When digital captures analogue
A continuous sine wave (the analogue signal) is sampled at a rate you control. When the sample rate clears the Nyquist rate (fs ≥ 2f), sinc interpolation reconstructs the original perfectly. Drop below it and a phantom low-frequency wave appears — aliasing — that the samples can no longer be distinguished from.
Nyquist rate: fs ≥ 2·f_max
Nyquist frequency: f_N = fs / 2
Aliased frequency: f_alias = |f − round(f/fs)·fs|
Sinc reconstruction: x(t) = Σ x[n]·sinc((t − n·T)/T), T = 1/fs
The wagon-wheel effect in old films is aliasing in action: at ~24 frames per second a fast wheel can appear to slow, stop, or spin backwards because its rotation exceeds the Nyquist rate of the camera.
What is the Nyquist sampling theorem? A continuous signal can be reconstructed perfectly from its samples only if the sampling rate fs is greater than twice the highest frequency present. That threshold, 2f, is the Nyquist rate.
What is the Nyquist frequency? It is half the sampling rate, fs/2 — the highest frequency representable unambiguously at a given sample rate. Anything above fs/2 aliases.
What is aliasing? When a signal is sampled too slowly (fs < 2f), the high-frequency component appears as a different, lower frequency in the samples and cannot be distinguished from a real low-frequency signal.
Fold the true frequency about multiples of the sample rate: f_alias = | f − round(f / fs) × fs |. The result always lies between 0 and the Nyquist frequency fs/2.
Human hearing tops out near 20 kHz, so Nyquist needs more than 40 kHz. 44.1 kHz leaves a transition band for the anti-aliasing filter while keeping the data rate modest.
It reconstructs the continuous signal by summing shifted, scaled sinc functions (sin(x)/x) centred on each sample. When Nyquist is satisfied, this is the exact reconstruction.
A low-pass filter applied before sampling that removes any content above the Nyquist frequency so nothing can alias. Real converters always use one.
No. Once frequencies above Nyquist fold down they overlap genuine low frequencies and the information is lost permanently. Aliasing must be prevented before sampling.
Sampling well above the Nyquist rate. It relaxes anti-aliasing filter requirements, spreads quantisation noise over a wider band, and is common in modern audio converters.
Film samples at about 24 frames per second. If a wheel rotates faster than the Nyquist rate for that frame rate, its motion aliases to a slower — or even reversed — apparent rotation: the wagon-wheel effect.