Differential signalling in free-space optical communication systems

In this paper, we review the differential signalling technique and investigate its implementation of in free-space optical (FSO) communication systems. The paper is an extended version of our previous works, where the effects of background noise, weak turbulence and pointing errors (PEs) were investigated separately. Here, for the first time, we present a thorough description of the differential signalling scheme including for combined effects. At first, we present an extension of the analysis of differential signalling to the case of moderate to strong atmospheric turbulence. Next, we investigate a more general case where both channel turbulence and PEs are taken into consideration. We provide closed-form expressions for the optimal detection threshold and the average bit-error-rate, and present a set of numerical results to illustrate the performance improvement offered by the proposed differential signalling under various turbulence and PEs conditions.


Introduction
The received signal in a free-space optical (FSO) communication system is subject to the deterministic and random factors associated with the atmospheric channel, such as geometric loss, particle scattering due to fog, smoke, low clouds, snow, and the influence of the atmospheric turbulence [1][2][3][4][5]. Additionally, pointing errors (PE) due to building sway and thermal expansions can further deteriorate the FSO link performance [6][7][8], whereas fog, smoke, rain, etc. result in a constant power loss; both turbulence and PE lead to random fluctuations of the amplitude and the phase of the received signal. In non-return-to-zero on-off keying (NRZ-OOK) intensity-modulation/direct-detection (IM/DD) based FSO systems, an optimal detection threshold level (DTL) at the receiver (Rx) should be used to distinguish the received '0' and '1' bits [3,4,9]. However, turbulence and PE can result in signal fading [7], which necessitates adaptive setting of the DTL [9]. Recently, a differential signalling scheme (DSS) or differential detection was proposed for on-off keying-(OOK)-based FSO links, which allows using a pre-fixed DTL under various channel conditions such as fog, smoke [9], weak turbulence [3], and PE [4].
In [23][24][25][26], no differential signalling was considered and the analysis reported cannot be applied to the FSO link with differential signalling in correlated channels.
The paper is organised as follows. In Section 2, the concept and theory behind the DSS technique are explained. Section 3 considers the case of the lossy channel as well as the background noise, whereas the influences of weak and moderate to strong turbulence regimes are discussed in Section 4. In Section 5, the channel with PE effect is investigated. The joint influence of turbulence and PE is introduced in Section 6. Section 7 presents an example of numerical results illustrating the performance improvement offered by DSS, and, lastly, Section 8 concludes the paper.

Differential Signalling Scheme
The proposed system block diagram is depicted in Figure 1. The NRZ-OOK signal S ∈ {0, 1} and its inverted version S are used for intensity-modulation of two optical sources at wavelengths of λ 1 and λ 2 , respectively. It is worth mentioning that by comparing S to the optimal DTL S thresh = E [S] where E[·] denotes the expected value, the original data bits (i.e., bit is "0" for S < S thresh and "1" elsewhere) can be retrieved. Note that the optimal DTL for S is also S thresh . The output intensities I i (i = 1, 2) of optical sources (OSs) are given by where Γ i denotes the electrical-to-optical conversion coefficient of i-th OS. Outputs of OSs are then passed through a beam combiner (BC) to ensure that both beams are transmitted over the L-length FSO channel. The optical signals y at the Rx are given by where 0 ≤ h i ≤ 1 denotes the channel coefficient (response) including the effects of geometrical and atmospheric losses, PE, and atmospheric turbulence. y is passed through a 50/50 beam splitter (BS) and optical filters (OF) with the centre wavelengths of λ 1 and λ 2 prior to being collected by an optical Rx. The generated photocurrents are amplified by transimpedance amplifiers (TIA) with outputs given by v 1 v 2 = 1 2 where i is the photodetector responsivity, G i is gain of TIA, n i is the additive white Gaussian noise (AWGN) with zero mean and variance σ 2 n,i = N 0 /2, and N 0 denotes the noise power spectral density. The combined output v t = v 1 − v 2 is given by where ∆n = n 1 − n 2 . A sampler, sampling at the centre of bit duration, and a threshold detector are used to regenerate the transmit data. From Label (4), the optimal DTL for v t is given by density. The combined output = − is given by where ∆ = − . A sampler, sampling at the centre of bit duration, and a threshold detector are used to regenerate the transmit data. From Label (4), the optimal DTL for is given by Figure 1. A schematic block diagram of differential signalling scheme (DSS). The labels OS, BC, BS, OR, OF, and TIA are optical source, beam combiner, beam splitter, optical Rx, optical filter, and transimpedance amplifier, respectively. is the bit duration.
where Mean(·) and Var(·) are the mean and variance, respectively, and ρ is the correlation coefficient between h 1 and h 2 . To recover the transmit bit stream, the optimal DTL should be set to the value given by Label (6a), with the variance given in Label (6).
In the electrical domain, S is represented by two distinct signal levels of v low and v high corresponding to bits "0" and "1", respectively. For S bits, "0" and "1" are recognised as v high and v low , respectively. This means that, for the link 1 in Figure 1, the received electrical signals corresponding to bits "0" and "1" are v low ηh + n 1 and v high ηh + n 2 , respectively. Therefore, the corresponding received differential signals for bits "0" and "1" are v low − v high ηh + ∆n and v high − v low ηh + ∆n, respectively. Note that the difference between the two bits is twice that of a single link, so without the loss of generality we replace the levels for bits "0" and "1" with 2v low ηh + ∆n and 2v high ηh + ∆n, respectively, and substitute the subtraction of channel responses by h DS , and rewrite Label (4) as v t = ηh DSS S + ∆n.
The BER expression for a NRZ-OOK FSO link with an equiprobable data transmission condition is P e = 0.5(P(e|1) + P(e|0)), provided that P(e|0) and P(e|1) are the conditional probabilities given as where Q(·) denotes the Gaussian Q-function defined as Q(x) = +∞ x exp −t 2 /2 dt, f h (·) is related to the channel condition, and the electrical signal-to-noise ratio (SNR) = (ηh) 2 /2N 0 . In the rest of the paper, we will show that, under different channel conditions, in an FSO system with DSS, the average of DTL is always zero and the variance of DTL has the minimum value if the channels are highly correlated. In addition, for highly correlated channels condition, we derive the BER expressions for different channel fading conditions.

Lossy Channel and Background Noise
As was mentioned earlier, h i can represent the total loss in the channel including geometrical loss, fog, or other particle induced scattering. The geometrical loss, which depends on the optical link configuration, is a constant quantity for a fixed point-to-point FSO link. On the other hand, losses due to fog/smoke, which are classified as slow fade events, can be assumed to be relatively constant over the channel for a given time. Under these conditions, both h 1 and h 2 can be considered constant, and therefore the DTL average and its variance can be determined using Label (6). Note that, since h i is constant, we can therefore substitute in Label (8) is the Dirac delta function. Thus, the averaging integral operation is simplified to the Gaussian Q-function.

Geometrical Loss
At the Rx, the received optical beam radius w Rx = w Tx + θ 0 L, and the geometrical loss is defined as [28] where w TX , and θ 0 are the laser beam radius, beam divergence angle at the transmitter (Tx), respectively. L is the link span and D Rx is the aperture diameter at the Rx. Note that, if the maximum Tx lens radius is known, then θ 0 = λ FSO πw Tx can be readily determined [29].

Fog/Smoke Attenuation
The fog loss is determined based on the visibility (Vis) of the channel, which is described by two well-known models of Kim and Kruse [28], and is defined as [30] Vis = 17 where λ 0 denotes the maximum sensitive wavelength for the human eye, which is normally set to 550 nm (i.e., the green colour). λ is the wavelength of the laser being used and β λ denotes the loss coefficient. Based on the Kim model, q is defined as [31] 1.6, Vis > 50 1.3, 6 < Vis < 50 1.6 × Vis + 0.34, 1 < Vis < 6 Vis − 0.5, 0.5 < Vis < 0.1 0, Vis < 0.1.
The relation between the total loss due to the absorption and scattering of light β λ and loss due to fog h fog is given by the Beer-Lambert law as [31] h f og = exp(−β λ L).
Since, for each individual channel, h i remains constant; therefore, we have Mean(h i ) = h i and Var(h i ) = 0. In addition, based on the assumption we made (i.e., h 1 ≈ h 2 ), the average of DTL from Label (6a) is zero. The total channel coefficient for a DSS based FSO system with geometrical and fog/smoke loss is h DSS = h geo × h fog . Therefore, using Label (9) and Label (12), the channel coefficient can be obtained for determining the BER expressions, which are summarised in Table 1.  (19). For k, x i and ω i refer to Appendix A.
Moderate to Strong Moderate to Strong Turbulence and Pointing Errors

Background Noise
Considering the background noise n bg,i (i.e., due to sun and artificial lights) [22] the optimal DTL in Label (5) can be reformulated as Therefore, the variance of DTL in Label (6b) is given as where ρ bg is the correlation coefficient between n bg,1 and n bg,2 . Using the differential signalling scheme proposed in [22] for the case of using two close wavelengths, here we have σ 2 bg,1 = σ 2 bg,2 and ρ bg ≈ 1. Assuming the same photodetector parameters at the receiver for the two wavelengths, we have Var(V thresh ) = 2σ 2 n . Under these conditions, the background noise can be effectively rejected after differential detection, and we can obtain the BER formula for the corresponding FSO system with DSS similar to the case in the previous subsection, as presented in Table 1. Note that this BER expression is valid as long as the Rx is not saturated by the ambient light [22].

Weak Turbulence
In FSO under weak turbulence, the fading of the optical signal can be modelled as h = h 0 exp(2X), where X is a distributed normal random variable (RV) with mean and variance of m h and σ 2 h , respectively [32]. This way, h has a lognormal probability distribution function (PDF) given as [32] For the log-normal distribution, we have where m h,i = −σ 2 h,i [32]; therefore, where ρ AT is the correlation coefficient between the turbulence influenced channels. To determine the strength of the turbulence, Rytov variance σ 2 R can be used [33]. For weak turbulence σ 2 R < 1 [34], the simplified form of Label (17b) can be used (see Table 1). In the case of a plane wave propagation through a turbulence channel, we have [33] where C 2 n (in m −2/3 ) is the refractive index structure parameter. The summation of two lognormal variables can be approximated by a lognormal variable [35,36]; therefore, h DSS = h 0 exp(2X DSS ), where X DSS is a Gaussian RV with mean m DSS and variance σ 2 DSS . We will adopt the same procedure as in Wilkinson's method [35] to estimate the required parameters of h DSS . To further normalize h DSS , we set the mean value to [27] where Var(h DS ) is given by Label (17b) replacing V thresh by h DSS and letting ηS thresh = 1. Once σ 2 DSS is determined, it is possible to specify the PDF of a DSS based FSO system using Label (8). The simplified BER expression for weak turbulence is summarised in Table 1. To obtain the closed form equations in weak turbulence, the Gauss-Hermite quadrature formula was used [37]; for details, see Appendix A.

Moderate to Strong Turbulence
For moderate (σ 2 R ≈ 1) to strong (σ 2 R > 1) turbulence regimes, we assume h = XY, where X and Y refer to large-scale and small-scale atmospheric turbulence effects, respectively [33]. Knowing that both X and Y are expressed by the gamma distribution, we model h by the PDF of gamma-gamma (GG) distribution as [38] where a ≥ 0 and b ≥ 0 are also known as the effective numbers of large-and small-scale turbulence cells, respectively [38,39]. K l (·), and Γ(·) denote the modified Bessel function of 2nd kind and order l, and the gamma function, respectively. The two parameters of a and b characterize the irradiance fluctuation PDF and are related to the atmospheric conditions, which are given by [33] For the plane wave propagation model and considering the Rx aperture diameter D Rx , the closed form expressions for σ 2 lnX and σ 2 lnY are given by [33] where d = kD 2 Rx /4L 0.5 [40]. Knowing that for GG distribution, we have [41,42] then it is easy to show that Thus, it is possible to formulate DTL average and variance of the DSS link with the closed-form expression as in Table 1. To derive the BER expression, the Bessel function in Label (20)  with Meijer-G function, which results in simplified BER expression in Table 1. See Appendix A for more details.

The Case of Pointing Errors
In a terrestrial FSO link, the joint geometrical loss and PE induced fading within the channel is given by [23] where r, A 0 and w eq correspond to beam displacement, the geometrical loss and equivalent beamwidth, respectively. Note that x 0 e −t 2 dt [23]. Note that, if the geometrical loss is taken into account by using Label (9), (25). The PE induced displacement has two major components: (i) the boresight r b -displacement between the beam centre and centre of the detector; and (ii) jitter r j -offset of the beam centre at detector plane [24]. The mean offset of PE r b represents a deviation originated from thermal expansion of buildings [8], whereas r j is a RV originated from building sway and vibrations [8]. From the statistical point of view, the jitter corresponds to the random variation of the optical beam footprint around the boresight direction with the jitter variance of σ 2 j [43]. It is shown in [8] that r in Label (25) has a Rician PDF as below with the mean and 2nd moment given by where s is the boresight displacement and I 0 (·) is the modified Bessel function of the 1st kind with order zero and γ = w eq /2σ j . Therefore, the variance of h i is Substituting Label (27) and Label (28) in Label (6) yields in the DTL average and variance expressions as given in Table 1. Note that we have used ρ PE to refer to the correlation between PE fading effects within the channels. By using the tail probability of the error events, we can derive the BER for this special scenario where only the PE is considered. The simplified BER expression is outlined in Table 1, whereas detailed derivation for the BER is given in Appendix A.
It is also possible to estimate the equivalent parameters of the DSS based FSO link. For this, we have adopted a single input single output (SISO) link with Rayleigh PE PDF and derived the parameters, which have the same PE variance as DSS. For the simplified case, where r b,1 = r b,2 = 0, assuming that γ 1 = γ 2 and A 1 = A 2 = A DSS , and simplifying Var(h DSS ) = Var(h 1 − h 2 ), the following equation is derived from which γ DSS of the equivalent PE PDF as For r b,1 = r b,2 = 0, Rician distribution becomes Rayleigh and we assume that the PDF of equivalent PE is also Rayleigh for comparison.

Weak Turbulence and PE
The PDF of h i = h a h p , where h a is the fading coefficient due to turbulence and h p represents the effect of PE under log-normal (weak turbulence) channel is given by where u a = The nth order moment of the above PDF is and therefore Therefore, in terms of both h a and h i , Label (32a) is re-written as Considering the assumption that h 1 ≈ h 2 , similar to the other channel conditions, the average of DTL will be zero. Then, knowing that Var determined using Label (23) and Label (27), which is already given in Label (32b). Finally, using Label (6) and Label (32b), Var(V thresh ) is obtained as given in Table 1. Deriving the expression for the BER in weak turbulence with the presence of PE provides a very complex closed-form expression, which is not convenient to provide here; however, the instruction for a numerical approach to calculate BER is given in Appendix A.

Moderate to Strong Turbulence and PE
Similar to the weak turbulence regime, the mean and variance for moderate-strong turbulence under the GG channel with the PE are given by Using Label (6), and Label (34b), Var(V thresh ) is determined as given in Table 1. Deriving the expression for the BER in the moderate to strong turbulence regime with the presence of PE again do not result in a simple closed-form expression, however, the instruction to simplify the numerical approach of BER is given in Appendix A.
Using Label (6), and Label (34b), Var(V thresh ) is determined as given in Table 1. Deriving the expression for the BER in the moderate to strong turbulence regime with the presence of PE again do not result in a simple closed-form expression, however, the instruction to simplify the numerical approach of BER is given in Appendix A.

Results and Discussion
In this section, we will present and further discuss the equations derived in the previous section. For each studied case, the variance of DTL as well as the BER of the link is obtained. The wavelengths adopted are 850 and 830 nm and we have assumed that both links have identical parameters. Furthermore, the normalised noise variance is set to 0.06. For the lossy channel and with the ambient light, where channels are inherently highly correlated, the variance of DTL and BER expressions remains the same as under the clear channel condition; therefore, here we focus on the other channel conditions of PE and turbulence. Regarding the series expressions with infinite upper limit in Table 1, it worth mentioning that we did not study the convergence of them. However, we considered between 30-50 terms to calculate the results. Figure 2 illustrates the results for weak turbulence. The predicted data shown are obtained using the equations given in Section 4.1. Using Label (17b), the predicted normalised variance of DTL against the turbulence for a range of correlation coefficients for SISO and DSS based schemes is shown in Figure 2a. Regardless of turbulence the variance of DTL approaches the minimum value in the cases of highly correlated channels. Note that, on the contrary, the variance of a SISO remains constant, whereas the DSS based scheme offers improved performance at higher turbulence levels. Figure 2b illustrates the BER as a function of SNR for the DSS system for a range turbulence levels. Also displayed are the simulated results, which show a close match with the predicted data. As can be observed, the DSS link, which is assumed to have a high correlation coefficient, display negligible deterioration at higher turbulence levels.  As discussed previously, in the case of moderate to strong turbulence, other models should be adopted to assess the link performance. Here, we have used a GG turbulence model as outlined in Section 6.2 to determine the variance of DTL and the BER of a DSS link as shown in Figure 3. As in the weak turbulence case, for higher levels of turbulence in highly correlated channels, DSS offers improved performance compared with SISO.  . The plots display the same profile as in Figure 2a with DSS offering improved performance at more significant PE. The benefit of highly correlated DSS compared to the SISO link with the same is more outstanding when reduces. Smaller can be interpreted as the reduced effect of PE on the optical beam. Using the predicted from Label (29), the variance of equivalent DSS PE ( , ) can be determined for a range of . Figure 4b depicts the jitter standard deviation against for a SISO link and an equivalent DSS link with = 10 cm, = 100 cm , and , = , = 10 cm . It is observed that the PE induced fading effect reduces with increasing value of . For instance, for an uncorrelated channel with = 0.3, the normalised variance is ~0.16, whereas, in highly correlated channels, the normalised variance is reduced to ~0.06. Also from both Label (29) and Figure 4b and for ≥ 0.5, we have ≤ . The predicted BER results against the SNR for a range of are displayed in Figure 4c showing a good match with the simulated data. Note that, for lower values of , the SNR penalty is rather high for a given BER.  Figure 4a depicts the variance of DTL versus the ρ PE for both SISO and DSS links for a range of equivalent beamwidth w eq . The plots display the same profile as in Figure 2a with DSS offering improved performance at more significant PE. The benefit of highly correlated DSS compared to the SISO link with the same w eq is more outstanding when w eq reduces. Smaller w eq can be interpreted as the reduced effect of PE on the optical beam. Using the predicted γ DSS from Label (29), the variance of equivalent DSS PE (σ j,DSS ) can be determined for a range of ρ PE . Figure 4b depicts the jitter standard deviation against ρ PE for a SISO link and an equivalent DSS link with a = 10 cm, w = 100 cm, and σ j,1 = σ j,2 = 10 cm. It is observed that the PE induced fading effect reduces with increasing value of ρ. For instance, for an uncorrelated channel with w eq = 0.3, the normalised variance is~0.16, whereas, in highly correlated channels, the normalised variance is reduced to~0.06. Also from both Label (29) and Figure 4b and for ρ PE ≥ 0.5, we have γ DSS ≤ γ 1 . The predicted BER results against the SNR for a range of w eq are displayed in Figure 4c showing a good match with the simulated data. Note that, for lower values of w eq , the SNR penalty is rather high for a given BER. For instance, at a BER of 10 −2 , the SNR penalty is~14 dB for w eq = 3 cm compared w eq = 7 cm. Note that, in Figure 4c, the different SNR vales are obtained by varying the value of noise variance σ 2 n for η = 1.
For instance, at a BER of 10 −2 , the SNR penalty is ~14 dB for = 3 cm compared = 7 cm. Note that, in Figure 4c, the different SNR vales are obtained by varying the value of noise variance for = 1.   Figure 5a shows the variance of DTL for simultaneous PE and weak turbulence as a function of the channel correlation coefficient for range of 0 to 1. Again, it is seen that, for higher turbulence levels and PE, the DSS system offers improved performance compared to the SISO. The BER versus the SNR for a range of for the combined effects of PE and weak turbulence is plotted in Figure 5b, showing performance degradation at higher levels of turbulence and PE. As expected from Figure 4c, larger improves the BER performance of DSS. Compared to the cases where only turbulence or PE exist in the channel, we observe a significant degradation in the performance. For instance, in a weak turbulence case for = 0.3 and SNR of 10 dB, the BER is less than 10 −3 . However, for the same turbulence situation, = 1, and SNR of 10 dB, the BER is less than 10 −2 . BER Analysis, w eq =3 Analysis, w eq =5 Analysis, w eq =7 Simulation, w eq =3 Simulation, w eq =5 Simulation, w eq =7 Figure 4. Single-input single-output (SISO) and differential signaling scheme (DSS) based free-space optical (FSO) links with pointing errors (PE): (a) the normalised variance of detection threshold level (DTL) versus the correlation coefficient of the channels; (b) the jitter standard deviation of the equivalent DSS PE vs. correlation coefficient ρ PE , for the SISO system with DSS and for a = 10 cm, and w = 100 cm, and (c) bit-error-rate (BER) against signal-to-noise ratio (SNR) for σ j = 1, s = 1, A 0 = 1, and different values of w eq with unit of cm. Figure 5a shows the variance of DTL for simultaneous PE and weak turbulence as a function of the channel correlation coefficient for range of 0 to 1. Again, it is seen that, for higher turbulence levels and PE, the DSS system offers improved performance compared to the SISO. The BER versus the SNR for a range of w eq for the combined effects of PE and weak turbulence is plotted in Figure 5b, showing performance degradation at higher levels of turbulence and PE. As expected from Figure 4c, larger w eq improves the BER performance of DSS. Compared to the cases where only turbulence or PE exist in the channel, we observe a significant degradation in the performance. For instance, in a weak turbulence case for σ 2 R = 0.3 and SNR of 10 dB, the BER is less than 10 −3 . However, for the same turbulence situation, w eq = 1, and SNR of 10 dB, the BER is less than 10 −2 . Appl. Sci. 2018, 7 Finally, the combined effects of PE and moderate to strong turbulence on the variance of DTL as well as the BER performance are illustrated in Figure 6a,b, respectively. For these cases, we can observe that even DDS is not mitigating the cannel fading efficiently. Therefore, a designer might consider additional methods such as using a larger aperture to improve the system performance by aperture averaging. The plots show further degradation in both the variance of DTL and the BER compared to Figure 5. The same trend of the larger and the better BER performance of DSS are also seen in Figure 6b. In a strong turbulence regime, PE are less dominant than turbulence. In all variance of DTL plots, one can see that once the channels are highly correlated, the normalised variance of DSS is reduced to the normalised variance of noise with the value of ~0.06. Therefore, under a highly correlated turbulence channel with significant channel fading, it is highly advantageous to implement a DSS FSO scheme in comparison to a SISO one. Finally, the combined effects of PE and moderate to strong turbulence on the variance of DTL as well as the BER performance are illustrated in Figure 6a,b, respectively. For these cases, we can observe that even DDS is not mitigating the cannel fading efficiently. Therefore, a designer might consider additional methods such as using a larger aperture to improve the system performance by aperture averaging. The plots show further degradation in both the variance of DTL and the BER compared to Figure 5. The same trend of the larger w eq and the better BER performance of DSS are also seen in Figure 6b. In a strong turbulence regime, PE are less dominant than turbulence. In all variance of DTL plots, one can see that once the channels are highly correlated, the normalised variance of DSS is reduced to the normalised variance of noise with the value of~0.06. Therefore, under a highly correlated turbulence channel with significant channel fading, it is highly advantageous to implement a DSS FSO scheme in comparison to a SISO one.

Conclusions
In this paper, we outlined the essentials of differential signalling based FSO systems subject to the both effects of pointing errors and turbulence. We provided a description comprehensive investigation of the system performance under different conditions of turbulence and/or pointing errors. We derived closed-form expressions for various channel conditions for the DSS based FSO system and showed predicted and simulated results for the variance of DTL and the BER for both SISO and DSS based FSO links. We showed that the DSS based link offers improved performance at higher turbulence and pointing error levels compared to the simple SISO link. Figure 6. Single-input single-output (SISO) and differential signaling scheme (DSS) based free-space optical (FSO) links with pointing errors (PE) and moderate to strong turbulence: (a) the normalised variance of detection threshold level (DTL) versus the correlation coefficient of the channels; and (b,c) bit-error-rate (BER) against signal-to-noise ratio (SNR) for different values of w eq with unit of cm and σ 2 R .

Conclusions
In this paper, we outlined the essentials of differential signalling based FSO systems subject to the both effects of pointing errors and turbulence. We provided a description comprehensive investigation of the system performance under different conditions of turbulence and/or pointing errors. We derived closed-form expressions for various channel conditions for the DSS based FSO system and showed predicted and simulated results for the variance of DTL and the BER for both SISO and DSS based FSO links. We showed that the DSS based link offers improved performance at higher turbulence and pointing error levels compared to the simple SISO link.

A.1. BER Expression in Weak Turbulence
When dealing with the BER expression in weak turbulence in Label (8), the mathematical simplification will result in an expression in the following form: By substituting h = h 0 exp(2X), where X has a normal random distribution, dh = 2hdx and setting the integral range, we have Next, substituting u = (x − µ)/ √ 2σ 2 , Label (A2) is re-written as [44], where k is the order of approximation, x i is the zero of the kth-order Hermite polynomial, and ω i denotes the kth-order weight factor.

A.2. BER Expression in Strong Turbulence
Given that the BER expression is where f h (x) is given in Label (20), and using the results given in [45], we can solve the integral in Label (A4), thus yielding where G m,n p,q (.|) is the Meijer-G function defined in [45].

A.3. BER Expression in PE
Let us consider the transmission of the OOK signal, where the received signal can be written corresponding to both states of s as The decision for the value of S is taken by comparing Y = y 1 − y 2 with Y th = 0. Considering two cases of (i) Y < 0 for S = 1; and (ii) Y > 0 for S = 0, the instantaneous probability of error is given by By substituting Label (A6a) and Label (A6b) in Label (A7a), we obtain Label (A7b); Label (A7c) is a consequence of the property that P{x < v} = 1 − P{x > v}, where P{x} stands for the probability of x. Since n i ∼ N 0, σ 2 n , i = 1, 2, it can be easily shown that n 1 − n 2 ∼ N 0, 2σ 2 n , and the PDF of n 1 − n 2 is given by From Label (A7), we have Substituting x √ 2σ n = y in Label (A9) and following some algebra, we have Hence, from Label (A7c), Label (A10), and Label (A11): Since 1 − Q(−x) = Q(x), therefore, Label (A12) can be rewritten as The PDF of h i (unconditional on the instantaneous value of r) under the boresight, PE (i.e., Rician distribution of r) is given by ([8] Equation (5)), where A = γ 2 Using Label (A14) and the relation Q in Label (A17), which results in Utilizing an identity ∞ 0 xe −αx 2 I 0 (x)dx = e 1/(4α) 2α (A20) in Label (A19) results in the power series based BER expression of the considered scheme.

A.4. BER Expression in Weak Turbulence with PE
The instantaneous BER of the considered scheme is given by Label (A13). The PDF of h i = h a h p for weak turbulence with boresight PE is evaluated as where f h a (x) is given in Label (15) and In addition, Since there is no closed-form available of the integral Label (A23), we evaluated this integral using MATLAB (R2016b, US). On substituting the result of Label (A23) into Label (A13), the average BER of the considered system can be obtained.

A.5. BER Expression for Moderate to Strong Turbulence and PE
Similar to weak turbulence with PE, the BER expression under moderate to strong turbulence with PE can be evaluated. Therefore, the PDF of h i = h a h p is evaluated as where f h a (x) and f h p (x) is given in Label (20) and Label (A22), respectively. In addition, Integral Label (A25) is determined by numerical integration using MATLAB. Substituting Label (A25) into Label (A13) gives an average BER of the considered system.