SherbyNet - /page/wikibook/entry.mu

<!DOCTYPE html>

<title>Quantization (signal processing)</title>

</head>

Quantization (signal processing)

</h1>

</header>

Quantization, in mathematics and <a href="Digital_signal_processing" title="Digital signal processing">digital signal processing</a>, is the process of mapping input values from a large set (often a continuous set) to output values in a (countable) smaller set, often with a finite <a href="Number_of_elements" class="mw-redirect" title="Number of elements">number of elements</a>. <a href="Rounding" title="Rounding">Rounding</a> and <a href="Truncation" title="Truncation">truncation</a> are typical examples of quantization processes. Quantization is involved to some degree in nearly all digital signal processing, as the process of representing a signal in digital form ordinarily involves rounding. Quantization also forms the core of essentially all <a href="Lossy_compression" title="Lossy compression">lossy compression</a> algorithms.

The difference between an input value and its quantized value (such as <a href="Round-off_error" title="Round-off error">round-off error</a>) is referred to as quantization error, noise or distortion. A device or <a href="Algorithm_function" class="mw-redirect" title="Algorithm function">algorithmic function</a> that performs quantization is called a quantizer. An <a href="Analog-to-digital_converter" title="Analog-to-digital converter">analog-to-digital converter</a> is an example of a quantizer.

<div class="mw-heading mw-heading2"><h2 id="Example">Example</h2></div>

For example, <a href="Rounding#Round_half_up" title="Rounding">rounding</a> a <a href="Real_number" title="Real number">real number</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> 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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta }</annotation>

</semantics>

</math><img src="./32769037c408874e1890f77554c65f39c523ebe2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \Delta }" loading="lazy"> can be expressed as

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q(x)=\Delta \cdot \left\lfloor {\frac {x}{\Delta }}+{\frac {1}{2}}\right\rfloor }</annotation>

</semantics>

</math><img src="./953f2b96d64a62e07c90e47cff07b22cfe2cdd85.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.012ex; height:6.176ex;" alt="{\displaystyle Q(x)=\Delta \cdot \left\lfloor {\frac {x}{\Delta }}+{\frac {1}{2}}\right\rfloor }" loading="lazy">,</dd></dl>

where the notation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lfloor \ \rfloor }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lfloor \ \rfloor }</annotation>

</semantics>

</math><img src="./a5c5f3c09db405aa8eeb516b1281a0af12c5633b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.645ex; height:2.843ex;" alt="{\displaystyle \lfloor \ \rfloor }" loading="lazy"> denotes the <a href="Floor_function" class="mw-redirect" title="Floor function">floor function</a>.

Alternatively, the same quantizer may be expressed in terms of the <a href="Ceiling_function" class="mw-redirect" title="Ceiling function">ceiling function</a>, as

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q(x)=\Delta \cdot \left\lceil {\frac {x}{\Delta }}-{\frac {1}{2}}\right\rceil }</annotation>

</semantics>

</math><img src="./589717e3717d0cced093c0af004dc20ffbe02f58.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.012ex; height:6.176ex;" alt="{\displaystyle Q(x)=\Delta \cdot \left\lceil {\frac {x}{\Delta }}-{\frac {1}{2}}\right\rceil }" loading="lazy">.</dd></dl>

(The notation <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lceil \ \rceil }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lceil \ \rceil }</annotation>

</semantics>

</math><img src="./7cd9da603e62fe9d5acb19777c56b59d13c3d2de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:2.645ex; height:2.843ex;" alt="{\displaystyle \lceil \ \rceil }" loading="lazy"> denotes the ceiling function).

The essential property of a quantizer is having a countable set of possible output values 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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta }</annotation>

</semantics>

</math><img src="./32769037c408874e1890f77554c65f39c523ebe2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \Delta }" loading="lazy"> is equal to 1. With <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta =1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta =1}</annotation>

</semantics>

</math><img src="./cf46f3fc2f930287a56caef6549a2909c3978fbd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.197ex; height:2.176ex;" alt="{\displaystyle \Delta =1}" loading="lazy"> or with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta }</annotation>

</semantics>

</math><img src="./32769037c408874e1890f77554c65f39c523ebe2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \Delta }" loading="lazy"> 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 <a href="Mean_squared_error" title="Mean squared error">mean squared error</a> produced by such a rounding operation will be approximately <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta ^{2}/12}">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta ^{2}/12}</annotation>

</semantics>

</math><img src="./34b5be83e88394057daa090c25107b1b57adb48a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.477ex; height:3.176ex;" alt="{\displaystyle \Delta ^{2}/12}" loading="lazy">.<a href="#cite_note-Sheppard-1">[1]</a><a href="#cite_note-Bennett-2">[2]</a><a href="#cite_note-OliverPierceShannon-3">[3]</a><a href="#cite_note-Stein-4">[4]</a><a href="#cite_note-GishPierce-5">[5]</a><a href="#cite_note-GrayNeuhoff-6">[6]</a> 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 <style data-mw-deduplicate="TemplateStyles:r1214402035">

/* start https://en.wikipedia.org/ */

.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num{display:block;line-height:1em;margin:0.0em 0.1em;border-bottom:1px solid}.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0.1em 0.1em}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);clip-path:polygon(0px 0px,0px 0px,0px 0px);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}

/* end https://en.wikipedia.org/ */

</style>⁠1/4⁠. In terms of <a href="Decibel" title="Decibel">decibels</a>, the noise power change is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle 10\cdot \log _{10}(1/4)\ \approx \ -6\ \mathrm {dB} .}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \scriptstyle 10\cdot \log _{10}(1/4)\ \approx \ -6\ \mathrm {dB} .}</annotation>

</semantics>

</math><img src="./6c8fd999c7f6991821f8d2fe533d63994d6de7ae.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.103ex; height:2.176ex;" alt="{\displaystyle \scriptstyle 10\cdot \log _{10}(1/4)\ \approx \ -6\ \mathrm {dB} .}" loading="lazy">

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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> and the reconstruction stage maps the index <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> to the reconstruction value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y_{k}}</annotation>

</semantics>

</math><img src="./4b2ab0248723a410cc2c67ce06ad5c043dcbb933.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.228ex; height:2.009ex;" alt="{\displaystyle y_{k}}" loading="lazy"> that is the output approximation of the input value. For the example uniform quantizer described above, the forward quantization stage can be expressed as

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k=\left\lfloor {\frac {x}{\Delta }}+{\frac {1}{2}}\right\rfloor }</annotation>

</semantics>

</math><img src="./e2dabc358baa4191891f0933adcfb4040bc67d33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:14.631ex; height:6.176ex;" alt="{\displaystyle k=\left\lfloor {\frac {x}{\Delta }}+{\frac {1}{2}}\right\rfloor }" loading="lazy">,</dd></dl>

and the reconstruction stage for this example quantizer is simply

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y_{k}=k\cdot \Delta }</annotation>

</semantics>

</math><img src="./4613add6fc8632fbf47861b25cec9fc8388f7339.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.153ex; height:2.509ex;" alt="{\displaystyle y_{k}=k\cdot \Delta }" loading="lazy">.</dd></dl>

This decomposition is useful for the design and analysis of quantization behavior, and it illustrates how the quantized data can be communicated over a <a href="Communication_channel" title="Communication channel">communication channel</a> – a source encoder can perform the forward quantization stage and send the index information through a communication channel, and a decoder can perform the reconstruction stage to produce the output approximation of the original input data. In general, the forward quantization stage may use any function that maps the input data to the integer space of the quantization index data, and the inverse quantization stage can conceptually (or literally) be a table look-up operation to map each quantization index to a corresponding reconstruction value. This two-stage decomposition applies equally well to <a href="Vector_quantization" title="Vector quantization">vector</a> as well as scalar quantizers.

<div class="mw-heading mw-heading2"><h2 id="Mathematical_properties">Mathematical properties</h2></div>

Because quantization is a many-to-few mapping, it is an inherently <a href="Non-linear" class="mw-redirect" title="Non-linear">non-linear</a> and irreversible process (i.e., because the same output value is shared by multiple input values, it is impossible, in general, to recover the exact input value when given only the output value).

The set of possible input values may be infinitely large, and may possibly be continuous and therefore <a href="Uncountable" class="mw-redirect" title="Uncountable">uncountable</a> (such as the set of all real numbers, or all real numbers within some limited range). The set of possible output values may be <a href="Finite_set" title="Finite set">finite</a> or <a href="Countably_infinite" class="mw-redirect" title="Countably infinite">countably infinite</a>.<a href="#cite_note-GrayNeuhoff-6">[6]</a> The input and output sets involved in quantization can be defined in a rather general way. For example, vector quantization is the application of quantization to multi-dimensional (vector-valued) input data.<a href="#cite_note-7">[7]</a>

<div class="mw-heading mw-heading2"><h2 id="Types">Types</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Analog-to-digital_converter">Analog-to-digital converter</h3></div>

An <a href="Analog-to-digital_converter" title="Analog-to-digital converter">analog-to-digital converter</a> (ADC) can be modeled as two processes: <a href="Sampling_(signal_processing)" title="Sampling (signal processing)">sampling</a> and quantization. Sampling converts a time-varying voltage signal into a <a href="Discrete-time_signal" class="mw-redirect" title="Discrete-time signal">discrete-time signal</a>, a sequence of real numbers. Quantization replaces each real number with an approximation from a finite set of discrete values. Most commonly, these discrete values are represented as fixed-point words. Though any number of quantization levels is possible, common word lengths are <a href="Audio_bit_depth" title="Audio bit depth">8-bit</a> (256 levels), 16-bit (65,536 levels) and 24-bit (16.8 million levels). Quantizing a sequence of numbers produces a sequence of quantization errors which is sometimes modeled as an additive random signal called quantization noise because of its <a href="Stochastic" title="Stochastic">stochastic</a> behavior. The more levels a quantizer uses, the lower is its quantization noise power.

<div class="mw-heading mw-heading3"><h3 id="Rate–distortion_optimization">Rate–distortion optimization</h3></div>

<a href="Rate%E2%80%93distortion_theory" title="Rate–distortion theory">Rate–distortion optimized</a> quantization is encountered in <a href="Source_coding" class="mw-redirect" title="Source coding">source coding</a> for lossy data compression algorithms, where the purpose is to manage distortion within the limits of the <a href="Bit_rate" title="Bit rate">bit rate</a> supported by a communication channel or storage medium. The analysis of quantization in this context involves studying the amount of data (typically measured in digits or bits or bit rate) that is used to represent the output of the quantizer and studying the loss of precision that is introduced by the quantization process (which is referred to as the distortion).

<div class="mw-heading mw-heading3"><h3 id="Mid-riser_and_mid-tread_uniform_quantizers">Mid-riser and mid-tread uniform quantizers</h3></div>

Most uniform quantizers for signed input data can be classified as being of one of two types: mid-riser and mid-tread. The terminology is based on what happens in the region around the value 0, and uses the analogy of viewing the input-output function of the quantizer as a <a href="Stairway" class="mw-redirect" title="Stairway">stairway</a>. Mid-tread quantizers have a zero-valued reconstruction level (corresponding to a tread of a stairway), while mid-riser quantizers have a zero-valued classification threshold (corresponding to a <a href="Stair_riser" class="mw-redirect" title="Stair riser">riser</a> of a stairway).<a href="#cite_note-Gersho77-9">[9]</a>

Mid-tread quantization involves rounding. The formulas for mid-tread uniform quantization are provided in the previous section.

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q(x)=\Delta \cdot \left\lfloor {\frac {x}{\Delta }}+{\frac {1}{2}}\right\rfloor }</annotation>

</semantics>

</math><img src="./953f2b96d64a62e07c90e47cff07b22cfe2cdd85.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:22.012ex; height:6.176ex;" alt="{\displaystyle Q(x)=\Delta \cdot \left\lfloor {\frac {x}{\Delta }}+{\frac {1}{2}}\right\rfloor }" loading="lazy">,</dd></dl>

Mid-riser quantization involves truncation. The input-output formula for a mid-riser uniform quantizer is given by:

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Q(x)=\Delta \cdot \left(\left\lfloor {\frac {x}{\Delta }}\right\rfloor +{\frac {1}{2}}\right)}</annotation>

</semantics>

</math><img src="./e294457483e180cf6b618167c48076fd31e58194.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:25.178ex; height:6.176ex;" alt="{\displaystyle Q(x)=\Delta \cdot \left(\left\lfloor {\frac {x}{\Delta }}\right\rfloor +{\frac {1}{2}}\right)}" loading="lazy">,</dd></dl>

where the classification rule is given by

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k=\left\lfloor {\frac {x}{\Delta }}\right\rfloor }</annotation>

</semantics>

</math><img src="./9db48424d238d4426587160971975b052e9d65bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.005ex; width:9.537ex; height:5.009ex;" alt="{\displaystyle k=\left\lfloor {\frac {x}{\Delta }}\right\rfloor }" loading="lazy"></dd></dl>

and the reconstruction rule is

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y_{k}=\Delta \cdot \left(k+{\tfrac {1}{2}}\right)}</annotation>

</semantics>

</math><img src="./df40d7422579750cb074a226966e8b9c8731280a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:16.781ex; height:3.509ex;" alt="{\displaystyle y_{k}=\Delta \cdot \left(k+{\tfrac {1}{2}}\right)}" loading="lazy">.</dd></dl>

Note that mid-riser uniform quantizers do not have a zero output value – their minimum output magnitude is half the step size. In contrast, mid-tread quantizers do have a zero output level. For some applications, having a zero output signal representation may be a necessity.

In general, a mid-riser or mid-tread quantizer may not actually be a uniform quantizer – i.e., the size of the quantizer's classification <a href="Interval_(mathematics)" title="Interval (mathematics)">intervals</a> may not all be the same, or the spacing between its possible output values may not all be the same. The distinguishing characteristic of a mid-riser quantizer is that it has a classification threshold value that is exactly zero, and the distinguishing characteristic of a mid-tread quantizer is that is it has a reconstruction value that is exactly zero.<a href="#cite_note-Gersho77-9">[9]</a>

<div class="mw-heading mw-heading3"><h3 id="Dead-zone_quantizers">Dead-zone quantizers</h3></div>

A dead-zone quantizer is a type of mid-tread quantizer with symmetric behavior around 0. The region around the zero output value of such a quantizer is referred to as the dead zone or <a href="Deadband" title="Deadband">deadband</a>. The dead zone can sometimes serve the same purpose as a <a href="Noise_gate" title="Noise gate">noise gate</a> or <a href="Squelch" title="Squelch">squelch</a> function. Especially for compression applications, the dead-zone may be given a different width than that for the other steps. For an otherwise-uniform quantizer, the dead-zone width can be set to any value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w}</annotation>

</semantics>

</math><img src="./88b1e0c8e1be5ebe69d18a8010676fa42d7961e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.664ex; height:1.676ex;" alt="{\displaystyle w}" loading="lazy"> by using the forward quantization rule<a href="#cite_note-10">[10]</a><a href="#cite_note-11">[11]</a><a href="#cite_note-SullivanIT-12">[12]</a>

<mrow>

<mrow>

<mrow>

<mrow>

<mfrac>

<mrow>

</mrow>

</mrow>

</mrow>

</mfrac>

</mrow>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k=\operatorname {sgn}(x)\cdot \max \left(0,\left\lfloor {\frac {\left|x\right|-w/2}{\Delta }}+1\right\rfloor \right)}</annotation>

</semantics>

</math><img src="./009afc81fefff3cd64b9a8d17bc18a714d49f547.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.505ex; width:39.832ex; height:6.343ex;" alt="{\displaystyle k=\operatorname {sgn}(x)\cdot \max \left(0,\left\lfloor {\frac {\left|x\right|-w/2}{\Delta }}+1\right\rfloor \right)}" loading="lazy">,</dd></dl>

where the function <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sgn} }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {sgn} }</annotation>

</semantics>

</math><img src="./ec838dfd8a4a659b2877f93a6b53f22fc7777d07.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.371ex; height:2.009ex;" alt="{\displaystyle \operatorname {sgn} }" loading="lazy">( ) is the <a href="Sign_function" title="Sign function">sign function</a> (also known as the signum function). The general reconstruction rule for such a dead-zone quantizer is given by

<msub>

</mrow>

</msub>

<mrow>

<mrow>

<mfrac>

</mfrac>

</mrow>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y_{k}=\operatorname {sgn}(k)\cdot \left({\frac {w}{2}}+\Delta \cdot (|k|-1+r_{k})\right)}</annotation>

</semantics>

</math><img src="./4a361f041f33dd3ceaa2f30b90671bd16d5bdb77.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.838ex; width:38.423ex; height:4.843ex;" alt="{\displaystyle y_{k}=\operatorname {sgn}(k)\cdot \left({\frac {w}{2}}+\Delta \cdot (|k|-1+r_{k})\right)}" loading="lazy">,</dd></dl>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r_{k}}</annotation>

</semantics>

</math><img src="./9b28e0e640d099f3676330bd4f604ae15c37bb4f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.137ex; height:2.009ex;" alt="{\displaystyle r_{k}}" loading="lazy"> is a reconstruction offset value in the range of 0 to 1 as a fraction of the step size. Ordinarily, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle 0\leq r_{k}\leq {\tfrac {1}{2}}}">

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle 0\leq r_{k}\leq {\tfrac {1}{2}}}</annotation>

</semantics>

</math><img src="./a5e4d14e88315ce3a9b5d58eed4cf10129b391b3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:11.155ex; height:3.509ex;" alt="{\displaystyle 0\leq r_{k}\leq {\tfrac {1}{2}}}" loading="lazy"> when quantizing input data with a typical <a href="Probability_density_function" title="Probability density function">probability density function</a> (PDF) that is symmetric around zero and reaches its peak value at zero (such as a <a href="Gaussian_distribution" class="mw-redirect" title="Gaussian distribution">Gaussian</a>, <a href="Laplacian_distribution" class="mw-redirect" title="Laplacian distribution">Laplacian</a>, or <a href="Generalized_Gaussian_distribution" class="mw-redirect" title="Generalized Gaussian distribution">generalized Gaussian</a> PDF). Although <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r_{k}}</annotation>

</semantics>

</math><img src="./9b28e0e640d099f3676330bd4f604ae15c37bb4f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.137ex; height:2.009ex;" alt="{\displaystyle r_{k}}" loading="lazy"> may depend on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> in general and can be chosen to fulfill the optimality condition described below, it is often simply set to a constant, such as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\tfrac {1}{2}}}">

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\tfrac {1}{2}}}</annotation>

</semantics>

</math><img src="./edef8290613648790a8ac1a95c2fb7c3972aea2f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:1.658ex; height:3.509ex;" alt="{\displaystyle {\tfrac {1}{2}}}" loading="lazy">. (Note that in this definition, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{0}=0}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y_{0}=0}</annotation>

</semantics>

</math><img src="./f952fc15fe931e15f2f4a766b3ce68dc52f64842.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:6.454ex; height:2.509ex;" alt="{\displaystyle y_{0}=0}" loading="lazy"> due to the definition of the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {sgn} }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \operatorname {sgn} }</annotation>

</semantics>

</math><img src="./ec838dfd8a4a659b2877f93a6b53f22fc7777d07.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.371ex; height:2.009ex;" alt="{\displaystyle \operatorname {sgn} }" loading="lazy">( ) function, so <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{0}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r_{0}}</annotation>

</semantics>

</math><img src="./fb12fcfddb65e3d1e6a044215f6e833f0cd4337b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.103ex; height:2.009ex;" alt="{\displaystyle r_{0}}" loading="lazy"> has no effect.)

A very commonly used special case (e.g., the scheme typically used in financial accounting and elementary mathematics) is to set <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w=\Delta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w=\Delta }</annotation>

</semantics>

</math><img src="./9956a3e4f12e1d7ed19e3bf9d0b7024fb3974225.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.698ex; height:2.176ex;" alt="{\displaystyle w=\Delta }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r_{k}={\tfrac {1}{2}}}">

<msub>

</mrow>

</msub>

<mfrac>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle r_{k}={\tfrac {1}{2}}}</annotation>

</semantics>

</math><img src="./2dd7cac862a69cf03af1e547d1359270cba0173d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:6.894ex; height:3.509ex;" alt="{\displaystyle r_{k}={\tfrac {1}{2}}}" loading="lazy"> for all <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">. In this case, the dead-zone quantizer is also a uniform quantizer, since the central dead-zone of this quantizer has the same width as all of its other steps, and all of its reconstruction values are equally spaced as well.

<div class="mw-heading mw-heading2"><h2 id="Noise_and_error_characteristics">Noise and error characteristics</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Additive_noise_model">Additive noise model</h3></div>

A common assumption for the analysis of quantization error is that it affects a signal processing system in a similar manner to that of additive <a href="White_noise" title="White noise">white noise</a> – having negligible correlation with the signal and an approximately flat <a href="Power_spectral_density" class="mw-redirect" title="Power spectral density">power spectral density</a>.<a href="#cite_note-Bennett-2">[2]</a><a href="#cite_note-GrayNeuhoff-6">[6]</a><a href="#cite_note-Widrow1-13">[13]</a><a href="#cite_note-Widrow2-14">[14]</a> The additive noise model is commonly used for the analysis of quantization error effects in digital filtering systems, and it can be very useful in such analysis. It has been shown to be a valid model in cases of high-resolution quantization (small <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta }</annotation>

</semantics>

</math><img src="./32769037c408874e1890f77554c65f39c523ebe2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \Delta }" loading="lazy"> relative to the signal strength) with smooth PDFs.<a href="#cite_note-Bennett-2">[2]</a><a href="#cite_note-MarcoNeuhoff-15">[15]</a>

Additive noise behavior is not always a valid assumption. Quantization error (for quantizers defined as described here) is deterministically related to the signal and not entirely independent of it. Thus, periodic signals can create periodic quantization noise. And in some cases, it can even cause <a href="Limit_cycle" title="Limit cycle">limit cycles</a> to appear in digital signal processing systems. One way to ensure effective independence of the quantization error from the source signal is to perform <a href="Dither" title="Dither">dithered</a> quantization (sometimes with <a href="Noise_shaping" title="Noise shaping">noise shaping</a>), which involves adding random (or <a href="Pseudo-random" class="mw-redirect" title="Pseudo-random">pseudo-random</a>) noise to the signal prior to quantization.<a href="#cite_note-GrayNeuhoff-6">[6]</a><a href="#cite_note-Widrow2-14">[14]</a>

<div class="mw-heading mw-heading3"><h3 id="Quantization_error_models">Quantization error models</h3></div>

In the typical case, the original signal is much larger than one <a href="Least_significant_bit" class="mw-redirect" title="Least significant bit">least significant bit</a> (LSB). When this is the case, the quantization error is not significantly correlated with the signal and has an approximately <a href="Uniform_distribution_(continuous)" class="mw-redirect" title="Uniform distribution (continuous)">uniform distribution</a>. When rounding is used to quantize, the quantization error has a <a href="Mean" title="Mean">mean</a> of zero and the <a href="Root_mean_square" title="Root mean square">root mean square</a> (RMS) value is the <a href="Standard_deviation" title="Standard deviation">standard deviation</a> of this distribution, given by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle {\frac {1}{\sqrt {12}}}\mathrm {LSB} \ \approx \ 0.289\,\mathrm {LSB} }">

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\frac {1}{\sqrt {12}}}\mathrm {LSB} \ \approx \ 0.289\,\mathrm {LSB} }</annotation>

</semantics>

</math><img src="./c72e40c7f9413dca8ed4f297a8c6845fb789afd6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:16.064ex; height:3.676ex;" alt="{\displaystyle \scriptstyle {\frac {1}{\sqrt {12}}}\mathrm {LSB} \ \approx \ 0.289\,\mathrm {LSB} }" loading="lazy">. When truncation is used, the error has a non-zero mean of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle {\frac {1}{2}}\mathrm {LSB} }">

<mfrac>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\frac {1}{2}}\mathrm {LSB} }</annotation>

</semantics>

</math><img src="./cfd195bb6e3343a60b349bd85cabd90b08578f09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.608ex; height:3.176ex;" alt="{\displaystyle \scriptstyle {\frac {1}{2}}\mathrm {LSB} }" loading="lazy"> and the RMS value is <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle {\frac {1}{\sqrt {3}}}\mathrm {LSB} }">

<mfrac>

<msqrt>

</msqrt>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\frac {1}{\sqrt {3}}}\mathrm {LSB} }</annotation>

</semantics>

</math><img src="./0f19eb3947c3d3dd9d73dd8025e4f750c5015466.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.671ex; width:5.72ex; height:3.676ex;" alt="{\displaystyle \scriptstyle {\frac {1}{\sqrt {3}}}\mathrm {LSB} }" loading="lazy">. Although rounding yields less RMS error than truncation, the difference is only due to the static (DC) term of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle {\frac {1}{2}}\mathrm {LSB} }">

<mfrac>

</mfrac>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \scriptstyle {\frac {1}{2}}\mathrm {LSB} }</annotation>

</semantics>

</math><img src="./cfd195bb6e3343a60b349bd85cabd90b08578f09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:4.608ex; height:3.176ex;" alt="{\displaystyle \scriptstyle {\frac {1}{2}}\mathrm {LSB} }" loading="lazy">. The RMS values of the AC error are exactly the same in both cases, so there is no special advantage of rounding over truncation in situations where the DC term of the error can be ignored (such as in AC-coupled systems). In either case, the standard deviation, as a percentage of the full signal range, changes by a factor of 2 for each 1-bit change in the number of quantization bits. The potential signal-to-quantization-noise power ratio therefore changes by 4, or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \scriptstyle 10\cdot \log _{10}(4)}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \scriptstyle 10\cdot \log _{10}(4)}</annotation>

</semantics>

</math><img src="./b09a0a45232f3ac39635982a95e3990b3b38c135.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.803ex; height:2.176ex;" alt="{\displaystyle \scriptstyle 10\cdot \log _{10}(4)}" loading="lazy">, approximately 6 dB per bit.

At lower amplitudes the quantization error becomes dependent on the input signal, resulting in distortion. This distortion is created after the anti-aliasing filter, and if these distortions are above 1/2 the sample rate they will alias back into the band of interest. In order to make the quantization error independent of the input signal, the signal is dithered by adding noise to the signal. This slightly reduces signal-to-noise ratio, but can completely eliminate the distortion.

<div class="mw-heading mw-heading3"><h3 id="Quantization_noise_model">Quantization noise model</h3></div>

Quantization noise is a <a href="Model_(abstract)" class="mw-redirect" title="Model (abstract)">model</a> of quantization error introduced by quantization in the ADC. It is a rounding error between the analog input voltage to the ADC and the output digitized value. The noise is non-linear and signal-dependent. It can be modeled in several different ways.

In an ideal ADC, where the quantization error is uniformly distributed between −1/2 LSB and +1/2 LSB, and the signal has a uniform distribution covering all quantization levels, the <a href="Signal-to-quantization-noise_ratio" title="Signal-to-quantization-noise ratio">Signal-to-quantization-noise ratio</a> (SQNR) can be calculated from

</mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {SQNR} =20\log _{10}(2^{Q})\approx 6.02\cdot Q\ \mathrm {dB} \,\!}</annotation>

</semantics>

</math><img src="./28d7ed262342c70036f3bde52bba94f1e19547fc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; margin-right: -0.387ex; width:36.373ex; height:3.176ex;" alt="{\displaystyle \mathrm {SQNR} =20\log _{10}(2^{Q})\approx 6.02\cdot Q\ \mathrm {dB} \,\!}" loading="lazy"></dd></dl>

where Q is the number of quantization bits.

The most common test signals that fulfill this are full amplitude <a href="Triangle_wave" title="Triangle wave">triangle waves</a> and <a href="Sawtooth_wave" title="Sawtooth wave">sawtooth waves</a>.

For example, a <a href="16-bit" class="mw-redirect" title="16-bit">16-bit</a> ADC has a maximum signal-to-quantization-noise ratio of 6.02 × 16 = 96.3 dB.

When the input signal is a full-amplitude <a href="Sine_wave" title="Sine wave">sine wave</a> the distribution of the signal is no longer uniform, and the corresponding equation is instead

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {SQNR} \approx 1.761+6.02\cdot Q\ \mathrm {dB} \,\!}</annotation>

</semantics>

</math><img src="./76bbd933a0dd004b25659cecaf61bf955372bd7a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-right: -0.387ex; width:29.347ex; height:2.509ex;" alt="{\displaystyle \mathrm {SQNR} \approx 1.761+6.02\cdot Q\ \mathrm {dB} \,\!}" loading="lazy"></dd></dl>

Here, the quantization noise is once again assumed to be uniformly distributed. When the input signal has a high amplitude and a wide frequency spectrum this is the case.<a href="#cite_note-16">[16]</a> In this case a 16-bit ADC has a maximum signal-to-noise ratio of 98.09 dB. The 1.761 difference in signal-to-noise only occurs due to the signal being a full-scale sine wave instead of a triangle or sawtooth.

For complex signals in high-resolution ADCs this is an accurate model. For low-resolution ADCs, low-level signals in high-resolution ADCs, and for simple waveforms the quantization noise is not uniformly distributed, making this model inaccurate.<a href="#cite_note-17">[17]</a> In these cases the quantization noise distribution is strongly affected by the exact amplitude of the signal.

The calculations are relative to full-scale input. For smaller signals, the relative quantization distortion can be very large. To circumvent this issue, analog <a href="Companding" title="Companding">companding</a> can be used, but this can introduce distortion.

<div class="mw-heading mw-heading2"><h2 id="Design">Design</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Granular_distortion_and_overload_distortion">Granular distortion and overload distortion</h3></div>

Often the design of a quantizer involves supporting only a limited range of possible output values and performing clipping to limit the output to this range whenever the input exceeds the supported range. The error introduced by this clipping is referred to as overload distortion. Within the extreme limits of the supported range, the amount of spacing between the selectable output values of a quantizer is referred to as its granularity, and the error introduced by this spacing is referred to as granular distortion. It is common for the design of a quantizer to involve determining the proper balance between granular distortion and overload distortion. For a given supported number of possible output values, reducing the average granular distortion may involve increasing the average overload distortion, and vice versa. A technique for controlling the amplitude of the signal (or, equivalently, the quantization step size <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta }</annotation>

</semantics>

</math><img src="./32769037c408874e1890f77554c65f39c523ebe2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.936ex; height:2.176ex;" alt="{\displaystyle \Delta }" loading="lazy">) to achieve the appropriate balance is the use of <a href="Automatic_gain_control" title="Automatic gain control">automatic gain control</a> (AGC). However, in some quantizer designs, the concepts of granular error and overload error may not apply (e.g., for a quantizer with a limited range of input data or with a countably infinite set of selectable output values).<a href="#cite_note-GrayNeuhoff-6">[6]</a>

<div class="mw-heading mw-heading3"><h3 id="Rate–distortion_quantizer_design">Rate–distortion quantizer design</h3></div>

A scalar quantizer, which performs a quantization operation, can ordinarily be decomposed into two stages:

<dl><dt>Classification</dt>

<dd>A process that classifies the input signal range into <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> non-overlapping <a href="Interval_(mathematics)" title="Interval (mathematics)">intervals</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{I_{k}\}_{k=1}^{M}}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{I_{k}\}_{k=1}^{M}}</annotation>

</semantics>

</math><img src="./5ceb3143e26ff4f2448fba56b28ec119c44ef38f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.626ex; height:3.176ex;" alt="{\displaystyle \{I_{k}\}_{k=1}^{M}}" loading="lazy">, by defining <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M-1}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M-1}</annotation>

</semantics>

</math><img src="./a0ff0c82e48914e34b3c3bd227cf4d09a2fb5eb7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:6.445ex; height:2.343ex;" alt="{\displaystyle M-1}" loading="lazy"> decision boundary values <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{b_{k}\}_{k=1}^{M-1}}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{b_{k}\}_{k=1}^{M-1}}</annotation>

</semantics>

</math><img src="./9199dd54e4e11c3a5f0d99b21e9ee70dc2b70ef9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.471ex; height:3.343ex;" alt="{\displaystyle \{b_{k}\}_{k=1}^{M-1}}" loading="lazy">, such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{k}=[b_{k-1}~,~b_{k})}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I_{k}=[b_{k-1}~,~b_{k})}</annotation>

</semantics>

</math><img src="./f3684925216f27532e6466f8856fb054798a4aee.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.23ex; height:2.843ex;" alt="{\displaystyle I_{k}=[b_{k-1}~,~b_{k})}" loading="lazy"> for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k=1,2,\ldots ,M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k=1,2,\ldots ,M}</annotation>

</semantics>

</math><img src="./ac248ad016fdafbb81762175f90559edf2f4af09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.289ex; height:2.509ex;" alt="{\displaystyle k=1,2,\ldots ,M}" loading="lazy">, with the extreme limits defined by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{0}=-\infty }">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{0}=-\infty }</annotation>

</semantics>

</math><img src="./128a2a0dde5bc288c7d8999d154187ea3a8b434d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.282ex; height:2.509ex;" alt="{\displaystyle b_{0}=-\infty }" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle b_{M}=\infty }">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle b_{M}=\infty }</annotation>

</semantics>

</math><img src="./39f6160923ba5d7344e3429e84f949e4577b50be.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.379ex; height:2.509ex;" alt="{\displaystyle b_{M}=\infty }" loading="lazy">. All the inputs <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> that fall in a given interval range <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I_{k}}</annotation>

</semantics>

</math><img src="./d658e7f6b34dd1d3025a7c9a72efba5b9f46475d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.112ex; height:2.509ex;" alt="{\displaystyle I_{k}}" loading="lazy"> are associated with the same quantization index <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy">.</dd>

<dt>Reconstruction</dt>

<dd>Each interval <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I_{k}}</annotation>

</semantics>

</math><img src="./d658e7f6b34dd1d3025a7c9a72efba5b9f46475d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.112ex; height:2.509ex;" alt="{\displaystyle I_{k}}" loading="lazy"> is represented by a reconstruction value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y_{k}}</annotation>

</semantics>

</math><img src="./4b2ab0248723a410cc2c67ce06ad5c043dcbb933.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.228ex; height:2.009ex;" alt="{\displaystyle y_{k}}" loading="lazy"> which implements the mapping <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in I_{k}\Rightarrow y=y_{k}}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in I_{k}\Rightarrow y=y_{k}}</annotation>

</semantics>

</math><img src="./d11bc710300f4775c00c72cf4a45ec3b6b53e2cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:16.378ex; height:2.509ex;" alt="{\displaystyle x\in I_{k}\Rightarrow y=y_{k}}" loading="lazy">.</dd></dl>

These two stages together comprise the mathematical operation of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y=Q(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y=Q(x)}</annotation>

</semantics>

</math><img src="./d1764d9f11a2d3d2d15d6b2a483eec179210d5e2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.231ex; height:2.843ex;" alt="{\displaystyle y=Q(x)}" loading="lazy">.

<a href="Entropy_coding" title="Entropy coding">Entropy coding</a> techniques can be applied to communicate the quantization indices from a source encoder that performs the classification stage to a decoder that performs the reconstruction stage. One way to do this is to associate each quantization index <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle k}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle k}</annotation>

</semantics>

</math><img src="./c3c9a2c7b599b37105512c5d570edc034056dd40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.211ex; height:2.176ex;" alt="{\displaystyle k}" loading="lazy"> with a binary codeword <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle c_{k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle c_{k}}</annotation>

</semantics>

</math><img src="./3d2f8052630e67b00d04e3487e1d68ed7070470b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.096ex; height:2.009ex;" alt="{\displaystyle c_{k}}" loading="lazy">. An important consideration is the number of bits used for each codeword, denoted here by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {length} (c_{k})}">

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {length} (c_{k})}</annotation>

</semantics>

</math><img src="./91859f2ea09cc0ae87099f4d508dc7146241cfd2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.236ex; height:2.843ex;" alt="{\displaystyle \mathrm {length} (c_{k})}" loading="lazy">. As a result, the design of an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">-level quantizer and an associated set of codewords for communicating its index values requires finding the values of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{b_{k}\}_{k=1}^{M-1}}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{b_{k}\}_{k=1}^{M-1}}</annotation>

</semantics>

</math><img src="./9199dd54e4e11c3a5f0d99b21e9ee70dc2b70ef9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.471ex; height:3.343ex;" alt="{\displaystyle \{b_{k}\}_{k=1}^{M-1}}" loading="lazy">, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{c_{k}\}_{k=1}^{M}}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{c_{k}\}_{k=1}^{M}}</annotation>

</semantics>

</math><img src="./a9acf6646441fbed725b59d8a09f5c1243dd738c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.61ex; height:3.176ex;" alt="{\displaystyle \{c_{k}\}_{k=1}^{M}}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{y_{k}\}_{k=1}^{M}}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{y_{k}\}_{k=1}^{M}}</annotation>

</semantics>

</math><img src="./9c8af3c644d08bd3d01bb32b308b4c0c04148b40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.742ex; height:3.176ex;" alt="{\displaystyle \{y_{k}\}_{k=1}^{M}}" loading="lazy"> which optimally satisfy a selected set of design constraints such as the bit rate <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

</semantics>

</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"> and distortion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D}</annotation>

</semantics>

</math><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy">.

Assuming that an information source <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle S}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle S}</annotation>

</semantics>

</math><img src="./4611d85173cd3b508e67077d4a1252c9c05abca2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.499ex; height:2.176ex;" alt="{\displaystyle S}" loading="lazy"> produces random variables <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> with an associated PDF <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>

</semantics>

</math><img src="./202945cce41ecebb6f643f31d119c514bec7a074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" loading="lazy">, the probability <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p_{k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p_{k}}</annotation>

</semantics>

</math><img src="./01084a31964201514f3e6bd0136989e11ea6e58a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:2.348ex; height:2.009ex;" alt="{\displaystyle p_{k}}" loading="lazy"> that the random variable falls within a particular quantization interval <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I_{k}}</annotation>

</semantics>

</math><img src="./d658e7f6b34dd1d3025a7c9a72efba5b9f46475d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.112ex; height:2.509ex;" alt="{\displaystyle I_{k}}" loading="lazy"> is given by:

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p_{k}=P[x\in I_{k}]=\int _{b_{k-1}}^{b_{k}}f(x)dx}</annotation>

</semantics>

</math><img src="./26424325c60e39665f71cb6c4881bb490b08e841.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; margin-left: -0.089ex; width:30.011ex; height:6.843ex;" alt="{\displaystyle p_{k}=P[x\in I_{k}]=\int _{b_{k-1}}^{b_{k}}f(x)dx}" loading="lazy">.</dd></dl>

The resulting bit rate <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

</semantics>

</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy">, in units of average bits per quantized value, for this quantizer can be derived as follows:

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</munderover>

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R=\sum _{k=1}^{M}p_{k}\cdot \mathrm {length} (c_{k})=\sum _{k=1}^{M}\mathrm {length} (c_{k})\int _{b_{k-1}}^{b_{k}}f(x)dx}</annotation>

</semantics>

</math><img src="./2cc60b0158fc37a302e5f29f7d9e18067481a173.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:52.387ex; height:7.343ex;" alt="{\displaystyle R=\sum _{k=1}^{M}p_{k}\cdot \mathrm {length} (c_{k})=\sum _{k=1}^{M}\mathrm {length} (c_{k})\int _{b_{k-1}}^{b_{k}}f(x)dx}" loading="lazy">.</dd></dl>

If it is assumed that distortion is measured by mean squared error,<a href="#cite_note-18">[a]</a> the distortion D, is given by:

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D=E[(x-Q(x))^{2}]=\int _{-\infty }^{\infty }(x-Q(x))^{2}f(x)dx=\sum _{k=1}^{M}\int _{b_{k-1}}^{b_{k}}(x-y_{k})^{2}f(x)dx}</annotation>

</semantics>

</math><img src="./2292fcf1093dc30c77e2f85e4ad930c2b695ec54.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:74.482ex; height:7.343ex;" alt="{\displaystyle D=E[(x-Q(x))^{2}]=\int _{-\infty }^{\infty }(x-Q(x))^{2}f(x)dx=\sum _{k=1}^{M}\int _{b_{k-1}}^{b_{k}}(x-y_{k})^{2}f(x)dx}" loading="lazy">.</dd></dl>

A key observation is that rate <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

</semantics>

</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"> depends on the decision boundaries <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{b_{k}\}_{k=1}^{M-1}}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{b_{k}\}_{k=1}^{M-1}}</annotation>

</semantics>

</math><img src="./9199dd54e4e11c3a5f0d99b21e9ee70dc2b70ef9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.471ex; height:3.343ex;" alt="{\displaystyle \{b_{k}\}_{k=1}^{M-1}}" loading="lazy"> and the codeword lengths <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{\mathrm {length} (c_{k})\}_{k=1}^{M}}">

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{\mathrm {length} (c_{k})\}_{k=1}^{M}}</annotation>

</semantics>

</math><img src="./e349fb4e6c0b6807ef136c6bb9c49ff43d0d595f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:15.75ex; height:3.176ex;" alt="{\displaystyle \{\mathrm {length} (c_{k})\}_{k=1}^{M}}" loading="lazy">, whereas the distortion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D}</annotation>

</semantics>

</math><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"> depends on the decision boundaries <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{b_{k}\}_{k=1}^{M-1}}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{b_{k}\}_{k=1}^{M-1}}</annotation>

</semantics>

</math><img src="./9199dd54e4e11c3a5f0d99b21e9ee70dc2b70ef9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.471ex; height:3.343ex;" alt="{\displaystyle \{b_{k}\}_{k=1}^{M-1}}" loading="lazy"> and the reconstruction levels <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{y_{k}\}_{k=1}^{M}}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{y_{k}\}_{k=1}^{M}}</annotation>

</semantics>

</math><img src="./9c8af3c644d08bd3d01bb32b308b4c0c04148b40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.742ex; height:3.176ex;" alt="{\displaystyle \{y_{k}\}_{k=1}^{M}}" loading="lazy">.

After defining these two performance metrics for the quantizer, a typical rate–distortion formulation for a quantizer design problem can be expressed in one of two ways:

<ol><li>Given a maximum distortion constraint <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D\leq D_{\max }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D\leq D_{\max }}</annotation>

</semantics>

</math><img src="./af7904fc86e20fc913effc17768cefd86d97df36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.238ex; height:2.509ex;" alt="{\displaystyle D\leq D_{\max }}" loading="lazy">, minimize the bit rate <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

</semantics>

</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"></li>

<li>Given a maximum bit rate constraint <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R\leq R_{\max }}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R\leq R_{\max }}</annotation>

</semantics>

</math><img src="./d5392b6623ce9c9c96329bfe254cf8e774675a78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:9.918ex; height:2.509ex;" alt="{\displaystyle R\leq R_{\max }}" loading="lazy">, minimize the distortion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D}</annotation>

</semantics>

</math><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"></li></ol>

Often the solution to these problems can be equivalently (or approximately) expressed and solved by converting the formulation to the unconstrained problem <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \min \left\{D+\lambda \cdot R\right\}}">

<mrow>

<mrow>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \min \left\{D+\lambda \cdot R\right\}}</annotation>

</semantics>

</math><img src="./df5a2218e7bc6b7531d68e36d2dcb69aa7474e1f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:16.15ex; height:2.843ex;" alt="{\displaystyle \min \left\{D+\lambda \cdot R\right\}}" loading="lazy"> where the <a href="Lagrange_multiplier" title="Lagrange multiplier">Lagrange multiplier</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>

</semantics>

</math><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"> is a non-negative constant that establishes the appropriate balance between rate and distortion. Solving the unconstrained problem is equivalent to finding a point on the <a href="Convex_hull" title="Convex hull">convex hull</a> of the family of solutions to an equivalent constrained formulation of the problem. However, finding a solution – especially a <a href="Closed-form_expression" title="Closed-form expression">closed-form</a> solution – to any of these three problem formulations can be difficult. Solutions that do not require multi-dimensional iterative optimization techniques have been published for only three PDFs: the uniform,<a href="#cite_note-19">[18]</a> <a href="Exponential_distribution" title="Exponential distribution">exponential</a>,<a href="#cite_note-SullivanIT-12">[12]</a> and <a href="Laplace_distribution" title="Laplace distribution">Laplacian</a><a href="#cite_note-SullivanIT-12">[12]</a> distributions. Iterative optimization approaches can be used to find solutions in other cases.<a href="#cite_note-GrayNeuhoff-6">[6]</a><a href="#cite_note-Berger72-20">[19]</a><a href="#cite_note-Berger82-21">[20]</a>

Note that the reconstruction values <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{y_{k}\}_{k=1}^{M}}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{y_{k}\}_{k=1}^{M}}</annotation>

</semantics>

</math><img src="./9c8af3c644d08bd3d01bb32b308b4c0c04148b40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.742ex; height:3.176ex;" alt="{\displaystyle \{y_{k}\}_{k=1}^{M}}" loading="lazy"> affect only the distortion – they do not affect the bit rate – and that each individual <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y_{k}}</annotation>

</semantics>

</math><img src="./4b2ab0248723a410cc2c67ce06ad5c043dcbb933.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.228ex; height:2.009ex;" alt="{\displaystyle y_{k}}" loading="lazy"> makes a separate contribution <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle d_{k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d_{k}}</annotation>

</semantics>

</math><img src="./b78f5b2abc48e63b987b6d7527caa5aa9b1bb512.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.298ex; height:2.509ex;" alt="{\displaystyle d_{k}}" loading="lazy"> to the total distortion as shown below:

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D=\sum _{k=1}^{M}d_{k}}</annotation>

</semantics>

</math><img src="./6759e8d74d80fd7800d16001df1e49c089d3ab73.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:11.062ex; height:7.343ex;" alt="{\displaystyle D=\sum _{k=1}^{M}d_{k}}" loading="lazy"></dd></dl>

where

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle d_{k}=\int _{b_{k-1}}^{b_{k}}(x-y_{k})^{2}f(x)dx}</annotation>

</semantics>

</math><img src="./15bdfe74aa22be46dd4522218f3bda5fc59ad0bc.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:26.416ex; height:6.843ex;" alt="{\displaystyle d_{k}=\int _{b_{k-1}}^{b_{k}}(x-y_{k})^{2}f(x)dx}" loading="lazy"></dd></dl>

This observation can be used to ease the analysis – given the set of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{b_{k}\}_{k=1}^{M-1}}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{b_{k}\}_{k=1}^{M-1}}</annotation>

</semantics>

</math><img src="./9199dd54e4e11c3a5f0d99b21e9ee70dc2b70ef9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.471ex; height:3.343ex;" alt="{\displaystyle \{b_{k}\}_{k=1}^{M-1}}" loading="lazy"> values, the value of each <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y_{k}}</annotation>

</semantics>

</math><img src="./4b2ab0248723a410cc2c67ce06ad5c043dcbb933.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.228ex; height:2.009ex;" alt="{\displaystyle y_{k}}" loading="lazy"> can be optimized separately to minimize its contribution to the distortion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D}</annotation>

</semantics>

</math><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy">.

For the mean-square error distortion criterion, it can be easily shown that the optimal set of reconstruction values <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{y_{k}^{*}\}_{k=1}^{M}}">

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{y_{k}^{*}\}_{k=1}^{M}}</annotation>

</semantics>

</math><img src="./a4a59687bce6f8cf8261d492815afe9f0f3f3894.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.742ex; height:3.176ex;" alt="{\displaystyle \{y_{k}^{*}\}_{k=1}^{M}}" loading="lazy"> is given by setting the reconstruction value <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle y_{k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y_{k}}</annotation>

</semantics>

</math><img src="./4b2ab0248723a410cc2c67ce06ad5c043dcbb933.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.228ex; height:2.009ex;" alt="{\displaystyle y_{k}}" loading="lazy"> within each interval <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I_{k}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle I_{k}}</annotation>

</semantics>

</math><img src="./d658e7f6b34dd1d3025a7c9a72efba5b9f46475d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.112ex; height:2.509ex;" alt="{\displaystyle I_{k}}" loading="lazy"> to the <a href="Conditional_expected_value" class="mw-redirect" title="Conditional expected value">conditional expected value</a> (also referred to as the <a href="Centroid" title="Centroid">centroid</a>) within the interval, as given by:

</mrow>

</mrow>

</msubsup>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle y_{k}^{*}={\frac {1}{p_{k}}}\int _{b_{k-1}}^{b_{k}}xf(x)dx}</annotation>

</semantics>

</math><img src="./34766e917f54bae886b389ad16227d5d7b91f9b7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:22.283ex; height:6.843ex;" alt="{\displaystyle y_{k}^{*}={\frac {1}{p_{k}}}\int _{b_{k-1}}^{b_{k}}xf(x)dx}" loading="lazy">.</dd></dl>

The use of sufficiently well-designed entropy coding techniques can result in the use of a bit rate that is close to the true information content of the indices <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{k\}_{k=1}^{M}}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{k\}_{k=1}^{M}}</annotation>

</semantics>

</math><img src="./de9e2048051738754e3e74d141def4edf9e8bbd5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.725ex; height:3.176ex;" alt="{\displaystyle \{k\}_{k=1}^{M}}" loading="lazy">, such that effectively

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {length} (c_{k})\approx -\log _{2}\left(p_{k}\right)}</annotation>

</semantics>

</math><img src="./141ccdb272b76a8c66e00c4fe2a76df0fc1d5df6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.623ex; height:2.843ex;" alt="{\displaystyle \mathrm {length} (c_{k})\approx -\log _{2}\left(p_{k}\right)}" loading="lazy"></dd></dl>

and therefore

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R=\sum _{k=1}^{M}-p_{k}\cdot \log _{2}\left(p_{k}\right)}</annotation>

</semantics>

</math><img src="./06761368ad66036d99a4b7bf173ceaf585c9323f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:22.443ex; height:7.343ex;" alt="{\displaystyle R=\sum _{k=1}^{M}-p_{k}\cdot \log _{2}\left(p_{k}\right)}" loading="lazy">.</dd></dl>

The use of this approximation can allow the entropy coding design problem to be separated from the design of the quantizer itself. Modern entropy coding techniques such as <a href="Arithmetic_coding" title="Arithmetic coding">arithmetic coding</a> can achieve bit rates that are very close to the true entropy of a source, given a set of known (or adaptively estimated) probabilities <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{p_{k}\}_{k=1}^{M}}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{p_{k}\}_{k=1}^{M}}</annotation>

</semantics>

</math><img src="./2772fbad6866aaf6f44334424688ee8ae3d32050.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.772ex; height:3.176ex;" alt="{\displaystyle \{p_{k}\}_{k=1}^{M}}" loading="lazy">.

In some designs, rather than optimizing for a particular number of classification regions <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">, the quantizer design problem may include optimization of the value of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> as well. For some probabilistic source models, the best performance may be achieved when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> approaches infinity.

<div class="mw-heading mw-heading3"><h3 id="Neglecting_the_entropy_constraint:_Lloyd–Max_quantization">Neglecting the entropy constraint: Lloyd–Max quantization</h3></div>

In the above formulation, if the bit rate constraint is neglected by setting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \lambda }">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \lambda }</annotation>

</semantics>

</math><img src="./b43d0ea3c9c025af1be9128e62a18fa74bedda2a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.355ex; height:2.176ex;" alt="{\displaystyle \lambda }" loading="lazy"> equal to 0, or equivalently if it is assumed that a fixed-length code (FLC) will be used to represent the quantized data instead of a <a href="Variable-length_code" title="Variable-length code">variable-length code</a> (or some other entropy coding technology such as arithmetic coding that is better than an FLC in the rate–distortion sense), the optimization problem reduces to minimization of distortion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle D}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D}</annotation>

</semantics>

</math><img src="./f34a0c600395e5d4345287e21fb26efd386990e6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.924ex; height:2.176ex;" alt="{\displaystyle D}" loading="lazy"> alone.

The indices produced by an <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">-level quantizer can be coded using a fixed-length code using <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R=\lceil \log _{2}M\rceil }">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R=\lceil \log _{2}M\rceil }</annotation>

</semantics>

</math><img src="./27e11c2290e53831cd6c28a3661b02e3c2c235f9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.783ex; height:2.843ex;" alt="{\displaystyle R=\lceil \log _{2}M\rceil }" loading="lazy"> bits/symbol. For example, when <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M=}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M=}</annotation>

</semantics>

</math><img src="./6a5897444f2fc12b87773bf24c2b4744789d5a8d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.896ex; height:2.176ex;" alt="{\displaystyle M=}" loading="lazy">256 levels, the FLC bit rate <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle R}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle R}</annotation>

</semantics>

</math><img src="./4b0bfb3769bf24d80e15374dc37b0441e2616e33.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.764ex; height:2.176ex;" alt="{\displaystyle R}" loading="lazy"> is 8 bits/symbol. For this reason, such a quantizer has sometimes been called an 8-bit quantizer. However using an FLC eliminates the compression improvement that can be obtained by use of better entropy coding.

Assuming an FLC with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> levels, the rate–distortion minimization problem can be reduced to distortion minimization alone. The reduced problem can be stated as follows: given a source <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> with PDF <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)}</annotation>

</semantics>

</math><img src="./202945cce41ecebb6f643f31d119c514bec7a074.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.418ex; height:2.843ex;" alt="{\displaystyle f(x)}" loading="lazy"> and the constraint that the quantizer must use only <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> classification regions, find the decision boundaries <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{b_{k}\}_{k=1}^{M-1}}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{b_{k}\}_{k=1}^{M-1}}</annotation>

</semantics>

</math><img src="./9199dd54e4e11c3a5f0d99b21e9ee70dc2b70ef9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:8.471ex; height:3.343ex;" alt="{\displaystyle \{b_{k}\}_{k=1}^{M-1}}" loading="lazy"> and reconstruction levels <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \{y_{k}\}_{k=1}^{M}}">

<msub>

</mrow>

</msub>

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \{y_{k}\}_{k=1}^{M}}</annotation>

</semantics>

</math><img src="./9c8af3c644d08bd3d01bb32b308b4c0c04148b40.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:7.742ex; height:3.176ex;" alt="{\displaystyle \{y_{k}\}_{k=1}^{M}}" loading="lazy"> to minimize the resulting distortion

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</msubsup>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msubsup>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</munderover>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle D=E[(x-Q(x))^{2}]=\int _{-\infty }^{\infty }(x-Q(x))^{2}f(x)dx=\sum _{k=1}^{M}\int _{b_{k-1}}^{b_{k}}(x-y_{k})^{2}f(x)dx=\sum _{k=1}^{M}d_{k}}</annotation>

</semantics>

</math><img src="./9a11a15b3c5710c31187e8dfd713f12ca0981a65.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -3.005ex; width:83.62ex; height:7.343ex;" alt="{\displaystyle D=E[(x-Q(x))^{2}]=\int _{-\infty }^{\infty }(x-Q(x))^{2}f(x)dx=\sum _{k=1}^{M}\int _{b_{k-1}}^{b_{k}}(x-y_{k})^{2}f(x)dx=\sum _{k=1}^{M}d_{k}}" loading="lazy">.</dd></dl>

Finding an optimal solution to the above problem results in a quantizer sometimes called a MMSQE (minimum mean-square quantization error) solution, and the resulting PDF-optimized (non-uniform) quantizer is referred to as a Lloyd–Max quantizer, named after two people who independently developed iterative methods<a href="#cite_note-GrayNeuhoff-6">[6]</a><a href="#cite_note-22">[21]</a><a href="#cite_note-23">[22]</a> to solve the two sets of simultaneous equations resulting from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\partial D/\partial b_{k}}=0}">

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\partial D/\partial b_{k}}=0}</annotation>

</semantics>

</math><img src="./a37a80a8ecb1ffae64b043e5392c9f98ec678753.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.07ex; height:2.843ex;" alt="{\displaystyle {\partial D/\partial b_{k}}=0}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\partial D/\partial y_{k}}=0}">

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\partial D/\partial y_{k}}=0}</annotation>

</semantics>

</math><img src="./1014f1ba7809ecb5830cb8a9d5e36f489480a9f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.212ex; height:2.843ex;" alt="{\displaystyle {\partial D/\partial y_{k}}=0}" loading="lazy">, as follows:

<mfrac>

<mrow>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\partial D \over \partial b_{k}}=0\Rightarrow b_{k}={y_{k}+y_{k+1} \over 2}}</annotation>

</semantics>

</math><img src="./c3f94675620ae34f529f89b7faf05351f9a0abb0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.338ex; width:27.533ex; height:5.843ex;" alt="{\displaystyle {\partial D \over \partial b_{k}}=0\Rightarrow b_{k}={y_{k}+y_{k+1} \over 2}}" loading="lazy">,</dd></dl>

which places each threshold at the midpoint between each pair of reconstruction values, and

<mfrac>

<mrow>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msubsup>

</mrow>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msubsup>

</mrow>

</mfrac>

</mrow>

<mfrac>

<msub>

</mrow>

</msub>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\partial D \over \partial y_{k}}=0\Rightarrow y_{k}={\int _{b_{k-1}}^{b_{k}}xf(x)dx \over \int _{b_{k-1}}^{b_{k}}f(x)dx}={\frac {1}{p_{k}}}\int _{b_{k-1}}^{b_{k}}xf(x)dx}</annotation>

</semantics>

</math><img src="./715d5b7d8dfad8e5096842f95b4b7f4189f334b2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -4.005ex; width:51.754ex; height:9.176ex;" alt="{\displaystyle {\partial D \over \partial y_{k}}=0\Rightarrow y_{k}={\int _{b_{k-1}}^{b_{k}}xf(x)dx \over \int _{b_{k-1}}^{b_{k}}f(x)dx}={\frac {1}{p_{k}}}\int _{b_{k-1}}^{b_{k}}xf(x)dx}" loading="lazy"></dd></dl>

which places each reconstruction value at the centroid (conditional expected value) of its associated classification interval.

<a href="Lloyd's_algorithm" title="Lloyd's algorithm">Lloyd's Method I algorithm</a>, originally described in 1957, can be generalized in a straightforward way for application to vector data. This generalization results in the <a href="Linde%E2%80%93Buzo%E2%80%93Gray_algorithm" title="Linde–Buzo–Gray algorithm">Linde–Buzo–Gray (LBG)</a> or <a href="K-means" class="mw-redirect" title="K-means">k-means</a> classifier optimization methods. Moreover, the technique can be further generalized in a straightforward way to also include an entropy constraint for vector data.<a href="#cite_note-ChouLookabaughGray-24">[23]</a>

<div class="mw-heading mw-heading3"><h3 id="Uniform_quantization_and_the_6_dB/bit_approximation">Uniform quantization and the 6 dB/bit approximation</h3></div>

The Lloyd–Max quantizer is actually a uniform quantizer when the input PDF is uniformly distributed over the range <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [y_{1}-\Delta /2,~y_{M}+\Delta /2)}">

<msub>

</mrow>

</msub>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [y_{1}-\Delta /2,~y_{M}+\Delta /2)}</annotation>

</semantics>

</math><img src="./28ac33ec6bc757d3e717fc9b35ee9b67676de2e5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:22.66ex; height:2.843ex;" alt="{\displaystyle [y_{1}-\Delta /2,~y_{M}+\Delta /2)}" loading="lazy">. 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 xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f(x)={\tfrac {1}{2X_{\max }}}}">

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f(x)={\tfrac {1}{2X_{\max }}}}</annotation>

</semantics>

</math><img src="./c0d075377e8f186bfd990d9ee8969b0a9714c9b5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.505ex; width:13.182ex; height:3.843ex;" alt="{\displaystyle f(x)={\tfrac {1}{2X_{\max }}}}" loading="lazy">, for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in [-X_{\max },X_{\max }]}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in [-X_{\max },X_{\max }]}</annotation>

</semantics>

</math><img src="./3cf4bd582eef8f8d55332145bed84a97829c283d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.737ex; height:2.843ex;" alt="{\displaystyle x\in [-X_{\max },X_{\max }]}" loading="lazy"> and 0 elsewhere.

The step size <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta ={\tfrac {2X_{\max }}{M}}}">

<mfrac>

<mrow>

<msub>

</mrow>

</msub>

</mrow>

</mfrac>

</mstyle>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta ={\tfrac {2X_{\max }}{M}}}</annotation>

</semantics>

</math><img src="./b101d88f0807c3bd3de76054ff190bd32fa9e6a6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.171ex; width:10.701ex; height:3.843ex;" alt="{\displaystyle \Delta ={\tfrac {2X_{\max }}{M}}}" loading="lazy"> and the signal to quantization noise ratio (SQNR) of the quantizer is

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

</mrow>

</mrow>

</msubsup>

</mrow>

</mrow>

</msubsup>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<mfrac>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

<mrow>

<msup>

</mrow>

</msup>

</mrow>

</mrow>

</mfrac>

</mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\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}</annotation>

</semantics>

</math><img src="./20a6923476ba7cb90a86fc06d0b9fa9e0f67a6bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -2.838ex; width:71.633ex; height:6.843ex;" alt="{\displaystyle {\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}" loading="lazy">.</dd></dl>

For a fixed-length code using <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

</semantics>

</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> bits, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M=2^{N}}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M=2^{N}}</annotation>

</semantics>

</math><img src="./2aaa5bdbbea857a913997a146a038f6b5242cfca.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.395ex; height:2.676ex;" alt="{\displaystyle M=2^{N}}" loading="lazy">, resulting in

</mrow>

<msub>

</mrow>

</msub>

<msup>

</mrow>

</msup>

</mrow>

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\rm {SQNR}}=20\log _{10}{2^{N}}=N\cdot (20\log _{10}2)=N\cdot 6.0206\,{\rm {dB}}}</annotation>

</semantics>

</math><img src="./63b07ba237bcf7826e8bb6692f3f39550ab38cd0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:54.839ex; height:3.176ex;" alt="{\displaystyle {\rm {SQNR}}=20\log _{10}{2^{N}}=N\cdot (20\log _{10}2)=N\cdot 6.0206\,{\rm {dB}}}" loading="lazy">,

or approximately 6 dB per bit. For example, for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

</semantics>

</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy">=8 bits, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">=256 levels and SQNR = 8×6 = 48 dB; and for <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

</semantics>

</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy">=16 bits, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">=65536 and SQNR = 16×6 = 96 dB. The property of 6 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 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 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 dB) based on the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Delta ^{2}/12}">

<msup>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Delta ^{2}/12}</annotation>

</semantics>

</math><img src="./34b5be83e88394057daa090c25107b1b57adb48a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.477ex; height:3.176ex;" alt="{\displaystyle \Delta ^{2}/12}" loading="lazy"> approximation.

At asymptotically high bit rates, the 6 dB/bit approximation is supported for many source PDFs by rigorous theoretical analysis.<a href="#cite_note-Bennett-2">[2]</a><a href="#cite_note-OliverPierceShannon-3">[3]</a><a href="#cite_note-GishPierce-5">[5]</a><a href="#cite_note-GrayNeuhoff-6">[6]</a> Moreover, the structure of the optimal scalar quantizer (in the rate–distortion sense) approaches that of a uniform quantizer under these conditions.<a href="#cite_note-GishPierce-5">[5]</a><a href="#cite_note-GrayNeuhoff-6">[6]</a>

<div class="mw-heading mw-heading2"><h2 id="In_other_fields">In other fields</h2></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output .hatnote{font-style:italic}.mw-parser-output div.hatnote{padding-left:1.6em;margin-bottom:0.5em}.mw-parser-output .hatnote i{font-style:normal}.mw-parser-output .hatnote+link+.hatnote{margin-top:-0.5em}@media print{body.ns-0 .mw-parser-output .hatnote{display:none!important}}

/* end https://en.wikipedia.org/ */

</style><div role="note" class="hatnote navigation-not-searchable">See also: <a href="Quantum_noise" title="Quantum noise">Quantum noise</a> and <a href="Quantum_limit" title="Quantum limit">Quantum limit</a></div>

Many physical quantities are actually quantized by physical entities. Examples of fields where this limitation applies include <a href="Electronics" title="Electronics">electronics</a> (due to <a href="Electron" title="Electron">electrons</a>), <a href="Optics" title="Optics">optics</a> (due to <a href="Photon" title="Photon">photons</a>), <a href="Biology" title="Biology">biology</a> (due to <a href="DNA" title="DNA">DNA</a>), <a href="Physics" title="Physics">physics</a> (due to <a href="Planck_limits" class="mw-redirect" title="Planck limits">Planck limits</a>) and <a href="Chemistry" title="Chemistry">chemistry</a> (due to <a href="Molecule" title="Molecule">molecules</a>).

<ul><li><a href="Beta_encoder" title="Beta encoder">Beta encoder</a></li>

<li><a href="Color_quantization" title="Color quantization">Color quantization</a></li>

<li><a href="Data_binning" title="Data binning">Data binning</a></li>

<li><a href="Discretization" title="Discretization">Discretization</a></li>

<li><a href="Discretization_error" title="Discretization error">Discretization error</a></li>

<li><a href="Least_count" title="Least count">Least count</a></li>

<li><a href="Posterization" title="Posterization">Posterization</a></li>

<li><a href="Pulse-code_modulation" title="Pulse-code modulation">Pulse-code modulation</a></li>

<li><a href="Quantile" title="Quantile">Quantile</a></li>

<li><a href="Quantization_(image_processing)" title="Quantization (image processing)">Quantization (image processing)</a></li>

<li><a href="Regression_dilution" title="Regression dilution">Regression dilution</a> – a bias in parameter estimates caused by errors such as quantization in the explanatory or independent variable</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Notes">Notes</h2></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output .reflist{margin-bottom:0.5em;list-style-type:decimal}@media screen{.mw-parser-output .reflist{font-size:90%}}.mw-parser-output .reflist .references{font-size:100%;margin-bottom:0;list-style-type:inherit}.mw-parser-output .reflist-columns-2{column-width:30em}.mw-parser-output .reflist-columns-3{column-width:25em}.mw-parser-output .reflist-columns{margin-top:0.3em}.mw-parser-output .reflist-columns ol{margin-top:0}.mw-parser-output .reflist-columns li{page-break-inside:avoid;break-inside:avoid-column}.mw-parser-output .reflist-upper-alpha{list-style-type:upper-alpha}.mw-parser-output .reflist-upper-roman{list-style-type:upper-roman}.mw-parser-output .reflist-lower-alpha{list-style-type:lower-alpha}.mw-parser-output .reflist-lower-greek{list-style-type:lower-greek}.mw-parser-output .reflist-lower-roman{list-style-type:lower-roman}

/* end https://en.wikipedia.org/ */

</style><div class="reflist reflist-lower-alpha">

<li id="cite_note-18"><a href="#cite_ref-18">^</a> Other distortion measures can also be considered, although mean squared error is a popular one.

</li>

</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output cite.citation{font-style:inherit;word-wrap:break-word}.mw-parser-output .citation q{quotes:"\"""\"""'""'"}.mw-parser-output .citation:target{background-color:rgba(0,127,255,0.133)}.mw-parser-output .id-lock-free.id-lock-free a{background:url("./mw/Lock-green.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-limited.id-lock-limited a,.mw-parser-output .id-lock-registration.id-lock-registration a{background:url("./mw/Lock-gray-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .id-lock-subscription.id-lock-subscription a{background:url("./mw/Lock-red-alt-2.svg")right 0.1em center/9px no-repeat}.mw-parser-output .cs1-ws-icon a{background:url("./mw/Wikisource-logo.svg")right 0.1em center/12px no-repeat}body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-free a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-limited a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-registration a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .id-lock-subscription a,body:not(.skin-timeless):not(.skin-minerva) .mw-parser-output .cs1-ws-icon a{background-size:contain;padding:0 1em 0 0}.mw-parser-output .cs1-code{color:inherit;background:inherit;border:none;padding:inherit}.mw-parser-output .cs1-hidden-error{display:none;color:var(--color-error,#d33)}.mw-parser-output .cs1-visible-error{color:var(--color-error,#d33)}.mw-parser-output .cs1-maint{display:none;color:#085;margin-left:0.3em}.mw-parser-output .cs1-kern-left{padding-left:0.2em}.mw-parser-output .cs1-kern-right{padding-right:0.2em}.mw-parser-output .citation .mw-selflink{font-weight:inherit}@media screen{.mw-parser-output .cs1-format{font-size:95%}html.skin-theme-clientpref-night .mw-parser-output .cs1-maint{color:#18911f}}@media screen and (prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .cs1-maint{color:#18911f}}

/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFSheppard1897" class="citation journal cs1 cs1-prop-long-vol"><a href="William_Fleetwood_Sheppard" title="William Fleetwood Sheppard">Sheppard, W. F.</a> (1897). <a rel="nofollow" class="external text" href="https://zenodo.org/record/1447738">"On the Calculation of the most Probable Values of Frequency-Constants, for Data arranged according to Equidistant Division of a Scale"</a>. Proceedings of the London Mathematical Society. s1-29 (1). Wiley: 353–380. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1112%2Fplms%2Fs1-29.1.353">10.1112/plms/s1-29.1.353</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0024-6115">0024-6115</a>.</cite>

</li>

<li id="cite_note-Bennett-2">^ <a href="#cite_ref-Bennett_2-0">a</a> <a href="#cite_ref-Bennett_2-1">b</a> <a href="#cite_ref-Bennett_2-2">c</a> <a href="#cite_ref-Bennett_2-3">d</a> W. R. Bennett, "<a rel="nofollow" class="external text" href="http://www.alcatel-lucent.com/bstj/vol27-1948/articles/bstj27-3-446.pdf">Spectra of Quantized Signals</a>", <a href="Bell_System_Technical_Journal" class="mw-redirect" title="Bell System Technical Journal">Bell System Technical Journal</a>, Vol. 27, pp. 446–472, July 1948.

</li>

<li id="cite_note-OliverPierceShannon-3">^ <a href="#cite_ref-OliverPierceShannon_3-0">a</a> <a href="#cite_ref-OliverPierceShannon_3-1">b</a> <cite id="CITEREFOliverPierceShannon1948" class="citation journal cs1">Oliver, B.M.; Pierce, J.R.; <a href="Claude_Shannon" title="Claude Shannon">Shannon, C.E.</a> (1948). "The Philosophy of PCM". Proceedings of the IRE. 36 (11): 1324–1331. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2Fjrproc.1948.231941">10.1109/jrproc.1948.231941</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0096-8390">0096-8390</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:51663786">51663786</a>.</cite>

</li>

<li id="cite_note-Stein-4"><a href="#cite_ref-Stein_4-0">^</a> Seymour Stein and J. Jay Jones, <a rel="nofollow" class="external text" href="https://books.google.com/books/about/Modern_communication_principles.html?id=jBc3AQAAIAAJ">Modern Communication Principles</a>, <a href="McGraw%E2%80%93Hill" class="mw-redirect" title="McGraw–Hill">McGraw–Hill</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-07-061003-3</bdi>, 1967 (p. 196).

</li>

<li id="cite_note-GishPierce-5">^ <a href="#cite_ref-GishPierce_5-0">a</a> <a href="#cite_ref-GishPierce_5-1">b</a> <a href="#cite_ref-GishPierce_5-2">c</a> <cite id="CITEREFGishPierce1968" class="citation journal cs1">Gish, H.; Pierce, J. (1968). "Asymptotically efficient quantizing". IEEE Transactions on Information Theory. 14 (5): 676–683. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2Ftit.1968.1054193">10.1109/tit.1968.1054193</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0018-9448">0018-9448</a>.</cite>

</li>

<li id="cite_note-GrayNeuhoff-6">^ <a href="#cite_ref-GrayNeuhoff_6-0">a</a> <a href="#cite_ref-GrayNeuhoff_6-1">b</a> <a href="#cite_ref-GrayNeuhoff_6-2">c</a> <a href="#cite_ref-GrayNeuhoff_6-3">d</a> <a href="#cite_ref-GrayNeuhoff_6-4">e</a> <a href="#cite_ref-GrayNeuhoff_6-5">f</a> <a href="#cite_ref-GrayNeuhoff_6-6">g</a> <a href="#cite_ref-GrayNeuhoff_6-7">h</a> <a href="#cite_ref-GrayNeuhoff_6-8">i</a> <cite id="CITEREFGrayNeuhoff1998" class="citation journal cs1"><a href="Robert_M._Gray" title="Robert M. Gray">Gray, R.M.</a>; Neuhoff, D.L. (1998). "Quantization". IEEE Transactions on Information Theory. 44 (6): 2325–2383. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2F18.720541">10.1109/18.720541</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0018-9448">0018-9448</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:212653679">212653679</a>.</cite>

</li>

<li id="cite_note-7"><a href="#cite_ref-7">^</a> <cite id="CITEREFAllen_GershoRobert_M._Gray1991" class="citation book cs1"><a href="Allen_Gersho" title="Allen Gersho">Allen Gersho</a>; <a href="Robert_M._Gray" title="Robert M. Gray">Robert M. Gray</a> (1991). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=DwcDm6xgItUC">Vector Quantization and Signal Compression</a>. <a href="Springer_Science%2BBusiness_Media" title="Springer Science+Business Media">Springer</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-7923-9181-4</bdi>.</cite>

</li>

<li id="cite_note-8"><a href="#cite_ref-8">^</a> Hodgson, Jay (2010). Understanding Records, p.56. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-1-4411-5607-5</bdi>. Adapted from Franz, David (2004). Recording and Producing in the Home Studio, p.38-9. Berklee Press.

</li>

<li id="cite_note-Gersho77-9">^ <a href="#cite_ref-Gersho77_9-0">a</a> <a href="#cite_ref-Gersho77_9-1">b</a> <cite id="CITEREFGersho1977" class="citation journal cs1"><a href="Allen_Gersho" title="Allen Gersho">Gersho, A.</a> (1977). "Quantization". IEEE Communications Society Magazine. 15 (5): 16–28. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2Fmcom.1977.1089500">10.1109/mcom.1977.1089500</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0148-9615">0148-9615</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:260498692">260498692</a>.</cite>

</li>

<li id="cite_note-10"><a href="#cite_ref-10">^</a> <cite id="CITEREFRabbaniJoshiJones2009" class="citation book cs1">Rabbani, Majid; Joshi, Rajan L.; Jones, Paul W. (2009). "Section 1.2.3: Quantization, in Chapter 1: JPEG 2000 Core Coding System (Part 1)". In Schelkens, Peter; Skodras, Athanassios; Ebrahimi, Touradj (eds.). <a rel="nofollow" class="external text" href="https://archive.org/details/jpegsuitethewile00sche">The JPEG 2000 Suite</a>. <a href="John_Wiley_%26_Sons" class="mw-redirect" title="John Wiley & Sons">John Wiley & Sons</a>. pp. <a rel="nofollow" class="external text" href="https://archive.org/details/jpegsuitethewile00sche/page/n73">22</a>–24. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-470-72147-6</bdi>.</cite>

</li>

<li id="cite_note-11"><a href="#cite_ref-11">^</a> <cite id="CITEREFTaubmanMarcellin2002" class="citation book cs1">Taubman, David S.; Marcellin, Michael W. (2002). "Chapter 3: Quantization". <a rel="nofollow" class="external text" href="https://archive.org/details/jpegimagecompres00taub">JPEG2000: Image Compression Fundamentals, Standards and Practice</a>. <a href="Kluwer_Academic_Publishers" class="mw-redirect" title="Kluwer Academic Publishers">Kluwer Academic Publishers</a>. p. <a rel="nofollow" class="external text" href="https://archive.org/details/jpegimagecompres00taub/page/n126">107</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-7923-7519-X</bdi>.</cite>

</li>

<li id="cite_note-SullivanIT-12">^ <a href="#cite_ref-SullivanIT_12-0">a</a> <a href="#cite_ref-SullivanIT_12-1">b</a> <a href="#cite_ref-SullivanIT_12-2">c</a> <cite id="CITEREFSullivan1996" class="citation journal cs1"><a href="Gary_Sullivan_(engineer)" title="Gary Sullivan (engineer)">Sullivan, G.J.</a> (1996). "Efficient scalar quantization of exponential and Laplacian random variables". IEEE Transactions on Information Theory. 42 (5): 1365–1374. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2F18.532878">10.1109/18.532878</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0018-9448">0018-9448</a>.</cite>

</li>

<li id="cite_note-Widrow1-13"><a href="#cite_ref-Widrow1_13-0">^</a> <cite id="CITEREFWidrow1956" class="citation journal cs1"><a href="Bernard_Widrow" title="Bernard Widrow">Widrow, B.</a> (1956). "A Study of Rough Amplitude Quantization by Means of Nyquist Sampling Theory". IRE Transactions on Circuit Theory. 3 (4): 266–276. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2Ftct.1956.1086334">10.1109/tct.1956.1086334</a>. <a href="Hdl_(identifier)" class="mw-redirect" title="Hdl (identifier)">hdl</a>:<a rel="nofollow" class="external text" href="https://hdl.handle.net/1721.1%2F12139">1721.1/12139</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0096-2007">0096-2007</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:16777461">16777461</a>.</cite>

</li>

<li id="cite_note-Widrow2-14">^ <a href="#cite_ref-Widrow2_14-0">a</a> <a href="#cite_ref-Widrow2_14-1">b</a> <cite id="CITEREFWidrow1961" class="citation journal cs1"><a href="Bernard_Widrow" title="Bernard Widrow">Widrow, B.</a> (1961). <a rel="nofollow" class="external text" href="https://web.archive.org/web/20110401025420/http://www-isl.stanford.edu/~widrow/papers/j1961statisticalanalysis.pdf">"Statistical analysis of amplitude-quantized sampled-data systems"</a> (PDF). Transactions of the American Institute of Electrical Engineers, Part II: Applications and Industry. 79 (6): 555–568. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2FTAI.1961.6371702">10.1109/TAI.1961.6371702</a>. Archived from <a rel="nofollow" class="external text" href="http://www-isl.stanford.edu/~widrow/papers/j1961statisticalanalysis.pdf">the original</a> (PDF) on 2011-04-01. Retrieved 2012-08-17.</cite>

</li>

<li id="cite_note-MarcoNeuhoff-15"><a href="#cite_ref-MarcoNeuhoff_15-0">^</a> <cite id="CITEREFMarcoNeuhoff2005" class="citation journal cs1">Marco, D.; Neuhoff, D.L. (2005). "The Validity of the Additive Noise Model for Uniform Scalar Quantizers". IEEE Transactions on Information Theory. 51 (5): 1739–1755. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2Ftit.2005.846397">10.1109/tit.2005.846397</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0018-9448">0018-9448</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:14819261">14819261</a>.</cite>

</li>

<li id="cite_note-16"><a href="#cite_ref-16">^</a> <cite id="CITEREFPohlman1989" class="citation book cs1">Pohlman, Ken C. (1989). <a rel="nofollow" class="external text" href="https://books.google.com/books?id=VZw6z9a03ikC&pg=PA37">Principles of Digital Audio 2nd Edition</a>. SAMS. p. 60. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780071441568</bdi>.</cite>

</li>

<li id="cite_note-17"><a href="#cite_ref-17">^</a> <cite id="CITEREFWatkinson2001" class="citation book cs1">Watkinson, John (2001). The Art of Digital Audio 3rd Edition. <a href="Focal_Press" title="Focal Press">Focal Press</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>0-240-51587-0</bdi>.</cite>

</li>

<li id="cite_note-19"><a href="#cite_ref-19">^</a> <cite id="CITEREFFarvardinModestino1984" class="citation journal cs1"><a href="Nariman_Farvardin" title="Nariman Farvardin">Farvardin, N.</a>; Modestino, J. (1984). "Optimum quantizer performance for a class of non-Gaussian memoryless sources". IEEE Transactions on Information Theory. 30 (3): 485–497. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2Ftit.1984.1056920">10.1109/tit.1984.1056920</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0018-9448">0018-9448</a>.</cite>(Section VI.C and Appendix B)

</li>

<li id="cite_note-Berger72-20"><a href="#cite_ref-Berger72_20-0">^</a> <cite id="CITEREFBerger1972" class="citation journal cs1"><a href="Toby_Berger" title="Toby Berger">Berger, T.</a> (1972). "Optimum quantizers and permutation codes". IEEE Transactions on Information Theory. 18 (6): 759–765. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2Ftit.1972.1054906">10.1109/tit.1972.1054906</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0018-9448">0018-9448</a>.</cite>

</li>

<li id="cite_note-Berger82-21"><a href="#cite_ref-Berger82_21-0">^</a> <cite id="CITEREFBerger1982" class="citation journal cs1"><a href="Toby_Berger" title="Toby Berger">Berger, T.</a> (1982). "Minimum entropy quantizers and permutation codes". IEEE Transactions on Information Theory. 28 (2): 149–157. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2Ftit.1982.1056456">10.1109/tit.1982.1056456</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0018-9448">0018-9448</a>.</cite>

</li>

<li id="cite_note-22"><a href="#cite_ref-22">^</a> <cite id="CITEREFLloyd1982" class="citation journal cs1">Lloyd, S. (1982). "Least squares quantization in PCM". IEEE Transactions on Information Theory. 28 (2): 129–137. <a href="CiteSeerX_(identifier)" class="mw-redirect" title="CiteSeerX (identifier)">CiteSeerX</a> <a rel="nofollow" class="external text" href="https://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.131.1338">10.1.1.131.1338</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2Ftit.1982.1056489">10.1109/tit.1982.1056489</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0018-9448">0018-9448</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:10833328">10833328</a>.</cite> (work documented in a manuscript circulated for comments at <a href="Bell_Laboratories" class="mw-redirect" title="Bell Laboratories">Bell Laboratories</a> with a department log date of 31 July 1957 and also presented at the 1957 meeting of the <a href="Institute_of_Mathematical_Statistics" title="Institute of Mathematical Statistics">Institute of Mathematical Statistics</a>, although not formally published until 1982).

</li>

<li id="cite_note-23"><a href="#cite_ref-23">^</a> <cite id="CITEREFMax1960" class="citation journal cs1">Max, J. (1960). "Quantizing for minimum distortion". IEEE Transactions on Information Theory. 6 (1): 7–12. <a href="Bibcode_(identifier)" class="mw-redirect" title="Bibcode (identifier)">Bibcode</a>:<a rel="nofollow" class="external text" href="https://ui.adsabs.harvard.edu/abs/1960ITIT....6....7M">1960ITIT....6....7M</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2Ftit.1960.1057548">10.1109/tit.1960.1057548</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0018-9448">0018-9448</a>.</cite>

</li>

<li id="cite_note-ChouLookabaughGray-24"><a href="#cite_ref-ChouLookabaughGray_24-0">^</a> <cite id="CITEREFChouLookabaughGray1989" class="citation journal cs1">Chou, P.A.; Lookabaugh, T.; <a href="Robert_M._Gray" title="Robert M. Gray">Gray, R.M.</a> (1989). "Entropy-constrained vector quantization". IEEE Transactions on Acoustics, Speech, and Signal Processing. 37 (1): 31–42. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1109%2F29.17498">10.1109/29.17498</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/0096-3518">0096-3518</a>.</cite>

</li>

</ol></div></div>

/* start https://en.wikipedia.org/ */

.mw-parser-output .refbegin{margin-bottom:0.5em}.mw-parser-output .refbegin-hanging-indents>ul{margin-left:0}.mw-parser-output .refbegin-hanging-indents>ul>li{margin-left:0;padding-left:3.2em;text-indent:-3.2em}.mw-parser-output .refbegin-hanging-indents ul,.mw-parser-output .refbegin-hanging-indents ul li{list-style:none}@media(max-width:720px){.mw-parser-output .refbegin-hanging-indents>ul>li{padding-left:1.6em;text-indent:-1.6em}}.mw-parser-output .refbegin-columns{margin-top:0.3em}.mw-parser-output .refbegin-columns ul{margin-top:0}.mw-parser-output .refbegin-columns li{page-break-inside:avoid;break-inside:avoid-column}@media screen{.mw-parser-output .refbegin{font-size:90%}}

/* end https://en.wikipedia.org/ */

</style><div class="refbegin" style="">

<ul><li><cite id="CITEREFSayood2005" class="citation cs2">Sayood, Khalid (2005), Introduction to Data Compression, Third Edition, Morgan Kaufmann, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-12-620862-7</bdi></cite></li>

<li><cite id="CITEREFJayantNoll1984" class="citation cs2">Jayant, Nikil S.; Noll, Peter (1984), Digital Coding of Waveforms: Principles and Applications to Speech and Video, Prentice–Hall, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-13-211913-9</bdi></cite></li>

<li><cite id="CITEREFGregg1977" class="citation cs2">Gregg, W. David (1977), Analog & Digital Communication, John Wiley, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-471-32661-8</bdi></cite></li>

<li><cite id="CITEREFSteinJones1967" class="citation cs2">Stein, Seymour; Jones, J. Jay (1967), Modern Communication Principles, <a href="McGraw%E2%80%93Hill" class="mw-redirect" title="McGraw–Hill">McGraw–Hill</a>, <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-07-061003-3</bdi></cite></li></ul>

</div>

<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>

<ul><li><cite id="CITEREFBernard_WidrowIstván_Kollár2007" class="citation book cs1">Bernard Widrow; István Kollár (2007). <a rel="nofollow" class="external text" href="http://www.mit.bme.hu/books/quantization/">Quantization noise in Digital Computation, Signal Processing, and Control</a>. Cambridge University Press. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>9780521886710</bdi>.</cite></li></ul>

/* start https://en.wikipedia.org/ */

.mw-parser-output .hlist dl,.mw-parser-output .hlist ol,.mw-parser-output .hlist ul{margin:0;padding:0}.mw-parser-output .hlist dd,.mw-parser-output .hlist dt,.mw-parser-output .hlist li{margin:0;display:inline}.mw-parser-output .hlist.inline,.mw-parser-output .hlist.inline dl,.mw-parser-output .hlist.inline ol,.mw-parser-output .hlist.inline ul,.mw-parser-output .hlist dl dl,.mw-parser-output .hlist dl ol,.mw-parser-output .hlist dl ul,.mw-parser-output .hlist ol dl,.mw-parser-output .hlist ol ol,.mw-parser-output .hlist ol ul,.mw-parser-output .hlist ul dl,.mw-parser-output .hlist ul ol,.mw-parser-output .hlist ul ul{display:inline}.mw-parser-output .hlist .mw-empty-li{display:none}.mw-parser-output .hlist dt::after{content:": "}.mw-parser-output .hlist dd::after,.mw-parser-output .hlist li::after{content:" · ";font-weight:bold}.mw-parser-output .hlist dd:last-child::after,.mw-parser-output .hlist dt:last-child::after,.mw-parser-output .hlist li:last-child::after{content:none}.mw-parser-output .hlist dd dd:first-child::before,.mw-parser-output .hlist dd dt:first-child::before,.mw-parser-output .hlist dd li:first-child::before,.mw-parser-output .hlist dt dd:first-child::before,.mw-parser-output .hlist dt dt:first-child::before,.mw-parser-output .hlist dt li:first-child::before,.mw-parser-output .hlist li dd:first-child::before,.mw-parser-output .hlist li dt:first-child::before,.mw-parser-output .hlist li li:first-child::before{content:" (";font-weight:normal}.mw-parser-output .hlist dd dd:last-child::after,.mw-parser-output .hlist dd dt:last-child::after,.mw-parser-output .hlist dd li:last-child::after,.mw-parser-output .hlist dt dd:last-child::after,.mw-parser-output .hlist dt dt:last-child::after,.mw-parser-output .hlist dt li:last-child::after,.mw-parser-output .hlist li dd:last-child::after,.mw-parser-output .hlist li dt:last-child::after,.mw-parser-output .hlist li li:last-child::after{content:")";font-weight:normal}.mw-parser-output .hlist ol{counter-reset:listitem}.mw-parser-output .hlist ol>li{counter-increment:listitem}.mw-parser-output .hlist ol>li::before{content:" "counter(listitem)"\a0 "}.mw-parser-output .hlist dd ol>li:first-child::before,.mw-parser-output .hlist dt ol>li:first-child::before,.mw-parser-output .hlist li ol>li:first-child::before{content:" ("counter(listitem)"\a0 "}

/* end https://en.wikipedia.org/ */

</style><style data-mw-deduplicate="TemplateStyles:r1236075235">

/* start https://en.wikipedia.org/ */

.mw-parser-output .navbox{box-sizing:border-box;border:1px solid #a2a9b1;width:100%;clear:both;font-size:88%;text-align:center;padding:1px;margin:1em auto 0}.mw-parser-output .navbox .navbox{margin-top:0}.mw-parser-output .navbox+.navbox,.mw-parser-output .navbox+.navbox-styles+.navbox{margin-top:-1px}.mw-parser-output .navbox-inner,.mw-parser-output .navbox-subgroup{width:100%}.mw-parser-output .navbox-group,.mw-parser-output .navbox-title,.mw-parser-output .navbox-abovebelow{padding:0.25em 1em;line-height:1.5em;text-align:center}.mw-parser-output .navbox-group{white-space:nowrap;text-align:right}.mw-parser-output .navbox,.mw-parser-output .navbox-subgroup{background-color:#fdfdfd}.mw-parser-output .navbox-list{line-height:1.5em;border-color:#fdfdfd}.mw-parser-output .navbox-list-with-group{text-align:left;border-left-width:2px;border-left-style:solid}.mw-parser-output tr+tr>.navbox-abovebelow,.mw-parser-output tr+tr>.navbox-group,.mw-parser-output tr+tr>.navbox-image,.mw-parser-output tr+tr>.navbox-list{border-top:2px solid #fdfdfd}.mw-parser-output .navbox-title{background-color:#ccf}.mw-parser-output .navbox-abovebelow,.mw-parser-output .navbox-group,.mw-parser-output .navbox-subgroup .navbox-title{background-color:#ddf}.mw-parser-output .navbox-subgroup .navbox-group,.mw-parser-output .navbox-subgroup .navbox-abovebelow{background-color:#e6e6ff}.mw-parser-output .navbox-even{background-color:#f7f7f7}.mw-parser-output .navbox-odd{background-color:transparent}.mw-parser-output .navbox .hlist td dl,.mw-parser-output .navbox .hlist td ol,.mw-parser-output .navbox .hlist td ul,.mw-parser-output .navbox td.hlist dl,.mw-parser-output .navbox td.hlist ol,.mw-parser-output .navbox td.hlist ul{padding:0.125em 0}.mw-parser-output .navbox .navbar{display:block;font-size:100%}.mw-parser-output .navbox-title .navbar{float:left;text-align:left;margin-right:0.5em}body.skin--responsive .mw-parser-output .navbox-image img{max-width:none!important}@media print{body.ns-0 .mw-parser-output .navbox{display:none!important}}

/* end https://en.wikipedia.org/ */

</style></div><div role="navigation" class="navbox" aria-labelledby="Digital_signal_processing96" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><style data-mw-deduplicate="TemplateStyles:r1239400231">

/* start https://en.wikipedia.org/ */

.mw-parser-output .navbar{display:inline;font-size:88%;font-weight:normal}.mw-parser-output .navbar-collapse{float:left;text-align:left}.mw-parser-output .navbar-boxtext{word-spacing:0}.mw-parser-output .navbar ul{display:inline-block;white-space:nowrap;line-height:inherit}.mw-parser-output .navbar-brackets::before{margin-right:-0.125em;content:"[ "}.mw-parser-output .navbar-brackets::after{margin-left:-0.125em;content:" ]"}.mw-parser-output .navbar li{word-spacing:-0.125em}.mw-parser-output .navbar a>span,.mw-parser-output .navbar a>abbr{text-decoration:inherit}.mw-parser-output .navbar-mini abbr{font-variant:small-caps;border-bottom:none;text-decoration:none;cursor:inherit}.mw-parser-output .navbar-ct-full{font-size:114%;margin:0 7em}.mw-parser-output .navbar-ct-mini{font-size:114%;margin:0 4em}html.skin-theme-clientpref-night .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}@media(prefers-color-scheme:dark){html.skin-theme-clientpref-os .mw-parser-output .navbar li a abbr{color:var(--color-base)!important}}@media print{.mw-parser-output .navbar{display:none!important}}

/* end https://en.wikipedia.org/ */

</style><div id="Digital_signal_processing96" style="font-size:114%;margin:0 4em"><a href="Digital_signal_processing" title="Digital signal processing">Digital signal processing</a></div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">Theory</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Detection_theory" title="Detection theory">Detection theory</a></li>

<li><a href="Discrete_time_and_continuous_time" title="Discrete time and continuous time">Discrete signal</a></li>

<li><a href="Estimation_theory" title="Estimation theory">Estimation theory</a></li>

<li><a href="Nyquist%E2%80%93Shannon_sampling_theorem" title="Nyquist–Shannon sampling theorem">Nyquist–Shannon sampling theorem</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Sub-fields</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Audio_signal_processing" title="Audio signal processing">Audio signal processing</a></li>

<li><a href="Digital_image_processing" title="Digital image processing">Digital image processing</a></li>

<li><a href="Speech_processing" title="Speech processing">Speech processing</a></li>

<li><a href="Statistical_signal_processing" class="mw-redirect" title="Statistical signal processing">Statistical signal processing</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Techniques</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Z-transform" title="Z-transform">Z-transform</a>

<ul><li><a href="Advanced_z-transform" title="Advanced z-transform">Advanced z-transform</a></li>

<li><a href="Matched_Z-transform_method" title="Matched Z-transform method">Matched Z-transform method</a></li></ul></li>

<li><a href="Bilinear_transform" title="Bilinear transform">Bilinear transform</a></li>

<li><a href="Constant-Q_transform" title="Constant-Q transform">Constant-Q transform</a></li>

<li><a href="Discrete_cosine_transform" title="Discrete cosine transform">Discrete cosine transform</a> (DCT)</li>

<li><a href="Discrete_Fourier_transform" title="Discrete Fourier transform">Discrete Fourier transform</a> (DFT)</li>

<li><a href="Discrete-time_Fourier_transform" title="Discrete-time Fourier transform">Discrete-time Fourier transform</a> (DTFT)</li>

<li><a href="Impulse_invariance" title="Impulse invariance">Impulse invariance</a></li>

<li><a href="Integral_transform" title="Integral transform">Integral transform</a></li>

<li><a href="Laplace_transform" title="Laplace transform">Laplace transform</a></li>

<li><a href="Post's_inversion_formula" class="mw-redirect" title="Post's inversion formula">Post's inversion formula</a></li>

<li><a href="Starred_transform" title="Starred transform">Starred transform</a></li>

<li><a href="Zak_transform" title="Zak transform">Zak transform</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Sampling_(signal_processing)" title="Sampling (signal processing)">Sampling</a></th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Aliasing" title="Aliasing">Aliasing</a></li>

<li><a href="Anti-aliasing_filter" title="Anti-aliasing filter">Anti-aliasing filter</a></li>

<li><a href="Downsampling_(signal_processing)" title="Downsampling (signal processing)">Downsampling</a></li>

<li><a href="Nyquist_rate" title="Nyquist rate">Nyquist rate</a> / <a href="Nyquist_frequency" title="Nyquist frequency">frequency</a></li>

<li><a href="Oversampling" title="Oversampling">Oversampling</a></li>

<li><a href="Sampling_rate" class="mw-redirect" title="Sampling rate">Sampling rate</a></li>

<li><a href="Undersampling" title="Undersampling">Undersampling</a></li>

<li><a href="Upsampling" title="Upsampling">Upsampling</a></li></ul>

</div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Data_compression_methods241" style="padding:3px"><table class="nowraplinks hlist mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Data_compression_methods241" style="font-size:114%;margin:0 4em"><a href="Data_compression" title="Data compression">Data compression</a> methods</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Lossless_compression" title="Lossless compression">Lossless</a> type</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Entropy_coding" title="Entropy coding">Entropy</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Adaptive_coding" title="Adaptive coding">Adaptive coding</a></li>

<li><a href="Arithmetic_coding" title="Arithmetic coding">Arithmetic</a></li>

<li><a href="Asymmetric_numeral_systems" title="Asymmetric numeral systems">Asymmetric numeral systems</a></li>

<li><a href="Golomb_coding" title="Golomb coding">Golomb</a></li>

<li><a href="Huffman_coding" title="Huffman coding">Huffman</a>

<ul><li><a href="Adaptive_Huffman_coding" title="Adaptive Huffman coding">Adaptive</a></li>

<li><a href="Canonical_Huffman_code" title="Canonical Huffman code">Canonical</a></li>

<li><a href="Modified_Huffman_coding" title="Modified Huffman coding">Modified</a></li></ul></li>

<li><a href="Range_coding" title="Range coding">Range</a></li>

<li><a href="Shannon_coding" title="Shannon coding">Shannon</a></li>

<li><a href="Shannon%E2%80%93Fano_coding" title="Shannon–Fano coding">Shannon–Fano</a></li>

<li><a href="Shannon%E2%80%93Fano%E2%80%93Elias_coding" title="Shannon–Fano–Elias coding">Shannon–Fano–Elias</a></li>

<li><a href="Tunstall_coding" title="Tunstall coding">Tunstall</a></li>

<li><a href="Unary_coding" title="Unary coding">Unary</a></li>

<li><a href="Universal_code_(data_compression)" title="Universal code (data compression)">Universal</a>

<ul><li><a href="Exponential-Golomb_coding" title="Exponential-Golomb coding">Exp-Golomb</a></li>

<li><a href="Fibonacci_coding" title="Fibonacci coding">Fibonacci</a></li>

<li><a href="Elias_gamma_coding" title="Elias gamma coding">Gamma</a></li>

<li><a href="Levenshtein_coding" title="Levenshtein coding">Levenshtein</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Dictionary_coder" title="Dictionary coder">Dictionary</a></th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Byte-pair_encoding" title="Byte-pair encoding">Byte-pair encoding</a></li>

<li><a href="LZ77_and_LZ78" title="LZ77 and LZ78">Lempel–Ziv</a>

<li><a href="Snappy_(compression)" title="Snappy (compression)">Snappy</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Other</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Delta_encoding" title="Delta encoding">Delta</a>

<ul><li><a href="Incremental_encoding" title="Incremental encoding">Incremental</a></li></ul></li>

<li><a href="Grammar-based_code" title="Grammar-based code">Grammar</a>

<li><a href="Sequitur_algorithm" title="Sequitur algorithm">Sequitur</a></li></ul></li>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Hybrid</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li>LZ77 + Huffman

<ul><li><a href="Deflate" title="Deflate">Deflate</a></li>

<li>LZ77 + ANS

<ul><li><a href="LZFSE" title="LZFSE">LZFSE</a></li></ul></li>

<li>LZ77 + Huffman + ANS

<ul><li><a href="Zstd" title="Zstd">Zstandard</a></li></ul></li>

<li>LZ77 + Huffman + context

<ul><li><a href="Brotli" title="Brotli">Brotli</a></li></ul></li>

<li>LZSS + Huffman

<li>LZ77 + Range

<li>LZHAM</li></ul></li>

<li>RLE + BWT + MTF + Huffman

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Lossy_compression" title="Lossy compression">Lossy</a> type</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Transform_coding" title="Transform coding">Transform</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Discrete_cosine_transform" title="Discrete cosine transform">Discrete cosine transform</a>

<li><a href="Wavelet_transform" title="Wavelet transform">Wavelet</a>

<ul><li><a href="Daubechies_wavelet" title="Daubechies wavelet">Daubechies</a></li>

<li><a href="Set_partitioning_in_hierarchical_trees" title="Set partitioning in hierarchical trees">SPIHT</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Predictive</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Adaptive_differential_pulse-code_modulation" title="Adaptive differential pulse-code modulation">ADPCM</a></li></ul></li>

<ul><li><a href="Algebraic_code-excited_linear_prediction" title="Algebraic code-excited linear prediction">ACELP</a></li>

<li>Motion

<ul><li><a href="Motion_compensation" title="Motion compensation">Compensation</a></li>

<li><a href="Motion_estimation" title="Motion estimation">Estimation</a></li>

<li><a href="Motion_vector" class="mw-redirect" title="Motion vector">Vector</a></li></ul></li>

<li><a href="Psychoacoustics" title="Psychoacoustics">Psychoacoustic</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Data_compression#Audio" title="Data compression">Audio</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Companding" title="Companding">Companding</a></li>

<li><a href="Convolution" title="Convolution">Convolution</a></li>

<li><a href="Dynamic_range" title="Dynamic range">Dynamic range</a></li>

<li><a href="Latency_(audio)" title="Latency (audio)">Latency</a></li>

<li><a href="Nyquist%E2%80%93Shannon_sampling_theorem" title="Nyquist–Shannon sampling theorem">Nyquist–Shannon theorem</a></li>

<li><a href="Sampling_(signal_processing)" title="Sampling (signal processing)">Sampling</a></li>

<li><a href="Silence_compression" title="Silence compression">Silence compression</a></li>

<li><a href="Sound_quality" title="Sound quality">Sound quality</a></li>

<li><a href="Speech_coding" title="Speech coding">Speech coding</a></li>

<li><a href="Sub-band_coding" title="Sub-band coding">Sub-band coding</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Audio_codec" title="Audio codec">Codec</a> parts</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Adaptive_differential_pulse-code_modulation" title="Adaptive differential pulse-code modulation">ADPCM</a></li>

<ul><li><a href="Algebraic_code-excited_linear_prediction" title="Algebraic code-excited linear prediction">ACELP</a></li>

<li><a href="Psychoacoustics" title="Psychoacoustics">Psychoacoustic model</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Image_compression" title="Image compression">Image</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Chroma_subsampling" title="Chroma subsampling">Chroma subsampling</a></li>

<li><a href="Coding_tree_unit" title="Coding tree unit">Coding tree unit</a></li>

<li><a href="Color_space" title="Color space">Color space</a></li>

<li><a href="Compression_artifact" title="Compression artifact">Compression artifact</a></li>

<li><a href="Image_resolution" title="Image resolution">Image resolution</a></li>

<li><a href="Macroblock" title="Macroblock">Macroblock</a></li>

<li><a href="Pixel" title="Pixel">Pixel</a></li>

<li><a href="Quantization_(image_processing)" title="Quantization (image processing)">Quantization</a></li>

<li><a href="Standard_test_image" title="Standard test image">Standard test image</a></li>

<li><a href="Texture_compression" title="Texture compression">Texture compression</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Methods</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Chain_code" title="Chain code">Chain code</a></li>

<li><a href="Deflate" title="Deflate">Deflate</a></li>

<li><a href="Fractal_compression" title="Fractal compression">Fractal</a></li>

<li><a href="Wavelet_transform" title="Wavelet transform">Wavelet</a>

<ul><li><a href="Daubechies_wavelet" title="Daubechies wavelet">Daubechies</a></li>

<li><a href="Set_partitioning_in_hierarchical_trees" title="Set partitioning in hierarchical trees">SPIHT</a></li></ul></li></ul>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Data_compression#Video" title="Data compression">Video</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">Concepts</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Display_resolution" title="Display resolution">Display resolution</a></li>

<li><a href="Film_frame" title="Film frame">Frame</a></li>

<li><a href="Frame_rate" title="Frame rate">Frame rate</a></li>

<li><a href="Video_compression_picture_types" title="Video compression picture types">Frame types</a></li>

<li><a href="Interlaced_video" title="Interlaced video">Interlace</a></li>

<li><a href="Video#Characteristics_of_video_streams" title="Video">Video characteristics</a></li>

<li><a href="Video_quality" title="Video quality">Video quality</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Video_codec" title="Video codec">Codec</a> parts</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<li><a href="Deblocking_filter" title="Deblocking filter">Deblocking filter</a></li>

<li><a href="Lapped_transform" title="Lapped transform">Lapped transform</a></li>

<li>Motion

<ul><li><a href="Motion_compensation" title="Motion compensation">Compensation</a></li>

<li><a href="Motion_estimation" title="Motion estimation">Estimation</a></li>

<li><a href="Motion_vector" class="mw-redirect" title="Motion vector">Vector</a></li></ul></li>

<li><a href="Wavelet_transform" title="Wavelet transform">Wavelet</a>

<ul><li><a href="Daubechies_wavelet" title="Daubechies wavelet">Daubechies</a></li>

</div></td></tr></tbody></table><div></div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%"><a href="Information_theory" title="Information theory">Theory</a></th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Compressed_data_structure" title="Compressed data structure">Compressed data structures</a>

<ul><li><a href="Compressed_suffix_array" title="Compressed suffix array">Compressed suffix array</a></li>

<li><a href="FM-index" title="FM-index">FM-index</a></li></ul></li>

<li><a href="Entropy_(information_theory)" title="Entropy (information theory)">Entropy</a></li>

<li><a href="Information_theory" title="Information theory">Information theory</a>

<ul><li><a href="Timeline_of_information_theory" title="Timeline of information theory">Timeline</a></li></ul></li>

<li><a href="Kolmogorov_complexity" title="Kolmogorov complexity">Kolmogorov complexity</a></li>

<li><a href="Prefix_code" title="Prefix code">Prefix code</a></li>

<li><a href="Rate%E2%80%93distortion_theory" title="Rate–distortion theory">Rate–distortion</a></li>

<li><a href="Redundancy_(information_theory)" title="Redundancy (information theory)">Redundancy</a></li>

<li><a href="Data_compression_symmetry" title="Data compression symmetry">Symmetry</a></li>

<li><a href="Smallest_grammar_problem" title="Smallest grammar problem">Smallest grammar problem</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Community</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Hutter_Prize" title="Hutter Prize">Hutter Prize</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">People</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Mark_Adler" title="Mark Adler">Mark Adler</a></li>

</div></td></tr></tbody></table></div>

<div class="navbox-styles"></div><div role="navigation" class="navbox" aria-labelledby="Noise_(physics_and_telecommunications)164" style="padding:3px"><table class="nowraplinks mw-collapsible autocollapse navbox-inner" style="border-spacing:0;background:transparent;color:inherit"><tbody><tr><th scope="col" class="navbox-title" colspan="2"><div id="Noise_(physics_and_telecommunications)164" style="font-size:114%;margin:0 4em"><a href="Noise_(spectral_phenomenon)" title="Noise (spectral phenomenon)">Noise</a> (physics and telecommunications)</div></th></tr><tr><th scope="row" class="navbox-group" style="width:1%">General</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Acoustic_quieting" title="Acoustic quieting">Acoustic quieting</a></li>

<li><a href="Distortion" title="Distortion">Distortion</a></li>

<li><a href="Active_noise_control" title="Active noise control">Noise cancellation</a></li>

<li><a href="Noise_control" title="Noise control">Noise control</a></li>

<li><a href="Noise_measurement" title="Noise measurement">Noise measurement</a></li>

<li><a href="Noise_power" title="Noise power">Noise power</a></li>

<li><a href="Noise_reduction" title="Noise reduction">Noise reduction</a></li>

<li><a href="Noise_temperature" title="Noise temperature">Noise temperature</a></li>

<li><a href="Phase_distortion" title="Phase distortion">Phase distortion</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Noise in...</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Noise" title="Noise">Audio</a></li>

<li><a href="Architectural_acoustics" title="Architectural acoustics">Buildings</a></li>

<li><a href="Noise_(electronics)" title="Noise (electronics)">Electronics</a></li>

<li><a href="Noise_pollution" title="Noise pollution">Environment</a></li>

<li><a href="Noise_regulation" title="Noise regulation">Government regulation</a></li>

<li><a href="Health_effects_from_noise" title="Health effects from noise">Human health</a></li>

<li><a href="Image_noise" title="Image noise">Images</a></li>

<li><a href="Noise_(radio)" class="mw-redirect" title="Noise (radio)">Radio</a></li>

<li><a href="Soundproofing" title="Soundproofing">Rooms</a></li>

<li><a href="Noise_and_vibration_on_maritime_vessels" title="Noise and vibration on maritime vessels">Ships</a></li>

<li><a href="Sound_masking" title="Sound masking">Sound masking</a></li>

<li><a href="Noise_barrier" title="Noise barrier">Transportation</a></li>

<li><a href="Noise_(video)" title="Noise (video)">Video</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Class of noise</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Additive_white_Gaussian_noise" title="Additive white Gaussian noise">Additive white Gaussian noise</a> (AWGN)</li>

<li><a href="Atmospheric_noise" title="Atmospheric noise">Atmospheric noise</a></li>

<li><a href="Background_noise" title="Background noise">Background noise</a></li>

<li><a href="Brownian_noise" title="Brownian noise">Brownian noise</a></li>

<li><a href="Burst_noise" title="Burst noise">Burst noise</a></li>

<li><a href="Cosmic_noise" title="Cosmic noise">Cosmic noise</a></li>

<li><a href="Flicker_noise" title="Flicker noise">Flicker noise</a></li>

<li><a href="Gaussian_noise" title="Gaussian noise">Gaussian noise</a></li>

<li><a href="Grey_noise" title="Grey noise">Grey noise</a></li>

<li><a href="Infrasound" title="Infrasound">Infrasound</a></li>

<li><a href="Jitter" title="Jitter">Jitter</a></li>

<li><a href="Johnson%E2%80%93Nyquist_noise" title="Johnson–Nyquist noise">Johnson–Nyquist noise</a> (thermal noise)</li>

<li><a href="Pink_noise" title="Pink noise">Pink noise</a></li>

<li><a href="Quantization_error" class="mw-redirect" title="Quantization error">Quantization error</a> (or q. noise)</li>

<li><a href="Shot_noise" title="Shot noise">Shot noise</a></li>

<li><a href="White_noise" title="White noise">White noise</a></li>

<li>Coherent noise

<ul><li><a href="Value_noise" title="Value noise">Value noise</a></li>

<li><a href="Gradient_noise" title="Gradient noise">Gradient noise</a></li>

<li><a href="Worley_noise" title="Worley noise">Worley noise</a></li></ul></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Engineering terms</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Channel_noise_level" title="Channel noise level">Channel noise level</a></li>

<li><a href="Circuit_noise_level" title="Circuit noise level">Circuit noise level</a></li>

<li><a href="Effective_input_noise_temperature" title="Effective input noise temperature">Effective input noise temperature</a></li>

<li><a href="Equivalent_noise_resistance" title="Equivalent noise resistance">Equivalent noise resistance</a></li>

<li><a href="Equivalent_pulse_code_modulation_noise" title="Equivalent pulse code modulation noise">Equivalent pulse code modulation noise</a></li>

<li><a href="Impulse_noise_(audio)" class="mw-redirect" title="Impulse noise (audio)">Impulse noise (audio)</a></li>

<li><a href="Noise_figure" title="Noise figure">Noise figure</a></li>

<li><a href="Noise_floor" title="Noise floor">Noise floor</a></li>

<li><a href="Noise_shaping" title="Noise shaping">Noise shaping</a></li>

<li><a href="Noise_spectral_density" title="Noise spectral density">Noise spectral density</a></li>

<li><a href="Noise%2C_vibration%2C_and_harshness" title="Noise, vibration, and harshness">Noise, vibration, and harshness</a> (NVH)</li>

<li><a href="Phase_noise" title="Phase noise">Phase noise</a></li>

<li><a href="Pseudorandom_noise" title="Pseudorandom noise">Pseudorandom noise</a></li>

<li><a href="Statistical_noise" class="mw-redirect" title="Statistical noise">Statistical noise</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Ratios</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Carrier-to-noise_ratio" title="Carrier-to-noise ratio">Carrier-to-noise ratio</a> (C/N)</li>

<li><a href="Carrier-to-receiver_noise_density" class="mw-redirect" title="Carrier-to-receiver noise density">Carrier-to-receiver noise density</a> (C/kT)</li>

<li><a href="DBrnC" class="mw-redirect" title="DBrnC">dBrnC</a></li>

<li><a href="Eb/N0" title="Eb/N0">Eb/N0</a> (energy per bit to noise density)</li>

<li><a href="Eb/N0#Relation_to_Es.2FN0" title="Eb/N0">Es/N0</a> (energy per symbol to noise density)</li>

<li><a href="Modulation_error_ratio" title="Modulation error ratio">Modulation error ratio</a> (MER)</li>

<li><a href="SINAD" class="mw-redirect" title="SINAD">Signal, noise and distortion</a> (SINAD)</li>

<li><a href="Signal-to-interference_ratio" title="Signal-to-interference ratio">Signal-to-interference ratio</a> (S/I)</li>

<li><a href="Signal-to-noise_ratio" title="Signal-to-noise ratio">Signal-to-noise ratio</a> (S/N, SNR)</li>

<li><a href="Signal-to-noise_ratio_(imaging)" title="Signal-to-noise ratio (imaging)">Signal-to-noise ratio (imaging)</a></li>

<li><a href="Signal-to-interference-plus-noise_ratio" title="Signal-to-interference-plus-noise ratio">Signal-to-interference-plus-noise ratio</a> (SINR)</li>

<li><a href="Signal-to-quantization-noise_ratio" title="Signal-to-quantization-noise ratio">Signal-to-quantization-noise ratio</a> (SQNR)</li>

<li><a href="Contrast-to-noise_ratio" title="Contrast-to-noise ratio">Contrast-to-noise ratio</a> (CNR)</li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Related topics</th><td class="navbox-list-with-group navbox-list navbox-even hlist" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="List_of_noise_topics" title="List of noise topics">List of noise topics</a></li>

<li><a href="Acoustics" title="Acoustics">Acoustics</a></li>

<li><a href="Colors_of_noise" title="Colors of noise">Colors of noise</a></li>

<li><a href="Interference_(communication)" title="Interference (communication)">Interference (communication)</a></li>

<li><a href="Noise_generator" title="Noise generator">Noise generator</a></li>

<li><a href="Spectrum_analyzer" title="Spectrum analyzer">Spectrum analyzer</a></li>

<li><a href="Thermal_radiation" title="Thermal radiation">Thermal radiation</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">Denoise methods</th><td class="navbox-list-with-group navbox-list navbox-odd hlist" style="width:100%;padding:0"><div style="padding:0 0.25em"></div><table class="nowraplinks navbox-subgroup" style="border-spacing:0"><tbody><tr><th scope="row" class="navbox-group" style="width:1%">General</th><td class="navbox-list-with-group navbox-list navbox-odd" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Low-pass_filter" title="Low-pass filter">Low-pass filter</a></li>

<li><a href="Median_filter" title="Median filter">Median filter</a></li>

<li><a href="Total_variation_denoising" title="Total variation denoising">Total variation denoising</a></li>

<li><a href="Wavelet#Wavelet_denoising" title="Wavelet">Wavelet denoising</a></li></ul>

</div></td></tr><tr><th scope="row" class="navbox-group" style="width:1%">2D (Image)</th><td class="navbox-list-with-group navbox-list navbox-even" style="width:100%;padding:0"><div style="padding:0 0.25em">

<ul><li><a href="Gaussian_blur" title="Gaussian blur">Gaussian blur</a></li>

<li><a href="Anisotropic_diffusion" title="Anisotropic diffusion">Anisotropic diffusion</a></li>

<li><a href="Bilateral_filter" title="Bilateral filter">Bilateral filter</a></li>

<li><a href="Non-local_means" title="Non-local means">Non-local means</a></li>

<li><a href="Block-matching_and_3D_filtering" title="Block-matching and 3D filtering">Block-matching and 3D filtering</a> (BM3D)</li>

<li><a href="Shrinkage_Fields_(image_restoration)" title="Shrinkage Fields (image restoration)">Shrinkage Fields</a></li>

<li><a href="Autoencoder#Denoising_autoencoder_(DAE)" title="Autoencoder">Denoising autoencoder</a> (DAE)</li>

<li><a href="Deep_Image_Prior" class="mw-redirect" title="Deep Image Prior">Deep Image Prior</a></li></ul>

</div></td></tr></tbody></table><div></div></td></tr></tbody></table></div></div><div><div class="zim-footer">

This article is issued from <a class="external text" title="Last edited on 2025-08-06" href="https://en.wikipedia.org/wiki/?title=Quantization_(signal_processing)&oldid=1304554335">Wikipedia</a>. The text is available under <a class="external text" href="https://creativecommons.org/licenses/by-sa/4.0/deed.en">Creative Commons Attribution-Share Alike 4.0</a> unless otherwise noted. Additional terms may apply for the media files.

</div>

</div></div>

</div>

</main>

</div>

</body></html>