Fourier transform of convolution proof

1. What is the Fourier transform of the convolution?
2. How do you prove the convolution theorem?
3. How the Fourier transform of the convolution of two functions calculated?
4. What is the significance of the convolution property of Fourier transform?

What is the Fourier transform of the convolution?

The convolution theorem (together with related theorems) is one of the most important results of Fourier theory which is that the convolution of two functions in real space is the same as the product of their respective Fourier transforms in Fourier space, i.e. f ( r ) ⊗ ⊗ g ( r ) ⇔ F ( k ) G ( k ) .

How do you prove the convolution theorem?

Proof of the convolution theorem

Note, in the equation below, that the convolution integral is taken over the variable x to give a function of u. The Fourier transform then involves an integral over the variable u. Now we substitute a new variable w for u-x. As above, the infinite integration limits don't change.

How the Fourier transform of the convolution of two functions calculated?

We've just shown that the Fourier Transform of the convolution of two functions is simply the product of the Fourier Transforms of the functions. This means that for linear, time-invariant systems, where the input/output relationship is described by a convolution, you can avoid convolution by using Fourier Transforms.

What is the significance of the convolution property of Fourier transform?

Furthermore, the convolution property highlights the fact that by decomposing a signal into a linear combination of complex exponentials, which the Fourier transform does, we can interpret the effect of a linear, time- invariant system as simply scaling the (complex) amplitudes of each of these exponentials by a scale ...