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[[Image:FloorQuantizer.png|right|frame|Quantization of ''x'' using ''Q(x)'' = floor((''Lx'') / ''L'').]]
In [[digital signal processing]], '''quantization''' is the process of approximating a continuous range of values (or a very large set of possible discrete values) by a relatively-small set of discrete symbols or integer values. More specifically, a [[signal (information theory)|signal]] can be multi-dimensional and quantization need not be applied to all dimensions. A discrete signal need not necessarily be quantized (a pedantic point, but true nonetheless and can be a point of confusion). ''See [[ideal sampler]].''
A common use of quantization is in the conversion of a [[continuous signal]] into a [[discrete signal]] by [[sample (signal)|sampling]] and then quantizing. Both of these steps are performed in [[analog-to-digital converter]]s with the quantization level specified by a number of [[bit]]s.
A specific example would be [[compact disc]] (CD) audio which is sampled at 44,100 [[Hz]] and quantized with 16 bits (2 [[byte]]s) which can be one of 65,536 (<math>2^{16}</math>) possible values per sample.
The simplest and best-known form of quantization is referred to as [[scalar]] quantization, since it operates on scalar (as opposed to multi-dimensional [[vector]]) input data. In general, a scalar quantization operator can be represented as
:<math>Q(x) = g(\
where:
where <math>x</math> is a real number, <math>i = \operatorname{round}(f(x))</math> is an integer, and <math>f(x)</math> and <math>g(i)</math> are arbitrary real-valued functions.▼
* <math>x</math> is a real number,
The integer value <math>i = \operatorname{round}(f(x))</math> is the representation that is typically stored or transmitted, and then the final interpretation is constructed using <math>g(i)</math> when the data is later interpreted.▼
* <math>\lfloor x \rfloor</math> is the [[floor function]], yielding the integer <math>i = \lfloor f(x) \rfloor</math>
▲
▲The integer value <math>i
If <math>x</math> is a real valued number between -1 and 1, a uniform quantization operator that uses ''M'' bits of precision to represent each quantization index can be expressed as▼
▲In computer audio and most other applications, a method known as ''uniform quantization'' is the most common. If <math>x</math> is a real valued number between -1 and 1, a uniform quantization operator that uses ''M'' bits of precision to represent each quantization index can be expressed as
:<math>Q(x) = \frac{\operatorname{round}(2^{M-1}x)}{2^{M-1}}</math>.▼
In this case the <math>f(x)</math> and <math>g(i)</math> operators are just multiplying scale factors (one multiplier being the inverse of the other). The value <math>2^{-(M-1)}</math> is often referred to as the ''quantization step size''. Using this quantization law and assuming that quantization noise is approximately [[uniform distribution (continuous)|uniformly distributed]] over the quantization step size (an assumption typically accurate for rapidly varying <math>x</math> or high <math>M</math>) and assuming that the input signal <math>x</math> to be quantized is approximately uniformly distributed over the entire interval from -1 to 1, the [[signal to noise ratio]] (SNR) of the quantization can be computed as
:<math>
\frac{S}{N_q} \approx 20 \
=
6.0206 M \ \operatorname{dB}</math>.
From this equation, it is often said that the SNR is approximately 6 [[decibel|dB]] per [[bit]].
In digital [[telephone|telephony]], two popular quantization schemes are the '[[A-law algorithm|A-law]]' (dominant in [[Europe]]) and '[[Mu-law algorithm|&
==Quantization and Data Compression==
Quantization plays a major part in [[lossy data compression]]. In many cases, quantization can be viewed as the fundamental element that distinguishes [[lossy data compression]] from [[lossless data compression]], and the use of quantization is nearly always motivated by the need to reduce the amount of data needed to represent a signal. In some compression schemes, like [[MP3]] or [[Vorbis]], compression is also achieved by selectively discarding some data, an action that can be analyzed as a quantization process (e.g., a vector quantization process) or can be considered a different kind of lossy process.▼
▲In many cases, quantization can be viewed as the fundamental element that distinguishes [[lossy data compression]] from [[lossless data compression]], and the use of quantization is nearly always motivated by the need to reduce the amount of data needed to represent a signal.
One example of a lossy compression scheme that uses quantization is [[JPEG]] image compression.
During JPEG encoding, the data representing an image (typically 8-bits for each of three color components per pixel) is processed using a [[discrete cosine transform]] and is then quantized and [[entropy encoding|entropy coded]]. By reducing the precision of the transformed values using quantization, the number of bits needed to represent the image can be reduced substantially.
For example, images can often be represented with acceptable quality using JPEG at less than 3 bits per pixel (as opposed the the typical 24 bits per pixel needed prior to JPEG compression).
Even the original representation using 24 bits per pixel requires quantization for its [[pulse-code modulation|PCM]] sampling structure.
In modern compression technology, the [[information entropy|entropy]] of the output of a quantizer matters more than the number of possible values of its output (the number of values being <math>2^M</math> in the above example).
==See also==
* [[Analog-to-digital converter]], [[Digital-to-analog converter]]
* [[Discrete]], [[Digital]]
* [[Dither]]
* [[Information theory]]
* [[Rate distortion theory]]
* [[Vector quantization]]
==External Links==
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