Jenkins, W.K. “Fourier Series, Fourier Transforms, and the DFT” Digital Signal Processing Handbook Ed. Vijay K. Madisetti and Douglas B. Williams Boca Raton: CRC Press LLC, 1999 (cid:13)c1999byCRCPressLLC 1 Fourier Series, Fourier Transforms, and the DFT 1.1 Introduction 1.2 FourierSeriesRepresentationofContinuousTime PeriodicSignals ExponentialFourierSeries(cid:15)TheTrigonometricFourierSeries (cid:15)ConvergenceoftheFourierSeries 1.3 TheClassicalFourierTransformforContinuousTime Signals Properties of the Continuous Time Fourier Transform (cid:15) FourierSpectrumoftheContinuousTimeSamplingModel(cid:15) FourierTransformofPeriodicContinuousTimeSignals(cid:15)The GeneralizedComplexFourierTransform 1.4 TheDiscreteTimeFourierTransform PropertiesoftheDiscreteTimeFourierTransform(cid:15)Relation- shipbetweentheContinuousandDiscreteTimeSpectra 1.5 TheDiscreteFourierTransform PropertiesoftheDiscreteFourierSeries(cid:15)FourierBlockPro- cessing in Real-Time Filtering Applications (cid:15) Fast Fourier TransformAlgorithms 1.6 FamilyTreeofFourierTransforms 1.7 SelectedApplicationsofFourierMethods FastFourierTransforminSpectralAnalysis(cid:15)FiniteImpulse ResponseDigitalFilterDesign(cid:15)FourierAnalysisofIdealand W.KennethJenkins PracticalDigital-to-AnalogConversion 1.8 Summary UniversityofIllinois, Urbana-Champaign References 1.1 Introduction Fouriermethodsarecommonlyusedforsignalanalysisandsystemdesigninmoderntelecommu- nications,radar,andimageprocessingsystems. ClassicalFouriermethodssuchastheFourierseries andtheFourierintegralareusedforcontinuoustime(CT)signalsandsystems,i.e.,systemsinwhich acharacteristicsignal,s.t/,isdefinedatallvaluesoft onthecontinuum−1 < t < 1. Amore recentlydevelopedsetofFouriermethods,includingthediscretetimeFouriertransform(DTFT)and thediscreteFouriertransform(DFT),areextensionsofbasicFourierconceptsthatapplytodiscrete time (DT) signals. A characteristic DT signal, sTnU, is defined only for values of n where n is an integerintherange−1 < n < 1. Thefollowingdiscussionpresentsbasicconceptsandoutlines importantpropertiesforboththeCTandDTclassesofFouriermethods,withaparticularemphasis ontherelationshipsbetweenthesetwoclasses. TheclassofDTFouriermethodsisparticularlyuseful (cid:13)c1999byCRCPressLLC asabasisfordigitalsignalprocessing(DSP)becauseitextendsthetheoryofclassicalFourieranalysis toDTsignalsandleadstomanyeffectivealgorithmsthatcanbedirectlyimplementedongeneral computersorspecialpurposeDSPdevices. TherelationshipbetweentheCTandtheDTdomainsischaracterizedbytheoperationsofsampling andreconstruction. Ifsa.t/denotesasignals.t/thathasbeenuniformlysampledeveryT seconds, thenthemathematicalrepresentationofsa.t/isgivenby X1 sa.t/D s.t/(cid:14).t −nT/ (1.1) nD−1 where(cid:14).t/isaCTimpulsefunctiondefinedtobezeroforallt 6D 0,undefinedatt D 0,andhas unitareawhenintegratedfromt D−1tot DC1. Becausetheonlyplacesatwhichtheproduct s.t/(cid:14).t−nT/isnotidenticallyequaltozeroareatthesamplinginstances,s.t/in(1.1)canbereplaced withs.nT/withoutchangingtheoverallmeaningoftheexpression. Hence,analternateexpression forsa.t/thatisoftenusefulinFourieranalysisisgivenby X1 sa.t/D s.nT/(cid:14).t −nT/ (1.2) nD−1 TheCTsamplingmodelsa.t/consistsofasequenceofCTimpulsefunctionsuniformlyspacedat intervalsofT secondsandweightedbythevaluesofthesignals.t/atthesamplinginstants,asdepicted inFig.1.1. Notethatsa.t/isnotdefinedatthesamplinginstantsbecausetheCTimpulsefunction itselfisnotdefinedatt D 0. However,thevaluesofs.t/atthesamplinginstantsareimbeddedas “areaunderthecurve”ofsa.t/,andassuchrepresentausefulmathematicalmodelofthesampling process. IntheDTdomainthesamplingmodelissimplythesequencedefinedbytakingthevalues ofs.t/atthesamplinginstants,i.e., sTnUDs.t/jtDnT (1.3) Incontrasttosa.t/,whichisnotdefinedatthesamplinginstants,sTnUiswelldefinedatthesampling instants,asillustratedinFig.1.2. Thus,itisnowclearthatsa.t/andsTnUaredifferentbutequivalent modelsofthesamplingprocessintheCTandDTdomains,respectively. Theyarebothusefulfor signalanalysisintheircorrespondingdomains. Theirequivalenceisestablishedbythefactthatthey haveequalspectraintheFourierdomain,andthattheunderlyingCTsignalfromwhichsa.t/and sTnUarederivedcanberecoveredfromeithersamplingrepresentation,providedasufficientlylarge samplingrateisusedinthesamplingoperation(seebelow). 1.2 Fourier Series Representation of Continuous Time Periodic Signals ItisconvenienttobeginthisdiscussionwiththeclassicalFourierseriesrepresentationofaperiodic timedomainsignal,andthenderivetheFourierintegralfromthisrepresentationbyfindingthelimit oftheFouriercoefficientrepresentationastheperiodgoestoinfinity. Theconditionsunderwhicha periodicsignals.t/canbeexpandedinaFourierseriesareknownastheDirichetconditions. They requirethatineachperiods.t/hasafinitenumberofdiscontinuities,afinitenumberofmaxima andminima,andthats.t/satisfiesthefollowingabsoluteconvergencecriterion[1]: Z T=2 js.t/jdt <1 (1.4) −T=2 Itisassumedinthefollowingdiscussionthatthesebasicconditionsaresatisfiedbyallfunctionsthat willberepresentedbyaFourierseries. (cid:13)c1999byCRCPressLLC FIGURE1.1: CTmodelofasampledCTsignal. FIGURE1.2: DTmodelofasampledCTsignal. 1.2.1 Exponential Fourier Series IfaCTsignals.t/isperiodicwithaperiodT,thentheclassicalcomplexFourierseriesrepresentation ofs.t/isgivenby X1 s.t/D anejn!0t (1.5a) nD−1 where!0 D2(cid:25)=T,andwheretheanarethecomplexFouriercoefficientsgivenby Z T=2 an D.1=T/ s.t/e−jn!0tdt (1.5b) −T=2 It is well known that for every value of t where s.t/ is continuous, the right-hand side of (1.5a) converges to s.t/. At values of t where s.t/ has a finite jump discontinuity, the right-hand side of(1.5a)convergestotheaverageofs.t −/ands.tC/,wheres.t−/(cid:17)lim(cid:15)!0s.t −(cid:15)/ands.tC/(cid:17) lim(cid:15)!0s.t C(cid:15)/. Forexample,theFourierseriesexpansionofthesawtoothwaveformillustratedinFig.1.3ischar- acterizedbyT D 2(cid:25),!0 D 1,a0 D 0,andan D a−n D Acos.n(cid:25)/=.jn(cid:25)/forn D 1;2;:::,. The coefficientsoftheexponentialFourierseriesrepresentedby(1.5b)canbeinterpretedasthespec- tralrepresentationofs.t/,becausethean-thcoefficientrepresentsthecontributionofthe(n!0)-th frequencytothetotalsignals.t/. Becausetheanarecomplexvalued,theFourierdomainrepresen- (cid:13)c1999byCRCPressLLC tationhasbothamagnitudeandaphasespectrum. Forexample,themagnitudeoftheanisplotted in Fig. 1.4 for the sawtooth waveform of Fig. 1.3. The fact that the an constitute a discrete set is consistentwiththefactthataperiodicsignalhasa“linespectrum,”i.e.,thespectrumcontainsonly integermultiplesofthefundamentalfrequency! . Therefore, theequationpairgivenby(1.5a) 0 and(1.5b)canbeinterpretedasatransformpairthatissimilartotheCTFouriertransformfor periodic signals. This leads to the observation that the classical Fourier series can be interpreted asaspecialtransformthatprovidesaone-to-oneinvertiblemappingbetweenthediscrete-spectral domainandtheCTdomain. Thenextsectionshowshowtheperiodicityconstraintcanberemoved toproducethemoregeneralclassicalCTFouriertransform,whichappliesequallywelltoperiodic andaperiodictimedomainwaveforms. FIGURE1.3: PeriodicCTsignalusedinFourierseriesexample. FIGURE1.4: MagnitudeoftheFouriercoefficientsforexampleofFigure1.3. 1.2.2 The Trigonometric Fourier Series Although Fourier series expansions exist for complex periodic signals, and Fourier theory can be generalizedtothecaseofcomplexsignals,thetheoryandresultsaremoreeasilyexpressedforreal- valuedsignals. Thefollowingdiscussionassumesthatthesignals.t/isreal-valuedforthesakeof simplifyingthediscussion. However,allresultsarevalidforcomplexsignals,althoughthedetailsof thetheorywillbecomesomewhatmorecomplicated. Forreal-valuedsignalss.t/,itispossibletomanipulatethecomplexexponentialformoftheFourier seriesintoatrigonometricformthatcontainssin.! t/andcos.! t/termswithcorrespondingreal- 0 0 (cid:13)c1999byCRCPressLLC valuedcoefficients[1]. ThetrigonometricformoftheFourierseriesforareal-valuedsignals.t/is givenby X1 X1 s.t/D bncos.n!0t/C cnsin.n!0t/ (1.6a) nD0 nD1 where!0 D2(cid:25)=T. Thebnandcnarereal-valuedFouriercoefficientsdeterminedby FIGURE1.5: PeriodicCTsignalusedinFourierseriesexample2. FIGURE1.6: FouriercoefficientsforexampleofFigure1.5. Z T=2 b D .1=T/ s.t/dt 0 −T=2 Z T=2 bn D .2=T/ s.t/cos.n!0t/dt; nD1;2;:::; (1.6b) −T=2 Z T=2 cn D .2=T/ s.t/sin.n!0t/dt; nD1;2;:::; −T=2 Anarbitraryreal-valuedsignals.t/canbeexpressedasasumofevenandoddcomponents,s.t/D s .t/ C s .t/, where s .t/ D s .−t/ and s .t/ D −s .−t/, and where s .t/ D even odd even even odd odd even Ts.t/Cs.−t/U=2ands .t/ D Ts.t/−s.−t/U=2. ForthetrigonometricFourierseries, itcanbe odd shownthats .t/isrepresentedbythe(even)cosinetermsintheinfiniteseries,s .t/isrepresented even odd bythe(odd)sineterms,andb istheDClevelofthesignal. Therefore,ifitcanbedeterminedby 0 inspectionthatasignalhasDClevel,orifitisevenorodd,thenthecorrectformofthetrigonometric (cid:13)c1999byCRCPressLLC seriescanbechosentosimplifytheanalysis. Forexample,itiseasilyseenthatthesignalshownin Fig.1.5isanevensignalwithazeroDClevel. Thereforeitcanbeaccuratelyrepresentedbythecosine serieswithbn D2Asin.(cid:25)n=2/=.(cid:25)n=2/;nD1;2;:::;asillustratedinFig.1.6. Incontrast,notethat thesawtoothwaveformusedinthepreviousexampleisanoddsignalwithzeroDClevel;thus,itcan becompletelyspecifiedbythesinetermsofthetrigonometricseries. Thisresultcanbedemonstrated bypairingeachpositivefrequencycomponentfromtheexponentialserieswithitsconjugatepartner, i.e.,cn Dsin.n!0t/Danejn!0t Ca−ne−jn!0t,wherebyitisfoundthatcn D2Acos.n(cid:25)/=.n(cid:25)/for thisexample. Ingeneralitisfoundthatan D.bn−jcn/=2fornD1;2;:::;a0 Db0,anda−n Dan(cid:3). ThetrigonometricFourierseriesiscommoninthesignalprocessingliteraturebecauseitreplaces complexcoefficientswithrealonesandoftenresultsinasimplerandmoreintuitiveinterpretation oftheresults. 1.2.3 Convergence of the Fourier Series TheFourierseriesrepresentationofaperiodicsignalisanapproximationthatexhibitsmeansquared convergencetothetruesignal. Ifs.t/isaperiodicsignalofperiodT,ands0.t/denotestheFourier seriesapproximationofs.t/,thens.t/ands0.t/areequalinthemeansquaresenseif Z T=2 MSE D js.t/−s.t/0j2dt D0 (1.7) −T=2 Even with (1.7) satisfied, mean square error (MSE) convergence does not mean that s.t/ D s0.t/ at every value of t. In particular, it is known that at values of t, where s.t/ is discontinuous, the Fourierseriesconvergestotheaverageofthelimitingvaluestotheleftandrightofthediscontinuity. For example, if t is a point of discontinuity, then s0.t / D Ts.t−/Cs.tC/U=2, where s.t−/ and 0 0 0 0 0 s.tC/weredefinedpreviously. (Notethatatpointsofcontinuity,thisconditionisalsosatisfiedby 0 thedefinitionofcontinuity.) BecausetheDirichetconditionsrequirethats.t/haveatmostafinite number of points of discontinuity in one period, the set St, defined as all values of t within one periodwheres.t/ 6D s0.t/,containsafinitenumberofpoints,andSt isasetofmeasurezerointhe formalmathematicalsense. Therefore, s.t/anditsFourierseriesexpansions0.t/areequalalmost everywhere,ands.t/canbeconsideredidenticaltos0.t/fortheanalysisofmostpracticalengineering problems. Convergencealmosteverywhereissatisfiedonlyinthelimitasaninfinitenumberoftermsare included in the Fourier series expansion. If the infinite series expansion of the Fourier series is truncatedtoafinitenumberofterms,asitmustbeinpracticalapplications,thentheapproximation willexhibitanoscillatorybehavioraroundthediscontinuity,knownastheGibbsphenomenon[1]. Lets0 .t/denoteatruncatedFourierseriesapproximationofs.t/,whereonlythetermsin(1.5a) N fromnD−N tonDN areincludedifthecomplexFourierseriesrepresentationisused,orwhere onlythetermsin(1.6a)fromnD0tonDN areincludedifthetrigonometricformoftheFourier seriesisused. Itiswellknownthatinthevicinityofadiscontinuityatt theGibbsphenomenon 0 causess0 .t/tobeapoorapproximationtos.t/. ThepeakmagnitudeoftheGibbsoscillationis13% N ofthesizeofthejumpdiscontinuitys.t−/−s.tC/regardlessofthenumberoftermsusedinthe 0 0 approximation. AsN increases,theregionthatcontainstheoscillationbecomesmoreconcentrated in the neighborhood of the discontinuity, until, in the limit as N approaches infinity, the Gibbs oscillationissqueezedintoasinglepointofmismatchatt . 0 Ifs0.t/isreplacedbysN0 .t/in(1.7),itisimportanttounderstandthebehavioroftheerrorMSEN asafunctionofN,where Z T=2 MSEN D js.t/−sN0 .t/j2dt (1.8) −T=2 (cid:13)c1999byCRCPressLLC AnimportantpropertyoftheFourierseriesisthattheexponentialbasisfunctionsejn!0t(orsin.n!0t/ and cos.n! t/ for the trigonometric form) for n D 0;(cid:6)1;(cid:6)2;::: (or n D 0;1;2;::: for the 0 trigonometricform)constituteanorthonormalset,i.e.,tnk D 1forn D k,andtnk D 0forn 6D k, where Z T=2 tnk D.1=T/ .e−jn!0t/.ejk!0t/dt (1.9) −T=2 AstermsareaddedtotheFourierseriesexpansion,theorthogonalityofthebasisfunctionsguarantees thattheerrordecreasesinthemeansquaresense,i.e.,thatMSEN monotonicallydecreasesasN is increased. Therefore,apractitionercanproceedwiththeconfidencethatwhenapplyingFourierseries analysismoretermsarealwaysbetterthanfewerintermsoftheaccuracyofthesignalrepresentations. 1.3 The Classical Fourier Transform for Continuous Time Signals TheperiodicityconstraintimposedontheFourierseriesrepresentationcanberemovedbytakingthe limitsof(1.5a)and(1.5b)astheperiodT isincreasedtoinfinity. Somemathematicalpreliminaries arerequiredsothattheresultswillbewelldefinedafterthelimitistaken. Itisconvenienttoremove the.1=T/factorinfrontoftheintegralbymultiplying(1.5b)throughbyT, andthenreplacing Tanbyan0 inboth(1.5a)and(1.5b). Because! 0 D2(cid:25)=T,asT increasestoinfinity,!0becomes infinitesimallysmall,aconditionthatisdenotedbyreplacing! with1!. Thefactor.1=T/in(1.5a) 0 becomes.1!=2(cid:25)/. Withthesealgebraicmanipulationsandchangesinnotation(1.5a)and(1.5b) takeonthefollowingformpriortotakingthelimit: X1 s.t/ D .1=2(cid:25)/ an0ejn1!t1! (1.10a) nD−1 Z T=2 an0 D s.t/e−jn1!tdt (1.10b) −T=2 ThefinalstepinobtainingtheCTFouriertransformistotakethelimitofboth(1.10a)and(1.10b) asT ! 1. Inthelimittheinfinitesummationin(1.10a)becomesanintegral,1!becomesd!, n1!becomes!,andan0 becomestheCTFouriertransformofs.t/,denotedbyS.j!/. Theresult issummarizedbythefollowingtransformpair,whichisknownthroughoutmostoftheengineering literatureastheclassicalCTFouriertransform(CTFT): Z 1 s.t/ D .1=2(cid:25)/ S.j!/ej!td! (1.11a) Z −1 1 S.j!/ D s.t/e−j!tdt (1.11b) −1 Often(1.11a\)iscalledtheFourierintegraland(1.11b)issimplycalledtheFouriertransform. The relationshipS.j!/DFfs.t/gdenotestheFouriertransformationofs.t/,whereFf(cid:1)gisasymbolic notationfortheFouriertransformoperator,andwhere!becomesthecontinuousfrequencyvariable after the periodicity constraint is removed. A transform pair s.t/ $ S.j!/ represents a one-to- oneinvertiblemappingaslongass.t/satisfiesconditionswhichguaranteethattheFourierintegral converges. From(1.11a)itiseasilyseenthatFf(cid:14).t −t 0/gDe−j!t0,andfrom(1.11b)thatF −1f2(cid:25)(cid:14).!− !0/g D ej!0t,sothat(cid:14).t −t0/ $ e−j!t0 andej!0t $ 2(cid:25)(cid:14).!−!0/arevalidFouriertransform (cid:13)c1999byCRCPressLLC pairs. UsingtheserelationshipsitiseasytoestablishtheFouriertransformsofcos.! t/andsin.! t/, 0 0 aswellasmanyotherusefulwaveformsthatareencounteredincommonsignalanalysisproblems. AnumberofsuchtransformsareshowninTable1.1. TheCTFTisusefulintheanalysisanddesignofCTsystems,i.e.,systemsthatprocessCTsignals. FourieranalysisisparticularlyapplicabletothedesignofCTfilterswhicharecharacterizedbyFourier magnitudeandphasespectra, i.e., byjH.j!/jandargH.j!/, whereH.j!/iscommonlycalled thefrequencyresponseofthefilter. Forexample,anidealtransmissionchannelisonewhichpasses asignalwithoutdistortingit. ThesignalmaybescaledbyarealconstantAanddelayedbyafixed time increment t , implying that the impulse response of an ideal channel is A(cid:14).t −t /, and its 0 0 correspondingfrequencyresponseisAe−j!t0. Hence,thefrequencyresponseofanidealchannelis specifiedbyconstantamplitudeforallfrequencies,andaphasecharacteristicwhichislinearfunction givenby!t . 0 1.3.1 Properties of the Continuous Time Fourier Transform TheCTFThasmanypropertiesthatmakeitusefulfortheanalysisanddesignoflinearCTsystems. Someofthemoreusefulpropertiesarestatedbelow. AmorecompletelistoftheCTFTpropertiesis giveninTable1.2. Proofsofthesepropertiescanbefoundin[2]and[3]. Inthefollowingdiscus- sionFf(cid:1)gdenotestheFouriertransformoperation, F−1f(cid:1)gdenotestheinverseFouriertransform operation,and(cid:3)denotestheconvolutionoperationdefinedas Z 1 f .t/(cid:3)f .t/D f .t −(cid:28)/f .(cid:28)/d(cid:28) 1 2 1 2 −1 1. Linearity(superposition): Ffaf .t/Cbf .t/gDaFff .t/gCbFff .t/g 1 2 1 2 (aandb,complexconstants) 2. Timeshifting: Fff.t −t0/gDe−j!t0Fff.t/g 3. Frequencyshifting: ej!0tf.t/DF−1fF.j.!−!0//g 4. Timedomainconvolution: Fff .t/(cid:3)f .t/gDFff .t/gFff .t/g 1 2 1 2 5. Frequencydomainconvolution: Fff .t/f .t/gD.1=2(cid:25)/Fff .t/g(cid:3)Fff .t/g 1 2 1 2 6. Timedifferentiation: −j!F.j!/DFfd.f.t//=dtg R 7. Timeintegration: Ff t f.(cid:28)/d(cid:28)gD.1=j!/F.j!/C(cid:25)F.0/(cid:14).!/ −1 TheabovepropertiesareparticularlyusefulinCTsystemanalysisanddesign,especiallywhenthe systemcharacteristicsareeasilyspecifiedinthefrequencydomain,asinlinearfiltering. Notethat properties1,6,and7areusefulforsolvingdifferentialorintegralequations. Property4providesthe basisformanysignalprocessingalgorithmsbecausemanysystemscanbespecifieddirectlybytheir impulseorfrequencyresponse. Property3isparticularlyusefulinanalyzingcommunicationsystems inwhichdifferentmodulationformatsarecommonlyusedtoshiftspectralenergytofrequencybands thatareappropriatefortheapplication. 1.3.2 Fourier Spectrum of the Continuous Time Sampling Model BecausetheCTsamplingmodelsa.t/,givenin(1.1),isinitsownrightaCTsignal,itisappropriate toapplytheCTFTtoobtainanexpressionforthespectrumofthesampledsignal: ( ) X1 X1 Ffsa.t/gDF s.t/(cid:14).t −nT/ D s.nT/e−j!Tn (1.12) nD−1 nD−1 Becausetheexpressionontheright-handsideof(1.12)isafunctionofej!T itiscustomarytodenote thetransformasF.ej!T/ D Ffsa.t/g. Laterinthechapterthisresultiscomparedtotheresultof (cid:13)c1999byCRCPressLLC TABLE1.1 SomeBasicCTFTPairs FourierSeriesCoefficients Signal FourierTransform (ifperiodic) CX1 CX1 akejk!0t 2(cid:25) ak(cid:14).!k!0/ ak kD−1 kD−1 ej!0t 2(cid:25)(cid:14).!C!0/ a1D1 akD0; otherwise cos!0t (cid:25)T(cid:14).!−!0/C(cid:14).!C!0/U a1Da−1D 21 akD0; otherwise sin!0t (cid:25)jT(cid:14).!−!0/−(cid:14).!C!0/U a1D−a−1D 21j akD0; otherwise a0(cid:18)D1; akD0; k6D0 (cid:19) x.t/D1 2(cid:25)(cid:14).!/ hasthisFourierseriesrepresentationforany choiceofT0>0 Periodicsquarewave 8 x.t/D<: 1; jtj<T1 T CX1 2sinkk!0T1(cid:14).!k!0/ !0(cid:25)T1sinc(cid:18)k!(cid:25)0T1(cid:19)D sinkk!(cid:25)0T1 0; T1<jtj(cid:20) 20 kD−1 and x.tCT /Dx.t/ 0 CX1 2(cid:25)CX1 (cid:18) 2(cid:25)k(cid:19) 1 (cid:14).t−nT/ T kD−1(cid:14) !− T akD T forallk nD−1 (cid:26) (cid:18) (cid:19) x.t/D 10;; jjttjj<>TT11 2T1sinc !(cid:25)T1 D 2sin!!T1 — (cid:18) (cid:19) ( W Wt sinWt 1; j!j<W sinc D X.!/D — (cid:25) (cid:25) (cid:25)t 0; j!j>W (cid:14).t/ 1 — 1 u.t/ C(cid:25)(cid:14).!/ — j! (cid:14).t−t0/ ej!t0 — e−atu.t/;Refag>0 1 — aCj! te−atu.t/;Refag>0 1 — .aCj!/2 .ntn−−11/We−atu.t/; .aC1j!/n — Refag>0 (cid:13)c1999byCRCPressLLC