Conditions for Lossless Scattering

We now present a theorem characterizing the condition of losslessness in terms of the eigenvalues and eigenvectors of the scattering matrix.

Theorem 3   A scattering matrix ${\mbox{\boldmath$A$}}\in {\cal C}^{N \times N}$ is lossless if and only if its eigenvalues lie on the unit circle and it admits a basis of $N$ linearly independent eigenvectors.

Proof: By definition (26), ${\mbox{\boldmath$A$}}$ is lossless if ${\mbox{\boldmath$A$}}^*{\mbox{\boldmath$\Gamma$}}{\mbox{\boldmath$A$}}= {\mbox{\boldmath$\Gamma$}}$, where ${\mbox{\boldmath$\Gamma$}}$ is a positive definite matrix. Therefore, ${\mbox{\boldmath$\Gamma$}}$ admits a Cholesky factorization ${\mbox{\boldmath$\Gamma$}}= {\mbox{\boldmath$U$}}^*{\mbox{\boldmath$U$}}$ where ${\mbox{\boldmath$U$}}$ is an upper triangular matrix which converts ${\mbox{\boldmath$A$}}$ to a unitary matrix via similarity transformation: ${\mbox{\boldmath$A$}}^*{\mbox{\boldmath$\Gamma$}}{\mbox{\boldmath$A$}}= {\mbox{... ... {{\mbox{\boldmath$A$}}}}^*{\tilde {\mbox{\boldmath$A$}}}={\mbox{\boldmath$I$}}$, where

$${\tilde {\mbox{\boldmath$A$}}}={\mbox{\boldmath$U$}}{\mbox{\boldmath$A$}}{\mbox{\boldmath$U$}}^{-1}$$ (60)

This shows the eigenvalues of every lossless scattering matrix lie on the unit circle. It readily follows from similarity to ${\tilde {\mbox{\boldmath$A$}}}$ that ${\mbox{\boldmath$A$}}$ admits $N$ linearly independent eigenvectors.

Conversely, assume $\vert\lambda_i\vert = 1$ for each eigenvalue $\lambda_i$ of ${\mbox{\boldmath$A$}}$, and that there exists a matrix ${\mbox{\boldmath$T$}}$ of linearly independent eigenvectors of ${{\mbox{\boldmath$A$}}}$. Then the matrix ${\mbox{\boldmath$T$}}$ diagonalizes ${\mbox{\boldmath$A$}}$ to give ${{\mbox{\boldmath$T$}}}^{-1}{\mbox{\boldmath$A$}}{\mbox{\boldmath$T$}}= {{\mbox... ...boldmath$A$}}^{*}{{{\mbox{\boldmath$T$}}}^{-1}}^{*} = {{\mbox{\boldmath$D$}}}^*$, where ${{\mbox{\boldmath$D$}}} = \hbox{diag}(\lambda_1,\dots,\lambda_N)$. Multiplying, we obtain ${\mbox{\boldmath$T$}}^{*}{{\mbox{\boldmath$A$}}}^{*}{{\mbox{\boldmath$T$}}^{-1}... ...x{\boldmath$A$}}= {{{\mbox{\boldmath$T$}}}^{-1}}^{*}{\mbox{\boldmath$T$}}^{-1}$. Thus, the condition of lossless scattering (26) is satisfied for ${\mbox{\boldmath$\Gamma$}}={{\mbox{\boldmath$T$}}^{-1}}^{*}{\mbox{\boldmath$T$}}^{-1}$ which is Hermitian and positive definite.

Theorem 3 can be extended to lossless junctions of lossy waveguides, as the reader can easily verify by applying the above proof to (27).

It is worth noting that most research in feedback delay networks for artificial reverberation has been concerned only with orthogonal feedback matrices [42,33] rather than on the more general class of matrices satisfying the losslessness condition (26) [78,106,79]. This is an excessive restriction since many of the matrices naturally arising from models of lossless physical junctions are not orthogonal. As can be seen above, physical scattering matrices are orthogonal only if we restrict attention to propagation in equal-admittance media [ ${\mbox{\boldmath$\Gamma$}}(z)={\mbox{\boldmath$I$}}$], or to propagation of normalized waves.