## 1 Introduction  
Many research groups worldwide, such as [1; 2; 3; 4; 5; 6; 7] work on different aspects of underwater wireless optical communications (UWOC). Their efforts include advanced simulation platforms and the design of receiver units for the UWOC Systems (Figure 1). One of the milestones in these projects is formulating a valid, operable, and consistent model for the water channel impulse response (CIR) function \(h_{c}(t)\) that accounts for underwater channels and the associated optical propagation impairments (absorption, single/multiple scattering, and scintillation). These impairments cause signal degradations, which are manifested as serious ISI noise. This causes severe limitations for the UWOC system [8] in terms of bit rate and communication linkspan. To overcome such challenges, many research groups reduced the link length to around 50 m [8] and the transmission rate to a low of a few Mbps [8]. Nevertheless, none of [1; 2; 3; 4; 5; 6; 7] provides an approach to derive a UWOC receiver filter (Figure 1) to enhance the receiver unit performance in the presence of ISI without restoring to reducing the bit rate or link span. The studies in [1; 2; 3; 4; 5; 6; 7] presented UWOC optical system performance calculations in terms of Bit Error Rate (BER), assuming that the impulse response of the receiver photodetector and its backend electronic circuitry (pre-amplifier, amplifier, and filter/equalizer) can be represented as a delta functions. Such an unrealistic assumption implies that the actual performance computation is at the input of the receiver photodetector rather than at the input of the receiver decision unit (Figure 1). Also, the work in [1; 2; 3; 4; 5; 6; 7] does not include any attempt to derive an optimum filter which requires an optimum Signal to Noise Ratio (SNR) or Q-Factor to secure an optimum BER for maximum channel link span. There is a large volume of work [9; 10; 11; 12; 13; 14; 15; 16; 17] based on complex CB and MCB to handle receiver design optimization for optical fiber communication. However, the main elements of receiver optimization methodology are the filter impulse response \(h_{f}(t)\), the photodetector internal gain (g), and the threshold for the decision unit \(F_{D}\) (Figure 1). In [9], the author discussed the receiver design based on directly minimizing the BER. In [10; 11; 12; 13; 14; 15; 16], the authors developed clever approaches to formulate a functional filter (\(E_{filter}\)) by applying upper CB or MCB on the BER for binary optical fiber systems. Typically, the derived \(E_{filter}\) encompasses the receiver filter function \(h_{f}(t)\), its derivative \(h^{\prime}_{f}(t)\), and \(F_{D}\). This implies that the function \(h_{f}(t)\) must be continuous, and its 1 st and 2 nd derivatives must exist. Consequently, \(E_{filter}\) satisfies the following variational calculus relation [18]:  
\[I=\int_{t_{1}}^{t_{2}}E_{filter}\Big{(}t,\ h_{c}(t),\ h_{f}(t),\ h^{\prime}_{f}(t ),\ F_{D}\Big{)}\ dt. \tag{1}\]  
Based on these mathematical foundations, the authors in [10; 11; 12; 13; 14; 15; 16] considered \(E_{filter}\) as an input for an Euler-Lagrange operator (\(J\)) = (\(\frac{\partial}{\partial t}-\frac{d}{dt}\frac{\partial}{\partial t}\)) [18] to obtain an extremal (minimum or maximum) value of \(E_{filter}\) by setting \(J(E_{filter})=0\). The extremal conditions are: for all \(\mathfrak{t}\in(t_{1},\ t_{2})\), if \(\frac{\partial^{2}}{\partial h^{\prime 2}}\ E_{filter}>0\), then the optimum \(h_{f}(t)\) minimizes \(E_{filter}\), if \(\frac{\partial^{2}}{\partial h^{\prime 2}}\ E_{filter}<0\) then \(h(t)\) maximizes \(E_{filter}\), and if \(\frac{\partial^{2}}{\partial h^{\prime 2}}\ E_{filter}=0\), then \(E_{filter}\) degenerates [18]. The authors [10; 11; 12; 13; 14; 15; 16; 17] considered realistic noise environment scopes, including the ISI and AGTN. Analytical solutions were obtained for simple scenarios (e.g., without AGTN, with white ADTN, and without ISI). For complex noise environments, they obtained numerical solutions for closed-form filter equations.