formulasearchengine - User contributions [en]

Kernel regression

2014-11-28T18:35:37Z

128.211.189.46: /* Script for example */

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Similarities between Wiener and LMS

2014-09-20T18:10:49Z

128.211.168.1:

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Ellipsoid

2014-03-03T22:55:20Z

128.211.178.12: /* Surface area */ <ref>

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Ellipse

2014-03-01T00:35:24Z

128.211.178.12: redir

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Henry's law

2014-02-17T20:55:53Z

128.211.178.2:

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Spacetime

2014-01-29T16:09:46Z

128.211.178.1: /* Privileged character of 3+1 spacetime */

{{About||static analysis in economics|Comparative statics|the technique of static correction used in [[exploration geophysics]]|Reflection seismology}}

{{Classical mechanics|cTopic=Branches}}
'''Statics''' is the branch of [[mechanics]] that is concerned with the analysis of loads ([[force]] and [[torque|torque, or "moment"]]) on [[physical system]]s in static equilibrium, that is, in a state where the relative positions of subsystems do not vary over time, or where components and structures are at a constant velocity. When in static equilibrium, the system is either at rest, or its [[center of mass]] moves at constant velocity.

By [[Newton's laws of motion|Newton's first law]], this situation implies that the net force and net torque (also known as moment of force) on every part of the system is zero. From this constraint, such quantities as [[stress (physics)|stress]] or [[pressure]] can be derived. The net forces equaling zero is known as the ''first condition for equilibrium,'' and the net torque equaling zero is known as the ''second condition for equilibrium.'' See [[statically determinate]].

==Vectors==
[[Image:Beam in static equilibrium2.svg|framed|Example of a beam in static equilibrium. The sum of force and moment is zero.]]
A scalar is a quantity, such as mass or temperature, which only has a magnitude. A vector is a quantity that has both a magnitude and a direction. There are several notations to identify a vector, including:
*A bold faced character '''V'''
*An underlined character V
*A character with an arrow over it <math>\overrightarrow{V}</math>.

Vectors can be added using the [[parallelogram law]] or the [[Euclidean vector|triangle law]]. Vectors contain components in orthogonal bases. Unit vectors i, j, and k are by convention [[Cartesian coordinate system|along the x, y, and z axes]].

==Force==
'''Force''' is the action of one body on another. A force is either a push or a pull. A force tends to move a body in the direction of its action. The action of a force is characterized by its magnitude, by the direction of its action, and by its point of application. Thus force is a vector quantity, because its effect depends on the direction as well as on the magnitude of the action.<ref>Meriam, James L., and L. Glenn Kraige. ''Engineering Mechanics'' (6th ed.) Hoboken, N.J.: John Wiley & Sons, 2007; p. 23</ref>

Forces are classified as either contact or body forces. A contact force is produced by direct physical contact; an example is the force exerted on a body by a supporting surface. A body force is generated by virtue of the position of a body within a force field such as a gravitational, electric, or magnetic field. An example of a body force is the weight of a body in the Earth's gravitational pull.<ref>''Engineering Mechanics'', p. 24</ref>

==Moment of a force==
In addition to the tendency to move a body in the direction of its application, a force can also tend to rotate a body about an axis. The axis may be any line which neither intersects nor is parallel to the [[line of action]] of the force. This rotational tendency is known as the ''moment'' ('''M''') of the force. Moment is also referred to as ''torque''.

===Moment about a point===
The magnitude of the moment of a force at a point O, is equal to the perpendicular distance from O to the line of action of F, multiplied by the magnitude of the force: M = F * d, where

F = the force applied 
d = the perpendicular distance from the axis to the line of action of the force. This perpendicular distance is called the moment arm.

The direction of the moment is given by the right hand rule, where counter clockwise (CCW) is out of the page, and clockwise (CW) is into the page. The moment direction may be accounted for by using a stated sign convention, such as a plus sign (+) for counterclockwise moments and a minus sign (−) for clockwise moments, or vice versa. Moments can be added together as vectors.

In vector format, the moment can be defined as the [[cross product]] between the radius vector, '''r''' (the vector from point O to the line of action), and the force vector, '''F''':<ref>{{cite book|last=Hibbeler|first=R. C.|title=Engineering Mechanics: Statics, 12th Ed.|year=2010|publisher=Pearson Prentice Hall|location=New Jersey|isbn=10: 0-13-607790-0}}</ref>

:<math>\textbf{M}_{O}=\textbf{r} \times \textbf{F}</math>

===Varignon's theorem===
[[Varignon's theorem]] states that the moment of a force about any point is equal to the sum of the moments of the components of the force about the same point.

==Equilibrium equations==
The static equilibrium of a particle is an important concept in statics. A particle is in equilibrium only if the resultant of all forces acting on the particle is equal to zero. In a rectangular coordinate system the equilibrium equations can be represented by three scalar equations, where the sums of forces in all three directions are equal to zero. An engineering application of this concept is determining the tensions of up to three cables under load, for example the forces exerted on each cable of a hoist lifting an object or of [[guy wires]] restraining a [[hot air balloon]] to the ground.<ref>{{cite book|last=Beer|first=Ferdinand|title=Vector Statics For Engineers|publisher=McGraw Hill|year=2004|isbn=0-07-121830-0}}</ref>

==Moment of inertia==
In classical mechanics, [[moment of inertia]], also called mass moment, rotational inertia, polar moment of inertia of mass, or the angular mass, (SI units kg·m²) is a measure of an object's resistance to changes to its rotation. It is the inertia of a rotating body with respect to its rotation. The moment of inertia plays much the same role in rotational dynamics as mass does in linear dynamics, describing the relationship between angular momentum and angular velocity, torque and angular acceleration, and several other quantities. The symbols I and J are usually used to refer to the moment of inertia or polar moment of inertia.

While a simple scalar treatment of the moment of inertia suffices for many situations, a more advanced tensor treatment allows the analysis of such complicated systems as spinning tops and gyroscopic motion.

The concept was introduced by [[Leonhard Euler]] in his 1765 book ''Theoria motus corporum solidorum seu rigidorum''; he discussed the moment of inertia and many related concepts, such as the principal axis of inertia.

==Solids==
Statics is used in the analysis of structures, for instance in [[architectural engineering|architectural]] and [[structural engineering]]. [[Strength of materials]] is a related field of mechanics that relies heavily on the application of static equilibrium. A key concept is the [[center of gravity]] of a body at rest: it represents an imaginary point at which all the [[mass]] of a body resides. The position of the point relative to the [[Foundation (engineering)|foundation]]s on which a body lies determines its [[Stability theory|stability]] in response to external forces. If the center of gravity exists outside the foundations, then the body is unstable because there is a torque acting: any small disturbance will cause the body to fall or topple. If the center of gravity exists within the foundations, the body is stable since no net torque acts on the body. If the center of gravity coincides with the foundations, then the body is said to be [[metastable]].

==Fluids==
[[Hydrostatics]], also known as [[fluid statics]], is the study of fluids at rest (i.e. in static equilibrium). The characteristic of any fluid at rest is that the force exerted on any particle of the fluid is the same at all points at the same depth (or altitude) within the fluid. If the net force is greater than zero the fluid will move in the direction of the resulting force. This concept was first formulated in a slightly extended form by [[France|French]] [[mathematician]] and [[philosopher]] [[Blaise Pascal]] in 1647 and became known as [[Pascal's Law]]. It has many important applications in [[hydraulics]]. [[Archimedes]], [[Abū Rayhān al-Bīrūnī]], [[Al-Khazini]]<ref name=Rozhanskaya-642>Mariam Rozhanskaya and I. S. Levinova (1996), "Statics", p. 642, in {{Harv|Morelon|Rashed|1996|pp=614–642}}: {{quote|"Using a whole body of mathematical methods (not only those inherited from the antique theory of ratios and infinitesimal techniques, but also the methods of the contemporary algebra and fine calculation techniques), Arabic scientists raised statics to a new, higher level. The classical results of Archimedes in the theory of the centre of gravity were generalized and applied to three-dimensional bodies, the theory of ponderable lever was founded and the 'science of gravity' was created and later further developed in medieval Europe. The phenomena of statics were studied by using the dynamic approach so that two trends - statics and dynamics - turned out to be inter-related within a single science, mechanics. The combination of the dynamic approach with Archimedean hydrostatics gave birth to a direction in science which may be called medieval hydrodynamics. [...] Numerous experimental methods were developed for determining the specific weight, which were based, in particular, on the theory of balances and weighing. The classical works of al-Biruni and al-Khazini may be considered the beginning of the application of experimental methods in [[medieval science]]."}}</ref> and [[Galileo Galilei]] were also major figures in the development of hydrostatics.

==See also==
{{Portal|Physics}}
*[[Analytical dynamics|Dynamics]]
*[[Mechanical equilibrium]]
*[[Solid mechanics]]
*[[Cremona diagram]]

==References==
<references />
*{{cite book|author=Beer, F.P. and Johnston Jr, E.R.|title=Statics and Mechanics of Materials|year=1992|publisher=McGraw-Hill, Inc}}
*{{cite book|author=Beer, Johnston, and Eisenberg|title=Vector Mechanics for Engineers: Statics, 9th Ed.|year=2009|publisher=ISBN 978-0-07-352923-3, McGraw Hill}}

==External links==
{{Commons category}}
{{Wiktionary}}
*[http://engineering-education.com/CATS-index.php Online test of statics conceptual knowledge (meant for teachers)]
*[http://oli.web.cmu.edu/openlearning/forstudents/freecourses/engineering-statics Free engineering Statics courseware with about 300 interactive exercises with hints and feedback] : Carnegie Mellon Open Learning Initiative
*[http://www.societyofrobots.com/mechanics_statics.shtml Statics for Robotics]
{{wikibooks|Statics}}
*[http://www.engin.brown.edu/courses/en3/Notes/Statics/Introduction/Introduction.pdf]
*[http://emweb.unl.edu/negahban/em223/intro.htm]

[[Category:Engineering]]
[[Category:Statics]]

Autoignition temperature

2014-01-11T14:26:04Z

128.211.253.133: /* Autoignition point of selected substances */

[[File:Quantization error.png|thumb|500px|The simplest way to quantize a signal is to choose the digital amplitude value closest to the original analog amplitude. The quantization error that results from this simple quantization scheme is a deterministic function of the input signal.]]

'''Quantization''', in mathematics and [[digital signal processing]], is the process of mapping a large set of input values to a (countable) smaller set – such as [[rounding]] values to some unit of precision. A device or [[algorithm function|algorithmic function]] that performs quantization is called a '''quantizer'''. The [[round-off error]] introduced by quantization is referred to as '''quantization error'''.

In [[analog-to-digital conversion]], the difference between the actual analog value and quantized digital value is called '''quantization error''' or '''quantization distortion'''. This error is either due to [[rounding]] or [[truncation]]. The error signal is sometimes modeled as an additional random signal called '''quantization noise''' because of its [[stochastic]] behaviour. 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 [[lossy compression]] algorithms.

==Basic properties and types of quantization==
[[File:2-bit resolution analog comparison.png|thumbnail|2-bit resolution with four levels of quantization compared to analog.<ref>Hodgson, Jay (2010). ''Understanding Records'', p.56. ISBN 978-1-4411-5607-5. Adapted from Franz, David (2004). ''Recording and Producing in the Home Studio'', p.38-9. Berklee Press.</ref>]]
[[File:3-bit resolution analog comparison.png|thumbnail|3-bit resolution with eight levels.]]

Because quantization is a many-to-few mapping, it is an inherently [[Nonlinear system|non-linear]] 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 [[uncountable]] (such as the set of all [[real number]]s, or all real numbers within some limited range). The set of possible output values may be [[finite set|finite]] or [[Countable set|countably infinite]]. 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.<ref>Allen Gersho and [[Robert M. Gray]], ''[http://books.google.com/books/about/Vector_Quantization_and_Signal_Compressi.html?id=DwcDm6xgItUC Vector Quantization and Signal Compression]'', [[Springer Science+Business Media|Springer]], ISBN 978-0-7923-9181-4, 1991.</ref>

There are two substantially different classes of applications where quantization is used:
* The first type, which may simply be called ''rounding'' quantization, is the one employed for many applications, to enable the use of a simple approximate representation for some quantity that is to be measured and used in other calculations. This category includes the simple rounding approximations used in everyday arithmetic. This category also includes [[analog-to-digital conversion]] of a signal for a digital signal processing system (e.g., using a sound card of a personal computer to capture an audio signal) and the calculations performed within most digital filtering processes. Here the purpose is primarily to retain as much signal fidelity as possible while eliminating unnecessary precision and keeping the dynamic range of the signal within practical limits (to avoid signal [[Clipping (signal processing)|clipping]] or [[arithmetic overflow]]). In such uses, substantial loss of signal fidelity is often unacceptable, and the design often centers around managing the approximation error to ensure that very little distortion is introduced.
* The second type, which can be called ''rate–distortion optimized'' quantization, is encountered in [[source coding]] for "lossy" data compression algorithms, where the purpose is to manage distortion within the limits of the bit rate supported by a communication channel or storage medium. In this second setting, the amount of introduced distortion may be managed carefully by sophisticated techniques, and introducing some significant amount of distortion may be unavoidable. A quantizer designed for this purpose may be quite different and more elaborate in design than an ordinary rounding operation. It is in this domain that substantial [[rate–distortion theory]] analysis is likely to be applied. However, the same concepts actually apply in both use cases.

The analysis of quantization 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''). The general field of such study of rate and distortion is known as ''[[rate–distortion theory]]''.

==Scalar quantization==
The most common type of quantization is known as ''scalar quantization''. Scalar quantization, typically denoted as <math>y=Q(x)</math>, is the process of using a quantization function <math>Q</math>( ) to map a scalar (one-dimensional) input value <math>x</math> to a scalar output value <math>y</math>. Scalar quantization can be as simple and intuitive as [[rounding]] high-precision numbers to the nearest integer, or to the nearest multiple of some other unit of precision (such as rounding a large monetary amount to the nearest thousand dollars). Scalar quantization of continuous-valued input data that is performed by an electronic [[sensor]] is referred to as ''[[analog-to-digital conversion]]''. Analog-to-digital conversion often also involves [[Sampling (signal processing)|sampling]] the signal periodically in time (e.g., at 44.1 [[Hertz|kHz]] for [[Compact disc|CD]]-quality audio signals).

== Rounding example ==
As an example, [[rounding]] a [[real number]] <math>x</math> 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>\Delta</math> can be expressed as
:<math>Q(x) = \sgn(x) \cdot \Delta \cdot \left\lfloor \frac{\left| x \right|}{\Delta}+\frac1{2}\right\rfloor</math>,
where the function <math>\sgn</math>( ) is the [[sign function]] (also known as the ''signum'' function).
For simple rounding to the nearest integer, the step size <math>\Delta</math> is equal to 1. With <math>\Delta = 1</math> or with <math>\Delta</math> equal to any other integer value, this quantizer has real-valued inputs and integer-valued outputs, although this property is not a necessity – a quantizer may also have an integer input domain and may also have non-integer output values. The essential property of a quantizer is that it has a countable set of possible output values that has fewer members than the set of possible input values. The members of the set of output values may have integer, rational, or real values (or even other possible values as well, in general – such as vector values or [[complex number]]s).

When the quantization step size is small (relative to the variation in the signal being measured), it is relatively simple to show<ref name=Sheppard>[[William Fleetwood Sheppard]], "On the Calculation of the Most Probable Values of Frequency Constants for data arranged according to Equidistant Divisions of a Scale", ''[[Proceedings of the London Mathematical Society]]'', Vol. 29, pp. 353–80, 1898.{{doi|10.1112/plms/s1-29.1.353}}</ref><ref name=Bennett>W. R. Bennett, "[http://www.alcatel-lucent.com/bstj/vol27-1948/articles/bstj27-3-446.pdf Spectra of Quantized Signals]", ''[[Bell System Technical Journal]]'', Vol. 27, pp. 446–472, July 1948.</ref><ref name=OliverPierceShannon>B. M. Oliver, J. R. Pierce, and [[Claude Shannon|Claude E. Shannon]], "The Philosophy of PCM", ''[[Proceedings of the IEEE|Proceedings of the IRE]]'', Vol. 36, pp. 1324–1331, Nov. 1948. {{doi|10.1109/JRPROC.1948.231941}}</ref><ref name=Stein>Seymour Stein and J. Jay Jones, ''[http://books.google.com/books/about/Modern_communication_principles.html?id=jBc3AQAAIAAJ Modern Communication Principles]'', [[McGraw–Hill]], ISBN 978-0-07-061003-3, 1967 (p. 196).</ref><ref name=GishPierce>Herbert Gish and John N. Pierce, "Asymptotically Efficient Quantizing", ''[[IEEE Transactions on Information Theory]]'', Vol. IT-14, No. 5, pp. 676–683, Sept. 1968. {{doi|10.1109/TIT.1968.1054193}}</ref><ref name=GrayNeuhoff>[[Robert M. Gray]] and David L. Neuhoff, "Quantization", ''[[IEEE Transactions on Information Theory]]'', Vol. IT-44, No. 6, pp. 2325–2383, Oct. 1998. {{doi|10.1109/18.720541}}</ref> that the [[mean squared error]] produced by such a rounding operation will be approximately <math>\Delta^2/ 12</math>.

Because the set of possible output values of a quantizer is countable, any quantizer can be decomposed into two distinct stages, which can be referred to as the ''classification'' stage (or ''forward quantization'' stage) and the ''reconstruction'' stage (or ''inverse quantization'' stage), where the classification stage maps the input value to an integer ''quantization index'' <math>k</math> and the reconstruction stage maps the index <math>k</math> to the ''reconstruction value'' <math>y_k</math> that is the output approximation of the input value. For the example uniform quantizer described above, the forward quantization stage can be expressed as
:<math>k = \sgn(x) \cdot \left\lfloor \frac{\left| x \right|}{\Delta}+\frac1{2}\right\rfloor</math>,
and the reconstruction stage for this example quantizer is simply <math>y_k = k \cdot \Delta</math>.

This decomposition is useful for the design and analysis of quantization behavior, and it illustrates how the quantized data can be communicated over a communication channel – a ''source encoder'' can perform the forward quantization stage and send the index information through a communication channel (possibly applying [[entropy coding]] techniques to the quantization indices), and a ''decoder'' can perform the reconstruction stage to produce the output approximation of the original input data. In more elaborate quantization designs, both the forward and inverse quantization stages may be substantially more complex. 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 [[vector quantization|vector]] as well as scalar quantizers.

== Mid-riser and mid-tread uniform quantizers ==
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 [[stairway]]. 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 ''[[Stair riser|riser]]'' of a stairway).<ref name=Gersho77>Allen Gersho, "Quantization", ''[[IEEE Communications Magazine|IEEE Communications Society Magazine]]'', pp. 16–28, Sept. 1977. {{doi|10.1109/MCOM.1977.1089500}}</ref>

The formulas for mid-tread uniform quantization are provided above.

The input-output formula for a mid-riser uniform quantizer is given by:
:<math>Q(x) = \Delta\cdot\left(\left\lfloor \frac{x}{\Delta}\right\rfloor + \frac1{2}\right)</math>,
where the classification rule is given by
:<math>k = \left\lfloor \frac{x}{\Delta} \right\rfloor</math>
and the reconstruction rule is
:<math>y_k = \Delta\cdot\left(k+\tfrac1{2}\right)</math>.

Note that mid-riser uniform quantizers do not have a zero output value – their minimum output magnitude is half the step size. When the input data can be modeled as a [[random variable]] with a [[probability density function]] (pdf) that is smooth and symmetric around zero, mid-riser quantizers also always produce an output ''[[Entropy (information theory)|entropy]]'' of at least 1 bit per sample.

In contrast, mid-tread quantizers do have a zero output level, and can reach arbitrarily low bit rates per sample for input distributions that are symmetric and taper off at higher magnitudes. For some applications, having a zero output signal representation or supporting low output entropy may be a necessity. In such cases, using a mid-tread uniform quantizer may be appropriate while using a mid-riser one would not be.

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 intervals 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.<ref name=Gersho77/>

Another name for a mid-tread quantizer is '''dead-zone quantizer''', and the classification region around the zero output value of such a quantizer is referred to as the ''dead zone''. The dead zone can sometimes serve the same purpose as a [[noise gate]] or [[squelch]] function.

==Granular distortion and overload distortion==
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>\Delta</math>) to achieve the appropriate balance is the use of ''[[automatic gain control]]'' (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).

==The additive noise model for quantization error==
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 [[white noise]] – having negligible correlation with the signal and an approximately flat [[power spectral density]].<ref name=Bennett/><ref name=GrayNeuhoff/><ref name=Widrow1>[[Bernard Widrow]], "A study of rough amplitude quantization by means of Nyquist sampling theory", ''IRE Trans. Circuit Theory'', Vol. CT-3, pp. 266–276, 1956. {{doi|10.1109/TCT.1956.1086334}}</ref><ref name=Widrow2>[[Bernard Widrow]], "[http://www-isl.stanford.edu/~widrow/papers/j1961statisticalanalysis.pdf Statistical analysis of amplitude quantized sampled data systems]", ''Trans. AIEE Pt. II: Appl. Ind.'', Vol. 79, pp. 555–568, Jan. 1961.</ref> 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>\Delta</math> relative to the signal strength) with smooth probability density functions.<ref name=Bennett/><ref name=MarcoNeuhoff>Daniel Marco and David L. Neuhoff, "The Validity of the Additive Noise Model for Uniform Scalar Quantizers", ''[[IEEE Transactions on Information Theory]]'', Vol. IT-51, No. 5, pp. 1739–1755, May 2005. {{doi|10.1109/TIT.2005.846397}}</ref> However, additive noise behaviour is not always a valid assumption, and care should be taken to avoid assuming that this model always applies. In actuality, the quantization error (for quantizers defined as described here) is deterministically related to the signal rather than being independent of it,<ref name=GrayNeuhoff/> and in some cases it can even cause [[limit cycle]]s to appear in digital signal processing systems.<ref name=Widrow2/>

One way to ensure effective independence of the quantization error from the source signal is to perform ''[[dither]]ed quantization'' (sometimes with ''[[noise shaping]]''), which involves adding random (or [[pseudo-random]]) noise to the signal prior to quantization.<ref name=GrayNeuhoff/><ref name=Widrow2/> This can sometimes be beneficial for such purposes as improving the subjective quality of the result, however it can increase the total quantity of error introduced by the quantization process.

==Quantization error models==
In the typical case, the original signal is much larger than one [[least significant bit|least significant bit (LSB)]]. When this is the case, the quantization error is not significantly correlated with the signal, and has an approximately [[uniform distribution (continuous)|uniform distribution]]. In the rounding case, the quantization error has a mean of zero and the [[root mean square|RMS]] value is the [[standard deviation]] of this distribution, given by <math>\scriptstyle {\frac{1}{\sqrt{12}}}\mathrm{LSB}\ \approx\ 0.289\,\mathrm{LSB}</math>. In the truncation case the error has a non-zero mean of <math>\scriptstyle {\frac{1}{2}}\mathrm{LSB}</math> and the RMS value is <math>\scriptstyle {\frac{1}{\sqrt{3}}}\mathrm{LSB}</math>. In the eight-bit ADC example, the RMS rounding error represents 0.113% of the full signal range.

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, noise with an amplitude of 2 least significant bits is added to the signal. This slightly reduces signal to noise ratio, but, ideally, completely eliminates the distortion. It is known as [[dither]].

==Quantization noise model==
[[File:quanterr.png|thumb|300px|Quantization noise for a 2-bit ADC operating at infinite [[sample rate]]. The difference between the blue and red signals in the upper graph is the quantization error, which is "added" to the quantized signal and is the source of noise.]]

Quantization noise is a [[Model (abstract)|model]] of quantization error introduced by [[quantization (signal processing)|quantization]] in the [[Analog-to-digital converter|analog-to-digital conversion]] (ADC) in
[[Communications system|telecommunication systems]] and [[Digital signal processing|signal processing]]. 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 modelled in several different ways.

In an ideal analog-to-digital converter, 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 [[Signal-to-quantization-noise ratio]] (SQNR) can be calculated from

:<math>\mathrm{SQNR} = 20 \log_{10}(2^Q) \approx 6.02 \cdot Q\ \mathrm{dB} \,\!</math>

Where Q is the number of quantization bits.

The most common test signals that fulfill this are full amplitude [[triangle wave]]s and [[sawtooth wave]]s.

For example, a [[16-bit]] ADC has a maximum signal-to-noise ratio of 6.02 × 16 = 96.3 dB.

When the input signal is a full-amplitude [[sine wave]] the distribution of the signal is no longer uniform, and the corresponding equation is instead

:<math> \mathrm{SQNR} \approx 1.761 + 6.02 \cdot Q \ \mathrm{dB} \,\!</math>

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.<ref>{{cite book
| last = Pohlman
| first =Ken C.
| title = Principles of Digital Audio 2nd Edition
| publisher = SAMS
| date = 1989
| page = 60
| url = http://books.google.com/books?id=VZw6z9a03ikC&pg=PA37&source=gbs_selected_pages&cad=0_1}} </ref> 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/sawtooth.

Quantization noise power can be derived from

:<math>\mathrm{N} = \frac {(\delta \mathrm{v})^2} { 12 } \mathrm{W} \,\!</math>

where <math>\delta \mathrm{v}</math> is the voltage of the level.

(Typical real-life values are worse than this theoretical minimum, due to the addition of [[dither]] to reduce the objectionable effects of quantization, and to imperfections of the ADC circuitry. On the other hand, specifications often use [[A-weighted]] measurements to hide the inaudible effects of [[noise shaping]], which improves the measurement.)

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.<ref>{{cite book
| last = Okelloto
| first = Tom
| title = The Art of Digital Audio 3rd Edition
| publisher = Focal Press
| date = 2001
| isbn = 0-240-51587-0}} </ref> In these cases the quantization noise distribution is strongly affected by the exact amplitude of the signal.

The calculations above, however, assume a completely filled input channel. If this is not the case - if the input signal is small - the relative quantization distortion can be very large. To circumvent this issue, analog [[dynamic range compression|compressors and expanders]] can be used, but these introduce large amounts of distortion as well, especially if the compressor does not match the expander. The application of such compressors and expanders is also known as [[companding]].

== Rate–distortion quantizer design ==
A scalar quantizer, which performs a quantization operation, can ordinarily be decomposed into two stages:
* '''Classification:''' A process that classifies the input signal range into <math>M</math> non-overlapping '''intervals''' <math>\{I_k\}_{k=1}^{M}</math>, by defining <math>M-1</math> '''boundary (decision)''' values <math> \{b_k\}_{k=1}^{M-1} </math>, such that <math> I_k = [b_{k-1}~,~b_k)</math> for <math>k = 1,2,\ldots,M</math>, with the extreme limits defined by <math> b_0 = -\infty</math> and <math> b_M = \infty</math>. All the inputs <math>x</math> that fall in a given interval range <math>I_k</math> are associated with the same quantization index <math>k</math>.
* '''Reconstruction:''' Each interval <math> I_k </math> is represented by a '''reconstruction value''' <math> y_k </math> which implements the mapping <math> x \in I_k \Rightarrow y = y_k </math>.

These two stages together comprise the mathematical operation of <math>y = Q(x)</math>.

[[Entropy coding]] 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>k</math> with a binary codeword <math>c_k</math>. An important consideration is the number of bits used for each codeword, denoted here by <math>\mathrm{length}(c_k)</math>.

As a result, the design of an <math>M</math>-level quantizer and an associated set of codewords for communicating its index values requires finding the values of <math> \{b_k\}_{k=1}^{M-1} </math>, <math>\{c_k\}_{k=1}^{M} </math> and <math> \{y_k\}_{k=1}^{M} </math> which optimally satisfy a selected set of design constraints such as the '''bit rate''' <math>R</math> and '''distortion''' <math>D</math>.

Assuming that an information source <math>S</math> produces random variables <math>X</math> with an associated [[probability density function]] <math>f(x)</math>, the probability <math>p_k</math> that the random variable falls within a particular quantization interval <math>I_k</math> is given by
:<math> p_k = P[x \in I_k] = \int_{b_{k-1}}^{b_k} f(x)dx </math>.

The resulting bit rate <math>R</math>, in units of average bits per quantized value, for this quantizer can be derived as follows:
:<math> 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 </math>.

If it is assumed that distortion is measured by [[mean squared error]], the distortion '''D''', is given by:
:<math> D = E[(x-Q(x))^2] = \int_{-\infty}^{\infty} (x-Q(x))^2f(x)dx = \sum_{k=1}^{M} \int_{b_{k-1}}^{b_k} (x-y_k)^2 f(x)dx </math>.

Note that other distortion measures can also be considered, although mean squared error is a popular one.

A key observation is that rate <math>R</math> depends on the decision boundaries <math>\{b_k\}_{k=1}^{M-1}</math> and the codeword lengths <math>\{\mathrm{length}(c_k)\}_{k=1}^{M}</math>, whereas the distortion <math>D</math> depends on the decision boundaries <math>\{b_k\}_{k=1}^{M-1}</math> and the reconstruction levels <math>\{y_k\}_{k=1}^{M}</math>.

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:
# Given a maximum distortion constraint <math>D \le D_\max</math>, minimize the bit rate <math>R</math>
# Given a maximum bit rate constraint <math>R \le R_\max</math>, minimize the distortion <math>D</math>

Often the solution to these problems can be equivalently (or approximately) expressed and solved by converting the formulation to the unconstrained problem <math>\min\left\{ D + \lambda \cdot R \right\}</math> where the [[Lagrange multiplier]] <math>\lambda</math> 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 [[convex hull]] of the family of solutions to an equivalent constrained formulation of the problem. However, finding a solution – especially a [[Closed-form expression|closed-form]] 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 probability distribution functions: the [[Uniform distribution (continuous)|uniform]],<ref>[[Nariman Farvardin]] and James W. Modestino, "Optimum Quantizer Performance for a Class of Non-Gaussian Memoryless Sources", ''[[IEEE Transactions on Information Theory]]'', Vol. IT-30, No. 3, pp. 485–497, May 1982 (Section VI.C and Appendix B). {{doi|10.1109/TIT.1984.1056920}}</ref> [[Exponential distribution|exponential]],<ref name=SullivanIT>[[Gary Sullivan (engineer)|Gary J. Sullivan]], "Efficient Scalar Quantization of Exponential and Laplacian Random Variables", ''[[IEEE Transactions on Information Theory]]'', Vol. IT-42, No. 5, pp. 1365–1374, Sept. 1996. {{doi|10.1109/18.532878}}</ref> and [[Laplace distribution|Laplacian]]<ref name=SullivanIT/> distributions. Iterative optimization approaches can be used to find solutions in other cases.<ref name=GrayNeuhoff/><ref name=Berger72>[[Toby Berger]], "Optimum Quantizers and Permutation Codes", ''[[IEEE Transactions on Information Theory]]'', Vol. IT-18, No. 6, pp. 759–765, Nov. 1972. {{doi|10.1109/TIT.1972.1054906}}</ref><ref name=Berger82>[[Toby Berger]], "Minimum Entropy Quantizers and Permutation Codes", ''[[IEEE Transactions on Information Theory]]'', Vol. IT-28, No. 2, pp. 149–157, Mar. 1982. {{doi|10.1109/TIT.1982.1056456}}</ref>

Note that the reconstruction values <math>\{y_k\}_{k=1}^{M}</math> affect only the distortion – they do not affect the bit rate – and that each individual <math>y_k</math> makes a separate contribution <math> d_k </math> to the total distortion as shown below:
:<math> D = \sum_{k=1}^{M} d_k </math>
where
:<math> d_k = \int_{b_{k-1}}^{b_k} (x-y_k)^2 f(x)dx </math>
This observation can be used to ease the analysis – given the set of <math>\{b_k\}_{k=1}^{M-1}</math> values, the value of each <math>y_k</math> can be optimized separately to minimize its contribution to the distortion <math>D</math>.

For the mean-square error distortion criterion, it can be easily shown that the optimal set of reconstruction values <math>\{y^*_k\}_{k=1}^{M}</math> is given by setting the reconstruction value <math>y_k</math> within each interval <math>I_k</math> to the conditional expected value (also referred to as the ''[[centroid]]'') within the interval, as given by:
:<math>y^*_k = \frac1{p_k} \int_{b_{k-1}}^{b_k} x f(x)dx</math>.

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>\{k\}_{k=1}^{M}</math>, such that effectively
:<math> \mathrm{length}(c_k) \approx -\log_2\left(p_k\right)</math>
and therefore
:<math> R = \sum_{k=1}^{M} -p_k \cdot \log_2\left(p_k\right) </math>.

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 [[arithmetic coding]] 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>\{p_k\}_{k=1}^{M}</math>.

In some designs, rather than optimizing for a particular number of classification regions <math>M</math>, the quantizer design problem may include optimization of the value of <math>M</math> as well. For some probabilistic source models, the best performance may be achieved when <math>M</math> approaches infinity.

== Neglecting the entropy constraint: Lloyd–Max quantization ==

In the above formulation, if the bit rate constraint is neglected by setting <math>\lambda</math> 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 [[variable-length code]] (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>D</math> alone.

The indices produced by an <math>M</math>-level quantizer can be coded using a fixed-length code using <math> R = \lceil \log_2 M \rceil </math> bits/symbol. For example when <math>M=</math>256 levels, the FLC bit rate <math>R</math> 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>M</math> 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>X</math> with [[probability density function|pdf]] <math>f(x)</math> and the constraint that the quantizer must use only <math>M</math> classification regions, find the decision boundaries <math>\{b_k\}_{k=1}^{M-1} </math> and reconstruction levels <math>\{y_k\}_{k=1}^M</math> to minimize the resulting distortion
:<math> D=E[(x-Q(x))^2] = \int_{-\infty}^{\infty} (x-Q(x))^2f(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 </math>.

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<ref name=GrayNeuhoff/><ref>Stuart P. Lloyd, "Least Squares Quantization in PCM", ''[[IEEE Transactions on Information Theory]]'', Vol. IT-28, pp. 129–137, No. 2, March 1982 {{doi|10.1109/TIT.1982.1056489}} (work documented in a manuscript circulated for comments at [[Bell Labs|Bell Laboratories]] with a department log date of 31 July 1957 and also presented at the 1957 meeting of the [[Institute of Mathematical Statistics]], although not formally published until 1982).</ref><ref>Joel Max, "Quantizing for Minimum Distortion", ''[[IEEE Transactions on Information Theory|IRE Transactions on Information Theory]]'', Vol. IT-6, pp. 7–12, March 1960. {{doi|10.1109/TIT.1960.1057548}}</ref> to solve the two sets of simultaneous equations resulting from <math> {\partial D / \partial b_k} = 0 </math> and <math>{\partial D/ \partial y_k} = 0 </math>, as follows:
:<math> {\partial D \over\partial b_k} = 0 \Rightarrow b_k = {y_k + y_{k+1} \over 2} </math>,
which places each threshold at the midpoint between each pair of reconstruction values, and
:<math> {\partial D \over\partial y_k} = 0 \Rightarrow y_k = { \int_{b_{k-1}}^{b_k} x f(x) dx \over \int_{b_{k-1}}^{b_k} f(x)dx } = \frac1{p_k} \int_{b_{k-1}}^{b_k} x f(x) dx </math>
which places each reconstruction value at the centroid (conditional expected value) of its associated classification interval.

[[Lloyd's algorithm|Lloyd's Method I algorithm]], originally described in 1957, can be generalized in a straighforward way for application to [[vector quantization|vector]] data. This generalization results in the [[Linde–Buzo–Gray algorithm|Linde–Buzo–Gray (LBG)]] or [[k-means clustering|k-means]] classifier optimization methods. Moreover, the technique can be further generalized in a straightforward way to also include an entropy constraint for vector data.<ref name=ChouLookabaughGray>Philip A. Chou, Tom Lookabaugh, and [[Robert M. Gray]], "Entropy-Constrained Vector Quantization", ''IEEE Transactions on Acoustics, Speech, and Signal Processing'', Vol. ASSP-37, No. 1, Jan. 1989. {{doi|10.1109/29.17498}}</ref>

== Uniform quantization and the 6 dB/bit approximation ==

The Lloyd–Max quantizer is actually a uniform quantizer when the input [[probability density function|pdf]] is uniformly distributed over the range <math>[y_1-\Delta/2,~y_M+\Delta/2)</math>. However, for a source that does not have a uniform distribution, the minimum-distortion quantizer may not be a uniform quantizer.

The analysis of a uniform quantizer applied to a uniformly distributed source can be summarized in what follows:

A symmetric source X can be modelled with <math> f(x)= \frac1{2X_{max}}</math>, for <math>x \in [-X_{max} , X_{max}]</math> and 0 elsewhere.
The step size <math>\Delta = \frac {2X_{max}} {M} </math> and the ''signal to quantization noise ratio'' (SQNR) of the quantizer is
:<math>{\rm SQNR}= 10\log_{10}{\frac {\sigma_x^2}{\sigma_q^2}} = 10\log_{10}{\frac {(M\Delta)^2/12}{\Delta^2/12}}= 10\log_{10}M^2= 20\log_{10}M</math>.

For a fixed-length code using <math>N</math> bits, <math>M=2^N</math>, resulting in
<math>{\rm SQNR}= 20\log_{10}{2^N} = N\cdot(20\log_{10}2) = N\cdot 6.0206\,\rm{dB}</math>,

or approximately 6 dB per bit. For example, for <math>N</math>=8 bits, <math>M</math>=256 levels and SQNR = 8*6 = 48 dB; and for <math>N</math>=16 bits, <math>M</math>=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 than 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>\Delta^2/12</math> approximation.

At asymptotically high bit rates, the 6 dB/bit approximation is supported for many source pdfs by rigorous theoretical analysis.<ref name=Bennett/><ref name=OliverPierceShannon/><ref name=GishPierce/><ref name=GrayNeuhoff/> Moreover, the structure of the optimal scalar quantizer (in the rate–distortion sense) approaches that of a uniform quantizer under these conditions.<ref name=GishPierce/><ref name=GrayNeuhoff/>


== Other fields ==
Many physical quantities are actually quantized by physical entities. Examples of fields where this limitation applies include [[electronics]] (due to electrons), [[optics]] (due to photons), [[biology]] (due to [[DNA]]), and [[chemistry]] (due to [[molecules]]). This is sometimes known as the "quantum noise limit" of systems in those fields. This is a different manifestation of "quantization error," in which theoretical models may be analog but physically occurs digitally. Around the [[quantum limit]], the distinction between analog and digital quantities vanishes.{{Citation needed|date=July 2009}}

==See also==
* [[Beta encoder]]
* [[Bit resolution]]
* [[Discretization error]]
* [[Posterization]]
* [[Pulse code modulation]]
* [[Regression dilution]] - a bias in parameter estimates caused by errors such as quantization in the explanatory or independent variable

==Notes==
{{Reflist|2}}

== References ==
*{{Citation |last=Sayood |first= Khalid|last2=|first2=|year= 2005 |title= Introduction to Data Compression, Third Edition |publisher= Morgan Kaufmann |isbn= 978-0-12-620862-7|doi=}}
*{{Citation |last=Jayant |first= Nikil S.|last2=Noll|first2=Peter|year= 1984 |title= Digital Coding of Waveforms: Principles and Applications to Speech and Video |publisher= Prentice–Hall |isbn=978-0-13-211913-9|doi=}}
*{{Citation |last=Gregg|first= W. David |year= 1977 |title= Analog & Digital Communication |publisher= John Wiley |isbn=978-0-471-32661-8
|doi=}}
*{{Citation |last=Stein |first= Seymour|last2= Jones|first2= J. Jay |year= 1967 |title= Modern Communication Principles |publisher= [[McGraw–Hill]] |isbn=978-0-07-061003-3|doi=}}

== External links ==
* [http://www.mit.bme.hu/books/quantization/ Quantization noise in Digital Computation, Signal Processing, and Control], Bernard Widrow and István Kollár, 2007.
* [http://www.techonline.com/community/related_content/20771 The Relationship of Dynamic Range to Data Word Size in Digital Audio Processing]
* [http://ccrma.stanford.edu/~jos/mdft/Round_Off_Error_Variance.html Round-Off Error Variance] — derivation of noise power of q²/12 for round-off error
* [http://www.ieee.li/pdf/essay/dynamic_evaluation_dac.pdf Dynamic Evaluation of High-Speed, High Resolution D/A Converters] Outlines HD, IMD and NPR measurements, also includes a derivation of quantization noise
* [http://www.dsplog.com/2007/03/19/signal-to-quantization-noise-in-quantized-sinusoidal/ Signal to quantization noise in quantized sinusoidal]

{{DSP}}
{{Compression Methods}}

{{Noise}}

{{DEFAULTSORT:Quantization (Signal Processing)}}
[[Category:Digital signal processing]]
[[Category:Computer graphic artifacts]]

[[Category:Digital audio]]
[[Category:Noise]]
[[Category:Signal processing]]
[[Category:Telecommunication theory]]

Shrikhande graph

2013-12-30T19:01:40Z

128.211.178.12: /* Properties */ copyedit for flow & brevity

{{New Testament manuscript infobox
|form = Papyrus
|number = '''3'''
|image =
|caption=
|name =
|sign = <math>\mathfrak{P}</math>3
|text = [[Gospel of Luke|Luke]] 7:36-45+10:38-42
|script = [[Greek language|Greek]]
|date = 6th/7th century
|found =
|now at = Vienna, [[Austrian National Library|Österr. Nationalbibliothek]], Pap. G. 2323
|cite =
|size = 24.5 x 11.5 cm (25x18)
|type = mixed
|cat = III
|hand =
|note =
}}
'''Papyrus 3''', designated by <math>\mathfrak{P}</math>3 (in the numbering Gregory-Aland), is a small fragment of fifteen verses from the [[Gospel of Luke]] dating to the 6th/7th century.<ref>[http://nttranscripts.uni-muenster.de/AnaServer?NTtranscripts+0+start.anv New Testament Transcripts Prototype]</ref> It is formed part of a [[lectionary]]. It is dated [[Palaeography|palaeographically]] to the 6th or 7th century.<ref name = Aland>{{Cite book
|last=Aland
|first=Kurt
|authorlink=Kurt Aland
|coauthors=[[Barbara Aland]]; Erroll F. Rhodes (trans.)
|title=The Text of the New Testament: An Introduction to the Critical Editions and to the Theory and Practice of Modern Textual Criticism
|publisher=[[William B. Eerdmans Publishing Company]]
|year=1995
|location=Grand Rapids
|page=96
|url=http://books.google.com/books?id=2pYDsAhUOxAC&printsec=frontcover&source=gbs_ge_summary_r&cad=0#v=onepage&q&f=false
|isbn=978-0-8028-4098-1}}</ref>

The Greek text-type of this codex is a mixed. [[Kurt Aland|Aland]] placed it in [[Categories of New Testament manuscripts#Category III|Category III]].<ref name = Aland/>

==Text==

Luke 7:36

: ΑΓΓ . . ΙΟ
:: ΤΟΝ Κ ̣̅ ΗΣου
: ΤΟΝ̣Τ̣Ω̣ΝΦΣΡΙΣ̣
: Ε̣Λ̣ΘΩΝΕ̣ΙΣΤ̣
: ΙΕΙΔΟΥΓΥΝΗΗΤΙΣΗ
: ΓΝΟΥΣΑΟΤΙ̣ΚΑΤΑΚ
: ΣΑΑΛΑΒΑΣΤΡΟΝΜΥΡ̣
: Ο̣ΔΑΣΑΥΤΟΥΚΛΑΣΙΟΥΣ̣
: ΥΣΠΟΔΑΣΑΥΤΟΥ ΚΑΙ
: ΕΞΕΜΑΞΕΝΚΑΙΚΑΤΕ
: ΗΛΙΦΕΝ ΤΩΜΥΡΩ
: ΑΣΑΥΤΟΝΕΙΠΕΝΕ . . . . ΤΩ
: ΗΣΕΓΙΓΝΩΣΚΕΝΑΝΤ̣ΙΣΚΑΙΠΟΤΑΠ
: ΤΑΙΑΥΤΟΥΟΤΙΑ.ΑΡΤΩΛΟ̣ΣΕΣΤΙΝ
: ΕΙΠΕΝ Ο ΙΣΠΡΟΣ̣Α.Τ̣Ο̣Ν̣ΣΙΜΩΝ
: ΔΕΔΕΔΑ . . . ΛΕΕΙ̣ΠΕΝΦΗΣΙΝΔΥΟ
: ΤΙΝΙΟΕΙΣΩ
: Η̣ΚΟΝ̣ΤΑΜΗ̣
: ΤΟΤΙΣΟΥ
: ΔΕΣΙΜΩ̣
: Π̣ΕΝΑΥΤΩ̣
: Ω̣Σ̣Ι̣ΜΩ

Luke 10:38

: ΝΤΟΥΑΓΙΟυΛΟυΚΑ
:: . . . . . . ΚΩΜΗ̣
::: ΡΕΥΕΣΘΑΙΑΥΤΟΥ
::: ΘΕΝΕΙΣΚΩΜΗΝΤΙΝ

==History==

The text of the manuscript was published by [[Karl Wessely]] in 1882.<ref>Karl Wessely, [http://www.archive.org/stream/wienerstudien11kircgoog#page/n208/mode/2up ''Evangelien-Fragmente auf Papyrus''], Wiener Studien 4 (1882), 198-214.</ref>

The manuscript is housed at the [[Austrian National Library]] (Pap. G. 2323).<ref name = Aland/>

==See also==
* [[List of New Testament papyri]]

==References==

{{Reflist}}

==Further reading==

* [[Karl Wessely]], [http://www.archive.org/stream/wienerstudien11kircgoog#page/n208/mode/2up ''Evangelien-Fragmente auf Papyrus''], Wiener Studien 4 (1882), 198-214.
* {{Cite book
| last = Gregory
| first = Caspar René
| authorlink = Caspar René Gregory
| title = Die griechischen Handschriften des Neuen Testament
| publisher = J.C. Hinrichs’sche Buchhandlung
| year = 1908
| location = Leipzig
| page = 45
| url = http://www.archive.org/stream/diegriechischen00greggoog#page/n55/mode/2up
| isbn = }}
* {{Cite book | author = Aland, K | author2 = Aland, B | year = 1995 | url = http://books.google.com/books?id=2pYDsAhUOxAC&pg=PA96&lpg=bd0v49Quux&dq=#v=onepage&q=&f=false | title = The Text of the New Testament | coauthors = Trans. Rhodes, EF | publisher = Wm. B. Eerdmans | page = 96 | isbn = 0-8028-4098-1}}

==External references==
* [http://nttranscripts.uni-muenster.de/AnaServer?NTtranscripts+0+start.anv New Testament Transcripts]
* {{Cite web|url=http://intf.uni-muenster.de/vmr/NTVMR/ListeHandschriften.php?ObjID=10003|title=Handschriftenliste|publisher=Institute for New Testament Textual Research|accessdate=13 August 2011|location=Münster}}

{{DEFAULTSORT:Papyrus 0003}}
[[Category:New Testament papyri]]
[[Category:6th-century biblical manuscripts]]
[[Category:Manuscripts of the Austrian National Library]]

Pairwise sorting network

2013-02-14T22:09:21Z

128.211.178.12: specify what we're counting

A '''binary expression tree''' is a specific application of a [[binary tree]] to evaluate certain expressions. Two common types of expressions that a binary expression tree can represent are [[algebra]]ic<ref name="brpreiss">{{cite web |url=http://www.brpreiss.com/books/opus5/html/page264.html#SECTION0010500000000000000000|title=Expression Trees|author=Bruno R. Preiss|year=1998|work= |publisher= |accessdate=December 20, 2010}}</ref>
and [[boolean algebra|boolean]]. These trees can represent expressions that contain both [[unary operation|unary]] and [[binary function|binary]] operators.<ref name="brpreiss"/>

In general, expression trees are special kind of binary trees. A binary tree is a tree in which all nodes contain zero, one or two children. This restricted structure simplifies the programmatic processing of Expression trees.

== Overview ==
[[File:Bt1.jpg|thumb|250px|right|Expression Tree]]
The leaves of a binary expression tree are operands, such as constants or variable names, and the other nodes contain operators. These particular trees happen to be binary, because all of the operations are binary, and although this is the simplest case, it is possible for nodes to have more than two children. It is also possible for a node to have only one child, as is the case with the unary minus operator. An expression tree, ''T'', can be evaluated by applying the operator at the root to the values obtained by recursively evaluating the left and right subtrees.<ref name="Gopal2010" />

=== Traversal ===
An algebraic expression can be produced from a binary expression tree by recursively producing a parenthesized left expression, then printing out the operator at the root, and finally recursively producing a parenthesized right expression. This general strategy (left, node, right) is known as an [[tree traversal|in-order travesal]].
An alternate traversal strategy is to recursively print out the left subtree, the right subtree, and then the operator. This traversal strategy is generally known as [[tree traversal|post-order traversal]].
A third strategy is to print out the operator first and then recursively print out the left and right subtree.<ref name="Gopal2010" />

These three standard depth-first traversals are representations of the three different expression formats: infix, postfix, and prefix. An infix expression is produced by the inorder traversal, a postfix expression is produced by the post-order traversal, and a prefix expression is produced by the pre-order traversal.<ref name="Gilberg" />
{{-}}

====Infix Traversal====
When an infix expression is printed, an opening and closing parenthesis must be added at the beginning and ending of each expression. As every subtree represents a subexpression, an opening parenthesis is printed at its start and the closing parenthesis is printed after processing all of its children.

Pseudocode:

<source lang="c">
Algorithm infix (tree)
/*Print the infix expression for an expression tree.
Pre : tree is a pointer to an expression tree
Post: the infix expression has been printed*/
if (tree not empty)
if (tree token is operator)
print (open parenthesis)
infix (tree left subtree)
print (tree token)
infix (tree right subtree)
if (tree token is operator)
print (close parenthesis)
end if
end if
end infix
</source>

====Postfix Traversal====
The postfix expression is formed by the basic postorder traversal of any binary tree. It does not require parentheses.

Pseudocode:

<source lang="c">
Algorithm postfix (tree)
/*Print the postfix expression for an expression tree.
Pre : tree is a pointer to an expression tree
Post: the postfix expression has been printed*/
if (tree not empty)
postfix (tree left subtree)
postfix (tree right subtree)
print (tree token)
end if
end postfix
</source>

====Prefix Traversal====
The prefix expression formed by prefix traversal uses the standard pre-order tree traversal. No parentheses are necessary.

Pseudocode:

<source lang="c">
Algorithm prefix (tree)
/*Print the prefix expression for an expression tree.
Pre : tree is a pointer to an expression tree
Post: the prefix expression has been printed*/
if (tree not empty)
print (tree token)
prefix (tree left subtree)
prefix (tree right subtree) and check if stack is not empty
end if
end prefix
</source>

== Construction of an Expression Tree ==
The evaluation of the tree takes place by reading the expression one symbol at a time. If the symbol is an operand, one-node tree is created and a pointer is pushed onto a [[Stack (abstract data type)|stack]]. If the symbol is an operator, the pointers are popped to two trees ''T1'' and ''T2'' from the stack and a new tree whose root is the operator and whose left and right children point to ''T2'' and ''T1'' respectively is formed . A pointer to this new tree is then pushed to the Stack.<ref>Mark Allen Weiss,''Data Structures and Algorithm Analysis in C,2nd edition'', Addison Wesley publications</ref>

=== Example ===
The input is: a b + c d e + * *
Since the first two symbols are operands, one-node trees are created and pointers are pushed to them onto a stack. For convenience the stack will grow from left to right.

[[File:B1t2.jpg|thumb|500px|center|Stack growing from Left to Right]]

The next symbol is a '+'. It pops the two pointers to the trees, a new tree is formed, and a pointer to it is pushed onto to the stack.

[[File:Bt3.jpg|thumb|350px|center|Formation of a New Tree]]

Next, c, d, and e are read. A one-node tree is created for each and a pointer to the corresponding tree is pushed onto the stack.

[[File:Bt4.jpg|thumb|400px|center|Creating One-Node Tree]]

Continuing, a '+' is read, and it merges the last two trees.

[[File:Bt5.jpg|thumb|400px|center|Merging Two Trees]]

Now, a '*' is read. The last two tree pointers are popped and a new tree is formed with a '*' as the root.

[[File:Bt6.jpg|thumb|350px|center|Forming a New Tree with a Root]]

Finally, the last symbol is read. The two trees are merged and a pointer to the final tree remains on the stack.<ref name="Gopal2010.353" />

[[File:Bt7.jpg|thumb|400px|center|Steps to Construct an Expression tree a b + c d e + * *]]

== Algebraic expressions ==
[[File:Binary Algebraic Expression Tree.JPG|thumb|250px|right|Binary algebraic expression tree equivalent to ((5 + z) / -8) * (4 ^ 2)]]
Algebraic expression trees represent expressions that contain [[number]]s, [[Variable (mathematics)|variables]], and unary and binary operators. Some of the common operators are × ([[multiplication]]), ÷ ([[Division (mathematics)|division]]), + ([[addition]]), − ([[subtraction]]), ^ ([[exponentiation]]), and - ([[negation]]). The operators are contained in the [[internal node]]s of the tree, with the numbers and variables in the [[leaf nodes]].<ref name="brpreiss"/> The nodes of binary operators have two [[child nodes]], and the unary operators have one child node.
{{-}}

== Boolean expressions ==
[[File:Binary Boolean Expression Tree.JPG|thumb|250px|right|Binary boolean expression tree equivalent to ((true <math>\lor</math> false) <math>\land</math> <math>\neg</math>false) <math>\lor</math> (true <math>\lor</math> false))]]
Boolean expressions are represented very similarly to algebraic expressions, the only difference being the specific values and operators used. Boolean expressions use ''true'' and ''false'' as constant values, and the operators include <math>\land</math> ([[Logical and|AND]]), <math>\lor</math> ([[Logical or|OR]]), <math>\neg</math> ([[Logical not|NOT]]).
{{-}}

== See also ==
[[Expression (mathematics)]]

== References ==
{{Reflist|refs=
<ref name="Gopal2010">Gopal, Arpita. ''Magnifying Data Structures''. PHI Learning, 2010, p. 352.</ref>
<ref name="Gopal2010.353">Gopal, Arpita. ''Magnifying Data Structures''. PHI Learning, 2010, p. 353.</ref>
<ref name="Gilberg">Richard F. Gilberg & Behrouz A. Forouzan. ''Data Structures: A Pseudocode Approach with C''. Thomson Course Technology, 2005, p. 280.</ref>
}}

[[Category:Binary trees]]

Voronoi diagram

2012-08-28T15:03:24Z

128.211.178.8: /* The simplest case */

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