Section 2 gives a generalized formulation of waveguide models: After a short introduction in classical terms, we consider the -component dual variables and , which can be associated with any physical wave quantities such as acoustic pressure and volume velocity, or voltage and current. This discrete-time multivariable formulation, while leading to results similar to those in classical electrical network theory [4], provides a general framework giving a unified treatment of various modeling problems.

In Section 3, we define the lossless junction of waveguide sections in terms of energy conservation. Such a formulation yields a class of lossless scattering junctions larger than that arising from the set of all physically meaningful scattering junctions.

Section 4 discusses power normalization of waveguide variables, and the results are applicable to both physical and non-physical scattering junctions.

In Section 5, we summarize formulas for physical, normalized, and unnormalized scattering junctions of -variable waveguide branches. We show how these junctions can be loaded with a lumped network and present an example in acoustic modeling.

In Section 6, using the physical interpretation of the computations, we give conditions for ensuring passivity of nonlinear and time-varying DWNs.

In Section 7, we present an algebraic analysis of the lossless junction, and the fundamental condition of losslessness is translated into the domain of eigenvalues and eigenvectors of the scattering matrix.

Section 8 shows how physical junctions can be implemented efficiently, both in the normalized and unnormalized cases. As a byproduct, two new three-multiply, three-add normalized lattice sections are derived. It is shown that the proposed implementations are robust with respect to coefficient quantization, and dynamic range requirements are addressed.

Section 9 considers an abstract geometric (as opposed to physically geometric) interpretation of physical scattering computations. The normalized scattering matrix is shown to be equivalent to a Householder reflection, while the unnormalized case can be seen as an oblique Householder reflection. Householder matrices are highly valued in numerical analysis because they conserve numerical dynamic range and thereby yield robust algorithms [34]. As a byproduct of this viewpoint, another new three-multiply, three-add normalized lattice section is derived which is especially well suited for implementation on general-purpose digital signal processing chips.

For completeness, Section 10 briefly covers the extension of lossless scattering and junction normalization to the case in which scattering matrices may contain complex rational transfer function elements. Such cases arise in the computational modeling of acoustic systems such as intersecting conical tubes and woodwind fingerholes. Junctions loaded by an arbitrary lumped impedance (e.g., a Helmholtz resonator mouthpiece model or complex violin bridge impedance), can also be handled this way.

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Center for Computer Research in Music and Acoustics (CCRMA), Stanford University

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