EECE 301 Signals & Systems Prof. Mark FowlerCh. 4: Fourier Analysis of D-T Signals4.1 Discrete-Time FT (DTFT)1/12EECE 301 Signals & SystemsProf. Mark FowlerNote Set #22• D-T Signals: Frequency-Domain Analysis• Reading Assignment: Section 4.1 of Kamen and Heck2/12Ch. 1 IntroC-T Signal ModelFunctions on Real LineD-T Signal ModelFunctions on IntegersSystem PropertiesLTICausalEtcCh. 2 Diff EqsC-T System ModelDifferential EquationsD-T Signal ModelDifference EquationsZero-State ResponseZero-Input ResponseCharacteristic Eq.Ch. 2 ConvolutionC-T System ModelConvolution IntegralD-T System ModelConvolution SumCh. 3: CT Fourier SignalModelsFourier SeriesPeriodic SignalsFourier Transform (CTFT)Non-Periodic SignalsNew System ModelNew SignalModelsCh. 5: CT Fourier SystemModelsFrequency ResponseBased on Fourier TransformNew System ModelCh. 4: DT Fourier SignalModelsDTFT(for “Hand” Analysis)DFT & FFT(for Computer Analysis)New SignalModelPowerful Analysis ToolCh. 6 & 8: Laplace Models for CTSignals & SystemsTransfer FunctionNew System ModelCh. 7: Z Trans.Models for DTSignals & SystemsTransfer FunctionNew SystemModelCh. 5: DT Fourier System ModelsFreq. Response for DTBased on DTFTNew System ModelCourse Flow DiagramThe arrows here show conceptual flow between ideas. Note the parallel structure between the pink blocks (C-T Freq. Analysis) and the blue blocks (D-T Freq. Analysis).3/12Ch. 4: Fourier Analysis of D-T SignalsIn this chapter we do for D-T signals what we did for C-T signals in Ch. 3:–Define a D-T FT (DTFT) for D-T signals and see that it works pretty much like the FT for C-T signals (CTFT)But… we also do something we can’t do for CTFT-based ideas:–Develop a computer-processing version of the DTFT… called the Discrete Fourier Transform (DFT) that will allow you to use the computer to numerically compute a “view” of the DTFTOrder of Coverage:– Sect. 4.1: DTFT & It’s Properties– Sect. 4.2 & 4.3: The DFT & DFT-Based Signal Analysis– Note: Section 4.4 is NOT covered– Sect. 4.5 provides some applications of DFT analysis… we’ll cover some other applications in class4/124.1 Discrete-Time FT (DTFT)5/12Recall: Sampling AnalysisImpulse GenCT LPFDACx(t)x[n] = x(nT)“Hold”Sample att = nTADCfX( f )B–BfFs2Fs–Fs–2Fs)(~tx )(ˆtx)(~fXAA/TAs long as Fs≥ 2B then we can clearly “see”…a view of X( f ) in )(~fXBut we “did” this using a FT of a signal inside the DAC…Is there some other way to do this by using the samples?6/12Impulse GenCT LPFDACx(t)x[n]“Hold”Sample att = nTADC)(~tx )(ˆtxMotivation for D-T Fourier Transform (DTFT)?????)(~fX7/12Recall Fourier Transform of )(~tx)()()()()(~ttxnTttxtxTnδδ=−=∑∞−∞=∑∞−∞==ktFjksetxTtxπ2)(1)(~∑∞−∞=+=kskFfXTfX )(1)(~FS of δT(t)FT & Mod. Proplike! looks )(~ what TellsωX8/12Take An Alternate Path to the DTFT!)()()()()(~ttxnTttxtxTnδδ=−=∑∞−∞=∑∞−∞==ktFjksetxTtxπ2)(1)(~∑∞−∞=+=ksFkXTX )2(1)(~πωωFS of δT(t)FT & Mod. Proplike! looks )(~ what TellsωX{}∑∑∞−∞=∞−∞=−=⎪⎭⎪⎬⎫⎪⎩⎪⎨⎧−=nnnTtnxnTtnxX)(][)(][)(~δδωFF∑∞−∞=−=nnTtnxtx )(][)(~δ∑∞−∞=−=nTjnenxXωω][)(~! )(~ compute how to TellsωXUses Samples!!9/12Re-Define to Get The DTFT!∑∞−∞=−=nTjnenxXωω][)(~∑∞−∞=Ω−=ΩnjnenxX ][)(Let Ω = ωT where T = 1/FsDTFT:Ω is called “D-T Frequency”unitdifferent a w.r.t.plotted""just thing...same really the are )( and )(~ ΩXXωω: rad/secΩ: rad/sampleΩ = ωT: (rad/sec) × (sec/sample) = rad/sample10/12DTFT X(Ω) shows…)(~fXΩ2π 4π–2π–4π)(ΩXA/Tπ–πDTFT of x[n]Only Need to Look Here!!!fX( f )B–BACTFT of x(t)fFs2Fs–Fs–2Fs)(~fXA/TFs/2–Fs/2CTFT of)(tx~“CTFT” X( f )which shows…If sampling was done right!!!11/12Physical Relationship of DTFTImpulse GenCT LPFDACx(t)x[n] = x(nT)“Hold”Sample att = nTADC)(~tx )(ˆtxfX( f )B–BACTFT of x(t)Ω2π 4π–2π–4π)(ΩXA/Tπ–πDTFT of x[n]fFs2Fs–Fs–2Fs)(~fXA/TCTFT of)(tx~Fs/2–Fs/212/12Motivating D-T System Analysis using DTFTD-TSystemDACx(t)x[n]ADCy(t)y[n]fX( f )B–BCTFT of x(t)fX( f )B–BCTFT of y(t)Ω)(ΩXπ–πDTFT of x[n]Ω)(ΩXπ–πDTFT of
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