/cdn.vox-cdn.com/assets/887098/20120117111418-1.jpeg "fourier transform")
The Fourier Transform is useful in engineering, sure, but it's a metaphor about finding the root causes behind an observed effect. In the smoothie world, imagine each person paid attention to a different ingredient: Adam looks for apples, Bob looks for bananas, and Charlie gets cauliflower (sorry bud). If a radio wave is our signal, we can use filters to listen to a particular channel. This "lossy compression" can drastically shrink file sizes (and why JPEG and MP3 files are much smaller than raw. If computer data can be represented with oscillating patterns, perhaps the least-important ones can be ignored. Maybe similar "sound recipes" can be compared (music recognition services compare recipes, not the raw audio clips). The crackle of random noise can be removed. 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. If earthquake vibrations can be separated into "ingredients" (vibrations of different speeds & amplitudes), buildings can be designed to avoid interacting with the strongest ones.
FOURIER TRANSFORM CODE
Don't get scared think of the examples as "Wow, we're finally seeing the source code (DNA) behind previously confusing ideas". Here's where most tutorials excitedly throw engineering applications at your face.
FOURIER TRANSFORM FULL
Collect the full recipe, listing the amount of each "circular ingredient".Apply filters to measure each possible "circular ingredient".The Fourier Transform finds the recipe for a signal, like our smoothie process: ( Really Joe, even a staircase pattern can be made from circles?)Īnd despite decades of debate in the math community, we expect students to internalize the idea without issue. This concept is mind-blowing, and poor Joseph Fourier had his idea rejected at first. The Fourier Transform takes a specific viewpoint: What if any signal could be filtered into a bunch of circular paths? The ingredients, when separated and combined in any order, must make the same result. Smoothies can be separated and re-combined without issue (A cookie? Not so much. Our collection of filters must catch every possible ingredient. We won't get the real recipe if we leave out a filter ("There were mangoes too!"). Adding more oranges should never affect the banana reading.įilters must be complete. The banana filter needs to capture bananas, and nothing else. We can reverse-engineer the recipe by filtering each ingredient. Well, imagine you had a few filters lying around: given a smoothie, how do we find the recipe? A recipe is more easily categorized, compared, and modified than the object itself.
You wouldn't share a drop-by-drop analysis, you'd say "I had an orange/banana smoothie". Why? Well, recipes are great descriptions of drinks.
In other words: given a smoothie, let's find the recipe. The Fourier Transform changes our perspective from consumer to producer, turning What do I have? into How was it made?
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. Onward!Ī math transformation is a change of perspective. This isn't a force-march through the equations, it's the casual stroll I wish I had. We'll save the detailed math analysis for the follow-up. If all goes well, we'll have an aha! moment and intuitively realize why the Fourier Transform is possible. Time for the equations? No! Let's get our hands dirty and experience how any pattern can be built with cycles, with live simulations. 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).Here's the "math English" version of the above: How do we get the smoothie back? Blend the ingredients.Why? Recipes are easier to analyze, compare, and modify than the smoothie itself.How? Run the smoothie through filters to extract each ingredient.What does the Fourier Transform do? Given a smoothie, it finds the recipe.Rather than jumping into the symbols, let's experience the key idea firsthand. Unfortunately, the meaning is buried within dense equations: The Fourier Transform is one of deepest insights ever made.
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