Data, Logic, and Computing: Winter 2024

Temporal Aliasing: The Wagon Wheel Effect

The wagon-wheel effect (also called the stroboscopic effect) is an optical illusion in which a wheel appears to rotate differently from its true rotation. The wheel can appear to rotate more slowly than the true rotation, it can appear stationary, or it can appear to rotate in the opposite direction from the true rotation.

Such an effect may be seen as just an illusion, but it should really be considered as a warning: if we do not know the actual physics of a phenomenon, sampling such a phenomenon with the wrong frequency may lead to misinterpretation: the discrete data are often not enough to capture the physics.

Illustration: a simple clock

Let us consider a simple clock with on red hand that indicates minutes. This hand behaves as expected, turning "clockwise", performing a complete rotation in 60 minutes, see below (hover on the image to generate the animation).

Now consider that you are tasked with generating a movie of the motion of the clock handle that can be stored on a computer. This amounts to taking pictures of the clock while the handle rotates, at fixed time intervals. The number of pictures you take per second defines the sampling rate in Hz. What should this sampling be? Does it matter? In the three images below, I show the resulting discrete "movies", when the time interval is 5 minutes (i.e. you take a picture every five minutes), 30 minutes, and 55 minutes, respectively. Hover on the images to see the resulting "movies".

Clearly, the choice of the time interval between each frame is not harmless! When this time interval is 5 minutes, the motion of the red hand is correctly captured, albeit a little jerky. When we take pictures every 55 minutes only, the hand seems to move counter-clockwise. If we did not know the actual behavior of a clock and this movie was the only information we had, we would not know that it is an artifact.

The Shannon-Nyquist theorem (often referred as the Nyquist theorem) tells us that an appropriate sampling rate has to be strictly greater than twice the largest frequency in the signal we are measuring. For the clock above, the hand moves at a speed of 1 cycle every 60 minutes; this means that we ought to capture its action at a rate strictly greater than 2 frames per 60 minutes. 5 minutes work, but 55 minutes would not: this is exactly what we observe. Note that if we take a rate of exactly 2 cycles per 60 minutes, i.e. if we take images avery 30 minutes, this would still not work correctly, as observed above (the motion of the handle is then ambiguous, as we are not sure if it moves in a clockwise or anti clockwise manner).