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The Fourier Transform: A Deeper Insight

The Fourier Transform is one of the deepest insights ever made. Unfortunately, its meaning is buried within dense equations. Rather than jumping into the symbols, let's experience the key idea firsthand.

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.

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).

A math transformation is a change of perspective. 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? 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.

The Fourier Transform takes a specific viewpoint: What if any signal could be filtered into a bunch of circular paths? This concept is mind-blowing. The Fourier Transform finds the recipe for a signal, like our smoothie process:

  1. Start with a time-based signal.
  2. Apply filters to measure each possible "circular ingredient".
  3. Collect the full recipe, listing the amount of each "circular ingredient".

The Fourier Transform is useful in engineering but it's a metaphor about finding the root causes behind an observed effect.

Think With Circles, Not Just Sinusoids: The Fourier Transform is about circular paths and Euler's formula is a clever way to generate one.

The Fourier Transform finds the set of cycle speeds, amplitudes, and phases to match any time signal. Our signal becomes an abstract notion that we consider as "observations in the time domain" or "ingredients in the frequency domain".

Discovering The Full Transform: The big insight is that our signal is just a bunch of time spikes. The Fourier Transform builds the recipe frequency-by-frequency.

Today's goal was to experience the Fourier Transform. We'll save the advanced analysis for next time. Happy math.

The original article: https://betterexplained.com/articles/an-interactive-guide-to-the-fourier-transform/