 Title Page 
 Acknowledgement 
 List of Tables 
 List of Illustrations 
 Table of Contents 
 Statistical noise models ... 
 Method of solution 
 Derivation of general density... 
 atmospheric noise model 
 Power series approximation as a... 
 Solution by method of integral... 
 Conclusions 
 Example of use of formulae 
 Power series approximation 
 Bibliography 
 Biography 
 Copyright 

Full Citation 
Material Information 

Title: 
A statistical model of atmospheric noise .. 

Physical Description: 
86 leaves : ; 28 cm. 

Language: 
English 

Creator: 
Barney, John Marshall, 1924 

Publication Date: 
1954 
Subjects 

Subject: 
Radio  Interference ( lcsh ) Electrical Engineering thesis Ph. D Dissertations, Academic  Electrical Engineering  UF 

Genre: 
bibliography ( marcgt ) nonfiction ( marcgt ) 
Notes 

Thesis: 
Dissertation (Ph. D.)  University of Florida, 1954. 

Bibliography: 
Bibliography: leaves 8385. 

General Note: 
Manuscript copy. 

General Note: 
Vita. 
Record Information 

Bibliographic ID: 
UF00085545 

Volume ID: 
VID00001 

Source Institution: 
University of Florida 

Rights Management: 
All rights reserved by the source institution and holding location. 

Resource Identifier: 
aleph  000559286 oclc  13454332 notis  ACY4735 

Table of Contents 
Title Page
Page i
Acknowledgement
Page ii
List of Tables
Page iii
List of Illustrations
Page iv
Table of Contents
Page v
Page vi
Statistical noise models  introduction
Page 1
Page 2
Page 3
Page 4
Method of solution
Page 5
Page 6
Derivation of general density functions
Page 7
Page 8
Page 9
Page 10
Page 11
Page 12
Page 13
Page 14
Page 15
Page 16
Page 17
Page 18
Page 19
Page 20
Page 21
Page 22
Page 23
Page 24
Page 25
Page 26
Page 27
Page 28
Page 29
Page 30
Page 31
atmospheric noise model
Page 32
Page 33
Page 34
Page 35
Page 36
Page 37
Page 38
Page 39
Page 40
Page 41
Page 42
Page 43
Page 44
Page 45
Power series approximation as a solution of general density function
Page 46
Page 47
Page 48
Page 49
Page 50
Page 51
Solution by method of integral transforms
Page 52
Page 53
Page 54
Page 55
Page 56
Page 57
Page 58
Page 59
Page 60
Page 61
Page 62
Page 63
Conclusions
Page 64
Example of use of formulae
Page 65
Page 66
Page 67
Page 68
Page 69
Page 70
Page 71
Page 72
Page 73
Page 74
Power series approximation
Page 75
Page 76
Page 77
Page 78
Page 79
Page 80
Page 81
Page 82
Bibliography
Page 83
Page 84
Page 85
Biography
Page 86
Page 87
Copyright
Copyright

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
NonStationary 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 amplitudemodulated receiver when
the input waveform consists of pulses of much shorter duration than
the reciprocal of the intermediatefrequency 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
rootmeansquare, 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(tP 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 xlf 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
onedimensional 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 "oisentgenerating"
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.
Nooaionary 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 rootmeansquare 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 = tt, 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 edimensional 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( XOIx,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 sorder 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 (xO)
T J=
and
(3.25) P (olx,I ,x) t,X: ) 7 (, xxto') (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 quasistationary 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_
=me 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 kfold multiple integral into a simple
kth power of a 3fold 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
D0 0
(3.37) / ) . j
e f d &i'" I J ^
(A.z 1476 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 tjT = 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 rootmeansquare 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(xr), 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(010) 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(101)
2. early Ho l anda Noise 0(10104) 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) Photomultiplier 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!
no
Lfnau 2.
oU
W1(x) is obtained by the inversion of equation (3.60):
(3.61) (A) =
X ;L aF T
n=(
I E cSxo),
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(Uu
.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 (pin)j t m
IX or "p'p n
\/v rr? man4
 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 shockexcited 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 ) wcdafIj
 34 
The last integral in (4.6) may be altered by making the substitution:
auV 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 ni 0
The value of the integral on the righthand 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 aTJ 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 5154
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 1L~iL i J i J i i i i 1 ii i i i
11
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(tui^ '
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 Xy 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"JA 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 characteristic 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 (xox) 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(xo.+ Y I cd,. iA _/ f
r'a 2Ji L.4WLu.) ) 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 ) + J1zi~
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 lognormal 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(15) =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 C5).
Eliminating F(1e)/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
doo
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 (e4)
yields
(6.20)
a I u L I cos(eO)
o 'f
Because of the periodic nature of the second integrand in (6.20):
ec aA co5e0
Sc. 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 zeroorder 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+
C5 C> K.
6.27) 5(4 f
CC
 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.
CC 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 amplitudemodulated
receiver, shockexcited 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 I77 J337 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
UaflAh
equation (I.15) becomes:
E
F,(u) =
nt
no
(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(xna) 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= (KO).
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 > eBIZl
g14 > X> goalt( .
h(xtBn)= o
 70 
The infinite integral can be broken into the sum of two integral;
one over the range OQX < BrBItl 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 BnBAlttoBr, 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(YLIti)
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 / (iti)wCndj .
Carrying out the integration over x in (1.26)
o 00 a(u '
( )/ )ca)1da FWjLtyl)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 (ao 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"at (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
(nI 
 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 ,ou 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 ollowinginge 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(I5(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(xo)(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
( Io)S Qr'o)
00 mrni P
&en! n
P+ehn >o
vni rnmp r7np
a. rI
'I
(I18 ya`(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
qI (OuE )
J au
;a
I
+ iu J L ]
0o
,(.u V 1
/ 1c3 eJ
OA
A, x "L u '
^ ( te)
 75 
X ax
f^^^ai^Ll ^_
^ L
,^_~ s
(11 .3) W, Cx) =
00 *
E Coo
& 447
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/
(117) 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
., Aar ,F
(11.12)
Applying equation (13), p. 255 in Bateman38 using the change of
variable y z2/cO in (II.12)
cCAALCp
(11.13)
~ao
= (2 X r/7) (. )
(11.8)
0L 3
ew
(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 LL?
t
h'.
t.
d t F 0, )t
0
[ ( 16 4r i.H
S(p Lz 3
0 17) nrx Li(Ij~l'I~('bi~r11
(.114)
dv = C
(II.16)
(11.17)
/ t 3
= aiffr) rKx
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 Co)< = 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, )
,~
/7AJ ^^^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( r7r
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 #41151.
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. McGrawHill, 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,
McGrawHill, New York, 1953.
85 
39. Noie study, Fabrication of Noise Measuring Equipment and Data
Collection an Collation Prora. Final Report, Contract
No. AF08(169)138, Mg. and Ind. Eap. Sta., University of
Florida, Gainesville.
40. Investigation of Atosheric Radio Noise Progress Report 1 3
Contract No. AF19(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^
UF Libraries:Digital Dissertation Project
Internet Distribution Conseni 4gFumeent
In reference to the following dissertation: . .. ,ouS
00 e3Y brilA
AUTHOR: Barney, John 
TITLE: A statistical modelof atmospheric oise .record number: 559286)
PUBLICATION DATE: 1954
I, (71 rne as copyright holder for the aforementioned
dissertation, hereby grant specific and limited archive and distribution rights to the Board of Trustees of
the University of Florida and its agents. I authorize the University of Florida to digitize and distribute
the dissertation described above for nonprofit, educational purposes via the Internet or successive
technologies.
This is a nonexclusive grant of permissions for specific offline and online uses for an indefinite term.
Offline uses shall be limited to those specifically allowed by "Fair Use" as prescribed by the terms of
United States copyright legislation (cf, Title 17, U.S. Code) as well as to the maintenance and
preservation of a digital archive copy. Digitization allows the University of Florida to generate image
and textbased versions as appropriate and to provide and enhance access using search software.
This grant of permissions prohibits use of the digitized versions for commercial use or profit.
ofCopyri ol
JTo h^9 ^ >Qr J? e t
Printed or Typed Name of Copyright Holder/Licensee
Printed or Typed Address of Copyright Holder/Licensee
Personal information blurred
d j,,'o0Y
Date of Signature
Please print, sign and return to:
5/30/08 3:10 PM
2 of 3

