Algebraic Properties of Lossless Junctions

Comparing (38) and (42) shows that the elements of the
normalized scattering matrix
in (42) are

(59) |

An elementary eigenvector analysis can be conducted using physical analogies. It is well known that a symmetric matrix has orthogonal eigenvectors. For the equal-impedance case, one eigenvector is always by symmetry: this corresponds to a collision of equal pressure waves at the junction, so the return scatter must be identical. This corresponds to the eigenvalue 1. For the -1 eigenvalues, a similar interpretation can be found: inject a unit pressure wave at all the branches but one, and ``pull out'' a pressure wave having magnitude at the remaining branch. In this case, the return scatter is inverted, since we have arranged that the pressure be zero at the junction and hence at each branch termination. In this way we can find eigenvectors analogous with which span the ()-dimensional subspace associated with the eigenvalue -1. Note that is orthogonal to this subspace, while the are not mutually orthogonal for .

For unequal impedances, a similar physical interpretation can be found for the eigenvectors. If we supply equal pressure waves to all branches at the junction, the reflected waves must be equal by symmetry, since , where is the junction pressure and all the are equal. Hence, remains an eigenvector corresponding to the eigenvalue 1. On the other hand, if we inject a unit pressure wave into all the branches but the th and ``pull out'' a pressure wave having magnitude at the th branch, then the junction pressure is again forced to zero by construction and the return scatter at any branch is the negative of the incoming wave on that branch. In this way we can find eigenvectors analogous with spanning the ()-dimensional subspace associated with the eigenvalue -1. In this case, none of the eigenvectors is necessarily orthogonal to the others.

The foregoing is an example of how physical intuition can help in finding algebraic properties of a given matrix in physical applications.

Another property of the scattering matrix
is that it is its
own inverse:
. This corresponds physically to the fact
that if the results of a scattering operation are fed back to the same
junction as incoming waves, the result must be the inverse of the
original scattering. An implication of this is that lossless
scattering networks can be run in *reverse,* i.e., by changing the
directions of all the delay lines and computing the junctions as
dictated by the wave impedances, the network will compute its own
inverse. If there are inputs and outputs, they must be interchanged.

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