STATISTICAL METHODS IN RADIO WAVE PROPAGATION Proceedings of a Symposium held at the University of California, Los Angeles. June 18-20, 1958 Edited by W. C. HOFFMAN SYMPOSIUM PUBLICATIONS DIVISION PERGAMON PRESS LONDON • OXFORD · NEW YORK · PARIS 1960 PERGAMON PRESS LTD. 4 & 5 Fitzroy Square, London WJ Headington Hill Hall, Oxford PERGAMON PRESS INC. 122 East 55th Street, New York 22, NY, 1404 New York Avenue, N,W., Washington, D.C. P.O. Box 47715, Los Angeles, California PERGAMON PRESS S.A.R.L. 24 Rue des Ιcoles, Paris PERGAMON PRESS G.m.b.H Am Salzhaus 4, Frankfurt Copyright 1960 PERGAMON PRESS INC. Library of Congress Card Number 59-10524 Printed in Northern Ireland at The Universities Press, Belfast PREFACE IN recent years there has appeared to exist an ever-growing feeling on the part of radio scientists and engineers in the propagation field that methods should be found to aid them in reducing and interpreting their data. Many felt that such methods ought to exist somewhere in modern statistical theory, but that exactly what this material might be and where it might be found were in general unknown. In point of fact, many such recently-developed techniques do exist and offer considerable potential value for radio propagation research. However, much of this material is buried in the recent statistical literature in such a way as to be relatively inaccessible to the non-specialist. For this reason the University of CaUfomia, Los Angeles, decided to hold a symposium* devoted to the subject of statistical methods in radio wave propagation. Such a symposium, bringing together the two widely-separated disciplines of modem statistical theory and propagation of radio waves should have considerable value in providing new and powerful tools for the radio scientist and research problems of interest and significance for the statistician. Not all of the important current statistical techniques are fully represented in the proceedings of this symposium, however. For example the statistical design of experiments and the analysis of variance and covariance are treated only in passing, for these subjects are so broad as to be deserving of separate treatments in their own right. To remedy this apparent omission to some extent a bibliography of standard reference works in statistics has been supphed at the end of this preface. It is behoved that the interested reader will be able to obtain sufl&cient material from these references to meet the needs of most practical situations. Statistical considerations arise in the field of radio wave propagation in two ways. The first occurs when the phenomena in question involve a combination of physical and statistical structures. For example, in the scattering of radio waves by a turbulent medium, the physical structure is imposed by Maxwell's equations but the dielectric "constant" appearing in these equations is a random function of space and time, thus giving rise to a statistical—or at least, probabihstic—structure. The second way in which statistical considerations enter is via uncontrolled factors in experimentation, e.g., the fine-structure of the troposphere in propagation of microwaves and the phenomenon of fading in ionospheric propagation. A laboratory type of controlled experiment, in which all factors are held constant except one, can seldom be obtained in radio wave propagation. Almost invariably, there exist factors beyond the control of the experimenter. If these factors are inaccessible to measurement as well, little can be done. However, if these uncontrolled factors can be measured, their existence may actually be advantageous, given the appropriate statistical techniques. As MooD put it, "Most experimental work today is based on the rule: 'Keep aU • A grant from the National Science Foundation in partial support of the symposium is gratefully acknowledged. viii PBEFACE variables constant but one', an ancient and erroneous dictum which guarantees a high degree of inefi&ciency. One well-designed experiment, taking account of all relevant factors, is worth dozens or even hundreds of experiments which study one factor at a time keeping the others constant." In line with the two types of statistical contexts indicated above, the papers in these proceedings are classified mainly into two broad categories: I. Those dealing primarily with statistical theory and methodology. II. Papers which emphasize radio propagation phenomena having a joint statistical and physical structure. A third category of considerable practical importance has been added: III. Instrumentation. None of these categories is sufνiciently distinctive to prevent some overlap, and this fact will be evident in nearly aU the papers in these pro­ ceedings. In particular it is often especially difficult to distinguish between papers falhng in the first of the above classes and in the second. The final session of the symposium was devoted to a round table discussion of the topic "The Current State of the Art in Statistics of Radio Wave Propagation and Probable Future Trends." A number of outstanding problems were singled out for attention in the course of the discussion. Several of the participants (W. S. AMENT, R. B. BANERJI, S. A. BOWHILL) were of the opinion that refinements and extensions of the Monte Carlo technique offer considerable promise for solving the complex problems in sampUng distributions and significance tests which often arise nowadays in radio propagation research. By the term '*Monte Carlo" is of course meant a samphng experiment based on a known duahty between the probability distribution governing the event of interest and that for another random event which can be more readily generated in the laboratory. The availabihty of large high-speed digital computers for this purpose makes the Monte Carlo approach much more feasible, and emphasizes the value of further research in the purely mathematical problems involved. Numerous new tables of random functions, with various correlation functions, will apparently be required. This led to the suggestion of some organization, a sort of "Random Numbers, Inc.", which would keep available digital computer programs for generating random functions for the interested experimenter, who would supply the appropriate correlation function at the time of computation. E. L. CROW suggested that one possible answer to the increasing complexity of statistical problems in the radio propagation field may be the use of non-para­ metric, or distribution-free, statistical methods. At present such techniques have been developed mainly for independent samples, and generalization to dependent observations would be required. In this connection W. C. HOFFMAN mentioned an old paper of A. M. MOOD {Annals of Math. Stat. 12 (1941), 268) dealing with the joint distribution of the medians in sampling from a multivariate population. CROW also mentioned the fact that the statistical design of experiments deserves more use among radio experimenters than it apparently receives at present, and it was agreed that some generahzation of presently-available experimental designs and hnear hypotheses seems indicated. The similarity between radio propagation experiments and meteorological investigations was brought out—both are ordinarily characterized by numerous uncontrolled factors—^and it was suggested that some guidance in experimental design might often be obtained from the corresponding PBEFACE ix meteorological situation. Ε. W. PIKE pointed out the value, in designing a good experiment, of precise formulation, in advance, of the hypothesis to be tested. The probabihstic situation in tropospheric forward-scattering came in for a good deal of discussion. W. S. AMENT and J. H. CHISHOLM discussed the role of multiple scattering in tropospheric forward scatter. AMENT stated that the apphcation of the weak law of large numbers to the multiple-scattering process would insure a stable signal. W. C. HOFFMAN suggested that a desirable approach here might involve the extension of J. KELLER'S theory of diffracted rays to the statistical case. E. W. PIKE noted that "we are all in a multi million dollar industry essentially manufacturing answers", and cited the numerous pitfalls awaiting the experimenter in radar. A good many of these can be detected in the course of analysis of the data by means of determining confidence intervals and what PIKE termed "guard computations". By this is meant the computation of several different functionals of the data and their checking against each other to ensure accuracy and quality control of the data. S. A. BOWHILL raised the question of why almost all auto-correlation functions encountered in ionospheric radio propagation are Gaussian in shape. He outhned two possible mechanisms which could account for the apparent favoring of Gaussian autocorrelograms. One was the convolution of a number of Gaussian functions, as might occur for instance in a receiver with a large number of bandpass stages. The other situation is that of a signal composed of an intermediate number of components, i.e., a function midway between the resultant of two interfering signals and the result of passing random noise through a bandpass filter. BOWHILL felt that a general characterization of the situations leading to a Gaussian auto-correlation function would be a valuable research contribution. The question of non-stationarity in stochastic processes (by which is meant the change of the underlying probability structure with time) came in for considerable mention. In fact this appears to be one of the outstanding current problems for radio scientists and engineers working in the propagation field. Several possible approaches to the non-stationary case were discussed, notably the classical method of estimation and possible removal of trend and cycles from the data, the recently-developed locally stationary stochastic processes of R. A. SILVERMAN, and an approach in terms of higher order multivariate moments developed by BLANC-LAPIERRE (see bibliography). J. H. CHISHOLM mentioned his method for detecting non-stationarity. This technique consists of breaking up the total record into a number of shorter sublengths, estimating the power spectrum for each sublength by the method of BLACKMAN and TUKEY (see bibho- graphy), and poohng those consecutive spectra as locally stationary which all fall within the same confidence band. A batching factor to adjust properly the width of the overall confidence band may be required. The question of the best method of estimating the power spectrum from a sample provoked a good deal of discussion. The relative advantages of transforming the estimated auto-correlation function according to an appropriate form of the Wiener-Kliintchine relation and of direct estimation of the spectrum were discussed by BANERJI and ROSENBLATT. The relative desirabihty of each approach came ÷ PBEFACE down to the type of computing equipment available and the possibihty of appropriate smoothing. Reference was made to the book of GREKANDER and ROSENBLATT (see bibliography) for a theoretical treatment of the matter. S. A. BOWHILL discussed the question of samphng fluctuations in the "tail" (i.e., the higher order lags) of the estimated auto-correlation function, and possible biases which might enter in the estimated spectrum. It was generally agreed that an appropriate weighting function could suflftciently reduce the deleterious effect of these higher order sampling fluctuations upon the quality of the estimate. S. A. BOWHILL outlined the present status of investigations of ionospheric winds and drifts, and enumerated what he felt were the outstanding remaining problems. The ionospheric drifts program in the U.S.A. is being carried out at a lower pace and a lower frequency than the very considerable IGY program in Europe. The statistics of ionospheric drifts are pretty well known, but the physical mechanisms are considerably more comphcated than was originally thought. For instance, the motion apparently consists of large-scale drifting ionospheric irregularities with random turbulent motions superimposed, each type having its own characteristic anisotropy. At 70 kc/s (the U.S.A. frequency) the drift motion is swamped by the random movements, while the situation is reversed at 2 mc/s. The drift pattern in the ionosphere is determined by auto-correlation measurements of the diffraction pattern on the ground, and the estimation problems associated with the sample auto-correlation function occur in their most acute form. BOWHILL further stated that one of the principal problems remaining is the nature of the irregularities at 80-100 kilometers altitude. These look entirely different at Low Frequency and at High Frequency, being reflected in the first case and scattered in the second, and chances for agreement between those working in each frequency range look pretty slim at present. R. B. BANEBJI stated that one of the outstanding statistical problems in the study of ionospheric drifts was the extension of the lag analysis of BBIGGS and PAGE (and its generalizations) to the anisotropic case. A second important question remaining to be investigated is the connection between the diffraction pattern observed at the ground and the diffraction pattern in the ionosphere. Bibliography of Selected Statistical Reference Works The references which follow are believed to provide a good cross-section of current statistical literature, especially as it pertains to the needs of the radio scientist. The references are listed under several headings, and in each category the more elementary treatments are usually given first. When appropriate, brief comments or descriptions are appended. GENERAL WORKS IN PROBABILITY AND STATISTICS 1. DIXON, W. J. and MASSEY, FRANK J., JR. Introduction to Statistical Analysis, 2nd ed., McGraw-HiD, 1957. 2. BENNETT, C. A. and FRANKLIN, N. L. Statistical Analysis in Chemistry and the Chemical Industry, Wiley, 1954. 3. PEARSON, E. S. and HARTLEY, H. O. Biometrika Tables for Statisticians, Camb. Univ. Press, 1954. 4. OSTLE, B. Statistics in Research, Iowa State College Press, 1954. 5. HALD, A. Statistical Theory With Engineering Applications and Statistical Tables and Formulas, Wiley, 1952. 6. HoBL, P. G. Introduction to Mathematical Statistics, 2nd ed., Wiley, 1957. 7. MOOD, A. M. Introduction to the Theory of Statistics, McGraw-Hill, 1950. 8. FELLER, W. An Introduction to Probability Theory and Its Applications, vol. I, 2nd ed., W٧ey, 1957. 9. CRAMER, H. Mathematical Methods of Statistics, Princeton Univ. Press, 1946. 10. RAO, C. R. Advanced Statistical Methods in Biometrie Research, Wiley, 1952. 11. KENDALL, M. G. Exercises in Theoretical Statistics, Hafner, 1954. 12. KENDALL, M. G. The Advanced Theory of Statistics, vols. I and Π, Grifiin, 1948. SAMPLING THEORY 1. COCHRAN, W. G. Sampling Techniques, Wiley, 1953. 2. DEMING, W. E. Some Theory of Sampling, Wiley, 1950. 3. HANSEN, M. H., HURWITZ, W. N. and MADOW, W. G. Sample Methods and Theory, vols. I and II, Wiley, 1953. DESIGN OF EXPERIMENTS 1. BROWNLBB, K. A. Industrial Experimentation, 4th ed.. Chemical Publishing Co., 1953. 2. DAVIBS, OWEN L. The Design and Analysis of Industrial Experiments, Hafner, 1954. 3. COCHRAN, W. G. and Cox, G. Μ. Experimental Designs, Wiley, 1950. 4. OsTLE, B. Statistics in Research, Iowa State College Press, 1954. 5. BENNETT, C. A. and FRANKLIN, N. L. Statistical Analysis in Chemistry and the Chemical Industry, Wiley, 1954. xii Bibliography of selected statistical reference works Distribution-free methods 1. DIXON, W. J. and MASSE γ, FRANK, J., JR. Introduction to Statistical Analysis, 2nd ed., McGraw-Hill, 1957. Excellent elementary treatment. 2. MOOD, A. M. Introduction to the Theory of Statistics, McGraw-Hill, 1950, Ch. 16. 3. VAN DER WAERDEN, B. L. Mathematische Statistik, Springer, 1957. 4. FRΔSER, D. A. S. Nonparametric Methods in Statistics, Wiley, 1957. Correlation and regression 1. QuENOUiLLE, M. H. Associated Measurements, Academic Press, 1952. 2. RAO, C. R. Advanced Statistical Methods in Biometrie Research, Wiley, 1952. 3. KENDALL, M. G. The Advanced Theory of Statistics, vols. I and II, Griffin, 1948. 4. WiLKS, S. S. Mathematical Statistics, Princeton Univ. Press, 1947. Time series and stochastic processes 1. WOLD, H. and JURΙEN, L. Demand Analysis, Wiley, 1953. Although the applications dealt with in this book are in economics, the first part of the book contains an excellent elementary treatment of stochastic processes. 2. WOLD, H. A Study in the Analysis of Stationary Time Series, 2nd ed., Almquist & Wiksell, 1954. An appendix by P. WHITTLE contains an excellent summary of recent developments in the field. 3. DAVENPORT, W. N., JR. and ROOT, W. L. Introduction to Random Signals and Noise, McGraw-Hill, 1958. 4. BLANC-LAPIERRE, A. and FORTET, R. Theorie des Fonctions Alιatoires, Masson, 1953. An excellent treatise, including many radio problems, in French. An appendix by J. KAMPE DE FΙRIET treats the mathematical theory of turbulence. 5. BARTLETT, M.S. An Introduction to Stochastic Processes, Camb. Univ. Press, 1955. A good, moderately elementary treatment, with many excellent applications. 6. WAX, N., editor. Selected Papers on Noise and Stochastic Processes, Dover, 1954. Contains a good selection of classical papers on the title subject. 7. DOOB, J. L. Stochastic Processes, Wiley, 1953. Estimation of power spectra 1. PBESS, H. and TUKEY, J. W. Power Spectral Methods of Analysis, Bell Telephone Laboratories Monograph 2606. 2. BLACKMAN, R. B. and TUKEY, J. W. "The Measurement of Power Spectra from the Point of View of Communication Engineering", Bell System Tech.J. 37 (1958). Pt. I, 185-282. Pt. II, 485-569. 3. GBENANDEB, U. and ROSENBLATT, M. Statistical Analysis of Stationary Time Series, Wiley, 1957. 4. GOODMAN, N. R. On the Joint Estimation of the Spectra, Gospectrum and Quadrature Spectrum of a Two-Dimensional Stationary Gaussian Process, Scientific Paper No. 10 of the New York Univ. Engineering Statistics Laboratory, 1957. Bibliography of selected statistical reference works xiii 5. BLANC-LAPIERRE, A. "Remarques sur Fanalyse harmonique des fonctions alιatoires". Revue Scientifique 85 (1947), 1027-1040. Contains a treatment of one possible approach to the non-stationary case. 6. SILVERMAN, R. A. Locally Stationary Random Processes, Div. Electromag. Res., Inst. Math. Sei., New York Univ. Res. Rep. No. MME-2 (1957). The m-Distribution—A General Fonnula of Intensity Distribution of Rapid Fading MiNORU NAKAGAMI Faculty of Engineering, Kobe University Kobe, Japan Abstract—This paper summarizes the principal results of a series of statistical studies in the last seven years on the intensity distributions due to rapid fading. The method of derivation and the principal characteristics of the m-distribution, originally found in our h.f. experiments and described by the author, are outlined. Its applicability to both ionospheric and tropospheric modes of propagation is fairly well confirmed by some observations. Its theoretical background is also discussed in detail. A theoretical interpretation of the log-normal distribution is given on the basis of this formula. An extremely simplified method is presented for estimating the improvement available from various systems of diversity reception. The mutual dependences between the m-formula and other basic distributions are fully discussed. Some generalized forms of the basic distributions are also investigated in relation to the m-formula. Two methods of approximating a given function with the m-distribution are shown. The joint distribution of two variables, each of which follows the m-distribution, is derived in two dijQferent ways. Based on this, some useful associated distributions are also discussed. 1. INTRODUCTION IN recent years, radio engineering requirements have become more stringent and necessitate not only more detailed information on median signal intensity, but also much more exact knowledge on fading statistics in both ionospheric and tropo­ spheric modes of propagation. Such circumstances have forced a large number of extensive experiments and numerous theoretical investigations to be performed on the intensity distribution of fading under various conditions. In order to describe closely the results of these comprehensive observations, diverse forms of the distribution have been presented up to now. Among them, the following three may be regarded, in view of practical uses, as the representatives: One is the Rayleigh distribution i)(Ä)=^.-<*'/"), (1) where Ω = i?^, time average of R^, This was, as is well known, derived theoretically by Lord RAYLEIGH (1880). Since PAWSEY'S (1935) experimental verification in h.f., many investigators have also confirmed the applicability of this form to fading in both modes of propagation, under scattering conditions at least. Another is the log-normal distribution Pix) = e-<^-W , (2) where χ denotes signal intensity in terms of db. This seems to have been first 3