F**ourier 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.

This post is part of a series on Image and Signal Processing. If you are looking for the Fourier Series, you may read the related note.

## Fourier spectrum

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
$c_n \in \R, \forall n \in \Z \implies$ there is *no* phase spectrum.

## Fourier Transform

The Fourier Transform is defined as follows:

Note that the exponent of
$e$ has a ** minus sign**: don’t forget it — it’d be a huge mistake!

If $f(t)$ is an energy signal, then $\mathcal{F}(f(t)) \downarrow$. 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
$e$ has a ** plus sign**: it’s the opposite of the “backwards” transform. Again: beware of the
$+$ and
$-$ signs, which can easily mess things up.

## Some properties

The Fourier Transform has some great properties that made it so useful to our world. They are described in the following.

### Linearity

### Time scaling

Values $a > 1$ (before applying the transform) speeds it up, so $\frac{1}{a} < 1$ (after applying the transform) slows it down.

Values $a < 1$ (before applying the transform) slows it down, so $\frac{1}{a} > 1$ (after applying the transform) speeds it up.

### Time shift

### Duality

Please read the note on even and odd functions if you don’t know what they are.