# Wave Digital Filters

In this section, we discuss the basic concepts and strategies for using wave digital filtering techniques. For a more detailed and complete discussion, see the following wave digital filter link.

## Background

• 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.

## General Approach

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

1. Express all physical quantities, such as forces and velocities, in terms of traveling-wave components.

2. Digitize the resulting traveling-wave system using the bilinear transform.

3. Combine elementary components (mass, springs, ...) using scattering junctions defined by either series or parallel connections.

## A Physical'' Derivation

• 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 R0, 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 R0 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) = FR+(s) + FR-(s),

and

and solving for F-(s) in terms of F+(s) and FR-(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 R0 is then determined so as to simplify the reflectance expression. Setting R0 = m in Sm(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

Sm(z) = -z-1,

with the bilinear tranform constant c = 1.

## Fundamental WDF Elements

1. From the previous discussion, a WDF mass element has R0 = m and a digitized reflectance given by

Sm(z) = -z-1.

2. In a similar way, the discrete-time reflectance for a WDF spring with R0 = k is found to be

Sk(z) = z-1.

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

## Connecting Wave Digital Elements

• 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.