1. Introduction

With the development of science and technology, marine applications such as underwater wireless sensor networks [1], autonomous underwater vehicles (AUV) [2], and underwater defense systems have attracted intensive attention. With the real-time communication needed in these marine applications, increasingly higher transmission speed is demanded in underwater wireless communication (UWC) systems. Traditional UWC mainly relies on acoustic systems, which have achieved several tens of kilometers of data transmission [3]. However, due to the small modulation bandwidth (only kHz) [4] and the slow propagation speed of acoustic waves in the underwater channel, the low communication data rate and high communication latency become key limitations. On the other hand, underwater radio frequency (RF) communication suffers from high attenuation leading to highly limited transmission distance (only a few meters to tens of meters) [5]. Hence, RF technologies are not adopted in the underwater environment.

To overcome the limitations of conventional UAC systems, the UOWC technology has been proposed and widely researched due to its great potential to achieve a high data rate reaching Gbps, thanks to the high carrier frequency and modulation bandwidth (exceeding MHz [6] and even GHz [7]). Furthermore, due to the high propagation speed of the light wave, the physical transmission latency in UOWC systems is also much lower than that in UAC systems. Typically, UOWC systems use light-emitting diodes (LEDs) or laser diodes (LDs) as transmitters and use photodiodes (PDs) or photomultiplier tubes (PMTs) as receivers. The direct line-of-sight (LOS) link is most widely used in UOWC due to its simplicity. However, a LOS link is not always possible due to the potential blockages by marine life, reefs, bubbles, and suspended particles [8]. Hence, the non-line-of-sight (NLOS) link which uses the water-air surface to reflect the signal and avoid obstacles is proposed in [9]. Since then, due to the advantage of solving the channel blockage issue, the NLOS UOWC system has attracted considerable attention [10–12].

In [13], the authors investigated a 20 MHz bandwidth NLOS UOWC system performance using MC simulations. Simulation results show that the received signal power in the NLOS link is much lower than the LOS link, when the signal incident angle to the water-air surface is beyond the critical angle due to the reflection loss. In [11] and [12], the authors presented an NLOS UOWC channel model using the vector radioactive transfer theory. Furthermore, in [14], the authors investigated the performance of the NLOS UOWC system with both LED and LD transmitters considering the impact of both transmitter and receiver geometrical configurations. Moreover, in our previous work [10], we presented an NLOS UOWC system model considering the impact of incident angles of both signal light and sunlight. However, in all of these previous works, the authors only considered the flat water surface to reflect the signal light for simplicity. In fact, natural water surfaces are almost impossible to be flat, especially the ocean surface. Hence, the NLOS UOWC system modeling and performance studies in these previous works are not accurate in practical environments.

To overcome this limit, some studies have further considered wavy surfaces in the NLOS UOWC link. For instance, in [9], the authors investigated the NLOS UOWC system considering reflective channels with wavy surfaces at different wind speeds. However, the wavy surface was modeled as the simple slope angle Gaussian distribution following the Gram-Charlier series. Moreover, in [15], the authors investigated the received signal power and path loss in the NLOS UOWC system considering a random air-sea interface. However, for simplicity, they considered random surface slopes following an isotropic Gaussian distribution measured from Cox and Munk [16]. In addition, the simple Gaussian distributed wavy surface was considered in recent air-water optical wireless communication studies [17,18]. Such simple Gaussian distribution wavy model neglects two important factors, the spatial correlation and the time relativity of wave. To overcome these limitations, the Pierson wave model was proposed in [19], which enabled the ocean wave model to progress from two-dimensional (Gram-Charlier wave model) to three-dimensional (Pierson wave model). Recently, a few works investigated the UOWC system performance considering the Pierson wave model. In [20], the authors investigated a dynamic water-to-air optical wireless communication system considering the Pierson wave model on the air-water interface. However, such an accurate wave surface model has only been considered in direct LOS UOWC systems.

To solve this issue, in this paper, we establish an accurate NLOS UOWC system model based on the Pierson wavy surface which contains both spatial correlation and time relativity information of wave incorporating wind speeds. MC simulations are used to investigate the impact of wavy surfaces under different wind speeds on the NLOS UOWC system. Results show that the wavy surface adversely affects the system performance, and hence, indicate that the NLOS UOWC system performance obtained under an ideal flat water surface is overestimated. To reduce the impact of ocean waves on the system performance, we further study the application of MIMO in the NLOS UOWC system. Results show that MIMO has the potential to reduce the probability and variance of the reduction in received signal power, improving the stability and reliability of the NLOS UOWC system under practical ocean waves. However, results also show that further increasing the number of receivers may not further improve the system performance, highlighting the need of detailed analysis in practical settings.

The remainder of the paper is organized as follows. In Section 2 we present the proposed NLOS UOWC system model incorporating the Pierson wave model under different wind speeds. In Section 3, we investigate the system performance through MC simulations. We also investigate the performance when further incorporating the MIMO principle. Then we apply the proposed model in practical environments, and key findings are discussed in Section 4. Finally, we conclude our work in section 5.

2. NLOS UOWC system with wavy surface

In this section, we first briefly present the Pierson wave model. Then we introduce the proposed NLOS channel model incorporating the wave model, where the transmitter and the receiver communicate using the signal light reflected by the wavy surface.

2.1 Pierson wave model

Pierson wave model can be viewed as a superposition of infinite simple Cosine and Sine waves which have different amplitudes, frequencies, and initial phases of $\alpha$ in the $(x,y)$ plane. The sea surface height $z$ at point $(x,y)$ at time $t$ can be expressed as [20,21]:

According to [20,21], the wave amplitude $a_{ij}$ can be expressed as:

Hence, the directional spectrum function of ocean waves $S(\omega,\alpha )$ can be expressed as [20,21]:

2.2 NLOS UOWC system model

We now establish the NLOS UOWC system model incorporating the above Pierson wave model. Here, the transmitter is modeled as a Lambertian source located in the coordinate origin $(x_0,y_0,z_0)$. As shown in Fig. 1, we differentiate the signal beam by the azimuthal angle $d\phi _{A}$ and zenithal angle $d\phi _{Z}$ of the central signal beam, which has a vector $\vec {n}_{LED}(\phi _{LED, A},\phi _{LED, Z})$ in the spherical coordinate system. Each divided signal optical beam carries a signal optical power of $P_{k,n}(\phi _{A,k},\phi _{Z,n})$. The total signal transmission power can be expressed as:

 

Fig. 1. Signal optical beam differential.

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After being radiated by the LED, each signal light propagates in the underwater channel. When the signal light with an initial propagation vector $\vec {v}_{signal(\phi _{A,k},\phi _{Z,n},\phi _{LED, A},\phi _{LED, Z})}$ reaches the wavy surface at the transit point $(x_t,y_t,z_t)$, according to (2.1), the norm vector at the transit point can be calculated as:

At the water surface, part of the light refracts to the air and the remaining reflects toward the receiver with a reflection vector $\vec {v}_{r}$, which causes reflection loss $L_{r}$. If the $u$th signal light after reflection can be detected by the PD as shown in Fig. 2, which is located at $(x_r,y_r,z_r)$ with the capture area radius $r_{PD}$ and norm vector $\vec {n}_{PD}(\theta _{PD, A},\theta _{PD, Z})$, we record its initial carrier optical power $P_u(\phi _{A, u},\phi _{Z, u})$, the signal transit point $(x_{t, u}, y_{t, u}, z_{t, u})$, the incident angle to wavy surface $\varphi _{t, u}$, and the arriving location at PD plane $(x_{r, u}, y_{r, u}, z_{r, u})$. Moreover, the optical filter transmittance $T_{f}(\lambda,\theta _{s, u})$ is also considered. It is related to the $u$th signal light incident angle $\theta _{s, u}$ [10], which can be calculated by:

 

Fig. 2. NLOS UOWC system considering wavy surfaces architecture.

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Hence, the $u$th signal light reflection loss $L_{r, u}$ can be expressed as [10–12]:

Moreover, when the signal light propagates through the underwater channel, general turbidity leads to the absorption and scattering loss, which can be expressed as [10–12]:

After passing through the optical filter, the $u$th signal power reaching the PD is:

3. Simulation and results

3.1 Simulation settings

Based on the theoretical framework described in the previous section, we investigate the performance of the NLOS UOWC system under different wind speeds. In the simulation, we first used the Pierson wave model to generate an $80 m \times 80 m$ ocean surface in the $(x,y)$ plane with wind speeds ranging from $0 m/s$ to $12.5 m/s$. It needs to be mentioned that we set a random time $t$ in every simulation. The key parameters used in the simulation are shown in Table 1.

Table 1. Pierson wave model parameters

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The waves generated following the Pierson model under different wind speeds are shown in Fig. 3. It is clear that the ocean surface is flat at the wind speed U of 0 m/s. Moreover, the wave height increases following the increment in wind speed. It is also clear that the ocean surfaces under the same wind can be different. This is mainly due to the random initial phase of the surface wave as described by (2.1). In addition, from the color bars that show the height of surfaces, it is obvious that the wave height under the same wind speed is distributed within a small range that follows the Pierson wave model described by (3).

 

Fig. 3. Pierson wave model under wind speed U from 0 m/s to 12.5 m/s.

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After generating wavy surfaces, we used MC simulations to investigate the received signal power and SNR in the NLOS UOWC system under different wave surfaces caused by different wind speeds. The NLOS UOWC system considered is shown in Fig. 4 and the flowchart of MC simulation is shown in Fig. 5. We generated $n_{ph}$ photons at the LED transmitter. The wavelength distribution and initial radiation direction followed the LED emission spectrum and Lambertian pattern. The initial power of each photon is $P_{ph}=\frac {P_{t}}{n_{ph}}$. Then, the photons propagated through a single travel distance $D_{single}=-\frac {log(N[0,1])}{c(\lambda )}$ in the underwater channel. If the photon arrived at the wavy ocean surface, we calculated and updated the optical power, location, and direction of the photon after reflection. Then the photon propagated the remaining part of $D_{single}$. If the photon reached the receiver within both the aperture area and the FOV, we recorded the parameters of the arriving photon and further calculated the optical filter transmittance $T_f$ following the photon incident angle. Finally, we recorded the power of each photon after passing through the optical filter. If the photon did not arrive, it was considered to interact with a particle in the medium, and it lost a scattering power, which is given by $L_{scattering}=1-\frac {a(\lambda )}{c(\lambda )}$ and propagated in a new scattering direction. The new propagation direction is determined by the new azimuthal angle $\zeta _{a}$ and scattering angle $\zeta _{s}$, which are shown in Fig. 4 and follow a random distribution $N[0,2\pi ]$ and the Henyey-Greenstein (HG) model, respectively [23]. Then the photon propagated another single travel distance $D_{single}$ until it reached the receiver.

 

Fig. 4. NLOS UOWC system setting in the Monte Carlo simulation.

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Fig. 5. NLOS UOWC system MC simulation flowchart.

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If the photon did not arrive at the wavy ocean surface after the initial propagation, it was also assumed to interact with a particle, which caused the photon to lose a scattering power and to propagate in a new direction until it arrived at the wavy ocean surface. After reflection, the photon was considered following the same process. It needs to be mentioned that the optical power of each photon was compared with a critical value $\xi$ before each scattering process, where we defined the photon as missing if the power is lower than $\xi$.

The key parameters of the transmitter, medium, and receiver used in the simulation are listed in Table 2. In the simulation, we considered a blue LED with the central wavelength $\lambda _c=472.5$ nm and the total transmitter power $P_t=1020$ mW [24]. The number of photons $n_{ph}$ generated in each simulation was $2\times 10^{7}$. Then we used a concentrating lens to reduce the LED beam semi-power angle $\phi _{1/2}$ to $30^\circ$, which was widely used in previous works [25,26]. We considered the clear ocean medium with the absorption coefficient $a(\lambda )=0.07$ m$^{-1}$, scattering coefficient $b(\lambda )=0.08$ m$^{-1}$, and attenuation coefficient $c(\lambda )=0.15$ m$^{-1}$ [23]. In addition, we considered a nine-layer blue pass thin-film optical filter [10] placed in front of the PD, where $r_{PD}=2$ inches and $\theta _{FOV} = 60^\circ$, to reduce the impact of solar background noise.

Table 2. Transmitter, medium, and receiver parameters

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In the system geometrical setting, variations in LED and PD depths, and LED direction angles directly affect the channel length and the corresponding communication performance. Additionally, the attenuation coefficient of water, coupled with LED depth, determines if the photon can arrive at the water surface. In our simulation, clear oceanic conditions are considered, and hence, we set the LED and PD at the same depth of $H_{LED}=H_{PD}=5$ m, and the horizontal distance between LED and PD at $D_{TR}=17.32$ m. It is worth mentioning that the LED and the PD orientation angles $\phi _{LED}$ and $\theta _{PD}$ are defined as $0^\circ$ when facing up vertically. These angles are defined as positive when rotating in a clockwise direction. Here, we set $\phi _{LED}=60^\circ$ and $\theta _{PD}=-60^\circ$.

We ran the simulation for 200 times under the same wind speed (i.e., 200 different wavy ocean surfaces generated) and recorded the received signal optical power. We considered different wind speeds $U$ of 0 m/s, 2.5 m/s, 5 m/s, 7.5 m/s, 10 m/s, and 12.5 m/s to investigate the influence of Pierson wavy surface under different wind speeds on the received signal optical power. In addition, we have also conducted simulations considering the previously studied sine/cosine wave model [18] under the same NLOS channel setting. This is achieved by setting $M=2$ and $N=1$ in the Pierson wave model denoted by Eq. (1). The received signal power is then studied accordingly at wind speeds of 2.5 m/s, 7.5 m/s, and 12.5 m/s. The results are shown in Fig. 6.

 

Fig. 6. Received signal power distribution under U from 0 m/s to 12.5 m/s.

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It is clear that with the flat ocean surface, the received signal power almost follows a normal distribution. Automatic beta distribution fitting from MATLAB shows that the received signal power with the highest probability is 0.0207 mW. Furthermore, when considering the ideal flat ocean surface, the UOWC system has the largest average received power and the smallest standard deviation over 200 simulations. With the increment in wind speed, the surface wave height starts to increase and the average received power starts to decrease. Moreover, the fitted curve shows that the probability of achieving the average power or higher decreases with the wind speed. This is mainly because the irregular wavy ocean surface reflects the signal light in different directions, reducing the probability of signal photons reaching the PD with limited aperture size and limited FOV. In addition, the variance of received signal power increases significantly with the wind speed, which shows the decrease in stability of NLOS UOWC system performance.

In addition, we can see that when the wind speed increases, the received signal power distribution changes from normal distribution to Beta distribution. In addition, the probability of weak signal reception increases, which means a higher communication failure possibility in practical applications. It is worth noting that although the probability of achieving the average power or higher decreases as the wind speed increases, the average received signal power does not decrease substantially. When the wind speed increases from 5 m/s to 12.5 m/s, the change in average power is minimal. This is mainly because the wave becomes stronger with increasing wind speed and the signal path becomes shorter when the wavy surface drops down. This causes a lower signal path loss which increases the probability of obtaining a stronger signal power. Due to the same reason, in terms of the maximum received signal power, an increasing trend is seen when the wind speed increases. For example, when the wind speed is 0 m/s or 12.5 m/s the maximum received signal power is about 0.03 mW and 0.06 mW, respectively. The received optical power obtained using the previous wave model under different wind speeds are shown in Fig. 6(g), Fig. 6(h), and Fig. 6(i). The sine wave model exhibits periodicity and hence, leads to reoccurring power patterns. This is characterized by a low standard deviation, which underestimates the variation. This simple model also underestimates the average power, especially at high wind speed as the stable high wave slopes appearing in some geometrical phases block the signal light. As a result, systems designed and optimized using the sine/cosine model may underutilize the actual capacity and at the same time, not budget sufficient margin for performance variations. Overall, the previous NLOS UOWC system [10,12] which only considered the flat water surface ignored the large communication quality variation under the strong winds in practical situations. On the other hand, our proposed model incorporating the Pierson wave model can better reflect the changes and patterns of the received signal light under different wind speeds. Moreover, compared with the previous NLOS UOWC system considering the simple slope angle Gaussian distribution [15], our proposed model can provide the system communication quality variation for a continuous period of time, which is explained in detail in Section 4.

3.2 Performance enhancement with MIMO

From the results shown in Section 3.1, it can be seen that the wavy surface affects the performance of NLOS UOWC systems significantly. Moreover, the large variance of the received signal power at strong winds significantly reduces the reliability of the underwater wireless link. To improve the system performance under strong wind, the MIMO principle exploring multiple channels to transmit signals with multiple transmitters and receivers provides a promising solution. Hence, based on the accurate NLOS UOWC system model built in Section 2, here we further study up to $2\times 4$ MIMO NLOS UOWC system as shown in Fig. 7. It needs to be mentioned that the same data is sent by multiple transmitters, leading to spatial diversity in MIMO.

 

Fig. 7. 2$\times$4 MIMO NLOS UOWC system considering wavy surfaces architecture.

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To investigate the enhancement enabled by MIMO, we set the wind speed $U = 12.5$ m/s. Then we use two LED transmitters and four PD receivers shown in Fig. 7 in the proposed NLOS UOWC system. Similar to the single-input-single-out (SISO) system geometrical setting, we set the transmitter central location $H_{Tx}$ and the receiver central location $H_{Rx}$ to 5 m, $D_{TR}$ to 17.32 m, and follow the same parameters for transmitters, medium, and receivers as shown in Table 2.

The locations of LEDs are shown on the left side of Fig. 7. We set $\Delta D_{LED}=8.966$ m and $D_{TR}=16.73$ m so that the signal path length is the same as the SISO NLOS UOWC case. Since the ocean wave period is tens of meters, we select a relatively large distance between the two transmitters to reduce the probability that the same surface wave affects multiple channels.

Moreover, we set the distance between each of the four receivers at $\Delta D_{PD, C}=0.5m$. Similarly, since the strong wave increases the probability of signal light being reflected off-center, we choose a relatively large distance between multiple receivers. We first activated one LED transmitter and two PD receivers (PD1 and PD3 in Fig. 7) in the NLOS UOWC system. Similar to the SISO case, we also ran the simulation for 200 times and recorded the received signal power. We adopted the Select Best (SB) recombination method at the receiver for subsequent signal processing.

In addition, we ran the same simulations when the number of activated receivers and transmitters changed from two to four and from one to two, respectively. When two transmitters were used, we reduced the emitted power of each LED by half ($P_{t1}=P_{t2}=510$ mW to keep the same total transmission power) and kept the same beam divergence for LED1 and LED2. As a comparison benchmark, we also investigate the MIMO system performance with the previous sine/cosine wave model [18]. The received signal power is shown in Fig. 8.

 

Fig. 8. Received signal power distribution under U = 12.5 m/s in the MIMO system.

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Compared with the SISO results shown in Fig. 8(a), when we increase the number of receivers both the received signal power with the highest probability and the average received signal power increase. This is mainly because the multiple PDs can collect the diverse signals reflected in different directions under the strong surface wave. However, although the probability of obtaining weak signal power is reduced, the standard deviation of received signal power remains large, which almost remains unchanged after increasing the number of receivers. This shows that the stability of system performance does not improve substantially with more receivers.

Results of the NLOS UOWC system with multiple transmitters shown in Fig. 8(d), Fig. 8(e), and Fig. 8(f). It is clear that the average received power decreases, and this is caused by the orientation angles of the receivers. It is worth mentioning that the received power with the highest probability increases when increasing the number of transmitters. This is mainly because the multiple signal paths in different directions enable the receiver to collect more diverse signals reflected in more directions under the strong surface wave. More importantly, the standard deviation of received signal power decreases significantly, showing the capability of improving the stability of the NLOS UOWC system under strong wavy surfaces. Compared with the results obtained using the sine/cosine wave surface model, it is clear that the average power and the variation are also underestimated, which further verifies that the proposed model considering the Pierson wave model can predict the communication capacity and performance variations more accurately. Overall, the MIMO principle can significantly enhance communication stability and reduce the probability of receiving weak signals.

Furthermore, we also investigated the impact of transmitter divergence, where the $2\times 4$ MIMO NLOS UOWC system under strong wavy surfaces (wind speed of 12.5 m/s) was considered. We first kept the LED1 beam divergence at $\phi _{1/2}=30^\circ$, and increased the LED2 beam divergence $\phi _{1/2}$ from $40^\circ$ to $60^\circ$. We repeated the simulation for 200 times. Moreover, we also increased the LED1 beam divergence $\phi _{1/2}$ from $40^\circ$ to $60^\circ$ while keeping the LED2 beam divergence at $\phi _{1/2}=60^\circ$. We also ran simulation for 200 times. The results are shown in Fig. 9.

 

Fig. 9. Received signal power distribution under U = 12.5 m/s in $2\times 4$ MIMO system under different transmitter divergence angles.

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It is clear from Fig. 9(a), Fig. 9(b), and Fig. 9(c) that when we kept one signal beam relatively concentrated and the other relatively divergent, both the average received signal power and the received signal power with the highest probability decrease slightly. This is mainly due to the smaller power density per unit area for the signal beam with a large divergence. Nevertheless, the use of a wide signal beam can effectively reduce the impact of wavy surfaces on the system stability, where the standard deviation of received signal power decreases sharply. In addition, when we kept one relatively divergent beam and increased the divergence of the other transmitter, the mean and standard deviation of received signal power as well as the received power with the highest probability all decrease, as shown by the results in Fig. 9(d), Fig. 9(e), and Fig. 9(f). Although it is desirable to have a low standard deviation to achieve a stable system performance, the probability of receiving a weak signal is greatly increased and the average power is significantly reduced. Hence, compared with the results shown in Fig. 9(c) and Fig. 8(a), a MIMO NLOS UOWC system with a combination of concentrated transmitter and more divergent transmitter provides a better option to reduce the impact of wavy water surfaces.

4. SNR and BER analysis

In addition to the received signal power, we also investigated the impact of wavy surfaces on the SNR and BER of the NLOS UOWC system. With the signal modeled by (16) and the solar noise modeled in our previous research [10], the SNR can be calculated as:

Based on SNR, the BER of the NLOS UOWC link can be further calculated. Here we consider the on-off-keying (OOK) symbol modulation that is widely used in UOWC systems [29], and the BER can be expressed as [30]:

We investigated the SNR and BER performance of the SISO, $2\times 4$ MIMO (beam divergence of two transmitters $\phi _{1/2}= 30^\circ$), and $2\times 4$ MIMO with beam shaping (LED1 $\phi _{1/2}=30^\circ$ and LED2 $\phi _{1/ 2}=60^\circ$) NLOS UOWC systems considering the wavy ocean surface under 12.5 m/s wind speed. In order to investigate the impact of ocean waves on the NLOS UOWC system over a continuous period of time, we considered the movement of the ocean wave during a period of 200 seconds. Here, we used the same system configuration as that described in Section 3. The bandwidth of the system was set to $B = 10$ MHz and PD responsivity was $\mathfrak {R} = 0.9$ [11].

Results are shown in Fig. 10. It is clear that compared with the SISO system, the MIMO system improves the SNR performance by nearly 2.5 dB (mulberry line). The main reason is the same as that presented in Section 3.2, where the spatial distribution of multiple channels can reduce the impact of ocean waves on the system, thereby increasing the probability of receiving high signal power, which further improves the SNR performance. Moreover, when we use one divergent transmitter together with one narrow-beam transmitter, the SNR decreases, but is still higher than the SISO system. However, the standard deviation of SNR becomes better, which is about 2.3 dB and 0.25 dB smaller than the SISO and MIMO systems, respectively. This is mainly because the divergent signal beam effectively reduces the impact of irregularly reflected signal lights caused by the wavy surface.

 

Fig. 10. SNR and BER under wind speed U = 12.5 m/s in the SISO, MIMO, and MIMO considering beam shaping NLOS UOWC system.

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In addition, we considered the forward error correction (FEC) limit of $3.8\times 10^{-3}$ in the BER analysis of the considered NLOS UOWC system [31]. It is clear that in the SISO system, due to strong surface waves caused by high wind speed, the BER in many time periods is higher than the FEC limit, leading to severely affected communication quality. On the other hand, when we adopt the MIMO system, the period of time with BER exceeding the FEC limit reduces significantly, showing better link reliability. The link reliability further improves with the MIMO system employing beam shaping.

From the results shown in Fig. 10, it can be seen that a larger number of receivers under the considered condition leads to better system performance. However, it is worth mentioning that more receivers do not necessarily lead to better performance. Here, we further investigate the impact of the number of receivers on the SNR and BER performance. We considered the same ocean wave under a wind speed of $12.5$ m/s during the same period of 200 seconds and used the same system configuration as that described in Section 3. The four extra receivers are located between the existing four receivers, and the distance from the center point is the same 0.5m. The results of the 1$\times$4, 1$\times$6, and 1$\times$8 NLOS UOWC systems are shown in Fig. 11.

 

Fig. 11. SNR and BER under wind speed U = 12.5 m/s in the 1$\times$4, 1$\times$6, and 1$\times$8 SIMO NLOS UOWC system.

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When the number of receivers increases from four to six, there is only a small improvement in the SNR performance. When the number of receivers further increases from six to eight, the average SNR remains almost unchanged. This is also reflected in system BER performance. This is because as the number of receivers increases, the spatial correlation between channels typically increases as well. Hence, the signals detected by different receivers become highly similar and the system performance is no longer improved. It is worth mentioning that we considered a relatively low-speed system that ignores the time domain dispersion. When it is considered, due to different channel lengths, the performance is further affected. In addition, we use different transmitters to send the same data. If different data streams are transmitted, further inter-channel interference needs to be considered. Hence, continuously increasing the number of receivers cannot linearly improve the performance of the NLOS UOWC system. The optimal number of receivers needs to be selected based on the channel and system status.

5. Conclusion

In this paper, we have established the accurate NLOS UOWC model considering practical ocean wavy surfaces, which are either ignored or oversimplified in previous studies. Moreover, we have investigated the performance of the NLOS UOWC system via our proposed model by using MC simulations. Results have shown that the wavy surfaces increase the probability of receiving a weak signal, and as the wind speed increases, the impact becomes larger. In addition, we have also studied the MIMO principle to enhance the performance of NLOS UOWC systems. Results have shown that the MIMO NLOS UOWC system can significantly reduce the probability of receiving a weak signal and reduce the standard deviation of the received signal power, which can improve both the data transmission performance and the reliability of the NLOS UOWC system. Therefore, our research shows the influence of ocean wavy surfaces on the NLOS UOWC system under different wind speeds and proves the effectiveness of the MIMO principle in reducing the influence of strong wavy surfaces, providing a guideline for the design and analysis of practical NLOS UOWC systems. Finally, we have investigated the SNR and BER performances in a practical system over a certain period of time. Results have confirmed that the MIMO setting has the potential to improve the SNR and BER performance, but the improvement does not simply scale up with the number of transceivers. It is worth mentioning that MIMO systems with different channel paths lead to time-domain signal spread. Here, we only focus on the enhancement of strong wave NLOS UOWC system performance enabled by the MIMO principle. Hence, we considered similar channel lengths and a relatively limited transmission data rate, leading to insignificant time domain impact. Furthermore, whilst extensive simulations have been conducted, the experimental verification of the accuracy of the proposed NLOS UOWC system model incorporating the Pierson wave model is highly desirable as the future work.