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Abstract
Recent research has shown that an accurate underwater channel characterization is necessary for underwater optical wireless communication (UOWC) in order to improve its current limitations related to the achievable data rate and the link distance, as required in undersea optical networks. This paper presents a new statistical model to characterize the scattering effect in terms of a fading never considered before. In this way, the probability density function of the scattering-induced fading channel is derived by means of a Gamma distribution by using only one degree of freedom in clear ocean and coastal waters. The developed fading model is employed to compute the performance of UOWC systems in terms of bit error rate and outage probability along with turbulence-induced fading modeled by a Weibull distribution. The results prove that smaller diversity order values are achieved when scattering-induced fading is the dominant effect, i.e., when the condition $\sigma_{s}^{2} > \frac {1}{\beta _{1}}$ is satisfied, where $\sigma_{s}^{2}$ and β1 are parameters related to the Gamma and Weibull distributions, respectively. Moreover, the optical power penalty due to scattering-induced fading is analytically evaluated in several turbulence conditions to provide a deeper insight. Optical power penalty values of up to 6 dB and 9 dB are achieved when compared with no scattering scenarios at moderate distances for clear ocean and coastal waters. As a key feature, scattering should be always considered in terms of fading for future designs of advanced UOWC systems. The analytical results are verified by Monte Carlo simulations.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Nowadays, the demand for high-speed and long-distance wireless links has positioned the underwater optical wireless communication (UOWC) technology as a solid alternative to the current acoustic and radio frequency (RF) systems [1]. High data rate, security, energy efficiency, and low latency are some of the advantages that such a technology can offer using the blue-green band of the visible light spectrum, for example, for underwater exploration activities, real-time video transmission, and novel Internet of Underwater Things (IoUT) applications, among others [2,3]. However, not everything mentioned above is an advantage. UOWC systems are mainly impaired by absorption and scattering, not only resulting in channel path loss and temporal dispersion [1,4,5], but also leading to an optical signal fading. In addition, recent research has indicated that rapid changes in the refractive index due to small variations in salinity, temperature and pressure can result in underwater oceanic turbulence, or simply turbulence, generating random fluctuations in the irradiance of the received optical beam [6]. Consequently, an entire research into the combined impact of scattering and turbulence in terms of fading is crucial to improve efficiently the UOWC systems design.
Some authors have studied numerically and experimentally the performance degradation of UOWC systems, taking into consideration the different degrading effects mentioned above as well as their combined impact on the overall performance [7–16]. At the same time, several statistical distributions have been developed to model the turbulent behavior of the UOWC channel under different oceanic conditions [17,18]. In the light of these studies, it can be noted that scattering is usually included in performance analysis of UOWC systems as a deterministic extra power loss and occasionally as a temporal dispersion by means of the Beers-Lambert’s law and channel impulse response of the system, respectively. Very recently, the bit error rate (BER) performance has been also analyzed in the presence of an additive noise caused by scattering when oceanic turbulence is neglected [19]. All these approaches show that the effect of scattering on UOWC systems can be modeled in different ways, presenting a quantified impact on the overall performance. However, to the best of the authors’ knowledge, the stochastic behavior of the UOWC channel due to scattering has not been considered yet. In other words, there is no work that has considered a potential fading effect on the irradiance of the received optical beam due to scattering in underwater environments. For instance, previous works in RF and terrestrial free-space optical (FSO) developed statistical models to consider the randomness of the path loss [20–22]. Particularly, the random nature of fog and dust was analyzed to find the probability distribution function (PDF) of the attenuation in [21,22]. Motivated by this, we observe that there remains a need for developing a more accurate channel model that can predict a potential scattering-induced fading. The stochastic behavior of scattering in terms of a fading effect is essential to characterize the underwater medium, as well as the study of the viability of the UOWC technology in turbid environments. This knowledge gap must be filled to provide with an accurate channel model that incorporates the combined effect of scattering-induced fading and oceanic turbulence to enable robust undersea optical technology. According to previous works, scattering plays a key role in underwater light propagation due to its degrading effect on subsea environments, especially when impurities of the water increase [1]. Therefore, more study of scattering is required to fully comprehend its influence on the optical beam propagation through seawater. As we show, the ability to handle a new fading effect due to the impurities of water leads to an unprecedented tractability of the subsea channel characterization and its commercial application.
In this paper, a statistical scattering-induced fading model is presented for the first time with the goal of providing with a more accurate UOWC channel model than the existing ones in the current literature which is urgently needed for academic and industrial purposes. In this way, we propose the Gamma distribution to model this new fading effect which is able to achieve an excellent goodness of fit when different transmitter sources such as laser diode (LD) and light emitting diode (LED) are assumed, and under different scattering conditions. The developed fading model is used to evaluate the performance of UOWC systems at high signal-to-noise-ratio (SNR) in terms of BER and outage probability (OP) under different undersea environments such as clear ocean and coastal waters. Furthermore, in order to make this performance analysis be closer to reality, the intensity random fluctuations caused by salinity-induced oceanic turbulence are also taken into consideration. This research certainly gives a valuable insight into the nature of this new fading effect, as well as its impact on communication systems design. It is concluded that the effect of scattering-induced fading can severely affect in terms of the BER and OP. A significant optical power penalty up to 6 dB and 9 dB is quantified when comparing with no scattering-induced fading scenarios for clear ocean and coastal waters at moderate distances less than 60 meters and 30 meters, respectively. In addition, an appreciable decrease of the diversity order imposed by scattering-induced fading is obtained in long link distances. The analytical results are verified by Monte Carlo simulations.
The remainder of the paper is organized as follows. The characterization of the developed scattering-induced fading model is described in Section 2. where different scattering scenarios are analyzed in detail. System and channel models are presented in Section 3. In Section 4, the developed fading model is used to study the UOWC system performance in terms of BER and OP under different scattering conditions. Numerical results and discussion are illustrated in Section 5. Finally, the paper is concluded in Section 6.
2. Characterization of scattering-induced fading model
2.1 Scattering channel simulation
As mentioned before, the UOWC channel is mainly affected by absorption, $a(\lambda )$, and scattering, $b(\lambda )$, which determine the propagation of light in water. These effects are characterized through the extinction coefficient as follows
where $\lambda$ represents the wavelength. A summary of the typical values of $a(\lambda )$, $b(\lambda )$, and $c(\lambda )$ are given in Table 1 for clear ocean and coastal waters when considering a wavelength of 532 nm [1].
Table 1. Extinction coefficient.
In order to analyze the random behavior of the UOWC channel, we use a Monte Carlo simulation tool, as presented in [5], to numerically evaluate the impact of absorption and scattering on the channel impulse response in horizontal UOWC links by sending on the order of $10^9$ photons. Furthermore, it should be noted that the Henyey-Greenstein (HG) phase function is adopted as scattering phase function due to its accurate in many field as astrophysics as well as atmospheric and oceanic optics [1,23]. Through this tool, we observe that the number of received photons changes as a function of a random variable. In addition, we observe that the photons fluctuations are directly related to the effects of absorption and scattering in each UOWC channel realization. To make sure that the mean of such an intensity is to be the unity, the optical intensity based on the received photons is normalized. For a better visual representation of statistical data, we simulate above 1000 realizations of the UOWC channel in different scattering conditions with the goal of quantifying the irradiance fluctuations of the received optical signal. After simulation, the data set is organized by a histogram to represent the PDF of the normalized received optical intensity.
2.2 Proposed scattering-induced fading model
From a statistical point of view, we model the intensity fluctuations, i.e., the scattering-induced fading through a random variable that follows a Gamma distribution due to its excellent behavior in many propagation problems related to optical wireless communications. For instance, in [24], the Gamma distribution is used to model large- and small-scale atmospheric turbulence. Thus, the proposed Gamma distribution is given by
In Figs. 1 and 2, the proposed PDF is computed via Eq. (2) for different transmitter sources such as LED and LD, respectively, when different UOWC link distances and types of water are considered. The rest of UOWC channel simulation parameters are summarized in Table 2. As can be observed, the proposed model fits accurately with its corresponding Monte Carlo simulation for the considered scenarios. On the one hand, it is confirmed that the scattering-induced fading is strongly dependent on the link distance and the type of water. For instance, in Fig. 1 it can be noted that scattering can induce considerable fluctuations for short link distances when Lambertian transmitter sources such as LEDs are considered. On the contrary, Fig. 2 indicates that scattering can be more severe for moderate and long link distances when an LD with a Gaussian beam shape is considered. According to these results, the use of LEDs may be unpractical when attempting to get long UOWC link distances owing to the impact of the scattering-induced fading and the temporal dispersion on the overall performance. On the other hand, in view of the depicted results it is observed that the histograms seem to approach to a negative exponential distribution as the strength of scattering-induced fading increases, especially in coastal water at long link distances. This phenomenon, which has not been quantified, presents a similar behavior to strong atmospheric turbulence models in terrestrial FSO systems [25]. Therefore, longer link distances than 20 m when using LED are not considered, as well as longer link distances than 60 m and 35 m for clear ocean and coastal waters, respectively, are not considered when using LD. In order to numerically evaluate the validity of the considered fading model, the coefficient of determination ($R^{2}$) is evaluated for all scenarios considered in Figs. 1 and 2. In the light of the goodness of fit, it can be concluded that the UOWC channel is modeled accurately by Eq. (2) regardless of the water type, transmitter source and link distance for the most practical UOWC applications, achieving a coefficient of determination above 0.9. It should be noted that this accuracy is achieved by using only one degree of freedom to model the fading effect.
Fig. 1. Gamma fitting distribution, $f_{h_{s}}(h_{s})$, and the corresponding normalized and simulated data histogram when different practical link distances are considered for (a) clear ocean and (b) coastal waters, as well as a green LED is used at the transmitter side.
Fig. 2. Gamma fitting distribution, $f_{h_{s}}(h_{s})$, and the corresponding normalized and simulated data histogram when different practical link distances are assumed for (a) clear ocean, and (b) coastal waters, as well as a green LD is used at the transmitter side.
Table 2. Simulated UOWC system parameters.
To gain further insight, it is instructive to express the parameter $\sigma _{s}^{2}$ as a function of link distance in order to thoroughly study the range of distances in which the scattering-induced fading can result in a severe degradation for potential submarine applications. According to our simulation results, it can be proved that $\sigma _{s}^{2}$ is highly distance-dependent and presents an exponential behavior as the link distance increases as follows
where $k_{1}$ and $k_{2}$ are the parameters to be solved from the Monte Carlo simulation, and $d$ is the link distance.In Fig. 3, $\sigma _{s}^{2}(d)$ is plotted as a function of link distance for clear ocean and coastal waters when a green LD is assumed. As can be observed, the analytical results match very well with Monte Carlo simulations, achieving an $R^{2}$ above 0.998 regardless of the water type.
Fig. 3. Strength of scattering-induced fading, $\sigma _{s}^{2}(d)$, as a function of link distance when a green LD is considered at the transmitter side. Values of $(k_1,k_2)=\{(1.452\times 10^{-6},0.209),(3.932 \times 10^{-5},0.304)\}$ for clear ocean and coastal waters, respectively.
3. System and channel models
3.1 System model
We consider a UOWC system based on intensity modulation and direct detection (IM/DD), using on-off keying (OOK) modulation due to its low cost and implementation complexity. In this way, the received electrical signal is given by
where $h$ is the fading coefficient of the channel, $R$ is the detector responsivity, $x$ is the transmitted optical intensity, and $z$ is additive white Gaussian noise with zero mean and variance $N_{0}/2$.3.2 Channel model
In order to provide an accurate channel model, we consider the effects of absorption and scattering along with the oceanic turbulence phenomenon. Hence, the fading coefficient, $h$, is modeled as the product of two statistically independent process given by
where $L$ is a deterministic factor that computes both oceanic path loss and geometric loss, $h_{s}$ is the scattering-induced fading, and $h_{o}$ is the oceanic turbulence-induced fading. The parameter $L$ can be calculated through the optical depth, $\tau _{od}$, as follows The optical depth and $L$ are related to the extinction coefficient and the link distance by the Beer-Lambert’s law as [26] where $F$ is a correcting factor in order to include the effect of scattering [5]. The parameter $F$ is actually the one that compensates for the underestimated power by the Beer Lambert’s law when scattering is assumed, i.e., the received power due to scattering.3.2.1 Oceanic turbulence-induced fading model
Many probabilistic distributions can be used to model the turbulence-induced fading effect. A commonly used model for salinity-induced turbulence is the Weibull distribution, which provides a good agreement with experimental measurement data for a wide range of turbulence scenarios, as presented in several salt water bodies [17,27]. In addition, the Weibull distribution has also been used in several reported articles in UOWC context in order to describe statistically oceanic turbulence [12–15]. Hence, the PDF of the considered turbulence-induced fading effect is obtained as follows
In order to obtain a simple and tractable expression for Eq. (11), the scintillation index can be approximated as a function of the link distance as in [14] as follows
where $\lambda _{1}$, $\lambda _{2}$, $\lambda _{3}$ are computed via curve adjustment. In order to analyze different oceanic turbulence scenarios, several values of $\lambda _{1}$, $\lambda _{2}$, $\lambda _{3}$ are summarized in Table 3 for different values of $w$, as well as for $\epsilon = 10^{-5} \textrm { m}^{2}/\textrm {s}^{3}$, and $\chi _{T}=10^{-7} \textrm { K}^{2}/\textrm {s}$.Table 3. Scintillation index parameters.
3.2.2 Composite fading channel
As mentioned before, the composite fading coefficient can be expressed as $h=L\cdot h_{s} \cdot h_{o}$. It can be noted that there is no related work that evaluates the performance of UOWC systems under a composite fading channel which is derived from the mixture between the Gamma and Weibull distributions. Hence, the PDF of the proposed fading channel is given by
where $f_{h|h_{s}}(h|h_{s})$ is the conditional probability given a scattering-induced fading state $h_{s}$. While the cumulative distribution function (CDF) of $f_{h}(h)$ was obtained in [30], its corresponding PDF would make this analysis highly complex to study the effect of the different channel parameters such as the link distance and the type of water on the system performance due to the fact that such a PDF appears in infinite summation-form. For the sake of clarity, we make use of an asymptotic analysis in order to offer insights as to what channel parameters determine the BER and the outage performance in the presence of our proposed composite fading channel. In this way, this asymptotic analysis undoubtedly favors the design of future transceivers for UOWC applications. Therefore, the expressions for $f_{h_{s}}(h_{s})$ and $f_{h_{o}}(h_{o})$ are approximated by a single polynomial term as in [31] as follows4. Performance analysis
4.1 Bit error rate performance
In this section, the BER performance of UOWC systems under our proposed fading channel is analyzed when adopting OOK signalling for the case of perfectly known channel state information at the receiver. For equally likely transmitted symbol, the conditional BER at the receiver is given by
where $Q(\cdot )$ represents the Gaussian-$Q$ function, $\gamma =P_{t}^{2}T_{b}/N_{0}$ is the normalized received electrical SNR in the absence of fading, $P_{t}$ is the average transmitted optical power, and $T_{b}$ is the bit period. Hence, $P_{b}$ is obtained by averaging over the PDF $f_{h}(h)$ as follows It must be mentioned that Eq. (21) might be mathematically intractable due to the fact that a closed-form expression of $f_{h}(h)$ appears in infinite summation-form, as commented before. Hence, we can use Eq. (19) to obtain an asymptotic BER expression at high SNR, as well as to provide with a better understanding about how the major system parameters impact on the BER performance.Firstly, to solve the integral in Eq. (21) we use the fact that $\textrm {erfc}(x)=2Q(\sqrt {2x})$ [32, Eq. (6.287)]. Then, by substituting Eq. (19) into Eq. (21), and making use of [33, Eq. (2.8.5.2)], the asymptotic closed-form expression for the average BER can be solved as follows
It can be deduced from the above expression that the BER performance at high SNR tends to $P_b\doteq \left (G_{c}\gamma \right )^{-G_{d}}$, where $G_{d}$ and $G_{c}$ denote diversity order and coding gain, respectively [31]. The diversity order determines the slope of the BER performance in a log-log scale at high SNR, and the coding gain determines the shift of the curve in decibeles. Thus, we can obtain the diversity order as follows
The diversity order allows us to determinate when scattering-induced fading or turbulence-induced fading dominates the BER performance at high SNR by means of the relationship between $\sigma _{s}^{2}$ and $\beta _{1}$. We say that scattering-induced fading dominates the BER performance at high SNR when $\sigma _{s}^{2} > \frac {1}{\beta _{1}}$, i.e., Eq. (22) tends to the first branch of the equation. Therefore, the diversity order depends only on the value of $\sigma _{s}^{2}$. On the other hand, oceanic turbulence-induced fading dominates the BER performance at high SNR when $\sigma _{s}^{2} < \frac {1}{\beta _{1}}$ holds. In this case, Eq. (22) tends to the second branch of the equation and consequently the diversity order depends entirely on $\beta _{1}$.4.2 Outage probability performance
In order to get a complete overview of the UOWC system performance in the presence of scattering-induced fading, the asymptotic outage performance is also analyzed in a similar way to the BER performance. The outage probability, OP, is defined as the probability that the instantaneous received SNR, $\gamma _{T}$, falls below a certain specified threshold, $\gamma _{th}$, as in [34]
where $\gamma _{T}=4\gamma h^{2}.$ By using this, the asymptotic OP can be written as5. Results and discussion
In this section, the proposed scattering-induced fading model is evaluated over oceanic turbulence channels and different scattering conditions. Without loss of generality, we assume a value of the oceanic path loss $L=1$ and a value of the detector responsivity $R=1$. As commented before, the main UOWC system parameters used for the channel simulation are summarized in Table 2. At the transmitter side, an LD with a transmit divergence angle of $\theta _{div}= 10$ mrad is assumed in the green-band. Furthermore, a receiver aperture diameter of $D=$ 5 mm and a receiver FOV of $180^{\circ }$ are assumed at the receiver side. On the other hand, UOWC link distances of $d =${50, 55, 62, 68, 70} m and $d =${24, 27, 30, 33, 35} m are assumed for clear ocean and coastal waters, respectively. The link distances have been selected according to technical specifications for commercial UOWC systems [35,36]. In order to represent several strength of temperature and salinity fluctuations as an example of different turbulence-induced fading scenarios, we compute different values of $w = \{-0.3, -1, -2, -3\}$ when $\chi _{T} = 10^{-7} \textrm { K}^{2}/\textrm {s}$ [6,28].
5.1 Performance analysis results
Firstly, the diversity order $G_{d}$ obtained in Eq. (23) is depicted as a function of the link distance for clear ocean and coastal waters in Fig. 4. Note that the value of the diversity order gain is obtained from the BER curve at high SNR regime. As can be seen, the effect of oceanic turbulence dominates the UOWC system performance at high SNR, i.e., the condition $\sigma _{s}^{2} > \frac {1}{\beta _{1}}$ is not satisfied when short and moderate link distances are assumed or under strong oceanic turbulence conditions. Within this context, higher diversity order gains are obtained. However, the link distances upon which the scattering imposes the diversity order, i.e., when the condition $\sigma _{s}^{2} > \frac {1}{\beta _{1}}$ is satisfied, are shorter in coastal water due to the effect of water turbidity.
Fig. 4. Diversity order $G_{d}$ as a function of the link distance $d$ (m), for clear ocean and coastal waters when different values of $w=\{-4,-3,-2,-1\}$ are considered.
Next, as an illustration of the asymptotic BER expression obtained in Eq. (22), the average BER of a UOWC system is computed and verified by Monte Carlo simulations. In Figs. 5(a) and 5(a), the average BER is plotted as a function of the electrical SNR $\gamma$ for clear ocean and coastal waters, respectively, assuming $w=-3$ and different link distances. Due to the long simulation time involved, BER simulation results up to $10^{-8}$ are plotted. As a reference, analogous scenarios are depicted in dashed line when scattering-induced fading is neglected. The corresponding asymptotic expression for the BER when scattering-induced fading is neglected, $P_{b}^{ns}$, can be derived by assuming $\sigma _{s}^{2}\to 0$ from the second branch of Eq. (22) by using [37, Eq. (06.05.25.0003.01)] as follows
Fig. 5. Average BER performance for (a) clear ocean, and (b) coastal waters over the composite fading channel when different link distances are assumed.
As depicted in Fig. 5, the asymptotic BER performance expression obtained in Eq. (22) is in good agreement with Monte Carlo simulation results, confirming the accuracy and usefulness of the asymptotic results. As can be observed, it is verified that the impact of scattering-induced fading on the performance of UOWC systems, intensifying its effect as the variance, i.e., the strength of the scattering-induced fading increases. As can be seen in Fig. 5(a), the degrading factor of scattering becomes significant when the link distance increases. As expected, a drastic performance degradation caused by scattering-induced fading is observed when link distances of $d=68$ m and $d=70$ m are assumed in clear ocean water. In other words, the effect of the scattering-induced fading at high SNR is more relevant than the turbulence-induced fading when increasing the link distance. This conclusion is qualitatively similar to those obtained in coastal water, as plotted in Fig. 5(b). Although the link distances considered in coastal water are shorter than in clear ocean water, the effect of scattering-induced fading impacts also notably on the UOWC system performance. This is consistent with the distances range studied in Fig. 4.
In Fig. 6, the outage performance is computed via Eq. (26) and verified by Monte Carlo simulations. Here, the OP analysis confirms the same conclusions which have been obtained in the above BER performance as well as the diversity order analysis, by emphasizing that scattering-induced fading impacts more severely on the performance of UOWC systems in turbid environments such as coastal water, as well as on long link distances in clear ocean water.
Fig. 6. Outage performance for (a) clear ocean, and (b) coastal waters over the composite fading channel when different link distances are assumed.
5.2 Optical power penalty results
As concluded in the previous subsection, practical UOWC links will deployed in those scenarios where oceanic turbulence-induced fading is dominating the BER performance at high SNR, i.e., when the relation $\sigma _{s}^{2} < \frac {1}{\beta _{1}}$ is satisfied and, hence, higher diversity order gains are achieved. Within this context, we compute the optical power penalty (OPP) in decibels produced by the effect of scattering-induced fading at high SNR. The OPP computes the additional power needed to obtain a given BER performance when scattering-induced fading is considered versus when scattering-induced fading is neglected. Thus, the OPP is obtained as the subtraction in decibels between $G_{c}$ from Eq. (22) and $G_{c}^{ns}$ from Eq. (27) when the effect of scattering-induced fading is neglected as follows
Fig. 7. Optical power penalty (OPP) as a function of the distance, $d$ (m), for clear ocean and coastal waters when different values of $w=\{-4,-3,-2,-1\}$ are considered.
6. Conclusion
In this paper, the performance of UOWC systems has been analyzed in the presence of a novel scattering-induced fading effect when turbulence-induced fading is also assumed. The developed channel model as well as the asymptotic BER and OP expressions are verified by Monte Carlo simulation for clear ocean and coastal waters.
Taking into account the results presented here, we can conclude that the scattering-induced fading must be included in an accurate UOWC system performance analysis since it presents a significant impact in terms of BER and outage performance when designing UOWC long-distance links. On the one hand, the scattering-induced fading effect was modeled accurately by a Gamma distribution under different kinds of water and distance-ranges. Because of scattering, the received optical beam presents random fluctuations on the irradiance, even more remarkable as the water gets turbid and the link distance increases. On the other hand, the developed BER and OP expressions for UOWC systems over the joint effect of scattering-induced fading and oceanic turbulence provide insight into the overall performance degradation. Our results indicate that the performance is mainly impaired at high SNR by oceanic turbulence for short and medium link distances, but not in long link distances, where the scattering-induced fading dominates the performance degradation. The impact of increasing the link distance has been also analyzed by evaluating the optical power penalty induced by scattering-induced fading, showing that a negligible increases of the link distance of the system causes a drastic degradation in terms of SNR, due to its exponential behavior with the link distance.
Our work provides a novel framework for more precise studies to assess the characterization of oceanic phenomena in the presence of scattering-induced fading for communication purposes, as well as to analyze the performance of UOWC systems not only in terms of BER and OP, but also in terms of channel capacity. Thus, our results can be applied directly to the development of future UOWC transceivers. We envision that our fading channel model may be extended by using other statistical oceanic turbulence models such as the lognormal distribution, as well as enhanced through the study of the chlorophyll-concentration depth profile to consider vertical links. As a future study, we plan to experimentally validate the proposed scattering-induced fading model under laboratory conditions.
Funding
Programa Operativo I+D+i FEDER Andalucía 2014-2020 (P18-RTJ-3343); Ministerio de Ciencia, Innovación y Universidades (PID2019-107792GB-I00).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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