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
- 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.
- From the previous discussion, a WDF mass element has R0 = m and a digitized reflectance given by
Sm(z) = -z-1.
- In a similar way, the discrete-time reflectance for a WDF spring with R0 = k is found to be
Sk(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
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