Scattering of Normalized Waves

We consider first diagonal impedance matrices, since they have an intuitive physical counterpart in lossless acoustic tubes. Under this assumption, we have the impedance matrix

$${\mbox{\boldmath$R$}}=\hbox{diag}[R_1,\dots,R_n]$$ (28)

and the admittance matrix

$${\mbox{\boldmath$\Gamma$}}=\hbox{diag}[\Gamma_1,\dots,\Gamma_n] = {\mbox{\boldmath$R$}}^{-1}.$$ (29)

By (19) and the assumption of a passive medium, the normalized pressure waves are uniquely defined as

$${\tilde p_i}^\pm=\frac{p_i^\pm}{\sqrt {R_i}}={p_i^\pm}{\sqrt {\Gamma_i}}$$ (30)

where the three signs above are all taken as ``$+$'' or all as ``$-$''. (Square roots in this paper are always taken as positive.) We can write also

$${\tilde {\mbox{\boldmath$p$}}}^\pm = {\mbox{\boldmath$\Gamma$}}^{\frac{1}{2}} {\mbox{\boldmath$p$}}^\pm$$ (31)

where ${\mbox{\boldmath$\Gamma$}}^{\frac{1}{2}}$ is the diagonal square root of ${\mbox{\boldmath$\Gamma$}}$. The junction of normalized waveguides is in this case represented by the scattering matrix

$${\tilde {\mbox{\boldmath$A$}}}={\mbox{\boldmath$\Gamma$}}^{\... ...{\mbox{\boldmath$A$}}{\mbox{\boldmath$\Gamma$}}^{-\frac{1}{2}}$$ (32)

As a side note, this equation is analogous to the relation between power-wave scattering matrices and voltage-wave scattering matrices as found in the WDF literature [27] for lumped circuit elements.

In the more general case, when the (complex) admittance matrix $\Gamma$ is not necessarily diagonal, but remains positive semidefinite as required for lossless propagation, we have that ${\mbox{\boldmath$\Gamma$}}$ admits a Cholesky factorization

$${\mbox{\boldmath$\Gamma$}}={\mbox{\boldmath$U$}}^*{\mbox{\boldmath$U$}}$$ (33)

where ${\mbox{\boldmath$U$}}$ is upper triangular. We can define the normalized pressure waves as

$${\tilde {\mbox{\boldmath$p$}}}^\pm = {\mbox{\boldmath$U$}}{\mbox{\boldmath$p$}}^\pm$$ (34)

The normalized scattering junction is obtained via the following similarity transformation on the unnormalized scattering matrix:

$${\tilde {\mbox{\boldmath$A$}}}={\mbox{\boldmath$U$}}{\mbox{\boldmath$A$}}{\mbox{\boldmath$U$}}^{-1}$$ (35)

Theorem 1   The scattering matrix of a normalized junction is unitary, i.e.,

$${\tilde {\mbox{\boldmath$A$}}}^*{\tilde {\mbox{\boldmath$A$}}}={\mbox{\boldmath$I$}}$$ (36)

Proof: From the condition of losslessness (26), we have ${\tilde {\mbox{\boldmath$A$}}}^* {\tilde {\mbox{\boldmath$A$}}} = {\mbox{\boldm... ...h$U$}}^*{\mbox{\boldmath$U$}}{\mbox{\boldmath$U$}}^{-1} = {\mbox{\boldmath$I$}}$

By an explicit computation of the matrix product in (36) we can show that the terms of any column of ${\tilde {\mbox{\boldmath$A$}}}$ are power complementary, i.e.,

$${\sum_{i=1}^{N}{\mid {\tilde a}_{i,k} \mid}^2}=1$$ (37)

for all $k$.