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== Example ==
:<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
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. 353–80, 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>
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