Generalized Complex Signal Power

The net complex power involved in the propagation can be defined as [4]

$${\cal P} = {\mbox{\boldmath$u$}}^* {\mbox{\boldmath$p$}}\, = \, ({\mbox{\bol... ...+ {\mbox{\boldmath$u$}}^-)^* ({\mbox{\boldmath$p$}}^++ {\mbox{\boldmath$p$}}^-)$$

$$= {{\mbox{\boldmath$u$}}^+}^{*}{\mbox{\boldmath$R$}}{\mbox{\boldmat... ...+- {{\mbox{\boldmath$u$}}^-}^{*}{\mbox{\boldmath$R$}}^*{\mbox{\boldmath$u$}}^-+$$

$${{\mbox{\boldmath$u$}}^-}^{*}{\mbox{\boldmath$R$}}{\mbox{\boldmat... ...^+- {{\mbox{\boldmath$u$}}^+}^{*}{\mbox{\boldmath$R$}}^*{\mbox{\boldmath$u$}}^-$$

$$\stackrel{\triangle}{=} ({\cal P}^+- {\cal P}^-) + ({\cal P}^\times - {\cal P}^{\times*})$$ (16)

where all quantities above may be functions of $z$, and ``$*$'' denotes paraconjugation (transposition and complex conjugation on the unit circle in the $z$ plane). The quantity ${\cal P}^+= {{\mbox{\boldmath$u$}}^+}^{*}{\mbox{\boldmath$R$}}{\mbox{\boldmath$u$}}^+$ is called right-going active power (or right-going average dissipated power⁴), while ${\cal P}^-= {{\mbox{\boldmath$u$}}^-}^{*}{\mbox{\boldmath$R$}}^* {\mbox{\boldmath$u$}}^-$ is called the left-going active power. The term ${\cal P}^+- {\cal P}^-$, the right-going minus the left-going power components, we call the net active power, while the term ${\cal P}^\times - {\cal P}^{\times*}$ is net reactive power. These names all stem from the case in which the matrix ${\mbox{\boldmath$R$}}(z)$ is positive definite for $\vert z\vert \geq 1$. In this case, both traveling components of the active power are real and positive, the active power itself is real, and the reactive power is purely imaginary.