The Wagon-wheel Effect, Stroboscopic Effects and Aliasing

Sometimes in movies one sees a stagecoach rambling through the countryside with some notorious highwayman (such as Dick Turpin) in hot pursuit.  The wheels appear to be going in the opposite direction at a very slow speed or not turning at all.  Images of any rotating object, including helicopter rotors or aircraft propellers, show this.  This phenomenon was mentioned one morning by YARS members during coffee and I decided to write an article about it.  One of our members mentioned a "stroboscopic effect" where, with a strobe light flashes at a sampling rate close to the period of motion causes this effect.  For example if a light flashes at 60 flashes per second (60 Hz) and the rotating object is rotating 60 revolutions per second (60 Hz), then the object will appear to be stationary.  If the object rotates at 61 revolutions per second, it will appear to slowly appear to move backwards.  At 50 revolutions per second it will appear to rotate slowly forwards. 

This effect can be used to measure the speed of moving objects simply by observing them.  It can also lead to irritation by persons sensitive to the effects, sometimes called "flicker".  This has nothing to do with a phenomenon called "rolling shutter" - which is merely a time delay artefact in a digital camera caused by how the camera scans an object over time, line by line (see https://www.youtube.com/watch?v=dNVtMmLlnoE ). 

  

Signal sampling representation.  The continuous or analog signal is sampled with discrete samples (as happens with an Analog to Digital Filter)

In signal processing, this effect is commonly known as "aliasing".  Aliasing causes different signals to become indistinguishable when sampled.  Sampling is the reduction of a continuous-time signal to a discrete time signal.  The set of samples is passed through a reconstruction filter to retrieve the original sample from the sampled set of data.  Sampling of a continuous signal is accomplished through an analog-to-digital converter (ADC).  Deviations from a perfect reconstruction are referred to as distortions.  Many things can cause distortion (aperture error, jitter, noise, slew rate limit error etc.) but the type of distortion of interest here is aliasing.

This animation shows the sampling of a sinusoid that is increasing in frequency.  Note when f=0.5fs the sampling is reconstructed frequency in the Fourier Transform plot (lower right image) matches the one above - reconstruction is good.
A Moiré Pattern created by two moving circles.  The constructive and destructive interference is creating an illusion thanks to aliasing

One effect of aliasing is seen in insufficient resolution and the generation of a Moiré pattern - a large scale interference pattern.  Images might seem counter-intuitive to signals, but consider them as just a two dimensional signal.  

Historically the term aliasing evolved from radio engineering because of the action of super-heterodyne receivers. When the receiver shifts multiple signals down to lower frequencies, from RF to IF by heterodyning, an unwanted signal, from an RF frequency equally far from the local oscillator (LO) frequency as the desired signal, but on the wrong side of the LO, can end up at the same IF frequency as the wanted one. If it is strong enough it can interfere with reception of the desired signal. This unwanted signal is known as an image or alias of the desired signal.

Example of Nyquist frequency and Nyquist rate.  

In signal processing there is something called the Nyquist frequency (in Hz).  It is a characteristic of a sampler.  It is one half the sampling rate (samples per second).  When the highest frequency (bandwidth) of a signal is less than the Nyquist frequency of the sampler, the resulting discrete-time sequence is said to be free of the distortion known as aliasing and the corresponding sample rate is said to be above the Nyquist rate for that particular signal. 

Frequency Domain Representation of Aliasing.  Solid red line is example of amplitude varying with frequency.  Red dashed lines are corresponding paths of aliases.

For example, in a typical sampling one chooses the highest frequency to be preserved and recreated based on expected content (voice, music etc.) and desired fidelity.  One inserts an anti-aliasing filter ahead of the sampler  The filter attenuates frequencies above that limit.  Finally one chooses a sampling rate (and Nyquist frequency) that provides an acceptable amount of aliasing.  In audio CDs, they are sampled at 44100 samples/second.  The Nyquist frequency is 22050 Hz.  The anti-aliasing filter must suppress higher frequencies but not affect frequencies in the audible hearing range for humans (0-20000 Hz or 0-20kHz).

 

Sources:

https://en.wikipedia.org/wiki/Aliasing 

https://en.wikipedia.org/wiki/Sampling_(signal_processing) 

https://en.wikipedia.org/wiki/Wagon-wheel_effect

https://michaelbach.de/ot/mot-wagonWheel/index.html

https://en.wikipedia.org/wiki/Nyquist%E2%80%93Shannon_sampling_theorem (mathematical foundations)