Convolution in frequency domain proof

1. What is convolution in frequency domain?
2. Why is convolution multiplication in frequency domain?
3. How do you prove the convolution theorem?
4. What is convolution in FFT?

What is convolution in frequency domain?

A convolution operation is used to simplify the process of calculating the Fourier transform (or inverse transform) of a product of two functions. When you need to calculate a product of Fourier transforms, you can use the convolution operation in the frequency domain.

Why is convolution multiplication in frequency domain?

We know that a convolution in the time domain equals a multiplication in the frequency domain. In order to multiply one frequency signal by another, (in polar form) the magnitude components are multiplied by one another and the phase components are added.

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.

What is convolution in FFT?

FFT convolution uses the principle that multiplication in the frequency domain corresponds to convolution in the time domain. The input signal is transformed into the frequency domain using the DFT, multiplied by the frequency response of the filter, and then transformed back into the time domain using the Inverse DFT.