Quantization (signal processing): Difference between revisions

Revision as of 20:14, 18 May 2005

Revision as of 21:04, 18 May 2005

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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.

== Mathematical description ==

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

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The integer value <math>i</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. The integer value <math>i</math> is sometimes referred to as the ''quantization index''.

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{\left\lfloor 2^{M-1}x \right\rfloor}{2^{M-1}}</math>.

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In digital [[telephone|telephony]], two popular quantization schemes are the '[[A-law algorithm|A-law]]' (dominant in [[Europe]]) and '[[Mu-law algorithm|&mu;-law]]' (dominant in [[North America]] and [[Japan]]). These schemes map discrete analog values to an 8-bit scale that is nearly linear for small values and then increases logarithmically as amplitude grows. Because the human ear's perception of [[loudness]] is roughly logarithmic, this provides a higher signal to noise ratio over the range of audible sound intensities for a given number of bits.

== Quantization and Datadata compression 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.

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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).

== Relation to quantization in nature ==

==See also==

At the most fundamental level, all [[physical quantity|physical quantities]] are quantized. This is a result of [[quantum mechanics]] (see [[Quantization (physics)]]). Signals may be treated as continuous for mathematical simplicity by considering the small quantizations as negligible.

In any practical application, this inherent quantization is irrelevant. First of all, it is overshadowed by [[signal noise]], the intrusion of extraneous phenomena present in the system upon the signal of interest. The second, which appears only in measurement applications, is the inaccuracy of instruments.

== Related topics ==

* [[Analog-to-digital converter]], [[Digital-to-analog converter]]

* [[Discrete]], [[Digital]]

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* [[Vector quantization]]

== External Linkslinks ==

[http://www.math.ucdavis.edu/~saito/courses/ACHA/44it06-gray.pdf Paper on mathematical theory and analysis of quantization]