Two steps are required when one wishes to

digitize an

analog signal. The first one is

sampling, and the second one is

quantization. During quantization, a given

sample of the signal (which can take a priori any value) is transformed into one of a finite

set of possible values. In general, the

dynamic range (i.e. the range of values for which the input signal will not be cropped or

saturated) is divided into a certain number of quantization levels having the same range (the quantization step). Since

binary numbers are habitually used to represent digitized signals, the number of quantization steps is most often a

power of two.

For example, if the dynamic range is *R* and *B* bits are used to represent the signal, then the quantization step will be

*Q* = *R*2^{-B}

The quatiziation noise *e* is simply the difference between the original signal and the quantized signal. Since most quantizers use a round-up scheme to determine the quantization level that will be used for each signal sample, the quantization noise is always limited to values between -*Q*/2 and +*Q*/2.

Typically, the quantization noise is assumed to be a random process for which the associated probability distribution is uniform between -*Q*/2 and +*Q*/2. Knowing this, one can easily compute its mean (E{*e*}=0) and variance (E{*e*^{2})=*Q*^{2}/12). The RMS error is then *Q*/sqrt(12).

The signal-to-noise ratio (SNR) due to quantization noise is thus, in dB,

20log_{10}(*R*/*Q*) = 20log_{10}(2^{B}) = *B*20log_{10}2

which is about 6*B* dB, that is, the SNR is about six times the number of bits used in the quantization, unless a scheme such as oversampling and/or noise shaping is used.

Quantization noise has in general very complicated statistical properties, unless the assumption of wide-amplitude, wide-band is made. That is, the quantization noise can be assumed to be a zero-mean, stationary white noise with uniform probablity density, and furthermore not correlated with the input signal, only if the signal spans the entire dynamic range and is crossing often all the quantization levels.