Synthesizing Natural Sounds Using Dynamic Models of Sound Attractors

Since the introduction of digital signal processing techniques, many approaches to the modeling and synthesis of natural sounds have been developed (Roads 1995; Tolonen, Välimäki, and Karjalainen 1998). Nevertheless, it is useful to investigate new approaches to the problem, because each approach can potentially enlarge the repertoire of sound processing methods. This article proposes a new kind of sound model based on the concept of system attractors and a method called dynamic modeling. Originally, this method had been developed to address the task of modeling chaotic systems given only a time series of samples from the output of the system (Principe, Rathie, and Kuo 1992; Haykin and Principe 1998). The basic idea that simple nonlinear dynamic systems can produce complex behavior dates back to Poincaré. However, the analytical investigation of nonlinear systems is complicated—even for very simple cases—and a clear understanding of the basic mechanisms that produce the complexity of nonlinear systems was established only after it was recognized that these systems can be investigated using computer models. Soon after initial investigations (Lorenz 1963), it was discovered that a new type of systems behavior existed: chaotic systems.

Chaotic systems are special cases of nonlinear dynamic systems, which are generally described in terms of state space models. A introduction of the basic relations of state space model is given in the following section. Note, however, that state space models are applicable to all kinds of physical systems and are widely used for modeling linear systems (Bay 1999). One result of the research that emerged from investigations of chaotic systems and that is of primary importance for the following application is the Reconstruction Theorem. This states that the behavior of a dynamic system in its state space can be reconstructed from a scalar time series of an observable variable of the system (Takens 1981; Sauer, Yorke, and Casdagli 1991). Obviously, this reconstruction covers only the subset of the state space and the parameter settings during the signal generation. For fixed system parameters, the subset generally consists of an attractor. After reconstructing the attractor, it can be used to train an adaptive system to copy the attractor dynamics of the original system. Despite the original intention, this approach to dynamic modeling is not restricted to stationary and chaotic dynamics but can be applied to model sound dynamics as well (Röbel 1993).

As will be described in more detail later, the proposed model can be interpreted as a nonlinear extension of the autoregressive filter of the subtractive synthesis model (Markel and Gray 1976). In order for the autoregressive filter to be stable, subtractive synthesis requires an excitation signal, and stable self-sustained oscillation cannot be achieved. With a nonlinear filter, however, the autoregressive model can be used as an oscillator. Here, the aforementioned relationship to the state space of the system yields an interpretation of the model as a dynamic system operating in a reconstruction of the state space of the musical instrument. In contrast to a recent physical modeling application of the theory of nonlinear dynamic systems (Rodet and Vergez 1999a, 1999b) that is intended to derive a basic functional model of musical instruments using simple nonlinear delay loop systems, dynamic modeling is a "black box" technique. Without special preparation, it does not provide any meaningful control parameters that can be used to interact with the model. Owing to the high level of abstraction necessary for the physical interpretation of the model, it cannot be used to understand the physical process of sound generation. Nevertheless, attractors are potential objects for signal description, similar to partials, but with a broader range of applications that even extends to chaotic systems. A related approach that employs simple synthetic attractors for sound synthesis has been proposed recently and independently (Monro 1993). However, the synthesis model proposed in that investigation is not intended to be used to model natural sounds. In another investigation, the reconstruction of sound attractors has been used for sound visualization (Monro and Pressing 1998).

This article is organized as follows. The first section provides a short review of the mathematical concepts of the theory...