A teaching model. Single sine wave, ideal instantaneous sampling. Real systems mix many frequencies and use anti-alias filters before the sampler.
Control Systems · signals

Sampling & Aliasing

Sample a wave too slowly and a phantom appears. See exactly when.
Nyquist: fₛ > 2f

Signal & sampler

Hz

What the sampler sees

true signal samples taken what you'd reconstruct
Readings
Field notes

The Nyquist rule & its ghosts

How it works

Why too-slow sampling lies

To capture a wave, you have to sample it more than twice per cycle. That threshold — fₛ > 2f — is the Nyquist criterion, and half the sampling rate (fₛ/2) is the Nyquist frequency, the highest frequency you can honestly record. Sample any faster wave than that and it doesn't just get lost: it masquerades as a lower frequency that was never there. That impostor is an alias.

Worked example

Sample a 900 Hz tone at 1000 Hz. The Nyquist frequency is only 500 Hz, so 900 Hz is over the limit. It folds down to |900 − 1000| = 100 Hz — you'd record and play back a 100 Hz hum that the original never contained.

It's the same effect as a car wheel appearing to spin backwards on film: the frame rate (sampling) is too slow for the wheel (signal), so your eye reconstructs a slower — even reversed — rotation.

What's the fold formula?

The apparent frequency is |f − fₛ·round(f/fₛ)| — the signal reflects ("folds") off multiples of the sampling rate and off the Nyquist frequency, landing somewhere between 0 and fₛ/2.

How do real systems avoid it?

With an anti-aliasing filter: an analogue low-pass filter before the sampler that removes anything above the Nyquist frequency, so nothing is left to fold down. CD audio samples at 44.1 kHz to cover hearing up to ~20 kHz with margin for the filter.

What happens exactly at f = fₛ/2?

You're right at the edge — two samples per cycle. In theory it's the limit; in practice phase luck means you can catch the peaks or the zero-crossings, so it's unreliable. Stay comfortably below fₛ/2.

Does a higher sample rate always help?

Up to a point — it raises the Nyquist frequency so more of your signal fits under it. But beyond covering your highest real frequency (plus filter margin), extra rate just makes bigger files without capturing anything new.