Real-time Digital Simulation of Guitar Amplifiers as Audio Effects

199 Pages·2012·3.04 MB·English

by  Jaromír Mačák

BRNO UNIVERSITY OF TECHNOLOGY
VYSOKÉ UČENÍ TECHNICKÉ V BRNĚ
FACULTY OF ELECTRICAL ENGINEERING AND COMMUNICATION
DEPARTMENT OF TELECOMMUNICATIONS
FAKULTA ELEKTROTECHNIKY A KOMUNIKAČNÍCH TECHNOLOGIÍ
ÚSTAV TELEKOMUNIKACÍ
REAL-TIME DIGITAL SIMULATION OF GUITAR
AMPLIFIERS AS AUDIO EFFECTS
DOCTORAL THESIS
DIZERTAČNI PRÁCE
AUTHOR Ing. JAROMÍR MAČÁK
AUTOR PRÁCE
BRNO UNIVERSITY OF TECHNOLOGY
VYSOKÉ UČENÍ TECHNICKÉ V BRNĚ
FACULTY OF ELECTRICAL ENGINEERING AND
COMMUNICATION
DEPARTMENT OF TELECOMMUNICATIONS
FAKULTA ELEKTROTECHNIKY A KOMUNIKAČNÍCH
TECHNOLOGIÍ
ÚSTAV TELEKOMUNIKACÍ
REAL-TIME DIGITAL SIMULATION OF GUITAR
AMPLIFIERS AS AUDIO EFFECTS
ČÍSLICOVÁ SIMULACE KYTAROVÝCH ZESILOVAČŮ JAKO ZVUKOVÝCH EFEKTŮ
V REÁLNÉM ČASE
DOCTORAL THESIS
DIZERTAČNI PRÁCE
AUTHOR Ing. JAROMÍR MAČÁK
AUTOR PRÁCE
SUPERVISOR Ing. JIŘÍ SCHIMMEL, Ph.D.
VEDOUCÍ PRÁCE
BRNO 2012
ABSTRACT
The work deals with the real-time digital simulation of guitar amplifiers considered as
nonlinear analog audio effects. The main aim is to design algorithms which are able
to simulate complex systems in real-time. These algorithms are mainly based on the
automated DK-method and the approximation of nonlinear functions. Quality of the
designed algorithms is evaluated using listening tests.
KEYWORDS
Nonlinear dynamic system, real-time digital signal processing, digital simulation, audio
effect, guitar amplifier.
ABSTRAKT
Práce se zabývá číslicovou simulací kytarových zesilovačů, jakož to nelineárních anal-
ogových hudebních efektů, v reálném čase. Hlavním cílem práce je návrh algoritmů,
které by umožnily simulaci složitých systémů v reálném čase. Tyto algoritmy jsou pre-
vážně založeny na automatizované DK-metodě a aproximaci nelineárních funkcí. Kvalita
navržených algoritmů je stanovana pomocí poslechových testů.
KLÍČOVÁ SLOVA
Nelineární setrvačné systémy, zpracování číslicových signálů v reálném čase, digitální
simulace, hudební efekt, kytarový zesilovač.
MAČÁK, Jaromír Real-time Digital Simulation of Guitar Amplifiers as Audio Effects:
doctoral thesis. Brno: Brno University of Technology, Faculty of Electrical Engineering
and Communication, Department of Telecommunications, 2012. 198 p. Supervised by
Ing. Jiří Schimmel, Ph.D.
DECLARATION
I declare that I have written my doctoral thesis on the theme of “Real-time Digital
Simulation of Guitar Amplifiers as Audio Effects” independently, under the guidance of
the doctoral thesis supervisor and using the technical literature and other sources of
information which are all quoted in the thesis and detailed in the list of literature at the
end of the thesis.
As the author of the doctoral thesis I furthermore declare that, as regards the creation
of this doctoral thesis, I have not infringed any copyright. In particular, I have not
unlawfully encroached on anyone’s personal and/or ownership rights and I am fully aware
of the consequences in the case of breaking Regulation S11 and the following of the
Copyright Act No 121/2000 Sb., and of the rights related to intellectual property right
and changes in some Acts (Intellectual Property Act) and formulated in later regulations,
inclusive of the possible consequences resulting from the provisions of Criminal Act No
40/2009 Sb., Section 2, Head VI, Part 4.
Brno ............... ..................................
(author’s signature)
ACKNOWLEDGEMENT
Firstly I would like to thank to my supervisor Jiří Schimmel for this interesting
topic and for his valuable advices and remarks which I was given during working on
this topic. I also thank to colleagues from my office for inspiring working environment.
Further I would like to thank to Audiffex company for the opportunity to
implement algorithms for the simulation of guitar analog effects in their products,
thanks to Lubor Přikryl, CEO of Audiffex, and to Vladimír Tichý, software developer
of Audiffex.
I have to thank to prof. Udo Zölzer and Martin Holters for the opportunity to
spend three amazing months at Helmut Schmidt University in Hamburg. This visit
inspired me during last stages of my research.
I also have to thank to Oliver Kröning for measuring the tubes from the simulated
preamp and to all who took part in the listening tests.
My greatest thanks belong to my family and girlfriend Dana for support and
patience during writing this thesis.
4
Faculty of Electrical Engineering
and Communication
Brno University of Technology
Purkynova 118, CZ-61200 Brno
Czech Republic
http://www.six.feec.vutbr.cz
ACKNOWLEDGEMENT
The research was performed in laboratories supported by the SIX project; the registration
number CZ.1.05/2.1.00/03.0072, the operational program Research and Development
for Innovation.
Brno ............... ..................................
(author’s signature)
CONTENTS
List of abbreviations 14
List of symbols and math operations 16
Introduction 19
1 State of the Art 20
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.2 Algorithms Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.2.1 Nodal Analysis Simulation Techniques . . . . . . . . . . . . . 22
1.2.2 Numerical Integration of Nonlinear Ordinary Differential Equa-
tions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.2.3 Simulation by Static Waveshaping and Digital Filter Design . 26
1.2.4 State Space Based Approach . . . . . . . . . . . . . . . . . . . 28
1.2.5 Nonlinear Wave Digital Filters . . . . . . . . . . . . . . . . . . 31
1.2.6 Volterra Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
1.3 Recent Advances . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
1.3.1 Advances in State-Space Modeling . . . . . . . . . . . . . . . 38
1.3.2 State-Space Approach for Parametric Circuits . . . . . . . . . 40
1.4 Basic Circuit Component Models for Real-time Audio Effect Simulation 42
1.4.1 Discretized Models of Capacitor and Inductor . . . . . . . . . 43
1.4.2 Diode Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
1.4.3 Transistor Model . . . . . . . . . . . . . . . . . . . . . . . . . 44
1.4.4 Tube Models . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
1.4.5 Transformer Core Models . . . . . . . . . . . . . . . . . . . . 46
2 Goals of Thesis 48
3 Circuit Analysis of Audio Effects 50
3.1 Simulation of Circuits with Audio Transformer . . . . . . . . . . . . . 50
3.1.1 Transformer Model . . . . . . . . . . . . . . . . . . . . . . . . 51
3.1.2 Basic Input Stage with Transformer . . . . . . . . . . . . . . . 55
3.1.3 Push-Pull Tube Amplifier . . . . . . . . . . . . . . . . . . . . 62
3.1.4 Automated Incorporation of Transformer Model into DK-method 66
3.2 Simulation of Circuits with Operational Amplifier . . . . . . . . . . . 72
3.2.1 Incorporation of Operational Amplifier Model into Automated
DK-method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.3 Further Considerations Regarding DK-Method . . . . . . . . . . . . . 81
3.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4 Approximation of Implicit Nonlinear Circuit Equations 86
4.1 Precomputation of Nonlinear Systems . . . . . . . . . . . . . . . . . . 86
4.1.1 Precomputation for approximation of nonlinear ODEs . . . . . 87
4.1.2 Precomputation for approximation of the state-space nonlinearity 89
4.2 Brief Overview of Function Approximation Techniques . . . . . . . . 91
4.3 Implementation and comparison of approximation of 1-D function . . 91
4.3.1 Non-uniform Grid Interpolation . . . . . . . . . . . . . . . . . 93
4.4 Approximation of N-D function . . . . . . . . . . . . . . . . . . . . . 99
4.4.1 Non-uniform grid interpolation . . . . . . . . . . . . . . . . . 103
4.4.2 Parallel evaluation of interpolations . . . . . . . . . . . . . . . 104
4.5 Customized approximation of transfer function . . . . . . . . . . . . . 106
4.5.1 Reshaping of transfer function . . . . . . . . . . . . . . . . . . 106
4.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5 Complex System Simulation 111
5.1 Modified Block-Wise Method . . . . . . . . . . . . . . . . . . . . . . 111
5.2 Guitar Tube Amplifier Simulation as a Case Study for Modified Block-
wise Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
5.2.1 Computational Complexity . . . . . . . . . . . . . . . . . . . . 123
5.2.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . 126
5.3 Decomposition of the DK-method nonlinear core . . . . . . . . . . . . 129
5.3.1 Precomputation . . . . . . . . . . . . . . . . . . . . . . . . . . 130
5.3.2 Further Look-up Table Size Reduction . . . . . . . . . . . . . 131
5.3.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . 135
5.4 DK-model Decomposition Using Connection Components . . . . . . . 138
5.5 Simulation of Circuit with Global Feedback . . . . . . . . . . . . . . 145
5.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150
6 Quality of Simulation of Audio Effect Circuits 152
6.1 Simulation of the Guitar Tube Preamp Engl E530 . . . . . . . . . . . 152
6.2 Subjective evaluation of guitar tube preamp simulation . . . . . . . . 159
6.2.1 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161
6.3 Subjective Comparison of Interpolation Techniques . . . . . . . . . . 162
6.4 Subjective Comparison of Output Transformer Model . . . . . . . . . 163
6.5 Audible Aliasing Distortion . . . . . . . . . . . . . . . . . . . . . . . 163
6.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
7 Conclusion 166
Author’s Publications 169
Bibliography 171
List of appendices 179
A Interpolation Techniques Comparison 180
B Implementation of Interpolation Formulas 182
C Incidence Matrices for Fender Type Preamp 184
D Incidence Matrices for Marshall Preamp 185
E Incidence Matrices for Marshall Preamp with the Decomposition 187
F K matrix for Marshall Preamp with the Decomposition 189
G Instructions for Listening Tests 190
H Answer Form for Listening Tests 192
I DVD content 195
LIST OF FIGURES
1.1 Block diagrams of Wiener (top), Hammerstein (middle) and Wiener-
Hammerstein nonlinear model. . . . . . . . . . . . . . . . . . . . . . . 35
2.1 Block diagram of guitar amplifier . . . . . . . . . . . . . . . . . . . . 48
3.1 Model of a transformer with two windings. . . . . . . . . . . . . . . . 51
3.2 Magnetic part of gyrator-capacitor transformer model. . . . . . . . . 53
3.3 A transformer connected with a symmetrical voltage signal source. . . 55
3.4 Output signal of the input stage circuit. . . . . . . . . . . . . . . . . 59
3.5 Hyseresis loop of transformer models used in input stage circuit. . . . 59
3.6 Solution of input stage nonlinear function with Frohlich model. . . . . 61
3.7 Solution of input stage nonlinear function with the GC model. . . . . 62
3.8 Circuit schematic for push-pull power tubes part of the tube power
amplifier. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.9 Ouput signals of the push-pull amplifier for different transformer models. 64
3.10 Distortion analysis of the push-pull amplifier with the linear model of
the output transformer. . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.11 Distortion analysis of the push-pull amplifier with the nonlinear model
of the output transformer. . . . . . . . . . . . . . . . . . . . . . . . . 65
3.12 Stamp of linear transformer model for conductance matrix . . . . . . 68
3.13 Stamp of nonlinear transformer model for the conductance matrix . . 71
3.14 Circuit schematic for the inverting amplifier (left) and comparator
(right). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.15 Transfer function of the inverting amplifier with model of OPA for
real-time processing. . . . . . . . . . . . . . . . . . . . . . . . . . . . 74
3.16 Transfer function of the inverting comporator with model of OPA for
real-time processing. . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.17 Solution of the nonlinear equation for the comparator circuit using
DK-method. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.18 Solution of the nonlinear equation for the comparator circuit using
equation (3.115). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
3.19 Solution of the nonlinear equation for the inverting amplifier. . . . . . 79
3.20 Circuit schematic of the simple LFO generator. . . . . . . . . . . . . 80
3.21 Transient analysis of the LFO generator simulated with the DK-method. 81
4.1 Circuit schematic for triode tube amplifier. . . . . . . . . . . . . . . . 88
4.2 Computational cost of interpolation algorithms. Measured for one
million of interpolations. . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.3 Circuit schematic of the nonlinear core of the Distortion effect. . . . . 95
4.4 Approximation of the transfer function of the diode clipper circuit. . . 97