Fourier Analysis
Build waveforms from sine waves. Builds on Standing Waves & Harmonics.
You want to build a perfect square wave (the kind you see on a digital oscilloscope) by adding together pure sine waves.
How many sine-wave harmonics do you need to reproduce a PERFECT square wave?
Infinitely many! A perfect square wave requires an infinite sum of odd harmonics: sin(ωt) + sin(3ωt)/3 + sin(5ωt)/5 + ... The sharp corners of a square wave contain infinitely high frequencies. With any finite number of terms, you’ll always see ringing near the edges (the Gibbs phenomenon) — an overshoot of about 9% that never goes away no matter how many terms you add!
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Add harmonics one at a time and watch the sum approach a square wave. Notice the ringing near the edges — that’s the Gibbs phenomenon. Even 50 harmonics can’t eliminate it! Toggle the spectrum view to see which frequencies are present.
f(t) = a₀/2 + Σ[aₙcos(nωt) + bₙsin(nωt)]
Any periodic signal, no matter how complex, is just a sum of simple sine waves. This turns complicated wave problems into simple algebra — solve each frequency separately, then add the solutions.
An EQ adjusts the volume of different frequency bands — it’s literally modifying the Fourier coefficients of the music signal. Boost the bass = increase the low-frequency components.
JPEG images use a 2D version of Fourier analysis (DCT). High-frequency details are discarded to reduce file size. That’s why over-compressed JPEGs look blurry — the sharp edges (high frequencies) have been removed.
Cosmologists decompose the CMB map into spherical harmonics (the spherical equivalent of Fourier modes). The power spectrum reveals the composition and geometry of the entire universe.
Speech recognition algorithms use Fourier transforms to identify the frequency content of your voice. Each vowel has a characteristic pattern of frequency peaks called formants.
“Fourier’s gift: any wave, no matter how wild, is just a choir of sine waves singing together. Decompose the choir, and you understand the wave.”
Fourier Lab
Build waveforms from sine waves. Add harmonics one at a time to construct square, triangle, and sawtooth waves. See the frequency spectrum update in real time. Experience the Gibbs phenomenon firsthand.
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Start building a square wave: add the 1st harmonic, then the 3rd (at 1/3 amplitude), then the 5th (at 1/5), and so on. Watch the sum get more and more square-like. Zoom into a corner — the Gibbs ringing persists even with 50+ terms! Now try building a triangle wave with odd harmonics at 1/n². Notice how much smoother it converges — smoother functions need fewer harmonics.
Sharp features require high frequencies. Smooth features only need low frequencies. The rate of convergence of a Fourier series tells you how ‘sharp’ the original function is. This deep connection between smoothness and frequency content underlies all of signal processing.