## I Introduction

Underwater wireless optical communication (UWOC) is becoming a dominant solution for high-throughput and large-data underwater communications thanks to its tremendous advantages, such as higher bandwidth, lower time latency, and better security, compared to the traditional acoustic communication systems. However, the UWOC channel severe impairments including absorption, scattering, and turbulence-induced fading hamper on the signal propagation and limit the viable communication range to typically less than [1, 2]. Therefore, removing this impediment and enabling UWOC for longer link ranges demand, firstly, comprehensive studies on these degrading effects and accurate channel modeling, and then designing UWOC systems that are robust with respect to these impairments and can reliably communicate over longer distances.

In the past few years, intensive research activities have been carried out to design more reliable and smart UWOC systems that can alleviate one or more of the channel main impairments. For example, in [3, 4, 5] multiple-input multiple-output (MIMO) transmission, with respect to the Monte Carlo (MC)-based fading-free impulse response modeling proposed in [6, 7, 8, 9] and lognormal fading model for weak turbulence [10, 11], has been employed for both diffusive light emitting diode (LED)-based and collimated laser-based UWOC links to mitigate turbulence-induced fading. Moreover, relying on the distance-dependency of all of the channel disturbing effects, the beneficial application of multi-hop serial relaying in alleviating all of the channel impairments has been explored in [12, 13]; the results show remarkable increase of the viable communication range for both point-to-point and optical code division multiple access (OCDMA)-based UWOC systems.

It is well understood that accurate channel modeling plays a key role on the precise evaluation of the performance of UWOC systems. In this context, while absorption and scattering are well studied for different water types and can be modeled using MC numerical simulations like [6, 7, 8, 14], the literature still lacks a comprehensive statistical study over turbulence-induced fading in UWOC channels which is the main scope of this paper.

In contrast to the underwater acoustic and radio frequency (RF) communications where multipath is the main source for signal fading, fading in wireless optical communications occurs as a result of random variations in the refractive index of the conveying medium which is referred to as turbulence. Optical turbulence in UWOC channels is mainly due to the random variations of the water temperature and salinity [15, 16, 17] while in free-space optics (FSO) random variations of the atmosphere pressure and temperature are its main reasons [10, 18, 19]. In addition to temperature- and salinity-induced turbulence, air bubbles can also cause random fluctuations on the received optical signal through the UWOC channel [20]. This also has some physical interpretations, e.g., when a diver is sending the information bits using optical signals, the surrounding air bubbles produced by the diver may cause fluctuations on the optical signal. In [21], the random fluctuations of the received optical signal through the UWOC channel with air bubbles has been experimentally studied and the accuracy of some of the most known statistical distributions in predicting the intensity random fluctuations has been examined. Subsequently, in a recent work [22], the performance of UWOC systems in the presence of different air bubble populations has experimentally been evaluated.

As explained before, in a typical UWOC channel, temperature- and/or salinity-induced turbulence is the main cause of fading on the optical signal. However, the accurate statistical distributions for such turbulence-induced fading in UWOC channels are not yet well investigated. In this context, the authors in [23, 24, 25] have experimented statistical distributions of UWOC channels with temperature gradient and in the absence of salinity random variations, however in very weak underwater turbulence regions characterized by the scintillation index values much less than unity (). In this paper, we run various scenarios to produce air bubbles and also induce temperature and salinity random variations across the optical beam propagation. Our scenarios cover a wide range of scintillation index values from weak to strong turbulence. We then examine the accuracy of different statistical distributions in terms of their goodness of fit and comment on the applicability of these distributions in each range of the scintillation index. We further obtain the fading coherence time to confirm that the UWOC turbulent channels are usually under slow fading.

The remainder of the paper is organized as follows. In Section II, we briefly review the preliminaries pertaining the UWOC channel with emphasis on the turbulence-induced fading. In Section III, we define the goodness of fit metric, and explain how the channel coherence time can experimentally be calculated to ascertain the slow fading nature of UWOC turbulent channels. In Section IV, we describe the statistical distributions under consideration with insights to their application in general wireless optical channels. Section V explains the system model and experimental set-up, Section VI presents the comprehensive experimental results for various scenarios, and Section VII concludes the paper and highlights some relevant future research directions.

## Ii UWOC Channel Overview

The channel modeling plays a critical role in ascertaining the performance of UWOC systems and predicting the outcomes of different techniques. Comprehensive studies on the channel modeling have revealed that the propagation of light under water is affected by three major degrading effects, i.e., absorption, scattering, and turbulence [26, 15]. In fact, during the propagation of photons through water, they may interact with water molecules and particles. In this case, the energy of each photon may be lost thermally, that is named absorption process and is characterized by absorption coefficient , where is the wavelength. And the direction of each photon may be altered which is defined as scattering process and is determined by scattering coefficient . Then the total energy loss of non-scattered light is described by extinction coefficient [26].

Many invaluable researches have been carried out in the past few years concerning on these two fundamental effects of UWOC channels, and thorough channel modeling based on MC numerical method with respect to absorption and scattering effects have been reported in several recent work [7, 8, 6, 5]. Subsequently, some other recent research activities have focused on proposing closed-form expressions for the characterization of the fading-free impulse response (FFIR) of both point-to-point and MIMO UWOC links based on MC numerical simulations [6, 14]. Although the above-mentioned FFIR takes into account both absorption and scattering effects, accurate and complete characterization of the UWOC channel impulse response requires turbulence-induced fading consideration as well. A fading coefficient can be multiplied by the channel FFIR to include the slow fading nature of UWOC channels [16, 10]. Although the recent mathematical works on the performance evaluation of UWOC systems [27, 11, 13, 3]

, inspired by the behaviour of optical turbulence in atmosphere, have considered lognormal probability density function (PDF), the accurate characterization of fading coefficients’ PDF demands a more specific investigation which will be experimentally carried out in this paper.

In order to measure the fading strength, it is common in the literature to define the scintillation index of a propagating light wave as [28, 15]

(1) |

where

is the instantaneous intensity at a point with position vector

, is the link length, anddenotes the expected value of the random variable (RV)

. There are lots of theoretical research activities toward mathematical investigation of the UWOC channel turbulence and how the scintillation index varies with the water specific parameters of turbulence [15, 27, 11, 29]. However, non of these prior works have specifically focused on the characterization of the UWOC fading statistical distribution which is a necessary task from the communication engineering point of view; this important study shapes the main body of our paper. We should emphasize that each statistical distribution has a set of constant parameters which directly relate to the scintillation index value; therefore, as it will be clarified in Section IV, these parameters vary with the fading strength.## Iii Evaluation Metrics

### Iii-a Goodness of Fit

In order to test a distribution’s fitness and evaluate the accordance of different statistical distributions with the experimental data, we use goodness of fit metric, also known as measure,

(2) |

in which is the sum of the square errors of the statistical distribution under consideration, i.e., , where is the number of bins of the acquired data histogram, and are respectively the measured and predicted probability values for a given received intensity level corresponding to the th bin. And is the sum of the squares of distances between the measured points and their mean, i.e., , where . Clearly, as the value of the measure for a given distribution approaches its maximum (i.e., ), the distribution better fits the measured data.

### Iii-B Coherence Time

In order to verify the slow fading nature of UWOC channels and corroborate the theoretical achievements in [16], we experimentally measure the channel coherence time for different channel conditions. To do so, we measure the time in which the temporal covariance coefficient of irradiance, defined as Eq. (3) [16, 28], remains above a certain threshold, e.g., a threshold.

(3) |

where with . Furthermore, , and is defined as the covariance of irradiance for two points and at different time instants and [28].

## Iv The Probability Density Functions Under Consideration

In this section, we overview the statistical distributions considered in this paper. It is worth mentioning that each statistical distribution contains a set of parameters which can be obtained such that they simultaneously satisfy two essential criteria. The first criteria is known as the fading normalization equality which means that the fading coefficient is normalized to emphasize that it neither amplifies nor attenuates the average power, i.e., . To do so during our experiments, we normalize the received vector from each channel realization, corresponding to the received optical power from a specific turbulent UWOC channel at each scenario, to its mean to make sure that the mean of the new vector is equal to one. Additionally, as the second criteria, we run an optimization procedure to obtain the distribution parameters such that they result into the best fit to the experimental data in terms of the goodness of fit.

### Iv-a Lognormal Distribution

Lognormal distribution is mainly used in the literature to describe the fluctuations induced by weak atmospheric turbulence, characterized by . In this case, the channel fading coefficient has the PDF of

(4) |

where and

are respectively the mean and variance of the Gaussian-distributed fading log-amplitude factor defined as

[10]. The fading normalization equality for the lognormal distribution leads to [10]. Therefore, lognormal distribution is a function of a single parameter which is related to the scintillation index value as [3].### Iv-B Gamma Distribution

The Gamma distribution with shape parameter

and scale parameter is expressed as(5) |

where is the Gamma function. For the Gamma distribution we note that and . Therefore, the Gamma distribution parameters are simply related to the scintillation index value as .

### Iv-C K Distribution

This distribution is mainly used in the FSO literature for strong atmospheric turbulence in which . The PDF of K distribution is given by [10]

(6) |

in which is the th-order modified Bessel function of the second kind and is a positive parameter which relates to the scintillation index value as . It is worth noting that the PDF of K distribution expressed in Eq. (6) already satisfies the fading normalization criteria, .

### Iv-D Weibull Distribution

The Weibull and exponentiated Weibull distributions were used in [30] to excellently describe the atmospheric turbulence in a wide range of scintillation index values. In this paper, we evaluate the accordance of these two distributions as good candidates for predicting the fluctuations in the received optical signal through a turbulent UWOC channel. The Weibull PDF is defined as [24]

(7) |

For the Weibull distribution we find that and . Therefore, the fading normalization equality results into the relation between and . Moreover, the scintillation index value can be obtained from the value of the parameter as .

### Iv-E Exponentiated Weibull Distribution

The PDF of a RV described by the exponentiated Weibull distribution with parameters , , and is given by

(8) |

The

th moment of the exponentiated Weibull PDF can be calculated as

[31, 32](9) |

where is a series defined as

(10) |

As it has been explained in [24, 32], Eq. (10) very quickly converges and considering the first terms suffices to acceptably approximate the infinite series of . Based on Eq. (9), the first moment for satisfying the fading normalization criteria and the second moment for the calculation of the scintillation index can be obtained. The resulted scintillation index formula for the exponentiated Weibull distribution is expressed as

(11) |

### Iv-F Gamma-Gamma Distribution

Such a statistical model, which factorizes the irradiance as the product of two independent random processes each with a Gamma PDF, has been widely employed in the FSO literature to model a wide range of atmospheric turbulence. The PDF of Gamma-Gamma distribution is expressed as [10];

(12) |

in which and are parameters related to the effective atmospheric conditions. The PDF in Eq. (12) is normalized so that it satisfies the equality . Furthermore, the scintillation index for Gamma-Gamma distribution is given by [10].

### Iv-G Generalized Gamma Distribution

In order to simultaneously cover the features of some of the statistical distributions, in this paper, we also evaluate the accordance of the generalized Gamma distribution, defined as

(13) |

in predicting the underwater turbulence-induced fading. In fact, the generalized Gamma distribution is a general case of some of the important statistical distributions in describing the optical turbulence. Particularly, when it reduces to the Weibull distribution, for it is equivalent to the Gamma distribution, and if

, the generalized Gamma distribution becomes the exponential distribution

[33]. A simple mathematical manipulation for the derivation of the moments shows that for the generalized Gamma distribution and . In this case, the scintillation index can be expressed as(14) |

Similar to the previous statistical distributions, in order to obtain the distribution parameters, namely , , and , we use an optimization procedure to simultaneously satisfy the equality and obtain the best goodness of fit.

## V Experimental Setup

Fig. 1 shows our experimental setup for collecting the received intensity samples. As it is shown, our experimental setup is implemented in a black-colored water tank with dimensions . At the transmitter side, a green laser diode with maximum output power of is driven by a MOS transistor to ensure a constant optical irradiance. Furthermore, to collimate and expand the transmitter beam, we have designed a beam-collimator (BC) using a plano-concave lens of focal length followed by a plano-convex lens with . The diameter of both of the lenses of the BC is . On the other hand, we use an ultraviolate-visible photodetector at the receiver side to capture the optical irradiance. The photodetector converts the received optical power to the electrical current according to the relation , where is the photodetector’s quantum efficiency, is the electron charge, is the Planck’s constant, and is the light frequency. Therefore, the photodetector’s output current is proportional to the received optical power and hence is a good representative of the received irradiance. We further amplify the photo-detected current and then sample and monitor the amplified signal by an HP Infiniium oscilloscope. For each test, we have collected samples with the sampling rate of . Additionally, in order to have a good accommodation with the conventional receivers in wireless optics, communicating through turbulent media, the receiver is equipped with a lens of diameter for the sake of aperture averaging. Both the transmitter laser and the receiver photodetector are sealed by placing in transparent boxes.

Various scenarios and sets of experiments are performed in this paper. The first scenario deals with the characterization of intensity fluctuation due to the random presence of air bubbles through the propagation path. In this case, in order to generate random air bubbles, a tunable air blower with the maximum blowing capacity of is employed. The received intensity is sampled for both the absence and presence of the transmitter BC and also the receiver aperture averaging lens (AAL). The second scenario, contains a set of various experiments concentrating on the statistical characterization of the underwater turbulence-induced fading due to the random temperature variations. To produce such random variations on the water temperature, we employ two different mechanisms. In the first method, we use three tunable water droppers to create three independent flow of hot water, with the temperature , to the cold water of tank with the temperature . As the second mechanism for generating random temperature variations, we insert three circular heating elements in the propagation path and pass the laser beam through the heaters. Moreover, various mixtures of these two methods for generating temperature variations are also considered in the presence of random air bubbles for the sake of generality. In the third set of experiments, in order to investigate the accordance of different statistical distributions in predicting the turbulence-induced fading due to the water random salinity variations, similar to the second scenario, we use three independently tunable water droppers to insert three flow of extremely salty water into the water tank.

## Vi Experimental Results

In this section, we take into account all of the three different scenarios mentioned in the previous section to experimentally evaluate the validity of various statistical distributions in predicting turbulence-induced fading in UWOC channels under a wide range of channel conditions. For each scenario, we have adjusted the transmitter laser power to ensure that a considerable power reaches the receiver. We further numerically calculate the temporal covariance coefficient of the received irradiance for various channel conditions to determine the fading coherence time as an important parameter in the evaluation of the performance of UWOC systems.

In order to cover a wide range of optical turbulence, we first run an abundant number of experiments each by slightly changing the air bubble population and hot or salty water flow rates, depending on the various scenarios considered in this paper. Then we choose from these many experimental results such that at least contain one from the small values, one from the values around the averages, and another from the highest possible scintillation index values we can get by tuning the air bubble population or hot/salty water flow rates at the highest values. In what follows, we first report few experimental results, pertaining to the first scenario, performed over the UWOC channels without the transmitter BC and receiver AAL to better elucidate the possibility of two-lobe statistical distributions we proposed recently in [21]. Then, in order to be commensurate with the practical link configurations, we only focus on the cases where the receiver AAL is employed with/without the transmitter beam collimator.

### Vi-a Intensity Fluctuation due to the Random Presence of Air Bubbles

In this subsection, we present the detailed results for a comprehensive set of experiments performed over UWOC channels under the random presence of air bubbles. Moreover, the effects of beam-collimator (and expander) in the transmitter side and aperture averaging in the receiver side are also investigated. Throughout this subsection, we denote the UWOC channels under air bubble random variations with , if the UWOC link is without BC at the transmitter side and AAL at the receiver side. On the other hand, if the UWOC link possesses only the receiver AAL, the scenario is denoted by . And if both of them exist in the configuration, we represent the scenario as . For all of these scenarios we run lots of experiments to cover a wide range of scintillation index, from weak to strong fading. The temporal domain fluctuations for each of the experiments are shown in Fig. 2, while the corresponding histograms of the normalized received optical power as well as the fitted statistical distributions are illustrated in Fig. 3. Moreover, detailed simulation results for the goodness of fit (GoF) values of each of the discussed statistical distributions in Sec. IV and their corresponding constant coefficients are listed in Table I.

Channel condition | Log-normal | Gamma | K dist. | Weibull | Exp. Weibull | Gamma-Gamma | Generalized Gamma | |
---|---|---|---|---|---|---|---|---|

Fresh water | ||||||||

Fig. 2(a) illustrates the received signal through a fresh water link without any artificially-induced turbulence. As it can be seen, for a UWOC channel with low link range (like our water tank) the received signal is approximately constant over a large period of time, i.e., fading has a negligible effect on the performance of low-range UWOC systems. As a consequence, the channel impulse response for such scenarios can be thoroughly described by a deterministic FFIR. Moreover, the corresponding histogram of the acquired data in Fig. 3(a) and the simulation results in the first row of Table I show that for such a channel condition the normalized received optical power is mainly confined around the mean of “

” with a negligible standard deviation and hence lots of the discussed statistical distributions in Sec. IV give an excellent match to the data histogram

^{1}

^{1}1Note that the parameter in the K distribution is positive and hence, based on the relation , the evaluation of K distribution is only valid for the channels with . Moreover, for very small values of , the parameter in the exponentiated Weibull and the parameters and in the Gamma-Gamma distribution have large values hampering the analytical evaluation of exponentiated Weibull and Gamma-Gamma distributions for such regimes of the scintillation index..

According to Fig. 2(a), due to the low link range available by the laboratory water tank, the received signal fluctuations are typically negligible. In other words, optical turbulence manifests its effect at longer link ranges, i.e., for UWOC channels with typically link ranges [15]; that is why we used some procedures to artificially induce fluctuations on the propagation path. Figs. 2(b)-(e) illustrate the received signal through a bubbly fresh water link with different concentrations of air bubbles, i.e., different fading strengths, when neither the transmitter BC nor the receiver AAL is employed in the link configuration (a link). As it is observed, the presence of air bubbles within the channel causes the received signal to severely fluctuate. This is mainly due to the random arrangement of air bubbles through the propagation path causing the propagating photons to randomly scatter in different directions and leave their direct path. The corresponding histograms of the acquired experimental data and also the accordance of different statistical distributions are depicted in Figs. 3(b)-(e). From these figures and the detailed numerical results in the second-fifth rows of Table I, we can observe that while a bubbly UWOC link under a relatively strong fading condition, like and , can be well described by many of the well-known statistical distributions (approximately all of the seven statistical distributions considered in this paper), for moderate fading conditions, like , non of the statistical distributions under consideration can acceptably model the fading behavior. The major reason can intuitively be deduced from the temporal representation of the received optical signal in Figs. 2(b)-(e). Based on these figures, the presence of air bubbles causes the received intensity to mainly lie either in large or small values. This is mainly because in the current link configuration neither the transmitter BC nor the receiver AAL is used and hence the received intensity is very sensitive to the beam scattering caused by the random presence of air bubbles. Therefore, as we showed in our previous experimental research [21], in such circumstances the typical single-lobe distributions cannot appropriately fit the experimental data and generally a two-lobe statistical distribution, such as the mixed exponential-lognormal distribution we have proposed in [21], is required to predict the statistical behavior of UWOC fading in all regions of the scintillation index.

In order to make the link geometry commensurate with the practical scenarios and in the meantime reduce the above-discussed sensitivity of the received optical signal to the beam scattering, hereafter, for all of the considered scenarios, we assume that the receiver is equipped with an AAL. Figs. 2(f)-(i) show the received signal through a bubbly fresh water link with different concentrations of air bubbles when just the receiver AAL is employed, i.e., a link. From the corresponding histograms of the acquired experimental data depicted in Figs. 3(f)-(i), one can observe that the application of AAL at the receiver side not only effectively reduces the fading strength measured by the scintillation index but also makes the fluctuations on the received optical signals through the turbulent UWOC channels predictable by the well-known single-lobe statistical distributions. In particular, based on the detailed numerical results in the sixth-ninth rows of Table I, we can observe that relatively all the seven statistical distributions can acceptably model the fading behavior; however, the generalized Gamma and exponentiated Weibull distributions, respectively, always give the best accordance to the acquired experimental data in terms of their GoF.

Before experimentally exploring the transmitter BC effect on the UWOC fading behavior, it is worth to first briefly clarify the functionalities of the transmitter BC and receiver AAL from the theoretical point of view. In fact, while an optical receiver with a very small aperture will result in a random signal, increasing the receiver aperture lens to larger than that of the spatial scale of the irradiance fluctuations, significantly reduces the signal fluctuations compared to a point receiver through averaging the fluctuations over the aperture which is well known as aperture averaging effect in the literature [34]. On the other hand, the expanded beam by the transmitter BC can be thought as of a collection of spatially separated rays each experiencing somewhat independent fluctuations when propagating through the random UWOC channel. Systematically speaking, AAL and BC attempt to somehow realize the receiver and transmitter diversity, respectively. Therefore, these two components together somehow perform in a similar fashion to MIMO transmission which significantly reduces the optical turbulence through spatial diversity by providing so-called independent paths from the transmitter to the receiver. Accordingly, we expect that these two components significantly reduce the turbulence strength measured by the scintillation index through the basic fundamentals of spatial diversity in optical turbulent channels [3]. In addition, since BC and AAL significantly reduce the deep fade probability (or equivalently, considerably reduce the link sensitivity to beam wandering), the quantized measured samples will no longer only accumulate either at the large or small values; instead, they will mainly tend to take the values around the mean value (normalized by one).

We should emphasize that because of the negligibility of background noise in UWOC systems deployed in typical water depths [4], the application of AAL sounds to be useful in most of the cases since it not only mitigates the turbulence impairments but also collects lots of the scattered photons onto its focal plane where the photo-detector is placed. On the other hand, BC usage at the transmitter side sometimes comes with the cost of reduced power reached to the receiver. In fact, when we expand the cross-section of the sharp optical beam of the laser using the transmitter BC, we expect the number of interactions between the propagating photons and water suspended particles and molecules significantly increase; this results in a remarkably higher number of absorption and scattering events and increases the total loss on the received optical power. However, depending on the channel condition, including the water type and turbulence strength, the aforementioned cost may be negligible. For example, in the turbid harbor water links the multiple-scattering effect of the channel is too much such that, even for the relatively short link range of m, the sharp beam of transmitter laser will be spread over a wide spatial area at the receiver plane; hence, the received pattern in both temporal and spatial domains (including the attenuation, delay spread, and spatial dispersion) tends to be similar to that of LED-based diffusive links [5]. This implies that for such link conditions, using either sharp laser or diffusive LED sources results in a similar received pattern over a real UWOC channel with a typical length, which itself signifies that using the transmitter BC will negligibly change the channel attenuation. Similar inferences can be derived for any other channel conditions.

Now, in order to experimentally investigate the effect of beam-collimator (and expander) on the statistics of the received optical signal through turbulent UWOC channels, as described in Sec. V, we also run a comprehensive set of experiments when the link geometry possesses such an integral part. Figs. 2(j)-(o) show the received signal through a bubbly fresh water link with different concentrations of air bubbles when both the transmitter BC and the receiver AAL are employed, i.e., a link. The corresponding histograms of the acquired experimental data depicted in Figs. 3(j)-(o) demonstrate that employing the transmitter BC in conjunction with the receiver AAL significantly reduces the fading strength such that in the highest amount of bubble concentration the scintillation index is considerably less than one. On the other hand, such a link configuration also remarkably causes the received intensity samples to concentrate around the mean of the acquired data, i.e., a relatively apposite behavior to links is observable such that the acquired data samples, with a good probability, are no longer only either at large or small values. As a consequence, and based on the detailed numerical results in the tenth-fifteenth rows of Table I, the fading in links can appropriately be characterized using simple single-lobe statistical distributions; however, once again, the generalized Gamma and exponentiated Weibull distributions, respectively, yield much better accordance to the acquired experimental data in terms of their GoF.

### Vi-B Turbulence-Induced Fading due to the Temperature Random Variations

Channel condition | Log-normal | Gamma | K dist. | Weibull | Exp. Weibull | Gamma-Gamma | Generalized Gamma | |
---|---|---|---|---|---|---|---|---|

Comments

There are no comments yet.