Computer representation of sound

Sound is a continuous analogue pressure wave, so it must be converted into a sequence of numbers before a computer can store it. This conversion is called digitisation or pulse code modulation (PCM) and has three steps: sampling, quantisation and coding.

 amplitude
   |    .--.        .--.
   |  .'    '.    .'    '.      o = samples taken at
   | o        o  o        o     equal time intervals
   +----|----|----|----|----> time
         T = 1/sampling rate

1. Sampling

The amplitude of the wave is measured at regular time intervals. The number of samples per second is the sampling rate (Hz). By the Nyquist theorem the sampling rate must be at least twice the highest frequency in the signal. Human hearing reaches about 20 kHz, so CD audio uses 44.1 kHz; telephone speech (up to about 4 kHz) uses 8 kHz.

2. Quantisation

Each sample is rounded to the nearest of a fixed number of levels. The number of bits per sample (resolution) decides the levels: 8 bits give 2^8 = 256 levels, 16 bits give 65,536 levels. The rounding causes quantisation noise, which falls as bits increase.

3. Coding

The quantised values are written as binary numbers (and optionally compressed) in a file such as WAV.

Example

Record 1 second of CD-quality stereo sound: 44,100 samples/s x 16 bits x 2 channels = 1,411,200 bits = 176,400 bytes per second. A 3 minute song therefore needs 176,400 x 180 = 31,752,000 bytes (about 31.8 MB).

For a 3-bit quantiser (8 levels, 0 to 7), samples of a wave at 3.2, 5.8, 6.9, 4.1 become 3, 6, 7, 4 and are stored as 011, 110, 111, 100.

Playback does the reverse: a DAC converts the numbers to voltages and a filter smooths them to an analogue wave.