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About me... Musician, Electrical Engineer, Mixing/Mastering Engineer Studied audio DSP at CCRMA 5+ years of making audio plugins, DAWs, etc. Not a great guitarist (but Im learning) 2 Klon Centaur Guitar pedal made by Bill

  2. About me... • Musician, Electrical Engineer, Mixing/Mastering Engineer • Studied audio DSP at CCRMA • 5+ years of making audio plugins, DAWs, etc. • Not a great guitarist (but I’m learning…) 2

  3. Klon Centaur Guitar pedal made by Bill Finnegan (MIT) from 1994-2000 3

  4. Virtual Analog Modelling Creatjng a digital emulatjon of a classic analog audio efgects. • Provide access to efgects that are old or rare. • Lower cost. • Convenience. • Improved understanding. 4

  5. “White-Box” Modelling Modelling a circuit through mathematjcal simulatjons of the physical interactjons of the component parts. • Nodal Analysis • Modifjed Nodal Analysis (MNA) • State-Space Formulatjon • Wave Digital Filters (WDF) • Port-Hamiltonian Formulatjon 5

  6. “White-Box” Modelling Advantages: Disadvantages: • Accurate modelling of • Ofuen computatjonally circuit behaviour (even in expensive (especially for extreme situatjons) real-tjme use) • Accurate modelling of • Requires knowledge of control parameters DSP, as well as physics, • Improved understanding of circuit theory, etc. the modelled efgect 6

  7. “Black-Box” Modelling Modelling a circuit by taking measurements, and designing a system to give a perceptually equivalent output. • Convolutjon with Impulse Response (for linear systems) • Volterra Series • Weiner-Hammerstein Method • Neural Networks 7

  8. “Black-Box” Modelling Advantages: Disadvantages: • Betuer for capturing • Diffjcult to include control “unique” behaviour parameters • Computatjonally cheaper • Minimal understanding of • Only requires background the efgect being modelled knowledge of DSP 8

  9. Difgerent Platforms Embedded Device: • Depends on the device Desktop Audio Plugin: (pedal, Eurorack module, multj-efgects processor) • Consumer-grade CPU • More powerful processors • Plenty of memory are more expensive • Have to share resources • Limited memory with other plugins • (Usually) don’t have to share resources 9

  10. Research Goals • Model sub-circuits from the Klon Centaur using difgerent modelling methods: • Nodal Analysis • Wave Digital Filters • Neural Networks • Create desktop and embedded implementatjons of the modelled efgect • Compare the advantages/disadvantages of each method 10

  11. Outline • Traditjonal Circuit Modelling • Nodal Analysis (Tone Stage sub-circuit) • Wave Digital Filters (FF-1 sub-circuit) • Neural Network Circuit Modelling • Recurrent Neural Network (Gain Stage sub-circuit) • Desktop and embedded implementatjons • Comparisons and Results 11

  12. Nodal Analysis 12

  13. Example Circuit: Tone Stage R23 C14 R24 RV2 R21 R22 − Vin Vout +4.5V + 13 Klon Centaur Tone Control Circuit

  14. Nodal Analysis: Continuous Time 1. Convert the circuit to the Laplace Domain, using the Laplace variable s = jω . The complex impedance of each principal circuit component is defjned as: Z C = 1 (1) Z R = R, Cs, Z L = Ls 14

  15. Nodal Analysis: Continuous Time 2. Form the Laplace domain transfer functjon. 1 V out ( s ) � � � � 1 1 1 1 1 C 14 + s + + (2) V in ( s ) = R 22 R 21+ Rv 2 b R 22 R 21+ Rv 2 b R 23+ Rv 2 a � � � � 1 1 s + − 1 1 1 C 14 + + R 23+ Rv 2 a R 24 R 24 R 21+ Rv 2 b R 23+ Rv 2 a 1 Maby, Solid State Electronic Circuits . 15

  16. Nodal Analysis: Discrete Time 3. Use a conformal map to map from the s-plane to z-plane (ofuen the bilinear transform). 2 1 − z − 1 s ← 2 (3) 1 + z − 1 T 2 Smith, Physical Audio Signal Processing . 16

  17. Nodal Analysis: Discrete Time 4. Implement the system as a digital fjlter. y [ n ] = b 0 x [ n ]+ b 1 x [ n − 1]+ b 2 x [ n − 2] − a 1 y [ n − 1] − a 2 y [ n − 2] (4) 17

  18. Discretization Considerations • Frequency warping • Stability 18

  19. Tone Stage Frequency Response 19

  20. Nodal Analysis Disadvantages: • Cannot be used to model Advantages: nonlinear circuits (can be extended with Modifjed • Simple and Nodal Analysis) computatjonally effjcient circuit models • Sometjmes diffjcult to compute parameter changes 20

  21. Wave Digital Filters 21

  22. Kirchofg Domain Circuits • Each circuit component has an impedance • Each component has a voltage across its terminals and current between • Components are connected in series/parallel confjguratjons (usually) 22

  23. Wave Domain Circuits Circuits are made up of wave ports with incident and refmected waves. Incident wave: (5) a = v + R 0 i Refmected wave: (6) b = v − R 0 i 23

  24. Wave Domain Circuits • Each circuit component is a “1-port element” that inputs incident and outputs refmected wave variables • Each series/parallel junctjon is an “N-port adaptor” that connects the 1-ports with a scatuering junctjon • Free parameter: port resistance 24

  25. Wave Digital Filters Wave Digital Filters (WDFs) were developed by Alfred Fetuweis in the 1970’s and 80’s. 3 • Digital simulatjon of circuits in the wave domain • Discretjze each circuit element independently • Create binary connectjon tree (BCT) between circuit elements 3 Fetuweis, “Wave digital fjlters: Theory and practjce”. 25

  26. Example Circuit: Feed-Forward Network 1 C3 R7 R19 + 4.5V C16 − Klon Centaur Feed-Forward Network 1 Circuit 26

  27. Example Circuit: Feed-Forward Network 1 R 7 R 19 S 2 S 3 V in S 1 P 1 V 4 . 5 C 3 C 16 WDF tree for the Klon Centaur Feed-Forward Network 1 Circuit. S and P nodes refer to series and parallel adaptors respectjvely. 27

  28. Time-Domain Response 28

  29. Wave Digital Filters Disadvantages: Advantages: • Cannot model circuits with • Modularity: circuit multjple nonlinearitjes or elements and topology can R -type topologies be alterred on-the-fmy • These types of circuits can • Each element can be be modelled using discretjzed with a difgerent R -adaptors, but with an conformal map increase in complexity 29

  30. Wave Digital Filters More informatjon: • Alfred Fetuweis, “Wave Digital Filters: Theory and Practjce”, Proceedings of the IEEE , vol. 74, no. 2, 1986 • Original reference for deriving WDF formalism • Kurt Werner, Virtual Analog Modeling of Audio Circuitry Using Wave Digital Filters , PhD. Thesis, Stanford University, 2016 • Great reference for deriving WDFs, including more recent advancements • Expands WDFs to handle R -type topologies and multjple nonlinearitjes 30

  31. Wave Digital Filters More informatjon: • François Germain, Non-oversampled physical modeling for virtual analog simulatjon , PhD. Thesis, Stanford University, 2019 • Example of independently discretjzing circuit elements with Alpha Transform • Jingjie Zhang and Julius Smith, “Real-tjme Wave Digital Simulatjon of Cascaded Vacuum Tube Amplifjers Using Modifjed Blockwise Method”, Proc. of the 21st Internatjonal Conference on Digital Audio Efgects , 2018 • Real-tjme simulatjon of an impressively large circuit 31

  32. Real-Time Neural Networks 32

  33. Black Box Modelling with Neural Nets Previous work: Damskägg et al., 2019 4 • Uses a WaveNet-style, “Temporal Convolutjonal Network” • Used to model distortjon pedal circuits • Also used to model tube amp distortjon 5 • Disadvantage: computatjonally expensive 4 Damskägg, Juvela, and Välimäki, “Real-Time Modeling of Audio Distortjon Circuits with Deep Learning”. 5 Damskägg et al., Deep Learning for Tube Amplifjer Emulatjon . 33

  34. Temporal Convolutional Networks Keith Bloemer: Smart Guitar Amp 6 6 https://github.com/keyth72/SmartGuitarAmp 34

  35. Temporal Convolutional Networks Christjan Steinmetz: Randomized Overdrive Neural Networks 7 7 https://github.com/csteinmetz1/ronn 35

  36. Black Box Modelling with Neural Nets Previous work: Parker et al., 2019 8 • Uses a deep, fully-connected “State Transitjon Network” • Approximates a state-space solutjon for nonlinear distortjon and fjlter circuits • Efgectjvely a “grey-box” model 8 Parker, Esqueda, and Bergner, “Modelling of Nonlinear State-Space Systems Using a Deep Neural Network”. 36

  37. State Transition Networks Natjve Instruments: Guitar Rig 6 Pro 9 9 https://blog.native-instruments.com/the-making-of-icm/ 37

  38. Black Box Modelling with Neural Nets Previous work: Wright et al., 2019 10 • Uses a single layer recurrent neural network • Used to model guitar distortjon circuits • Can also be used to model tjme-varying circuits 11 10 Wright, Damskägg, and Välimäki, “Real-Time Black-Box Modelling with Recurrent Neural Networks”. 11 Wright and Välimäki, “Neural Modelling of Time-Varying Efgects”. 38

  39. Recurrent Neural Network Advantages of using RNNs to model distortjon circuits: • Makes sense (recurrent units can be distortjon efgects) • Computatjonally effjcient • Can include circuit control parameters 39

  40. Recurrent Neural Network Output y [ n ] Fully Connected Layer Input x [ n ] Current State h [ n ] Recurrent Layer z − 1 Previous State h [ n − 1] 40

  41. Recurrent Neural Network Recurrent layer: Gated Recurrent Unit (7) z [ n ] = σ ( W z x [ n ] + U z h [ n − 1] + b z ) (8) r [ n ] = σ ( W r x [ n ] + U r h [ n − 1] + b r ) (9) c [ n ] = tanh ( W c x [ n ] + r [ n ] ◦ U c h [ n − 1] + b c ) (10) h [ n ] = z [ n ] ◦ h [ n − 1] + (1 − z [ n ]) ◦ c [ n ] 41

  42. Example Circuit: Centaur Gain Stage 42

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