Back in the days when it was socially acceptable to admit to owning a Nintendo Wii (no U to be seen back then), I was a huge fan of playing Super Paper Mario when I could borrow it from a friend (Fig. 1). The storyline was typical, Mario and friends travel through a number of dangerous worlds, defeating baddies, collecting goodies, and facing off the final boss at the end. I confess I never defeated the game (but no spoilers here!), as the save system was in such a way that I would persistently load into the final play-through with minimal health and entirely forget the controls each time I tried. Perhaps the smart thing would have been to restart all over again. But this is not important.

Super Paper Mario was a game that let you flip between 2D and 3D. This flipping ability let’s the player access an additional layer of the world that previously was hidden. You could make changes in the 3D world that affected the way the 2D world worked, by moving things in and out of the plane, and sometimes seeing passages that prevently didn’t exist. The 2D and 3D worlds were intimately related. This is my convoluted and off the cuff explanation for Fourier transforms (I’ll find out after this once I reread the Wikipedia entry for the umpteenth time).

Imagine you’re dealing with this convoluted passage way, trying to navigate your way to a Pure Heart. You flip into the 3D space and you see that there is a frequency, a regularity, or perhaps everything is just a little lower than you’d like. You bonk some blocks with your head, dodge some goombas, and flip back into 2D. There it is! A straight path to the exit. It was there all along, but you had to move things in the 3D world to access it.

In my world, this is exactly what Fourier transforms are for. You take a signal that seems rather confusing, and want to see some of it’s characteristics in the frequency domain. Once you’ve flipped into frequency space, you can see all the details that make up your time signal. This might give you some clues to your final destination. But you also might need to tweak a few things to clean up the interpretation. By applying a function that can only be done in the frequency space, you have affected your time domain as well. You can see the results of your work by flipping back into the time domain.

Fourier transforms can be applied to more than time signals, and work for any sort of complicated situation you have. Say you want to recreate a complex squiggle in 2D, but want to make it exclusively out of single curves in 3D, you can do that. There’s more to it, but let’s say I have a strong itch to find a second-hand Wii and give Super Paper Mario another shot. Bleck!