the Nyquist frequency
/ NYE-kwist /
If you check a clock only once every hour, you can tell the hour hand is moving but you have no hope of following the second hand — it spins many times between your glances. Sampling has the same ceiling: the rate at which you look sets a hard limit on the fastest wiggle you can possibly track. The Nyquist frequency is that ceiling — the highest frequency a given sampling rate can honestly represent.
If you sample at f_s samples per second, the Nyquist frequency is exactly f_s / 2. The reason is intuitive: to capture an oscillation you need at least two samples per cycle — one to catch a peak and one a trough — so the fastest cycle you can pin down repeats every two samples, which is f_s / 2 cycles per second. In a length-N DFT this corresponds to the middle bin, index N/2, the fastest pattern the data can represent (alternating +1, -1, +1, -1...). Every real frequency from 0 up to f_s / 2 maps to its own DFT bin; anything faster has nowhere distinct to go and instead folds back below f_s / 2 as an alias, which is why the Nyquist frequency is also called the folding frequency.
The Nyquist frequency is the practical design number behind every digitizer. Audio sampled at 44.1 kHz has a Nyquist limit of 22.05 kHz, just above human hearing; a medical sensor sampling at 1000 Hz can faithfully report rhythms only up to 500 Hz. The companion rule is the Nyquist RATE — the minimum SAMPLING rate (twice the highest frequency present) you must use to avoid aliasing — which is the same number seen from the other side. Engineers usually sample comfortably above it and add an anti-aliasing filter, because a signal with content right at Nyquist is borderline and easily corrupted.
A CD samples at 44,100 Hz, so its Nyquist frequency is 22,050 Hz — chosen to sit just above the roughly 20,000 Hz upper limit of human hearing. Frequencies below 22,050 Hz are captured cleanly; anything above is removed by an anti-aliasing filter before sampling so it cannot fold down into the audible band.
Half the sampling rate is the hard ceiling on representable frequency.
Sampling at exactly twice a frequency is not enough in practice: a sine sampled precisely at its zero crossings reads as all zeros. The sampling theorem requires the rate to be STRICTLY greater than twice the highest frequency, with margin to spare.