## Abstract

Although free space optical (FSO) communication is a promising technique for deep space communication and it can help in the rapid development of space exploration missions, it encounters coronal turbulence during superior solar conjunction. To improve the bit error rate (BER) performance of FSO communication system under the influence of coronal turbulence, a hybrid modulation scheme, *L*-PPM-MSK-SIM–which is a combination of pulse position modulation (PPM), minimum shift keying (MSK), and sub-carrier intensity modulation (SIM) techniques–is proposed in this study. Considering various noise sources, both the BER and channel capacity of the communication system are evaluated under the lognormal (LN) turbulence channel. Our simulation results demonstrate that the BER performance with the *L*-PPM-MSK-SIM scheme is superior to that with *L*-PPM and BPSK-SIM schemes. In addition, the parameters of the coronal turbulence and FSO communication system have a tremendous influence on the link BER and channel capacity. Moreover, our results also revel that thermal noise is more predominant than the short noise and background noise for the BER performance of deep space FSO communication.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

There are currently a large number of deep space exploration projects that aim to promote our understanding of the universe’s evolution and to even ultimately expand the living space of human beings [1]. Considering the influence of ultra-long communication distance, large path loss, long transmission delay, solar scintillation, and so on, the communication strategy has a crucial role in guaranteeing that the data be successfully communicated between the Earth and probe. Compared with mature radio frequency (RF) technology, which has been well-employed in NASA’s deep space network, free space optical (FSO) communication is a complementary alternative for deep space communication and has received considerable attention [2–4]. The significant advantage in license-free frequency spectrum, high speed transmission, cost-effectiveness, and so on, further stimulates the growing application of FSO in deep space projects, such as Mars Laser Communications Demonstration [5], Mars Polar Lander [6], Mars Orbiter Laser Altimeter [7] and so on.

Although FSO communication is a promising application in deep space exploration, it faces a major challenge during superior solar conjunction, when the Sun lies between the Earth and the probe [4]. Under this circumstance, the FSO communication link will pass through the solar corona and encounter the turbulent solar wind plasma that is erupted from the Sun. This causes the signal intensity to dramatically fluctuate due to the variation of the refractive index in coronal turbulence during this period, which is referred to as solar scintillation [8, 9]. The bit error rate (BER) of the FSO communication system obviously increases at this time. Consequently, the link performance is degraded and the communication is even interrupted. The influence of coronal turbulence on Earth-to-probe links has been reported in reports from both the astronomy and communication communities [10–12]. Therefore, the question of how to enhance the link performance under the influence of coronal turbulence on a deep space FSO communication system during superior solar conjunction has been a challenging topic.

Several mitigation technologies, such as RF/FSO relay communication, spatial diversity, error correction coding, etc., have been reported in recent literature [2,4]. In addition, the modulation technique has been conducted in a FSO communication system when the optical waves propagate in atmospheric, oceanic, and coronal turbulence channels [13–15]. The results indicate that the influence of these turbulence on the FSO communication can be effectively reduced by adopting an appropriate modulation and demodulation schemes. Considering the deep space exploration scenario with FSO communication, many modulation schemes have been proposed with two main criteria: excellent BER performance and power efficiency [16,17].

As a well known binary modulation scheme, the on-off keying (OOK) technique is popular in FSO communication due to its simplicity and low cost [18]. In addition, several efficient detection methods of OOK modulation are proposed due to slow fading property of the FSO link [19]. Therefore, it has been deployed in the satellite communication based on the intensity modulation/direct detection (IM/DD) [20]. However, the requirement of adaptive threshold to optimally operate in the atmospheric turbulence is a major disadvantage. Consequently, this system first needs the knowledge of the turbulence state and noise level, which can be challenge when designing a system for real application. Alternatively, the pulse position modulation (PPM) technique has been proposed and widely employed due to the needless adaptive threshold [21]. In addition, the high average power efficiency of the PPM has led to research into its potential application in satellite optical communication [2]. Meanwhile, other derivative PPM techniques have been proposed to enhance power efficiency in price paid for bandwidth efficiency, such as *L*-PPM, which divides each symbol interval into *L* time slots. In [22], a close form BER formula was derived for the Earth-to-satellite FSO communication with an *L*-PPM scheme. The packet error rate of the ground-to-satellite laser up-link communication with the OOK and *L*-PPM schemes have been analyzed and compared in [23]. In our previous study [24], the PPM scheme has been applied to overcome the effect of coronal turbulence on BER of the optical link and shown excellent performance in Earth-to-Mars communication. Optical sub-carrier intensity modulation (SIM) has the advantages of not requiring an adaptive threshold as used by the OOK scheme and it has a higher bandwidth-efficient than PPM, which have led it to be applied in a FSO communication system under atmospheric and oceanic turbulence. The influence of the atmosphere turbulence on binary phase-shift keying (BPSK) SIM FSO communication for satellite-to-ground links is studied in [25], which further proves to be a robust scheme–even with strong background noise. In [26], the differential phase-shift keying (DPSK) SIM scheme has gained significant interest because of its 3 dB power efficiency improvement when compared to OOK modulation. However, the increased complexity for implementing SIM-based FSO communication system at the transmitter and receiver makes it unsuitable for the design of deep space FSO communication. Benefit from the narrow power spectrum and constant amplitude at each bit transition instant, minimum shift keying (MSK) technology has been well adopted in various FSO communication scenarios [27]. Besides, It has been demonstrated that the MSK technology exhibit exceptional performance in combined with the other modulation schemes [28].

According to this analysis, a modulation scheme with perfect BER performance and excellent power efficiency is urgently required to meet the rapid development of deep space exploration with FSO communication. Inspired by advantages of these modulation schemes, we propose an innovative hybrid modulation scheme, that combines the *L*-PPM, MSK, and SIM techniques to further improve the BER performance. The formulas of the BER and channel capacity for the deep space FSO communication system under coronal turbulence are derived based on the proposed *L*-PPM-MSK-SIM scheme. Then, numerical evaluations are carried out, which prove that the proposed *L*-PPM-MSK-SIM scheme achieves better BER performance than the BPSK-SIM and 2-PPM schemes under lognormal (LN) turbulent channel.

The rest of this paper is organized as follows. In Section 2, the configuration of the deep space FSO communication with *L*-PPM-MSK-SIM scheme is outlined first. The close form expressions for the BER and channel capacity are then given based on the proposed scheme under LN coronal turbulence channel. Our theoretical analysis results are discussed and presented in Section 3. Finally, Section 4 summarizes the conclusions.

## 2. System model and formulation

The signal transmitted from an Earth station will pass through the atmosphere, ionosphere, deep space, and even solar corona during superior solar conjunction, and it will finally be received by the probe during deep space exploration [4]. In contrast from the traditional deep space communication under S/X/Ka bands, a hybrid RF/FSO communication system is a promising scheme for the future deep space exploration and is employed here. In this system, the signal is first received by a relay satellite orbiting around the Earth by RF and it is then transmitted to the spacecraft via an optical wave. For simplicity, only the effect of coronal turbulence on the FSO communication link is considered in this study.

As we have discussed in Section 1, the combined scheme with *L*-PPM and MSK-SIM is proposed in this study to achieve higher BER performance for the deep space FSO communication during superior solar conjunction. The diagram of the communication system with *L*-PPM-MSK-SIM scheme is given in Fig. 1.

#### 2.1. Deep space FSO communication system with L-PPM-MSK-SIM scheme

As shown in Fig. 1, the information with a block of log_{2}*L* data bits is first converted and modulated into the *L*-PPM symbol format in the transmitter. Note that *L* denotes the average length of the input symbols. Then, the parallel-to-serial conversion is used to convert the *L*-PPM codeword into an array of data. The generated array is then transmitted into the sub-carrier signal with the MSK modulation schemes. Next, the signal is transmitted into a DC bias to ensure the bias current is larger than the threshold current, and finally passes through the turbulent channel as derived by the laser. Therefore, the modulated signal with *L*-PPM-MSK-SIM can be expressed as

*I*denotes the intensity of the

_{k}*k*-th code.

*m*and

*ω*represent the optical modulation index and the carrier frequency, respectively. The symbol interval is given by

*T*= log

_{s}_{2}

*L*/(

*R*) and

_{b}L*R*is the data bit rate.

_{b}*c*and

_{k}*θ*denote the codeword of

_{k}*L*-PPM and initial phase of the

*k*-th code element, respectively. In addition, the rectangular pulse of one time slot duration is

*h*(

*t*).

As we have analyzed in our former studies [4, 24], the optical waves commute between the relay satellite and the probe immediately comes into contact with the coronal turbulence during superior solar conjunction. Meanwhile, the beam will suffer from spreading, distortion, and even pointing instability that are induced by the temporal and spatial changes in the refractive index of the solar corona, as depicted in Fig. 1. This phenomenon is referred to as solar scintillation. In this study, we focus on the effect of weak solar wind turbulence on the propagation of the optical waves. Therefore, the lognormal (LN) turbulence distribution model is used here to characterize the coronal turbulence channel because it is well accepted for the weak turbulence level [15,21]. The probability density function of the optical wave intensity can be written as

*I*and

*I*

_{0}represent the received average irradiance in the turbulent medium and in the absence of the turbulence, respectively. The log intensity variance of the coronal turbulence, ${\sigma}_{l}^{2}$, which also is named as Rytov variance, normally dependent on the characteristics of the coronal channel. For the weak turbulence, the Rytov variance has a close relation with the variance of amplitude fluctuations 〈

*χ*

^{2}〉, and it can be simplified as: ${\sigma}_{l}^{2}=4\u3008{\chi}^{2}\u3009$ [17].

Considering the extra long transmission distance, the Rytov variance of a plane wave propagation through the coronal turbulence has been derived in our previous research and is rewritten as [15]

*r*is the classical electron radius.

_{e}*p*,

*L*, and

_{o}*η*represent the spectral index, the outer scale of the coronal turbulence, and the relative solar wind density fluctuations ratio, respectively. The optical wave propagation distance is

*L*. In addition, Γ (·) is the Gamma function and 〈·〉 represents the ensemble average. Note that the solar wind density,

_{link}*N*(

_{e}*r*), is a crucial parameter for the variance of the amplitude fluctuations, which is related to the heliocentric distance,

*r*, as shown in Fig. 1 and can be expressed as [29,30]

*R*denotes the radius of the Sun.

_{sun}We further let *K* = (*I*_{1} + *I*_{2} + *I*_{3} + · · · + *I _{N}*)/

*I*

_{0}be the sum of

*N*LN random variables, which obeys $K~N\left({m}_{K},{\sigma}_{K}^{2}\right)$. Here,

*m*and ${\sigma}_{K}^{2}$ denote the mean value and intensity variance of

_{K}*Z*. These two parameters can be achieved as

Therefore, the PDF of *K* can be finally expressed as

The distorted signal will then reach the receiver side. It will be first filtered by the optical band pass filter (OBPF) as depicted in Fig. 1. The APD photo detector is then applied to convert the optical signal into an electrical signal, and this signal is filtered by the electrical band pass filter (EBPF). Then, the signal is passed into the MSK demodulator, serial-to-parallel system, and PPM decoder successively. Because the photo currents are normally proportional to the modulated signal, the photo current generated by the APD and MSK demodulator for one symbol stream can be expressed as

*R*and

*G*denote the responsivity of the photo detector and the average gain of APD, respectively.

*α*is the channel attenuation. As we have analyzed in Section 1, the loss in the deep space communication link mainly consists of coronal attenuation and geometric loss. It is noteworthy that both the geometric loss and the coronal attenuation effects have been studied and it has been proven that the coronal attenuation results from the solar wind plasma is very small in the ultra-long transmission link [1,31]. In this study,

*α*is chosen as a constant value. In addition,

*n*(

*t*) in Eq. (7) represents receiver noise.

To give a comprehensive analysis of the impact of the noise on the FSO communication system, the thermal noise, shot noise, and background noise are all taken into consideration in this study [17]. The thermal fluctuations of the electrons in the receiver circuit lead to thermal noise. The variance of thermal noise is given as

where*k*denotes the Boltzmann constant.

_{B}*T*and

_{e}*F*are the temperature and amplifier noise figure of the receiver circuit, respectively. Δ

_{n}*B*=

_{n}*R*/2 is the effective noise bandwidth.

_{b}Induced by the fluctuations of the photon count, the shot noise occurs and is normally modeled as a stationary Gaussian random process. Because the APD photo detector is used in this paper, as shown in Fig. 1, the variance of the APD shot noise can be written as

where*q*is the electron charge,

*F*=

_{A}*ζG*+ (1 −

*ζ*) (2 − 1/

*G*) denotes the excess noise factor, and

*ζ*is the ionization factor.

The background noise occurs due to radiation produced by the environmental condition and is received by the FSO communication system, which will eventually deteriorate the receiver’s sensitivity. Its variance is obtained as

*W*(

*λ*),

*N*(

*λ*), and

*FOV*represent the spectral radiance of the sky, spectral radiant emittance of the Sun, photo detector field of view angle in radians, respectively. Δ

*λ*is the bandwidth of the OBPF.

Note that the thermal noise, shot noise, and background noise can be considered as an independent Gaussian random processes. Therefore, the resultant noise variance for the APD detector can be obtained simply by adding individual variances as

#### 2.2. BER performance and channel capacity analysis of the communication system

In this subsection, we devote to derive the BER and channel capacity formulas for the deep space FSO communication system with the hybrid *L*-PPM-MSK-SIM scheme. For the sake of calculation, the subcarrier phase, *θ _{k}*, is simplified as zero. Therefore, the signal at the output of MSK demodulator for one symbol period can be recast as

*n*(

_{s}*t*) is an additive white Gaussian noise with variance of ${\sigma}_{\mathit{ns}}^{2}={\sigma}_{L\u2013\text{PPM}}^{2}/2$. Note that ${\sigma}_{L\u2013\text{PPM}}^{2}={N}_{0}{R}_{b}L/(2{\text{log}}_{2}L)$ denotes the variance of the

*L*-PPM system and

*N*

_{0}represents the double-sided power spectral density of the Gaussian noise.

We further assume an equivalent probable data transmission for both conditional and unconditional BER calculation over the LN channel model. Therefore, the conditional BER for the *L*-PPM-MSK-SIM can be written as [18]

*Q*(·) is the Gaussian

*Q*function with $Q(x)={\int}_{x}^{\infty}\frac{1}{\sqrt{2\pi}}\text{exp}\left(-{t}^{2}/2\right)dt$.

By averaging the conditional BER in Eq. (13) over the PDF of the LN channel in Eq. (6), the unconditional BER for the deep space FSO communication system can be obtained as

By making the change of variable $x=\left(\text{ln}K-{m}_{K}\right)/\left(\sqrt{2}{\sigma}_{K}\right)$ and substituting it into Eq. (14), we have

Note that this equation cannot be simplified to a close form expression, but it can be approximated by the following Gauss-Hermite quadrature integration [17]

*w*and

_{i}*x*represent the corresponding weight factors and zero points of the

_{i}*n*-th order Hermite polynomial He

*(*

_{n}*x*), respectively. The accuracy of this approximate transformation is directly dependent on the

*n*value. In this paper, we chose

*n*= 20 in the subsequent simulations, which has been proven to meet the accuracy requirement very well [32].

By substituting Eq. (16) into Eq. (15), the tractable expression of the unconditional BER can be finally simplified as

By inserting Eqs. (8), (9), and (10) into Eq. (11), the variance of the total noise with the zero points of Hermite polynomial can be expressed as

Apart from the BER, the channel capacity is another key parameter for performance analysis in the deep space FSO communication system. The average channel capacity per unit bandwidth is normally considered as a random variable, which varies with signal-to-noise rate (SNR), *γ* = (*RI*)^{2}/*N*_{0}, and is expressed as

*p*(

_{γ}*γ*) is the PDF of

*γ*, and it can be achieved by the PDF,

*p*(

*K*), with respect to the random variable of

*γ*.

Note that the mean value, *m _{γ}*, and variance value, $4{\sigma}_{l}^{2}$, are related as ${m}_{\gamma}=\mathit{ln}(\gamma )-{\sigma}_{l}^{2}$. By substituting Eq. (20) into Eq. (19) and letting

*x*= ln

*γ*, the average capacity of deep space FSO communication system under LN turbulence channel can be obtained as

*φ*(

*x*) = ln[1 + exp(

*x*)],

*f*(

*x*) is the PDF of normal random variable

*X*, which obeys $X~N\left({\mu}_{x},{\sigma}_{x}^{2}\right)$. To obtain the close form solution for the average channel capacity, a 3-point estimate of random variable is used in this study [33]. According to the previous research [33], this PDF of

*X*can be further expressed as a sum of three discrete probability mass points located at ${\mu}_{x}-\sqrt{3}{\sigma}_{x}^{2}$,

*μ*, ${\mu}_{x}+\sqrt{3}{\sigma}_{x}^{2}$.

_{x}Therefore, the approximate channel capacity can be obtain by substituting Eq. (21) into Eq. (22) and is finally expressed as

## 3. Simulation and discussion

In this section, the performance of deep space FSO communication system based on the *L*-PPM-MSK-SIM scheme has been investigated. The BER performance with the proposed scheme is compared with the other schemes first. After that, both the BER and channel capacity are taken into consideration to evaluate the impact of various parameters under LN channel model. Note that the parameters used for this communication system were taken from the astronomy report and from NASA’s deep space exploration report [12,34]. For the sake of conciseness, and without loss of generality, some of the key parameters are given in Table 1 unless otherwise specified. Note that the value of FOV is one of the crucial parameters for the effect of the background noise and has been studied in [35]. In this paper, a large FOV value is adopted to meet the real deep space communication.

To evaluate the BER performance with the *L*-PPM-MSK-SIM scheme, both the analytical and simulation results are carried out for *L* = 2 under different wavelengths, *λ* = 850 nm and *λ* = 1064 nm. Both the 2-PPM and BPSK-SIM schemes are adopted for comparison. Note that the data rate, *R _{b}*, and the SNR,

*γ*, are kept constant for the sake of fair comparisons between these modulation schemes. The simulation results are depicted in Fig. 2. Note that the results for different schemes working with

*λ*=850 nm are represented by the lines and symbols with blue color. To the contrary, the lines and symbols with red color mean the schemes with

*λ*=1064 nm. Besides, both the theoretical analysis and numerical simulation results are represented by various lines and symbols, respectively.

As shown in Fig. 2, it is obvious that the BER performance with 2-PPM-MSK-SIM scheme is much better than that of BPSK-SIM and 2-PPM schemes. The quantitative result of Fig. 2 is given in Table 2 for a constant BER value of 10^{−10}. It is observed that the required average irradiance, *I _{o}*, for 2-PPM-MSK-SIM is −6.6 dBm, whereas it is −1.8 dBm and 3.6 dBm to obtain a standard BER of 10

^{−10}for 2-PPM and BPSK-SIM schemes at the wavelength of

*λ*= 850 nm. Similarly, at the larger wavelength of

*λ*= 1064 nm, the required average irradiance for 2-PPM-MSK-SIM, 2-PPM, and BPSK-SIM schemes are −5.2 dBm, 0.7 dBm, and 6.9 dBm, respectively. Therefore, this demonstrates that the 2-PPM-MSK-SIM scheme avoids the power penalty of 4.8 dB and 10.2 dB at

*λ*= 850 nm in comparison with 2-PPM and BPSK-SIM schemes, respectively. Note that this advantage of 2-PPM-MSK-SIM scheme is more obvious at

*λ*= 1064 nm, but is not repeated for the sake of simplicity. In addition, the BER decreases as the wavelength decreases. This phenomenon revels that the effect of the coronal turbulence can be restrained by decreasing the wavelength.

Apart from these phenomena, the results of theoretical analysis of these modulation schemes are fairly consistent with the numerical simulation and under different wavelengths, as shown in Fig. 2 and Table 2. In addition, the error between the theoretical analysis and numerical simulation of the 2-PPM-MSK-SIM is smaller than the other two schemes. These phenomena indicate that the theoretical results of the proposed 2-PPM-MSK-SIM scheme are correct. This can be attributed to the fact that the proposed modulation scheme takes advantage of the other modulation techniques. Therefore, its theoretical analysis results are close to the numerical simulation when compared with the other techniques. In addition, the BER performance of the deep space FSO communication system can be significantly improved by the *L*-PPM-MSK-SIM over the LN turbulence channel during superior solar conjunction.

According to this analysis, the *L*-PPM-MSK-SIM scheme is used next to evaluate the impacts of some crucial parameters on the BER and channel capacity due to its favorable performance. The influence of the average length of symbol *L* on the BER and channel capacity is demonstrated in Fig. 3. This scheme requires the average irradiance of 1.2 dBm at *L* = 2 for a standard BER of 10^{−10}; however, it increases to 4.3 dBm at *L* = 4, 6.7 dBm at *L* = 8, and 9.4 dBm at *L* = 16. Therefore, it can be concluded that the BER performance decreases with the increase of the average length of symbol. The same conclusion also can be obtained for the channel capacity, as shown in Fig. 3(b). For example, at the constant average irradiance, *I _{o}* = 5 dBm, the channel capacity with

*L*= 4 is less 1.9 bits/s/Hz than at

*L*= 2. The channel capacities with

*L*= 8 and

*L*= 16 are less 1.1 bits/s/Hz and 2.4 bits/s/Hz than

*L*= 4, respectively. These phenomenon can be explained from the following physical mechanism. With the increase of

*L*, the modulation order, log

_{2}

*L*, is enlarged. Therefore, plenty of data are changed by the parallel-to-serial module at the transmitter, while some of the data cannot be effectively transmitted. After the signal passes through the coronal turbulence and reaches the receiver, the data will also be converted by the serial-to-parallel module and judged by the PPM demodulator according to the maximal photo current. During this process, the error probability increases due to the large modulation order and the channel capacity decreases.

Figure 4 demonstrates the impact of the heliocentric distance, *r*, on the BER performance and channel capacity. Note that some parameters are set as *L* = 2 and *I _{o}* = −5 dBm. From Fig. 4, it is evident that the BER decreases with the increase of

*r*by 2-PPM-MSK-SIM scheme. This conclusion has also been achieved in our previous research with the 2-PPM scheme [24] and can be explained as that the intensity of the coronal turbulence decreases along with the increase of heliocentric distance due to the decrease of the solar wind density according to Eq. (4). Meanwhile, it will have less impact on the BER performance and also enhance the channel capacity. Note that the channel capacity in the case with no coronal turbulence is also depicted in Fig. 4(b) for comparison.

The influence of some other parameters on the BER of deep space FSO communication system with 2-PPM-MSK-SIM scheme under coronal turbulence channel is shown in Fig. 5. Note that the color bar on the right-hand of Fig. 5 denotes the normalized BER value. As shown, the normalized BER decreases with the decrease of the wavelength, which has been depicted and analyzed in Fig. 2. In addition, it will also decrease along with the increase of spectral index. With the decrease of solar wind density fluctuation ratio and increase of outer scale, the BER obviously decreases. This is reasonable because the intensity of the coronal turbulence decrease due to the small fluctuation ratio and the focusing effect of the outer scale is more prominent during this period. As shown in Fig. 5(c)–5(d), the larger gain, responsivity, and APD load resistance will induce less BER. However, the larger bit rate results in more BER. This variation tendency can be explained because the effective sensitivity of the photo detector is improved with a larger *R*, *G*, and *R _{L}*. In addition, more transmission bandwidths are needed to meet the large bit rate; therefore, it will induce high noise power and finally result in larger BER.

The channel capacity at the constant average irradiance, *I _{o}* = 5 dBm, using the 2-PPM-MSK-SIM scheme under the effect of spectral index, wavelength, bit rate, and APD load resistance is given in Table 3. The simulation results also revel that decreasing the wavelength is an effective way to increase the channel capacity, and the performance is obviously improved under the larger spectral index. In addition, the capacity will decrease with the increase of the bit rate and the decrease of the load resistance. Note that the influence of the fluctuation ratio, outer scale, responsivity, and gain on the capacity are ignored for the sake of simplicity.

Figure 6 illustrates the BER given by Eqs. (17) and (18) against the average irradiance for different noise limiting conditions. For a constant average irradiance value, such as *I _{o}* = −15 dBm, the BER induced by the thermal noise is larger than that resulting from short noise and background noise. In addition, the thermal noise predominantly induces BER when compared with the BER caused by the total noise. Therefore, we can conclude that the performance of deep space FSO communication system with

*L*-PPM-MSK-SIM scheme is mainly limited by thermal noise. Moreover, in a thermal noise-limited condition, the communication system will still need approximately 10 dB of average irradiance compared with the background noise.

## 4. Conclusion

In this paper, the BER performance and channel capacity for optical waves propagation through the coronal turbulence with LN channel are evaluated with the hybrid *L*-PPM-MSK-SIM technique. The results reveal that the proposed *L*-PPM-MSK-SIM technique achieves better BER performance than the traditional *L*-PPM and BPSK-SIM schemes, both in numerical simulation and theoretical analysis. In addition, the performance of the deep space FSO communication system is highly dependent upon the coronal turbulence parameters. Both the BER and channel capacity can be optimized by proper selection of the system parameters, such as the wavelength, bit rate, APD load resistance, and so on. Moreover, we found that thermal noise has a dominant role in BER performance when compared with the shot noise and background noise. Thus, the *L*-PPM-MSK-SIM scheme can help to improve the performance of a deep space FSO communication system during superior solar conjunction.

## Funding

Natural National Science Foundation (NSFC) (61801181, 61831008); Open Research Fund of Key Laboratory for Information Science of Electromagnetic Waves (MoE), Fudan University (EMW201902); The Verification Platform of Multi-tier Coverage Communication Network for Oceans (PCL2018KP002).

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