Convolution.docx

Convolution.docx

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Convolution

Convolution

Convolutionisthemostimportantandfundamentalconceptinsignalprocessingandanalysis.Byusingconvolution,wecanconstructtheoutputofsystemforanyarbitraryinputsignal,ifweknowtheimpulseresponseofsystem. 

Howisitpossiblethatknowingonlyimpulseresponseofsystemcandeterminetheoutputforanygiveninputsignal?

Wewillfindoutthemeaningofconvolution.

RelatedTopics:

 WindowFilters 

Download:

 conv1d.zip, conv2d.zip

∙Definition

∙ImpulseFunctionDecomposition

∙ImpulseResponse

∙BacktotheDefinition

∙Convolutionin1D

∙C++ImplementationforConvolution1D

∙Convolutionin2D

∙SeparableConvolution2D

∙C++AlgorithmforConvolution2D

Definition

First,let'sseethemathematicaldefinitionofconvolutionindiscretetimedomain.Laterwewillwalkthroughwhatthisequationtellsus. 

（Wewilldiscussindiscretetimedomainonly.） 

where x[n] isinputsignal, h[n] isimpulseresponse,and y[n] isoutput.*denotesconvolution.Noticethatwemultiplythetermsof x[k] bythetermsofatime-shifted h[n] andaddthemup. 

Thekeystoneofunderstandingconvolutionislaidbehindimpulseresponseandimpulsedecomposition.

ImpulseFunctionDecomposition

Inordertounderstandthemeaningofconvolution,wearegoingtostartfromtheconceptofsignaldecomposition.Theinputsignalisdecomposedintosimpleadditivecomponents,andthesystemresponseoftheinputsignalresultsinbyaddingtheoutputofthesecomponentspassedthroughthesystem.

Ingeneral,asignalcanbedecomposedasaweightedsumofbasissignals.Forexample,inFourierSeries,anyperiodicsignal（evenrectangularpulsesignal）canberepresentedbyasumofsineandcosinefunctions.Buthere,weuseimpulse（delta）functionsforthebasissignals,insteadofsineandcosine.

Examinethefollowingexamplehowasignalisdecomposedintoasetofimpulse（delta）functions.Sincetheimpulsefunction, δ[n] is1at n=0,andzerosat n≠0. x[0] canbewrittento 2·δ[n].And, x[1] willbe 3·δ[n-1],because δ[n-1] is1at n=1 andzerosatothers.Insameway,wecanwrite x[2] byshifting δ[n] by2, x[2] =1·δ[n-2].Therefore,thesignal, x[n] canberepresentedbyadding3shiftedandscaledimpulsefunctions.

Ingeneral,asignalcanbewrittenassumofscaledandshifteddeltafunctions; 

ImpulseResponse

Impulseresponseistheoutputofasystemresultingfromanimpulsefunctionasinput. 

Anditisdenotedas h[n]. 

Ifthesystemis time-invariant,theresponseofatime-shiftedimpulsefunctionisalsoshiftedassameamountoftime.

Forexample,theimpulseresponseof δ[n-1] is h[n-1].Ifweknowtheimpulseresponse h[n],thenwecanimmediatelygettheimpulseresponse h[n-1] byshifting h[n] by+1.Consequently, h[n-2] resultsfromshifting h[n]by+2.

Ifthesystemis linear （especiallyscalarrule）,ascaledininputsignalcausesanidenticalscalingintheoutputsignal. 

Forinstance,theimpulseresponseof 3·δ[n] isjustmultiplyingby3to h[n].

Iftheinputhas3components,forexample, a·δ[n]+ b·δ[n-1]+ c·δ[n-2],thentheoutputissimplya·h[n]+ b·h[n-1]+ c·h[n-2].Thisiscalledadditivepropertyof linear system,thus,itisvalidonlyonthelinear system. 

BacktotheDefinition

Bycombiningthepropertiesofimpulseresponseandimpulsedecomposition,wecanfinallyconstructtheequationofconvolution.In linear and time-invariantsystem,theresponseresultingfromseveralinputscanbecomputedasthesumoftheresponseseachinputactingalone. 

Forexample,ifinputsignalis x[n]=2·δ[n]+3·δ[n-1]+1·δ[n-2],thentheoutputissimply y[n]=2·h[n]+3·h[n-1]+1·h[n-2]. 

Therefore,wenowclearlyseethatiftheinputsignalis 

thentheoutputwillbe 

.Noteonecondition;convolutionworksonthe linear and timeinvariantsystem.

Tosummarize,asignalisdecomposedintoasetofimpulsesandtheoutputsignalcanbecomputedbyaddingthescaledandshiftedimpulseresponses.

Furthermore,thereisanimportantfactunderconvolution;theonlythingweneedtoknowaboutthesystem'scharacteristicsistheimpulseresponseofthesystem, h[n].Ifweknowasystem'simpulseresponse,thenwecaneasilyfindouthowthesystemreactsforanyinputsignal.

Convolutionin1D

Let'sstartwithanexampleofconvolutionof1dimensionalsignal,thenfindouthowtoimplementintocomputerprogrammingalgorithm.

x[n]={3,4,5} 

h[n]={2,1}

x[n] hasonlynon-zerovaluesat n=0,1,2, andimpulseresponse, h[n] isnotzeroat n=0,1.Otherswhicharenotlistedareallzeros.

Input:

 x[n]

ImpulseResponse:

 h[n]

Onethingtonotebeforewemoveon:

Trytofigureoutyourselfhowthissystembehaves,byonlylookingattheimpulseresponseofthesystem.Whentheimpulsesignalisenteredthesystem,theoutputofthesystemlookslikeamplifierandechoing.Atthetimeis0,theintensitywasincreased（amplified）bydoubleandgraduallydecreasedwhilethetimeispassed.

Fromtheequationofconvolution,theoutputsignal y[n] willbe 

. 

Let'scomputemanuallyeachvalueof y[0], y[1], y[2], y[3], ...

Output:

 y[n]

Noticethat y[0] hasonlyonecomponent, x[0]h[0],andothersareomitted,becauseothersareallzerosat k≠0.Inotherwords, x[1]h[-1]= x[2]h[-2]=0.Theaboveequationsalsoomitthetermsiftheyareobviouslyzeros. 

If n islargerthanandequalto4, y[n] willbezeros.Sowestopheretocompute y[n]andaddingthese4signals（y[0], y[1], y[2], y[3]）producestheoutputsignaly[n]={6,11,14,5}.

Let'slookmorecloselytheeachoutput.Inordertoseeapatternclearly,theorderofadditionisswapped.Thelasttermcomesfirstandthefirsttermgoestothelast.And,allzerotermsareignored.

y[0]= x[0]·h[0] 

y[1]= x[1]·h[0]+ x[0]·h[1] 

y[2]= x[2]·h[0]+ x[1]·h[1] 

y[3]= x[3]·h[0]+ x[2]·h[1]

Canyouseethepattern?

Forexample, y[2] iscalculatedfrom2inputsamples; x[2] and x[1],and2impulseresponsesamples; h[0] and h[1].Theinputsamplestartsfrom2,whichissameasthesamplepointofoutput,anddecreased.Theimpulseresponsesamplestartsfrom0andincreased.

Basedonthepatternthatwefound,wecanwriteanequationforanysampleoftheoutput;

where i isanysamplepointandkisthenumberofsamplesinimpulseresponse. 

Forinstance,ifanimpulseresponsehas4samples,thesampleofoutputsignalat9is; 

y[9]= x[9]·h[0]+ x[8]·h[1]+ x[7]·h[2]+ x[6]·h[3] 

Noticethefirstsample, y[0] hasonlyoneterm.Basedonthepatternthatwefound, y[0] iscalculatedas:

y[0]= x[0]·h[0]+ x[-1]·h[1].Because x[-1] isnotdefined,wesimplypadittozero.

C++ImplementationforConvolution1D

Implementingtheconvolutionalgorithmisquitesimple.Thecodesnippetisfollowing;

for（i=0;i

{

y[i]=0;//settozerobeforesum

for（j=0;j

{

y[i]+=x[i-j]*h[j];//convolve:

multiplyandaccumulate

}

}

However,youshouldconcernseveralthingsintheimplementation. 

Watchouttherangeofinputsignal.Youmaybeoutofboundofinputsignal,forexample,x[-1],x[-2],andsoon.Youcanpadzerosforthoseundefinedsamples,orsimplyskiptheconvolutionattheboundary.Theresultsattheboththebeginningandendedgescannotbeaccurateanyway. 

Second,youneedroundingtheoutputvalues,iftheoutputdatatypeisintegerandimpulseresponseisfloatingpointnumber.And,theoutputvaluemaybeexceededthemaximumorminimumvalue. 

Ifyouhaveunsigned8-bitintegerdatatypeforoutputsignal,therangeofoutputshouldbebetween0and255.Youmustcheckthevalueisgreaterthantheminimumandlessthanthemaximumvalue. 

Downloadthe1Dconvolutionroutineandtestprogram. 

conv1d.zip

Convolutionin2D

2Dconvolutionisjustextensionofprevious1Dconvolutionbyconvolvingbothhorizontalandverticaldirectionsin2dimensionalspatialdomain.Convolutionisfrequentlyusedforimageprocessing,suchassmoothing,sharpening,andedgedetectionofimages. 

Theimpulse（delta）functionisalsoin2Dspace,so δ[m,n] has1where m and n iszeroandzerosat m,n≠0.Theimpulseresponsein2Disusuallycalled"kernel"or"filter"inimageprocessing. 

Thesecondimageis2Dmatrixrepresentationofimpulsefunction.Theshadedcenterpointistheoriginwherem=n=0.

Onceagain,asignalcanbedecomposedintoasumofscaledandshiftedimpulse（delta）functions; 

Forexample, x[0,0] is x[0,0]·δ[m,n], x[1,2] is x[1,2]·δ[m-1,n-2],andsoon.Notethatthematricesarereferencedhereas[column,row],not[row,column].Mishorizontal（column）directionandNisvertical（row）direction.

And,theoutputof linear and timeinvariantsystem canbewrittenbyconvolutionofinputsignal x[m,n],andimpulseresponse, h[m,n]; 

Noticethatthekernel（impulseresponse）in2Discenteroriginatedinmostcases,whichmeansthecenterpointofakernelis h[0,0].Forexample,ifthekernelsizeis5,thenthearrayindexof5elementswillbe-2,-1,0,1,and2.Theoriginislocatedatthemiddleofkernel.

Examineanexampletoclarifyhowtoconvolvein2Dspace. 

Let'ssaythatthesizeofimpulseresponse（kernel）is3x3,andit'svaluesarea,b,c,d,... 

Noticetheorigin（0,0）islocatedinthecenterofkernel. 

Let'spickasimplestsampleandcomputeconvolution,forinstance,theoutputat（1,1）willbe;

Itresultsinsumof9elementsofscaledandshiftedimpulseresponses.Thefollowingimageshowsthegraphicalrepresentationof2Dconvolution.

2DConvolution

Noticethatthekernelmatrixisflippedbothhorizontalandverticaldirectionbe