|Vol. 29 Issue 2 Reviews||Reviews > Publications >|
|Lutz Trautmann and Rudolf Rabenstein: Digital Sound Synthesis by Physical Modeling Using the Functional Transformation Method|
Hardcover, 2003, ISBN 0-306-47875-7, 226 pages; Kluwer Academic/Plenum
Publishers, New York, 233 Springer Street, New York, New York 10013, USA;
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by Bob L. Sturm
This book is derived from over 20 articles written by Lutz Trautmann and Rudolf Rabenstein on a new physical modeling technique they term functional transformation method (FTM). Rather than assembling these articles into one volume, the authors have written a textbook that provides an excellent presentation of the FTM, as well as the physics of musical instruments, and current physical modeling techniques. Far from being an introduction to physical modeling, this text becomes very dense after the third chapter. But with patience, and much review of partial differential equations (PDEs) and transforms, I have found this text to be very rewarding and the FTM exciting.
The book begins with a brief chapter about basic sound synthesis techniques that have been used to simulate existing musical instruments. These include wavetable, granular, additive, and subtractive synthesis, as well as frequency modulation. The authors say all these fundamental methods tend to simulate directly the perceived sound rather than the sound production mechanisms of real vibrating structures (p. 13). And thus comes the drive for constructing a physical model to simulate the acoustics rather than the perceived sound of an instrument.
The book continues with a chapter on the physics of musical instruments, of which strings and membranes are covered. The musical instrument is presented as a combination of excitation and resonator, the coupling between which depends on the accuracy needed. First addressed are terminated strings, either fixed or coupled with impedances. Then the kettledrum is presented. The final section of this chapter derives in-depth physical descriptions of the string, membrane, and resonant bodies. To get through this chapter required more than a cursory review of The Physics of Musical Instruments, by Neville H. Fletcher and Thomas D. Rossing (New York: Springer-Verlag, 1998). The authors use of vector PDEs to describe the systems, though elegant and concise, makes scratch paper a necessity.
Having derived very complete systems of PDEs describing these instruments, to implement them digitally the equations need to be discretized. Chapter 4 addresses several methods for doing so, including finite difference methods (FDM) and digital waveguides (DWG) in the time domain, and modal synthesis (MS) in the frequency domain. First, the FDM is used to discretize scalar and vector PDEs for transverse vibrating dispersive strings. Next the DWG method is demonstrated for lossy and non-lossy strings. The DWG is then extended to two dimensions for simulating a membrane. The final section deals with MS, a frequency-based approach to modeling, and the method most related to FTM.
The fifth chapter, the largest and most important in the book, presents the FTM as a direct solution of the initial-boundary-value problem, described by the system of PDEs derived earlier. Because the FTM provides an analytical solution, the need to discretize or approximate a system of PDEs is circumvented. Similar to how a Laplace transform can turn an ordinary differential equation into a simple algebraic relation with no discretization, the FTM uses the Laplace and Sturm-Liouville transforms (no relation to the author!) to turn PDEs into algebraic equations.
The FTM process is as follows. A Laplace transform is performed on the initial-boundary-value problem to remove the temporal derivatives from the system, and include the initial conditions as additive terms. Next the spatial derivatives are removed, and the boundary conditions are included, using a Sturm-Liouville transform. This is far from a trivial operation, where the transformation kernel depends on the PDE. These transformations turn the initial-boundary-value problem into a multi-dimensional transfer function model.
As an example, the following generalized system described by equations 1–3, is derived from the physics of the model, and its initial and boundary conditions.
Equation 1 presents the non-homogenous PDE with operators for time , spatial , and mixed derivatives , and an excitation function , which could be non-linear. Equation 2 gives the initial conditions (e.g. the pluck shape of a string), and equation 3 provides the boundary conditions (e.g. non-rigid termination). Through the transformation detailed above, this system of equations becomes:
where the transformed output , is just a sum of transfer functions, , multiplying the transformed inputs—excitation force , initial conditions , and boundary conditions . This multi-dimensional transfer function model thus embodies equations 1–3 as an algebraic relationship.
It is from this point that the equation is discretized from the s-domain to the z-domain. The transfer functions become second-order resonant filters and embody the precise modal characteristics of the system. A spatial discretization is performed when taking the inverse Sturm-Liouville transform at a point of interest, for example the pickup position of a guitar. An important point to remember is that any position on the structure can be used, which is not a possibility with other synthesis methods without significantly altering the model (e.g. spatial resolution of MS). At the end of this step we are left with a transfer function model in the z-domain, which then requires an inverse z-transform to return to the time-domain, and hopefully a signal that sounds like the modeled instrument!
The final portion of this chapter deals with the application of the FTM to many different systems: 1-dimensional vibrating strings coupled and excited in various ways, plates and membranes of simple shapes, and three-dimensional resonant bodies. It is interesting to note that to realize each of these systems the number of modes necessary increases exponentially as the spatial dimensions increase. For the 1-dimensional string simulated at a sampling rate of 44.1 kHz, around 100 modes are sufficient (p. 151); for the 2-dimensional vibrating plate over 1,000 modes are needed (p. 178); and for a 3-dimensional resonant body 460,000 modes are needed to cover all possible frequencies (p. 185)!
The final chapters compare the FTM with other physical modeling methods. It is shown that the spectrum of a nylon guitar string simulated using FDM is very different from that simulated using FTM. The discretization of the FDM (via truncated Taylor series) causes high errors throughout the spectrum, especially in the high frequencies. The FTM however simulates all modes precisely, as well as the dispersion effects. Though the DWG method gives the most efficient implementation of a 1-dimensional vibrating string, it not only approximates the modes, but also cannot distinguish between inner and external losses. Using the FTM energy dispersion can come from inner losses (e.g. elasticity), damping due to the air, and non-ideal string termination. Furthermore, the accuracy of the DWG model depends on the filter order being used; for more accurate higher partials the filter orders must be increased. For FTM the highest order filter necessary is second order for each mode; only a higher number of simulated partials need to be simulated for better accuracy.
The method closest to the FTM is MS, since both techniques operate in the frequency domain. The difference between the two resides in the approach to the model. It has been shown that the FTM provides a continuous description of a system. MS, however, models a system using not only springs and masses that approximate the shape and interconnections of an object, but also using a limited number of measured vibrations from a specimen. Though MS might create a model more similar to the one measured, it provides no way of adjusting its physical parameters. For instance a Tibetan prayer bowl modeled with MS cannot be transformed into steel, filled with water, or changed in shape and size. This is completely possible with FTM, which I believe is its most attractive feature.
There are a few things that are presently hindrances to FTM. Because the entire approach is based on a mathematical model, it is restricted to shapes that are relatively simple, such as strings, tubes, and rectangular or circular plates. Basically any shape that can be described mathematically is an option. Another drawback is its computational load. It is painfully clear that real-time FTM synthesis is currently possible only for 1-dimensional systems. In the future, this will not be a limitation. To create an accurate physical model with the lowest computational load the authors suggest combining the FTM with the DWG method. This amounts to little more than using FTM to design filters to use with the DWG method. In essence it is a DWG model with parameters derived using the FTM.
Only so far into the book can one read without wondering how the FTM sounds. The authors have provided some examples on the Web. Mr. Trautmann has written a Java applet that demonstrates the FTM for a vibrating string (www.nt.e-technik.uni-erlangen.de/~traut/SoundSynth/Applet.html; ability to read German will prove helpful). Many parameters can be changed including length, pickup position, number of simulated modes, material, damping effects, and tension. More exciting examples are available as demonstrations of a new plug-in (www.creamware.de/index.php?lang=eng&seite=noahexfinal; demonstrations are available at the bottom). The Spanish guitar example sounds at first synthetic, but when excited by strumming becomes quite realistic. The only thing missing is of course the extrinsic sound of the fingers on the textured strings.
In conclusion, this book provides a thorough presentation of the FTM. The authors give many examples and comparisons, but limited by space are restricted to terse derivations. Due to the dense and highly technical nature of the book, I do not recommend it to someone who is not at least acquainted with PDEs. Otherwise this text provides an excellent presentation of FTM. It will remain an important part of my library.