Digital Sound Synthesis by Physical Modeling Using the Functional Transformation Method

243 Pages·2003·11.717 MB·English

by  Lutz Trautmann, Rudolf Rabenstein (auth.)

Digital Sound Synthesis
by Physical Modeling
U sing the Functional
Transformation Method
Digital Sound Synthesis
by Physical Modeling
Us ing the Functional
Transformation Method
Lutz Trautmann Rudolf Rabenstein
and
Telecommunication Laboratory, LMS
Erlangen, Germany
Springer-Science+Business Media, LLC
Library of Congress Cataloging-in-Publication Data
Trautmann, Lutz.
Digital sound synthesis by physical modeling using the functional transformation method
/Lutz Trautman and Rudolf Rabenstein.
p. cm.
Includes bibliographical references and index.
ISBN 978-1-4613-4900-6 ISBN 978-1-4615-0049-0 (eBook)
DOI 10.1007/978-1-4615-0049-0
1. Frequency synthesizers. 2. Transformations (Mathematics) 3. Sound-Recording and
reproducing-Digital techniques-Mathematics. 4. Vibration-Mathematical models. I.
Rabenstein, Rudolf. 1I. Title.
TK7872.F73T73 2003
621.3815' 486-dc22
2003054470
ISBN 978-1-4613-4900-6
©2003 Springer Science+Business Media NewYork
Originally published by Kluwer Academic/Plenum Publishers, New York in 2003
Softcover reprint of the hardcover 1s t edition 2003
10987654321
A C.I.P. record for this book is available from the Library of Congress
All rights reserved
No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form
or by any means, electronic, mechanicaI, photocopying, microfilming, recording, or otherwise,
witbout written permission from tbe Publisher, with the exception of any material supplied
specifically for the purpose of being entered and executed on a computer system, for exclusive
use by the purchaser of the work
Foreword
This book considers signal processing and physical modeling meth
ods for sound synthesis. Such methods are useful for example in mu
sic synthesizers, computer sound cards, and computer games. Physical
modeling synthesis has been commercialized for the first time about 10
years ago. Recently, it has been one of the most active research topics
in musical acoustics and computer music.
The authors of this book, Dr. Lutz Trautmann and Dr. Rudolf
Rabenstein, are active researchers and inventors in the field of sound
synthesis. Together they have developed a new synthesis technique,
called the functional transformation method, which can be used for pro
ducing musical sound in real time. Before this book, they have published
over 20 papers on the topic in journals and conference proceedings. In
this excellent textbook, the results are combined in a single volume. I
believe that this will be considered an important step forward for the
whole community.
The functional transformation method is proposed as a new way of de
signing physically based synthesis models for musical instruments. The
derivation of the method uses an elegant technique, the Sturm-Liouville
transformation, which is rarely used in acoustic signal processing. The
resulting signal processing structure for modeling linear systems is sim
ilar to that used in modal synthesis, that is, a parallel connection of
second-order filters. However, the functional transformation method of
fers certain advantages over the modal synthesis technique: most impor
tantly, it avoids the frequency errors, which occur due to discretization in
the modal synthesis. The functional transformation method also allows
nonlinear interconnections of structures, which is important for many
musical systems.
The computational cost of an implementation that uses a second-order
r"sonator for each vibrating mode can be seen as a disadvantage. Fortu-
v
vi DIGITAL SOUND SYNTHESIS USING THE FTM
nately, computers are very fast and are getting faster all the time. Even
today, a polyphonic real-time synthesizer based on the functional trans
formation has been realized with a multi-processor system. It will be an
interesting future research task to develop structures that decrease the
computational load of the functional transformation method. However,
as computers keep on getting faster, the computational load will not be
an issue after some years.
In addition to introducing the new physical modeling technique, this
book gives a brief general and historical overview of the technical field
of sound synthesis. Basic wavetable, granular, additive, subtractive, and
FM synthesis techniques and some of their modifications are explained.
The fundamental physics of musical instruments are also covered. Two
musical structures, a vibrating string and a drum membrane, are dis
cussed in detail, and the partial differential equations related to these
systems are derived.
Some former physical modeling methods are tackled in detail: the
finite difference, the digital waveguide, and the modal synthesis meth
ods. The theoretical basis of the modal synthesis method is elaborated
with care, because it is closely related to the new method. Finally,
this text compares the new method with the previous ones. The dig
ital waveguide and the finite difference method simulate vibrations in
the time domain. The functional transformation method essentially has
a frequency-domain point of view, just like the modal synthesis. All
these methods can be derived from the wave equation or its extensions,
and are thus just different viewpoints to the same physical reality. This
comparison shows clearly that the new method has properties different
from those of earlier methods. It is superior in many aspects while it
may be weaker in some others, which is typical to all methods. It can
be said that the functional transformation method is truly a novel and
interesting method for physical modeling of musirpJ instruments.
Espoo, Finland, April 6, 2003
Vesa Valimaki, professor of audio signal processing
Helsinki University of Technology
Department of Electrical and Communications Engineering
Laboratory of Acoustics and Audio Signal Processing
Espoo, Finland
List of Figures
3.1 Illustration of a spatially I-D initial-boundary-value
problem. 18
3.2 Construction of a guitar. 20
3.3 String vibration filtered at the bridge positions. 22
3.4 Strings terminated by separated impedance functions. 23
3.5 Strings terminated by an impedance network. 24
3.6 Construction of a kettle drum. 26
3.7 Forces on a string segment for the derivation of
longitudinal string vibrations. 31
3.8 Forces and torques on a string segment for the
derivation of torsional vibrations. 36
3.9 Forces and bending moments on a string segment
for the derivation of transversal string vibrations. 41
3.10 Bowing force for transversal ~~cl torsional excited
strings. 50
3.11 Forces and bending moments on a rectangular mem-
brane segment for the derivation of bending mem-
brane vibrations. 53
4.1 Illustration of a spatially I-D initial-boundary-value
problem discretized by FDM. 67
4.2 Dependencies of the new calculated grid point on
previous grid points in the FDM simulating the
transversal vibrating lossy string. 70
4.3 FDM simulation of a transversal vibrating lossy
and dispersive guitar string. 72
vii
viii DIGITAL SOUND SYNTHESIS USING THE FTM
4.4 Arrangement of the staggered grid points for FDM
simulations with CDA of the wave equation in vec-
tor form. 74
4.5 FDM simulation of a longitudinal vibrating guitar
string with boundary conditions of third kind. 75
4.6 Basic DWG stringed instrument model. 77
4.7 Basic DWG string model. 79
4.8 Illustration of a spatially I-D initial-boundary-value
problem simulated with the DWG. 81
4.9 DWG simulation of the guitar nylon 'B' string vi-
bration. 82
4.10 2-D rectangular DWM with scattering junctions
between both polarizations. 84
4.11 Analytically calculated frequencies and frequencies
used by the MS for a longitudinal vibrating guitar string. 88
4.12 Realization of the basic MS algorithm. 90
5.1 General procedure of the FTM solving initial-bound-
ary-value problems defined in form of PDEs. 101
5.2 Multidimensional transfer function model derived
from scalar PDEs. 111
5.3 Basic structure of the FTM simulations derived
from scalar PDEs. 117
5.4 Illustration of the spatially 1-D initial-boundary-
value problem simulated with the FTM. 118
5.5 Multidimensional transfer function model derived
from vector PDEs. 127
5.6 Basic structure of the FTM simulations derived
from vector PDEs. 129
5.7 General procedure of the FTM to solve initial-
boundary-value problems with nonlinear excitation
functions. 131
5.8 MD implicit equation derived from scalar PDEs
with a nonlinear excitation force. 134
5.9 Basic structure of FTM simulations derived from
scalar PDEs with a nonlinear excitation force. 137
5.10 lVID implicit equation derived from scalar PDEs
with solution-dependent coefficients. 142
5.11 Basic structure of FTM simulations derived from
scalar PDEs with solution-dependent coefficients. 143
Figures ix
5.12 FTM simulation of a transversal vibrating guitar string. 150
5.13 FTM simulation of a longitudinal vibrating guitar string. 154
5.14 FTM simulation of a longitudinal vibrating guitar
string with boundary conditions of third kind. 158
5.15 General procedure of the FTM to solve coupled
initial-boundary-value problems given in form of
two vector PDEs. 160
5.16 FTM simulation of two interconnected longitudi-
nal vibrating guitar strings. 162
5.17 Discrete system of a struck string with a piano hammer. 165
5.18 FTM simulation of a hammer-string interaction in
a piano. 166
5.19 Recursive system realization of one mode of the
transversal vibrating string. 168
5.20 FTM simulation of a slapped bass string. 170
5.21 Discrete system of a plucked string with inherent
tension-modulated nonlinearities. 173
5.22 FTM simulation of a tension-modulated vibrating string. 173
5.23 FTM simulation of the tension-modulated vibrat
ing string with slap force. 175
5.24 FTM simulation of reverberation plate vibrations. 179
5.25 FTM simulation of circular drum head vibrations. 181
5.26 Dimensions of a simplified cuboid piano body. 182
5.27 FTM simulation of impulse responses within a res
onant body of an upright piano. 185
6.1 Magnitude spectra of the simulated nylon 'B' gui-
tar string 192
6.2 Combination of the FTM ami the DWG. 201
6.3 Magnitude response of the loss filter for a DWG model. 203
6.4 Phase delay of the dispersion filter. 205
6.5 Deflection and spectrum of the example nylon gui
tar string, simulated with the FTM and with the
DWG. 207
List of Tables
3.1 Comparison of three simplifying models for the
subdivision of stringed instruments. 25
3.2 Parameters and variables used for the derivation of
PDEs describing longitudinal, torsional and transver-
sal string vibrations. 30
3.3 Parameters used for the derivation of PDEs de-
scribing bending membrane vibrations. 52
3.4 Parameters used for the derivation of PDEs de-
scribing resonant body vibrations. 58
3.5 Coefficients of the different initial-boundary-value
problems in the unified scalar notation. 61
4.1 Physical parameters of a typical nylon 'B' guitar string. 72
5.1 Physical parameters used for the simulation of the
hammer-string interaction in a piano and the fret-
string interaction in a slapped bass. 164
5.2 Physical parameters used for the simulation of the
vibrating quadratic reverberation plate and the cir-
cular drum head of a kettle drum. 177
5.3 Summary of the computational complexities of dif-
ferent systems simulated with FTM. 187
6.1 Computational cost of the DWG and the FTM
simulations for a nylon 'B' guitar string simulated
with Is = 44.1 kHz. 199
xi
List of symbols
Operators
x·y scalar product between x and y
s
* convolution with respect to the temporal frequency
variable s
t
* convolution with respect to time
0-
1 inverse operation of (.)
(. )d discretized function of (.)
OH
hermitian of (.)
OT
transposed of (.)
0* conjugate complex of (.)
\7 gradient or divergence operation
fb,B, gb,B, boundary operators for vector PDEs
f -
adjoint boundary operators for vector PDEs
b,B, gb,B
fb,s,B {}, gb,s,B {} boundary operators for scalar PDEs
fi initial operator for vector PDEs
ii,s {} initial operator for scalar PDEs
D first-order temporal differentiation
t
DXl first-order spatial differentiation
DO
temporal differential operator
Im{} imaginary part of a complex function
LNLO nonlinear scalar spatial differential operator
LwO scalar spatial differential operator
Lw,s{} self-adjoint scalar spatial differential operator
O(T) higher order terms (depending linearly on T)
ReO real part of a complex function
W{} scalar differential operator containing mixed deri
vatives
WDO part of W {} containing temporal derivatives
xiii