Quantization (signal processing)

In digital signal processing, quantization is the process of approximating a continuous signal by a set of discrete symbols or integer values; that is, converting an analog signal to a digital one. In general, a quantization operator can be represented as

File:FloorQuantizer.png

Quantization of x using Q(x) = floor(Lx)/L.

${\displaystyle Q(x)=\operatorname {round} (f(x))}$

where x is a real number, Q(x) an integer, and f(x) is an arbitrary real-valued function that controls the "quantization law" of the particular coder.

In computer audio, a linear scale is most common. If x is a real valued number between -1 and 1, the quantization operator can therefore be alternately expressed as,

${\displaystyle Q(x)=\operatorname {floor} (2^{M-1}x)/2^{M-1}}$

where floor() returns the highest integer less than or equal to x and M is the number of bits used to quantize the value. Using this quantization law and assuming that quantization noise is uniformly distributed (accurate for rapidly varying xx or high M), the signal to noise ratio can be approximated as

${\displaystyle {\begin{matrix}{\frac {S}{N_{q}}}\end{matrix}}\approx (6.02M+1.76)dB}$.

From this equation, it is often said that the SNR is approximately 6dB per bit.

In digital telephony, two popular quantization schemes are the 'A-law' (dominant in Europe) and 'µ-law' (dominant in North America and Japan). These schemes map discrete analog values to an 8 bit scale that is nearly linear for small values and then increase logarithmically as amplitude grows. Because the human ear's perception of loudness is roughly logarithmic, this provides providing a higher signal to noise ratio over the range of audible sound intensities for a given number of bits.

See also:

• Information theory
• Rate distortion theory
• Vector quantization