Sampling is no more than periodic measurement, and it will be shown here that there is no theoretical need for sampling to be audible. Practical equipment may, of course be less than ideal, but, given good engineering practice, the ideal may be approached quite closely.
Audio sampling must be regular, because the process of timebase correction prior to conversion back to analog assumes a regular original process as was shown in Section 1. The sampling process originates with a pulse train which is shown in FGR. 3(a) to be of constant amplitude and period. The audio waveform amplitude-modulates the pulse train in much the same way as the carrier is modulated in an AM radio transmitter. One must be careful to avoid over-modulating the pulse train as shown in (b) and this is achieved by applying a DC offset to the analog waveform so that silence corresponds to a level half-way up the pulses as at (c). Clipping due to any excessive input level will then be symmetrical.
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FGR. 2 Since sampling and quantizing are orthogonal, the order in which they
are performed is not important. In (a) sampling is performed first and the
samples are quantized. This is common in audio convertors. In (b) the analog
input is quantized into an asynchronous binary code. Sampling takes place when
this code is latched on sampling clock edges. This approach is universal in
video convertors.
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FGR. 3 The sampling process requires a constant-amplitude pulse train as shown
in (a). This is amplitude modulated by the waveform to be sampled. If the input
waveform has excessive amplitude or incorrect level, the pulse train clips
as shown in (b). For an audio waveform, the greatest signal level is possible
when an offset of half the pulse amplitude is used to centre the waveform as
shown in (c).
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FGR. 4 (a) Spectrum of sampling pulses. (b) Spectrum of samples. (c) Aliasing
due to sideband overlap. (d) Beat-frequency production. (e) 4x oversampling.
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In the same way that AM radio produces sidebands or images above and below the carrier, sampling also produces sidebands although the carrier is now a pulse train and has an infinite series of harmonics as shown in FGR. 4(a). The sidebands repeat above and below each harmonic of the sampling rate as shown in (b).
The sampled signal can be returned to the continuous-time domain simply by passing it into a low-pass filter. This filter has a frequency response which prevents the images from passing, and only the baseband signal emerges, completely unchanged. If considered in the frequency domain, this filter can be called an anti-image filter; if considered in the time domain it can be called a reconstruction filter.
If an input is supplied having an excessive bandwidth for the sampling rate in use, the sidebands will overlap (FGR. 4(c)) and the result is aliasing, where certain output frequencies are not the same as their input frequencies but instead become difference frequencies (d)). It will be seen from FGR. 4 that aliasing does not occur when the input frequency is equal to or less than half the sampling rate, and this derives the most fundamental rule of sampling, which is that the sampling rate must be at least twice the highest input frequency. Sampling theory is usually attributed to Shannon who applied it to information theory at around the same time as Kotelnikov in Russia. These applications were pre-dated by Whittaker. Despite that it’s often referred to as Nyquist's theorem.
Whilst aliasing has been described above in the frequency domain, it can be described equally well in the time domain. In FGR. 5(a) the sampling rate is obviously adequate to describe the waveform, but at (b) it’s inadequate and aliasing has occurred.
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FGR. 5 In (a) the sampling is adequate to reconstruct the original signal. In
(b) the sampling rate is inadequate, and reconstruction produces the wrong
waveform (dashed). Aliasing has taken place.
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Aliasing is commonly seen on television and in the cinema, owing to the relatively low frame rates used. With a frame rate of 24Hz, a film camera will alias on any object changing at more than 12Hz. Such objects include the spokes of stagecoach wheels. When the spoke-passing frequency reaches 24Hz the wheels appear to stop. Aliasing does, however, have useful applications, including the stroboscope, which makes rotating machinery appear stationary, the sampling oscilloscope, which can display periodic waveforms of much greater frequency than the sweep speed of the tube normally allows and the spectrum analyzer.
One often has no control over the spectrum of input signals and in practice it’s necessary also to have a low-pass filter at the input to prevent aliasing. This anti-aliasing filter prevents frequencies of more than half the sampling rate from reaching the sampling stage.