Decomposing into Frequencies

Real Analysis · Axiom Academy

Every complex sound is secretly a sum of pure tones — Fourier analysis reveals the hidden frequencies, and that one idea runs MP3, JPEG, and your equalizer. A musical chord looks like a tangled wave, yet it's just a few pure tones added together. Pull that wave apart into its frequencies and you can build it , read it , and even shrink it — the same three moves that compress audio and images. Stack pure tones into a signal Each slider is one pure sine — a harmonic at a whole-number multiple of the 440 Hz fundamental. Add them up and a single rich waveform appears: that's f(t) = Σ Aₙ·sin(2πn·f₀t), the sum of its partials. Now view the SAME signal in the frequency domain. Each bar is one harmonic; its height is that frequency's share of the total energy (by Parseval, share ∝ Aₙ²). The shape of these bars — not the loudness — is what identifies the sound. Compress it — keep the big frequencies, drop the small This is exactly how MP3 and JPEG shrink a file: keep the few loud frequency components, throw the quiet ones away. Drag how many to keep — the reconstruction (red) lays right over the original (grey), and the readout shows how little you actually lost.

This is the written version of the interactive lesson above. See the full Real Analysis course.