1. Introduction

Free-space optical (FSO) communication has attracted considerable attention due to those merits of easy deployment, license-free spectrum, high security, and high-speed transmission rates [1–3]. Recently, with the advent in coherent detection technology that allows for background noise rejection, high sensitivity, and improved spectral efficiency, there has been a regain of interest in coherent FSO communications with coherent homodyne/heterodyne receivers [4–8]. For instance, a homodyne binary phase shift keying (BPSK) transmission at 5.6 Gbps has been established between two islands [5,6]. However, the atmospheric turbulence in FSO channel induces not only amplitude fluctuation but also phase distortion. Therefore, the efficiency of coherent detection and the link performance are severely degraded [5,7,9,10].

To model the FSO channel, various statistical models have been proposed, such as Log-Normal distribution, $K$ distribution, Gamma-Gamma (G-G) distribution, and the Exponentiated Weibull (EW) distribution [2,11–14]. Most of them focus on intensity modulation and direct detection (IM/DD) systems or ideal coherent systems with the assumption of perfect coherent detection where only amplitude fluctuation is considered. Nevertheless, in coherent FSO systems, the optical signal is required to be coherently mixed with a local oscillator (LO) laser to provide shot-noise-limited (SNL) performance. The wavefront matching between the signal and LO is largely deteriorated by both the turbulence-induced amplitude fluctuation and phase distortion [8,15,16]. Even those models could provide sufficient insights into the amplitude fluctuation, the effect of the turbulence-induced phase distortion could hardly be considered or could only be accounted for under certain restrictions. In addition, to cope with the phase distortions, phase compensations of different Zernike modes are usually required in coherent FSO systems. In order to model the composite effects of turbulence-induced amplitude fluctuation, and phase distortion, together with the compensation for phase distortions under varied turbulence cases, the modified Rician distribution has thus been introduced. Its capability and advantage has not only been verified theoretically by Monte Carlo simulation [16] and but also demonstrated experimentally in both maritime and terrestrial coherent FSO links [17]. Furthermore, it has been further employed for the analysis of the coherent FSO relay links [18], etc.

In addition, FSO systems also suffer from pointing errors (PEs) due to such as building sway and vibrations of transceivers [19,20]. For the G-G fading channel, the combined effect of turbulence and PEs has been modeled towards SISO links [20]. However, it is still quite challenging for multi-input multi-output (MIMO) links with PEs due to the mathematical complexity of the composite channel probability density function (PDF) [21,22]. On the other hand, for the Rician channel, the impact of PEs is seldom analyzed. The difficulty arises from the fact that the Rician PDF contains a modified Bessel function of the first kind and precludes from modeling the composite channel distribution of closed form. Conventionally, the Rician PDF could be expanded in Taylor series, which, however, holds only for the condition of negligible-to-small PEs for SISO links [23]. In general, the divergence of the transmitter is rather small (1-2.5 mrad in typical) and the receiver diameter is usually around 5-20 cm level [19,21]. This way, the alignment could be largely deteriorated by wind loads, thermal expansions, and platform vibrations, leading to severe pointing errors (1-5 mrad or larger) and performance degeneration. So far, coherent FSO system with large range of PEs (negligible-to-moderate, and strong-to-severe) has gone largely overlooked in SISO links. Furthermore, analysis of coherent FSO link with PEs in MIMO systems is even more difficult due to the complex high-order integration for BER analysis [22].

To this end, we have been largely motivated to analysis the performance of homodyne BPSK in both SISO and SIMO links over Rician turbulence channel with PEs. To model the composite effects, an exact analytical channel PDF applicable for all range of PEs and different values of SNR is formulated in terms of Meijer-G function via expansion of the Rician model in series of exponential terms. This leads to the derivation of an exact analytical BER expression for SISO links. Thanks to the moment generating function (MGF), the BER expression for (maximal ratio combining) MRC based SIMO links is established in a single integral which can be directly calculated with high accuracy. As it enables both analytical and numerical verifications for almost all different link conditions, the BER performance in both SISO and SIMO links are investigated. On top of that, from an engineering point of view, high-SNR approximated channel model, asymptotic BER, coding and diversity gains are also derived and assessed. The relation between the PEs and the modal compensations are carefully discussed, revealing their impacts on BER performances. Our modeling, in conjunction with the analysis approaches, is thus verified and can be adapted for a wide range of coherent FSO system.

2. System description and channel model

2.1 Coherent communication system

To simplify the analysis, a homodyne coherent BPSK FSO link is considered [8,15]. After being mixed with the LO, the received optical signal is down-converted. The corresponding photo-current can be expressed as

2.2 Channel models

For the coherent detection, the efficiency of coherent mixing and wavefront matching are deteriorated by the turbulence-induced amplitude fluctuation and phase distortion, degrading the SNR and system performance. The PDF of the intensity fading $\left ( {{h_a}} \right )$ is known to follow a modified Rician distribution as

To determine ${<h_a>}$ and $K$, both the turbulence amplitude fluctuation and the turbulenced-indeced phase distortion should be considered. The log-amplitude variance $\sigma _I^2$ can be described by a scintillation index (SI), i.e., $SI = \exp \left ( {4\sigma _I^2} \right ) - 1$. To compensate the turbulence-induced phase distortions, either zonal or modal compensation should be deployed. The residual phase variance after compensation of $J$ Zernike terms is given by $\sigma _{\phi }^{2}=C_{J}\left (D / r_{o}\right )^{5/3}$ [15,16]. The coefficient $C_J$ is determined by the number of compensated modes, e.g., the $J = 1, 3, 6$ describes the compensation for piston, tip-tilt, and astigmatism, respectively [8,15,23]. And the receiver aperture diameter $D$ is normalized by the wavefront coherence diameter $r_0$, which describes the spatial correlation of phase fluctuations in the receiver plane [15,16,23,24].

The atmospheric attenuation, ${h_l}$, is often deterministic and can be expressed in terms of the visibility and weather condition [21]. As it acts as a scaling factor, with proper approximation, it is always assumed to be equal to 1 [20,21].

Regarding the PE, a general model of misalignment-induced fading given in [19] is considered. Assuming a Gaussian spatial intensity profile of beam waist radius ${\omega _z}$, on the receiver plane at distance $z$ from the transmitter and a circular receive aperture of radius $D$, the PDF of ${h_p}$ is characterized by

In a typical FSO system with PEs, the normalized pointing error deviation $(\sigma _{\mathrm {s}}/D)$ is assumed to be between 1 and 10, and the normalized beamwith $(\omega _{\mathrm {Z}}/D)$ could be 5 and 10 [19–21]. When $\omega _{\mathrm {Z}}/D =10$, the value of $\xi$ is 5 and 0.5 for negligible PEs and severe PEs, respectively, which is in accordance with previous reports [25]. The smaller $\xi$ is, the stronger impact of the PE it implies. Therefore, we believe that it is appropriate to select $\xi = 6.7$ for negligible PEs, $\xi = 0.7$ for strong PEs, and $\xi = 0.4$ for severe PEs in our analysis.

2.3 Composited channel statistical model

Using the individual PDFs discussed above, the unconditional PDF of the composite fading channel is given by [20]

2.3.1 Series representation of the Rician PDF

As aforementioned, an analytical expression for the composite channel could bring convenience in the performance analysis. To this end, the ${f_{{h_a}}}\left ( {{h_a}} \right )$ can be tightly approximated by applying Taylor series onto the ${I_0}$ [26, eq. (8.445)], i.e., $I_{0}\left [2 \sqrt {\frac {(1+K) K}{\left \langle h_{a}\right \rangle } h_{a}}\right ]=\sum _{i=1}^{\infty } \frac {1}{(i-1) ! \Gamma (i)}\left (\frac {(1+K) K}{\left \langle h_{a}\right \rangle } h_{a}\right )^{i-1}$. It has been shown that this expansion converges when $i$ is large. Moreover, the convergence of this series is dependent on the channel state and becomes slow until the truncated number $N = \sqrt {\frac {{\left ( {1 + K} \right )K}}{{{\left \langle h_{a}\right \rangle }}}{h_a}}$ [26–28]. To achieve sufficient convergence for almost all parameter values, a large value of $N$, e.g., $N = 20$, is selected in our analysis. In this case, the Rician PDF could be tightly approximated and number of terms for the PDF estimation is largely reduced. Being analytically tractable, it converges as follows

2.3.2 Analytical expression of the composite channel PDF

Based on the above Rican PDF, now we can substitute Eqs. (5) and (6) into the integral in Eq. (4), the resulting ${f_h}\left ( h \right )$ reads:

Then, an analytical expression of the composite PDF can be derived in terms of the Meijer-G function $G_{p,q}^{m,n}\left [ . \right ]$, defined in [26, eq. (8.350.2)]. By utilizing the $\Gamma (a, b)=G_{1,2}^{2,0}\left [\left .b\right |_{0, a} ^{1}\right ]$ [31, eq. (06.06.26.0005.01)] and the relationship [26, eq. (9.31.5)], it allows us to further simplify the composite PDF as

3. Performance analysis for SISO channel

3.1 Exact analytical BER

The exact BER can be obtained by averaging the conditional BER ${P_e}\left ( {\gamma _{o} {|}h} \right )$ over the composite channel PDF as

Then we alternatively apply the Q-function in the Meijer-G form ${erfc}\left (\sqrt {\gamma _{0}h}\right )=\frac {1}{\sqrt {\pi }} \boldsymbol {G}_{1,2}^{2,0}\left [\gamma _{0} h\right |_{0,0.5} ^{1}]$ [32, eq. (12)]. Along with the composite channel PDF the integral of the product of two Meijer-G functions [26, eq. (7.811.1)], the exact BER can be analytically expressed as

3.2 High-SNR approximated PDF and asymptotic BER

To get more insights into the coherent FSO system, BER performance in high SNR regime is further evaluated. In typical, high-SNR performance is mainly determined by the behaviors of channel PDF near the origin [29]. As such, the composite PDF in Eq. (9) can be approximated by a single polynomial term as $P_{h \rightarrow 0}(h)=a h^{b}+o\left (h^{b}\right )$ at $h \to 0$, where $o({h^b})$ could be neglected as ${\lim _{h \to 0}}o\left ( {h^{\textrm{b}}} \right )/h = 0$. So as to obtain the expression of ${P_{h \to 0}}\left ( h \right )$, the Meijer-G function in the composite PDF is expanded around $h \to 0$ [31, eq. (07.34.06.0006.01)], as follows

3.3 Pointing error induced SNR loss factor

When $\xi > 1$ is satisfied, it is found that the diversity gain is independent of negligible-to-moderate PEs, as typically for some FSO sceneries [21,33] . In other words, this requires the equivalent beam radius at the receiver is at least twice the pointing error displacement standard deviation. It is thus convenient to compare the BER performance with varied $\xi$. The impact of PEs can be translated into a sensitivity loss, relative to the case without PEs, as follows

4. Performance analysis for SIMO links

In this section, the maximal-ratio combining (MRC) scheme is employed when our model is extended to the analysis for SIMO links.

4.1 Exact BER expression

A typical SIMO link is assumed to include L identical but independent apertures. Different from the SISO link, traditional PDF-based BER analysis is almost mathematically intractable for the the SIMO link. Because it involves a L-fold integral defined by the joint multivariate PDF of the SNR in each branch [22]. When the composite channel is considered in the SIMO link, MGF-based approach is applied to further simplify the BER analysis. In this way, the L-fold integral can be reduced into a single-fold integral [34]. The total SNR with MRC is the sum of the individual SNR at each branch, i.e., ${{{\gamma }}_{MRC}} = \mathop \sum \limits_{l = 1}^L {\gamma _l}$ with ${\gamma _l}$ being the SNR in the $l^{th}$-branch [24,35]. Then, the MGF of ${{{\gamma }}_{MRC}}$ is the product of the MGF of each ${\gamma _l}$ and the BER can be simplified to a single finite integral as

4.2 Asymptotic BER expression

In the high SNR regime, the MGF can be asymptotically expanded using the dominant term of the $G_{2,2}^{2,1}\left [z | \begin {array}{l}{a_{1}, a_{2}} \\ {b_{1}, b_{2}}\end {array}\right ]$. Depending on $\xi$, the MGF in high SNR regime $\left ( {\textrm{s} \to 0} \right )$ is simplified as

Meanwhile, it is further manifested that the diversity gain is only determined by the receiver branches ($L$) under the condition of negligible-to-moderate PEs, as the same way as the SISO link. In this case, it could be deduced that the diversity gain could be linearly increased by distributing more branches in SIMO links. However, when the impact of PE becomes stronger, i.e., ${\xi < 1}$ , the diversity gain is partially reduced. For instance, when $\xi = 0.4$ , the diversity gain is largely suppressed to merely $16\ \%$.

5. Numerical results and discussion

In this section, detailed analyses on the performance of the previously derived SISO and SIMO FSO systems are discussed. The optical wavelength is assumed to be ${{\lambda }} = 1550\, \textrm{nm}$. For a typical FSO system, a source-to-destination distance of $1 km$ and receiver diameter of $D = 10\ \textrm {cm}$ is assumed. Without loss of generality, the weak, moderate, and strong turbulence is characterized by $SI = 0.1 ,0.33$ , and 1.6, respectively. The path loss factor ${h_l} = 1$ is considered. As for the impacts of PEs, the $\xi$ is varied within $\xi = 6.7,1.4,0.7,0.4$ with ${A_0} = 1$. For instance, generally speaking, ${{\xi }} = 6.7$ represents negligible PE while $\xi = 0.4$ stands for severe PE.

The combined effects of Rician turbulence and PEs on the BER performance for a SISO-FSO link is verified in Fig. 1. Without loss of generality, we consider two cases: weak $\left ( {\textrm{SI} = 0.1} \right )$ and strong turbulence $\left ( {\textrm{SI} = 1.6} \right )$. When the impact of PEs is as large as $\xi = 0.4$, the BER for all SNRs remain above $1.0 \times {10^{-2}}$ under weak turbulence and also above $1.0 \times {10^{-1}}$ under strong turbulence. In this case, the severe PE leads to a noise floor and the BER could hardly be improved by increasing SNRs. For moderate PE, namely $\xi = 1.4$ , the BER at $SNR = \ 30 \ \textrm {dB}$ becomes $6.0\times {10^{-4}}$ and $2.0\times {10^{-2}}$ under weak and strong turbulence, respectively. Within moderate PE regime, even for a large change from $\xi = 6.7$ to $\xi = 1.4$, there is no significant degradations in the BER performance. This indicates that the system seems to be quite tolerable for negligible-to-moderate PEs. According to the discussion in section 3.3, the sensitivity loss due to the PEs could be easily estimated. For instance, at $BER = {10^{-4}}$, the SNR loss is approximated around 3 dB for $\xi = 1.4$ than that for $\xi = 6.7$. As shown in Fig. 1, the exact BER closely matches the numerical results for all the SNRs and the PEs. Moreover, the asymptotic BER approaches the exact BER in the high SNR regime.

 

Fig. 1. BER performance of coherent FSO-SISO link in the composite channel with Rician turbulence and PEs, under (a) weak turbulence $SI = 0.1$, (b) strong turbulence $SI = 1.6$. To represent the impacts of PEs from negligible to severe, the $\xi$ is varied as $\xi = 6.7,1.4,0.7,0.4$. Analytical results (solid lines), asymptotic results (dashed lines), and numerical simulation (*) results are both included.

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As the turbulence-induced phase distortions could be improved by modal compensation, here we further investigate the BER performance under different Zernike modal compensations $(J = 1, 3, 6)$ when various PEs are considered. In Fig. 2, the condition of negligible PE $\left ( {\xi = 6.7} \right )$ is firstly assessed. Compared with the case for $J = 1$, the BER at $SNR = \ 30 \ \textrm {dB}$ is improved from the original $2.0 \times {10^{-3}}$ to $4.0 \times {10^{-7}}$ for $J = 3$, and further to $6.0 \times {10^{-9}}$ for $J = 6$. All these results are in accordance with the previous reports [8,35] that low order corrections such as tip-tilt correction $(J = 3)$ are able to improve the BER performance significantly under the condition of $D/{r_0}=2$. When the impacts of PEs are considered, it is obvious that the modal compensation can effectively suppress the impacts of PEs in low-to-moderate SNR regime. On the other hand, degradation in BER performance can be found in high SNR regime for variations in PEs from $\xi = 6.7$ to $\xi = 0.4$. In comparison with the case for $J = 1$, the deterioration of PEs on BER performance now shifts to high SNR regime when modal compensations are applied. This is mainly due to the fact that the mean value $\left \langle {{h_a}} \right \rangle$ gradually approaches the unity and the Rician factor $K$ also becomes stronger when higher order modal compensation is applied. As for a coherent FSO system, the BER performance is mainly dominated by the phase distortion, therefore, the modal compensation is more effective in the small-to-moderate SNR regime. On the contrary, in the high SNR regime, the BER is gradually deteriorated by the PEs, as the misalignment induces extra intensity fading.

 

Fig. 2. BER performance under various compensation modes $(J = 1, 3, 6)$ over the composite Rician turbulence and PE fading channel. The $J = 1, 3, 6$ describes compensation for piston, tip-tilt, astigmatism, respectively. To represent general turbulence, moderate turbulence of $SI = 0.33$ is considered.

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The impact of PEs for the MRC-based SIMO link is further investigated in Fig. 3, where two cases are included, Fig.3 (a) without modal compensation, and Fig.3 (b) with $J = 3$ compensated mode in each aperture. To represent a general FSO scenario, moderate turbulence with $SI = 0.33$ is assumed. For the sake of fair comparison, aperture area of each receiver is assumed to be $1/L$ times the aperture area of a SISO link [24,35], ${D_l} = D/\sqrt L ,0 \ll l \le L$ , where $D$ denotes the aperture diameter in SISO link.

 

Fig. 3. Comparison of BER performance for MRC-SIMO link in cases of different number of receiver branches $(L = 1, 2, 4)$ under moderate turbulence $(SI = 0.33)$ with varied PEs. Two situations are involved, (a) without and (b) with modal compensation $(J = 3)$. The BER for the SISO link $(L = 1)$ is depicted as a benchmark.

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In general, it can be inferred by Fig. 3(a) that, in the SIMO system, due to the presence of PEs, the more branches are employed, the greater impairment on the BER improvement can be observed. In particular, for negligible PE, the BER could be improved from $2.0 \times {10^{-3}}$ at $SNR = \ 30 \ \textrm {dB}$ for a single receiver, to $1.0 \times {10^{-5}}$ for $L = 2$, and further to $1.0 \times {10^{-10}}$ for $L = 4$. Similar tendencies are observed in the case of moderate PE [red curves in Fig. 3(a)] which is manifested as a quasi-linear BER improvement. This cross-verifies the relation unveiled between the diversity gain and number of branches as previously indicated by Eq. (29). Nevertheless, with the presence of strong-to-severe PEs, such trendency has been largely retard. For instance, in the case of $\xi = 0.7$ [(pink curves in Fig. 3(a)], the BER at $SNR = \ 30 \ \textrm {dB}$ is now only $2.0 \times {10^{-2}}$ for a single receiver and the improvement is narrowed to $2.0 \times {10^{-3}}$ for $L = 2$, and to merely $1.0 \times {10^{-6}}$ for $L = 4$. This tendency is more significant for $\xi = 0.4$ which is, however, not shown to keep the figure clear. Clearly, for all the cases, the asymptotic BERs (dashed curves) are sufficiently tight with the exact ones in the high SNR regime.

In Fig. 3(b), the effect of modal compensation is verified in the SIMO linkwith varied PEs. As can be inferred by the results from the SISO link, the effect of low-order modal compensation is already sufficient. Thus the modal compensation of $J = 3$ is investigated without losing generality. Not surprisingly, the model compensation further facilitates the BER improvement from small to large SNR cases. In addition, the impairments due to the PEs are largely alleviated. This is confirmed as the beginning of the PE-induced BER deterioration is now observed at a higher SNR compared to the case without modal compensation. It should be also noted that the boundary of the BER is almost limited by the PEs and could be further improved with higher-order modal compensations.

Therefore, to ensure BER performance, PEs existing in any practical SISO and SIMO links should be considered and suppressed. Besides, the impairments due to PEs could also be reduced with other techniques, such as optimizing transmitter beamwidth. As only low-ordered modal compensations can provide significant BER improvements and suppression onto the impacts of PEs, deployments of multiple receivers with modal compensations are suggested for achieving sufficient BER performances.

6. Conclusion

We investigate the BER performances for coherent FSO system in both SISO and MRC-based SIMO links, within which the combined effects of Rican turbulence and PEs are both modeled and considered. Different from other channels models, the modified Rician distribution is manifested by its capability to model turbulence amplitude fluctuation, optical phase distortions and compensation for a practical coherent receiver. For the ease of analysis, a novel mathematically traceable expression for the composite channel PDF is first formulated, which allows accounting for a wide range of PEs, namely, negligible-to-severe PEs. Based on this model, an exact analytical BER expression is derived for SISO links while the extension to MRC-based SIMO links is achieved through a simple one-dimension integral of the MGF. To provide in-depth engineering insights for both types of links, high-SNR approximated channel PDF, asymptotic BER expressions, coding and diversity gains, are explored and discussed in detail. For the SISO link, it is instructive to find that the impact of PEs dominates the BER performance but can be efficiently mitigated by the modal compensation in low-to-moderate SNR regime. With higher-order of compensations, the PEs induced BER deterioration is retard as it becomes obvious only at higher SNRs compared with the non-compensated cases. For SIMO links, quite similar tendencies could also be observed while the impact of PEs becomes more severe with the increase of the number of branches. In high SNR regime, thanks to the asymptotic BER evaluation, the SNR penalty due to the PEs is estimated for both types of links. Both analytical and numerical investigations are carried out, constituting a cross-verification for the effectiveness of our modeling. The proposed modeling provides an accurate and more complete description for the Rician channel with the presence of a wide range of PEs with respect to the coherent FSO system. Therefore, it offers an efficient and useful approach for practical performance analysis, enabling further system optimization from an application point of view.