# Fourier Transform

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.

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:

$\mathcal{F}(f(t)) = F(\mu) = \int_{-\infty}^{+\infty} f(t) e^{-j 2 \pi \mu t} dt$

Fourier Transform

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:

$\mathcal{F}^{-1}(F(\mu)) = f(t) = \int_{-\infty}^{+\infty} F(\mu) e^{j 2 \pi \mu t} d\mu$

Inverse Fourier Transform

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

$a_1 f_1(t) + a_2 f_2(t) \stackrel{\mathcal{F}}{\longrightarrow} a_1 F_1(\mu) + a_2 F_2(\mu)$

### Time scaling

$z(t) = f(at) \stackrel{\mathcal{F}}{\longrightarrow} Z(\mu) = \frac{1}{a} F(\frac{\mu}{a})$

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

$\mathcal{F}(f(t - t_0)) = F(\mu) e^{-j 2 \pi \mu t_0}$

### Duality

$f(t) \stackrel{\mathcal{F}}{\longrightarrow} F(\mu) \\ F(t) \stackrel{\mathcal{F}}{\longrightarrow} f(-\mu) (= f(\mu) \text{ if f is even})$

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

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