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,

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

where 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 x 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