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<!DOCTYPE html>
<html>
<head> <title>Quantization</title>
<meta charset="UTF-8" />
<meta name="generator" content="TeX4ht (http://www.cse.ohio-state.edu/~gurari/TeX4ht/)" />
<link rel="stylesheet" type="text/css" href="index.css" />
<script type="text/javascript"
src="https://cdn.mathjax.org/mathjax/latest/MathJax.js?config=TeX-AMS-MML_HTMLorMML"
></script>
<style type="text/css">
.MathJax_MathML {text-indent: 0;}
</style>
</head><body
>
<div class="maketitle">
<h2 class="titleHead"><a
href="https://github.com/vicente-gonzalez-ruiz/quantization" >Quantization</a></h2>
<div class="author" ><span
class="ecrm-1200">Vicente Gonz</span><span
class="ecrm-1200">á</span><span
class="ecrm-1200">lez Ruiz</span></div><br />
<div class="date" ><span
class="ecrm-1200">November 17, 2020</span></div>
</div>
<h3 class="sectionHead"><span class="titlemark">1 </span> <a
id="x1-10001"></a><a
href="https://en.wikipedia.org/wiki/Quantization_(signal_processing)" >Basics</a></h3>
<hr class="figure" /><div class="figure"
>
<!--l. 12--><p class="noindent" ><div style="text-align:center;"> <img width=500 src="graphics/cuantif.svg" /> </div> <a
id="x1-1001r1"></a>
<a
id="x1-1002"></a>
<br /> </p><div class="caption"
><span class="id">
Figure 1: : </span><span
class="content">The quantization procedure.
</span></div><!--tex4ht:label?: x1-1001r1 -->
</div><hr class="endfigure" />
<ul class="itemize1">
<li class="itemize">A quantizer discretizes the amplitude of a <a
href="https://en.wikipedia.org/wiki/Pulse-amplitude_modulation" >PAM signal</a> <!--l. 23--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><mrow
><mi
>s</mi><mrow ><mo
class="MathClass-open">(</mo><mrow><mi
>n</mi><msub><mrow
><mi
>T</mi></mrow><mrow
><mi
>s</mi></mrow></msub
></mrow><mo
class="MathClass-close">)</mo></mrow></mrow></math>
(where <!--l. 23--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><mi
>s</mi></math>
is an analog signal, <!--l. 24--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><mi
>n</mi> <mo
class="MathClass-rel">∈</mo> <mi
>ℤ</mi></math>
and <!--l. 24--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><msub><mrow
><mi
>T</mi></mrow><mrow
><mi
>s</mi></mrow></msub
></math>
is a sampling period), producing an analog <a
href="https://en.wikipedia.org/wiki/Pulse-code_modulation" >PCM signal</a> <!--l. 27--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><mi
>s</mi><mrow ><mo
class="MathClass-open">[</mo><mrow><mi
>n</mi></mrow><mo
class="MathClass-close">]</mo></mrow></math>
(see Fig. <a
href="#x1-1001r1">1<!--tex4ht:ref: fig:cuantif --></a>). Therefore, quantization maps the values of samples (that are
defined in a continous space) into a discrete set of values <span class="cite">[<a
href="#Xvetterli1995wavelets">3</a>]</span>.
</li>
<li class="itemize">The quantization process can be modeled as <table class="equation"><tr><td> <a
id="x1-1003r1"></a>
<!--l. 34--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="block" class="equation">
<mi
>s</mi><mrow ><mo
class="MathClass-open">[</mo><mrow><mi
>n</mi></mrow><mo
class="MathClass-close">]</mo></mrow> <mo
class="MathClass-rel">=</mo> <mi
>s</mi><mrow ><mo
class="MathClass-open">(</mo><mrow><mi
>n</mi><msub><mrow
><mi
>T</mi></mrow><mrow
><mi
>s</mi></mrow></msub
></mrow><mo
class="MathClass-close">)</mo></mrow> <mo
class="MathClass-bin">+</mo> <mi
>e</mi><mrow ><mo
class="MathClass-open">(</mo><mrow><mi
>n</mi><msub><mrow
><mi
>T</mi></mrow><mrow
><mi
>s</mi></mrow></msub
></mrow><mo
class="MathClass-close">)</mo></mrow><mo
class="MathClass-punc">,</mo>
</math></td><td class="eq-no">(1)</td></tr></table>
<!--l. 37--><p class="noindent" >being <!--l. 37--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><mrow
><mi
>e</mi><mrow ><mo
class="MathClass-open">(</mo><mrow><mi
>n</mi><msub><mrow
><mi
>T</mi></mrow><mrow
><mi
>s</mi></mrow></msub
></mrow><mo
class="MathClass-close">)</mo></mrow></mrow></math>
an unpredectible quantization error (also called, quantization noise). Quantization
produces a loss of information in the signal because we cannot recover
<!--l. 39--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><mi
>s</mi><mrow ><mo
class="MathClass-open">(</mo><mrow><mi
>n</mi><msub><mrow
><mi
>T</mi></mrow><mrow
><mi
>s</mi></mrow></msub
></mrow><mo
class="MathClass-close">)</mo></mrow></math> from
<!--l. 39--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><mi
>s</mi><mrow ><mo
class="MathClass-open">[</mo><mrow><mi
>n</mi></mrow><mo
class="MathClass-close">]</mo></mrow></math> without
<!--l. 40--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><mi
>e</mi><mrow ><mo
class="MathClass-open">(</mo><mrow><mi
>n</mi><msub><mrow
><mi
>T</mi></mrow><mrow
><mi
>s</mi></mrow></msub
></mrow><mo
class="MathClass-close">)</mo></mrow></math>.
</p></li>
<li class="itemize">Quantizers are defined from their set of
<!--l. 43--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><mrow
><msub><mrow
><mi
>d</mi></mrow><mrow
><mi
>i</mi></mrow></msub
><mo
class="MathClass-punc">;</mo><mi
>i</mi> <mo
class="MathClass-rel">∈</mo> <mi
>ℤ</mi></mrow></math> (<span
class="ecti-1000">decision levels</span>)
and <!--l. 44--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><mrow
><msub><mrow
><mi
>r</mi></mrow><mrow
><mi
>i</mi></mrow></msub
><mo
class="MathClass-punc">;</mo><mi
>i</mi> <mo
class="MathClass-rel">∈</mo> <mi
>ℤ</mi></mrow></math> (<span
class="ecti-1000">representation</span>
<span
class="ecti-1000">levels</span>). <!--l. 45--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><mrow ><mo
class="MathClass-open">{</mo><mrow><msub><mrow
><mi
>d</mi></mrow><mrow
><mi
>i</mi></mrow></msub
></mrow><mo
class="MathClass-close">}</mo></mrow></math>
and <!--l. 45--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><mrow ><mo
class="MathClass-open">{</mo><mrow><msub><mrow
><mi
>r</mi></mrow><mrow
><mi
>i</mi></mrow></msub
></mrow><mo
class="MathClass-close">}</mo></mrow></math>
must be finite.
</li>
<li class="itemize">Another design parameter of quantizers is the <span
class="ecti-1000">decision boundaries</span>
<!--l. 49--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><msub><mrow
><mi
>d</mi></mrow><mrow
><mstyle
class="text"><mtext >min</mtext></mstyle></mrow></msub
></math> and
<!--l. 49--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><msub><mrow
><mi
>d</mi></mrow><mrow
><mstyle
class="text"><mtext >max</mtext></mstyle></mrow></msub
></math>,
(<!--l. 50--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><msub><mrow
><mi
>d</mi></mrow><mrow
><mstyle
class="text"><mtext >min</mtext></mstyle></mrow></msub
> <mo
class="MathClass-rel"><</mo> <msub><mrow
><mi
>d</mi></mrow><mrow
><mstyle
class="text"><mtext >max</mtext></mstyle></mrow></msub
></math>)
that define the so called <span
class="ecti-1000">low and high overload regions</span>, respectively, as <table class="equation"><tr><td>
<a
id="x1-1004r2"></a>
<!--l. 52--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="block" class="equation">
<mi
>s</mi><mrow ><mo
class="MathClass-open">[</mo><mrow><mi
>n</mi></mrow><mo
class="MathClass-close">]</mo></mrow> <mo
class="MathClass-rel">=</mo> <mrow ><mo
class="MathClass-open">{</mo><mrow><mtable align="axis" style=""
equalrows="false" columnlines="none" equalcolumns="false" class="array"><mtr><mtd
class="array" columnalign="left"> <msub><mrow
><mi
>r</mi></mrow><mrow
><mstyle
class="text"><mtext >min</mtext></mstyle></mrow></msub
><mo
class="MathClass-punc">,</mo> </mtd><mtd
class="array" columnalign="left"> <mstyle
class="text"><mtext >if </mtext><mstyle
class="math"><mi
>s</mi><mrow ><mo
class="MathClass-open">(</mo><mrow><mi
>n</mi><msub><mrow
><mi
>T</mi></mrow><mrow
><mi
>s</mi></mrow></msub
></mrow><mo
class="MathClass-close">)</mo></mrow> <mo
class="MathClass-rel"><</mo> <msub><mrow
><mi
>d</mi></mrow><mrow
><mstyle
class="text"><mtext >min</mtext></mstyle></mrow></msub
></mstyle><mtext ></mtext></mstyle> </mtd>
</mtr> <mtr><mtd
class="array" columnalign="left"> <msub><mrow
><mi
>r</mi></mrow><mrow
><mstyle
class="text"><mtext >max</mtext></mstyle></mrow></msub
><mo
class="MathClass-punc">,</mo> </mtd><mtd
class="array" columnalign="left"> <mstyle
class="text"><mtext >if </mtext><mstyle
class="math"><mi
>s</mi><mrow ><mo
class="MathClass-open">(</mo><mrow><mi
>n</mi><msub><mrow
><mi
>T</mi></mrow><mrow
><mi
>s</mi></mrow></msub
></mrow><mo
class="MathClass-close">)</mo></mrow> <mo
class="MathClass-rel">></mo> <msub><mrow
><mi
>d</mi></mrow><mrow
><mstyle
class="text"><mtext >max</mtext></mstyle></mrow></msub
></mstyle><mtext ></mtext></mstyle> </mtd>
</mtr> <mtr><mtd
class="array" columnalign="left"> <mi
>s</mi><mrow ><mo
class="MathClass-open">(</mo><mrow><mi
>n</mi><msub><mrow
><mi
>T</mi></mrow><mrow
><mi
>s</mi></mrow></msub
></mrow><mo
class="MathClass-close">)</mo></mrow> <mo
class="MathClass-bin">+</mo> <mi
>e</mi><mrow ><mo
class="MathClass-open">(</mo><mrow><mi
>n</mi><msub><mrow
><mi
>T</mi></mrow><mrow
><mi
>s</mi></mrow></msub
></mrow><mo
class="MathClass-close">)</mo></mrow><mo
class="MathClass-punc">,</mo> </mtd><mtd
class="array" columnalign="left"> <mstyle
class="text"><mtext >otherwise</mtext></mstyle><mo
class="MathClass-punc">.</mo> </mtd></mtr><!--ll--></mtable>
</math></td><td class="eq-no">(2)</td></tr></table>
</li>
<li class="itemize">For the sake of simplicity, lets denote the inputs of the quantizer as
<!--l. 61--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><mi
>x</mi><mrow ><mo
class="MathClass-open">(</mo><mrow><mo
class="MathClass-rel">=</mo> <mrow ><mo
class="MathClass-open">{</mo><mrow><mi
>s</mi><mrow ><mo
class="MathClass-open">(</mo><mrow><mi
>n</mi><msub><mrow
><mi
>T</mi></mrow><mrow
><mi
>s</mi></mrow></msub
></mrow><mo
class="MathClass-close">)</mo></mrow></mrow><mo
class="MathClass-close">}</mo></mrow></mrow><mo
class="MathClass-close">)</mo></mrow></math> and the
outputs as <!--l. 61--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><mi
>y</mi><mrow ><mo
class="MathClass-open">(</mo><mrow><mo
class="MathClass-rel">=</mo> <mrow ><mo
class="MathClass-open">{</mo><mrow><mi
>s</mi><mrow ><mo
class="MathClass-open">[</mo><mrow><mi
>n</mi></mrow><mo
class="MathClass-close">]</mo></mrow></mrow><mo
class="MathClass-close">}</mo></mrow></mrow><mo
class="MathClass-close">)</mo></mrow></math>.
The performance of the quantizer can be measured as the distance between the input
<!--l. 63--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><mi
>x</mi></math> and the
output <!--l. 63--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><mi
>y</mi></math>,
and typically the squared error is used: <table class="equation-star"><tr><td>
<!--l. 65--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="block" class="equation">
<mi
>d</mi><mrow ><mo
class="MathClass-open">(</mo><mrow><mi
>x</mi><mo
class="MathClass-punc">,</mo><mi
>y</mi></mrow><mo
class="MathClass-close">)</mo></mrow> <mo
class="MathClass-rel">=</mo> <mo
class="MathClass-rel">|</mo><mi
>x</mi> <mo
class="MathClass-bin">−</mo> <mi
>y</mi><msup><mrow
><mo
class="MathClass-rel">|</mo></mrow><mrow
><mn>2</mn></mrow></msup
><mo
class="MathClass-punc">.</mo>
</math></td></tr></table>
<!--l. 68--><p class="noindent" >Thus, the MSE is </p><table class="equation"><tr><td> <a
id="x1-1005r3"></a>
<!--l. 69--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="block" class="equation">
<mstyle
class="text"><mtext >MSE</mtext></mstyle> <mo
class="MathClass-rel">=</mo> <mi
>E</mi><mrow ><mo
class="MathClass-open">(</mo><mrow><mo
class="MathClass-rel">|</mo><mi
>x</mi> <mo
class="MathClass-bin">−</mo> <mi
>y</mi><msup><mrow
><mo
class="MathClass-rel">|</mo></mrow><mrow
><mn>2</mn></mrow></msup
></mrow><mo
class="MathClass-close">)</mo></mrow> <mo
class="MathClass-rel">=</mo><munder class="msub"><mrow
><mo
> ∑</mo>
</mrow><mrow
><mi
>i</mi></mrow></munder
><msubsup><mrow
><mo
> ∫
<!--nolimits--></mo><!--nolimits--></mrow><mrow
><msub><mrow
><mi
>x</mi></mrow><mrow
><mi
>i</mi><mo
class="MathClass-bin">−</mo><mn>1</mn></mrow></msub
></mrow><mrow
><msub><mrow
><mi
>x</mi></mrow><mrow
><mi
>i</mi></mrow></msub
>
</mrow></msubsup
><msup><mrow
><mrow ><mo
class="MathClass-open">(</mo><mrow><mi
>x</mi> <mo
class="MathClass-bin">−</mo> <msub><mrow
><mi
>y</mi></mrow><mrow
><mi
>i</mi></mrow></msub
></mrow><mo
class="MathClass-close">)</mo></mrow></mrow><mrow
><mn>2</mn></mrow></msup
><msub><mrow
><mi
>f</mi></mrow><mrow
>
<mi
>X</mi></mrow></msub
><mrow ><mo
class="MathClass-open">(</mo><mrow><mi
>x</mi></mrow><mo
class="MathClass-close">)</mo></mrow><mi
>d</mi><mi
>x</mi><mo
class="MathClass-punc">,</mo>
</math></td><td class="eq-no">(3)</td></tr></table>
<!--l. 72--><p class="noindent" >where <!--l. 72--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><msub><mrow
><mi
>f</mi></mrow><mrow
><mi
>X</mi></mrow></msub
><mrow ><mo
class="MathClass-open">(</mo><mrow><mi
>x</mi></mrow><mo
class="MathClass-close">)</mo></mrow></math>
is the <a
href="https://en.wikipedia.org/wiki/Probability_density_function" >probability density function (PDF)</a> of
<!--l. 74--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><mi
>x</mi></math> and
<!--l. 74--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><mi
>E</mi><mrow ><mo
class="MathClass-open">(</mo><mrow><mo
class="MathClass-bin">⋅</mo></mrow><mo
class="MathClass-close">)</mo></mrow></math>
the <a
href="https://en.wikipedia.org/wiki/Expected_value" >expectation</a>.
</p>
</li></ul>
<h3 class="sectionHead"><span class="titlemark">2 </span> <a
id="x1-20002"></a>Scalar quantization</h3>
<ul class="itemize1">
<li class="itemize">In a scalar quantizer, each input sample
<!--l. 81--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><msub><mrow
><mi
>x</mi></mrow><mrow
><mi
>i</mi></mrow></msub
></math> is individually
quantized as <!--l. 82--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><msub><mrow
><mi
>y</mi></mrow><mrow
><mi
>i</mi></mrow></msub
></math>,
and the quantization error, determined by <table class="equation-star"><tr><td>
<!--l. 83--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="block" class="equation">
<mrow ><mo
class="MathClass-open">{</mo><mrow><msub><mrow
><mi
>e</mi></mrow><mrow
><mi
>i</mi></mrow></msub
></mrow><mo
class="MathClass-close">}</mo></mrow> <mo
class="MathClass-rel">=</mo> <mrow ><mo
class="MathClass-open">{</mo><mrow><msub><mrow
><mi
>y</mi></mrow><mrow
><mi
>i</mi></mrow></msub
></mrow><mo
class="MathClass-close">}</mo></mrow><mo
class="MathClass-bin">−</mo><mrow ><mo
class="MathClass-open">{</mo><mrow><msub><mrow
><mi
>x</mi></mrow><mrow
><mi
>i</mi></mrow></msub
></mrow><mo
class="MathClass-close">}</mo></mrow><mo
class="MathClass-punc">,</mo>
</math></td></tr></table>
<!--l. 86--><p class="noindent" >can be modeled as a noise process that: (1) is uncorrelated to the input
<!--l. 87--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><mi
>x</mi></math>,
(2) is white, and (3) follows a uniform distribution.
However, this can only be done when the quantization step
<!--l. 90--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><mi
>Δ</mi> <mo
class="MathClass-rel"><</mo><mo
class="MathClass-rel"><</mo> <mstyle
xlink:type="simple" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="https://en.wikipedia.org/wiki/Standard_deviation" class="label" ><msub><mrow
><mi
>σ</mi></mrow><mrow
><mi
>x</mi></mrow></msub
></mstyle><!--endlabel--></math> <span class="cite">[<a
href="#Xvetterli1995wavelets">3</a>]</span>.
</p>
</li></ul>
<!--l. 94--><p class="noindent" >
</p>
<h3 class="sectionHead"><span class="titlemark">3 </span> <a
id="x1-30003"></a>Uniform (lineal) scalar quantization</h3>
<ul class="itemize1">
<li class="itemize">In an uniform quantizer (see Fig. <a
href="#x1-1001r1">1<!--tex4ht:ref: fig:cuantif --></a>), the quantization step
<!--l. 99--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><mrow
><mi
>Δ</mi></mrow></math>
satisfies that the input range is divided into intervals of constant size <table class="equation"><tr><td>
<a
id="x1-3001r4"></a>
<!--l. 101--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="block" class="equation">
<mi
>Δ</mi> <mo
class="MathClass-rel">=</mo> <msub><mrow
><mi
>d</mi></mrow><mrow
><mi
>i</mi><mo
class="MathClass-bin">+</mo><mn>1</mn></mrow></msub
> <mo
class="MathClass-bin">−</mo> <msub><mrow
><mi
>d</mi></mrow><mrow
><mi
>i</mi></mrow></msub
> <mo
class="MathClass-rel">=</mo> <msub><mrow
><mi
>r</mi></mrow><mrow
><mi
>i</mi><mo
class="MathClass-bin">+</mo><mn>1</mn></mrow></msub
> <mo
class="MathClass-bin">−</mo> <msub><mrow
><mi
>r</mi></mrow><mrow
><mi
>i</mi></mrow></msub
><mo
class="MathClass-punc">.</mo>
</math></td><td class="eq-no">(4)</td></tr></table>
</li>
<li class="itemize">We define the number of decision levels as <table class="equation"><tr><td> <a
id="x1-3002r5"></a>
<!--l. 110--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="block" class="equation">
<mi
>Q</mi> <mo
class="MathClass-rel">=</mo> <mfrac><mrow
><msub><mrow
><mi
>d</mi></mrow><mrow
><mstyle
class="text"><mtext >m</mtext></mstyle><mi
>a</mi><mi
>x</mi></mrow></msub
> <mo
class="MathClass-bin">−</mo> <msub><mrow
><mi
>d</mi></mrow><mrow
><mstyle
class="text"><mtext >m</mtext></mstyle><mi
>i</mi><mi
>n</mi></mrow></msub
></mrow>
<mrow
><mi
>Δ</mi></mrow></mfrac> <mo
class="MathClass-punc">.</mo>
</math></td><td class="eq-no">(5)</td></tr></table>
<hr class="figure" /><div class="figure"
><div style="text-align:center;"> <img width=600 src="graphics/midrise.svg" /> </div> <a
id="x1-3003r2"></a>
<a
id="x1-3004"></a>
<br /> <div class="caption"
><span class="id">
Figure 2: : </span><span
class="content">An uniform midrise quantizer (see the <a
href="https://nbviewer.jupyter.org/github/vicente-gonzalez-ruiz/quantization/blob/master/graphics/midrise.ipynb" >notebook</a>).
<!--l. 122--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><mi
>Δ</mi> <mo
class="MathClass-rel">=</mo> <mn>1</mn></math>
and <!--l. 122--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><mi
>Q</mi> <mo
class="MathClass-rel">=</mo> <mn>1</mn><mn>3</mn></math>
(the decision boundaries have been ignored). The decision levels
(<!--l. 122--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><mi
>x</mi></math>)
are <!--l. 122--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><mrow ><mo
class="MathClass-open">{</mo><mrow><mo
class="MathClass-rel">⋯</mo><mspace width="0.3em" class="thinspace"/><mo
class="MathClass-punc">,</mo><mo
class="MathClass-bin">−</mo><mn>3</mn><mo
class="MathClass-punc">,</mo><mo
class="MathClass-bin">−</mo><mn>2</mn><mo
class="MathClass-punc">,</mo><mo
class="MathClass-bin">−</mo><mn>1</mn><mo
class="MathClass-punc">,</mo><mn>0</mn><mo
class="MathClass-punc">,</mo><mn>1</mn><mo
class="MathClass-punc">,</mo><mn>2</mn><mo
class="MathClass-punc">,</mo><mn>3</mn><mo
class="MathClass-punc">,</mo><mo
class="MathClass-rel">⋯</mo><mspace width="0.3em" class="thinspace"/></mrow><mo
class="MathClass-close">}</mo></mrow></math>
and the representation levels (<!--l. 122--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><mi
>y</mi></math>)
are <!--l. 122--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><mrow ><mo
class="MathClass-open">{</mo><mrow><mo
class="MathClass-rel">⋯</mo><mspace width="0.3em" class="thinspace"/><mo
class="MathClass-punc">,</mo><mo
class="MathClass-bin">−</mo><mn>2</mn><mo
class="MathClass-punc">.</mo><mn>5</mn><mo
class="MathClass-punc">,</mo><mo
class="MathClass-bin">−</mo><mn>1</mn><mo
class="MathClass-punc">.</mo><mn>5</mn><mo
class="MathClass-punc">,</mo><mo
class="MathClass-bin">−</mo><mn>0</mn><mo
class="MathClass-punc">.</mo><mn>5</mn><mo
class="MathClass-punc">,</mo><mn>0</mn><mo
class="MathClass-punc">.</mo><mn>5</mn><mo
class="MathClass-punc">,</mo><mn>1</mn><mo
class="MathClass-punc">.</mo><mn>5</mn><mo
class="MathClass-punc">,</mo><mn>2</mn><mo
class="MathClass-punc">.</mo><mn>5</mn><mo
class="MathClass-punc">,</mo><mo
class="MathClass-rel">⋯</mo><mspace width="0.3em" class="thinspace"/></mrow><mo
class="MathClass-close">}</mo></mrow></math>.
</span></div><!--tex4ht:label?: x1-3003r3 -->
</div><hr class="endfigure" />
<hr class="figure" /><div class="figure"
><div style="text-align:center;"> <img width=600 src="graphics/midtread.svg" /> </div> <a
id="x1-3005r3"></a>
<a
id="x1-3006"></a>
<br /> <div class="caption"
><span class="id">
Figure 3: : </span><span
class="content">An uniform midtread quantizer (see the <a
href="https://nbviewer.jupyter.org/github/vicente-gonzalez-ruiz/quantization/blob/master/graphics/midtread.ipynb" >notebook</a>).
<!--l. 133--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><mi
>Δ</mi> <mo
class="MathClass-rel">=</mo> <mn>1</mn></math>
and <!--l. 133--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><mi
>Q</mi> <mo
class="MathClass-rel">=</mo> <mn>1</mn><mn>2</mn></math>
(the decision boundaries have been ignored). The decision levels
(<!--l. 133--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><mi
>x</mi></math>)
are <!--l. 133--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><mrow ><mo
class="MathClass-open">{</mo><mrow><mo
class="MathClass-rel">⋯</mo><mspace width="0.3em" class="thinspace"/><mo
class="MathClass-punc">,</mo><mo
class="MathClass-bin">−</mo><mn>2</mn><mo
class="MathClass-punc">.</mo><mn>5</mn><mo
class="MathClass-punc">,</mo><mo
class="MathClass-bin">−</mo><mn>1</mn><mo
class="MathClass-punc">.</mo><mn>5</mn><mo
class="MathClass-punc">,</mo><mo
class="MathClass-bin">−</mo><mn>0</mn><mo
class="MathClass-punc">.</mo><mn>5</mn><mo
class="MathClass-punc">,</mo><mn>0</mn><mo
class="MathClass-punc">.</mo><mn>5</mn><mo
class="MathClass-punc">,</mo><mn>1</mn><mo
class="MathClass-punc">.</mo><mn>5</mn><mo
class="MathClass-punc">,</mo><mn>2</mn><mo
class="MathClass-punc">.</mo><mn>5</mn><mo
class="MathClass-punc">,</mo><mo
class="MathClass-rel">⋯</mo><mspace width="0.3em" class="thinspace"/></mrow><mo
class="MathClass-close">}</mo></mrow></math>
and the representation levels (<!--l. 133--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><mi
>y</mi></math>)
are <!--l. 133--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><mrow ><mo
class="MathClass-open">{</mo><mrow><mo
class="MathClass-rel">⋯</mo><mspace width="0.3em" class="thinspace"/><mo
class="MathClass-punc">,</mo><mo
class="MathClass-bin">−</mo><mn>2</mn><mo
class="MathClass-punc">,</mo><mo
class="MathClass-bin">−</mo><mn>1</mn><mo
class="MathClass-punc">,</mo><mo
class="MathClass-bin">−</mo><mn>0</mn><mo
class="MathClass-punc">,</mo><mn>1</mn><mo
class="MathClass-punc">,</mo><mn>2</mn><mo
class="MathClass-punc">,</mo><mo
class="MathClass-rel">⋯</mo><mspace width="0.3em" class="thinspace"/></mrow><mo
class="MathClass-close">}</mo></mrow></math>.
</span></div><!--tex4ht:label?: x1-3005r3 -->
</div><hr class="endfigure" />
<hr class="figure" /><div class="figure"
><div style="text-align:center;"> <img width=600 src="graphics/deadzone.svg" /> </div> <a
id="x1-3007r4"></a>
<a
id="x1-3008"></a>
<br /> <div class="caption"
><span class="id">
Figure 4: : </span><span
class="content">An uniform deadzone quantizer (see the <a
href="https://nbviewer.jupyter.org/github/vicente-gonzalez-ruiz/quantization/blob/master/graphics/deadzone.ipynb" >notebook</a>).
<!--l. 144--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><mi
>Δ</mi> <mo
class="MathClass-rel">=</mo> <mn>1</mn></math>
and <!--l. 144--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><mi
>Q</mi> <mo
class="MathClass-rel">=</mo> <mn>1</mn><mn>2</mn></math>
(the decision boundaries have been ignored). The decision levels
(<!--l. 144--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><mi
>x</mi></math>)
are <!--l. 144--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><mrow ><mo
class="MathClass-open">{</mo><mrow><mo
class="MathClass-rel">⋯</mo><mspace width="0.3em" class="thinspace"/><mo
class="MathClass-punc">,</mo><mo
class="MathClass-bin">−</mo><mn>3</mn><mo
class="MathClass-punc">,</mo><mo
class="MathClass-bin">−</mo><mn>2</mn><mo
class="MathClass-punc">,</mo><mo
class="MathClass-bin">−</mo><mn>1</mn><mo
class="MathClass-punc">,</mo><mn>1</mn><mo
class="MathClass-punc">,</mo><mn>2</mn><mo
class="MathClass-punc">,</mo><mn>3</mn><mo
class="MathClass-punc">,</mo><mo
class="MathClass-rel">⋯</mo><mspace width="0.3em" class="thinspace"/></mrow><mo
class="MathClass-close">}</mo></mrow></math>
and the representation levels (<!--l. 144--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><mi
>y</mi></math>)
are <!--l. 144--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><mrow ><mo
class="MathClass-open">{</mo><mrow><mo
class="MathClass-rel">⋯</mo><mspace width="0.3em" class="thinspace"/><mo
class="MathClass-punc">,</mo><mo
class="MathClass-bin">−</mo><mn>2</mn><mo
class="MathClass-punc">,</mo><mo
class="MathClass-bin">−</mo><mn>1</mn><mo
class="MathClass-punc">,</mo><mo
class="MathClass-bin">−</mo><mn>0</mn><mo
class="MathClass-punc">,</mo><mn>1</mn><mo
class="MathClass-punc">,</mo><mn>2</mn><mo
class="MathClass-punc">,</mo><mo
class="MathClass-rel">⋯</mo><mspace width="0.3em" class="thinspace"/></mrow><mo
class="MathClass-close">}</mo></mrow></math>.
</span></div><!--tex4ht:label?: x1-3007r3 -->
</div><hr class="endfigure" />
</li>
<li class="itemize">Under the premise that <!--l. 149--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><mi
>e</mi></math>
is uniform, and considering that
<!--l. 150--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><msub><mrow
><mi
>y</mi></mrow><mrow
><mi
>i</mi></mrow></msub
> <mo
class="MathClass-rel">=</mo> <mrow ><mo
class="MathClass-open">(</mo><mrow><msub><mrow
><mi
>x</mi></mrow><mrow
><mi
>i</mi><mo
class="MathClass-bin">−</mo><mn>1</mn></mrow></msub
> <mo
class="MathClass-bin">+</mo> <msub><mrow
><mi
>x</mi></mrow><mrow
><mi
>i</mi></mrow></msub
></mrow><mo
class="MathClass-close">)</mo></mrow><mo
class="MathClass-bin">∕</mo><mn>2</mn></math> (something quite
reasonable when <!--l. 150--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><mi
>x</mi></math>
can also be considered as uniform) the average quantization error is
<!--l. 152--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><mfrac><mrow
><mi
>Z</mi></mrow>
<mrow
><mn>4</mn></mrow></mfrac> </math>
(<!--l. 152--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><mfrac><mrow
><mi
>Z</mi></mrow>
<mrow
><mn>2</mn></mrow></mfrac> </math> is the
meximum and <!--l. 152--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><mn>0</mn></math>
is the minimum), and for this particular case <table class="equation"><tr><td> <a
id="x1-3009r6"></a>
<!--l. 154--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="block" class="equation">
<mstyle
class="text"><mtext >MSE</mtext></mstyle> <mo
class="MathClass-rel">=</mo> <mfrac><mrow
><mn>1</mn></mrow>
<mrow
><mi
>Δ</mi></mrow></mfrac><msubsup><mrow
><mo
>∫
<!--nolimits--></mo><!--nolimits--></mrow><mrow
><mo
class="MathClass-bin">−</mo><mi
>Δ</mi><mo
class="MathClass-bin">∕</mo><mn>2</mn></mrow><mrow
><mi
>Δ</mi><mo
class="MathClass-bin">∕</mo><mn>2</mn></mrow></msubsup
><msup><mrow
><mi
>e</mi></mrow><mrow
><mn>2</mn></mrow></msup
><mi
>d</mi><mi
>e</mi> <mo
class="MathClass-rel">=</mo> <mfrac><mrow
><msup><mrow
><mi
>Δ</mi></mrow><mrow
><mn>2</mn></mrow></msup
></mrow>
<mrow
><mn>1</mn><mn>2</mn></mrow></mfrac> <mo
class="MathClass-punc">.</mo>
</math></td><td class="eq-no">(6)</td></tr></table>
</li>
<li class="itemize">Uniform quantizers are used in most A/D (analogic/digital) converters, were
it is expected the generation of uniformely distributed sequences of
samples.
</li></ul>
<!--l. 167--><p class="noindent" >
</p>
<h4 class="subsectionHead"><span class="titlemark">3.1 </span> <a
id="x1-40003.1"></a>Using codewords (encoding)</h4>
<ul class="itemize1">
<li class="itemize">Depending on the number of <!--l. 172--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><mrow
><mi
>Q</mi></mrow></math>
different possible values (or <span
class="ecti-1000">bins</span>) for <!--l. 173--><math
xmlns="http://www.w3.org/1998/Math/MathML"
display="inline" ><mrow
><mi
>s</mi><mrow ><mo