"An Interactive Guide To The Fourier Transform"

"The Fourier Transform changes our perspective from consumer to producer, turning What did I see? into How was it made?

In other words: given a smoothie, let's find the recipe.

Why? Well, recipes are great descriptions of drinks. You wouldn't share a drop-by-drop analysis, you'd say "I had an orange/banana smoothie". A recipe is more easily categorized, compared, and modified than the object itself.

So... given a smoothie, how do we find the recipe?...

The Fourier Transform takes a specific viewpoint: What if any signal could be filtered into a bunch of circular paths?

Whoa. This concept is mind-blowing, and poor Joseph Fourier had his idea rejected at first. (Really Joe, even a staircase pattern can be made from circles?)

And despite decades of debate in the math community, we expect students to internalize the idea without issue. Ugh. Let's walk through the intuition...

If sound waves can be separated into ingredients (bass and treble frequencies), we can boost the parts we care about, and hide the ones we don't. The crackle of random noise can be removed. Maybe similar "sound recipes" can be compared (music recognition services compare recipes, not the raw audio clips)."

http://betterexplained.com/articles/an-interactive-guide-to-the-fourier-transform/

Fourier transforms are so cool. They are one of those aspects of math test have a certain poetry, and that give you an intuition that can be applied more generally. Sometimes when I'm trying to understand abstract social stuff going on in the world, I think about how it's really composed of a bunch of different phenomena that are adding together and obscuring each other. Some of them are independent of each other, and could be cracked to make the rest of the problem clearer and more manageable.

(credit to JT)