Fourier Transform
Definition
The mathematical prism that splits any signal into its pure frequencies — how noise-canceling headphones know exactly what to cancel.
Technical definition
F(ω) = ∫f(t)e^(−iωt)dt decomposes a time signal into frequency components; the inverse transform rebuilds it. It is the basis of spectral analysis.
Example
A guitar-tuner app Fourier-transforms your plucked string's sound to find the dominant frequency and tell you whether you are sharp or flat.
Common confusion
The Fourier transform does not create frequencies that aren't there — it only reveals what the signal already contains. Garbage in, garbage spectrum out.