An Intuitive Understanding of Fourier Transform

Fourier transform is really widely-used in digital signal processing. I use it for frequency domain analysis in my research papers, but actually I always have little knowledge about how it works and what is the theory behind.

\[F(\omega) = \int_{-\infty}^{\infty}f(t)e^{-i\omega t}\,dt\]

This is a very common definition of Fourier transform, there are also a plenty of resources on the internet explaining it and using it. However, the more intuitive explanation is its inverse transform:

\[f(t) = \frac{1}{2\pi} \int_{-\infty}^{\infty}F(\omega)e^{i\omega t}\,d\omega\]

Now, we begin with the simplest form of moving around a circle 1:

\[f(t) = Re^{i\omega t}\]

Here, $R$ is the radius of circle and $\omega$ can be seen as the angular frequency. Now, if we have two circles, one at the end of the other, the position turns to:

\[f(t) = R_1 e^{i \omega_1 t} + R_2 e^{i \omega_2 t}\]

Now, we can have three, four or infinitely-many circles at the meantime:

\[f(t) = \int_{-\infty}^{\infty}R(\omega)e^{i \omega t}\,d\omega\]

We can trace any path using a system of wheels, that is, wheels on wheels on wheels…

Therefore, you can trace any time-dependent path using infinitely circles of different radii and frequencies.

Specially, if your path closes on itself, the Fourier transform turns out to simplify to a Fourier series:

\[f(t) = \sum_{k=-\infty}^{\infty}c_k e^{ik\omega_0t}\]

where $\omega_0$ is the angular frequency of the slowest circle.

  1. If you have confusion about this representation, think about Euler’s formula: $e^{it}={\cos}t+i{\sin}t$. ↩