Average symbol error probability and channel capacity of the underwater wireless optical communication systems over oceanic turbulence with pointing error impairments

Zhiru Lin; Guanjun Xu; Guanjun Xu; Qinyu Zhang; Qinyu Zhang; Zhaohui Song

doi:10.1364/OE.457043

1. Introduction

The demand for the underwater communication has exploded with the expansion of human activities in the oceanic environment, such as underwater exploration, offshore oilfield exploitation, and disaster prevention, etc. [1–3]. The radio frequency communication and underwater acoustic communication have been extensively employed in the ocean but encounter bottleneck due to the disadvantages, such as insecurity, limited bandwidth, and low transmission speed. Motivated by the merits of large data rate and high security, underwater wireless optical communication (UWOC) is perceived as a technology with great potential in the oceanic environment.

Although the UWOC systems have been deployed in many underwater scenarios, they suffer serious channel deterioration, resulting from the complicated underwater environment. Pioneering studies have pointed out three critical impacts: the turbulence-induced channel fading by the random variation of the scintillation index, the misalignment effect due to water movement, and the absorption and scattering impacts caused by the interaction of the optical waves with the water [4–6]. The overall performance of the UWOC systems will be seriously degraded and the optical link is even interrupted [7,8]. Therefore, it is essential to evaluate the performance of the UWOC systems operating in the ocean environment.

Considering the influence of oceanic turbulence caused by the fluctuations of salinity and temperature in the ocean, various scintillation index models for the optical waves propagating in oceanic turbulence were proposed with the aid of the Rytov theory and the extended Huygens-Fresnel principle [9–11]. Ata et al. [12] proposed the scintillation index model for the plane waves and spherical waves in underwater turbulence and proved that the spherical waves have smaller index than the plane waves. The authors in [13] also presented this phenomenon and further revealed that the eddy diffusivity ratio of the oceanic turbulence has a significant impact on the scintillation index. In addition, the oceanic turbulence’s thermal expansion and random movement induce significant pointing error impairments on the UWOC systems [14,15]. The authors in [16] considered both the influence of oceanic turbulence and pointing errors on the optical link and proved that pointing errors can significant degrade link performance.

With the help of the scintillation index models, the performance metrics of the UWOC systems, such as the symbol error probability (SEP) and channel capacity, were evaluated, considering various fading conditions. Rubén et al. calculated the channel capacity and mutual information in an actual UWOC scenario considering path loss and salinity-induced oceanic turbulence effects [17]. The bit error ratio (BER) of the UWOC systems for vertical links was also derived under the Gamma-Gamma (GG) fading [18], and the diversity order of the system was determined by both the pointing errors and the channel fading parameters. The Lognormal (LN) distribution model was used to study the BER of the UWOC systems in weak oceanic turbulence in [19]. In contrast, the GG distribution model was employed in [20], to characterize the strong oceanic turbulence. In [21], the authors further considered the average channel capacity under the composited channel fading model of absorption, scattering, and oceanic turbulence effects.

Several irradiance probability density function (PDF) models have been extensively studied, such as LN and GG models [22,23]. Note that the LN distribution is more suitable for weak oceanic turbulence. Under the assumption of the LN model, an upper bound on the average BER of the UWOC system operating in weak oceanic turbulence was studied by Baykal [24]. In [25], the power series expression of the beam spreading function was used to investigate the BER and capacity over LN oceanic turbulence channels. In contrast to the LN model, the GG distribution model is suitable for strong turbulence. He et al. [26] studied the influence of various parameters on the BER under different anisotropic oceanic turbulence based on the assumption of the GG oceanic turbulence model. In [27], the average BER and outage probability for a partially coherent laser UWOC system was analyzed in the presence of GG oceanic turbulence. A new generalized distribution, referred to as the Málaga distribution model, was proposed to model unbounded optical wavefront irradiance under various turbulent conditions. Therefore, the LN, generalized K, GG distribution models can all be presented as special cases of the Málaga model. Balsells et al. expressed the average BER in the Málaga distribution model [28]. In addition, the Málaga distribution model was used to analyze the channel capacity over oceanic turbulence [29].

Considering these influence factors on the UWOC systems, some modulation techniques, such as on-off keying (OOK) modulation and pulse position modulation (PPM) were used to improve the system performance. The performance of the optical waves in weak oceanic turbulence, using the LN fading model, was studied with the OOK modulation in [16]. In [30] and [31], the authors further proved that the UWOC systems with the PPM technique achieve better performance than with the OOK modulation since it has advantages in enhancing power efficiency and does not require an adaptive threshold. Utilizing the $M$-ary PPM technique in the UWOC systems, the performance the UWOC systems in weak oceanic turbulence was evaluated by Peppas et al. [32]. Additionally, Gökçe et al. studied the performance of the Gaussian beam propagating in strong oceanic turbulence conditions, using the $M$-ary PPM technique [33].

According to the above analysis, both the LN and GG fading models were used, assuming that the optical waves propagate in weak and strong oceanic turbulence, respectively. However, the investigation of the UWOC systems in a weak-to-moderate turbulence is still in its infancy, to the best of our knowledge. In addition, the power limitation in the real UWOC systems (i.e., limited average power and limited peak power) is a crucial problem but was ignored in many studies. Moreover, more investigations on the combined effects of absorption, oceanic turbulence, and pointing error impairments should be conducted to reveal these effects on the UWOC systems.

To fill these gaps, we focus on the combined effect of the path loss, pointing errors, and oceanic turbulence on the UWOC systems. Specifically, the Málaga distributed fading model is used in this study, which can well characterize the turbulence fading under weak-to-moderate conditions. Furthermore, the $M$-ary PPM technique is employed, to decrease the average SEP of the UWOC systems. Finally, the channel capacity under peak power-limitation and average power-limitation is evaluated.

The structure of this paper is organized as follows. The system and channel models are described in Section 2. In Section 3, the average SEP and the average channel capacity are studied with the aid of the PPM technique. The simulation results and detail discussions are presented in Section 4, followed with concluding remarks in Section 5.

2. System and channel models

Figure 1 presents the diagram of the UWOC systems with $M$-ary PPM. At the transmitter, there is a serial-to-parallel converter and a $M$-ary PPM modulator. The modulated signal is converted into an optical signal by the laser driver and further passes through the oceanic turbulence with a transmission distance of $L$. Then, the optical signal is detected by the photodiode (PD) and is finally converted into an electrical signal for demodulating.

Fig. 1. Block diagram of the UWOC systems with $M$-ary PPM.

Download Full Size | PDF

With the $M$-ary PPM technique, the transmitted signal $x$ is partitioned into $M$ plots, i.e., $x = (x_0, x_1,\ldots, x_{M-1})$. It is assumed that $x$ is distorted by both additive white Gaussian noise (AWGN) and the fading coefficient $h$. Therefore, the received signal is given by

(1)$$y_{k}=h g x_{k}+n_{k}, k=0, \ldots, M-1,$$

where $g$ is the PD responsibility [in amperes per watt (A/W)]. Data symbols are assigned into the slots in $x$, therefore, only one of them has the amplitude of $MP_T$, while the others are zero. Here, $P_T$ represents the average power of the transmitted optical signal and $n_k$ denotes the AWGN with mean zero and variance $\sigma _{n}^{2}$.

The fading coefficient $h$, characterizing the variation of the propagation channel, is comprised of pass loss $h_l$, pointing errors $h_p$, and oceanic turbulence $h_o$. Hence, the fading coefficient $h$ can be formulated as

(2)$$h=h_{l} h_{p} h_{o}.$$

2.1 Turbulence fading model

In this study, the Málaga distribution model is employed to characterize the turbulent channel. It should be noted that this is a generalized fading model which has been proved with wide applications, from weak to moderate turbulence [34]. The probability density function (PDF) of the Málaga distribution model can be expressed as [35]

(3)$$f_{h_{o}}\left(h_{o}\right)=A \sum_{m=1}^{\beta} a_{m} h_{o}^{\frac{\alpha+m}{2}-1} K_{\alpha-m}\left(2 \sqrt{\frac{\alpha \beta h_{o}}{b_{g} \beta+P^{\prime}}}\right),$$

with

(4a)$$ A=\frac{2 \alpha^{\frac{a}{2}}}{b_{g}^{\left(1+\frac{a}{2}\right)} \Gamma(\alpha)}\left(\frac{b_{g} \beta}{b_{g} \beta+P^{\prime}}\right)^{\beta+\frac{\alpha}{2}}, $$

(4b)$$ a_{m} =\left(\begin{array}{c} \beta-1 \\ m-1 \end{array}\right) \frac{\left(b_{g} \beta+P^{\prime}\right)^{\left(1-\frac{m}{2}\right)}}{(m-1) !}\left(\frac{P^{\prime}}{b_{g}}\right)^{m-1}\left(\frac{\alpha}{\beta}\right)^{\frac{m}{2}}, $$

where $b_g$ represents the average power of the scattered components received by the off-axis, while the average power from the coherent contributions is denoted by $P^{\prime }=P+2 b \rho +2 \sqrt {2 b \rho P} \cos \left (\phi _{A}-\phi _{B}\right )$. Here, $P$ is the average power of the line-of-sight (LOS) component, $\phi _{A}$ and $\phi _{B}$ are the deterministic phases of the LOS and the coupled to LOS scatter terms, respectively, $\mathrm {K}_{\alpha -m}(\cdot )$ is the modified Bessel function of the second kind with order $\alpha -m$ and $\Gamma (\cdot )$ is the Gamma function. Note that the Málaga distribution model has been applied to characterize the fading of oceanic turbulence under weak-to-moderate conditions [29,36].

The parameters $\alpha$ and $\beta$ represent the effective numbers of small-scale and large-scale cells, respectively, which are strongly related to the oceanic turbulence. The exact expression of these parameters for spherical wave propagation in oceanic turbulence can be further written as [23,37]

(5)$$\begin{aligned} \alpha= & \left\{\operatorname { e x p } \left\{0.04 \sigma_{I}^{2}\left(\frac{8.56 Q_{l}}{8.56+Q_{l}+0.195 \sigma_{I}^{2} Q_{l}^{7 / 6}}\right)^{7 / 6}\right.\right.\\ & \times\left[1+1.753\left(\frac{8.56}{8.56+Q_{l}+0.195 \sigma_{I}^{2} Q_{l}^{7 / 6}}\right)^{1 / 2}\right.\\ & \left.\left.-0.252\left(\frac{8.56}{8.56+Q_{l}+0.195 \sigma_{I}^{2} Q_{l}^{7 / 6}}\right)^{7 / 12}\right\}-1\right\}^{{-}1} \end{aligned}$$

(6)$$\beta=\left(\exp \left[\frac{0.51 \sigma_{I}^{2}}{\left(1+0.69 \sigma_{I}^{12 / 5}\right)^{7 / 6}}\right]-1\right)^{{-}1},$$

where $Q_l=10.89d_0/k_0\eta ^{2}$, $\sigma _{I}^{2}$ is the scintillation index of the spherical waves in an ocean environment and is given by [13]

(7)$$\begin{aligned} \sigma_{I}^{2} & =2 \pi k_{0}^{2} d_{0} C_{0} \alpha_{1}^{2} \frac{\chi_{T}}{\omega^{2}} \varepsilon^{-\frac{1}{3}} \int_{0}^{1} \int_{0}^{\infty} \kappa^{-\frac{8}{3}}\left\{1-\cos \left[\frac{d_{0} \kappa^{2}}{k_{0}}\left(\zeta-\zeta^{2}\right)\right]\right\} \\ & \cdot\left[1+C_{1}(\kappa \eta)^{\frac{2}{3}}\right]\left(\omega^{2} \exp \left({-}A_{T} \delta\right)+d_{r} \exp \left({-}A_{s} \delta\right)-\omega\left(d_{r}+1\right) \exp \left({-}A_{T S} \delta\right)\right) d \kappa d \zeta \end{aligned},$$

where $k_0 = 2\pi /\lambda$ is the wave number, $\lambda$ is the wavelength, and $d_0$ is the transmission distance. The symbols $C_0$ and $C_1$ are equal to 0.72 and 2.35, respectively. In addition, $\alpha _1$ is the thermal expansion coefficient, $\chi _{T}$ is the dissipation rate of mean-squared temperature, $\varepsilon$ is the dissipation rate of turbulent kinetic energy per unit mass of fluid and $\eta$ denotes the Kolmogorov microscale length. Moreover, $\kappa$ is the magnitude of the spatial frequency and $\zeta$ is the normalized link distance, $\delta =1.5 C_{1}^{2}(\kappa \eta )^{4 / 3}+C_{1}^{3}(\kappa \eta )^{2}$. We assume $\lambda =0.53$ $\mu \rm {m}$, $d_{0}=20$ m, $\alpha _{1}=2.56 \times 10^{-4}$ $l / \mathrm {deg}$, $\chi _{T}=1 \times 10^{-5}$ $\mathrm {K}^{2} \mathrm {S}^{-3}$, $\varepsilon =1 \times 10^{-2}$ $\mathrm {m}^{2} \mathrm {s}^{-3}$, $\eta =1 \times 10^{-4}$ m when the salinity is $35 \%$ and temperature is 20 $^{\circ }$C [38,39] (i.e., $A_{T}=1.863 \times 10^{-2}$, $A_{s}=1.9 \times 10^{-4}$ and $A_{T S}=9.41 \times 10^{-3}$). The temperature-salt index, $\omega$, represents the contribution of temperature and salinity fluctuations to the refractive index distribution, which ranges from $-5$ to 0 [40]. When $\omega$ tends to $-5$, it refers to temperature-induced refraction, and it refers to salinity-induced refraction when $\omega$ tends to 0. The eddy diffusivity ratio $d_r$ is a function of $\omega$ and is given by

(8)$$d_{r} \approx\left\{\begin{array}{c} \frac{|\omega|}{|\omega|-\sqrt{|\omega|(|\omega|-1)}}, \quad|\omega| \geq 1 \\ 1.85|\omega|-0.85, \quad 0.5 \leq|\omega|<1 \\ 0.15|\omega|, \quad|\omega|<0.5 \end{array}\right. .$$

2.2 Pointing errors model

In the UWOC systems, pointing error occurs due to the random movements of the transceiver caused by ocean waves and currents. Therefore, a unified pointing errors model is required to evaluate the effect of the pointing errors. In this study, the fading coefficient for the pointing errors is given by [18]

(9)$$h_{p} \approx A_{0} e^{{-}2 s^{2} / w_{e}^{2}},$$

where ${A_0}{\rm {\ =\ }} {\rm erf}\left ( v \right )$ is the fraction of the collected power at $s = 0$, and $s$ is the random radial displacement of the receiver detection. In Eq. (9), $w_e$ denotes the equivalent beam width, $w_{e}=w_{z}\left [\sqrt {\pi } \operatorname {erf}(v) /\left (2 v e^{-v^{2}}\right )\right ]^{1 / 2}$ and $v=\sqrt {\pi } r / \sqrt {2} w_{z}$ . In addition, $r/w_z$ is the radius of the receiver aperture normalized by the beam waist.

By considering independent identical Gaussian distributions $\sigma _{s}^{2}$ with horizontal and vertical swing variances, the PDF of $s$ can be described by a Rayleigh distribution as [41]

(10)$$f_{s}(s)=\frac{s}{\sigma_{s}^{2}} \exp \left(-\frac{s^{2}}{2 \sigma_{s}^{2}}\right).$$

Making mathematics manipulations between Eqs. (9) and (10), the PDF of $h_p$ is expressed as

(11)$$f_{h_{p}}\left(h_{p}\right)=\frac{\xi^{2}}{A_{0}^{\xi^{2}}} h_{p}^{\xi^{2}-1}, 0 \leq h_{p} \leq A_{0},$$

where $\xi =w_{e} /\left (2 \sigma _{s}\right )$ is the ratio between the equivalent beam radius at the receiver and the standard deviation pointing error displacement at the receiver.

2.3 Path loss model

The path loss in ocean environment is determined by both absorption and scattering. Here, $a(\lambda )$ and $b(\lambda )$ are used, respectively, to denote the absorption coefficient and scattering coefficient. The total attenuation can be described by the extinction coefficients $c(\lambda )$, which is expressed as $c(\lambda )=a(\lambda )+b(\lambda )$ [42]. Some typical values of $a(\lambda )$, $b(\lambda )$, and $c(\lambda )$ for different types of waters are presented in Table 1 [43].

Table 1. The typical values of absorption, scattering and extinction coefficients in different water types.

View Table | View all tables in this article

Since the multiple scatter beam can also be received by the detector in many practical applications, but was ignored in most works [44], a modified expression for path loss that takes the contribution of scattered rays into account is used in this study [18]

(12)$$h_{l}=10 \log _{10}\left(\left(\frac{r}{\theta_{1 / e} d_{0}}\right)^{2} e^{{-}c(\lambda) d_{0}\left(\frac{r}{\theta_{1 / e }d_{0}}\right)^{T}}\right),$$

where $r$ is the receiver aperture diameter and $\theta _{1/e}$ is the full width transmitter beam divergence angle. The correction coefficient $T$ is determined via data fitting to simulation data, which increases with the increase of water turbidity [45].

2.4 Composited fading model for the UWOC systems

The composited channel fading distribution of $h = h_l h_p h_o$ can be derived by utilizing

(13)$$f_{h}(h)=\int_{\frac{h}{A_{0} h_l}}^{\infty} f_{h \mid h_{o}}\left(h \mid h_{o}\right) f_{h_{o}}\left(h_{o}\right) d h_{o},$$

where $f_{h \mid h_{o}}\left (h \mid h_{o}\right )$ is the conditional probability of turbulence fading coefficient and is written as

(14)$$f_{h \mid h_{o}}\left(h \mid h_{o}\right)=\frac{1}{h_{o} h_{l}} f_{h_{p}}\left(\frac{h}{h_{o} h_{l}}\right).$$

By substituting Eq. (14) into Eq. (11) and then inserting the result into Eq. (13), we have the PDF of $h$ as

(15)$$f_{h}(h)=\frac{\xi^{2} A}{A_{0}^{\xi^{2}}} h^{\xi^{2}-1} h_{l}^{-\xi^{2}} \sum_{m=1}^{\beta} a_{m} \int_{\frac{h}{A_{0} h_l}}^{\infty} h_{o}^{\frac{\alpha+m}{2}-1-\xi^{2}} \mathrm{~K}_{\alpha-m}\left(2 \sqrt{\frac{\alpha \beta h_o}{b_g \beta+P^{\prime}}}\right) d h_{o},$$

where the second kind of the $v$-th order modified Bessel function $\mathrm {K}_{v}(\cdot )$ in Eq. (15) can be further expressed as the Meijer’s G function with the help of $\mathrm {K}_{a-b}(2 \sqrt {x})=\frac {1}{2} x^{-\frac {a+b}{2}} G_{0,2}^{2,0}(x \mid a, b)$ [35], and we have

(16)$$f_{h}(h)=\frac{\xi^{2} A}{2 A_{0}^{\xi^{2}}} h^{\xi^{2}-1} h_{l}^{-\xi^{2}} \sum_{m=1}^{\beta} a_{m} \int_{\frac{h}{A_{0} h_{l}}}^{\infty} h_{o}^{\frac{\alpha+m}{2}-1-\xi^{2}} G_{0,2}^{2,0}\left(\frac{\alpha \beta h_{o}}{b_g \beta+P^{\prime}} \mid \frac{\alpha-m}{2},-\frac{\alpha-m}{2}\right) d h_{o}.$$

By employing [46, Eq. (0734.21.0085.01)] and simplifying the Meijer’s G function with the relationship $x^{r} G_{p,q}^{m, n}\left (x \mid \begin {array}{l} a_{p} \\ c_{q} \end {array}\right )=G_{p,q}^{m, n}\left (x \mid \begin {array}{l} a_{p}+r \\ c_{q}+r \end {array}\right )$, the composited PDF of the UWOC systems considering the effects of path loss, pointing errors, and oceanic turbulence can be finally recast as

(17)$$f_{h}(h)=\frac{\xi^{2} A}{2} h^{{-}1} \sum_{m=1}^{\beta} a_{m}\left(\frac{\alpha \beta}{b_g \beta+P^{\prime}}\right)^{-\frac{a+m}{2}} G_{3,0}^{1,3}\left(\frac{\alpha \beta}{b_g \beta+P^{\prime}} \frac{h}{A_{0} h_{l}} \mid \begin{array}{c} \xi^{2}+1 \\ \xi^{2}, \alpha, m \end{array}\right).$$

3. System performance

3.1 Average symbol error probability

We considered the UWOC systems with the $M$-ary PPM scheme and the intensity modulation and direct detection (IM/DD) technique in this study. The average SEP is given by

(18)$$p_{s}=\int_{0}^{\infty} p_{s}(h) f_{h}(h) d h,$$

where the conditional SEP $p_{s}(h)$ can be obtained as [47]

(19)$$p_{s}(h) \approx \sum_{k=1}^{M-1} \frac{({-}1)^{k+1}}{2^{k} \sqrt{k+1}}\left(\begin{array}{c} M-1 \\ k \end{array}\right) p_{s}\left(k, h\right),$$

with

(20)$$p_{s}(k, h)=e^{{-}k h^{2} M^{2} \gamma^{2} /[2(k+1)]}.$$

where $\gamma$ represents the average electrical signal-to-noise ratio (SNR).

Applying the identity $e^{-x}=G_{1,0}^{0,1}\left (\left.x\right |_{0} ^{-}\right )$ [48], the conditional SEP can be rewritten as

(21)$$p_{s}(h) \approx \sum_{k=1}^{M-1} \frac{({-}1)^{k+1}}{2^{k} \sqrt{k+1}}\left(\begin{array}{c} M-1 \\ k \end{array}\right) G_{0,1}^{1,1}\left(\left.\frac{k M^{2} \gamma^{2} h^{2}}{2(k+1)}\right|_{0} ^{-}\right).$$

Inserting Eq. (21) into Eq. (18), the average SEP of the considered UWOC systems can be further recast as

(22)$$\begin{aligned} p_{s}= & \frac{\xi^{2} A}{2} \sum_{m=1}^{\beta} a_{m}\left(\frac{\alpha \beta}{b_{g} \beta+P^{\prime}}\right)^{-\frac{\alpha+m}{2}} \cdot \sum_{k=1}^{M-1} \frac{({-}1)^{k+1}}{2^{k} \sqrt{k+1}}\left(\begin{array}{c} M-1 \\ k \end{array}\right) \\ & \cdot \int_{0}^{\infty} h^{{-}1} G_{0,1}^{1,0}\left(\left.\frac{k M^{2} \gamma^{2} h^{2}}{2(k+1)}\right|_{0} ^{-}\right) G_{1,3}^{3,0}\left(\frac{\alpha \beta}{b_{g} \beta+P^{\prime}} \frac{h}{A_{0} h_{l}} | \begin{array}{c} \xi^{2}+1 \\ \xi^{2}, \alpha, m \end{array}\right) d h \end{aligned}.$$

In order to facilitate the calculation, we use the integration theorem for Meijer’s G functions of Eq. (07.34.21.0013.01) in [46] and the approximate average SEP is established as

(23)$$\begin{aligned} {p_s} = & \frac{{{\xi ^2}A}}{\pi }\sum_{k = 1}^{M - 1} {\sum_{m = 1}^\beta {\frac{{{{\left( { - 1} \right)}^{k + 1}}{2^{\alpha + m - k - 4}}{a_m}}}{{\sqrt {k + 1} }}} } \left( {\begin{array}{c} {M - 1}\\ k \end{array}} \right){\left( {\frac{{\alpha \beta }}{{{b_g}\beta + {P'}}}} \right)^{ - \frac{{\alpha + m}}{2}}}\\ & \cdot G_{5,2}^{0,5}\left( {\frac{{8k{M^2}{\gamma ^2}}}{{k + 1}}\frac{{A_0^2h_l^2\left( {{b_g}\beta + {P'}} \right)}}{{{\alpha ^2}{\beta ^2}}}\left| {\begin{array}{c} {\frac{{2 - {\xi ^2}}}{2},\frac{{1 - \alpha }}{2},\frac{{2 - \alpha }}{2},\frac{{1 - m}}{2},\frac{{2 - m}}{2}}\\ {0, - \frac{{{\xi ^2}}}{2}} \end{array}} \right.} \right) \end{aligned}.$$

3.2 Average channel capacity

As another performance metric for the UWOC systems, the average channel capacity in this study is defined as the ergodic capacity, since the instantaneous channel capacity exhibits its random behavior that related to turbulence conditions. Therefore, the average channel capacity is given by

(24)$$C=\mathbb{E}[C(h)]=\int_{0}^{\infty} C(h) f_{h}(h) dh,$$

where $\mathbb {E}$ is the expectation operation.

In consideration of actual realization, the optical peak power and the average power must be constrained by

(25a)$$ \operatorname{Pr}\left[X>A_{1}\right] =0, $$

(25b)$$ \mathbb{E}[X] \leq \varepsilon_{1}, $$

where $\operatorname {Pr}$ stands for probability function, $A_1$ and $\varepsilon _1$ represent the allowed peak power and the allowed average power, respectively.

According to [49], we consider the following three cases in this study:

• Case I: both an average-power and a peak-power constraint are imposed with $a \in \left (0, \frac {1}{2}\right )$.
• Case II: both an average-power and a peak-power constraint are imposed with $a \in \left [\frac {1}{2}, 1\right ]$.
• Case III: only an average-power constraint is imposed.

where $a=\frac {\varepsilon _{1}}{A}$ denotes the ratio between the allowed average-power and the allowed peak-power.

For the UWOC systems with limited average power and limited peak power using IM/DD technique, the lower-bound channel capacity with $0<a < \frac {1}{2}$ is given by

(26)$$C_{{\rm I}}(h)=\frac{1}{2} \log _{2}\left[1+\gamma^{2} h^{2} \frac{e^{2 a \mu^{*}}}{2 \pi e}\left(\frac{1-e^{-\mu^{*}}}{\mu^{*}}\right)^{2}\right],$$

where $\mu ^{*}$ always exists and is the unique solution to $a=\frac {1}{\mu ^{*}}-\frac {e^{-\mu ^{*}}}{1-e^{-\mu ^{*}}}$.

Using the following identity $\log _{2}(1+x)=\frac {1}{\ln 2} G_{1,2}^{2,2}\left (\left.x\right |_{1,0} ^{1,1}\right )$ [50] and substituting Eqs. (17) and (26) into Eq. (24) with some mathematics manipulations, the average channel capacity for Case I can be simplified into a closed-form expression as

(27)$$\begin{aligned} {C_{\rm I}} & = \frac{{{\xi ^2}A}}{{32\pi \ln 2}}\sum_{m = 1}^\beta {{a_m}} {\left( {4\frac{{{b_g}\beta + {P'}}}{{\alpha \beta }}} \right)^{\frac{{\alpha + m}}{2}}}\\ & \cdot G_{8,4}^{1,8}\left( {\frac{{8{e^{2{\alpha _1}{\mu ^*} - 1}}}}{\pi }{{\left( {\frac{{1 - {e^{ - {\mu ^*}}}}}{{{\mu ^*}}}} \right)}^2}{{\left( {\frac{{{b_g}\beta + {P'}}}{{\alpha \beta }}} \right)}^2}A_0^2h_l^2{\gamma ^2}\left| {\begin{array}{c} {1,1,\frac{{1 - {\xi ^2}}}{2},\frac{{2 - {\xi ^2}}}{2},\frac{{1 - \alpha }}{2},\frac{{2 - \alpha }}{2},\frac{{1 - m}}{2},\frac{{2 - \alpha }}{2}}\\ {1,0,\frac{{ - {\xi ^2}}}{2},\frac{{1 - {\xi ^2}}}{2}} \end{array}} \right.} \right) \end{aligned}.$$

For $\frac {1}{2} \le a \le 1$, the lower-bound average channel capacity under the limitation of Case II can be expressed as

(28)$$C_{\rm II}(h)=\frac{1}{2} \log _{2}\left(1+\frac{1}{2 \pi e} \gamma^{2} h^{2}\right).$$

Following the same calculations for the average capacity of the UWOC systems in Case I, the average capacity in Case II is obtained as

(29)$$\begin{aligned} {C_{\rm II}} & = \frac{{{\xi ^2}A}}{{32\pi \ln 2}}\sum_{m = 1}^\beta {{a_m}} {\left( {4\frac{{{b_g}\beta + {P'}}}{{\alpha \beta }}} \right)^{\frac{{\alpha + m}}{2}}}\\ & \cdot G_{8,4}^{1,8}\left( {\frac{8}{{\pi e}}{{\left( {\frac{{{b_g}\beta + {P'}}}{{\alpha \beta }}} \right)}^2}A_0^2h_l^2{\gamma ^2}\left| {\begin{array}{c} {1,1,\frac{{1 - {\xi ^2}}}{2},\frac{{2 - {\xi ^2}}}{2},\frac{{1 - \alpha }}{2},\frac{{2 - \alpha }}{2},\frac{{1 - m}}{2},\frac{{2 - \alpha }}{2}}\\ {1,0,\frac{{ - {\xi ^2}}}{2},\frac{{1 - {\xi ^2}}}{2}} \end{array}} \right.} \right) \end{aligned}.$$

For only the average-power limitation in Case III, the average channel capacity is lower-bound by

(30)$$C_{\rm III}(h)=\frac{1}{2} \log _{2}\left(1+\frac{e}{2 \pi} \gamma^{2} h^{2}\right).$$

Similar to previous calculations for Case I and II, the average channel capacity in Case III can be finally achieved as

(31)$$\begin{aligned} {C_{\rm III}} & = \frac{{{\xi ^2}A}}{{32\pi \ln 2}}\sum_{m = 1}^\beta {{a_m}} {\left( {4\frac{{{b_g}\beta + {P'}}}{{\alpha \beta }}} \right)^{\frac{{\alpha + m}}{2}}}\\ & \cdot G_{8,4}^{1,8}\left( {\frac{{8e}}{\pi }{{\left( {\frac{{{b_g}\beta + {P'}}}{{\alpha \beta }}} \right)}^2}A_0^2h_l^2{\gamma ^2}\left| {\begin{array}{c} {1,1,\frac{{1 - {\xi ^2}}}{2},\frac{{2 - {\xi ^2}}}{2},\frac{{1 - \alpha }}{2},\frac{{2 - \alpha }}{2},\frac{{1 - m}}{2},\frac{{2 - \alpha }}{2}}\\ {1,0,\frac{{ - {\xi ^2}}}{2},\frac{{1 - {\xi ^2}}}{2}} \end{array}} \right.} \right) \end{aligned}.$$

4. Numerical results and discussions

In this section, the influences of the system and channel parameters on the UWOC systems are analyzed. Some relevant parameters used for the simulation are adopted from [35,51] and listed in Table 2. The Monte Carlo (MC) simulation with $10^6$ independent runs in MATLAB are used to verify the accuracy of the our derived expressions.

Table 2. Parameters used in the simulations.

View Table | View all tables in this article

The average SEP versus average electrical SNR under Lognormal, Gamma-Gamma, and Málaga models is demonstrated in Fig. 2. Compared to the Gamm-Gamma distribution model, the spherical wave has a better fit in the case of strong oceanic turbulence $\left (\alpha =2.296, \beta =2\right )$ with the assumption of the Málaga model. In addition, for the UWOC system covering weak oceanic turbulence $\left (\alpha =8, \beta =4\right )$, the analytical results under the assumption of the Lognormal model do not fit well with MC results. Therefore, we conclude that the Málaga distribution model has higher computational accuracy than the other models [28].

Fig. 2. Average SEP versus average electrical SNR under different oceanic turbulence model.

Download Full Size | PDF

The average SEP of the UWOC systems, as a function of the average electrical SNR for three different $\alpha$ and $\beta$ values are presented in Fig. 3, where $\left (\alpha =2.296, \beta =2\right )$, $\left (\alpha =4.2, \beta =3\right )$, and $\left (\alpha =8, \beta =4\right )$ represent for strong, moderate, and weak turbulence conditions, respectively. For the UWOC systems in weak turbulence, the average SEP is smaller than that in the moderate and strong turbulence. We conclude that the average SEP of the UWOC systems is strongly related to oceanic conditions.

Fig. 3. Average SEP versus average electrical SNR under different oceanic turbulence conditions.

Download Full Size | PDF

In addition, the average channel capacity under different turbulence conditions is shown in Fig. 4. The average channel capacity increases as the average electrical SNR increases. In addition, the average channel capacity under weak turbulence is higher than that under moderate and strong turbulence, which means the UWOC systems perform better in weak oceanic turbulence. Moreover, the UWOC systems have the highest and lowest capacity in Case III and I, respectively. Finally, the analytical and MC results reach a good match as the SNR increases.

Fig. 4. Average channel capacity versus average electrical SNR under different oceanic turbulence conditions.

Download Full Size | PDF

The effect of the dissipation rates of the mean-squared temperature, $\chi _{T}$, and salinity-temperature contribution factor, $\omega$, on the average SEP is demonstrated in Fig. 5. Obviously, the average SEP increases along with an increase of $\chi _{T}$. There is a complex change in the variation of the average SEP as $\omega$ increases from $-5$ to $-0.1$. This variation tendency can be divided into three parts. The average SEP increases slowly with the increase of $\omega$ when temperature dominates. Then, a rapid increase in average SEP occurs when $\omega$ reaches $-1$. However, the average SEP decreases with the increase of $\omega$ when the salinity dominates. These phenomena can be explained by the definition of the scintillation index and its related terms, as shown in Eqs. (7) and (8).

Fig. 5. Average SEP versus the rate of dissipation of the mean-squared temperature and the salinity-temperature contribution factor.

Download Full Size | PDF

Figure 6 shows the influence of the mean-squared temperature dissipation rate, $\chi _{T}$, and salinity-temperature contribution factor, $\omega$, on the average channel capacity under three power limitation cases. Figures 6 (a), (b), and (c) represent the limitations under Case I, II, and III, respectively. Obviously, the average channel capacity decreases as the mean-squared temperature dissipation rate increases. However, the salinity-temperature contribution factor induces a complicated variation for the average channel capacity. Note that the color bar on the left of each subfigure denotes the capacity value. The average channel capacity in Fig. 6 increases when $\omega$ is small. Then, there is a noticeable color change towards cool colors in these three figures. These phenomena mean that the average channel capacity decreases slowly when the turbulence is dominated by temperature. However, the average channel capacity decreases rapidly with the increase of $\omega$ when salinity dominates. Finally, the average channel capacity increases when $\omega$ tends to $0$.

Fig. 6. Average channel capacity versus the rate of dissipation of the mean-squared temperature and the salinity-temperature contribution factor. (a): Case I, (b): Case II, and (c): Case III.

Download Full Size | PDF

The average SEP under different normalized beam widths and modulation orders is illustrated in Fig. 7. The analytical results match well with the MC results with the increase of SNR. In addition, the average SEP is reduced with the increases of the average SNR and the modulation order. For instance, the average SEP with modulation order $M = 16$ is $5.21 \times 10^{-5}$ when the normalized beam widths is 2 and SNR is 30 dB, while the SEP is $1.22 \times 10^{-4}$ with $M = 4$. This can be explained by the fact that the achieved power increases as the modulation order increases, which leads to a higher SNR [52]. Moreover, a sharp decrease in the average SEP is presented in Fig. 7, with an increase in the normalized beam widths.

Fig. 7. Average SEP versus average electrical SNR for different modulation orders and different values of normalized beam widths.

Download Full Size | PDF

Figure 8 illustrates the impact of normalized beam widths on the average channel capacity of the UWOC systems under three power limitations. The analytical and MC simulation results have an exact match, which validates the accuracy of our derivations. As the average electrical SNR increases, the average channel capacity increases. Obviously, the average channel capacity in Case III are undoubtedly the largest one since only the average-power is limited. For contrast, Case I has the smallest capacity. These results are strongly related to $a$, which is the ratio between the allowed average-power and the allowed peak-power in Eq. (26) and the average electrical SNR in Eqs. (28) and (30). More clearly, the average channel capacity for Case I, II, and III under the SNR of 20, 30, and 40 dB are listed in Table 3. Obviously, the jitter has a significant impact on the average channel capacity. The difference in average channel capacity under different jitters is decreased and becomes stable as the SNR increases. In addition, the difference between Case I and III is smaller than that between Case II and III under the same SNR and jitter conditions.

Fig. 8. Average channel capacity versus average electrical SNR for different values of normalized beam widths in three cases.

Download Full Size | PDF

Table 3. The values of the average channel capacity in Case I, II, and III.

View Table | View all tables in this article

Note that Fig. 8 and Table 3 both show that using a narrow beam width can increase the power of the received signal, resulting in better system performance. In other words, the smaller $W_z/r$ is better for the UWOC systems. However, the transmitter beam vibrates and can easily move from the LOS in the direction of the receiver [53], when the beam width is narrow, which further induces communication interruption due to the misalignment effect. Therefore, it is necessary to select an optimized beam width for the real UWOC systems.

The effects of the noise on the average SEP and the average channel capacity are depicted in Fig. 9 and Fig. 10, respectively. Three typical noises are considered here: thermal noise, background noise, and quantum noise, which are expressed as $\sigma _{\text {te}}^{2}=\frac {4 k_{1} T_{e} R_{b}}{R R_{L} I_{o}^{2}}$, $\sigma _{\text {bg }}^{2}=\frac {2 q I_{B g} R_{b}}{R I_{o}^{2}}$ and $\sigma _{\text {quantum }}^{2}=\frac {2 q R_{b}}{R I_{o}}$. Therefore, the total noise is the combination of these noises. As shown in Fig. 9, the analytical results are almost the same as the simulation results. The average SEP decreases with the increase in the average receiving irradiance for the thermal noise, background noise, quantum noise and total noise. Note that the thermal noise dominates when $I_o$ is less than $-35$ dBm. However, the quantum noise dominates when the average receiving irradiance increases to $-25$ dBm. This is mainly resulted by the denominator of the noise expressions.

Fig. 9. Average SEP versus average receiving irradiance for different types of noise.

Download Full Size | PDF

Fig. 10. Average channel capacity versus average receiving irradiance for different types of noise under three capacity limitation cases.

Download Full Size | PDF

The analytical results in Fig. 10 also provide a perfect match to the simulation results for the average capacity. The channel capacity of the considered UWOC systems with thermal noise is smaller than that with quantum noise when $-50 \le I_o \le -30$ dBm. However, this phenomenon reverse when $I_o > -30$ dBm. In other words, the influence of thermal noise on the average channel capacity dominates at low irradiance but is replaced by the quantum noise at a high irradiance. In addition, the descending order of the average channel capacity under the power limitations is Case III, II, and I. For instance, the average channel capacity in Case III under the thermal noise, background noise and quantum noise are 19, 23, and 20 bps/Hz for $I_o = -35$ dBm, but decreases to 17, 22, and 19 bps/Hz in Case II, and 15, 20, and 17 bps/Hz in Case I.

The effect of different water conditions on the average SEP is shown in Fig. 11. The pure ocean water, clear ocean water, coastal ocean water, and turbid harbor water are considered in this study. The corresponding extinction coefficients are 0.056, 0.150, 0.305, and 2.170 $\rm {m^{-1}}$, respectively [54]. Notably, the analytical results in Fig. 11 match well with the numerical results with the increase of SNR, which verifies the correctness of our derivations. In addition, increasing the average electrical SNR can significantly decrease the average SEP. For instance, the average SEP of pure ocean water is $9 \times 10^{-3}$, while increases to $5 \times 10^{-3}$, $3 \times 10^{-3}$, and $5 \times 10^{-2}$ for clear ocean water, coastal ocean water, and turbid harbor water, respectively, when the average electrical SNR is 30 dB. These value decrease to $5 \times 10^{-5}$, $1 \times 10^{-5}$, $9 \times 10^{-4}$, and $1 \times 10^{-3}$, respectively, when the average electrical SNR is 45 dB. Moreover, the UWOC systems in pure ocean water has the lowest SEP than the other three types of water conditions.

Fig. 11. Average SEP versus average electrical SNR under different types of water.

Download Full Size | PDF

Figure 12 presents the influence of different types of water on the average channel capacity under three limitation cases. The analytical and MC results also coincide. It is observed that the UWOC systems in pure ocean water has the largest capacity, while it has the smallest value in turbid harbor water. This can be explained by the fact that the extinction coefficient of pure ocean water is much lower than that of turbid harbor water. This means that the optical signal has larger attenuation in turbid water, resulting in the bigger average SEP and the smaller average channel capacity.

Fig. 12. Average channel capacity versus average electrical SNR for different types of water under three cases.

Download Full Size | PDF

5. Conclusion

In this paper, a composited channel model for the UWOC systems is established considering the effects of oceanic turbulence, pointing error, and path loss. A closed-form expression of the average SEP with the $M$-ary PPM modulation is proposed. In addition, the average channel capacity, considering the average-power constraint and peak-power constraint, is analyzed. Monte Carlo simulation is used to verify the correctness of the results. Besides, the simulation results reveal that the influence of thermal noise is dominant at low SNR, while the influence of quantum noise is higher at high SNR. Furthermore, this study proves that the UWOC systems under the average-power constraint has a larger capacity than that with the limitations of both the average-power and peak-power constraints. All these results pave the way for the design of the UWOC systems.

Funding

Young Elite Scientist Sponsorship Program by CAST; Shanghai Space Innovation Fund (SAST2020-054); The Major Key Project of PCL (PCL2021A03-1).

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

1. X. Cui, X. Yin, H. Chang, Y. Guo, Z. Zheng, Z. Sun, G. Liu, and Y. Wang, “Analysis of an adaptive orbital angular momentum shift keying decoder based on machine learning under oceanic turbulence channels,” Opt. Commun. 429, 138–143 (2018). [CrossRef]

2. B. Luo, G. Wu, L. Yin, Z. Gui, and Y. Tian, “Propagation of optical coherence lattices in oceanic turbulence,” Opt. Commun. 425, 80–84 (2018). [CrossRef]

3. Q. Liang, Y. Zhang, and D. Yang, “Effects of turbulence on the vortex modes carried by quasi-diffracting free finite energy beam in ocean,” J. Mar. Sci. Eng. 8(6), 458 (2020). [CrossRef]

4. H. M. Oubei, E. Zedini, R. T. ElAfandy, A. Kammoun, M. Abdallah, T. K. Ng, M. Hamdi, M.-S. Alouini, and B. S. Ooi, “Simple statistical channel model for weak temperature-induced turbulence in underwater wireless optical communication systems,” Opt. Lett. 42(13), 2455–2458 (2017). [CrossRef]

5. Y. Li, S. Wang, C. Jin, Y. Zhang, and T. Jiang, “A survey of underwater magnetic induction communications: Fundamental issues, recent advances, and challenges,” IEEE Commun. Surv. Tutorials 21(3), 2466–2487 (2019). [CrossRef]

6. Y. Wu, Y. Zhang, Y. Li, and Z. Hu, “Beam wander of gaussian-schell model beams propagating through oceanic turbulence,” Opt. Commun. 371, 59–66 (2016). [CrossRef]

7. Z. Zeng, S. Fu, H. Zhang, Y. Dong, and J. Cheng, “A survey of underwater optical wireless communications,” IEEE Communications Surveys Tutorials 19(1), 204–238 (2017). [CrossRef]

8. X. Sun, C. H. Kang, M. Kong, O. Alkhazragi, Y. Guo, M. Ouhssain, Y. Weng, B. H. Jones, T. K. Ng, and B. S. Ooi, “A review on practical considerations and solutions in underwater wireless optical communication,” J. Lightwave Technol. 38(2), 421–431 (2020). [CrossRef]

9. Y. Ata and Y. Baykal, “Effect of anisotropy on bit error rate for an asymmetrical gaussian beam in a turbulent ocean,” Appl. Opt. 57(9), 2258 (2018). [CrossRef]

10. X. Luan, P. Yue, and X. Yi, “Scintillation index of an optical wave propagating through moderate-to-strong oceanic turbulence,” J. Opt. Soc. Am. A 36(12), 2048 (2019). [CrossRef]

11. Y. Li, Y. Zhang, and Y. Zhu, “Oceanic spectrum of unstable stratification turbulence with outer scale and scintillation index of gaussian-beam wave,” Opt. Express 27(5), 7656–7672 (2019). [CrossRef]

12. Y. Ata and Y. Baykal, “Scintillations of optical plane and spherical waves in underwater turbulence,” J. Opt. Soc. Am. A 31(7), 1552–1556 (2014). [CrossRef]

13. M. Elamassie, M. Uysal, Y. Baykal, M. Abdallah, and K. Qaraqe, “Effect of eddy diffusivity ratio on underwater optical scintillation index,” J. Opt. Soc. Am. A 34(11), 1969–1973 (2017). [CrossRef]

14. C. Gabriel, M. A. Khalighi, S. Bourennane, P. Léon, and V. Rigaud, “Misalignment considerations in point-to-point underwater wireless optical links,” in 2013 MTS/IEEE OCEANS - Bergen, (2013), pp. 1–5.

15. H. Zhang and Y. Dong, “Link misalignment for underwater wireless optical communications,” in 2015 Advances in Wireless and Optical Communications (RTUWO), (2015), pp. 215–218.

16. Y. Li, Y. Zhang, and Y. Zhu, “Capacity of underwater wireless optical links with pointing errors,” Opt. Commun. 446, 16–22 (2019). [CrossRef]

17. R. Boluda-Ruiz, P. Salcedo-Serrano, B. Castillo-Vázquez, A. Garcia-Zambrana, and J. Garrido-Balsells, “Capacity of underwater optical wireless communication systems over salinity-induced oceanic turbulence channels with isi,” Opt. Express 29(15), 23142–23158 (2021). [CrossRef]

18. M. Elamassie and M. Uysal, “Vertical underwater vlc links over cascaded gamma-gamma turbulence channels with pointing errors,” in 2019 IEEE International Black Sea Conference on Communications and Networking (BlackSeaCom), (2019), pp. 1–5.

19. X. Yi, Z. Li, and Z. Liu, “Underwater optical communication performance for laser beam propagation through weak oceanic turbulence,” Appl. Opt. 54(6), 1273–1278 (2015). [CrossRef]

20. Y. Fu and Y. Du, “Performance of heterodyne differential phase-shift-keying underwater wireless optical communication systems in gamma-gamma-distributed turbulence,” Appl. Opt. 57(9), 2057–2063 (2018). [CrossRef]

21. Z. Zou, W. Ping, W. Chen, A. Li, H. Tian, and L. Guo, “Average capacity of a uwoc system with partially coherent gaussian beams propagating in weak oceanic turbulence,” J. Opt. Soc. Am. A 36(9), 1463 (2019). [CrossRef]

22. P. N. Ramavath, S. A. Udupi, and P. Krishnan, “Co-operative rf-uwoc link performance over hyperbolic tangent log-normal distribution channel with pointing errors,” Opt. Commun. 469, 125774 (2020). [CrossRef]

23. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40(8), 1554–1562 (2001). [CrossRef]

24. Y. Baykal, “Bit error rate of pulse position modulated optical wireless communication links in oceanic turbulence,” J. Opt. Soc. Am. A 35(9), 1627–1632 (2018). [CrossRef]

25. P. Saxena and M. R. Bhatnagar, “A simplified form of beam spread function in underwater wireless optical communication and its applications,” IEEE Access 7, 105298–105313 (2019). [CrossRef]

26. F. He, Y. Du, J. Zhang, W. Fang, B. Li, and Y. Zhu, “Bit error rate of pulse position modulation wireless optical communication in gamma-gamma oceanic anisotropic turbulence,” Acta Phys. Sin. 68(16), 164206 (2019). [CrossRef]

27. S. Zhou, Z. Si, D. Ding, J. Wang, Y. Li, and C. Bai, “On performance of partially coherent laser underwater wireless optical communication over gamma-gamma ocean turbulence channel,” in 2021 IEEE International Conference on Signal Processing, Communications and Computing (ICSPCC), (2021), pp. 1–5.

28. A. Jurado-Navas, J. M. G. Balsells, J. F. Paris, M. Castillo-Vázquez, and A. Puerta-Notario, “General analytical expressions for the bit error rate of atmospheric optical communication systems,” Opt. Lett. 36(20), 4095–4097 (2011). [CrossRef]

29. G. Xu and J. Lai, “Average capacity analysis of the underwater optical plane wave over anisotropic moderate-to-strong oceanic turbulence channels with the málaga fading model,” Opt. Express 28(16), 24056–24068 (2020). [CrossRef]

30. M. Chen, S. Zhou, and T. Li, “The implementation of ppm in underwater laser communication system,” in 2006 International Conference on Communications, Circuits and Systems, (2006), pp. 1901–1903.

31. T. Xu, X. Sun, and D. Wen, “Implementation of pulse position modulated blue-green laser communication system,” in Sixth International Symposium on Precision Engineering Measurements and Instrumentation, (SPIE, 2010), pp. 1302–1307.

32. K. P. Peppas, A. C. Boucouvalas, and Z. Ghassemloy, “Performance of underwater optical wireless communication with multi-pulse pulse-position modulation receivers and spatial diversity,” IET Optoelectron. 11(5), 180–185 (2017). [CrossRef]

33. M. C. Gökçe, Y. Baykal, and Y. Ata, “Performance analysis of m-ary pulse position modulation in strong oceanic turbulence,” Opt. Commun. 427, 573–577 (2018). [CrossRef]

34. A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, A. Puerta-Notario, and J. Awrejcewicz, “A unifying statistical model for atmospheric optical scintillation,” Numer. simulations of physical and engineering processes 181, 181–205 (2011). [CrossRef]

35. A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, M. Castillo-Vázquez, and A. Puerta-Notario, “Impact of pointing errors on the performance of generalized atmospheric optical channels,” Opt. Express 20(11), 12550–12562 (2012). [CrossRef]

36. G. Xu, Z. Song, and Q. Zhang, “Outage probability and channel capacity of an optical spherical wave propagating through anisotropic weak-to-strong oceanic turbulence with málaga distribution,” J. Opt. Soc. Am. A 37(10), 1622–1629 (2020). [CrossRef]

37. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser beam scintillation with applications, vol. 99 (SPIE press, 2001).

38. E. B. Kraus and J. A. Businger, Atmosphere-ocean interaction, vol. 27 (Oxford University Press, 1994).

39. N. Ramsing and J. Gundersen, “Seawater and gases: tabulated physical parameters of interest to people working with microsensors in marine systems,” Techn Rep MPI Mar Microbiology Bremen (1994).

40. P. Yue, M. Wu, X. Yi, Z. Cui, and X. Luan, “Underwater optical communication performance under the influence of the eddy diffusivity ratio,” J. Opt. Soc. Am. A 36(1), 32–37 (2019). [CrossRef]

41. A. A. Farid and S. Hranilovic, “Outage capacity optimization for free-space optical links with pointing errors,” J. Lightwave Technol. 25(7), 1702–1710 (2007). [CrossRef]

42. C. Wang, H.-Y. Yu, and Y.-J. Zhu, “A long distance underwater visible light communication system with single photon avalanche diode,” IEEE Photonics J. 8(6), 1–8 (2016). [CrossRef]

43. M. Ali, D. N. Jayakody, Y. Chursin, S. Affes, and S. Dmitry, “Recent advances and future directions on underwater wireless communications,” Archives of Computational Methods in Engineering 26, 1–34 (2019). [CrossRef]

44. M. Elamassie, F. Miramirkhani, and M. Uysal, “Channel modeling and performance characterization of underwater visible light communications,” in 2018 IEEE International Conference on Communications Workshops (ICC Workshops), (2018), pp. 1–5.

45. M. Elamassie, F. Miramirkhani, and M. Uysal, “Performance characterization of underwater visible light communication,” IEEE Transactions on Communications 67(1), 543–552 (2019). [CrossRef]

46. I. Wolfram, “Research, mathematica edition: Version 8.0,” [Online]. Available: http://functions.wolfram.com.

47. W. Gappmair, S. Hranilovic, and E. Leitgeb, “Performance of ppm on terrestrial fso links with turbulence and pointing errors,” IEEE Communications Letters 14(5), 468–470 (2010). [CrossRef]

48. V. S. Adamchik and O. I. Marichev, “The algorithm for calculating integrals of hypergeometric type functions and its realization in reduce system,” in Proceedings of the International Symposium on Symbolic and Algebraic Computation, (1990), p. 212–224.

49. A. Lapidoth, S. M. Moser, and M. A. Wigger, “On the capacity of free-space optical intensity channels,” IEEE Trans. Inf. Theory 55(10), 4449–4461 (2009). [CrossRef]

50. I. Gradshteyn and I. Ryzhik, Table of Integrals, Series, and Products (Academic Press, 2007).

51. V. George, N. Hector, G. Wilfried, S. Harilaos, and T. George, “Df relayed subcarrier fso links over malaga turbulence channels with phase noise and non-zero boresight pointing errors,” Appl. Sci. 8(5), 664 (2018). [CrossRef]

52. A. E. Morra, H. S. Khallaf, H. M. H. Shalaby, and Z. Kawasaki, “Performance analysis of both shot- and thermal-noise limited multipulse ppm receivers in gamma–gamma atmospheric channels,” J. Lightwave Technol. 31(19), 3142–3150 (2013). [CrossRef]

53. S. Arnon, “Optimization of urban optical wireless communication systems,” IEEE Trans. Wireless Commun. 24(5), 626–629 (2003). [CrossRef]

54. C. Gabriel, M. A. Khalighi, S. Bourennane, P. Leon, and V. Rigaud, “Channel modeling for underwater optical communication,” in 2011 IEEE GLOBECOM Workshops (GC Wkshps), (2011), pp. 833–837.

Type of water	Absorption Coefficient $a (λ)$	Scattering Coefficient $b (λ)$	Extinction Coefficient $c (λ)$
Pure Ocean Water	0.053	0.003	0.056
Clear Ocean Water	0.114	0.037	0.151
Coastal Ocean Water	0.179	0.219	0.398
Turbid Harbor Water	0.295	1.875	2.170

Parameter, Symbol	Value (unit)	Parameter, Symbol	Value (unit)
Detection aperture radius, $r$	$0.05 (m)$	Elementary charge, $q$	$1.602 \times 10^{- 19} (C)$
Beam waist radius at $1$ , $W_{z}$	$(0.1, 0.25, 0.4) (m)$	Boltzmann constant, $k_{l}$	$1.38 \times 10^{- 23} (J / K)$
Jitter standard deviation, $σ_{s}$	0.15 $(m)$	Symbol rate, $R_{b}$	$1.55 \times 10^{8} (b p s)$
Photodetector responsivity, $R$	$1$	Background radiation irradiance, $I_{b g}$	$\begin{array}{l} 4 / π \times {0.6}^{2} \times 10^{- 6} \\ + 5.5 \times 10^{- 5} \end{array} (A)$
Receive circuit load resistance, $R_{L}$	$50 (Ω)$	The ratio between the allowed average-power and the allowed peak-power, $a$	$0.1$
Temperature, $T_{e}$	$300 (K)$		$0.1$

Type of water	Absorption Coefficient $a (λ)$	Scattering Coefficient $b (λ)$	Extinction Coefficient $c (λ)$
Pure Ocean Water	0.053	0.003	0.056
Clear Ocean Water	0.114	0.037	0.151
Coastal Ocean Water	0.179	0.219	0.398
Turbid Harbor Water	0.295	1.875	2.170

Parameter, Symbol	Value (unit)	Parameter, Symbol	Value (unit)
Detection aperture radius, $r$	$0.05 (m)$	Elementary charge, $q$	$1.602 \times 10^{- 19} (C)$
Beam waist radius at $1$ , $W_{z}$	$(0.1, 0.25, 0.4) (m)$	Boltzmann constant, $k_{l}$	$1.38 \times 10^{- 23} (J / K)$
Jitter standard deviation, $σ_{s}$	0.15 $(m)$	Symbol rate, $R_{b}$	$1.55 \times 10^{8} (b p s)$
Photodetector responsivity, $R$	$1$	Background radiation irradiance, $I_{b g}$	$\begin{array}{l} 4 / π \times {0.6}^{2} \times 10^{- 6} \\ + 5.5 \times 10^{- 5} \end{array} (A)$
Receive circuit load resistance, $R_{L}$	$50 (Ω)$	The ratio between the allowed average-power and the allowed peak-power, $a$	$0.1$
Temperature, $T_{e}$	$300 (K)$		$0.1$

Average symbol error probability and channel capacity of the underwater wireless optical communication systems over oceanic turbulence with pointing error impairments

Abstract

1. Introduction

2. System and channel models

2.1 Turbulence fading model

2.2 Pointing errors model

2.3 Path loss model

2.4 Composited fading model for the UWOC systems

3. System performance

3.1 Average symbol error probability

3.2 Average channel capacity

4. Numerical results and discussions

5. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (12)

Tables (3)

Equations (33)

Optics Express