Impulse Response and Convolution

1 : Signal ProcessingImpulse Response and ConvolutionMarch 10, 2020 The Signals and System AbstractionDescribe asystem(physical, mathematical, or computational) by the wayit transforms aninput signalinto anoutput is particularly useful for systems that arelinear and input into additive parts and sum the responses to the [n]y[n]n=++++=n 1 0 1 2 3 4 5nnnn 1 0 1 2 3 4 5nSuperpositionBreak input into additive parts and sum the responses to the [n]y[n]n=++++=n 1 0 1 2 3 4 5nnnn 1 0 1 2 3 4 5nSuperposition works because the system system is linear if its Response to a weighted sum of inputs is equal tothe weighted sum of its responses to each of the [n]y1[n]andsystemx2[n]y2[n]the system is linear ifsystem x1[n] + x2[n] y1[n] + y2[n]is true for all and and all input into additive parts and sum the responses to the [n]y[n]

2 N=++++=n 1 0 1 2 3 4 5nnnn 1 0 1 2 3 4 5nSuperposition works if the system input into additive parts and sum the responses to the [n]y[n]n=++++=n 1 0 1 2 3 4 5nnnn 1 0 1 2 3 4 5nReponses to parts are easy to compute if system system is time-invariant if delaying the input to the system simply delaysthe output by the same amount of [n]y[n]the system is time invariant ifsystemx[n n0]y[n n0]is true for allnand input into additive parts and sum the responses to the [n]y[n]n=++++=n 1 0 1 2 3 4 5nnnn 1 0 1 2 3 4 5nSuperposition is easy if the system ResponseIf a system is linear and time-invariant (LTI), its input-output relation iscompletely specifiedby the system s unit-sample responseh[n].1. One can always find the unit-sample Response of a [n]h[n]2.

3 Time invariance implies that shifting the input simply shifts the [n k]h[n k]3. Homogeneity implies that scaling the input simply scales the [k] [n k]x[k]h[n k]4. Additivity implies that the Response to a sum is the sum of [n] = k= x[k] [n k]y[n] = k= x[k]h[n k] (x h)[n]The output of an LTI system canalwaysbe found by convolving:(x h)[n].ConvolutionResponse of an LTI system to an arbitrary [n]y[n]y[n] = k= x[k]h[n k] (x h)[n]This operation is of Convolutiony[n] = k= x[k]h[n k] 2 1 0 1 2 3 4 5nx[n]h[n] 2 1 0 1 2 3 4 5nStructure of Convolutiony[0] = k= x[k]h[0 k] 2 1 0 1 2 3 4 5nx[n]h[n] 2 1 0 1 2 3 4 5nStructure of Convolutiony[0] = k= x[k]h[0 k] 2 1 0 1 2 3 4 5kx[k]h[k] 2 1 0 1 2 3 4 5kStructure of Convolutiony[0] = k= x[k]h[0 k] 2 1 0 1 2 3 4 5kx[k]h[k]h[ k] flip 2 1 0 1 2 3 4 5k 2 1 0 1 2 3 4 5kStructure of Convolutiony[0] = k= x[k]h[0 k] 2 1 0 1 2 3 4 5kx[k]h[k]h[0 k] shift 2 1 0 1 2 3 4 5k 2 1 0 1 2 3 4 5kStructure of Convolutiony[0] = k= x[k]h[0 k] 2 1 0 1 2 3 4 5kx[k]h[k]h[0 k]h[0 k]

4 Multiply 2 1 0 1 2 3 4 5k 2 1 0 1 2 3 4 5k 2 1 0 1 2 3 4 5kStructure of Convolutiony[0] = k= x[k]h[0 k]kx[k]h[k]h[0 k]h[0 k]x[k]h[0 k] multiply 2 1 0 1 2 3 4 5kk 2 1 0 1 2 3 4 5k 2 1 0 1 2 3 4 5kStructure of Convolutiony[0] = k= x[k]h[0 k]kx[k]h[k]h[0 k]h[0 k]x[k]h[0 k] sum k= kk 2 1 0 1 2 3 4 5k 2 1 0 1 2 3 4 5kStructure of Convolutiony[0] = k= x[k]h[0 k]kx[k]h[k]h[0 k]h[0 k]x[k]h[0 k] k= kk 2 1 0 1 2 3 4 5k 2 1 0 1 2 3 4 5k= 1 Structure of Convolutiony[1] = k= x[k]h[1 k]kx[k]h[k]h[1 k]h[1 k]x[k]h[1 k] k= kk 2 1 0 1 2 3 4 5k 2 1 0 1 2 3 4 5k= 2 Structure of Convolutiony[2] = k= x[k]h[2 k]kx[k]h[k]h[2 k]h[2 k]x[k]h[2 k] k= kk 2 1 0 1 2 3 4 5k 2 1 0 1 2 3 4 5k= 3 Structure of Convolutiony[3] = k= x[k]h[3 k]kx[k]h[k]h[3 k]h[3 k]x[k]h[3 k] k= kk 2 1 0 1 2 3 4 5k 2 1 0 1 2 3 4 5k= 2 Structure of Convolutiony[4] = k= x[k]h[4 k]kx[k]h[k]h[4 k]h[4 k]x[k]h[4 k] k= kk 2 1 0 1 2 3 4 5k 2 1 0 1 2 3 4 5k= 1 Structure of Convolutiony[5] = k= x[k]h[5 k]kx[k]h[k]h[5 k]h[5 k]x[k]h[5 k] k= kk 2 1 0 1 2 3 4 5k 2 1 0 1 2 3 4 5k= 0 Check Yourself1 1 Which plot shows the result of the Convolution above?

5 None of the aboveCheck Yourself1 1 Express mathematically:((23)nu[n]) ((23)nu[n])= k= ((23)ku[k]) ((23)n ku[n k])=n k=0(23)k (23)n k=n k=0(23)n=(23)nn k=01= (n+ 1)(23)nu[n]= 1,43,43,3227,8081, ..Check Yourself1 1 Which plot shows the result of the Convolution above? none of the aboveUnit-Sample ResponseThe unit-sample Response is acompletedescription of a [n]h[n]It can be used to determine the Response toanyother [n] nh[n]Givenh[n]one can compute the Response to any arbitrary input [n] = (x h)[n] k= x[k]h[n k]Continuous-Time SystemsSuperposition and Convolution are of equal importance for CT ResponseA CT system is completely characterized by itsimpulse Response , muchas a DT system is completely characterized by its unit-sample have worked with the Impulse (Dirac delta) function (t) s defined in a limit as (t)represent a pulse of width and height1 so that its area (t) 1 Then (t) = lim 0p (t)t (t)The Impulse function can be used to break an arbitrary inputx(t)

6 Intotime-based components, much as [k]is used for discrete-time ResponseAn arbitrary CT signal can be represented by an infinite sum of infinitesimalimpulses (which define an integral).Approximate an arbitrary signalx(t)(blue) as a sum of pulsesp (t)(red).tx(t)x (t) = k= x(k )p (t k ) and the limit ofx (t)as 0will approximatex(t).lim 0x (t) = lim 0 k= x(k )p (t k ) x( ) (t )d The result in CT is much like the result for DT:x(t) = x( ) (t )d x[n] = m= x[m] (n m) Impulse ResponseIf a system is linear and time-invariant (LTI), its input-output relation iscompletely specifiedby the system s Impulse responseh(t).1. One can always find the Impulse Response of a (t)h(t)2. Time invariance implies that shifting the input simply shifts the (t )h(t )3.

7 Homogeneity implies that scaling the input simply scales the ( ) (t )x( )h(t )4. Additivity implies that the Response to a sum is the sum of (t) = x( ) (t )d y(t) = x( )h(t )d (x h)(t)The output of an LTI system canalwaysbe found by convolving:(x h)(t). Impulse ResponseThe Impulse Response is acompletedescription of a (t)h(t)It can be used to determine the Response toanyother (t) th(t)Givenh(t)one can compute the Response to any arbitrary input (t) = (x h)(t) x( )h(t )d Comparison of CT and DT ConvolutionConvolution of CT signals is analogous to Convolution of DT :y[n] = (x h)[n] = k= x[k]h[n k]CT:y(t) = (x h)(t) = x( )h(t )d Check Yourselfte tu(t) te tu(t)Which plot shows the result of the Convolution above? none of the aboveCheck YourselfWhich plot shows the result of the following Convolution ?

8 Te tu(t) te tu(t)(e tu(t)) (e tu(t))= e u( )e (t )u(t )d = t0e e (t )d =e t t0d =te tu(t)tCheck Yourselfte tu(t) te tu(t)Which plot shows the result of the Convolution above? none of the aboveProperties of ConvolutionCommutivity:(x y)(t) = (y x)(t)(x y)(t) x(t )y( )d let =t (x y)(t) = x( )y(t )( d )= x( )y(t )d = (y x)(t)h(t)x(t)(x h)(t)x(t)h(t)(h x)(t) = (x h)(t)Properties of ConvolutionAssociativity.((x y) z)(t) =(x (y z))(t)((x y) z)(t) ( x(t )y( )d )z( )d let = + ((x y) z)(t) = ( x(t )y( )d )z( )d = x(t )( y( )z( )d )d =(x (y z))g(t)h(t)(g h)(t)(x g)(t)x(t)((x g) h)(t)x(t)(x (g h))(t)Properties of ConvolutionDistributivity over addition.(x (g+h))(t) = (x g)(t) + (x h)(t)(x (g+h))= x(t )(g( ) +h( ))d = x(t )g( )d + x(t )h( )d = (x g)(t) + (x h)(t)g(t)h(t)g(t)+h(t)+x(t)(x (g+h)(t))x(t)(x (g+h))(t)ConvolutionConvolution is an importantcomputational : characterizing LTI systems Determine the unit-sample responseh(t).

9 Calculate the output for an arbitrary input using Convolution :y(t) = (x h)(t) = x(t )h( )d Applications of ConvolutionConvolution is an importantconceptual tool:it provides an importantnew way tothinkabout the behaviors of systems: microscopes and from even the best microscopes are perfect lens transforms a spherical wave of light from the target into aspherical wave that converges to the is inversely related to the diameter of the perfect lens transforms a spherical wave of light from the target into aspherical wave that converges to the is inversely related to the diameter of the perfect lens transforms a spherical wave of light from the target into aspherical wave that converges to the is inversely related to the diameter of the can be represented by convolving the image with the optical point-spread-function (3D Impulse Response ).

10 Targetimage =Blurring is inversely related to the diameter of the 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 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 Space TelescopeHubble Space Telescope (1990-) Space TelescopeWhy build a space telescope?Telescope images are blurred by the telescope lenses AND by (x,y)hd(x,y)XYatmosphericblurringblur due tomirror sizeht(x,y) = (ha hd)(x,y)XYground-basedtelescopeHubble Space TelescopeTelescope blur can be respresented by the Convolution of blur due to at-mospheric turbulence and blur due to mirror size.