Simple Visual Demonstration of the Aliasing Effect

In many technical domains such as audio or control systems engineering, an analogue signal representing a physical quantity has to be sampled in order to permit a digital representation; this sampling process can produce aliasing effects if some principles are not observed. The theory behind discrete-time sampling and the aliasing effect is generally taught in engineering courses and requires some mathematical basics, e.g. the Nyquist–Shannon sampling theorem. At times, however, I find myself in a situation where I need to explain to colleagues or friends such concepts without using mathematics. The purpose of this short story is an attempt to explain aliasing and the importance of sampling signals at least above a certain frequency, the Nyquist frequency, in order to avoid aliasing.

Event sequence sampled above (top) and below (bottom) Nyquist frequency; the latter creating aliasing.

Suppose you are watching a person bouncing up and down on a trampoline at a certain frequency, say 1Hz, once a second. Further suppose, you are looking through a shutter that opens briefly 8 times a second (at 8Hz = 8 x the bouncing frequency of 1Hz, or 8 samples within the bouncing period of one second). What would you see? You would see the person at different positions over a one second cycle allowing you to follow the motion.

Person bouncing at 1Hz shown for 3 cycles, sampled at 8 Hz: motion at 1Hz can be seen.

Now imagine, the shutter opens four times a second (4Hz = 4 x the bouncing frequency of 1Hz, or 4 samples within the bouncing period of one second), what would you see? Most likely, you would see the person in four distinct positions, say up, middle, and down, up, still allowing you to infer the complete motion.

Person bouncing at 1Hz shown for 3 cycles, sampled at 4 Hz: motion at 1Hz can still be seen.

Now imagine, the shutter opens exactly twice a second (2Hz = 2 x the bouncing frequency of 1Hz, or 2 samples within the bouncing period of one second), what would you see? Most likely, you would see the person in two distinct positions, say up and down, still allowing you to infer the motion (depending whereabouts in the cycle you catch the person).

Person bouncing at 1Hz shown for 3 cycles, sampled at 2 Hz : motion at 1Hz can just about be seen.

What if the shutter opened say 4 times in 3 seconds (4/3 Hz = 4/3 x the bouncing frequency of 1Hz, or 4 samples within 3 times the bouncing period of one second, i.e. in three seconds)? You would start to see a impossibly slow motion, as an artefact, as if the gravitational constant had changed or you were watching somebody bouncing on Mars. The apparent motion happens at 1/3Hz, a result of aliasing.

Person bouncing at 1Hz shown for 3 cycles, sampled at 4/3Hz < 2Hz: artificial motion is produced.

What if the shutter opened only once a second, (1Hz = 1 x the bouncing frequency of 1Hz, or 1sample within the bouncing period of one second)? You would see the person as if by magic held in a constant position, completely defying gravity (the position will depend whereabouts in the cycle you catch the person). The apparent motion happens at 0Hz, again a result of aliasing.

Person bouncing at 1Hz shown for 3 cycles, sampled at 1Hz < 2Hz: no motion at all.

Without going into further details, the shutter needs to open at least twice per bouncing period; in other words, the sampling frequency has to be at least twice the bouncing frequency. More generally speaking, the sampling frequency has to be at least twice the largest frequency found in the physical quantity that has to be sampled. Conversely, if the sampling frequency is below twice the largest frequency found in the physical quantity that has to be sampled, the higher frequencies will be shadowed into lower frequencies; aliasing occurs.