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## Material Information- Title:
- A statistical model of atmospheric noise ..
- Creator:
- Barney, John Marshall, 1924-
- Publication Date:
- 1954
- Language:
- English
- Physical Description:
- 86 leaves : ; 28 cm.
## Subjects- 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 ) - Genre:
- 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.
## Record Information- Source Institution:
- University of Florida
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- University of Florida
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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.
- Resource Identifier:
- 000559286 ( ALEPH )
13454332 ( OCLC ) ACY4735 ( NOTIS )
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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) 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 (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 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^ 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 non-exclusive grant of permissions for specific off-line and on-line uses for an indefinite term. Off-line 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 text-based 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 |

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