Problem #4

UCI. MAE200B, winter 2006. by nasser Abbasi.

Problem: Prove the convolution theorem for the Fourier transform

Solution

Some definitions first. The Fourier transform is defined as (using textbook definition)

and the inverse fourier transform is

And the convolution is defined as

Start the proof by showing that , this implies that

So, take the fourier transform of the convolution, we obtain

Let , hence , and so . When , and the above can be written as

Exchange the integrals order we obtain

Break the exponential into product of 2 exponential

We can now move the marked terms outside the inner integral since they do not depend on

But by definition and similarly , hence eq (1) can be written as

This results means that

This completes the proof. (We should also proof the other direction since this is an IFF problem, i.e. starting from take its inverse fourier transform we would get the convolution). But I think this proof done here should be sufficient.