## 3 Optical Receiver Filter Equation-MGF Optimisation Procedure  
In this section, we are presenting the derivation of the receiver filter \(h_{f}(t)\) equation based on the MGF of \(r_{sig}(t)\). We should note here that the output of \(h_{f}(t)\) constitutes the input to the decision unit (Figure 3). For clarification purposes, the mathematical manipulations are mapped to three use cases with ascending noise complexity. The first use case (I) handles the signal \(r_{sig}(t)\) without ISI degradation and additive thermal Gaussian noise (ATGN). However, the computation covers the status (I) without ISI (which resembles the path-loss channel) and (2) with ISI. The second use case (II) covers the signal \(r_{sig}(t)\) scenarios with white AGTN and with and without ISI. The third use case (III) deals with \(r_{sig}(t)\), which includes colored AGTN and ISI. The derivations of the three use cases are based on (a) the \(r_{sig}\) includes the ISI term expressed in Equation (6) and (b) the modulation format is binary OOK, which covers the return-to-zero (RTZ) and non-return-to-zero (NRZ). During the derivation, we use two types of MGFs: ordinary MGF and central MGF. Such naming is based on [28], and it helps distinguish the formulation attributes of the underlying MGF. However, the statistical entities of the MGF Design model can be defined as follows: (I) the system, (II) the system's attributes, (III) the system structures, and (IV) the connected sequence of underlying processes within the system. Hence, our MGF optimization model encapsulates the following: (I) the system is the Optical Receiver System (Rx), (II) the system's attributes include \(\lambda(t)\) in Equation (6), \(h_{c}(t)\) given in [8], \(r_{th}(t)\) which is the AGTN in Equation (8a), and \(h_{f}(t)\) which is the solution under consideration, (III) the system structure layout is shown in Figure 3, and (IV) the connected sequence of processes are: (IV-I) The convolution process of the transmitted digital OOK optical signal \(s(t)\) in Equation (2) with the propagation channel \(h_{c}(t)\) given in [8]. The outcome of this convolution is the received optical signal \(\lambda(t)\) in Equation (6c), (IV-II) the photodetector counting process which yields \(N(t)\) of photoelectrons that form the electronic signal rsig(t) in Equation (4), where the photoelectron count \(N(t)\) is governed by incident photon intensity \(\lambda(t)\), (IV-III) at the backend of the photodetector an electronic pre-amplifier/amplifier process that causes the addition of the AGTN in Equation (8a), and (IV-IV) the electronic equalization/filtering process with an impulse response function \(h_{f}(t)\) that represents the solution of Equation (19), which we are trying to optimize. While the authors in [10; 11; 12; 13; 14; 15; 16] used CB to conclude the receiver functional \(\psi\), we use a new Rx model to create the \(\psi\) based on an MGF, which describes \(r_{f}(t)\). This model includes \(r_{opt}(t)\) as an input, \(r_{f}(t)\) as output, and \(F_{D}\) as a selection reference value. Hence, the central MGF can be used by considering \(F_{D}\) as a reference value for \(\lambda(t)\) defined in Equation (6). However, the statistical entities MGF or LMGF can be used to describe the system's attributes, structures, and the connected sequence of underlying processes. The filter output signal \(r_{f}(t)\) is sampled at regular time intervals, and each resulting sample value \(r^{*}(kT)\) is compared with a fixed \(F_{D}\). Based on a built-in hypothesis role \(\{H_{i}\}\) where \(i=[0,\,1]\), the receiver decision unit concludes a decision related to the input \(r_{f}(t)\) being above \(F_{D}\) or below \(F_{D}\). The pivotal variables in the MGF optimization procedure are (a) the received optical intensity \(\lambda(t)\) given in Equation (6) and (b) the electronic filter \(h_{f}(t)\) shown in Figure 3. We should remember that \(\lambda(t)\) formula in Equation (6) includes the convolution of the transmitted optical pulse \(h_{p}(t)\) and water channel impulse response \(h_{c}(t)\).