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Schnelle Transformationen. Eine Medienarchäologische Und Humboldt-Universität zu Berlin Kultur-, Sozial- und Bildungswissenschaftliche Fakultät Institut für Musikwissenschaft und Medienwissenschaft Masterarbeit zur Erlangung des akademischen Grades Master of Arts (M. A.) im Fach Medienwissenschaft Schnelle Transformationen Eine medienarchäologische und objektorientierte Untersuchung von Fourier-Transformationsalgorithmen Fast Transformations. A Media Archaeological and Object-Oriented Analysis of Fourier Transform Algorithms Eingereicht von: Johannes Maibaum [email protected] am 15. November 2016 in Berlin Tag der mündlichen Prüfung: 15. Dezember 2016 1. Gutachter: Prof. Dr. Wolfgang Ernst 2. Gutachter: Dr. Stefan Höltgen Dieses Werk ist unter einer Creative Commons Lizenz vom Typ Namensnennung-Nicht kom- merziell 3.0 Deutschland zugänglich. Um eine Kopie dieser Lizenz einzusehen, konsultieren Sie http://creativecommons.org/licenses/by-nc/3.0/de/ oder wenden Sie sich brieflich an Creative Commons, Postfach 1866, Mountain View, California, 94042, USA. Inhaltsverzeichnis 1 Einleitung 1 2 Die Frage nach dem Algorithmus 7 3 Objektorientierte Ontologie 18 3.1 Perspektivwechsel: System- und Grundoperationen .............. 23 3.2 Philosophisches „Zimmern“: Bogosts Carpentry ................ 30 4 Die Fourier-Transformation 40 4.1 Das Problem der schwingenden Saite ...................... 41 4.2 Fouriers Algorithmus zur Koeffizientenbestimmung .............. 48 4.3 Eine Grundoperation in Fouriers Algorithmus ................. 55 4.4 Modulationen im Gehör und im Radio ...................... 61 5 Die schnelle Fourier-Transformation 73 5.1 Algorithmische Beschleunigungen ........................ 79 5.2 Divide and Conquer ................................ 90 6 Fazit 103 Abbildungsverzeichnis 110 Literaturverzeichnis 112 1 Einleitung Unter dem Namen Digital Humanities wird seit einigen Jahren ein Forschungszweig eta- bliert, dessen Protagonisten – Geisteswissenschaftler und Informatiker – sich die Fähigkei- ten von Digitalcomputern zunutze machen wollen, um kulturwissenschaftliche Analysen auf den technomathematischen Stand der Dinge zu bringen. Dazu werden Algorithmen auf rie- sige Datensätze kultureller Güter wie Bilder, Comics, Filme oder Klänge angesetzt, um einen vollständigeren Überblick zu erlangen, als es die menschliche Lebenserwartung und Analy- sefähigkeit erlauben würde: If Google can analyse billions of Web pages and over a trillion links several times each day, we should be able to do better than simply consider a handful of images and generalize fromthem – even if we don’t have the same resources. […] Instead of being fuzzy and blurred, our horizon (knowledge of a cultural field as a whole) can become razor sharp and at the same time aquirea new depth (being able to sort and cluster millions of artifacts along multiple dimensions).1 Diesem Enthusiasmus stehen andere Stimmen aus der Kulturwissenschaft gegenüber, die sich gerade angesichts der eingesetzten Algorithmen skeptisch äußern und warnen, dass al- gorithmische Akteure vermeintlich unbemerkt klassische geisteswissenschaftliche Arbeits- kraft ersetzen.2 Eine differenziertere Position nimmt hingegen Katherine Hayles ein, die schreibt, dass „the tension between algorithmic analysis and hermeneutic close reading“ – der angenommene Unterschied zwischen menschlich interpretierendem und algorithmi- schem, mathematisch kalkulierendem Lesen – „should not be overstated. Very often the rela- tionship is configured not so much as an opposition as a synergetic interaction.“3 Die Digital Humanities sollten demnach die Stärken von Menschen und Algorithmen gewinnbringend kombinieren. 1 Lev Manovich. „How to Compare One Million Images?“ In: Understanding Digital Humanities. Hrsg. von David Michael Berry. Houndmills und New York: Palgrave Macmillan, 2012, S. 249–278, hier S. 252. 2 Mit der Beobachtung, dass „Algorithmen einen Spielbericht über ein kleines Baseball-Team aus Illinois […] verfasst hatten“, beginnt z. B. die essayistische Untersuchung von Mercedes Bunz. Die stille Revolution. Wie Algorithmen Wissen, Arbeit, Öffentlichkeit und Politik verändern, ohne dabei viel Lärm zu machen. Berlin: Suhrkamp, 2012, S. 11. 3 Nancy Katherine Hayles. „How We Think: Transforming Power and Digital Technologies“. In: Understand- ing Digital Humanities. Hrsg. von David Michael Berry. Houndmills und New York: Palgrave Macmillan, 2012, S. 42–66, hier S. 48. 1 Dieses vielschichtige Verhältnis der Digital Humanities zu Algorithmen stellt die Medi- entheorie vor die Aufgabe, algorithmisches Wirken tiefergehender zu analysieren. Dieser Problematik nähert sich die vorliegende Arbeit mittels einer spezifischen Methode: „Medi- enarchäologie hat […] die Genealogie jener Apparaturen zum Gegenstand, die jenseits der Hermeneutik – parergonal – an Bildern und Texten am Werk sind; […].“4 So als zu untersu- chende, medienarchäologische Apparaturen verstanden, stehen zwei Algorithmen im Fokus dieser Arbeit: die Fourier-Transformation und ihre „schnelle“ Variante, die fast Fourier trans- form oder kurz FFT. Diese Auswahl hat zwei Gründe. Erstens ist die Fourier-Transformation ein Paradebeispiel für jene Analysealgorithmen, die auch in den Digital Humanities zum Einsatz kommen. In der digitalen Signalverarbei- tung (DSP) gelingt es mit ihrer Hilfe, Zeitsignale in ein Frequenzspektrum zu transformie- ren oder umgekehrt zu einem gegebenen Spektrum ein entsprechendes Zeitsignal zu erzeu- gen. Zeitsignale sind der genuine output klassischer technischer Medien wie beispielsweise dem Radio oder dem Fernsehen: kontinuierlich variierende, meist elektrische Spannungen, die an Lautsprechern oder Bildröhren in für menschliche Ohren hörbare Klänge und für menschliche Augen sichtbare, bewegte oder unbewegte Bilder transformiert werden kön- nen. Das Zusammengesetztsein dieser Zeitsignale aus manchmal unendlich vielen einzelnen sinusförmigen Grundschwingungen unterschiedlicher Frequenz, eben jenes Spektrum, ist menschlichen Sinnesorganen in der Regel nicht zugänglich. Erreichbar ist diese Auflösung in Schwingungsanteile jedoch auf einem mathematischem Weg, den Fourier-Transformatio- nen zu beschreiten erlauben. Mit der Zeit- und der Frequenzanalyse sind die zwei techno- mathematischen Methoden kommunikationstheoretischer Signalanalyse aufgezählt, die zur Mitte des 20. Jahrhunderts, kurz nach Ende des Zweiten Weltkriegs, bekannt sind: „Hitherto, communication theory was based on two alternative methods of signal analysis. One is the description of the signal as a function of time; the other is Fourier analysis.“5 Fourier-Trans- formationen verarbeiten Informationen im Sinne Claude Shannons – kontinuierliche Signa- le oder diskrete Daten6 – und übersetzen oder übertragen sie bidirektional zwischen den zwei Darstellungsweisen, dem Zeit- und dem Frequenzbereich. Fourier-Transformationen 4 Wolfgang Ernst. Das Gesetz des Gedächtnisses. Medien und Archive am Ende (des 20. Jahrhunderts). Berlin: Kadmos, 2007, S. 32. 5 Dennis Gábor. „Theory of Communication“. In: Journal of the Institution of Electrical Engineers 93.26 (Nov. 1946), S. 429–457, hier S. 429. 6 Diese Unterscheidung trennt die beiden Teile seiner mathematischen Kommunikationstheorie: „In this final installment of the paper we consider the case where the signals or the messages or both are continuously variable, in contrast with the discrete nature assumed until now.“ Claude Elwood Shannon. „A Mathematical Theory of Communication“. In: Bell System Technical Journal 27.3/4 (Juli/Okt. 1948), S. 379–432, 623–656, hier S. 623. 2 sind demnach „Medientechnologien“ ganz nach Friedrich Kittlers technischer Definition.7 Zweitens ist „the Fourier transformation [sic …] continuously an important reference point for German media archaeologists“.8 So taucht sie etwa in Arbeiten von Kittler, Bernhard Siegert und Wolfgang Ernst auf. Diese drei Theoretiker eint der Ansatz, mithilfe der Fourier- Transformation – in Siegerts Worten – einen „Riß […] im Denken der Repräsentation“ zu umschreiben, den schon erwähnten Unterschied zwischen dem Kontinuierlichen oder dem Analogen und dem Diskreten oder dem Digitalen.9 Siegert setzt diesen Riss historisch im 18. und 19. Jahrhundert an, in der Zeit als der französische Mathematiker Jean Baptiste Joseph Fourier sein Kalkül entwickelte: Seit Fourier […] hört das Rauschen des Meeres auf, die Grenze der Analyse (des Wahrnehmbaren und des Sagbaren) zu bezeichnen; mit Fourier werden unendliche Summen von Oszillationen zum Medium der Analyse selbst. Die Analyse operiert nun nicht mehr über dem Abgrund des Realen als dem Reich des Nichtrepräsentierbaren, sondern im Realen/Reellen. Fouriers Analyse macht nicht mehr wie Leibniz’ Analyse vor dem Unbewußten als dem Ununterscheidbaren halt, sie begibt sich vollständig hinein.10 Mithilfe der Fourier-Transformation wird erstmals der gesamte Bereich real auftretender Si- gnale nach der Zeit und der Frequenz mathematisch analysierbar und somit Teil des Archivs, des Aussagbaren, wie es Michel Foucault definiert.11 Und dies sowohl in kontinuierlichen als auch in diskreten Frequenzspektren. Zur Unterscheidung dieser beiden Fälle wird in der Ma- thematik zwischen Fourier-Transformationen und Fourier-Reihen differenziert, die beide von Fourier entwickelt wurden.12 Erstere transformieren eine kontinuierliche Zeit-Funktion in ein unendliches, kontinuierliches Spektrum, letztere eine kontinuierliche Zeit-Funktion in ein ebenso unendliches, aber diskretes Spektrum harmonischer Frequenzen.13 In der folgenden Arbeit ist, wenn von der klassischen Fourier-Transformation die Rede ist,
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