Digital Sound Synthesis by Physical Modeling Using the Functional Transformation Method (review)

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 ad-dressed 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 initialboundary 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...