Tài liệu 01 Fourier Series, Fourier Transforms, and the DFT doc

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 [...]... Although the DFFT is similar in concept to the classical CT Fourier series, the formal properties of the DFFT [5] serve to clarify the effects of frequency domain sampling and time domain aliasing These effects are obscured in the classical treatment of the CT Fourier series because the emphasis is on the inherent “line spectrum” that results from time domain periodicity The DFFT is useful for the analysis... baseband spectrum does not overlap with the higher-order replicas) and the CT signal can be exactly recovered from its samples by extracting the baseband spectrum of S(ej ω ) with an ideal low-pass filter that recovers the original CT spectrum by removing all spectral replicas outside the baseband and scaling the baseband by a factor of T 1.5 The Discrete Fourier Transform To obtain the discrete Fourier. .. significance lies in the fact that it is the most general form that represents the point at which Fourier and Laplace transform concepts become the same Identifying this connection reinforces the notion that Fourier and Laplace transform concepts are similar because they are derived by placing different constraints on the same general form 1.4 The Discrete Time Fourier Transform The discrete time Fourier transform... Fourier transform (DFT) the continuous frequency domain of the DTFT is sampled at N points uniformly spaced around the unit circle in the z-plane, i.e., at the points c 1999 by CRC Press LLC FIGURE 1.8: Illustration of the relationship between the CT and DT spectra ωk = (2π k/N ), k = 0, 1, , N − 1 The result is the DFT pair defined by (1.20a) and (1.20b) The signal s[n] is either a finite length... [6] Whether the DFT and the DFS are considered identical or distinct is not very important in this discussion The important point to be emphasized here is that the DFT treats s[n] as though it were a single period of a periodic sequence, and all signal processing done with the DFT will inherit the consequences of this assumed periodicity 1.5.1 Properties of the Discrete Fourier Series Most of the properties... convolution implemented by the DFT will correspond to the desired linear convolution if the block length of the DFT, NDFT , is chosen sufficiently large so that NDFT ≥ N + M − 1 and both h[n] and s[n] are padded with zeroes to form blocks of length NDFT 1.5.2 Fourier Block Processing in Real-Time Filtering Applications In some practical applications either the value of M is too large for the memory available,... discussed in the previous sections The family tree of CT Fourier transform is shown in Fig 1.10, where the most general, and consequently the most powerful, Fourier transform is the classical complex Fourier transform (or equivalently, the bilateral Laplace transform) Note that the complex Fourier transform is identical to the bilateral Laplace transform, and it is at this level that the classical... because the periodic waveform created by the FFT may have sharp discontinuities at the boundaries of the blocks This effect is minimized by removing the mean of the data (it can always be reinserted) and by windowing the data so the ends of the block are smoothly tapered to zero A good rule of thumb is to taper 10% of the data on each end of the block using either a cosine taper or one of the other common... ){X[((−k))N ]} h[n] The output y(n) in response to input s[n] is given by N −1 h[k]s[n − k] y[n] = (1.21) k=0 where y(n) is obtained by transforming h[n] and s[n] into H [k] and S[k] using the DFT, multiplying the transforms point-wise to obtain Y [k] = H [k]S[k], and then using the inverse DFT to obtain y[n] = DFT 1 {Y [k]} If s[n] is a finite sequence of length M, then the results of the circular convolution... in the sense that they have the same Fourier domain representation A list of common DTFT pairs is presented in Table 1.3 Just as the CT Fourier transform is useful in CT signal system analysis and design, the DTFT is equally useful in the same capacity for DT systems It is indeed fortuitous that Fourier transform theory can be extended in this way to apply to DT systems In the same way that the CT Fourier . (1.21) wherey(n)isobtainedbytransformingh[n]ands[n]intoH[k]andS[k]usingtheDFT,multiplying thetransformspoint-wisetoobtainY[k]=H[k]S[k],andthenusingtheinverseDFTtoobtain y[n] =DFT −1 {Y[k]}.Ifs[n]isafinitesequenceoflengthM,thentheresultsofthecircular convolutionimplementedbytheDFTwillcorrespondtothedesiredlinearconvolutioniftheblock lengthoftheDFT,N DFT ,ischosensufficientlylargesothatN DFT ≥N+M−1andbothh[n] ands[n]arepaddedwithzeroestoformblocksoflengthN DFT . 1.5.2. 1 0,n= N, ,N DFT − 1  (1.22) TheDFTisthenusedtoobtainY pad [n] =DFT{ h pad [n]} DFT{ s k [n]},andy pad [n]=IDFT{Y pad [n]}. Fromthe y pad [n] array the values