On systematic wave digital modeling of passive hyperbolic partial differential equations

Abstract

Wave digital models are known to be most suitable for the digital emulation of passive physical systems. Starting from a system description in the form of a suited partial differential equation (PDE), a reference circuit having a one-to-one corresponding wave digital structure has to be synthesized. This synthesis transfers the principle of action at proximity and the passivity of the physical system into algorithms, which offer full parallelism, passivity, and robustness against signal and coefficient quantizations. Finding a reference circuit to a given PDE is an open problem, which has only partially been solved. In this paper, we present a schematic reference circuit synthesis for a special class of hyperbolic PDEs. In order to cut down the complexity of the synthesis, we introduce additional design parameters and decompose the PDE into an ordinary differential equation (ODE) and hyperbolic PDE subsystems. The parameters can be exploited to optimize the reference circuit and its corresponding wave digital structure. The method is applied to the Sine-Gordon equation from the soliton theory and verified by simulation. Figure 1. Steps in wave digital emulation.In the first step, the physical system has to be represented by a mathematical model in the form of passive hyperbolic partial differential equations (PDEs), which reflect its essential characteristics, passivity, and principle of action at proximity. These properties are transferred into a multidimensional (MD) electrical reference circuit, representing the PDEs (e.g. in its loop or node equations). Replacing every circuit element with its wave digital correspondence, one obtains a wave digital structure free of delay-free directed loops. An implementation of this structure yields the resulting wave digital algorithm. During the whole synthesis, the passivity and exclusively local nature of the underlying physical phenomena are retained and finally transferred to the algorithm, which is passive and robust against signal and coefficient quantizations and allows for massive parallelism.The crux of this procedure is the synthesis of a reference circuit, which demands profound background knowledge of the wave digital concept, as well as some essential network theory from the developer. Hence, a schematic approach that allows for computer-aided circuit syntheses is desirable. A reference circuit synthesis based on admittance matrices has been presented in [3]. However, this approach is restricted to linear constant PDEs and steady-state solutions only. In contrast, the syntheses of electrical circuits presented in [4, 5] consider ordinary differential equations (ODEs) only, but allow for the modeling of certain nonlinearities [4]. A first extension of this approach to PDEs is derived in [6], but the resulting circuits are usually unsuited as reference circuits for wave digital modeling and an elaborate post-processing is necessary. For some established examples, this post-processing has manually been performed in [7]. Under further restrictions to the PDEs, fi...