Generalized transmit laser selection for vertical underwater wireless optical communications over Gamma-Gamma turbulence channels

Qi Zhang; Dianwu Yue; Xianying Xu

doi:10.1364/OE.500860

1. Introduction

Underwater wireless communication (UWC) is required in various marine development and maritime applications, such as marine environmental monitoring, oceanographic data collection, seabed resources investigation, port and shipping security. UWC refers to transmitting data in an unguided water environment through the use of wireless carriers, i.e., acoustic waves, radio frequency waves and optical waves [1–4]. Although underwater acoustic communication (UAC) is the most widely used technology in UWC, it still has several intrinsic technical limitations. The major disadvantages with UAC are low transmission data rate (typically on the order of Kbps), severe communication delay (typically in seconds) and bulky acoustic transceivers [1–3]. For underwater radio frequency communication (URFC), the fatal limitation is its short link range due to high absorption [1]. Compared with UAC and URFC, underwater wireless optical communication (UWOC) has the lowest link delay, the highest transmission rate [5] and the lowest implementation cost. It provides a reliable solution for high-speed and large-capacity underwater data transmission [6,7].

In addition to the absorption and scattering of optical signals by seawater, the performance of UWOC systems is mainly limited by ocean turbulence. Turbulence refers to the rapid fluctuation of the refractive index in an underwater environment caused by temperature and salinity fluctuation, and results in channel fading. The statistical distribution of underwater turbulence channel fading was experimentally studied in [8–12]. Lognormal and Gamma-Gamma (GG) probability density function (PDF) match the measurement results of weak turbulence and medium/strong turbulence [8,9,12] respectively.

Typical link configurations of UWOC systems include horizontal and vertical links. The turbulence intensity of the horizontal link is fixed in the transmission range, while the turbulence intensity of the vertical link varies with the depth of seawater, which leads to the difficulty of the vertical link.

The horizontal link has been well studied [13–18]. In [13], a multi-hop UWOC system with equidistant relays is studied. The influence of anisotropic weak-to-strong oceanic turbulence on the performance of UWOC systems over Malaga distribution channels is investigated in [14]. In [15], the model of information capacity for UWOC links with pointing errors and the carrier of perfect Laguerre-Gaussian beam in absorbed and weakly turbulent seawater is modeled. The bit error rate performance of UWOC systems over GG turbulence [16,17] and generalized gamma turbulence [18] is analyzed. However, the results of [13–18] are based on horizontal links with fixed turbulence intensity and are not applicable to vertical underwater links.

Since there are some realistic applications for requiring to transmit data between communication buoys and underwater network nodes, such as marine sensor networks, which are used to monitor earthquakes and tsunamis, or deployed in some critical sea areas as military monitoring networks. In these cases, UWOC channels are mostly vertical links. In [19], the authors modeled the vertical underwater link as a multi-layer cascaded fading channel where fading coefficients associated with different layers were modeled as independent and non-identical distributed for the first time. The bit error rate [20] and outage probability [21] performance of vertical stratified underwater optical links in the presence of GG turbulence conditions were investigated. [22] analyzed outage probability of a vertical underwater wireless optical link subject to lognormal weak turbulence. To capture the effects of air bubbles and temperature gradients on channel statistics, [23] considered each layer of vertical link with a mixture exponential-generalized Gamma distribution. [24] unified the performance analysis for the multi-layer vertical underwater link using generalized Gamma, exponential-generalized Gamma, exponentiated Weibull and GG oceanic turbulence models. Considering the fact that the buoy attached to the transmitter will fluctuate and oscillate due to wind and waves, [25] ignored the influence of turbulence and studied the effect of sea waves on vertical underwater optical links. The research on the vertical link is not sufficient. The existing works [19–25] of the vertical link mainly focus on the theoretical analysis of vertical channel model, the research on the design of vertical UWOC systems is still limited. In order to reduce the effects of turbulence fading, appropriate system design scheme for fading-mitigation is needed.

To combat the degrading effects of underwater turbulence, multiple input multiple output (MIMO) technology in UWOC was proposed to extract the spatial diversity [26–28]. In the case of MIMO, there are multiple optical links on both the transmission and reception sides, which significantly increases the complexity and cost of the system [29]. Additionally, in optical MIMO, channels are highly correlated because multiple sources transmit signals simultaneously [29]. As an alternative, transmit laser selection (TLS) can be considered [29–32]. In a typical TLS system, the transmitter is equipped with multiple laser sources, and only one source is selected to be active to transmit signals in a signaling cycle, while the remaining sources remain silent. Compared with the MIMO scheme, the channel of the TLS scheme is uncorrelated due to transmission from a single source [29]. The TLS scheme can achieve full diversity while significantly reducing the complexity of the transmitter and decoding complexity.

The comparison of some TLS research literatures is shown in Table 1. [29] proposed the free space optical (FSO) cooperative communication system with TLS under GG atmospheric turbulence and analyzed the outage probability and average symbol error probability. [30] considered TLS for a diver-to-diver underwater optical link with lognormal turbulence and derived a closed-form expression for asymptotical bit error rate. In [31], for the horizontal underwater link in the lognormal weak ocean turbulence environment, the advanced scheme of UWOC cooperative communication with TLS is designed, which can enhance the performance of conventional UWOC communication. In these TLS systems, the source with the highest instantaneous received signal-to-noise ratio (SNR) is always selected, which is based on idealized assumptions. In practice, there are various reasons that may lead to the selection of another source which is not optimal, such as imperfect channel estimation and erroneous feedback due to feedback delay and channel degradation [32]. Since such practical operation of TLS systems, this paper considers a vertical UWOC system with generalized TLS (GTLS) in GG ocean turbulence channels, where the ${n^{{\rm {th}}}}$ ($n = 1,\ 2,\ \cdots,\ N$) best laser source is selected among the available $N$ lasers. Furthermore, there is very little research on GTLS for turbulence channels. [32] considered a FSO system with GTLS and studied the asymptotic bit error rate performance over lognormal atmospheric turbulence channels. However, these results are not applicable to vertical GG turbulence UWOC. To the best of our knowledge, the analysis of GTLS for vertical UWOC with GG turbulence channels has not been done.

Table 1. Comparison of literatures on TLS

View Table

Against the above background, we present for the first time the analysis of a vertical UWOC system with GTLS in multi-layer cascaded GG turbulence channels in this study. We derive the exact closed-form outage probability expression of the GTLS system based on Meijer G-function and Gauss hypergeometric function. Diversity order expression is proposed and the asymptotical diversity order at high SNR values is investigated. Outage probability and diversity order of the vertical UWOC system under different conditions, such as the number of channel layers (the vertical link distances), the source index selected by the transmitter, and the number of laser sources, are analyzed. Compared to the system without GTLS (i.e. $N=1$), the proposed system has lower outage probability and higher diversity order.

2. System and channel models

As illustrated in Fig. 1, we consider a vertical UWOC system with $N$ laser sources (${S_i}$, $i = 1,\ 2,\ \cdots,\ N$) and a single photodiode detector (PD). The transmitter with $N$ lasers is deployed on a buoy. PD is deployed on the node of underwater sensor network to receive optical signals. We assume that the transmitter and PD remain relatively stationary and aligned. ${d_0}$ is the depth of the transmitter from the sea surface. The transmission range of the vertical underwater link is ${d_T}$. Based on [19], we model the vertical underwater link as a multi-layer cascaded channel. The thickness of the ${k^{{\rm {th}}}}$ layer is ${d_k}$ ($k = 1,\ 2,\ \cdots,\ K$, $K$ is the number of channel layers), and the thickness of all layers is equal. Then, the total link range becomes ${d_T} = \sum \nolimits _{k = 1}^K {{d_k}}$.

The receiver estimates the channel fading coefficient, which determines the instantaneous received SNR value and selects the index of the optimal laser source with the largest channel coefficient. Through the feedback channel, it sends the index to the transmitter, and the transmitter switches to the optimal source accordingly. In the ideal case, the optimal source is the source that gives the largest value of the fading coefficient. However, in practice, channel estimation and/or feedback errors may lead to sub-optimal or worse choices.

Considering moderate to strong turbulence conditions, the channel fading coefficients of different layers are modeled as independent GG distributions with different parameters. In a certain layer, the fading coefficients from different laser sources follow independent distribution with the same parameters. Therefore, for all laser sources, the channel fading coefficients of the end-to-end links are independent and identically distributed (i.i.d).

Let ${h_1},\ {h_2},\ \cdots,\ {h_n},\ \cdots,\ {h_N}$ represent i.i.d GG cascade channel fading coefficients arranged in decreasing order. ${h_n} = \prod \nolimits _{k = 1}^K {{h_{{n_k}}}}$ is the ${n^{{\rm {th}}}}$ highest fading coefficient associated with the ${n^{{\rm {th}}}}$ optimal laser source. ${{h_{{n_k}}}}$ ($k = 1,\ 2,\ \cdots,\ K$) denotes the multiplicative fading coefficient of the ${k^{{\rm {th}}}}$ layer. Let $x$ denote the transmission signal from the selected source with average electric transmission power of ${P_{{\rm {te}}}}$. The received signal can be expressed as

(1)$$y = \rho r{h_c}{h_n}x + {n_0},$$

where $\rho$ is the electro-optical conversion efficiency of the laser diode, $r$ is the responsivity of the photodiode, ${n_0} \sim {\cal N}\left ( {0,\sigma _n^2} \right )$ is the additive white Gaussian noise (AWGN) and ${h_c}$ represents the channel path loss caused by absorption and scattering effects. In order to simplify the analysis model, we ignore the geometric loss and use Beer-Lambert’s law [33], then ${h_c}$ can be simply expressed as

(2)$${h_c} = \exp \left( { - c{d_T}} \right),$$

where $c$ is the extinction coefficient which is the sum of absorption and scattering coefficients. The spacing between sources is on the order of centimeters and much smaller than the vertical link distance which is on the order of tens of meters. Therefore, the channel path loss can be considered identical for all channels [30]. In our channel model, we ignore the pointing errors caused by link misalignment, because the pointing errors can be compensated with proper tracking techniques or its effect remains at a negligible level for short distances [32]. In such cases, we need to consider only the impact of turbulence fading.

Under the GG ocean turbulence model, the PDF of channel turbulence fading coefficient ${h_{{t_k}}}$ of ${k^{{\rm {th}}}}$ layer for a general vertical link is given as [12]

(3)$${f_{{h_{{t_k}}}}}\left( {{h_{{t_k}}}} \right) = \displaystyle\frac{{2{{\left( {{\alpha _k}{\beta _k}} \right)}^{{{\left( {{\alpha _k} + {\beta _k}} \right)} \mathord{\left/{\vphantom {{\left( {{\alpha _k} + {\beta _k}} \right)} 2}} \right.} 2}}}}}{{\Gamma \left( {{\alpha _k}} \right)\Gamma \left( {{\beta _k}} \right)}}{h_{{t_k}}}^{{{\left( {{\alpha _k} + {\beta _k}} \right)} \mathord{\left/{\vphantom {{\left( {{\alpha _k} + {\beta _k}} \right)} 2}} \right.} 2} - 1}{{\rm{K}}_{{\alpha _k} - {\beta _k}}}\left( {2\sqrt {{\alpha _k}{\beta _k}{h_{{t_k}}}} } \right),$$

where $\Gamma \left ( \cdot \right )$ is Gamma function, ${{\rm K}_p}\left ( \cdot \right )$ is modified Bessel function of the second kind with order $p$, ${\alpha _k}$ and ${\beta _k}$ are the effective number of large-scale and small-scale cells of the scattering process for the ${k^{{\rm {th}}}}$ layer, respectively. Under the assumption of Gaussian beam propagation, they are given as [34]

(4)$${\alpha _k} = {\left[ {\exp \left( {\frac{{0.49\sigma _{{B_k}}^2}}{{{{\left( {1 + 0.56\left( {1 + {\Theta _k}} \right)\sigma _{{B_k}}^{12/5}} \right)}^{7/6}}}}} \right) - 1} \right]^{ - 1}},$$

(5)$${\beta _k} = {\left[ {\exp \left( {\frac{{0.51\sigma _{{B_k}}^2}}{{{{\left( {1 + 0.69\sigma _{{B_k}}^{12/5}} \right)}^{5/6}}}}} \right) - 1} \right]^{ - 1}},$$

where ${\sigma _{{B_k}}^2}$ is the Rytov variance of the ${k^{{\rm {th}}}}$ layer, which can be calculated as [35]

(6)$$\begin{array}{l} \sigma _{{B_k}}^2 = 8{\pi ^2}k_{\rm{0}}^2{d_k}\displaystyle\int_0^1 {\int_0^\infty {\kappa {\Phi _{{n_k}}}\left( \kappa \right)} \exp \left( { - \displaystyle\frac{{{\Lambda _k}{d_k}{\kappa ^2}{\zeta ^2}}}{{{k_{\rm{0}}}}}} \right)\exp \left( { - \displaystyle\frac{{D_R^2{\kappa ^2}{\zeta ^2}}}{{16}}} \right)} \\ \quad \quad \times \left\{ {1 - \cos \left[ {\displaystyle\frac{{{d_k}{\kappa ^2}}}{{{k_{\rm{0}}}}}\zeta \left( {1 - {{\bar \Theta }_k}\zeta } \right)} \right]} \right\}d\kappa d\zeta \end{array}.$$

In Eq. (6), ${k_0} = {{2\pi } \mathord {\left /{\vphantom {{2\pi } \lambda }} \right.} \lambda }$ is the wavenumber with wavelength $\lambda$, $\kappa$ is the magnitude of the spatial frequency, ${D_R}$ is the aperture diameter of the receiver, $\zeta$ is the normalized path coordinate. Let ${\Lambda _{{0_k}}} = {{2{d_k}} \mathord {\left /{\vphantom {{2{d_k}} {{k_{\rm {0}}}}}} \right.} {{k_{\rm {0}}}}}\omega _{\rm {0}}^2$ and ${\Lambda _k} = {{2{d_k}} \mathord {\left /{\vphantom {{2{d_k}}{{k_{\rm {0}}}}}} \right.} {{k_{\rm {0}}}}}\omega _{{d_k}}^2$ denote respectively the Fresnel ratio at the transmitter and receiver, where ${\omega _0}$ is the spot size of the Gaussian beam and ${\omega _{{d_k}}} \approx {\omega _0}{\left [ {1 + {{\left ( {{{\lambda {d_k}} \mathord {\left /{\vphantom {{\lambda {d_k}} {\pi \omega _0^2}}} \right.} {\pi \omega _0^2}}} \right )}^2}} \right ]^{{1 \mathord {\left /{\vphantom {1 2}} \right.} 2}}}$ is the beam waist of the Gaussian beam. ${\Theta _{{0_k}}} = {\left [ {\left ( {{{{\Lambda _{{0_k}}}} \mathord {\left /{\vphantom {{{\Lambda _{{0_k}}}} {{\Lambda _k}}}} \right.} {{\Lambda _k}}}} \right ) - \Lambda _{{0_k}}^2} \right ]^{{1 \mathord {\left /{\vphantom {1 2}} \right.} 2}}}$ and ${\Theta _k} = {{{\Theta _{{{\rm {0}}_k}}}} \mathord {\left /{\vphantom {{{\Theta _{{{\rm {0}}_k}}}} {\left ( {\Lambda _{{{\rm {0}}_k}}^2 + \Theta _{{{\rm {0}}_k}}^2} \right )}}} \right.} {\left ( {\Lambda _{{{\rm {0}}_k}}^2 + \Theta _{{{\rm {0}}_k}}^2} \right )}}$ are the beam curvature parameter at the transmitter and receiver respectively and ${{\bar \Theta }_k} = 1 - {\Theta _k}$ denotes the complementary parameter. ${{\Phi _{{n_k}}}\left ( \kappa \right )}$ is the spatial power spectrum of turbulent fluctuations of the seawater refraction index for the ${k^{{\rm {th}}}}$ layer and given as [36]

(7)$$\begin{array}{l} {{\Phi _{{n_k}}}\left( \kappa \right) = {{\left( {4\pi {\kappa ^2}} \right)}^{ - 1}}{C_0}\left( {\displaystyle\frac{{A_k^2{\chi _T}}}{{w_k^2}}} \right){\varepsilon ^{ - \frac{1}{3}}}{\kappa ^{ - \frac{5}{3}}}\left[ {1 + {C_1}{{\left( {\kappa {\eta _k}} \right)}^{\frac{2}{3}}}} \right]\left[ {w_k^2\exp \left( { - {C_0}C_1^{ - 2}P_{{T_k}}^{ - 1}{\delta _k}} \right)} \right.}\\ {\quad \quad \quad \;\; + \;{d_{{r_k}}}\exp \left( { - {C_0}C_1^{ - 2}P_{{S_k}}^{ - 1}{\delta _k}} \right) - \left. {{w_k}\left( {{d_{{r_k}}} + 1} \right)\exp \left( { - 0.5{C_0}C_1^{ - 2}P_{T{S_k}}^{ - 1}{\delta _k}} \right)} \right]} \end{array}.$$

In Eq. (7), the constants ${C_0} = 0.72$ and ${C_1} = 2.35$, ${\chi _T}$ is the dissipation rate of mean-squared temperature, $\varepsilon$ is the dissipation rate of turbulent kinetic energy per unit mass of fluid and ${\delta _k} = 1.5C_1^2{\left ( {\kappa {\eta _k}} \right )^{{4 \mathord {\left /{\vphantom {4 3}} \right.} 3}}} + C_1^3{\left ( {\kappa {\eta _k}} \right )^2}$. Here, ${\eta _k} = {\left ( {{{v_k^3} \mathord {\left /{\vphantom {{v_k^3} \varepsilon }} \right.} \varepsilon }} \right )^{{1 \mathord {\left /{\vphantom {1 4}} \right.} 4}}}$ denotes the Kolmogorov microscale length, where ${v_k}$ is the kinematic viscosity. ${P_{{T_k}}}$, ${P_{{S_k}}}$ and ${P_{T{S_k}}}$ are Prandtl numbers for temperature, salinity and temperature salt coupling, respectively. ${P_{{S_k}}} \approx {P_{{T_k}}} \times {10^2}$ and ${P_{T{S_k}}}$ is the one-half of the harmonic mean of ${P_{{T_k}}}$ and ${P_{{S_k}}}$. The relative strength of temperature and salinity fluctuations for the ${k^{{\rm {th}}}}$ layer is defined as

(8)$${w_k} =\displaystyle \frac{{{A_k}{{\left( {{{d{T_0}} \mathord{\left/{\vphantom {{d{T_0}} {dz}}} \right.} {dz}}} \right)}_k}}}{{{B_k}{{\left( {{{d{S_0}} \mathord{\left/{\vphantom {{d{S_0}} {dz}}} \right.} {dz}}} \right)}_k}}},$$

where ${{A_k}}$ and ${{B_k}}$ denote thermal expansion coefficient and saline contraction coefficient respectively, ${{{\left ( {{{d{T_0}} \mathord {\left /{\vphantom {{d{T_0}} {dz}}} \right.} {dz}}} \right )}_k}}$ and ${{{\left ( {{{d{S_0}} \mathord {\left /{\vphantom {{d{S_0}} {dz}}} \right. } {dz}}} \right )}_k}}$ are the temperature and salinity differences between top and bottom boundaries for the ${k^{{\rm {th}}}}$ layer. According to ${w_k}$, the eddy diffusivity ratio ${{d_{{r_k}}}}$ can be calculated as [19]

(9)$${d_{{r_k}}} \approx \left\{ {\begin{array}{l}{\left| {{w_k}} \right| + {{\left| {{w_k}} \right|}^{{1 \mathord{\left/{\vphantom {1 2}} \right.} 2}}}{{\left( {\left| {{w_k}} \right| - 1} \right)}^{{1 \mathord{\left/{\vphantom {1 2}} \right.} 2}}},}\\ {1.85\left| {{w_k}} \right| - 0.85,}\\ {0.15\left| {{w_k}} \right|.}\end{array}} \right.\begin{array}{l}{\left| {{w_k}} \right| \ge 1}\\ {0.5 \le \left| {{w_k}} \right| \le 1}\\ {\left| {{w_k}} \right| < 0.5} \end{array}$$

3. Performance analysis of vertical UWOC systems with GTLS

3.1 Outage probability analysis

For the vertical cascaded channel, the PDF of the total turbulent fading coefficient ${h_t} = \prod \nolimits _{k = 1}^K {{h_{{t_k}}}}$ is given as [19]

(10)$${f_{{h_t}}}\left( {{h_t}} \right) =\displaystyle \frac{{{P_{\alpha \beta }}}}{{{P_{\Gamma \left( \alpha \right)\Gamma \left( \beta \right)}}}}G_{0,2K}^{2K,0}\left( {{P_{\alpha \beta }}{h_t}\left| {\begin{array}{c}\cdots \\ \bf{a} \end{array}} \right.} \right),$$

(11)$${\bf{a}} = \left[ {{\alpha _1} - 1, \ldots ,{\alpha _K} - 1,{\beta _1} - 1, \ldots ,{\beta _K} - 1} \right].$$

In Eq. (10), $G_{p,q}^{m,n}\left ( { \cdot \left | \cdot \right.} \right )$ is Meijer G-function, ${P_{\alpha \beta }} = \prod \nolimits _{k = 1}^K {\left ( {{\alpha _k}{\beta _k}} \right )}$ and ${P_{\Gamma \left ( \alpha \right )\Gamma \left ( \beta \right )}} = \prod \nolimits _{k = 1}^K {\left ( {\Gamma \left ( {{\alpha _k}} \right )\Gamma \left ( {{\beta _k}} \right )} \right )}$. The cumulative distribution function (CDF) of ${{h_t}}$ can be calculated as

(12)$$\begin{array}{l} {{F_{{h_t}}}\left( {{h_t}} \right) = \displaystyle\int_0^{{h_t}} {{f_{{h_t}}}\left( h \right)dh} }\\ {\qquad \quad\;\; = \displaystyle\frac{{{P_{\alpha \beta }}}}{{{P_{\Gamma \left( \alpha \right)\Gamma \left( \beta \right)}}}}\int_0^{{h_t}} {G_{0,2K}^{2K,0}\left( {{P_{\alpha \beta }}h\left| {\begin{array}{c} \cdots \\ {\bf{a}} \end{array}} \right.} \right)dh} } \end{array}.$$

Using the integration of Meijer G-functions of Eq. (07.34.21.0084.01) in [37], we can express Eq. (12) as

(13)$${F_{{h_t}}}\left( {{h_t}} \right) =\displaystyle\frac{{{P_{\alpha \beta }}{h_t}}}{{{P_{\Gamma \left( \alpha \right)\Gamma \left( \beta \right)}}}}G_{1,2K + 1}^{2K,1}\left( {{P_{\alpha \beta }}{h_t}\left| {\begin{array}{c}0\\ {{\bf{a}}, - 1} \end{array}} \right.} \right).$$

The properties of Meijer G-function of Eq. (07.34.16.0001.01) in [37] is used within Eq. (13), it can be simplified as

(14)$${F_{{h_t}}}\left( {{h_t}} \right) =\displaystyle\frac{1}{{{P_{\Gamma \left( \alpha \right)\Gamma \left( \beta \right)}}}}G_{1,2K + 1}^{2K,1}\left( {{P_{\alpha \beta }}{h_t}\left| {\begin{array}{c}1\\ {{\bf{b}},0} \end{array}} \right.} \right),$$

(15)$${\bf{b}} = \left[ {{\alpha _1}, \ldots ,{\alpha _K},{\beta _1}, \ldots ,{\beta _K}} \right].$$

Based on the order statistics of Eq.(2.2.2) in [38], PDF of the ${n^{{\rm {th}}}}$ highest fading coefficient ${h_n}$ can be expressed as

(16)$${f_{{h_n}}}\left( {{h_n}} \right) = \displaystyle\frac{{N!}}{{\left( {N - n} \right)!\left( {n - 1} \right)!}}{\left[ {{F_{{h_t}}}\left( {{h_n}} \right)} \right]^{N - n}}{\left[ {1 - {F_{{h_t}}}\left( {{h_n}} \right)} \right]^{n - 1}}{f_{{h_t}}}\left( {{h_n}} \right).$$

Let $\gamma = h_n^2\bar \gamma$ denotes the instantaneous SNR where $\bar \gamma = {{{\rho ^2}{r^2}h_c^2{P_{{\rm {te}}}}} \mathord {\left /{\vphantom {{{\rho ^2}{r^2}h_c^2{P_{{\rm {te}}}}} {\sigma _n^2}}} \right.} {\sigma _n^2}}$ is the average SNR (ASNR). The PDF of $\gamma$ can be given as

(17)$$\begin{array}{l} {f_\gamma }\left( \gamma \right) = {f_{{h_n}}}\left( {\sqrt {\displaystyle\frac{\gamma }{{\bar \gamma }}} } \right)\left| {\displaystyle\frac{{d{h_n}}}{{d\gamma }}} \right|. \end{array}$$

Using Eq. (16) and Eq. (17), the PDF expression of $\gamma$ can be written as

(18)$${f_\gamma }\left( \gamma \right) = \displaystyle\frac{{N!}}{{2\left( {N - n} \right)!\left( {n - 1} \right)!\sqrt {\bar \gamma \gamma } }}{\left[ {{F_{{h_t}}}\left( {\sqrt {\displaystyle\frac{\gamma }{{\bar \gamma }}} } \right)} \right]^{N - n}}{\left[ {1 - {F_{{h_t}}}\left( {\sqrt {\displaystyle\frac{\gamma }{{\bar \gamma }}} } \right)} \right]^{n - 1}}{f_{{h_t}}}\left( {\sqrt {\displaystyle\frac{\gamma }{{\bar \gamma }}} } \right).$$

The CDF of $\gamma$ is found by

(19)$${F_\gamma }\left( \gamma \right) = \displaystyle\int_0^\gamma {{f_\gamma }\left( s \right)ds} .$$

Replacing Eq. (18) in Eq. (19), we have

(20)$${F_\gamma }\left( \gamma \right) = \displaystyle\frac{{N!}}{{\left( {N - n} \right)!\left( {n - 1} \right)!}}\int_0^\gamma {{{\left[ {{F_{{h_t}}}\left( {\sqrt {\displaystyle\frac{s}{{\bar \gamma }}} } \right)} \right]}^{N - n}}{{\left[ {1 - {F_{{h_t}}}\left( {\sqrt {\displaystyle\frac{s}{{\bar \gamma }}} } \right)} \right]}^{n - 1}}{f_{{h_t}}}\left( {\sqrt {\displaystyle\frac{s}{{\bar \gamma }}} } \right)\left( {\displaystyle\frac{1}{{2\sqrt {\bar \gamma s} }}} \right)ds} .$$

Then we apply $\left ( {{1 \mathord {\left /{\vphantom {1 {\left ( {2\sqrt {\bar \gamma s} } \right )}}} \right. } {\left ( {2\sqrt {\bar \gamma s} } \right )}}} \right )ds = d\left ( {\sqrt {{s \mathord {\left /{\vphantom {s {\bar \gamma }}} \right. } {\bar \gamma }}} } \right )$ in Eq. (20), we obtain

(21)$${F_\gamma }\left( \gamma \right) = \displaystyle\frac{{N!}}{{\left( {N - n} \right)!\left( {n - 1} \right)!}}\int_0^{\sqrt {\frac{\gamma }{{\bar \gamma }}} } {{{\left[ {{F_{{h_t}}}\left( {\sqrt {\displaystyle\frac{s}{{\bar \gamma }}} } \right)} \right]}^{N - n}}{{\left[ {1 - {F_{{h_t}}}\left( {\sqrt {\displaystyle\frac{s}{{\bar \gamma }}} } \right)} \right]}^{n - 1}}{f_{{h_t}}}\left( {\sqrt {\displaystyle\frac{s}{{\bar \gamma }}} } \right)d\sqrt {\displaystyle\frac{s}{{\bar \gamma }}} } .$$

Utilizing Eq. (22), we can express Eq. (21) as Eq. (23).

(22)$${f_{{h_t}}}\left( {\sqrt {\displaystyle\frac{s}{{\bar \gamma }}} } \right)d\sqrt {\displaystyle\frac{s}{{\bar \gamma }}} = d{F_{{h_t}}}\left( {\sqrt {\displaystyle\frac{s}{{\bar \gamma }}} } \right).$$

(23)$${F_\gamma }\left( \gamma \right) = \displaystyle\frac{{N!}}{{\left( {N - n} \right)!\left( {n - 1} \right)!}}\int_0^{{F_{{h_t}}}\left( {\sqrt {\frac{\gamma }{{\bar \gamma }}} } \right)} {{{\left[ {{F_{{h_t}}}\left( {\sqrt {\displaystyle\frac{s}{{\bar \gamma }}} } \right)} \right]}^{N - n}}{{\left[ {1 - {F_{{h_t}}}\left( {\sqrt {\displaystyle\frac{s}{{\bar \gamma }}} } \right)} \right]}^{n - 1}}d{F_{{h_t}}}\left( {\sqrt {\displaystyle\frac{s}{{\bar \gamma }}} } \right)} .$$

To solve the integration in Eq. (23), we use Eq. (3.194.1) in [39], and the CDF expression can be calculated as

(24)$$\begin{array}{l} {{F_\gamma }\left( \gamma \right) = \displaystyle\frac{{N!}}{{\left( {N - n + 1} \right)!\left( {n - 1} \right)!}}{{\left[ {{F_{{h_t}}}\left( {\sqrt {\displaystyle\frac{\gamma }{{\bar \gamma }}} } \right)} \right]}^{N - n + 1}}}\\ {\quad \quad \quad \times {\;_2}{F_1}\left( {1 - n,N - n + 1;N - n + 2;{F_{{h_t}}}\left( {\sqrt {\displaystyle\frac{\gamma }{{\bar \gamma }}} } \right)} \right)} \end{array},$$

where ${}_2{F_1}\left ( \cdot \right )$ is Gauss hypergeometric function.

The outage probability is defined as the probability that the instantaneous SNR $\gamma$ is lower than the threshold SNR ${\gamma _{th}}$. It can be given by

(25)$${P_{{\rm{out}}}} = P\left( {\gamma \le {\gamma _{th}}} \right) = {F_\gamma }\left( {{\gamma _{th}}} \right).$$

Substituting Eq. (14) and Eq. (24) into Eq. (25), the closed-form expression of outage probability can be obtained as

(26)$$\begin{array}{l} {{P_{{\rm{out}}}} = \displaystyle\frac{{N!}}{{\left( {N - n + 1} \right)!\left( {n - 1} \right)!P_{\Gamma \left( \alpha \right)\Gamma \left( \beta \right)}^{N - n + 1}}}{{\left[ {G_{1,2K + 1}^{2K,1}\left( {{P_{\alpha \beta }}\sqrt {\displaystyle\frac{{{\gamma _{th}}}}{{\bar \gamma }}} \left| {\begin{array}{c}1\\ {{\bf{b}},0} \end{array}} \right.} \right)} \right]}^{N - n + 1}}}\\ {\qquad \times {\;_2}{F_1}\left( {1 - n,N - n + 1;N - n + 2;\displaystyle\frac{1}{{{P_{\Gamma \left( \alpha \right)\Gamma \left( \beta \right)}}}}G_{1,2K + 1}^{2K,1}\left( {{P_{\alpha \beta }}\sqrt {\displaystyle\frac{{{\gamma _{th}}}}{{\bar \gamma }}} \left| {\begin{array}{c}1\\ {{\bf{b}},0} \end{array}} \right.} \right)} \right)} \end{array}.$$

3.2 Diversity order analysis

In this section, we consider the diversity order (DO) of the vertical UWOC system with GTLS. Here, diversity order is negative slope of the outage probability curve on a log-log scale, which determines how fast the outage probability changes with SNR. Diversity order is defined as [21]

(27)$${\rm{DO}}\left( {\bar \gamma } \right) ={-} \displaystyle\frac{{\partial \ln {P_{{\rm{out}}}}\left( {\bar \gamma } \right)}}{{\partial \ln \bar \gamma }}.$$

Substituting Eq. (26) into Eq. (27), we have

(28)$${\rm{DO}}\left( {\bar \gamma } \right) ={-} \left( {N - n + 1} \right)\underbrace {\displaystyle\frac{{\partial \ln \varphi \left( {\bar \gamma } \right)}}{{\partial \ln \bar \gamma }}}_{X\left( {\bar \gamma } \right)} - \underbrace {\displaystyle\frac{{\partial \ln \psi \left( {\bar \gamma } \right)}}{{\partial \ln \bar \gamma }}}_{Y\left( {\bar \gamma } \right)},$$

where

(29)$$\varphi \left( {\bar \gamma } \right) = G_{1,2K + 1}^{2K,1}\left( {{P_{\alpha \beta }}\sqrt {\displaystyle\frac{{{\gamma _{th}}}}{{\bar \gamma }}} \left| {\begin{array}{c}1\\ {{\bf{b}},0} \end{array}} \right.} \right),$$

(30)$$\psi \left( {\bar \gamma } \right) = {}_2{F_1}\left( {1 - n,N - n + 1;N - n + 2;\displaystyle\frac{1}{{{P_{\Gamma \left( \alpha \right)\Gamma \left( \beta \right)}}}}\varphi \left( {\bar \gamma } \right)} \right).$$

In Eq. (28), we can express $X\left ( {\bar \gamma } \right )$ as

(31)$$X\left( {\bar \gamma } \right) = \displaystyle\frac{{\partial \ln \varphi \left( {\bar \gamma } \right)}}{{\partial \ln \bar \gamma }} = \displaystyle\frac{{\bar \gamma \varphi '\left( {\bar \gamma } \right)}}{{\varphi \left( {\bar \gamma } \right)}},$$

where ${\varphi '\left ( {\bar \gamma } \right )}$ is the first derivative of ${\varphi \left ( {\bar \gamma } \right )}$ with respect to ${\bar \gamma }$. We utilize the derivative of Meijer G-function of Eq. (07.34.20.0001.01) in [37] along with the chain rule of differentiation, ${\varphi '\left ( {\bar \gamma } \right )}$ can be expressed as

(32)$$\varphi '\left( {\bar \gamma } \right) ={-} \displaystyle\frac{1}{2}{P_{\alpha \beta }}\gamma _{th}^{\frac{1}{2}}{{\bar \gamma }^{ - \frac{3}{2}}}G_{2,2K + 2}^{2K,2}\left( {{P_{\alpha \beta }}\sqrt {\displaystyle\frac{{{\gamma _{th}}}}{{\bar \gamma }}} \left| {\begin{array}{c}{ - 1,0}\\ {{\bf{a}},0, - 1} \end{array}} \right.} \right).$$

Using the properties of Meijer G-function of Eq. (07.34.03.0001.01) in [37], Eq. (32) can be simplified as

(33)$$\varphi '\left( {\bar \gamma } \right) ={-} \displaystyle\frac{1}{2}{P_{\alpha \beta }}\gamma _{th}^{\frac{1}{2}}{{\bar \gamma }^{ - \frac{3}{2}}}G_{0,2K}^{2K,0}\left( {{P_{\alpha \beta }}\sqrt {\displaystyle\frac{{{\gamma _{th}}}}{{\bar \gamma }}} \left| {\begin{array}{c}\cdots \\ {\bf{a}} \end{array}} \right.} \right).$$

Replacing Eq. (29) and Eq. (33) in Eq. (31), we obtain $X\left ( {\bar \gamma } \right )$ as

(34)$$X\left( {\bar \gamma } \right) ={-} \displaystyle\frac{1}{2}{P_{\alpha \beta }}\gamma _{th}^{\frac{1}{2}}{{\bar \gamma }^{ - \frac{1}{2}}}\displaystyle\frac{{G_{0,2K}^{2K,0}\left( {{P_{\alpha \beta }}\sqrt {\displaystyle\frac{{{\gamma _{th}}}}{{\bar \gamma }}} \left| {\begin{array}{c}\cdots \\ {\bf{a}} \end{array}} \right.} \right)}}{{G_{1,2K + 1}^{2K,1}\left( {{P_{\alpha \beta }}\sqrt {\displaystyle\frac{{{\gamma _{th}}}}{{\bar \gamma }}} \left| {\begin{array}{c}1\\ {{\bf{b}},0} \end{array}} \right.} \right)}}.$$

Then, utilizing Eq. (07.34.16.0001.01) in [37], Eq. (34) can be simplified as

(35)$$X\left( {\bar \gamma } \right) ={-} \frac{1}{2}\frac{{G_{0,2K}^{2K,0}\left( {{P_{\alpha \beta }}\sqrt {\displaystyle\frac{{{\gamma _{th}}}}{{\bar \gamma }}} \left| {\begin{array}{c}\cdots \\ \bf{b} \end{array}} \right.} \right)}}{{G_{1,2K + 1}^{2K,1}\left( {{P_{\alpha \beta }}\sqrt {\displaystyle\frac{{{\gamma _{th}}}}{{\bar \gamma }}} \left| {\begin{array}{c}1\\ {{\bf{b}},0} \end{array}} \right.} \right)}}.$$

${Y\left ( {\bar \gamma } \right )}$ can be expressed as

(36)$$Y\left( {\bar \gamma } \right) = \displaystyle\frac{{\partial \ln \psi \left( {\bar \gamma } \right)}}{{\partial \ln \bar \gamma }} = \displaystyle\frac{{\bar \gamma \psi '\left( {\bar \gamma } \right)}}{{\psi \left( {\bar \gamma } \right)}},$$

where ${\psi '\left ( {\bar \gamma } \right )}$ is the first derivative of ${\psi \left ( {\bar \gamma } \right )}$ with respect to ${\bar \gamma }$. We utilize the derivative of Gauss hypergeometric function of Eq.(07.23.20.0010.01) in [37] along with the chain rule of differentiation, ${\psi '\left ( {\bar \gamma } \right )}$ can be expressed as

(37)$$\psi '\left( {\bar \gamma } \right) = \displaystyle\frac{{\left( {1 - n} \right)\left( {N - n + 1} \right)}}{{\left( {N - n + 2} \right){P_{\Gamma \left( \alpha \right)\Gamma \left( \beta \right)}}}}\varphi '\left( {\bar \gamma } \right){}_2{F_1}\left( {2 - n,N - n + 2;N - n + 3;\displaystyle\frac{1}{{{P_{\Gamma \left( \alpha \right)\Gamma \left( \beta \right)}}}}\varphi \left( {\bar \gamma } \right)} \right).$$

Substituting Eq. (33) into Eq. (37), we get

(38)$$\begin{array}{l} {\psi '\left( {\bar \gamma } \right) ={-} \displaystyle\frac{{\left( {1 - n} \right)\left( {N - n + 1} \right){P_{\alpha \beta }}\gamma _{th}^{\frac{1}{2}}{{\bar \gamma }^{ - \frac{3}{2}}}}}{{2\left( {N - n + 2} \right){P_{\Gamma \left( \alpha \right)\Gamma \left( \beta \right)}}}}G_{0,2K}^{2K,0}\left( {{P_{\alpha \beta }}\sqrt {\displaystyle\frac{{{\gamma _{th}}}}{{\bar \gamma }}} \left| {\begin{array}{c}\cdots \\ {\bf{a}} \end{array}} \right.} \right)}\\ {\qquad \;\;{ \times _2}{F_1}\left( {2 - n,N - n + 2;N - n + 3;\displaystyle\frac{1}{{{P_{\Gamma \left( \alpha \right)\Gamma \left( \beta \right)}}}}\varphi \left( {\bar \gamma } \right)} \right)} \end{array}.$$

Inserting Eq. (30) and Eq. (38) into Eq. (36) and using Eq. (07.34.16.0001.01) in [37], ${Y\left ( {\bar \gamma } \right )}$ can be written as

(39)$$\begin{array}{l} Y\left( {\bar \gamma } \right) ={-} \displaystyle\frac{{\left( {1 - n} \right){{\left( {N - n + 1} \right)}_2}{F_1}\left( {2 - n,N - n + 2;N - n + 3;\displaystyle\frac{1}{{{P_{\Gamma \left( \alpha \right)\Gamma \left( \beta \right)}}}}\varphi \left( {\bar \gamma } \right)} \right)}}{{2\left( {N - n + 2} \right){P_{\Gamma \left( \alpha \right)\Gamma \left( \beta \right)}}_2{F_1}\left( {1 - n,N - n + 1;N - n + 2;\displaystyle\frac{1}{{{P_{\Gamma \left( \alpha \right)\Gamma \left( \beta \right)}}}}\varphi \left( {\bar \gamma } \right)} \right)}}\\ \quad\quad\;\times\; G_{0,2K}^{2K,0}\left( {{P_{\alpha \beta }}\sqrt {\displaystyle\frac{{{\gamma _{th}}}}{{\bar \gamma }}} \left| {\begin{array}{c}\cdots \\ {\bf{b}} \end{array}} \right.} \right) \end{array}.$$

Replacing Eq. (35) and Eq. (39) in Eq. (28), we obtain the closed-form expression of diversity order as

(40)$$\begin{array}{l} {{\rm{DO}}\left( {\bar \gamma } \right) = \displaystyle\frac{{N - n + 1}}{2}\displaystyle\frac{{G_{0,2K}^{2K,0}\left( {{P_{\alpha \beta }}\sqrt {\displaystyle\frac{{{\gamma _{th}}}}{{\bar \gamma }}} \left| {\begin{array}{c}\cdots \\ {\bf{b}} \end{array}} \right.} \right)}}{{G_{1,2K + 1}^{2K,1}\left( {{P_{\alpha \beta }}\sqrt {\displaystyle\frac{{{\gamma _{th}}}}{{\bar \gamma }}} \left| {\begin{array}{c}1\\ {{\bf{b}},0} \end{array}} \right.} \right)}} + \displaystyle\frac{{\left( {1 - n} \right)\left( {N - n + 1} \right)}}{{2\left( {N - n + 2} \right){P_{\Gamma \left( \alpha \right)\Gamma \left( \beta \right)}}}}}\\ {\qquad \;\;\;\; \times\; \displaystyle\frac{{_2{F_1}\left( {2 - n,N - n + 2;N - n + 3;\displaystyle\frac{1}{{{P_{\Gamma \left( \alpha \right)\Gamma \left( \beta \right)}}}}G_{1,2K + 1}^{2K,1}\left( {{P_{\alpha \beta }}\sqrt {\displaystyle\frac{{{\gamma _{th}}}}{{\bar \gamma }}} \left| {\begin{array}{c}1\\ {{\bf{b}},0} \end{array}} \right.} \right)} \right)}}{{_2{F_1}\left( {1 - n,N - n + 1;N - n + 2;\displaystyle\frac{1}{{{P_{\Gamma \left( \alpha \right)\Gamma \left( \beta \right)}}}}G_{1,2K + 1}^{2K,1}\left( {{P_{\alpha \beta }}\sqrt {\displaystyle\frac{{{\gamma _{th}}}}{{\bar \gamma }}} \left| {\begin{array}{c}1\\ {{\bf{b}},0} \end{array}} \right.} \right)} \right)}}}\\ {\qquad \;\;\;\; \times\; G_{0,2K}^{2K,0}\left( {{P_{\alpha \beta }}\sqrt {\displaystyle\frac{{{\gamma _{th}}}}{{\bar \gamma }}} \left| {\begin{array}{c}\cdots \\ {\bf{b}} \end{array}} \right.} \right)} \end{array}.$$

In the following, we consider asymptotically high SNR to derive the diversity order, i.e., asymptotical diversity order (ADO), which is defined as

(41)$$ \mathrm{ADO}=\underset{\bar{\gamma} \rightarrow \infty}{\mathrm{DO}(\bar{\gamma})}=-\frac{\partial \ln \left(P_{\substack{\mathrm{out} \\ \bar{\gamma} \rightarrow \infty}}(\bar{\gamma})\right)}{\partial \ln \bar{\gamma}} . $$

Under the assumption of high SNR, i.e., $\bar \gamma \to \infty$, the inner argument of Gauss hypergeometric function in Eq. (26) goes to zero, by using the properties of Gauss hypergeometric function of Eq. (07.23.02.0001.01) in [37], the hypergeometric function term in Eq. (26) approaches 1. Then the expression of outage probability can be simplified as

(42)$$\mathop {{P_{{\rm{out}}}}\left( {\bar \gamma } \right)}_{\bar \gamma \to \infty } = \displaystyle\frac{{N!}}{{\left( {N - n + 1} \right)!\left( {n - 1} \right)!P_{\Gamma \left( \alpha \right)\Gamma \left( \beta \right)}^{N - n + 1}}}{\left[ {G_{1,2K + 1}^{2K,1}\left( {{P_{\alpha \beta }}\sqrt {\displaystyle\frac{{{\gamma _{th}}}}{{\bar \gamma }}} \left| {\begin{array}{c}1\\ {{\bf{b}},0} \end{array}} \right.} \right)} \right]^{N - n + 1}} .$$

In Eq. (42), we have ${P_{\alpha \beta }}\sqrt {{{{\gamma _{th}}} \mathord {\left /{\vphantom {{{\gamma _{th}}} {\bar \gamma }}} \right. } {\bar \gamma }}} \to 0$. Utilizing Eq. (07.34.06.0006.01) in [37] and the recursive property of Gamma function, i.e., $\Gamma \left ( {1 + x} \right ) = x\Gamma \left ( x \right )$, Eq. (42) can be written as

(43)$$\mathop {{P_{{\rm{out}}}}\left( {\bar \gamma } \right)}_{\bar \gamma \to \infty } = \displaystyle\frac{{N!}}{{\left( {N - n + 1} \right)!\left( {n - 1} \right)!P_{\Gamma \left( \alpha \right)\Gamma \left( \beta \right)}^{N - n + 1}}}{\left[ {\sum\nolimits_{k = 1}^{2K} {\displaystyle\frac{{\prod\nolimits_{j = 1,j \ne k}^{2K} {\Gamma \left( {{b_j} - {b_k}} \right)} }}{{{b_k}}}P_{\alpha \beta }^{{b_k}}\gamma _{th}^{\frac{{{b_k}}}{2}}{{\bar \gamma }^{ - \frac{{{b_k}}}{2}}}} } \right]^{N - n + 1}}.$$

According to Eq. (43), when $\bar \gamma \to \infty$, the highest order term of ${\bar \gamma }$ dominates the outage probability. Ignoring the lower order terms, we obtain

(44)$$ \begin{aligned} \underset{\bar{\gamma} \rightarrow \infty}{P_{\text {out }}(\bar{\gamma})}= & \frac{N !}{(N-n+1) !(n-1) ! P_{\Gamma(\alpha) \Gamma(\beta)}^{N-n+1}} \\ & \times\left[\frac{\prod_{j=1, b_j \neq \min (\mathbf{b})}^{2 K} \Gamma\left(b_j-\min (\mathbf{b})\right)}{\min (\mathbf{b})} P_{\alpha \beta}^{\min (\mathbf{b})} \gamma_{t h}^{\frac{\min (\mathbf{b})}{2}}\right]^{N-n+1} \bar{\gamma}^{-\frac{N-n+1}{2} \min (\mathbf{b})} \end{aligned} $$

Substituting Eq. (44) into Eq. (41), we get the expression of asymptotical diversity order as

(45)$$\begin{array}{l} {\rm{ADO}} = \displaystyle\frac{{N - n + 1}}{2}\min \left( {\bf{b}} \right)\\ \qquad\;\; = \displaystyle\frac{{N - n + 1}}{2}\min \left( {{\alpha _1}, \ldots ,{\alpha _K},{\beta _1}, \ldots ,{\beta _K}} \right) \end{array}.$$

Based on Eq. (45), we obtain the results that the asymptotical diversity order depends on the number of laser sources in the transmitter, the selected source index and the minimum turbulence parameter in all cascade channel layers.

4. Simulation results

In this section, we verify the exact closed-form expression for outage probability derived in Eq. (26) through simulation. We further give numerical results and verify the closed-form expressions for diversity order and asymptotical diversity order in Eq. (40) and Eq. (45), respectively. Unless otherwise stated, we consider threshold SNR of ${\gamma _{th}} = 10\ {\rm { dB}}$, detector aperture diameter of ${D_R} = 5\ {\rm { cm}}$, noise variance of $\sigma _n^2 = - 100\ {\rm { dBm}}$, the electro-optical conversion efficiency of $\rho {\rm { = 0}}{\rm {.5 }}\ {{\rm {W}} \mathord {\left /{\vphantom {{\rm {W}} {\rm {A}}}} \right. } {\rm {A}}}$, the responsivity of the photodiode of $r = 0.28\ {\rm { }}{{\rm {A}} \mathord {\left /{\vphantom {{\rm {A}} {\rm {W}}}} \right. } {\rm {W}}}$, the extinction coefficient of $c = 0.15\ {\rm { }}{{\rm {m}}^{ - 1}}$, the spot size of Gaussian beam of ${\omega _0} = 0.01\ {\rm { m}}$, wavelength of $\lambda {\rm { = 530 nm}}$, the dissipation rate of mean-squared temperature of ${\chi _T} = 5 \times {10^{ - 3}}\ {\rm { }}{{\rm {K}}^2}{{\rm {s}}^{ - 3}}$ and the dissipation rate of turbulent kinetic energy per unit mass of fluid of $\varepsilon = 1 \times {10^{ - 3}}\ {\rm { }}{{\rm {m}}^2}{{\rm {s}}^{ - 3}}$. We use the temperature and salinity profiles of the Pacific Ocean at high latitudes given by Fig. 1 in [19], thermal expansion coefficient ${A_k}$ and saline contraction coefficient ${B_k}$ can be computed by TEOS-10 toolbox (MATLAB Oceanographic Toolbox of International Thermodynamic Equation of Seawater-2010) [40], kinematic viscosity ${v_k}$ and Prandtl number for temperature ${P_{{T_k}}}$ can be computed by the toolbox ’Seawater Thermophysical Properties Library’ [41].

Fig. 1. GTLS vertical UWOC system block diagram.

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In Fig. 2, we investigate the outage probability considering the vertical underwater link with the number of channel layers of $K = 1,\ {\rm { }}2,\ {\rm { }}3,\ {\rm { }}4$. We assume a layer thickness of 20 m (${d_k} = 20\ {\rm {m}}$) and the link distances are 20 m, 40 m, 60 m and 80 m, respectively. Since the transmitter is placed on a buoy at the sea surface, we set ${d_0} = 0\ {\rm {m}}$. The Rytov variance of each layer can be calculated as $\sigma _{{B_1}}^2 = 1.78$, $\sigma _{{B_2}}^2 = 1.64$, $\sigma _{{B_3}}^2 = 1.48$ and $\sigma _{{B_4}}^2 = 1.29$. According to Eq. (4) and Eq. (5), the Gamma-Gamma ocean turbulence parameters are $\left ( {{\alpha _1},{\rm { }}{\beta _1}} \right ) = \left ( {4.03,{\rm { }}1.81} \right )$, $\left ( {{\alpha _2},{\rm { }}{\beta _2}} \right ) = \left ( {4.05,{\rm { }}1.88} \right )$, $\left ( {{\alpha _3},{\rm { }}{\beta _3}} \right ) = \left ( {4.09,{\rm { }}2.00} \right )$ and $\left ( {{\alpha _4},{\rm { }}{\beta _4}} \right ) = \left ( {4.17,{\rm { }}2.17} \right )$. We consider the GTLS system with $N = 5$ laser sources and the optimal laser source is selected, i.e., $n = 1$. As a benchmark, the system without GTLS (i.e. $N = 1$) is also included. It is observed from Fig. 2 that our derived closed-form expression for outage probability in Eq. (26) is consistent with the simulation results, which verifies the accuracy of the derivation. It is also observed that the outage probability of the GTLS system is lower than that of the system without GTLS, and the outage probability worsens as the number of layers (the link distance) increases. For example, assuming that the target outage probability of the GTLS system is ${P_{{\rm {out}}}} = {10^{ - 3}}$, the required ASNR for the single-layer case is $\bar \gamma = 18.75\ {\rm {dB}}$. For the two-layer case, the required ASNR is 25 dB. For the cases of three- and four-layer, the required ASNR increases to 30 dB and 35 dB. As the number of link layers increases, the required ASNR increases by 6.25 dB, 11.25 dB and 16.25 dB compared with the single-layer case, respectively. If the target outage probability of the GTLS system is reduced to ${P_{{\rm {out}}}} = {10^{ - 6}}$, it can be achieved just for single-layer and two-layer cases within the range of ASNR considered. For the system without GTLS, the outage probability is always higher than ${10^{ - 3}}$.

Fig. 2. Outage probability with different numbers of channel layers (different vertical link distances). Here, $n = 1$.

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In Fig. 3, we consider a vertical underwater link with the number of channel layers of $K = 2$ and the GTLS system with $N = 5$ laser sources. As a benchmark, the system without GTLS (i.e. $N = 1$) is also considered. Outage probability is obtained based on Eq. (26) and, as expected, it is well matched with the simulation results. Our results further quantify the outage probability performance deterioration associated with choosing the ${n^{{\rm {th}}}}$ optimal source. It is observed from Fig. 3 that the outage probability increases with the increase of the source index of $n$. For example, ASNR of $\bar \gamma = 25\ {\rm {dB}}$ is required in order to achieve a target outage probability of ${P_{{\rm {out}}}} = {10^{ - 3}}$ for the ideal case where the optimal source is always selected ($n = 1$). The required ASNR climbs to 31.65 dB and 39.15 dB indicating an extra ASNR of 6.65 dB and 14.15 dB if the second optimal source ($n = 2$) and third optimal source ($n = 3$) is selected due to incorrect selection of source index. The targeted outage probability can not be achieved for the fourth optimal source ($n = 4$) and fifth optimal source ($n = 5$) within the range of ASNR shown. It can also be observed that even if the third optimal source ($n = 3$) is selected in the GTLS system, its outage probability performance is still better than that of the system without GTLS.

Fig. 3. Outage probability with different source indexes. Here, $K = 2$.

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In Fig. 4, we analyze the effect of the number of laser sources on the outage probability performance of GTLS system. We consider the number of laser sources of $N = 2,\ {\rm { }}3,\ {\rm { }}4,\ {\rm { }}5,\ {\rm { }}6$, the number of channel layers of $K = 2$ and assume that the second optimal source is selected for all cases ($n = 2$). Similar to the previous analysis, the derived outage probability expression provides a good match to simulation results. It can be also shown from Fig. 4 that the outage probability increases with the decrease of the number of laser sources of $N$. For instance, ASNR of $\bar \gamma = 27.80\ {\rm {dB}}$ is required for $N = 6$ in order to achieve a target outage probability of ${P_{{\rm {out}}}} = {10^{ - 3}}$. The ASNR increases to 31.65 dB and 36.65 dB, an extra ASNR of 3.85 dB and 8.85 dB is required for $N = 5$ and $N = 4$. However, in the case of $N = 3$ and $N = 2$, the target outage probability can not be obtained within the range of ASNR used. The results show that increasing the number of laser sources within the allowable range of hardware cost is a useful scheme to improve the performance of GTLS system.

Fig. 4. Outage probability with different numbers of laser sources. Here, $n = 2$ and $K = 2$.

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In Fig. 5, we present the diversity order considering the vertical underwater link with different numbers of channel layers. We consider the GTLS system with $N = 5$ laser sources and the system without GTLS (i.e. $N = 1$). As a benchmark, we further include the diversity order that is given by numerical evaluation in Eq. (27). It can be observed that our derived closed-form expression for the diversity order in Eq. (40) provides an exact match to the diversity order in Eq. (27). This proves the correctness of our derivation. At a lower ASNR level, the diversity order increases with the increase of ASNR and decreases with the increase of the number of layers. For example, the diversity order of the GTLS system at 50 dB is obtained as 4.44, 3.44, 2.69 and 2.18 for single-, two-, three- and four-layer cases. At a higher ASNR level, the diversity order does not change significantly with the increase of ASNR. The diversity order with different numbers of channel layers reaches the same upper limit of the diversity order, i.e., the asymptotical diversity order obtained by Eq. (45). For the GTLS system, the asymptotical diversity order depends on the number of sources of $N$, the selected source index of $n$ and the minimum turbulence parameter ${{\beta _1}}$. As given by Eq. (45), we obtain ${\rm {ADO\ =\ }}{{\left ( {N - n + 1} \right ){\beta _1}} \mathord {\left /{\vphantom {{\left ( {N - n + 1} \right ){\beta _1}} 2}} \right. } 2} = 4.525$. For the system without GTLS, the diversity order is lower than that of the GTLS system and the asymptotical diversity order only depends on the minimum turbulence parameter, i.e., ${\rm {ADO}} = {{{\beta _1}} \mathord {\left /{\vphantom {{{\beta _1}} 2}} \right. } 2} = 0.905$. It should be noted that although the systems with different numbers of channel layers obtain the same asymptotical diversity order, the diversity order in the actual range of ASNR is different. This is because the convergence to full diversity becomes slower as the number of layers increases.

Fig. 5. Diversity order with different numbers of channel layers (different vertical link distances). Here, $n = 1$.

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In Fig. 6, we investigate the diversity order for a vertical underwater link with the number of channel layers of $K = 2$ and consider two cases. For Case 1, we analyze the effect of the source index on the diversity order of the GTLS system with $N = 7$. It is observed from Case 1 that the diversity order decreases if the sub-optimal source is selected. For example, the diversity order at 50 dB can be achieved as 4.81 for the ideal situation where the optimal source is selected ($n = 1$). It decreases to 4.12 and 3.43 if the second optimal source ($n = 2$) and third optimal source ($n = 3$) is selected. At high ASNR values, the diversity order reaches the asymptotical diversity order which is given by Eq. (45) as 6.335, 5.430 and 4.525 for $n = 1,\ 2$ and 3. For Case 2, we consider the second optimal source is always selected ($n = 2$) and analyze the effect of the number of laser sources on the diversity order. It is shown from Case 2 that the diversity order increases with the increase of the number of laser sources of $N$. For instance, the diversity order at 50 dB is 0.68 for $N = 2$, and it increases to 1.37, 2.06 and 2.74 for $N = 3,\ 4$ and 5. At high ASNR values, the diversity order also reaches the asymptotical diversity order which is obtained by Eq. (45) as 0.905, 1.810, 2.715 and 3.620 for $N = 2,\ 3,\ 4$ and 5. For Case 1, the source index of $n$ increases by 1, and the asymptotical diversity order decreases by ${{{\beta _1}} \mathord {\left /{\vphantom {{{\beta _1}} 2}} \right. } 2}$. For Case 2, the number of sources of $N$ increases by 1, the asymptotical diversity order increases by ${{{\beta _1}} \mathord {\left /{\vphantom {{{\beta _1}} 2}} \right. } 2}$. The asymptotical diversity order is an integral multiple of ${{{\beta _1}} \mathord {\left /{\vphantom {{{\beta _1}} 2}} \right. } 2}$, which is also consistent with the analysis shown in Eq. (45).

Fig. 6. Diversity order analysis of GTLS system. Here, $K = 2$. For Case 1, $N = 7$ and $n = 1,\ {\rm { }}2,\ {\rm { }}3$. For Case 2, $n = 2$ and $N = 2,\ {\rm { }}3,\ {\rm { }}4,\ {\rm { }}5$.

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5. Conclusion

In this paper, we have considered a vertical UWOC system with GTLS where the ${n^{{\rm {th}}}}$ ($n = 1,\ {\rm { 2, }}\ \cdots {\rm {, }}\ N$) optimal laser source is selected among the $N$ lasers of transmitter. In multi-layer cascaded Gamma-Gamma ocean turbulence channels, we have derived the exact closed-form outage probability expression in terms of Meijer G and Gauss hypergeometric functions. Then, based on the outage probability result, the diversity order expression and the asymptotical diversity order at high ASNR values have been analyzed. Finally, the simulation results are presented to confirm the accuracy of our derivations. Analytical and simulation results showed that compared to the system without GTLS (i.e. $N=1$), the GTLS system performs better in outage probability and diversity order performance. The increase of the number of channel layers (vertical link distances) and the selection of sub-optimal sources deteriorates the outage probability and diversity order performance, while the increase of the number of laser sources improves the performance of GTLS system.

In our results, the proposed GTLS scheme over underwater vertical turbulent channel improves outage probability and diversity order performance, it also has some limitations that need further study. First, our analysis is based on Gamma-Gamma turbulence model, and there are more general turbulence models such as the Malaga distribution and the exponential Weibull distribution. Second, in order to facilitate the system analysis to draw insightful conclusions, we do not consider the channel estimation method and the design of feedback channel. Third, we have theoretically derived and simulated the performance of GTLS system, but there is still a lack of experimental verification. Finally, we assume that the transmitter and receiver nodes are relatively stationary and ignore the effect of node movement on system performance. In the future, we will focus on the above limitations to further consider the system model and analyze the system performance.

Funding

National Natural Science Foundation of China (61971081).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are available in Ref. [40,41].

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Literature	System	Channel model
[29]	FSO system with TLS	atmospheric GG turbulence
[30,31]	UWOC system with TLS	underwater horizontal lognormal turbulence
[32]	FSO system with GTLS	atmospheric lognormal turbulence

Generalized transmit laser selection for vertical underwater wireless optical communications over Gamma-Gamma turbulence channels

Abstract

1. Introduction

2. System and channel models

3. Performance analysis of vertical UWOC systems with GTLS

3.1 Outage probability analysis

3.2 Diversity order analysis

4. Simulation results

5. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (6)

Tables (1)

Equations (45)

Optics Express