Vitold Belevitch
Encyclopedia
Vitold Belevitch was a Belgian mathematician and electrical engineer of Russian extraction who produced some important work in the field of electrical network theory. Born to parents fleeing the Bolshevik
s, he settled in Belgium where he worked on early computer construction projects. Belevitch is responsible for a number of circuit theorems and introduced the now well-known scattering parameters
.
Belevitch had an interest in languages and found a mathematical derivation of Zipf's law. He also published on machine languages. Another field of interest was transmission lines, where he published on line coupling. He worked on telephone conferencing and introduced the mathematical construct of the conference matrix
.
, now incorporated into Russia, but at the time part of Finland. Belevitch's parents were Russian and his mother was an ethnic Pole. They were attempting to flee from their home in Petrograd (St Petersburg) in Russia to escape the Bolshevik revolution, which Belevitch's father opposed. Belevitch's heavily pregnant mother succeeded in crossing the border into Finland and continued on to Helsinki
after Vitold was born, where the birth was registered. She headed for Helsinki because her husband's father was principal of the Russian school there. Belevitch's father was arrested before he could follow and was deported to Siberia
, where he died without ever seeing his son.
In 1926 Belevitch, while still a small child, emigrated with his mother to Belgium.
at Namur
. In 1937, aged 16, he enrolled at the Université Catholique de Louvain
where he studied electrical and mechanical engineering, graduating in 1942. Belevitch gained his PhD in applied sciences at the same university in 1945. His sponsor was Charles Manneback and his second advisor was Wilhelm Cauer
, the founder of the field of network synthesis.
From 1953 until 1985 Belevitch lectured at the university. He taught circuit theory and other mathematical subjects related to electrical science. In 1960 he became a special professor (buitengewoon hoogleraar). Although Belevitch worked as an electrical engineer, his primary interest was mathematics, especially algebra. There was a tradition in Belgium of the most gifted mathematicians entering engineering rather than pure mathematics or physics. Belevitch showed his mathematical leanings by preferring the use of blackboard and chalk to any audio-visual
aids during lectures. He even lectured in this way when presenting the opening lecture to a large audience at an international conference at the IEE
in London.
headquartered in Brussels
but, along with their other European holdings, sold to International Telephone and Telegraph (ITT) in 1925. At BTMC Belevitch became head of the transmission department. It was here that he came into contact with Wilhelm Cauer who became a great influence on him. Cauer was one of the leading circuit theorists of the day and at the time worked for Mix & Genest
in Berlin
, a sister company under the ITT umbrella. Cauer died during the Second World War but Belevitch long after continued to consider his works to be the highest authority on matters of circuit theory.
From 1951 Belevitch was involved in the design of electronic computers which BTMC were developing for the Belgian government. The purpose of this program was to "catch up" with the advances made in the English-speaking world during the war. It resulted in the construction of the Machine mathématique IRSIA-FNRS.IRSIA-FRNS: named after the two Belgian government departments sponsoring the project.
IRSIA: Institut pour l’Encouragement de la Recherche Scientifique dans l’Industrie et dans l’Agriculture (Institute for Promotion of Scientific Research in Industry and Agriculture)
FNRS: Fonds National de la Recherche Scientifique (National Fund for Scientific Research) From 1952 Belevitch represented the electrical engineering aspect of this project. In 1955 Belevitch became director of the Belgian Computing Centre (Comité d'Étude et d'Exploitation des Calculateurs Électroniques) in Brussels which operated this computer for the government. Initially, only the 17-rack prototype was operational. One of the first tasks to which it was put was the calculation of Bessel function
s. The full 34-rack machine was moved from Antwerp and put into service in 1957. Belevitch used this machine to investigate transcendental function
s.
In 1963 Belevitch became head of the newly formed Laboratoire de Recherche MBLEMBLE: Manufacture Belge de Lampes Electriques, a company originally selling light bulbs under the Mazda
brand, later entirely absorbed into Philips. See MBLE on Dutch Wikipedia. (later Philips Research Laboratories Belgium) under the Philips
director of research Hendrik Casimir
in Eindhoven. This facility specialised in applied mathematics for Philips and was heavily involved in computing research. Belevitch stayed in this post until his retirement in November 1984.
Belevitch died on 26 December 1999. He is survived by a daughter, but not his wife.
, modulators, coupled lines
, and non-linear circuits. He was on the editorial board of the International Journal of Circuit Theory from its foundation in 1973. He also made major contributions in information theory
, electronic computers, mathematics and linguistics
.
Belevitch dominated international conferences and was prone to asking searching questions of the presenters of papers, often causing them some discomfort. The organiser of one conference at Birmingham University in 1959 made Belevitch the chairman of the session in which the organiser gave his own presentation. It seems he did this to restrain Belevitch from asking questions. Belevitch stopped attending conferences in the mid-1970s with the exception of the IEEE International Symposium on Circuits and Systems in Montreal
in 1984 in order to receive the IEEE Centennial Medal.
to describe the scattering
behaviour of particles and waves. The "S" in S-parameter stands for scattering in recognition of this. (called repartition matrix by Belevitch). This work was reproduced in part in a later paper by Belevitch, Transmission Losses in 2n-terminal Networks. Belgium was occupied by Nazi Germany
for most of World War II
and this prevented Belevitch from any communication with American colleagues. It was only after the war that it was discovered that the same idea, under the scattering matrix name, had independently been used by American scientists developing military radar
s. The American work by Montgomery, Dicke
and Purcell
was published in 1948. Belevitch in his work had applied scattering matrices to lumped
circuits and was certainly the first to do so, whereas the Americans were concerned with the distributed
circuits used at microwave
frequencies in radar.
Belevitch produced a textbook, Classical Network Theory, first published in 1968 which comprehensively covered the field of passive one-port, and multiport circuits. In this work he made extensive use of the now-established S parameters from the scattering matrix concept, thus succeeding in welding the field into a coherent whole. The eponymous Belevitch's theorem
, explained in this book, provides a method of determining whether or not it is possible to construct a passive, lossless circuit from discrete elements (that is, a circuit consisting only of inductor
s and capacitor
s) that represents a given scattering matrix.
in 1950, so called because they originally arose in connection with a problem Belevitch was working on concerning telephone conferencing. However, they have applications in a range of other fields as well as being of interest to pure mathematics. Belevitch was studying setting up telephone conferencing by connecting together ideal transformers. It turns out that a necessary condition for setting up a conference with n telephone ports and ideal signal loss is the existence of an n×n conference matrix. Ideal signal loss means the loss is only that due to splitting the signal between conference subscribers – there is no dissipation within the conference network.
The existence of conference matrices is not a trivial question, they do not exist for all values of n. Values of n for which they exist are always of the form 4k+2 (k integer) but this is not, by itself, a sufficient condition. Conference matrices exist for n of 2, 6, 10, 14, 18, 26, 30, 38 and 42. They do not exist for n of 22 or 34. Belevitch obtained complete solutions for all n up to 38 and also noted that n=66 had multiple solutions.
Belevitch first introduced the great factorization theorem in which he gives a factorization of paraunitary matrices. Paraunitary matrices occur in the construction of filter banks used in multirate digital systems. Apparently, Belevitch's work is obscure and difficult to understand. A much more frequently cited version of this theorem was later published by P. P. Vaidyanathan.
and the etymology of Indo-European languages
.
Belevitch wrote a book on human and machine languages in which he explored the idea of applying the mathematics of information theory
to obtain results regarding human languages. The book highlighted the difficulties for machine understanding of language for which there was some naive enthusiasm amongst cybernetics
researchers in the 1950s.
Belevitch also wrote a paper, On the Statistical Laws of Linguistic Distribution, which gives a derivation for the well-known empirical
relationship, Zipf's law. This law, and the more complex Mandelbrot law, provide a relationship between the frequency of word occurrence in languages and the word's rank. In the simplest form of Zipf's law, frequency is inversely proportional to rank. Belevitch expressed a large class of statistical distributions (not only the normal distribution) in terms of rank and then expanded each expression into a Taylor series
. In every case Belevitch obtained the remarkable result that a first order truncation of the series resulted in Zipf's law. Further, a second-order truncation of the Taylor series resulted in Mandelbrot's law. This gives some insight into the reason why Zipf's law has been found experimentally to hold in such a wide variety of languages.
of linear control system
s. A system is controllable if it can be moved from one state to another through the system state space
in a finite time by application of control inputs. This test is known as the Popov-Belevitch-Hautus, or PBH, test. There is also a PBH test for determining the observability
of a system – that is, the ability to determine the state of a system in finite time solely from the system's own outputs.
The PBH test was originally discovered by Elmer G. Gilbert
in 1963, but Gilbert's version only applied to systems that could be represented by a diagonalizable matrix
. The test was subsequently generalised by Vasile M. Popov
(in 1966), Belevitch (in Classical Network Theory, 1968) and Malo Hautus in 1969.
(IEEE) and was vice-chair of the Benelux
section when it was formed in 1959. He was awarded the IEEE Centennial Medal
, and in 1993, the Society Award (now called Mac Van Valkenburg Award) of the IEEE Circuits and Systems Society
. He was also a member of the North Rhine-Westphalian Academy of Sciences.
Belevitch received an honorary doctoral degree in 1975 from the Technical University of Munich
, and another from the École Polytechnique Fédérale de Lausanne
, Switzerland, in 1978. He was also rewarded with Belgian royal medals.
Since 2003, the IEEE Circuits and Systems Society has instituted a Vitold Belevitch award for work in circuit theory. The award is presented biennially at the European Conference on Circuit Theory & Design.
Bolshevik
The Bolsheviks, originally also Bolshevists , derived from bol'shinstvo, "majority") were a faction of the Marxist Russian Social Democratic Labour Party which split apart from the Menshevik faction at the Second Party Congress in 1903....
s, he settled in Belgium where he worked on early computer construction projects. Belevitch is responsible for a number of circuit theorems and introduced the now well-known scattering parameters
Scattering parameters
Scattering parameters or S-parameters describe the electrical behavior of linear electrical networks when undergoing various steady state stimuli by electrical signals....
.
Belevitch had an interest in languages and found a mathematical derivation of Zipf's law. He also published on machine languages. Another field of interest was transmission lines, where he published on line coupling. He worked on telephone conferencing and introduced the mathematical construct of the conference matrix
Conference matrix
In mathematics, a conference matrix is a square matrix C with 0 on the diagonal and +1 and −1 off the diagonal, such that CTC is a multiple of the identity matrix I...
.
Early life
Belevitch was born 2 March 1921 in Terijoki, KareliaKarelia
Karelia , the land of the Karelian peoples, is an area in Northern Europe of historical significance for Finland, Russia, and Sweden...
, now incorporated into Russia, but at the time part of Finland. Belevitch's parents were Russian and his mother was an ethnic Pole. They were attempting to flee from their home in Petrograd (St Petersburg) in Russia to escape the Bolshevik revolution, which Belevitch's father opposed. Belevitch's heavily pregnant mother succeeded in crossing the border into Finland and continued on to Helsinki
Helsinki
Helsinki is the capital and largest city in Finland. It is in the region of Uusimaa, located in southern Finland, on the shore of the Gulf of Finland, an arm of the Baltic Sea. The population of the city of Helsinki is , making it by far the most populous municipality in Finland. Helsinki is...
after Vitold was born, where the birth was registered. She headed for Helsinki because her husband's father was principal of the Russian school there. Belevitch's father was arrested before he could follow and was deported to Siberia
Siberia
Siberia is an extensive region constituting almost all of Northern Asia. Comprising the central and eastern portion of the Russian Federation, it was part of the Soviet Union from its beginning, as its predecessor states, the Tsardom of Russia and the Russian Empire, conquered it during the 16th...
, where he died without ever seeing his son.
In 1926 Belevitch, while still a small child, emigrated with his mother to Belgium.
Education
Belevitch was educated in French in Belgium, until July 1936 at the Notre-Dame de la Paix CollegeFacultés universitaires Notre-Dame de la Paix
The University of Namur or Facultés Universitaires Notre-Dame de la Paix , in Namur , is a Jesuit, Catholic private university in the French Community of Belgium...
at Namur
Namur (city)
Namur is a city and municipality in Wallonia, in southern Belgium. It is both the capital of the province of Namur and of Wallonia....
. In 1937, aged 16, he enrolled at the Université Catholique de Louvain
Université catholique de Louvain
The Université catholique de Louvain, sometimes known, especially in Belgium, as UCL, is Belgium's largest French-speaking university. It is located in Louvain-la-Neuve and in Brussels...
where he studied electrical and mechanical engineering, graduating in 1942. Belevitch gained his PhD in applied sciences at the same university in 1945. His sponsor was Charles Manneback and his second advisor was Wilhelm Cauer
Wilhelm Cauer
Wilhelm Cauer was a German mathematician and scientist. He is most noted for his work on the analysis and synthesis of electrical filters and his work marked the beginning of the field of network synthesis...
, the founder of the field of network synthesis.
From 1953 until 1985 Belevitch lectured at the university. He taught circuit theory and other mathematical subjects related to electrical science. In 1960 he became a special professor (buitengewoon hoogleraar). Although Belevitch worked as an electrical engineer, his primary interest was mathematics, especially algebra. There was a tradition in Belgium of the most gifted mathematicians entering engineering rather than pure mathematics or physics. Belevitch showed his mathematical leanings by preferring the use of blackboard and chalk to any audio-visual
Audio-visual
The term Audio-Visual may refer to works with both a sound and a visual component, the production or use of such works, or to equipment used to create and present such works...
aids during lectures. He even lectured in this way when presenting the opening lecture to a large audience at an international conference at the IEE
Institution of Electrical Engineers
The Institution of Electrical Engineers was a British professional organisation of electronics, electrical, manufacturing, and Information Technology professionals, especially electrical engineers. The I.E.E...
in London.
Career
After graduating in 1942, Belevitch joined the Bell Telephone Manufacturing Company (BTMC) in Antwerp, originally part of the International Bell Telephone CompanyInternational Bell Telephone Company
The International Bell Telephone Company of Brussels, Belgium, was created in 1879 by the National Bell Telephone Company of Boston, Massachusetts, United States, initially to sell imported telephones and switchboards in Continental Europe....
headquartered in Brussels
Brussels
Brussels , officially the Brussels Region or Brussels-Capital Region , is the capital of Belgium and the de facto capital of the European Union...
but, along with their other European holdings, sold to International Telephone and Telegraph (ITT) in 1925. At BTMC Belevitch became head of the transmission department. It was here that he came into contact with Wilhelm Cauer who became a great influence on him. Cauer was one of the leading circuit theorists of the day and at the time worked for Mix & Genest
Mix & Genest
Mix & Genest was a 1879 branch of the ITT Corporation.Mix & Genest was founded on the 1st of October 1879 by the businessman Wilhelm Mix and the engineer Werner Genest in Berlin-Schöneberg. The company was very successful and became one of the pioneers in low voltage devices. Among the products...
in Berlin
Berlin
Berlin is the capital city of Germany and is one of the 16 states of Germany. With a population of 3.45 million people, Berlin is Germany's largest city. It is the second most populous city proper and the seventh most populous urban area in the European Union...
, a sister company under the ITT umbrella. Cauer died during the Second World War but Belevitch long after continued to consider his works to be the highest authority on matters of circuit theory.
From 1951 Belevitch was involved in the design of electronic computers which BTMC were developing for the Belgian government. The purpose of this program was to "catch up" with the advances made in the English-speaking world during the war. It resulted in the construction of the Machine mathématique IRSIA-FNRS.IRSIA-FRNS: named after the two Belgian government departments sponsoring the project.
IRSIA: Institut pour l’Encouragement de la Recherche Scientifique dans l’Industrie et dans l’Agriculture (Institute for Promotion of Scientific Research in Industry and Agriculture)
FNRS: Fonds National de la Recherche Scientifique (National Fund for Scientific Research) From 1952 Belevitch represented the electrical engineering aspect of this project. In 1955 Belevitch became director of the Belgian Computing Centre (Comité d'Étude et d'Exploitation des Calculateurs Électroniques) in Brussels which operated this computer for the government. Initially, only the 17-rack prototype was operational. One of the first tasks to which it was put was the calculation of Bessel function
Bessel function
In mathematics, Bessel functions, first defined by the mathematician Daniel Bernoulli and generalized by Friedrich Bessel, are canonical solutions y of Bessel's differential equation:...
s. The full 34-rack machine was moved from Antwerp and put into service in 1957. Belevitch used this machine to investigate transcendental function
Transcendental function
A transcendental function is a function that does not satisfy a polynomial equation whose coefficients are themselves polynomials, in contrast to an algebraic function, which does satisfy such an equation...
s.
In 1963 Belevitch became head of the newly formed Laboratoire de Recherche MBLEMBLE: Manufacture Belge de Lampes Electriques, a company originally selling light bulbs under the Mazda
Mazda (light bulb)
Mazda was a trademarked name created by the Shelby Electric Company for incandescent light bulbs. The name was used from 1909 through 1945 in the United States by Shelby and later General Electric; Mazda brand light bulbs were made for decades after 1945 outside the USA...
brand, later entirely absorbed into Philips. See MBLE on Dutch Wikipedia. (later Philips Research Laboratories Belgium) under the Philips
Philips
Koninklijke Philips Electronics N.V. , more commonly known as Philips, is a multinational Dutch electronics company....
director of research Hendrik Casimir
Hendrik Casimir
Hendrik Brugt Gerhard Casimir FRS was a Dutch physicist best known for his research on the two-fluid model of superconductors in 1934 and the Casimir effect Hendrik Brugt Gerhard Casimir FRS (July 15, 1909 in The Hague, Netherlands – May 4, 2000 in Heeze) was a Dutch physicist best known...
in Eindhoven. This facility specialised in applied mathematics for Philips and was heavily involved in computing research. Belevitch stayed in this post until his retirement in November 1984.
Belevitch died on 26 December 1999. He is survived by a daughter, but not his wife.
Works
Belevitch is best known for his contributions to circuit theory, particularly the mathematical basis of filtersFilter (signal processing)
In signal processing, a filter is a device or process that removes from a signal some unwanted component or feature. Filtering is a class of signal processing, the defining feature of filters being the complete or partial suppression of some aspect of the signal...
, modulators, coupled lines
Crosstalk (electronics)
In electronics, crosstalk is any phenomenon by which a signal transmitted on one circuit or channel of a transmission system creates an undesired effect in another circuit or channel...
, and non-linear circuits. He was on the editorial board of the International Journal of Circuit Theory from its foundation in 1973. He also made major contributions in information theory
Information theory
Information theory is a branch of applied mathematics and electrical engineering involving the quantification of information. Information theory was developed by Claude E. Shannon to find fundamental limits on signal processing operations such as compressing data and on reliably storing and...
, electronic computers, mathematics and linguistics
Linguistics
Linguistics is the scientific study of human language. Linguistics can be broadly broken into three categories or subfields of study: language form, language meaning, and language in context....
.
Belevitch dominated international conferences and was prone to asking searching questions of the presenters of papers, often causing them some discomfort. The organiser of one conference at Birmingham University in 1959 made Belevitch the chairman of the session in which the organiser gave his own presentation. It seems he did this to restrain Belevitch from asking questions. Belevitch stopped attending conferences in the mid-1970s with the exception of the IEEE International Symposium on Circuits and Systems in Montreal
Montreal
Montreal is a city in Canada. It is the largest city in the province of Quebec, the second-largest city in Canada and the seventh largest in North America...
in 1984 in order to receive the IEEE Centennial Medal.
Scattering matrix
It was in his 1945 dissertation that Belevitch first introduced the important idea of the scattering matrixIt is now recognised that the S-parameters used in circuit theory are in all essentials comparable with the scattering matrices used by physicists in scattering theoryScattering theory
In mathematics and physics, scattering theory is a framework for studying and understanding the scattering of waves and particles. Prosaically, wave scattering corresponds to the collision and scattering of a wave with some material object, for instance sunlight scattered by rain drops to form a...
to describe the scattering
Scattering
Scattering is a general physical process where some forms of radiation, such as light, sound, or moving particles, are forced to deviate from a straight trajectory by one or more localized non-uniformities in the medium through which they pass. In conventional use, this also includes deviation of...
behaviour of particles and waves. The "S" in S-parameter stands for scattering in recognition of this. (called repartition matrix by Belevitch). This work was reproduced in part in a later paper by Belevitch, Transmission Losses in 2n-terminal Networks. Belgium was occupied by Nazi Germany
Nazi Germany
Nazi Germany , also known as the Third Reich , but officially called German Reich from 1933 to 1943 and Greater German Reich from 26 June 1943 onward, is the name commonly used to refer to the state of Germany from 1933 to 1945, when it was a totalitarian dictatorship ruled by...
for most of World War II
World War II
World War II, or the Second World War , was a global conflict lasting from 1939 to 1945, involving most of the world's nations—including all of the great powers—eventually forming two opposing military alliances: the Allies and the Axis...
and this prevented Belevitch from any communication with American colleagues. It was only after the war that it was discovered that the same idea, under the scattering matrix name, had independently been used by American scientists developing military radar
Radar
Radar is an object-detection system which uses radio waves to determine the range, altitude, direction, or speed of objects. It can be used to detect aircraft, ships, spacecraft, guided missiles, motor vehicles, weather formations, and terrain. The radar dish or antenna transmits pulses of radio...
s. The American work by Montgomery, Dicke
Robert H. Dicke
Robert Henry Dicke was an American physicist who made important contributions to the fields of astrophysics, atomic physics, cosmology and gravity.-Biography:...
and Purcell
Edward Mills Purcell
Edward Mills Purcell was an American physicist who shared the 1952 Nobel Prize for Physics for his independent discovery of nuclear magnetic resonance in liquids and in solids. Nuclear magnetic resonance has become widely used to study the molecular structure of pure materials and the...
was published in 1948. Belevitch in his work had applied scattering matrices to lumped
Lumped element model
The lumped element model simplifies the description of the behaviour of spatially distributed physical systems into a topology consisting of discrete entities that approximate the behaviour of the distributed system under certain assumptions...
circuits and was certainly the first to do so, whereas the Americans were concerned with the distributed
Distributed element model
In electrical engineering, the distributed element model or transmission line model of electrical circuits assumes that the attributes of the circuit are distributed continuously throughout the material of the circuit...
circuits used at microwave
Microwave
Microwaves, a subset of radio waves, have wavelengths ranging from as long as one meter to as short as one millimeter, or equivalently, with frequencies between 300 MHz and 300 GHz. This broad definition includes both UHF and EHF , and various sources use different boundaries...
frequencies in radar.
Belevitch produced a textbook, Classical Network Theory, first published in 1968 which comprehensively covered the field of passive one-port, and multiport circuits. In this work he made extensive use of the now-established S parameters from the scattering matrix concept, thus succeeding in welding the field into a coherent whole. The eponymous Belevitch's theorem
Belevitch's theorem
Belevitch's theorem is a theorem in electrical network analysis due to the Russo-Belgian mathematician Vitold Belevitch . The theorem provides a test for a given S-matrix to determine whether or not it can be constructed as a lossless rational two-port network.Lossless implies that the network...
, explained in this book, provides a method of determining whether or not it is possible to construct a passive, lossless circuit from discrete elements (that is, a circuit consisting only of inductor
Inductor
An inductor is a passive two-terminal electrical component used to store energy in a magnetic field. An inductor's ability to store magnetic energy is measured by its inductance, in units of henries...
s and capacitor
Capacitor
A capacitor is a passive two-terminal electrical component used to store energy in an electric field. The forms of practical capacitors vary widely, but all contain at least two electrical conductors separated by a dielectric ; for example, one common construction consists of metal foils separated...
s) that represents a given scattering matrix.
Telephone conferencing
Belevitch introduced the mathematical concept of conference matricesConference matrix
In mathematics, a conference matrix is a square matrix C with 0 on the diagonal and +1 and −1 off the diagonal, such that CTC is a multiple of the identity matrix I...
in 1950, so called because they originally arose in connection with a problem Belevitch was working on concerning telephone conferencing. However, they have applications in a range of other fields as well as being of interest to pure mathematics. Belevitch was studying setting up telephone conferencing by connecting together ideal transformers. It turns out that a necessary condition for setting up a conference with n telephone ports and ideal signal loss is the existence of an n×n conference matrix. Ideal signal loss means the loss is only that due to splitting the signal between conference subscribers – there is no dissipation within the conference network.
The existence of conference matrices is not a trivial question, they do not exist for all values of n. Values of n for which they exist are always of the form 4k+2 (k integer) but this is not, by itself, a sufficient condition. Conference matrices exist for n of 2, 6, 10, 14, 18, 26, 30, 38 and 42. They do not exist for n of 22 or 34. Belevitch obtained complete solutions for all n up to 38 and also noted that n=66 had multiple solutions.
Other work on circuits
Belevitch wrote a comprehensive summary of the history of circuit theory. He also had an interest in transmission lines, and published several papers on the subject. They include papers on skin effects and coupling between lines ("crosstalk") due to asymmetry.Belevitch first introduced the great factorization theorem in which he gives a factorization of paraunitary matrices. Paraunitary matrices occur in the construction of filter banks used in multirate digital systems. Apparently, Belevitch's work is obscure and difficult to understand. A much more frequently cited version of this theorem was later published by P. P. Vaidyanathan.
Linguistics
Belevitch was educated in French but continued to speak Russian to his mother until she died. In fact, he was able to speak many languages, and could read even more. He studied SanskritSanskrit
Sanskrit , is a historical Indo-Aryan language and the primary liturgical language of Hinduism, Jainism and Buddhism.Buddhism: besides Pali, see Buddhist Hybrid Sanskrit Today, it is listed as one of the 22 scheduled languages of India and is an official language of the state of Uttarakhand...
and the etymology of Indo-European languages
Indo-European languages
The Indo-European languages are a family of several hundred related languages and dialects, including most major current languages of Europe, the Iranian plateau, and South Asia and also historically predominant in Anatolia...
.
Belevitch wrote a book on human and machine languages in which he explored the idea of applying the mathematics of information theory
Information theory
Information theory is a branch of applied mathematics and electrical engineering involving the quantification of information. Information theory was developed by Claude E. Shannon to find fundamental limits on signal processing operations such as compressing data and on reliably storing and...
to obtain results regarding human languages. The book highlighted the difficulties for machine understanding of language for which there was some naive enthusiasm amongst cybernetics
Cybernetics
Cybernetics is the interdisciplinary study of the structure of regulatory systems. Cybernetics is closely related to information theory, control theory and systems theory, at least in its first-order form...
researchers in the 1950s.
Belevitch also wrote a paper, On the Statistical Laws of Linguistic Distribution, which gives a derivation for the well-known empirical
Empirical
The word empirical denotes information gained by means of observation or experimentation. Empirical data are data produced by an experiment or observation....
relationship, Zipf's law. This law, and the more complex Mandelbrot law, provide a relationship between the frequency of word occurrence in languages and the word's rank. In the simplest form of Zipf's law, frequency is inversely proportional to rank. Belevitch expressed a large class of statistical distributions (not only the normal distribution) in terms of rank and then expanded each expression into a Taylor series
Taylor series
In mathematics, a Taylor series is a representation of a function as an infinite sum of terms that are calculated from the values of the function's derivatives at a single point....
. In every case Belevitch obtained the remarkable result that a first order truncation of the series resulted in Zipf's law. Further, a second-order truncation of the Taylor series resulted in Mandelbrot's law. This gives some insight into the reason why Zipf's law has been found experimentally to hold in such a wide variety of languages.
Control Systems
Belevitch played a part in developing a mathematical test for determining the controllabilityControllability
Controllability is an important property of a control system, and the controllability property plays a crucial role in many control problems, such as stabilization of unstable systems by feedback, or optimal control....
of linear control system
Control system
A control system is a device, or set of devices to manage, command, direct or regulate the behavior of other devices or system.There are two common classes of control systems, with many variations and combinations: logic or sequential controls, and feedback or linear controls...
s. A system is controllable if it can be moved from one state to another through the system state space
State space (controls)
In control engineering, a state space representation is a mathematical model of a physical system as a set of input, output and state variables related by first-order differential equations...
in a finite time by application of control inputs. This test is known as the Popov-Belevitch-Hautus, or PBH, test. There is also a PBH test for determining the observability
Observability
Observability, in control theory, is a measure for how well internal states of a system can be inferred by knowledge of its external outputs. The observability and controllability of a system are mathematical duals. The concept of observability was introduced by American-Hungarian scientist Rudolf E...
of a system – that is, the ability to determine the state of a system in finite time solely from the system's own outputs.
The PBH test was originally discovered by Elmer G. Gilbert
Elmer G. Gilbert
Elmer G. Gilbert is an American aerospace engineer and a Professor Emeritus of Aerospace Engineering at the University of Michigan.He is a member of the National Academy of Engineering and a recipient of the 1994 IEEE Control Systems Award Elmer G. Gilbert is an American aerospace engineer and a...
in 1963, but Gilbert's version only applied to systems that could be represented by a diagonalizable matrix
Diagonalizable matrix
In linear algebra, a square matrix A is called diagonalizable if it is similar to a diagonal matrix, i.e., if there exists an invertible matrix P such that P −1AP is a diagonal matrix...
. The test was subsequently generalised by Vasile M. Popov
Vasile M. Popov
Vasile Mihai Popov is a leading systems theorist and control engineering specialist. He is well known for having developed a method to analyze stability of nonlinear dynamical systems, now known as Popov criterion.- Biography :...
(in 1966), Belevitch (in Classical Network Theory, 1968) and Malo Hautus in 1969.
IEEE and Honours
Belevitch was a Fellow of the Institute of Electrical and Electronics EngineersInstitute of Electrical and Electronics Engineers
The Institute of Electrical and Electronics Engineers is a non-profit professional association headquartered in New York City that is dedicated to advancing technological innovation and excellence...
(IEEE) and was vice-chair of the Benelux
Benelux
The Benelux is an economic union in Western Europe comprising three neighbouring countries, Belgium, the Netherlands, and Luxembourg. These countries are located in northwestern Europe between France and Germany...
section when it was formed in 1959. He was awarded the IEEE Centennial Medal
IEEE Centennial Medal
The IEEE Centennial Medal was a medal minted and awarded in 1984 to celebrate the Centennial of the founding of the Institute of Electrical and Electronics Engineers in 1884.The medal obverse shows 1884 in calligraphic writing and 1984 in an LCD font...
, and in 1993, the Society Award (now called Mac Van Valkenburg Award) of the IEEE Circuits and Systems Society
IEEE Circuits and Systems Society
The IEEE Circuits and Systems Society is a society of the IEEE. It is also known by the acronym IEEE CASS. In the hierarchy of IEEE, the Circuits and Systems Society is one of close to 40 technical societies organized under the IEEE's Technical Activities Board.From the , the field of interest of...
. He was also a member of the North Rhine-Westphalian Academy of Sciences.
Belevitch received an honorary doctoral degree in 1975 from the Technical University of Munich
Technical University of Munich
The Technische Universität München is a research university with campuses in Munich, Garching, and Weihenstephan...
, and another from the École Polytechnique Fédérale de Lausanne
École polytechnique fédérale de Lausanne
The École polytechnique fédérale de Lausanne is one of the two Swiss Federal Institutes of Technology and is located in Lausanne, Switzerland.The school was founded by the Swiss Federal Government with the stated mission to:...
, Switzerland, in 1978. He was also rewarded with Belgian royal medals.
Since 2003, the IEEE Circuits and Systems Society has instituted a Vitold Belevitch award for work in circuit theory. The award is presented biennially at the European Conference on Circuit Theory & Design.
Selected publications
Belevitch was a prolific publisher with around 4000 pages of scientific output. He was publishing throughout his career right up to, and beyond, his retirement in 1984.Books
- Langage des Machines et Langage Humain, Bruxelles: Office de publicité, 1956 (in French).
- Théorie des Circuits de Télécommunication, Louvain: Librairie Universitaire, 1957 (in French).
- Théorie des Circuits Nonlinéaires en Régime Alternatif: Redresseurs, Modulateurs, Oscillateurs, Louvain: Uystpruyst, 1959 (in French).
- Classical Network Theory, San Francisco: Holden-Day, 1968 .
Journal articles
- "Extension of Norton's method of impedance transformation to band-pass filters", Electrical Communication, vol.24, no.1, pp. 59–65, March 1947.
- "Insertion loss and effective phase shift in composite filters at cut-off frequencies", Electrical Communication, vol.24, no.2, pp. 192–194, June 1947.
- "Transmission losses in 2n-terminal networks", Journal of Applied Physics, vol.19, No. 7, pp. 636–638, 1948.
- "Development in the design of filters", Electrical Communication, vol.26, no.1, pp. 84–98, March 1949.
- "Theory of 2n-terminal networks with application to conference telephony", Electrical Communication, vol.27, no.3, pp. 231–244, September 1950.
- "Topics in the design of insertion loss filters", IRE Transactions on Circuit Theory, vol.2, iss.4, pp. 337–346, December 1955.
- "Théorie de l'information et statistique linguistique", Académie Royale de Belgique: Bulletin de la Classe des Sciences, series 5, vol.42, pp. 419–436, 1956. (in French)
- "On the statistical laws of linguistic distribution", Annales de la Société Scientifique de Bruxelles, vol.3, iss.73, pp. 310–326, 1959 . (in French)
- "Summary of the history of circuit theory", Proceedings of the IRE, vol.50, iss.5, pp. 848–855, May 1962.
- "Factorization of scattering matrices with applications to passive network synthesis", Philips Research Reports, vol.18, pp. 275–317, 1963 .
- "Interpolation matrices", Philips Research Reports, vol.25, pp. 337–369, 1970 .
- "On the realizability of non-rational positive real functions", International Journal of Circuit Theory and Applications, vol.1, iss.1, pp. 17–30, March 1973.
- "Theory of the proximity effect in multiwire cables", Philips Research Reports, vol.32, pp. 16–43, 96-177, 1977 .
- "The Gauss hypergeometric ratio as a positive real function", SIAMSociety for Industrial and Applied MathematicsThe Society for Industrial and Applied Mathematics was founded by a small group of mathematicians from academia and industry who met in Philadelphia in 1951 to start an organization whose members would meet periodically to exchange ideas about the uses of mathematics in industry. This meeting led...
Journal on Mathematical Analysis, vol.13, pp. 1024–1040, 1982. - "Ideal rectifier n-ports are solvable", International Journal of Circuit Theory and Applications, vol.15, iss.2, pp. 171–180, April 1987.