Fourier transform and energy of a convolution

1. How is convolution related to Fourier transform?
2. What are four important properties of Fourier transform?
3. What does the Fourier transform do?

How is convolution related to Fourier transform?

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 ) .

What are four important properties of Fourier transform?

The important properties of Fourier transform are duality, linear transform, modulation property, and Parseval's theorem.

What does the Fourier transform do?

The Fourier transform gives us insight into what sine wave frequencies make up a signal. You can apply knowledge of the frequency domain from the Fourier transform in very useful ways, such as: Audio processing, detecting specific tones or frequencies and even altering them to produce a new signal.