## 1 Introduction
In [15; 16], the authors extended the receiver approach to include the m-ary optical receiver system and coherent homodyne detection optical receiver [16]. This extension adds credibility to the CB optimization methodology as a reliable receiver design methodology. The teams [10; 11; 12; 13; 14; 15; 16; 17] assumed that the channel impulse response \(h_{c}(t)\) is a delta function for optical fiber, which is not the case for UWOC. While the CB approach [10; 11; 12; 13; 14; 15; 16] could be applied to the UWOC system, it includes complicated steps in deriving the CB filter equation, optimization, and implementations for linear detection systems. Also, this approach minimizes the chance of obtaining a  
Figure 1: Typical Optical Communication System [8]. closed-form \(E_{filter}\). Against such a background, we propose a simple and effective MGF receiver filter modelling methodology in this paper. This new approach can benefit the CB method while avoiding its pitfalls. The work [17] derived a Bayes optimal receiver using the Karhunen-Loeve expansion and upper CB for system performance. This work does not offer an easier way, but it handles the aspects of filter optimization using different mathematical criteria. Figure 2 shows the main four underwater channel models for \(h_{c}(t)\)[8]. All underwater CIR models are rooted in the primary Double Gamma Function (DGF) channel model. These CIR models differ based on the scope of propagation impairments for the underlying water type [8]. The primary UWOC channel impairments are absorption, scattering, and scintillation. The last two have pronounced impacts on the temporal and spatial profile of the transmitted optical pulse, causing ISI and power distribution distortion on the surface of the photodetector [8]. In this communication, we utilize a DGF model to describe the temporal profile of the \(h_{c}(t)\). The DGF water channel model provides enough realistic scope of the UWOC channel deficiencies. Also, the DGF model parameters for Coastal and Harbor waters are available [8]. Figure 2 shows the temporal profiles of the UWOC propagation channel models for Coastal and Harbor waters. Figure 2: Water channel impulse responses \(h_{c}(t)\)_of_ four channel models [DGF, WDGF, BP, CEAPF] and the corresponding received optical pulses \(r_{opt}(t)\) for a transmitted optical pulse \(s(t)\) with pulse width = 0.25, Bit rate: 200 Mbit, water link span: 100 m.  
The work in [8] provides a good survey for the UWOC research activities. Yet, UWOC technology, from a communication engineering perspective, is an evolving field with ongoing research and advancements. The summary of the main directions of explorations and accomplishments that made notable contributions in the state-of-the-art UWOC systems are (I) Modulation techniques and Wavelength selection: Researchers in [19; 20] have been developing efficient modulation techniques for underwater optical channels. These techniques aim to address challenges such as signal attenuation, scattering, and absorption in water. Also, choosing appropriate wavelengths for communication is crucial in underwater environments. The researchers have explored different parts of the electromagnetic spectrum to find wavelengths that offer better UWOC transmission characteristics [21], (II) Multiple Input Multiple Output (MIMO) Transmission [21]: MIMO systems have been investigated to enhance the capacity and reliability of UWOC links by utilizing multiple transmitters and receivers simultaneously, (III) Advanced Signal Processing [20; 22] and Energy-Efficient Communication Protocols [23]: Signal processing algorithms have been developed to mitigate the impact of underwater channel distortions at a limited scale, including methods for equalization, error correction, and adaptive modulation constructed on the advancement in optical fiber communication. Also, developing energy-efficient communication protocols [23] is essential for UWOC systems, especially those relying on battery-powered devices.