The Sampling Theorem — below the line it's exact; above it, a masquerade

time · the source, the comb & the reconstruction

frequency · |X[k]| via the imported FFT

f_s = 10.0 Hz · Nyquist 5.0 Hz

f₁ = 2.0 Hz

a₁ = 1.0

f₂ = 3.5 Hz

a₂ = 0.5

presets

band-limited

the wagon wheel

exact alias

show

sinc lobes

ghost alias

free-drag (Hann)

Sample a signal at rate f_s and every frequency below the Nyquist line f_s/2 is carried exactly. Whittaker–Shannon threads a sinc through the samples and recovers the band-limited source to a residual of ~5e-14 — the live max |error| from the self-test, not the bare word "perfect." (The finite-window plain sum caps near 1e0; the periodic Dirichlet kernel removes that truncation — that is why it's the proof.)

Above the line, it's a masquerade. An undersampled cosine and its fold produce the byte-for-byte identical sample vector (equal to machine-ε, in fact strictly ===), so the samples cannot tell them apart. The reconstruction provably does not rebuild the source — it rebuilds the alias, exactly. Run the Butterfly's imported FFT and watch the spectral line land at the wrong bin.

below Nyquist — the line is EXACTf_max < f_s/2

The control — f_s is load-bearing

Drag f_s down. The gold comb thins; the reconstruction (green) lies on the true source (teal) until f_s crosses 2·f_max — then green peels off teal and snaps to the ghost (red). The comb never moves: same samples, different signal. That snap is the exhibit.

Coherent ↔ free-drag

Presets are coherent (f·N/f_s ∈ ℤ ⇒ one clean spectral stem; N=256). Free-drag breaks coherence, so the spectrum is computed with a light Hann window to tame leakage; the readout NAMES the active mode so the picture never lies. (Display windowing is separate from the byte-exact claim — that leg owns the integer-cycle LUT.)

What the FFT says

The audio — muted by default

—true tone f

—what the samples carry

Sound off

Sound on

The alias being audible is the one thing the eye-proof can't deliver. The two buttons play the true tone and the tone the samples actually carry — the ear hears the ghost. Honours the estate-wide ws:pref:muted key; first sound waits for a click.

Legend

the true source (teal = truth)

the sample comb (gold = kept)

the sinc reconstruction (green = rebuilt)

the ghost alias (red = the lie)

the f_s/2 Nyquist mirror (violet)

↔ the cross

← The Butterfly · the transform the radix-2 FFT this bench runs — imported, no re-typed math. Sampling folds in time; convolution wraps in frequency: the same fold, other domain. ← The Butterfly's Voice · where the teeth first bit the Voice shows leakage (a smear, f_s=48k far below Nyquist — no folding); this shows aliasing (a fold). Leakage = smear, aliasing = fold — two faces of one sampling ceiling.