Digital sampling
Digital sampling, PCM sampling, or just sampling is the process of representing a signal waveform as a series of numbers which represent the measurement of the sound's amplitude, taken at regular intervals. This process, also commonly referred to as PCM is widely used in modern audio and video systems, including television and telephone networks.
Strictly speaking, the process of sampling must be regarded as separate from the process of digitising. Sampling produces a series of values which may be represented in various ways - the output from the process can be a series of analog pulses (Pulse-height modulation) or a series of fixed amplitude pulses (Pulse position modulation or Pulse-width modulation. Most commonly though, the samples are represented by binary numbers, in a process known as PCM, an acronym for Pulse-code modulation, because they are then amenable to storage and processing in digital systems such as computers.
The basic theory of digital sampling in relation to audio and video is widely misrepresented, based quite simply on the totally wrong idea that the samples are the signal. This misconception is understandable, given that it is indeed possible to listen to digital samples directly (as is done in some cheap players) or to view video samples directly (as is done in most standard (non-HD) televisions) but this must be regarded as a 'cheap and cheereful' approach, and misses out a vital component of basic sampling theory - the Reconstruction filter.
While it is obvious to anyone that sampling an origianl waveform and then presenting the sample values as joined-up segments will produce an approximation to the original, especially if the original only changes slowly between samples, this is not what sampling is really about.
What Nyquist realised was that if an original signal is filtered (band-limited) to remove all frequencies above what we call the Nyquist frequency, then it is possible to reproduce the exact (band-limited) waveform by processing the samples in a 'Reconstruction filter which is simply another low-pass filter with a cut-off frequency uqual to the Nyquist frequency. There is no approximation, no distortion, what goes in comes out, apart from any components above the Nyquist frequency.
Errors resulting from the Nyquist limitation
This is only literally true if the two filters employed are 'Brick-wall filters, in other words they cut off totally above the Nyquist frequency. Even if such filters were realisable in practice basic theory says that they would have infinite delay - they would take forever to produce any output. This must not be seen as an obstacle to perfect reproduction though. By designing with a 'Guard band it is possible to use imperfect filters to obtain output that is as accurate as we care to make it (within the bandwidth limitation).
Quantising errors resulting from the process of digitisation
In digital sampling, the accuracy of the resulting waveform is also affected by the stepwise nature of the digitising process, resulting in what is referred to as 'Quantisation error. This error, which occurs from sample to sample, is not necessarily random, but may be correlated with the signal, producing serious audible distortion in audio systems that do not take steps to eliminate it. Some early CD's suffered from Quantising distortion which was especially audible on quiet piano notes, adding a granular noise that sounded like 'sand in the speakers'. It could also be heard as spurious tones accompanying higher frequencies. Quantising distortion soon became a thing of the past though, with a better understanding of the process of 'Dither' which involved adding a low level of noise to the signal before sampling in order to randomise the individual sample errors and hence 'de-correlate' the resultant errors from the signal, so that all that was heard was noise (hiss).
Digital sampling in Audio
Dither essential on 16-bit, not needed on 24-bit. Seminal paper by Lipshitz. Gaussian vs TPDF dither. Noise shaping. Digital dither needed in converting eg 24 to 16-bit.
The myth of 'superaudio' 96k and 192k not yet shown to be necessary. 16-bit gives considerably lower noise than the ambient and mic noise that can be acheived on almost all practical recordings (-66 to -68dB ITU-R 468 weighted).
Digital sampling in Video
Dither not needed - contouring tolerable. 12-bit for best results
- Work in progress