Modal Representations and Computational Complexity in Physical Modeling Sound Synthesis Applications

Modal representations have been used successfully in the context of musical sound synthesis based on linear and time-invariant physical models. It is also particularly simple, from such a representation, to deduce estimates of computational complexity in a discrete time simulation from modal density: the operation count, as well as memory requirements may be related directly to the audio sample rate and the defining parameters of the model under consideration. It is intuitively obvious, but interesting nonetheless to note that for general LTI systems, these estimates hold not merely for modal representations, but for any numerical method, including time domain techniques: there are fundamental limits on how much computational work is required to simulate a given instrument, and one should not expect to be able to do better. Special cases, including digital waveguides, are discussed in this light. Questions are raised regarding the possibility of deducing similar performance bounds using nonlinear modal representations. INTRODUCTION Sound synthesis based on distributed physical models of musical instruments has given rise to a number of different frameworks. Among these are lumped networks of masses and springs [1], modal decompositions [2], and time-domain numerical methods [3], including those based around the use of scattering structures such as digital waveguides [4] and wave digital filters [5]. All of these methods are, at the most basic level, attempts at a numerical simulation of an underlying model problem described by a set of partial differential equations (PDEs). These methods are of varying degrees of generality: modal synthesis methods, for example, are best suited to linear problems, and waveguide methods to the special but very important case of the 1D wave equation---time domain methods and lumped networks are capable of handling more generally nonlinear systems as well, though the problem of numerical stability can become quite severe. It is sometimes claimed in the literature [6] that, for a given underlying system, one method is more efficient than another, in terms of an over-all arithmetic operation count, or memory usage. Though such claims can be valid with regard to special cases (which will be discussed in this paper), one might expect that for general systems, there are fundamental limits on computational cost which are governed by the system parameters and geometry alone, as well as the audio sample rate. It is difficult to arrive at such bounds in the general case of nonlinear systems, but for linear systems, of great interest in musical acoustics, it is possible to garner much important information through an examination of modal distributions. DISTRIBUTED MODELS IN MUSICAL ACOUSTICS Of the linear distributed systems which appear in musical acoustics, by far the most important family is that given by ( ) 0 2 2 2 2 = ∇ − + ∂ ∂ u t u p d γ (Eq. 1)