Fourier transformation for pedestrians

207 Pages·2006·1.701 MB·English

by  Tilman Butz

FourierTransformationforPedestrians
T. Butz
Fourier Transformation
for Pedestrians
With117Figures
123
ProfessorDr.TilmanButz
Universita¨tLeipzig
Fakulta¨tfu¨rPhysikundGeowissenschaften
Linne´str.5
04103Leipzig,Germany
e-mail:[email protected]
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ISBN-103-540-23165-XSpringerBerlinHeidelbergNewYork
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To Renate, Raphaela, and Florentin
Preface
Fourier1 Transformation for Pedestrians. For pedestrians? Harry J. Lipkin’s
famous “Beta-decay for Pedestrians” [1], was an inspiration to me, so that’s
why. Harry’s book explains physical problems as complicated as helicity
and parity violation to “pedestrians” in an easy to understand way. Dis-
crete Fourier transformation, by contrast, only requires elementary algebra,
something any student should be familiar with. As the algorithm2 is a lin-
ear one, this should present no pitfalls and should be as “easy as pie”. In
spite of that, stubborn prejudices prevail, as far as Fourier transformations
are concerned, viz. that information could get lost or that you could end up
trusting a hoax; anyway, who’d trust something that is all done with “smoke
andmirrors”.Theaboveprejudicesoftenarecausedbynegativeexperiences,
gainedthroughimproperuseofready-madeFouriertransformationprograms
or hardware. This book is for all who, being laypersons – or pedestrians –
are looking for a gentle and also humorous introduction to the application
of Fourier transformation, without hitting too much theory, proofs of exis-
tence and similar things. It is appropriate for science students at technical
colleges and universities and also for “mere” computer–freaks. It’s also quite
adequate for students of engineering and all practical people working with
Fourier transformations. Basic knowledge of integration, however, is recom-
mended.Ifthisbookcanhelptoavoidprejudicesorevendoawaywiththem,
writing it has been well worthwhile. Here, we show how things “work”. Gen-
erallywediscusstheFouriertransformationinonedimensiononly.Chapter1
introduces Fourier series and, as part and parcel, important statements and
theorems that will guide us through the whole book. As is appropriate for
pedestrians, we’ll also cover all the “pits and pitfalls” on the way. Chapter 2
covers continuous Fourier transformations in great detail. Window functions
willbedealtwithinChap.3inmoredetail,asunderstandingthemisessential
toavoidthedisappointmentcausedbyfalseexpectations.Chapter4isabout
discreteFouriertransformations,withspecialregardtotheCooley–Tukeyal-
gorithm(FastFourierTransform,FFT).Finally,Chap.5willintroducesome
1 JeanBaptisteJosephFourier(1768–1830),Frenchmathematicianandphysicist.
2 Integration and differentiation are linear operators. This is quite obvious in the
discrete version (Chap. 4) and is, of course, also valid when passing on to the
continuous form.
VIII Preface
useful examples for the filtering effects of simple algorithms. From the host
of available material we’ll only pick items that are relevant to the recording
and preprocessing of data, items that are often used without even thinking
about them. This book started as a manuscript for lectures at the Technical
University of Munich and at the University of Leipzig. That’s why it’s very
muchatextbookandcontainsmanyworkedexamples–toberedone“manu-
ally” –aswellasplentyofillustrations. Toshowthatatextbook(originally)
written in German can also be amusing and humorous, was my genuine con-
cern,becausededicationandassiduityontheirownarequiteinclinedtostifle
creativity and imagination. It should also be fun and boost our innate urge
to play. The two books “Applications of Discrete and Continuous Fourier
Analysis” [2] and “Theory of Discrete and Continuous Fourier Analysis” [3]
had considerable influence on the makeup and content of this book, and are
to be recommended as additional reading for those “keen on theory”.
This English edition is based on the third, enlarged edition in German
[4]. In contrast to this German edition, there are now problems at the end
of each chapter. They should be worked out before going to the next chap-
ter. However, I prefer the word “playground” because you are allowed to go
straight to the solutions, compiled in the Appendix, should your impatience
get the better of you. In case you have read the German original, there I
apologised for using many new-German words, such as “sampeln” or “wrap-
pen”; I won’t do that here, on the contrary, they come in very handy and
make the translator’s job (even) easier. Many thanks to Mrs U. Seibt and
Mrs K. Schandert, as well as to Dr. T. Reinert, Dr. T. Soldner, and espe-
cially to Mr H. Go¨del (Dipl.-Phys.) for the hard work involved in turning a
manuscript into a book. Mr St. Jankuhn (Dipl.-Phys.) did an excellent job
in proof-reading and computer acrobatics.
Last but not least, special thanks go to the translator who managed to
convert the informal German style into an informal (“downunder”) English
style.
Recommendations, queries and proposals for change are welcome. Have
fun while reading, playing and learning.
Leipzig,
September 2005 Tilman Butz
Preface of the Translator
More than a few moons ago I read two books about Richard Feynman’s life,
and that has made a lasting impression. When Tilman Butz asked me if I
could translate his “Fourier Transformation for Pedestrians”, I leapt at the
chance–mywayofgettingabitmoreintoscience.Duringtherathermechan-
ical process of translating the German original, within its TEX-framework, I
made sure I enjoyed the bits for the pedestrians, mere mortals like myself.
Of course I’m biased, I’ve known the author for many years – after all he’s
my brother.
Hamilton, New Zealand,
September 2005 Thomas-Severin Butz
Contents
Introduction.................................................. 1
1 Fourier Series............................................. 3
1.1 Fourier Series .......................................... 3
1.1.1 Even and Odd Functions .......................... 3
1.1.2 Definition of the Fourier Series ..................... 4
1.1.3 Calculation of the Fourier Coefficients............... 6
1.1.4 Fourier Series in Complex Notation................. 11
1.2 Theorems and Rules .................................... 13
1.2.1 Linearity Theorem................................ 13
1.2.2 The First Shifting Rule ........................... 14
1.2.3 The Second Shifting Rule ......................... 17
1.2.4 Scaling Theorem ................................. 21
1.3 Partial Sums, Bessel’s Inequality, Parseval’s Equation ....... 21
1.4 Gibbs’ Phenomenon .................................... 24
1.4.1 Dirichlet’s Integral Kernel ......................... 24
1.4.2 Integral Notation of Partial Sums .................. 26
1.4.3 Gibbs’ Overshoot................................. 27
Playground ................................................ 30
2 Continuous Fourier Transformation....................... 33
2.1 Continuous Fourier Transformation ....................... 33
2.1.1 Even and Odd Functions .......................... 33
2.1.2 The δ-Function .................................. 34
2.1.3 Forward and Inverse Transformation ................ 35
2.1.4 Polar Representation of the Fourier Transform ....... 40
2.2 Theorems and Rules .................................... 42
2.2.1 Linearity Theorem................................ 42
2.2.2 The First Shifting Rule ........................... 42
2.2.3 The Second Shifting Rule ......................... 43
2.2.4 Scaling Theorem ................................. 44
2.3 Convolution, Cross Correlation, Autocorrelation, Parseval’s
Theorem .............................................. 46
2.3.1 Convolution ..................................... 46
2.3.2 Cross Correlation................................. 55
XII Contents
2.3.3 Autocorrelation .................................. 56
2.3.4 Parseval’s Theorem............................... 57
2.4 Fourier Transformation of Derivatives ..................... 58
2.5 Pitfalls................................................ 60
2.5.1 “Turn 1 into 3” .................................. 60
2.5.2 Truncation Error ................................. 63
Playground ................................................ 66
3 Window Functions........................................ 69
3.1 The Rectangular Window ............................... 69
3.1.1 Zeros ........................................... 70
3.1.2 Intensity at the Central Peak ...................... 70
3.1.3 Sidelobe Suppression.............................. 71
3.1.4 3 dB-Bandwidth ................................. 72
3.1.5 Asymptotic Behaviour of Sidelobes ................. 73
3.2 The Triangular Window (Fejer Window) .................. 73
3.3 The Cosine Window .................................... 74
3.4 The cos2-Window (Hanning)............................. 75
3.5 The Hamming Window ................................. 77
3.6 The Triplet Window .................................... 78
3.7 The Gauss Window..................................... 79
3.8 The Kaiser–Bessel Window .............................. 80
3.9 The Blackman–Harris Window ........................... 81
3.10 Overview over Window Functions......................... 84
3.11 Windowing or Convolution? ............................. 87
Playground ................................................ 88
4 Discrete Fourier Transformation .......................... 89
4.1 Discrete Fourier Transformation.......................... 89
4.1.1 Even and Odd Series and Wrap-around ............. 89
4.1.2 The Kronecker Symbol or the “Discrete δ-Function” .. 90
4.1.3 Definition of the Discrete Fourier Transformation..... 92
4.2 Theorems and Rules .................................... 96
4.2.1 Linearity Theorem................................ 96
4.2.2 The First Shifting Rule ........................... 96
4.2.3 The Second Shifting Rule ......................... 97
4.2.4 Scaling Rule/Nyquist Frequency.................... 98
4.3 Convolution, Cross Correlation, Autocorrelation, Parseval’s
Theorem .............................................. 99
4.3.1 Convolution ..................................... 100
4.3.2 Cross Correlation................................. 103
4.3.3 Autocorrelation .................................. 104
4.3.4 Parseval’s Theorem............................... 104
4.4 The Sampling Theorem ................................. 105
4.5 Data Mirroring......................................... 109
Contents XIII
4.6 Zero-padding .......................................... 112
4.7 Fast Fourier Transformation (FFT) ....................... 118
Playground ................................................ 126
5 Filter Effect in Digital Data Processing ................... 131
5.1 Transfer Function ...................................... 131
5.2 Low-pass, High-pass, Band-pass, Notch Filter .............. 132
5.3 Shifting Data .......................................... 139
5.4 Data Compression ...................................... 141
5.5 Differentiation of Discrete Data .......................... 141
5.6 Integration of Discrete Data ............................. 143
Playground ................................................ 147
Appendix: Solutions .......................................... 149
References.................................................... 197
Index......................................................... 199