1. Introduction

With the scarcity of radio-frequency spectrum, free-space optical (FSO) communication covering ultra-wide unlicensed spectrum has attracted growing attention [1,2]. For remarkable features such as high capacity and feasible deployment, the FSO technique promisingly offers high-capacity connectivity to 5G/6G communication [3,4]. Nevertheless, the near-ground FSO links are susceptible to random fluctuations induced by atmospheric turbulence, the dominant factor degrading transmission performance [1]. The joint of the forward error correction (FEC) and spatial diversity [5–7] has been demonstrated to mitigate the fading. The low-dense parity-check (LDPC) coded multiple-input multiple-output (MIMO) FSO communication system first appeared in [5], where LDPC codes and spatial diversity reception could combat the turbulence fading when the sub-channels are independent. In practice, the deterioration caused by spatially correlated (SC) fading is not negligible, especially in moderate-strong turbulence channels [8]. The polar-coded MIMO-FSO systems with SC fading using on-off key (OOK) modulations were introduced in [7], where polar codes surpassed LDPC codes in the same conditions. However, the OOK modulations cannot meet the capacity requirements of future optical communication. Unipolar M-ary pulse amplitude modulations (M-PAM) are the candidates to achieve a near-capacity in FSO systems [9], and the non-uniform distributions of M-PAM are capable of further increasing the capacity of FSO systems [10–12].

Considering the combination with FECs, non-uniform symbols can be generated mainly through the following approaches: 1) Adding distribution matchers. Before channel encoding, the constant composition distribution matcher (CCDM) or the enumerative sphere shaping (ESS) techniques are usually employed to map uniform bits to non-uniform symbols [10–15]. In contrast to CCDM, the complexity of ESS is lower, but ESS requires a lookup table, and the latency caused by serial computation cannot be ignored [15]. Additionally, the corresponding inverse distribution matcher is required for the receiver to recover the original data bits, inevitably increasing the decoding complexity; 2) Relying on FECs. The many-to-one schemes can realize probabilistic shaping by mapping ambiguity, but iterative decoding is performed due to the ambiguity [16]. Although the ambiguous bits can be placed on the frozen index of polar codes to avoid iterative decoding, the inter-leaver and the inverse inter-leaver are still required in [17], and the non-uniform probability distributions are restricted to be dyadic. However, the shaped polar codes can generate non-uniform distributions, which do not affect the decoding complexity [18,19] and are friendly to the receiver with no additional shaping decoder. Nevertheless, these schemes are currently designed for bipolar systems, which cannot be directly extended to unipolar FSO systems. Meanwhile, the shaping gains of non-uniform signaling are greatly diminished by moderate-strong turbulence in the single-input single-output (SISO) systems [10,12], posing challenges for FSO communication.

In this paper, we focus on the probabilistically shaped polar-coded MIMO-FSO systems to combat turbulence fading and improve transmission performance. The shaping-polar encoder with modified Maxwell-Boltzmann (MMB) distributions/ exponential distributions (ED) is designed for probabilistic shaping, which uses the pre-encoder to form shaping bits at the transmitter and cannot increase the decoding complexity at the receiver. It is worth noting that the ED formats make it possible to form non-pairwise distributions in FSO systems. Besides, the achievable information rate (AIR) of MIMO-FSO systems is deduced as the function of the number of transceivers, probabilistic distributions, and atmospheric turbulence (AT) fading with or without SC fading. The AIR of MIMO-FSO systems is calculated to determine the number of shaping bits for the shaping-polar encoder. The generated distributions of the encoder can approach the theoretically optimal ones by adjusting the amount of shaping bits. Regarding the AIR and the bit error rate (BER) performance, the shaped MIMO-FSO systems are comparable to the uniform ones with one more physical receiver.

The rest of this paper is organized as follows. In section 2, the principle of the probabilistically shaped polar-coded MIMO-FSO systems is introduced, where the shaping-polar encoder with MMB distributions/ED is proposed. In section 3, the AIR of the MIMO-FSO system is evaluated, and the optimization of AIR is to determine the number of shaping bits for the shaping-polar encoder. The simulation results and discussion on BER performance are presented in section 4. Finally, several conclusions are drawn in section 5.

2. Principle of probabilistically shaped polar-coded MIMO-FSO systems

The block diagram of the probabilistically shaped polar-coded MIMO-FSO systems is depicted in Fig. 1. At the transmitter, three typical shapes of probabilistic distributions can be achieved by the shaping-polar encoder, i.e., uniform, pairwise, and non-pairwise. Here, the pairwise distributions denote that the probabilities of two symbols in pairs are equal, and the non-pairwise distributions indicate that different symbols have different probabilities. Then, the bit-interleaved coding modulation (BICM) is used for PAM4 mapping due to its relatively simple process with low complexity. In the diversity system, the transmitted data is repeatedly coded and sent to the Q transmitters for transmission, where the data on each channel is identical. Each transmitter is composed of a laser source, an optical modulator, an optical amplifier and a fiber collimator. The equal-gain combining (EGC) is performed on the received signals from ${\; }R$ photodetectors. Finally, the bit metric decoding (BMD) and the successive cancellation list (SCL) decoding are used to recover the information bits. In this paper, the “$\textrm{Q} \times \textrm{R}$ system” denotes that there are Q transmitters and R receivers in MIMO-FSO systems.

 

Fig. 1. Block diagram of the probabilistically shaped polar-coded $Q \times R$ systems over FSO channels. BICM: Bit-interleaved coded modulation, PAM: Pulse amplitude modulation, PD: Photodetector, (a) the structure of transmitters, and (b) the structure of receivers.

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2.1 Shaping-polar encoder with MMB/exponential distributions

The shaped polar codes were initially proposed for bipolar symbols in additive white Gaussian noise (AWGN) channels and kept symmetric and pairwise [18]. For conventional polar codes, the occurrence probability of bit “1” in the codeword ${\boldsymbol c}$ is 0.5, i.e., ${P_{\boldsymbol c}}(1 )= 0.5$, while the shaped polar codes utilizing the pre-encoder (e.g., the SCL decoder) can make ${P_{\boldsymbol c}}(1 )\ne 0.5$. Let ${{\boldsymbol G}_N}$ represent the polar transform of length $\textrm{N}$, and the polar codeword generated by the input ${\boldsymbol u}$ is given as ${\boldsymbol c} = \; {\boldsymbol u}{{\boldsymbol G}_N}$. If the codedword ${\boldsymbol c}{\; }$ is transmitted over the binary symmetric channel with crossover probability p and noise ${\boldsymbol u}{{\boldsymbol G}_N}$, an all-zero vector ${\; }{\boldsymbol c} \oplus {\boldsymbol u}{{\boldsymbol G}_N} = 0$ can be obtained at the receiver. The shaping bits, as the temporary information bits, are derived by the SCL decoder, allowing ${P_{\boldsymbol c}}(1 )$ to approach p after polar encoding.

When shaping the PAM4 signaling, two length-$N$/2 independent polar codewords ${{\boldsymbol c}_1}$ and ${{\boldsymbol c}_2}\; $ with ${P_{{{\boldsymbol c}_1}}}(1 )= {p_1}$ and ${P_{{{\boldsymbol c}_2}}}(1 )= {p_2}$ are required to form the length-$N$ codeword with target distributions, where ${p_1}$ and ${p_2}$ are the occurrence probability of bit “1” in the sub-codeword ${{\boldsymbol c}_1}$ and ${{\boldsymbol c}_2}$, respectively. If there is no pre-encoder before the polar encoder, the value of ${p_i}$ will be 0.5. As shown in Fig. 2(a), only frozen bits (black lines) and information bits (red lines) are included in polar codes, and the eye diagram is uniformly distributed. The gray mapping ${{\boldsymbol b}_1}{{\boldsymbol b}_2}$ is adopted for BICM, and ${{\boldsymbol b}_1}$ is the less significant bit-level. Based on the polar transform [20], the relation of ${{\boldsymbol c}_1}{{\boldsymbol c}_2}$ and ${{\boldsymbol b}_1}{{\boldsymbol b}_2}$ is given by

In this paper, the probability mass function of constellation symbols is defined as ${{\boldsymbol P}_s} = [{{P_{s0}},{P_{s1}},{\; } \ldots ,{\; }{P_{s({M - 1} )}}} ]$, where M is the modulation order and ${P_{s({m - 1} )}}$ is the mth element of ${{\boldsymbol P}_s}$. For instance, the probability of the symbol “3” is

 

Fig. 2. Different structures of shaping-polar encoders for unipolar symbols, (a) no pre-encoder, (b) single pre-encoder, and (c) double pre-encoders.

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The other expressions of ${P_{sm}}$ are listed in the third row of Table 1.

Table 1. The relationship between the ${{\boldsymbol P}_{\boldsymbol s}}$ with MMB distributions/ED and ${{\boldsymbol p}_{\boldsymbol i}}$ for PAM4

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Then, the problems that need to be solved are how to generate appropriate distributions. Although the Maxwell-Boltzmann (MB) distributions with the parameter v are almost capacity-achieving in AWGN channels [13], the MB distributions cannot be directly extended to FSO as symbols in FSO systems are unipolar. The MMB distributions were proposed to address the non-symmetry [10], as listed in the fourth row of Table 1 (the normalized parameter is omitted for simplicity). To ensure that ${{\boldsymbol P}_s}\; $ satisfies MMB distributions, the relation between ${p_1},{p_2}$ and the parameter v should be

The structure of Fig. 2(b) can correspondingly meet the requirement of Eq. (3), where the shaping bits are presented by blue lines, and the generated distributions are pairwise. The other feasible alternative is to adopt the ED formats in FSO systems [12,21], which is listed in the fifth row in Table 1. In this case, the relationship between ${p_1},{p_2}$ and v is expressed as

It is noteworthy that the ED formats make it possible to generate non-pairwise distributions by polar codes as described in Fig. 2(c), whereas the MMB distributions do not work. The structure of Fig. 2(c) can be realized by ED formats, where double pre-encoders are needed before the polar encoder. The shaping bits (blue lines) of Fig. 2(c) appear not only in the second half but also in the first half. In the following sections, the structure of the single pre-encoder shaping, shown in Fig. 2(b), is denoted as the SPS scheme. Moreover, the structure of double pre-encoders shaping, depicted in Fig. 2(c), is denoted as the DPS scheme. The optimization of the number of shaping bits will be illustrated in section 3, which is associated with the AIR of MIMO-FSO systems.

2.2 MIMO-FSO channel model

In FSO channels, the irradiance fluctuations can be modeled as a Gamma-Gamma distribution over a wide range of turbulence conditions [1]. The probability distribution function (PDF) of the fading g is written as

The parameters $\alpha $ and $\beta $ denote the large- and small-scale eddies, respectively, which are the function of Rytov variance. Different Rytov variances can represent the intensity of fading. The novation of $\Gamma ({\cdot} ){\; }$ denotes the gamma function, and ${K_v}({\cdot} )$ is the modified Bessel function of the second kind with the order $v$. The received signal of the $r$th PD at t time slot is

Here ${\boldsymbol E}[g ]= 1$ and ${\boldsymbol E}[{{g^2}} ]= 1 + 1/\alpha + 1/\beta + 1/({\alpha \beta } )$ according to [22], and the signal-to-noise ratio (SNR) at the $r$th PD can be expressed as $SNR = {\boldsymbol E}[{y_r^2} ]/{\sigma ^2}$, which is related to ${{\boldsymbol P}_s}$. In fact, the required spacing between the apertures or between the beams to ensure uncorrelated fading is usually too large to be provided, especially in moderate and strong turbulence conditions [8]. An efficient way to simulate the inevitable SC fading ${g_{rq}}\; $ based on the gamma-gamma distribution was introduced in our previous work [7], utilizing the Chorisky decomposition to achieve the SC fading with different correlation coefficients $ \rho $. In this paper, the intensity of SC fading is characterized by the values of $ \rho $, where $0 \le \rho \le 1.$

2.3 BDM for probabilistically shaped MIMO-FSO systems

The EGC method, as an efficient MIMO combination scheme, is performed on the received signals. Therefore, the reliability of symbols related to the $j$th received signal is given as

3. Optimal distributions of shaped polar-coded MIMO-FSO systems

3.1 Achievable information rate of MIMO-FSO systems

Since the BMD is a mismatched decoding method, there is a gap between the achievable rate of BMD and the mutual information of channels [14]. In this paper, the achievable rate of BMD is considered as the target to be optimized, which is regarded as the AIR of MIMO-FSO systems. Now we explain how to calculate the ${R_{BMD}}$ in MIMO-FSO systems.

For SISO AWGN systems, the ${R_{BMD}}$ is the function of both ${B_{j,i}}$ and ${L_{j,i}}$ in [21, Eq. (8)], where ${B_{j,i}}\; $ is the $i$th bit level of the $j$th transmitted symbol's binary mapping and only related to the transmitted symbols. In terms of ${L_{j,i}}$, the calculation of ${L_{j,i}}$ in MIMO-FSO systems has been discussed in section 2.3, containing the SC fading between the transceivers. Different from the AWGN channels, the operator of ensemble averaging is necessary to be performed on the AT fading $(g )$ in the condition of Gaussian noise realization $(y|g)$ for FSO systems. Thus the ${R_{BMD}}$ in MIMO-FSO channels can be written as Eq. (10),

Here $H(X )$ is the entropy of transmitted symbols, and the operation of ${\boldsymbol E}[. ]$ denotes ensemble averaging. From Eq. (8), (9), and (10), the ${R_{BMD}}$ of MIMO-FSO systems is associated with the number of transceivers, the non-uniform distribution ${{\boldsymbol P}_s}$ and the channel conditions. When the channel conditions and the number of transceivers are fixed, the channel capacity can be improved by optimizing ${{\boldsymbol P}_s}$.

3.2 Optimization of the number of shaping bits

In this section, the optimization of capacity is discussed to obtain the maximum achievable rate and the corresponding input distribution for any given SNR over different turbulence channels. Firstly, the average electrical power constraint ${P_{av}}{\; }$ is considered because a) electrical-to-optical conversion is unavoidable in FSO communication, fundamentally limited by the electrical power; b) the peak power constraint can be ignored, as the optical amplifier is usually applied to adjust the average launch power in FSO systems [23]. Assuming that the probabilistic distributions satisfy the ED formats, the optimization problem of ${R_{BMD}}$ in MIMO-FSO systems can be given by

The $n{s_i}$ usually needs to be adjusted until the generated distribution ${{\boldsymbol Q}_s}$ is close to the target one ${\boldsymbol P}_\textrm{s}^{\ast }$, of which the Kullback-Leibler (KL) divergence should be small, e.g., ${10^{ - 4}}$.

The AIR of PAM4 with optimal ${v^\ast }$ over strong AT channels in MIMO-FSO systems is described in Fig. 3. Depending on the number of pre-encoders, three shapes of distribution can be generated, as illustrated in section 2.1, namely uniform, pairwise and non-pairwise. As shown in Fig. 3, the capacity of $4 \times 4$ systems (red curves) outperforms that of $3 \times 3$ systems (blue curves), let alone SISO systems. The non-pairwise distributions yield larger capacity than uniform or pairwise distributions in FSO systems. In Fig. 3(a), given uncorrelated fading, the uniform $4 \times 4$ systems or non-pairwise $3 \times 3$ systems are superior to the uniform SISO systems by 11.95dB at 1 bit/ channel use (bpcu), while the shaping gains of non-pairwise $4 \times 4$ systems are 13.55dB under the same situations, surpassing those of uniform $4 \times 5$ systems. From Fig. 3(b), it can be found that as the correlation coefficient $\rho $ increases to 0.8, the shaping gains decline by about 1.5dB, implying that the SC fading between the receivers can reduce the channel capacity. In summary, the AIR of non-pairwise $4 \times 4$ systems is comparable to the uniform MIMO systems with one more physical receiver, even in the strong AT channels with a high correlation coefficient.

 

Fig. 3. The AIR performance of PAM4 symbols with optimal ${v^\ast }$ in $3 \times 3$ or $4 \times 4$ systems over strong AT channels, (a) $\rho = 0$, and (b)$ \rho = 0.8$.

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Take the $4 \times 4$ systems at 1bpcu as an example. In Fig. 4(a), if the intensity of turbulence or the correlation coefficient increases, the optimal ${v^\ast }$ of both DPS schemes and SPS schemes will decrease, especially under the strong AT with high correlation coefficients situations. The initial number of shaping bits can be calculated by Eq. (12), and the generated distributions ${{\boldsymbol Q}_s}$ either pairwise or non-pairwise can be obtained by the shaping-polar encoder of Fig. 2. For example, as shown in Fig. 4(b), the generated distributions ${{\boldsymbol Q}_s}$ (red/ navy regions) are very close to the target ones ${\boldsymbol P}_\textrm{s}^{\ast }$ (pink/ light blue regions) under strong AT fading with $\rho = 0.8$, regardless of the MMB distributions or ED formats. In this paper, the degree of closeness is denoted by the KL divergence, and the number of shaping bits is adjusted to ensure that the KL divergence is less than ${10^{ - 4}}$. Meanwhile, it can be seen from Fig. 4(b) that symbols with smaller energy are more likely to occur. When the transmission power is fixed at 1, the $\Delta $ values of the SPS and DPS schemes are 0.553 and 0.644, respectively. Compared to the SPS schemes, the $\Delta $ of the DPS schemes is larger, which will be more beneficial to decode at the receiver. When the codeword length of polar codes is 1024, the shaping bits required in $4 \times 4{\; }$ systems at 1bpcu are listed in Table 2.

 

Fig. 4. The optimization of $4 \times 4$ systems over different AT channels at 1 bpcu, (a) the optimal ${v^\ast }$ for different shaping-polar encoders, and (b) the optimal ${\boldsymbol P}_\textrm{s}^{\ast }$ and the generated ${{\boldsymbol Q}_s}$.

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Table 2. The total number of shaping bits under different turbulence conditions

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4. Simulations and results

In order to evaluate the BER performance of the shaped polar-coded MIMO-FSO systems, Monte Carlo simulations are performed through MATLAB 2020 and Visual Studio 2013, following the setup depicted in Fig. 1. The Rytov variance is used to control the intensity of turbulence, and the values of 0.2, 1.6, and 3.5 denote weak, moderate, and strong, respectively. Since the capacity of $4 \times 4$ systems is the largest in Fig. 3, the following simulations are based on the $4 \times 4$ systems. The optimal ${v^\ast }$ varies with the AT fading and SC fading, and the corresponding values of $ns_i^\ast $ are listed in Table 2. In terms of the settings for polar codes, the codeword length is 1024, and the code rate is equal to 0.5 at 1bpcu. The construction of polar codes draws on the standard of 5G channel coding [24]. The simple SCL decoder is used as the pre-encoder, and the cyclic redundancy check (CRC) aided SCL (CA-SCL) is employed at the receiver [25–27], where the length of the list and CRC is equal to 8 for the fast decoding. At last, a total of 300,000 simulations are performed with a maximum of 100 error blocks.

The BER performance of SPS and DPS schemes is investigated in Fig. 5. When there is uncorrelated fading, the required SNR of the uniform SISO systems (light yellow region) to reach $BER = {10^{ - 5}}$ grows from 15.2dB to 22dB as the turbulence increases. However, the required SNR for $4 \times 4$ systems (grey region) to reach the threshold of BER only ranges from 7.6dB to 8.3dB in the same conditions. It indicates that the spatial diversity techniques can perform better than SISO systems to combat the turbulence fading. Although more transceivers can offer better BER performance, these physical transceivers cannot be infinite considering the space, cost, and other factors. The proposed shaped-MIMO schemes can improve transmission performance under the inherent MIMO systems. As described in Fig. 5, compared to the uniform SISO system over strong AT channels, the corresponding SPS and DPS scheme can achieve 14.8 dB and 15.3dB shaping gains at $BER = {10^{ - 5}}$, respectively. In addition, the SPS scheme outperforms the uniform MIMO system by 1.3dB in the weak AT channels, and the shaping gains of the DPS scheme are up to 1.9dB.

 

Fig. 5. BER performance of shaped $4 \times 4$ systems over weak, moderate, and strong AT channels without SC fading, (a) single pre-encoder shaping, and (b) double pre-encoders shaping

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The comparisons of BER performance for different numbers of transceivers over different AT channels are depicted in Fig. 6. The results show that the shaped $4 \times 4$ systems surpass the uniform $4 \times 5$ systems by 0.34dB and 0.94dB in weak AT conditions, as the blue curves shown in Fig. 6. Even in the strong AT channels, the BER performance of the DPS schemes is still better than the uniform $4 \times 5$ systems, as presented in the red curves of Fig. 6(b), while the performance of the SPS schemes is slightly inferior. As analyzed in section 3, the constellation spacing of the DPS schemes is larger than that of the SPS schemes. Thus the better BER performance of DPS schemes is observed in Fig. 6(b) when the turbulence increases.

 

Fig. 6. The comparisons of BER performance for shaped $4 \times 4$ systems and uniform $4 \times 5$ systems without SC fading, (a) single pre-encoder shaping, and (b) double pre-encoders shaping.

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In terms of the SC fading, the BER performance of DPS schemes is evaluated for the correlation coefficients of 0, 0.4, and 0.8, denoting no correlation, weak correlation, and strong correlation. In Fig. 7(a), the uniform 4×5 systems perform better than the uniform 4×4 systems when the AT fading is weak, but they are both inferior to the DPS schemes under different SC fading conditions. From Figs. 7(b)(c), when the correlation coefficients are 0.8, the shaped MIMO systems can achieve about 1.04∼ 1.17dB shaping gains than the uniform ones over moderate-strong AT channels, which are shrunk up to 0.56dB compared to the weak AT conditions of Fig. 7(a). Generally, the shaping gains based on shaped 4×4 systems range from 1.04∼1.86 dB under different AT fading with SC fading situations. The BER performance of shaped MIMO systems outperforms the uniform MIMO systems with one more physical receiver.

 

Fig. 7. BER performance of DPS $4 \times 4$ systems with different SC fading, (a) weak AT, (b) moderate AT, and (c) strong AT.

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5. Conclusions

This paper proposes the probabilistically shaped polar-coded MIMO-FSO systems relying on FECs to combat the two types of fading, i.e., SC fading and AT fading. The proposed shaping-polar encoder can generate three typical shapes of distribution with different numbers of pre-encoders, where the shaping schemes with non-pairwise distributions are demonstrated to be more suitable for FSO systems than other formats. The exponential distributions are employed to make it possible to generate non-pairwise distributions for FSO systems, while the MMB formats do not work. The results show that the AIR of the shaped $4 \times 4$ systems can exceed that of the uniform $4 \times 5$ systems even under strong AT conditions. In terms of the BER performance, compared to uniform SISO systems, more than 15 dB shaping gains can be obtained by shaped $4 \times 4$ systems, which outperform the uniform ones by about 1∼1.9 dB under different channels conditions. Even in the strong AT channels with high SC fading, the transmission performance of the shaped $4 \times 4$ systems with non-pairwise distributions is comparable to that of the uniform MIMO systems with one more physical receiver.