Generalized Lossless Scattering

A scattering junction is said to be lossless if ${\cal P}^+= {\cal P}^-$, 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

$${\mbox{\boldmath$\Gamma$}}(z) = {\mbox{\boldmath$A$}}^* {\mbox{\boldmath$\Gamma$}}(z) {\mbox{\boldmath$A$}}$$ (26)

where ${\mbox{\boldmath$\Gamma$}}(z)$ is a generalized wave admittance which is para-Hermitian and analytic for $\vert z\vert>1$⁶. In this case the matrix ${\mbox{\boldmath$A$}}$ is said to be a lossless scattering matrix. We refer to (26) as the condition of losslessness.

When wave propagation occurs in a medium which is passive but not necessarily lossless, the condition for lossless scattering becomes

$${\mbox{\boldmath$\Gamma$}}^*(1/z^*) = {\mbox{\boldmath$A$}}^*{\mbox{\boldmath$\Gamma$}}(z) {\mbox{\boldmath$A$}}$$ (27)

Notice that (26) is closely related to the well-known

Lyapunov equation [43] 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].