Q. What is 'aliasing' and what's the cause of it?

Published in SOS January 2006
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With reference to A-D/D-A converters, what exactly is an 'alias'? How and when do they occur?

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Technical Editor Hugh Robjohns replies: An alias occurs when a signal above half the sample rate is allowed into, or created within, a digital system. It's the anti-aliasing filter's job to limit the frequency range of the analogue signal prior to A-D conversion, so that the maximum frequency does not exceed half the sampling rate — the so-called Nyquist limit.

qa B1 A-D diag 1.s
Figure 1: The D-A converter's low-pass filter, set at half the sample rate, removes the upper and lower images while keeping the wanted audio.
Figure 1: The D-A converter's low-pass filter, set at half the sample rate, removes the upper and lower images while keeping the wanted audio.

qa B2 A-D diag 2.s
Figure 2: When the 10kHz signal overloads the A-D converter, the resulting third harmonic at 30kHz creates an alias at 18kHz which will be allowed through by the low-pass filter.
Figure 2: When the 10kHz signal overloads the A-D converter, the resulting third harmonic at 30kHz creates an alias at 18kHz which will be allowed through by the low-pass filter.

Aliasing can occur either because the anti-alias filter in the A-D converter (or in a sample-rate converter) isn't very good, or because the system has been overloaded. The latter case is the most common source of aliasing, because overloads result in the generation of high-frequency harmonics within the digital system itself (and after the anti-aliasing filter).

The sampling process is a form of amplitude modulation in which the input signal frequencies are added to and subtracted from the sample-rate frequency. In radio terms, the sum products are called the upper sideband and the subtracted products are called the lower sideband. In digital circles they are just referred to as the 'images'.

These images play no part in the digital audio process — they are essentially just a side-effect of sampling — but they must be kept well above the wanted audio frequencies so that they can be removed easily without affecting the wanted audio signal. This is where all the trouble starts. The upper image isn't really a problem, but if the lower one is allowed too low, it will overlap the wanted audio band and create 'aliases' that cannot be removed.

Let's consider what occurs if we put a 10kHz sine-wave tone into a 48kHz sampled digital system. The sampling process will generate additional signal frequencies at 58kHz (48 + 10) and 38kHz (48 - 10). Both of these images are clearly far above half the sample rate (24kHz), so can be easily removed with a low-pass filter, which is the reconstruction filter on the output of the D-A converter, leaving the wanted audio (the 10kHz tone) perfectly intact. See Figure 1, above.

However, consider what happens if our 10kHz tone is cranked up too loud and overloads the A-D converter's quantising stage. If you clip a sine wave, you end up with something approximating a square wave, and the resulting distortion means that a chain of odd harmonics will be generated above the fundamental. So our original 10kHz sine wave has now acquired an unwanted series of strong harmonics at 30kHz, 50kHz and so on.

Note that these harmonics were generated in the overloaded quantiser and after the input anti-aliasing filter that was put there to stop anything above half the sample rate getting in to the system. By overloading the converter, we have generated 'illegal' high-frequency signals inside the system itself and, clearly, overloading the quantiser breaks the Nyquist rule of not allowing anything over half the sample rate into the system.

Considering just the third harmonic at 30kHz for the moment, the sampling modulation process means that this will 'mirror' around the sample rate just as before, generating additional signal frequencies at 78kHz (48 + 30) and 18kHz (48 - 30). The 18kHz product is clearly below half the sample rate, and so will be allowed through by the reconstruction filter. This is the 'alias'. We started with a 10kHz signal, and have ended up with both 10kHz and 18kHz (see Figure 2, above). Similarly, the 50kHz harmonic will produce a 2kHz frequency, resulting in another alias.

Note that, unlike an analogue system, in which the distortion products caused by overloads always follow a normal harmonic series, in a digital system aliasing results in the harmonic series being 'folded back' on itself to produce audible signals that are no longer harmonically related to the source.

In the simplistic example I've explained, we have ended up with aliases at 2kHz and 18kHz that have no obvious musical relationship to the 10kHz source. This is why overloading a digital system sounds so nasty in comparison to overloading an analogue system.

I hope this brief explanation helps to clear up the topic of aliasing for you. 

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