2.2 Capacity

From [1], the capacity of soft-decision $M$-PPM on the Webb+Gaussian channel in bits per channel use is¹

$$C = \log_2 M - E_{{\mathbf v} \vert {\mathbf x_1}} \log_2 \sum_{j... ...ight) \,\, \phi\left(v_j;{\frac{\beta_0\rho_+}{\rho_0 }\Delta},\beta_0\right)},$$ (5)

where the four fundamental parameters are

$$\rho_0 \triangleq {(m_1-m_0)^2\over\sigma_0^2+\sigma'^2}$$ (6)

$$\rho_+ \triangleq {(m_1-m_0)^2\over\sigma_1^2-\sigma_0^2}$$ (7)

$$\Delta \triangleq \delta_1^2-\delta_0^2$$ (8)

$$\beta_0 \triangleq {\sigma_0^2\over\sigma_0^2 + \sigma'^2}$$ (9)

and where the components of ${\mathbf v}$ are distributed as

$$v_j \sim \left\{ \begin{array}{ll} WG(\sqrt{\rho_0 },\frac{\rho_0... ...0,1,\frac{\beta_0\rho_+\Delta}{\rho_0 },\beta_0) & j\neq 1. \end{array} \right.$$ (10)

The capacity can be rewritten as

$$C = \log_2M - E_{{\mathbf v}\vert{\mathbf x}_1} \log_2\left[ \sum_{j=1}^M L(v_j)/L(v_1)\right],$$ (11)

where

$$L(v_j) = {\phi\left(\sqrt{\frac{\rho_+}{\rho_0 +\rho_+}}(v_j-\sqr... ...ight) \over \phi\left(v_j;{\frac{\beta_0\rho_+}{\rho_0 }\Delta},\beta_0\right)}$$ (12)

is the likelihood function for $v_j$.