2011 • 162 Pages • 2.26 MB • English

Posted April 14, 2020 • Uploaded
by nader.bridie

PREVIEW PDF

Page 1

Page 2

This page intentionally left blank

Page 3

A Student’s Guide to Fourier Transforms Fourier transform theory is of central importance in a vast range of applications in physical science, engineering and applied mathematics. Providing a concise introduction to the theory and practice of Fourier transforms, this book is invaluable to students of physics, electrical and electronic engineering and computer science. After a brief description of the basic ideas and theorems, the power of the technique is illustrated through applications in optics, spectroscopy, electronics and telecommunications. The rarely discussed but important ﬁeld of multi-dimensional Fourier theory is covered, including a description of Computerized Axial Tomography (CAT) scanning. The book concludes by discussing digital methods, with particular attention to the Fast Fourier Transform and its implementation. This new edition has been revised to include new and interesting material, such as convolution with a sinusoid, coherence, the Michelson stellar interferometer and the van Cittert–Zernike theorem, Babinet’s principle and dipole arrays. j . f . james is a graduate of the University of Wales and the University of Reading. He has held teaching positions at the University of Minnesota, The Queen’s University, Belfast and the University of Manchester, retiring as Senior Lecturer in 1996. He is a Fellow of the Royal Astronomical Society and a member of the Optical Society of America and the International Astronomical Union. His research interests include the invention, design and construction of astronomical instruments and their use in astronomy, cosmology and upper-atmosphere physics. Dr James has led eclipse expeditions to Central America, the central Sahara and the South Paciﬁc Islands. He is the author of about 40 academic papers, co-author with R. S. Sternberg of The Design of Optical Spectrometers (Chapman & Hall, 1969) and author of Spectrograph Design Fundamentals (Cambridge University Press, 2007).

Page 4

Page 5

A Student’s Guide to Fourier Transforms with Applications in Physics and Engineering Third Edition J. F. JAMES

Page 6

cambridge university press Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sa˜o Paulo, Delhi, Tokyo, Mexico City Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521176835 C J. F. James 2011 This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1995 Second edition 2002 Third edition 2011 Printed in the United Kingdom at the University Press, Cambridge A catalogue record for this publication is available from the British Library ISBN 978 0 521 17683 5 Paperback Cambridge University Press has no responsibility for the persistence or accuracy of URLs for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate.

Page 7

Contents Preface to the ﬁrst edition page ix Preface to the second edition xi Preface to the third edition xiii 1 Physics and Fourier transforms 1 1.1 The qualitative approach 1 1.2 Fourier series 2 1.3 The amplitudes of the harmonics 4 1.4 Fourier transforms 8 1.5 Conjugate variables 10 1.6 Graphical representations 11 1.7 Useful functions 11 1.8 Worked examples 18 2 Useful properties and theorems 20 2.1 The Dirichlet conditions 20 2.2 Theorems 22 2.3 Convolutions and the convolution theorem 22 2.4 The algebra of convolutions 30 2.5 Other theorems 31 2.6 Aliasing 34 2.7 Worked examples 36 3 Applications 1: Fraunhofer diffraction 40 3.1 Fraunhofer diffraction 40 3.2 Examples 44 3.3 Babinet’s principle 54 3.4 Dipole arrays 55 v

Page 8

vi Contents 3.5 Polar diagrams 58 3.6 Phase and coherence 58 3.7 Fringe visibility 60 3.8 The Michelson stellar interferometer 61 3.9 The van Cittert–Zernike theorem 64 4 Applications 2: signal analysis and communication theory 66 4.1 Communication channels 66 4.2 Noise 68 4.3 Filters 69 4.4 The matched ﬁlter theorem 70 4.5 Modulations 71 4.6 Multiplex transmission along a channel 77 4.7 The passage of some signals through simple ﬁlters 77 4.8 The Gibbs phenomenon 81 5 Applications 3: interference spectroscopy and spectral line shapes 86 5.1 Interference spectrometry 86 5.2 The Michelson multiplex spectrometer 86 5.3 The shapes of spectrum lines 91 6 Two-dimensional Fourier transforms 97 6.1 Cartesian coordinates 97 6.2 Polar coordinates 98 6.3 Theorems 99 6.4 Examples of two-dimensional Fourier transforms with circular symmetry 100 6.5 Applications 101 6.6 Solutions without circular symmetry 103 7 Multi-dimensional Fourier transforms 105 7.1 The Dirac wall 105 7.2 Computerized axial tomography 108 7.3 A ‘spike’ or ‘nail’ 112 7.4 The Dirac fence 114 7.5 The ‘bed of nails’ 115 7.6 Parallel-plane delta-functions 116 7.7 Point arrays 118 7.8 Lattices 119 8 The formal complex Fourier transform 120

Page 9

Contents vii 9 Discrete and digital Fourier transforms 127 9.1 History 127 9.2 The discrete Fourier transform 128 9.3 The matrix form of the DFT 129 9.4 A BASIC FFT routine 133 Appendix 137 Bibliography 141 Index 143

A student’s guide to Fourier transforms : with applications in physics and engineering

2002 • 147 Pages • 1.28 MB

1. Problems of Fourier series and Fourier transforms used in engineering applications. 2

2016 • 150 Pages • 6.56 MB

Transforms and Applications Handbook, Third Edition (Electrical Engineering Handbook)

2010 • 914 Pages • 17.61 MB

Convolutions of hypercomplex Fourier transforms with applications in image processing

2014 • 86 Pages • 2.76 MB

Transforms and Applications Handbook, Third Edition

2010 • 914 Pages • 15.95 MB

Integral Transforms in Science and Engineering

1979 • 489 Pages • 4.46 MB

Integral Transforms in Science and Engineering.

2016 • 493 Pages • 19.5 MB

Fourier Transforms in Spectroscopy

2001 • 271 Pages • 4.17 MB

Fourier Transforms In Spectroscopy

2001 • 261 Pages • 3.89 MB

Fourier Transforms In Spectroscopy

2004 • 261 Pages • 3.89 MB

The Fourier Transform and its Applications - Stanford Engineering

2007 • 428 Pages • 30.06 MB

A New Twist to Fourier Transforms

2004 • 239 Pages • 12.41 MB

Fourier transforms. Principles and applications

2014 • 774 Pages • 7.23 MB

Fourier Transforms: Principles and Applications

2014 • 774 Pages • 7.65 MB

Fourier-Mukai and Nahm Transforms in Geometry and Mathematical Physics

2009 • 434 Pages • 3.05 MB

Fourier-Mukai and Nahm transforms in geometry and mathematical physics

2009 • 434 Pages • 1.95 MB