Editing Quantization (signal processing) (section)

===Uniform quantization and the 6&nbsp;dB/bit approximation===
The Lloyd–Max quantizer is actually a uniform quantizer when the input PDF is uniformly distributed over the range <math>[y_1-\Delta/2,~y_M+\Delta/2)</math>. However, for a source that does not have a uniform distribution, the minimum-distortion quantizer may not be a uniform quantizer. The analysis of a uniform quantizer applied to a uniformly distributed source can be summarized in what follows:

A symmetric source X can be modelled with <math> f(x)= \tfrac1{2X_{\max}}</math>, for <math>x \in [-X_{\max} , X_{\max}]</math> and 0 elsewhere.
The step size <math>\Delta = \tfrac {2X_{\max}} {M} </math> and the ''signal to quantization noise ratio'' (SQNR) of the quantizer is
:<math>{\rm SQNR}= 10\log_{10}{\frac {\sigma_x^2}{\sigma_q^2}} = 10\log_{10}{\frac {(M\Delta)^2/12}{\Delta^2/12}}= 10\log_{10}M^2= 20\log_{10}M</math>.

For a fixed-length code using <math>N</math> bits, <math>M=2^N</math>, resulting in
<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 = 8×6 = 48&nbsp;dB; and for <math>N</math>=16 bits, <math>M</math>=65536 and SQNR = 16×6 = 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.

At asymptotically high bit rates, the 6&nbsp;dB/bit approximation is supported for many source PDFs by rigorous theoretical analysis.<ref name=Bennett/><ref name=OliverPierceShannon/><ref name=GishPierce/><ref name=GrayNeuhoff/> Moreover, the structure of the optimal scalar quantizer (in the rate–distortion sense) approaches that of a uniform quantizer under these conditions.<ref name=GishPierce/><ref name=GrayNeuhoff/>
<!-- I don't think that was proved by anyone else before it was done by Gish & Pearce in '68. For example, was it done by Koshelev in '63? (I don't think so) Zador in '66? (I don't know - probably not) Goblick & Holsinger in '67? (I don't see it in that paper.) -->