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Proof of Convolution Theorem

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7/27/2019 Proof of Convolution Theorem

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TheConvolutionTheorem

TheConvolutionTheoremstates:

! ! ! ! ! ! ! ! !"#! ! ! ! ! ! !(!)Proof:

PartI:ProofoftheShiftTheoremorshift-invariance:

Firstweprovetheshiftproperty,thatis,thattheFourierTransform(F),asa

system,isshift-invariant:

Thatis,ifthen

Thatmeansthataddingashift(x-a)doesntchangethespectrum,F(u),itjust

addsalinearphase,whichamountstomultiplicationby!!!!!"#.MultiplyingF(u)

by!!!!!"#fordifferentatranslatesorshiftsf(x)bya.

Letx=x-aanddx=dx

Then! ! !

!=

! ! !!= ! !! !!!!![! !

!!! ]

!"

!

!!

Now!!!!!! !!!!

= !!!!!!" !!!!!!"! !!"!!!!!!!" !"!!"#$%%:

So! ! ! ! = !!!!!!"!(!)whichistheShiftTheorem

Therefore,theFourierTransform,asaSystem,isshift-invariant.

Significance:Shiftingortranslatingaspatialfunctionadistancex=aaddsalinear

phase=uatotheoriginalphase.Conversely,alinearphasefilterproducesatranslationoftheimage.Themagnitudespectrumremainsunchanged.

Ff(x) F(u) Ff(x-a) !

!!!!"#

F(u)
7/27/2019 Proof of Convolution Theorem

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