QUESTION #490

# What is it aliasing? When it occurs?

Aliasing is a signal processing term. Aliasing occurs when a system is measured at an insufficient sampling rate. It is perhaps best explained through example.

Imagine a disk (or paper plate) with a dot near the edge. If the disk began rotating at one revolution per minute, you could observe the angular velocity by looking at it. Now close your eyes. If you open your eyes every 15 seconds and observe the dot, you can still measure direction of rotation and speed. Every 30 seconds and it becomes difficult to determine the rotation of the plate (this is the NYQUIST FREQUENCY). Look every 75 seconds and the plate appears to be rotating opposite to its true rotation. This is aliasing. The same thing happens when a digital measurement device does not sample a signal often enough.

The most common example comes from old westerns. If you watch a movie with a stage coach, the wagon wheels appear to move backwards once a certain speed is reached. The wagon wheel is the same as a paper plate from example one with several dots around the perimeter (wheel spokes) instead of one.

Aliasing occurs when you sample a signal (anything which repeats a cycle over time) too slowly (at a frequency comparable to or smaller than the signal being measured), and obtain an incorrect frequency and/or amplitude as a result.

As an example, let's say I have a ball that has three positions: 0 (middle), +1 (up), and -1 (down). Let's say that ball starts at 0 at time=0, goes to +1, then to 0, then to -1, then back to 0, and each of these 4 movements takes 1 second. Since it takes 4 seconds to complete this cycle, the frequency of this motion would be (1 cycle)/(4 seconds) = .25 Hz.

Now let's say you want to measure this frequency, but you only look at the ball once every 5 seconds. At t=0, you see it at 0, at t=5 you see it at 1, at t=10 you see it at 0, at t=15 you see it at -1, and finally at t=20 you see it at zero after what appears to be a complete cycle. You will then say that the frequency appears to be (1 cycle)/(20 seconds) = .05 Hz. But this is wrong!!! Why? Because you weren't looking often enough, and you missed a bunch of the movement, resulting in you 'measuring' a totally different frequency. That's aliasing.

But aliasing can get much more devious than this. What happens if you decided to look every 2 seconds (or 4 or 6 for that matter)? Right! Every time you looked the ball would be at 0, and you'd miss the motion entirely!

Or how about if you look every two seconds, but you start looking half a second after t=0? The first time you look, the ball is halfway up to +1 (say +0.5). The next time you look (at t=2.5), the ball is halfway to the bottom (at -0.5). The third time you look, the ball is back at +0.5, so it looks like you've been through a complete cycle, and you actually measure the frequency correctly in this case (1 cycle / 4 seconds), but it looks like the ball is only going between +0.5 and -0.5. In this case aliasing has caused you to incorrectly measure the amplitude of the motion.

So now that we've seen how much aliasing can mess with our ability to measure things, how do you get around it? Well, when you want to measure a system / signal that is oscillating, the rule of thumb is as follows. First, you figure out what kinds of frequencies you expect to see in your system. If we were going to be measuring the position of a ball that I was physically moving up and down, we definitely wouldn't be worried about it going up and down ten billion times per second (actually we could probably rule out anything larger than like 10 times per second). Then you take the highest frequency you think you could see and make sure you are measuring the system at least 10 times that frequency. (In the example above, say we guessed the fastest I could have moved the ball was 20 cycles per second. We would then measure at least 200 times per second, and we obviously wouldn't have missed anything.)