An Interactive Guide to the Fourier Transform

Summary

The Fourier Transform is one of deepest insights ever made. Unfortunately, the meaning is buried within dense equations: Yikes. Rather than jumping into the symbols, let's experience the key idea firsthand. Here's a plain-English metaphor: What does the Fourier Transform do? Given a smoothie, it finds the recipe. How? Run the smoothie through filters to extract each ingredient. Why? Recipes are easier to analyze, compare, and modify than the smoothie itself. How do we get the smoothie back? Blend the ingredients. Here's the "math English" version of the above: The Fourier Transform takes a time-based pattern, measures every possible cycle, and returns the overall "cycle recipe" (the amplitude, offset, & rotation speed for every cycle that was found). Time for the equations? No! Let's get our hands dirty and experience how any pattern can be built with cycles, with live simulations. If all goes well, we'll have an aha! moment and intuitively realize why the Fourier Transform is possible. We'll save the detailed math analysis for the follow-up. This isn't a force-march through the equations, it's the casual stroll I wish I had. Onward! VIDEO From Smoothie to Recipe A math transformation is a change of perspective. We change our notion of quantity from "single items" (lines in the sand, tally system) to "groups of 10" (decimal) depending on what we're counting. Scoring a game? Tally it up. Multiplying? Decimals, please. The Fourier Transform changes our perspective from consumer to producer, turning What do I have? 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? Well, imagine you had a few filters lying around: Pour through the "banana" filter. 1 oz of bananas are ex...