Quantization (signal processing): Difference between revisions

Revision as of 21:31, 14 December 2020

Revision as of 03:17, 4 January 2021

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{{shortShort description|Process of mapping a continuous set to a countable set}}

{{Use American English|date=April 2019}}

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Because quantization is a many-to-few mapping, it is an inherently [[Nonlinear system|non-linear]] and irreversible process (i.e., because the same output value is shared by multiple input values, it is impossible, in general, to recover the exact input value when given only the output value).

The set of possible input values may be infinitely large, and may possibly be continuous and therefore [[uncountable]] (such as the set of all [[real number]]s, or all real numbers within some limited range). The set of possible output values may be [[finite set|finite]] or [[Countable set|countably infinite]].<ref name=GrayNeuhoff/> The input and output sets involved in quantization can be defined in a rather general way. For example, [[vector quantization]] is the application of quantization to multi-dimensional (vector-valued) input data.<ref>{{cite book |author1=[[Allen Gersho]] |author-link=Allen Gersho |author2=Robert M. Gray |author-link2=Robert M. Gray |url=https://books.google.com/books?id=DwcDm6xgItUC |title=Vector Quantization and Signal Compression |publisher=[[Springer Science+Business Media|Springer]] |isbn=978-0-7923-9181-4 |date=1991}}</ref>

==Types==

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===Additive noise model===

A common assumption for the analysis of [[quantization error]] is that it affects a signal processing system in a similar manner to that of additive [[white noise]] – having negligible correlation with the signal and an approximately flat [[power spectral density]].<ref name=Bennett/><ref name=GrayNeuhoff/><ref name=Widrow1>[[Bernard Widrow]], "A study of rough amplitude quantization by means of Nyquist sampling theory", ''IRE Trans. Circuit Theory'', Vol. CT-3, pp. 266–276, 1956. {{doi|10.1109/TCT.1956.1086334}}</ref><ref name=Widrow2>[[Bernard Widrow]], "[http://www-isl.stanford.edu/~widrow/papers/j1961statisticalanalysis.pdf Statistical analysis of amplitude quantized sampled data systems]", ''Trans. AIEE Pt. II: Appl. Ind.'', Vol. 79, pp. 555–568, Jan. 1961.</ref> The additive noise model is commonly used for the analysis of quantization error effects in digital filtering systems, and it can be very useful in such analysis. It has been shown to be a valid model in cases of high resolution quantization (small <math>\Delta</math> relative to the signal strength) with smooth probability density functions.<ref name=Bennett/><ref name=MarcoNeuhoff>Daniel Marco and David L. Neuhoff, "The Validity of the Additive Noise Model for Uniform Scalar Quantizers", ''[[IEEE Transactions on Information Theory]]'', Vol. IT-51, No. 5, pp. 1739–1755, May 2005. {{doi|10.1109/TIT.2005.846397}}</ref>

Additive noise behavior is not always a valid assumption. Quantization error (for quantizers defined as described here) is deterministically related to the signal and not entirely independent of it. Thus, periodic signals can create periodic quantization noise. And in some cases it can even cause [[limit cycle]]s to appear in digital signal processing systems. One way to ensure effective independence of the quantization error from the source signal is to perform ''[[dither]]ed quantization'' (sometimes with ''[[noise shaping]]''), which involves adding random (or [[pseudo-random]]) noise to the signal prior to quantization.<ref name=GrayNeuhoff/><ref name=Widrow2/>