Those often work by disk integrators
Not sure if this one in particular does, but the ones I've seen work by disk integrators. Essentially to do a Fourier analysis you need to calculate the integral of (x*sin(w*t)) with w being the frequency and t the time.
This can be done by disk (or sphere) integrators. They work like this:
You have a disk (let's assume it's horizontal) which can turn, for example it can follow your input signal. If you input signal goes up, it'll turn in one direction, if it goes down, it'll turn into the other direction, if it remains the same, the disk will stop.
On top of that disk, there's another, smaller disk mounted on an axle which can move to the left and right. The small disk pushes against the larger one in a way so the small disk turns with it.
Imagine the big disk revolves in one direction at one speed. If you move the small disk from left to right on it's axle, it'll turn in one direction on the left side, then gradually get slower as it approaches the center of the big disk where it will stop, before going on turning into the other direction at increasing speed. If you are a mechanic you can calculate that the speed of the small disk is proportional to the speed of the large disk multiplied by its position.
Do that twice for every frequency, once for the real part, once for the imaginary one, and you'll have a nice fourier analysis.