Appendix A: The Digital Waveguide Transformer

A useful technique for junction normalization is by insertion of ideal transformers. First we define the ideal transformer in the $2 \times 2$ case ($N$=2) and apply it to the normalization of junctions where the impedance matrix is diagonal. Then we show a generalization to the $N \times N$ case and a non-diagonal impedance matrix.

The ideal cascade transformer is a lossless 2-port which scales up pressure by some factor and scales down velocity by the same factor without generating scattering reflections. Signal power is conserved. Since a transformer introduces a discontinuity in both pressure and velocity, it is not physical (although there are physical approximations used in microwave engineering). A conical acoustic tube section can be regarded as an approximate transformer, as can horn loudspeakers.

Consider the general two-port diagram in Fig. 11. Since a 2-port transformer, by definition, preserves signal power, we have

$$p_1 u_1 = (p_1^+ + p_1^-)(p_1^+ - p_1^-)/R_1$$

$$= -(p_2^+ + p_2^-)(p_2^+ -p_2^-)/R_2$$

$$= - p_2 u_2$$ (94)

which is satisfied by

$$\begin{array}{l} p_1^- = {\sqrt \frac{R_1}{R_2}}p_2^+ \stack... ...frac{R_2}{R_1}}p_1^+ \stackrel{\triangle}{=}g p_1^+ \end{array}$$ (95)

where $g$ is defined as the ``turns ratio'' since it corresponds to the voltage stepping ratio in electric transformers. Regarding the 2-port transformer as a special kind of two-waveguide junction, its scattering matrix is

$$\Sigma=\left[ \begin{array}{ll}0 & {\sqrt \frac{R_1}{R_2}}... ... \begin{array}{ll}0 & \frac{1}{g}\\ g & 0 \end{array} \right]$$ (96)

Figure 11: The ideal 2-port transformer.

The ideal 2-port transformer is depicted in Fig. 11.

In the case of a junction of $N$ waveguides, we can normalize the junction by making all the waveguide impedances equal. This is accomplished by inserting $N$ 2-port transformers, each of them coupling the original branch impedance at the junction with a unit impedance at each waveguide. This corresponds to the diagonal similarity transformation of (32). Of course, we can choose one of the original branch impedances as reference impedance, so that inserting $(N-1)$ 2-port transformers will suffice.

The physical interpretation of ideal transformers can be extended more generally. We define the generalized ideal transformer as a $2N$-port satisfying the equations [63]

$$\begin{array}{rcr} {\mbox{\boldmath$p$}}_1 & = & {\mbox{\boldmath... ...ox{ }} {\mbox{\boldmath$u$}}_1 % (format optimization experiment) \end{array}{}$$ (97)

where ${\mbox{\boldmath$p$}}_1$ and ${\mbox{\boldmath$u$}}_1$ are respectively the $N$-component vectors of pressure and velocity at the first group of $N$ ports, and ${\bf p}_2$ and ${\mbox{\boldmath$u$}}_2$ are the corresponding vectors for the second group of $N$ ports. ${\mbox{\boldmath$T$}}$ is an $N \times N$ matrix called the turns-ratio matrix.

It is easy to check that the definition (98) satisfies the conservation of power:

$${{\mbox{\boldmath$u$}}_1}^*{\mbox{\boldmath$p$}}_1 = - {{\mb... ...}}_2 = - { {\mbox{\boldmath$u$}}_2}^* {\mbox{\boldmath$p$}}_2$$ (98)

To normalize a junction in the general case of non-diagonal impedance matrix, we can use a transformer having turns-ratio matrix

$${\mbox{\boldmath$T$}}= {{\mbox{\boldmath$U$}}^{-1}}^*$$ (99)

where ${\mbox{\boldmath$U$}}$ is the Cholesky factor of (33). The resulting scattering matrix is therefore given by (35).

We can say that the matrix ${\mbox{\boldmath$U$}}$ of (33) acts as an ideal transformer on the vector of all $N$ waveguide variables.

An important application of the ideal transformer is to decouple scattering junctions from their attached waveguide branches. That is, by modulating the transformer ``turns ratios,'' we may arbitrarily modulate the scattering parameters without also modulating the signal power stored in the waveguide branches.