1. Introduction

Free space optical (FSO) communication has been regarded as one of the next-generation wireless communication solutions with the advantages of flexible deployment, high bandwidth, and low cost [1], which can provide massive connectivity for the Internet of Things (IoT) and 5 G communications. Intensity modulation/direct detection (IM/DD) has been widely used in FSO systems, such as on-off keying (OOK), pulse position modulation (PPM), and pulse amplitude modulation (PAM), where PAM has higher spectral efficiency and throughput [2]. Nevertheless, the availability and reliability of FSO severely suffer from adverse environmental factors, especially atmospheric turbulence [3]. The forward error correction (FEC) techniques, e.g., low-density parity check (LDPC) codes and polar codes, are widely used in FSO to combat turbulence [4,5]. Coded modulation can combine high-order modulations with FEC to make transmission reliable, and the bit-interleaved coded modulation (BICM) is a typical parallel structure with low computational complexity [6]. Recently, non-uniform signals have been proven to increase the channel capacity over uniformly distributed (UD) signals on AWGN channels. Probabilistic amplitude shaping (PAS) is a typical joint of coded modulation and probabilistic shaping (PS) [7], but it cannot be directly extended to FSO due to the asymmetry of optical signals. In [8–10], numerical simulations of LDPC or polar coded PS PAM schemes have been proposed to deal with the unipolar signals to optimize FSO channel capacity, where the pairwise distributions are suitable for PAM8 with negligible performance loss. Furthermore, the experiment of [11] for the first time shows that LDPC coded unipolar PS PAM4/8 could achieve the power margin over a 2.5 m free-space link with atmospheric attenuation. However, the influence of atmospheric turbulence was not discussed in [11].

Previous FSO experiments of [12–14] implied that channel fading induced by atmospheric turbulence is instantaneous. Hence, pilot data should be inserted into the transmitted frame to offer dynamic channel state information (CSI), and the accuracy of estimated CSI will affect the demodulation performance. On the other hand, although PS techniques can obtain more power margin as the entropy of the transmitted data decreases, PAM formats require a higher power budget than OOK signals to ensure transmission quality [2]. Therefore, digital signal processing (DSP) algorithms are essential for bandwidth-efficient FSO communication. For instance, nonlinear or linear equalization algorithms are usually employed for high-order modulations to mitigate the impairments and reduce the power sensitivity over free-space links [11,15,16]. However, those classical DSP algorithms may suffer a performance penalty for strongly shaped constellations due to the imperfect channel estimation of symbols with low probabilities [17]. It is significant for conventional DSP to be modified to accommodate shaped constellations.

In this paper, the transmission of polar coded probabilistically shaped PAM8 with systematic interleaver (SIL) and non-uniformly distributed pilot over weak turbulence channels is investigated. At the transmitter, the constant composition distribution matcher (CCDM) is used to map uniform bits to non-uniform symbols. An efficient SIL is proposed for BICM-PS-PAM8 over weak turbulence channels, maintaining the desired probability mass function (PMF) and matching the 5 G channel coding standard of polar codes. In order to deal with the dynamic channel fading induced by turbulence and avoid DSP failure for shaped constellations, the pilot data is used to provide perfect CSI for demodulation, where the PMFs of the pilot are identical to those of the transmitted data. At the receiver, the effect of channel fading is included in the noise variance of the normalized data, which does not require to be estimated individually. Afterward, the combination of nonlinear and linear equalizers aims to reduce the sensitivity, where the step size of the decision-directed least mean square (DDLMS) equalizer can vary with the distance between the received signals and estimated amplitude levels. In our experiment, weak turbulence can be generated by controlling the temperature gradients inside an FSO chamber. The observed channel fading of the received electrical signals obeys the typical log-normal (LN) turbulence model, where the log-intensity variance is from 0.017 to 0.037. The transmission performance shows that the proposed polar coded PS system outperforms others in combating turbulence over weak turbulence channels.

The rest of this paper is organized as follows. Section II introduces the principle of the systematic interleaver and the process of DSP in FSO systems. The detailed experimental setup is illustrated in Section III, and the experimental results are shown in Section IV. Finally, the conclusion is drawn in Section V.

2. Principles

2.1 Systematic interleaver for PS-FSO system

Polar code is the first channel coding proven to reach the Shannon limit [18], where the sub-channels have different reliability by channel polarization, and information bits are usually assigned to high-reliability indexes. Let ${G_N}$ denote the polar transform of $N$-length, and the codeword $\mathbf{c}$ of input vector $\mathbf{u}$ is written by

Here, ${F^{ {\otimes} n}}$ represents the $n$-th Kronecker power of polar kernel F. For binary channel coding, the polar construction $Q_0^N = \{{Q_0^N,Q_1^N, \ldots ,Q_{N - 1}^N} \}$ is given in the 5 G channel coding standard of [19], where $W(Q_i^N)$ denotes the reliability of $Q_i^N$ and $W(Q_i^N)$ < $W(Q_{i + 1}^N)$. However, the optimal polar construction can be achieved by MC simulation for high-order modulations, but it is time-consuming for practical systems, especially for time-variant channels [10].

To improve the performance of polar coded high-order modulations, polar interleavers for unsystematic polar codes were proposed to make modulation polarization well [20,21]. The interleaver firstly divides the codeword into $m = lo{g_2}M$ columns according to sub-channels reliability, where M is the modulation order. Then, bits in each row with the highest reliability are mapped to the most significant bit (MSB), and those with the lowest reliability are the least significant bit (LSB) to make modulation polarization. However, it cannot be directly extended to PS-FSO systems: i) it is difficult for the unsystematic polar code to maintain the non-uniform PMF after encoding, while the systematic polar code can be used to protect the PMF of CCDM [9,10]. ii) extra process of interleaving is required before the systematic encoder. Otherwise, the desired pairwise PMF cannot be achieved after polar interleaver.

Assuming that p bits are punctured and ${n_s} = ({N - p} )/m$ symbols are generated after PAM8 mapping, the transmission rate (bits/channel use, bpcu) can be expressed as

As depicted in Fig. 1, ${k_u} + {k_\gamma }$ is the total number of information bits for UD schemes, while ${k_u}$ and ${k_\gamma }$ denote the number of bits fed into CCDM and shaping redundancy bits for PS schemes, respectively. Compared to UD systems, the SIL for PS systems has extra interleaving before encoding. As shown in Fig. 1(b), ${k_u}$ uniform information bits are mapped to PS PAM4 (${b_2}{b_1}$) signals by CCDM, where ${b_2}$ denotes the MSB. Then, these shaped bits and ${k_\gamma }$ shaping redundancy bits are fed into SIL-I, systematic encoder and SIL-II modules. Finally, parity bits and shaping redundancy bits are mapped to LSB (${b_0}$) to implement the target PMF of PAM8 shown in Fig. 1 (b), where the PMF of PS PAM8 is referred to as a pairwise distribution and the probabilities of two consecutive symbols are equal.

 

Fig. 1. Transmitter of systematic polar coded PAM8 in FSO systems for (a) uniform and (b) non-uniform symbols. (c) The process of systematic interleaver (SIL) for the PS scheme.

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The detailed process of SIL for PS PAM8 in FSO systems is presented in Fig. 1(c). The sequence $\mathbf{t}$ to be encoded consists of shaping bits (${b_2}{b_1}$), shaping redundancy bits (${k_\gamma }$), cyclic redundancy check (CRC) bits, and frozen bits (Fro). Then, it is first interleaved by SIL-I as

Here, ${D_N}(i )= \,Q_{N + 1 - i}^N$ and $W({{D_N}(i )} )$>$W({{D_N}({i + 1} )} )$. After systematic encoding, the codeword $\mathbf{c}$ is shown in Fig. 2, and the process of the SIL-II module is

 

Fig. 2. Dynamic DSP for PS-FSO systems. The ideal and actual PDF of (a) UD and (b) PS PAM8. (c) The PMF of the pilot for each frame, and (d) the process of DSP in FSO systems.

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2.2 Dynamic DSP for PS-FSO system

Given the influence of turbulence, the discrete-time channel model in FSO systems can be expressed as

Here, $\textrm{E}[I ]$ is the expectation of fading I and ${I_0}$ denotes the intensity fluctuation without turbulence. The parameter $\sigma _l^2$ represents the log-intensity variance and can be estimated by the parameters of propagation distance and temperature gradients [13,14]. Since channel fading changes more slowly than the bit rate in FSO systems [8], $I(k )$ can be regarded as a block fading process I over L symbols, and formula (6) can be rewritten as

Thus, the channel fading can be characterized by the noise variance ${\hat{\sigma }^2}$ of normalized data without being estimated individually. The values of ${\hat{\sigma }^2}$ dynamically fluctuates in turbulence conditions, and the corresponding ESNRs will vary with the strength of turbulence.

Figure 2 presents the actual PDF of received UD or PS PAM8 signals after normalization over a 4m-free space link. As shown in Fig. 2(a) and (b), the actual PDF (blue area) of UD signals is close to the ideal one (red dashed line), while the actual peaks of PS signals are offset for symbols with low probabilities, i.e., the actual amplitude levels of PS signals are different from the ideal ones. If the ideal PDF of PS signals is used for demodulation, symbols with low probabilities will lead to more errors. Let set $\hat{A}$ represent the estimated amplitude levels after equalization. According to the maximum a posteriori probability (MAP) detection, the optimum decision threshold $\tau $ between the two symbols ${a_r}$ and ${a_{r + 1}}$ of the set $ A$ at fading $ I$ satisfies

In Fig. 2(a), the actual PDF of UD signals approach the ideal PDF, so the optimal decision threshold of $\hat{A}$ has little difference from that of A. However, for PS systems, if ${a_r}$ and ${a_{r + 1}}$ of set A are used to calculate the decision threshold, the threshold $\tau ^{\prime}$ of Eq. (10) will not be optimal, as the red area of extra errors ${P_e}({\tau^{\prime}} )- {P_e}(\tau )> 0$ shown in Fig. 2(b). Therefore, to provide perfect CSI for symbols with low probabilities in practical system, the pilot and the transmitted data are designed to be independent and identically distributed, i.e., the pilot of UD signals is uniform, and that of PS signals is non-uniform, as eye diagrams are shown in Fig. 2(c).

Regarding Fig. 2(d), the dynamic DSP in FSO systems consists of normalization, Volterra nonlinear equalization (VNLE), linear equalization (DDLMS) and bitwise LLRs calculation. The amplitude levels $\hat{A}$ and noise variance ${\hat{\sigma }^2}$ are dynamically estimated by the pilot for the DDLMS-type equalization and LLRs calculation. The update of conventional DDLMS is expressed as [22]

In order to provide precise symbols decisions for demodulation and decoding, a double-step size DDLMS is proposed, where the step size is related to the distance between signals and the adjacent amplitude levels, and the update process can be rewritten as

Here, $\rho $ is the coefficient and satisfies $0 < \rho < 1$, and D is the distance between the two amplitude levels adjacent to $y(t )$. As shown in Fig. 2(d), if $y(t )$ is within the interval $[{{{\hat{a}}_r},\; {{\hat{a}}_{r + 1}}} ]$, D will be equal to ${D_r} = \; {\hat{a}_{r + 1}} - \; {\hat{a}_r}$, where $r = 1,2, \ldots ,\; M - 1$. The value of ${\mu _1}$ is set to be larger than that of ${\mu _2}$ as signals within $\rho D$ are closer to $d(t )$ than others, as shown in Fig. 2(d).

In terms of the bitwise LLRs calculation, the bit metric decoder is used for BICM to recover data bits when decoding. The LLR in the $l$th bit level of the kth equalized signal $y(k )$ is

3. Experiment setup

The schematic of polar coded PS PAM8 in weak turbulence links is illustrated in Fig. 3. At the transmitter, CCDM is employed to map uniform bits to non-uniform PAM4 symbols. As described in Section 2.1, PAM8 signals with pairwise distributions are generated after interleaving and encoding and shaped by a root-raised cosine (RRC) filter with a roll-off factor of 0.1. Each frame comprises 100 codewords with $N = 1024$ and 4.6% pilot data (1649 symbols) with the same PMF as the transmitted data. Then, these offline data are loaded into the arbitrary waveform generator (AWG, Keysight M8195A) with a 3dB-bandwidth of 25 GHz operating at 64GSa/s. The transmission rate ${R_t}$ is fixed to be 1.5bpcu or 1.8bpcu, and the code rate of UD PAM8 signals is 1/2 or 3/5. Considering 25% shaping redundancy, i.e., ${k_\gamma } = 85$, the code rate of PS schemes is 3/4, and the PMF of 1.5bpcu and 1.8bpcu are [0.3079,0.3079, 0.1510,0.1510,0.03666,0.03666,0.0044,0.0044] and [0.2507,0.2507,0.1628,0.1628,0.0674, 0.0674,0.0191,0.0191], respectively. Since the symbol spacing is inversely proportional to the symbol entropy at the same average power, the peak-to-peak voltages of UD signals, PS signals at 1.8bpcu and 1.5bpcu are set to 480 mv, 788 mv, and 1000 mv, within the linear interval of electro-absorption modulated laser (EML). Then, the output 25GBd PAM8 signals of AWG are modulated by a 30 GHz EML at 1550 nm, where an erbium-doped optical fiber amplifier (EDFA) is placed after the EML to make systems limited by the average power according to [23]. The variable optical attenuator (VOA) is employed to control the transmitted optical power (${P_t}$).

 

Fig. 3. Schematic of polar coded PS PAM8 over weak turbulence channels.

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In the free space links, optical fiber collimators (FC, Thorlabs C40APC-C) are applied to transmit optical signals with about 3.2 dB atmospheric attenuation. An acrylic chamber is utilized to imitate weak turbulence with the dimension of $4 \times 0.4 \times 0.4{\textrm{m}^3}$, of which the holes are equally spaced to form temperature gradients. In order to generate different turbulence conditions, independent fans blow hot or cold air (off, cold, hot I, hot II) in the direction perpendicular to light beam propagation. Limited by the length of the FSO chamber, only weak turbulence cases can be produced. The values of log-intensity variance can be calculated by equations in [24] using the recorded temperature profiles (°C) listed in Table 1.

Table 1. Different turbulence conditions

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At the receiver, signals are detected by a 35 GHz photodiode (PD, PD-1000) and captured by a real-time digital storage oscilloscope (DSO, Tektronix DPO75902SX) operating at 100GSa/s for the offline process. After match filtering and synchronization, the received data are fed into the dynamic DSP module shown in Fig. 2(d). The normalized data is processed by an optimized second-order VNLE with 105 taps for the 1st order and 15 taps for the 2nd order to compensate for the nonlinearities of channels. Then, a 15-tap DDLMS or double-step size DDLMS is utilized to provide more precise symbol decisions for demodulation. Then, bitwise LLRs are calculated and de-interleaved before decoding, which is the inverse process of Fig. 1(c). For polar codes, the CRC-aided successive cancellation list (CA-SCL) algorithm is used for decoding [25], where the length of CRC bits and the list is set to eight. In addition, LDPC with basic graph 1 (BG1) is employed in our experiment to make comparisons, of which the length of the LDPC codeword is 1036. Given the computation complexity, the number of iterations of the belief-propagation (BP) algorithm is eight for the LDPC decoder [13]. Finally, the inverse CCDM is applied to recover information bits for shaping signals. At last, the evaluation of BER performance is counted by 400 frames (${\sim} 3.6 \times {10^7}$ symbols).

4. Experiment results

In Fig. 4, the post-FEC BER performance of polar code (PC) with SIL and LDPC code for uniform and non-uniform signals after VNLE over a 4 m free-space link is first investigated. As shown in Fig. 4(a), the BER performance of polar coded UD signals with SIL is superior to LDPC codes and significantly has advantages over conventional polar codes without interleaver at different transmission rates, indicating that SIL makes modulation polarization well to improve transmission performance. A bit exchange scheme of [10] is proposed to maintain the PMF of non-uniform signals for polar codes, and we compare it with the proposed SIL. It can be seen in Fig. 4(b) that the BER performance of polar coded PS signals with the interleaver of [10] or proposed SIL outperforms LDPC codes. Furthermore, polar coded PS scheme with SIL is better than that with an interleaver of [10] by 0.58 dB at ${R_t} = 1.5bpcu$ and $BER = {10^{ - 4}}$, and the gains increase to 1 dB at ${R_t} = 1.8bpcu$ and $BER = {10^{ - 3}}$. Furthermore, for UD signals or PS signals, polar codes with SIL can always achieve 1.0 dB coding gains over LDPC codes at ${R_t} = 1.5bpcu$ and $BER = {10^{ - 4}}$.

 

Fig. 4. Transmission performance over no turbulence links. The post-FEC BER performance of (a) UD PAM8 and (b) PS PAM8, which is polar code with SIL based (solid, filled rectangle), without SIL based (dashed, half-filled rectangle), and LDPC code based (solid, triangle). Eye diagrams of (i)∼(ii) UD signals and (v)∼(viii) PS signals.

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On the other hand, it can be found that PS schemes are friendly to a power-limited system. The eye diagrams of UD or PS PAM8 are depicted in Fig. 4 (i)∼(viii), where Fig. 4 (i)∼(iv) are UD signals at ${P_t} ={-} 2dBm,\; - 1dBm,\; 0dBm\; and\; 1dBm$, Fig. 4(v)∼(vi) are PS signals at ${R_t} = 1.5bpcu$ and ${P_t} ={-} 3dBm,\; - 2dBm$ and Fig. 4(vii)∼(viii) are PS signals at ${R_t} = 1.8bpcu$ and ${P_t} ={-} 1dBm,\; 0dBm$. It implies that more energy is needed to support the eye-opening for symbols with higher entropy. For example, the eye diagram of PS PAM8 is clear at -2dBm, while 1dBm transmitted average power is required for UD PAM8 at 1.5bpcu.

In Fig. 5, comparisons of imperfect CSI (iCSI) based on the standard constellations and perfect CSI (pCSI) estimated by the designed pilot are evaluated, where PC denotes the polar coded scheme with SIL. Here, the iCSI means that the equalized amplitude level $\hat{a}$ is not estimated, and the traditional PAM constellation is directly employed to calculate the LLRs, i.e., $\hat{a} = a$, where $a \in A = \{{0,1, \ldots ,M - 1} \}$. Regarding the pCSI, the amplitude level $\hat{a} \in \hat{A}$ of pCSI is estimated by the pilot data after equalization, which can better characterize the symbol spacing after transmission. As presented in Fig. 5(a), the performance loss between iCSI and pCSI for UD signals at different transmission rates is negligible because the PDF of UD PAM8 is close to the ideal PDF shown in Fig. 2 (a). On the contrary, the post-FEC BER performance of PS signals in Fig. 5 (b) has difficulty decreasing if the iCSI is used to decode. Hence, it is significant to obtain perfect CSI for probabilistic shaping constellations to obviate the performance penalty due to symbols with low probabilities.

 

Fig. 5. Performance of perfect and imperfect CSI after VNLE. The schemes are perfect CSI based (solid) and imperfect CSI based (dashed) for (a) UD PAM8 and (b) PS PAM8.

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In terms of equalization, an optimized second-order VNLE with 105 taps for the 1st order and 15taps for the 2nd order is used to compensate for the nonlinearities of transmission systems, which can complete the convergence of the mean square error (MSE) in the first stage. Afterward, a 15-tap linear DDLMS is utilized to provide precise symbol decisions for final demodulation. The MSE convergence of DDLMS for UD signals and PS signals at ${P_t} ={-} 3dBm\,$ with different step sizes is shown in Fig. 6, where ${\mu _2} = 0.0001$ is the benchmark step size and $\rho = D/3$. In Fig. 6 (a), the MSE convergence of UD signals with double-step size (${\mu _1} + {\mu _2}$) does not change significantly and ${\mu _1}$ is approximately equal to ${\mu _2}$. In Fig. 6 (b), it can be seen that the MSE of PS signals with double-step size is between that of ${\mu _1}$ and ${\mu _2}$, where it is close to the MSE of ${\mu _2}$ in the beginning iterations and approaches that of ${\mu _1}$ during the subsequent iterations, where ${\mu _1}$ is 10 times larger than ${\mu _2}.$

 

Fig. 6. The MSE of (a) UD and (b) PS signals after DDLMS with different step sizes.

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In Fig. 7, the post-FEC BER performance of the proposed double-step size DDLMS is almost the same as that of conventional DDLMS when transmitting UD signals, while the double-step size DDLMS can save more power than regular DDLMS with step size $\mu $ = 0.0001 for PS signals. To better analyze the effects of step size for DDLMS, the BER performance of $\mu = 0.001$ for PS signals is evaluated. As the orange and blue dash curves shown in Fig. 7, the post-FEC BER performance can be further reduced as $\mu $ increases for either polar coded s or LDPC coded systems, but there is still a performance gap between the orange dash curves and the red solid curves. In addition, as presented in Fig. 7(a), the required ${P_t}$ of polar codes with SIL to reach $BER = {10^{ - 4}}$ can be reduced by up to 0.72 dB and 0.78 dB for UD and PS signals after linear equalization, respectively. Under the same conditions, there are 0.78 dB and 0.83 dB power budgets saved for LDPC codes, as shown in Fig. 7(b). Furthermore, it can be seen that the shaping gains of polar codes or LDPC codes can approach up to 2.0 dB at 1.5bpcu.

 

Fig. 7. The post-FEC BER performance of different equalizers at 1.5bpcu. Equalization is VNLE based (violet, rectangle), VNLE + DDLMS based (orange/blue, circle), and VNLE + double-step size DDLMS based (red, rhombus), coded by (a) polar codes with SIL and (b) LDPC codes.

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The channel fading of different turbulence is shown in Fig. 8, which is counted before normalization as the process of optical-to-electrical conversion is regarded as linear. The pilot data of each frame is used to estimate the mean of amplitude levels, which is regarded as the channel fading of the current frame. As shown in Fig. 8(a)∼(d), channel fading is stable over no turbulence channel but varies with the temperature gradients inside the FSO chamber. The observed results are well matched with the LN turbulence model, indicating that the experimental setup of the weak turbulence is successful. The ESNRs of UD signals after VNLE under different turbulence conditions at ${P_t} = 0dBm$ is shown in Fig. 8(e)∼(f), the means of which are about 20 dB, but the variances of which intensely fluctuate as the strength of turbulence increases.

 

Fig. 8. The histograms of channel fading and ESNR of cases listed in Table 1 at ${\textrm{P}_t} = 0\textrm{dBm}$. no tur: (a) and (e), tur1: (b) and (f), tur2: (c) and (g), tur3: (d) and (f).

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The post-FEC BERperformance of different schemes to combat turbulence is shown in Fig. 9. The settings of double-step size DDLMS are adjusted to ${\mu _1} = 5 \times {10^{ - 4}}$ for PS signals under weak turbulence conditions. As shown in Fig. 9(a)∼(b), the BER performance of polar codes (orange and red curves) is always better than that of LDPC codes (purple and blue curves), regardless of whether uniform or non-uniform symbols are transmitted over weak turbulence channels. Moreover, LDPC or polar coded PS schemes outperform codes UD schemes in combating turbulence. For instance, the BER of polar coded UD schemes is ${10^{ - 3}}$ in the tur1 case, but that of polar coded PS schemes can decrease to ${10^{ - 5}}$, where two orders of magnitude improvement in BER can be achieved. As different equalization schemes shown in Fig. 9(a)∼(b), the BER performance of UD signals or PS signals is close to each other as turbulence increases, while the joint of VNLE and proposed DDLMS (solid curves) always outperforms others. It can be found for PS signals of Fig. 9(b) that when the BER of VNLE is around ${10^{ - 3}}$, the conventional DDLMS and the proposed double-step size DDLMS can further reduce the BER to ${10^{ - 4}}$, especially in slightly weak turbulence regions.

 

Fig. 9. The post-FEC BER performance of (a) UD PAM8 and (b) PS PAM8 using different equalization schemes in weak turbulence. (c) The Q factor performance.

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To better describe the performance over weak turbulence channels, the Q factor can be calculated in terms of BER as

The corresponding Q factors of polar codes and LDPC codes with the joint of VNLE and proposed double-step size DDLMS equalization under different weak turbulence are presented in Fig. 9(c). With the assistance of hybrid equalization, polar codes can achieve 0.43∼0.56 dB and 0.65∼1.3 dB coding gains over LDPC codes for UD and PS PAM8 signals, respectively. Although shaping gains decrease as turbulence increases, the Q factor of the proposed polar coded PS schemes can be improved by 0.49∼1.74 dB compared to uniform coded schemes under weak turbulence conditions.

5. Conclusion

In this study, the experiment of polar coded probabilistic shaped 25GBd PAM8 signals over weak turbulence channels is evaluated for the first time. An efficient and low-complexity SIL is proposed for systematic polar codes to make modulation polarization well, and 1.0 dB coding gains can always be achieved at different transmission rates compared to LDPC codes. On the other hand, the dynamic DSP is essential to accommodate time-varying channel fading and strongly shaped constellations, where the identically distributed pilot is used for dynamic channel estimation and provides perfect CSI to reduce symbol decision errors. In addition, the double-step size DDLMS after VNLE can further improve transmission performance for both UD and PS signals, achieving more than 0.7 dB power margin over a 4 m free-space link. Given a fixed transmitted power, the proposed polar coded PS scheme with SIL and hybrid equalization can bring 0.49∼1.74 dB improvement to the Q factor in weak turbulence regimes, which has advantages over other schemes in combating turbulence and realizing bandwidth-efficient FSO communication.