Quantization (signal processing): Difference between revisions

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The essential property of a quantizer is having a countable-set of possible output-values members smaller 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>{{cite journal | last=Sheppard | first=W. F. |author-link=William Fleetwood Sheppard| title=On the Calculation of the most Probable Values of Frequency-Constants, for Data arranged according to Equidistant Division of a Scale | journal=Proceedings of the London Mathematical Society | publisher=Wiley | volume=s1-29 | issue=1 | year=1897 | issn=0024-6115 | doi=10.1112/plms/s1-29.1.353 | pages=353–380| url=https://zenodo.org/record/1447738 }}</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>{{cite journal | last1=Oliver | first1=B.M. | last2=Pierce | first2=J.R. | last3=Shannon | first3=C.E. |author-link3=Claude Shannon| title=The Philosophy of PCM | journal=Proceedings of the IRE | publisher=Institute of Electrical and Electronics Engineers (IEEE) | volume=36 | issue=11 | year=1948 | issn=0096-8390 | doi=10.1109/jrproc.1948.231941 | pages=1324–1331| s2cid=51663786 }}</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>{{cite journal | last1=Gish | first1=H. | last2=Pierce | first2=J. | title=Asymptotically efficient quantizing | journal=IEEE Transactions on Information Theory | publisher=Institute of Electrical and Electronics Engineers (IEEE) | volume=14 | issue=5 | year=1968 | issn=0018-9448 | doi=10.1109/tit.1968.1054193 | pages=676–683}}</ref><ref name=GrayNeuhoff>{{cite journal | last1=Gray | first1=R.M. |author-link=Robert M. Gray| last2=Neuhoff | first2=D.L. | title=Quantization | journal=IEEE Transactions on Information Theory | publisher=Institute of Electrical and Electronics Engineers (IEEE) | volume=44 | issue=6 | year=1998 | issn=0018-9448 | doi=10.1109/18.720541 | pages=2325–2383| s2cid=212653679 }}</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{{sfrac|1|4}}. In terms of [[decibels]], the noise power change is <math>\scriptstyle 10\cdot \log_{10}(1/4)\ \approx\ -6\ \mathrm{dB}.</math>
 
Because the set of possible output values of a quantizer is countable, any quantizer can be decomposed into two distinct stages, which can be referred to as the ''classification'' stage (or ''forward quantization'' stage) and the ''reconstruction'' stage (or ''inverse quantization'' stage), where the classification stage maps the input value to an integer ''quantization index'' <math>k</math> and the reconstruction stage maps the index <math>k</math> to the ''reconstruction value'' <math>y_k</math> that is the output approximation of the input value. For the example uniform quantizer described above, the forward quantization stage can be expressed as
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<math>{\rm SQNR}= 20\log_{10}{2^N} = N\cdot(20\log_{10}2) = N\cdot 6.0206\,\rm{dB}</math>,
 
or approximately 6&nbsp;dB per bit. For example, for <math>N</math>=8 bits, <math>M</math>=256 levels and SQNR = 8&times;68×6 = 48&nbsp;dB; and for <math>N</math>=16 bits, <math>M</math>=65536 and SQNR = 16&times;616×6 = 96&nbsp;dB. The property of 6&nbsp;dB improvement in SQNR for each extra bit used in quantization is a well-known figure of merit. However, it must be used with care: this derivation is only for a uniform quantizer applied to a uniform source. For other source PDFs and other quantizer designs, the SQNR may be somewhat different from that predicted by 6&nbsp;dB/bit, depending on the type of PDF, the type of source, the type of quantizer, and the bit rate range of operation.
 
However, it is common to assume that for many sources, the slope of a quantizer SQNR function can be approximated as 6&nbsp;dB/bit when operating at a sufficiently high bit rate. At asymptotically high bit rates, cutting the step size in half increases the bit rate by approximately 1 bit per sample (because 1 bit is needed to indicate whether the value is in the left or right half of the prior double-sized interval) and reduces the mean squared error by a factor of 4 (i.e., 6&nbsp;dB) based on the <math>\Delta^2/12</math> approximation.