A Nonlinear Analysis Framework for Electronic Synthesizer Circuits
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A Nonlinear Analysis Framework for Electronic Synthesizer Circuits Fran¸cois Georges Germain Department of Music Research McGill University Montreal, Canada October 2011 A thesis submitted to McGill University in partial fulfillment of the requirements for the degree of Master of Arts. c 2011 Fran¸cois Georges Germain i Abstract This thesis presents a theoretical and experimental study of the nonlinear behaviour of analog synthesizers’ effects. The goal of this thesis is to evaluate and complete current research on nonlinear system modelling, both in and out of the field of music technology. The cases of single-input and multiple-input effects are considered. We first present an electronic analysis of the circuits of common examples of analog effects such as Moog’s lowpass filter and Bode’s ring modulator, extracting the equations of each system. We then discuss the results of experiments made on these systems in order to extract qualitative information about the distortion found in the system input-output relationship. Secondly, we look at the literature for methods used to model single-input nonlinear systems, and we investigate the opportunities to extend these techniques to multi-input systems. We focus on two different modelling approaches. The black-box approach seeks to model the input-output transfer function of the system as closely as possible without any particular assumption on the system. The circuit modelling approach uses the knowledge of electronic component behaviour to extract a transfer function from the known circuit of the system. The results of both approaches are compared to our experiments in order to evaluate their accuracy, identify flaws and, when possible, suggest potential improvements of the methods. ii R´esum´e Cette th`ese pr´esente une ´etude th´eorique et exp´erimentale du comportement nonlin´eaire des effets de synth´etiseurs analogiques. Elle vise `a´evaluer et compl´eter les recherches actuelles sur la mod´elisation des syst`emes non-lin´eaires, `ala fois dans le domaine de la technologie de la musique et en dehors. Les cas des effets `aune ou plusieurs entr´ees sont examin´es. Pour ce faire, nous pr´esentons d’abord une analyse ´electronique des circuits de plusieurs exemples usuels d’effets analogiques tels que le filtre passe-bas de Moog ou le modulateur en anneau de Bode. Les ´equations r´egissant chaque syst`eme en sont d´eriv´ees. Nous discutons ensuite le r´esultat d’exp´eriences men´ees sur ces syst`emes pour extraire une caract´erisation qualitative de la distorsion pr´esente dans le rapport entr´ee-sortie du syst`eme. Dans un second temps, nous examinons les m´ethodes de mod´elisation des syst`emes non-lin´eaires `aune entr´ee trouv´ees dans la litt´erature, et nous explorons les possibilit´es d’extension de ces techniques aux syst`emes `aplusieurs entr´ees. Deux approches de mod´elisation sont abord´ees. L’approche boˆıte noire vise `amod´eliser la fonction de transfert entr´ee- sortie du syst`eme aussi fid`element que possible sans hypoth`ese sur la structure du syst`eme. L’approche de la mod´elisation du circuit utilise quant `aelle la connaissance du comporte- ment des composants ´electroniques pour extraire une fonction de transfert `apartir du circuit (connu) du syst`eme. Les r´esultats associ´es aux deux approches sont compar´es `anos exp´eriences pour ´evaluer leur performance, et identifier des lacunes et, quand c’est possible, des opportunit´es d’am´elioration de ces m´ethodes. iii Acknowledgments I gratefully acknowledge McGill University and the Schulich School of Music for their financial aid, as well as the Centre for Interdisciplinary Research in Music Media and Technology (CIRMMT) and the Digital Composition Studios (DCS) for their material support. Many thanks to my directors, Professor Philippe Depalle and Professor Marcelo M. Wanderley, for their support, advice and encouragement. Special thanks to Carolina Brum, Darryl Cameron, Yves M´ethot (CIRMMT) and Richard McKenzie for their technical as- sistance in setting up my experiments, and to my colleagues Vincent Fr´eour, Joseph Thi- bodeau and Jason A. Hockman for their suggestions. I am grateful to Doctor Thomas H´elie from IRCAM for sharing data about his research. I also wish to thank Gabriel Vigliensoni, Junk Suk Lee and Hani Bakhshaei for accom- panying all along the days (and sometimes the nights) in the laboratory, all the members of the lab soccer and basketball teams, and the professors and students of Music Technology in general for all the discussions we had about graduate work, research and life through these two years. Finally, many thanks to my parents G´erard and Yvette for their never-ending support all along my graduate work at McGill. iv Contents 1 Introduction 1 1.1 Projectoverview ................................ 1 1.2 Analogsynthesishistory ............................ 2 1.3 Analogsynthesizerseffects . 3 1.4 Systemdescription ............................... 3 1.5 Nonlinearanalysis................................ 5 1.6 Distortion characterization . 6 1.7 Single-input and multiple-input nonlinear models . 8 1.8 Thesis’layout.................................. 9 2 Synthesizer circuits 10 2.1 Lowpassfiltersinsynthesizers . 10 2.1.1 Ladderfilters .............................. 10 2.1.2 Alternative filter designs . 11 2.1.3 TheMoogladderfilter ......................... 11 2.2 Ringmodulators ................................ 15 2.2.1 Diode-basedringmodulator . 15 2.2.2 Transistor-based ring modulator . 17 2.3 Circuitasymmetries .............................. 19 2.3.1 Hoffmann-Burchardi’sexperiment . 19 2.3.2 Asymmetriesinsymmetricstages . 20 2.4 Summary .................................... 23 3 Experimental study 24 3.1 Remarks..................................... 25 Contents v 3.2 Experimentsonlowpassfilters. .. .. 26 3.2.1 Setupformeasurementsattheoutlets . 26 3.2.2 Setupformeasurements onthecircuitboard . 27 3.2.3 Cutofffrequencyextraction . 27 3.2.4 Distortionmeasurement . .. .. 28 3.3 Moogfilter.................................... 28 3.3.1 Input-outputbehaviourforaDCinput . 28 3.3.2 Cutofffrequencycontrol . 30 3.3.3 Harmonic distortion of the Moog filter . 33 3.3.4 Influence of the frequency cutoff setting on the filter harmonic distortion 34 3.3.5 Influence of the input amplitude on the filter harmonic distortion . 36 3.4 Experimentsonringmodulators. 37 3.5 KorgMS-50ringmodulator .......................... 38 3.6 Moogerfoogerringmodulator ......................... 40 3.7 Summary .................................... 44 4 Nonlinear modelling using black-box approach 45 4.1 Lineartheory .................................. 45 4.2 Volterraseries.................................. 47 4.2.1 SISOmodel ............................... 47 4.2.2 Laplace transform and Z-transform of the expansion . ... 49 4.2.3 MISOmodel............................... 49 4.2.4 Mathematicalproperties . 50 4.2.5 Practicaluse .............................. 51 4.3 Block-basedmodels............................... 52 4.3.1 SISOmodels .............................. 52 4.3.2 MISOHammersteinmodels . .. .. 53 4.3.3 Numberofcoefficients ......................... 54 4.4 Summary .................................... 54 5 Identification methods 56 5.1 Excitationsignals................................ 56 5.1.1 Sinusoidsandswept-sines . .. .. 56 Contents vi 5.1.2 Pseudo-randomandrandomsignals . 57 5.2 LTIsystemidentification............................ 58 5.3 Orderseparation ................................ 61 5.3.1 Method ................................. 61 5.3.2 Use.................................... 62 5.4 Correlation-basedmethods. 62 5.4.1 Functionalexpansions ......................... 63 5.4.2 Hammersteinmodel .......................... 67 5.4.3 Wienermodel.............................. 68 5.4.4 PolynomialHammersteinmodel . 69 5.4.5 Extensiontotwo-inputsystems . 71 5.5 Explicit least-squares method . 73 5.5.1 Kernelorthogonalization . 73 5.5.2 Functionalexpansions ......................... 73 5.5.3 PolynomialHammersteinmodel . 73 5.5.4 Extensiontotwo-inputsystems . 77 5.5.5 Computationcost............................ 77 5.6 Swept-sine-basedmethod. 77 5.6.1 Excitationsignal ............................ 77 5.6.2 Inversesignal .............................. 78 5.6.3 Extraction of the impulse responses . 78 5.6.4 Extensiontotwo-inputsystems . 78 5.7 Summary .................................... 79 6 Applications to synthesizer circuits 80 6.1 Implementation of the identification methods . .. 80 6.2 Moog ladder filter numerical models . 81 6.2.1 Circuit modelling . 81 6.2.2 Backwardfinitedifference . .. .. 81 6.2.3 TruncatedVolterraseries. 82 6.3 Finitedifferencemodelstudy ......................... 84 6.3.1 Sinusoidal excitation . 85 6.3.2 Even-orderdistortion.......................... 85 Contents vii 6.3.3 Distortion ................................ 87 6.3.4 Nonlinear identification . 88 6.4 Nonlinear identification of the Moog ladder filter . 90 6.5 Nonlinear identification of ring modulators . 96 6.6 Summary .................................... 100 7 Conclusion and Future Work 102 7.1 Summary .................................... 102 7.2 Futurework................................... 103 7.3 Conclusion.................................... 104 A Electronic components 106 A.1 Linearelements ................................. 106 A.2 Diode ...................................... 106 A.3 Transistor .................................... 108 A.4 Operationalamplifier.............................. 109 B Studied synthesizers’ effects 111 B.1 Moogeffects................................... 111