Fourier Transform

Published May 19, 2022  -  By Marco Garosi

Image and signal processing

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 cn∈R,∀n∈Z  ⟹  c_n \in \R, \forall n \in \Z \implies there is no phase spectrum.

Fourier Transform

The Fourier Transform is defined as follows:

F(f(t))=F(μ)=∫−∞+∞f(t)e−j2πμtdt \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 ee has a minus sign: don’t forget it — it’d be a huge mistake!

If f(t)f(t) is an energy signal, then F(f(t))↓\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:

F−1(F(μ))=f(t)=∫−∞+∞F(μ)ej2πμtdμ \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 ee 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

a1f1(t)+a2f2(t)⟶Fa1F1(μ)+a2F2(μ) 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)⟶FZ(μ)=1aF(μa) z(t) = f(at) \stackrel{\mathcal{F}}{\longrightarrow} Z(\mu) = \frac{1}{a} F(\frac{\mu}{a})

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

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

Time shift

F(f(t−t0))=F(μ)e−j2πμt0 \mathcal{F}(f(t - t_0)) = F(\mu) e^{-j 2 \pi \mu t_0}

Duality

f(t)⟶FF(μ)F(t)⟶Ff(−μ)(=f(μ) if f is even) 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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