An Intuitive Understanding of Fourier Transform
Fourier transform is really widelyused 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.
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:
Now, we begin with the simplest form of moving around a circle ^{1}:
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:
Now, we can have three, four or infinitelymany circles at the meantime:
We can trace any path using a system of wheels, that is, wheels on wheels on wheels…
Therefore, you can trace any timedependent 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:
where $\omega_0$ is the angular frequency of the slowest circle.
 Fourier Transform Clarified
 Fourier transform for dummies
 The Scientist and Engineer’s Guide to Digital Signal Processing
 Ptolemy and Homer (Simpson)

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