Elsevier

Optics Communications

Volume 505, 15 February 2022, 127476
Optics Communications

Ergodic capacity and error performance of spatial diversity UWOC systems over generalized gamma turbulence channels

https://doi.org/10.1016/j.optcom.2021.127476Get rights and content

Abstract

In this paper, we study the ergodic capacity (EC) and average bit error rate (BER) of spatial diversity underwater wireless optical communications (UWOC) over the generalized gamma (GG) fading channels using quadrature amplitude modulation (QAM) direct current-biased optical orthogonal frequency division multiplexing (DCO-OFDM). We derive closed-form expressions of the EC and BER for the spatial diversity UWOC with the equal gain combining (EGC) at receivers based on the approximation of the sum of independent identical distributed (i.i.d) GG random variables (RVs). Numerical results of EC and BER for QAM DCO-OFDM spatial diversity systems over GG fading channels are presented. The numerical results are shown to be closely matched by the Monte Carlo simulations, verifying the analysis. The study clearly shows the adverse effect of turbulence on the EC & BER and advantage of EGC to overcome the turbulence effect.

Introduction

Underwater wireless optical communication (UWOC) is considered as an attractive complementary solution to underwater acoustic communication due to its high data rate, low latency and low implementation cost [1]. However, there are several challenges in implementing the UWOC system due to the adverse channel condition caused by absorption, scattering and turbulence in the oceanic environment [2]. There have been several attempts to establish a realistic channel model that can accurately predict the performance of the UWOC [1], [2], [3], [4], [5], [6], [7], [8], [9]. The attenuation due to absorption and scattering is well understood and mostly modeled by the radiative transfer equation [3], ray-tracing simulations [4], and a closed-form expression by fitting simulation data [5]. Underwater optical turbulence (UOT) caused by the variations in the refractive index due to temperature fluctuation, salinity variations and the presence of air bubbles adversely affect the link performance [6], [7]. Hence, there has been significant attention to accurately characterize the UWOC turbulence channel to predict the performance and establish mitigation techniques [8], [9].

The early studies on the UOT channel adopted lognormal distribution which is commonly used to describe atmospheric turbulence in free-space optical (FSO) system. However, the mechanisms of turbulence generation in underwater and FSO channels are different and hence the lognormal distribution does not accurately model the UOT in all the conditions [7]. Subsequently, several studies of UOT have been carried out taking into account of several fading causes. In [10], a generalized gamma distribution was proposed to describe the temperature-induced weak turbulence. In [11], it was verified that Weibull distribution matches the measured data under all turbulence channels caused by salinity gradient. However, it was shown in [12] that the measured data in the presence of air bubbles do not fit well by a single-lobe distribution and required a two-lobe statistical model. Taking into account both air bubbles and temperature gradient, an exponential-generalized gamma (EGG) distribution model was proposed in [7] which excellently matches the experimental data under all channel conditions. Using beam expander-and-collimator (BEC) at the transmitter side and/or aperture averaging lens (AAL) at the receiver side, it was shown in [13] that the GG and exponential Weibull distributions provide an excellent agreement with the measured data in UWOC channels with random temperature and salinity variations in the presence of air bubbles, covering a wide range of scintillation index values from weak to strong turbulence. In fact, GG distribution is a general case of some of the important statistical distributions in modeling fading such as exponential, Rayleigh, gamma, Weibull, Nakagami-m and lognormal [13], [14].

It was established that turbulence-induced fading considerably increases the average bit error rate (BER) and hence results in a large power penalty to achieve desired BER performance. This, in turn, either significantly reduces the communication range or decreases the maximum achievable data rate. To mitigate fading impairments, several techniques have been proposed, including (a) error control coding in conjunction with interleaving [15], [16], (b) maximum likelihood sequence detection [17], (c) multi-hop relaying transmission [18] and (d) spatial diversity [19], [20]. The first three approaches have several practical limitations, namely large-size interleaves, high computational complexity and high cost, but spatial diversity is not only the most practical and effective to mitigate the fading but also reduces the possibility of temporal blockage by multiple apertures at the transmitter and/or the receiver [21], [22]. The spatial diversity scheme involves repetition coding at the transmitter and linear diversity combining (LDC) technique at the receiver. There are three well-known LDC techniques, namely maximal ratio combining (MRC), equal gain combining (EGC) and selection combining (SC). Though MRC offers the best performance, it requires prior channel knowledge which is very difficult to obtain especially for a time-varying channel [21], [22]. Furthermore, the performance of EGC can match very close to MRC for most practical cases [23]. Therefore, EGC is a more realistic option for practical implementation due to its simplicity and low complexity [24].

The selection of the modulation scheme also affects the performance of UWOC in the oceanic turbulence channel. The intensity modulation and direct detection with baseband modulation such as On–Off​ keying (OOK) and pulse position modulation (PPM) are frequently used for UWOC due to their simplicity and low cost. Though simple to implement, OOK needs adaptive detection to achieve optimal performance in UOT channel [25]. PPM scheme, on the other hand, requires a very tight pulse and symbol synchronization and has inferior bandwidth efficiency compared to OOK. The subcarrier modulation (SCM) scheme is an alternative to the baseband modulation [26]. The SCM permits the use of high-order constellations such as M-ary phase-shift keying (M-PSK) and M-QAM by modulating an RF signal onto the intensity of the optical beam [27]. The OFDM is a special case of SCM, where the carriers are orthogonal over one symbol period. OFDM offers high spectral efficiency, resistance to inter-symbol interference and frequency selective fading [28], [29]. To meet the real and unipolar requirement of optical intensity modulation, several modified OFDMs were proposed, such as DCO-OFDM, asymmetrically-clipped optical OFDM (ACO-OFDM), and asymmetrically clipped DC-biased Optical OFDM (ADO-OFDM). DCO-OFDM is usually adopted in high bit rate communication due to its high bandwidth utilization.

Although GG distribution has been shown to adequately model fading channels, there are very limited studies to evaluate the performance of various spatial diversity schemes over GG fading channels. Aalo et al. derived the average symbol error rate performance of multilevel modulation schemes for wireless communication systems in GG channels with diversity receivers [30]. Sagias et al. derived and evaluated union upper bounds for the outage and the average bit error probability for ECG receivers [31]. In addition, Costa in [32] used another GG RV to approximate the sum of independent identical distributed (i.i.d) GG RVs) and then derived the analytical expressions of outage probability and average BER for the spatial-diversity wireless communication systems using noncoherent frequency shift keying and noncoherent differential phase-shift keying. However, to the best knowledge of the authors, there is no prior work to evaluate the ergodic capacity and average BER of QAM-OFDM for spatial diversity UWOC systems over GG fading channels. Hence, in this paper, we propose and study DCO-OFDM with a diversity scheme and EGC reception to mitigate the fading and achieve high spectral efficiency. We derive a closed-form expression of average BER based on the approximation of the sum of GG RVs and Gauss–Laguerre quadrature integrals. We also derive the EC in terms of Fox’s functions. Numerical results for average BER and EC of QAM DCO-OFDM with spatial diversity in the UWOC systems over GG fading channels are presented. The results are further validated by Monte-Carlo (MC) simulations. To the best of our knowledge, this is the first study of comprehensive mathematical derivation of the average BER and EC for DCO-OFDM under turbulent UWOC.

The rest of the paper is organized as follows: Section 2 describes the UWOC channel in the presence of UOT and DCO-OFDM with EGC diversity systems. In Section 3, we derive the average BER and EC. In Section 4, numerical and simulation results are presented, compared and discussed. Finally, Section 5 concludes this paper.

Section snippets

UWOC system and channel model

A block diagram of the proposed spatial diversity UWOC systems with DCO-OFDM scheme is shown in Fig. 1. We consider one transmitter and N receiving apertures (the total area is the same regardless of N) and standard modulation/ demodulation procedure of DCO-OFDM as described in [33]. The information bits are modulated employing L-QAM scheme (where log2(L) > 0 is an integer), followed by a serial to parallel (S/P) converter. Before the inverse fast Fourier transform (IFFT) operation, Hermitian

Performance analysis

To obtain the pdf of γEGCeff, the pdf of the weighted sum should be evaluated. Another GG variable has been proposed to approximate the sum of multiple GG RVs. Thus, based on the pdf of the EGC output SNR, analytical expressions for average BER and EC with the EGC diversity scheme over the GG fading channels can be derived which are presented in forms of Fox’s H-function.

Results and discussion

In this section, numerical results for the average BER and EC based on the analytical expressions and MATHEMATICA are presented for different diversity orders under various GG fading conditions. To verify the analytical results, the results of the MC simulations, using MATLAB, are also presented. Based on the study in [13], two fading channel scenarios are considered and their corresponding parameters of GG RVs are shown in Table 1.

Fig. 2 depicts the pdf of the sum of four i.i.d GG-distributed

Conclusion

In this paper, the spatial diversity with EGC was proposed to combat turbulence-induced fading impairments in UWOC systems over GG channels. Based on the approximation of the sum of GG RVs, the average BER and EC for QAM-DCO-OFDM were evaluated. Using Gauss–Laguerre quadrature integral, a closed-form expression of the average BER is derived. In addition, with the help of transforms and integral formulas involving Fox’s H function, the EC is given in the forms of Fox’s H function. Based on the

CRediT authorship contribution statement

Hongyan Jiang: Conceptualization, Methodology, Software, Validation, Investigation, Writing – original draft, Funding acquisition. Hongbing Qiu: Supervision. Ning He: Investigation, Funding acquisition. Wasiu Popoola: Validation, Writing. Zahir Ahmad: Supervision, Validation, Writing. Sujan Rajbhandari: Conceptualization, Methodology, Software, Validation, Investigation, Writing – original draft, Writing – review and editing, Supervision.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Funding

This research was funded in part by the National Natural Science Foundation of China (grant number 61961008 and 61761014), in part by the Dean Project of Guangxi Key Laboratory of Wireless Broadband Communication and Signal Processing (grant number GXKL06200127), and in part by the project of improving basic ability of scientific research for middle-aged and young scholars of higher education of Guangxi (grant number 2020KY05028). All authors have read and agreed to the published version of

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