Digital Representations of Sound

Time-Domain Sampled Representation
Sound in the analog world is said to be continuous in both time and amplitude. The sound’s analog amplitude can be measured to an arbitrary degree of accuracy, and measurements can be taken at any point in time. A digital signal is different; the signal is defined only at certain points in time, and the signal may take on only a finite number of values.
To sample a signal means to examine it at some point in time. Sampling usually happens
at equally separated intervals, at a rate called the sample frequency, determined by the
Sampling Theorem. If a signal contains frequency components up to some frequency f, then
the sample frequency must be at lease 2f in order to reconstruct the signal properly. In
practical systems, the sample frequency must be higher than 2f. In the early days of digital
audio, sample frequencies at 44.1 kHz and 48 kHz were adopted to handle the full 20-kHz
range of human hearing. The sample frequency of 32 kHz is also common. In multimedia
systems for the PC market, submultiples of 44.1 (22.05 kHz, 11.025 kHz) are often found.
The highest frequency that can be handled (i.e., one-half the sample rate) is often called the Nyquist frequency.
To quantise a signal means to determine the signal’s value to some arbitrary degree of accuracy. The digital signal is defined only at the times where the vertical bars occur. The height of each vertical bar can take on only certain values, shown by dashed lines, which are sometimes higher and sometimes lower than the original signal. If the height of each bar is translated into a digital number, then the signal is said to be represented by pulse-code modulation, or PCM. Pohlmann gives a good overview of other digital representations of sampled signals. (In the world of digital music, a sampled and quantised signal shown by the vertical bars in figure 1.9 (a) is called simply a sampled signal, to distinguish it from signals synthesised, for example, with frequency modulation; this is discussed below in more detail.)