Time on the left — a two-tone signal, the sinusoid it is tested against, their product with its area shaded — and frequency on the right, where that area becomes one bar. Evidence: a square wave's partial sums converging while the Gibbs overshoot refuses to; the ladder is sample rates on a log scale.
Which fruits went in, and how much of each?
Imagine someone hands you a smoothie and asks what's in it. You can't see the fruit any more — it's all mixed together. But if you taste carefully you can say: mostly banana, a little strawberry, no mango at all. The Fourier transform does that for sounds and wiggles: it tells you which simple notes were blended to make the whole.
Every messy wiggle is a recipe of simple waves — the transform reads the recipe.
X(f) = ∫ x(t) · e^(−i2πft) dt
Seismograph
ground
100 HZ
Telephone
voice
8 KHZ
CD audio
1982
44.1 KHZ
Radar ADC
RF
1 GHZ
Bars are log-scale — each step is 10×; the spectrum reaches half of each (Nyquist)