Real-time-capable Emulation of Nonlinear Analog Circuits by Means of Novel Wave Digital Filters

波浪数字滤波器的概念是由Alfred Fettweis（1926-2015）提出的，它提供了一种优雅的方法，以独特的方式将模拟电路和递归数字结构相互关联。其基本思想是将电路的每个电子元件单独离散，并仅基于其局部定义。然后，根据参考电路中模拟元件的接线，将得到的数字构建块连接起来，形成一个数字结构，即波数字滤波器。这种模拟电路的物理模型能够准确、实时地再现其动态特性。由于模块化的结构原则，一些独特的积极性质，如无源性，稳定性，鲁棒性和参考系统的拓扑结构被保留在相应的波数字滤波器。在这里，不是使用通常的基尔霍夫量电压和电流，而是利用波动变量来实现易于计算的信号流图。不幸的是，可直接转换为波数字结构的参考电路的集合是有限的：即使是非常简单的非线性网络或包含更复杂的电路，环形拓扑也不能通过经典波数字滤波器表示，因为引入了不可计算的无延迟环路。这个未解决的问题使得波数字概念在通用电路仿真领域的应用变得不可能，尽管它具有所有有利的特性。所提出的研究项目的目标是抵消这些实现问题，并允许基于波数字滤波器的实时和通用电路仿真。这里选择的科学方法是基于应用程序的一种新颖的迭代方法，该方法利用了波数字滤波器的收缩特性。这导致了完全模块化的数字结构，以非常高的精度模拟底层模拟参考结构。该方法的理论基础是如此丰富，以至于可以在数学上保证对所有类型的参考电路（例如所有二极管电路和任意拓扑）的收敛性。对于其他具有重大技术相关性的电路，例如高度非线性的晶体管电路，尚缺少类似的证明，因此需要在拟议项目的范围内进行开发。此外，还需要回答与高效实时结构设计相关的进一步问题。这包括通过构建或分析该方法相对于非常大的电路的缩放行为来最小化迭代次数的实现原则。预期的结果对科学界和实际应用都有很大的兴趣。

The concept of Wave Digital Filters, introduced by Alfred Fettweis (1926-2015), provides an elegant approach to interrelate analog circuits and recursive digital structures in a unique fashion. The underlying idea is to discretize each electric component of the circuit individually and solely based on its local definition. The resulting digital building blocks then are connected in accordance with the wiring of the analog components in the reference circuit, forming a digital structure, the Wave Digital Filter. This physical model of the analog circuit is able to recreate its dynamic properties accurately and in real-time. Due to the modular construction principles, a number of unique positive properties like passivity, stability, robustness and the topology of the reference system are preserved in the corresponding Wave Digital Filter. Here, instead of using the usual Kirchhoff quantities voltage and current, wave variables are utilized to achieve a readily computable signal flow graph. Unfortunately, the set of reference circuits that are directly translatable into a Wave Digital structure is limited significantly: Even very simple nonlinear networks or circuits containing more complex, ring-like topologies are not representable by means of classic Wave Digital Filters, as non-computable delay-free loops are introduced. This unsolved problem rendered the application of the Wave Digital concept to the field of generic circuit simulation impossible in spite of all its favourable properties.The objective of the proposed research project is to counteract these realization problems and to allow for a real-time capable and generic circuit simulation based on Wave Digital Filters. The scientific approach chosen here is based on a novel iteration method of the applicants that exploits the contractivity property of Wave Digital Filters. This results in completely modular digital structures which emulate the underlying analog reference structure with very high accuracy. The theoretic fundament of the method is so substantial that convergence can even be guaranteed mathematically for whole classes of reference circuits (e.g. all diode circuits and arbitrary topologies). For other circuits of great technical relevance, for example highly nonlinear transistor circuits, an analogous proof is still missing and therefore is to be developed within the scope of the proposed project. Additionally, further questions are to be answered which relate to the design of highly efficient real-time structures. This includes implementation principles that minimize the number of iterations by construction or analyses of the scaling behaviour of the approach with respect to very large circuits. The expected results are of great interest for both the scientific community and practical applications.