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5. Sampling issues

5.3 Frequency range

Where the dynamic range of a signal chain in a networked audio system is mainly determined by the bit depth, the frequency range is mainly determined by the sampling rate.

The process of sampling an audio signal every sample interval is visualised in figure 505: the waveform of the audio signal is chopped into discrete samples with a high sample rate (figure 505a) and a low sample rate (figure 505b). It can be assumed that for high frequencies to be reproduced by a sampling system, a high sample rate has to be used. However, a high sample rate implicates that electronic circuits have to be designed to operate with high frequencies, the data transfer and DSP involve a high data rate (bandwidth), and storage requires a high storage capacity. For efficiency reasons, it is necessary to determine exactly how high a sampling rate must be to capture a certain frequency bandwidth of an audio signal, without too much costs for the design of electronic circuits, digital transport, DSP and storage.

To determine the minimum sampling rate to capture an audio signal’s full bandwidth, figure 506 presents the sampling process both in the time domain and in the frequency domain. The most important observation when looking at the process in the frequency domain is that the sampling signal is a pulse train with a fundamental harmonic at the sampling frequency, and further odd harmonics at 3, 5, 7 etc. times the fundamental frequency. Multiplying the audio waveform with the sample pulse train results in the original waveform, plus sum and difference artefacts around every harmonic of the sample waveform. (Figure 506 only shows the 1st and 3rd harmonic).

In figure 506c, the human auditory system’s frequency limit is indicated with the dotted line B. If the difference artefact of the 1st harmonic of the sample pulse train at the sample rate fs lies completely above B, then all artefacts fall outside of the hearing range - so the reproduction of the audio signal is accurate inside the boundaries of the audio universe. However, if either the audio signal has a full bandwidth (B) and fs falls below 2B (figure 507a), or if fs is twice the full bandwith B and the audio signal includes frequency components higher than B (figure 507b), the reproduction is no longer accurate because the difference artefacts of the first harmonic fall in the audible range. This phenomenon is called aliasing.

The conclusion is that 1) fs must be at least twice the frequency limit of the human auditory system (20 kHz) to be able to reproduce an accurate audio signal, and 2) the bandwidth of the audio signal must be less than half of fs . The second conclusion is called the Nyquist-Shannon sampling theorem(*5O).

The first conclusion leads to the system design parameter of selecting a sample frequency of at least 40 kHz to allow an accurate representation of signals up to the hearing limit of 20 kHz. But the second conclusion makes it very difficult to do so, because most audio signals that come into a digital audio system - eg. the output of a microphone - possess frequency components above 20 kHz. With a sampling frequency of 40 kHz, the frequencies above 20 kHz must be attenuated by an analogue anti aliasing (low pass) filter with a brickwall slope to prevent them from entering the audible hearing range.

In the analogue world, its impossible to design a brickwall filter with infinitive roll-off. The solution is to select a slightly higher sampling frequency to allow a less steep roll-off of the low pass filter. But still, a high roll-off slope creates phase shifts in the high frequency range - so to ensure phase linearity in the analogue anti aliasing filter, the sampling rate should be as high as possible. With the conception of the CD standard, Philips and Sony settled on a 44.1 kHz sampling rate as the optimal compromise between analogue filter design and CD storage capacity - leaving only 2 kHz for the roll-off, so the analogue filters on the first CD players where very tight designs. For broadcast and live applications, 48 kHz became the standard, allowing for slightly less tight anti aliasing filter designs. Some years later, with the availability of faster digital processing, the sampling rate could be set to a much higher level to allow much simpler analogue filters to be used - the processing to prevent aliasing could then be performed in the digital domain. This concept is called ‘oversampling’, now used by virtually all manufacturers of A/D and D/A converters(*5P).

All professional digital audio systems on the market today are capable of using a sample rate of at least 44.1 kHz with oversampling A/D and D/A converters, ensuring accurate full bandwidth conversion and processing of audio signals.

>>5.4 Timing issues

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