The Wheel That Spins Backwards
Film a car accelerating and there's a point where the wheels appear to stop, then rotate backwards while the car keeps moving forward. Everyone has seen it. It's not a camera fault.
The camera samples 24 or 25 times per second. The wheel is rotating faster than that. Between one frame and the next, a spoke moves nearly a full rotation — and the camera has no way to know it went all the way round rather than moving slightly backwards.
This is aliasing, and it's the same phenomenon that makes digital audio require a filter before the converter, makes an oscilloscope show a signal that isn't there, and makes a badly-designed sensor report a completely wrong reading.
The Sampling Theorem
The rule is precise: to reconstruct a signal, you must sample at more than twice its highest frequency component.
That threshold — twice the highest frequency — is the Nyquist rate. Half your sampling rate is the Nyquist frequency, the highest signal frequency you can represent.
CD audio: 44,100 Hz sampling → 22,050 Hz Nyquist frequency
48 kHz audio: 48,000 Hz sampling → 24,000 Hz Nyquist frequency
Cinema film: 24 Hz sampling → 12 Hz Nyquist frequency
Human hearing extends to roughly 20 kHz in young adults. CD's 44.1 kHz rate gives a 22.05 kHz Nyquist frequency, leaving a small margin above the audible range — which is exactly why that number was chosen rather than something rounder.
Note the word "more than." Sampling at exactly twice the frequency isn't sufficient. Sample a sine wave at precisely twice its frequency and, depending on phase, you might catch it at every zero crossing and record nothing at all.
What Aliasing Actually Does
When a frequency above Nyquist is sampled, it doesn't vanish. It folds back into the representable range and appears as a different, lower frequency — one that was never in the original signal.
The folded frequency is:
apparent = |actual − (n × sample rate)|
for whichever integer n brings the result into the range below Nyquist.
Sample a 30 kHz tone at 44.1 kHz:
|30,000 − 44,100| = 14,100 Hz
A 30 kHz tone — inaudible — becomes a clearly audible 14.1 kHz tone that wasn't in the original recording. And it's not a harmonic or a distortion product with any musical relationship to the source. It's an arbitrary artefact.
This is what makes aliasing worse than simply losing high frequencies. Losing them is a limitation. Aliasing actively adds false content.
Once aliased, it cannot be removed. The false frequency is now indistinguishable from a real signal at that frequency. No amount of downstream filtering recovers the original, because the information needed to separate them is gone.
Anti-Alias Filtering
The solution is to remove out-of-range frequencies before sampling, using an analogue low-pass filter ahead of the converter.
The filter has an awkward job. It must pass everything below Nyquist and block everything above it, and real filters don't have infinitely steep transitions. A very steep filter introduces phase distortion and ringing; a gentle one lets some content through to alias.
This tension is a large part of the argument for higher sample rates. Record at 96 kHz and your Nyquist frequency is 48 kHz, giving the filter a wide transition band well above anything audible. The extra bandwidth isn't there because anyone can hear 40 kHz — it's there so the filter can be gentle in the region that matters.
Oversampling takes the same approach digitally. Sample at a very high rate with a simple analogue filter, then apply a precise digital filter and reduce the rate. Digital filters can be far steeper without phase problems, so this gets you the best of both.
Beyond Audio
The same maths governs anything sampled.
Images. Photograph a finely striped shirt and you may get moiré — colourful interference patterns that aren't in the fabric. The sensor's pixel grid is sampling a spatial frequency above its Nyquist limit. Some cameras include an optical low-pass filter over the sensor to blur slightly and prevent this; others omit it for sharpness and accept occasional moiré.
Oscilloscopes. A scope sampling at 1 GS/s cannot faithfully display a 700 MHz signal. It will display something, and that something may look like a stable low-frequency waveform. Engineers have chased phantom signals for hours before checking the sample rate.
Rotating machinery. A stroboscope flashing at a frequency close to a shaft's rotation rate makes it appear stationary or slowly rotating. Useful when deliberate; dangerous when accidental, since a machine that looks stopped under fluorescent lighting may be spinning at speed.
Sensor data logging. Log a vibration sensor at 10 Hz on a machine with a 60 Hz component and you'll record a slow oscillation that doesn't exist. Condition monitoring systems are specified with sample rates well above the highest frequency of interest for exactly this reason.
Video. Anything periodic filmed at a fixed frame rate is subject to it — helicopter rotors, propellers, spoked wheels, spinning fans.
Working Out the Numbers
The Frequency Converter handles conversions between Hz, kHz, MHz, GHz and RPM:
- Enter the value.
- Select the source unit.
- Read the equivalents.
The RPM conversion is the one that catches people out when calculating sample rates for mechanical systems:
Hz = RPM / 60
A shaft at 3,000 RPM rotates at 50 Hz. But the frequencies of interest are rarely the rotation rate itself — a shaft with 12 fan blades produces a blade-passing frequency of 12 × 50 = 600 Hz, and bearing defect frequencies can be higher still.
Sample at several times the highest frequency you care about, not twice. Nyquist is the absolute floor for reconstruction. In practice, sampling at 5 to 10 times the highest frequency of interest gives you margin for filter roll-off, makes the data easier to interpret visually, and improves amplitude accuracy.
Practical Tips
Identify your highest frequency of interest first, then choose a rate. Working backwards from an arbitrary rate is how aliasing happens.
Always filter before sampling, not after. Post-sampling filtering cannot undo aliasing.
Suspect aliasing when a measurement shows a frequency that has no physical explanation. Change the sample rate and re-measure. A real signal stays at the same frequency; an alias moves.
Watch out for periodic light sources when observing rotation. LED and fluorescent lighting flickers at mains-related frequencies and can produce a stroboscopic effect.
Remember that the theorem assumes a band-limited signal. Real signals often have content above the band you assumed, which is exactly why the filter is mandatory rather than optional.
Higher sample rates aren't automatically better for storage. They cost proportionally more space and processing. The benefit is filter headroom and processing margin, not additional audible content.
FAQ
What is the Nyquist frequency? Half the sampling rate — the highest signal frequency that can be correctly represented at that rate.
Why is CD audio 44.1 kHz? It gives a Nyquist frequency of 22.05 kHz, comfortably above the upper limit of human hearing, with margin for the anti-alias filter's transition band. The specific number derives from early digital recording equipment that stored audio on video tape.
Can aliasing be removed afterwards? No. The aliased frequency is indistinguishable from a genuine signal at that frequency once sampled.
Why do wheels appear to spin backwards on film? The frame rate is below twice the rotation frequency, so the wheel's position between frames is ambiguous and the brain interprets the smaller apparent movement — which may be backwards.
Is 96 kHz audio better than 48 kHz? It gives more filter headroom and processing margin. Whether the difference is audible in a finished recording is debated; the technical benefits during production are less controversial.
How much above Nyquist should I sample? For measurement work, aim for 5–10 times your highest frequency of interest rather than the theoretical minimum of just over 2.
The Takeaway
Sampling doesn't just fail gracefully when the signal is too fast — it fails deceptively, producing convincing readings that were never there. Knowing your Nyquist frequency and filtering below it is the difference between a measurement you can trust and one that quietly lies.
Convert between Hz, kHz, MHz, GHz and RPM free with the Frequency Converter at sadiqbd.com — no sign-up, instant results.
