## 4 Results and Discussion
### MGF Model Validation  
In terms of mathematical formulation, first, we must emphasise that any model should preserve the dynamics of the statistical processes and maintain their impact boundaries. In our case, the statistical process is the photodetector DSPP process, which has been depicted in Equations (5) and (6). This is what [10-16] based their design optimisation approaches on. Based on this mathematical fabric, the Rx model should produce a causal equaliser/filter unit that considers the DSPP. The DSPP is described in terms of MGF and LMGF through Equations (7)-(10). We need to verify that our proposed MGF model for the Rx system is reliable in this sense. We must apply our model for a photodetector with a direct convolutional process, not a transformational-convolutional DSPP process [27]. Also, we utilise the use case II of a \(r_{sig}\) signal with a white AGTN noise. For such a configuration, the LMGF takes the following form [10,15]:  
\[L_{r_{f}(m)}=(-1)^{m}\int\lambda_{m}\;h_{f}\;dt+\;\frac{N_{0}}{4}\int\;\;\mathrm {h}_{f}^{2}\;\;dt. \tag{24}\]  
Now we apply the same steps from Equations (12)-(14), then the equalizer/filter equation in Equation (14) takes the following form:  
\[\lambda_{0}-\lambda_{1}+2N_{0}h_{f}=0\;. \tag{25}\]  
The general analytical solution of Equation (25) is:  
\[h_{f}=\frac{\lambda_{1}-\lambda_{0}}{2N_{0}}. \tag{26}\]  
Equation (26) represents a whitened matched filter for a deterministic signal corrupted by a white AGTN noise. While Equation (26) differs from the DSPP-based Equation (18) above, it is the same as that published by [10,13] using CB and MCB, respectively. This observation means that CB and MCB approach optimization steps inadvertently masked the built-in DSPP in the optical receiver design, which is assumed by [10,13]. In other words, CB and MCB do not produce a causal filter unit. On the contrary, our model does, which makes the MGF Model more reliable than the CB and MCB method. If we consider RTZ OOK, Equation (17) takes the same format reported in [13] for the same use case. However, for the use case I of the \(r_{sig}\) without AGTN, the MCB filter impulse response in [13] is similar to Equation (15) above. For the use case III \(r_{sig}\) signal with a colored AGTN noise, Equation (22) agrees with that published by [13,15]. Second, regarding optimisation criteria, the CB and MCB optimization in [10-16] is based on minimizing the BER for the decision unit. Our model's optimization is based on directly maximizing the received processed signal energy presence around the pulse center to elevate the signal-to-noise ratio (SNR) at decision time. The two approaches are essentially the same due to the probability of error association with SNR through the error function (ERF). The outcome SNR is shown in the eye diagram of each use case's optimum filter out \(r_{f}\). When comparing our proposed optimum filter with the raised cosine filter (RCF), which is a reference industry standard, it is essential to mention that RCF access bandwidth (BW) is proportional to the roll-off factor (\(\beta\)). This means that we need a lower \(\beta\) at the run time to reduce the access BW. This reduction in \(\beta\) could mean less ability for the filter to reduce the ISI at high and moderate bit rates. At a low bit rate, the RCF is fine. This is inherently linked to the RCF's undershoot cycles being tied up to the bit period boundaries, causing unnecessary higher flooring in the eye diagram. Our optimum filter for ISI use-cases shown in Figures 4f and 5f for Coastal, Figures 6f and 7f for Harbour, shows practically no floor for the Coastal scenarios and limited floor depth for the Harbour scenarios. The reason behind this is that for the corresponding ISI use cases in Figures 4b-7b, we can notice that the undershoot cycle crossing is not tied up to the bit boundaries like the RCF. Such behavior means our proposed filter is optimized when necessary and is not fixed to fixed bit boundaries. Figure 4: Use Case I. \(r_{\text{sig}}(t)\) without AGTN.