Editing Quantization (signal processing) (section)

==Example==
For example, [[Rounding#Round half up|rounding]] a [[real number]] <math>x</math> to the nearest integer value forms a very basic type of quantizer – a ''uniform'' one.  A typical (''mid-tread'') uniform quantizer with a quantization ''step size'' equal to some value <math>\Delta</math> can be expressed as

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

where the notation <math> \lfloor \ \rfloor </math> denotes the [[floor function]].

Alternatively, the same quantizer may be expressed in terms of the [[ceiling function]], as 
:<math>Q(x) = \Delta \cdot  \left\lceil \frac{x}{\Delta} - \frac{1}{2} \right\rceil</math>.

(The notation <math> \lceil \ \rceil </math> denotes the ceiling function).

The essential property of a quantizer is having a countable set of possible output values smaller than the set of possible input values. The members of the set of output values may have integer, rational, or real values. For simple rounding to the nearest integer, the step size <math>\Delta</math> is equal to 1. With <math>\Delta = 1</math> or with <math>\Delta</math> equal to any other integer value, this quantizer has real-valued inputs and integer-valued outputs.

When the quantization step size (Δ) is small relative to the variation in the signal being quantized, it is relatively simple to show that the [[mean squared error]] produced by such a rounding operation will be approximately <math>\Delta^2/ 12</math>.<ref name=Sheppard>{{cite journal | last=Sheppard | first=W. F. |author-link=William Fleetwood Sheppard| title=On the Calculation of the most Probable Values of Frequency-Constants, for Data arranged according to Equidistant Division of a Scale | journal=Proceedings of the London Mathematical Society | publisher=Wiley | volume=s1-29 | issue=1 | year=1897 | issn=0024-6115 | doi=10.1112/plms/s1-29.1.353 | pages=353–380| url=https://zenodo.org/record/1447738 }}</ref><ref name=Bennett>W. R. Bennett, "[http://www.alcatel-lucent.com/bstj/vol27-1948/articles/bstj27-3-446.pdf Spectra of Quantized Signals]", ''[[Bell System Technical Journal]]'', Vol. 27, pp. 446–472, July 1948.</ref><ref name=OliverPierceShannon>{{cite journal | last1=Oliver | first1=B.M. | last2=Pierce | first2=J.R. | last3=Shannon | first3=C.E. |author-link3=Claude Shannon| title=The Philosophy of PCM | journal=Proceedings of the IRE | publisher=Institute of Electrical and Electronics Engineers (IEEE) | volume=36 | issue=11 | year=1948 | issn=0096-8390 | doi=10.1109/jrproc.1948.231941 | pages=1324–1331| s2cid=51663786 }}</ref><ref name=Stein>Seymour Stein and J. Jay Jones, ''[https://books.google.com/books/about/Modern_communication_principles.html?id=jBc3AQAAIAAJ Modern Communication Principles]'', [[McGraw–Hill]], {{ISBN|978-0-07-061003-3}}, 1967 (p. 196).</ref><ref name=GishPierce>{{cite journal | last1=Gish | first1=H. | last2=Pierce | first2=J. | title=Asymptotically efficient quantizing | journal=IEEE Transactions on Information Theory | publisher=Institute of Electrical and Electronics Engineers (IEEE) | volume=14 | issue=5 | year=1968 | issn=0018-9448 | doi=10.1109/tit.1968.1054193 | pages=676–683}}</ref><ref name=GrayNeuhoff>{{cite journal | last1=Gray | first1=R.M. |author-link=Robert M. Gray| last2=Neuhoff | first2=D.L. | title=Quantization | journal=IEEE Transactions on Information Theory | publisher=Institute of Electrical and Electronics Engineers (IEEE) | volume=44 | issue=6 | year=1998 | issn=0018-9448 | doi=10.1109/18.720541 | pages=2325–2383| s2cid=212653679 }}</ref> Mean squared error is also called the quantization ''noise power''.  Adding one bit to the quantizer halves the value of Δ, which reduces the noise power by the factor {{sfrac|1|4}}. In terms of [[decibel]]s, the noise power change is <math>\scriptstyle 10\cdot \log_{10}(1/4)\ \approx\ -6\ \mathrm{dB}.</math>

Because the set of possible output values of a quantizer is countable, any quantizer can be decomposed into two distinct stages, which can be referred to as the ''classification'' stage (or ''forward quantization'' stage) and the ''reconstruction'' stage (or ''inverse quantization'' stage), where the classification stage maps the input value to an integer ''quantization index'' <math>k</math> and the reconstruction stage maps the index <math>k</math> to the ''reconstruction value'' <math>y_k</math> that is the output approximation of the input value.  For the example uniform quantizer described above, the forward quantization stage can be expressed as
:<math>k = \left\lfloor \frac{x}{\Delta} + \frac{1}{2}\right\rfloor</math>,
and the reconstruction stage for this example quantizer is simply 
:<math>y_k = k \cdot \Delta</math>.

This decomposition is useful for the design and analysis of quantization behavior, and it illustrates how the quantized data can be communicated over a [[communication channel]] – a ''source encoder'' can perform the forward quantization stage and send the index information through a communication channel, and a ''decoder'' can perform the reconstruction stage to produce the output approximation of the original input data. In general, the forward quantization stage may use any function that maps the input data to the integer space of the quantization index data, and the inverse quantization stage can conceptually (or literally) be a table look-up operation to map each quantization index to a corresponding reconstruction value. This two-stage decomposition applies equally well to [[vector quantization|vector]] as well as scalar quantizers.