## 3 Optical Receiver Filter Equation-MGF Optimisation Procedure
### MGF-Based Model for Time-Invariant Rx System
#### 3.2.1 Use Case I: r\({}_{sig}\)(t) without AGTN Noise and ISI  
This use case involves a linear time-invariant Rx without AGTN noise. This means we are considering a path-loss channel, which implies that we are considering just the absorption aspect of the seawater impairments. In this work, we assume that the optical receiver under consideration is deploying a PIN photodetector (i.e., the internal gain \(g=1\)). The LMGF in Equation (10) for such a scenario reduces to the following [11; 15]:  
\[L_{r_{f(m)}}=(-1)^{m}\int\lambda_{m}(t)(Exp[\ h_{f}(t)]-1)dt. \tag{11a}\]  
Equation (11a) is the first term of Equation (10), which represents \(r_{sig}(t)\), which explicitly includes the ISI in addition to the 0 th transmitted bit. A linear Rx implicitly is an invariant system with respect to the insertion of a time shift to \(\lambda_{m}(t)\)_or_\(h_{f}(t)\). This means that:  
\[\int\lambda_{0}(t\pm\Delta)(Exp[h_{f}(t)]-1)dt =\int\lambda_{0}(t)(Exp[h_{f}(t\pm\Delta)]-1)dt,\ \text{For m }=0, \tag{11b}\] \[\int\lambda_{1}(t\pm\Delta)(Exp[-h_{f}(t)]-1)dt =\int\lambda_{1}(t)(Exp[-h_{f}(t\pm\Delta)]-1)dt,\ \text{For m }=1. \tag{11c}\]  
If we take a very small \(\Delta\), then it will be a valid approximation to take the first two terms of [PERSON]'s series of \(\lambda(t\pm\Delta)\) and \(h_{f}(t\ \pm\Delta)\). From Equation (10), using the notation of m \(\in\{0,1\}\), we can rewrite the relationships in Equation (11) as follows:  
\[\int\lambda^{\prime}_{m}\ (Exp[(-1)^{m}h_{f}\ ]\ \ -1)dt+(-1)^{m}\ \int \lambda_{m}\ h^{\prime}_{f}\ \ Exp[(-1)^{m}h_{f}]\ dt=0,\] (12a) or in a more compact (shorthand) form \[\int d\Big{[}\lambda_{m}\ (Exp[(-1)^{m}h_{f}\ ]-1)\Big{]}\ dt=0,\] (12b) where \[\lambda_{m}\equiv\lambda_{m}(t),\lambda^{\prime}_{m}\ \equiv\ \lambda^{\prime}_{m}(t),\ h_{f}\equiv h_{f}(t),\ \text{and}\ h^{\prime}_{f}\ \equiv\ h^{\prime}_{f}(t)\]. The above equation shows that the system constraint could be applied directly to the integrand of the LMGF. Thisis possible because \(L_{rf(m)}\) is a continuous function. By applying Equation (12) for each bit value, i.e., m = {0, 1}, and execute the integrating in Equation (12), then we obtain:  
\[\lambda_{1}\ \ (Exp[-h_{f}]\ \ \ -1)+C_{1}=0,\ \text{For ON state} \tag{13a}\] \[\lambda_{0}\ \ (Exp[\ \ h_{f}]\ \ \ -1)+C_{0}=0,\ \text{For ON state} \tag{13b}\]  
where \(C_{m}\) is the constant of integration. The boundary values of \(\lambda_{m}(t=\pm\infty)\ \sim\ \varphi_{dc}\) for both bit states and the boundary values for \(h_{f}(t=\pm\infty)\ \sim\ 0\). Therefore, we can safely say that C\({}_{1}\) = C\({}_{2}\) = 0. Equation (13) describes two independent states in one time-invariant system. Hence, we can assemble both Equations (13a) and (13b) to describe the states of the underlying overall Rx system. Therefore, we can rewrite Equation (13) as follows:  
\[\lambda_{0}\ \ Exp[2\ h_{f}]-(\lambda_{1}+\lambda_{0})\ Exp[h_{f}]+\lambda_{1}=0. \tag{14}\]  
The above quadratic equation describes the Rx system model without AGTN noise. This model equation includes the following operating attributes: \(\lambda_{m}\) through Equation (6) includes transmitted pulse shape \(h_{p}\), propagation channel impulse response \(h_{c}\), photodetector quantum efficiency (\(\eta\)), and photodetector impulse response \(h_{d}\), also Equation (14) includes the filter impulse response \(h_{f}\). In Equation (14), the decision threshold \(F_{D}\) is seemingly missing. To include \(F_{D}\) in the Rx model, we need to use the central moment [28] definition of \(r_{f}\) in Equation (7) instead of the ordinary moment. Including \(F_{D}\) requires the replacement of \(\lambda_{m}\) in Equation (7) with \((\lambda_{m}+(-1)^{\text{m}}\ F_{D})\). Equation (14) has an analytical solution which is given as follows:  
\[h_{f}=ln\left[\frac{\lambda_{1}}{\lambda_{0}}\right]. \tag{15}\]  
Equation (15) represents a Maximum Likelihood Filter (MLF) when considering the signal component without the thermal noise.