# Problem Involving Aliasing and Critical Fusion Frequency Concepts

I have been trying to study a course from MIT on Biomedical Signal Processing and I thought that the best way to learn would be to solve some problems. Although they might be out of my depth, I am not looking for so much as quick answers as ways to understand the problem and how to remedy my knowledge gaps in this area. Specifically, I would like to solve this problem:

If you have ever seen a Western, you must have noticed that, as the stagecoach changes
speed, the wheel appears to reverse direction even though the coach moves forward. This
is an example of aliasing. The sampling mechanism is the movie camera capturing images
at a frequency of 24 frames/second. The eye acts as a lowpass filter, in the sense that
successive images reaching the eye at a rate exceeding 10 images/second appear as if they
are superimposed. This is called the critical fusion frequency.
Answer the following questions assuming that the wagon wheel is 1.6 m in diameter and has
12 spokes.

(a) Express the true angular velocity of the wheel (in radians/second) as a function of the
coach velocity.

(b) At what set of coach velocities does the wheel appear to reverse direction?

(c) Determine an expression for the apparent angular velocity of the wheel as a function of
the coach velocity.

(d) How many spokes does the wheel appear to have the first time it appears to reverse
direction? The second time? Explain the difference between these two cases.


I think I understand aliasing, whereby because of the low frequency of the sampling-rate it causes the output signal to be of lower frequency than it actually is, and in order to solve this problem a low-pass filter is required. However, I am struggling to begin to tackle this problem as I am stuck at the first question of expressing the true angular velocity of the wheel as a function of coach velocity.

Therefore, I would be much obliged if someone could provide me with some hints or some sort of a guide as to how I should tackle these problems.

Edit: I think I overthought (a) and it looked to be pretty simple. So that would mean angular velocity of the wheel is simply the velocity of the coach divided by the radius of the wheel which in this case is 0.8 m.

• Hi! Show some efforts please... What's your attempt ? – Fat32 Nov 27 '19 at 19:56
• Part (a) is a story problem from high-school algebra, if that helps. Only they left out a critical parameter, which you will need to assume (hint, if the wheel diameter just happens, totally by random chance to be something convenient divided by $\pi$, the arithmetic will be simpler). – TimWescott Nov 27 '19 at 20:18