Error Characterization of Differential Detection and Non-Differential Detection for MIMO UWOC Systems in Seawater Turbulent Channels

Abstract

Ocean turbulence is an important factor affecting the development of underwater wireless optical communication (UWOC). To improve the error characteristics of the underwater optical communication system, we propose a differential detection-based multi-receiver-multi-transmitter (MIMO) underwater laser communication transmission method. Additionally, we derive the expressions for calculating the average BER of the MIMO underwater wireless optical communication system with differential detection and non-differential detection in the case that the two transmitted beams are completely uncorrelated. The error characteristics of the MIMO system are simulated and analyzed from the perspective of ocean turbulence intensity and link distance. The simulation results show that the differential detection method has a lower average BER compared to the non-differential detection method in the case of moderate-to-strong ocean turbulence. In addition, the differential detection methods do not have the error floor effect, and non-differential detection methods have the error floor effect. The more the turbulence intensity affects the average BER of the MIMO UWOC system with the increase of the communication link distance, the more obvious is the effect of the turbulence intensity on the average BER of the MIMO UWOC system. Accordingly, the simulation analysis shows that the differential detection method is more suitable for the construction of communication links under long-distance and medium-strong turbulence.

1. Introduction

As an important strategic space for sustainable human development, the ocean not only provides sufficient natural gas, oil, and other material energy for people’s lives but is also an important strategic high point for military development in the world today [1]. The development of devices, such as marine environmental monitors, underwater high-resolution imaging systems, and autonomous underwater vehicles (AUVs), has increased the need for real-time communication and high-capacity data transmission between underwater end devices [2]. Space laser communication is gradually becoming the main way to solve the problem of underwater high-capacity data transmission because of its strong anti-interference capability [3], high transmission capacity [4], high rate [5], and fast deployment [6]. However, the reasons affecting the development of UWOC are the scattering and absorption of light by seawater and ocean turbulence, where ocean turbulence subject to changes in salinity, temperature, and seawater density is the main factor affecting beam propagation [7]; turbulence especially causes scintillation effects, resulting in the received light energy showing violent jitter, which will seriously degrade the performance of UWOC systems [8]. To reduce the impact of ocean turbulence on underwater laser communication systems, researchers in various countries have conducted extensive research.

Considering the current similarity between ocean turbulence and atmospheric turbulence, the analysis of the propagation characteristics of ocean turbulence on light waves is mainly based on the classical atmospheric turbulence research results in free optical communication. Ruijie Li et al. [9] took Malaga (M) turbulence channel as an example and analyzed the average error probability with pointing error under atmospheric turbulence channel and also analyzed the BER closure expressions for different modulation techniques under different turbulence conditions. Ansari et al. [10] analyzed the basic functional expressions for pointing error, average BER, and traversal capability based on the Malaga turbulence model to illustrate the impact of atmospheric turbulence intensity and pointing error severity on the performance of communication systems. M. Khalighi et al. [11] artificially reduces the effect of background radiation on terrestrial free-space optical communication systems. They proposed using dual wavelength transmitting data and detecting the data by differential detection mode at the receiving end. In addition, they derived the integral function relationship between the signal-to-noise ratio of differential detection and the average BER. Toselli et al. [12] proposed to reduce the effect of flicker index on the error characteristics of communication systems by using a combination of adaptive optics and aperture averaging.

However, the seawater channel is a complex time-varying channel whose random ups and downs of refractive index are affected by the random changes of seawater temperature and salinity as well as the special characteristics of underwater laser communication wavelengths and detector device performance. In particular, the effect of ocean turbulence on light has a very important role in the design of underwater laser communication systems; thus, the needs of ocean turbulence and high-speed data transmission should be considered comprehensively.

Rahman et al. [13] conducted modeling simulations for multilayer vertical underwater links and analyzed the performance of underwater optical communication systems using ocean turbulence models, such as generalized gamma (GG), exponential GG (EGG), exponentiated Weibull (EW), and gamma-gamma (ΓΓ) as well as derived expressions for the signal-to-noise ratio of the probability density function and cumulative distribution function. Yuqing Fu et al. [14] investigated the effect of aperture averaging on the average BER of DPSK UWOC systems in the case of moderately strong turbulence based on planar and spherical wave transmission models. Meanwhile, Gökçe, Muhsin Caner et al. [15] quantified the improvement of the aperture averaging of Gaussian beams on the average BER of communication systems under weak turbulence conditions. I. Yahya Baykal et al. [16] analyzed the upper bound on the average BER of ppm-owc links operating in ocean turbulence, and simulations verified that the error performance of PPM is better than OOK modulation. Hema, R. et al. [17] proposed a DC-biased optical orthogonal frequency division multiplexing technique with MIMO for underwater wireless optical channels and simulated and analyzed the communication performance in terms of signal-to-noise ratio, channel capacity, and BER under weak turbulence conditions. Jianying Wang et al. [18] analyzed the effect of pointing error and beam spread scenarios on the average BER of MIMO underwater communication systems using the Malaga turbulent channel as an example, although many scholars have analyzed methods to improve the average BER performance in oceanic turbulent channels. However, none of the above literature have analyzed the different detection performance of MIMO UWOC systems under medium-strength turbulence [19]. Medium-strength turbulence is an important component that affects seawater turbulence; thus, it is necessary to analyze the system performance of MIMO UWOC systems from the point of view of practical application and enrichment of theory.

The gamma-gamma model can better simulate the medium-intensity ocean turbulence. This paper analyzes the error characteristics of two MIMO UWOC systems, differential detection, and non-differential detection based on the gamma-gamma turbulence model. The expressions for the average BER calculation of differential detection and non-differential detection MIMO UWOC under the influence of gamma-gamma turbulence are pushed in the case of completely uncorrelated received signals. Based on the above derived expressions, the effects of turbulence intensity and communication distance on the BER of MIMO UWOC are simulated and analyzed, while the error floor effect of MIMO UWOC detection and reception system is simulated and analyzed, and the average BER performances of differential and non-differential detection in MIMO UWOC are compared.

The novelty of this paper is reflected in the following two aspects:

• The expressions for calculating the average BER of two MIMO UWOC systems with differential detection and non-differential detection are derived based on the gamma-gamma turbulence model when the two transmitted beams are completely uncorrelated.
• Simulation demonstrates that the differential detection method has a lower average BER compared with the non-differential detection method from the perspective of ocean turbulence intensity and communication link distance. Additionally, the differential detection method does not have an error flattening effect, while the non-differential detection method has an error flattening effect.

2. UWOC System Modeling

2.1. MIMO UWOC System Transmission Model

The composition of the MIMO-based underwater laser communication system is shown in Figure 1. An example of a differential detection system is shown in Figure 1a. Firstly, the signal-processing board is used to divide the band modulation information into the original signal, $S$ and the inverted signal, $\overline{S}$, which are then modulated to the lasers with wavelengths, ${\lambda }_1$ and ${\lambda }_2$, respectively. Affected by the absorption scattering of the water channel, usually underwater optical communication uses emission wavelengths of 450 nm and 532 nm. We designed the transmitting antennas of the two beams to be a few centimeters or closer to each other to ensure that the two beams have the same turbulence impact after passing through the ocean turbulence channel. After passing through the ocean turbulence channel with communication distance, L, to reach the laser communication receiver, let the optical signal reaching the receiver be $y_1$ and $y_2$. First through the optical filter, filter out the background noise and another transmitter light signal after receiving the optical antenna focus to the detector target surface. The photodetector converts the optical signal to the electrical signal, and then the output signal is subtracted and judged to recover the original data transmission information.

The non-differential detection system is shown in Figure 1b. Non-differential detection is different from differential detection in that it does not require the transmit information to be inverted. The two transmit signals are of the same amplitude and same phase. After the modulator is loaded on the laser, which makes the optical signal of the same modulation information that is emitted, the laser emission is completed by the transmitting optical antenna. After the ocean turbulence channel, the receiving optical antenna will focus the spatial light to the detector target surface, and the detector output electrical signal will be superimposed and judged to recover the original data transmission information.

Based on the above theory, it is known that the output signals of the two detectors in the MIMO system are

$$V_1={\tau }_1{R}_1{y}_1+n_1={\tau }_1{R}_{1}SI{h}_1+n_1,$$

(1)

$$V_2={\tau }_2{R}_2{y}_2+n_2={\tau }_2{R}_2\overline{S}I{h}_2+n_2,$$

(2)

where $S$ is the signal after sending data “1” or “0”. $\overline{S}$ is the signal after sending data inverse when $S$ is 1 and $\overline{S}$ is 0. $I$ is the light intensity when the transmission signal is “1”. $n_1$ and $n_2$ are the noise of the two detectors, respectively. The noise satisfies the Gaussian white noise with mean value 0 and variance $N_0/2$. $h_1$ and $h_2$ are the random attenuation of the optical signal by ocean turbulence. $R_1$ and $R_2$ are the detector photoelectric conversion efficiency. ${\tau }_1$ and ${\tau }_2$ are the optical loss of the system. Since the detectors used in the MIMO laser communication system are all the same, the photoelectric conversion efficiency of the detectors can be simplified to $R_1=R_2=R$, and the optical loss of the system can be simplified to ${\tau }_1={\tau }_2=\tau$.

It follows that the output signal of the differential detection system is

$$V_R=V_1-V_2={\tau }_1{R}_{1}SI{h}_1-{\tau }_2{R}_2\overline{S}I{h}_2+n_1-n_2=\{\begin{array}{ll}{\tau }_1{R}_{1}I{h}_1+n_1-n_2,& S=1\\ -{\tau }_2{R}_{2}I{h}_2+n_1-n_2,& S=0\end{array},$$

(3)

The output signal of the non-differential detection system is

$$\begin{array}{c}{V}_R=V_1+V_2={\tau }_1{R}_{1}SI{h}_1+{\tau }_2{R}_{2}SI{h}_2+n_1+n_2\\ =\{\begin{array}{l}{\tau }_1{R}_{1}I{h}_1+{\tau }_2{R}_{2}I{h}_2+n_1+n_2,S=1\\ n_1+n_2,S=0\end{array}\end{array}.$$

(4)

2.2. UWOC Turbulence Channel Model

According to the literature [20,21,22,23], it is known that the gamma-gamma turbulence model can accurately describe the probability density function of the signal light intensity decay under the medium-intensity ocean turbulence effect [24]; thus, the probability density function is expressed as

$$f\left(h_1\right)=\frac{2\left({\alpha }_1{\beta }_1\right)}{\Gamma \left({\alpha }_1\right)\Gamma \left({\beta }_1\right)}{h}_1^{\frac{{\alpha }_1+{\beta }_1}{2}-1}{K}_{{\alpha }_1-{\beta }_1}\left(2\sqrt{{\alpha }_1{\beta }_1{h}_1}\right),$$

(5)

$$f\left(h_2\right)=\frac{2\left({\alpha }_2{\beta }_2\right)}{\Gamma \left({\alpha }_2\right)\Gamma \left({\beta }_2\right)}{h}_2^{\frac{{\alpha }_2+{\beta }_2}{2}-1}{K}_{{\alpha }_2-{\beta }_2}\left(2\sqrt{{\alpha }_2{\beta }_2{h}_2}\right),$$

(6)

where $K_v(·)$ is a Bessel function of the second kind with order $v$. $\Gamma (·)$ is expressed as Gamma function. If the optical radiation at the receiving end is a plane wave, $\alpha$ and $\beta$ in the intensity probability density function are related to ocean conditions [25], which are defined as:

$$\alpha ={\left[\exp \left(\frac{0.49{\sigma }_l^2}{{\left(1+1.11{\sigma }_l^{12/5}\right)}^{7/6}}\right)-1\right]}^{-1},$$

(7)

$$\beta ={\left[\exp \left(\frac{0.51{\sigma }_l^2}{{\left(1+0.69{\sigma }_l^{12/5}\right)}^{5/6}}\right)-1\right]}^{-1},$$

(8)

where ${\sigma }_l^2$ is the Rytov variance of the laser intensity undulation in ocean turbulence. Because of the close distance between the two transmitting antennas, the two light transmissions can be in the same dimensional and weather conditions. Since the value of the turbulence structure constant, $C_n^2$, depends on environmental factors, such as also the height and weather conditions, it is known that both beams pass through the same turbulence, $C_{n_1}^2=C_{n_2}^2=C_n^2$. For a horizontally propagating optical field in a turbulent medium, the refractive index structure constant, $C_n^2$, is a constant value, and the Rytov variance of the plane wave can be expressed as

$${\sigma }_l^2=1.23C_n^2{k}^{7/6}{L}_p^{11/6}$$

(9)

where k is the wave vector k = 2π⁄λ and λ is the laser carrier wavelength. $L_p$ is the link distance. The design chooses laser emission wavelengths of 450 nm and 532 nm because the distance between the two transmitters and the receiver is the same, and thus, it can be assumed that they pass through the same underwater turbulence channel; we have ${\alpha }_1={\alpha }_2=\alpha$ and ${\beta }_1={\beta }_2=\beta$. Then the gamma-gamma turbulence model of the MIMO system can be simplified as

$$f\left(h_1\right)=\frac{2{\left(\alpha \beta \right)}^{\frac{\alpha +\beta }{2}}}{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{h}_1^{\frac{\alpha +\beta }{2}-1}{K}_{\alpha -\beta }\left(2\sqrt{\alpha \beta h_1}\right),$$

(10)

$$f\left(h_2\right)=\frac{2{\left(\alpha \beta \right)}^{\frac{\alpha +\beta }{2}}}{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{h}_2^{\frac{\alpha +\beta }{2}-1}{K}_{\alpha -\beta }\left(2\sqrt{\alpha \beta h_2}\right).$$

(11)

3. MIMO System Error Characterization

The UWOC system is designed using intensity modulation/direct detection (IM/DD) without considering external factors, such as receiver alignment problems in the communication system, background light in the channel, and other interference between devices. Assuming that the communication system is affected only by the fading and additive Gaussian white noise under the transmission path of the turbulent channel, the instantaneous BER equation can be expressed as

$$P_e=P\left(0\right)P\left(e/0\right)+P\left(1\right)P\left(e/1\right),$$

(12)

where $P\left(0\right)$ and $P\left(1\right)$ are the probabilities of sending data 1 and 0 with a mean value of 0.5. $P\left(e/0\right)$ is the probability that a “0” pulse is incorrectly judged as a “1”. $P\left(e/1\right)$ is the probability that a “1” pulse is incorrectly judged as a “0”.

3.1. Derivation of the Average BER at Differential Detection

If the sent data amplitude satisfies the normal distribution, then the probability of sending a “1” code satisfies $N\left[\left(\tau RI{h}_1\right),N_0\right]$, and $N_0$ is the variance. The probability of sending a “0” code satisfies $N\left[\left(\tau RI{h}_2\right),N_0\right]$ so that the instantaneous error rate of a wrong decision is

$$P\left(e/0\right)={\displaystyle \underset{v_{th}}{\overset{\infty }{\int }}\frac{1}{\sqrt{2\pi N_0}}}\exp \left[-\frac{{\left(x+\tau RI{h}_2\right)}^2}{2N_0}\right]dx=\frac{1}{2}erfc\left(\frac{v_{th}+\tau RI{h}_2}{\sqrt{2N_0}}\right),$$

(13)

$$P\left(e/1\right)={\displaystyle \underset{-\infty }{\overset{v_{th}}{\int }}\frac{1}{\sqrt{2\pi N_0}}}\exp \left[-\frac{{\left(x-\tau RI{h}_1\right)}^2}{2N_0}\right]dx=\frac{1}{2}erfc\left(\frac{\tau RI{h}_1-v_{th}}{\sqrt{2N_0}}\right),$$

(14)

where $erfc\left(x\right)=\frac{2}{\sqrt{\pi }}{\displaystyle \underset{x}{\overset{\infty }{\int }}{e}^{-{\eta }^2}d\eta }$ is the complementary error function. Then, the average BER of the MIMO system can be expressed as

$$\langle P_e\rangle ={\displaystyle \underset{0}{\overset{\infty }{\int }}{\displaystyle \underset{0}{\overset{\infty }{\int }}f\left(h_1,h_2\right)P_e}}d{h}_{1}d{h}_2,$$

(15)

where $f\left(h_1,h_2\right)$ is the joint probability density of $h_1$ and $h_2$. For two signals that are completely uncorrelated, then, $h_1$ and $h_2$. obey the same gamma-gamma distribution, so $h_1=h_2=h$.When the judgment threshold is an over-zero judgment, the instantaneous BER at which an erroneous judgment occurs can be simplified as

$$P\left(e/0\right)=P\left(e/1\right)=\frac{1}{2}erfc\left(\frac{\tau RIh}{\sqrt{2N_0}}\right).$$

(16)

Then, the average BER of differential detection is simplified as

$$\langle P_e\rangle ={\displaystyle \underset{0}{\overset{\infty }{\int }}f\left(h\right)P_e}dh.$$

(17)

Bringing Equations (10)–(12) and (16) into Equation (17) yields

$$\langle P_e\rangle =\frac{{\left(\alpha \beta \right)}^{\frac{\alpha +\beta }{2}}}{\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{\displaystyle \underset{0}{\overset{\infty }{\int }}{h}^{\frac{\alpha +\beta }{2}-1}{K}_{\alpha -\beta }\left(2\sqrt{\alpha \beta h}\right)}erfc\left(\frac{\tau RIh}{\sqrt{2N_0}}\right)dh,$$

(18)

Based on the performance of the Meijer G function $K_v(·)=\frac{1}{2}{G}_{0,2}^{2,0}\left[\frac{x^2}{4}|\begin{array}{c}-\\ v/2,-v/2\end{array}\right]$, $erfc\left(\sqrt{x}\right)=\frac{1}{\sqrt{\pi }}{G}_{1,2}^{2,0}\left[x|\begin{array}{c}1\\ 0,1/2\end{array}\right]$ and Equation (21) in the literature [26], Equation (18) can be reduced (see Appendix A for details of the reduction process).

$$\langle P_e\rangle =\frac{2^{\alpha +\beta -3}}{{\pi }^{3/2}\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{G}_{5,2}^{2,4}\left(\frac{8{\tau }^2}{{\alpha }^2{\beta }^2}{\gamma }_0|\begin{array}{c}\frac{1-\alpha }{2},\frac{2-\alpha }{2},\frac{1-\beta }{2},\frac{2-\beta }{2},1\\ 0,\frac{1}{2}\end{array}\right),$$

(19)

3.2. Derivation of Average BER for Non-Differential Detection

If the sent data amplitude satisfies the normal distribution, then the probability of sending a “1” code satisfies $N\left[\left(\tau RI{h}_1+\tau RI{h}_2\right),N_0\right]$, and $N_0$ is the variance so that the instantaneous error rate of a wrong decision is:

$$P\left(e/0\right)={\displaystyle \underset{v_{th}}{\overset{\infty }{\int }}\frac{1}{\sqrt{2\pi N_0}}}\exp \left[-\frac{x^2}{2N_0}\right]dx=\frac{1}{2}erfc\left(\frac{v_{th}}{\sqrt{2N_0}}\right),$$

(20)

$$\begin{array}{ll}P\left(e/1\right)& ={\displaystyle \underset{-\infty }{\overset{v_{th}}{\int }}\frac{1}{\sqrt{2\pi N_0}}}\exp \left[-\frac{{\left(x-\tau RI{h}_1-\tau RI{h}_2\right)}^2}{2N_0}\right]dx\\ & =\frac{1}{2}erfc\left(\frac{\tau RI{h}_1+\tau RI{h}_2-v_{th}}{\sqrt{2N_0}}\right)\end{array},$$

(21)

When the judgment threshold uses over-zero judgment, the instantaneous BER at which an erroneous judgment occurs can be simplified as

$$P\left(e/0\right)=\frac{1}{2}erfc\left(\frac{0}{\sqrt{2N_0}}\right)=\frac{1}{2},$$

(22)

$$P\left(e/1\right)=\frac{1}{2}erfc\left(\frac{\tau RI{h}_1+\tau RI{h}_2}{\sqrt{2N_0}}\right),$$

(23)

Because $h_1=h_2=h$, then the instantaneous BER at which an error judgment occurs $P\left(e/1\right)$ can be simplified as

$$P\left(e/1\right)=\frac{1}{2}erfc\left(\frac{2\tau RIh-v_{th}}{\sqrt{2N_0}}\right),$$

(24)

Then, the average BER of non-differential detection is simplified as

$$\langle P_e\rangle ={\displaystyle \underset{0}{\overset{\infty }{\int }}f\left(h\right)P\left(0\right)P\left(e/0\right)+f\left(h\right)P\left(1\right)P\left(e/1\right)}dh,$$

(25)

Based on the performance of the Meijer G function, the average BER formula for non-differential detection can be obtained by bringing Equations (10), (22) and (24) into Equation (25) (see Appendix B for details of the derivation process).

$$\begin{array}{ll}\langle P_e\rangle & =\frac{\alpha \beta }{4\Gamma \left(\alpha \right)\Gamma \left(\beta \right)\left(1-\alpha \right)\left(1-\beta \right)}\\ & +\frac{2^{\alpha +\beta -4}}{{\pi }^{3/2}\Gamma \left(\alpha \right)\Gamma \left(\beta \right)}{G}_{5,2}^{2,4}\left(\frac{8{\tau }^2}{{\alpha }^2{\beta }^2}{\gamma }_0|\begin{array}{c}\frac{1-\alpha }{2},\frac{2-\alpha }{2},\frac{1-\beta }{2},\frac{2-\beta }{2},1\\ 0,\frac{1}{2}\end{array}\right)\end{array}.$$

(26)

4. Numerical Simulation of MIMO UWOC System

The BER performance of the MIMO UWOC system is simulated and analyzed, and the parameters used in the simulation process are shown in

Table 1. MIMO UWOC Simulation Parameters.

Parameter	Value	Unit
Wavelength, $\lambda$	450, 532	nm
Turbulence structure constant, $C_n^2$	$8.5\times {10}^{-15}$, $1.5\times {10}^{-14}$, $5.0\times {10}^{-14}$	${\mathrm{m}}^{-2/3}$
Photoelectric conversion efficiency, $R$	1	A/W
Link distance, $L$ Optical loss rate, $\tau$	1000, 3000	m
	0.8	%
Signal to ratio, ${\gamma }_0$	0–100	dB

. The distance between the two beams is set to be similar in the simulation process so their ocean turbulence attenuation coefficients obey the gamma-gamma function with the same parameters.

4.1. Analysis of Ocean Turbulence Intensity on the Average BER of MIMO UWOC Systems

To analyze the error characteristics of marine turbulence on the MIMO communication system, the average BER versus SNR curves of MIMO are simulated for weak, medium, and strong turbulence, $C_n^2=8.5\times {10}^{-15}$, $1.5\times {10}^{-14}$, $5.0\times {10}^{-14}$, at a link distance of 1 km. The simulation results are shown in Figure 2.

According to the simulation results, the average BER of DD and NDD MIMO communication systems increases gradually with the increase of turbulence intensity at the same SNR. For the same average, BER = $2.28\times {10}^{-7}$, the required SNR of DD under weak turbulence, $C_n^2=8.5\times {10}^{-15}$, is 35 dB less than that under medium turbulence $C_n^2=1.5\times {10}^{-14}$ of 45 dB.

In the case of $C_n^2=8.5\times {10}^{-15}$, the average BER of the MIMO system with NDD is better than that of differential detection when the SNR is less than 33 dB. When the signal-to-noise ratio is greater than 33 dB, the average BER of DD is better than that of NDD.

4.2. Analysis of Link Distance on the Average BER of MIMO UWOC Systems

The turbulence structure constant is set to $C_n^2=1.5\times {10}^{-14} {\mathrm{m}}^{-2/3}$, and the average BER of differential detection and non-differential detection MIMO systems with link distances of $L=1\mathrm{km}$ and $2\mathrm{km}$ are analyzed, respectively. The simulation results are shown in Figure 3, the longer the link distance, the worse the average BER of the MIMO system. The reason for the above phenomenon is that when the transmission distance increases, the probability of detecting the deep fading of the received signal light intensity increases, and the fading of the light intensity makes the signal light smaller than the background noise at the receiver side, which makes the judgment of the data incorrect so the link distance has a greater impact on the average BER of the DD MIMO system.

4.3. Analysis of Error Floor Effect of MIMO UWOC System

The error-flooring effect is that when the signal-to-noise ratio is raised to a certain value, the BER is maintained at a fixed level without decreasing. In order to analyze whether the differential detection system can overcome the error floor effect, the average BER curve of DD with the signal-to-noise ratio in the 0–120 dB range is simulated, and the simulation results are shown in Figure 4.

The simulation results show that the average BER of the DD MIMO communication system keeps decreasing with the increase of SNR; thus, it is concluded that there is no error-flattening effect in the differential detection MIMO communication system. In the weak turbulence case of $C_n^2=8.5\times {10}^{-15}$, the average BER of the non-differential detection system does not decrease with the increase of SNR after the SNR increases to 40 dB.

5. Conclusions

In this paper, the error characteristics of the MIMO underwater wireless optical communication system are analyzed, and the average BER expressions for the two cases of differential detection and non-differential detection are derived based on the gamma-gamma turbulence model. The average BER performance of the MIMO UWOC system is simulated and analyzed in terms of both ocean turbulence intensity and communication distance. The simulation results show that when the signal-to-noise ratio increases to 40 dB, the non-differential detection system shows the error floor effect, while the differential detection system does not have the error floor effect. The longer the communication link distance, the more obvious the effect of ocean turbulence on the average BER of the MIMO system, and the average BER of the differential detection is better than that of the non-differential detection system. Therefore, the differential detection provides theoretical support for the establishment of long-range UWOC links under moderately strong turbulence.