The Discrete Fourier Transform

... (s) The Fourier transform equals the Laplace transform evaluated along the “jω axis” in the s plane, i.e., along the line s = jω, for which σ =0.Also, the Laplace transform is obtainable from the ... obtainable from the Fourier transform viaanalytic continuation. The usefulness of the Laplace transform relativeto the Fourier transform is exactly analogous to that of the z transform outlined ... chapter derives various Fourier theorems for the case of the DFT. Included are symmetry relations, the shift theorem, convo-lution theorem, correlation theorem, power theorem, and theoremspertaining...

... (1.19b)whereω=ωTisthenormalizedDTfrequencyaxisexpressedinradians.NotethatS(ejωT)=S(ejω)consistsofaninfinitenumberofreplicasoftheCTspectrumS(jω),positionedatintervalsof(2π/T)ontheωaxis(oratintervalsof2πontheωaxis),asillustratedinFig.1.8.NotethatifS(jω)isbandlimitedwithabandwidthωc,andifTischosensufficientlysmallsothatωs>2ωc,thentheDTspectrumisacopyofS(jω)(scaledby1/T)inthebaseband.Thelimitingcaseofωs=2ωciscalledtheNyquistsamplingfrequency.WheneveraCTsignalissampledatorabovetheNyquistrate,noaliasingdistortionoccurs(i.e.,thebasebandspectrumdoesnotoverlapwiththehigher-orderreplicas)andtheCTsignalcanbeexactlyrecoveredfromitssamplesbyextractingthebasebandspectrumofS(ejω)withanideallow-passfilterthatrecoverstheoriginalCTspectrumbyremovingallspectralreplicasoutsidethebasebandandscalingthebasebandbyafactorofT.1.5 TheDiscreteFourierTransformToobtainthediscreteFouriertransform(DFT)thecontinuousfrequencydomainoftheDTFTissampledatNpointsuniformlyspacedaroundtheunitcircleinthez-plane,i.e.,atthepointsc1999byCRCPressLLCFIGURE1.8:IllustrationoftherelationshipbetweentheCTandDTspectra.ωk=(2πk/N),k=0,1, ... Illinois,Urbana-Champaign1.1Introduction1.2FourierSeriesRepresentationofContinuousTimePeriodicSignalsExponentialFourierSeries•TheTrigonometricFourierSeries•Convergence of the Fourier Series1.3TheClassicalFourierTransformforContinuousTimeSignalsPropertiesoftheContinuousTimeFourierTransform• Fourier ... Spectrum of the Continuous Time Sampling Model•FourierTransform ofPeriodicContinuousTimeSignals• The Generalized Complex Fourier Transform 1.4TheDiscreteTimeFourierTransformProperties ofthe Discrete...

... because the Standard window reduces the bias but at the same time increases the variance. On the otherhand, the use of the Rectangular window makes aFTE biased even for high SNR.In Fig. 21 the ... for SNR below 10 dB the FTEwith the Standard window offers less bias than the TFBMPM. Nevertheless, the rmse obtained with the TFBMPM is less than the one computed using the Standard window ... reduction in the side lobes. The Standard windowachieves the largest reduction of the bias, but it doesso at the expense of broadening the main lobe, whichhiÅͭ1, 0 £ i £ N 0 10, otherwise;(5.2.4)results...