A scattering junction is said to be lossless if
i.e., the active power absorbed at the junction is zero. In the case
of lossless propagation in a passive medium, it is easy to verify
from (19), (23) and (24) that
this is ensured if and only if
When wave propagation occurs in a medium which is passive but not
necessarily lossless, the condition for lossless scattering becomes
Notice that (26) is closely related to the well-known
Lyapunov equation  for testing the asymptotic stability of discrete-time linear systems.
The definition of lossless junctions employed here includes junctions arising from the connection of physical waveguides, and extends the formal treatment to non-physical junctions as well. In certain applications, it can be useful to use non-physical scattering matrices having a particular structure which increases the computational efficiency, or which gives the desired behavior of the system [78,79].
In DWNs, waveguide sections separating scattering junctions are normally implemented using delay lines. Since in any physical simulation there will be delay in both directions, the delay lines isolate the scattering junctions, allowing parallel computation of the junctions. This is an advantage over traditional ladder/lattice filters in which delay elements are present in only one flow direction [57,92], thus preventing both a direct pipelined computation and a physical interpretation. On the other hand, having the delays along only one direction is easier to work with when the problem is to realize a ladder/lattice filter having a prescribed transfer function [57,112].