What is Audio Frequency Aliasing?
Audio Aliasing is an effect which occurs when converting an analogue signal into a digital one with an insufficient sampling frequency.
The result of this effect is that the high-frequency components of that analogue signal will not be correctly interpreted, and the digital signal will not be an accurate copy of the analogue one.
Analogue to Digital conversion.
When analogue signals are digitised and turned into digital signals, the analogue signal is sampled at regularly occurring points in time, or in other words, the instantaneous amplitude of the analogue signal is recorded to create a digital copy of the analogue signal.
This happens very quickly in audio signals, for example, CD audio is sampled at 44.1 kHz (44,100 samples per second).
Aliasing occurs when a signal is sampled at an insufficient rate. Two audio signals can become indistinguishable from each other once they have been sampled and converted– they have become aliases of each other.
Nyquist Theorem.
The Nyquist sampling theorem states that:
“To avoid aliasing, the sampling frequency must be at least twice that of the highest frequency which is to be represented“. If we use the example of CD audio, a sampling frequency of 44.1 kHz means that the highest frequency which can be represented without aliasing is 22.05 kHz. For CD audio this is sufficient as the upper limit of human hearing is around 15 to 20 kHz depending on the individual.
Where Anti-Aliasing can fail.
Aliasing can occur either because the anti-alias filter in the A-D converter (or in a sample-rate converter) doesn’t have a steep enough roll-off, or alternatively because the system has been overloaded. Distortion caused by overloading the input or conversion circuitry is the most common source of aliasing, because overloads result in the generation of multiple high-frequency harmonics within the digital system itself after the anti-aliasing filtering.
Sampling images.
The sampling process is similar to 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‘.
Unwanted Effects.
These images play no part in the digital audio process — they are essentially just a side-effect of sampling. However they must be kept well above the wanted audio frequencies so that they can be removed easily without affecting the quality of the required audio signals. This is where all the can trouble begin. The upper image isn’t really a problem – that’s easily filtered out, but if the lower one is too low in frequency, it will mix with the audio we do want and because the frequencies are similar, this will create ‘aliases‘ that cannot be removed.
Unwanted guests you can’t get rid of.
This is what the aliases turn into… that guest at the party who causes bad feelings and will not leave. Once aliasing effects are there there is no way you can filter them out without causing even more audio degradation.
Example of Aliasing.
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 = (48kHz + 10kHz) and 38kHz = (48kHz – 10kHz). Both of these images are clearly far above half the sample rate, and outside the audible range for humans (24kHz), and be easily removed by a low-pass filter, the “reconstruction filter” on the output of the D-A converter, which leaves the wanted audio (the 10kHz tone) perfectly intact.
However, consider what happens if our 10kHz tone is too high in amplitude, and exceeds the input range of 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 ‘unwanted’ 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.
10kHz Overload in a sampling system.
When the 10kHz signal overloads the A-D converter, there is a strong third harmonic at 30kHz which creates an alias at 18kHz which will be allowed through by the low-pass filter. How does this happen?
Considering just the third harmonic at 30kHz (otherwise things get very complex, very quickly), the sampling modulation process means that this will create ‘mirror‘ signals around the sample rate just as before, generating additional signal frequencies at 78kHz (48kHz + 30kHz) and 18kHz (48kHz – 30kHz). 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. You can see how things get out of hand very, very quickly when D/A or A/D converters are either run at too low a sampling frequency, or have an input that exceeds their maximum input voltage.
Note that, unlike an analogue system, in which the distortion products caused by overloads always follow a normal harmonic series, and can even give quite a pleasant sound, (consider tape saturation on an old reel to reel recorder, or soft clipping in a valve amplifier) overloading, or incorrect clock frequencies in a digital system aliasing result in the harmonic series being “folded back or mirrored” on itself to produce audible signals that are no longer harmonically related to the source (they are referred to as “Inharmonics”).
In this very basic example, 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.