Aliasing

Aliasing is a common problem in signal processing, both visually and aurally. If you’ve ever watched the spokes on a wheel turn in a movie, you may have wondered why they sometimes appear to go backward or even stand still. The reason for this has to do with how film works. Typically, the illusion of continuity is created by rapidly juxtaposing many still photographs, for instance, \( 24 \) per second. (This number is usually called the frame rate.) In other words, one photograph takes place in every \( \frac{1}{24} \) seconds. Let’s assume that we are photographing a car whose wheels turn at a rate of \( \frac{1}{4} \) revolution every \( \frac{1}{24} \) seconds. Then on film, the car will appear to be moving forward as normal. But suppose instead that the car is traveling three times faster — that is, its wheels are turning at a rate of \( \frac{3}{4} \) revolution every \( \frac{1}{24} \) seconds. When we observe the film, without sufficient visual context, it will actually appear as though the wheels are turning backward \( \frac{1}{4} \) revolution every \( \frac{1}{24} \) seconds. In other words, the car was moving so fast that the frame rate couldn’t keep up. In fact, even if the car had only been moving twice as fast — that is, at a rate of \( \frac{1}{2} \) revolution every \( \frac{1}{24} \) seconds — we wouldn’t be able to distinguish whether its motion was forward or backward on film.

Thus, if we want to accurately capture the motion of a wheel on film, we actually need to choose a frame rate at least twice the rate of the wheel. (This is the essence of the Nyquist-Shannon sampling theorem.) So a rate of \( 24 \) frames per second can accommodate any wheel spinning at a rate slower than \( 12 \) revolutions per second. If we take the average circumference of a tire to be about \( 67 \) inches, then \( 12 \) revolutions per second corresponds to \( 67 \) feet per second, or approximately \( 45 \) mph. Hence, any motion in excess of \( 45 \) mph may appear distorted by aliasing without a faster frame rate.

All these arguments apply to audio processing as well! In digital signal processing, a continuous sound is sampled by gathering rapid data about discrete instants of the sound, similar to taking rapid photographs of a visual event. The sampling rate (analogous to the frame rate) indicates just how much data are gathered in one second. For example, one common choice of sampling rate is \( 44100 \) samples per second. But if the sampling rate does not exceed twice the frequency of a signal, then the signal may be subject to aliasing. This results in low-frequency signals which were not present in the analog sound manifesting themselves in the digital version. For example, a tone with a frequency of \( 43660 \) Hz could be heard instead as a tone with frequency \( 44100 – 43660 = 440 \) Hz, a concert A.

While aliasing is usually a drawback of digital processing that engineers try to avoid, it sometimes has an interesting sound. Here’s a quick little track that demonstrates the concept:

The program is written so that the frequency slides perpetually higher; however, you will eventually start hearing a quasi-random scattering of different pitches due to aliasing.

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