Performance analysis of OSSK-UWOC systems considering pointing errors and channel estimation errors

Guixuan Ding; Guixuan Ding; Xing Du; Xing Du; Hao Du; Hao Du; Sheng Wang; Sheng Wang; Hui Feng; Hui Feng; Guoning Xu; Guoning Xu; Zhenyang Xiong; Zhenyang Xiong; Zhongzhen Jia; Yongxiang Li

doi:10.1364/OE.506993

1. Introduction

Underwater wireless optical communication (UWOC) is an active research field because of its capability to overcome the constraints of traditional wireless communication technologies [1]. In the past decades, UWOC presented the potential value of the application for marine surveying, scientific research, and other related areas. The swift growth of UWOC can be ascribed to the advanced digital signal processing techniques [2], enhanced digital communications [3], cost-effective transceiver components [4], and innovative light source modulation methods [5]. Although UWOC holds great potential, the communication range of the UWOC systems is restricted due to the attenuation effects of the UWOC channel, such as absorption, scattering, turbulence disturbing, and alignment error.

Underwater optical turbulence (UOT) is a crucial factor that affects the performance of optical signal transmission in seawater [6]. Recently, numerous statistical models have been proposed to explore the fading caused by the UOT [7–17]. These models comprise the log-normal distribution [7], Weibull distribution [8], generalized gamma distribution [9], and gamma-gamma distribution [10]. In the cases of moderate and strong fluctuations in UOT, the gamma-gamma distribution has been widely utilized. In particular, the spatial multiplexing multiple-input multiple-output (MIMO) technology, as an efficient approach to overcome the UOT fading issue and the signal transmission delay, is extensively used [11–14]. However, the practical implementation of spatial multiplexing MIMO technology encounters obstacles such as the inter-channel interference and inter-antenna synchronization issues. The spatial modulation (SM), as a novel MIMO technique, addresses the limitations posed by the other MIMO techniques. It is applied to the field of wireless optical communication and is known as optical spatial modulation (OSM) [15]. The OSM employs the spatial data from the transmitter to encode information, which limits the link expenses and eases the receiver intricacy. Thus, underwater OSM technology has been widely studied and deployed. Huang examined the BER performance of UWOC-MIMO using spatial modulation in weak UOT conditions [16]. Bhowal presented an amplified and cooperative UWOC using optical improved quadrature spatial modulation (OIQSM), which results in performance improvement [17]. The researches on OSSK technology as a basic, low-complexity OSM technique has made some progress. But according to our literature review, there has been limited research on the deployment of OSSK in the UOT environment.

Meanwhile, the performance of UWOC systems may be restricted due to the pointing errors [18]. The previous studies have investigated the effect of such errors on UWOC systems. Fu performed an objective evaluation of the typical BER performance of rectangular QAM-UWOC in the presence of pointing errors with strong ocean turbulence [19]. And, an analysis of diversity in multi-aperture UWOC systems with pointing errors using exponential generalized gamma channels was conducted by Bansal [20]. Lin studied the mean symbol error probability and channel capacity of an UWOC system affected by oceanic turbulence, whilst accounting for impairments stemming from pointing errors [21]. Additionally, the impact of channel estimation errors on the efficiency of the UWOC system should not be neglected. A number of studies have been done on this research topic. Huang derived theoretical formulas to obtain the BER by considering complete and incomplete channel state information [22]. Feng assessed the performance of the communication system using a less precise Gamma-Gamma channel model [23]. Priyalakshmi conducted a study on the estimation of channels and the correction of errors in UWOC systems that involves the vertical non-line-of-sight channels [24].

To the best of our knowledge, a performance analysis of OSSK-UWOC systems over the Gamma-Gamma turbulence channel considering the channel estimation errors and pointing errors has not been reported. In this paper, we investigate the BER performance of an OSSK-UWOC system under a channel with the combined effect of the above three factors for the first time. The contributions of this work can be summarized as follows: (i) We propose a new expression for the PDF, considering turbulence-induced fading, channel estimation and pointing errors. (ii) Using the joint PDF combining these three factors, we derive a new analytical expression for the difference in attenuation coefficients between the two channels. (iii) The resulting expression can be used for deriving the average BER for the OSSK-UWOC system.

The rest structure of this paper is organized as follows: in Section 2, the system and channel model of the OSSK-UWOC system are proposed. In Section 3, the statistical characteristics are analyzed. In Section 4, the numerical results are discussed. And in Section 5, the main findings are concluded.

2. System and channel models

2.1 OSSK-UWOC system schematic

For an OSSK-UWOC system comprising N_t laser diodes (LDs) and N_r photodetectors (PDs), the system model can be displayed in Fig. 1(a). The binary bitstream information entering the system is divided into groups of length m = log₂N_t. To choose the serial number ${x_j}$ of the activated laser, each group is then mapped to a constellation vector $\boldsymbol{X} = [{{x_1},{x_2}, \cdots ,{x_{{N_t}}}} ]$. The symbols that are agreed upon by the transceiver and receiver are transmitted on the j-th laser. At this moment, the un-activated lasers do not transmit any information. The SE of the OSSK-UWOC system is log₂N_t bpcu. The optical signal emitted by the laser travels through the UWOC channel then detected by a certain detector on the receiver. The received signal ($\boldsymbol{Y} \in {\mathrm{\Re }^{{N_r} \times 1}}$) can be expressed as follows:

(1)$$\boldsymbol{Y} = \sqrt {{E_b}} \boldsymbol{HX} + \boldsymbol{n}$$

where, ${E_b} = {\eta ^2}P_t^2/N_r^2$, η is the photoelectric conversion efficiency, P_t denotes the peak power, n represents the ${N_r}$ dimensional Gaussian white noise vector (mean is 0 and variance is N₀/2). $\boldsymbol{H} = {[{{h_i}_j} ]_{{N_r} \times {N_t}}}$ is the N_t× N_r dimensional UWOC channel coefficient matrix, and h_ij is the UWOC channel fading coefficient. The channel coefficient matrix is a linear expression of three random factors (i.e., h = h_l h_a h_p), h_l refers to the path loss, h_a is the fading due to UOT, and h_p refers to the fading due to pointing errors (subscript ij omitted for convenience).

Fig. 1. The diagram of OSSK-UWOC system.

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To de-map the transmitted binary information, the receiver’s detector must estimate the activated laser’s serial number. Here, the decoder decodes the received signal according to the known channel information by using maximum likelihood criteria:

(2)$${\hat{x}_j} = \arg \min ||{\boldsymbol{Y} - \eta {P_t}\boldsymbol{H}} ||_\textrm{F}^\textrm{2}$$

where ${\hat{x}_j}$ is the estimated LD serial number.

2.2 Underwater optical turbulence

The moderate and strong UOT fading h_a is described by gamma-gamma distribution which is obtained as the product of gamma distribution modeling small-scale UOT effects and gamma distribution modeling large-scale UOT effects. The PDF of h_a is given by [7]:

(3)$${f_{{h_a}}}({{h_a}} )= \frac{{2{{(\alpha \beta )}^{(\alpha + \beta )/2}}}}{{\Gamma (\alpha )\Gamma (\beta )}}{({h_a})^{(\alpha + \beta )/2 - 1}} \times {K_{\alpha - \beta }}\left[ {2\sqrt {\alpha \beta {h_a}} } \right],{h_a}\mathrm{\geqslant }0$$

where, ${K_m}({\cdot} )$ denotes the modified Bessel function of the 2^nd kind of order m; $\Gamma ({\cdot} )$ is the Gamma function; α and β are the large-scale and small-scale coefficients describing the scintillation of the light wave, whose values are related to the scintillation coefficients. When the Gaussian beam is used for signal transmission and the aperture is used for signal acceptance, the coefficients α and β are obtained from the following equation [25]:

(4.1)$$\alpha = {\left[ {\exp \left( {\frac{{0.196\sigma_l^2}}{{{{({1 + 0.18{d^2} + 0.186\sigma_l^{12/5}} )}^{7/6}}}}} \right) - 1} \right]^{ - 1}}$$

(4.2)$$\beta = {\left[ {\exp \left( {\textrm{ }\frac{{0.204\sigma_l^2{{({1 + 0.23\sigma_l^{12/5}} )}^{ - 5/6}}}}{{1 + 0.9{d^2} + 0.207{d^2}\sigma_l^{12/5}}}\textrm{ }} \right) - 1} \right]^{ - 1}}$$

where, $\sigma _l^2 = 1.23C_n^2{k^{7/6}}{L^{11/6}}$ is the Rytov variance, $d = \sqrt {k{D^2}/4L} $, k = 2π / λ is the wave number, $C_n^2$ is the refractive index structure constant, D is the detector acceptance size, and L is the path distance from the transmitter to the receiver.

2.3 Pointing errors

For the OSSK-UWOC system with N_r PDs at the receiver, the pointing errors caused by both random jitters and line-of-sight errors should be fully considered. The random jitters refer to the random changes in the angle between the actual target line at the transmitter and the axis at the receiver due to platform vibration, mechanical noise, water flow disturbance, etc. Line-of-sight errors refer to the errors caused by the axial deviation of the receiver from the line-of-sight of the optical communication system, including the additional line-of-sight errors caused by machining and assembly, and the fixed line-of-sight errors inherent in the spatial distribution of the detector.

The schematic of the UWOC system in the presence of pointing errors is shown in Fig. 1(b). Assuming that N_r detectors (N_r = 3) are uniformly distributed on a circle of radius r₀ and that the coordinates of the additional pointing error in the cross section of the receiver is $({x^{\prime},y^{\prime}} )$, the centre coordinates of the m-th detector $({\mu_x^m,\mu_y^m} )$ are $({{r_0}cos{\theta_m} - x^{\prime},{r_0}sin{\theta_m} - y^{\prime}} )$, where ${\theta _m} = 2\pi ({m - 1} )/{N_r}$. The line-of-sight deviation of the m-th detector is:

(5)$$\begin{aligned} {s_m} &= \sqrt {{{({\mu_x^m} )}^2} + {{({\mu_y^m} )}^2}} \\ &= \sqrt {r_0^2 + {{x^{\prime}}^2} + {{y^{\prime}}^2} - 2{r_0}({x^{\prime}\cos {\theta_m} + y^{\prime}\sin {\theta_m}} )} \end{aligned}$$

In the presence of line-of-sight errors, it is generally assumed that the horizontal and pitch errors of the randomly jittered pointing errors r_m of the m-th receiving aperture are independent of each other and follow a Gaussian distribution with the same variance $r_x^m\sim N({\mu_x^m,\sigma_s^2} )$, $r_y^m\sim N({\mu_y^m,\sigma_s^2} )$, which ${r_m} = |{{\boldsymbol{r}_{\boldsymbol{m}}}} |= {({{{({r_x^m} )}^2} + {{({r_y^m} )}^2}} )^{1/2}}$ follows the Rice distribution:

(6)$${f_{{r_m}}}({{r_m}} )= \frac{{{r_m}}}{{\sigma _s^2}}\exp \left[ { - \frac{{({r_m^2 + s_m^2} )}}{{2\sigma_s^2}}} \right]{I_0}\left( {\frac{{{r_m}{s_m}}}{{\sigma_s^2}}} \right)$$

where, ${I_0}({\cdot} )$ denotes the modified Bessel function of the kind zero.

Considering the transmission of the beam in turbulence, h_p and r_m obey the Gaussian distribution, and the PDF of h_p can be obtained [26]

(7)$$\begin{aligned} {f_{{h_p}}}({h_p}) &= \frac{{{\varphi ^2}}}{{A_1^{{\varphi ^2}}}}\exp \left( { - \frac{{{s_m}^2}}{{2\sigma_s^2}}} \right){h_p}^{{\varphi ^2} - 1}\\ & \times {I_0}\left( {\frac{{{s_m}}}{{{\sigma_s}}}\sqrt { - 2{\varphi^2}\ln \left( {\frac{{{h_p}}}{{{A_1}}}} \right)} } \right),0 \le {h_p} \le {A_1} \end{aligned}$$

where, A₁ is the received power component at r_m= 0; $\varphi = {w_{Zeq}}/2{\sigma _s}$ is the ratio of the equivalent beam radius at the receiver to the standard deviation of the pointing errors displacement at the receiver; $w_{zeq}^2 = w_z^2\sqrt \pi \textrm{erf}(\nu )/2\nu \textrm{exp}( - {\nu ^2})$ is the equivalent beam width of the receiving field; $\textrm{erf}({\cdot} )$ is the error function; $\nu = \sqrt \pi R/\sqrt 2 {w_z}$; w_z is beam width of the source field; and $R = D/2$ is the radius of the receiving aperture.

2.4 Composite channel model

The channel fading coefficient h can be expressed as a linear product of three factors: seawater attenuation, turbulence effect, and pointing errors. h_l denotes the loss of light intensity caused by seawater attenuation and is a constant obeying Beers-Lambert law. Without loss of generality let h_l= 1. Thus, the PDF of h considering the UOT fading h_a and the pointing errors h_p can be expressed as:

(8)$${f_h}(h) = \int_{h/{A_1}}^\infty {{f_{{h_a}}}({h_a}){f_{{h_p}}}(\frac{h}{{{h_a}}})} \frac{1}{{{h_a}}}\textrm{d}{h_a}$$

where, ${f_{{h_p}}}({h / {{h_a}}})$ is the conditional PDF in the presence of UOT fading.

For simplicity of derivation, the PDF of h_p in Eq. (7) is approximated using the improved Rayleigh distribution model [27]:

(9)$${f_{{h_p}}}({h_p}) = \frac{{\varphi _{\textrm{mod}}^2}}{{A_{\textrm{mod}}^{\varphi _{\textrm{mod}}^2}}}{({{h_p}} )^{\varphi _{\textrm{mod}}^2}},0 \le {h_p} \le {A_{\textrm{mod}}}$$

where, ${\varphi _{\textrm{mod}}} = {w_{_{Zeq}}}/({2{\sigma_{\textrm{mod}}}} )$ is the improved φ, $\sigma _{\textrm{mod}}^2 = {[{{{({3\mu_x^2\sigma_s^4 + 3\mu_y^2\sigma_s^4 + 2\sigma_s^6} )} / 2}} ]^{1/3}}$.

Substituting Eq. (9) into Eq. (8) yields the PDF of h under the joint effect as:

(10)$${f_h}(h )= \frac{{\alpha \beta \varphi _{\textrm{mod}}^2}}{{{A_{\textrm{mod}}}\Gamma (\alpha )\Gamma (\beta )}}G_{1,3}^{3,0}\left( {\frac{{\alpha \beta h}}{{{A_{\textrm{mod}}}}}\left|{\begin{array}{c} {\varphi_{\textrm{mod}}^2}\\ {\varphi_{\textrm{mod}}^2 - 1,\alpha - 1,\beta - 1} \end{array}} \right.} \right)$$

where, $G_{p,q}^{m,n}({\cdot} )$ is the Meijer’s G function.

2.5 Channel estimation errors

In practical UWOC communication systems, channel estimation errors are inevitable. Therefore, the channel gain with estimation error (i.e., imprecise channel gain) is modeled as the sum of the estimated channel gain and the estimation error. The imprecise channel gain can be expressed as [28–30]:

(11)$$\hat{h} = \rho h + \sqrt {1 - {\rho ^2}} \varepsilon$$

where, $\rho \in [{0,\textrm{ }1} ]$, the coefficient ρ represents the accuracy of channel estimation. ρ=1 indicates no estimation error. The channel estimation error $\varepsilon$ is an Gaussian random variable independent of h with zero-mean and variance ${\sigma ^2}$.

Based on Eq. (11), the imprecise channel gain results from the adding two variables. Here, setting $X = \rho h$, $Y = \sqrt {1 - {\rho ^2}} \varepsilon$, and $Z\textrm{ = }\hat{h}$. Thus, employing the convolution theorem ${f_Z}(z) = \int_0^\infty {{f_X}} (x){f_Y}(z - x)dx$, one can express the PDF of Z:

(12)$${f_Z}(z) = \frac{{\alpha \beta \varphi _{\textrm{mod}}^2\,\exp \left( { - \frac{{{z^2}}}{{2{\sigma^2}(1 - {\rho^2})}}} \right)}}{{A_{\bmod }^{}\Gamma (\alpha )\Gamma (\beta )\rho {{(1 - {\rho ^2})}^{1/2}}\sigma }}{I_A}$$

where,

(13)$${I_A} = \int_0^\infty {\exp } \left( { - \frac{{{x^2}}}{{2{\sigma^2}(1 - {\rho^2})}}} \right)\exp \left( {\frac{{zx}}{{{\sigma^2}(1 - {\rho^2})}}} \right)G_{1,3}^{3,0}\left( {\frac{{\alpha \beta h}}{{{A_{\textrm{mod}}}}}\left|{\begin{array}{c} {\varphi_{\textrm{mod}}^2}\\ {\varphi_{\textrm{mod}}^2 - 1,\alpha - 1,\beta - 1} \end{array}} \right.} \right)\textrm{d}x\textrm{ }$$

To obtain a simplified expression, with the help of [Ref. 31, Eq. (07.34.21.0013.01) and Eq. (07.34.16.0001.01)], the integral term I_A is converted to a series expansion and subsequently substituted into Eq. (12). Therefore, Eq. (12) can be regularized as follows:

(14)$${f_Z}(z )= {B_1}{E_1}\sum\limits_{k = 0}^\infty {\frac{{{2^{0.5k + \alpha + \beta - 3}}{z^k}{G_k}}}{{k!{\sigma ^k}{{(1 - {\rho ^2})}^{k/2}}}}}$$

where, ${B_1} = {{\alpha \beta \varphi _{\textrm{mod}}^2} / {\Gamma (\alpha )\Gamma (\beta )}}$, ${E_1} = {{\exp \left( { - \frac{{{z^2}}}{{2{\sigma^2}(1 - {\rho^2})}}} \right)} / {\sqrt 2 {\pi ^{3/2}}\sigma }}{(1 - {\rho ^2})^{1/2}}$, ${G_k} = G_{6,3}^{1,6}$ $\left[ {\frac{{8A_{\bmod }^2{\rho^2}}}{{{\sigma^2}(1 - {\rho^2}){\alpha^2}{\beta^2}}}\left| {\begin{array}{*{20}{c}} {\frac{{1 - \varphi_{\textrm{mod}}^2}}{2},\frac{{2 - \varphi_{\textrm{mod}}^2}}{2},\frac{{1 - \alpha }}{2},\frac{{2 - \alpha }}{2},\frac{{1 - \beta }}{2},\frac{{2 - \beta }}{2}}\\ {\frac{k}{2},\frac{{ - \varphi_{\textrm{mod}}^2}}{2},\frac{{1 - \varphi_{\textrm{mod}}^2}}{2}} \end{array}} \right.} \right]$.

3. Statistical characteristics analysis

In this section, we analyze the statistical depiction of the pair-wise error probability (PEP) in an UWOC system under UOT fading, considering channel estimation errors, and pointing errors. The PEP forms a crucial foundation for computing theoretical upper bounds on the spatial modulation BER of the system. The PEP, indicating the likelihood of the transmitter signal ${x_{{l_1}}}$ being mistakenly identified as ${x_{{l_2}}}$, is denoted as $\textrm{P}({{l_1} \to {l_2}} )$ [32]:

(15)$$\textrm{P}({l_1} \to {l_2}) = Q\left( {\frac{1}{{{N_r}}}\sqrt {\frac{{\bar{\gamma }{{\log }_2}{N_t}}}{2}\sum\limits_{r = 1}^{{N_r}} {{{|{{h_{r{l_1}}} - {h_{r{l_2}}}} |}^2}} } } \right)$$

where, $Q({\cdot} )$ represents the Gaussian Q-function, $\bar{\gamma } = {E_b}{T_b}/({N_0}{\log _2}{N_t})$ stands for the average SNR, and ${T_b}$ denotes the interval between symbols. Notably, ${h_{r{l_1}}}$ and ${h_{rl2}}$ are attenuation coefficients that correspond to the imprecise channel gains between the r-th PD and the ${l_1}$-th and ${l_2}$-th LDs, the expression for the PDF of which is shown in Eq. (14), respectively.

To calculate the PEP, the PDF of the difference in attenuation coefficients between the two channels should be obtained. Therefore, let $\left| {{h_{r{l_1}}} - {h_{r{l_2}}}} \right| = {a_r}$. The PDF of a_r can be derived using [Ref. 33, Eq. (3.462)]:

(16)$${f_{{A_r}}}({{a_r}} )= {B_2}{E_2}\sum\limits_{k = 0}^\infty {\sum\limits_{n = 0}^k {\left( \begin{array}{l} n\\ k \end{array} \right)\frac{{{2^{k + 2\alpha + 2\beta - 6}}{a_r}^{k - n}G_k^2}}{{{\sigma ^{2k}}{{(1 - {\rho ^2})}^k}{{({k!} )}^2}}}} } {\left[ {\frac{2}{{{\sigma^2}(1 - {\rho^2})}}} \right]^{ - \frac{\zeta }{2}}}\Gamma (\zeta ){\textrm{D}_\zeta }\left( {\frac{{{a_r}}}{{\sqrt {2{\sigma^2}(1 - {\rho^2})} }}} \right)$$

where, ${B_2} = {{\varphi _{\textrm{mod}}^2} / {{\Gamma ^2}(\alpha ){\Gamma ^2}(\beta )}}$, ${E_2} = {{\exp \left( { - \frac{{3{a^2}}}{{8{\sigma^2}(1 - {\rho^2})}}} \right)} / {2\pi {\sigma ^2}}}(1 - {\rho ^2})$, $\zeta = n + k + 1$, ${\textrm{D}_\zeta }(\cdot )$ is parabolic trigonometric function.

To the best of our current knowledge, the closed form of Eq. (15), derived using Eq. (16), has not been reported in literature, which poses an obstacle to the BER analysis of the OSSK-UWOC system. Nevertheless, the analysis can be notably simplified through the approximation of the PDF of a_r. We were pleased to discover that the Nakagami distribution can be applied to fit the PDF of a_r appropriately. Figure 2 and Table 1 provide the relevant evidence. The expression for Nakagami distribution is as follows:

(17)$${f_R}(r )= \frac{{2{r^{2\mu - 1}}}}{{\Gamma (\mu ){\Omega ^\mu }}}\exp \left( { - \frac{{{r^2}}}{\Omega }} \right)$$

Applying the random variable transformation to the asymptotic PDF of the Nakagami distributed random variable results in the asymptotic PDF of γ_r. The PDF of the electrical SNR (${\gamma _r} = a_r^2\bar{\gamma }$) can be obtained from the PDF of a_r:

(18)$$\begin{aligned} {f_{{\gamma _r}}}({{\gamma_r}} )&= \frac{1}{{2\sqrt {{\gamma _r}\bar{\gamma }} }}{f_{{A_r}}}({{a_r}} )\left|{_{{a_r} = \sqrt {{{{\gamma_r}} / {\bar{\gamma }}}} }} \right.\\& = \frac{{{\gamma _r}^{\mu - 1}}}{{{{\bar{\gamma }}^\mu }\Gamma (\mu ){\Omega ^\mu }}}\exp \left( { - \frac{{{\gamma_r}}}{{\Omega \bar{\gamma }}}} \right) \end{aligned}$$

Then, the moment generating function (MGF) of γ_r needs to be calculated using the standard definition and employing the identity [Ref. 33, Eq. (3.381.4)]:

(19)$$\begin{aligned} {M_{{\gamma _r}}}({ - s} )&= \int_0^\infty {\exp ({ - s{\gamma_r}} )} {f_{{\gamma _r}}}({{\gamma_r}} )\textrm{d}{\gamma _r}\\ &= \frac{{{{({\Omega \bar{\gamma }s + 1} )}^\mu }}}{{{{({\Omega \bar{\gamma }} )}^{2\mu }}}} \end{aligned}$$

Fig. 2. Comparison of simulated cdfs of a_r and its predicted approximation.

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Table 1. RMSE for a_r and its predicted approximation

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We can obtain a precise upper limit for the BER in the OSSK-UWOC system by implementing the commonly recognized method of boundary joint analysis [34]:

(20)$$\textrm{BER}_{\textrm{OSSK}}^{(u)} \le \frac{1}{{{N_t}{{\log }_2}({N_t})}}\sum\limits_{{l_1} = 1}^{{N_t}} {\sum\limits_{{l_2} = 1}^{{N_t}} {{d_p}({x_{{l_1}}},{x_{{l_2}}})\textrm{P}({l_1} \to {l_2})} }$$

Due to the fact that the channels under examination are independently and identically distributed and the MGF of their difference is not dependent on the transmitter indices (n and k), the Q-function in the PEP can be disregarded in the dual summations [35]. Therefore, the dual summations can be substituted with the subsequent equation:

(21)$$\frac{1}{{{N_t}{{\log }_2}({N_t})}}\sum\limits_{{l_1} = 1}^{{N_t}} {\sum\limits_{{l_2} = 1}^{{N_t}} {{d_p}} } ({x_{{l_1}}},{x_{{l_2}}}) = \frac{{{N_t}}}{2}$$

When the probability distribution of the elements of the channel matrix H is established, the MGF can be used to calculate the average PEP:

(22)$$\textrm{APEP}({l_1} \to {l_2}) = \frac{1}{\pi }\int_0^{\pi /2} {{M_{{\gamma _{sm}}}}} \left( {\frac{{{{\log }_2}{N_t}}}{{4N_r^2{{\sin }^2}\theta }}} \right)\textrm{d}\theta$$

Applying the [Ref. 33, Eq. (3.681)] to bring Eq. (19) into Eq. (22), yields that

(23)$$\begin{aligned} \textrm{APEP}({l_1} \to {l_2}) &= \frac{1}{\pi }{\left( {\frac{{4N_r^2}}{{\Omega \bar{\gamma }{{\log }_2}{N_t}}}} \right)^{\mu {N_r}}}B\left( {\frac{1}{2} + \mu {N_r},\frac{1}{2}} \right)\\ &\times F\left( {\mu {N_r},\frac{1}{2} + \mu {N_r};1 + \mu {N_r}; - \frac{{4N_r^2}}{{\Omega \bar{\gamma }{{\log }_2}{N_t}}}} \right) \end{aligned}$$

where, $B({\cdot} )$ is the beta function, $F({\cdot} )$ is the Gauss hypergeometric function.

Bringing Eqs. (21) and (23) into Eq. (20), we finally get the average BER expression for OSSK, which can be represented as:

(24)$$\begin{aligned} \textrm{ABER} &= \frac{{{N_t}}}{{2\pi }}{\left( {\frac{{4N_r^2}}{{\Omega \bar{\gamma }{{\log }_2}{N_t}}}} \right)^{\mu {N_r}}}B\left( {\frac{1}{2} + \mu {N_r},\frac{1}{2}} \right)\\ &\times F\left( {\mu {N_r},\frac{{2\mu {N_r} + 1}}{2};\mu {N_r} + 1; - \frac{{4N_r^2}}{{\Omega \bar{\gamma }{{\log }_2}{N_t}}}} \right) \end{aligned}$$

4. Numerical results and discussion

In this section, we discuss the numerical results acquired through analytical derivation in Sec. 3 and valid the proposed theory by numerical simulations. Additionally, employing equations for α and β provided in [25], we calculate the large-scale and small-scale coefficients for the Gaussian beam under moderate UOT (α = 6.4, β = 11.4) and strong UOT (α = 2.5, β = 6.9). In addition to the notes presented in the follow figures, the following simulation parameters are utilized in Table 2.

Table 2. Parameters used for simulation

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Figure 3 shows the effect of the number of lasers (N_t= 2, 4, 8) on the BER of the system under moderate and strong underwater optical turbulence. As the number of lasers increases, the SE increases from 1 bpcu to 3 bpcu, but the BER performance decreases. When BER = 2 × 10⁻³, along with N_t increasing from 4 to 8, the signal-to-noise losses are 3.5 and 6 dB for moderate and strong UOT, respectively. Thus, the OSSK-UWOC system makes a trade-off between SE and BER performance. Furthermore, for a given strength of both UOTs, the moderate UOT model provides a marginal coding efficiency of 1.5—2 dB compared to the strong UOT model.

Fig. 3. Effect of the number of lasers on the BER under moderate and strong UOT.

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As illustrated in Fig. 4, we then compare the effect of different numbers of detectors (N_r = 2, 3, 4) on the BER of the system when N_t = 2. The resulting increase in the number of detectors is able to reduce the effect of UOT on the BER of the system. It can be seen from Fig. 4 that the BER of the system with N_r = 4 is significantly reduced for the same SNR case compared with N_r = 2. However, there is an upper limit to the use of detectors. This is to ensure that the beam at the receiver covers as many detectors as possible. In addition, we also compare the differences between moderate UOT and strong UOT, where the medium UOT model provides a marginal coding efficiency of about 8 dB compared to the strong UOT model. For the moderate UOT, with a SNR of 36 dB, the BER is 9.2 × 10⁻³ for N_r= 2, while it decreases to 1.5 × 10⁻³ for N_r= 4. In the case of strong UOT, the change in BER is insignificant, going from 4.2 × 10⁻² to 2.5 × 10⁻².

Fig. 4. Effect of different numbers of detectors on the BER.

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To investigate the effect of the pointing errors on the performance of the OSSK-UWOC system, we focus on the normalized beam width(w_z/R) and the normalized displacement standard deviation (σ_s/R) in the pointing errors. Figure 5 shows the effect of different normalized beam widths w_z/R on the system performance at moderate UOT condition. Fixing σ_s/R = 2 and varying w_z/R to 5, 10 and 15 results in an increase in beam waist width at the receiver. It can be seen that a narrower beam waist improves the system performance, this is because a narrower beam waist means that the light intensity is more concentrated with less power loss. However, a narrow beam waist is highly susceptible to misalignment effects, resulting in loss of line-of-sight, which can disrupt the system link and reduce system reliability. Techniques such as adaptive optics, multi-transceiver systems and wavelength diversity can be used to minimise the effect of beam waist on UWOC system performance. In addition, we analyze the effect of the PD numbers on the BER for different values of w_z/R. The smaller the PD numbers, the smaller the effect on BER of changing the w_z/R value. Further, the BER advantage of N_r= 4 over N_r= 2 is more readily apparent at 41.5 dB.

Fig. 5. Effect of the normalized beam width on the performance of the OSSK-UWOC.

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Figure 6 represents the system BER performance under the combined effect of both the normalized displacement standard deviation σ_s/R and the channel estimation errors. Varying the value of σ_s/R has little effect on the BER performance, and the two curves are almost coincident for σ_s/R = 2, 6, indicating that the OSSK-UWOC system is insensitive to the random variations of light intensity at the receiver caused by UOT or random interference. In addition, as the correlation coefficient of the channel estimation error decreases, the BER performance subsequently decreases. For example, when BER = 10⁻³, ρ increases from 0.7 to 0.9, and the SNR loses to 1.5 dB.

Fig. 6. BER performance under the combined effect of both the normalized displacement standard deviation and the channel estimation errors.

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Figure 7 aims to investigate how the number of lasers and detectors affects the OSSK-UWOC system in the presence of channel estimation errors. Initially, an unanticipated result is found when comparing the BERs under low channel estimation error (ρ = 0.95) and high channel estimation error (ρ = 0.7). Under the moderate UOT condition with N_r= 4 and N_t= 2, the system with high channel estimation error (ρ = 0.7) shows the 4 dB increase in SNR gain compared with the system with low channel estimation error (ρ = 0.95). Under the strong UOT condition, the system presents roughly 1.5 dB advantages. The OSSK-UWOC system^{\prime}s upper BER limit is positively correlated with the differential statistics between the two sub-channels (${h_{r{l_1}}}$ and ${h_{rl2}}$). The value of ${a_r}$ decreases as the channel estimation error increases (decrease in ρ), which makes the BER performance of the system at ρ = 0.7 better than at ρ = 0.95. It should also be noted that the variation of detectors number has a greater effect on the system BER performance than the variation of lasers number. As demonstrated in the six curves in Fig. 7(a) and (b), adjusting the number of detectors results in a significant change in the BER performance, from N_r= 2 to N_r= 4 with BER performance seeing an improvement of 10⁻² to 10⁻³ orders of magnitude. However, altering the value of ρ or the number of lasers does not yield a significant difference in the BER of the system, since it maintains the variation of 10⁻¹ magnitude orders.

Fig. 7. BER performance under the combined effect of both the channel estimation errors and PD and LD numbers (a) moderate UOT; (b) strong UOT.

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5. Conclusion

In this paper, the BER analysis of the OSSK-UWOC system is performed considering pointing errors, gamma-gamma turbulence fading and channel estimation errors. This study reveals a significant variation in the impact of the quantities of lasers and detectors on the system, with an augmentation in the lasers count leading to an enhancement in the spectral efficiency, albeit at the expense of a deterioration in the BER performance. Conversely, the augmentation of detectors mitigates the impact of channel fading on the BER of the system. The BER performance experiences an improvement ranging from 10⁻² to 10⁻³ orders of magnitude when transitioning from N_r= 2 to N_r= 4. In addition, it is advantageous to employ a narrow beam waist in the OSSK system to optimize its performance. The implementation of OSSK in the UWOC systems provides unanticipated and intriguing results, wherein the presence of channel estimation error confers a BER performance advantage upon the system. Specifically, a system characterized by high channel estimation error (ρ = 0.7) demonstrated a 4 dB enhancement in SNR gain in contrast to the system with low channel estimation error (ρ = 0.95). In future research endeavors, it is advisable to allocate attention to the channel modeling and in-depth BER analysis in UWOC system.

Funding

Natural Science Foundation of Hainan Province (622MS105); National Natural Science Foundation of China (U23A20336); Youth Innovation Promotion Association of the Chinese Academy of Sciences (2020133).

Disclosures

The authors declare no conflicts of interest.

Data availability

All data generated or analyzed during this study are included in this article.

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$A_{mod}$	$φ_{mod}$	$α$	$β$	$ρ$	$μ$	$Ω$	RMSE
0.0076	1.2270	2	1.4	0.9	0.5122	0.1512	0.71%
0.0076	1.2270	9.1	3.0	0.8	0.5088	0.1456	0. 73%
0.0076	1.2270	4	1.9	0.9	0.5107	0.1452	0. 77%

Parameters	Value
Light wavelength λ / nm	532
Communication link L / m	10
Radius of the receiving aperture D / mm	3
Normalized beam width w_z/R	8
Normalized displacement standard deviation σ_s/R	3
Channel estimation coefficient ρ	0.9

$A_{mod}$	$φ_{mod}$	$α$	$β$	$ρ$	$μ$	$Ω$	RMSE
0.0076	1.2270	2	1.4	0.9	0.5122	0.1512	0.71%
0.0076	1.2270	9.1	3.0	0.8	0.5088	0.1456	0. 73%
0.0076	1.2270	4	1.9	0.9	0.5107	0.1452	0. 77%

Parameters	Value
Light wavelength λ / nm	532
Communication link L / m	10
Radius of the receiving aperture D / mm	3
Normalized beam width w_z/R	8
Normalized displacement standard deviation σ_s/R	3
Channel estimation coefficient ρ	0.9

Performance analysis of OSSK-UWOC systems considering pointing errors and channel estimation errors

Abstract

1. Introduction

2. System and channel models

2.1 OSSK-UWOC system schematic

2.2 Underwater optical turbulence

2.3 Pointing errors

2.4 Composite channel model

2.5 Channel estimation errors

3. Statistical characteristics analysis

4. Numerical results and discussion

5. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (7)

Tables (2)

Equations (25)

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