Performance analysis of one- and two-way relays for underwater optical wireless communications

Anirban Bhowal; Rakhesh Singh Kshetrimayum

doi:10.1364/OSAC.1.001400

1. Introduction

Underwater communication refers to a type of wireless communication in which the message is transmitted underwater without any guided media by carriers like radio frequency (RF), acoustic, or optical waves. This type of communication suffers from severe scattering and absorption effects [1]. The salinity of the water and presence of other living organisms in the aquatic environment also plays an important role in affecting the performance of the underwater system. Underwater communication by acoustic and RF carriers is bandwidth limited. So to switch over to higher data rates, we can opt for underwater optical wireless communication (UOWC). UOWC has certain benefits such as low implementation costs, low latency, and high data speeds. UOWC is similar to terrestrial free space optical (FSO) communication [2]. But in the underwater environment, the emitted photons after collision with water molecules and other suspended particles undergo energy loss and change in direction for optical signals. The absorption coefficient $a (λ)$ and scattering coefficient $b (λ)$ are considered to take these effects into account [3].

The acoustic form of underwater communication has some disadvantages. The frequency of operation is very low leading to poor data rates. Also, delay time in acoustic links is large (typically in seconds). Acoustic transceivers are also expensive and not energy efficient. The RF method suffers from the problem of short link range. It also requires expensive energy inefficient transceivers and large antennas. Speed of light is much greater than acoustic waves. Hence one can opt for UOWC as a possible alternative. UOWC links are immune to link latency and also provide more security [4,5].

The channel model for UOWC is complex due to the nature of the underwater environment. Water flow, temperature, salinity, and turbidity of water influence the performance and lifetime of UOWC equipments. The devices also need to be energy efficient also due to a lack of power source in the aquatic environment. In the underwater environment, coloured dissolved organic material (CDOM) and chlorophyll are present, which are capable of absorbing lights. As a result, the turbidity of water increases and the propagation of light is affected. The concentration of CDOM also changes with ocean depth variations, thereby varying the corresponding light attenuation coefficients. Rapid changes in the refractive index of water occur due to varying temperature and pressure in the ocean currents, which is known as turbulence [6].

In UOWC, underwater wireless sensor nodes communicate with each other. The system consists of multiple distributed nodes such as seabed sensors, relay buoys, remotely operated underwater vehicles (ROV), and autonomous underwater vehicles (AUV). These nodes can perform sensing, signal processing and communicate with each other helping in data exchange in the aquatic environment. The sensors situated at the bottom of the ocean beds transmit data through optical or acoustic links to the ROVs and AUVs. These ROVs and AUVs in turn transfer the data to submarines and ships. The data processing centre situated above the aquatic surface exchange data with the submarines and ships through RF/FSO links. A system diagram is illustrated in Fig. 1.

Device-to-device (D2D) UOWCs can be of use in military applications, environmental monitoring, offshore exploration, and disaster precaution. For this, cooperative communication deploying the amplify and forward (AF) technique at the relay can be beneficial. Monte Carlo based numerical simulations [7,3] of the channel in UOWC has been performed in literature, which has shown that the fading free impulse response (FFIR) of the channel, with absorption and scattering effects taken into account, cannot be adopted like FSO channel links. The tremendous amount of multiple scattering of the propagating light causes temporal dispersion on the channel FFIR and consequently on the received optical signal. Turbulence induced fading must be considered while designing the channel. The channel FFIR is multiplied by a positive fading coefficient to characterize the fading induced channel turbulence. The turbulence effect of UOWC can be best described by log-normal channel model [8], whereas in FSO communication, log-normal and gamma-gamma models are used to describe the atmospheric turbulence. An incoherent UOWC-FSO system deploys the technique of intensity modulation of optical sources like an LED/laser at the transmitting side and direct detection by a photo-detector at the receiving side [9]. The modulation scheme used is binary phase shift keying (BPSK).

Attenuation phenomena in sea water can be best described by Beer-Lambert’s Law. But in this work, the blue/green region of the visible spectrum has been chosen, as the 450-550 nm spectral range has the least attenuation and scattering coefficient. In UOWC systems, receivers with large aperture are generally deployed due to lack of background noise under water, thereby leading to an effectively weak turbulence effect. The distance of communication between UOWC devices is typically within the range of 100 m thereby negating the effect of Pointing and misalignment errors. So we are considering only turbulence induced fading, which can be best described using log-normal channel model due to weak turbulence in sea water. Since the range of UOWC communication is very limited (restricted to 100 m [10]), we have to opt for one-way relay (OWR) or two-way relay (TWR) based communication to increase the effective link range. Usually deep underwater ocean explorations will need longer distance communications since the ocean depths are generally in kilometers. Hence, D2D relay based communication becomes a necessity. OWR will allow either transmission or receiving of a message in a particular instant of time, whereas TWR enables simultaneous receiving and transmission of messages.

In this paper, the performance of the UOWC cooperative system (unidirectional and bidirectional relay) has been analyzed for log-normal underwater fading channel for the first time. The contributions of this paper are as follows. The closed form expressions for outage probability lower bounds of OWR and TWR based UOWC are derived. The closed form expression for ASEP of OWR based UWOC is obtained by approximating log-normal underwater fading channel by means of a mixture of gamma distributions. The closed form expressions for outage probability and ASEP for OWR and TWR based UWOC are validated by Monte Carlo simulations. Such a study will be very useful for deployment of D2D underwater wireless communication systems.

This paper is organized as follows. The channel model is described in Section 2. Section 3 vividly explains the system model of TWR and OWR based UOWC. The end-to-end probability distribution function (PDF) and cumulative distributive function (CDF) for such systems are provided. The calculation of ASEP for different modulation schemes is described in Section 4. Analysis using a mixture of gamma distributions is provided in Section 5. Section 6 analyses the results, while Section 7 summarizes and concludes the paper.

2. Channel model

Let the random variable $h \geq 0$ be the signal envelope following log-normal distribution. Then the PDF of $h$ can be written as

f (h) = \frac{1}{h \sqrt{2 π σ^{2}}} exp (- \frac{{(l n (h) - μ)}^{2}}{2 σ^{2}})

where

μ

and

σ^{2}

are the mean and variance of the random variable

Y = l n (h)

. Let

γ = {(h_{k})}^{2}

where k

{S R, R D}

. SN denotes the source node, RN denotes the destination node, and DN denotes the destination node as depicted in Fig. 2. SR signifies the channel path from SN to RN whereas RD is the channel path from RN to DN. We will use the PDF of received SNR

γ

given in [11]:

f_{γ} (γ) = \frac{1}{\sqrt{32 π} γ σ} exp (\frac{- {(l n (\frac{γ}{\bar{γ}}) + 8 σ^{2})}^{2}}{32 σ^{2}})

where

\bar{γ}

and

σ^{2}

represent the average SNR and log-normal fading variance of the UOWC links. The CDF can be expressed as [11]:

F_{γ} (γ) = 1 - \frac{1}{2} e r f c (\frac{l n (\frac{γ}{\bar{γ}}) + 8 σ^{2}}{\sqrt{32} σ})

where erfc(.) denotes the complementary error function given as

e r f c (x) = \frac{2}{\sqrt{π}} \int_{x}^{\infty} e^{- t^{2}} d t

.

The scintillation index denoted by $σ_{I}^{2}$ is used to describe the fluctuations of refractive index for turbulent seawater. Its value indicates the nature of turbulence. For example $σ_{I} >$ 1 indicates strong turbulence while $0 < σ_{I} < 1$ denotes weak turbulence. In weak turbulence environments we assume that $σ^{2}$ =- $μ$ and $σ_{I}^{2} = e^{4 σ^{2}} - 1$ [12]. It may be noted as $σ^{2}$ increases, $σ_{I}^{2}$ also increases. The parameter $σ^{2}$ has been calculatedusing other parameters like wavelength, link distance, and refractive index constant depending on the type of sea water and turbulence. The water type can be pure sea water, clear ocean water, coastal ocean water, etc., which is denoted by the term $ω_{0}$ . We have considered coastal ocean water for our analysis, which will have absorption and scattering effects due to high concentrations of plankton, detritus, and minerals. The parameters are listed in Table 1.

Table 1. Parameter values

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3. System model

The model which is used for UOWC can be explained in Fig. 1. The ROVs and sensors need to interact through optical wireless links, leading to the necessity of D2D communication. The simplest form of the three-node cooperation model, comprising SN, RN, and DN nodes, has been considered, as shown in Fig. 2. For such dual-hop relay systems, there is no direct link between SN and DN. One can extend this analysis to a multi-hop relay system [13] for further extending the link range. For transmitting purposes, LEDs/lasers are used, and for the receiving side, a photodetector of responsivity $R_{e}$ is used.

3.1. Outage probability for one-way relay system (UOWC_OWR)

Figure 1 Practical system model for UOWC (dashed line represent bidirectional optical links).

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Figure 2 System model for OWR based UOWC.

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The received signal $r_{S R}$ at RN at the end of the first time slot can be written as

r_{S R} = \sqrt{E R_{e}} h_{S R} x + n_{S R}

where x denotes the transmitted signal and

n_{S R}

is the zero mean complex Gaussian random variable with variance

N_{0} / 2

. Time slots are the smallest division of a communication channel that is assigned to particular users in a communication system. Sothe nodes can transmit information in a particular time slot only. The total energy required for transmission in the two time slots by SN and RN is 2E. The received signal

r_{R D}

at DN at the end of second time slot can be written as:

r_{R D} = \sqrt{c E R_{e}} h_{S R} h_{R D} x + n_{R D}

where

n_{R D}

is the zero mean complex Gaussian random variable with variance

N_{0} / 2

. For AF relaying the variable gain c can be defined as [14,15]

c = \frac{R_{e} E / N_{0}}{1 + R_{e} | h_{S R} |^{2} E / N_{0} + R_{e} | h_{R D} |^{2} E / N_{0}}

The signal to noise ratio (SNR) obtained as output at DN for OWR AF based systems, can be written as

γ_{S R D} = \frac{γ_{S R} γ_{R D}}{1 + γ_{S R} + γ_{R D}}

where,

γ_{S R} = \frac{R_{e} | h_{S R} |^{2} E}{N_{0}} γ_{R D} = \frac{R_{e} | h_{R D} |^{2} E}{N_{0}}

Since it is difficult to obtain $γ_{S R D}$ , the high SNR case is considered and approximated as

γ_{S R D} = \frac{γ_{S R} γ_{R D}}{1 + γ_{S R} + γ_{R D}} ≅ \frac{1}{2} \frac{2}{\frac{1}{γ_{S R}} + \frac{1}{γ_{R D}}} = γ_{a p p r o x}

An upper bound can be obtained by the inequality

γ_{a p p r o x} < γ_{u p} = m i n (γ_{S R}, γ_{R D})

A tight lower bound on CDF of $γ_{S R D}$ is given by a closed form expression [16]

F_{l o w e r} (γ) = P (γ_{S R} < γ) + P (γ_{R D} < γ) - P (γ_{S R} < γ) P (γ_{R D} < γ)

The outage probability can be defined as: $P_{o u t} (γ_{t h}) = P_{r} (m i n (γ_{S R}, γ_{R D}) < γ_{t h})$

= 1 - \frac{1}{4} e r f c (\frac{l n (\frac{2^{R} - 1}{R_{e} {\bar{γ}}_{S R}}) + 8 σ_{S R}^{2}}{\sqrt{32} σ_{S R}}) e r f c (\frac{l n (\frac{2^{R} - 1}{R_{e} {\bar{γ}}_{R D}}) + 8 σ_{R D}^{2}}{\sqrt{32} σ_{R D}})

where

γ_{t h} = 2^{R} - 1

,

{\bar{γ}}_{S R} = E / N_{0}

,

{\bar{γ}}_{R D} = E / N_{0}

(assuming that SNR at both SN and RN nodes are equal) and R is the target data rate in bps/Hz. The CDF of lower bound can be written as:

F_{l o w e r} (γ) = 1 - \frac{1}{4} e r f c (\frac{l n (\frac{γ}{R_{e} {\bar{γ}}_{S R}}) + 8 σ_{S R}^{2}}{\sqrt{32} σ_{S R}}) e r f c (\frac{l n (\frac{γ}{R_{e} {\bar{γ}}_{R D}}) + 8 σ_{R D}^{2}}{\sqrt{32} σ_{R D}})

The corresponding PDF can be obtained by differentiating (13) and the final expression can be written as

\begin{array}{l} f_{l o w e r} (γ) = \frac{1}{8 \sqrt{2 π} γ} [\frac{1}{σ_{S R}} exp (\frac{- {(l n (\frac{γ}{R_{e} {\bar{γ}}_{S R}}) + 8 σ_{S R}^{2})}^{2}}{32 σ_{S R}^{2}}) e r f c (\frac{(l n (\frac{γ}{R_{e} {\bar{γ}}_{R D}}) + 8 σ_{R D}^{2})}{\sqrt{32} σ_{R D}}) \\ + \frac{1}{σ_{R D}} exp (\frac{- {(l n (\frac{γ}{R_{e} {\bar{γ}}_{R D}}) + 8 σ_{R D}^{2})}^{2}}{32 σ_{R D}^{2}}) e r f c (\frac{(l n (\frac{γ}{R_{e} {\bar{γ}}_{S R}}) + 8 σ_{S R}^{2})}{\sqrt{32} σ_{S R}})] \end{array}

3.2. Outage probability for two way relay system (UOWC_TWR)

Figure 3 System model for UOWC_TWR.

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For the TWR system in Fig. 3, both the nodes (SN and DN) send information to the relay. All the nodes work in full duplex mode; i.e., they can send and receive information in the same time slot. The SNR at DN can be expressed as [17]:

γ_{S R D} = \frac{γ_{S R} γ_{R D}}{1 + γ_{S R} + 2 γ_{R D}}

where the terms have been explained earlier. For the high SNR case, it can be approximated as

γ_{S R D} = \frac{γ_{S R} γ_{R D}}{1 + γ_{S R} + 2 γ_{R D}} ≅ \frac{1}{2} \frac{2}{\frac{2}{γ_{S R}} + \frac{1}{γ_{R D}}} = γ_{a p p r o x}

A tight lower bound on CDF of $γ_{S R D}$ is given by a closed form expression as in the earlier case:

F_{l o w e r} (γ) = P {\frac{1}{2} m i n (γ_{S R}, 2 γ_{R D}) < γ}

The outage probability can be defined as

\begin{matrix} P_{o u t} (γ_{t h}) = P {\frac{1}{2} m i n (γ_{S R}, 2 γ_{R D}) < γ_{t h}} \\ = 1 - \frac{1}{4} e r f c (\frac{l n (\frac{2 * 2^{R} - 1}{R_{e} {\bar{γ}}_{S R}}) + 8 σ_{S R}^{2}}{\sqrt{32} σ_{S R}}) e r f c (\frac{l n (\frac{2^{R} - 1}{R_{e} {\bar{γ}}_{R D}}) + 8 σ_{R D}^{2}}{\sqrt{32} σ_{R D}}) \end{matrix}

The CDF of lower the bound for UOWC_TWR system can be written as

\begin{matrix} F_{l o w e r} (γ) = 1 - \frac{1}{4} e r f c (\frac{l n (\frac{2 * γ}{R_{e} {\bar{γ}}_{S R}}) + 8 σ_{S R}^{2}}{\sqrt{32} σ_{S R}}) e r f c (\frac{l n (\frac{γ}{R_{e} {\bar{γ}}_{R D}}) + 8 σ_{R D}^{2}}{\sqrt{32} σ_{R D}}) \end{matrix}

Similarly the PDF of the lower bound for this system can be expressed as

\begin{array}{l} f_{l o w e r} (γ) = \frac{1}{8 \sqrt{2 π} γ} [\frac{1}{σ_{S R}} exp (\frac{- {(l n (\frac{2 γ}{R_{e} {\bar{γ}}_{S R}}) + 8 σ_{S R}^{2})}^{2}}{32 σ_{S R}^{2}}) e r f c (\frac{(l n (\frac{γ}{R_{e} {\bar{γ}}_{R D}}) + 8 σ_{R D}^{2})}{\sqrt{32} σ_{R D}}) \\ + \frac{1}{σ_{R D}} exp (\frac{- {(l n (\frac{γ}{R_{e} {\bar{γ}}_{R D}}) + 8 σ_{R D}^{2})}^{2}}{32 σ_{R D}^{2}}) e r f c (\frac{(l n (\frac{2 γ}{R_{e} {\bar{γ}}_{S R}}) + 8 σ_{S R}^{2})}{\sqrt{32} σ_{S R}})] \end{array}

4. ASEP calculation

The average ASEP for q-ary modulation scheme over the entire fading channel can be written as [18]

P_{e} ≅ \int_{0}^{\infty} a Q (\sqrt{e γ}) f_{l o w e r} (γ) d γ

The closed form expression cannot be evaluated. So the integral is calculated using software like Mathematica. In this paper, the binary phase shift keying (BPSK) modulation scheme has been used. The values of $a$ and $e$ depend on the constellation scheme and have been taken from Eq. (7) of [19]. The ASEP for UOWC_OWR has been calculated using Eq. (21) where the value of $f_{l o w e r} (γ)$ has been obtained using Eq. (14). Similarly ASEP for UOWC_TWR systems can be calculated using Eq. (21) and Eq. (20).

It can be observed from the ASEP expressions that log-normal distribution poses a lot of problems for analyzing UOWC system performance, as the distribution contains intractable integrals. Therefore closed form expressions for ASEP cannot be obtained, and thus log-normal distribution can be approximated by a mixture of gamma (MG) distribution [20], which offers ease of calculation.

5. PDF, CDF, and error analysis using MG distributions

The PDF of the MG distribution can be written as [21]

f_{M G} (x) = \sum_{i = 1}^{n} P_{i} \frac{x^{α_{i} - 1} β_{i}^{α_{i}}}{T (α_{i})} e^{- x β_{i}} Ω

where the number of mixing coefficients is denoted by n, T denotes the gamma function,

P_{i}

is the mixing coefficient of the

i^{t h}

gamma distribution having properties like

P_{i} > 0

and

\sum_{i = 1}^{n} P_{i} = 1

. The shape and scale parameters of the

i^{t h}

gamma distribution for

x > 0

are

α_{i}

and

β_{i}

, respectively. The CDF of the MG distribution can be obtained by integrating the PDF and can be written as

F_{M G} (x) = \sum_{i = 1}^{n} P_{i} (1 - \frac{1}{T (α_{i})} G_{1, 2}^{2, 0} [\frac{1}{0, α_{i}} | β_{i} x])

where

G_{p, q}^{m, n} (: | z)

is the Meijer G function [22,23]. The outage probability for UOWC_OWR systems can be calculated as

P_{o u t} = \sum_{i = 1}^{n} P_{i} (1 - \frac{1}{T (α_{i})} G_{1, 2}^{2, 0} [\frac{1}{0, α_{i}} | \frac{β_{i} γ_{t h}}{R_{e} \bar{γ}}])

where

γ_{t h}

=

2^{R_{d}} - 1

for UOWC_OWR systems,

R_{d}

is the data rate.

Now the summation terms are treated separately and error analysis is done for a particular term as the same analysis holds true for all other terms. Finally all the error terms are added up to get the final error term. For AF based UOWC systems, end-to-end CDF of the entire system is computed and then end-to-end PDF is computed by differentiating the CDF. For ASEP calculation of UOWC_OWR, Eq. (21) is used, the Q function is expressed in terms of the Meijer G function, and after taking assumptions like ${\bar{γ}}_{S R}$ = ${\bar{γ}}_{R D}$ = $\bar{γ}$ , we can write overall ASEP as

\begin{array}{l} P_{t o t a l} ≅ {\sum^{​}}^{}^{}_{n}^{i = 1} P_{i} (1 - P_{i}) \frac{1}{T (α_{i}) \sqrt{π}} G_{2, 2}^{2, 1} (\frac{1 - α_{i}, 1}{0, \frac{1}{2}} | \frac{\bar{γ} R_{e}}{β_{i}}) \\ + {\sum^{​}}^{}^{}_{n}^{i = 1} \frac{P_{i}^{2}}{{(T (α_{i}))}^{2} \sqrt{π}} G_{1, 0 : 1, 2 : 1, 2}^{0, 1 : 2, 0 : 2, 0} (\frac{1 - α_{i}}{-} | \frac{1}{0, α_{i}} | \frac{1}{0, \frac{1}{2}} | 1, \frac{\bar{γ} R_{e}}{β_{i}}) \end{array}

The derivation of ASEP is given in Appendix A, while the asymptotic analysis for outage probability is shown in Appendix B. Thus it can be seen that the ASEP and outage probability expressions are now available in closed form. Asymptotic analysis can also be performed for the Meijer G function in an ASEP expression. The Meijer G function has to be expressed in terms of basic mathematical functions. For this, the argument in the Meijer G function is first inverted using Eq. (6.2.2) in [24]. Then the Meijer G function at very high values of its argument can be written as described in Eq. (41) in [25].

6. Results

We have considered the average SNRs and the fading coefficients of both the SN-RN and RN-DN links to be equal. For outage probability calculation, the threshold data rate (R) has been assumed as 3 bps/Hz. Link range (L) is the distance between SN and RN and between RN and DN. RN has been assumed to be equidistant from SN and DN.A self-explanatory block diagram of Monte Carlo simulation has also been depicted in Fig. 4 for the benefit of readers. It can be seen from Fig. 5 that outage probability performance of the system improves with a decrease in $σ^{2}$ value for which the link distance and turbulence is less. Note that the outage probability lower bounds of OWR and TWR based UOWC are calculated using Eq. (12) and (18) respectively. It has been reported in [26] that the $σ^{2}$ and $σ_{I}^{2}$ increases with the link distance. So smaller value of $σ^{2}$ may be interpreted for shorter link distance. $σ^{2}$ has been varied as 0.04, 0.09, and 0.16 for both one way and two way relay. TWR based UOWC suffers from self-interference effect as the nodes are transmitting and receiving simultaneously. So it shows more outage than its counterpart. The analytical results have been validated by means of Monte Carlo simulations as well. For example, to achieve an outage probability value of 0.1, the OWR based UOWC system requires 25 dB of SNR while TWR based UOWC requires 27 dB of SNR. Thus there is a SNR gain of 2dB for one way relaying method. This has been clearly illustrated by means of Table 2 for $σ^{2} = 0.09$ where the SNRs required to achieve different outage probability values have been depicted.

Figure 4 Block diagram of Monte Carlo simulation model.

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Figure 5 Outage probability for amplify and forward relay based UOWC.

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Table 2. Comparison of lower bounds on outage probability for $σ^{2} = 0.09$

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MG distribution is the best fit for log-normal distribution as shown by simulation results in Fig. 6. It can be seen from the PDF curves that higher components of MG distributions closely match those of log-normal curve but as components increase, computational complexity also grows. So there is a trade-off between complexity and error. 5-MG distribution shows an almost perfect match with log-normal distribution and hence has been considered for further system analysis. The parameters chosen obtained using the expectation maximization algorithm [20] are $α_{i} = 4.9987, 6.99, 8.996, 0.9938, 6.9977$ , $β_{i} = 3.884, 7.351, 9.716, 1.166, 13.075$ and $P_{i} = 0.3021, 0.107, 0.1720, 0.1061, 0.3119$ .

Figure 6 Comparison between log-normal and different MG distributions.

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In Fig. 7, the outage probability for an OWR based UOWC system has been verified using 5-MG distributions. The analytical results of outage probability for an OWR based UWOC are calculated using Eq. (12). $σ^{2}$ =0.09 and $s i g m a^{2}$ =0.16 have been considered for this case. The 5-MG analytical results (obtained from Eq. (24)) have also been verified by means of Monte Carlo simulations. The results obtained are in close agreement. The results have been justifiedusing asymptotic analysis as well, obtained using Eq. (34). In the case of higher SNR values, the aysmptotic results gradually merge with the analytical results, thereby justifying our analysis.

Figure 7 Outage probability for OWR based UOWC using 5-MG distributions for different values of $σ$ (solid line represents analytical results, dashed line represents asymptotic results, and * represents simulation results).

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The error performance of the system for the BPSK modulation scheme can be observed from Fig. 8. The analytical results for log-normal distribution are obtained using Eq. (21). As mentioned above, no closed form expression is available, so the integral is numerically evaluated using Mathematica. The closed form expressions are obtained for SER (using Eq. (25)) by approximating log-normal distribution by MG distributions. It is evident that lesser values of $σ^{2}$ give better performance. In this case, $σ^{2}$ has been varied as 0.09 and 0.16. ASEP results obtained from 5-MG distributions closely match with the results obtained from log-normal distribution.

Thus the outage probabilities of OWR and TWR AF based UOWC systems have been compared for different parameters. AF relaying with variable gain has been used to evaluate the outage probabilities for log-normal fading channels. But since the ASEP expressions are intractable in nature, closed form expressions have been obtained using 5-MG distribution approximation. To verify that our approximation is correct, log-normal distribution has initially been verified by 5-MG distribution approximation for different values. Then the outage probability and ASEP of OWR AF based UOWC systems have been calculated using 5-MG distribution, and it can be observed that the results obtained using 5-MG distribution are in close agreement to the the results obtained using log-normal distribution, thereby validating our analysis. Our novelty lies in the fact that closed form expressions for ASEP have been obtained for OWR based UOWC systems using 5-MG distribution to approximate log-normal channels.

Figure 8 ASEP of OWR based UOWC (Solid line represents analytical results using log-normal distribution, dashed line represents analytical results using 5-MG distribution, and * represents simulation results using log-normal distribution).

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7. Conclusion

In this paper, the mplify and forward technique has been used to improve the performance of OWR and TWR UOWC systems. System performance has been analyzed in terms of outage probability and ASEP and has been verified by Monte Carlo simulations. The simulation and analytical results are in close agreement with each other, thereby validating our analysis. The closed form expression for outage probability lower bounds of OWR and TWR based UOWC are provided. The closed form expressions for ASEP have been obtained for the first time for D2D communication systems using MG distribution. In the future, we will extend this analysis for multi-hop D2D relays for UWOC systems.

Appendix A ASEP analysis using MG distribution

The end-to-end PDF of the UOWC_OWR system can be defined as

\begin{array}{l} f_{E 2 E} (γ) = f (γ_{S R}) + f (γ_{R D}) - f (γ_{S R}) F (γ_{R D}) - f (γ_{R D}) F (γ_{S R}) \\ = f (γ_{S R}) (1 - F (γ_{R D})) + f (γ_{R D}) (1 - F (γ_{S R})) \end{array}

For $γ_{S R}$ = $γ_{R D}$ = $γ$ ,

\begin{array}{l} f_{E 2 E} (γ) = 2 f (γ) (1 - F (γ)) \\ = 2 f (γ) {\sum^{}}_{n}^{i = 1} [(1 - P_{i}) + \frac{P_{i}}{T [α]} G_{1, 2}^{2, 0} (\frac{1}{0, α} | \frac{γ β_{i}}{\bar{γ} R_{e}})] \end{array}

where the values of $f (γ)$ and $F (γ)$ are taken from Eq. (22) and Eq. (23). Now the overall ASEP for BPSK modulation can be written as

P_{e} = \int_{0}^{\infty} f_{E 2 E} (γ) Q (\sqrt{2 γ}) d γ

Now,

P_{e} = f_{1} + f_{2}

where

f_{1} = \int_{0}^{\infty} 2 f (γ) \sum_{i = 1}^{n} (1 - P_{i}) Q (\sqrt{2 γ}) d γ

and

f_{2} = \int_{0}^{\infty} 2 f (γ) \sum_{i = 1}^{n} \frac{P_{i}}{T (α_{i})} G_{1, 2}^{2, 0} (\frac{1}{0, α} | \frac{γ β_{i}}{\bar{γ} R_{e}}) Q (\sqrt{2 γ}) d γ

. Writing the Q function in terms of the Meijer G function,

Q (\sqrt{2 x}) = \frac{1}{2 \sqrt{π}} G_{1, 2}^{2, 0} (\frac{1}{0, 1 / 2} | x)

So we can write

f_{1}

as

\begin{array}{l} f_{1} = \int_{0}^{\infty} \sum_{i = 1}^{n} \frac{P_{i} (1 - P_{i}) γ^{α_{i} - 1} β_{i}^{α_{i}}}{T (α_{i})} e^{- β_{i} γ} \frac{1}{\sqrt{π}} G_{1, 2}^{2, 0} (\frac{1}{0, 1 / 2} | γ) d γ \\ = \sum_{i = 1}^{n} P_{i} (1 - P_{i}) \int_{0}^{\infty} \frac{β_{i}^{α_{i}}}{T (α_{i}) \sqrt{π}} γ^{α_{i} - 1} G_{0, 1}^{1, 0} (\frac{-}{0} | \frac{β_{i} γ}{\bar{γ} R_{e}}) G_{1, 2}^{2, 0} (\frac{1}{0, 1 / 2} | γ) d y \\ = \sum_{i = 1}^{n} \frac{P_{i} (1 - P_{i})}{T (α_{i}) \sqrt{π}} G_{2, 2}^{2, 1} (\frac{1 - α_{i}, 1}{0, 1 / 2} | \frac{\bar{γ} R_{e}}{β_{i}}) \end{array}

\begin{array}{l} f_{2} = \int_{0}^{\infty} \sum_{i = 1}^{n} \frac{P_{i} γ^{α_{i} - 1} β_{i}^{α_{i}}}{T (α_{i}) T (α_{i})} e^{- β_{i} γ} \frac{1}{\sqrt{π}} P_{i} G_{1, 2}^{2, 0} (\frac{1}{0, α_{i}} | \frac{β_{i} γ}{\bar{γ} R_{e}}) G_{1, 2}^{2, 0} (\frac{1}{0, 1 / 2} | γ) d γ \\ = \int_{0}^{\infty} \sum_{i = 1}^{n} \frac{P_{i} γ^{α_{i} - 1} β_{i}^{α_{i}}}{T (α_{i}) T (α_{i})} G_{0, 1}^{1, 0} (\frac{-}{0} | \frac{β_{i} γ}{\bar{γ} R_{e}}) \frac{1}{\sqrt{π}} P_{i} G_{1, 2}^{2, 0} (\frac{1}{0, α_{i}} | \frac{β_{i} γ}{\bar{γ} R_{e}}) G_{1, 2}^{2, 0} (\frac{1}{0, 1 / 2} | γ) d γ \\ = \sum_{i = 1}^{n} \frac{P_{i}^{2}}{{(T (α_{i}))}^{2} \sqrt{π}} G_{1, 0 : 1, 2 : 1, 2}^{0, 1 : 2, 0 : 2, 0} (\frac{1 - α_{i}}{-} | \frac{1}{0, α_{i}} | \frac{1}{0, \frac{1}{2}} | 1, \frac{\bar{γ} R_{e}}{β_{i}}) \end{array}

By combining the terms

f_{1}

and

f_{2}

we can write the overall ASEP as expressed in Eq. (25).

Appendix B Asymptotic analysis for outage probability

For asymptotic analysis of outage probability, we can write outage probability (from Eq. (24)) as

P_{o u t} = \sum_{i = 1}^{n} P_{i} (1 - \frac{1}{T (α_{i})} G_{1, 2}^{2, 0} [\frac{1}{0, α_{i}} | \frac{β_{i} γ_{t h}}{R_{e} \bar{γ}}])

The argument in the Meijer G function is first inverted using Eq. (6.2.2) in [24] and the expression is written as:

P_{o u t} = \sum_{i = 1}^{n} P_{i} (1 - \frac{1}{T (α_{i})} G_{2, 1}^{0, 2} [\frac{1, 1 - α_{i}}{0} | \frac{R_{e} \bar{γ}}{β_{i} γ_{t h}}])

Let $z = \frac{R_{e} \bar{γ}}{β_{i} γ_{t h}}$ . Then the Meijer G function at very high values of its argument can be written as described in Eq. (41) in [25]. So the overall asymptotic expression is written as

P_{o u t} = \sum_{i = 1}^{n} P_{i} (1 - \frac{1}{T (α_{i})} \sum_{k = 1}^{2} z^{a_{k} - 1} \frac{\prod_{l = 1, l \neq k}^{2} T (a_{k} - a_{l})}{\prod_{l = 1}^{1} T (a_{k} - b_{l})})

where

a = [1, 1 - α_{i}]

,

b = [0]

.

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Water type	Link range L (m)	$σ$	Scintillation index $σ_{I}^{2}$
Coastal ( $ω_{0}$ =0.55)	20	0.2	0.43
	22	0.3	0.52
	25	0.4	0.7

Water type	Link range L (m)	$σ$	Scintillation index $σ_{I}^{2}$
Coastal ( $ω_{0}$ =0.55)	20	0.2	0.43
	22	0.3	0.52
	25	0.4	0.7

Performance analysis of one- and two-way relays for underwater optical wireless communications

Abstract

1. Introduction

2. Channel model

3. System model

3.1. Outage probability for one-way relay system (UOWC_OWR)

3.2. Outage probability for two way relay system (UOWC_TWR)

4. ASEP calculation

5. PDF, CDF, and error analysis using MG distributions

6. Results

7. Conclusion

Appendix A ASEP analysis using MG distribution

Appendix B Asymptotic analysis for outage probability

References

Cited By

Figures (8)

Tables (2)

Equations (34)

OSA Continuum

$P_{o u t}$	SNR for various techniques (in dB)
$P_{o u t}$	UOWC_OWR	UOWC_TWR
0.5	19	21
0.1	25	27
0.05	27	29

$P_{o u t}$	SNR for various techniques (in dB)
$P_{o u t}$	UOWC_OWR	UOWC_TWR
0.5	19	21
0.1	25	27
0.05	27	29