Sampling and the Nyquist Theorem
Two different sine waves passing through identical samples. That picture is why the factor of two is not arbitrary, and why the anti-aliasing filter cannot be digital.
Skip to the animationA signal band-limited to f_max and sampled faster than 2·f_max can be reconstructed exactly from its samples — but sampled any slower, a higher frequency becomes indistinguishable from a lower one, which is irreversible and must be prevented by an analogue filter before the sampler.
The theorem
A continuous signal is defined at every instant, which is infinitely many numbers. Sampling keeps only the values at regular intervals. The natural assumption is that information is lost — and Nyquist's result is that under one condition, nothing is.
The reconstruction is exact, not approximate. It is a theorem with a proof, and it is the entire reason digital audio, video and telephony are possible.
Aliasing
Undersample and a lower-frequency wave passes through exactly the same points. Nothing in the sampled data distinguishes them, so the high frequency masquerades as a low one.
Aliasing is irreversible. The ambiguity is in the data itself, so no amount of processing afterwards can resolve it — which is the single most important practical consequence of the theorem.
Two samples per cycle is the minimum that can register a cycle's existence. The condition is strictly f_s > 2f_max: at exactly twice, samples landing on zero crossings would read nothing.
Standard rates and why
| System | Sample rate | Band limit | Margin |
|---|---|---|---|
| Telephone speech | 8 kHz | 3.4 kHz | Generous — the filter is cheap |
| CD audio | 44.1 kHz | 20 kHz | 10% for the anti-alias filter |
| Professional audio | 48 / 96 / 192 kHz | 20 kHz | Oversampled, for gentler filters |
| Video (ITU-R 601) | 13.5 MHz | 5.75 MHz | Chosen to suit both 525- and 625-line systems |
The filters
The anti-aliasing filter must be analogue and must precede the sampler. Filtering afterwards is useless, because aliased components are indistinguishable from real ones by then. It is the one part of a digital signal chain that cannot itself be digital.
At the other end, a real DAC holds each sample until the next, producing a staircase with high-frequency images. A reconstruction filter smooths it, and the hold's predictable amplitude droop is compensated for.
Oversampling pushes the images further from the band so a gentler, cheaper filter suffices — which is why high sample rates persist beyond what hearing requires.
Bandpass sampling
The theorem is usually stated with f_max, but what actually matters is the occupied bandwidth. A 2 MHz signal centred at 100 MHz can be sampled at 5 MHz and allowed to alias down to baseband undamaged, provided the band does not straddle a multiple of f_s/2.
Every software-defined radio relies on this. It is how a cheap converter receives signals far above its own sample rate — aliasing used deliberately, as a frequency-translation tool.
The numbers you will be asked for
- Nyquist criterion
f_s > 2 · f_max
- Nyquist frequency
f_N = f_s / 2
the highest representable frequency
- Alias frequency
f_alias = |f_signal − n·f_s|
- Bandpass sampling
f_s > 2B
B is the occupied bandwidth, not f_max
- Sample-and-hold droop
sinc(π f / f_s)
Watch it work
Check yourself
question 1 / 4
One question at a time. Pick an answer to see why it is right or wrong, then move on — there is no score to keep and nothing is saved.