Equivalence of Digital Waveguide and Finite Difference Schemes

It was shown in §H.4.3 that the digital waveguide (DW) model for the ideal vibrating string performs the same ``state transitions'' as the more standard finite-difference time-domain (FDTD) scheme (also known as the ``leapfrog'' recursion). This appendix, initially published in [455], further establishes that the solution spaces of the two schemes are isomorphic. That is, a linear, one-to-one transformation is derived which converts any point in the state-space of one scheme to a unique point in the other scheme. Since boundary conditions and initial values are more intuitively transparent in the DW formulation, the simple means of converting back and forth can be useful in initializing and constructing boundaries for FDTD simulations, as we will see.

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Subsections

• Introduction
  • Finite Difference Time Domain (FDTD) Scheme
  • Digital Waveguide (DW) Scheme
  • FDTD and DW Equivalence
  
  
• State Transformations
• Excitation Examples
  • Localized Displacement Excitations
  • Localized Velocity Excitations
  • More General Velocity Excitations
  • Additive Inputs
  • Interpretation of the Time-Domain KW Converter
  
  
• State Space Formulation
  • FDTD State Space Model
  • DW State Space Model
    • DW Displacement Inputs
    • DW Non-Displacement Inputs
    • Input Locality
  • Boundary Conditions
    • Resistive Terminations
    • Boundary Conditions as Perturbations
    • Reactive Terminations
    • Interior Scattering Junctions
  • Lossy Vibration
  • State Space Summary
  
  
• Computational Complexity
• Summary
• Future Work
• Acknowledgments

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.