A wheel turning twenty-three times a second, filmed at twenty-four
It looks like one slow turn backwards. Not roughly, and not
because the camera is bad: the frames it records are exactly the frames a wheel
turning once a second the other way would have given it. Sampling throws away the
difference between a wave and every wave a whole multiple of the sampling rate away,
and what it throws away it cannot get back. A 30 kHz tone recorded at 44.1 kHz comes
back as 14.1 kHz, which is not silence and not ultrasound — it is a note you can
hear, that was never played.
Two waves, one set of numbers
Here are two sine waves at different frequencies, and the moments a recorder looks at them. Every sample is the same number for both. There is no measurement you could make on the recording that would tell you which one was in the room.
The wheel
A spoked wheel is a wave: a spoke passing a point is a cycle. Film it and every frame lands somewhere on that wave, and the wheel you see is whichever rotation fits the frames with the least turning between them. Past half the frame rate it fits better backwards.
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Where the fold is
Plot what you record against what was there and the line goes up to half the sampling rate, then folds and comes back down, and does it again, for ever. Every frequency on the way up has a partner on the way down that is indistinguishable from it — the fold is the whole story, and half the sampling rate is where it happens.
What this page checks about itself
The identity at the bottom of all of this is that
sin(2π(f + k·fs)n/fs) = sin(2πfn/fs) for whole k and
n, because the difference between the angles is a whole number of
turns. The checks compare the sample values themselves rather than the formula.