Fourier Series
Every periodic signal is a chord of sine waves
Watch the epicycle stack trace the curve — showing how f(t) = Σ Aₙ·sin(nωt + φₙ) builds a waveform one spinning circle at a time. Drag a spectrum stem to set amplitude, rotate its handle to set phase, or load a preset (square, sawtooth, AM, FM…) to see the frequency-domain signature behind each shape.Fourier Series
Reference notes — DFT · modulation · where this shows up in AI
DFT — the inverse of this app
This page does synthesis: you pick the spectrum, it builds the wave. The Discrete Fourier Transform does analysis: given N samples x[k], it recovers each stem viaX[n] = Σₖ x[k]·e^(−i2πnk/N).
|X[n]| is the stem height, arg X[n] is the handle angle. The FFT is just a fast
(O(N log N)) way to compute it. Synthesis and analysis are exact inverses — nothing is lost either
way.
Modulation — AM vs FM
AM multiplies a carrier by the message: multiplication in time = shifting in frequency, so the message reappears as two mirror sidebands at ωc±ωm (3 lines total). FM wobbles the carrier's frequency instead: even one sine message spawns an infinite Bessel-weighted combJₙ(β) of sidebands. That's why FM needs more bandwidth but
keeps a constant envelope — noise that clips amplitude doesn't touch the information.
Where AI uses this
Convolution theorem: convolution in time = multiplication in frequency — the trick behind fast CNN layers and why filters are "frequency selectors". Transformer positional encoding is literally this app: position t is encoded assin/cos(ωₙt) at geometrically spaced ωₙ, a Fourier basis for "where am I in the
sequence".
Fourier features / SIREN nets use sin(Wx+φ) activations so networks can learn
high-frequency detail — a learned Fourier series. And spectral decay ⇄ smoothness (square vs
triangle
here) is the same idea as spectral bias: neural nets learn low frequencies first.
This tool does synthesis, not analysis: you set an amplitude and phase for each harmonic, and it sums them into a waveform in real time with a hand-rolled canvas renderer — no charting library. The sections below build up the math the stage and spectrum panels are actually showing; the "Reference notes" accordion inside the Explorer tab covers the DFT, AM/FM modulation, and where this shows up in AI, so that ground isn't repeated here.
1. A Fourier series is a sum of harmonically-related circles
Any periodic signal $f(t)$ with period $T$ (fundamental frequency $\omega = 2\pi/T$) can be written as a sum of sines and cosines at integer multiples of $\omega$: $$f(t) = A_0 + \sum_{n=1}^{\infty} A_n\sin(n\omega t + \varphi_n)$$ Each term is called a harmonic: $n=1$ is the fundamental (same frequency as the signal itself), $n=2$ is the first overtone at double the frequency, and so on. This tool lets you build that sum by hand, one generator at a time, and watch the partial sum converge toward whatever shape you're targeting as you add more harmonics.
2. Each term is the Unit Circle Explorer, spinning
A single term $A_n\sin(n\omega t+\varphi_n)$ is exactly the $y$-coordinate of a point going around a circle of radius $A_n$ at angular speed $n\omega$, starting at phase $\varphi_n$ — the same $(\cos\theta,\sin\theta)$ construction as the Unit Circle Explorer, just with the angle now advancing over time instead of being dragged by hand. Stacking generators tip-to-tail (each circle's center riding on the tip of the previous one) is the classical "epicycle" picture, and it's a literal drawing of the sum in the formula above, not just an animation trick.
3. Two equivalent descriptions of the same signal
The stage plots $y(t)$ against time — the time domain. The spectrum plots $|A_n|$ against frequency $n\omega$ — the frequency domain. Both fully determine the signal; neither carries more information than the other. Dragging a spectrum stem up or down changes that harmonic's radius $A_n$; rotating its handle changes its starting phase $\varphi_n$. Every one of those edits shows up immediately in the time-domain stage, because the stage is nothing more than the spectrum, summed.
4. Power splits by harmonic (Parseval's theorem)
The "Power computed two ways" readout is Parseval's theorem made visible: the average power of the signal, measured directly in the time domain, equals the sum of each harmonic's own power measured in the frequency domain: $$\langle y(t)^2 \rangle = \sum_n \tfrac{1}{2}A_n^2$$ That's why the two numbers in the tool always agree — they're the same total, computed two different ways. It also means the "Power by harmonic" bar chart is a literal breakdown of where the signal's energy lives, harmonic by harmonic, with nothing double-counted or left over.
5. Sharp corners need infinitely many harmonics
A perfectly smooth wave (a single sine) needs exactly one harmonic. A signal with a sharp corner or jump — the square wave and sawtooth presets — needs infinitely many, because no finite sum of smooth sine waves can reproduce a true discontinuity exactly. Truncating the sum still overshoots the jump by a fixed fraction no matter how many harmonics you add (the Gibbs phenomenon); you can see the ripples this leaves behind by adding generators to the square-wave preset one at a time. The rate the harmonic amplitudes decay with $n$ is a direct readout of smoothness: a sharp corner decays slowly ($1/n$), while a signal with no corners at all decays much faster.
6. Pedagogical flow
- Load "Pure sine" and confirm there's exactly one generator, one spectrum stem, and the two power numbers already match.
- Load "Square wave" and add generators one at a time: watch the corners sharpen but never quite settle, and watch the power-by-harmonic bars fall off slowly.
- Drag one stem's phase handle a quarter turn and watch the time-domain stage shift — the amplitude spectrum alone never tells the whole story.
- Compare "Square wave" against "Triangle": the triangle's corners are gentler (continuous, but not smooth), and its harmonics decay noticeably faster.