Thought Toys · Strategy & computation · Exhibit 46
A wheel turns one way, but on film it can crawl, freeze, or turn the other way. It isn't the wheel playing tricks — it's the camera looking too rarely. Sample any signal fewer than twice per cycle and it comes back as a slower, wrong one that you can't tell apart from the real thing.
The wheel — truly turning, and how the strobe sees it truestrobed
Same samples, two waves that fit them true fsamplesreconstructed
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The disc really rotates at f turns per second — the faint spoke. The strobe only glimpses it fs times a second, freezing the amber spoke at each glimpse and holding it until the next. String those glimpses together and the amber wheel seems to rotate at its own rate — the apparent frequency. When you photograph often enough (fs above 2f, the Nyquist rate — twice the highest frequency present) the amber wheel tracks the real one. Drop below it and the glimpses land a little short of a full turn each time, so the wheel appears to drift backward — the wagon-wheel effect from old Westerns and any wheel under mains lighting.
The lower plot is the same story as a wave: the amber dots are your samples, and the cyan curve is the slowest wave that threads through them. Above Nyquist it lies exactly on the true wave; below, it's a genuinely different, slower wave — and no measurement of the dots alone could ever tell the two apart. That indistinguishability is the whole reason sampling has a hard floor: CD audio samples at 44,100/s to clear the ~20,000 Hz ceiling of hearing, and cameras add shutter tricks so spokes and helicopter blades don't lie. Once you've undersampled, the true frequency isn't hiding in the data — it is simply gone.
improve/verify/46-aliasing.js): the phase-wrap and this closed form agree to 1e-15 off the
half-integer tie set; above Nyquist the apparent frequency is exactly f; the aliased magnitude
always lands in [0, fs/2]; and cosine samples of f, of
f+fs, and of the fold are bit-for-bit identical. Negative
controls: a best-fit reconstruction recovers the true frequency above Nyquist but returns
the alias below it (aliasing is real, not a display trick); f <
fs < 2f gives a strictly negative apparent frequency (backward);
fs = f freezes it; and a very fast sampler never aliases — oversampling is
always safe.
Also in Strategy & computation: The secretary problem →
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