EECE 301 Signals & Systems Prof. Mark Fowler“Computing” CT Convolution1/20EECE 301 Signals & SystemsProf. Mark FowlerNote Set #11• C-T Systems: “Computing” Convolution• Reading Assignment: Section 2.6 of Kamen and Heck2/20Ch. 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).3/20)()()()()]()([tvtxtvtxtvtxdtd∗=∗=∗C-T convolution propertiesMany of these are the same as for DT convolution. We only discuss the new ones here. See the next slide for the othersderivative⎥⎦⎤⎢⎣⎡∗=∗⎥⎦⎤⎢⎣⎡=∫∫∫∞−∞−∞−tttdhtxthdxdyλλλλλλ)()()()()(The properties of convolution help perform analysis and design tasks that involve convolution. For example, the associative property says that (in theory) we can interchange to order of two linear systems… in practice, before we can switch the order we need to check what impact that might have on the physical interface conditions.1. Derivative Property:2. Integration Property Let y(t) = x(t)*h(t), then4/20Convolution PropertiesThese are things you can exploit to make it easier to solve convolution problems1.Commutativity⇒ You can choose which signal to “flip”)()()()( txththtx∗=∗2. Associativity⇒ Can change order → sometimes one order is easier than another )())()(())()(()( twtvtxtwtvtx∗∗=∗∗3. Distributivity⇒ may be easier to split complicated system h[n] into sum of simple ones⇒ we can split complicated input into sum of simple ones (nothing more than “linearity”)OR4. Convolution with impulses)()()(ττδ−=−∗txttx)()()()())()(()(2121thtxthtxththtx∗+∗=+∗5/20“Computing” CT Convolution-For D-T systems, convolution is something we do for analysis and for implementation (either via H/W or S/W).-For C-T systems, wedo convolution for analysis…nature does convolution for implementation.If we are analyzinga given system (e.g., a circuit) we may need to compute a convolution to determine how it behaves in response to various different input signalsIf we are designinga system (e.g., a circuit) we may need to be able to visualize how convolution works in order to choose the correct type of system impulse response to make the system work the way we want it to.We’ll learn how to perform “Graphical Convolution,” which is nothing more than steps that help you use graphical insight to evaluate the convolution integral.6/20Steps for Graphical Convolution x(t)*h(t)1. Re-Write the signals as functions of τ: x(τ) and h(τ)2. Flip just one of the signals around t = 0 to get either x(-τ) or h(-τ) a. It is usually best to flip the signal with shorter durationb. For notational purposes here: we’ll flip h(τ) to get h(-τ) 3. Find Edgesof the flipped signal a. Find the left-hand-edge τ-value of h(-τ): call it τL,0b. Find the right-hand-edge τ-value of h(-τ): call it τR,04. Shift h(-τ) by an arbitrary value of t to get h(t - τ) and get its edgesa. Find the left-hand-edge τ-value of h(t - τ) as a function of t: call it τL,t• Important: It will always be… τL,t= t + τL,0 b. Find the right-hand-edge τ-value of h(t - τ) as a function of t: call it τR,t• Important: It will always be… τR,t= t + τR,0 Note: I use τ for what the book uses λ... It is not a big deal as they are just dummy variables!!!τττdthxty )()()( −=∫∞∞−Note: If the signal you flipped is NOT finite duration, one or both of τL,tand τR,twill be infinite (τL,t= –∞ and/or τR,t= ∞)7/205. Find Regions of τ-Overlapa. What you are trying to do here is find intervals of t over which the product x(τ) h(t - τ) has a single mathematical form in terms of τb. In each region find: Interval of t that makes the identified overlap happenc. Working examples is the best way to learn how this is doneTips: Regions should be contiguous with no gaps!!!Don’t worry about < vs. ≤ etc.6. For Each Region: Form the Product x(τ) h(t - τ) and Integratea. Form product x(τ) h(t - τ)b. Find the Limits of Integration by finding the interval of τ over which the product is nonzeroi. Found by seeing where the edges of x(τ) and h(t - τ) lieii. Recall that the edges of h(t - τ) are τL,tand τR,t , which often depend on the value of t• So… the limits of integration maydepend on tc. Integrate the product x(τ) h(t - τ) over the limits found in 6b i. The result is generally a function of t, but is only valid for the interval of t found for the current regionii. Think of the result as a “time-section” of the output y(t)Steps Continued8/207. “Assemble” the output from the output time-sections for all the regionsa. Note: you do NOT add the sections togetherb. You define the output “piecewise”c. Finally, if possible, look for a way to write the output in a simpler formSteps Continued9/20x(t)t22h(t)t13Example: Graphically Convolve Two Signals∫∫∞∞−∞∞−ττ−τ=ττ−τ=dthxdtxhty)()()()()(By “Properties of Convolution”…these two forms are EqualThisis why we can flip eithersignalBy “Properties of Convolution”…these two forms are EqualThisis why we can flip eithersignalConvolve these two signals:10/20x(τ)22h(-τ)3Step #2: Flip h(τ) to get h(-τ)ττ–1x(τ)22h(τ)13Step #1: Write as Function of τττUsually Easier to Flip the Shorter Signal0011/20x(τ)22h(-τ)3Step #3: Find Edges of Flipped Signalτ0τ–10τR,0 = 0τL,0 = –112/20Motivating Step #4: