- Background
- General Approach
- A ``Physical'' Derivation
- Fundamental WDF Elements
- Connecting Wave Digital Elements

- Wave digital filters (WDF) were initially developed by Alfred Fettweis in the late 1960s for digitizing lumped electrical circuits.
- Wave digital filter techniques involve the use of traveling-wave variables in the context of lumped system analysis.
- Very simple expressions can be developed for various physical components, such as masses, springs, and dashpots. These components can then be interconnected using adaptors, which are scattering junction interfaces.
- WDF techniques are convenient when trying to eliminate delay-free loops because of the flexibility available in defining system wave impedances.

A WDF model of a system can be developed from its continuous-time, differential equation representation as follows:

- Express all physical quantities, such as forces and velocities, in terms of traveling-wave components.
- Digitize the resulting traveling-wave system using the bilinear transform.
- Combine elementary components (mass, springs, ...) using scattering junctions defined by either series or parallel connections.

- A physical component, such as a mass or spring, or an electrical component, such as an inductor or capacitor, is considered to be represented by an impedance
*R*(*s*). - Attached to this impedance is a waveguide of infinitesimal length having wave impedance
*R*_{0}, as diagrammed in Fig. 15. - The interface to the element is described in terms of a scattering junction, as illustrated in Fig. 16, where physical variables of force (and velocity) are decomposed into traveling-wave components.
- The waveguide impedance
*R*_{0}is arbitrary because it has been introduced only to facilitate the interconnection of fundamental WDF elements. - At the junction, incoming velocities (or currents) must sum to zero. Forces (or voltages) on either side of the junction must be equal. These two properties define a parallel junction.
- The WDF element can now be represented in terms of a reflectance as seen by the infinitesimal waveguide section:

This relationship can be found by expressing the junction properties above in terms of traveling-wave components,

*F*^{+}(*s*) +*F*^{-}(*s*) =*F*_{R}^{+}(*s*) +*F*_{R}^{-}(*s*),

and

and solving for*F*^{-}(*s*) in terms of*F*^{+}(*s*) and*F*_{R}^{-}(*s*). - As an example, the driving-point impedance of a mass is given by
*R*(*s*) =*m s*, which implies a reflectance of

- The arbitrary waveguide impedance
*R*_{0}is then determined so as to simplify the reflectance expression. Setting*R*_{0}=*m*in*S*_{m}(*s*) results in

- Finally, the reflectance expression is digitized using the bilinear transform by making the substitution

- For the case of the WDF mass element, this reduces to

*S*_{m}(*z*) = -*z*^{-1},

with the bilinear tranform constant*c*= 1.

- From the previous discussion, a WDF mass element has
*R*_{0}=*m*and a digitized reflectance given by

*S*_{m}(*z*) = -*z*^{-1}.

- In a similar way, the discrete-time reflectance for a WDF spring with
*R*_{0}=*k*is found to be

*S*_{k}(*z*) =*z*^{-1}.

- A WDF dashpot (or resistance) is defined to have and a reflectance

- WDF elements are interconnected using parallel or series adaptors.
- Parallel adaptors are used for physical connections where forces are equal and velocities sum to zero.
- Series adaptors are used for physical connections where velocities are equal and forces sum to zero.
- The details of these interconnections will not be covered here, but the interested reader is referred to Julius Smith's complete wave digital filter analysis.

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