Pseudopower and Pseudopassivity

Fettweis [42] defines the instantaneous pseudopower absorbed by a port with port resistance $R$ (real) at time step $n$ in terms of the discrete input and output wave quantities as

$$w_{inst}(n) = \frac{1}{R}\left(a^{2}(n)-b^{2}(n)\right) = 4\left(\underline{a}^{2}(n)-\underline{b}^{2}(n)\right)$$ (2.18)

which, when the transformation (2.14) is inverted, gives

$$w_{inst}(n) = 4v(n)i(n)$$

This discrete power definition coincides with the standard definition of power in classical network theory from (2.2), aside from the factor of 4, which is of no consequence if definition (2.18) is applied consistently throughout a wave digital network.

For a real LTI $N$-port, in an exponential state of complex frequency $z$, the steady-state average pseudopower may be written in terms of the $N\times 1$ vectors $\hat{\underline{{\bf a}}}$ and $\hat{\underline{{\bf b}}}$ which contain the power-normalized complex amplitudes $\hat{\underline{a}}_{j}$ and $\hat{\underline{b}}_{j}$, for $j=1,\hdots,N$ as

$$\bar{w} = 4\left(\hat{\underline{{\bf a}}}^{*}\hat{\underline{{\bf a}}}-\hat{\underline{{\bf b}}}^{*}\hat{\underline{{\bf b}}}\right)$$

The steady-state reflectance $\underline{{\bf S}}(z^{-1})$ is defined by

$$\hat{\underline{{\bf b}}} = \underline{{\bf S}}\,\hat{\underline{{\bf a}}}$$

and gives

$$\bar{w} = 4\left(\hat{\underline{{\bf a}}}^{*}({\bf I}_{N}-\underline{{\bf S}}^{*}\underline{{\bf S}})\hat{\underline{{\bf a}}}\right)$$

where ${\bf I}_{N}$ is the $N\times N$ identity matrix. For pseudopassivity [42], we require, then (recalling that the bilinear transform (2.11) maps the right half $s$-plane to the exterior of the unit circle in the $z$ plane) that

$$\underline{{\bf S}}^{*}(z^{-1})\underline{{\bf S}}(z^{-1})\leq {\bf I}_{N} \hspace{0.3in}\vert z\vert\geq 1$$ (2.19)

$\underline{{\bf S}}(z^{-1})$ is sometimes called a bounded real matrix. If (2.19) holds with equality for $\vert z\vert=1$, then it is called lossless bounded real (LBR) [193]. In general, to bounded real matrix reflectances there correspond positive real matrix impedances, and vice versa. In terms of voltage wave quantities, we have for a wave digital $N$-port, that

$$\hat{{\bf a}} = 2{\bf R}^{\frac{1}{2}}\hat{\underline{{\bf a}}}\hspace{0.5in}\hat{{\bf b}} = 2{\bf R}^{\frac{1}{2}}\hat{\underline{{\bf b}}}$$

where ${\bf R}^{\frac{1}{2}}$ is the diagonal square root of the matrix containing the $N$ port resistances $R_{1},\hdots,R_{N}$ on its diagonal. We then have

$${\bf S} = {\bf R}^{\frac{1}{2}}\underline{{\bf S}}{\bf R}^{-\frac{1}{2}}$$

for the voltage wave scattering matrix ${\bf S}$ and thus we require

$${\bf S}^{*}(z^{-1}){\bf R^{-1}}{\bf S}(z^{-1})\leq {\bf R}^{-1} \hspace{0.3in}\vert z\vert\geq 1$$ (2.20)

for passivity. For one-ports, the requirements (2.20) and (2.19) are the same.

Also note that we have, by applying the power wave variable definitions (2.16), and the discrete impedance relation $\hat{{\bf v}} = {\bf Z}\hat{{\bf i}}$ (which is identical to the analog relation from (2.1), except that we now have ${\bf Z} = {\bf Z}(z^{-1})$), that

$${\bf S} = ({\bf ZR}^{-1}+{\bf I})^{-1}({\bf ZR}^{-1}-{\bf I})$$ (2.21)

If the $N$-port is not LTI, then it is possible to apply a similar idea to the expression for the instantaneous pseudopower, from (2.18) in order to derive a passivity condition [46]; In this case, pseudopassivity has also been called incremental pseudopassivity [125].