Table Of ContentFourier Series and Transforms
A Computer Illustrated Text
FOURIER SERIES
AND
TRANSFORMS
R D Harding
University of Cambridge
First published 1985 by Taylor & Francis Group
Published 2019 by CRC Press
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> Contents
Preface vii
Introduction 1
Part 1 Sampling and Resolution 3
Part 2 Periodic Functions, Harmonics and Fourier Series 7
Part 3 Fourier Transforms 22
Appendix 1 The FFT Algorithm 46
Appendix 2 Proof of the DFT Inverse Relations 48
Appendix 3 Program Outlines 50
Program User's Notes Apple Pascal Version 64
Program User's Notes BBC Basic Version 71
Index 75
> Preface
There are several topics in mathematics at advanced secondary school
level or early university course level whose teaching I have always thought
would benefit by being more extensively illustrated than is possible in
conventional texts. Many such topics are of interest not only to specialist
mathematicians but also to scientists and engineers, and from many years
experience of teaching them, what seems to be lacking from the normal
treatment is a visual and qualitative feel for the properties of the
mathematics.
The widespread and rapid growth of cheap microcomputers means that
it is possible to try to remedy this lack through the use of computer
graphics. In this computer illustrated text normal figures and diagrams are
replaced with screen displays which can be produced dynamically by the
reader, offering a far greater variety of illustrations than would be possible
with printed diagrams alone. Of course I have had to assume that the
reader has access to, and is reasonably familiar with one of the computers
and its programming language which I have chosen to write for.
In principle the idea could be applied to an entire textbook but it is felt
that to start with it is better to concentrate on a few topics for which this
treatment seems especially useful, and also to concentrate on imparting an
overall feel for these topics since it is not being suggested that existing texts
are inadequate in technical detail. This text is therefore meant to be
complementary to a standard textbook or lecture course.
Some may wonder whether it would not be better to have the computer
reproduce the text as well as the diagrams. There certainly are such
packages available, but unless the topic is either very elementary or
capable of being discussed very briefly indeed, the present state of
computer technology does not allow these to run on a cheap home
vn
viii PREFACE
microcomputer such as the BBC Microcomputer or the Apple II. Another
problem is that mathematics requires a wide range of special symbols and
again the technology is not widely and cheaply available which would
allow these symbols to be shown on a screen. There is also the fact that a
printed text can be more easily browsed through.
These considerations have led to the design of this monograph, which it
is hoped will be the first of several such titles. As it is experimental,
comments and criticisms will be particularly welcome. I hope it will make
learning more interesting and enjoyable for you, the reader. Good luck.
RDM
March 1984
> Acknowledgments
The author gratefully acknowledges the hospitality and help of the
Algebra Institute, University of California, Santa Barbara, where much of
this text was written.
The author also acknowledges Acornsoft Ltd for permission to use
routines from Graphs and Charts on the BBC Microcomputer.
