EECE 301 Signals & Systems Prof. Mark Fowler5.4 SamplingSummary of Sampling1EECE 301 Signals & SystemsProf. Mark FowlerNote Set #21• C-T to D-T Conversion: Sampling of C-T signals• Reading Assignment: Section 5.4 of Kamen and Heck• We study this now for two reasons: The analysis uses C-T Frequency-Domain System Analysis Methods Next we will study Fourier Transform ideas for D-T signals and this gives a good transition2Ch. 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).35.4 SamplingThe Connection Between:Continuous Time& Discrete Time Warning: I don’t really like how the book covers this! It is not that it is wrong… it just fails to make the correct connection between the mathematics and physical reality!!!! Follow these notes and you’ll get it!!!4• Sampled & Digitized music on a Compact Disc– What ensures that we can “perfectly” reconstruct the music signal from its samples???!!!!0 0.2 0.4 0.6 0.8 1-2-1012TimeSignal Value-1012Signal ValueAmpCode“Burn” bits into CDRecordCreates a sequence of samples (i.e., a sequence of numbers)PlayLaserSensorDecodeReconstruct AmpMicrophoneSpeaker= Original?TimeSample &DigitizeSampling is Key Part of CD Scheme5• Systems that use Digital Signal Proc. (DSP) generally – get a continuous-time signal from a sensor–a cont.-time system modifies the signal– an “analog-to-digital converter” (ADC) samples the signal to create a discrete-time signal–A discrete-time system to do the Digital Signal Processing – and then (if desired) convert back to analog using a “digital-to-analog converter (DAC)AnalogElectronicsADCDSPComputerSampling is Key Part of Many SystemsC-TSignalC-TSystemD-TSystemC-TSignalD-TSignalD-TSignalDACC-TSignal6ADCIf Sampling is “Valid”…We Should be Able to “Perfectly” Reconstruct from SamplesC-TSignalD-TSignalDACC-TSignalx[n]x(t)Clock)(ˆtx)()(ˆtxtx=Can we make:???If we can… then we can process the samples x[n]as an alternative to processing x(t)!!!70 0.2 0.4 0.6 0.8 1-2-1012TimeSignal Value0 0.2 0.4 0.6 0.8 1-2-1012TimeSignal ValuePractical Sampling-Reconstruction Set-UpPulse GenCT LPFDigital-to-Analog Converter(DAC))(ˆtxx(t)x [n] = x (nT)“Hold”Sample att = nTAnalog-to-Digital Converter (ADC)T = Sampling Interval Fs = 1/T = Sampling Rate)(~txClock att = nT8• You learn the circuits in an electronics class• Here we focus on the “why,” so we need math models• We start in a little different place than the book but we end up with the same result (but a little easier to see how/why))()(][ nTxtxnxnTt===Note: the book uses an “impulse sampling” model for the ADC…but that has no connection to a physicalADC… we’ll see later that it doeshave a physical connection to the physical DAC!Math Model for Sampling (ADC)• Math Modeling the ADC is easy….– x[n] = x(nT) , so the nthsample is the value of x(t) at t = nT9Math Model for Reconstruction (DAC)Pulse Genh(t)H(ω))(ˆtxx[n])(~tx• Math Model for the DAC consists of two parts:– converting a DT sequence (of numbers) into a CT pulse train– “smoothing” out the pulse train using a lowpass filterCT LPF∑∞−∞=−=nnTtpnTxtx )()()(~)(~txt)(txT 2T 3T-T-2T)()(~)(ˆ)()(~)(ˆωωωHXXthtxtx=∗=)(tpt“Prototype” Pulse10“Impulse Sampling” Model for DAC∑∞−∞=−=nnTtnTxtx )()()(~δ)()( ttpδ=∑∞−∞=−=nnTttxtx )()()(~δNow we have a good model that handles quite well what REALLYhappens inside a DAC… but we simplify it !!!!Why???? 1. Because delta functions are EASY to analyze!!!2. Because it leads to the best possible results (see later!)3. We can easily account for real-life pulses later!!To Ease Analysis: Use)()( ttpδ=In this form… this is called the “Impulse Sampled” signal.Now.. Using property of delta function we can also write…11Impulse Genh(t)H(ω))(~txDACx(t)x[n] = x(nT)“Hold”Sample att = nTADCSampling Analysis (p. 1)Analysis will be done using the Impulse Sampling Math Model)(ˆtx)(txt)()()()()(~ttxnTttxtxTnδδ=−=∑∞−∞=)(tTδtT 2T 3T-T-2T“Impulse Train”Note: we are using the “impulse sampling” model in the DAC not the ADC!!!Impulse Sampled Signal)()()(~ttxtxTδ=tT 2T 3T-T-2T12Sampling Analysis (p. 2)Goal = Determine Under What Conditions We Get:Reconstructed CT Signal = Original CT Signal)()(ˆtxtx=Approach: 1. Find the FT of the signal2. Use Freq. Response of Filter to get3. Look to see what is needed to make )(~tx)()(~)(ˆωωωHXX =)()(ˆωωXX =13Sampling Analysis (p. 3)Step #1: Hmmm… well δT(t) is periodic with period Tso we COULD expand it as a Fourier series:∑∞−∞==ktFjkkTsectπδ2)(Period = T secFund. Freq = Fs = 1/T HzSo… what are the FS coefficients???[]TeTetTetTcttFjkTTtFjkTTtFjkTksss11)(1)(1022/2/22/2/2=====−−−−−∫∫πππδδOnly one delta inside a single period By sifting property of the delta function!!!So… an alternate model for δT(t) is∑∞−∞==ktFjkTseTtπδ21)(14Original FTSampling Analysis (p. 4)So we now have….∑∑∞−∞=∞−∞==⎥⎥⎦⎤⎢⎢⎣⎡==ktFjkktFjkTssetxTeTtxttxtxππδ22)(11)()()()(~By frequency shift property of FT… each term is a frequency shifted version of the original signal!!!So using the frequency shift property of the FT gives:∑∞−∞=+=kskFfXTfX )(1)(~Extremely Important Result… the basis of all understanding of sampling!!![] ++++++−+−+=