Editing Quantization (signal processing) (section)

===Quantization noise model===
[[File:Frequency spectrum of a sinusoid and its quantization noise floor.gif|thumb|300px|Comparison of quantizing a sinusoid to 64 levels (6 bits) and 256 levels (8 bits). The additive noise created by 6-bit quantization is 12 dB greater than the noise created by 8-bit quantization. When the spectral distribution is flat, as in this example, the 12 dB difference manifests as a measurable difference in the noise floors.]]

Quantization noise is a [[Model (abstract)|model]] of quantization error introduced by quantization in the ADC. It is a rounding error between the analog input voltage to the ADC and the output digitized value. The noise is non-linear and signal-dependent. It can be modeled in several different ways.

In an ideal ADC, where the quantization error is uniformly distributed between −1/2 LSB and +1/2 LSB, and the signal has a uniform distribution covering all quantization levels, the [[Signal-to-quantization-noise ratio]] (SQNR) can be calculated from

:<math>\mathrm{SQNR} = 20 \log_{10}(2^Q) \approx 6.02 \cdot Q\ \mathrm{dB} \,\!</math>

where Q is the number of quantization bits.

The most common test signals that fulfill this are full amplitude [[triangle wave]]s and [[sawtooth wave]]s.

For example, a [[16-bit]] ADC has a maximum signal-to-quantization-noise ratio of 6.02 × 16 = 96.3&nbsp;dB.

When the input signal is a full-amplitude [[sine wave]] the distribution of the signal is no longer uniform, and the corresponding equation is instead

:<math> \mathrm{SQNR} \approx  1.761 + 6.02 \cdot Q \ \mathrm{dB} \,\!</math>

Here, the quantization noise is once again ''assumed'' to be uniformly distributed.  When the input signal has a high amplitude and a wide frequency spectrum this is the case.<ref>{{cite book
  | last = Pohlman
  | first =Ken C.
  | title = Principles of Digital Audio 2nd Edition
  | publisher = SAMS
  | date = 1989
  | page = 60
  | isbn =9780071441568
 | url = https://books.google.com/books?id=VZw6z9a03ikC&pg=PA37}}</ref>  In this case a 16-bit ADC has a maximum signal-to-noise ratio of 98.09&nbsp;dB.  The 1.761 difference in signal-to-noise only occurs due to the signal being a full-scale sine wave instead of a triangle or sawtooth.

For complex signals in high-resolution ADCs this is an accurate model. For low-resolution ADCs, low-level signals in high-resolution ADCs, and for simple waveforms the quantization noise is not uniformly distributed, making this model inaccurate.<ref>{{cite book
  | last = Watkinson
  | first = John
  | title = The Art of Digital Audio 3rd Edition
  | publisher = [[Focal Press]]
  | date = 2001
  | isbn = 0-240-51587-0}}</ref> In these cases the quantization noise distribution is strongly affected by the exact amplitude of the signal.

The calculations are relative to full-scale input. For smaller signals, the relative quantization distortion can be very large. To circumvent this issue, analog [[companding]] can be used, but this can introduce distortion.