``Traveling Waves'' in Lumped Systems
One of the topics in classical network theory is the reflection and transmission, or scattering formulation for lumped networks . Lumped scattering theory also serves as the starting point for deriving wave digital filters (the subject of Appendix F). In this formulation, forces (voltages) and velocities (currents) are replaced by so-called wave variables
Dropping the time argument, since it is always `(t)', we see that
These are the basic relations for traveling waves in an ideal medium such as an ideal vibrating string or acoustic tube. Using voltage and current gives elementary transmission line theory.
Reflectance of an Impedance
Let denote the driving-point impedance of an arbitrary
continuous-time LTI system. Then, by definition,
where and denote the Laplace transforms
of the applied force and resulting velocity, respectively.
The wave variable decomposition in Eq.(C.74) gives
We may call the reflectance of impedance relative to . For example, if a transmission line with characteristic impedance were terminated in a lumped impedance , the reflection transfer function at the termination, as seen from the end of the transmission line, would be .
Properties of Passive Impedances
Digital Waveguide Filters