magnitude and phase of Fourier transform

The Fourier transform is often represented in terms of its magnitude and phase :

where

Intuition

We can decompose the exponential term in the Fourier Transform using the Euler’s formula:

If we put the real part and imaginary part to a complex plane, than

• the magnitude of the result vector is the magnitude of the Fourier transform
• the angle is the phase of the Fourier transform, ¹

Footnotes

1. The intuition behind Fourier and Laplace transforms I was never taught in school - YouTube ↩