Aliasing
Raise a tone past Nyquist and watch it fold back down. Not distortion — two signals becoming indistinguishable.
Skip to the animationSampling below the Nyquist rate makes a high-frequency component produce exactly the same samples as a lower-frequency one, so the two become indistinguishable — which is ambiguity rather than distortion, and is why no processing after the sampler can undo it.
Watching a tone fold
Sample at 10 kHz. A 2 kHz tone gives five samples per cycle and is plainly recoverable. At 4 kHz there are only 2.5 samples per cycle and it *looks* inadequate — but the theorem guarantees exact reconstruction, so sparse is not insufficient.
At exactly 5 kHz the phase decides everything: sample at the peaks and the amplitude is right, sample at the zero crossings and every sample is zero, indistinguishable from silence. That is why the condition is f_s > 2f_m strictly.
At 7 kHz the samples are identical to those of a genuine 3 kHz tone. The alias is |f − k·f_s| for whichever integer k brings it into the band.
Why 'distortion' is the wrong word
Nothing was corrupted. A real 3 kHz tone and a real 7 kHz tone produce the same set of numbers, so the mapping from signals to samples has stopped being one-to-one.
The information was destroyed at the instant of sampling, so there is nothing left to repair. This is exactly why the anti-aliasing filter must be analogue and must come before the converter — a digital filter afterwards has only the ambiguous samples.
Where you have already seen it
- Wagon wheels turning backwards — film samples at 24 fps and the spokes alias.
- Moiré on a striped shirt — the sensor's pixel grid samples the fabric spatially.
- A stroboscope freezing a fan — sampling at the rotation rate aliases it to zero.
- Jagged edges in rendered images — spatial aliasing, which is what graphics *anti-aliasing* is named after.
All four are one phenomenon: something periodic, sampled too slowly. Graphics anti-aliasing filters before sampling for exactly the reason an audio converter does.
The defence
- 1Put an analogue low-pass filter before the converter, cutting off below
f_s/2. - 2Or oversample — sample far above Nyquist so a gentle analogue filter suffices — then filter and decimate digitally.
Oversampling relaxes the analogue filter; it does not remove it. Something must still stop energy above the final Nyquist frequency from reaching the sampler, which is why every ADC datasheet specifies an input filter however fast the part is.
The numbers you will be asked for
- Alias frequency
f_alias = |f − k·f_s|
For the integer k that brings it into 0 to f_s/2.
- Folding
f_s/2 acts as a mirror
Frequencies above it reflect back down.
- Condition to avoid it
f_s > 2·f_m
Strictly greater; the equality case is ambiguous.
- Oversampling ratio
OSR = f_s / (2·f_m)
Larger OSR allows a gentler analogue filter.
Advantages and disadvantages
Advantages
- Entirely preventable, by a filter that costs very little.
- Predictable — the alias frequency is given by a formula.
- Deliberate undersampling turns it into a way of shifting bands down.
- The same analysis explains visual and temporal artefacts, not just audio.
Disadvantages
- Irreversible once it has happened.
- The required filter is analogue, so it cannot be improved in software later.
- A perfect brick-wall filter is impossible, so some margin is always spent.
- It is easy to introduce accidentally by decimating without filtering first.
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.