# Cooperative Terrestrial-Underwater FSO System: Design and Performance Analysis
## 2 System Model and Definitions  
This section describes the new hybrid communication system. As shown in Figure 1, the hybrid FSO-TU system has two different hops or links, both of which are considered point-to-point optical links: the terrestrial link and the underwater link. Both links operate with an intensity modulation with a direct detection (IM/DD) system. Therefore, intensity modulation (IM) varies the amplitude of the optical signal transmitted on both links to carry the information, so that a certain optical power is emitted to transmit a \"1\" and no power is emitted to transmit a \"0\". This traditional modulation scheme is known as on-off keying (OOK) and is chosen here because of its low cost and simplicity of implementation [25]. At the receiver, the optical signal is then directly detected (DD) in a photodetector, which converts it into an electrical signal proportional to the received power. We note that, as seen below, in such harsh environments, high levels of attenuation and signal degradation are expected. As shown in Figure 1, the terrestrial link consists of an optical transmitter located on land and a receiver located at sea level on a structure that is assumed to be sufficiently stable to be unaffected by the characteristic maritime movements that could greatly degrade the performance of the link. The signal current induced in the terrestrial receiver can be described as follows:  
\[i_{1}=xR_{1}h_{1}+i_{n1}, \tag{1}\]  
where \(R_{1}\) represents the detector responsivity measured in A/W, with \(x\) denoting the transmitted intensity measured in W, and with \(h_{1}\) being the channel attenuation factor. Furthermore, it is assumed that the detector current noise signal, \(i_{n1}\), is modelled as a zero-mean additive white Gaussian noise (AWGN) with variance \(\sigma_{1}^{2}\). The transmitted signal is taken as symbols drawn equiprobably from an OOK constellation such that \(x\in\{0,2P_{1}\}\), and \(P_{1}\) is the average transmitted optical power. The signal-to-noise ratio (SNR) corresponding to the terrestrial link is defined as  
\[\gamma_{1}=\frac{P_{1}R_{1}h_{1}}{\sigma_{1}}, \tag{2}\]  
Figure 1: AF-based FSO-TU system. and can also be expressed in a simpler way as \(\gamma_{1}=\gamma_{a}h_{1}\) assuming that \(\gamma_{a}=(P_{1}R_{1})/\sigma_{1}\). The scheme of Figure 1 depicts the terrestrial receiver working as a relay node in an amplify and forward (AF) scheme, amplifying the signal that is later transmitted underwater. This amplification can compensate for part of the losses that occur when the optical signal propagates through the atmospheric medium. Once the signal is amplified at the relay node, it is transmitted under the seawater, resulting in an underwater link. This second link is assumed to be short range with a depth of up to several tens of meters. After covering this distance, the optical signal is picked up by a receiver in a submerged station, in an autonomous underwater vehicle (AUV) or in a submarine as illustrated in Figure 1. There, the signal current induced in the underwater detector can then be expressed as  
\[i_{2}=g{i_{1}}R_{2}h_{2}+i_{n2}, \tag{3}\]  
where \(g\) is the relay gain, and \(h_{2}\) is the attenuation factor of the underwater channel. In addition, \(R_{2}\) and \(i_{n2}\) are defined in the same way as \(R_{1}\) and \(i_{n1}\), described in Equation (1), but in this case corresponding to the underwater channel. For the sake of simplicity, the ambient ocean noise, \(i_{n2}\), is assumed to follow an AWGN model with variance \(\sigma_{n}^{2}\). However, this assumption can be improved by adding an impulsive noise [26; 27; 28] since the AWGN model ignores the impulsive occurrence of electromagnetic interference, ocean noise or noise caused by humans using other machines, especially for shallow water scenarios. Additional noise sources, such as water bubbles caused by ship propellers inside ports and harbors, will be further investigated and published elsewhere.