Editing Quantization (signal processing) (section)

===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.