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Fourier Transforms in Spectroscopy . J. Kauppinen, J. Partanen

ISBNs: 3-527-40289-6 (Hardcover); 3-527-60029-9 (Electronic)

Jyrki Kauppinen, Jari Partanen

Fourier Transforms in Spectroscopy

Fourier Transforms in Spectroscopy . J. Kauppinen, J. Partanen

ISBNs: 3-527-40289-6 (Hardcover); 3-527-60029-9 (Electronic)

Jyrki Kauppinen, Jari Partanen

Fourier Transforms

in Spectroscopy

Berlin Weinheim New York Chichester

Brisbane Singapore Toronto

Fourier Transforms in Spectroscopy . J. Kauppinen, J. Partanen

ISBNs: 3-527-40289-6 (Hardcover); 3-527-60029-9 (Electronic)

Authors:

Prof. Dr. Jyrki Kauppinen

Department of Applied Physics

University of Turku

Finland

e-mail: jyrki.kauppinen@utu.fi

e-mail: jari.partanen@utu.fi

This book was carefully produced. Nevertheless, authors and publisher do not warrant the

information contained therein to be free of errors. Readers are advised to keep in mind that

statements, data, illustrations, procedural details or other items may inadvertently be

inaccurate.

1st edition, 2001

with 153 figures

Library of Congress Card No.: applied for

A catalogue record for this book is available from the British Library.

Die Deutsche Bibliothek - CIP Cataloguing-in-Publication-Data

A catalogue record for this publication is available from Die Deutsche Bibliothek

ISBN 3-527-40289-6

Printed on acid-free paper.

reproduced in any form - by photoprinting, microfilm, or any other means - nor transmitted or

translated into machine language without written permission from the publishers. Registered

names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be

considered unprotected by law.

Printing: Strauss Offsetdruck GmbH, D-69509 Mörlenbach. Bookbinding: J. Schäffer GmbH &

Co. KG, D-67269 Grünstadt.

Printed in the Federal Republic of Germany.

WILEY-VCH Verlag Berlin GmbH

Bühringstraße 10

D-13086 Berlin

Federal Republic of Germany

Department of Applied Physics

Dr. Jari Partanen

Fourier Transforms in Spectroscopy . J. Kauppinen, J. Partanen

ISBNs: 3-527-40289-6 (Hardcover); 3-527-60029-9 (Electronic)

Preface

How much should a good spectroscopist know about Fourier transforms? How well should a

professional who uses them as a tool in his/her work understand their behavior? Our belief

is, that a profound insight of the characteristics of Fourier transforms is essential for their

successful use, as a superﬁcial knowledge may easily lead to mistakes and misinterpretations.

But the more the professional knows about Fourier transforms, the better he/she can apply all

those versatile possibilities offered by them.

On the other hand, people who apply Fourier transforms are not, generally, mathemati-

cians. Learning unnecessary details and spending years in specializing in the heavy math-

ematics which could be connected to Fourier transforms would, for most users, be a waste

of time. We believe that there is a demand for a book which would cover understandably

those topics of the transforms which are important for the professional, but avoids going into

unnecessarily heavy mathematical details. This book is our effort to meet this demand.

We recommend this book for advanced students or, alternatively, post-graduate students

of physics, chemistry, and technical sciences. We hope that they can use this book also

later during their career as a reference volume. But the book is also targeted to experienced

professionals: we trust that they might obtain new aspects in the use of Fourier transforms by

reading it through.

Of the many applications of Fourier transforms, we have discussed Fourier transform

spectroscopy (FTS) in most depth. However, all the methods of signal and spectral processing

explained in the book can also be used in other applications, for example, in nuclear magnetic

resonance (NMR) spectroscopy, or ion cyclotron resonance (ICR) mass spectrometry.

We are heavily indebted to Dr. Pekka Saarinen for scientiﬁc consultation, for planning

problems for the book, and, ﬁnally, for writing the last chapter for us. We regard him as a

leading specialist of linear prediction in spectroscopy. We are also very grateful to Mr. Matti

Hollberg for technical consultation, and for the original preparation of most of the drawings

in this book.

Jyrki Kauppinen and Jari Partanen

Turku, Finland, 13th October 2000

Fourier Transforms in Spectroscopy . J. Kauppinen, J. Partanen

ISBNs: 3-527-40289-6 (Hardcover); 3-527-60029-9 (Electronic)

Contents

1Basic deﬁnitions

1.1

Fourier series . . . .............................. 11

1.2

Fourier transform ............................... 14

1.3

Dirac’s delta function . . . . . ........................ 17

2General properties of Fourier transforms

2.1

Shift theorem . . . .............................. 24

2.2

Similarity theorem . . . . . . . ....................... 25

2.3

Modulation theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.4

Convolution theorem . . . . . . ....................... 26

2.5

Power theorem . . . ............................. 28

2.6

Parseval’s theorem . . . . . . ........................ 29

2.7

Derivative theorem . . . . . . ........................ 29

2.8

Correlation theorem . . . . . . ........................ 30

2.9

Autocorrelation theorem . . . . ....................... 31

3Discrete Fourier transform

3.1

Effect of truncation . . . . . . . ....................... 36

3.2

Effect of sampling . .............................. 39

3.3

Discrete spectrum............................... 43

4Fast Fourier transform (FFT)

4.1

Basis of FFT . . . . .............................. 49

4.2

Cooley–Tukey algorithm . . . . . ...................... 54

4.3

Computation time . .............................. 56

5Other integral transforms 61

5.1 Laplace transform . . . . . . . ........................ 61

5.2 Transfer function of a linear system . . ................... 66

5.3 z transform . ................................. 73

6Fourier transform spectroscopy (FTS)

6.1

Interference of light . . . . . . ........................ 77

6.2

Michelson interferometer . . . ........................ 78

6.3

Sampling and truncation in FTS ....................... 83

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