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Abstract
Although underwater wireless optical communication (UWOC) has the advantages of high speed, low latency, and high confidentiality, the transmission of light in water will be affected by the absorption and scattering of particles, which will lead to the aggravation of channel path loss as well as channel pulse spreading, finally causing false codes. Therefore, how to analyze the channel impulse response (CIR) effectively is a key task in channel modeling. In this paper, we consider a two-way underwater vertical line-of-sight (LOS) communication system model, based on the inherent optical property (IOP) model of chlorophyll, using the Kopelevich phase function containing water depth information, the CIR curves under different water types and transceiver configurations are plotted using the Monte-Carlo Simulation (MCS). The obtained simulation results are fitted with the double gamma function (DGF) model and the Gaussian model, respectively. The Gaussian model exhibits better properties than the DGF model in each water condition. Based on the closed-form expression of the CIR obtained from the Gaussian model, we solve for the bit error rate (BER) and 3-dB bandwidth of the system under different settings. The conclusions obtained can be used for the design and optimization of underwater vertical channels.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
With the development of the global economy, land resources can no longer meet human needs. Two-thirds of the Earth’s surface is covered by oceans, therefore, there is a growing interest in the study of ocean communication systems. Underwater wireless communication (UWC) refers to the transmission of data through wireless carriers in an underwater environment, which can be radio frequency (RF) waves, sound waves, and light waves. Due to the bandwidth limitation of RF systems and underwater acoustic communication systems, underwater wireless optical communication (UWOC) technology has aroused researchers’ attention in recent years with its advantages of higher data transmission rate, lower link delay, and good confidentiality [1,2].
Although UWOC systems have the above significant advantages, the presence of organic and inorganic particles in seawater can cause the transmission of light in underwater environments to be affected by absorption and scattering, ultimately leading to pulse spreading and increased path loss [3]. In this case, it is necessary to effectively evaluate UWOC system characteristics such as path loss and channel impulse response (CIR) to accurately calculate the link budget when designing a UWOC system. UWOC link attenuation can be modeled using the radiation transfer equation (RTE), which is an equation for the attenuation of electromagnetic wave transmission, and it analyzes the transmission process of light in a medium [4]. RTE can be solved by analytical methods or numerical methods, but RTE is a complex integral differential equation involving multiple variables, so it is difficult to find an exact analytical solution [5]. Only a few analytical RTE models have been proposed in recent years. Jaruwatanadilok [6] devised an analytical solution for RTE employing the modified Stokes vector. Compared with analytical solutions, numerical methods are preferred to solve the RTE. The most common numerical method for solving RTE is Monte-Carlo simulation (MCS).
MCS is a probabilistic method for simulating underwater light propagation losses by sending and tracking large numbers of photons, which is flexible, easy to program, and provides precise solutions. In recent times, a number of researchers have employed MCS approach to solve the RTE or study the characterization of UWOC channels. Li et al. [7] built a Monte Carlo simulator to simulate the impulse response of UWOC channels. The authors also used this simulator to evaluate the 3-dB bandwidth of the system for different link ranges and water conditions [8]. Xu et al. proposed a new approach to solve the radiation transmission equation and path loss using a laser diode (LD) light source, which not only combines MCS with a partially pruned deep neural network (PPDNN) but also uses the Fournier–Forand (FF) phase function that has hardly been used by previous researchers [9]. The researchers mentioned above have only studied UWOC channels, but the analysis of the UWOC system performance is more worthy of study as well as important. Cox et al. [10] used Petzold measurements and the Henyey-Greenstein (HG) phase function to simulate the MCS system loss of the communication link for different receiver aperture sizes and fields of view (FOV) and simulated the power loss between the receiver and the transmitter. Gabriel et al. [11] plotted the curves of CIR and quantified the channel time dispersion by considering different receiver aperture sizes. Based on [11,12] used the Two-Term Henyey-Greenstein (TTHG) model for photon scattering, evaluated the CIR of optical channels under different conditions, and also used simple on-off keying (OOK) modulation to evaluate the performance of a typical UWOC system in terms of bit error rate (BER).
In the above work, researchers have focused on horizontal scenarios, in which link attenuation is always modeled as constant values. This theory may apply to horizontal links, however, there are many vertical communication scenarios in practical applications such as port security, environmental monitoring, and tactical surveillance, an autonomous underwater vehicle (AUV) is often necessary to be arranged underwater to exchange information with a buoy or a ship on the surface of the sea. To meet the needs of current and future maritime applications, we consider a system model for a vertical UWOC link. In this case, chlorophyll in phytoplankton is the dominant component affecting the optical properties in shallow seawater. Chlorophyll concentration varies stratified with seawater depth, therefore, the effect of chlorophyll on channel attenuation must be considered when communicating vertically. Some previous workers have already investigated the variation of chlorophyll with depth. In [13], a one-parameter model based on chlorophyll concentration which follows Gaussian distribution with depth was used to calculate how light in seawater attenuates with depth. Based on [13,14] further investigated the effect of the vertical component on the propagation of light in UWOC links, and the results showed that the attenuation reaches its maximum when the beam passes through the region of the highest chlorophyll concentration. Based on this chlorophyll model, [15] studied the outrage probability of vertical UWOC links, however, the analysis of CIR was not mentioned in this work, and only the downlink communication was considered in their work.
In recent years, several researchers have used different models including double gamma function (DGF) [16], Beta Prime (BP) [17], a combination of exponential and arbitrary power function (CEAPF) [18], and so on to fit the UWOC impulse response, resulting in a closed expression for the CIR to solve for the system BER and 3-dB bandwidth.
Based on previous studies, in this paper we mainly compare the fitting accuracy of the DGF model and the Gaussian model for CIR under different channel conditions and system settings, which results in a closed-form expression for the model with the highest fitting accuracy, and compare the uplink and downlink communications based on this expression. For each scenario, we derived the system BER and 3-dB bandwidth to evaluate the system performance.
The rest of this paper is organized as follows: Section 2 introduces the system model of this study; Section 3 describes the channel model of this work and compares the advantages and disadvantages of different phase functions to explain the reasons for choosing the Kopelevich phase function; Section 4 introduces the detailed process of MCS; Section 5 analyzes the numerical results of the simulation and summarizes the effects of different factors on the link attenuation; The performance (BER and 3-dB bandwidth) of the system is analyzed in Section 6; Section 7 draws the conclusions of this paper.
2. System model
We consider a system model for a typical vertical communication scenario, which is shown in Fig. 1. A transmitter (Tx) and a receiver (Rx) are located both at the bottom of a ship or buoy and at the top of the AUV. For Tx, the light source uses a laser beam with its irradiance distribution obeying Gaussian distribution. Tx communicates with Rx through a scattered vertical line-of-sight (LOS) link with a stratified variation of chlorophyll concentration. L indicates the communication link distance, and different FOVs are set at the Rx end.
Two typical communication scenarios are considered in our work. The first one is to send a message from an AUV located underwater to a ship or buoy floating on the surface (uplink), where Tx is set on the AUV with coordinates (0, 0, L), and the Rx is deployed at the bottom of the buoy at the coordinate origin; The other scenario is data transferred from the buoy to the AUV (downlink), where Tx is mounted at the coordinate origin at the bottom of the buoy, and Rx is located at the top of the AUV with coordinates (0, 0, L). The vertical depth goes from L to 0 for uplink communication and from 0 to L for downlink.
3. Chlorophyll-dependent inherent optical properties (IOPs) model
The IOPs of seawater depend only on the properties of the medium and can give a comprehensive description of seawater’s optical properties. The IOP that determines the propagation of light in seawater can be summarized as two parts: absorption and scattering, and the two main properties of these two parts are the absorption coefficient and the volume scattering function. The absorption and scattering properties of seawater are mainly determined by the particles in seawater, of which phytoplankton, among organic particles, are the main ones that determine the optical properties of most seawater. Their chlorophyll affects the absorption and scattering of light and therefore plays a dominant role in determining the attenuation properties of seawater when its concentration is high [3]. Chlorophyll concentration varies with depth and the effect of chlorophyll concentration on light attenuation must be taken into account when a vertical link is considered, so accurate modeling of the chlorophyll concentration distribution with depth is necessary.
3.1 Absorption model
The absorption coefficient of the chlorophyll-based IOP model is formulated as follows [19]:
3.2 Scattering model
The volume scattering function $\beta \left ( \theta \right )$ (${{\text {m}}^{\text {-1}}}\cdot \text {s}{{\text {r}}^{\text {-1}}}$ ) can be expressed as the product of the scattering phase function (SPF) $\widetilde {\beta }\left ( \theta \right )$ ($\text {s}{{\text {r}}^{\text {-1}}}$ ) and the scattering coefficient $b\left ( \lambda \right )$ (${{\text {m}}^{\text {-1}}}$ ):
In our work, the total scattering coefficient can be expressed as $b\left ( \lambda,z \right )={{b}_{w}}\left ( \lambda \right )+b_{s}^{0}\left ( \lambda \right ){{C}_{s}}\left ( z \right )+b_{l}^{0}\left ( \lambda \right ){{C}_{l}}\left ( z \right )$, where ${{b}_{w}}\left ( \lambda \right )=0.005826{{\left ( 400/\lambda \right )}^{4.322}}$ is the pure water scattering coefficient, $b_{s}^{0}\left ( \lambda \right )=1.1513{{\left ( 400/\lambda \right )}^{1.7}}$ is the small particle scattering coefficient, and $b_{l}^{0}\left ( \lambda \right )=0.3411{{\left ( 400/\lambda \right )}^{0.3}}$ is the large particle scattering coefficient [19].
The composite SPF based on the chlorophyll-dependent IOPs consists of two parts: scattering of pure water and scattering of particles, and it can be expressed as follows:
Fig. 2. Comparison curves of different SPFs, where (a) uses absolute value transverse coordinates and (b) uses logarithmic transverse coordinates.
Table 1. Summary of common SPFs’ characteristics.
3.3 Impulse response modeling
In this part, we will present two models for fitting CIR: the DGF model and the Gaussian model.
DGF were first adopted to model the impulse response in clouds by Mooradian and Geller [26]. The form of such a function can be written as
The Gaussian function was proposed by mathematician Carolus Fridericus Gauss, and its mathematical expression is as follows:
where $a\left ( a>0 \right ),b$ and $c$ are real constants and $t$ is the time scale.After determining the fitting model, the parameters in the model can be computed according to the nonlinear least square criterion [27] by
where $p,q,r$ are the fitting parameters to be solved, ${{h}_{FM}}(t)$ denotes the function of the fitting model, which can be ${h}_{DGF}(t)$ or ${h}_{Gaussian}(t)$. ${{h}_{MC}}(t)$ is the CIR data obtained by MCS, and $\arg \min (\cdot )$ is the operator to return the argument of the minimum. Eq. (9) will be solved by the curve fitting tool in MATLAB.4. Monte-Carlo system model
In this section, MCS is used to statistically model the propagation and scattering behavior of photons in the vertical direction. The key idea is to simulate the multiple scattering process as a succession of elementary events whose probability laws are known [28]. Figure 3 shows the block diagram of our Monte Carlo simulation method, whose details are described next.
Our light source is Gaussian, and the number of photons and scattering times are initialized before the propagation process begins: set the number of photons $i=1$, the number of scattering times $k=1$, the initial weight ${{w}_{0}}=1$, the weight threshold ${{w}_{t}}={{10}^{-4}}$, and the rouletting threshold $\rho =10$. For the propagation process of each photon, the following sampling values need to be generated according to the corresponding distribution functions: the random step size $r$, the scattering angle $\theta$, and azimuthal angle $\varphi$, where regarding the distribution of the scattering angle, we use the forward lookup table (FLT) sampling method to generate the scattering angle, the main procedures of it are described as follows [29]: 1)The angular cosine is divided into several subintervals, and then the Kopelevich phase function is numerically integrated over −1 to 1 with $\cos \theta$ as the independent variable to obtain its cumulative distribution function (CDF) sampling value, which ranges between 0 and 1, and thus a lookup table of CDF values and corresponding scattering angle values is obtained. 2) A random number between 0 and 1 is generated, and the corresponding scattering angle cosine is calculated by searching the corresponding interval through the bisection search algorithm, in this way, its corresponding scattering angle ${{\theta }_{1}}$ can be obtained. We also draw the generated scattering angle by FLT method with its corresponding generated random number (histogram) and the CDF of the Kopelevich phase function (solid red line) in Fig. 4, which shows good fitness.
Fig. 4. Fitting diagram of Kopelevich-CDF (red solid line) and normalized histogram of scattering angle generated by FLT method
After that, update the photon coordinates and its direction cosine and judge whether the photon is located in the receiver FOV, if yes, we judge whether the photon is located in the receiver aperture, if yes, mark the photon as received, if the photon is not in the FOV or not in the receiver aperture, update the photon weight to ${{w}_{1}}$, if the photon weight is greater than the threshold ${{w}_{t}}$, then $k=k+1$, perform the next scattering event and repeat the above steps, the otherwise, a roulette judgment is performed [10]: for each subsequent optical event, a uniform random variable is compared to $\rho$ and the photon’s propagation is terminated if $\mathbb {R}>1/\rho$, else the new weight of photon is set as $\rho w$ and its scattering process will continue, where $\mathbb {R}$ is a random number chosen from a uniform distribution on [0, 1], $w$ is the old weight, and $\rho =10$ is the rouletting threshold. This approach reduces the amount of unnecessary computation in the system while maintaining the total energy of the system. The above process is repeated until the last photon completes its transmission. If a photon is marked as received, then we record the propagation distance ${{d}_{i}}$ of the photon and the received power ${{P}_{r}}$ for solving for path loss and CIR. In our simulation, ${{P}_{r}}$ is equal to the sum of the weights of all received photons. The path loss can be calculated by $PL=10\lg ({{P}_{t}}/{{P}_{r}})$, where ${{P}_{t}}$ is the transmit power, and it equals to the number of emitted photons (the initial weights of all photons are set to 1). CIR can be calculated by the following steps: 1) Calculate the propagation time of the received photons by $\frac {{{d}_{i\left ( i=1,2,3\cdots \right )}}}{v}$, where $v$ represents the speed of light in water; 2) After all photon propagation time statistics are completed, the maximum ${{t}_{\max }}$ and minimum ${{t}_{\min }}$ of the time are obtained. Take $\left [ {{t}_{\min }},{{t}_{\max }} \right ]$ as an interval, which is uniformly divided into ${{N}_{t}}$ time blocks noted as $\left \{ {{T}_{j}};j=1,2,3,\ldots,{{N}_{t}} \right \}$, then count the total intensity of the photons arriving in each time block marked as $\left \{ H[j];j=1,2,3,\ldots,{{N}_{t}} \right \}$; 3) The intensity is normalized by the lengths of the corresponding time intervals and the $h\left [ j \right ]$ is obtained in units of $\text {W/}{{\text {m}}^{\text {2}}}$.
5. Simulation results and discussions
In this section, we provide some simulation results mainly to study the characteristics of the vertical underwater system. We consider a LOS configuration where the transmitter and the receiver are perfectly aligned. We analyze the characteristics of the system we consider in three aspects, i.e., discussions under different water conditions, discussions under different system configurations, and performance evaluation. The following typical parameters are set in our simulation by default: the wavelength $\lambda$ = 532 nm, a beam width of $W$ = 1 mm, a beam divergence angle of ${{\theta }_{0}}$ = 4.2$^{\circ }$ and 42$^{\circ }$, FOV = 50$^{\circ }$ and 150$^{\circ }$, and a link distance of L = 5 m, 10 m, and 30 m. The water condition is S2, S4, S6, and S8. Unless otherwise specified, our communication direction is downlink.
5.1 Discussions under different water conditions
In this section, we compare the path loss, number of photon scattering, and CIR under four different water conditions when the system transceiver configuration is ${{\theta }_{0}}$ = 4.2$^{\circ }$ and FOV = 150$^{\circ }$, and we specify the link range as 10 m and 30 m.
5.1.1 Path Loss
Figure 5(a) shows the curves of path loss with various link distances when the HG, FF, and Kopelevich phase functions are used for communication under different conditions, respectively. It can be seen that the path loss under the S2 water condition is smaller than that under the S8 water condition when the link distance is the same regardless of which phase function is used. The plausible explanation is that higher chlorophyll concentration causes greater link attenuation. When under the same water condition, it can be seen that the path loss obtained using the Kopelevich phase function is the largest, followed by the FF phase function, and the HG phase function has the smallest path loss. This is because the HG phase function underestimates the backward scattering and the FF phase function only considers a set of particles of a specific size while the real ocean has particles of different sizes. More importantly, both the HG and FF phase functions do not consider the effect of chlorophyll on channel attenuation, therefore, both the HG and FF phase functions underestimate the path loss of the UWOC vertical channel. In contrast, the Kopelevich phase function classifies the suspended particles in seawater according to their scales, and the scattering degree of particles of different sizes is different, which is closer to the actual seawater situation.
Fig. 5. Variation of path loss with link distance using (a) different SPFs, and (b) water conditions.
Figure 5(b) shows the system path loss with the variation of the link distance L in four water types environments. Obviously, the path loss of all four water types shows an increasing trend as L increases. It can be seen that at the same distance, S8 has the largest path loss, followed by S6 which is slightly larger than S4, and when the link distance is 5m-15m, the corresponding path loss results of S4 and S6 are almost the same, and S2 is the smallest. This is consistent with the ranking of chlorophyll concentrations in these four water types, which indicates that the lower the chlorophyll concentration, the more photons are received and thus the smaller the path loss. Besides, we can find from Fig. 5(b) that the difference in path loss among the four water types increases with distance increasing, indicating that the higher the chlorophyll concentration of the water type, the more sensitive its path loss changes with the link range. We also plot the path loss calculated using the model of Beer-Lambert [30] (dashed line) and compare it with the MCS method we use. From the figure, we can see that the path loss obtained by Beer’s law is obviously higher than that obtained by MCS because it only considers a single scattering process and always overestimates the power loss. Furthermore, Beer-Lambert law can’t be used to estimate the CIR.
5.1.2 CIR
Figure 6 depicts the CIR curves for four different water types, and all curves are obtained when ${{\theta }_{0}}$ = 4.2$^{\circ }$ and FOV = 150$^{\circ }$. Figure 6(a) to Fig. 6(d) represents S2, S4, S6, and S8 in order, and the CIR at link range of 10 m and 30 m are compared under each water condition. We see that CIR curves for L=10 m always show higher peaks and narrower temporal broadening than those for L=30 m under any water conditions. This phenomenon implies that short propagation distance leads to low path loss and intersymbol interference (ISI), which is associated with good channel conditions for underwater optical communications. Taking S2 as an example, the peak of CIR of L=10 m reaches about 1250 W/${{\text {m}}^{2}}$, and the temporal broadening is about $0.9\times {{10}^{-7}}$ s, while the CIR maximum is only about 700 W/${{\text {m}}^{2}}$ when L=30 m, and the temporal broadening is about $1.75\times {{10}^{-7}}$ s, which is larger than that of L=10 m. Comparing the four sets of curves, it can be observed that under the same distance, the intensity of the CIR curve of S2 is the largest, reaching 1250 W/${{\text {m}}^{2}}$, followed by S4 and S6, and the peak of CIR of S8 is the smallest, only 270 W/${{\text {m}}^{2}}$. This can be explained by the fact that the higher the chlorophyll concentration in the water, the more serious the attenuation caused to the link, thus weakening the intensity of the CIR.
Fig. 6. Curves of CIR at different depths under the same water condition: (a) S2, (b) S4, (c) S6, and (d) S8.
5.1.3 Number of photon scattering
Figure 7 plots the number of photons received at the receiving end of the system for scattering times of n = 1, n = 2, and n = 4, respectively. As in the previous analysis on path loss, here we also investigate the scattering times for four water conditions. We compared the variation of the received photon number under different water conditions with different scattering times. ${{10}^{5}}$ photons are sent from transmitter under each channel condition. The results show that the received photon number gradually decreases with increasing communication distance for n = 1. The basic reason is that the longer distance leads to a longer path of light propagation, which results in increased path loss and photons need to be scattered more times to reach the receiver.
Fig. 7. Statistics of the number of received photons at different distances and scattering times: (a) S2, (b) S4, (c) S6, and (d) S8.
Comparing the four water conditions we can find that for the same system configurations and the same n, the water condition of S8 receives the least number of photons, S6 and S4 receive more photons, and S2 receives the most photons. By comparing the variation of the path loss in Fig. 5(b), we can observe that this is because the path loss is the largest for S8, followed by S6, and S4, and then the smallest for S2 for the same distance condition. The above results show that the variation of the scattering times is basically consistent with the variation of the path loss: with the increase of the communication distance, the path loss gradually increases, and the number of multiple scattering photons also increases. Besides, under the water condition which reaches the largest pass loss, the received photon number is also the least.
5.2 Discussions under different system configurations
In this section, we selected two representative water conditions: high chlorophyll concentration S8 and low chlorophyll concentration S2 for simulation. Under these two water conditions, the curves of path loss and CIR are obtained with three different system transceiver settings: (1) ${{\theta }_{0}}$ = 4.2$^{\circ }$, FOV = 50$^{\circ }$; (2) ${{\theta }_{0}}$ = 4.2$^{\circ }$, FOV = 150$^{\circ }$; (3) ${{\theta }_{0}}$ = 42$^{\circ }$, FOV = 150$^{\circ }$. After determining the configuration with the best performance, we fitted the CIR obtained with two typical models.
5.2.1 Path Loss
Figure 8 shows the system path loss for the three system transceiver configurations, and it can be seen that the path loss increases with the increase of communication distance for all three system configurations. Comparing the three curves under the same channel condition, we can find that the path loss reaches the largest and the system performance is the worst when ${{\theta }_{0}}$ = 42$^{\circ }$, FOV = 150$^{\circ }$, while the path loss is approximately the same when ${{\theta }_{0}}$ = 4.2$^{\circ }$, FOV = 50$^{\circ }$ and ${{\theta }_{0}}$ = 4.2$^{\circ }$, FOV = 150$^{\circ }$. It can be concluded that the smaller the emitted beam dispersion angle, the more photons received, the smaller the path loss, and thus the better the system performance. However, the change of FOV has little effect on the path loss of the system. when the beam dispersion angle is fixed.
Fig. 8. Path loss curves under different system configurations for the same water condition: (a) S2, (b) S8.
5.2.2 CIR
We compared the CIR under different transceiver settings with the lowest chlorophyll concentration water S2 and the highest chlorophyll concentration water S8 with a link distance of 10 m. As shown in Fig. 9, the beam scattering angles of 4.2$^{\circ }$ and 42$^{\circ }$ are small and large beam scattering angle configurations, respectively, and FOV=50$^{\circ }$ and FOV=150$^{\circ }$ represent narrow receiver FOV and wide receiver FOV configurations, respectively. Comparing each subplot, it can be found that the CIR intensity of the narrow transceiver configuration under the two water types is the largest and the spreading is the smallest, the narrow beam dispersion angle wide receptive FOV configuration is the second, and the CIR performance of the wide transceiver configuration is the worst. This can be explained by that a concentrated transmitting light source can make the beam more concentrated so that more photons can propagate along the line of sight and thus be received by the receiver. In addition, narrow FOV receivers will ignore photons with long propagation times and large arrival angles, thus decreasing the temporal dispersion. Besides, comparing the CIR results of the two different water types, it can be found that the CIR intensity under S2 is higher than that under S8, which is due to the higher chlorophyll concentration, resulting in more severe link attenuation.
5.2.3 Fitting of CIR
Based on the above analysis, we learn that the system performs best with a transceiver configuration of ${{\theta }_{0}}$ = 4.2$^{\circ }$ and FOV = 50$^{\circ }$. We choose two models to fit the CIR data obtained by MCS under this system configuration. Figure 10 shows the fitting results under different water conditions when the link distance is 5m and 10m. Observing the two subplots, it can be seen that the CIR intensity of L = 5m is greater than that of L = 10m for the same water conditions due to the greater attenuation caused by a longer link range. In addition, the R-squares of all scenarios above are given in Table 2. It can be seen that the R-squares were all greater than 0.99 when fitted with the Gaussian model while the R-square of the DGF model is only 0.7024 for the best situation, which indicates that the Gaussian model fits better than the DGF model for short link distances. As can be seen in Fig. 10, the DGF model is clearly not as good as the Gaussian model for any water type. This is mainly due to the shorter link distance and therefore less dispersion in time, so the DGF model may break down, which is consistent with the conclusions obtained in the literature [17].
Fig. 10. CIR in different water types (MC and DGF denote Monte-Carlo and double gamma function respectively): (a) 5 m, (b) 10 m.
Table 2. R-squares of DGF and Gaussian models under different channel environments.
6. Performance evaluation
In this part, we evaluate the performance of the vertical LOS UWOC system based on the optimal CIR closed-form expression obtained above, where reliability is measured by BER and effectiveness is evaluated by 3-dB bandwidth.
6.1 BER performance
We chose S2 with the lowest chlorophyll concentration and S8 with the highest chlorophyll concentration as a representative of clear water and turbid water, respectively. Under the two water conditions, we evaluated the BER changes under different link distances and signal-to-noise (SNR) ratios for both uplink and downlink communication. The BER calculation platform was built through Simulink, and the process was shown in Fig. 11. The transmitter first sends a pseudo-random sequence $s\left ( t \right )$ and then uses a simple OOK modulation scheme to turn it into a zeroed square wave signal ${{x}_{OOK}}(t)$ with a period of T=${{10}^{-6}}$s, where the probability of sending 0 and 1 code is equal. Besides, the transmitter signal power is normalized to 1 W. $h(t)$ is the CIR obtained by fitting the Gaussian model. Due to the existence of thermal noise at the receiver, which is additive Gaussian white noise (AWGN), we use the AWGN module with an SNR ${{\gamma }_{n}}$ to simulate the noise at the receiver (marked as $n(t)$ in Fig. 11). Because the transmitter power is always controlled as 1 W, the change of ${{\gamma }_{n}}$ can control the magnitude of the noise power. Finally $y\left ( t \right )={{x}_{OOK}}(t)*h(t)+n(t)$ is obtained as the output of the considered channel, where $*$ represents the convolution operator. For each evaluation, we emitted at least ${{10}^{6}}$ bits.
Figures 12 plots the BER performance versus ${{\gamma }_{n}}$ with various link ranges and water types. We calculate the BER performance when ${{\theta }_{0}}$ = 4.2$^{\circ }$ and FOV = 50$^{\circ }$ for transmitting data in different communication directions. From Fig. 12, we can observe that the BER decreases as ${{\gamma }_{n}}$ increases. And it can be seen that under S2 water condition when L = 5 m and the communication direction is uplink, an SNR of 8 dB was achieved with a BER of $3.2\times {{10}^{-3}}$, which is below the forward error correction (FEC) limit of $3.8\times {{10}^{-3}}$. Meanwhile, for the link distance of 10 m, a maximum SNR of 14 dB was obtained with a BER of $8.94\times {{10}^{-7}}$. This is because when the CIR has temporal dispersion, the more components the signal takes up, the fewer error bits are caused by the spreading of CIR. In addition, comparing the two subplots we can also see that the BER of the uplink is less than that of the downlink at the same link distance. This is because the chlorophyll concentration is a function of the depth [31]. Overall, shallow seas have high chlorophyll concentrations while deep seas have low levels. For uplink, the transmitted photons first encounter the low chlorophyll concentration region that is associated with low absorption and scattering. Therefore, they can travel a fair long distance along the LOS path. This leads to a relatively low path loss. On the other hand, as photons transmit downward, they experience significantly high chlorophyll concentration at the beginning. Most of them are absorbed or scattered in other directions, which prevents them from propagating further. So, the downlink will have greater path loss as compared to the uplink. Besides, it can be seen that the BER in S2 water condition is smaller than that in S8, indicating that the higher chlorophyll concentration leads to larger link attenuation thus aggravating the ISI caused by temporal dispersion and eventually leading to higher BER. The BER of the system can be reduced by using the equalization technique, which will be our future work.
Fig. 12. BER performance versus SNR under the same water condition with various link distances: (a) S2, (b) S8.
6.2 3-dB bandwidth
We also obtained the 3-dB bandwidth of the system using the closed expression of CIR. Based on the above analysis of BER, we can find that when the SNR is 12 dB, the BER is smaller than ${{10}^{-3}}$ in all cases. Therefore, in this part, assuming that the channel has AWGN with a SNR of 12 dB, we derived the variation of channel bandwidth with communication distance under two representative water conditions and two communication directions.
Using the closed-form expression of the impulse response, we calculated the Fourier transform expression of the Gaussian model through Mathematics as follows:
Then the 3-dB channel bandwidth can be computed by solving the following equation:
As shown in Fig. 13, we compare the 3-dB bandwidth of the uplink and downlink systems when the water conditions are S2 and S8 and the link distance is 5 m-20 m. It can be found that when the water conditions are certain, the bandwidth decreases with the increase of the link distance for both uplink and downlink cases, and the bandwidth of the uplink system is larger than that of the downlink system for the same link range. However, there is no specific trend in the influence of water conditions on bandwidth for the same link distance. It can also be seen that when the water condition is S8 and the communication direction is uplink, a bandwidth of 22.05 MHz was obtained with a link distance of 5 m, and the bandwidth is 6.74 MHz when the link distance is 20 m. It concludes that the vertical UWOC systems can’t work over 22 Mbps in this scenario due to the possible ISI.
Fig. 13. 3-dB bandwidth in different communication directions and at different link distances: (a) S2, (b) S8.
7. Conclusion
In this paper, we consider an underwater vertical LOS communication scenario based on particle scattering using the chlorophyll-dependent IOP model, where the Kopelevich phase function containing water depth information is selected. The path loss and CIR of the underwater vertical channel are obtained by MCS, the CIR curves under different channel conditions and transceiver configurations are plotted, and the data simulated by MCS are fitted with the DGF model and the Gaussian model, respectively. The results show that the Gaussian model performs better than the DGF model in each water condition with its R-squares can reach more than 0.99. On this basis, we solve the BER and 3-dB bandwidth of the system, and the results show that the BER is positively correlated with the communication distance and turbidity of the water, and negatively correlated with the SNR. In addition, the performance difference between the uplink and downlink is verified, and the results show that the performance of the uplink is better than that of the downlink under the same conditions. The experimental verification and the distribution that the parameters of the Gaussian model obey will be our further work. Furthermore, the effect of turbulence is not considered in our work, however, light is inevitably affected by turbulence when it is transmitted in a real underwater environment, so it is also the focus of our future work to study the characteristics of a channel that exists both particle scattering and turbulence.
Funding
National Key Research and Development Program of China (2022YFC2808101).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
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