Effect of Receiver’s Tilted Angle on the Capacity for Underwater Wireless Optical Communication

Abstract

This paper focuses on the effect of the receiver’s tilted angle on the capacity of a clear ocean water Underwater Wireless Optical Communication (UWOC) system. To achieve this goal, the relationship between the channel capacity and the receiver’s tilted angle is investigated. First, we propose a double-exponential fading model with pointing error which can more accurately depict the channel of clear ocean water UWOC instead of the traditional Beer’s law model. Based on this channel model, we present the close-form expression of the capacity bounds of the UWOC system. Then, an optimization problem is formulated to improve the capacity by tilting the receiving plane. Both theoretical analyses and simulation results verify that the capacity bounds of UWOC can be enhanced dramatically by tilting the receiver plane at an optimal angle. Thus, in practice, we can provide an effective design strategy for a UWOC system.

1. Introduction

Underwater wireless optical communication (UWOC) has attracted considerable attention due to its high band-width and large data rates compared with conventional acoustic communications [1,2,3]. However, in UWOC, the received signal suffers severe attenuation effects caused by the optical properties of the water channel, namely, absorption and scattering, which is defined as channel loss in [4,5] and degrades the system performance. Besides this, for successful wireless optical communication, the optical beam needs to be highly directive. Practically, however, due to the misalignment between the transmitter and receiver, the so-called pointing loss is incurred [5,6]. Obviously, the pointing loss effect will impair the system performance dramatically.

So far, many studies have been conducted on the effect of misalignment of the transmitter and receiver on the received intensity [4,5,6,7,8]. However, in the above literature, the receiver plane was fixed and could not be tilted, which greatly limited the performance of the UWOC system. Although [9,10] allowed for the tilting or movement of the receiver plane, no further consideration was given to theoretically optimize the performance of UWOC by tilting the receiver plane to overcome the pointing loss.

Channel capacity is an important indicator to evaluate the system performance of a communication link. For a visible light communication (VLC) system, the closed-form expression of the tight bounds on the capacity are presented in [11,12]. However, for UWOC, analyzing capacity is still a work in progress [13,14] because the underwater channel model is difficult to be depicted by a closed-form expression. Generally, Beer’s law adopted in [7,8,9] is used to describe the underwater channel loss and to evaluate the system performance; however, it overlooks the indirect path. Compared with the traditional Beer’s law, a double-exponential model originally used in [15,16] can more accurately depict the UWOC channel loss, but neither of them considered the pointing loss between the transmitter and receiver due to the receiver misalignment.

Based on the research results on the channel loss in [8,15] and the tight bounds on the capacity in [11], and considering the pointing loss, in this paper, we propose a new double-exponential channel model with a pointing error angle existing in a practical environment to thoroughly study the capacity of a UWOC system. Moreover, we consider the optimizing method of the capacity performance of the UWOC system by tilting the receiver plane. Numerical and simulation results both verify the effects of tilting the receiver plane on the capacity enhancement.

The remainder of the paper is organized as follows. The double-exponential with pointing error angle UWOC model is presented in Section 2. In Section 3, we derive the theoretical expressions of the capacity bounds for the UWOC channel. In Section 4, the optimization problem is formulated to improve the capacity by tilting the receiver plane; Section 5 presents numerical and simulation results of the above-mentioned optimization problems; and finally, we conclude the paper in Section 6.

2. System Model

The UWOC system model is illustrated in Figure 1. Point light source O is assumed as the origin of the 3D coordinates; the receiver moves within a quarter circle plane. (For analysis simplicity, only the 1/4 circle plane is considered, and the quadrant where the receiver is located is set as the first quadrant). Let the coordinates of the light source and receiver be [0,0,0] and $[x_0,y_0,z_0]$, respectively. The field of view (FOV) of the receiver is 180°, and d is the distance between the light source and the receiver.

We also assume that the beam of the light source is always aimed at the receiver. For the receiver, the vector from the receiver to the light source is defined as ${\overrightarrow{V}}_{or}$. ${\overrightarrow{V}}_n$ is the unit normal vector perpendicular to the plane of the receiver. $\beta$ is the incidence angle, which is also the pointing error angle in [5]. $\theta$ is the receiver’s tilted angle. To further simplify the problem analysis, we have set the Z axis, vectors ${\overrightarrow{V}}_{or}$ and ${\overrightarrow{V}}_n$, as co-planar. In this way, the relationship between them can be listed in the dashed box in the upper-right corner of Figure 1.

Assuming that the transceiver is located in a clear ocean water situation, and considering the channel fading of the UWOC and the noise term of the receiver, the received current signal is a combination of the transmitted signal and the path loss in the link; referring to Cox’s link equations considering the path loss [8], the received signal has the following expression

$$y=L\tau P_{s}x+n,$$

(1)

where $\tau =\eta e/hv$ is the photoelectric transformation coefficient, in which $\eta$ is the quantum efficiency, $h$ is Planck’s constant, $v$ is the frequency of light waves in seawater, and $e$ is the electron charge. $P_s$ is the transmitting power; $x\in \{0,1\}$ is the transmitting bit of the on-off keying (OOK) intensity modulation. $n$ is the background noise of the receiver and can be simulated as Gaussian white noise with a mean value of zero and variance of ${\sigma }^2$. In (1), path loss L can be described as

$$L={\tau }_{channel}\cdot {\tau }_{po\mathrm{int}},$$

(2)

where ${\tau }_{channel}$ is the channel loss, which is from absorption and scattering. Compared with the conventional Beer’s law, the double-exponential channel loss model can more accurately depict the channel fading in a clear ocean water type [15,16]. Therefore, it can be modified and depicted as

$${\tau }_{channel}=C_1{e}^{C_{2}d}+C_3{e}^{C_{4}d},$$

(3)

where $C_1$, $C_2$, $C_3$, and $C_4$ are the fitting coefficients obtained by Monte Carlo simulations. However, when misalignment deployment of the receiver and transmitter occurs, the pointing loss must be considered. As shown in [6], it can be expressed as

$${\tau }_{po\mathrm{int}}=\cos \beta ,$$

(4)

where $\beta$ is the pointing error angle.

Insert (3) and (4) into (2); the double-exponential channel loss model with pointing error is written as

$$L=\left(C_1{e}^{C_{2}d}+C_3{e}^{C_{4}d}\right)\cos \beta .$$

(5)

According to the knowledge of spatial analytic geometry, the pointing loss $\cos \beta$ can be expressed in the following form

$$\cos \beta =\frac{\langle {\overrightarrow{V}}_n,{\overrightarrow{V}}_{or}\rangle }{\Vert {\overrightarrow{V}}_n\Vert \cdot \Vert {\overrightarrow{V}}_{or}\Vert },$$

(6)

where the vector ${\overrightarrow{V}}_n=[\cos \phi \sin \theta ,\sin \phi \sin \theta ,\cos \theta ]$, and ${\overrightarrow{V}}_{or}=[-x_0,-y_0,-z_0]$; substituting both of them into (6), it simplifies to

$$\cos \beta =\frac{1}{d}\left[(-x_0)\cos \phi \sin \theta +(-y_0)\sin \phi \sin \theta +(-z_0)\cos \theta \right],$$

(7)

where $\phi$ is the azimuth angle formed by the positive direction of the X axis and the projection of ${\overrightarrow{V}}_n$ on the horizontal plane. In fact, since ${\overrightarrow{V}}_n$, ${\overrightarrow{V}}_{or}$ and $\overrightarrow{z}$ are coplanar, $\phi$ is totally determined by the coordinates of the receiver, as shown in Figure 2 below. As can be seen from Figure 2, the azimuth angle $\phi$ can be expressed as

$$\phi ={\tan }^{-1}\frac{y_0}{x_0},$$

(8)

As shown in (7), the pointing loss should be a function with respect to the tilted angle $\theta$ at a fixed distance. If we tilt the receiver plane, the pointing loss will change accordingly.

3. Capacity Bounds for UWOC

According to the work of Wang et al. [11], combined with the path loss in UWOC mentioned in (2), the lower and upper bounds on the channel capacity for the UWOC are expressed as

$$C_{Low}=\frac{1}{2}{\log }_2\left(1+\frac{e}{2\pi }{\left(\frac{L{P}_s\tau }{2\sigma }\right)}^2\right),$$

(9)

$$C_{Upp}={\log }_2\left(\frac{{\left(\frac{\sqrt{e}L{P}_s\tau }{2}\right)}^{{\mu }^{*}}}{{\pi }^{\frac{{\mu }^{*}}{2}}{\sigma }^{{\mu }^{*}}{({\mu }^{*})}^{2{\mu }^{*}}{(1-{\mu }^{*})}^{\left(\frac{3(1-{\mu }^{*})}{2}\right)}}\right),$$

(10)

where ${\mu }^{*}\in [0,1]$ is the solution to the equation

$${\mu }^4-\frac{1}{\pi }{\left(\frac{L{P}_s\tau }{2\sigma }\right)}^2{(1-\mu )}^3=0,$$

(11)

Based on (5), (7), and (9) mentioned above, it can be seen that when the distance $d$ is fixed, the lower bound on the channel capacity is a unary function with respect to the receiver’s tilted angle $\theta$. Changing the receiver’s tilted angle $\theta$ will make it possible to obtain the optimal low bound on the capacity of UWOC at a given distance.

4. Optimization Problem: Raising and Solving

Therefore, the above description turns into one mathematical optimization problem. To this end, the optimization problem of capacity is first given, then the optimization issue is further proved to be a simple convex optimization problem, and lastly the theoretical expression of the optimal tilted angle is obtained.

4.1. Description of the Capacity Optimization Problem

Taking the maximum lower bound of UWOC capacity as the optimization target and considering the limit of the receiver’s tilted angle, the optimization problem can be formulated as

$$\underset{s.t.0\le \theta \le \pi /2}{\underset{\theta }{\max }{C}_{Low}}\begin{array}{c}\end{array}$$

(12)

4.2. Solution of the Optimization Problem

Based on (9), the channel capacity of UWOC is a unary increasing function with respect to the path loss $L$ with pointing error. Therefore, (12) is equal to

$$\underset{s.t.0\le \theta \le \pi /2}{\underset{\theta }{\max }L}$$

(13)

Substituting (7) into (5), the expression of path loss $L$ with receiver’s tilted angle $\theta$ is

$$L=\left(C_1{e}^{C_{2}d}+C_3{e}^{C_{4}d}\right)\cdot \left[(-x_0)\cos \phi \sin \theta +(-y_0)\sin \phi \sin \theta +(-z_0)\cos \theta \right]/d$$

(14)

Further taking the first and second derivatives of $L$ with respect to θ we obtain

$$\frac{dL}{d\theta }=\left(C_1{e}^{C_{2}d}+C_3{e}^{C_{4}d}\right)\cdot \left[(-x_0)\cos \phi \cos \theta +(-y_0)\sin \phi \cos \theta +(-z_0)(-\cos \theta )\right]/d,$$

(15)

and

$$\begin{array}{c}\frac{d^{2}L}{d{\theta }^2}=-\left(C_1{e}^{C_{2}d}+C_3{e}^{C_{4}d}\right)\cdot \left[(-x_0)\cos \phi \sin \theta +(-y_0)\sin \phi \sin \theta +(-z_0)\cos \theta \right]/d=-L\\ \end{array}$$

(16)

Since the channel fading $L$ is nonnegative, the second derivative of $L$ with respect to $\theta$ is less than or equal to 0, indicating that the objective function $L$ is a convex function with respect to $\theta$. In other words, there is a value of $\theta$ that maximizes $L$ and thus maximizes the capacity. Since (13) is convex and the constraint is also convex, it is a convex optimization problem. With the first derivative of $L$ with respect to $\theta$, the optimal tilted angle ${\theta }_0$ for capacity to reach its maximum is

$${\theta }_0=\arctan \left[\frac{x_0\cos \phi +y_0\sin \phi }{z_0}\right]=\arctan \left[\frac{d^{\prime }}{z_0}\right],$$

(17)

where $d^{\prime }$ is the distance between the receiver and the projection point of the light source on the X–Y circular plane. Combining Figure 2 and (17), it is easy to see that the pointing error angle $\beta$ is equal to $0^{°}$ while the system capacity reaches the maximum, that is, the optimal tilted angle ${\theta }_0$ is the case where ${\overrightarrow{V}}_n$ and the incident light are fully aligned.

5. Numerical Simulations and Analyses

In this section, we will investigate how the distance between the light source and receiver d and the receiver’s tilted angle $\theta$ affects the capacity bounds. We will verify that pointing error $\cos \beta$ can be eliminated by tilting the receiver plane, hence improving the capacity of UWOC. The simulation parameters are listed in

Table 1. Main Simulation Parameters.

Parameters	Value
$C_1$	$0.9790\times {10}^{-4}$
$C_2$	$-0.1499$
$C_3$	$4.4162\times {10}^{-4}$
$C_4$	$-0.2876$
Fitting error	$2.1831\times {10}^{-4}$
Transmit optical power	0.3 W

.

Figure 3 depicts the capacity bounds of UWOC against the tilted angle $\theta$ when distance d between the light source and receiver is set to 19.25 m, 20.5 m, 21.75 m, and 23 m. As shown in Figure 3, we can see that the change trends of the curves of the capacity bounds with respect to the tilted angle $\theta$ are similar. In particular, the capacity bounds are monotonously decreasing functions at the distance of 19.25 m. Other capacity bounds increase first and then decrease with the increase in the tilted angle. For the given distance, there exists an optimal tilted angle for each curve. For example, at the distance of 21.75 m, when the tilted angle is set to 25 degrees, the maximum of the lower bound on capacity can be achieved. As a whole, as the distance increases, the optimal tilted angle becomes higher.

We also compared the capacity bounds at different distances at a certain tilted angle value. When the tilted angle is small (below 40 degrees), the capacity bounds decrease dramatically as the distance increases. However, as the tilted angle becomes large, the capacity bounds decrease slightly as the distance increases. As a whole, when the tilted angle is small, distance is a major influential factor only on the capacity bounds; when the tilted angle increases, the tilted angle rather than distance becomes a major factor.

We also notice that at a short distance, the capacity bounds vary more evidently than at a long distance. This can be explained as follows: when the distance is short, tilting the receiver plane slightly can almost make up for the pointing error loss; thus, the optimal capacity can be achieved. With the increase in distance, the corresponding optimal tilted angle gradually increases, which means that when the receiver is farther away from the light source, the receiver plane needs to be deflected at a larger angle to overcome the adverse effect of the pointing error.

Figure 4 shows the curved surface distribution of the UWOC channel capacity’s lower bound over the horizontal circular surface shown in Figure 1 under two situations of untilted and optimally tilted receiver planes. In the simulation, the point of light source O is assumed as the origin of the optical axis Z. The distance between the light source and the receiver plane is set to 19.25 m. The receiver moves on a quarter circle plane with a radius of 12.6 m. As can be seen from Figure 4a,b, irrespective of whether the receiver plane is tilted, the capacities on both boundaries of circles are the worst, and the optimal capacity can be reached when the receiver is located directly below the light source. This result can be explained that as distance increases, the path loss will gradually increase and lead to capacity performance deterioration.

In addition, the capacities are the same on the circumference of a circle in the X–Y plane with a radius a fixed distance from the optical axis Z. This is because the points on this circle are the same distance from the light source. Comparing Figure 4b with Figure 4a, we can find that after tilting the receiver plane with an optimal angle, the lower bound of capacity increases in the whole X–Y plane. For example, when d is set to 23 m, the lower bound of capacity $C_{Low}$ 0.593 bit/s when the receiver plane is not tilted, while $C_{Low}$ reaches 0.7479 bit/s after tilting with the optimal angle. This shows that through tilting the receiver plane, the receiver can overcome the adverse effect of the pointing error, further enhancing the capacity performance.

Figure 5 shows the distribution of the optimal tilted angle when the receiver is located at different positions of the X–Y circular plane. The light source is set at the origin of the optical axis. As can be seen from Figure 5, when the receiver is located directly below the source, it is obvious that the optimal tilted angle is 0°. When the receiver moves in the X–Y plane, the optimal tilted angle becomes gradually larger as the distance between the light source and receiver increases. When the receiver is positioned at the boundary, the optimal tilted angle will reach the maximum of 33.2°. This means that we need to tilt the receiver plane to a greater angle to overcome the adverse effect of the pointing error. In addition, similar to the results of Figure 4b, the optimal tilted angles are the same on the circumference of any circle in the X–Y plane.

6. Conclusions

This paper focuses on the effect of a receiver’s tilted angle on the channel capacity of a clear ocean water UWOC system. We propose a new double-exponential channel model with pointing error angle existing in a practical environment to optimize the capacity of the UWOC system. Then, we derive the close-form expressions of the capacity bounds for the UWOC system. Based on the results, the optimal receiver’s tilted angle to maximize the lower bound of the capacity is obtained. The simulation results suggest that in practice, in order to achieve the optimal performance of a UWOC system, we can tilt the receiver plane at an optimal angle at a given communication distance or configure the optimized communication distance under a fixed tilted angle.