EECE 301 Signals & Systems Prof. Mark FowlerConvolution for C-T systemsIn what form will we know h(t)?Big Picture1/11EECE 301 Signals & SystemsProf. Mark FowlerNote Set #10• C-T Systems: Convolution Representation• Reading Assignment: Section 2.6 of Kamen and Heck2/11Ch. 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).3/11Convolution for C-T systemsWe saw for D-T systems:-Definition of Impulse Response h[n]-How TI & Linearity allow us to use h[n] to write an equation that gives the output due to input x[n] (That equation is Convolution)C-T LTIICs = 0δ(t)h(t)ttδ(t)h(t)If system is causal, h(t) = 0 for t < 0The same ideas arise for C-T systems!(And the arguments to get there are very similar… so we won’t go into as much detail!!)Impulse Response: h(t) is what “comes out” when δ(t) “goes in”4/11Note: In D-T systems, δ[n] has a height of 1 In C-T systems, δ(t) has a “height of infinity” and a “width of zero”-So, in practice we can actually make δ[n]-But we cannot actually make δ(t)!!How do we know or get the impulse response h(t)?1. It is given to us by the designer of the C-T system.2. It is measured experimentally-But, we cannot just “put in δ(t)”-There are other ways to get h(t) but we need chapter 3 and 5 information first3. Mathematically analyze the C-T system -Easiest using ideas in Ch. 3, 5, 6, & 85/11In what form will we know h(t)?Our focus is here1. h(t) known analytically as a function -e.g. h(t) = e-2tu(t)2. We may only have experimentally obtained samples:- h(nT) at n = 0, 1, 2, 3, … , N-1Now we can…Use h(t) to find the zero-state response of the system for an inputh(t)x(t) y(t)C-T LTIICs = 0λλλdthxty )()()( −=∫∞∞−y(t)=x(t)*h(t)CONVOLUTIONNotation:Following similar ideas to the DT case we get that:6/11Example 2.14 Output of RC Circuit with Unit Step Inputx(t)y(t)We have seen that this circuit is modeled by the following Differential Equation: )(1)(1)(txRCtyRCdttdy=+Problem: Find the zero-state response of this circuit to a unit step input… i.e., let x(t) = u(t) and find y(t) for the case of the ICs set to zero (for this case that means y(0-) = 0). The book considers a different case… x(t) is a pulseSo… we need to solve this Diff. Eq. for the case of x(t) = u(t).The previous slides told us that we can use convolution…7/11In Chapter 6 we will learn how to find the impulse response by applying the Laplace Transform to the differential equation. The result is:)(1)(0,00,1)()/1()/1(tueRCthortteRCthtRCtRC−−=⎪⎩⎪⎨⎧<≥=But… to do that we need to know the impulse response h(t) for this system (i.e., for this differential equation)!!!8/11x(t)y(t) = x(t)*h(t) t1vx(t) = u(t)*For our step input:t1vx(t) = u(t)This is the convolution we need to do…9/11∫∞∞−−==λλλdtxhthtxty )()()(*)()(Plug in given forms for h(t) and x(t)[]∫∞∞−−−=λλλλdtuueRCRC)()()/1(1This is the general form for convolutionThis makes the integrand = 0 whenever t- λ < 0 or in other words whenever λ > t.And… it is 1 otherwise.This makes the integrand = 0 whenever λ < 0. And… it is 1 otherwise.⎪⎪⎩⎪⎪⎨⎧≤>=∫−0,00,)(0)/1(1ttdetytRCRCλλSo exploiting these facts we see that the only thing the unit steps do here is to limit the range of integration…(Note that if t < 0 then the integrand is 0 for all λ )So… to find the output for this problem all we have to do is evaluate this integral to get a function of t10/11t (sec)y(t) (volts)Output for RC = 1Once we can compute this kind of integral in general we can find out what the output looks like for any given input![][][][]tRCtRCtRCtRCRCtRCRCeeeeRCede)/1(0)/1(0)/1(0)/1(10)/1(11−−−−−−=−−−=−=−=∫λλλλThis integral is the easiest one you learned in Calc I!!!⎪⎩⎪⎨⎧≤>−=−0,00,1)()/1(ttetytRCRecall Time-Constant Rules: ► 63% after 1 TC► ≈100% after 5 TCs11/11Big PictureFor a LTI C-T system in zero state we no longer need the differential equation model…-Instead we need the impulse response h(t) & convolutionNew alternative model!Differential EquationConvolution& Impulse resp.Equivalent Models (for zero state)Compare to “Big Picture” for DT Case…
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