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# 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 via analog-to-digital conversion. In general, a quantization operator can be represented as

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

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,

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

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

$\frac{S}{N_q} \approx (6.02M + 1.76)dB$.

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 a higher signal to noise ratio over the range of audible sound intensities for a given number of bits.