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Oversampling is the process of sampling a signal at a frequency higher than the target sampling frequency (which is usually the Nyquist frequency). Usually, oversampling is employed to assist in anti-aliasing.

In digital audio recording and playback, oversampling is used to compensate for the lack of good analog low-pass filters. In essence, this allows us to improve the quality of the filtering by augmenting a low-quality analog filter with a high-quality digital filter. It's much easier to make a good digital filter than to make a good analog filter.

Oversampling for A/D Conversion

Here is a concrete example of how oversampling can help with digitization (analog-to-digital conversion). Suppose you want the final sampling rate to be 48kHz. This means that, according to Shannon's Sampling Theorem, the input analog signal must be bandlimited to 24kHz. This is usually achieved by applying an analog low-pass filter to the analog signal to eliminate all frequency components above 24kHz.

Ideally, this filter should be a sinc function. In the frequency domain, the filter is a box function -- the value of the filter is 1 in the range of 0kHz to 24kHz (the pass band), a sudden jump to 0 at 24kHz, and 0 thereafter. However, in practice, it's impossible to create this ideal filter. Therefore, we have to do with an approximation -- an approximation which attenuates some frequencies below 24kHz while not completely eliminating those above 24kHz. Instead of a sudden jump from 1 to 0 at 24kHz, the filter has a gradual slope from 1 to 0 centered around that frequency (this range of frequencies is called the transition band).

Here's where oversampling is useful. We push up the cutoff frequency of the analog filter above 24kHz -- say, to 48kHz. This way, even with the imperfect analog filter we have, no frequency below 24kHz is attenuated and all frequencies above 72kHz are completely eliminated. We then oversample the signal at 96kHz. Frequencies in the range 48kHz to 72kHz are folded into the frequency range 24kHz to 48kHz, but this is OK. Now we filter that digitized signal again -- using a digital filter this time. The digital filter serves two purposes: (1) to filter out all frequencies above 24kHz, and (2) to reduce the sampling frequency to 48kHz (resampling). In effect, the digital filter takes a weighted average of consecutive samples. There's no aliasing because the final sampling rate is the Nyquist frequency. Now we have the signal digitized at 48kHz like we wanted.

Oversampling for D/A Conversion

Oversampling is also useful for the reverse process of digital-to-analog conversion. This process is the exact mirror of oversampling analog-to-digital conversion. First, remember from signal processing theory that the sampled 48kHz digital signal has aliases centered at frequencies which are integer multiples of 48Hz. To convert this back into an analog signal, we have to get rid of those aliases with a low-pass analog filter. Again, ideally we would like a perfect sinc filter which eliminates frequencies above 24kHz, but we don't have one. Oversampling comes to the rescue.

First, the 48kHz digital signal is interpolated by using another digital filter to a sampling frequency of 96kHz. This moves the first alias image to be centered at 96kHz instead of at 48kHz. Since the relevant signal is bandlimited to 24kHz, there are no frequency components in the 24kHz to 72kHz range. Now we can use the analog filter we used in the A/D conversion to get rid of the components above 72kHz.

Oversampling and Quantization Noise

Oversampling is also useful in dealing with another artifact of digitization: quantization noise. Consider a non-oversampled digitization at 48kHz. It is easy to show that quantization noise shows up as white noise in the frequency region from 0kHz to 24kHz. If you oversample and digitize at 96kHz, the total amount of noise doesn't change, but it is spread out in the region from 0kHz to 48kHz. The noise in the range 0kHz to 24kHz is reduced by a factor of 2. The digital decimation filter eliminates the noise in the 24kHz to 48kHz range. In general, a factor of 2 in the oversampling rate leads to a factor of 2 in reduction of the quantization noise in the digitized signal.

More sophisticated implementations employ the technique of noise shaping. In this technique, the frequency distribution of the quantization noise is modified so that more of the power is distributed in a higher frequency range. That high-frequency noise can then be eliminated with a low-pass digital filter. See the writeup on sigma-delta ADC for a concrete application of this technique.

Related Ideas

A related concept in computer graphics is called supersampling. Unlike oversampling, supersampling does not involve analog filtering or quantization.

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