Every wiggle, sound, image, and vibration in the universe can be rebuilt from pure sine waves. The Fourier transform is the machine that finds the recipe — it takes a signal apart into the frequencies hiding inside it. Play with the instruments below to feel how it works, then see where it powers the technology around you.
1 The core idea — building a signal
Any shape is just sines stacked up
Before we take signals apart, let's build one. Add a few sine waves of different speeds and heights. On the left you see them sum into one complex wave (time domain). On the right is the recipe — one bar per ingredient (frequency domain). Hit Square wave to watch a stack of sines approximate a sharp edge.
Time domain — the signal
Frequency domain — the recipe
summed signalindividual sine componentsfrequency bars = amplitudes
The big idea: the right-hand chart contains the exact same information as the left, just written differently. Knowing "how much of each frequency" is enough to perfectly rebuild the wave. The Fourier transform is simply the dictionary that translates left ⇄ right.
2 The transform itself — taking signals apart
Feed it a mystery wave, get back its frequencies
Now the reverse. Here a real Fast Fourier Transform (FFT) runs in your browser on each waveform and reveals its hidden frequency content. Notice the patterns: a square wave is built only from odd harmonics; a sawtooth uses all of them. The transform discovers this with no prior knowledge of the shape.
Input signal f(t)
FFT magnitude spectrum |F(ω)|
3 Bring your own signal
Now do it on your data
Speak into your mic or snap a photo with your webcam. The left panels show what the computer literally receives — just numbers. The right panels show the same data after a live FFT. Watch your voice become a fingerprint; watch a picture become a frequency map.
🎤 Live voice — your signal in real time
Waveform — what the mic captures
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FFT spectrum — frequencies right now
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Spectrogram — frequencies over time (scrolls right → left)
Try this: hum a steady note → tight horizontal bands at your pitch and its harmonics. Whistle → a single thin moving line. Say "shhh" → a fat cloud at the top of the spectrogram (lots of high frequencies). Say "aaah" → multiple stacked lines (your vocal formants).
📷 Pictures are signals too — 2D FFT
Original (grayscale) — pixels as numbers
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|F(u,v)| — 2D FFT magnitude (log, centered)
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🔬 Reconstruction — inverse FFT from the top-K coefficients only
drag the slider to reconstruct the image from a handful of coefficients
This is the JPEG/MP3 trick. Slide left → keep only the brightest few FFT bins → image becomes a blurry blob (just the lowest frequencies). Slide right → add more bins → edges and texture snap back in. Around 1–2% of the coefficients you already get something recognizably "your face" — the rest is just diminishing returns. This is how a 2 MB photo becomes a 200 KB JPEG.
∑ The common thread
One sentence to remember it by
In every example above — building a wave, cleaning noise, compressing a photo, tuning your bass, reading an MRI — the move is identical: switch a signal from "what it does over time" to "what frequencies it's made of," do something easy in that view, and (if needed) switch back.
The forward transform analyzes; the inverse transform rebuilds. Continuous signals use the integral form below, while everything in this page used its digital cousin, the discrete FFT.