Wave Digital Filters

A Wave Digital Filter (WDF) [136] is a particular kind of digital filter based on physical modeling principles. Unlike most digital filter types, every delay element in a WDF can be interpreted physically as holding the current state of a mass or spring (or capacitor or inductor). WDFs can also be viewed as a particular kind of finite difference scheme having unusually good numerical properties [55]. (See Appendix N for an introduction to finite difference schemes.)

Wave digital filters were developed initially by Alfred Fettweis [134] in the late 1960s for digitizing lumped electrical circuits composed of inductors, capacitors, resistors, transformers, gyrators, circulators, and other elements of classical network theory [136]. The WDF approach is based on the traveling-wave formulation of lumped electrical elements introduced by Belevitch [33], and derived in §L.6.

A WDF is constructed by interconnecting simple discrete-time models of individual masses, springs, and dashpots (or inductors, capacitors, and resistors). The rules for interconnecting the elementary models are based on scattering theory (discussed in §H.8). This is a direct result of the fact that all signals explicitly computed may be physically interpreted as traveling wave components of physical variables.

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Subsections

• Wave Digital Elements
  • A Physical Derivation of Wave Digital Elements
    • Reflectance of a General Lumped Waveguide Termination
    • Reflectances of Elementary Impedances
    • Choosing Waveguide Impedance to Simplify Element Reflectance
    • Digitizing Elementary Reflectances via the Bilinear Transform
  • Summary of Wave Digital Elements
  • Wave Digital Mass
  • Wave Digital Spring
  • Wave Digital Dashpot
  • Limiting Cases
  • Unit Elements
  
  
• Adaptors for Wave Digital Elements
  • Two-Port Parallel Adaptor for Force Waves
    • Compatible Port Connections
  • General Parallel Adaptor for Force Waves
    • Alpha Parameters
    • Reflection Coefficient, Parallel Case
    • Physical Derivation of Reflection Coefficient
    • Reflection Free Port
  • Two-Port Series Adaptor for Force Waves
  • General Series Adaptor for Force Waves
    • Beta Parameters
    • Reflection Coefficient, Series Case
    • Physical Derivation of Series Reflection Coefficient
    • Series Reflection Free Port
  
  
• Wave Digital Modeling Examples
  • ``Piano hammer in flight''
    • Extracting Physical Quantities
  • Force Driving a Mass
    • A More Formal Derivation of the Wave Digital Force-Driven Mass
  • Force Driving a Spring against a Wall
  • Spring and Free Mass
  • Mass and Dashpot in Series
    • Checking the WDF against the Analog Equivalent Circuit
  • Wave Digital Mass-Spring Oscillator
    • Oscillation Frequency
    • Linearly Growing State Variables
    • Why this can't happen in reality
    • Energy-Preserving Parameter Changes (Mass-Spring Oscillator)
    • Exercises in Wave Digital Modeling

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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