Sampling and Aliasing Visualiser

What is aliasing?

Aliasing is what happens when a signal is sampled too slowly to be recorded faithfully: the samples fit a lower frequency exactly as well as the true one, so anything rebuilt from them comes out at that lower frequency, the alias. A sine of frequency f sampled at a rate fs is recorded at f_alias = |f − k fs|, where k is the whole number nearest to f/fs.

Every alias lies between 0 and fs/2, the Nyquist frequency, and a tone is recorded as itself only if it is already below fs/2. Above it the tone is not lost or blurred. It is replaced, cleanly and with no warning, by a different tone.

The reason is that a sine repeats. Sampling takes a value at each instant tₙ = n/fs, and between one sample and the next a tone at f and a tone at f − k fs differ by exactly k whole cycles, which no sample can see: sin(2π f n/fs) = sin(2π (f − k fs) n/fs) for every whole number k. So a 1 Hz sine and an 11 Hz sine sampled 10 times a second give identical numbers, and a 9 Hz sine gives the same numbers upside down, which is a 1 Hz sine running backwards. Once the numbers are all that is left, nothing can say which tone made them.

Claude Shannon set this out as the sampling theorem in 1949, in “Communication in the Presence of Noise”, building on Harry Nyquist’s 1928 work on telegraph signalling, and E. T. Whittaker in 1915 and Vladimir Kotelnikov in 1933 reached the same result independently. The folding of the spectrum behind the alias formula is worked through in chapter 4 of Oppenheim and Schafer’s Discrete-Time Signal Processing.

Reading the visualiser

The main trace shows the true signal as a solid line, the samples as amber stems and dots, and the recovered signal dashed: the one sine below fs/2 that passes through every sample, which is what an ideal reconstruction returns. Below the Nyquist frequency the dashed line lies on the signal. Above it, the dashed line goes its own slower way and still meets every dot.

Play sweeps a playhead across the window, taking each sample as it passes, at half a second of real time per sample interval at 1x. The slow pace is deliberate: a screen redraws at its own frame rate, which is a second sampling, and could alias a fast picture itself. The window always spans between 8 and 20 sample intervals, in round lengths such as 0.5 s, 1 s and 2 s.

Below the trace on the left is the same tone as a wheel: the spoke turns once per cycle and its height is the signal. Each amber dot on the rim is where a sample caught the spoke, the dashed spoke turns at the speed the samples suggest, as a camera filming the wheel would show, and the arrow inside the rim is the step between the last two frames as the eye reads it.

On the right is the spectrum after sampling, with the band from 0 to fs/2 that a reconstruction keeps shaded, and under the stage the folding diagram plots alias frequency against signal frequency. Both are explained below. Of the readouts:

• Apparent rotation is the alias with its sign kept, f − k fs: negative means the wheel seems to turn backwards.
• Nyquist rate is twice the highest frequency present, which the sampling rate has to exceed, and Samples per cycle is fs/f.

The Start from buttons load the cases this page works through. The units menu switches the whole tool between hertz and seconds, kilohertz and milliseconds, or megahertz and microseconds. With the scene focused, the up and down arrow keys change the signal frequency by 0.5 and the left and right keys change the sampling rate.

Worked example: a 9 Hz sine sampled at 10 Hz

The visualiser opens on a 9 Hz sine sampled 10 times a second. By hand:

• The Nyquist frequency is fs/2 = 10/2 = 5 Hz, and 9 Hz is above it, so the tone aliases.
• The nearest whole multiple of the rate: f/fs = 9/10 = 0.9, so k = 1.
• The signed alias is f − k fs = 9 − 10 = −1 Hz, so f_alias = 1 Hz, and the minus sign says it runs backwards.
• Check it on a sample. The first after t = 0 is at t = 0.1 s, where sin(2π × 9 × 0.1) = sin(324°) = −0.5878, and the backwards 1 Hz sine gives −sin(2π × 1 × 0.1) = −sin(36°) = −0.5878. The next is sin(648°) = −sin(72°) = −0.9511 from both.
• On the wheel, the spoke turns 9/10 = 0.9 of a turn between frames, which the eye reads as 0.9 − 1 = −0.1 of a turn, 36° backwards a frame: one turn a second the wrong way, the −1 Hz of the readout.
• To record 9 Hz faithfully the rate has to be more than 2 × 9 = 18 Hz. At 10 Hz there are only 10/9 = 1.111 samples per cycle, against the more than 2 needed.

The readouts agree: an alias frequency of 1 Hz, a Nyquist frequency of 5 Hz, an apparent rotation of −1 Hz, backwards, a Nyquist rate of 18 Hz and 1.111 samples per cycle. Eleven samples fit the 1 s window, and the dashed sine through them completes exactly one cycle, falling first where a sine that starts at zero would rise.

The Nyquist frequency and the sampling theorem

The sampling theorem says that a signal with no frequency above B is fixed completely by samples taken more than 2B times a second, and can be rebuilt exactly from them. Two names come out of it that are easy to mix up. The Nyquist frequency, fs/2, belongs to the sampler: it is the highest frequency a given rate can record. The Nyquist rate, 2B, belongs to the signal: it is the rate a sampler has to beat to record it. A CD samples at 44.1 kHz, so its Nyquist frequency is 22.05 kHz, and music that stops at 20 kHz has a Nyquist rate of 40 kHz.

The inequality is strict, and the Exactly at Nyquist preset shows why. A 5 Hz sine sampled at 10 Hz is caught twice a cycle, and every sample is sin(πn + φ) = (−1)ⁿ sin φ. With a phase of 0 the samples all land on zero crossings and the tone vanishes completely. At 90° they land on the peaks, alternating +1 and −1, and the tone survives. At 30° they are ±0.5, so the reconstruction returns a tone of half the amplitude. Exactly twice a cycle, the samples cannot tell amplitude from phase.

Just below the limit the samples suffice in principle but look strange: at 4.9 Hz and 10 Hz they alternate in sign under an envelope that swells and fades every 5 seconds, a slow beat at fs − 2f = 0.2 Hz, the gap between 4.9 Hz and its image at 5.1 Hz, and only a long run of them rebuilds the tone.

Why aliases fold back

The folding diagram under the stage shows the pattern for every frequency at once. The alias rises with the signal frequency up to fs/2, falls back to 0 at fs, rises again to fs/2 at 1.5 fs, and so on, like a strip of paper folded at every multiple of fs/2. That is where the Nyquist frequency’s other name, the folding frequency, comes from. Frequencies at the same height on the zigzag are indistinguishable once sampled: at 10 Hz, tones of 4, 6, 14 and 16 Hz all come out at 4 Hz.

The spectrum strip shows the same thing from the other side. Sampling a tone at f puts copies of it at every k fs ± f, and for 9 Hz at 10 Hz the strip up to 20 Hz holds lines at 1, 9, 11 and 19 Hz. A reconstruction filter passes only the band from 0 to fs/2, and exactly one copy always lands in it, which is the alias.

The CD audio preset makes it concrete. A 30 kHz tone is far above hearing, but sampled at 44.1 kHz it lands at |30 − 1 × 44.1| = 14.1 kHz, well inside the audible band, as a whistle that was never in the room. CD audio covers sound up to 20 kHz, which leaves the 2.05 kHz between 20 kHz and the 22.05 kHz Nyquist frequency for a filter to remove everything higher before the sampler sees it.

The wagon wheel effect

Film is sampling in time. A camera takes 24 frames a second, and a wheel turning 22 times a second moves 22/24 = 0.917 of a turn between frames. The eye joins each frame to the next by the shortest move that fits, 0.917 − 1 = −0.083 of a turn, which is 30° backwards a frame. So on screen the wheel turns backwards twice a second, 22 − 24 = −2 turns a second, while the carriage rolls forwards. At exactly 24 turns a second it stands still, and at 25 it creeps forwards once a second.

The Wagon wheel on film preset loads the 22-turn case. Watch the solid spoke race almost all the way round between frames while the amber dots, and the dashed spoke the camera sees, step back 30° at a time. A real wheel has identical spokes, so its picture repeats several times a turn and the frequency that folds is the spoke rate: a wheel with 12 spokes turning twice a second passes 24 spokes a second and stands still on film.

Controlled, the effect is useful. A stroboscope flashing at the frequency of a vibration freezes it, which is how a timing light reads an engine’s ignition timing. A school ripple tank, like the one in the Wave Interference Simulator, is watched under a stroboscope for the same reason: flashing at the dipper’s frequency catches each wave one cycle on, so the pattern looks frozen, and flashing slightly slower makes it creep forwards.

Two tones that land in the same place

Aliasing can also stack one tone on top of another. Load Two tones that cancel: 3 Hz and 7 Hz of equal strength, both starting at zero, sampled at 10 Hz. The 3 Hz tone is below the 5 Hz Nyquist frequency and stays put, while the 7 Hz tone folds to 7 − 10 = −3 Hz, the same frequency upside down. Every sample is sin(0.6πn) + sin(1.4πn) = 0, so the samples record silence, while between them the signal swings as far as ±2.

That is why aliasing cannot be undone afterwards: once two frequencies share the same samples, no processing can separate them. Everything that prevents aliasing has to happen before the sampler.

Stopping aliasing before it starts

The cure is an anti-aliasing filter: a low-pass filter in front of the sampler that removes everything at and above fs/2, so nothing is left to fold. No real filter cuts off like a wall, so systems sample faster than twice the highest frequency they care about and give the filter room to fall. The Bode Plot and Filter Visualiser shows how gradually a low-pass filter’s gain falls above its corner frequency. A first-order filter loses only 20 dB per decade, so audio converters use filters of much higher order, or oversample and filter digitally before bringing the rate down.

Any instrument that samples can be caught out. A digital oscilloscope lowers its sample rate at slow timebase settings so that the record fits its memory, and a 1 MHz sine sampled 1.1 million times a second is drawn as a clean, steady 100 kHz sine. Changing the timebase is the usual check, because a real signal keeps its frequency when the sample rate changes and an alias does not. For how a scope turns a signal into a trace, see the Oscilloscope Simulator.

What this model leaves out

• Samples that take no time. Each sample here is an instant. A real converter averages over a short aperture or holds each value until the next, which slightly weakens the highest frequencies.
• Exact values and perfect timing. There is no quantisation, where a real converter rounds each sample to one of a fixed number of levels, 65,536 for 16-bit CD audio, and no clock jitter. Both add a little noise.
• Pure tones that last for ever. Real signals have harmonics, and each one folds on its own: a 3 kHz square wave has odd harmonics, and its 15th, at 45 kHz, lands at 0.9 kHz on a 44.1 kHz sampler, below the fundamental itself. The Fourier Series Visualiser builds a square wave from those harmonics one at a time.
• An ideal reconstruction. The recovered sine is what a perfect low-pass filter would return from an endless run of samples. Real filters and short records fall short of it, most of all close to fs/2.

Common mistakes

• Mixing up the Nyquist frequency and the Nyquist rate. One is half the sampling rate and belongs to the sampler. The other is twice the highest frequency in a signal and belongs to the signal.
• Thinking aliasing starts above the sampling rate. It starts above half of it: a 6 Hz tone sampled at 10 Hz comes out at 4 Hz.
• Sampling at exactly twice the frequency. That is the limit, not a safe rate. A 5 Hz tone sampled at 10 Hz can come out as nothing at all.
• Filtering after sampling. By then an alias looks exactly like a real tone. The filter has to come before the sampler, and the highest frequency present sets the rate, harmonics and noise included.
• Believing a slow wobble in logged data. A logger reading a 50 Hz mains hum 49 times a second records a steady 1 Hz oscillation that is not there.