Question: When did variation first appear?

Leonard Euler

Euler used the results obtained in the study of specific models of solid bodies - and in the dynamics of a solid body (it is in the memoir mentioned that the equations of dynamics of a body with a fixed point, referred to arbitrary Cartesian axes [54]), both in hydrodynamics and in the theory of elasticity.


Question: When did variation first appear?

Leonard Euler

In the theory of elasticity, a number of Euler's studies are devoted to the theory of bending beams and rods; at the same time, in his early works (1740s) he deals with the problem of longitudinal bending of an elastic rod, composing and solving differential equation of the bent axis of the rod [2].


Question: When did variation first appear?

Leonard Euler

In 1757, in his work “On the Load of Columns” [79], Euler for the first time in history received a formula for determining the critical load during compression of an elastic rod, marking the beginning of the theory of stability of elastic systems [7].


Question: When did variation first appear?

Leonard Euler

This formula found practical application much later, almost a hundred years later, when railway construction began in many countries (primarily England), which required calculations for the strength of railway bridges; it was at this time that engineers adopted, after some clarification, the Euler model [54] [80].


Question: When did variation first appear?

Leonard Euler

Hydrodynamics


Question: When did variation first appear?

Leonard Euler

Euler is — along with D. Bernoulli and J. L. Lagrange, one of the founders of analytical hydrodynamics; here he was credited with creating the theory of motion of an ideal fluid (that is, a liquid that does not have viscosity) and solving a number of specific tasks of fluid mechanics [55].


Question: When did variation first appear?

Leonard Euler

In his work “Principles of Motion of Liquids” [81] (1752; published nine years later), he, applying his equations for the dynamics of the elementary material volume of a continuous medium to the model of an incompressible ideal liquids, for the first time obtained equations of motion for such a liquid, as well as (for the general three-dimensional case [82]) the equation of continuity.


Question: When did variation first appear?

Leonard Euler

Studying the vortex-free motion of an incompressible liquid, Euler introduced a function


Question: When did variation first appear?

Leonard Euler

S


Question: When did variation first appear?

Leonard Euler

{\ displaystyle S}


Question: When did variation first appear?

Leonard Euler

(later named G. Helmholtz velocity potential) and showed that it satisfies the partial differential equation — this is how the equation now known as the Laplace equation entered science [54].


Question: When did variation first appear?

Leonard Euler

Euler significantly summarized the results of this work in the treatise “General Principles of the Movement of Liquids” [83] (1755).


Question: When did variation first appear?

Leonard Euler

Here, for the case of a compressible ideal fluid, he presented (almost in modern designations) the equation of continuity and the equation of motion (three scalar differential equations, which in vector notation corresponds to the Euler equation — the basic equation of hydrodynamics of an ideal fluid [84]).


Question: When did variation first appear?

Leonard Euler

Euler noted that to close this system of four equations, a determining relationship is needed that makes it possible to express pressure


Question: When did variation first appear?

Leonard Euler

p


Question: When did variation first appear?

Leonard Euler

{\ displaystyle p}


Question: When did variation first appear?

Leonard Euler

(which Euler called “elasticity”) as a function of density


Question: When did variation first appear?

Leonard Euler

q


Question: When did variation first appear?

Leonard Euler

{\ displaystyle q}


Question: When did variation first appear?

Leonard Euler

and “other properties


Question: When did variation first appear?

Leonard Euler

r


Question: When did variation first appear?

Leonard Euler

{\ displaystyle r}


Question: When did variation first appear?

Leonard Euler

, which affects elasticity” (in fact, temperature was meant) [2] [54].


Question: When did variation first appear?

Leonard Euler

Discussing the possibility of the existence of non-potential movements of an incompressible fluid, Euler gave the first concrete example of its vortex flow, and for potential movements of such a liquid, he received the first integral is a special case of the now known Lagrange-Cauchy integral [2].


Question: When did variation first appear?

Leonard Euler

In the same year, Euler's memoir “General Principles of the State of Equilibrium of Liquids” [85] also belongs, which contained a systematic presentation of the hydrostatics of an ideal fluid (including the derivation of the general equation equilibrium of liquids and gases) and a barometric formula for an isothermal atmosphere was derived [2].


Question: When did variation first appear?

Leonard Euler

In these works, Euler, writing down the equations of motion and equilibrium of a liquid, took the Cartesian coordinates of the current position of the material particle as independent spatial variables — Euler variables (D'Alembert used such variables for the first time in hydrodynamics [52]).


Question: When did variation first appear?

Leonard Euler

Later, in the work “On the principles of fluid movement. Section Two” [86] (1770) Euler also introduced the second form of hydrodynamics equations, in which Cartesian coordinates of the position of a material particle in the initial were taken as independent spatial variables moment in time (now known as Lagrange variables) [87].


Question: When did variation first appear?

Leonard Euler

Optics


Question: When did variation first appear?

Leonard Euler

Euler collected the main achievements in this area in a three-volume book “Dioptrica” (Latin: Dioptrica, 1769—1771).


Question: When did variation first appear?

Leonard Euler

Among the main results: rules for calculating the optimal characteristics of refractors, reflectors and microscopes, calculating the greatest brightness of the image, the largest field of view, the shortest tool length , the highest magnification, eyepiece characteristics [23].


Question: When did variation first appear?

Leonard Euler

Newton argued that the creation of an achromatic lens is fundamentally impossible.


Question: When did variation first appear?

Leonard Euler

Euler argued that a combination of materials with different optical characteristics could solve this problem.


Question: When did variation first appear?

Leonard Euler

In 1758, Euler, after a long controversy, managed to convince the English optician John Dollond of this, who then made the first achromatic lens, combining two lenses made of glasses with each other of various composition [88], and in 1784, Academician F. Epinus built the world's first achromatic microscope in St. Petersburg [89].


Question: When did variation first appear?

Leonard Euler

Astronomy


Question: When did variation first appear?

Leonard Euler

Euler worked extensively in the field of celestial mechanics.


Question: When did variation first appear?

Leonard Euler

One of the urgent tasks at that time was to determine the parameters of the orbit of a celestial body (for example, a comet) from a small number of observations.


Question: When did variation first appear?

Leonard Euler

Euler significantly improved numerical methods for this purpose and practically applied them to determining the elliptical orbit of the comet in 1769; Gauss relied on these works, who gave the final solution tasks [90].


Question: When did variation first appear?

Leonard Euler

Euler laid the foundations for the perturbation theory, later completed by Laplace and Poincare [90].


Question: When did variation first appear?

Leonard Euler

He introduced the fundamental concept of osculating elements of the orbit and derived differential equations that determine their change over time.


Question: When did variation first appear?

Leonard Euler

He built a theory of precession and nutation of the Earth's axis, predicted the “free movement of the poles” of the Earth, discovered a hundred years later by Chandler [91].


Question: When did variation first appear?

Leonard Euler

In 1748-1751, Euler published a complete theory of light and parallax aberration.


Question: When did variation first appear?

Leonard Euler

In 1756, he published the differential equation of astronomical refraction, studied the dependence of refraction on air pressure and temperature at the observation site.


Question: When did variation first appear?

Leonard Euler

These results had a huge impact on the development of astronomy in subsequent years [90].


Question: When did variation first appear?

Leonard Euler

Euler presented a very accurate theory of the motion of the moon, developing for this purpose a special method of variation of orbital elements.


Question: When did variation first appear?

Leonard Euler

Subsequently, in the 19th century, this method was expanded, used in the model of the motion of large planets and is still used today.


Question: When did variation first appear?

Leonard Euler

The Mayer tables, calculated on the basis of Euler's theory (1767), were also suitable for solving the urgent problem of determining longitude at sea, and the English Admiralty paid Mayer and Euler for it special award [90].


Question: When did variation first appear?

Leonard Euler

Euler's main works in this area are:


Question: When did variation first appear?

Leonard Euler

“Theory of Moon Motion”, 1753;


Question: When did variation first appear?

Leonard Euler

“Theory of planets and comets motion”, 1774;


Question: When did variation first appear?

Leonard Euler

“New Theory of Moon Motion”, 1772.


Question: When did variation first appear?

Leonard Euler

Euler studied the gravitational field of not only spherical, but also ellipsoidal bodies, which was a significant step forward [92].


Question: When did variation first appear?

Leonard Euler

He also pointed out for the first time in science the centuries-old displacement of the inclination of the plane of the ecliptic (1756), and at his suggestion, a slope at the beginning of 1700 has since been adopted as a reference [90].


Question: When did variation first appear?

Leonard Euler

He developed the foundations of the theory of motion of the moons of Jupiter and other highly compressed planets [91].


Question: When did variation first appear?

Leonard Euler

In 1748, long before P. N. Lebedev, Euler hypothesized that the tails of comets, auroras and zodiac light have a common source of the effect of solar radiation on the atmosphere or matter of celestial bodies [90].


Question: When did variation first appear?

Leonard Euler

Music theory


Question: When did variation first appear?

Leonard Euler

Throughout his life, Euler was interested in musical harmony, trying to give it a clear mathematical justification.


Question: When did variation first appear?

Leonard Euler

The purpose of his early work, The Experience of New Music Theory (Tentamen novae theoriae musicae, 1739), was to attempt to mathematically describe how pleasant (euphonic) music differs from unpleasant ( dissonant) [93].


Question: When did variation first appear?

Leonard Euler

At the end of chapter VII of Experience, Euler arranged the intervals according to the “degrees of pleasantness” (gradus suavitatis), while the octave was ranked in the II (most pleasant) class, and diaschism - to the last, XXVII class (the most dissonant interval); some classes (including the first, third, sixth) in the Euler amenities table were missed [94].


Question: When did variation first appear?

Leonard Euler

There was a joke about this work that it had too much music for mathematicians and too much math for musicians [92].


Question: When did variation first appear?

Leonard Euler

In his declining years, in 1773, Euler read a report at the St. Petersburg Academy of Sciences, in which he finally formulated his lattice representation of the sound system; this idea was metaphorically designated by the author as a “mirror of music” (Latin: speculum musicae).


Question: When did variation first appear?

Leonard Euler

The following year, Euler's report was published as a short treatise De harmoniae veris principiis per speculum musicum repraesentatis (“On the True Foundations of Harmony Presented through Speculum musicae”) [95].


Question: When did variation first appear?

Leonard Euler

Under the name “sound network” (German: Tonnetz), it was widely used in German musical theory of the XIX century.


Question: When did variation first appear?

Leonard Euler

Other areas of expertise


Question: When did variation first appear?

Leonard Euler

In 1749, Euler published a two-volume monograph “Marine Science, or a Treatise on Shipbuilding and Shipbuilding”, in which he applied analytical methods to practical problems of shipbuilding and navigation at sea, such as the shape of ships, issues of stability and balance, methods for controlling the movement of the ship [7].


Question: When did variation first appear?

Leonard Euler

General theory of ship stability A. N. Krylova relies on “Marine Science” [93].


Question: When did variation first appear?

Leonard Euler

Euler's scientific interests also included physiology; in particular, he applied methods of hydrodynamics to the study of the principles of blood movement in blood vessels.


Question: When did variation first appear?

Leonard Euler

In 1742, he sent an article on the flow of liquids in elastic tubes (considered as models of vessels), and in December 1775 he presented his memoir “Fundamentals of Determining blood movement through the arteries” (Principia pro motu sanguines per arteria determinando).


Question: When did variation first appear?

Leonard Euler

In this work, the physical and physiological principles of blood movement caused by periodic contractions of the heart were analyzed.


Question: When did variation first appear?

Leonard Euler

Treating blood as an incompressible liquid, Euler found a solution to his equations of motion for the case of rigid tubes, and in the case of elastic tubes, he limited himself only to obtaining general equations of finite movements [2].


Question: When did variation first appear?

Leonard Euler

Students


Question: When did variation first appear?

Leonard Euler

One of the main tasks assigned to Euler upon his arrival in Russia was the training of scientific personnel.


Question: When did variation first appear?

Leonard Euler

Among Euler's direct students [5]:


Question: When did variation first appear?

Leonard Euler

M. E.


Question: When did variation first appear?

Leonard Euler

Golovin, academician of mathematics.


Question: When did variation first appear?

Leonard Euler

P. B.


Question: When did variation first appear?

Leonard Euler

Aminokhodtsev, academician of astronomy.


Question: When did variation first appear?

Leonard Euler

S. K. Kotelnikov, academician of mathematics.


Question: When did variation first appear?

Leonard Euler

A. AND. Lexel is a talented astronomer and mathematician.


Question: When did variation first appear?

Leonard Euler

S. ME.


Question: When did variation first appear?

Leonard Euler

Rumovsky, academician of astronomy.


Question: When did variation first appear?

Leonard Euler

N. AND. Fuss, academician of mathematics.


Question: When did variation first appear?

Leonard Euler

AND. A. Euler, Leonard Euler's oldest son, is a talented mathematician.


Question: When did variation first appear?

Leonard Euler

memento


Question: When did variation first appear?

Leonard Euler

Named after Euler:


Question: When did variation first appear?

Leonard Euler

many concepts in mathematics and other sciences, see: list of objects named after Leonard Euler;


Question: When did variation first appear?

Leonard Euler

Euler Crater on the Moon;


Question: When did variation first appear?

Leonard Euler

Asteroid 2002 Euler;


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Leonard Euler

The summit of Euler Peak in the Pamirs;


Question: When did variation first appear?

Leonard Euler

International Mathematical Institute n.a.


Question: When did variation first appear?

Leonard Euler

Leonard Euler of the Russian Academy of Sciences, founded in 1988 in Leningrad;


Question: When did variation first appear?

Leonard Euler

Leonard Euler Gold Medal of the USSR Academy of Sciences and the Russian Academy of Sciences;


Question: When did variation first appear?

Leonard Euler

, awarded annually to Canadians for achievements in this field of mathematics since 1993;


Question: When did variation first appear?

Leonard Euler

Leonard Euler International Charitable Foundation for the Support of Mathematics [96] [C 8];


Question: When did variation first appear?

Leonard Euler

Street in Almaty.


Question: When did variation first appear?

Leonard Euler

The complete collection of Euler's works, published since 1909 by the Swiss Society of Naturalists, has not yet been completed; 75 volumes were planned to be published, of which 73 [97] were published:


Question: When did variation first appear?

Leonard Euler

29 volumes in mathematics;


Question: When did variation first appear?

Leonard Euler

31 volumes on mechanics and astronomy;


Question: When did variation first appear?

Leonard Euler

13 — in physics.


Question: When did variation first appear?

Leonard Euler

Eight additional volumes will be devoted to Euler's scientific correspondence (more than 3,000 letters) [98].


Question: When did variation first appear?

Leonard Euler

In 1907, Russian and many other scientists celebrated the 200th anniversary of the great mathematician, and in 1957, the Soviet and Berlin Academies of Sciences dedicated solemn sessions to his 250th anniversary.


Question: When did variation first appear?

Leonard Euler

On the eve of Euler's 300th anniversary (2007), an international anniversary forum was held in St. Petersburg and a film about Euler's life was shot [C 9].


Question: When did variation first appear?

Leonard Euler

In the same year, in St. Petersburg, at the entrance to the Euler International Institute, a monument to Euler by sculptor A. G. Dyoma [C 10].


Question: When did variation first appear?

Leonard Euler

The authorities of St. Petersburg, however, rejected all proposals to name a square or street after the scientist; there is still not a single Euler Street in Russia [99].


Question: When did variation first appear?

Leonard Euler

Personal qualities and ratings


Question: When did variation first appear?

Leonard Euler

According to contemporaries, Euler was good-natured by nature, gentle, and almost did not quarrel with anyone [7].


Question: When did variation first appear?

Leonard Euler

Even Johann Bernoulli, whose difficult character was experienced by his brother Jacob and son Daniel, was invariably warm to him.


Question: When did variation first appear?

Leonard Euler

To complete his life, Euler needed only one thing — the possibility of regular mathematical creativity.


Question: When did variation first appear?

Leonard Euler

He could even work intensively “with a baby on his knees and with a cat on his back” [100].


Question: When did variation first appear?

Leonard Euler

At the same time, Euler was cheerful, sociable, loved music, philosophical conversations [101].


Question: When did variation first appear?

Leonard Euler

Academician P. P. Pekarsky, relying on the testimonies of Euler's contemporaries, recreated the image of the scientist as follows: “Euler had a great art not to flaunt his studies, hide his superiority and be at the level each and every one. Always an even mood, gentle and natural fun, some mockery mixed with good-natured, the conversation was naive and playful — all this made the conversation with him as pleasant as it was attractive” [6].


Question: When did variation first appear?

Leonard Euler

According to his contemporaries, Euler was very religious [102].


Question: When did variation first appear?

Leonard Euler

According to Condorcet, every night Euler gathered his children, servants and disciples who lived with him for prayer.


Question: When did variation first appear?

Leonard Euler

He read them a chapter from the Bible and sometimes accompanied them with a sermon [103].


Question: When did variation first appear?

Leonard Euler

In 1747, Euler published a treatise in defense of Christianity against atheism, “Protecting Divine Revelation from the Attacks of the Freethinkers” [104].


Question: When did variation first appear?

Leonard Euler

Euler's fascination with theological reasoning caused a negative attitude towards him (as a philosopher) of his famous contemporaries, D'Alembert and Lagrange [5].


Question: When did variation first appear?

Leonard Euler

Frederick II, who considered himself a “freethinker” and corresponded with Voltaire, said that Euler “smells like a priest” [5].


Question: When did variation first appear?

Leonard Euler

Euler was a caring family man, willingly helped colleagues and young people, and generously shared his ideas with them.


Question: When did variation first appear?

Leonard Euler

There is a known case when Euler delayed his publications on the calculus of variations so that the young and then unknown Lagrange, who independently came to the same discoveries, could publish them first [25 ].


Question: When did variation first appear?

Leonard Euler

Lagrange has always admired Euler both as a mathematician and as a person; he said, “If you really love mathematics, read Euler” [105].


Question: When did variation first appear?

Leonard Euler

“Read, read Euler, he is our common teacher,” Laplace liked to repeat (French: Lisez Euler, lisez Euler, c'est notre maître à tous.) [106].


Question: When did variation first appear?

Leonard Euler

Euler's works were studied with great benefit by the “king of mathematicians” Karl Friedrich Gauss, and almost all famous scientists of the XVIII-XIX centuries.


Question: When did variation first appear?

Leonard Euler

D'Alembert, in one of his letters to Lagrange [107], calls Euler “this devil” (French: se diable d'homme), as if wanting to express, according to commentators [5], that what Euler did exceeds human strength.


Question: When did variation first appear?

Leonard Euler

M. IN. Ostrogradsky said in a letter to N. N. Fussu: “Euler created a modern analysis, one enriched it more than all his followers combined, and made it the most powerful tool of the human mind” [5].


Question: When did variation first appear?

Leonard Euler

Academician S. AND. Vavilov wrote: “Together with Peter I and Lomonosov, Euler became a good genius of our Academy, who determined its glory, its strength, its productivity” [5].


Question: When did variation first appear?

Leonard Euler

Accommodation addresses


Question: When did variation first appear?

Leonard Euler

In Berlin


Question: When did variation first appear?

Leonard Euler

From 1743 to 1766, Euler lived in a house at Behrenstraße 21/22.


Question: When did variation first appear?

Leonard Euler

The house has been preserved, and a memorial plaque is installed on it [108].


Question: When did variation first appear?

Leonard Euler

In Saint Petersburg


Question: When did variation first appear?

Leonard Euler

Since 1766, Euler lived in an apartment building at the address: Nikolaevskaya Embankment, 15 (with a break caused by a strong fire).


Question: When did variation first appear?

Leonard Euler

In Soviet times, the street was renamed the “Lieutenant Schmidt Embankment”.


Question: When did variation first appear?

Leonard Euler

A memorial plaque has been installed on the house, and now it houses a secondary school [C 11].


Question: When did variation first appear?

Leonard Euler

Stamps, coins, banknotes


Question: When did variation first appear?

Leonard Euler

In 2007, the Central Bank of the Russian Federation issued a commemorative coin [C 12] to commemorate the 300th anniversary of the birth of L. Euler.


Question: When did variation first appear?

Leonard Euler

The portrait of Euler was also placed on the Swiss 10-French banknote (6th series) and postage stamps of Switzerland, Russia and Germany.


Question: When did variation first appear?

Leonard Euler

Postage stamp of the USSR,


Question: When did variation first appear?

Leonard Euler

1957 year


Question: When did variation first appear?

Leonard Euler

Postage stamp of the GDR dedicated to Leonard Euler, 1983, 20 pfennig (Michael 2825, Scott 2371)


Question: When did variation first appear?

Leonard Euler

Silver coin of Russia 2007


Question: When did variation first appear?

Leonard Euler

Swiss banknote with a portrait of young Euler


Question: When did variation first appear?

Leonard Euler

Mathematical Olympiads


Question: When did variation first appear?

Leonard Euler

Many facts in geometry, algebra and combinatorics, proved by Euler, are widely used in Olympiad mathematics.


Question: When did variation first appear?

Leonard Euler

On April 15, 2007, an Internet Olympiad for schoolchildren in mathematics was held, dedicated to the 300th anniversary of the birth of Leonard Euler, supported by a number of organizations [C 13].


Question: When did variation first appear?

Leonard Euler

Since 2008, the Leonard Euler Mathematical Olympiad has been held for eighth graders, designed to partially replace them with the loss of the regional and final stages of the All-Russian Mathematical Olympiads for 8th grade [C 14].


Question: When did variation first appear?

Leonard Euler

Some of Euler's famous descendants


Question: When did variation first appear?

Leonard Euler

Historians have found only more than a thousand direct descendants of Leonard Euler.


Question: When did variation first appear?

Leonard Euler

The eldest son Johann Albrecht became a major mathematician and physicist.


Question: When did variation first appear?

Leonard Euler

Carl's second son was a famous doctor.


Question: When did variation first appear?

Leonard Euler

The youngest son Christopher was later a lieutenant general of the Russian army and commander of the Sestroretsk Arms Plant.


Question: When did variation first appear?

Leonard Euler

All Euler's children took Russian citizenship (Euler himself remained a Swiss citizen all his life [9]).


Question: When did variation first appear?

Leonard Euler

As of the end of the 1980s, historians counted about 400 living descendants, about half of them lived in the USSR [109].


Question: When did variation first appear?

Leonard Euler

Here is a brief family tree of some of the known descendants of Euler (the surname is given if it is not “Euler”).


Question: When did variation first appear?

Leonard Euler

Leonard Euler


Question: When did variation first appear?

Leonard Euler

1707—1783 Ivan Leontyevich


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Leonard Euler


Question: When did variation first appear?

Leonard Euler

Leontyevich [110] [111]


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Leonard Euler

1740—1790 Christopher


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Leonard Euler

1743—1808Anna Charlotte


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Leonard Euler

Wilhelmina


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Leonard Euler

1773—1871 Albertina Benedict


Question: When did variation first appear?

Leonard Euler

Filippina Louisa


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Leonard Euler

1766-1829 Leonty


Question: When did variation first appear?

Leonard Euler

Karlovich


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Leonard Euler


Question: When did variation first appear?

Leonard Euler

Khristoforovich


Question: When did variation first appear?

Leonard Euler

1773—1849 Pavel


Question: When did variation first appear?

Leonard Euler

Khristoforovich


Question: When did variation first appear?

Leonard Euler

1786—1840 Fedor


Question: When did variation first appear?

Leonard Euler

Khristoforovich [112]


Question: When did variation first appear?

Leonard Euler

1784—1835 Collins


Question: When did variation first appear?

Leonard Euler

Eduard Davydovich


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler

Pavel Nikolaevich


Question: When did variation first appear?

Leonard Euler

1798-1855 Leonty


Question: When did variation first appear?

Leonard Euler

Leontyevich


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler

Aleksandrovich


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Leonard Euler

1819—1872 Nikolai


Question: When did variation first appear?

Leonard Euler

Pavlovich


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler

Aleksandrovich


Question: When did variation first appear?

Leonard Euler


Question: When did variation first appear?

Leonard Euler

Other Euler descendants include N. AND. Hecker V. F. Hecker and I. R. Hecker V. E.


Question: When did variation first appear?

Leonard Euler

Skalon, E.


Question: When did variation first appear?

Leonard Euler

N. Berendts.


Question: When did variation first appear?

Leonard Euler

Descendants include many scientists, geologists, engineers, diplomats, doctors, and there are also nine generals and one admiral [9].


Question: When did variation first appear?

Leonard Euler

Bibliography


Question: When did variation first appear?

Leonard Euler

Check date values in: |date= (help)


Question: When did variation first appear?

Leonard Euler

Experience of a new music theory (fragments of the treatise) //Music Academy, 1995, No. 1, pp.140-146.


Question: When did variation first appear?

Leonard Euler

in Latin


Question: When did variation first appear?

Leonard Euler

See also


Question: When did variation first appear?

Leonard Euler

List of sites named after Leonard Euler


Question: When did variation first appear?

Leonard Euler

The history of mathematical notation


Question: When did variation first appear?

Leonard Euler

Astronomical Observatory of the St. Petersburg Academy of Sciences


Question: When did variation first appear?

Leonard Euler

Notes


Question: When did variation first appear?

Leonard Euler

Used literature


Question: When did variation first appear?

Leonard Euler

Literature


Question: When did variation first appear?

Leonard Euler

External link in |chapter= (help)


Question: When did variation first appear?

Leonard Euler

Cite journal requires |journal= (help) CS1 maint: multiple names: authors list (link) CS1 maint: date and year (link)


Question: When did variation first appear?

Leonard Euler

Cite journal requires |journal= (help) CS1 maint: date and year (link)


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Leonard Euler

CS1 maint: multiple names: authors list (link)


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Leonard Euler

Burya A.


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Leonard Euler

— P. 605—607.


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Leonard Euler

Check date values in: |date= (help)


Question: When did variation first appear?

Leonard Euler

References


Question: When did variation first appear?

Leonard Euler

(so called. Enestrom's pointer)


Question: When did variation first appear?

Leonard Euler

Check date values in: |accessdate= (help)


Question: When did variation first appear?

Leonard Euler

(in English)


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Leonard Euler


Question: When did variation first appear?

Leonard Euler

Euler's works


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Leonard Euler

Check date values in: |accessdate= (help); URL—WikiLink conflict (help)


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Leonard Euler

in the MacTutor archive


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Leonard Euler


Question: When did variation first appear?

Leonard Euler

Category:Leonard Euler


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Leonard Euler

Leonard


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Leonard Euler

Category:University of Basel alumni


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Leonard Euler

Category:Astronomers alphabetically


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Leonard Euler

Category:Swiss astronomers


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Leonard Euler

Category:Astronomers of the Russian Empire


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Leonard Euler

Category:German astronomers


Question: When did variation first appear?

Leonard Euler

Category: Mathematicians in number theory


Question: When did variation first appear?

Leonard Euler

Category:Music theorists


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Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

An analog-to-digital converter [1] [2] [3] (ADC, English: Analog-to-Digital converter, ADC) is a device that converts an input analog signal into a discrete code (digital signal).


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

The reverse conversion is carried out using a digital-to-analog converter (DAC, DAC).


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

As a rule, an ADC is an electronic device that converts voltage into a binary digital code.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

However, some non-electronic devices with digital output should also be classified as ADCs, such as some types of angle-to-code converters.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

The simplest single-bit binary ADC is the comparator.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Permission


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

The ADC resolution — the minimum change in the magnitude of the analog signal that can be converted by this ADC — is due to its bit depth.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

In the case of a single measurement without taking into account noise, the resolution is directly determined by the bit depth of the ADC.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

The bit depth of the ADC characterizes the number of discrete values that the converter can output at the output.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

In binary ADCs, it is measured in bits, in ternary ADCs it is measured in trits.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

For example, a binary 8-bit ADC can output 256 discrete values (0... 255) because


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

{\ displaystyle 2^ {8} =256}


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

, a ternary 8 bit ADC is capable of producing 6561 discrete values because


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

{\ displaystyle 3^ {8} =6561}


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

The voltage resolution is equal to the voltage difference corresponding to the maximum and minimum output code divided by the number of output discrete values.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

For example:


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Example 1


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Input Range = 0 to 10 Volts


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

The bit width of the binary ADC is 12 bits: 212 = 4096 quantization levels


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Binary ADC voltage resolution: (10-0) /4096 = 0.00244 volts = 2.44 mV


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

The bit depth of the ternary ADC 12 tri: 312 = 531,441 quantization levels


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Ternary ADC voltage resolution: (10-0) /531441 = 0.0188 mV = 18.8 μV


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Example 2


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Input range = −10 to +10 volts


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

The bit width of the binary ADC is 14 bits: 214 = 16384 quantization levels


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Binary ADC voltage resolution: (10- (-10)) /16384 = 20/16384 = 0.00122 volts = 1.22 mV


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

The bit depth of the ternary ADC 14 tri: 314 = 4,782,969 quantization levels


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Ternary ADC voltage resolution: (10- (-10)) /4782969 = 0.00418 mV = 4.18 μV


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

In practice, ADC resolution is limited by the signal-to-noise ratio of the input signal.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

With a high noise intensity at the ADC input, it becomes impossible to distinguish between adjacent levels of the input signal, that is, the resolution deteriorates.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

At the same time, the achievable resolution is described by the effective bit depth (English: effective number of bits, ENOB), which is less than the actual bit depth of the ADC.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

When converting a highly noisy signal, the low bits of the output code are almost useless, as they contain noise.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

To achieve the stated bit depth, the signal-to-noise ratio of the input signal must be approximately 6 dB for each bit depth (6 dB corresponds to a twofold change in the signal level).


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Conversion types


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

According to the method of algorithms used, ADCs are divided into:


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Serial direct conversions


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Sequential approximation


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Serial with sigma-delta modulation


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Parallel single-stage


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Parallel two- and more steps (conveyor)


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

The first two types of ADCs imply the mandatory use of a sampling and storage device (UHF) in their composition.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

This device is used to store the analog value of the signal for the time required to perform the conversion.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Without it, the result of converting a serial ADC will be unreliable.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Integral sequential approximation ADCs are produced, both containing UHC in their composition and requiring external UHC.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Linear ADCs


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Most ADCs are considered linear, although analog-to-digital conversion is essentially a nonlinear process (since the operation of mapping continuous space to discrete is an operation nonlinear).


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

The term linear as applied to an ADC means that the range of input values mapped to the output digital value is linearly related to that output value, that is, the output value k is achieved when the input range is from


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

m (k+b)


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

upto


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

m (k + 1+b),


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

where m and b are some constants.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

The b constant is usually 0 or −0.5.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

If b = 0, the ADC is called a quantizer with a non-zero step (mid-rise), but if b = −0.5, then the ADC is called a quantizer with zero in the center of the quantization step (mid-tread).


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Nonlinear ADCs


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

If the probability density of the amplitude of the input signal had a uniform distribution, the signal-to-noise ratio (as applied to quantization noise) would be the maximum possible.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

For this reason, before amplitude quantization, the signal is usually passed through an inertialess transducer, the transfer function of which repeats the distribution function of the signal itself.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

This improves the reliability of signal transmission, as the most important areas of the signal amplitude are quantized with the best resolution.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Accordingly, in digital-to-analog conversion, it will be necessary to process the signal with a function inverse to the distribution function of the original signal.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

This is the same principle used in companders used in tape recorders and various communication systems, it is aimed at maximizing entropy.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

(Don't confuse compander with compressor!)


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

For example, a voice signal has a Laplace amplitude distribution.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

This means that the amplitude neighborhood of zero carries more information than areas with a larger amplitude.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

For this reason, logarithmic ADCs are often used in voice transmission systems to increase the dynamic range of transmitted values without changing the quality of signal transmission in the region of small amplitudes.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

8-bit a-law or μ-law logarithmic ADCs provide a wide dynamic range and have high resolution in the most critical range of small amplitudes; linear ADCs with similar quality the transmission would have to have a bit width of about 12 bits.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Features


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

The transfer characteristic of the ADC is the dependence of the numerical equivalent of the output binary code on the value of the input analog signal.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

They talk about linear and nonlinear ADCs.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

This division is conditional.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Both gear characteristics are stepped.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

But for “linear” ADCs, it is always possible to draw such a straight line that all transfer characteristic points corresponding to the input values


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

δ


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

k


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

{\ displaystyle\ delta\ cdot 2^ {k}}


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

(where


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

δ


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

{\ displaystyle\ delta}


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

is the sampling step, k is in the range 0..


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

N, where N is the ADC bit depth), were equidistant from it.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Accuracy


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

There are several sources of ADC error.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Quantization errors and (assuming that the ADC should be linear) nonlinearity are inherent in any analog-to-digital conversion.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

In addition, there are so-called aperture errors, which are the result of jitter (English: jitter) of the clock generator, they occur when the signal as a whole (and not a single sample is converted ).


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

These errors are measured in units called MHR —.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

In the above example of an 8-bit binary ADC, the error in 1 MZR is 1/256 of the full signal range, that is, 0.4%, in a 5-digit ternary ADC, the error in 1 MZR is 1/243 of the full range signal, that is, 0.412%, in an 8-triple ternary ADC, the error in 1 MHR is 1/6561, that is, 0.015%.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Quantization errors


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Quantization errors are a consequence of the ADC's limited resolution.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

This shortcoming cannot be eliminated by any type of analog-to-digital conversion.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

The absolute value of the quantization error at each sample is in the range from zero to half of the MPR.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

As a rule, the amplitude of the input signal is much larger than the MPR.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

In this case, the quantization error is not correlated with the signal and has a uniform distribution.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Its root-mean-square value coincides with the standard deviation of the distribution, which is


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

L


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

S


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

B


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

L


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

S


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

B


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

{\ displaystyle {1\ over {\ sqrt {12}}}\ mathrm {LSB}\ approach 0.289\\ mathrm {LSB}}


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

In the case of an 8-bit ADC, this would be 0.13% of the full signal range.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Nonlinearity


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

All ADCs have errors related to nonlinearity, which are a consequence of the physical imperfection of the ADC.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

This leads to the fact that the transfer characteristic (in the sense above) is different from the linear one (more precisely, from the desired function, since it is not necessarily linear).


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Errors can be reduced by calibrating [4].


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

An important parameter describing nonlinearity is integral nonlinearity (INL) and differential nonlinearity (DNL).


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Aperture error (jitter)


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Let us digitize a sinusoidal signal


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

x


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

t


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

A


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

sin


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

π


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

f


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

t


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

{\ displaystyle x (t) =A\ sin 2\ pi f_ {0} t}


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Ideally, counts are taken at regular intervals.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

However, in reality, the time of taking the sample is subject to fluctuations due to jitter of the clock jitter.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Assuming that the uncertainty of the point in time of taking the order


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Δ


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

t


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

{\ displaystyle\ Delta t}


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

, we get that the error caused by this phenomenon can be assessed as


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

E


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

a


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

p


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

x


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

t


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Δ


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

t


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

A


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

π


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

f


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Δ


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

t


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

{\ displaystyle E_ {ap}\ leq |x' (t)\ Delta t|\ leq 2A\ pi f_ {0}\ Delta t}


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

The error is relatively small at low frequencies, but at higher frequencies it can increase significantly.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

The effect of the aperture error can be ignored if its magnitude is relatively small compared to the quantization error.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Thus, it is possible to set the following requirements for jitter of the clock signal front:


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Δ


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

t


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

q


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

π


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

f


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

{\ displaystyle\ Delta t< {\ frac {1} {2^ {q}\ pi f_ {0}}}


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

where


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

q


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

{\ displaystyle q}


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

— ADC bit depth.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

ADC bit widthMaximum input frequency44.1 kHz192 kHz1 MHz10 MHz100 MHz828.2 ns6.48 ns1.24 ns124 ps12.4 ps107.05 ns1.62 ns311 ps31.1 ps3.11 ps121.76 ns405 ps77.7 ps7.77 ps777 fs14441 ps101 ps19.4 ps1.94 ps194 fs16110 ps25.3 ps4.86 ps486 fs48.6 fs1827.5 ps6.32 ps1.21 ps121 fs12.1 fs24430 fs98.8 fs19.0 fs1.9 fs190 ac


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

From this table, we can conclude that it is advisable to use an ADC of a certain bit depth, taking into account the restrictions imposed by clock jitter.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

For example, it is pointless to use a precision 24-bit ADC to record sound if the clock distribution system is unable to provide ultra-low uncertainty.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

In general, the quality of the clock signal is extremely important for more than this reason.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

For example, from the AD9218 (Analog Devices) chip description:


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Any high speed ADC is extremely sensitive to the quality of the sampling clock provided by the user.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

A track-and-hold circuit is essentially a mixer.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Any noise, distortion, or timing jitter on the clock is combined with the desired signal at the analog-to-digital output.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

That is, any high-speed ADC is extremely sensitive to the quality of the digitizing clock frequency supplied by the user.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

The sampling and storage circuit is essentially a mixer (multiplier).


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Any noise, distortion, or jitter of the clock frequency is mixed with the useful signal and fed to the digital output.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Sampling rate


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

An analog signal is a continuous function of time, and in an ADC it is converted into a sequence of digital values.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Therefore, it is necessary to determine the frequency at which digital values are sampled from an analog signal.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

The frequency at which digital values are produced is called the sampling rate of the ADC.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

A continuously changing signal with a limited spectral band is digitized (that is, the signal values are measured at a time interval T — the sampling period), and the original signal can be is accurately restored from discrete values in time by interpolation.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

The accuracy of the recovery is limited by the quantization error.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

However, according to the Kotelnikov-Shannon theorem, accurate recovery is possible only if the sampling rate is higher than twice the maximum frequency in the signal spectrum.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Since real ADCs cannot perform analog-to-digital conversion instantaneously, the analog input value must be kept constant at least from the beginning to the end of the conversion process (This time interval is called the conversion time).


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

This problem is solved by using a special circuit at the input of the ADC — a sampling-storage device (UHF).


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

UHF, as a rule, stores the input voltage on a capacitor that is connected to the input via an analog switch: when the key is closed, the input signal is sampled (the capacitor is charged to the input voltage), when opened, storage.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Many ADCs, made in the form of integrated circuits, contain built-in UHF.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Spectra overlay (aliasing)


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

All ADCs work by sampling input values at fixed intervals.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Therefore, the output values are an incomplete picture of what is being input.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Looking at the output values, there is no way to determine how the input signal behaved between samples.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

If it is known that the input signal changes quite slowly relative to the sampling rate, then we can assume that the intermediate values between the samples are somewhere between the values of these samples.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

If the input signal changes rapidly, no assumptions can be made about the intermediate values of the input signal, and therefore it is impossible to uniquely restore the shape of the original signal.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

If the sequence of digital values produced by the ADC is converted back to analog somewhere by a digital-to-analog converter, it is desirable that the received analog signal be the most accurate copy of the original signal.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

If the input signal changes faster than its readings are made, it is impossible to recover the signal accurately, and a false signal will be present at the DAC output.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

False frequency components of the signal (absent in the spectrum of the original signal) are called alias (false frequency, side low-frequency component).


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

The frequency of false components depends on the difference between the signal frequency and the sampling rate.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

For example, a 2 kHz sine wave sampled at 1.5 kHz would be played back as a 500 Hz sine wave.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

This problem is known as frequency overlap (aliasing).


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

To prevent spectra overlapping, the signal supplied to the ADC input must be passed through a low-pass filter to suppress spectral components whose frequency exceeds half the frequency discretization.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

This filter is called anti-aliasing filter, and its use is extremely important when building real ADCs.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

In general, the use of an analog input filter is interesting not only for this reason.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

It would seem that the digital filter, which is usually used after digitization, has incomparably better parameters.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

But if there are components in the signal that are significantly more powerful than the useful signal and far enough away from it in frequency to be effectively suppressed by an analog filter, such the solution allows you to maintain the dynamic range of the ADC: if the interference is 10 dB stronger than the signal, it will waste, on average, three bits of bit depth.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Although spectra overlapping is in most cases an undesirable effect, it can be used for good.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

For example, this effect makes it possible to avoid converting the frequency down when digitizing a narrowband high-frequency signal (see the mixer).


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

To do this, however, the analog input stages of the ADC must have significantly higher parameters than is required for standard use of the ADC at the main (video or lower) harmonic.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

This also requires effective filtering of out-of-band frequencies to the ADC, since after digitization there is no way to identify and/or filter out most of them.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Pseudorandom signal mixing (dither)


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Some characteristics of the ADC can be improved by using the method of mixing a pseudorandom signal (English: dither).


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

It consists in adding a small amplitude random noise (white noise) to the analog input signal.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

The noise amplitude, as a rule, is chosen at the level of half of the MPR.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

The effect of this addition is that the MPR state randomly changes between states 0 and 1 with a very small input signal (without adding noise, the MPR would be in state 0 or 1 for a long time).


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

For a signal with mixed noise, instead of simply rounding the signal to the nearest digit, there is a random rounding up or down, with the average time during which the signal is rounded to one or a different level depends on how close the signal is to this level.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Thus, the digitized signal contains information about the amplitude of the signal with a resolution better than the MZR, that is, the effective bit depth of the ADC increases.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

The negative side of the technique is the increase in noise in the output signal.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

In fact, the quantization error is smeared over several adjacent samples.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

This approach is more desirable than simply rounding to the nearest discrete level.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

As a result of using the method of mixing a pseudorandom signal, we have a more accurate reproduction of the signal over time.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Small changes in the signal can be restored from pseudorandom jumps of the MSR by filtering.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

In addition, if the noise is determined (the amplitude of the added noise is known exactly at any time), then it can be subtracted from the digitized signal, having previously increased its bit depth, thereby almost completely get rid of the added noise.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Sound signals of very small amplitudes, digitized without a pseudo-random signal, are perceived by ear very distorted and unpleasant.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

When mixing a pseudo-random signal, the true signal strength is represented by the average value of several consecutive samples.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

However, since 2009, due to the reduction in the cost of 24-bit ADCs, which even without dither have a dynamic range of more than 120 dB, which is several orders of magnitude higher than the full hearing range perceived by a person, this technology has lost its relevance in sound engineering.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

At the same time, it is used in RF and microwave technology, where the bit rate of the ADC is usually low due to the high sampling rate.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

A similar process, also called dither or error diffusion, is used to represent halftones of images in computer graphics at low bits per pixel.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

At the same time, the image becomes noisy, but visually perceives it more realistically than the same image obtained by simple quantization.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Resampling


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

As a rule, signals are digitized at the minimum required sampling rate for reasons of economy, while the quantization noise is white, that is, its power spectral density is uniform distributed throughout the band.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

If you digitize a signal with a sampling rate much higher than the Kotelnikov-Shannon theorem, and then digitally filter it to suppress the spectrum outside the frequency band of the original signal, the signal-to-noise ratio will be better than using the entire band.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Thus, it is possible to achieve an effective resolution greater than the bit depth of the ADC.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Oversampling can also be used to mitigate the requirements for the steepness of the transition from the passband to the suppression band of the anti-aliasing filter.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

To do this, the signal is digitized, for example, at twice the frequency, then digitally filtered, suppressing the frequency components outside the band of the original signal, and finally lowering the frequency discretization by decimation.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

ADC types


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

The following are the main ways to build an electronic ADC:


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Direct conversion ADC


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Direct-Conversion (Flash) ADCs, which are fully parallel to ADCs, contain one comparator for each discrete input signal level.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

At any given time, only comparators corresponding to levels below the input signal level produce an overshoot signal at their output.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Signals from all comparators go either directly to a parallel register, then the code is processed by software, or to a hardware logic encoder that generates the desired digital code in hardware depending on the code at the encoder input.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Data from the encoder is recorded in a parallel register.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

The sampling rate of parallel ADCs generally depends on the hardware characteristics of analog and logic elements, as well as on the required sampling rate.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Direct conversion parallel ADCs are the fastest, but usually have a resolution of 8 bits or more, like digital oscilloscopes, as they entail high hardware costs (


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

n


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

{\ displaystyle 2^ {n} -1=2^ {8} -1=255}


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

comparators).


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

ADCs of this type have a very large chip size of the chip, high input capacitance, and can produce short-term errors at the output.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

They are often used for video or other high-frequency signals, and are also widely used in industry to monitor rapidly changing processes in real time.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Professional models can have a resolution of up to 14 bits or higher [5].


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Subbranging Direct-Conversion (Flash) ADCs [6] are partially serial ADCs.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

R. Staffin and R. Lohman R. were proposed in 1956 (Staffin and R. Lohman, “Signal Amplitude Quantizer”, U.S. Patent 2,869,079, Filed December 19, 1956, Issued January 13, 1959) [7].


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Slightly reducing the speed can significantly reduce the number of op-amps to


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

k


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

n


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

k


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

k


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

{\ displaystyle k\ cdot (2^ {n/k} -1) +k-1}


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

, where n is the number of bits of the output code, and k is the number of parallel direct conversion ADCs.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

With 8 bits and 2 ADCs, 31 op-amps will be required.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Two (k=2) or more sub-band steps are used.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

When k=2, the converter is called Half-Flash (Subbranging) ADC.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Second, third, etc. ADCs are used to reduce the quantization error of the first ADC by digitizing this error.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

The first step is to perform a coarse (low resolution) conversion.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Next, the difference between the input signal and the analog signal corresponding to the coarse conversion result (from the auxiliary DAC to which the coarse code is applied) is determined.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

In the second step, the difference found is multiplied by


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

n


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

k


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

{\ displaystyle 2^ {n/k}}


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

and is undergoing the following transformation.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

The resulting code is combined with the coarse code to obtain the full output digital value.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

This type of ADC is slower than parallel direct-conversion ADCs, has a high resolution and a small body size.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

To increase the speed of the output digitized data stream in parallel-serial direct-conversion ADCs, the pipeline operation of parallel ADCs is used.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Pipelined Subbranging Direct-Conversion (Flash) ADC) [8] is used in parallel-serial direct conversion ADCs, in contrast to the usual operation mode of parallel- sequential direct conversion ADC, in which data is transmitted after full conversion, during pipeline operation, partial transformation data is transmitted as soon as it is ready until the end complete transformation.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

In 1966, Kinniment et al.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

proposed an architecture for a parallel-serial direct conversion ADC with recirculation (Recirculating ADC Architecture) [9].


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

This architecture uses a single subrange parallel direct-conversion ADC.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Subbranging Direct-Conversion (Flash) ADCs, fully serial ADCs (k=n), slower than parallel direct conversion ADCs and slightly slower parallel-serial direct-conversion ADCs, but even more (up to


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

n


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

n


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

n


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

n


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

n


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

n


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

n


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

{\ displaystyle n\ cdot (2^ {n/n} -1) +n-1=n\ cdot (2^ {1} -1) +n-1=2n-1}


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

, where n is the number of bits of the output code, and k is the number of parallel direct conversion ADCs) reduce the number of op-amps (at 8 bits, 15 op-amps will be required: 8 comparators per op amp and 7 subtractors multipliers by 2 by OH) [10].


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Ternary ADCs of this type are approximately 1.5 times faster than binary ADCs of the same type commensurate in terms of the number of levels and hardware costs [11].


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Sequential approximation ADC


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

A sequential approximation ADC or a bitwise balancing ADC contains a comparator, an auxiliary DAC and a sequential approximation register.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

The ADC converts an analog signal into a digital signal in N steps, where N is the bit depth of the ADC.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

At each step, one bit of the required digital value is determined, starting from the PPP (High Significant Digit) and ending with the MSR (Lowest Significant Digit).


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

The sequence of actions for determining the next bit is as follows.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

The auxiliary DAC is set to the analog value formed from the bits already defined in the previous steps; the bit to be defined in this step is set to 1, the lower bits are set to 0.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

The value obtained at the auxiliary DAC is compared with the analog input value.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

If the input signal value is greater than the AUX DAC value, the bit being determined is set to 1, otherwise 0.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Thus, determining the final digital value resembles a binary search.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

ADCs of this type have both high speed and good resolution.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

However, in the absence of a storage sampling device, the error will be significantly greater (imagine that after digitizing the largest bit, the signal begins to change).


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Differential encoding ADC


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

Differential coding ADCs (English: delta-encoded ADC) contain a reversible counter, the code from which goes to the auxiliary DAC.


Question: What is the simplest single-bit binary analog-to-digital converter?

Analog-to-digital converter

The input signal and the signal from the auxiliary DAC are compared on the comparator.