Non-constant scattering matrices and applications

The formulation of sections 3 and 4 can be restated for the more general case in which the scattering matrix has elements which are functions of $z$.

In this case the scattering relationship is

$${\mbox{\boldmath$p$}}^-(z)={\mbox{\boldmath$A$}}(z) {\mbox{\boldmath$p$}}^+(z)$$ (86)

In the case of lossless propagation, the scattering junction is said to be lossless if

$${\mbox{\boldmath$\Gamma$}}(z) = {\mbox{\boldmath$A$}}^*(1/z^*) {\mbox{\boldmath$\Gamma$}}(z) {\mbox{\boldmath$A$}}(z)$$ (87)

In the more general case of passive propagation, the scattering junction is said to be lossless if

$${\mbox{\boldmath$\Gamma$}}^*(1/z^*) = {\mbox{\boldmath$A$}}^*(1/z^*) {\mbox{\boldmath$\Gamma$}}(z) {\mbox{\boldmath$A$}}(z)$$ (88)

By adding (89) to its conjugate transpose we get

$${\mbox{\boldmath$\Gamma$}}(z) + {\mbox{\boldmath$\Gamma$}}^... ...+ {\mbox{\boldmath$\Gamma$}}^*(1/z^*)]{\mbox{\boldmath$A$}}(z)$$ (89)

and we see that the construction of a normalized scattering matrix proceeds from a spectral factorization (if it exists) of the real part of the admittance matrix:

$${\mbox{\boldmath$\Gamma$}}(z) + {\mbox{\boldmath$\Gamma$}}^*(1/z^*) = {\mbox{\boldmath$U$}}^*(1/z^*) {\mbox{\boldmath$U$}}(z)$$ (90)

When the medium is passive we have ${\mbox{\boldmath$\Gamma$}}(z) + {\mbox{\boldmath$\Gamma$}}^*(1/z^*) \geq {\mbox{\boldmath$0$}}$, and a spectral factorization does exist. The normalized scattering matrix is then

$${\tilde {\mbox{\boldmath$A$}}}(z) = {\mbox{\boldmath$U$}}(z) {\mbox{\boldmath$A$}}(z) {\mbox{\boldmath$U$}}^{-1}(z)$$ (91)

and it is para-unitary, i.e.,

$${\tilde {\mbox{\boldmath$A$}}}^*(1/z^*) {\tilde {\mbox{\boldmath$A$}}}(z) = {\mbox{\boldmath$I$}}\, .$$ (92)

A real-coefficient matrix ${\tilde {\mbox{\boldmath$A$}}}(z)$ which is analytic for $\vert z\vert \geq 1$ and which satisfies (93) is called Lossless Bounded Real [63,112]. The condition (93) implies that the elements of any column of ${\tilde {\mbox{\boldmath$A$}}}(z)$ are power complementary, i.e.,

$${\sum_{i=1}^{N}{\mid {\tilde a}_{i,k}(e^{j \omega}) \mid}^2} = 1\, .$$ (93)

In fixed-point implementations, the condition (94) can be satisfied as shown in [71,83,114]. Given a set of $N$ power-complementary filters { ${\tilde a}_{1,k}(z) \dots {\tilde a}_{N,k}(z)$}, they can be implemented by means of $N$ allpass filters under mild conditions [114], and the property of power complementarity is structurally induced in the sense that coefficient quantization cannot alter it.