Prove the commutative properties of the convolution integral f * g = g * f

Dillard 2020-11-14 Answered
Prove the commutative properties of the convolution integral f * g = g * f
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Expert Answer

Arham Warner
Answered 2020-11-15 Author has 102 answers
Jeffrey Jordon
Answered 2021-09-29 Author has 2087 answers

a) First prove the commutative property as follows:

(fg)=0tf(τ)g(tτ)dτ

Now change the integration variable as follows

Let σ=tτ Then dσ=dt

(fg)(t)=t0f(tσ)f(σ)(1)dσ

=t0f(tσ)g(σ)dσ

=(gf)(t)

Therefore, (fg)=(gf)

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