Published May 19, 2022 - By Marco Garosi
Fourier Transform (also know as FT) is a mathematical transform, named after French mathematician Joseph Fourier, that allows us to decompose a periodic function (signal) on time or on space into functions depending on temporal frequency or on spatial frequency.
The Fourier Transform produces a spectrum made up of two parts:
- phase spectrum, and
- amplitude spectrum.
For periodic signals:
- the amplitude spectrum is symmetric to the y-axis;
- the phase spectrum is antisymmetrical to the y-axis.
If there is no phase spectrum.
The Fourier Transform is defined as follows:
Note that the exponent of has a minus sign: don’t forget it — it’d be a huge mistake!
If is an energy signal, then . Other signals “support” the Fourier Transform as well, of course.
Inverse Fourier Transform
There’s also an inverse Fourier Transform — that is, it takes a function depending on spacial or temporal frequency and transforms it back to a function depending on time or space. Here it is:
Note that the exponend of has a plus sign: it’s the opposite of the “backwards” transform. Again: beware of the and signs, which can easily mess things up.
The Fourier Transform has some great properties that made it so useful to our world. They are described in the following.
Values (before applying the transform) speeds it up, so (after applying the transform) slows it down.
Values (before applying the transform) slows it down, so (after applying the transform) speeds it up.
Please read the note on even and odd functions if you don’t know what they are.