We have seen how, before we sample a signal, it is necessary to apply a filter to it (which in the case of audio signals has a 20 KHz bandwidth). This ensures that aliasing frequencies aren't present when the analogue signal is reconstructed from the samples. Ideally this filter should be a rectangle and therefore it ought to have infinite slopes [Slope rate ] on its two faces in order to avoid including too many frequencies that go beyond the 20 KHz mark in the filtered signal. But, as we have just seen, this isn't possible in the real world, therefore it will have a slope that is as high as possible, but not infinite.

This last fact has various implications. The first is that it is undoubtedly more expensive to create a filter with such a steep slope. The second is a physical implication. To really get to grips with it, let's turn to an empirical description of the phenomenon. Let's imagine a signal as being made up of electrons that are brought to a halt by the low-pass filter if their speed becomes excessive (if the signal's frequency exceeds the filter's cut-frequency). The impact that occurs between the electrons and such a steep barrier could generate chaotic scattering of the electrons, which in the audible band will be perceived as hissing at high frequencies. The solution consists in using filters that have slopes that are not as steep, so that the electrons may meet a softer front on impact with the filter. However, a softer front inevitably moves the cut-frequency towards the right thus including frequencies that are outside the audible band into the signal (and we'd be "back to square one" with the aliasing-frequency problems!). At this point we'd turn to oversampling, namely, the audio signal gets sampled at a frequency that is higher than the classic 44.1 KHz frequency. In the following diagram we can observe how a filter applied to a signal that has been sampled normally compares with one that has been oversampled:

Notice how a 20 KHz band-signal is filtered by a filter with a 22 KHz cut-frequency and is sampled at a 44.1 KHz frequency. By applying a four-fold oversampling operation, namely, by using a sampling frequency which is 4x44.1KHz equal to 176.4 KHz, we can use an anti-aliasing filter with a far softer slope. This operation, as we can see, results in shifting the hissing phenomenon into a frequency range around the newly formed cut-frequency which is located well beyond the audible threshold. This way the phenomenon still remains, but being outside the ears' audible range, it sounds like it has been removed. The third major implication related to oversampling is the reduction of quantization errors. In this case too, the signal of the band we are dealing is far greater, therefore quantization noise (present all over the signal's band) and which never changes, gets distributed uniformly over the entire new band, which is now larger than the initial signal's band (in four-fold oversampling specifically, the band has an 88.1 KHz frequency, therefore 4 times the audio signal's band). This results in a reduction of the average quantization noise error. More specifically: we get a 3dB reduction of the quantization error for each oversampling octave. In other words, every time we double the sampling frequency (44.1KHz -> 88.2KHz -> 176.4KHz) what we get is a 3dB reduction at the first doubling and of a further 3 dB at the second doubling (clearly the overall reduction is not 6dB; we mustn't forget that dB's are added up and subtracted using logarithmic formulae).

The kind of oversampling we have been looking at so far is carried out on analogue signals. There is however yet a different kind of oversampling called *digital oversampling*, which also results in "spreading" the quantization error over the entire spectrum. This takes place by adding new samples that have been calculated by mathematical interpolation. This means that we can add one (or more) virtual samples in between two real samples, the virtual sample being calculated, for example, as an average of the two. This softens the sampled wave-form, which will have lower steps, as illustrated in the following diagram:

Even though it resolves some of our problems, oversampling still works out to be rather taxing both in terms of the amount of memory needed for storing samples and for the complexity of the necessary circuitry.

Some machines such as ADATs (Alesis Digital Audio Tape - a digital recording system which uses high quality VHS videotape as its support
[ADAT
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) allow for a 128 x 44.1 KHz oversampling. It becomes evident that this entails enormous amounts of memory for memorizing the samples. In this case the samples are so crammed up that only one bit per sample gets stored and it indicates with a 0 or a 1 whether the sample has a larger or smaller amplitude than the previous one. This process is called *decimation* and it guarantees pretty good accuracy in representing the samples without losing the advantages given by the oversampling process.

Example of oversampling