term for this type of noise at the present

time is "white" noise, the term *white" being derived from the flat

energy spectrum of this type of noise which resembles that of white

light. It is also oomonly referred to as "fluctuation" noise.

Some of the authors who wrote during the later thirties and early

forties verifying this conclusion were Bell Landon Janskys Harris,
8 9
Thompson and North, and Williams? As this type of noise could be

described only in terms of statistical parameters such as average,


* Superscript numerals refer to Bibliography.








-3-


root-mean-square, first amplitude probability density function, etc.,

meters were designed to measure these paramrnala The first meter

designed to measure the first amplitude probability distribution

function was built by Peterson1 and distributions of amplitude as a

function of time for fluctuation noise were measured by Landon1 sev-

eral years later.

In 1944 8. O. Rice published the first of a series of papers in

the Bell System Technical Journal entitled "The Mathematical Analysis

of Random Noise,12'13'14 This series was the first comprehensive

study of the overall problem of describing fluctuation noise and was

destined to become a classicw in the field. Much of the ensuing

theory of random noise and other types of noise originating in elec-

trical circuits is based on the results and methods presented in

these papers. A short list of the wide number of subjects with which

these papers deal is given below:

1. Fourier series representation of noise current)

2. Probability distribution of noise current;

3. Correlation functions of noise current;

4. Power spectrum of noise current;

5. Distribution of envelope of noise current;

6. Expected number of maxima per second;

7. Expected number of zero crossings per second;

8. Characteristic functions of the noise current;

9. Amount of noise in the output of a nonlinear detector
when either a noise voltage ar a noise voltage plus a
signal is fed into the input of such a device.








-4-

Many other analytical approaches to different noise problems

appeared about this time, some of the more outstanding contributions

being papers by Middletonl5'16 Bennett7 Nortb Van Viec9 ac20

Seigert, Bagasaine2 and Hamburger.3 A comprehensive review of the

many studies of atmospheric noise levels over the world was compiled

by Burgess and Thomas24 during this period.















CHAPTER II


METHOD OF SOLUTION


The particular problem with which we are concerned is the devel-

opment of the analytical expressions for the amplitude density func-

tions for atmospheric noise, making use of some particular type of

time function. The fundamental approach used has been to start with

the most general case, a nonstationary random process, and develop

the density functions for such a process. The development used here

follows that of Rice12,13'1 and Middleton25 although carried out in

more detail. The restraint introduced by a stationary process results

in a simplification of the general equations. A further restriction

is made concerning the independence of events and the expressions

to be used for the atmospheric noise model evolved. The time func-

tion of the individual noise waves appears explicitly in these ex-

pressions but the character of this time function has not yet been

defined.

As a particular case an exponential and a triangular time func-

tion are chosen and the resulting expressions evaluated wherever

possible. These expressions can be evaluated only under certain lim-

ited conditions, so that other methods must be found to solve these

equations.


-5-








-6-


An approximate solution is obtained by means of a power series

expansion and the validity of the results is discussed.

Finally a different approach to the solution of the resulting

equations is suggested and a method of applying this procedure is

outlined*















CHAPTER III


DERIVATION OF GENERAL DENSITY FUNCTIONS


Probability Theory and tando Protesses

To facilitate the understanding of the mathematical tools used

in the development of this statistical model a brief discussion of

som of the fundamental properties of random variables will be given.

A set of functions may be specified by giving one or more prop-

erties which members of the set always possess and which other func-

tions do not* If a measure is given which evaluates the probability

that a member of the set will have a prescribed configuration, the

set is called an ensemble. Such an ensemble of functions, which for

the present discussion may be considered functions of time, defines

a random process.

A random process f(t) does not depend in a precise way on the

variable to If a function of the enslable is chosen at random, the

probability that its value at time t lies between x and xtdx is

given by dx times a function 1L(x;t) which will be referred to as

the first probability density function. It follows that
40

(3.1) A [ j (t)j


-7-








-8-


and in general


(3.2) Av9. F [F(t f \oo(X t) dV


In the same way the probability that f(t) will have a value between

x1 and x -i dl at a time t1 and a value between x2 and x2+ dx+

at a time t2 later is equal to dxldx2 times the second probability

density function, W2(xl,tlx2,t2). Higher ordered probability den-

sity functions are defined in a similar manner. The complete set

of density functions describes the random process.

A random process f(t) is said to be stationary if the probability

densities of f(t-P T) are the same as those of f(t). The random

process in this case is invariant under translations in time and the

probability densities are written WI(x), W2(xl,x2;T), et sea.

Frequently a random process may consist of a combination of

several variables. Such a combination of variables might be denoted

by (flf2). The first probability density function is then a function

of two variables. The probability that f- lies between x1 and xl-f dxi

and that f2 lies between x2 and x2 -f- dx2 is Wl(xlx2)dxldx2* If f1

and f2 are statistically independent then the first probability den-

aity function of (f1f2) is equal to the product of the probability

densities of fl and f2 respectively. This process can be continued

for the higher ordered probability density functions of more than one

variable.

With the probability density functions, the characteristic funo-







-9-


tions may be associated. The characteristic function, Fl(u), of the

one-dimensional random variable f(t) is defined by

co
(3.3) Lu x -L L X \ d


The probaLblity density function may be expressed in terms of the

characteristic functions by application of the Fourier inversion

principle:


(3.4) E)-L LL- FL)Jd


If f(t) is ergodic (that is if f(t) is stationary and if there is no

stationary subset of the functions of the ensemble with probability

unequal to zero or unity), the characteristic function may be found

as a tine average:



(3.5) (7u..) 2T -- o T t
-T

It will be observed by inspection of equation (3.3) that the moments

of the distribution of f(t) may be obtained as coefficients in a

Taylor's Series expansion of the characteristic function. A similar

definition using the real variable is known as the "oisent-generating"
26
function for this reason.

The notion of the characteristic function may be generalized to

distributions in several dimensions. Thus the characteristic function









-10 -


of f(t), f(t + T) is



(3.6) h x )dx,dx
v.o --o

A rigorous discussion of theorems concerning characteristic functions

and probability density functions is given by Cramer.


Noo-aionary Randoam Prease General Model

As described above all of the statistical properties of a random

wave may be obtained once the set of density functions, Wl,..*.,Wn

which describe the process are known, 'or most physical systems,

the first two density functions are sufficient to describe the pro-

0ess, as they enable one to calculate the following properties of

Ohe random wave:

1. The average or steady state value;

2. The mean square amplitude or root-mean-square value;

3. The correlation function;

a. The mean density or power spectrum;

4. The first amplitude distribution function;

5. The second amplitude distribution function;

6. All moments of the first and second amplitude
distribution functions.

The first two density functions are given the following inter-

pretation,


(x ,,X,,.-'xn, L )d ~x ''"' n. -= the joint probability that
(3.7) x,,...,x, lie in the ranges
x,+dx,,...,x+ dx, at the
time t,.


_ ~








-11 -


and
the joint probability that x,,...x, simulta-
(/,X j n* tj'X.. n ) --_ neously fall in the interval x,+dxI,...,x+dx,,,
at time t and that x:,...,x. simultaneously
(3.8) foil in the interval x +dx,.. .,x',+dx, at the
time t, later.

For stationary processes one has to consider only the time interval

T = t-t, between observations, as the choice of the tim origin is

then arbitrary. Thus one can obtain W1 from W2 for stationary pgo-

cesses by letting the time difference, T, approach infinity. Per-

forming such a limiting process yields,



(3.9) Lm. l (x,. .. ,;, XT)= (xI(^ 3rW >*X)


This may be justified by reasoning that for large values of T there

is no longer any correlation between the values xljx2,... ,r oecurr-

ing at time t1 and x,,...,x occurring at a time t2 later* For

linear systems the first density function described above, (3.7),

is sufficient; however, for nonlinear systems such as encountered

in communications systems the second order density function, (3.8),

is required. Furthermore, in nonlinear cases it is often necessary

to use the characteristic function of the second density function

in order to obtain any solution.

The nature of the process to be considered here is restricted to

the condition that it can be described in terms of three parameters,

amplitude variation, ak, time of occurrence, tk and width, rk. Thus

the random nature of the phenomena can be described in terms of the








- 12 -


kth time function xk(t) = xk(t;ak,tkrk), where the random variables

ak, tk and rk have a joint density function k(aktkrk). Using this
type of time expression we desire to develop the first and second
order e-dimensional joint probability density functions:


(3.10) \A/(,x ....Xs;I,) dx -dx dx
T-

and


(3.11) ,(X,, xlx ) t "t. ..dx, dx ; dx.


which are the probabilities that the various x (j = 1,2,...,s) of
the nonstationary system lie in the ranges xl, x4+ dx1; x2, x2 +- dx2;
...; x,, Xs + dxs at the time t and xj, xq dxj; x, x2- dS--; ..."

xs, x + dxs at a tile t2, where 0 tl t 2- T and s 1 for the
first density function and > 2 for the second order density function.
The "s" functions (xj) may be distinct but not necessarily independent.
Each of the xj is the resultant of exactly "K" events in the interval

(0,T) so that we amy write

,( j) k(i) Kli) O() pqj) 0i ocj. k,) K(q
(3.12) X. =X .t a; a i a i -r t **/Lj J -.


where j = 1,2,..,.
j'm 1 for the first order density function
j'= 1,2 for the second order density function.









- 13 -


Here k(j) denotes the kth event in the series k = O,1,...,K for the

th function xj. We write k(j) to distinguish different values of

k which may occur for different values of J. These are denoted by

superscripts on the random variables, aj, rj, and tj. The subscript

"J" on each of the parameters likewise distinguish between the dif-

ferent possible statistical properties among the "J" resultant waves

xj. In this paper the term event will be used to specify a particu-

lar type of waveform occurring in the period of time (0,T). Taking

into account all possible numbers of events (K = 0,1,..., Co), that

can occur in this times

00

3* 3) (x V t, X) T k=o

and


(3.14) r


The light hand side of equations (3.13) or (3.14) could be written as

probability of exactly K events in the period (O,T)] Lconditional
K=O probability that if there are exactly I events in (0,T) then
x1 lies in the interval (xl,zxl + dx1), x2 lies in the interval
(x2,2 + dx2),.... x, lies in the interval (xz,x + dxs) at the
time t ; xj lies in the interval (x!,xI + dx ... and x' lies
in the interval (xl,x,-+ d34) at th tie t where 0 1t
The conditional probabilities can be obtained in terms of the

characteristic function for the "s" random variables. These condi-

tional probabilities areas









-14 -


5
.0o x .

(3.15) P( XO-Ix,5. t1 ) Icz1 n )
-Ota _-

and


00 io ,, ao
fd., (du, du ...( d
-)-W/ J17 Z r
C .. o Oo

(3.16) P(KI t, t'' ;xL Ut u _
LI
T,




The definition of the characteristic function yields



(3.17) Cu,,.-., t, ) = =

SSTATITICAL AVG.
and



(3.18) a 's i
SSTATIsTiCAL AVG-.

The characteristic ftiations can be written as an s-order product

of the densities of the random variables aj, rj, and tj



(3.19) F;= 7 ai%4)d(,i, )
;L1








- 15 -


5
I LzA 4j


j[aj- : a j j **, "^, ;, S J4't"
06ri^) Ki I o(j) J n G j K1

w [a a j ....d dt,) L I

Ot k. jd Ci


The integration in (3.19) and (3.20) is performed over all allowed
values of the random parameters as listed above in equation (3.21).

The restriction is made that for each strip of K events and any one

member of x the parameters ak ) W, and t are governed by
the sam probability laws; however these distributions wj amy differ
for different members of the set. It is assumed that there is no
correlation between the different random parameters, and the basic
probability WI(K)T for the oecurrenee of K events in (0,T) is iden-

tical for the j members of the xj of the set. If wj w for all j
of the set, then equations (3.19) and (3.20) reduce tot



/


and



SA it,


where


(3.21) ^(ajs-^rj) ,^^) =








- 16 -


and





(3.23) i((p",,, ^ t, ) =i W





When K = 0 (no events in the interval (0,T)), the xj vanish and the
characteristic function becomes unity (as the integral of the density
function over all values is unity by definition), and the conditional
probabilities given by equations (3.15) and (3.16) become:

S
(3.24) F(ol x, ,) = l j (x-O)
T J=-

and


(3.25) P (olx,I ,x) t,X: ) 7 (, xxt-o') (x-)


where 6(xj 0) = Dirac delta singular funotion. The resultant X
of exactly K events in (O,T) is taken to be a linear superposition
of elementary impulses e. Therefore, the following expression can
be written for (xj) s


r K >(iI r 1 i) 0( a eL'- ,L
(3.26) X^^;j [t i ^








- 17 -


where the random anplitudes, ak(j), random widths, r k(j), and random

times of occurrences tk(j), are for the elementary pulses. Equation

(3.26) shows that the statistical properties of the resultant (xj),

clearly depend upon the model structure and on the distribution

w(aj,rjt ) of the random parameters.

This concudes the development of a general model of a non-

stationary or quasi-stationary phenomenon. The next section will

deal with additional modifications allowable in considering atmos-

pheric noise phenomena.


Stationary Random Processes General Model

Atmospheric noise is classified as nonperiodic, overlapping,

impuls; noise. The following assumptions are made in order to

obtain a mathematical model that might prove tractable to stand-

ard methods of the calculus and the theory of probability. It may

be assumed that all elementary impulses, *e, are identical in shape,

that they can be linearly superimposed, and that their amplitudes,
k(J) k(j)
a durations, r are random while their times of occurrence,

t(J) are independent random quantities.

It has been shown by Hurwitz and Kac28 that these assumptions

yield a Poisson distribution for the "KN events as given by:

N/V

(3.27) (v)T


where R = the average number of impulses arriving in the period (O,T)

for the ensemble of strips (0,T).








-18 -


Equation (3.27) is to be substituted for the 1(K)T appearing in

equations (3.13) and (3.14). Equation (3.27) gives the probability

that out of an infinite number of similarly prepared systems, each of

duration T, and in each one of which there may be (0,l,..,K,..,Oo)

impulses, the interval dll contain exactly "K" events. Equation (3.27)

can be written in a somewhat different forms

-nT
(3.28) \JI (nT)


where,

n = average number of events per unit time
Siim. (K,+,l- .-.+K 1 N_
=m-e mT T
a = the number of similarly prepared strips.

By substituting (3.28) into (3.25) and (3.26) after making use of

(3.19) and (3.20) the characteristic function for this Poisson case
is obtained

00




(3.29)



Kl1
n=7 vTj








- 19 -


and


(3.30) Fa ',. 4,'",,s;t,/...,U. ;c =
r


Snr), 6


Equations (3.29) and (3.30) can be simplified by the assumption

that the ak(), rk(j), and tk(J) are independent and have identical

distributions for all "k". This condition of independence permits
the factoring of the three k-fold multiple integral into a simple
kth power of a 3-fold integral. Carrying out this process for (3,29):



r.zoT 0 r. j2 P
t0 -00 0


(3.31)


7- f TT
(reT) AK



where A dn da c a, h, ) T,,
0 -_


Combining (AnT)k and expanding,








- 20 -


-nT L -nT nTA
-nT I nTA r ... = ( E
L' -, T


(3.32) f(,, .,L t -I)r
I -


S a ,A, To


However;



(3.33)


nT= N


a Vd


d.u da ca,%T)dJT = 1
0- a'


therefore equation (3.32) becomes



(3.34) eF(x,..,Upt,-)= exp. /- -.
I Ti -00 T


Following the same procedure for (3.30):


(3.35) f (u,,- t,: t =


/ eY -jc a Twca )
So
- Aa .,^^a ^'A"o]^
[N 2^e









-21 -


Substituting (3.34) into (3.13) and making use of (3.15) and (3.29):





(o k o a

(3.*36) (,,-..x5 L) it 1


n (3.2) into (31) and making use of (3.16) and (3.30)




Substituting (3.25) into (3.U) and making use of (3.16) and (3.30):


oo o 00 00


D-0 0


(3.37) / ) -. j
e- f d &i'" I J ^



(A.-z 147-6 Qt J
Ap- j-


The expressions given by equations (3.36) and (3.37) can be

simplified by making use of the general properties of the individual

pulses, *j. First it is assumed that there are no disturbances out-

side the interval (0,T), therefore the limits of integration of TO

may be extended to plus and minus infinity. Next it is required that









- 22 -


the integration over To be convergent for all T. This condition is

always satisfied for physically realizable pulses, as they must pos-

sees finite energy. Further simplification becomes possible by mak-

ing the following substitutions:



(3.38) Xz= (t,-7o) t= 6(t.-t3, where j= a,... s ,


here B is a shape factor of the individual pulse which can be de-

scribed in terms of the mean duration of the set of pulses by the

relation


(3.39)


-
B J


Solving equation (3.38) for tj To'


(3.40) T,--
B


therefore


(3.41)


-t t- tj---T = xj
B 13


t -T = _x_+


(3.42)


or- t a =








-23 -


Substituting (3.42) into the expression for ej, the new variable hj
is defined:


(3.43) hj e5 J


The substitution of (3.43) into the second parts of (3.36) and (3.37)
yields:


(3.44) a :A Z- -h (t E
0 -*00 -CO

and

00 r 000 L Qi^^ ^^ LSJ^^^ A,
(3.45) /A a iwflt-|)[6a Y i 4 ^


For stationary processes the probability density, w(a,r,t1 x/B),
is independent of the choice of the tiie origin, with the consequence
that setting t$ 1 0, reduces this expression to w(a,r, -x/B). Fur-
thermore this density function is symmetrical in time, since there
can be no distinction statistically between "forward* or "backward"
in tiae if the process is stationary. Therefore:


(3.46) U(a, tV -)= ) Lu(aJ1,-) = (-)


Substituting (3.46) and (3.44) into (3.36) the first probability
density function for the stationary case is









- 24 -


5
j- '




Here the characteristic function is given by



(3.48) Fc(,,..JU) ) ex p. { ifda wi a)l '= -I Jx



Substituting (3.46) and (3.45) into (3.37) the second probability

density function for the statiorary case becomes:



.r 00 ,L )

o ,0

(3.49) 5'(.,3 ,Xd'). S
r(L~ 1 -Li,





where the characteristic function is given by




: o -


(3.s) F(LC,...U,;:...u;;~) e~F. ;fu;j~rtj~ h ia~c~h(Y~j~5

L-









- 25 -


Because the process is assumed stationary, the interval length (O,T)

no longer enters explicitly. If the periods of time, or epochs ti

(and therefore x), are assumed to be uniformly distributed in the in-

terval (O,T) and independent of the durations and amplitudes of the

various impulses, the density function w(a,r,x/B) becomes:



(3.51) (a )= } .
-l-


The limit of equation (3.51) as T approaches infinity is:



(3.52) Lim. [j1 LLjCa,f1)/-J"OO or


where 6 = the average number of pulses per second times the mean du-

ration of the pulses = a dimensionless parameter whose magnitude de-

termines the character of the noise.

It is to be recalled that if a, r, and To are independent,



(3.53) LO(a,,T i) U= ja^ )LuCnu) .


Substituting (3.52) and (3.53) into (3.47), (3.48), (3.49) and (3.50)

yields:


f for

-00 -0









- 26 -


Co 00co


S3.. T 2, ) y -
-' 00 w o f a
00L


f U, C ~l.. () ;


5


J -l

J


0 -0


(3.57)fc ( ..,,U ; ,'..., 1j jT ) j- p'

x -4 x
X y


Equations (3.54), (3.55), (3.56), and (3.57) are the general


equations for the stationary model.


(3.56) (,, .. ;X 4,', )


_


+" oo 00 A ^ (x++> I

Swcm' pDca)da r6 i1 -1 4x
(3.55) ex p*.~ = A t I
L so -03







-27 -


Discussion 2L aPodd eGsult
It is obvious that the character of the distribution functions

of xj depends upon the amount of overlapping among the individual im-

pulses. For heavy overlapping, the distributions become the well-

known "normal" distribution of several variables. That is, the values

of amplitudes of the order of the root-mean-square have a significant

probability of recurring. Slight overlapping causes appreciable gaps

between the pulses, so that small or zero amplitudes are most likely

to occur. For the normal or nearly normal case (considerable over-

lapping), the precise form of the elementary, independent transients,

and their individual statistics are unimportant as far as the nature

of the distribution is concerned. This is true because there are

such a large number of pulses (in any short interval of time At),

that their individuality is lost in the combined effect. (This fol-

lows from the Central Limit Theorem of probability).9 For the case

of widely spaced pulses, however, the shape and statistical properties

of the individual pulses are critical in determining the form of the

probability densities, W, and W2. It is this dependence upon indi-

vidual pulse shape that makes the explicit evaluation of the density

functions so much more difficult. For little or no overlapping, one

needs merely to apply conventional methods to a single representa-

tive pulse.

As the type of Poisson noise depends upon the "density" of im-

pulses in any given time interval, it can be seen from equations

(3.48), (3.50) and the preceding argument that the parameter













(3.58) [ ;LaYv.wner o pulses persecori

essentially determines the statistical character of the noise. That
is X determines the class of the noise, impulse type static, nearly
normal random noise, or fluctuation noise. Table 1 on page 29 lists
a variety of physical situations to which the Poisson model applies,

the order of magnitude of *, and the general nature of the densities
describing the random process.
Before continuing the development of the statistical model for
atmospheric noise it would be well to cite at this point an example
to indicate how the expressions developed previously can be applied
in solving a relatively simple problem,


btample
Consider the case of a train of overlapping rectangular pulses,
where the amplitudes are distributed according to the Gaussian laws


(3.59)- E
c) 9 E
,2 2
where a2 a = the variance, h(xr) = U(x) U(x-r), and the

durations have any meaningful value. The characteristic function is
obtained by substituting (3.59) into (3.55)8


* A complete solution of this problem is given in Appendix I.








TABLE I


Phenomenon: Magnitude Character of
Poisson Noise of Distributions

1. Impulsive Ranom Noie 0(0-10) Depends upon individ-
(a) Static; ignition noise; ual pulse shape and
solar interference 0(10 ) pulse statistics.
Strong dependence on
(b) Underwater sound; re- magnitude of Y.
flections from random-
ly oriented objects
moving relative to -1
observer 0(10 )

(c) Speech model. 0(10-1)


2. early Ho l anda Noise 0(10-104) No nal distribution
(a) Heavy atmospheric static Ylth one or more
correction terms*
(b) Precipitation noise These are of order
",dor Y~- de-
(c) Clutter, sea waves, etc. pending on whether
or not the third
(d) Underwater sound moments exist.
Noticeable to weak
(e) Window (not densely dependence on magni-
sown) tude of Y

(f) Solar static; sun-
spot conditions.


3. NoPal Bandom Noe 0(104-.o) Normal distribution;
(a) Shot noise ignorable correction
terms. ( enters
(b) Photo-multiplier noise only as a scale fac-
tor for the probabil-
(c) Thermal noise ity densities, whose
form now does not de-
(d) Clutter (scattering pend on y )
from water droplets)

(e) Barkhausen noise

(f) Window, electronic
interference, inherent
tube noise.


-29-







- 30 -


F(u =l 6


00

exp.~ c -03


(3.60)


l n!
n-o


Lfnau- 2.
oU


W1(x) is obtained by the inversion of equation (3.60):


(3.61) (A) =


X ;L aF T-

n=(


-I- E cSx-o),


Similarly substituting (3.59) into (3.57) the characteristic function

of the second density function becomes


pt 4m4n fl~

E- ptm n! n
P,m.,n=o




xp. L (r,, +n)-t auL (P++n)- rpVA' + n(U-u
.Z


Inversion of (3.62) gives the second order density functions


C- d


(3.62) F ( U) T=













S(-o) t'-o)




/ P ny C^^]
P Pn>o


-C (pi-n)j t -m
IX or- "p'p n

\/v rr? man-4


- 31 -


(3.63) (^r^7 T) ( < ,,}

aTTr a--
















CHAPTER IV


ATMOSPHERIC NOISE MODEL


In Chapter III the general expressions to be used for the density

functions of atmospheric noise are derived. To select an actual type

of waveform for the atmospheric noise model it is necessary to con-

sider the physical situation which this model is to represent. This

physical situation is the output envelope of a normal amplitude-

modulated receiver which is being shock-excited by a random series

of pulses at the input, the pulses being the electromagnetic energy

emanating from lightning flashes. Under these conditions the output

waveform consists of a train of exponential impulses with a varying

degree of overlapping, depending upon the density of the input pulses.

Two analytical time functions that can be used to approximate this

output waveform are an exponential time pulse and a triangular time

pulse. Each of these cases will be examined in detail in this chapter.


xPonmental TJime Pulase General ModalL

The time function is given by the relations


(4.1) cx,) = E 6 o
(4.1) c = 6 U 0 sX o0


- 32 -








- 33 -


and its dwdth is defined as the point at which the amplitude is one-

tenth of its original value; or at the value


(4.2) X a.3 B


For ease of mathematical manipulation the original time function

will be redefined to exist only over this range, that is

h(x, = E, .3
(4.3)
0= o> > .3
/I
However equation (3.39) of Chapter III gives


(4.4) B = or = .


This enables us to write the time function of (4.3) in a slightly

different forms


(4.5) hc
0>^ A --.
= "-

Substituting the ti"e function given by (4.5) into the expression

for the characteristic function, (3.55)s
-Xjl
(r6 o 'ac. 1.
u(4.6) =- exp.4 J JwIdr ) wcdaf-Ij







- 34 -


The last integral in (4.6) may be altered by making the substitution:

au-V A Go
(4.7) E = A .
n=o


and noting that

XAI -nxT-
(4.8) n x TL *


These manipulations reduce the integral of (4.6) to the form

-XA. Al -nxI= Z 3


/ n=o c
0

The integral in equation (4.9) has the value of 2.3/r2 for the

case n = 0 and the coefficient of the integral is unity, therefore

the value of (4.9) at n = 0 is also zero. This reduces the integral

to the form;

00 -nxJ A

(4.10) I -2 r -f
0 n-i 0

The value of the integral on the right-hand side of equation (4.10)

for any value of n is given by:


0 -nx -a.3i-







- 35 -


La..3rt
as E 41 for positive integral values of n.
Repeated application of (4.11) in (4.10) yields

(4.12) -x

n

Substituting equation (4.12) in the equation for the characteristic
function, (4.6)t


(4.13) F, = ep. CL. yfXf-_ ,


Inpection of the second integral of (4.13) shows that it can be
evaluated without defining the density function governing the a's.
This is shown to be true be referring to the fundamental definition
of moments, equation (3.1), thus


(4.14) J a)lw(da =


and equation (4.13) can be written


(4.15) r(L) e exp.< Z- n_~. n! f 00 (


The density fun tion of the widths of the pulses, r, is as.
sumed to follow the Rayleigh law:







- 36 -


-ft
(4.16) LC(J) = -


However, the substitution of this particular density function in

(4.15) requires that the range of integration of the variable be re-
cdced, as the integral becomes indeterminate if the limits of zero

and infinity are maintained. This change of the limits of integra-

tion is not too disturbing, as, for any physical receiving system with
a finite input, the pulse width can be neither zero nor infinite.

The zero constraint is set by the finite bandwidth of the receiver
and the infinite contraint by the finite energy criteria for any

physical pulse. The maximum allowable width is designated as r2 and
the minimum allowable width r1. This restriction of the range of
the variable changes the density function given in (4.16) as the area

under the integral must equal unity. The new density function ist




(4.17) __nl) A E e




Substituting (4.17) into (4.15)
( 80x 02-

(4.I) F(r) = Ip -^ l aT-J fci







- 37 -


The integral can be evaluated by a change of variable, y2 = r2/2 0c ,
which reduces it to the form






The values of this expression are tabulated0 and for brevity (4.19)
will be written


(4.20) f E i


where Ei(v) is the exponential integral evaluated at v.
Substituting (4.20) into (4.18) the expression for the first
characteristic function ist


(4.21) 7(a)= exp.cl C (I


where the constant C is given by ai L *




The coentant X is not included in the grouping of the other
constants as its range of values is dictated by the density of the
noise bursts as discussed in detail in Chapter III.

The first density function is found by inversion of the charac-
teristic function, (4.21)t









- 38 -


-oo




potentiall Tm Pulse Particular Results

The first density function given by equation (4.22) cannot be

evaluated in its present form. It is necessary to consider the dif-

Cerent density functions governing the distribution of peak amplitudes,

w(a), and determine the manner in which the moments vary. This crite-

rion determines whether the series in the exponent of equation (4.22)

will converge or not. If the series is divergent, (4.22) cannot be

evaluated as it is the courier Transform of the base a to this series,

and a necessary condition for a Fourier Transform to exist is that the

function whose transform is being calculated most converge.

An example at the series in the exponent of (4.22) diverging is

given if the peak amplitudes of the individual pulses are assumed to

be distributed exponentially;

-a

(4.23) ujw ,



The nth moment of a is



(4.24) alL
0







- 39 -


Substituting (4.24) into (4.22)

a,
0 n



However,

00
((udn o [a + LLL]
(4.26) n -
n--

which is divergent for all values of u
Transform of the base e to this exponent does not exist, and as

(4.25) is the Fourier Transform of this function it cannot be eval-

uated.


Gxponential Ti Pulse A Discrete Solution

If the density function of peak amplitudes is such that aF = kn

(k = a positive real number, not necessarily integral), then (4.22)

can be evaluated. This example is considered here, as the integral

of the resulting density function is a good approximation to the

measured atmospheric noise distribution function for small values of

the variable39,40

The relationship given above for the moments reduces (4.22) tot



(4.27) U/() -- j 6 E dec.
v' ." Ia 7 )







- 40 -


LU
Expanding E in a series similar to (4.7) and (4.8) the density

function (4.27) becomes:




--o
_-'c" oo
(4.28) )ck 00"


= 0



Inspection of (4.28) shows that this is a discrete density

function, having values only at integral values of x. To compare

this density function with the measured atmospheric noise distri-

bution it is necessary to integrate (4.28) with respect to x, as

the distribution &fnction is the integral of the density function.

Carrying out this integration



(4.2) x .] 1D U(X-) (Ck)n


where the factor D is a normalizing constant.

A graph of a measured atmospheric distribution with the cal-

culated values of equation (4.29) shown thereon is given in Fig-

ure 1 on page 41. Although this is a discrete type of distribution

function, while the measured distribution function is continuous,

it is a good approximation for mall values of the variable and may

prove useful in studies over this range of atmospheric noise values.




AHRL 5-1-54


2 3 4 5 6 8 10


2 3 4 5 6 8 10~


2 3 4 5 6 8 103


2 3 4 5 6 8 I04


0.2 -







0. 1 1-L~iL i J i J i i i i 1 ii i i- i
1-1
0.5 Tfi 11
.5- ;.- -- .. 1 -- -- -- --


2










2 --- .- ----- --- i I--: -- -
i 4








0 -- ---t ::
o -! F -. .- i i "1i *- i_ i i- ii t I
70






70 pgEt ATMOSPHERIC NOISE DISTRIBUTION
so ---- --,-







Date Time Freq ENB
T=- 6/25/54 116 135 KC 0.144 KC
90 --4- -i. -i !---- -.-4-- -








o -Theoretical points
95
98 ----------- -
.. i.. ;
Z__- ,2 ATMOSPHERIC NOISE DISTRIBUTION

..... tDate Time Freq ENB
6/25/54 9116 135 KC 0.144 KC


........ .... it I- o -Theoretical points

-4. ./

-

A f i- Is I 1 1 1 _1 1 14f


2 3 4 5 6 8 102 2 3 4 5 6 8 10,
MICROVOLTS PER METER INPUT TO ANTENNA
FIGURE I


2 3 4 5 6 8 104


2 3 4 5 6 8 10








- 42 -


Triangular Time Pulse General Model
The time function is given by the relation:


(4.30) ( [u(t-ui-^ '


where k is a positive real constant having values between one and zero.
The factor kr determines the width of the positively increasing time
function (slope), and the factor (1 kr) determines the width of the
decreasing time function. Substituting the time function (4.30) into
the expression for the characteristic function, equation (3.55):






(4.31) F,(> XP (Ct e *j
xfe^ dx 'k *- j
La x O





The integral over the range of x can be reduced to the form given in

(4.31) by the definition of the time pulse, (4.30), That is, the time
function is zero outside of the range zero to one and the step func-
tion notation enables each portion of the time function to be treated
individually. Straightforward integration of the last three integrals
in (4.31) gives:







- 43 -


4 < x ia X-y r-
(4.32) ^ df 1 X -
0

Substituting equation (4.32) in the equation for the characteristic
function


(4.33) r() --p J e7[Y^^ f djax.uOJL u.Q
[ o o L J


Inspection of (4.33) shows that the first exponential contains
integrals of the density functions alone, and as the integral of the
density function must eqal unity by definition, then the particular
form of the density function has no effect upon the integral. In the

second exponential term of (4.33) the variable r does not appear in

any of the integrals except as a density function, so the actual form
of the density function of r has no effect upon this term. Applying
the above reasoning (4.33) can be written:


(4.34) AF(LA P Y"J-A C


The density function of the peak amplitudes of the pulses, a, is

assumed to follow the Rayleigh laws

(35)
(4.35) acu --c
wca) = ~d







-44-


Substituting (4.35) into (4.34) the integral expression becomes


(4.36) o ( L da 6 f d.


The second integral in (4.36) is the normal error integral over one-
half of its range without the normalizing factor -L j therefore,


(4.37) J L


The second integral of (4.36) is the characteristic function of
the normal error curve without the proper normalizing factor;
therefore,


i t E"V) da I. -
(4.38) a.j --i I L


The characteristic function is obtained by substituting (4.38)
and (4.37) into (4.36) and then substituting this result into (4.34):


(4.39) F ) 6 eup.J J


The first density function is found by inversion of the charac-
teristic function, (4.39);


(4.40) .1} :A --
W a )ITTT









45 -



The expression for the first density function as given by (4.40)

cannot be evaluated directly. Approximations to this solution will

be discussed in detail in the next chapter.
















CHAPTER V


POWER SERIES APPROXIMATION AS A SOLUTION

OF GENERAL DENSITY FUNCTION


The two examples considered in Chapter IV indicate the magni-

tude of the problem of evaluating the density function if the ampli-

tude of the time pulse varies during the duration of the pulse.

Under certain conditions, to be discussed in detail later in this

chapter, it is possible to obtain a series expansion for the density

functions in powers of the parameter X For a rapidly convergent

series the first few terms often can be evaluated to obtain a good

approximation to the density function.


General Macression

The general series expansion for Wl(x) and W2(x) is obtained

directly by inversion of the equations for the char-acteristic funo-

tions, (3.55) and (3.57) after a substitution has been made. In this

chapter the series expansion for the characteristic function of the

first density function is derived. The higher ordered functions are

obtained by a similar process.

Define the characteristic function by the relations


- 46 -








- 47 -


(5.1) f> ) (u)


Comparison of (5.1) vdtb the originally derived expression for the
characteristic function (3.55) shoes that


00 fO \ aw k(ATZ) ,
(5.2) = cU(^Xjr a)daf[C6 -I]J


Substituting (5.1) in the general expression for the first density
function:


QO 00 u
) -L j U- !


(5.3) 1o f L '1
Sx (x-ox) t / j
-G








For (5.3) to be useful, the series should converge rapidly. That is
n should not assume values of more than three or four, or the work
of evaluating the expression becomes so great that graphical solutions
of the original integral are probably just as satisfactory.








-48 -


Solution for Triang arime Pulse
To indicate the manner in which (5.3) can be used, an approxima-
tion for the first density function derived for a triangular time pulse,

(4.40) is evaluated, For this particular function D(u) is given by:


(5.4) DO L,


Substituting (5.4) into the general expression for the first order
density function, (5.3):



MrIL U. A' n
CO

Carrying out the expansion through n = 3 in (5.5),

L I ao IAA -J"%
,L^) = k(x-o.+ Y I --c-d,.- -iA _/ f
r'a 2Ji L.4W-Lu.) ) I-) +


~- o




4 o D +
II .3



T, &) 3 f
oo Io









- 49 -


The details of evaluating (5.6) are given in Appendix II. The value

of the density function is found to bes


(K)= -o) t ___r


I Ji 4 X i]
- t .(i^ j
a 2.Lr

F, .)v


-t.~ ,,g I
3 ~rn


2..-4 2
2.^L' X,
^
'92 4V ) + J-1zi~
Y 17


(5.7)


+ y3 (1
1.~'


F3






-'Vl^


LL
13
1- 3 F, a
^,^'-*>-s,

+ 3^^


1"


where IFl(a,b;y) is the confluent hypergeometric function.0


y'(njh


[,X


[.^'^)~y



f^^)":








-50-


To compare this expression with the measured results obtained for

atmospheric noise, it is necessary to integrate (5.7). Carrying

out this integration (see Appendix II)i


P(,K^) V i X -t XLL[6-

3 y CT


(a
- < (


F,(1 1- ) -t () iKlk


-LIF
&l ) ',
3iw ]'


(5.8)


tI-
yV


F,3
tU-(,.,-.L ;



2. ra










- 51 -


The comparison between the measured atmospheric distribution func-

tions and (5.8) is not very good. The major difficulty is the restric-

tion introduced by the condition that for x O, P(x) = 0, which gives

the relationship between aC and



(5.9) aa t-


Another term or two in the power series would alter this expres-

sion and probably produce a much closer correlation between the theo-

retical expression and the measured distribution. It is felt that

this is a satisfactory type of analytical expression as the measured

atmospheric noise distribution is log-normal in character, and the

integral ofthis density function is expressed in terms of the con-

fluent hypergeometric function also. The labor required to obtain

these additional terms would be considerable, and for this reason

they are not included in this paper.
















CHAPTER VI


SOLUTION BT METHOD OF INTEGRAL TRANSFORMS


In Chapter IV it is shown that if the amplitude of the time pulse

varies with the duration of the pulse, the method of Fourier Trans-

forms, which is used to obtain the probability density functions,

cannot be used to solve the problem. This point is substantiated by
25
Middleton in his paper on phenomenological models. The method of

series expansion given in Chapter V is limited in many cases of prac-

tical interest, as the series does not coverage rapidly. This chapter

is concerned with an alternate method of solving problems of this type.

For pulses of the form chosen in Chapter III the characteristic

functions are either products of exponentials or are exponentials

raised to exponential powers. As these types of expressions are not

amenable to the standard methods used for evaluating Fourier Trans-

forms, it is necessary to find some type of transformation which will

alter the expression to be integrated in such a way that the process

of integration can be carried out. The problem is to find some type

of integral transformation which will change the form of the integral

from one which defies the ordinary methods of the calculus to a more

tractable form.


- 52 -







- 53 -


Theory f Integral Transforms

The method of integral transformation is used to solve this prob-

lem, therefore a review of the pertinent factors of integral transforma-

tions is presented here. The most common transform in use in the Blec-

trical Engineering field today is probably the Laplace Transform:

00 -st
(6.1) L ) = (t) dt


where s is the complex frequency and t is time. A more general class

of this same type of function may be developed using the relationship:

00
(6.2) (cp) = f x) f p ) xc


where K(ppx) is a known function of the two variables p and x, and

the integral of (6.2) is convergent. Under these conditions the func-

tion I(p) is called the integral transform of the function f(x) by the

kernel K(p,x). From this definition it is obvious that the kernel,


(6.3) K = P


yields the Laplace transform as given in (6.1). The other two most

commonly used kernels are:


* A list of definitions used in this chapter is given on page 62.








- 54 -


(6.4) K(Px) = x


which gives the Mellin Transformation, and


(6.5) (p,) = x (Px)


where J (pm) = Bessel function of the first kind of order v, which

yields the Hankel Transformation. By a change of variable the kernels

for the Fourier Sine, Cosine, and Exponential Transformations may be

developed. The classical method of doing this is in terms of a re-

quirement of the Mellin Transform, M(a)31 It should be mentioned, as
a matter of historical interest, that the first systematic investiga-

tion of the problem of integral transforms was done by Mellin in 189632

One of the important properties of the kernel is that it is a
linear operator. That is, if the function to be transformed is aml-

tiplied by a constant the operator does not affect the constant; or

if a sum of functions is to be transformed, they can be transformed

singularly and the results added algebraically.

The operator transforming a function into its integral transform
is denoted bys


Nl(.) = 1p)


(6.6)







- 55 -


Assume that for every fintion B(p), belonging to a certain class of
function of the variable p, the equation


(6.7) N() = Bcp)


is satisfied by one, and only one function, f(x). Then it can be
proved that there exists a linear operator N-. called the inverse
of N, such that the equations


(6.8) /Vo.) = B p), (x~N N-(B)


are equivalent.3 The problem is the determination of these inverse
operators for some special oases of the operator N. That is, the
following integral equations are to be solved:




(6.9) I-() = J(o)(ph




and

b
(6.10) } ) = I( H(P) x)Jp
a








-56-


A formula of the type given by (6.10) which expresses the function f(x)
in terms of its integral transform (6.9) is called an inversion formula.
A necessary condition for the integral equation (6.9) to have a
solution of the form (6.10) is that the Mellin Transforms K(s), H(s)

of the functions K(x), H(x) should satisfy the functional equation


(6.u) (s) H(1-5) =1 -


This can oe proved by direct substitution. The definition of the
Mellin Transform is


0 0 O
o


(6.12) cx(5)






where q = px. Similarly




S)X l (P?
(6.13)

f(>~~ l^ hH*)
0








- 57 -


Letting s' a 1 a in (6.13)s


(6.4) ('- F 5) = M(s)H C-5).



Eliminating F(1-e)/M(s) from (6.12) and (6.14) yields (6.11). It is

to be noted here that the form of the kernel was assumed to be a prod-

uct of p and x. This does not greatly restrict the usefulness of these

theories.

Relations Between certain Transform
and Fourier Transforms

The change ac variable necessary to convert Fourier Transforms

into other types of transformations is the problem we wish to consider.

That sich a transformation is possible in the case of the two trans-

formations mentioned above, the Mellin and Hankel, will be proved.

In the case of the Mellin Transformation, make the change of vari-

able, x = ey


(6.15) M(,5 ,.) f COXM 5
0 -00 -00


Equation (6.15) in the case s = -is is the Fourier Exponential Trans-

form of the variables (eY,-is)


(6.16) F(d) y)
o _O
00









-58 -


This relationship can be written rlibolleally


(6.17)


where M [f(x); s = Mellin transform of the variables x and a, and
F [f(.z)j -i] = Fourier transform of the variables ex and -ia.
The relationship between the courier Bxponential Transform and
the rankel Transform is not so simple, but is more useful in many
instances. This relation between transforms can be shown by con-
sidering the Fourier Transform of the two variables p and q$


(6.18)


-4
F-(qs) = r (Je) a 4 dx j
d-oo


Making the following shags of variable in (6.18)


X= ACo e
(6.19) n
S=t s in e


p= u.coS
SU. 5I0


jdx = JLcAe

A +Cb = nuco5 (e-4)


yields


(6.20)


a I u L I cos(e-O)

o 'f


Because of the periodic nature of the second integrand in (6.20):


ec aA co5e-0


S-c. r cosee =

0


(6.a)


o


FC,) =


/I I[ ) r[5 = f 0;-45)








- 59 -


However inspection of (6.19) shows that u = (p2+ q2), so that F(p,q)

is actually a function of the single variable u only, and may be
written


(6.22) F() =) To J


The general expression for the Hankel transformation is


0
(6.23) H x)z = I X i W J ) X


therefore (6,22) is seen to be the zero-order transform of the funo-
tion f(r). Symbolically:


(6.24) Hl5.> ]= f [ ]


This type of relationship can be extended to Fourier Exponential
Transforms of n variables31
There are many other types of integral transforms which have
been explored and discussed in the literature, most of which, how-
ever do not transform from the Fourier Exponential Transformation
very readily. For a discussion of the theory of such transforms and
their proofs the reader is referred to either of two very creditable
works: "Fourier Transforms" by I. N. Sneddon31 or "The Theory of
Fourier Integralsf by E. C. Titchmarsh34 The latter reference is








-60-


mathematically thorough, while the former is concerned with practical

applications to the solution of boundary value problems.


Applications to Atmospheric Noise Model

It is not the purpose of this chapter to find a solution by this

particular method, but to indicate if such a solution might be possi-

ble. If standard known transformations will yield a solution, they

are to be applied, and for this reason the applicability of the two

transformations discussed previously are considered.

Before applying the Mellin transforms to obtain a solution for

a particular problem, it is necessary to list the limitations before

a function can have a Mellin Transform and its Inverse. The limita-

tions can be stated by one condition, that is the integral



(6.25), o



must be bounded, i.e. converge. If this is so, the following re-

lationships are valid:

0o

(6.26) /fls) J -' (x)dK
0


and
C+
C-5 C> K.
6.27) 5(4 f

C-C







- 61 -


Equation (6.26) is the direct Mellin Transform and equation (6.27) is

the inverse Mellin Transform.

The probability density function obtained for exponential time

pulses consisted of the Fourier Transform of an infinite product of

exponential functions. This type of expression is complicated rather

than simplified by the change of variable necessary to relate the

Fourier Exponential Transform to the Mellin Transform, therefore the

Mellin Transform does not offer a method of solving this particular

problem.

The probability density function obtained for triangular time

pulses consists of the product of two exponentials, one to the in-

verse power of the variable times an exponential to the variable to

the second power and the other to the inverse power of the variable.

As in the case discussed above, the change of variable relating the

two transformations does not simplify this expression. Therefore,

for the two particular cases which were considered, the Mellin Trans-

form does not offer any simpler type of solution. Although these are

only two particular examples, the general form of the characteristic

function used in this analysis makes the applicability of Mellin

Transforms remote. That is, the general form of the characteristic

function embodied here is that of an exponential raised to a multiple

integral of several variables. That this could ever lead to any type

of solution other than



(6.28)









- 62 -


is remote. Therefore further investigation of the Mellin Transform

does not appear justified.

As the Hankel Transform deals with the second order density func-

tions which were not calculated for the two cases, the applicability

of this particular type of transformation cannot be investigated.

The general form of the characteristic function given by (6.28)

indicates that some type of logarithmic transformation should lead to

a solution of the problem, or an exponential to the exponential type

of transformation might also serve the purpose.


Definitions of Transforms

The Fourier EIponential Transforms areas



(6.29) F -
-00


and



(6.30) f(x) r u)-
-oO


where u and x are real variables.

The Mellin Transforms aret


(6.31) fA )-
Jo








- 63 -


and
CHOO

(6.32) 5(X) = Xs s Xs.
C-C L

where a is the complex variable.

The Hankel Transforma are:
0J


(6.33) J(5x)Jx)kdx



and


Y) 5x) (S) S
(6.34) '00
















CHAPTER VII


CONCLUSIONS


A general statistical model of the first and second probability

density functions of the output envelope of an amplitude-modulated

receiver, shock-excited by atmospheric noise, is developed. This

model is used to obtain explicit expressions for two particular time

pulses, exponential and triangular. The resulting integral equations

are not amenable to the ordinary methods of the calculus, and various

methods of evaluating the equations are investigated.

One discrete solution is obtained which, although different from

the continuous distribution of atmospheric noise as measured in the

laboratory, gives a good approximation for small values of the vari-

able. Another, obtained by a'series expansion of the integral equa-

tion, offers considerable promise for a continuous type of solution.

As a final attack upon the problem, the method of integral trans-

formations is considered. Although no general method can be derived,

as a different transformation is required for each different time

function, it does appear that this method of solution might be satis-

factory.

These studies indicate that the method of series expansion should

be extended and considerable study made of possible transforms that

might prove applicable to the solution of this type of problem.

64 -














APPENDIX I


EXAMPLE OF USE OF FORMULAE

Consider the case of a train of overlapping rectangular pulses

in which the amplitudes are distributed according to the Gaussian laws


(I.I) u(a)- E-


2 -2
where, a- a a 2 0 the variance, h(x,r) = U(x/B) U(x/B r) =
the time function of the pulse and th durations have any meaningful
value. Noting that

h(x;r.) = I -
(I.2)
-0 o>x>Br


and substituting (I.2) into (3.55),



(x F((A)= () p.^n"c [6=7^ )^

(1.3) 3 J c)

( -CJ


- 65 -






- 66 -


However from (3.39)1


(i.*) En (= I


Therefore substituting (1.4) into (I.3)


(1.5)


F(U) = ex p.


-do f -)
nC^ l~c~d^^.


Making use of the fundamental requirement of a density function that
the area under the curve by unity:


F exLjLuJca)da


(1.6)


Sxp Ja
ex ]f. L


Performing the following change of variable in (I.6)


(1.7) = a i d = a, da =


gives the expression


F (u.)


(1.8)


-Y
SE6 ex p.


(06
r r00
/ir0 L

I;_s C


- j








- 67 -


Those familiar with probability functions will immediately rec-

ognize the integral term in (1.8) as the characteristic function of

the normal variable z. For those not so familiar with these concepts

the proof of this is carried out below. The first step is to substi-

tute an infinite sum for the expression eiuZs

o0

(I.9)
n=o


Considering only the integral portion of (1.8) and substituting (1.9)



(I.10) --- 6 f =fr a
r n=o -a
-00

The nth term of this series is given by:

= Ja n n; 7 r -

at


The power of the variable z in the integral is given as 2n, as for

all odd powers the integral is equal to zero. However, the nth term

is derived in such a manner that n can have all positive integer val-

ues. Applying (1.11) to (I.10)1


ai.l---" 2 ..
n=o -co n=o
(1.12)


jr I-77 J3-37 T^P.








-68 -


Let



(1.13)


then (I.12) beoomDs:


b~ (L

-00


(..27 -T-
Lj3w- o~


00o n

= n !
^b ^


& ~63
a


Substituting (1.14) into (I.8),


_-
F~cc) e x


(1.15)


{Yic l~


Equation (1.15) may be put in a somewhat more useful form by making

another series expansion for the exponential:


e p. s6 S Le6


and noting that


(1.17)


L
rt & i5 -K']0


UaflA-h


equation (I.15) becomes:


E


F,(u) =


nt
n-o


(1.16)


0 A


(I.18)








-69-


The first density rfuntion, Wl(x), is obtained by the inversion
of (I.18):


(1.19)


yf -lac(x-na) -a nu
\ n .- d _.
W; ^ ) -: 6 Z -^ ^


Inspection shows that (1.19) is of the same form as (1.8) and so the
same method may be utilized which yields:


(I.') Cx)= 3 x-

00
The delta function occurs for n = 0 as J du= (K-O).
The second order density function is found by inversion of the
second order characteristic function:


(I.2.)


F (ic Tr) ex p


The time functions are now:

h(, ) Uh-()-U(-) =

(1.22) h(,n) = o
h(o.at, x) U( +) U(- B-. 1


Wmf() Art fW (a) da

ia, hix,)t a' h4x)e t,n h
co6


4 xA Bre


- 61_ii/_. X -6t
-5lt > x > e-BIZ-l

-g14 > X> goalt( .


h(x-tBn)= o







- 70 -


The infinite integral can be broken into the sum of two integral;

one over the range OQX < Br-BItl where neither function is zero, and

the other over the range where one function or the other can be zero,

i.e. no overlapping. As the pulse is symmetrical, rather than integrate

from -B\tIoOand Bn-BAlttoBr, simply integrate from 0 to Br. Outside

of these ranges the integral is zero (see Figure 2, area A area B).


h (x,r)


-Bill


Figure 2


As r is always greater than zero, then


(1.23)


(1.24)


Ern Bit/ > 0


S7 lIt


Therefore the lower limit of the range of integration over r becomes I t/.

Applying the above reasoning and (1.24) to (I.21):







- 71 -


(1.25) F(uLU')T) = exp.


-r iti a a


-o J [ -t

" 0 It l/


The second part of (1.25) is the product of two sanctions, similar

to (1.3), therefore:
S(YL-Iti)
FU e ,0 L Ee l=i) ]
e .ada (r A (G iac I
(I.26) F(u~ ) )- ep. .
)C o


The integration of (1.26) is facilitated by the use of the normalized

correlation function of the elementary tine impulse:


(1.27)


(t) = B / (i-ti)wCndj .


Carrying out the integration over x in (1.26)



o 00 a(u '
( )/ )ca)1da FWj-Ltyl)Ar -1d .
(1.28) F ( LT,)= LT f )x- p.^ -0 IN








- 72 -


Substituting (I.27) into (I.23)


(1.29) F(U))T) = f, FL')T


-y (t) o ')

~ olc~i r -


Substituting the value of w(a), equation (I.1), into (1.29) and pro-

ceeding as before


(i.3o) W F o) u (a-o 6
n=o


where F1(u) and F1(u') are given by (1.18).

sions for Fl(u) and Fl(u') into (I.30):


(1.31) F(Ut 16)zM
m, p= o


Substituting the expres-


tiZtmin) auk'fptn)
z 6


_c mTr^tP"a-t (P+N


The second density function is found by applying the inversion
theorem to (1.31), i.e. using (3.56):

C -LuaE-(urx' 'i(nr') -
e E I


(1.32)


P7n,po


S[a+ q(t)]


Vj, (Ix w (r
(n-I -







- 73 -


Again for m n p = 0 the integral yields delta functions. Let


St r(tD]
L ,ip>
men,",p>0


(1.33)


,m! n!
! n! p!


Separating variables in (1.32) and substituting (I.33):


rA ,o-u ax- j(mrn) --
AfE d-
-ao
-Y [J4(t)]
D(1.34) C T=( (x'-o) +0 u'1'-a(pt lf o -' aAnj'n]
/ L d u




Considering only the second integral and completing the squares

Pefrmn 00 f -ollowin-ginge -of TVar4bl u I73
(1.35) E (p) f ^-J


Performing the following change of variable upon (1.35):


(1.36)


results ins


u1nEX L'-3(P+d


(1.37) 6
V/P-+ -


1fn
(qua-)
Zarnln


jv.


v= cL'~Tn~


Sun
,/p+-n


00

- V b(I-5(pPr4d







-74 -


Equation (I.37) is of the form treated previously, (1.8), therefore:


Substituting (1.38) for the value of the second integral in (I.34)s


cx,r, T., = a S-(tE S(x-o)(x'-o)


(1.39)


+AE 4(pA4n))a
,/p-+


M ux(pti)-nx'-ain(p+n)]
lt 0


The integral in (1.39) is of the same form as (1.8), therefore:


Ex- >n p+L,
(I.40) a[mnrnp +np]
/ 4 rnp mr p
S p+n
Substituting (1.33) and (1.40) into (1.39) yields the second probability
density functions


(1.41) W(CxAITo)z


( I-o)S Qr'-o)
00 mrni P
&--en! n
P+ehn >o




vni rnmp r7np


a. rI


'I


(I18 y-a`(fQ)J (o4< ac)
df.38)n
\/+n


-Cx' (>Pet n)]
;(p-^n)t-'














APPENDIX II


POWER SERIES APPROXIMATION

The function D(u) in the case of triangular time pulses is
given by:


(IDI.L -


Substituting (II.1) in the general expression for the first density
func tion:


ncl -o 6 a
(II.2) (x0 (x-) Z Vau:x 6a du.
V (X) 1(:


Expanding (11.2) and collecting like powers of K I


f(_--- j
-00


- /FK
q-I (-OuE )
J au
;a


I


+ -iu -J L ]
-0o
,-(.u -V- 1
-/ 1-c3 eJ
-OA


A-, x -"L u '
^ -(- te)


- 75 -


-X- ax
f^^^-ai^-Ll ^_


^ L



,^_~ s


(11 .3) W, Cx) =


00 *
E- Coo
& 4-4-7


4 i'Vrlr-







- 76 -


A fundamental theorem of courier Transfonns states


F riFc[)] oLT (uCxdx
(n F [ L O-- J _
(II*- ) -CO


where f(x) is the transform of F(u). Considering the integrals of
equation (13) one at a time:

(.5) iLL
00 -L'u.fei
co a

The lower limit on the integral over x has been made zero, as x is
the envelope of ths linearly rectified voltage and it cannot have
a negative value.






The result of the first integration in (11.6) is obtained from
equation (1), Table 28 in Bierens de Haan36 The second integral
is evaluated by using the error function, equation (3), p. 387

in Bate with the chane of variable t y/


(11-7) j E OL7 (-








- 77-


~LLAK -(.gL
du.
:- 2..2


- 2C77T)
d~-


?C 92.zdJ
0


Applying equation (7), p. 253 in Bateman38


i F7(- J -


'.
F 3
-E


Applying equation (13), p. 255 in Bateman38 using the change of

variable y = i2/o- in (II.9):


(Ii ,LU.X U -
(II.10) d UL
(- D L





("*") LU,.


2(.7) G
- W^(z


F, Q,






F,^i~ i


Applying equation (7), p. 253 in Bateman3q


.-, A-ar ,F


(11.12)


Applying equation (13), p. 255 in Bateman38 using the change of

variable y z2/cO- in (II.12)


-cCAA-LCp


(11.13)
~ao


= (2 X r/-7) (. )


(11.8)


0L 3
e-w


(11.9)









-78-


S(-W
--00


( L7) = Xo
0
(m170l 91di 1r'


. o I ti '
f 3 U
-' -t 1


= a(l) d-


o


(11.15)


- ac/(r) cr


This result is obtained by two successive integration by parts after

a change of variable t = y2/0-'


^t u.-t







-( C.-;A
- n L-L?


-t
h'.
-t.


d t F 0, )t


0


[ ( 16 4r i.H


S(p Lz 3
0 17) nr-x Li(Ij~l'I~('bi~r11


(.114)


dv = C


(II.16)


(11.17)
/ t 3


= aiffr) r-Kx


f & [,6 F, ( -/1.








-79 -


Substituting (II.5), (II.6), (11.7), (II.10), (II.13), (II.1), (11.15),

(11.16) and (11.17) into (II.3),


x( ) a- ) + 1 ['1 t 2r


-3t


'nj E ar~ de


I FJ
~ r fi2
;IZ'^


1?
i.


r6a^2. -w)+i^2
f3a








-f1 6r i+ 3L 0 VI
-^.~~1.6 [,^,<,''^4r ;'' 3' 3 -'
3 I




V r


(11.18)


4
,;,


4+ I .







- 80 -


To determine if equation

tion to the atmospheric noise

grate (II.18) with respect to

by term


(II.18) is a satisfactory approxima-

distribution, it is necessary to inte-

x. Carrying out this integration term


(11.19) J c C-o)< = I .
o






0


1('


(12.a) #,) 1
T- RI 0 k~


( 1.23) oIff fF, (-,
-- -







- 81 -


o


L L
[,a1 3J+ K
+8~ -6 3


-sr a (J '9


- ,
F3 -




3


F, )


,~


/7A-J ^^^4


pT &
r


y,1I


(11.25)


3 ( ; +

_X"L


y r)/-


(11.24)


- 3 (U)


F,( )L \a







- 82 -


Substituting equations (II.19) through (11.25) into (II.18) and
collecting like terms


PCw) =I + x -c-
^

Lx 3 3 -
+I~3c~~~ 1,12, ~~-v '
( 'JUI'l La 3 j r c


+- Ly-
+ ;rr{C LiFC (AxY1


fS^ / '- + F,( 1-
XL ^ FW^) r=13 i ,V.
3r 3

24- F( r7-r
r5- E I 2l(


F3 .


A-,


3LfE
+srE


6r 2)
[fw-


4, X,- ,
^^'c]



.1*it^ j


(11.26)


+ t ('11
4.r(Jf


t-


Ilin~"


F ., ) < < ,I,-- j
















BIBLIOGRAPHY


1. Schottky, W., Ann, ea Phvsik. v. 57, p. 541, 1918.

2. Hull, A. W., and Williams, N. H., Physical Rev., v. 25, p. 173, 1925.

3. Nyquist, H., Physical Re.., v. 32, p. 110, 1928.

4. Bell, D. A., Jour. IL.. Lct. iE., v. 82, p. 522, 1938.

5. Landon, V. D., Proc. "g, v. 24, p. 1514, 1936.

A. Jansky, K. G., Proc. ILE v. 27, p. 763, 1939.

7. Harris, W. A., R Baeviel. v. 5, p. 505, 1941.

8. Thompson, B. J. and North, D. 0., RCA review, v. 5, p. 371, 1941.

9. Williams, F. C., Inst. *of gel gap&., v. 14, p. 325, 1939.

10. Peterson, H. 0., P_ I&, v. 23, p. 128, 1935.

U1. Landon, V. D., Proc. IE, v. 29, p. 50, 1941.

12. Rice, S. 0., Bell Sr stam Teh Jour., v. 23, p. 282, 1944.

13. Rice, S. O., Bell Stem IT ch Lc. v. 24, p. 46, 1945.

14. Rice, S. 0., Bel System Te p Jo=t., v. 27, p. 109, 1948.

15. Middleton, D., Jgur. Ab*i, .X., v. 17, P. 778, 1946.

16. Middleton, D., Quart. Apl. jth,, v. 7, p. 128, 1949.

17. Bennett, W. R., Jour. A.. Acou~ =t, s, v. 15, p. 165, 1944.

18. North, D. 0., Paper read before IRE, Jan. 23, 1944.

19. Van Vleck, J. H., RRL Report #411-5-1.

20. Kac, U., Bu1. A e. MaI l Soc., v. 49, P. 314, 1943.


- 83 -








-84 -


21. Seigert, A. J. F., and Kac,, ,M &ur A4ZP MhM., v. 18, p. 383,
1947.

22. Ragassine, J. R., ~EPr. IS v. 30, p. 277, 1942.

23. Hamburger, G. L., WiMleAs j g., v. 25, p. 44, 1948.

24. Burgess, R. B., and Thomas, H. A., Paper R.R.B./090, Radio Division,
National Physical Laboratory, Teddington, England.

25. Middleton, D., Jour. 4p. EPha,, v. 22, p. 1143, 1951.

26. Kendall, M. The Advapced Theory oL Statistics. Charles Griffin
and Co., London, p. 90, 1945.

27. Cramer, H., Mathematical Methods a Statistica. Princeton Univer-
sity Press, Princeton, 1951.

28. Hurwitz, H. and Kac, M., AIE. Mtb, sars., v. 15, p. 173, 1944.

29. Upensky, J. V., Introduction to Mathematical Probability. McGraw-
Hill, New York, p. 131, 1937.

30. Whittaker, E. T., and Watson, G. N., A Course of Modern Analysis.
Univ. Press, Cambridge, p. 337, 1950.

31. Sneddon, I. N., Fourier Transforms. McGraw-Hill, New York, 1951.

32. Mellin, H., Ata 8 c aS. Fennioaes v. 21, p. 1, 1896.

33. Curry, E. B., Aa. k. Mon~Ail v. 50, P. 365, 1943.

34. Titchmarsh, E. C., Introduction to the Theory of Fourier Integrals,
Claredon Press, Oxford, 1948.

35. Campbell, G. A., and Foster, R. M., Fourier Integrals for Practical
Applications. Monograph, Bell Telephone System, 1942.

36. Bierena De Haan, Nouvelles Tables D' Integres Definies. Stechert
and Co., New York, 1939.

37. Bateman, H., and Staff, Tables f j.Interal Transforms, v. 1, McGraw-
Hill, New York, 1954.

38. Bateman, H. and Staff, Higher Transcendental Functions, v. 1,
McGraw-Hill, New York, 1953.










-85 -


39. Noie study, Fabrication of Noise Measuring Equipment and Data
Collection an Collation Prora. Final Report, Contract
No. AF-08(169)-138, Mg. and Ind. Eap. Sta., University of
Florida, Gainesville.

40. Investigation of Atosheric Radio Noise Progress Report 1 3
Contract No. AF-19(604)-876, Eig and Ind. Exp. Sta., University
of Florida, Gainesville.


1


..,..
















BIOGRAPHY


John Marshall Barney was born in Baltimore, Maryland, on December 13,

1924. He began his undergraduate studies at North Carolina State College

in 1943 while in the Armed Services. After receiving his discharge from

the Armed Services he attended the University of Florida where he re-

ceived the degree of Bachelor of Electrical Engineering in 1948.

In 1950 he received the degree of Master of Science in Electrical

Engineering from the Massachusetts Institute of Technology, and since

then has done work leading to the degree of Doctor of Philosophy at the

University of Florida. The major field of study was electrical engi-

neering, with minors in physics and mathematics.

While at the Massachusetts Institute of Technology, Mr. Barney was

employed as a teaching assistant. Since 1950 he has been on the staff

of the Electrical Engineering Department at the University of Florida and

has been actively engaged in studies of atmospheric noise.

He is a member of the Sigma Tau honorary engineering fraternity.


- 86-












This dissertation was prepared under the direction of the chairman

of the candidate's supervisory committee and has been approved by all

members of the committee. It was submitted to the Dean of the College of

Engineering and to the Graduate Council and was approved as partial ful-

fillment of the requirements for the degree of Doctor of Philosophy.

August 9, 1954





Dean, College of Engineering




Dean, Graduate School


SUPERVISORY COIALITTEEt


Chairman







^^.(Qu^





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AUTHOR: Barney, John -
TITLE: A statistical modelof atmospheric oise .record number: 559286)
PUBLICATION DATE: 1954





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