Editing Quantization (signal processing) (section)

==Types==
[[File:2-bit resolution analog comparison.png|thumbnail|2-bit resolution with four levels of quantization compared to analog<ref>Hodgson, Jay (2010). ''Understanding Records'', p.56. {{ISBN|978-1-4411-5607-5}}. Adapted from Franz, David (2004). ''Recording and Producing in the Home Studio'', p.38-9. Berklee Press.</ref>]]
[[File:3-bit resolution analog comparison.png|thumbnail|3-bit resolution with eight levels]]

===Analog-to-digital converter===
An [[analog-to-digital converter]] (ADC) can be modeled as two processes: [[Sampling (signal processing)|sampling]] and quantization.  Sampling converts a time-varying voltage signal into a [[discrete-time signal]], a sequence of real numbers.  Quantization replaces each real number with an approximation from a finite set of discrete values.  Most commonly, these discrete values are represented as fixed-point words. Though any number of quantization levels is possible, common word lengths are [[audio bit depth|8-bit]] (256 levels), 16-bit (65,536 levels) and 24-bit (16.8&nbsp;million levels). Quantizing a sequence of numbers produces a sequence of quantization errors which is sometimes modeled as an additive random signal called '''quantization noise''' because of its [[stochastic]] behavior.  The more levels a quantizer uses, the lower is its quantization noise power.

===Rate–distortion optimization===
''[[Rate–distortion theory|Rate–distortion optimized]]'' quantization is encountered in [[source coding]] for lossy data compression algorithms, where the purpose is to manage distortion within the limits of the [[bit rate]] supported by a communication channel or storage medium. The analysis of quantization in this context involves studying the amount of data (typically measured in digits or bits or bit ''rate'') that is used to represent the output of the quantizer and studying the loss of precision that is introduced by the quantization process (which is referred to as the ''distortion'').

===Mid-riser and mid-tread uniform quantizers===
Most uniform quantizers for signed input data can be classified as being of one of two types: ''mid-riser'' and ''mid-tread''. The terminology is based on what happens in the region around the value 0, and uses the analogy of viewing the input-output function of the quantizer as a [[stairway]]. Mid-tread quantizers have a zero-valued reconstruction level (corresponding to a ''tread'' of a stairway), while mid-riser quantizers have a zero-valued classification threshold (corresponding to a ''[[Stair riser|riser]]'' of a stairway).<ref name=Gersho77>{{cite journal | last=Gersho | first=A. |author-link=Allen Gersho| title=Quantization | journal=IEEE Communications Society Magazine | publisher=Institute of Electrical and Electronics Engineers (IEEE) | volume=15 | issue=5 | year=1977 | issn=0148-9615 | doi=10.1109/mcom.1977.1089500 | pages=16–28| s2cid=260498692 }}</ref>

Mid-tread quantization involves rounding. The formulas for mid-tread uniform quantization are provided in the previous section.

:<math>Q(x) = \Delta \cdot  \left\lfloor \frac{x}{\Delta} + \frac{1}{2} \right\rfloor</math>,

Mid-riser quantization involves truncation. The input-output formula for a mid-riser uniform quantizer is given by:
:<math>Q(x) = \Delta\cdot\left(\left\lfloor \frac{x}{\Delta}\right\rfloor + \frac1{2}\right)</math>,
where the classification rule is given by
:<math>k = \left\lfloor \frac{x}{\Delta} \right\rfloor</math>
and the reconstruction rule is
:<math>y_k = \Delta\cdot\left(k+\tfrac1{2}\right)</math>.

Note that mid-riser uniform quantizers do not have a zero output value – their minimum output magnitude is half the step size. In contrast, mid-tread quantizers do have a zero output level. For some applications, having a zero output signal representation may be a necessity.

In general, a mid-riser or mid-tread quantizer may not actually be a ''uniform'' quantizer – i.e., the size of the quantizer's classification [[interval (mathematics)|intervals]] may not all be the same, or the spacing between its possible output values may not all be the same. The distinguishing characteristic of a mid-riser quantizer is that it has a classification threshold value that is exactly zero, and the distinguishing characteristic of a mid-tread quantizer is that is it has a reconstruction value that is exactly zero.<ref name=Gersho77/>

===Dead-zone quantizers===
A '''dead-zone quantizer''' is a type of mid-tread quantizer with symmetric behavior around 0. The region around the zero output value of such a quantizer is referred to as the ''dead zone'' or ''[[deadband]]''. The dead zone can sometimes serve the same purpose as a [[noise gate]] or [[squelch]] function. Especially for compression applications, the dead-zone may be given a different width than that for the other steps. For an otherwise-uniform quantizer, the dead-zone width can be set to any value <math>w</math> by using the forward quantization rule<ref>{{cite book| first1=Majid |last1=Rabbani |first2=Rajan L. |last2=Joshi |first3=Paul W. |last3=Jones |editor1-first=Peter |editor1-last=Schelkens |editor2-first=Athanassios |editor2-last=Skodras |editor3-first=Touradj |editor3-last=Ebrahimi |title=The JPEG 2000 Suite | url=https://archive.org/details/jpegsuitethewile00sche | url-access=limited |publisher=[[John Wiley & Sons]] |date=2009 |isbn=978-0-470-72147-6 |chapter=Section 1.2.3: Quantization, in Chapter 1: JPEG 2000 Core Coding System (Part 1) |pages=[https://archive.org/details/jpegsuitethewile00sche/page/n73 22]–24}}</ref><ref>{{cite book| first1=David S. |last1=Taubman |first2=Michael W. |last2=Marcellin |title=JPEG2000: Image Compression Fundamentals, Standards and Practice | url=https://archive.org/details/jpegimagecompres00taub | url-access=limited |publisher=[[Kluwer Academic Publishers]] |date=2002 |isbn=0-7923-7519-X |chapter=Chapter 3: Quantization |page=[https://archive.org/details/jpegimagecompres00taub/page/n126 107]}}</ref><ref name=SullivanIT/>
:<math>k = \sgn(x) \cdot \max\left(0, \left\lfloor \frac{\left| x \right|-w/2}{\Delta}+1\right\rfloor\right)</math>,
where the function {{no break|<math>\sgn</math>(&nbsp;)}} is the [[sign function]] (also known as the ''signum'' function). The general reconstruction rule for such a dead-zone quantizer is given by
:<math>y_k = \sgn(k) \cdot\left(\frac{w}{2}+\Delta\cdot (|k|-1+r_k)\right)</math>,
where <math>r_k</math> is a reconstruction offset value in the range of 0 to 1 as a fraction of the step size. Ordinarily, <math>0 \le r_k \le \tfrac1{2}</math> when quantizing input data with a typical [[probability density function]] (PDF) that is symmetric around zero and reaches its peak value at zero (such as a [[Gaussian distribution|Gaussian]], [[Laplacian distribution|Laplacian]], or [[generalized Gaussian distribution|generalized Gaussian]] PDF). Although <math>r_k</math> may depend on <math>k</math> in general and can be chosen to fulfill the optimality condition described below, it is often simply set to a constant, such as <math>\tfrac1{2}</math>. (Note that in this definition, <math>y_0 = 0</math> due to the definition of the {{no break|<math>\sgn</math>(&nbsp;)}} function, so <math>r_0</math> has no effect.)

A very commonly used special case (e.g., the scheme typically used in financial accounting and elementary mathematics) is to set <math>w=\Delta</math> and <math>r_k=\tfrac1{2}</math> for all <math>k</math>. In this case, the dead-zone quantizer is also a uniform quantizer, since the central dead-zone of this quantizer has the same width as all of its other steps, and all of its reconstruction values are equally spaced as well.