6.003: Signals and Systems Lecture 10 October 15, 200916.003: Signals and SystemsConvolutionOctober 15, 2009Multiple Representations of CT and DT SystemsVerbal descriptions: preserve the rationale.Difference/differential equations: mathematically compact.y[n] = x[n] + z0y[n − 1] ˙y(t) = x(t) + s0y(t)Block diagrams: illustrate signal flow paths.+Rz0X Y+As0X YOperator representations: analyze systems as polynomials.YX=11 − z0RYX=A1 − s0ATransforms: representing diff. equations with algebraic equations.H(z) =zz − z0H(s) =1s − s0ConvolutionRepresenting a system by a single signal.Responses to arbitrary signalsAlthough we have focused on responses to simple signals (δ[n], δ(t))we are generally interested in responses to more complicated signals.How do we compute responses to a more complicated input signals?No problem for difference equations / block diagrams.→ use step-by-step analysis.Check YourselfWhat is y[3]?+ +R RX Yx[n]ny[n]nSuperpositionBreak input into additive parts and sum the responses to the parts.nx[n]y[n]n=++++=n−1 0 1 2 345nnnn−1 0 1 2 345nSuperposition works if the system is linear.6.003: Signals and Systems Lecture 10 October 15, 20092LinearityA system is linear if its response to a weighted sum of inputs is equalto the weighted sum of its responses to each of the inputs.Givensystemx1[n] y1[n]andsystemx2[n] y2[n]the system is linear ifsystemαx1[n] + βx2[n] αy1[n] + βy2[n]is true for all α and β.SuperpositionBreak input into additive parts and sum the responses to the parts.nx[n]y[n]n=++++=n−1 0 1 2 345nnnn−1 0 1 2 345nReponses to parts are easy to compute if system is time-invariant.Time-InvarianceA system is time-invariant if delaying the input to the system simplydelays the output by the same amount of time.Givensystemx[n] y[n]the system is time invariant ifsystemx[n − n0] y[n − n0]is true for all n0.SuperpositionBreak input into additive parts and sum the responses to the parts.nx[n]y[n]n=++++=n−1 0 1 2 345nnnn−1 0 1 2 345nSuperposition is easy if the system is linear and time-invariant.Structure of SuperpositionIf a system is linear and time-invariant (LTI) then its output is thesum of weighted and shifted unit-sample responses.systemδ[n] h[n]systemδ[n − k] h[n − k]systemx[k]δ[n − k] x[k]h[n − k]systemx[n] =∞Xk=−∞x[k]δ[n − k]∞Xk=−∞x[k]h[n − k]ConvolutionResponse of an LTI system to an arbitrary input.LTIx[n] y[n]y[n] =∞Xk=−∞x[k]h[n − k] ≡ (x ∗ h)[n]This operation is called convolution.6.003: Signals and Systems Lecture 10 October 15, 20093NotationConvolution is represented with an asterisk.∞Xk=−∞x[k]h[n − k] ≡ (x ∗ h)[n]It is customary (but confusing) to abbreviate this notation:(x ∗ h)[n] = x[n] ∗ h[n]NotationDo not be fooled by the confusing notation.Confusing (but conventional) notation:∞Xk=−∞x[k]h[n − k] = x[n] ∗ h[n]x[n] ∗ h[n] looks like an operation of samples; but it is not!x[1] ∗ h[1] 6= (x ∗ h)[1]Convolution operates on signals not samples.Unambiguous notation:∞Xk=−∞x[k]h[n − k] ≡ (x ∗ h)[n]The symbols x and h represent DT signals.Convolving x with h generates a new DT signal x ∗ h.Structure of Convolutiony[2] =∞Xk=−∞x[k]h[2 − k]kx[k] h[k]h[2 − k]h[2 − k]x[k]h[2 − k]∗∞Xk=−∞kk−2−1 0 1 2 345k−2−1 0 1 2 345kStructure of Convolutiony[3] =∞Xk=−∞x[k]h[3 − k]kx[k] h[k]h[3 − k]h[3 − k]x[k]h[3 − k]∗∞Xk=−∞kk−2−1 0 1 2 345k−2−1 0 1 2 345kCheck Yourself∗Which plot shows the result of the convolution above?1. 2.3. 4.5. none of the aboveConvolutionRepresenting an LTI system by a single signal.h[n]x[n] y[n]Unit-sample response h[n] is a complete description of an LTI system.Given h[n] one can compute the response y[n] to any arbitrary inputsignal x[n]:y[n] = (x ∗ h)[n] ≡∞Xk=−∞x[k]h[n − k]6.003: Signals and Systems Lecture 10 October 15, 20094CT ConvolutionConvolution of CT signals is analogous to convolution of DT signals.DT: y[n] = (x ∗ h)[n] =∞Xk=−∞x[k]h[n − k]CT: y(t) = (x ∗ h)(t) =Z∞−∞x(τ)h(t − τ)dτConvolutionConvolution is an important computational tool.Example: characterizing LTI systems• Determine the unit-sample response h[n].• Calculate the output for an arbitrary input using convolution:y[n] = (x ∗ h)[n] =Xx[k]h[n − k]Applications of ConvolutionConvolution is an important conceptual tool: it provides an impor-tant new way tothink about the behaviors of systems.Example systems: microscopes and telescopes.MicroscopeImages from even the best microscopes are blurred.MicroscopeBlurring can be represented by convolving the image with the optical“point-spread-function” (3D impulse response).targetimage∗=Blurring is inversely related to the diameter of the lens.MicroscopeMeasuring the “impulse response” of a microscope.Image diameter ≈ 6 times target diameter: target → impulse.6.003: Signals and Systems Lecture 10 October 15, 20095MicroscopeImages at different focal planes can be assembled to form a three-dimensional impulse response (point-spread function).MicroscopeBlurring along the optical axis is better visualized by resampling thethree-dimensional impulse response.MicroscopeBlurring is much greater along the optical axis than it is across theoptical axis.MicroscopeThe point-spread function (3D impulse response) is a useful way tocharacterize a microscope. It provides a direct measure of blurring,which is an important figure of merit for optics.Hubble Space TelescopeHubble Space Telescope (1990-)http://hubblesite.orgHubble Space TelescopeWhy build a space telescope?Telescope images are blurred by the telescope lenses AND by at-mospheric turbulence.ha(x, y) hd(x, y)X Yatmosphericblurringblur due tomirror sizeht(x, y) = (ha∗ hd)(x, y)X Yground-basedtelescope6.003: Signals and Systems Lecture 10 October 15, 20096Hubble Space TelescopeTelescope blur can be respresented by the convolution of blur dueto atmospheric turbulence and blur due to mirror size.−2 −1 0 1 2θ−2 −1 0 1 2θ−2 −1 0 1 2θ−2 −1 0 1 2θ−2 −1 0 1 2θ−2 −1 0 1 2θha(θ)ha(θ)hd(θ)hd(θ)ht(θ)ht(θ)∗∗==d = 12cmd = 1m[arc-seconds]Hubble Space TelescopeThe main optical components of the Hubble Space Telescope aretwo mirrors.http://hubblesite.orgHubble Space TelescopeThe diameter of the primary mirror is 2.4 meters.http://hubblesite.orgHubble Space TelescopeHubble’s first pictures of distant stars (May 20, 1990) were moreblurred
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