◄ WORLD V · SONNY 5DART 302 · a helldive at the net

NYQUIST–SHANNON sample fast enough and the continuous signal is perfectly recoverable from its dots

The sampling theorem is the bridge between the analog and the digital: a signal containing no frequency above B is completely determined by samples taken at rate > 2B (the Nyquist rate), and can be reconstructed exactly by summing shifted sinc pulses. Sample too slowly and different frequencies become indistinguishable — aliasing — and a high tone masquerades as a low one (the wagon-wheel effect, the moiré on a screen). The whole of digital audio, imaging, and communication rests on it. Recovery is caused by the band-limit: it is that structure — no energy above B — that makes the dots enough to rebuild the curve.

THE TECHNIQUE band-limited to B ⇒ samples above 2B reconstruct exactly; below ⇒ aliasing

A 7 Hz tone. The demo shows it reconstructs exactly above the Nyquist rate (14 Hz) but aliases to a lower frequency when undersampled: live demo







  
HISTORY & CREDIT WKS theorem · Nyquist = the rate

  
“Sampling always loses the in-between.” — not if the signal is band-limited and you sample above twice its bandwidth: then the samples are the whole story, and the continuous curve is recovered exactly. The band-limit is the guarantee. cited

  
1915–49 · the reconstruction theorem is Whittaker–Kotel'nikov–Shannon (WKS) — E.T. Whittaker (1915, the sinc interpolation), Kotel'nikov (1933), Shannon (1949). Harry Nyquist (1928) established the rate (2B pulses through bandwidth B, hence the “Nyquist rate”) but did not state the sampling/reconstruction theorem.
the fact · band-limited to B, sample above 2B, reconstruct exactly by sinc interpolation.
now · all digital audio, imaging, radio, and data acquisition.

  
Above the Nyquist rate the dots are the signal; below it, frequencies collide and a tone lies about its pitch. Recovery guaranteed by the band-limit — the structure the samples ride on. Nyquist-Shannon






  
RECOMMEND FOR I-13 reconstruction vs aliasing at the Nyquist rate, computed

  
On the canonical compiler, a 7 Hz tone has Nyquist rate 14; sampled at 10 (below) it aliases to |7−10|=3 Hz; sampled above 14 it reconstructs as 7:

  
$ i13 run rec_nyquist.i13   # 7 Hz tone, various sample rates
nyquist_rate = 14     -- must sample above 2*7
alias_under = 3       -- at fs=10 (< 14): 7 Hz masquerades as 3 Hz
recovered_ok = 7      -- above the Nyquist rate: exact reconstruction

  
Recommend: Nyquist–Shannon is reconstruction gated by a structural precondition, computed by i13 as a threshold. It reports the Nyquist rate 14, the aliased frequency 3 when undersampled at 10, and exact recovery 7 above the rate. The recovery is caused by the band-limit: only because the signal has no energy above B are the samples sufficient to rebuild it — violate that and the reconstruction returns the wrong signal (the alias), not a degraded one. It is the continuous cousin of the batch's discrete reconstructions: the right structure makes the parts (samples) enough for the whole (the curve).