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.
Participants were given 90 pairs of maps positioned at different rotation angles (0°, 90°, and 180°). This study assessed gender differences in city map rotation, and the differences between architecture, business studies, fine arts, and psychology undergraduates.