Probability And Information Theory, With Applications To Radar.pdf-资料库

O T H ER T I T L ES P U B L I S H ED IN T HE S E R I ES (FORMERLY PERGAMON SCIENCE SERIES ELECTRONICS A ND W A V E S) Vol. 1 Signal Noise and Resolution in Nuclear Counter Amplifiers by A. B. GILLESPIE Vol. 2 Scintillation Counters by J. B. B I R KS Vol. 4 Physics and Applications of Secondary Electron Emission Vol. 5 Millimicrosecond Pulse Techniques (2nd edition) by H. B R U I N I NG 6 Introduction by I. A. D. L E W IS and F. H. W E L LS Computers to Electronic Analogue Vol. by C. A. A. W A SS Vol. 7 Scattering and Diffraction of Radio Waves by J. R. M E N T Z ER Vol. 8 Space-charge Waves and Slow Electromagnetic Waves by A. H. W. B E CK Vol. 9 Statistical Theory of Signal Detection Vol. 10 Laplace Transforms by C A RL W. H E L S T R OM for Electronic Engineers by J. G. H O L B R O OK Vol. 11 Frequency Modulation Theory—Application to Microwave Links by J. FAGOT and P H. M A G NE Vol. 12 Theory of Microwave Valves by S. D. G V O Z D O V ER Vol. 13 Electronic Computers by A. I. K I T OV and N. A. K R I N I T S K II Vol. 14 Topics in Engineering Logic by M. N A D L ER Vol. 15 Environmental Testing Techniques by G. W. A. D U M M ER and N. B. G R I F F IN Vol. 16 Fundamentals of Microwave Electronics by V. N. SHEVCHIK Vol. 17 Static Electromagnetic Vol. 18 Problems Changers Frequency by L. L. ROZHANSKII in the Design and Development of 750 MW by V. P. ANEMPODISTOV, E. G. K A S H A R S K II and U R U S OV Turbogenerators I. D. Vol. 19 Controlled-Delay Devices by S. A. DOGANOVSKII and V. A. I V A N OV Vol. 20 High Sensitivity Counting Techniques by D. E. W A TT and D. R A M S D EN Vol. 21 Asynchronised Synchronous Machines by M. M. B O T V I N N IK Vol. 22 Sampling Systems Theory and its Application, Vol, 1

PROBABILITY AND INFORMATION THEORY, WITH APPLICATIONS TO RADAR By P. M. W O O D W A R D, M.A. Mathematics Division, Royal Radar Establishment, Ministry of Aviation SECOND EDITION P E R G A M ON P R E SS O X F O RD • L O N D ON • E D I N B U R GH • N EW Y O RK P A R IS • F R A N K F U RT

PERGAMON PRESS LTD. Headington Hill Hall, Oxford 4 & 5 Fitzroy Square, London W .l PERGAMON PRESS (SCOTLAND) LTD. 2 & 3 Teviot Place, Edinburgh 1 PERGAMON PRESS INC. 122 East 55th Street, New York 22, N.Y. GAUTHIER-VILLARS ED. 55 Quai des Grands-Augustins, Paris 6 PERGAMON PRESS G.m.b.H. Kaiserstrasse 75, Frankfurt am Main Copyright 1953 Pergamon Press Ltd. First Published 1953 Second Impression 1957 Reprinted 1963 Second Edition 1964 Library of Congress Catalog Card Number 64-7502 Printed in Great Britain by Latimer Trend & Co. Ltd., Whitstable.

EDITOR'S PREFACE THE aim of these monographs is to report upon research carried out in electronics and applied physics. Work in these fields continues to expand rapidly, and it is recognised that the collation and dis semination of information in a usable form is of the greatest im portance to all those actively engaged in them. The monographs will be written by specialists in their own subjects, and the time required for publication will be kept to a minimum in order that these accounts of new work may be made quickly and widely available. Wherever it is practical the monographs will be kept short in length to enable all those interested in electronics to find the essentials necessary for their work in a condensed and concentrated form. D. W. F RY AUTHOR'S PREFACE THE first two chapters of this short monograph are concerned with established mathematical techniques rather than with fresh ideas. They provide the code in which so much of the mathematical theory of electronics and radar is nowadays expressed. Information theory is the latest extension of this code, and I hope that it will not be considered improper that I have tried in Chapter 3 to summarise so much of C. E. SHANNON'S original work, which already exists in book-form (The Mathematical Theory of Communication, by CLAUDE SHANNON and WARREN WEAVER). The account which is given in Chapter 3 may perhaps spur the reader who has not studied the original literature into doing so. Chapters 4 and 5 deal with some of the fascinating problems, which have been discussed so often in recent years, of detecting signals in noise. The present approach was suggested to me by SHANNON'S work on communication theory and is based on inverse probability; it is my opinion that of all statistical methods, this one comes closest to expressing intuitive notions in the precise language ix

X PREFACE of mathematics. Chapters 5, 6 and 7 are devoted to radar, which is simple enough (ideally) to lend itself to fairly exact mathematical treatment along the lines suggested in the previous chapters. This material is based on papers which have appeared in technical journals, Chapter 6 in particular being a revised account of work originally carried out at T.R.E. in partnership with I.L.DAVIES. It was this work which led to the present monograph, but it is hoped that the first four chapters—originally conceived as an introduction to the special problems of radar—may find an independent usefulness for the reader whose interests are not so narrowly confined. I have to thank the Chief Scientist, Ministry of Supply, for per mission to publish this book. Malvern March, 1953. P. M. W. PREFACE TO SECOND EDITION OVER the last ten years, authoritative works of monumental propor tions on radar and communication theory have appeared, with bibliographies almost as long as the text of this little Monograph. To attempt, in the second edition, a major enlargement of what was only an extended research paper would be to destroy its purpose, and I have thus left the original text substantially unchanged. One new chapter is now added, describing in the briefest possible terms, the way in which direct probabilities are usually applied to the decision problem in radar design. To research workers, however, I would suggest that Chapter 7 still poses some interesting practical problems. P. M. W. Malvern February, 1964

1 AN INTRODUCTION TO PROBABILITY THEORY 1.1 THE RULES OF PROBABILITY IF n possibilities are equally likely and exactly m of them have some attribute A, we say that the probability of A is mjn. Strictly, this is not a definition of probability because it assumes that the notion of equally likely possibilities is understood in the first place. From a purely mathematical point of view, however, no definition is required. All we need is a set of rules for adding and multiplying probabilities, which are then taken as the basic postulates of the theory. But the study of probability is made easier and the rules become intuitive rather than arbitary when from the beginning there is an obvious practical interpretation, and this the opening remark supplies. Since probability is a fraction of equally likely possibilities, it is often helpful to set these out in tabular form. Thus A A A B B B BG (1) signifies that P(A), the probability of A, is f and so on. It is immedi ately evident that the probability of A or B is P(A) + P(B). This is the sum rule and it applies only when A and B cannot simul taneously be true, in other words, when they are mutually exclusive. When all mutually exclusive attributes have been taken into account, their probabilities will naturally add up to unity. It frequently happens that two sets of attributes, each mutually exclusive among themselves, have to be considered together. Sup pose, for instance, that we have eight pencils, three red (^4), four black (B) and one blue (C). The scheme (1) represents the equally likely possibilities when one pencil is selected randomly. But the same pencils may also be hard (J) and soft (K) as follows, A A A B B B BC J J K J J J KK 1 )

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