Yikes. Rather than deciphering it symbol-by-symbol, let’s experience the idea (hear the song then read the sheet music, dig?).
Let’s start with 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.
Next, we’ll refine the analogy into “math-English”:
The Fourier Transform extracts each “cycle ingredient” from a time-based signal (the cycle strength, delay & speed), resulting in a final “cycle recipe”