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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Responses to arbitrary signals Example.
+ +R
R
1
0 01
x[n] y[n] n n
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Responses to arbitrary signals Example.
+ +R
R
1
1 02
x[n]n
y[n]n
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Responses to arbitrary signals Example.
+ +R
R
1
1 13
x[n]n
y[n]n
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Responses to arbitrary signals Example.
+ +R
R
0
1 12
x[n]n
y[n]n
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Responses to arbitrary signals Example.
+ +R
R
0
0 11
x[n]n
y[n]n
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Responses to arbitrary signals Example.
+ +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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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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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
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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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
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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Structure of Convolution
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
Structure of Convolution
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Structure of Convolution
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
Structure of Convolution
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Structure of Convolution
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
Structure of Convolution
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Structure of Convolution
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
Structure of Convolution
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Structure of Convolution
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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Structure of Convolution
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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
Structure of Convolution
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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
Structure of Convolution
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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
Structure of Convolution
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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
Structure of Convolution
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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
Structure of Convolution
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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
Structure of Convolution
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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
Structure of Convolution
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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
Blurring is inversely related to thediameterof the lens.
Microscope Blurring can be represented by convolving the image with the optical
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Blurringcanberepresentedbyconvolvingtheimagewiththeopticalpoint-spread-function (3D impulse response).
target image
=
Blurring is inversely related to thediameterof the lens.
Microscope Blurring can be represented by convolving the image with the optical
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Blurringcanberepresentedbyconvolvingtheimagewiththeopticalpoint-spread-function (3D impulse response).
target image
=
Blurring is inversely related to thediameterof the lens.
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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).
Courtesy of Anthony Patire. Used with permission.
Microscope Blurring along the optical axis is better visualized by resampling the
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Blurringalong theopticalaxis isbettervisualizedby resampling thethree-dimensional impulse response.
Courtesy of Anthony Patire. Used with permission.
Microscope Blurring ismuchgreateralong theopticalaxis than it isacross the
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g g g p
opticalaxis.
Courtesy of Anthony Patire. Used with permission.
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?
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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.
http://hubblesite.org
Hubble Space Telescope Thediameterof theprimarymirror is2.4meters.
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http://hubblesite.org
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
http://hubblesite.org
Hubble Space Telescope Theparabolicmirrorwasground4 m tooflat!
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http://hubblesite.org
Hubble Space Telescope CorrectiveOpticsSpaceTelescopeAxialReplacement (COSTAR):
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eyeglassesfor
Hubble!
Hubble COSTAR
Hubble Space Telescope Hubble imagesbeforeandafterCOSTAR.
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before after
http://hubblesite.org
Hubble Space Telescope Hubble imagesbeforeandafterCOSTAR.
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before
after
http://hubblesite.org
Hubble Space Telescope Images fromground-based telescopeandHubble.
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http://hubblesite.org
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.
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6.003 Signals and Systems
Spring 2010
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