Table Of ContentFourier Transforms in Spectroscopy. J. Kauppinen, J. Partanen
Copyright © 2001 Wiley-VCH Verlag GmbH
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
Copyright © 2001 Wiley-VCH Verlag GmbH
ISBNs: 3-527-40289-6 (Hardcover); 3-527-60029-9 (Electronic)
Jyrki Kauppinen, Jari Partanen
Fourier Transforms
in Spectroscopy
Berlin (cid:1) Weinheim (cid:1) New York (cid:1) Chichester
Brisbane (cid:1) Singapore (cid:1) Toronto
Fourier Transforms in Spectroscopy. J. Kauppinen, J. Partanen
Copyright © 2001 Wiley-VCH Verlag GmbH
ISBNs: 3-527-40289-6 (Hardcover); 3-527-60029-9 (Electronic)
Authors:
Prof. Dr. Jyrki Kauppinen Dr. Jari Partanen
Department of Applied Physics Department of Applied Physics
University of Turku University of Turku
Finland Finland
e-mail: [email protected] e-mail: [email protected]
Thisbookwascarefullyproduced.Nevertheless,authorsandpublisherdonotwarrantthe
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statements, data, illustrations, procedural details or other items may inadvertently be
inaccurate.
1st edition, 2001
with 153 figures
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Fourier Transforms in Spectroscopy. J. Kauppinen, J. Partanen
Copyright © 2001 Wiley-VCH Verlag GmbH
ISBNs: 3-527-40289-6 (Hardcover); 3-527-60029-9 (Electronic)
Preface
HowmuchshouldagoodspectroscopistknowaboutFouriertransforms? Howwellshoulda
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
successfuluse,asasuperficialknowledgemayeasilyleadtomistakesandmisinterpretations.
ButthemoretheprofessionalknowsaboutFouriertransforms,thebetterhe/shecanapplyall
thoseversatilepossibilitiesofferedbythem.
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
thosetopicsofthetransformswhichareimportantfortheprofessional,butavoidsgoinginto
unnecessarilyheavymathematicaldetails. Thisbookisourefforttomeetthisdemand.
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
laterduring theircareerasareference volume. Butthebookis alsotargetedtoexperienced
professionals: wetrustthattheymightobtainnewaspectsintheuseofFouriertransformsby
readingitthrough.
Of the many applications of Fourier transforms, we have discussed Fourier transform
spectroscopy(FTS)inmostdepth. However,allthemethodsofsignalandspectralprocessing
explainedinthebookcanalsobeusedinotherapplications,forexample,innuclearmagnetic
resonance(NMR)spectroscopy,orioncyclotronresonance(ICR)massspectrometry.
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
leadingspecialistoflinearpredictioninspectroscopy. WearealsoverygratefultoMr. Matti
Hollbergfortechnicalconsultation, andfortheoriginalpreparationofmostofthedrawings
inthisbook.
JyrkiKauppinenandJariPartanen
Turku,Finland,13thOctober2000
Fourier Transforms in Spectroscopy. J. Kauppinen, J. Partanen
Copyright © 2001 Wiley-VCH Verlag GmbH
ISBNs: 3-527-40289-6 (Hardcover); 3-527-60029-9 (Electronic)
Contents
1 Basicdefinitions 11
1.1 Fourierseries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2 Fouriertransform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.3 Dirac’sdeltafunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2 GeneralpropertiesofFouriertransforms 23
2.1 Shifttheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2.2 Similaritytheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.3 Modulationtheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.4 Convolutiontheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.5 Powertheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
2.6 Parseval’stheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.7 Derivativetheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.8 Correlationtheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.9 Autocorrelationtheorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3 DiscreteFouriertransform 35
3.1 Effectoftruncation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.2 Effectofsampling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3 Discretespectrum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
4 FastFouriertransform(FFT) 49
4.1 BasisofFFT. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2 Cooley–Tukeyalgorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.3 Computationtime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
5 Otherintegraltransforms 61
5.1 Laplacetransform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
5.2 Transferfunctionofalinearsystem . . . . . . . . . . . . . . . . . . . . . 66
5.3 ztransform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73
6 Fouriertransformspectroscopy(FTS) 77
6.1 Interferenceoflight . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
6.2 Michelsoninterferometer . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6.3 SamplingandtruncationinFTS . . . . . . . . . . . . . . . . . . . . . . . 83
8 0Contents
6.4 Collimatedbeamandextendedlightsource . . . . . . . . . . . . . . . . . 89
6.5 Apodization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
6.6 ApplicationsofFTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
7 Nuclearmagneticresonance(NMR)spectroscopy 109
7.1 Nuclearmagneticmomentinamagneticfield . . . . . . . . . . . . . . . . 109
7.2 PrinciplesofNMRspectroscopy . . . . . . . . . . . . . . . . . . . . . . . 112
7.3 ApplicationsofNMRspectroscopy . . . . . . . . . . . . . . . . . . . . . . 115
8 Ioncyclotronresonance(ICR)massspectrometry 119
8.1 Conventionalmassspectrometry . . . . . . . . . . . . . . . . . . . . . . . 119
8.2 ICRmassspectrometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
8.3 FouriertransformsinICRmassspectrometry . . . . . . . . . . . . . . . . 124
9 DiffractionandFouriertransform 127
9.1 FraunhoferandFresneldiffraction . . . . . . . . . . . . . . . . . . . . . . 127
9.2 Diffractionthroughanarrowslit . . . . . . . . . . . . . . . . . . . . . . . 128
9.3 Diffractionthroughtwoslits . . . . . . . . . . . . . . . . . . . . . . . . . 130
9.4 Transmissiongrating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
9.5 Gratingwithonlythreeorders . . . . . . . . . . . . . . . . . . . . . . . . 137
9.6 Diffractionthrougharectangularaperture . . . . . . . . . . . . . . . . . . 138
9.7 Diffractionthroughacircularaperture . . . . . . . . . . . . . . . . . . . . 143
9.8 Diffractionthroughalattice . . . . . . . . . . . . . . . . . . . . . . . . . . 144
9.9 LensandFouriertransform . . . . . . . . . . . . . . . . . . . . . . . . . . 145
10 Uncertaintyprinciple 155
10.1 Equivalentwidth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155
10.2 Momentsofafunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
10.3 Secondmoment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
11 Processingofsignalandspectrum 165
11.1 Interpolation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
11.2 Mathematicalfiltering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
11.3 Mathematicalsmoothing . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
11.4 Distortionand(S/N)enhancementinsmoothing . . . . . . . . . . . . . . 184
11.5 Comparisonofsmoothingfunctions . . . . . . . . . . . . . . . . . . . . . 190
11.6 Eliminationofabackground . . . . . . . . . . . . . . . . . . . . . . . . . 193
11.7 Eliminationofaninterferencepattern . . . . . . . . . . . . . . . . . . . . 194
11.8 Deconvolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
12 Fourierself-deconvolution(FSD) 205
12.1 PrincipleofFSD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
12.2 Signal-to-noiseratioinFSD . . . . . . . . . . . . . . . . . . . . . . . . . 212
12.3 Underdeconvolutionandoverdeconvolution . . . . . . . . . . . . . . . . . 217
12.4 Bandseparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 218
12.5 Fouriercomplexself-deconvolution . . . . . . . . . . . . . . . . . . . . . 219
9
12.6 Even-orderderivativesandFSD . . . . . . . . . . . . . . . . . . . . . . . 221
13 Linearprediction 229
13.1 Linearpredictionandextrapolation . . . . . . . . . . . . . . . . . . . . . . 229
13.2 Extrapolationoflinearcombinationsofwaves . . . . . . . . . . . . . . . . 230
13.3 Extrapolationofdecayingwaves . . . . . . . . . . . . . . . . . . . . . . . 232
13.4 Predictabilityconditioninthespectraldomain . . . . . . . . . . . . . . . . 233
13.5 Theoreticalimpulseresponse . . . . . . . . . . . . . . . . . . . . . . . . . 234
13.6 Matrixmethodimpulseresponses . . . . . . . . . . . . . . . . . . . . . . 236
13.7 Burg’simpulseresponse . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
13.8 Theq-curve . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
13.9 Spectrallinenarrowingbysignalextrapolation . . . . . . . . . . . . . . . 242
13.10 Imperfectimpulseresponse . . . . . . . . . . . . . . . . . . . . . . . . . . 243
13.11 TheLOMEPlinenarrowingmethod . . . . . . . . . . . . . . . . . . . . . 248
13.12 Frequencytuningmethod . . . . . . . . . . . . . . . . . . . . . . . . . . . 250
13.13 Otherapplications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
13.14 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258
Answerstoproblems 261
Bibliography 265
Index 269
Fourier Transforms in Spectroscopy. J. Kauppinen, J. Partanen
Copyright © 2001 Wiley-VCH Verlag GmbH
ISBNs: 3-527-40289-6 (Hardcover); 3-527-60029-9 (Electronic)
Index
B(x),transmissionfunction,128 Autocorrelation,31
fN,criticalsamplingfrequency,42 Autocorrelationtheorem,31
H∞,inverseLaplacetransform,64
k,wavenumber,77 Background,193
K,resolutionenhancementfactor,206 Bandlimitedwhitenoise,170
s,complexangularfrequency,61 Bandseparation,218
wN =ei2π/N,53 Band-passfiltering,172
W(cid:11)x(ν),instrumentalfunctioninFTS,84 Beamdeflectiontechnique,105
L
WT(f), instrumental function of truncation, Besselfunctions,144
36 Binaryfilters,148
W(cid:11)t, instrumentalfunctionofdiscretetrans- Binaryinversion,54
T
form,39 Boxcarfunction,36
W(cid:16), instrumental function of an extended Burg’sformula,240
source,93 Burg’simpulseresponse,239
q-curve,241
Cauchy–Schwarzinequality,161
ztransform,73
Centerofmass,159
,Fouriertransform,14
(cid:1)−1,inverseFouriertransform,14 Combfunction,133
(cid:1) ComplexamplitudesofaFourierseries,12
0,Laplacetransform,63
(cid:1)−1,inverseLaplacetransform,61 Complexangularfrequency,61
(cid:1)0 Complexconjugate,28
∞,Laplacetransform,64
(cid:1)−∞1,inverseLaplacetransform,64 Convolution,26
(cid:1)δ(t),Dirac’sdeltafunction,17 Convolutiontheorem,26
ofdiscreteFouriertransform,45
(cid:5)L(x),triangularfunction,99 oftheztransform,76
ν,wavenumber,77
oftheLaplacetransform,74
(cid:4)2T(t),boxcarfunction,36 Cooley–Tukeyalgorithm,54
(x,d),combfunction,133
∗(cid:1),convolution,26 Correlationtheorem,30
Cosinetransform,23
,crosscorrelation,30
(cid:1)∧ Criticalcomputinginterval,43
=,correspondencerelation,18 Criticalsamplingfrequency,42
Criticalsamplinginterval,41
Aliasing,42 Crosscorrelation,30
Amplitudespectrum,23
AmplitudesofaFourierseries,12 deBrogliewavelength,156
Aperturebroadening,92 Deapodization,248
Apodization,99 Deconvolution,196
Apodizationfunction,99 Derivativetheorem,29
270 Index
oftheLaplacetransform,66 Gyromagneticratio
Derivatives,221 ofanucleus,109
Diffraction
Fraunhofer,127 Hankeltransform,152
Fresnel,127
ordersof,134 ICR,ioncyclotronresonance,121
Diffractiongrating,136 imaginglens,147
Dirac’sdeltafunction,17 Impulsefunction,17
DirectFouriertransform,56 Impulseresponse,67,173,234
Dirichletconditions,11 Burg’smethod,239
Discontinuitycorrection,176 matrixmethod,236
Discretecomplexamplitudespectrum,12 theoretical,234
DiscreteFouriertransform,35 Instrumentalfunction
Dispersionspectrum,104 inFTS,84
DispersiveFT-IRspectrometry,104 ofanextendedsource,93
ofdiscretetransform,39
Eigenfunction oftruncation,36
ofanoperator,158 Instrumentalresolution,94
Equivalentwidth Interferencerecord,81
ofafunction,155 Interferogram,81
ExcitationofNMR,114 Interferometer
Michelson,79
FastFouriertransform,49 Twyman–Green,89
FCSD, Fourier complex self-deconvolution, Interpolation,165
219 degreeof,166
FFT,fastFouriertransform,49 Ioncyclotronfrequency,121
FID,freeinductiondecay,115 Ioncyclotronresonance,121
Filterfunction,172
Filtering Laplacetransform,63
mathematical,170 inverse,61
Fouriercomplexself-deconvolution,219 Larmorangularfrequency,110
Fourierself-deconvolution,205 Larmorfrequency,115
Fourierseries,11 Larmorprecession,110
Fouriertransform,14 Lattice,144
inverse,14 Levinson–Durbinrecursion,239
Fouriertransformpair,14 Linearsystem,66
Fouriertransformspectroscopy,77 Lorentzianlineshape,182
Fraunhoferdiffraction,127 Low-passfilter,148
Freeinductiondecay,115
Fresneldiffraction,127 Magneticmoment
FSD,Fourierself-deconvolution,205 ofanucleus,109
FTS,Fouriertransformspectroscopy,78 Magnetronfrequency,123
FundamentalfrequencyofaFourierseries,12 Massspectrometer,119
Matrixmethodimpulseresponse,236
Gibbsphenomenon,12 Meansquare,160
Grating Meansquaredeviation,160
diffraction,136 Michelsoninterferometer,79
transmission,132 Modulationtheorem,26
Gulfs,250 Molecularspectroscopy,100
Index 271
Moment Signal,23
ofinertia,160 Similaritytheorem,25
oforderk,158 Sincfunction,36
Sinetransform,23
NMR,nuclearmagneticresonance,112 Slit,128
Nuclearmagneticresonance,112 Smoothing
Nyquistfrequency,42 mathematical,180
Smoothingfunction,173
One-pointextrapolation,229
Smoothingparameter,187
Optimalsmoothing,181
Spatialcoordinate,130
Optimaltruncation
Spatialfiltering,148
ofthesignal,95
Spectralorders,39
Overdeconvolution,217
Spectrum,23
Squarewave,12
Parseval’stheorem,29
Standarddeviation,160
Phasespectrum,23
Stepresponse,67
Photoacousticspectroscopy,105
SWIFT, stored-waveform inverse Fourier
Planck’sconstant,100
transform,124
Powerspectrum,31
Powertheorem,28
Theoreticalimpulseresponse,234
Predictabilitycondition,233
Theoreticalresolution,86
Probabilitydensity,157
Time-resolvedspectrometry,105
Probabilityinterpretation,157
Transferfunction,66,172
RCcircuit Transformlens,145
differentiating,71 Transformplane,145
integrating,68 Transmissionfunction,128
RCLcircuit,72 Transmissiongrating,132
Reflectioncoefficient,239 Trappingfrequency,122
Resolution Triangularfunction,99
instrumental,94 Truncation,36
theoretical,86 Twyman–Greeninterferometer,89
Resolutionenhancementfactor,206
rmsvalue,160 Uncertaintyprinciple,156
Rootmeansquare,160 Underdeconvolution,217
Undersampling,42
Samplingtheorem,48 Unitstep,61
Scalingtheorem,25
Selectionrules,101 Variance,160
Shifttheorem,24
oftheLaplacetransform,74 Wavenumber,77
