Wave Digital Filters

A *Wave Digital Filter* (WDF) [137] 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 D for an
introduction to finite difference schemes.) WDFs have been applied
often in music signal processing
[397,341,557,364,351,558,556,56,529,525,486].

Wave digital filters were developed initially by Alfred Fettweis
[136] in the late 1960s for digitizing lumped
*electrical* circuits composed of inductors, capacitors,
resistors, transformers, gyrators, circulators, and other elements of
classical network theory [137]. The WDF approach is
based on the traveling-wave formulation of lumped electrical elements
introduced by Belevitch [34].

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 §C.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.

- Wave Digital Elements
- A Physical Derivation of Wave Digital Elements
- 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
- General Parallel Adaptor for Force Waves
- Two-Port Series Adaptor for Force Waves
- General Series Adaptor for Force Waves

- Wave Digital Modeling Examples
- ``Piano hammer in flight''
- Force Driving a Mass
- Force Driving a Spring against a Wall
- Spring and Free Mass
- Mass and Dashpot in Series
- Wave Digital Mass-Spring Oscillator
- Oscillation Frequency
- DC Analysis of the WD Mass-Spring Oscillator
- WD Mass-Spring Oscillator at Half the Sampling Rate
- Linearly Growing State Variables in WD Mass-Spring Oscillator
- A Signal Processing Perspective on Repeated Mass-Spring Poles
- Physical Perspective on Repeated Poles in Mass-Spring System
- Mass-Spring Boundedness in Reality
- Energy-Preserving Parameter Changes (Mass-Spring Oscillator)
- Exercises in Wave Digital Modeling

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