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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 c  1999byCRCPressLLC 1 Fourier Series, Fourier Transforms, and the DFT W. Kenneth Jenkins University of Illinois, Urbana-Champaign 1.1Introduction 1.2FourierSeriesRepresentationofContinuousTime PeriodicSignals ExponentialFourierSeries • TheTrigonometricFourierSeries • Convergence of the Fourier Series 1.3TheClassicalFourierTransformforContinuousTime Signals PropertiesoftheContinuousTimeFourierTransform • Fourier Spectrum of the Continuous Time Sampling Model • FourierTransform ofPeriodicContinuousTimeSignals • The Generalized Complex Fourier Transform 1.4TheDiscreteTimeFourierTransform Properties ofthe Discrete Time FourierTransform • Relation- ship between the Continuous and Discrete Time Spectra 1.5TheDiscreteFourierTransform Properties of the Discrete Fourier Series • Fourier Block Pro- cessing in Real-Time Filtering Applications • Fast Fourier Transform Algorithms 1.6FamilyTreeofFourierTransforms 1.7SelectedApplicationsofFourierMethods Fast Fourier Transform in Spectral Analysis • Finite Impulse Response Digital Filter Design • Fourier Analysis of Ideal and Practical Digital-to-Analog Conversion 1.8Summary References 1.1 Introduction Fourier methods are commonly used for signal analysis and system design in modern telecommu- nications, radar, and image processing systems. ClassicalFourier methods such as the Fourier series andtheFourier integral areusedforcontinuoustime(CT)signalsandsystems,i.e.,systemsinwhich a characteristic signal, s(t),isdefinedatallvaluesoft on the continuum −∞ <t<∞ .Amore recentlydevelopedsetofFouriermethods,includingthediscretetimeFouriertransform(DTFT)and the discrete Fourier transform (DFT),are extensions of basic Fourier concepts that apply to discrete time (DT) signals. A characteristic DT signal, s[n], is defined only for values of n where n is an integer in the range −∞ <n<∞. The following discussion presents basic concepts and outlines important properties forboth the CT and DT classes ofFouriermethods, with aparticular emphasis ontherelationshipsbetweenthese twoclasses. The classofDT Fouriermethodsisparticularly useful c  1999 by CRC Press LLC asabasisfordigitalsignal processing(DSP)becauseit extendsthetheory of classical Fourieranalysis to DT signals and leads to many effective algorithms that can be directly implemented on general computers or special pur pose DSP devices. TherelationshipbetweentheCTandtheDTdomainsischaracterizedbytheoperationsofsampling and reconstruction. If s a (t) denotes a signal s(t) that has been uniformly sampled every T seconds, then the mathematical representation of s a (t) is given by s a (t) = ∞  n=−∞ s(t)δ(t − nT ) (1.1) where δ(t) is a CT impulse function defined to be zero for all t = 0, undefined at t = 0, and has unit area when integ rated from t =−∞to t =+∞. Because the only places at which the product s(t)δ(t−nT ) isnotidenticallyequaltozeroareatthesamplinginstances,s(t)in(1.1)canbereplaced with s(nT ) without changing the overall meaning of the expression. Hence, an alternate expression for s a (t) that is often useful in Fourier analysis is given by s a (t) = ∞  n=−∞ s(nT )δ(t − nT ) (1.2) The CT sampling model s a (t) consists of a sequence of CT impulse functions uniformly spaced at intervalsofT secondsandweightedbythevaluesofthesignals(t)atthesamplinginstants,asdepicted in Fig. 1.1. Note that s a (t) is not defined at the sampling instants because the CT impulse function itself is not defined at t = 0. However, the values of s(t) at the sampling instants are imbedded as “area under the curve” of s a (t), and as such represent a useful mathematical model of the sampling process. In the DT domain the sampling model is simply the sequence defined by taking the values of s(t) at the sampling instants, i.e., s[n]=s(t)| t=nT (1.3) Incontrast to s a (t), which is notdefinedatthesamplinginstants,s[n] iswelldefinedatthesampling instants, as illustratedin Fig. 1.2. Thus, it is nowclearthats a (t) ands[n] aredifferent but equivalent models of the sampling process in the CT and DT domains, respectively. They are both useful for signal analysis in their corresponding domains. Their equivalenceis established by the fact that they have equal spectra in the Fourier domain, and that the underlying CT signal from which s a (t) and s[n] are derived can be recovered from either sampling representation, provided a sufficiently large sampling rate is used in the sampling operation (see below). 1.2 Fourier Series Representation of Continuous Time Periodic Signals It is convenient to begin this discussion with the classical Fourier series representation of a p eriodic timedomainsignal,andthenderive the Fourierintegralfromthisrepresentationbyfinding the limit of the Fourier coefficient representationas the period goes toinfinity. The conditionsunderwhicha periodic signal s(t) can be expanded in a Fourier series are known as the Dirichet conditions. They require that in each period s(t) has a finite number of discontinuities, a finite number of maxima and minima, and that s(t) satisfies the following absolute convergence criterion [1]:  T/2 −T/2 |s(t)| dt < ∞ (1.4) Itis assumed in the following discussion that these basic conditions are satisfied byall functions that will be represented by a Fourier ser ies. c  1999 by CRC Press LLC FIGURE1.1:CTmodelofasampledCTsignal. FIGURE1.2:DTmodelofasampledCTsignal. 1.2.1 ExponentialFourierSeries IfaCTsignals(t)isperiodicwithaperiodT,thentheclassicalcomplexFourierseriesrepresentation ofs(t)isgivenby s(t)= ∞  n=−∞ a n e jnω 0 t (1.5a) whereω 0 =2π/T,andwherethea n arethecomplexFouriercoefficientsgivenby a n =(1/T)  T/2 −T/2 s(t)e −jnω 0 t dt (1.5b) Itiswellknownthatforeveryvalueoftwheres(t)iscontinuous,theright-handsideof(1.5a) convergestos(t).Atvaluesoftwheres(t)hasafinitejumpdiscontinuity,theright-handside of(1.5a)convergestotheaverageofs(t − )ands(t + ),wheres(t − )≡lim →0 s(t−)ands(t + )≡ lim →0 s(t+). Forexample,theFourierseriesexpansionofthesawtoothwaveformillustratedinFig.1.3ischar- acterizedbyT=2π,ω 0 =1,a 0 =0,anda n =a −n =Acos(nπ)/(jnπ)forn=1,2, ,.The coefficientsoftheexponentialFourierseriesrepresentedby(1.5b)canbeinterpretedasthespec- tralrepresentationofs(t),becausethea n -thcoefficientrepresentsthecontributionofthe(nω 0 )-th frequencytothetotalsignals(t).Becausethea n arecomplexvalued,theFourierdomainrepresen- c  1999byCRCPressLLC tationhasbothamagnitudeandaphasespectrum.Forexample,themagnitudeofthea n isplotted inFig.1.4forthesawtoothwaveformofFig.1.3.Thefactthatthea n constituteadiscretesetis consistentwiththefactthataperiodicsignalhasa“linespectrum,”i.e.,thespectrumcontainsonly integermultiplesofthefundamentalfrequencyω 0 .Therefore,theequationpairgivenby(1.5a) and(1.5b)canbeinterpretedasatransformpairthatissimilartotheCTFouriertransformfor periodicsignals.ThisleadstotheobservationthattheclassicalFourierseriescanbeinterpreted asaspecialtransformthatprovidesaone-to-oneinvertiblemappingbetweenthediscrete-spectral domainandtheCTdomain.Thenextsectionshowshowtheperiodicityconstraintcanberemoved toproducethemoregeneralclassicalCTFouriertransform,whichappliesequallywelltoperiodic andaperiodictimedomainwaveforms. FIGURE1.3:PeriodicCTsignalusedinFourierseriesexample. FIGURE1.4:MagnitudeoftheFouriercoefficientsforexampleofFigure1.3. 1.2.2 TheTrigonometricFourierSeries AlthoughFourierseriesexpansionsexistforcomplexperiodicsignals,andFouriertheorycanbe generalizedtothecaseofcomplexsignals,thetheoryandresultsaremoreeasilyexpressedforreal- valuedsignals.Thefollowingdiscussionassumesthatthesignals(t)isreal-valuedforthesakeof simplifyingthediscussion.However,allresultsarevalidforcomplexsignals,althoughthedetailsof thetheorywillbecomesomewhatmorecomplicated. Forreal-valuedsignalss(t),itispossibletomanipulatethecomplexexponentialformoftheFourier seriesintoatrigonometricformthatcontainssin(ω 0 t)andcos(ω 0 t)termswithcorrespondingreal- c  1999byCRCPressLLC valued coefficients [1]. The trigonometric form of the Fourier series for a real-valued signal s(t) is given by s(t) = ∞  n=0 b n cos(nω 0 t) + ∞  n=1 c n sin(nω 0 t) (1.6a) where ω 0 = 2π/T .Theb n and c n are real-valued Fourier coefficients determined by FIGURE 1.5: Periodic CT signal used in Fourier series example 2. FIGURE 1.6: Fourier coefficients for example of Figure 1.5. b 0 = (1/T )  T/2 −T/2 s(t)dt b n = (2/T )  T/2 −T/2 s(t)cos(nω 0 t)dt, n = 1, 2, , (1.6b) c n = (2/T )  T/2 −T/2 s(t)sin(nω 0 t)dt, n = 1, 2, , An arbitrary real-valued signal s(t) can be expressed as a sum of even and odd components, s(t) = s even (t) + s odd (t),wheres even (t) = s even (−t) and s odd (t) =−s odd (−t), and where s even (t) = [s(t) + s(−t)]/2 and s odd (t) =[s(t) − s(−t)]/2. For the trigonometric Fourier series, it can be shownthats even (t)isrepresentedbythe(e ven)cosinetermsintheinfiniteseries,s odd (t)isrepresented by the (odd) sine terms, and b 0 is the DC level of the signal. Therefore, if it can be determined by inspectionthata signalhasDClevel,orifit isevenorodd,thenthecorrectformof thetrigonometric c  1999 by CRC Press LLC seriescanbechosentosimplifytheanalysis.Forexample,itiseasilyseenthatthesignalshownin Fig.1.5isanevensignalwithazeroDClevel.Thereforeitcanbeaccuratelyrepresentedbythecosine serieswithb n =2Asin(πn/2)/(πn/2),n=1,2, ,asillustratedinFig.1.6.Incontrast,notethat thesawtoothwaveformusedinthepreviousexampleisanoddsignalwithzeroDClevel;thus,itcan becompletelyspecifiedbythesinetermsofthetrigonometricseries.Thisresultcanbedemonstrated bypairingeachpositivefrequencycomponentfromtheexponentialserieswithitsconjugatepartner, i.e.,c n =sin(nω 0 t)=a n e jnω 0 t +a −n e −jnω 0 t ,wherebyitisfoundthatc n =2Acos(nπ)/(nπ)for thisexample.Ingeneralitisfoundthata n =(b n −jc n )/2forn=1,2, ,a 0 =b 0 ,anda −n =a ∗ n . ThetrigonometricFourierseriesiscommoninthesignalprocessingliteraturebecauseitreplaces complexcoefficientswithrealonesandoftenresultsinasimplerandmoreintuitiveinterpretation oftheresults. 1.2.3 ConvergenceoftheFourierSeries TheFourierseriesrepresentationofaperiodicsignalisanapproximationthatexhibitsmeansquared convergencetothetruesignal.Ifs(t)isaperiodicsignalofperiodT,ands  (t)denotestheFourier seriesapproximationofs(t),thens(t)ands  (t)areequalinthemeansquaresenseif MSE=  T/2 −T/2 |s(t)−s(t)  | 2 dt=0 (1.7) Evenwith(1.7)satisfied,meansquareerror(MSE)convergencedoesnotmeanthats(t)=s  (t) ateveryvalueoft.Inparticular,itisknownthatatvaluesoft,wheres(t)isdiscontinuous,the Fourierseriesconvergestotheaverageofthelimitingvaluestotheleftandrightofthediscontinuity. Forexample,ift 0 isapointofdiscontinuity,thens  (t 0 )=[s(t − 0 )+s(t + 0 )]/2,wheres(t − 0 )and s(t + 0 )weredefinedpreviously.(Notethatatpointsofcontinuity,thisconditionisalsosatisfiedby thedefinitionofcontinuity.)BecausetheDirichetconditionsrequirethats(t)haveatmostafinite numberofpointsofdiscontinuityinoneperiod,thesetS t ,definedasallvaluesoftwithinone periodwheres(t)=s  (t),containsafinitenumberofpoints,andS t isasetofmeasurezerointhe formalmathematicalsense.Therefore,s(t)anditsFourierseriesexpansions  (t)areequalalmost everywhere,ands(t)canbeconsideredidenticaltos  (t)fortheanalysisofmostpracticalengineering problems. Convergencealmosteverywhereissatisfiedonlyinthelimitasaninfinitenumberoftermsare includedintheFourierseriesexpansion.IftheinfiniteseriesexpansionoftheFourierseriesis truncatedtoafinitenumberofterms,asitmustbeinpracticalapplications,thentheapproximation willexhibitanoscillatorybehavioraroundthediscontinuity,knownastheGibbsphenomenon[1]. Lets  N (t)denoteatruncatedFourierseriesapproximationofs(t),whereonlythetermsin(1.5a) fromn=−Nton=NareincludedifthecomplexFourierseriesrepresentationisused,orwhere onlythetermsin(1.6a)fromn=0ton=NareincludedifthetrigonometricformoftheFourier seriesisused.Itiswellknownthatinthevicinityofadiscontinuityatt 0 theGibbsphenomenon causess  N (t)tobeapoorapproximationtos(t).ThepeakmagnitudeoftheGibbsoscillationis13% ofthesizeofthejumpdiscontinuitys(t − 0 )−s(t + 0 )regardlessofthenumberoftermsusedinthe approximation.AsNincreases,theregionthatcontainstheoscillationbecomesmoreconcentrated intheneighborhoodofthediscontinuity,until,inthelimitasNapproachesinfinity,theGibbs oscillationissqueezedintoasinglepointofmismatchatt 0 . Ifs  (t)isreplacedbys  N (t)in(1.7),itisimportanttounderstandthebehavioroftheerrorMSE N asafunctionofN,where MSE N =  T/2 −T/2 |s(t)−s  N (t)| 2 dt (1.8) c  1999byCRCPressLLC AnimportantpropertyoftheFourierseriesisthattheexponentialbasisfunctionse jnω 0 t (orsin(nω 0 t) andcos(nω 0 t)forthetrigonometricform )forn=0,±1,±2, (orn=0,1,2, forthe trigonometricform)constituteanorthonormalset,i.e.,t nk =1forn=k,andt nk =0forn=k, where t nk =(1/T)  T/2 −T/2 (e −jnω 0 t )(e jkω 0 t )dt (1.9) AstermsareaddedtotheFourierseriesexpansion,theorthogonalityofthebasisfunctionsguarantees thattheerrordecreasesinthemeansquaresense,i.e.,thatMSE N monotonicallydecreasesasNis increased.Therefore,apractitionercanproceedwiththeconfidencethatwhenapplyingFourierseries analysismoretermsarealwaysbetterthanfewerintermsoftheaccuracyofthesignalrepresentations. 1.3 TheClassicalFourierTransformforContinuous TimeSignals TheperiodicityconstraintimposedontheFourierseriesrepresentationcanberemovedbytakingthe limitsof(1.5a)and(1.5b)astheperiodTisincreasedtoinfinity.Somemathematicalpreliminaries arerequiredsothattheresultswillbewelldefinedafterthelimitistaken.Itisconvenienttoremove the(1/T)factorinfrontoftheintegralbymultiplying(1.5b)throughbyT,andthenreplacing Ta n bya  n inboth(1.5a)and(1.5b).Becauseω 0 =2π/T,asTincreasestoinfinity,ω 0 becomes infinitesimallysmall,aconditionthatisdenotedbyreplacingω 0 withω.Thefactor(1/T)in(1.5a) becomes(ω/2π).Withthesealgebraicmanipulationsandchangesinnotation(1.5a)and(1.5b) takeonthefollowingformpriortotakingthelimit: s(t)= (1/2π) ∞  n=−∞ a  n e jnωt ω (1.10a) a  n =  T/2 −T/2 s(t)e −jnωt dt (1.10b) ThefinalstepinobtainingtheCTFouriertransformistotakethelimitofboth(1.10a)and(1.10b) asT→∞.Inthelimittheinfinitesummationin(1.10a)becomesanintegral,ωbecomesdω, nωbecomesω,anda  n becomestheCTFouriertransformofs(t),denotedbyS(jω).Theresult issummarizedbythefollowingtransformpair,whichisknownthroughoutmostoftheengineering literatureastheclassicalCTFouriertransform(CTFT): s(t)= (1/2π)  ∞ −∞ S(jω)e jωt dω (1.11a) S(jω)=  ∞ −∞ s(t)e −jωt dt (1.11b) Often(1.11a\)iscalledtheFourierintegraland(1.11b)issimplycalledtheFouriertransform.The relationshipS(jω)=F{s(t)}denotestheFouriertransformationofs(t),whereF{·}isasymbolic notationfortheFouriertransformoperator,andwhereωbecomesthecontinuousfrequencyvariable aftertheperiodicityconstraintisremoved.Atransformpairs(t)↔S(jω)representsaone-to- oneinvertiblemappingaslongass(t)satisfiesconditionswhichguaranteethattheFourierintegral converges. From(1.11a)itiseasilyseenthatF{δ(t−t 0 )}=e −jωt 0 ,andfrom(1.11b)thatF −1 {2πδ(ω− ω 0 )}=e jω 0 t ,sothatδ(t−t 0 )↔e −jωt 0 ande jω 0 t ↔2πδ(ω−ω 0 )arevalidFouriertransform c  1999byCRCPressLLC pairs. UsingtheserelationshipsitiseasytoestablishtheFouriertransformsofcos(ω 0 t)andsin(ω 0 t), as well as many other useful waveforms that are encountered in common signal analysis problems. A number of such transforms are shown in Table 1.1. The CTFT is useful in the analysis and design of CT systems, i.e., systems that process CT signals. FourieranalysisisparticularlyapplicabletothedesignofCTfilterswhicharecharacterizedbyFourier magnitude and phase spectra, i.e., by |H(jω)| and arg H(jω),whereH(jω) is commonly called the frequency response of the filter. For example, an idealtransmissionchannel isone which passes a signal without distorting it. The signal may be scaled by a real constant A and delayed by a fixed time increment t 0 , implying that the impulse response of an ideal channel is Aδ(t − t 0 ), and its corresponding frequency response is Ae −jωt 0 . Hence, the frequency response of an ideal channel is specifiedbyconstantamplitudeforallfrequencies,anda phasecharacteristicwhichislinearfunction given by ωt 0 . 1.3.1 Properties of the Continuous Time Fourier Transform The CTFT has many properties that make it useful for the analysis and design of linear CT systems. Some of the more useful properties are stated below. Amore complete list of the CTFT properties is giveninTable1.2. Proofs of these properties can be found in [2] and [3]. In the following discus- sion F {·} denotes the Fourier transform operation, F −1 {·} denotes the inverse Fourier transform operation, and ∗ denotes the convolution operation defined as f 1 (t) ∗ f 2 (t) =  ∞ −∞ f 1 (t − τ)f 2 (τ ) dτ 1. Linearity (superposition): F {af 1 (t) + bf 2 (t)}=aF {f 1 (t)}+bF{f 2 (t)} (a and b, complex constants) 2. Time shifting: F{f(t − t 0 )}=e −jωt 0 F {f(t)} 3. Frequency shifting: e jω 0 t f(t)= F −1 {F(j(ω− ω 0 ))} 4. Time domain convolution: F {f 1 (t) ∗ f 2 (t)}=F {f 1 (t)}F {f 2 (t)} 5. Frequency domain convolution: F{f 1 (t)f 2 (t)}=(1/2π)F {f 1 (t)}∗F {f 2 (t)} 6. Time differentiation: −jωF(jω) = F{d(f (t))/dt} 7. Time integration: F{  t −∞ f(τ)dτ}=(1/j ω)F (j ω ) + πF(0)δ(ω) The above properties are particularly useful in CT system analysis and design, especially when the system characteristics are easily specified in the frequency domain, as in linear filtering. Note that properties 1, 6, and7 areuseful for solving differential or integralequations. Property 4 provides the basis for many signal processing algorithms because many systems can be specified directly by their impulseorfrequencyresponse. Property3isparticularlyusefulinanalyzingcommunicationsystems inwhichdifferentmodulationfor matsarecommonlyusedtoshiftspectralenergytofrequencybands that are appropriate for the application. 1.3.2 Fourier Spectrum of the Continuous Time Sampling Model Because the CT sampling model s a (t), given in (1.1), is in its ownright a CT signal, it is appropriate to apply the CTFT to obtain an expression for the spectrum of the sampled signal: F {s a (t)}=F  ∞  n=−∞ s(t)δ(t − nT )  = ∞  n=−∞ s(nT )e −jωTn (1.12) Becausetheexpressionontheright-hand sideof(1.12)isafunctionofe jωT itiscustomarytodenote the transform as F(e jωT ) = F {s a (t)}. Later in the chapter this result is compared to the result of c  1999 by CRC Press LLC TABLE1.1 SomeBasicCTFTPairs FourierSeriesCoefficients Signal FourierTransform (ifperiodic) +∞  k=−∞ a k e jkω 0 t 2π +∞  k=−∞ a k δ(ω k ω 0 )a k e jω 0 t 2πδ(ω+ω 0 ) a 1 =1 a k =0, otherwise cosω 0 tπ[δ(ω−ω 0 )+δ(ω+ω 0 )] a 1 =a −1 = 1 2 a k =0, otherwise sinω 0 t π j [δ(ω−ω 0 )−δ(ω+ω 0 )] a 1 =−a −1 = 1 2j a k =0, otherwise x(t)=12πδ(ω) a 0 =1,a k =0,k=0  hasthisFourierseriesrepresentationforany choiceof T 0 >0  Periodicsquarewave x(t)=    1, |t|<T 1 0,T 1 <|t|≤ T 0 2 +∞  k=−∞ 2sinkω 0 T 1 k δ(ω k ω 0 ) ω 0 T 1 π sinc  kω 0 T 1 π  = sinkω 0 T 1 kπ and x(t+T 0 )=x(t) +∞  n=−∞ δ(t−nT) 2π T +∞  k=−∞δ  ω− 2πk T  a k = 1 T forallk x(t)=  1, |t|<T 1 0, |t|>T 1 2T 1 sinc  ωT 1 π  = 2sinωT 1 ω — W π sinc  Wt π  = sinWt πt X(ω)=  1, |ω|<W 0, |ω|>W — δ(t) 1— u(t) 1 jω +πδ(ω) — δ(t−t 0 )e jωt 0 — e −at u(t),Re{a}>0 1 a+jω — te −at u(t),Re{a}>0 1 (a+jω) 2 — t n−1 (n−1)! e −at u(t), Re{a}>0 1 (a+jω) n — c  1999byCRCPressLLC [...]... Syst., vol CAS-33, no 7, pp 732–734, July 1986 [6] Oppenheim, A.V and Schafer, R.W., Digital Signal Processing, Englewood Cliffs, NJ: PrenticeHall, 1975 [7] Blahut, R.E., Fast Algorithms for Digital Signal Processing, Reading, MA: Addison-Wesley, 1985 [8] Deller, J.R., Jr., Tom, Dick, and Mary discover the DFT, IEEE Signal Processing Mag., vol 11, no 2, pp 36–50, Apr 1994 [9] Burrus, C.S and Parks, T.W.,... modern digital signal processing, was obtained from more classical forms of the Fourier transform by simultaneously discretizing the time and frequency domains The DFT, together with the remarkable computational efficiency provided by the fast Fourier transform (FFT) algorithm, has contributed to the resounding success that engineers and scientists have experienced in applying digital signal processing. .. DFFT is useful for the analysis and design of digital filters that are produced by frequency sampling techniques 1.4.2 Relationship between the Continuous and Discrete Time Spectra Because DT signals often originate by sampling CT signals, it is important to develop the relationship between the original spectrum of the CT signal and the spectrum of the DT signal that results First, c 1999 by CRC Press... occurred In general, an analog signal should always be prefiltered with an CT low-pass filter prior to sampling so that aliasing distortion does not occur Figure 1.16 shows the frequency response of a fifth-order elliptic analog low-pass filter that meets industry standards for prefiltering speech signals These signals are subsequently sampled at an 8-kHz sampling rate and transmitted digitally across telephone... Unfortunately, the postfilter Hp (j ω) cannot be implemented perfectly, and, therefore, the actual reconstructed signal always contains some distortion in practice that arises from errors in approximating the ideal postfilter Figure 1.17 shows a digital processor, complete with analog-todigital and digital- to-analog converters, and the accompanying analog pre- and postfilters necessary for proper operation... both continuous time (CT) and discrete time (DT) signals and systems Emphasis was placed on illustrating how these various c 1999 by CRC Press LLC FIGURE 1.16: A fifth-order elliptic analog anti-aliasing filter used in the telecommunications industry with an 8-kHz sampling rate FIGURE 1.17: Analog pre- and postfilters required at the analog to digital and digital to analog interfaces forms of the Fourier... have the same spectrum 1.3.3 Fourier Transform of Periodic Continuous Time Signals We saw earlier that a periodic CT signal can be expressed in terms of its Fourier series The CTFT can then be applied to the Fourier series representation of s(t) to produce a mathematical expression for the “line spectrum” characteristic of periodic signals ∞ F{s(t)} = F n=−∞ an ej nω0 t ∞ = 2π an δ(ω − nω0 ) (1.13) n=−∞... ed., Englewood Cliffs, NJ: Prentice-Hall, 1974 [2] Oppenheim, A.V., Willsky, A.S., and Young, I.T., Signals and Systems, Englewood Cliffs, NJ: Prentice-Hall, 1983 [3] Bracewell, R.N., The Fourier Transform, 2nd ed., New York: McGraw-Hill, 1986 [4] Oppenheim, A.V and Schafer, R.W., Discrete-Time Signal Processing, Englewood Cliffs, NJ: Prentice-Hall, 1989 [5] Jenkins, W.K and Desai, M.D., The discrete-frequency... harmonics These effects produce artifacts in the spectral domain that must be carefully monitored to assure that an accurate spectrum is obtained from FFT processing 1.7.2 Finite Impulse Response Digital Filter Design A common method for designing FIR digital filters is by use of windowing and FFT analysis In general, window designs can be carried out with the aid of a hand calculator and a table of well-known... overlaps the previous block by N − 1 points Overlap-Add Processing This algorithm is similar to the previous one except that the kth input block is defined as sk [n] = s[n + kL], 0, n = 0, , L − 1 n = L, , NDFT − 1 (1.24) where L = NDFT − N + 1 The filter function hpad [n] is augmented with zeroes, as before, to create hpad [n], and the DFT processing is executed as before In each block ypad [n] . Jenkins, W.K. “Fourier Series, Fourier Transforms, and the DFT” Digital Signal Processing Handbook Ed. Vijay K. Madisetti and Douglas B. Williams Boca Raton:. useful c  1999 by CRC Press LLC asabasisfordigitalsignal processing( DSP)becauseit extendsthetheory of classical Fourieranalysis to DT signals and leads to many effective