Underwater wireless optical communication (UWOC) has been widely considered a supplement to traditional underwater acoustic communication. A real-time UWOC video delivery system was developed in a laboratory water tank based on a field-programmable gate array (FPGA) with binary frequency shift keying (2FSK) modulation. The system achieved full-duplex communication by using the transmission control protocol (TCP) and forward error correction (FEC). A high-power 445 nm lightemitting diode (LED) array was adopted to enhance the transmitted optical power and increase the transmission link distance. We present an underwater optical channel model that considers the effects of both geometry and channel loss, especially considering the impact of the refractive index of the optical medium and the non-line-of-sight (NLOS) links formed by water surface reflection. MATLAB was used to simulate this channel model and predict the received optical power distribution on the receiving plane. Additionally, we propose improved calculation methods for the consumed electrical power and transmitted optical power of the LED array. We also investigate the relationship between the optimum avalanche gain of an avalanche photodiode (APD) and the signal-to-noise ratio (SNR). This full-duplex system achieved a 1 Mbps data transmission rate at an SNR of 10.1 dB and a distance of 10 m for an underwater link. In addition, when the optical power of the LED array is enhanced, the link range is predicted to be 14.5 m with an attenuation coefficient of 0.056 /m. INDEX TERMS Binary frequency shift keying, full-duplex, FPGA, high-power LED array, optical link model, Reed-Solomon code, underwater wireless optical communication. I. INTRODUCTION The ocean is the cradle of life. Approximately 71% of the Earth's surface is covered by the ocean. The vast marine resources on Earth are indispensable for many aspects of life. It is necessary to exploit and utilize those resources with underwater wireless communication (UWC) technology, which has considerable potential in facilitating the use of underwater vehicles, devices, observatories, and sensors. Underwater wired communication uses fiber optic or copper cable, which is expensive, inflexible, and vulnerable to marine life, making it largely infeasible for use in underwater mobile systems. Acoustic waves, radio frequency (RF) waves, and optical waves are three primary physical information carriers for underwater wireless information transmission [1]-[3]. Acoustic waves involve mechanical waves with relatively little attenuation underwater (0.1-4 dB/km), and thus, they can cover long distances up to dozens of kilometers. However, acoustic waves have a low propagation speed (1500 m/s) and limited bandwidth (kHz), which leads to a multipath phenomenon, large time latency, and bulky antennas [2]. These characteristics hinder the application of acoustic waves in real-time and bandwidth-intensive scenarios. RF waves are another carrier that can provide a high data rate (Mbps), high bandwidth (MHz), and high speed...

Underwater wireless optical communication (UWOC) is a promising technology that can be a candidate to improve the communication capacity and speed in aquatic media. The aim of this study is to examine the performance of a silicon photomultiplier (SiPM) array-based multiple-input multiple-output (MIMO) UWOC system. A SiPM is a modern solid-state photodetector with extremely high sensitivity up to the single-photon level or a photon-counting ability, which helps in detecting extremely weak light signals after long-distance underwater channel attenuation. We clarify the basic characteristics and photon-counting detection mode of a SiPM. In particular, the photocount of a SiPM is approximated by a Gaussian distribution, and theoretical analysis shows that only 13.3 photons need to be detected during “1” symbol period to achieve a bit error rate of 10−3 in an ambient light environment. Moreover, a SiPM also has a better analog mode detection ability than an avalanche photodiode (APD) and realizes 2 Mbps analog communication owing to its unique array structure and high photon detection efficiency. Furthermore, MIMO, i.e., spatial diversity, is applied as an effective method to relax the link alignment, improve the system performance, and alleviate the effect of optical turbulence. In our experiment, with a photon-counting 6×3 MIMO scheme, an energy per bit of 7.38×10−9 J/bit is achieved at a scintillation index of 4.66×10−3 in a 10 m water tank with 1 Mbps on-off-keying (OOK) modulation. To the best of our knowledge, this is the first study on a MIMO-UWOC system based on the photon-counting mode of a SiPM array. This UWOC system combines the advantages of SiPMs and the MIMO scheme and has the potential to realize long-distance UWOC under optical turbulence.

Let (R, m) be a local ring of prime characteristic p and q a varying power of p. We study the asymptotic behavior of the socle of R/I [q] where I is an m -primary ideal of R. In the graded case, we define the notion of diagonal F -threshold of R as the limit of the top socle degree of R/m [q] over q when q → ∞. Diagonal F -threshold exists as a positive number (rational number in the latter case) when:(1) R is either a complete intersection or R is F -pure on the punctured spectrum;(2) R is a two dimensional normal domain. In the latter case, we also discuss its geometric interpretation and apply it to determine the strong semistability of the syzygy bundle of (x d , y d , z d ) over the smooth projective curve in P 2 defined by x n + y n + z n = 0. The rest of this paper concerns a different question about how the length of the socle of R/I [q] vary as q varies. We give explicit calculations of the length of the socle of R/m [q] for a class of hypersurface rings which attain the minimal Hilbert-Kunz function. We finally show, under mild conditions, the growth of such length function and the growth of the second Betti numbers of R/m [q] differ by at most a constant, as q → ∞.