Open Access
Add to BibSonomy
Abstract
Our method, using Nikishov’s spectral model, assesses weak turbulence disturbance in underwater wireless optical communication (UWOC) systems. It computes scintillation indices and generates turbulence coefficients. It simulates optical signals at the detector, determining the bit error rate (BER) via decision demodulation. Analysis reveals that the slope of the relationship curve between the receiving antenna aperture and the scintillation index is maximized to 0.0377 when the receiving antenna aperture is 18 mm. For an average BER of 1E-6, required average signal-to-noise ratios (SNRs) are 9.84, 10.21 and 13.01 dB. Comparisons reveal <4% relative error and 0.071% root-mean-square error in average BER vs. average SNR curves, validating time-domain semi-physical UWOC simulations.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The increasing need for underwater communications and increasing civil, military, and scientific research activities in the ocean, has spurred rapid development in submarine observation networks, unmanned underwater vehicles, and other new underwater exploration communication technologies [1–4]. In particular, high-speed underwater communication has received widespread attention [5,6]. Compared with radiofrequency and acoustic-based systems, UWOC is a new technology that provides high bandwidth, low latency, and high security [7,8]. Seawater has low loss characteristics under blue and green light (i.e. the transmission window of 450–550 nm), which are the basis for the development of UWOC [9,10].
Turbulence is a crucial factor impacting UWOC link performance and its broader adoption. While laser propagation in atmospheric turbulence has been extensively studied both theoretically and experimentally, exploration of laser propagation in complex oceanic turbulence remains relatively unexplored yet essential. This propagation resembles the physical mechanism of atmospheric turbulence, where the water’s refractive index fluctuates with random changes in temperature and salinity. Underwater optical turbulence-induced light intensity scintillation causes severe fluctuations in light energy at the receiver, significantly impairing communication performance. Therefore, understanding the signal disturbance distribution caused by the underwater optical turbulence channel and assessing system performance under such disturbances are imperative for UWOC implementation. These findings are crucial for advancing marine resource development, marine military security, and marine scientific observation.
Andrews et al. [11] proposed lognormal and gamma–gamma distribution models to accurately describe the probability density functions of weak and medium/strong turbulences, respectively, in Gaussian light intensity scintillation. These models have provided mathematical frameworks for subsequent studies on turbulence. Vali et al. [12] used the Monte Carlo method to present a novel turbulence model for UWOC applications based on the variation of refractive index in a horizontal link. The model simulates each photon interaction with the medium, updating its current position and direction to describe photon detection at the receiving antenna aperture. The received optical power values are obtained for all the channel realisations according to the number of received photons. However, the mathematical formula of the physical model is complex, requiring extended calculation times. Cai et al. [13] integrated the Beer–Lambert law with a Monte Carlo phase screen into a stepwise propagation framework to establish integrated multiparameter UWOC. This model allows the simultaneous evaluation of underwater absorption, scattering, and turbulence effects. However, their analysis of pressure, temperature, and salinity effects on the bit error rate was conducted separately. They substituted a probability density function to represent turbulence’s disturbance effect, but did not quantitatively analyse turbulence in the time domain. Vali et al. [14] injected water along the optical beam propagation path within a pool to create uniform turbulence and obtain the probability density function (PDF) of the received optical power. Different turbulence strengths were created by changing the temperature and the injected water flow rate. Experiments showed that the PDF of experimental data had a good fit with the lognormal distribution under different turbulence intensities. However, the experiment required bulky equipment and complicated experimental conditions. To our knowledge, current underwater turbulence experiments are conducted in controlled settings, employing diverse testing instruments and data processing systems to mimic various turbulence conditions. Controlling turbulence by adjusting numerous parameters is complex and necessitates bulky experimental apparatuses. In addition, underwater experiments have high requirements for system integrity and waterproofing. Modelling underwater turbulence is essential for establishing UWOC links. Therefore, it is imperative to study underwater turbulence characteristics and their impact on communication system performance, considering both practical applications and solid theoretical bases.
We developed a digital simulation method to model underwater weak-turbulence channels that simplifies the experimental process. We then analysed the variation in turbulence scintillation at different receiving antenna apertures and propagation distances for 450 nm wavelength light. Furthermore, we simulated the scintillation caused by underwater weak-turbulence, which influences the performance (i.e. BER) of an UWOC system and established the relationship between average BER (<BER>) and average SNR (<SNR>). By comparing the results of our simulation model with theoretical calculations, we validated the model’s feasibility. The proposed method of digitizing weak turbulence disturbance signals provides a new semi-physical simulation method for UWOC system performance experiments, laying a foundation for their development and analysis.
2. Mechanistic analysis of turbulence effects
2.1 Turbulence disturbance
In UWOC, factors such as temperature and pressure affect water, resulting in refractive index fluctuations and turbulence. This turbulence, in turn, causes variations in light intensity during beam transmission, degrading communication quality. In this study, we examined an UWOC system with a circular receiver antenna aperture, as illustrated in Fig. 1. A signal generation module produced a signal that was used to modulate an emission laser sent through a transmitting antenna. The optical signal reached the receiving end after passing through turbulence. Here, the optical receiving antenna captured the laser signal, which was then sent to a detector. This laser signal was then converted into an electrical signal by the detector, and the analogue signal was digitized by an analogue-to-digital converter. Finally, the baseband signal was recovered following filtering and demodulation in the signal processing module.
The scintillation index ($\sigma _I^2$) is defined as the variance of the normalized light intensity of the light wave as follows:
Under weak turbulence conditions, the scintillation index of the plane wave on the receiving antenna aperture is defined as [15]
In Eq. (4), we consider the refractive-index power spectrum of underwater turbulence with locally homogeneous isotropic and stable stratification described by the Nikishov spectral model [16]:
Light intensity scintillation, resulting from fluctuations in light energy, can cause the received signal intensity to decay below the detection threshold. This random variation in light intensity, or jitter, can be modelled using the PDF. The lognormal distribution is the predominant statistical model for characterizing weak turbulence, and its PDF is defined as follows:
Mathematical expectation $E(H )$ and variance $var(H )$ are computed as follows:
From Eqs. (1), (2) and (6), we have
Hence, the PDF of the light-intensity-normalized turbulence disturbance coefficient for weak turbulence follows a lognormal distribution:
2.2 Communication error characterisation
We assume an intensity modulation/direct detection UWOC link with on–off keying (OOK) non-return-to-zero modulation in an underwater turbulence channel. At the receiving end of the communication system, the receiving light intensity corresponding to the $k$-th data transmission interval, namely, $[{({k - 1} ){T_b},k{T_b}} ]$ with ${T_\textrm{b}}$ being the transmission data duration, can be written as [17]
where h is the disturbance coefficient induced by turbulence and assumed to remain constant over a large number of transmission bits, ${s_k}$ represents the data transmitted with optical power ${P_t}$ under OOK non-return-to-zero modulation. For the transmission bit 1 or 0, the corresponding emission optical powers, ${P_t}$, are denoted as ${P_1}$ and ${P_0}$, respectively. In this study, we assumed zero optical power (i.e. ${P_0} = 0$) when transmitting digital signal code 0. Furthermore, ${n_k}$ is additive white Gaussian noise independent of the signal.In the underwater weak-turbulence, if OOK modulation is used, <BER > ($\langle {R_{\textrm{BE}}}\rangle$) is given by
where $P({e|h} )$ is the BER conditional on channel coefficient h and ${f_h}(h )$ is the PDF of h. where $P(e|h,{s_k} = 1)$ is the error probability conditional on h and ${s_k} = 1$.When noise is signal-independent, the optimal maximum likelihood decision threshold is given by
Hence, the optimal threshold is contingent on channel coefficient h. For optimal maximum likelihood detection, the receiver needs to continuously estimate changes in h and adjust the threshold accordingly.
From Eqs. (13) and (14), we have
<SNR > ($\langle {R_{\textrm{SN}}} \rangle$) of underwater turbulence is expressed as [12]
From Eqs. (12) and (15), we obtain
3. Modelling
3.1 Coefficient modelling
According to the Box–Muller transformation [18], random variables with lognormal distribution $H\sim \textrm{Lognormal}({\mu ,{\sigma^2}} )$ can be constructed as follows: For a lognormal distribution, $\ln h$ and $\ln y$ are assumed to have mean $\mu $ and follow independent normal distributions with variance ${\sigma ^2}$. Let $p({\ln h} )$ and $p({\ln y} )$ be their density functions. Then,
As H and Y are independent, their joint probability density satisfies
$H$ and Y are expressed in polar coordinates as $H = R\cos \theta $ and $Y = R\sin \theta $. Then,
This yields distribution functions ${P_R}$ and ${P_\theta }$ of R and $\theta $, respectively:
Clearly, Y follows a uniform distribution on [0, 2π]. Let
Then, its inverse function is given by
When Z and $\varphi $ adhere a uniform distribution within the range [0,1], the distribution function of R becomes ${F_R}(r )$ and that of $\theta $ is ${F_\theta }(\Phi )$. Therefore, two random variables, ${U_1}$ and ${U_2}$, which also follow a uniform distribution on [0,1] can be chosen such that
that is, $R = \sqrt { - 2{\sigma ^2}\ln ({U_1})} $, $\theta = 2\pi {U_2}$. Hence, we obtain the first two expressions for $\ln h$ and $\ln y$ as follows:Therefore, it follows that
3.2 Simulation principle
The Monte Carlo method is a mathematical technique rooted in probability theory and statistics and establishes a correlation between the problem being solved and a specific probability model. As per the mathematical model correlating $\langle {R_{\textrm{BE}}}\rangle$ and $\langle {R_{\textrm{SN}}} \rangle$, Fig. 2 describes the Monte Carlo turbulence model for underwater turbulence. First, high-speed signals are generated by the transmitter. As these signals pass through the underwater channel, turbulence introduces light intensity disturbances, essentially generating random numbers following a lognormal distribution, thus simulating the jitter effect of turbulence on transmitted optical power. The turbulence-affected signal then reaches the receiver, where inherent white Gaussian noise from the photoelectric device is superimposed onto it. Data recovery at the receiving end is achieved by selecting an appropriate decision threshold. Finally, an algorithm is employed to calculate < BER > and determine its relationship with < SNR > .
3.3 Implementation
We use the NI LabVIEW software to simulate Eqs. (7) and (28) and generate a program flowchart for turbulence disturbance coefficient h, as shown in Fig. 3.
The relationship between < BER > and < SNR > along with the relative error and root-mean-square error (RMSE) are obtained as described in Algorithm 1. Turbulence disturbance coefficient h is used as the basis for signal processing according to Eq. (11). A square-wave signal with 105 samples is employed to generate 7000 values of turbulence disturbance coefficient h following a lognormal distribution. Subsequently, the optimal decision threshold of the signal captured by the receiver through the underwater channel is selected, and < BER > at the receiver is analysed. According to Eq. (14), the optimal threshold depends on channel disturbance (i.e. $h$). Thus, for optimal maximum likelihood detection, the receiver should continuously estimate the variations in h and adjust the threshold accordingly.
Algorithm 1. Simulation implementation.
4. Results and discussion
4.1 Theoretical analysis of UWOC system performance under a weak-turbulence channel
To obtain the scintillation index of plane wave transmission in underwater turbulence, Eq. (3) was used to numerically simulate scintillation index variation with propagation distance under different receiving apertures. The parameters required for the simulation and their values are listed in Table 1.
Table 1. Numerical simulation parameters.
Based on the abovementioned parameters, Fig. 4 illustrates the simulation of the scintillation index in underwater turbulence and related variables. The propagation distance is considerably less than the threshold distance (the distance at which the scintillation index reaches 1), indicating weak turbulence [19]. Consequently, we selected $L = 6\; \textrm{m}$ and $L = 12\; \textrm{m}$ for subsequent analysis.
Fig. 4. Plane wave scintillation index according to propagation distance at various antenna apertures. (a) The change of receiving antenna aperture and (b) the change of propagation distance.
Figure 4(a) shows that a larger receiving antenna aperture can reduce the scintillation index caused by underwater turbulence and suppress the impact of turbulence on system performance. As the propagation distance increased, the effect of improvement was further enhanced. At propagation distances of 6 m and 12 m, the slope was greatest when the receiving aperture was 18 mm with values of 0.00472 and 0.0377, respectively, clearly indicating the improvement to turbulence scintillation. Subsequently, the improvement gradually weakened. This showed that a larger receiving antenna aperture did not necessarily result in stronger channel improvement. Therefore, in practical applications, choosing an appropriate receiving antenna aperture must not only satisfy the performance requirements of the system, but also take into account the weight and size of the system.
Figure 4(b) shows the curve of scintillation index variation with propagation distance corresponding to different receiving antenna apertures. When the apertures were 18 mm and 60 mm, the scintillation index increased from $1.050E - 3$ to $8.403E - 3$ and $1.142E - 4$ to $9.134E - 4$ as the propagation distance increased from 6 m to 12 m, respectively. The difference in scintillation index increased by $6.554E - 3$. It can be seen that when the aperture of the receiving end increased, the scintillation index increased exponentially with increase in propagation distance.
To analyse the error characteristics of the UWOC system under different turbulence intensities, Eq. (16) and Eq. (17) were implemented in the LabView software to simulate its geometry. The relationship between < BER > under OOK modulation and < SNR > at different scintillation indices is shown in Fig. 5. When the < BER > was $1E - 6$, the < SNR > required for scintillation indices $\sigma _I^2 = 9.134E - 4$, $\sigma _I^2 = 1E - 2$, and $\sigma _I^2 = 1E - 1$ were 9.84 dB, 10.21 dB, and 13.01 dB, respectively. It is evident that as turbulence intensity increases, the UWOC system’s requirement for < SNR > also increases. Notably, at very high turbulence intensities, enhancing < SNR > is insufficient, as the system’s < BER > remains elevated, rendering communication infeasible.
4.2 Digital simulation of underwater weak-turbulence and communication system
According to the analysis in Section 4.1, during digital simulation design of the system, we chose the receiving antenna diameter $D = 60\; \textrm{mm}$. In the case of underwater weak-turbulence, when the propagation distance is 12 m, the scintillation index was $\sigma _I^2 = 9.134E - 4$.
The perturbation effect of weak turbulence on light intensity follows a lognormal distribution. According to the analysis from Eq. (2) and Eq. (8) in Section 2.1, the variance and scintillation index are equal to $9.134E - 4$ and the expectation $E(H )= 1$. The turbulence disturbance coefficients were generated according to the program flowchart shown in Fig. 3, obtaining the time-domain graph shown in Fig. 6(a). Here, the horizontal axis is time and the vertical axis is the turbulence disturbance coefficient h. As seen from the graph, the maximum turbulence disturbance coefficient was 1.158 and the minimum was 0.878.
Fig. 6. Simulation of turbulence disturbance signals adhering to a lognormal distribution. (a) Time-domain sample graph and (b) sample distribution.
Figure 6(b) shows the probability distribution of turbulence disturbance coefficients. It can be seen that the generated turbulence disturbance coefficients followed a lognormal distribution, validating the use of the Box–Muller transformation to generate a lognormal-distributed turbulence model. The turbulence disturbance coefficients obtained by formula calculation and simulation statistics were mainly distributed around 1, with average values of 1.006 and 1.018, respectively. Therefore, the performance of the UWOC system under the lognormal-distribution turbulence disturbance channel was further analysed based on this transformation.
According to Eq. (11), a communication system model was established. The original signal was a square-wave signal with a data rate of 10 Mbps and power of 2 mW, as shown in Fig. 7(a). The signal after reaching the receiving end through the underwater weak-turbulence channel is shown in Fig. 7(b). The signal was affected by weak turbulence disturbance as shown in Fig. 6(a) and superimposed with Gaussian white noise from the receiver. In the intercepted signal, the turbulence increased the optical signal power by 8.08%. The decision threshold was calculated according to Eq. (14), before the received signal was demodulated and the baseband signal obtained as shown in Fig. 7(c), where the BER was derived by comparison with the original data.
Fig. 7. Communication system transmission signal simulation. (a) Original square wave signal, (b) receiver signal and (c) demodulated baseband signal.
Simulation was then carried out to determine the relationship between BER and < BER > and SNR and < SNR>, resulting in the relationship curve between < BER > and < SNR > of the UWOC system shown by the blue line in Fig. 8. The relationship curve between < BER > and< SNR > obtained according to Eq. (17) is shown by the red line in Fig. 8. The RMSE of the two curves obtained by the simulated and theoretical data is 0.071% and the average relative error is 1.00%. As can be seen from Fig. 9, the relative errors of both curves were both below 4%. Thus, there was a good agreement between the two curves.
Fig. 8. Comparison of < BER > and < SNR > derived from theoretical data and Monte Carlo simulation data.
The results show that before conducting underwater experiments for UWOC systems, a simulation analysis of communication performance can be conducted using the communication reception model. The method of generating digital signals representing weak turbulence disturbance enables semi-physical simulation. This approach not only enhances simulation reliability but also simplifies and eases the complexities associated with water entry tests.
5. Conclusion
In conclusion, we investigated receiver light intensity fluctuations in UWOC systems and proposed a digital simulation method for underwater weak-turbulence disturbance signals. By creating an isotropic underwater weak turbulence channel scintillation model, we conducted simulation analyses for the following parameters: $\lambda = 450\; \textrm{nm}$, $\omega ={-} 2.5$, $\eta = {10^{ - 3}}\; m$, $\varepsilon = {10^{ - 5}}\; {m^2}/{s^3}$, ${\chi _T} = {10^{ - 5}}\; {K^2}/s$. Increasing the receiving aperture mitigates turbulence-induced scintillation. At 6 m and 12 m propagation distances, the slope peaked at a receiving aperture of 18 mm, with values of 0.00472 and 0.0377, respectively, highlighting the improvement in turbulence scintillation. However, the scintillation index still increased with increasing propagation distance. The scintillation index variance between 18 mm and 60 mm increased by $6.554E - 3$ as the propagation distance increased from 6 m to 12 m. Based on this analysis, our simulation design for the communication system resulted in a 60 mm receiving antenna aperture and 12 m propagation distance. Combining the generated weak turbulence digital signal with the communication reception model, the relationship curve between < BER > and < SNR > was obtained. The RMSE for the curves obtained by simulated and theoretical data was 0.071% and the average relative error was 1.00%, indicating good agreement between them. The proposed technique for generating weak turbulence disturbance digital signals offers a novel approach for UWOC system experimentation through semi-physical simulation.
Funding
National Key Research and Development Program of China (2021YFA0718804, 2022YFB2903402, 2022YFB3902500); Foundation of Equipment Pre-research Area (62602010125); Education Department of Jilin Province (JJKH20220745KJ).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are available from the corresponding authors upon reasonable request.
References
1. T. Sawa, “Research and development for practical use of underwater optical wireless communication,” in 2023 IEEE Underwater Technology (UT) (2023), pp. 1–5.
2. D. H. Li, Z. J. Ding, C. L. Jing, et al., “Development and status of International Seabed Observation Network,” Ocean Dev. Manage. 37(11), 13–18 (2020).
3. X. Liu, S. Yi, X. Zhou, et al., “34.5 m underwater optical wireless communication with 2.70 Gbps data rate based on a green laser diode with NRZ-OOK modulation,” Opt. Express 25(22), 27937–27947 (2017). [CrossRef]
4. I. Alimi, A. Shahpari, A. Sousa, et al., “Challenges and opportunities of optical wireless communication technologies,” in Optical Communication Technology (Intech, 2017). [CrossRef]
5. T. R. Ren, X. N. Yu, S. F. Tong, et al., “Design and evaluation of high-sensitivity underwater optical communication transceiver based on digital signal processing,” Chin. J. Laser 49(4), 0406005 (2022). [CrossRef]
6. X. Yang, Z. Tong, Y. Dai, et al., “100 m full-duplex underwater wireless optical communication based on blue and green lasers and high sensitivity detectors,” Opt. Commun. 498, 127261 (2021). [CrossRef]
7. H. M. Oubei, C. Shen, A. Kammoun, et al., “Light based underwater wireless communications,” Jpn. J. Appl. Phys. 57(8S2), 08PA06 (2018). [CrossRef]
8. M. F. Ali, D. N. Jayakody, Y. A. Chursin, et al., “Recent advances and future directions on underwater wireless communications,” Arch. Comput. Methods Eng. 27(5), 1379–1412 (2020). [CrossRef]
9. S. Q. Duntley, “Light in the sea,” J. Opt. Soc. Am. 53(2), 214–233 (1963). [CrossRef]
10. G. D. Gilbert, T. R. Stoner, and J. L. Jernigan, “Underwater experiments on the polarization, coherence, and scattering properties of a pulsed blue-green laser,” in Underwater Photo-Optical Instrumentation Applications (1966), pp. 8–14.
11. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, “Theory of optical scintillation,” J. Opt. Soc. Am. A 16(6), 1417–1429 (1999). [CrossRef]
12. Z. Vali, A. Gholami, Z. Ghassemlooy, et al., “Modeling turbulence in underwater wireless optical communications based on Monte Carlo simulation,” J. Opt. Soc. Am. A 34(7), 1187–1193 (2017). [CrossRef]
13. R. Cai, M. Zhang, D. Dai, et al., “Analysis of the underwater wireless optical communication channel based on a comprehensive multiparameter model,” Appl. Sci. 11(13), 6051 (2021). [CrossRef]
14. Z. Vali, A. Gholami, Z. Ghassemlooy, et al., “Experimental study of the turbulence effect on underwater optical wireless communications,” Appl. Opt. 57(28), 8314–8319 (2018). [CrossRef]
15. L. C. Andrews, R. L. Phillips, and C. Y. Hopen, Laser Beam Scintillation with Applications, vol. 99 (SPIE Press, 2001).
16. V. V. Nikishov and V. I. Nikishov, “Spectrum of turbulent fluctuations of the sea-water refraction index,” Int. J. Fluid Mech. Res. 27(1), 82–98 (2000). [CrossRef]
17. M. T. Dabiri, S. M. Sadough, and M. A. Khalighi, “FSO channel estimation for OOK modulation with APD receiver over atmospheric turbulence and pointing errors,” Opt. Commun. 402, 577–584 (2017). [CrossRef]
18. G. E. P. Box and M. E. Muller, “A note on the generation of random normal deviates,” Ann. Math. Stat. 29(2), 610–611 (1958). [CrossRef]
19. J. R. Yao, M. Elamassie, and O. Korotkova, “Spatial power spectrum of natural water turbulence with any average temperature, salinity concentration, and light wavelength,” J. Opt. Soc. Am. A 37(10), 1614–1621 (2020). [CrossRef]
