2.1 Channel model

Each slot statistic $y_i$ at the output of an APD is a Webb+Gaussian distributed random variable [1]:

$$y_i \sim \left\{ \begin{array}{ll} W(m_1,\sigma_1^2,\delta_1^2) +... ...2,\delta_0^2) + N(m',\sigma'^2) & \mbox{nonsignaling slot,} \end{array} \right.$$ (1)

where $W(m,\sigma^2,\delta^2) = m + \sigma W(0,1,\delta^2)$ is a shifted, scaled version of the zero-mean, unit-variance Webb random variable $W(0,1,\delta^2)$ that has probability density

$$\phi(w;\delta^2) \triangleq \frac{1}{\sqrt{2\pi}} (1 + w/\delta)^{-3/2} e^{-{w^2}/{2(1+w/\delta)} } ~, ~~ w> -\delta$$ (2)

and where $N(m,\sigma^2) = m + \sigma N(0,1)$ is a shifted, scaled version of the zero-mean, unit-variance Gaussian random variable $N(0,1)$ that has probability density

$$\phi(x) \triangleq \frac{1}{\sqrt{2\pi}} e^{-x^2/2}.$$ (3)

The Webb+Gaussian random variable $W(m,\sigma^2,\delta^2)+N(m',\sigma'^2)$ can also be written as a shifted, scaled version of a composite zero-mean, unit-variance random variable, as

$$W(m,\sigma^2,\delta^2)+N(m',\sigma'^2) \triangleq WG(m+m',\sigma^2+\sigma'^2,\delta^2,\beta)$$

$$= m+m' + \sqrt{\sigma^2+\sigma'^2}\left(\sqrt{\beta} W(0,1,\delta^2) + \sqrt{1-\beta} N(0,1)\right),$$

where $\beta = \sigma^2/(\sigma^2+\sigma'^2)$. The term in parentheses is a zero-mean, unit-variance random variable, $WG(0,1,\delta^2,\beta)$, with probability density

$$\phi(x;\delta^2,\beta) = \int_{-\delta}^\infty \frac{1}{\sqrt{\be... ...right) \frac{1}{\sqrt{1-\beta}}~\phi\left({{x-w}\over\sqrt{1-\beta}}\right) dw.$$ (4)