Error probability analysis of OOK and variable weight MPPM coding schemes for underwater optical communication systems affected by salinity turbulence

A. Jurado-Navas; N. González Serrato; J. M. Garrido-Balsells; M. Castillo-Vázquez

doi:10.1364/OSAC.1.001131

1. Introduction

The underwater environment is challenging for all modes of communications, which requires careful consideration of distinct tradeoffs between data rate and range. For example, acoustic waves provide a long operational range of the order of kilometers but at very low data rates (orders of kilobits/second). Underwater Wireless Optical Communication (UWOC) systems are attracting considerable attention by the scientific community in the last years because it is viable to deploy real applications with high-bandwidth around 1 Gigabit per seconds, Gbps, and beyond [1–5]. However the performance of these communication systems is directly affected by the optical attenuation coefficient (combined effect of water and other effects as the absorption due to phytoplankton and detritus [6]) because of the great attenuation of light propagating through water. Although the communications range can be as short as several meters (turbid water) or some few hundred of meters depending on water quality, the aforementioned accomplished bandwidths are significantly higher than those with acoustic methods, for instance, so applications such as short-range communications among unmanned underwater vehicles, submarines, ships, buoys, docking stations, and divers can be feasible. For these reasons, and because UWOC transceivers benefits from their low costs and small volumes, many potential applications of UWOC systems have been released in the last years [7] for environmental monitoring (especially with the ever-increasing global climate change), offshore exploration, disaster precaution, and military operations.

In this paper, we focus our study in the impact of underwater optical turbulence on the performance of UWOC systems. Turbulence in UWOC is normally caused by ocean currents which induce random variations in the water temperature and pressure and, therefore, in its refractive index [8], analogous to the dynamics predicted for atmospheric optical channels. Factors as turbidity [9] or salinity [10] of water are also completely related to those refractive index variations, affecting the performance in these systems. Hence, this random space-time redistribution of the refractive index causes a variety of effects on the propagating optical wave in regards to its temporal irradiance fluctuations, commonly described as scintillation [11–13]. Thus, many researches [14,15] are based on the results of atmospheric free-space optical channel models assuming that similarity between underwater and atmospheric optical turbulence.

In order to characterize the underwater optical turbulence-induced fading, several probabilistic models have been proposed in the bibliography: the lognormal distribution [15] used for weak fluctuation regime; the double Gamma model [9], which fits well with turbid seawater such as coastal and harbor water; or the Weibull function model, recently proposed in [16] to characterize the fading of salinity induced turbulent UWOC channels. As indicated in [16], the Weibull model is preferred to the exponentiated Weibull one when considering the effect of salinity gradient induced turbulence (specially interesting in transition areas between seas and oceans where different masses of water meet), while the latter one is better suited to characterize fading in temperature-induced turbulent underwater wireless optical links. In addition, as indicated in [17], the Weibull distribution gives a good fit to the probability density function (PDF) when the receiver aperture is much grater than the coherence radius of the medium, something feasible considering the short-range links involved in UWOC systems. Furthermore, the Weibull distribution offers an excellent fit for experimental data under all turbulence conditions.

In this work we analyze the performance of an UWOC system in terms of average bit error rate (ABER) under turbulence induced by salinity gradient. Then we present a novel closed-form expression for the ABER of the system when employing any generalized coding technique. As an illustrative example, we offer the behavior of these systems when a variable-weight multiple pulse-position modulation (vw-MPPM) is employed [18]. Finally, and as a reference, we also provide the performance of UWOC systems when a simple on-off keying (OOK) signalling technique is used. We must remark that, through this paper, we neglect scattering and ISI effects to focus on the effect of turbulence.

2. Channel and system models

As indicated, turbulence in UWOC occurs as a result of random small index-of-refraction fluctuations along the propagation path. These variations in underwater medium are mainly induced by random salinity, density and temperature variations of the underwater environment [5], and leads to fluctuations in the intensity of the signal at the receiver. This phenomenon is called scintillation and degrades the performance of UWOC. Due to the dynamic nature of underwater environment, there exists no specific model for underwater turbulence unlike in the case of atmospheric optical communications. But as the physical mechanisms of underwater and atmospheric turbulences are similar, and considering the specific power spectrum derived in [19] for the fluctuations of turbulent seawater refractive index, the Rytov method and its expansions can be applied including the modulation process employed to derive the PDF of irradiance in atmospheric optical environments [17,20,21]. Accordingly, the Weibull distribution has been recently used in [16, 23] to describe the PDF of the irradiance fluctuations due to salinity induced turbulence. That Weibull distribution was directly adapted from [17] and its accuracy is corroborated since UWOC systems typically employ large receiving apertures due to the negligibility of background noise under water, which commonly leads to an effectively reduced turbulence [22]. The Weibull PDF is defined as:

f_{I} (I) = \frac{K}{λ} {(\frac{I}{λ})}^{K - 1} exp [- {(\frac{I}{λ})}^{K}],

with K > 0 being the shape parameter related to the scintillation index of the irradiance fluctuations and λ > 0 is the scale parameter related to the mean value of the irradiance [17]. Assuming E[I] = 1 then, according to [24],

λ = \frac{1}{Γ (1 + 1 / K)}

since E[I] = λΓ(1 + 1/K) and E[I²] = λ²Γ(1 + 2/K), whereas the scintillation index is given by:

σ_{I}^{2} \frac{Γ (1 + 2 / K)}{Γ {(1 + 1 / K)}^{2}} - 1 \approx K^{- 11 / 6}

where the approximated expression for K is only valid for weak and moderate turbulence regime [24]. In addition, in Eq. (3), Γ(·) represents the gamma function.

Regarding the system model, we consider an on-off keying (OOK) modulation format in an intensity modulation with direct detection (IM/DD) scheme. Accordingly, each bit symbol is transmitted by pulsing the light source either on (logic “1”) or off (logic “0”) during each bit time. Thus, the time-dependent photocurrent at the detector output is written as

i = i_{S} + i_{N} = α {RIP}_{t} + i_{N},

where α represents the propagation loss factor [25,26], with R denoting the detector responsivity, I is the normalized scintillation induced irradiance with E[I] = 1, and with I following the Weibull statistical distribution shown in Eq. (1). On the other hand, P_t denotes the average of transmitted optical power, i_S represents the signal current in the detector when a transmitted “1” is sent whilst i_N is the detector current noise signal caused by a zero-mean additive white Gaussian noise with variance

σ_{N}^{2}

.

3. Average bit error rate (ABER) for uncoding OOK format

In this section we derive the analytical expression for the ABER associated with an UWOC IM/DD system using an OOK signaling technique. For this purpose, and as a previous step, the associated conditional BER (CBER) is firstly calculated for a given electrical signal-to-noise ratio (SNR) when analyzing an AWGN channel in the ideal case of absence of turbulence (namely SNR₀), assuming each transmitted symbol equally likely to be sent. In addition, the conditional bit error probabilities when the transmitted bit is “0” or “1” are assumed to be equal. Hence, and from [27], the CBER of IM/DD with AWGN channel using OOK is expressed as

P_{b} (e | I) = \frac{1}{2} erfc (\frac{i_{S_{0}} I}{2 \sqrt{2} σ_{N}}) = \frac{1}{2} erfc (\frac{{SNR}_{0} I}{2 \sqrt{2}}),

where SNR₀ = i_S₀/σ_N, where i_S₀ = αRP_t denotes the signal current in absence of turbulence-induced fading, with α representing the attenuation coefficient associated to the medium. Therefore, the ABER, P_b(e), can be obtained by averaging P_b(e|I) over the PDF of the irradiance, f_I(I). Hence:

P_{b} = \int_{0}^{\infty} \frac{1}{2} erfc (\frac{{SNR}_{0} \cdot I}{2 \sqrt{2}}) f_{I} (I) d I .

In Eq. (6), the PDF of the optical irradiance is defined according to the Weibull model, as indicated in Eq. (1). Now, by using the integration by parts formula for solving Eq. (6), we can obtain that

P_{b} = {(P_{b} (e | I) F_{I} (I)) |}_{0}^{\infty} - \int_{0}^{\infty} \frac{d}{d I} [P_{b} (e | I)] F_{I} (I) d I .

Since P_b(e|∞) = 0 and F_I(0) = 0 (note that negative values for the optical irradiance are not allowed), then the last expression can be reduced to:

P_{b} = - \int_{0}^{\infty} \frac{d}{d I} [\frac{1}{2} erfc (\frac{{SNR}_{0} \cdot I}{2 \sqrt{2}})] F_{I} (I) d I,

where the cumulative distribution function (CDF) for the irradiance, I, is directly obtained by integrating Eq. (1) as:

F_{I} (I) = 1 - exp [- {(\frac{I}{λ})}^{K}] .

Thus, we can employ [28, Ec. (06.27.13.0005.01)] to derive an expression for the derivative of P_b(e|I) with respect to I:

\frac{d}{d I} [P_{b} (e | I)] = - \frac{{SNR}_{0}}{2 \sqrt{2 π}} exp [- {(\frac{{SNR}_{0} I}{2 \sqrt{2}})}^{2}] .

Next we introduce the last result in Eq. (8) in order to solve the resulting integral. To this aim, a generalized Gauss-Laguerre quadrature [29] is proposed, defined by:

\int_{0}^{\infty} x^{β} e^{- x} f (x) d x = \sum_{i = 1}^{n} H_{i} f (x_{i}) + E_{n},

where β is a constant, x_i is the i–th zero of the Laguerre polynomial,

L_{n}^{β} (x)

, H_i is the corresponding weight coefficient and E_n is the truncation error. If the normalization of the Laguerre polynomials is chosen so that

L_{n}^{β} = \sum_{m = 0}^{n} (\begin{array}{l} n + β \\ n - m \end{array}) \frac{{(- x)}^{m}}{m!} .

then, according to [29], the weight coefficients are given by

H_{i} = \frac{Γ (n + β + 1) x_{i}}{n! {(n + 1)}^{2} {[L_{n + 1}^{β} (x_{i})]}^{2}}, (i = 1, 2, \dots, n) .

Hence, Eq. (8) can be rewritten as

P_{b} = \frac{1}{2 \sqrt{π}} \int_{0}^{\infty} x^{- \frac{1}{2}} exp (- x) (1 - exp [- {(\frac{2 \sqrt{2}}{λ {SNR}_{0}} \sqrt{x})}^{K}]) d x

after having used the following change of variables:

x = {(\frac{{SNR}_{0}}{2 \sqrt{2}})}^{2} I^{2}; d x = 2 {(\frac{{SNR}_{0}}{2 \sqrt{2}})}^{2} I d I .

Finally, we can apply Eq. (11) to solve Eq. (14) as:

P_{b} = \frac{1}{2 \sqrt{π}} \sum_{i = 1}^{n} H_{i} {1 - exp [- {(\frac{2 \sqrt{2}}{λ {SNR}_{0}} \sqrt{x_{i}})}^{K}]}

where β = −1/2 directly obtained by comparing Eq. (14) with Eq. (11) so the weight coefficients, H_i, given in Eq. (13) can be directly calculated.

As a remarking comment, we can repeat all these steps to derive a closed-from expression for the exact ABER of the UWOC system using an exponentiated Weibull distribution instead of the Weibull one shown in Eq. (1). The analytical procedure is shown in the Appendix. For the sake of simplicity, this paper is focused on the Weibull distribution since, as we commented above, large receiving apertures are commonly employed in UWOC, which leads to an effectively reduced turbulence accurately modeled by the simplest Weibull distribution.

4. vw-MPPM

Through this section, we want to show how to derive analytical closed-from expressions for the exact ABER of UWOC systems employing any generic coding scheme. As a representative example of a generic coding technique, we analyze the use of variable weight multiple pulse-position modulation (vw-MPPM), successfully proposed by the authors in indoor [18] and outdoor [30] wireless communications.

Typically, in wireless optical communications affected by turbulence, a modulation technique that increases the peak-to-average optical power ratio (PAOPR) parameter is preferred as it provides better performance in the optical link, overcoming the imposed distortion when a system bandwidth constraint is required. The basic design criterion is to keep the average optical power transmitted at a constant level. Furthermore, rate-adaptive transmission schemes are preferred in order to make the communication suitable to the adverse channel conditions, depending on the available signal-to-noise ratio (SNR), until a sufficiently low error probability can be attained. In this respect, the rate-adaptive transmission scheme using block coding of variable Hamming weight is a very good alternative to maximize the link performance, achieving a high rate adaptability by simply changing the coding translation matrix. This coding technique is based on multiple pulse-position modulation (MPPM) where codewords with different Hamming weights are allowed. This fact minimises the presence of pulses at the optical signal leading to an increment in PAOPR so vw-MPPM can be seen as an improved version of both conventional classical scheme based on pulse-position modulation (PPM) and MPPM in terms of link performance. We must remark that PPM can achieve higher-energy information efficiency than OOK [31]; whilst the MPPM technique has been used in wireless optical systems to reduce the average transmitted optical power while maintaining the same performance [32]. And, as commented, vw-MPPM improves such aforementioned classic coding techniques.

The coding process consists of a translation procedure between the input data alphabet, C_K, with k-bit codewords, and the coded alphabet, C̃_N, a subset of C_N, comprising 2ⁿ possible n-bit codewords. For the proper choice of C̃_N, codewords with different Hamming weight are proposed, leading to a block coding with a variable amount of pulses. Hence, if C_n,w is the block code consisting of all possible codewords of length n with a Hamming weight of w, the code C̃_N is defined using the following codes C_n,w:

{\tilde{C}}_{N} = (\cup_{i = 0}^{x - 1} C_{n, i}) \cup {\tilde{C}}_{n, x},

with C̃_n,x representing the codewords subset of C_n,x used in C̃_N. Thus, the coding table associated with C̃_N consists of all the possible codewords with Hamming weight i, 0 ≤ i ≤ (x − 1), together with a number of x-weighted codewords given by

2^{k} - \sum_{i = 0}^{x - 1} (\begin{matrix} n \\ i \end{matrix})

where

(\begin{matrix} n \\ i \end{matrix})

is the number of codewords of C_n,i. In this sense, the rate associated to the block code is given by k/n.

As vw-MPPM is a nonlinear block coding scheme, the standard methods based on the characteristic functions of linear block codes are not suitable to obtain closed-form expressions of CBER. For this reason, a novel alternative based on a hyperexponential fitting technique was proposed in [33] to achieve a successful CBER approximated expression, given by

CBER (I, γ_{0}) = P_{b} (e | I) \approx a exp [- b {(γ_{0} I^{2})}^{c}]

with γ₀ being the electrical SNR in absence of turbulence, and where the hyperexponential fitting parameters are a, b, c ∈ ℜ⁺. In Table 1 we show the hyperexponential fitting parameters for most relevant vw-MPPM code rates [18].

Table 1. Hyperexponential Fitting Parameters a, b and c in Absence of Turbulence

View Table

Thus, the ABER, P_b(e), is again obtained by averaging P_b(e|I) over f_I(I), in the form:

P_{b} = \int_{0}^{\infty} a exp [- b {(γ_{0} I^{2})}^{c}] f_{I} (I) d I .

For the sake of uniformity with the case of uncoding OOK format, we replace γ₀ by SNR₀ in Eq. (20). Thus, ${SNR}_{0} = \sqrt{(γ_{0})}$ is defined as i_S₀/σ_N, with i_S₀ = αRP_t. In Eq. (20), the PDF of the optical irradiance is defined according to the Weibull model, as was indicated in Eq. (1). Then we can apply the integration by parts formula given in Eq. (7) for solving Eq. (20), where the derivative of the approximated CBER given in Eq. (19) with respect to I is calculated as:

\frac{d}{d I} [a exp [- b {({({SNR}_{0} I)}^{2})}^{c}]] = - 2 a b c \cdot {SNR}_{0}^{2 c} I^{2 c - 1} exp [- b {({({SNR}_{0} I)}^{2})}^{c}]

Then, by repeating the same process than for the case of uncoding OOK format [Eqs. (11)–(13)], we can obtain the closed-form expression for the ABER of vw-MPPM block coding scheme. Hence:

P_{b} = a \sum_{i = 1}^{n} H_{i} \cdot {1 - exp [- {(\frac{x_{i}^{\frac{1}{2 c}} σ_{N}}{b^{\frac{1}{2 c}} R P_{t} λ})}^{κ}]}

where H_i is the weight coefficient given in Eq. (13), where β = 0 in such aforementioned Eq. (13). That value can be calculated in a straightforward manner by comparing Eq. (14) with Eq. (20).

5. Results and discussion

In this section, we present some Monte Carlo numerical results for different turbulence conditions (weak, weak-to-moderate, moderate-to-strong and strong) used to corroborate the validity of the ABER expressions proposed in this paper, showing its immediate applicability in performance studies. To accomplish the Monte Carlo simulations, first we generate a vector containing a sequence of irradiance following a Weibull PDF by using the well-known acceptance-rejection algorithm [34]. The length of such sequence is of 3 × 10⁶ samples, and they are employed to achieve error probabilities of up to 10⁻⁶ with respect to a same constant vector of SNR₀ values.

Figure 1 shows the simulation results of average BER for uncoding OOK format and for weak, weak-to-moderate, moderate-to-strong and strong atmospheric turbulence conditions. These four situations are described in [35] from experimental measurements, where the corresponding values for the irradiance variance are, respectively: $σ_{I}^{2} = 0.0496$ , 0.1015, 0.7885 and 1.0652. In addition, the values of such optical irradiance variances are also taken from the acquired data presented in [35]. From such values of $σ_{I}^{2}$ , and considering Eqs. (2) and (3), we can obtain the values for λ and κ, respectively, required for representing the analytical results given by Eq. (16). The average BER curves achieved with Eq. (16) and particularized for those turbulence environments are represented as solid lines in Fig. 1. Furthermore, numerical data generated via Monte-Carlo simulation are also represented in Fig. 1 as circles. The coincidence between simulation data and the analytically obtained curves of average BER is completely corroborated, showing the high accuracy between the simulation results and the closed-form expression derived in this paper in Eq. (16).

Fig. 1 Analytical (solid lines average bit error rate and Monte Carlo simulation results (circles) vs SNR₀ for uncoding OOK formats under weak ( $σ_{I}^{2} = 0.0496$ ), weak-to-moderate ( $σ_{I}^{2} = 0.1015$ ), moderate-to-strong ( $σ_{I}^{2} = 0.7885$ ) and strong ( $σ_{I}^{2} = 1.0652$ ) salinity induced turbulence. Scenarios and values of irradiance variances taken from acquired data presented in [35]. As a reference, the ideal AWGN channel is depicted in dashed line.

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As it can be checked, solving the integral involving the ABER in Eq. (6) by using the generalized Gauss-Laguerre quadrature rule noticeably reduces the time required by a computer to perform such operation avoiding the use of simulation techniques. Accordingly, it is not required to process millions of samples to solely obtain a single point in the ABER curve; on the contrary, the analytical solution allows us to obtain the complete ABER curve in a straightforward manner. Even with all that, numerical results are really useful to validate the closed-form expression proposed in Eq. (6) with high accuracy. As a remarking comment, we have employed n = 20 coefficients to solve Eq. (16).

On another note, the results shown in Fig. 1 are completely expected. In this regard, an increase in the intensity of the turbulence-induced fading affecting the transmitted optical signal (due to a higher seawater salinity random variations across the optical beam propagation) implies a worse link performance in terms of average bit error rate. For example, for SNR₀ = 15 dB, P_b ≈ 10⁻⁶ for the case of weak turbulence regime; whilst the ABER grows up to 5.5 × 10⁻⁵, 1.6 × 10⁻² and 2.7 × 10⁻² for the other analyzed cases of weak-to-moderate, moderate-to-strong and strong turbulence regimes, respectively, while maintaining the same level of SNR₀.

As a final remark, for ABERs lower than 10⁻⁶ we observe that the numerical results slightly diverge from the analytical results. The reason is, precisely, that we are employing an insufficient number of samples in the Monte Carlo simulation (3 millions of samples, as we commented above) for those lower ABERs that may not provide a convenient characterization of a sufficient statistic at those levels of performance.

Regarding the vw-MPPM case analyzed in Section 4, similar conclusions can be extracted. Through Figss 2–4 it is shown that the combination of hyperexponential fitting method of conditional BER and the Gauss-Laguerre quadrature proposed in this paper yields to the development of a closed-form analytical and easily computable solution of the average BER given in Eq. (20) when a Weibull salinity induced oceanic turbulence is assumed. As expected, numerical results, represented by circles, confirm the theoretical results anticipated by the proposed expression in Eq. (22) for four different turbulence conditions (weak, weak-to-moderate, moderate-to-strong and strong), showing its immediate applicability in performance studies of the vw-MPPM coding scheme in UWOC systems.

Fig. 2 Analytical (solid lines) average bit error rate and Monte Carlo simulation results (circles) vs SNR₀ for a vw-MPPM format with code rate of 2/3 under weak ( $σ_{I}^{2} = 0.0496$ ), weak-to-moderate ( $σ_{I}^{2} = 0.1015$ ), moderate-to-strong ( $σ_{I}^{2} = 0.7885$ ) and strong ( $σ_{I}^{2} = 1.0652$ ) salinity induced turbulence. Scenarios and values of irradiance variances taken from acquired data presented in [35]. As a reference, the ideal AWGN channel is depicted in dashed line.

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Fig. 3 Analytical (solid lines) average bit error rate and Monte Carlo simulation results (circles) vs SNR₀ for a vw-MPPM format with code rate of 6/12 under weak ( $σ_{I}^{2} = 0.0496$ ), weak-to-moderate ( $σ_{I}^{2} = 0.1015$ ), moderate-to-strong ( $σ_{I}^{2} = 0.7885$ ) and strong ( $σ_{I}^{2} = 1.0652$ ) salinity induced turbulence. Scenarios and values of irradiance variances taken from acquired data presented in [35]. As a reference, the ideal AWGN channel is depicted in dashed line.

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Fig. 4 Analytical (solid lines) average bit error rate and Monte Carlo simulation results (circles) vs SNR₀ for a vw-MPPM format with code rate of 9/36 under weak ( $σ_{I}^{2} = 0.0496$ ), weak-to-moderate $σ_{I}^{2} = 0.1015$ ), moderate-to-strong ( $σ_{I}^{2} = 0.7885$ ) and strong ( $σ_{I}^{2} = 1.0652$ ) salinity induced turbulence. Scenarios and values of irradiance variances taken from acquired data presented in [35]. As a reference, the ideal AWGN channel is depicted in dashed line.

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In particular, Figs. 2–4 depict the ABER as a function of the signal-to-noise-ratio, SNR₀, for coding rates of R = 2/3, 1/2 and 1/4, respectively, and the four studied turbulence regimes. From this figures, it also observed that the higher the rate reduction, the better the ABER improvement. Note that, as expected, in any of these figures, the higher the turbulence strength, the worse the ABER obtained. Again, an excellent fitting between numerical and simulated results for the ABER is observed. For example, for an error probability of 10⁻³ and weak turbulence regime, it is required a SNR₀ of 3.61 dB for a code rate of 2/3; while such requirement decreases to SNR₀ = −2.77 dB for the code rate of 1/4. If we consider this latter code rate (1/4), an error probability of 10⁻³ is achieved for weak-to-moderate, moderate-to-strong and strong turbulence when SNR₀ = −0.82, 15.07 and 19.59 dB, respectively.

Furthermore, we can compare the case of vw-MPPM with the previous uncoding OOK format. In this latter case, a signal-to-noise ratio of 9.53 dB is required to achieve an ABER of 10⁻³, when the weak turbulence ( $σ_{I}^{2} = 0.0496$ ) is considered, exceeding in around 6 dB the worse case in vw-MPPM for the same turbulence condition.

As we comment above, this paper is focused on analyzing the effect of salinity gradient induced turbulence where, commonly, a Weibull model is preferred to characterize such process. Additionally, receiver apertures in UWOC systems are typically much grater than the coherence radius of the mediums (short-range links) reinforcing the decision of including a Weibull distribution in the analytical treatment derived in this paper. Nevertheless, it is possible to obtain similar expressions when an exponentiated Weibull PDF is assumed, as shown in Appendix.

As a final comment, the hyperexponential fitting method analyzed in Section 4 allows the generalization of the derived expressions to any coding scheme for which the a, b, and c parameters can be first calculated.

6. Concluding remarks

To summarize, simple closed-form expressions for the ABER in UWOC systems employing any generic coding scheme with different rate-adaptive coding are derived. A Weibull PDF was considered to analyze the link performance affected by salinity induced turbulence as an alternative to other commonly used models as, for instance, the lognormal or the generalized Gamma. The Weibull distribution has been recently shown as a very good alternative to model the effect of salinity gradients, of great interest specially in transition areas between seas and oceans. However, the Weibull distribution is not analytically tractable when deriving an expression for the ABER in an UWOC IM/DD system affected by turbulence and employing an OOK format. Even more difficult when a generalized coding technique is considered to be studied. In this paper we detail a simple procedure to calculate a closed-form expression by means of a generalized Gauss-Laguerre quadrature and the CDF for the transmitted optical irradiance. Moreover, a generalization of the derived expressions to any coding scheme can be accomplished by employing the hyperexponential fitting method for which its a, b, and c parameters must be previously obtained. For all cases considered in this paper (OOK and vw-MPPM), the proposed ABER expressions have been corroborated by Monte Carlo simulations for different turbulence conditions and several code rates achieving an excellent fitting between analytical and simulated results.

To sum up, the results presented in this paper are of great of relevance to study areas of strategic interest for a variety of potential applications including environmental issues, natural disaster precautions, or military operations among others. Accordingly, they show its immediate applicability in performance studies of any coding scheme in UWOC systems. Following [16], the Weibull model shows, in addition, an excellent accuracy when predicting the experimental data for any turbulence regime. For this reason, it is even more interesting to have analytical expressions as the ones presented in this paper, to anticipate the performance of an UWOC link.

Appendix: Derivation of ABER for exponentiated Weibull turbulence

Consider the exponentiated Weibull PDF and CDF defined, respectively, as:

f_{I} (I) = \frac{α K}{λ} {(\frac{I}{λ})}^{K - 1} exp [- {(\frac{I}{λ})}^{K}] {1 - exp [- {(\frac{I}{λ})}^{K}]}^{α - 1},

F_{I} (I) = {1 - exp [- {(\frac{I}{λ})}^{K}]}^{α} .

as indicated in [17], with K > 0 being the shape parameter related to the scintillation index of the irradiance fluctuations and λ > 0 is the scale parameter related to the mean value of the irradiance [17], as in the Weibull case. The additional parameter, α, is an extra shape parameter that provides more versatility in the shape of the tails, and it is dependent on the receiver aperture size. When α = 1, Eq. (23) reduces to Eq. (1).

Now, we can repeat the same steps indicated from Eq. (5) to Eq. (8). Next, we introduce Eq. (24) into Eq. (8) and we apply Eq. (10) to Eq. (13). Hence, Eq. (8) can be now written as:

P_{b} = \frac{1}{2 \sqrt{π}} \int_{0}^{\infty} x^{- \frac{1}{2}} exp (- x) {(1 - exp [- {(\frac{2 \sqrt{2}}{λ {SNR}_{0}} \sqrt{x})}^{K}])}^{α} d x

After using the same change of variables shown in Eq. (15), we apply again Eq. (11) to solve Eq. (25) as:

P_{b} = \frac{1}{2 \sqrt{π}} {\sum_{i = 1}^{n} H_{i} {1 - exp [- {(\frac{2 \sqrt{2}}{λ {SNR}_{0}} \sqrt{x_{i}})}^{K}]}}^{α}

where β = −1/2 directly obtained by comparing Eq. (25) with Eq. (11), so Eq. (13) can be directly calculated.

References

1. A. Khalighi, C. Gabriel, and V. Rigaud, “Optical communication system for an underwater wireless sensor network,” EGU General Assembly 14, 2685 (2012).

2. L. Mullen, B. Cochenour, and A. Laux, “Spatial and temporal dispersion in high bandwidth underwater laser communication links,” Proc. IEEE Military Commun. Conf. (2008), pp. 1–7.

3. The Sonardyne Site, “BlueComm Underwater Optical Communications”. Sonardyne International Ltd. [retrieved 18 July 2018], http://www.sonardyne.com/products/all-products/instruments/1148-bluecomm-underwater-optical-modem.html.

4. C. Wang, H.-Y. Yu, and Y.-J. Zhu, “A long distance underwater visible light communication system with single photon avalanche diode,” IEEE Photon. J. 8, 1–11 (2016). [CrossRef]

5. H. Kaushal and G. Kaddoum, “Underwater optical wireless communication,” IEEE Access 4, 1518–1547 (2016). [CrossRef]

6. J. R. Apel, Principles of Ocean Physics (Academic Press, 1987).

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

8. L. J. Johnson, F. Jasman, R. J. Green, and M. S. Leeson, “Recent advances in underwater optical wireless communications,” Underwater Technol. 32, 167–175 (2014). [CrossRef]

9. S. Tang, Y. Dong, and X. Zhang, “Impulse response modeling for underwater wireless optical communication links,” IEEE Trans. Commun. , 62, 226–234 (2014). [CrossRef]

10. L. J. Johnson, R. J. Green, and M. S. Leeson, “Underwater optical wireless communications: depth-dependent beam refraction,” Appl. Opt. 53, 7273–7277 (2014). [CrossRef] [PubMed]

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

12. M. C. Gökçe and Y. Baykal, “Scintillation analysis of multiple-input single-output underwater optical links,” Appl. Opt. 55, 6130–6136 (2016). [CrossRef] [PubMed]

13. O. Korotkova, N. Farwell, and E. Shchepakina, “Light scintillation in oceanic turbulence,” Waves Random Complex Media 22, 260–266 (2012). [CrossRef]

14. F. Hanson and M. Lasher, “Effects of underwater turbulence on laser beam propagation and coupling into single-mode optical fiber,” Appl. Opt. 49, 3224–3230 (2010). [CrossRef] [PubMed]

15. W. Liu, Z. Xu, and L. Yang, “SIMO detection schemes for underwater optical wireless communication under turbulence,” Photon. Res. 3, 48–53 (2015). [CrossRef]

16. H. M. Oubei, E. Zedini, R. T. ElAfandy, A. Kammoun, T. K. Ng, M.-S. Alouini, and B. S. Ooi, “Efficient Weibull channel model for salinity induced turbulent underwater wireless optical communications,” in Opto-Electronics and Communications Conference (OECC) and Photonics Global Conference (PGC) (2017). [CrossRef]

17. R. Barrios and F. Dios, “Exponentiated Weibull distribution family under aperture averaging for Gaussian beam waves,” Opt. Express 2, 13055–13064 (2012). [CrossRef]

18. J. M. Garrido-Balsells, A. García-Zambrana, and A. Puerta-Notario, “Variable weight MPPM technique for rate-adaptive optical wireless communications,” IET Electron. Lett. 42, 43–44 (2006). [CrossRef]

19. V. V. Nikishov and V. I. Nikishov, “Spectrum of turbulent fluctuations of the sea-water refraction index,” Int. J. Fluid Mech. Res. 27, 82–98 (2000). [CrossRef]

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

21. A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, and A. Puerta-Notario, “A unifying statistical model for atmospheric optical scintillation,” in Numerical Simulations of Physical and Engineering Processes (In-Tech, 2011), pp. 181–206.

22. M. V. Jamali, A. Mirani, A. Parsay, B. Abolhassani, P. Nabavi, A. Chizari, P. Khorramshahi, S. Abdollahramezani, and J. A. Salehi, “Statistical studies of fading in underwater wireless optical channels in the presence of air bubble, temperature, and salinity random variations (long version),” arXiv:1801.07402 (2018)

23. M. P. Bernotas and C. Nelson, “Probability density function analysis for optical turbulence with applications to underwater communications systems,” in “SPIE Defense+ Security. International Society for Optics and Photonics” (2016), paper 98270D.

24. R. R. Parenti and R. J. Sasiela, “Distribution models for optical scintillation due to atmospheric turbulence,” MIT Lincoln Laboratory Technical Report TR-1108 (2005).

25. C. Gabriel, M. Khalighi, S. Bourennane, P. Leon, and V. Rigaud, “Monte-Carlo-based channel characterization for underwater optical communication systems,” J. Opt. Commun. Netw. 5, 1–12 (2013). [CrossRef]

26. C. Gussen, P. Diniz, M. Campos, W. Martins, F. Costa, and J. Gois, “A survey of underwater wireless communication technologies,” J. Commun. Information Systems , 31, 242–255 (2016). [CrossRef]

27. L. C. Andrews and R. L. Phillips, Laser Bean Propagation through Random Media (SPIE Press, 2005). [CrossRef]

28. Wolfram, http://functions.wolfram.com/

29. P. Concus, D. Cassatt, G. Jaehnig, and E. Melby, “Tables for the evaluation of $\int_{0}^{\infty} x^{β} e^{- x} f (x) d x$ by Gauss-Laguerre quadrature,” Am. Math. Soc. 17, 245–256 (1963).

30. A. Jurado-Navas, J. M. Garrido-Balsells, M. Castillo-Vázquez, and A. Puerta-Notario, “Closed-form expressions for the lower-bound performance of variable weight multiple pulse-position modulation optical links through turbulent atmospheric channels,” IET Commun. 6, 390–397 (2012). [CrossRef]

31. J.M. Kahn and J.R. Barry, “Wireless infrared communications,” Proc. IEEE 85, 265–298 (1997). [CrossRef]

32. H. Park and J. R. Barry, “Trellis-coded multiple-pulse-position modulation for wireless infrared communications,” IEEE Trans. Commun. 52, 643–651 (2004). [CrossRef]

33. J. M. Garrido-Balsells, A. Jurado-Navas, J. Francisco Paris, M. Castillo-Vázquez, and A. Puerta-Notario, “Closed-form BER analysis of variable weight MPPM coding under gamma-gamma scintillation for atmospheric optical communications,” Opt. Lett. 37, 719–721 (2012). [CrossRef]

34. R. Y. Rubinstein, Simulation and the Monte Carlo Method (Wiley, 1981). [CrossRef]

35. M. V. Jamali, P. Khorramshahi, and R. Ramírez, “Statistical distribution of intensity fluctuations for underwater wireless optical channels in the presence of air bubbles,” in Iran Workshop on Proceedings Communication and Information Theory (IWCIT) (2016).

Code rate	a	b	c

2/3	0.3870	2.7150	0.8990
6/12	0.6364	6.8958	0.8871
9/36	0.7246	42.4424	0.8600

Error probability analysis of OOK and variable weight MPPM coding schemes for underwater optical communication systems affected by salinity turbulence

Abstract

1. Introduction

2. Channel and system models

3. Average bit error rate (ABER) for uncoding OOK format

4. vw-MPPM

5. Results and discussion

6. Concluding remarks

Appendix: Derivation of ABER for exponentiated Weibull turbulence

References

Cited By

Figures (4)

Tables (1)

Equations (26)

OSA Continuum