- 1 Introduction to conversion
- 2 Sampling and aliasing
- 3 Reconstruction
- 4 Filter design
- 5 Choice of sampling rate
- 6 Sample and hold
- 7 Sampling clock jitter
- 8 Aperture effect
- 9 Quantizing
- 10 Quantizing error
- 11 Introduction to dither
- 12 Requantizing and digital dither
- 13 Dither techniques
- Rectangular pdf dither
- Triangular pdf dither
- Gaussian pdf dither
- 14 Basic digital-to-analog conversion
- 15 Basic analog-to-digital conversion
- 16 Alternative convertors
- 17 Oversampling
- 18 Oversampling without noise shaping
- 19 Noise shaping
- 20 Noise-shaping ADCs
- 21 A one-bit DAC
- 22 One-bit noise-shaping ADCs
- 23 Operating levels in digital audio
Section 1 introduced the fundamental characteristic of digital audio which is that the quality is independent of the storage or transmission medium and is determined instead by the accuracy of conversion between the analog and digital domains. This section will examine in detail the theory and practice of this critical aspect of digital audio.
Introduction to conversion
Any analog audio source can be characterized by a given useful bandwidth and signal-to-noise ratio. If a well-engineered digital channel having a wider bandwidth and a greater signal-to-noise ratio is put in series with such a source, it is only necessary to set the levels correctly and the analog signal is then subject to no loss of information whatsoever.
The digital clipping level is above the largest analog signal, the digital noise floor is below the inherent noise in the signal and the low- and high frequency response of the digital channel extends beyond the frequencies in the analog signal.
The digital channel is a 'wider window' than the analog signal needs and its extremities cannot be explored by that signal. As a result there is no test known which can reliably tell whether or not the digital system was or was not present, unless, of course, it is deficient in some quantifiable way.
The wider-window effect is obvious on certain Compact Discs which are made from analog master tapes. The CD player faithfully reproduces the tape hiss, dropouts and HF squashing of the analog master, which render the entire CD mastering and reproduction system transparent by comparison.
On the other hand, if an analog source can be found which has a wider window than the digital system, then the digital system will be evident due to either the reduction in bandwidth or the reduction in dynamic range. No analog recorder comes into this category, but certain high quality capacitor microphones can slightly outperform many digital audio systems in dynamic range and considerably outperform the frequency range of human hearing.
The sound conveyed through a digital system travels as a stream of bits. Because the bits are discrete, it is easy to quantify the flow, just by counting the number per second. It is much harder to quantify the amount of information in an analog signal (from a microphone, for example) but if this were done using the same units, it would be possible to decide just what bit rate was necessary to convey that signal without loss of information, i.e. to make the window just wide enough. If a signal can be conveyed without loss of information, and without picking up any unwanted signals on the way, it will have been transmitted perfectly.
The connection between analog signals and information capacity was made by Shannon, in one of the most significant papers in the history of this technology, and those parts which are important for this subject are repeated here. The principles are straightforward, and offer an immediate insight into the relative performances and potentials of different modulation methods, including digitizing.
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FGR. 1 To receive eight different levels in a signal unambiguously, the peak-to-peak
noise must be less than the difference in level. Signal-to-noise ratio must
be at least 8:1 or 18 dB to convey eight levels. This can also be conveyed
by three bits (23 = 8). For 16 levels, SNR would have to be 24 dB, which would
be conveyed by four bits.
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FGR. 1 shows an analog signal with a certain amount of super imposed noise, as is the case for all real audio signals. Noise is defined as a random superimposed signal which is not correlated with the wanted signal. To avoid pitfalls in digital audio, this definition must be adhered to with what initially seems like pedantry. The noise is random, and so the actual voltage of the wanted signal is uncertain; it could be anywhere in the range of the noise amplitude. If the signal amplitude is, for the sake of argument, sixteen times the noise amplitude, it would only be possible to convey sixteen different signal levels unambiguously, because the levels have to be sufficiently different that noise will not make one look like another. It is possible to convey sixteen different levels in all combinations of four data bits, and so the connection between the analog and quantized domains is established.
The choice of sampling rate (the rate at which the signal voltage must be examined to convey the information in a changing signal) is important in any system; if it is too low, the signal will be degraded, and if it is too high, the number of samples to be recorded will rise unnecessarily, as will the cost of the system. Here it will be established just what sampling rate is necessary in a given situation, initially in theory, then taking into account practical restrictions. By multiplying the number of bits needed to express the signal voltage by the rate at which the process must be updated, the bit rate of the digital data stream resulting from a particular analog signal can be determined.
There are a number of ways in which an audio waveform can be digitally represented, but the most useful and therefore common is pulse code modulation or PCM which was introduced in Section 1. The input is a continuous-time, continuous-voltage waveform, and this is converted into a discrete-time, discrete-voltage format by a combination of sampling and quantizing. As these two processes are orthogonal (a 64-dollar word for at right angles to one another) they are totally independent and can be performed in either order. FGR. 2(a) shows an analog sampler preceding a quantizer, whereas (b) shows an asynchronous quantizer preceding a digital sampler. Ideally, both will give the same results; in practice each has different advantages and suffers from different deficiencies. Both approaches will be found in real equipment.
The independence of sampling and quantizing allows each to be discussed quite separately in some detail, prior to combining the processes for a full understanding of conversion.