Mit Convolution

7/27/2019 Mit Convolution

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6.003: Signals and Systems Convolution

March2,2010


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Mid-term Examination #1 Tomorrow,Wednesday,March3,No recitations tomorrow.Coverage: RepresentationsofCTandDTSystems

Lectures17Recitations18Homeworks14

Homework4willnotcollectedorgraded. Solutionsareposted. Closedbook: 1pageofnotes (81

211 inches; frontandback).

Designedas1-hourexam; twohours tocomplete.

7:30-9:30pm.


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Multiple Representations of CT and DT Systems

Verbaldescriptions:

preserve

the

rationale.

Difference/differential equations: mathematicallycompact.

y[n] = x[n] + z0y[n1] y(t) = x(t) + s0y(t)Block diagrams: illustrate signalflowpaths.

X + Y XR

A Y+s0z0

Operator representations: analyze systemsaspolynomials.Y 1 Y A= =X 1z0R X 1s0A

Transforms: representingdiff. equationswithalgebraicequations. z 1H(z) = H(s) = zz0 ss0


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Convolution Representinga systembya single signal.


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Responses to arbitrary signals Althoughwehave focusedon responses to simple signals ([n], (t))wearegenerallyinterestedinresponsestomorecomplicatedsignals.Howdowecomputeresponsestoamorecomplicated inputsignals?Noproblem fordifferenceequations/blockdiagrams.use step-by-stepanalysis.


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Check Yourself

Example: Find y[3]+ +

R RX Y

when the input isx[n]

n1. 1 2. 2 3. 3 4. 4 5. 5

0. noneof theabove


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Responses to arbitrary signals Example.

+ +R

R

0

0 00

x[n] y[n] n n


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+ +R

R

1

0 01

x[n] y[n] n n


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+ +R

R

1

1 02

x[n]n

y[n]n


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+ +R

R

1

1 13

x[n]n

y[n]n


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+ +R

R

0

1 12

x[n]n

y[n]n


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+ +R

R

0

0 11

x[n]n

y[n]n


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+ +R

R

0

0 00

x[n]n

y[n]n


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Alternative: Superposition Break input intoadditivepartsand sum the responses to theparts.

x[n]n y[n]n= n

++

++=

n1 0 1 2 3 4 5 n

n

n

n1 0 1 2 3 4 5


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Superposition Break input intoadditivepartsand sum the responses to theparts.

x[n]n y[n]n= n

++

++=

n1 0 1 2 3 4 5 n

n

n

n1 0 1 2 3 4 5

Superpositionworks if the system is linear.


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Linearity Asystem is linear if itsresponsetoaweightedsumof inputs isequalto theweighted sumof its responses toeachof the inputs.Given

system y1[n]x1[n] and

systemx

2[n

] y

2[n

]

the system is linear if x1[n] + x2[n] y1[n] + y2[n]system

is true forall and .


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x[n]n y[n]n= n

++

++=

n1 0 1 2 3 4 5 n

n

n

n1 0 1 2 3 4 5

Superpositionworks if the system is linear.


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x[n]n y[n]n= n

++

++=

n1 0 1 2 3 4 5 n

n

n

n1 0 1 2 3 4 5

Reponses topartsareeasy tocompute if system istime-invariant.


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Time-Invariance Asystem istime-invariant ifdelayingthe inputtothesystemsimplydelays theoutputby the sameamountof time.Given

x[n] system y[n]the system is time invariant if

x[nn0] system y[nn0]is true forall n0.

S iti


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x[n]n y[n]n= n

++

++=

n1 0 1 2 3 4 5 n

n

n

n1 0 1 2 3 4 5

Superposition iseasy if the system is linearand time-invariant.


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Convolution


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Convolution ResponseofanLTI system toanarbitrary input.

x[n] LTI y[n]

y[n] = x[k]h[nk](xh)[n]k=

Thisoperation iscalledconvolution.

Notation


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Notation Convolution is representedwithanasterisk.

x[k]h[nk](xh)[n]

k=It iscustomary (butconfusing) toabbreviate thisnotation:

(xh)[n] = x[n]h[n]


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Structure of Convolution


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y[n] = x[k]h[nk]

k=x[n] h[n]

n n

21 0 1 2 3 4 5 21 0 1 2 3 4 5



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y[0] = x[k]h[0k]

k=x[n] h[n]

n n

21 0 1 2 3 4 5 21 0 1 2 3 4 5



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y[0] = x[k]h[0k]

k=x[k] h[k]

k k

21 0 1 2 3 4 5 21 0 1 2 3 4 5



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y[0] = x[k]h[0k]

k=x[k] flip h[k]

k k

21 0 1 2 3 4 5 21 0 1 2 3 4 5h[k]

k21 0 1 2 3 4 5



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y[0] = x[k]h[0k]

k=x[k] shift h[k]

k k

21 0 1 2 3 4 5 21 0 1 2 3 4 5h[0k]

k21 0 1 2 3 4 5


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y[0]= x[k]h[0k] k=

x[k] multiply h[k]

k k21 0 1 2 3 4 5

h[0k] h[0k]k k

21 0 1 2 3 4 5x[k]h[0k]

k21 0 1 2 3 4 5



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y[0]= k=

x[k]h[0k]

x[k] sum h[k]

h[0k] h[0k]k

k

k k21 0 1 2 3 4 5

x[k]h[0k]

kk=21 0 1 2 3 4 5



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y[0] = k=

x[k]h[0k]

x[k] h[k]

h[0k] h[0k]k

k

k k21 0 1 2 3 4 5

x[k]h[0k]

k = 1 k=21 0 1 2 3 4 5



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y[1] = k=

x[k]h[1k]

x[k] h[k]

h[1k] h[1k]k

k

k k21 0 1 2 3 4 5

x[k]h[1k]

k = 2 k=21 0 1 2 3 4 5



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y[2] = k=

x[k]h[2k]

x[k] h[k]

h[2k] h[2k]k

k

k k21 0 1 2 3 4 5

x[k]h[2k]

k = 3 k=21 0 1 2 3 4 5



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y[3] = k=

x[k]h[3k]

x[k] h[k]

h[3k] h[3k]k

k

k k21 0 1 2 3 4 5

x[k]h[3k]

k = 2 k=21 0 1 2 3 4 5



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y[4] = k=

x[k]h[4k]

x[k] h[k]

h[4k] h[4k]k

k

k k21 0 1 2 3 4 5

x[k]h[4k]

k = 1 k=21 0 1 2 3 4 5



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y[5] = k=

x[k]h[5k]

x[k] h[k]

h[5k] h[5k]k

k

k k21 0 1 2 3 4 5

x[k]h[5k]

k = 0 k=21 0 1 2 3 4 5

Check Yourself


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Which

plot

shows

the

result

of

the

convolution

above?

1. 2.

3. 4.5. noneof theabove

Check Yourself


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Expressmathematically: n n k nk2 2 2 2

u[n] u[n] = u[k] u[nk]3 3 3 3k=

n

k nk

2 2=

3 3k=0n n 2 n 2 n

= = 13 3

k=0 k=0 2 n

= (n + 1) u[n]3 4 4 32 80

= 1,3

,3

,27, 81 , . . .

Check Yourself


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Which

plot

shows

the

result

of

the

convolution

above?

3

1. 2.

3. 4.5. noneof theabove


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CT Convolution


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The same sortof reasoningapplies toCT signals. x(t)

tx(t)= lim x(k)p(tk)

0k

where p(t)1

t

As 0, k, d,andp(t)(t): x(t) x()(t)d


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CT Convolution


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ConvolutionofCTsignalsisanalogoustoconvolutionofDTsignals.

DT: y[n] = (xh)[n] = x[k]h[nk]k=

CT: y(t) = (xh)(t) = x()h(t)d

Check Yourself


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t

etu(

t)

t

etu(

t)

Whichplot shows the resultof theconvolutionabove?

1.t 2. t

3.t 4. t

5. noneof theabove

Check Yourself


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Whichplot shows the resultof the followingconvolution? etu(t) etu(t)

t

t

etu(t) etu(t) = eu()e(t)u(t)d t t t t= e e(t)d =e d =te u(t)

0 0

t

Check Yourself


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t

etu(t)

t

etu(t)

Whichplotshowstheresultoftheconvolutionabove? 4

1.t 2. t

3.t 4. t

5. noneof theabove

Convolution


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Convolution isan importantcomputational tool.Example: characterizingLTI systems Determine theunit-sample response h[n].Calculate

the

output

for

an

arbitrary

input

using

convolution:

y[n] = (xh)[n] = x[k]h[nk]


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Microscope A f t l t f h i l f li ht f th t t


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A perfect lens transforms a sphericalwave of light from the targetintoa sphericalwave thatconverges to the image.

target image

Blurring is inversely related to thediameterof the lens.

Microscope A perfect lens transforms a spherical wave of light from the target


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A perfect lens transforms a sphericalwave of light from the targetintoa sphericalwave thatconverges to the image.

target image


Microscope Blurring can be represented by convolving the image with the optical


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Blurringcanberepresentedbyconvolvingtheimagewiththeopticalpoint-spread-function (3D impulse response).

target image

=


Microscope Blurring can be represented by convolving the image with the optical


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Blurringcanberepresentedbyconvolvingtheimagewiththeopticalpoint-spread-function (3D impulse response).

target image

=



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Microscope Measuring the impulse response of a microscope


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Measuring the impulse response ofamicroscope. Imagediameter6 times targetdiameter: target impulse.

Courtesy of Anthony Patire. Used with permission.

Microscope Images at different focal planes can be assembled to form a three-


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Images at different focal planes can be assembled to form a threedimensional impulse response (point-spread function).


Microscope Blurring along the optical axis is better visualized by resampling the


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Blurringalong theopticalaxis isbettervisualizedby resampling thethree-dimensional impulse response.


Microscope Blurring ismuchgreateralong theopticalaxis than it isacross the


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g g g p

opticalaxis.


Microscope Thepoint-spread function (3D impulse response) isausefulway to


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p p ( p p ) y

characterizeamicroscope. Itprovidesadirectmeasureofblurring,which isan importantfigureofmerit foroptics.

Hubble Space Telescope HubbleSpaceTelescope (1990-)


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p p ( )

http://hubblesite.org

Hubble Space Telescope Whybuilda space telescope?
http://hubblesite.org/http://hubblesite.org/


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Telescope images are blurred by the telescope lenses AND by atmospheric turbulence.

ha(x,y) hd(x,y)X Yatmospheric blurdue toblurring mirror size

ht(x,y) = (hahd)(x,y)X Yground-based

telescope

Hubble Space Telescope Telescope blur can be respresented by the convolution of blur due


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toatmospheric

turbulence

and

blur

due

to

mirror

size.

ha() hd() ht()

d= 12cm=

2 1 0 1 2 2 1 0 1 2 2 1 0 1 2

ha() hd() ht()d= 1m

=2 1 0 1 2 2 1 0 1 2 2 1 0 1 2

[arc-seconds]

Hubble Space Telescope Themain optical components of the Hubble Space Telescope are


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twomirrors.


Hubble Space Telescope Thediameterof theprimarymirror is2.4meters.


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Hubble Space Telescope Hubbles first pictures of distant stars (May 20, 1990) weremore


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blurred thanexpected.

expected

earlyHubble

point-spread imageoffunction distant star


Hubble Space Telescope Theparabolicmirrorwasground4 m tooflat!


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Hubble Space Telescope CorrectiveOpticsSpaceTelescopeAxialReplacement (COSTAR):


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eyeglassesfor

Hubble!

Hubble COSTAR

Hubble Space Telescope Hubble imagesbeforeandafterCOSTAR.


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before after


Hubble Space Telescope Hubble imagesbeforeandafterCOSTAR.


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before

after


Hubble Space Telescope Images fromground-based telescopeandHubble.


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Impulse Response: Summary The impulse response is a complete description of a linear, time-


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invariantsystem.

One can find the output of such a system by convolving the inputsignalwith the impulse response.The impulse response is an especially useful description of sometypes of systems, e.g., optical systems,whereblurring is an importantfigureofmerit.

MIT OpenCourseWarehttp://ocw.mit.edu
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6.003 Signals and Systems

Spring 2010

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