581:. The accuracy of predictions of response from such designs depended on accurate impedance matching between sections. This could be achieved with sections entirely internal to the filter but it was not possible to perfectly match to the end terminations. For this reason, image filter designers incorporated end sections in their designs of a different form optimised for an improved match rather than filtering response. The choice of form of such sections was more a matter of designer experience than design calculation. Network synthesis entirely did away with the need for this. It directly predicted the response of the filter and included the terminations in the synthesis.
588:. Whereas network analysis asks what is the response of a given network, network synthesis on the other hand asks what are the networks that can produce a given desired response. Cauer solved this problem by comparing electrical quantities and functions to their mechanical equivalents. Then, realising that they were completely analogous, applying the known
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was published. The original manuscript to the second volume was destroyed as a result of the war. Although Cauer was able to reproduce this work, he was not able to publish it and it too was lost during the war. Some time after his death, however, his family arranged for the publication of some of
713:
Most of the above work is contained in Cauer's first and second monographs and is largely a treatment of one-ports. In his habilitation thesis Cauer begins to extend this work by showing that a global canonical form cannot be found in the general case for three-element kind multiports (that is,
425:
Cauer seems to have got on very poorly with his German colleagues. According to Rainer Pauli, his correspondence with them was usually brief and business-like, rarely, if ever, discussing issues in depth. By contrast, his correspondence with his
American and European acquaintances was warm,
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networks. He is considered the founder of the field and the publication of his principal work in
English was enthusiastically greeted, even though this did not happen until seventeen years later (in 1958). Prior to network synthesis, networks, especially filters, were designed using the
700:
Cauer's work was initially ignored because his canonical forms made use of ideal transformers. This made his circuits of less practical use to engineers. However, it was soon realised that Cauer's
Tchebyscheff approximation could just as easily be applied to the rather more useful
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is realisable as an impedance network. The second is to find the canonical (minimal) forms of these functions and the relationships (transforms) between different forms representing the same transfer function. Finally, it is not, in general, possible to find an exact
33:
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solution to an ideal transfer function - such as zero attenuation at all frequencies below a given cutoff frequency and infinite attenuation above. The third task is therefore to find approximation techniques for achieving the desired responses.
629:
He discovered the necessary and sufficient condition for realisability of a one-port impedance. That is, those impedance expressions that could actually be built as a real circuit. In later papers he made generalisations to multiport
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by Soviet soldiers as a hostage. Soviet intelligence was actively looking for scientists they could use in their own researches and Cauer was on their list of people to find but it would seem that this was unknown to his executioners.
167:. Prior to his work, electronic filter design used techniques which accurately predicted filter behaviour only under unrealistic conditions. This required a certain amount of experience on the part of the designer to choose suitable
612:
impedances. The transfer function between a voltage and a current amounting to the expression for the impedance itself. A useful network can be produced by breaking open a branch of the network and calling that the output.
308:, applying probability theory to telephone switching. He also worked on timer relays. He had two telecommunications-related publications during this period on "Telephone switching systems" and "Losses of real inductors".
439:
250:
district of Berlin. The building still exists, but is now a primary school, the Ludwig Cauer
Grundschule. He later attended the Mommsen Gymnasium, Berlin. His father, also Wilhelm Cauer, was a
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became a major obstacle to Cauer's work from 1933 onwards. The anti-Jewish hysteria of the time forced many academics to leave their posts, including the director of the
Mathematics Institute,
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By 1935 Cauer had three children whom he was finding increasingly difficult to support, which prompted him to return to industry. In 1936 he temporarily worked for the aircraft manufacturer
1337:
Cauer himself only proved necessity for this condition. Later, at MIT, Cauer supervised the doctoral thesis of O. Brune (1931) which proved sufficiency of the condition now called
721:
to all symmetric 2-ports, that is 2-ports which are electrically symmetrical but not necessarily topologically symmetrical, finding a number of canonical circuits. He also studied
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in Newark, New Jersey but then returned to Göttingen with the intention of building a fast analogue computer there. However, he was unable to obtain funding due to the depression.
647:
He generalised Foster's ladder realisation to filters which included resistors (Foster's were reactance only) and discovered an isomorphism between all two-element kind networks.
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in the US which must have been of enormous help when Cauer embarked on a study of filter design. Bell were at the forefront of filter design at this time with the likes of
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551:) to protect them from the expected fall of Berlin to the Russians, Cauer, against advice, returned to Berlin. His body was located after the end of the war in a
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346:, at the Institute of Applied Mathematics and Mechanics of the Technische Hochschule Charlottendorg. This paper is the beginning of modern network synthesis.
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brought him into contact with leading
American engineers in the field of filters. This proved useful when Cauer was unable to feed his children during the
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Cauer discovered that all solutions for the realisation of a given impedance expression could be obtained from one given solution by a group of
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According to Cauer, there are three major tasks that network synthesis has to address. The first is the ability to determine whether a given
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262:). Cauer became interested in mathematics at the age of thirteen and continued to demonstrate that he was academically inclined as he grew.
714:
networks containing all three R, L and C elements) for the generation of realisation solutions, as it can be for the two-element kind case.
426:
technically deep and often included personal family news and greetings. This correspondence went beyond his
American contacts and included
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and ideal transformers could be dispensed with. From then on network synthesis began to supplant image design as the method of choice.
1074:
829:*Cauer, W, "Untersuchungen ĂĽber ein Problem, das drei positiv definite quadratische Formen mit Streckenkomplexen in Beziehung setzt",
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to 2-element LC n-ports (1931) and showed that all equivalent LC networks could be derived from each other by linear transformations.
246:, an institution founded by his great-grandfather, Ludwig Cauer. This school was located on Cauerstrasse, named after Ludwig, in the
225:
was published after his death. Cauer's legacy continues today, with network synthesis being the method of choice for network design.
650:
He identified the canonical forms of filter realisation. That is, the minimal forms, which includes the ladder networks obtained by
622:
Following on from Foster, Cauer generalised the relationship between the expression for the impedance of a one-port network and its
198:. He studied early computer techniques in the US prior to returning to Germany. According to Wilhelm Cauer's son Emil the rise of
1553:
476:
Vow of allegiance of the
Professors of the German Universities and High-Schools to Adolf Hitler and the National Socialistic State
1123:
837:, pp86–132, 1931. On a problem where three positive definite quadratic forms are related to one-dimensional complexes (in German)
395:
Cauer met, and had strong contacts with, many of the key researchers in the field of filter design at Bell Labs. These included
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can be viewed as a special case of elliptic filters and can be arrived at using the same approximation techniques. So can the
585:
1245:
191:
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footing, providing tools that could produce exact solutions to a given specification for the design of an electronic filter.
959:
Brune, O, "Synthesis of a finite two-terminal network whose driving-point impedance is a prescribed function of frequency",
285:, and his first publication (1923) was in this field. For reasons that are not clear, he changed his field after this to
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458:
335:, is a milestone in filter theory and inspired Cauer to generalise this approach into what has now become the field of
331:
that Cauer had much correspondence and it was his work that Cauer recognised as being of such importance. His paper,
1010:
461:(VDE, the German Electrical Engineers Society). He left the VDE, however, in 1942 after a serious falling out with
384:
who was building machines for the solution of mathematical problems. Essentially, these were what we would now call
722:
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259:
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Proceedings of the
Fourteenth International Symposium of Mathematical Theory of Networks and Systems (MTNS2000)
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1501:
926:, Akad. Verlags-Gesellschaft Becker und Erler, Leipzig, 1941. Theory of Linear AC Circuits, Vol I (in German)
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503:
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772:, pp63–72, 1929. On a class of functions represented by truncated Stieltjes continued fractions (in German)
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to design filters. Cauer's application of
Tchebyscheff polynomials resulted in the filters now known as
948:, Akademie-Verlag, Berlin, 1960. Theory of Linear AC Circuits, Vol II (published posthumously in German)
764:
Cauer, W, "Über eine Klasse von Funktionen, die die Stieljesschen Kettenbrüche als Sonderfall enthält",
540:
his papers as the second volume, based on surviving descriptions of the intended contents of volume II.
242:, on 24 June 1900. He came from a long line of academics. His early grammar school (gymnasium) was the
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243:
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1474:, October 1935. Recommended by R. Pauli as the most profound treatise on Cauer's theory (in French).
506:, it stifled his future career. Thus he gained the title of professor but was never given a chair.
388:: Cauer was interested in using them to solve linear systems to aid in filter designs. His work on
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Cauer, W, "Die Verwirklichung der Wechselstromwiderstände vorgeschriebener Frequenzabhängigkeit",
457:
After leaving the Technical Institute for Mix & Genest, Cauer sought to become active in the
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221:. However, his family succeeded in reconstructing much of this from his notes and volume II of
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754:, pp355–388, 1926. The realisation of impedances of prescribed frequency dependence (in German)
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The relationship of Mix & Genest with Bell gave Cauer an easy path to collaboration with
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The manuscripts for some of Cauer's most important unpublished works were destroyed during
8:
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269:. He married Karoline Cauer (a relation) in 1925 and eventually fathered six children.
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Cauer started off in a field completely unrelated to filters; from 1922 he worked with
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E. Cauer, W. Mathis, and R. Pauli, "Life and Work of Wilhelm Cauer (1900 – 1945)",
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911:, pp185–192, 228–235, 1940. Quadripoles with prescribed insertion loss (in German)
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490:. Although Cauer was not Jewish, it became known that he had a Jewish ancestor,
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Mathis, W and Cauer, E, University of Hannover, 2002. A PowerPoint presentation.
881:, pp1–44, 1933. An interpolation problem of positive-real functions (in German)
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and in 1930 took his family to the USA where he had obtained a scholarship (a
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Cauer, W, "Ein Interpolationsproblem mit Funktionen mit positivem Realteil",
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848:, pp157–174, 1932. Ideal transformers and linear transformations (in German)
502:. While this revelation was not sufficient to have Cauer removed under the
159:. He is most noted for his work on the analysis and synthesis of electrical
761:, pp268–274, 1927. On the variables of some passive quadripoles (in German)
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900:, N.F., pp1–33, 1934. Equivalence of 2-poles without resistors (in German)
1019:(in German), vol. 3, Berlin: Duncker & Humblot, pp. 179–180
717:
Cauer extended the work of Bartlett and Brune on geometrically symmetric
266:
126:
114:
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Cauer, W, "The Poisson integral for functions with positive real part",
801:
Sitzungsberichte d. PreuĂź. Akademie d. Wissenschaften, phys-math. Klasse
1456:, "A recent contribution to the design of electrical filter networks".
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Microwave Filters, Impedance-Matching Networks, and Coupling Structures
894:
Nachrichten d. Gesellschaft d. Wissenschaften Göttingen, math-phys. Kl.
692:(maximally flat) filter, although this was an independent discovery by
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transitions for a given maximum attenuation variation. The well known
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Sitzungsberichte d. PreuĂź. Akademie d.Wissenschaften, phys-math Klasse
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Cauer, W, "Ideale Transformatoren und lineare Transformationen",
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Cauer treated network synthesis as being the inverse problem of
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Briefly, Cauer served in the German army in the final stages of
1464:, pp150–211, 1931–32. A comparison of the methods of Cauer and
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The realisation of impedances of specified frequency dependence
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Cauer, W, "Vierpole mit vorgeschriebenem Dämpfungsverhalten",
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Cauer, W, "Äquivalenz von 2n-Polen ohne Ohmsche Widerstände",
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of victims of Russian executions. Cauer had been shot dead in
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in New York making major contributions. However, it was with
548:
568:
The major part of Cauer's legacy is his contribution to the
364:
Cauer found that he could not support his family during the
171:
to include in the design. Cauer placed the field on a firm
202:
in Germany stifled Cauer's career because he had a remote
870:, pp369–394, 1932. On positive-real functions (in German)
373:
796:, pp425–433, 1930. Telephony filter circuits (in German)
757:
Cauer, W, "Ăśber die Variablen eines passiven Vierpols",
676:, or sometimes Cauer filters, which have optimally fast
32:
933:, McGraw-Hill, New York, 1958. (published posthumously)
818:, VDI-Verlag, Berlin, 1931. Filter circuits (in German)
195:
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Cauer, W, "Die Siebschaltungen der Fernmeldetechnik",
349:
In 1927 Cauer went to work as a research assistant at
1478:
Wilhelm Cauer: His Life and the Reception of his Work
1036:
Wilhelm Cauer: His Life and the Reception of his Work
946:
Theorie der linearen Wechselstromschaltungen, Vol. II
1470:
Julia, R, "Sur la Theorie des Filtres de W. Cauer",
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Cauer, W, "Ăśber Funktionen mit positivem Realteil",
766:
Jahresberichte der Dt. Mathematikervereinigung (DMV)
543:
After taking his children to stay with relatives in
289:. He graduated in applied physics in 1924 from the
924:
Theorie der linearen Wechselstromschaltungen, Vol.I
905:
Telegraphen-, Fernsprech-, Funk- und Fernsehtechnik
803:, pp673–681, 1931. A reactance theorem (in German)
163:and his work marked the beginning of the field of
1112:
1510:
1574:Academic staff of Technische Universität Berlin
342:In June 1926 Cauer presented his thesis paper,
1564:People executed by the Soviet Union by firearm
1413:, "Summary of the History of Circuit Theory",
524:and then became director of the laboratory of
254:and a professor of railway engineering at the
16:German mathematician and scientist (1900–1945)
392:was completed in 1931 while still in the US.
1534:Massachusetts Institute of Technology alumni
1426:Innovation and the Communications Revolution
1008:
789:Journal of Applied Mathematics and Mechanics
535:In 1941, the first volume of his main work,
361:and became an external university lecturer.
223:Theorie der linearen Wechselstromschaltungen
151:(24 June 1900 – 22 April 1945) was a German
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186:. His work for a German subsidiary of the
1559:German people executed by the Soviet Union
1428:, Institute of Electrical Engineers, 2002
931:Synthesis of Linear Communication Networks
783:, pp272–282, 1929. Quadripoles (in German)
465:, previously his PhD supervisor and ally.
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206:ancestor. Cauer was murdered during the
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1053:official site (in German), accessed and
418:For a short while, Cauer worked for the
233:
1549:German civilians killed in World War II
1124:MacTutor History of Mathematics Archive
238:Wilhelm Adolf Eduard Cauer was born in
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440:Lignes Télégraphiques et Téléphoniques
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1544:German Army personnel of World War I
1263:Academic Genealogy of Mathematicians
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1172:Foster, R M, "A reactance theorem",
842:Elektrische Nachrichtentechnik (ENT)
777:Elektrische Nachrichtentechnik (ENT)
608:Initially, the work revolved around
563:
1393:
353:'s Institute of Mathematics at the
300:He then spent a period working for
192:German economic crisis of the 1920s
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1539:20th-century German mathematicians
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1266:, page 60, World Scientific, 2010
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473:In November 1933 Cauer signed the
459:Verband Deutscher Elektrotechniker
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14:
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1236:Kemp, Dr. Peter Heinrich (2000).
1067:"Ludwig-Cauer-Grundschule Berlin"
799:Cauer, W, "Ein Reaktanztheorem",
696:arrived at by a different method.
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1174:Bell System Technical Journal
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450:and Hungarian graph theorist
295:Technische Universität Berlin
260:Technische Universität Berlin
98:Technische Universität Berlin
537:Theory of Linear AC Circuits
366:economic crisis of the 1920s
7:
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725:2-ports. He also extended
494:, who had been a banker to
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357:. In 1928 he obtained his
10:
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1289:Royal Aeronautical Society
1073:(in German), accessed and
244:Kaiserin Augusta Gymnasium
1529:Harvard University alumni
1472:Bull. Soc. Franc. Electr.
1240:(in German). p. 78.
875:Mathematische Zeitschrift
748:Archiv fĂĽr Elektrotechnik
442:in Paris, mathematicians
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1403:, Perpignan, June 2000.
1285:The Aeronautical Journal
1283:K.C. Garner, "Reviews",
1129:University of St Andrews
1051:Ludwig Cauer Grundschule
1016:Neue Deutsche Biographie
432:General Electric Company
1439:Matthaei, Young, Jones
1300:Mathis et al., pp.83-84
1287:, volume 63, page 375,
1071:Architektur Bild Archiv
670:Chebyshev approximation
355:University of Göttingen
293:in Charlottenburg (now
258:in Charlottenburg (now
1421:, pp848–855, May 1962.
1415:Proceedings of the IRE
853:Bull. Amer. Math. Soc.
775:Cauer, W, "Vierpole",
642:affine transformations
579:image impedance method
370:Rockefeller fellowship
306:Bell Telephone Company
287:electrical engineering
188:Bell Telephone Company
184:electrical engineering
1009:Piloty, Hans (1957),
864:Mathematische Annalen
831:Mathematische Annalen
291:Technische Hochschule
256:Technische Hochschule
234:Early life and family
182:but soon switched to
1483:3 March 2016 at the
1115:Robertson, Edmund F.
590:Lagrangian mechanics
482:The rising force of
194:and he moved to the
1382:E. Cauer et al., p6
1364:E. Cauer et al., p5
1312:E. Cauer et al., p4
1238:Meisenheimer Jugend
1226:E. Cauer et al., p9
1214:E. Cauer et al., p8
1154:E. Cauer et al., p3
1113:O'Connor, John J.;
1103:E. Cauer et al., p2
988:Tchebyscheff filter
694:Stephen Butterworth
420:Wired Radio Company
333:A reactance theorem
1458:Journ. Math. Phys.
1407:19 September 2008.
1180:, pp259–267, 1924.
967:, pp191–236, 1931.
961:J. Math. and Phys.
859:, pp713–717, 1932.
656:continued fraction
557:Berlin-Marienfelde
386:analogue computers
380:. He worked with
378:Harvard University
304:, a branch of the
283:general relativity
180:general relativity
76:Berlin-Marienfelde
1443:McGraw-Hill 1964.
1373:Cauer, 1927, 1933
1350:Cauer, 1929, 1931
1247:978-3-89811-587-2
686:Chebyshev filters
624:transfer function
597:transfer function
570:network synthesis
564:Network synthesis
405:Sidney Darlington
337:network synthesis
165:network synthesis
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133:Doctoral students
105:Scientific career
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329:Ronald M. Foster
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252:Privy Councillor
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122:Doctoral advisor
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849:
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797:
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773:
762:
755:
734:
731:
710:
707:
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664:
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648:
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635:Transformation
633:
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470:
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372:) to study at
323:in Boston and
274:
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208:fall of Berlin
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95:
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72:(aged 44)
66:
62:
61:
52:
41:
37:
36:
28:
27:
24:
15:
9:
6:
4:
3:
2:
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1499:
1498:Wilhelm Cauer
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1398:
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1379:
1370:
1361:
1359:
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1347:
1340:
1339:positive-real
1334:
1325:
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1307:
1297:
1290:
1286:
1280:
1273:
1269:
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1199:
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1136:29 July 2012.
1135:
1132:Accessed and
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1077:29 July 2012.
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1057:29 July 2012.
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663:Approximation
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617:Realisability
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428:A.C. Bartlett
423:
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382:Vannevar Bush
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153:mathematician
150:
149:Wilhelm Cauer
141:
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68:22 April 1945
67:
63:
59:
58:German Empire
55:
42:
38:
34:
29:
25:Wilhelm Cauer
22:
19:
1506:
1471:
1461:
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1418:
1414:
1411:Belevitch, V
1400:
1389:Bibliography
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1042:
1035:
1034:Emil Cauer:
1030:
1021:; (
1014:
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733:Publications
716:
712:
709:Further work
699:
668:He used the
607:
594:
583:
567:
545:Witzenhausen
542:
536:
534:
516:
508:
496:Frederick II
492:Daniel Itzig
481:
474:
472:
456:
434:in Wembley,
424:
417:
397:Hendrik Bode
394:
389:
363:
359:habilitation
348:
343:
341:
332:
310:
299:
279:Max von Laue
276:
264:
237:
222:
216:
177:
173:mathematical
148:
147:
104:
70:(1945-04-22)
50:24 June 1900
18:
1524:1945 deaths
1519:1900 births
1328:Cauer, 1926
814:*Cauer, W,
690:Butterworth
452:DĂ©nes KĹ‘nig
436:Roger Julia
267:World War I
127:Georg Hamel
115:Mathematics
84:Nationality
1513:Categories
1434:0852962185
1272:9814282294
994:References
944:Cauer, W,
929:Cauer, W,
922:Cauer, W,
723:antimetric
658:expansion.
553:mass grave
448:Georg Pick
413:Otto Zobel
325:Otto Zobel
214:soldiers.
46:1900-06-24
1424:Bray, J,
1163:Bray, p62
978:Black box
652:Stieltjes
630:networks.
520:works in
513:at their
504:race laws
317:Bell Labs
157:scientist
78:, Germany
1481:Archived
1134:archived
1075:archived
1055:archived
972:See also
682:stopband
678:passband
610:one-port
511:Fieseler
469:Nazi era
313:AT&T
169:sections
1500:at the
868:vol 106
835:vol 105
719:2-ports
602:element
600:finite-
574:passive
515:Fi 156
500:Prussia
430:of the
219:the war
161:filters
1462:vol 11
1432:
1419:vol 50
1341:or PR.
1270:
1244:
1178:Vol. 3
965:vol 10
909:vol 29
879:vol 38
857:vol 38
794:vol 10
770:vol 38
752:vol 17
530:Berlin
522:Kassel
517:Storch
484:Nazism
463:Wagner
409:Foster
273:Career
212:Soviet
204:Jewish
200:Nazism
111:Fields
88:German
54:Berlin
1466:Zobel
1291:1959.
898:vol 1
846:vol 9
781:vol 6
549:Hesse
1430:ISBN
1268:ISBN
1242:ISBN
547:(in
411:and
376:and
155:and
65:Died
40:Born
680:to
654:'s
572:of
528:in
498:of
438:of
374:MIT
297:).
281:on
210:by
1515::
1460:,
1417:,
1355:^
1317:^
1305:^
1219:^
1207:^
1185:^
1176:,
1141:^
1127:,
1121:,
1117:,
1082:^
1069:,
1049:,
1013:,
1001:^
963:,
907:,
896:,
877:,
866:,
855:,
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833:,
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750:,
479:.
454:.
446:,
415:.
407:,
403:,
399:,
339:.
196:US
56:,
1436:.
1274:.
1250:.
1025:)
644:.
626:.
48:)
44:(
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