## Abstract

Point-to-point underwater optical wireless communication (UOWC) links are mainly impaired by scattering due to impurities and turbidity in the open water, resulting in a significant inter-symbol interference (ISI) that limits seriously both channel capacity and the maximum practical information rate. This paper conducts, for the first time, the channel capacity analysis of UOWC systems in the presence of ISI and salinity-induced oceanic turbulence when the undersea optical channel is accurately modeled by linear discrete-time filtering of the input symbols. In this way, novel upper and lower bounds on channel capacity and mutual information are developed for non-uniform on-off keying (OOK) modulation when different constraints are imposed on the channel input. The results show that the capacity-achieving distribution, which is computed through numerical optimization, is discrete and depends on the optical signal-to-noise-ratio (SNR). Moreover, a non-uniform input distribution significantly improves the channel capacity of such systems affected by ISI and oceanic turbulence, especially at low optical SNR. Monte Carlo techniques are employed to test the developed bounds for different undersea optical channels with one, two and three casual ISI coefficients.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Point-to-point underwater optical wireless communication (UOWC) has become consolidated as a promising candidate to provide long-distance and high-speed links for a wide variety of scientific and industrial purposes related to ocean observation that is essential for the Green Deal, as well as for transmitting large amounts of data through seawater with high security [1,2]. However, the dispersive nature of the UOWC channel leads to an undesirable effect of inter-symbol interference (ISI) due to impurities in the seawater, not only limiting the achievable transmission rate severely, but also reducing link reliability [3]. To this impairment we may add the fact that small variations in salinity and temperature can lead to an optical signal fading called oceanic turbulence, degrading the overall performance of such systems [1,2]. Therefore, finding the capacity and the maximum practical information rate that can be used over the undersea optical channel with ISI in different practical scenarios such as clear ocean and coastal waters is a great challenge from an information-theoretical viewpoint.

The performance of UOWC systems has been extensively investigated for the past ten years mainly in terms of the bit error rate (BER) and outage probability by including different degrading factors such as absorption and scattering, temperature- and salinity-induced oceanic turbulence following different statistical distributions such as log-Normal, Weibull and exponential-generalized Gamma to name a few, as well as angular pointing errors due to water flows and random sea surface slopes [4–10] (and references therein). These reports demonstrated that an accurate channel model is crucial to permit a complete, realistic system design. Simultaneously, numerous experiments have been conducted to measure the performance of UOWC systems under laboratory conditions such as in-pool testing that further prove their feasibility to provide long-distance and high-speed undersea links [11–13]. In [11], spread spectrum techniques are used to achieve a 42 m link when employing a laser diode (LD) in the blue-green band. A data rate of 5 Gbps was reported at a distance of 50 m using a discrete multi-tone transmission [12]. By employing on-off keying (OOK) modulation, a data rate of 2.7 Gbps was reported at a distance of 34.5 m [13]. Both BER and outage performance have been analyzed to some extent, however channel capacity analysis has not received enough attention yet and there is not much work done in this regard [14–19]. Moreover, these analysis are based on the Shannon’s information theory and, hence, they are not simple to approach in practice. In [14], the average channel capacity for a buoy-based multiple-input/multiple-output UOWC system is analyzed in the presence of log-Normal oceanic turbulence. In [15], a pointing error model inherited from terrestrial free-space optical (FSO) communication systems is used to investigate the capacity over log-Normal oceanic turbulence channels. Similar capacity results are obtained in [16] under strong oceanic turbulence conditions. In [17,18], channel capacity is analyzed over oceanic turbulence channels by using the Málaga distribution. The impact of adaptive optics on channel capacity has been also studied under weak oceanic turbulence conditions [19]. As a general viewpoint, all these reports focus on examining the influence of oceanic turbulence on channel capacity, concluding that such an oceanic turbulence reduces considerably the capacity. These investigations into channel capacity, however, do not take into account the impact of ISI. In other words, the temporal dispersion produced by scattering is not taken into account when analyzing the channel capacity of UOWC systems. In contrast to terrestrial FSO systems, the dispersive nature of the undersea optical channel impacts drastically on the overall performance. As concluded in [20,21], UOWC links result in being noisy channels with finite memory that can be modeled by linear discrete-time filtering of the input symbols. Thus, analyzing how the effect of ISI degrades channel capacity of UOWC systems is certainly of real interest and needs to be quantified for communication purposes. Historically, the study of capacity of the discrete-time Gaussian channel in the presence of ISI has been addressed where both channel capacity and information rate have been estimated under different constraints imposed on the channel input [22–26].

In this paper, new upper and lower bounds on the capacity of undersea optical channels corrupted not only by white Gaussian noise, but also by scattering-induced ISI and salinity-induced oceanic turbulence are presented in the context of UOWC systems using non-uniform signalling. The capacity derivation, which is found by using the discrete-time Fourier transform (DFT) of the equivalent discrete-time impulse response of the system, is obtained by bounding the mutual information when non-negative and average optical power constraints are imposed on the channel input. Additionally, when a binary input constraint is imposed on the channel input, new upper and lower bounds on channel capacity and mutual information are also provided for different undersea optical channels with one, two and three casual ISI coefficients. Both channel capacity and mutual information are then computed for practical UOWC scenarios. The obtained results demonstrate that channel capacity is clearly affected not only by oceanic turbulence as concluded in the current literature, but also by scattering-induced ISI. Monte Carlo simulation results are further illustrated to test the derived bounds.

The balance of this paper is organized as follows. The system and channel models under study are described in Section 2 where the DFT of the equivalent discrete-time impulse response of the system is modeled analytically. In Section 3, the channel capacity analysis is performed where different constraints are imposed on the channel input. In Section 4, numerical results for channel capacity and mutual information (MI) are discussed under different severity of scattering and turbulence conditions. Finally, Section 5 concludes the paper.

## 2. System and channel models

Let us consider a single-input/single-output UOWC system using non-uniform OOK modulation which is based on intensity-modulation and direct-detection (IM/DD) due to its low cost and implementation simplicity [27,28]. Here, a green LD operating at 532 nm is used at the transmitter side, as well as a circular receiver aperture is used at the receiver side. A direct consequence of this modulation scheme is that the input signal is non-negative as it is proportional to the light intensity. Additionally, for safety and practical reasons, both the average optical power and the amplitude of the transmitted intensity are limited [29]. Thus, the input-output relation of the discrete-time model of a pulse amplitude modulation (PAM) IM/DD channel with ISI and oceanic turbulence, as shown in Fig. 1, can be expressed as

where $R$ is the photodetector responsivity in amps per watt and is considered to be unity without loss of generality, $P_{t}$ is the average optical power, $T_{b}$ is the bit period, $h$ represents the oceanic path loss and fading due to oceanic turbulence, $X_{k}\in \{0,1\}$ are independent identically distributed (i.i.d.) random variables following a Bernoulli distribution with parameter $p$ that represent the bits to be transmitted, the symbol * denotes convolution, and $c_{k}$ is a finite-length sequence of real coefficients that represents the equivalent discrete-time impulse response of the system that is derived as where $g(t)$ is the transmitter filter, $\mathbb {C}(t)$ is the fading-free channel impulse response with unity area [21], and $f(t)$ is the matched filter, i.e, $f(t)=g(-t)$. Non-return to zero (NRZ) OOK modulation is considered here, so that $g(t)$ is a rectangular pulse of duration $T_{b}$. Lastly, $Z_{k}$ are i.i.d. Gaussian random variables with zero mean and variance $\sigma _{n}^{2}=N_{0}/2$, i.e., $Z_{k}\sim N\left (0,\sigma _{n}^{2}\right )$ which are computed as $Z_{k}=\left . z(t)*f(t)\right |_{t=kT_{b}}$. For convenience, the UOWC channel output samples are expressed asIn order to derive new bounds on the capacity of UOWC channels with ISI in the next section, it is required to find the transfer function of the equivalent discrete-time impulse response $c_{k}$ which is computed from the DFT as

where $j=\sqrt {-1}$. Note that $C(\omega )$ is periodic in $\omega$ (normalized frequency) with period $2\pi$. According to this study, $c_{k}$ presents a decaying exponential, demonstrating that the energy of the ISI coefficients decreases progressively. Hence, the frequency response of the UOWC system under study can be well approximated by $C(\omega )\simeq a\left (1-b e^{-j\omega }\right )^{-1}$ with $(a,b)\in \mathbb {R}^{+}$ and $0<b<1$, whose magnitude of the frequency response, $|C(\omega )|$, is readily obtained as where $a$ can be computed as $a=c_{0}$, and $b$ represents the impulse response tail as $b=\sum _{n=1}^{M}c_{n}$. As can be seen in greater detail in Appendix A, the approximation of the DFT of $c_{k}$ is able to achieve a value of the coefficient of determination R-square (R$^{2}$) above 0.95 in almost all simulated cases. In Fig. 2, some examples of the magnitude of the frequency response of different UOWC channels are plotted when using a green LD with a transmit divergence angle of $\theta =10$ mrad operating at $\lambda =523$ nm with $\lambda$ being the wavelength, as well as a photodetector diameter of $D=20$ cm and a receiver field-of-view (FOV) value of $180^{\circ }$. Note that the results of the DFT of $c_{k}$ obtained from Monte Carlo simulations are depicted with a solid line, whereas the results of the approximate DFT of $c_{k}$ defined in Eq. (5) are depicted with a dashed line. The DFT for an ISI-less UOWC channel, i.e., $M=0$ and $c_{0}=1$, is also displayed in cyan color as a reference. From this figure, remarkable differences can be observed when comparing clear ocean water with coastal water. For instance, in clear ocean water, the effect of scattering is not significant and, hence, $|C(\omega )|$ is practically flat $\forall$ $|\omega |\leq \pi$, allowing a quasi ISI-free transmission through seawater at moderate distances. Besides, the Nyquist criterion for no ISI is satisfied in this scenario [30]. These results are further aligned with early reported works [20,21]. However, this conclusion cannot be drawn from coastal waters since scattering is much more severe due to a higher concentration of planktonic matter, among others, compared to clear ocean water [20]. Therefore, it may be anticipated that the ISI will present a greater impact on channel capacity for coastal water than for clear ocean water. These channels will be used to obtain upper and lower bounds on channel capacity of the UOWC system under study. It is also noteworthy to mention that the channels depicted in Fig. 2 are not unique and, hence, other UOWC channels with ISI may take place, depending on different aspects related to the communication system itself such as the characteristics of the transmitter and receiver, the symbol period, the link distance, as well as other major system parameters.#### 2.1 Statistical fading model

Oceanic path loss and salinity-induced oceanic turbulence are represented by $h=L h_{o}$. The parameter $L$ is computed from a modified version of the Beer Lambert’s law to include scattering and geometric loss for a narrow LD with a Gaussian profile, as presented in [21, Eq. (11)], as

where $D$ is the photodetector diameter, $\theta$ is the transmit divergence angle, $F$ is a correcting factor to include scattering [21], and $d$ is the link distance. The extinction coefficient, $\alpha _{T}$, is defined as the sum of absorption ($\alpha _{1}$) and scattering ($\alpha _{2}$) [3], whose values for a wavelength of 532 nm are summarized in Table 1.Oceanic turbulence, $h_{o}$, is accurately modeled according to the Weibull distribution [6,31], whose probability density function (PDF) with parameters $\beta _{1}$ and $\beta _{2}$ is given by

The above PDF is currently being adopted to describe statistically oceanic turbulence that is present in the open waters and oceans with salinity gradient [6,7,21,31,32]. Some mathematical expressions were developed in [6] for $\beta _{1}$ and $\beta _{2}$, yielding

where $\sigma _{h_{o}}^{2}$ is the scintillation index of oceanic turbulence. Considering a plane-wave propagation as a limiting case of a collimated Gaussian beam propagation through seawater, $\sigma _{h_{o}}^{2}$ is defined, as in [33], as## 3. Channel capacity analysis

In this section, upper and lower bounds on channel capacity of UOWC systems in the presence of ISI and salinity-induced oceanic turbulence are obtained when different constraints are imposed on the input channel. The basic for the discrete-time Gaussian channel with ISI developed in the previous section will be used to compute both channel capacity and mutual information for such communication systems.

#### 3.1 Upper bound on channel capacity

In order to gain a better understanding of this analysis, let us start defining the capacity for a memoryless channel, i.e, when ISI does not take place or equivalently when $M=0$ and $c_{0}=1$. According to [37, Chapter 10], the capacity in bits per channel use for a discrete-time memoryless Gaussian channel is formulated as follows

where $I\left (X;Y|h\right )$ is defined as the conditional mutual information between the input $X$ and the output $Y$ given an oceanic turbulence state $h$. Notice that the capacity is defined as the maximum average mutual information which is found over all possible input distributions $p(x)$ that satisfy the average optical power constraint of $P_{t}$. As stated in Section 1, we compute, for the first time, the channel capacity of UOWC systems for non-uniform OOK modulation, as previously adopted in the context of terrestrial FSO communication systems through atmospheric turbulence with no ISI [38–40]. In this way, the non-negativity constraint is also satisfied due to the fact that a unidimensional signal space is considered where one constellation point is 0 and the other one is the Euclidean distance $d_{E}=(1/p)P_{t}\sqrt {T_{b}}$. This approach implies that $\mathbb {E}[X^{2}]\leq pd_{E}^{2}$. Thus, knowing that the capacity is obtained when the input $X$ is a zero-mean Gaussian random variable, i.e., $X\sim N\left (0,\sigma _{X_{el}}^{2}\right )$ [30,37], the conditional mutual information is upper bounded byIn order to solve the above integral, we can use the fact that both $\exp (-z)$ and $\ln (1+z)$ can be expressed in terms of the Meijer’s G-function $G_{p,q}^{m,n}[\cdot ]$ [41, Eq. (7.811)] as $\exp (-z)=G_{0,1}^{1,0}\left [z|0\right ]$ [42, Eq. (01.03.26.0004.01)] and $\ln (1+z)=G_{2,2}^{1,2}\left [z|_{1,0}^{1,1}\right ]$ [42, Eq. (01.04.26.0003.01)], respectively. As a result, an approximate closed-form expression for the upper bound on average MI is written in terms of the well-known H-Fox function $H_{p,q}^{m,n}[\cdot ]$ [43, Eq. (1.1)] as follows

The above approximation can be interpreted as an upper bound on average MI of an ISI-less UOWC channel where the parameter $a$ acts as a power degradation factor due to the ISI. The accuracy of such an expression will be checked in the next Section where a numerical integration for Eq. (14) will be also discussed when no approximations are used. Finally, an upper bound on channel capacity in bits per channel use is found numerically by maximizing the above closed-form expression over the input distribution $p$ as follows

For comparison purposes, the classical AWGN channel capacity in the absence of ISI and oceanic turbulence is obtained as#### 3.2 Channel capacity with a binary input constraint

Now, we compute numerically the mutual information of the UOWC channel under study using non-uniform OOK modulation when a binary input constraint is imposed. Thus, the input-output relation of the UOWC system with ISI and oceanic turbulence was defined in Eq. (3) as

where $A=(1/p)\gamma$, $X_{k}\in \{0,1\}$ are i.i.d. random variables following a Bernoulli distribution with parameter $p$, and $Z_{k}\sim N(0,1)$ due to the fact that the above statistical channel model is normalized by replacing $Y_{k}$ with $Y_{k}/\sigma _{n}$. In this way, the conditional mutual information for the UOWC channel with ISI subject to a binary input constraint is defined, as in [44, Section 4.6], asNote that $f_{Y}\left (y|x_{i},x_{i-1},\ldots ,x_{i-M},h\right )$ depends on the probabilities assignment on the ISI sequence $\mathbb {X}_{r}=\{x_{i-1},\ldots ,x_{i-M}\}$ where $P(\mathbb {X}_{r})$ is the probability of the sequence $\mathbb {X}_{r}$. Since $M$ is the length of the channel impulse response tail, $2^{M}$ is the number of possible ISI sequences to be considered due to the fact that $x_{i-n}$ takes on the values 0 and 1. For instance, if $M=2$ and $\mathbb {X}_{r}=\{x_{i-1}=1,x_{i-2}=0\}$, then $P(\mathbb {X}_{r})=p(1-p)$, and so on. Thus, the average MI, $I^{b}(X;Y)$, can be obtained by averaging Eq. (20) over the PDF of $h$ in Eq. (7) as follows

Finally, the capacity in bits per channel use is found numerically by maximizing the above average MI over the input distribution $p$ as follows

It should be noted that a closed-form expression for $C^{b}$ is not known. In other words, it is not possible to obtain a solution for $I^{b}(X;Y)$ and $C^{b}$ formulas given in Eq. (20) and (23) and, hence, they will be evaluated in the next Section via numerical optimization methods available in standard mathematical software packages, as well as the Gaussian upper bound on channel capacity $C^{G}$ obtained in Eq. (17). In particular, the above expression is really time-consuming and technically difficult to compute. Furthermore, the complexity of the above expression grows with $M$ as it increases the number of possible ISI sequences. Motivated by this, we provide now useful upper and lower bounds on mutual information subject to a binary input constraint that result in being tighter as the ISI becomes less significant.

Firstly, the upper bound is obtained in terms of the average MI of a memoryless UOWC channel with input $X_{k}$ and output $Y_{k}=\sqrt {\rho _{UB}} A X_{k} + Z_{k}$, where $\rho _{UB}$ is understood as a power improvement due to the memory caused by the ISI coefficients [24,25]. Thus, the factor $\rho _{UB}$ is derived from the norm of $c_{k}$ as follows

Secondly, the lower bound is also obtained in terms of the average MI of a memoryless UOWC channel with input $X_{k}$ and output $Y_{k}=\rho _{LB} A X_{k} + Z_{k}$. Unlike $\rho _{UB}$, the factor $\rho _{LB}$ is understood as a power degradation due to the memory caused by the ISI coefficients. Thus, $\rho _{LB}$ is obtained, as in [24,25], as follows

Finally, the average MI, $I^{b}(X;Y)$, can be upper and lower bounded in terms of the average MI of a memoryless UOWC channel as Note that the above bounds are computed from Eqs. (20) and (21) for an ISI-less UOWC channel, i.e., when $M=0$ and $c_{0}=1$.## 4. Numerical results and discussion

sIn this section, the new bounds on channel capacity and mutual information of UOWC systems with ISI over oceanic turbulence channels in bits/channel use are computed and tested by Monte Carlo simulations. In particular, UOWC channels with one, two and three ISI coefficients i.e., ISI memory of degree one, two and three, are analyzed. The major UOWC system parameters are summarized in Table 2 where different link distances are considered for clear ocean and coastal waters as practical scenarios according to previous reported papers [11–13], and available commercial systems such as those provided by Sonardyne for link distances up to 150 m [45]. At the transmitter side, a green LD is used with a transmit divergence angle of $\theta =10$ mrad while, at the receiver side, a photodetector diameter of $D=20$ cm and a FOV of $180^{\circ }$ are assumed. Regarding oceanic turbulence, the scintillation index expressed in Eq. (9) can be calculated numerically or using the approximate closed-form expression developed in [9, Appendix A], as well as the parameters $\beta _{1}$ and $\beta _{2}$ are calculated from Eq. (8). Scintillation index values of $\sigma _{h_{o}}^{2}=\{0.1,0.4,0.6,0.8,1,1.5\}$ are computed for link distances of $d=\{15,25,30,35,40,50\}$ m, respectively, which model different salinity-induced oceanic turbulent conditions when $w=-1$, $\epsilon =10^{-5}$ $m^{2}/s^{3}$, and $\chi _{T}=10^{-7}$ $K^{2}/s$ [34,35]. Without loss of generality, all figures plot channel capacity and mutual information for a normalized path loss of $L=1$. Nonetheless, path loss values of 29.24 dB, 37.91 dB and 46.10 dB are computed in clear ocean water for link distances of 30 m, 40 m and 50 m, respectively, as well as for the setup outlined in Table 2. Analogously, path loss values of 28.78 dB, 47.92 dB and 64.91 dB are also computed in coastal water for link distances of 15 m, 25 m and 35 m, respectively. These path loss values would induce a right shift in decibels of the current channel capacity and mutual information curves.

In Figs. 3 and 4, the average MI, $I^{b}(X;Y)$, obtained in Eq. (22) is plotted as a function of the optical SNR $\gamma$[dB] when different non-uniform input distributions of $p=\{0.1,0.25,0.5\}$ and different symbol rate of $R_{b}=\{1,2\}$ Gbps are considered. Additionally, the upper and lower bounds on average MI, i.e., $I_{UB}^{b}(X;Y)$ and $I_{LB}^{b}(X;Y)$, are depicted, as well as the channel capacity, $C^{b}$, obtained in Eq. (23). Note that all $I_{UB}^{b}(X;Y)$ results are plotted with a black dotted line, whereas all $I_{LB}^{b}(X;Y)$ results are plotted with a green dotted line. For comparison purposes, both the average channel capacity and the average MI for ISI-less UOWC channels are also included by using a dashed line. On the one hand, it is verified that the effect of salinity-induced oceanic turbulence cannot be ignored when analyzing both channel capacity and mutual information of UOWC systems, particularly in open waters, as previously concluded using other performance metrics such as BER and outage probability [14–19]. On the other hand, as can be seen in Fig. 3, all results are overlapped due to the fact that scattering is not significant in clear ocean water. In such a case, the impulse response tail equals 1, i.e., $M=1$. According to these results, the inequality represented in Eq. (26) becomes an equality, concluding that the impact of ISI can be neglected in this kind of water at moderate link distances. On the contrary, in Fig. 4, the ISI degrades both average channel capacity and average MI by inducing an optical SNR penalty of a few decibels with respect to the case of an ISI-less channel. This optical SNR penalty depends mainly on scattering, link distance and data rate. In coastal water, it is observed that the upper bound, $I_{UB}^{b}(X;Y)$, results in being much tighter and useful than the lower one, $I_{LB}^{b}(X;Y)$, relaxing the computational load as required to compute Eq. (22). As expected, this upper bound is even tighter as the ISI becomes less significant. Interestingly, we have quantified what it was already anticipated qualitatively in Section 2 from the DFT of $c_{k}$ with regard to the impact of ISI on channel capacity. It is observed that the ISI severely degrades channel capacity when the level of impurities of water increases, as noted in Section 2.

In Fig. 5, the Gaussian upper bound on channel capacity, $C^{G}$, obtained in Eq. (17), as well as the channel capacity, $C^{b}$, obtained in Eq. (23) subject to a binary constraint are plotted as a function of the optical SNR $\gamma$[dB]. Both channel capacity derivations are computed by using numerical optimization methods that are available in standard mathematical software packages. Firstly, it is corroborated that the Gaussian upper bound, $C^{G}$, which is obtained via maximization of the approximate average mutual information as derived in terms of the H-Fox function in Eq. (17) achieves a high accuracy for the whole range of optical SNR considered in both figures. Such capacity results match very well with the Gaussian upper bound, $C^{G}$, obtained via Monte Carlo simulations from Eq. (14) when no approximations (labeled $C^{G}$ with no approx.) are used in all practical UOWC scenarios analyzed. For comparison purposes, these figures also plot the average MI for uniform distribution with $p=0.5$ when the same average optical power as the non-uniform distribution is assumed. Additionally, these figures plot the classical AWGN channel capacity, $C_{AWGN}^{G}$, obtained in Eq. (18) as a reference. As expected, the ISI does not make channel capacity increase, quite the contrary. The combined effect of scattering-induced ISI and oceanic turbulence impacts drastically on channel capacity of the UOWC system. Moreover, all the average mutual information and capacity results tend asymptotically to the binary entropy function $H_{B}(p)$ as the optical SNR increases where values of $H_{B}(p)=\{1,0.811,0.468\}$ are computed for input distributions of $p=\{0.5,0.25,0.1\}$, respectively.

Finally, it is concluded that the capacity-achieving distribution, which is obtained through numerical optimization, is discrete and can be expressed analytically as $f_{X}(x)=(1-p)\delta (x)+p\delta (x-d_{E})$ with $\delta (\cdot )$ being the Dirac delta function, and $p$ being obtained numerically for each optical SNR value. It is highlighted that this non-uniform input distribution notably improves the channel capacity of the UOWC system in the presence of ISI and oceanic turbulence particularly at low optical SNR, as also corroborated in the context of terrestrial FSO systems with no ISI [39,40]. In fact, this key feature can be also observed from another interesting viewpoint as illustrated in Fig. 6. In such a figure, the average MI, $I^{b}(X;Y)$, is plotted as a function of the non-uniform input distribution $p$ when different optical SNR values of $\gamma =\{-5,10\}$ dB are considered in order to corroborate that the optimum values of $p$ fundamentally change with each value of the optical SNR and do not depend on the severity of the ISI. Similar conclusions can be drawn from this figure to those previously obtained when comparing clear ocean and coastal waters.

## 5. Conclusion

In this paper, novel insights into the combined impact of scattering-induced ISI and salinity-induced oceanic turbulence on channel capacity and mutual information have been provided when different constraints were imposed on the channel input.

On the one hand, a new upper bound on channel capacity subject to non-negative and average optical power constraints has been found by using the DFT of the equivalent discrete-time impulse response of the UOWC system when the channel inputs are Gaussian random variables with fixed average optical power. In the light of this research, it was verified that channel capacity is clearly affected not only by salinity-induced oceanic turbulence, but also by scattering-induced ISI. Both degrading factors cannot be neglected from an information-theoretical viewpoint. On the other hand, it is concluded that, in clear ocean water, channel capacity subject to a binary input constraint can be well approximated by the channel capacity with no ISI where the derived upper and lower bounds converge on the same bound under negligible scattering conditions. However, in coastal water, different conclusions are obtained. The effect of scattering is dominating the capacity, and the originated ISI incurs a significant optical SNR penalty in terms of rate when comparing with an ISI-less channel. Despite the fact that the lower bound on channel capacity is not the tightest possible in coastal water, the upper one results in being remarkably tight. Furthermore, it is able to achieve a substantial computational reduction in general terms since channel capacity is computed from the channel capacity of the UOWC system with no ISI by adding a power coefficient due to the memory introduced by the ISI. Finally, due to the similarities between undersea and terrestrial free-space communications, it is thought that non-uniform signalling algorithms developed for terrestrial FSO systems to generate linear binary codes might be implemented in the context of UOWC systems with ISI.

In the near future, the use of passband modulation schemes and multiple inputs and/or multiple outputs structures should be explored to try to mitigate the impact of scattering-induced ISI and oceanic turbulence on channel capacity and practical information rates while narrow-beam LD are used to enable fast, long-distance undersea communications.

## A. Results of equivalent discrete-time impulse response

In this appendix, we show some results of the equivalent discrete-time response of different practical UOWC scenarios for clean ocean (Table 3) and coastal (Table 4) waters. Moreover, the value of each length of the channel impulse response tail, $M$, is also listed in both Tables. The parameters $a$ and $b$ from which the DFT is modeled for each scenario are also provided along with the corresponding value of the coefficient of determination R$^{2}$. Note that $\mathbb {C}(t)$ is evaluated via Monte Carlo simulation based on photon tracing which makes use of the Henyey-Greenstein phase function as the scattering angular distribution in seawater, as described in greater detail in [21]. All these results have been obtained by considering the system parameters summarized in Tables 1 and 2. The results outlined in Table 3 for clear ocean water demonstrate that, under these scattering conditions, the equivalent discrete-time channel can be modeled as a linear filter with only one coefficient, i.e., $M=1$, achieving a value of R$^{2}$ above 0.97 when different data rate values of $R_{b}=\{0.5,1,2\}$ Gbps are considered. In other words, the impact of ISI on channel capacity under clear ocean water conditions can be neglected when the data rate is below a few Gbps. However, under coastal water conditions, the situation is quite different since the effect of scattering becomes significant even when the data rate is not too high, i.e., as $R_{b}$ approaches infinity, the value of $M$ also approaches infinity, as deduced from Table 4. In this case, the equivalent discrete-time channel can be modeled as a linear filter with more than one coefficient, achieving a value of R$^{2}$ above 0.87 for data rate values of $R_{b}=\{0.5,1,2\}$ Gbps.

## Funding

Programa Operativo I+D+i FEDER Andalucía 2014-2020 (P18-RTJ-3343); Spanish MICINN (PID2019-107792GB-I00).

## Disclosures

The authors declare no conflicts of interest.

## Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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