2.3 Relationship of Fundamental and Physical Parameters

The capacity of soft-decision $M$-ary PPM on the Webb+Gaussian channel given by Eq. (11) is a real-valued function $C({\mathbf a})$, where ${\mathbf a} = (\rho_0 , \rho_+, \Delta, \beta_0)$ is the vector of fundamental parameters. The fundamental parameter vector ${\mathbf a}$ can be expressed in terms of physical parameters

$$P = \{\eta, \bar n_s, \bar n_b, F, I_b, I_s, T_s, T, R_L, G\}$$

by

$$\rho_0 \triangleq {(m_1-m_0)^2\over\sigma_0^2+\sigma'^2} = {G^2\eta^2{\bar n}_s^2 \over \bar FG^2(\eta {\bar n}_b + {I_bT_... ...ext{-}}}) + {I_sT_s\over e_{\text{-}}} + {2\kappa TT_s\over R_Le_{\text{-}}^2}}$$ (13)

$$\rho_+ \triangleq {(m_1-m_0)^2\over\sigma_1^2-\sigma_0^2} = {\eta\bar n_s\over F}$$ (14)

$$\Delta \triangleq \delta_1^2-\delta_0^2 = {\eta\bar n_sF\over(F-1)^2}$$ (15)

$$\beta_0 \triangleq {\sigma_0^2\over\sigma_0^2+\sigma'^2} = {1\over 1+{\sigma'^2\over\sigma_0^2}} = {1\over 1+ { {I_sT_s\over e_{\text{-}}} +{2\kappa TT_s\over R_Le_{\text{-}}^2} \over FG^2(\eta {\bar n}_b + {I_bT_s\over e_{\text{-}}}) } }$$ (16)

Expressions of parameters $m_0$, $m_1$, $\sigma_0^2$, $\sigma_1^2$, $\sigma'^2, \delta_0^2, \delta_1^2$ in terms of of the physical parameters can be found in, e.g., [4,5,1].