Quantization (signal processing): Difference between revisions

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== Example ==
As anFor 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>,
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where the notation <math> \lfloor \ \rfloor </math> denotes the [[floor function]].
 
The essential property of a quantizer is that it hashaving a countable -set of possible output -values thatmembers has fewer memberssmaller 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>[[William Fleetwood Sheppard]], "On the Calculation of the Most Probable Values of Frequency Constants for data arranged according to Equidistant Divisions of a Scale", ''[[Proceedings of the London Mathematical Society]]'', Vol. 29, pp. 353&ndash;80, 1898.{{doi|10.1112/plms/s1-29.1.353}}</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>B. M. Oliver, J. R. Pierce, and [[Claude Shannon|Claude E. Shannon]], "The Philosophy of PCM", ''[[Proceedings of the IEEE|Proceedings of the IRE]]'', Vol. 36, pp. 1324–1331, Nov. 1948. {{doi|10.1109/JRPROC.1948.231941}}</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>Herbert Gish and John N. Pierce, "Asymptotically Efficient Quantizing", ''[[IEEE Transactions on Information Theory]]'', Vol. IT-14, No. 5, pp. 676–683, Sept. 1968. {{doi|10.1109/TIT.1968.1054193}}</ref><ref name=GrayNeuhoff>[[Robert M. Gray]] and David L. Neuhoff, "Quantization", ''[[IEEE Transactions on Information Theory]]'', Vol. IT-44, No. 6, pp. 2325–2383, Oct. 1998. {{doi|10.1109/18.720541}}</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 ¼. In terms of [[decibels]], the noise power change is <math>\scriptstyle 10\cdot \log_{10}(1/4)\ \approx\ -6\ \mathrm{dB}.</math>