Signal Analysis

Differential Equations · Axiom Academy

Signal Analysis: Audio Frequency Decomposition Every sound — a voice, a chord, a chirp — is really several pure tones added together. Pull one apart and watch its frequencies fall out. A musical note is never just one frequency: it's a fundamental plus a stack of harmonics riding on top of it. Fourier's insight — — says any repeating signal decomposes this way, and it's the reason MP3s, noise-canceling headphones, and Siri all work. Drag the fundamental frequency and watch the waveform change — the combined signal is nothing more than three sine waves stacked on top of each other. Toggle a harmonic off and its spike vanishes from the spectrum — this bar chart IS the Fourier decomposition: one bar per frequency, height = amplitude. Shape the tone — which one is "bass"? An equalizer's bass-boost knob amplifies ONE part of the spectrum. Pick a component, drag the boost, and watch the waveform — only one choice actually deepens the bass. Every waveform above is in action — a signal split into frequencies you can see and hear separately . This same move — splitting a function into simple oscillating pieces — is exactly how Fourier series solve the heat and wave equations : instead of one hard PDE, you solve infinitely many easy ones and add the results back together.

This is the written version of the interactive lesson above. See the full Differential Equations course.