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TheDiscreteFourierTransformandItsUse5.35.01Fall201122September2011

Contents1 Motivation 12 TheContinuousFourierTransform 4

2.1 A Brief Introduction to Linear Algebra . . . . . . . . . . . . . . . 42.2 TheWaveFormulationoftheFourierTransform . . . . . . . . . 4

3 TheDiscreteFourierTransform(DFT) 63.1 An Efficient Algorithm Exists . . . . . . . . . . . . . . . . . . . . 6

4 UsingtheFFT inMatlab 74.1 A Mixture of Sinusoids . . . . . . . . . . . . . . . . . . . . . . . . 94.2 A Finite-Length Pulse . . . . . . . . . . . . . . . . . . . . . . . . 12

5 ConclusionsandFurtherDirections 131 MotivationInphysicsandengineering,wefrequentlyencountersignalsthatis,continuousstreams of data with a fixed sampling raterepresenting some information ofinterest. For example, consider the waves generated by an AM or FM transmitter,thecrashingofwavesonabeach,apieceofmusic,orraindropshittingthe ground. In each case, we can define some event to monitor, such as thenumber of raindrops hitting a certain region, and record that data over time.Butjust having this data does not tell us much, because we are not used tounravelingsuchcomplicatedandseeminglynoisypatterns. Forexample,belowisthewaveform(signalintensity,44.1kHzsampling)ofanexcerptfromasong:

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At first glance, we can see that some patterns emerge. For example, thereappearstobeastrongsignaleveryfewthousandsampleswithmorecomplicateddetailsinbetween,butbeyondthatwecannoteasilysaymuchaboutthissignal.But what if we can reduce these data to something more managable, such astherhythmofeachinstrument? Certainly,wecouldsitand listentothemusicto discern which instruments play at which frequency if we want to recreatethe song, but this requires significant time and effort by a dedicated observer.What if we want to automate the process for practical reasons, such as voicetranscriptionorsongidentification? Inthiscase,wewouldneedtoestablishanautomated,well-characterizedmethodforanalyzingthesignal.

One such tool is the Fourier transform, which converts a signal varying intime,space,oranyotherdimensionintoonevaryinginaconjugatedimension.For example, a time-varying signal may be converted into a signal varying infrequency,

such

that

we

are

able

to

view

monochromatic

time

oscillations

as

singlepeaksinthefrequencysignal. Appliedtoourcomlexmusicwaveform,wesee:

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Suddenly,westartseeingthatafewkeyrhythmsandinstrumentsarepresentin our clip, asrepresented by the tallestpeaks. Instead ofworrying about thewholesignal,wecanfocusourattentiononjustthemostpromienentcontributors to the signal, allowing us to address practically these signals, rather thanwallowing inavastseaofnumbers.

But what about applications to physical chemistry? We mentioned earliertheconceptofinterferometry,thetechniqueofexaminingtheintensityofasignalat various points in space as a method for determining the spatial frequenciespresent in the incoming light, and therefore the temporal frequencies. This ismathematicallynodifferentfromtheclipofmusic: wesamplethe intensityatdifferent positions, plot the signal, and use a Fourier transform to discern thecontributionofeachfrequencyof light. Theresult isaspectrumoffrequencies(orwavelengths,dependingonwhichwayweprefertolabelthehorizontalaxis),which is then available for interpretation using the laws of physics. The twosituations are very different, yet both can be viewed in a sufficiently similarwaythatthe mathematicsgoverningthebehaviorofthetwo systemsbecomesidentical. Therecognitionoftheseisomorphismsinthestructureofphysicalandmathematical phenomena revolutionizes science and technology, because theypermitustosolvemanyproblemsusingidenticaltechniques,meaningthattheworkputintounderstandingatechniqueinonecontextsuddenlypaysoffinallfields.

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2 TheContinuousFourierTransform2.1 ABriefIntroduction toLinearAlgebraBefore delving intothemechanicsoftheFouriertransformas implemented ona computer, it is important to see the origin of the technique and how it isconstructed. Therefore, we will start with the continuous Fourier transform,seek an understanding of its structure, and exploit that understanding to seehowwemighttransformdiscretesignals,asnearlyallrealdataare.

To begin with, recall the concept of a vector. We typically encounter suchstructuresasarrows in Euclideanspacewhichareendowedwithdirectionandmagnitude, and we consider two vectors equal when these two properties arethesameforeach. Youmayalsobefamiliarwiththeideathatavectorinspacecan be expressed as the sum of other vectors, such as the i,j, and k vectors,representingthex,y,andz axes,respectively. Thesevectorsareorthonormal,that is, each has a magnitude of 1 and sits at a right angle (is orthogonal) totheothers. So,togethertheyformwhatwecallanorthonormalbasisset,whichmeansthatallvectorsinthisthree-dimensionalspacecanbeexpressedassomelinear combination of the basis vectors. More concisely, for basic vectors ej,constant coefficients cj, and some vectorv in anN-dimensional vector spacespannedby{ej}:

NNv= cjej (1)

j=1With a bit of math, we can define some fundamental behavior for abstractvectorsandvectorspaces,includingtheabstractideaofaninnerproduct,whichyou may know as a dot product. Briefly, an inner product acts a measureof the spatial overlap of two vectors, and therefore can be used as a test fororthogonality. Theinnerproductofavectorwithitselfgivesthemagntitudeofthevectorsquared,andsowecanexpressanyvectorforsomebasic{ej}as:

NNvejv= ej (2)

ejejj=1

So,foranyvectorinafinitevectorspace,wehaveamethodforexpressingthevectorasasumoveranybasis,meaningthatwearefreetochooseonewhichisconvenienttoworkwith. Forcontinuousvectorspaceswemustbealittlemorecarefulwithoutdefinitions,butthebasic ideasstillhold.2.2 TheWaveFormulationoftheFourierTransformRecalltheFourierseries,whichexpressesanintegrablefunctionwithperiod2asasumofsinusoidalfunctions:

Na0f(t) = + ajcos(jt) +bjsin(jt) (3)

2j=1

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We can view the varioussinusoids and the constant coefficient as vectors, anddefinethe innerproductasan integralovertheperiod(withaconstant factorfornormalization): 1

f(t)g(t) = f(t)g(t)dt (4)

Under this operation, we can see that each sine, cosine, and the constant aremutuallyorthogonal,andwithabitmoreworkwecanprovethattheyspanallintegrableperiodicfunctions. Perhapsmoreimportantly,thismethodallowsusasimplewaytocomputetheexpansionofthefunctiononthebasicset,freeingus to note the contribution of each frequency. Building on this, the extensionto an arbitrary integrable function involves the relaxation of the restriction ofperiodicityandallowanywavefrequency inourbasisset. Thebasicstructureof the space has changed somewhat, but the inner product we defined holdsif we extend the limits to be infinite and introduce a factor to correct for vol

1

ume . Therefore, forf(t) to be expressed as a sum of waves with frequenciesandcoefficientsF(),theprojectionoff(t)ontothewavebasissetwilllooksomethinglike:

1F() = f(t)cos(t)dt (5)

2 But,this isnotquitecomplete,asweneedtoconsiderfunctionswhicharenotattheirmaximumatt=0. So,wecouldwriteasimilarexpressionforsin(t),orwecouldbemoreclever.

Considerthecomplexnumbers,whichwecanexpressinanumberofequivalentways:

iz=a+ib=|z|e =|z|cos() +i|z|sin() (6)So,whatifwerecastourviewofwavesinlightofthis? Wecanexpresscompactlythe

idea

of

a

wave

propagating

along

an

axis

by

viewing

the

wave

as

extending

ontothecomplexaxis,asasortofhelixoffixeddiameterwhichcoilsaroundtheaxisofpropagation. Since(inmostreal-worldexamples)ouroriginalwaveisarealfunction,wecanjusttaketherealpartofthiswave function,which,withcomplex coefficients allowed, becomes some sum of sine and cosine functions,leading to the phase offset we were originally seeking. So, we can express themorecompletetransform:

1itdtF() = f(t)e (7)

2 whereweallowF()totakeoncomplexvalues,toaccountforthisphaseshift2.Wecanalso invertthisexpression:

1itdtf(t) = F()e (8)2

ThiswecalltheFouriertransformandits inverse.1This isrelatedtotheconversionbetweenunits inthetransform, fromangular frequency

toregular frequency.2Re((a+ib)eit)=Re((a+ib)cos(t) + (iab)sin(t))=acos(t)bsin(t)

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3 TheDiscreteFourierTransform(DFT)ThecontinuousFouriertransform is itselfagreat featofmathematics,andanenormousnumberofinterestingproblemscanbeexpressedandsolvedefficientlyinthisform. However,there isoneproblem: whilewecanworkoncontinuousintervalswithoutissueinmathematics,therealworldtendstobealittlemorediscrete. Wecannotreallyhaveapplesonthetable,unlessyouarethesortofpersonwholikestoworkinunusualbases,andlikewisetheclockmechanismsintimepiecesandcomputerscountusingsomedivisionofseconds. Similarly,computermemory isdivided intodiscreteblocks,andwestoreourstreamsofdatausing these discrete blocks. In such a world, it becomes highly impractical tothinkaboutvaryingcontinuouslyoverfrequencies,butwhatifweinsteadsomehowchooseonlyalimitednumberoffrequencies,suchthatwecanapproximatethetrueresultinawaywhichbalancespracticalityandaccuracy?3.1 AnEfficientAlgorithmExistsThegoodnewsis,wecan. AnalgorithmforcomputingtheFouriertransformofadiscretefunction(thediscreteFouriertransformorDFT)efficientlydoesexist,andexploitsthemathematicalstructureofthepolynomials,whichcanalsobeviewedusingthelinearalgebraframeworkweused. Usingcertaintricks,thefastFouriertransform(FFT)canbeusedtocalculatetheDFTmuchmorerapidly,with the running time required being O(nln(n)) instead of O(n2), wheren isthe signal length and O represents the way the running time of the operationscaleswiththeparamatern. Whatthismeansisthatdoublingthesignalonlyslightly more than doubles the time needed to compute the transform, ratherthan quadrupling it, and so we are able to work with rather large signals in amorereasonable lengthoftime.

ThedetailsoftheFFTare fartoogorytobespelledouthere,butthebasic idea is that we can express the inner product of each reference frequency

it).with the signal as a sum over complex divisions of unity (e These products can in turn be expressed as a matrix product of with the signal vector, meaning our problem becomes a matter of efficiently multiplying a matrix with a vector. This can be broken down into subproblems by quarteringthe matrix repeatedly, which is how we achieve the reduction to O(nln(n))growth. A full derivation of the method can be found in any good book onalgorithm design. See, for example, chapter 30 of the second edition of Carmen,Leiserson,Rivest,andSteinsIntroduction toAlgorithms. Weissteingivesa rather technical overview of the mathematics of the algorithm on his web-site at http://mathworld.wolfram.com/FastFourierTransform.html, withplenty

of

references

to

more

complete

manuscripts

on

the

subject.

To

see

how the algorithm is implemented on a computer, the GNU Scientific Library(http://www.gnu.org/software/gsl/) isopen-source and includes a fewdifferentFFTalgorithms.

6
http://mathworld.wolfram.com/FastFourierTransform.htmlhttp://www.gnu.org/software/gslhttp://mathworld.wolfram.com/FastFourierTransform.htmlhttp://www.gnu.org/software/gsl


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4 UsingtheFFT inMatlabNowthatwehaveseenabitofwheretheFouriertransformcomesfromandhavesatisfied ourselves that an efficient method for computing the DFT of a signalexists, we can exploit our knowledge for practical gain. Normally, this wouldrequirethatweactuallysitdownandprogramacomputertoexecutetheFFT,butthegoodfolkswritingmathematicspackagessuchasMatlab,Mathematica,theGNUScientificLibrary,andcountlessothershavealreadydonethisforus.As a result, any numerical analysis package worth using should include thisfunction,buttodaywewillbeusingMatlab.

WhyMatlab? Forone,Matlabactsasafriendly interfacetothenumericalworkhorse that is Fortran. Another good reason is that MIT pays for us tohavealicensetousethesoftware,andsowemightaswellusesomethingthatiswell-written,hasaniceenoughinterface,andcostsnothingtouse(atleastuntilgraduation). Therearealternativesfromtheopensourcecommunitywhicharefreeregardlessofwhoyouare(GSL,forone),butmostoftheserequirealittlemore computing expertise than is practical for this class3. In short, Matlabprovides us the most power for the least effort required to understand how acomputer works, leaving us free to worry more about our methods than theirlow-levelimplementation.

So, how do we use Matlab? There are a number of texts on this subject,including Mathworks tutorial on their product and dozens of Matlab forxtextbooks, wherex isyourfieldofstudy. So,herewewillcoveronlythemostimportantaspectsoftheoperationofthesoftware.

The first step is to download and install Matlab, following the directionsavailableathttp://ist.mit.edu/services/software/math/tools. Packagesare available for Windows, Macintosh, and Linux, all of whichhave somewhatdifferent interfaces and commands shortcuts. However, the basic functionalityof the package remains the same, so we will talk mostly about the Matlabcommandsofinterest.

Once you install and run the package, you should be greeted with an interpreter with all sorts of subwindows. The one we care about most is in thecenter, as this is our command line interface to the Matlab engine. We willenterourcommandsandseeresultsherewhenworkinginteractively,andwhenrunningscriptsanderrorsorresultswillalsobedisplayedhere.

Matlabsdatastructureisthearray,whichcanhaveessentiallyanynumberofdimensions,limitedbythememoryinyourcomputer. Wefillthisarraywitha single data type, such as 32-bit integers or floating-point numbers, and canaccesstheelementsbychoosingtheappropriateindices. Wecanevenuseaniceshorthandtocreatecertaintypesofarrays:

3If you find that you are interested in physical chemistry more thanjust as a hobby, itwould be wise to acquire a working knowledge of computer science fundamentals, such asthosetaught inAbelsonandSussmansStructureandInterpretationofComputerPrograms,aswellasasolidunderstandingofalgorithmicdesign. Evenifyouneverendupwritingmuchcode yourself, the discipline necessary to understand the analysis will prove highly useful inexperimentaldesign.

7
http://ist.mit.edu/services/software/math/toolshttp://ist.mit.edu/services/software/math/tools


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>> 1:10ans =

1 2 3 4 5 6 7 8 9 10Inthearrayabove,wecreatedanarrayofascendingnumbers,bydefaultseparatedby 1. Wecanalsochangethespacingof thearray elements by insertingathirdterm:>> 1:0.5:3ans =

1.0000 1.5000 2.0000 2.5000 3.0000Notice that we now haveanarrayoffloating-pointnumbers, rather than integers. The representation of floating point numbers on a computer is approximate,sowhenperformingcalculationswemustmakesurethattheprecisionofthe representation(32-bit, 64-bit, etc) is sufficient for our needs4. In thiscasewearehappywithjustafewsignificantfigures,sothestandard32-bitfloatingpointnumberisadequate.

Nowthatwecancreatearrays,wecanstoreandaddressthem(asemicolonsuppressestheoutputofacommand):>> a = 1:10;>> a(2:5)ans =

2 3 4 5Wecanevencreatemultidimensionalarraysandaddresseachdimensionseparately:>>

a

=

rand(3,3)

a =

0.9649 0.9572 0.14190.1576 0.4854 0.42180.9706 0.8003 0.9157

>> a(:,3)ans =

0.14190.42180.9157

Here, the : is used as shorthand for the whole dimension. So, we have askedMatlab for every row of the third column, but wecouldeasily mix and matchthe

indices

to

get

the

exact

window

desired.

TherealstrengthofMatlabisthatitcontainssomehigh-qualityalgorithms

for calculating quantities of great interest to us. For example, we can rapidly4If you will ever have to work with computers in anything more than a passive capac

ity, learn about machine structures and their implications. Knowing the limitations of yourinstruments isabsolutelyessentialtoperforminggoodexperimentalwork.

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calculatetheeigenvaluesandeigenvectorsoflargematrices,takeFouriertransformsoflongsignals,performstatisticalanalysisof largedatasets,makeplotsofthousandsofdatapoints,andsoon. ThebreadthofwhatMatlabcando isdocumented inthemanualforthepackage,althoughmanybooksalsoexisttointroduceareadertosomeofthemostrelevantcommands. Fornow,letusturnourattentiontoasimplescriptfortakingaFouriertransformandplottingit.4.1 AMixtureofSinusoidsHere isabriefscripttocomputeasignal,take itsFouriertransform,andtheninvertthetransformtoshowthatwerecovertheoriginalsignal:Fs = 1000; % frequency of samplingT = 1/Fs; % sample timeL = 1000; % length of samplingt = (0:L-1)*T;% mix up the signalfrequencies = [1, 120; 0.7, 100; 0.9, 60];data = frequencies(:,1)*sin(2*pi*frequencies(:,2)*t);data = data + 0.5*(1-2*rand(1,L));subplot(3,1,1);plot(Fs*t,data);xlabel(t/ms);ylabel(intensity);NFFT = 2^nextpow2(L);Y = fft(data, NFFT)/L;f = Fs/2*linspace(0,1,NFFT/2+1);subplot(3,1,2);plot(f,2*abs(Y(1:NFFT/2+1)));xlabel(frequency/Hz);ylabel(|F(\omega)|);subplot(3,1,3);recovered = L*ifft(Y);plot(Fs*t,recovered(1:L),b,

Fs*t,data-recovered(1:L), r);xlabel(t/ms);ylabel(intensity);

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Letuswalkthroughthescriptandfigureoutwhateachpartdoes. First,wesetsomeparameterstogivemeaningtothearrays,whichareotherwiseunitless5:Fs = 1000; % frequency of samplingT = 1/Fs; % sample timeL = 10000; % length of samplingt = (0:L-1)*T;Next, we create the signal by amplifying the value of each sinusoid by someamplitude,asstoredastwocolumnsinourarrayfrequencies.% mix up the signalfrequencies = [1, 120; 0.7, 100; 0.9, 60];data = frequencies(:,1)*sin(2*pi*frequencies(:,2)*t);Nowthatwehavecreatedthesignal,weintroducesomerandomnoise:data = data + 0.5*(1-2*rand(1,L));Now, data contains values calculated for various values of t according to thefunction:

f(t)=sin(2(120)t) + 0.7sin(2(100)t) + 0.9sin(2(60)t) +(t) (9)Next,weplotthesignalasafunctionoftime:subplot(3,1,1);plot(Fs*t,data);xlabel(t/ms);ylabel(intensity);NowwetaketheFFTofthedata,scalingthelengthofthesignaltoapowerof2 to optimize the accuracy of the result (see a description of the algorithm tounderstandwhythisworks):NFFT = 2^nextpow2(L);Y = fft(data, NFFT)/L;f = Fs/2*linspace(0,1,NFFT/2+1);PlottheFFTbelowthesignal,usingthemagnitudeofeachcoefficienttoshowthe contribution of the frequency only, ignoring the phase shift. Normally theFFT gives us a double-sided result, but we ignore the high-frequency signalsbecausetheyareamirrorofthe lowerfrequencies inamplitude:subplot(3,1,2);

plot(f,2*abs(Y(1:NFFT/2+1)));xlabel(frequency/Hz);ylabel(|F(\omega)|);

5Toacomputer,numbersdonotinherentlyhaveunits. Itisuptoustokeeptrackofthemourselvesandverifythattheyare, in fact,reasonablevalues.

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Finally, perform the inverse of the FFT, take the difference with the orignalsignal,andplotboth:subplot(3,1,3);recovered = L*ifft(Y);plot(Fs*t,recovered(1:L),b,

Fs*t,data-recovered(1:L), r);xlabel(t/ms);ylabel(intensity);Runningthisscriptgivesusaniceplotwithsomeusefulinformationanddemonstratesthatthealgorithmdoes,infact,work:

Note that our signal peaks, while clearly present, are broadened. This is anartifactofthediscretenatureofourtransform,becausewecannotperformaninfinitely precise inner product to obtain an infinitessimally wide peak. Thisfinite resolutioncan be improved by increasingour sampling frequency,but insome cases this becomes impractical or impossible. It is an important skill torecognizetheselimitationsbeforecommittingtimeandefforttoanexperiment,so play around with the script to understand the limitations of this idealizedsituation. For example, try adding a high-frequency component to the signal,suchas550Hz. Wheredoesthisappearonthefrequencyspectrum?

Nowthatwehaveaworkingscript,wecanputinanysignalwewant,scalethe units appropriately, and see the result. For example, we can replace oursinusoids with sawtooth, square, or even a signal of our choosing, such as an

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interferogram. AlgorithmsevenexisttoperformamultidimensionalFFT,sowecanusethissametechniquetostudydiffractionpatternsincrystalsorsolutionstogaininsightastotheaveragestructureofamaterial6.4.2 AFinite-LengthPulseBelowisascriptforgeneratingapulseofagivenlength,timeoffset,andcentralfrequency,andforcomputingitsFFT:Fs = 10^9; % Sampling frequencyT = 1/Fs; % Sample timeL = 3*10^-6; % Length of signal in timeL_counts = ceil(L/T); % length of signal in samplest = (0:L_counts-1)*T; % Time vectorpw = 1*10^-6; % time that our packet existspw_counts = ceil(pw/T); % length of the packet in samplesps = 1*10^-6; % offset from zero for the packetps_counts = ceil(ps/T); % offset of the packet in samplesf = 15*10^6; % packet frequencyy = sin(2*pi*(t-ps)*f);y(1:(ps_counts-1)) = 0;y((ps_counts+pw_counts):L_counts) = 0;subplot(2,1,1);plot(t*10^6,y)xlabel(Time/\mus);ylabel(Intensity);NFFT = 2^nextpow2(L_counts); % Next power of 2 from length of yY = fft(y,NFFT)/L_counts;f = Fs/2*linspace(0,1,NFFT/2+1);% Plot single-sided amplitude spectrum.subplot(2,1,2);plot(f,2*abs(Y(1:NFFT/2+1)))xlabel(Frequency/Hz)ylabel(|F(\omega)|)Andanexampleresult:

6SometheoreticalexamplesarediscussedbyChandler inhisIntroduction toModernStatisticalMechanics,andGuiniersX-RayDiffraction hasamorecompletediscussion.

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5

ThesesortsofpulsesareimportantinNMRspectroscopy,whereyoumighthear them referred to as or/2 pulses. Think about these ideas when performingyourNMRexperimentsforthisclass.

Conclusionsand

Further

Directions

Because of the general way we can apply the Fourier transform to a class ofproblems, it is considered one of the most practical and beautiful products ofmathematics. Becauseofitsefficiency,thetransformhasbeenusedasabuildingpoint forother techniquesvitalto classicalcomputation, signalanalysis, high-frequencydesign,precisioninstrumentation,andavarietyofotherfields. Evenemerging fields make use of the Fourier transform: Shors quantum factoringalgorithm7 isfundamentallyaFouriertransformofasignalcomposedofvaluescomputed in a cyclic group, and most efficient quantum algorithms employ aquantum Fourier transform to achieve their complexity reduction. Coherentcontrolof lightforspectroscopyandcommunication isgovernedbythemixingof optical frequencies, and a number of important projects are being carriedout

to

understand

the

behavior

of

increasingly

short

and

well-defined

pulses

of

light in various systems. No matter where you look in physics, engineering,andcomputerscience,theFouriertransformisanabsolutelyessentialtool,andlearningtorecognizesituationswhereit isapplicable isaskillwhichwillservewellovertime.

7http://arxiv.org/abs/quant-ph/9508027

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