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Chapter 2: Wave Digital Filters

Chapter 2 is intended as a review of lumped wave digital filters, minus any discussion of filtering applications, since we will only be looking at simulation applications in the remainder of the thesis. Because these concepts are used extensively throughout the sequel, the reader is advised to begin here, even though discussion of numerical methods for PDE solving does not begin in earnest until the next chapter. We follow the standard development (as in, say, Fettweis's comprehensive review paper [46] which is the chief reference for this chapter) and begin with a brief introduction to the theory of electrical -port devices [12], and, in particular, the key concept of *passivity*, which later plays a pivotal role as the stability criterion for multidimensional simulation networks. We then review the basics of the lumped wave digital discretization procedure, involving the use of a passivity-preserving continuous-to-discrete spectral bilinear transformation (the trapezoid rule in the time domain) and the transformation to wave variables. The wave digital counterparts of the standard circuit elements (capacitors, inductors, resistors, transformers, etc.) are then introduced, as are adaptors, which are simply the wave variable counterparts to Kirchoff's series and parallel connection rules. The chapter is concluded with a brief description of finite word-length arithmetic properties of WDFs, and a look at some specialized vector elements that will later come in handy (and are in fact necessary) for the simulation of some elastic dynamic systems. It is important to keep in mind that though we only discuss lumped elements and networks in Chapter 2, the basic set of construction rules (essentially classical electrical network theory) remains unchanged when we move to a multidimensional setting in the next chapter.

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Stefan Bilbao
2002-01-22