Secrecy performance of FSO communication systems with non-zero boresight pointing errors

Free space optical (FSO) communication is a promising candidate for the next generation (5G and beyond) wireless communication systems, due to its merits (i.e. low latency, high data rate, and license-free band, among others). However, atmospheric turbulence (AT) as well as pointing error (PE) are two of the main challenges with FSO communication that affect its performance. Here, the exact closed-form expression of the average secrecy capacity and secrecy outage probability under the composite effect of AT and non-zero boresight PEs is evaluated. For all the regimes of the AT (weak to strong), a generalised Malaga distribution is used to model the channel fading gain of the FSO link. The expressions are generalised and valid for all turbulence, and are applicable for intensity modulation direct detection as well as heterodyne detection techniques.


INTRODUCTION
Free space optical (FSO) communication has many advantages such as the license-free band, high data rate, low latency, and quick deployability, among others [1]. Despite its aforementioned merits the performance of FSO system is limited by atmospheric turbulence (AT) and pointing error (PE). As the solar radiation reaches the surface of the Earth, the air near the Earth's surface has higher temperature compared to the air at relatively higher altitudes. The warmer air rises to mix turbulently with the surrounding cooler air which results in the random temperature fluctuations. This temperature variation leads to inhomogeneities in the medium thereby resulting in the formation of discrete cells or eddies of different sizes and refractive indices. The interaction of the transmitted laser beam with this turbulent path generates random fluctuations in the amplitude and phase of the received signal. This phenomenon is called AT which results in corresponding intensity fluctuations and leads to system performance degradation, especially in long distance transmission of about several kilometers [1][2][3].
In addition, there might be a degradation in the performance due to slight misalignment between the transmitter (Tx) and the receiver (Rx), caused by for example, swaying buildings, vibrations, and thermal expansion of the building, which leads to beam pointing errors (PEs). PEs represent the horizontal and This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited. © 2020 The Authors. IET Communications published by John Wiley & Sons Ltd on behalf of The Institution of Engineering and Technology vertical displacement along the laser beam; both these components are assumed to follow independent Gaussian distribution [4]. PE has two components: jitter and boresight. Jitter is the random offset of the beam and the detector plane mainly caused by dynamic wind load, weak earthquake, and swaying buildings. Boresight refers to the fixed displacement between the beam center and the detector center caused by thermal expansion [5]. It was established in [6] that PE is a serious issue in urban areas and the effect of PE was experimentally demonstrated.
Conventionally, FSO systems utilise intensity modulation direct detection (IM/DD) due to the low cost and ease of implementation, contrary to heterodyne detection (HD) technique which has a higher cost and is relatively difficult to implement. In HD technique, a robust oscillator field mixes with the received signal which provides much better spatial and frequency selectivity than IM/DD. Furthermore, HD technique can recover the information from amplitude, phase, or polarisation from the received signal, which improves the spectral efficiency and FSO system performance . Another main advantage of HD technique is that it can overcome thermal noise [7,8]. Nevertheless, both detection techniques are used as per application and cost.
Inherently FSO communication is more secure than radio frequency (RF) communication because the optical beams are more directional compared to the RF beams thus less subject to eavesdropping/intercepting. However, optical beams can be still intercepted. In [9], two possible cases to intercept the laser beam have been mentioned. In the first scenario, when the eavesdropper is located close to the Tx, the potential eavesdropper cannot intercept the beam without partially blocking the line-of-sight (LOS) between the legitimate Tx and Rx due to the narrowness of the laser beam near the Tx. In order to intercept the beam, a sufficiently sophisticated device such as a beam splitter is required by the eavesdropper to collect fractional power of the transmitted beam. In the second scenario, the eavesdropper is located near the legitimate Rx. The laser beam usually experiences divergence due to optical diffractions. Thus, the eavesdropping is possible in FSO systems when the location of eavesdropper lies in the divergence region of the laser beam. This is one of the most probable ways of eavesdropping in FSO communication systems. In recent few years, research on the physical layer security (PLS) has been increased significantly [10].
Data encryption is one way to secure the data, but it leads to higher bandwidth consumption as well as increased complexity in upper layer [9]. PLS is a complimenting solution to securing the communication in the presence of eavesdroppers [9,11]. In this regard, PLS is widely considered as a promising technique to enhance secrecy in next generation (5G and beyond) communication systems [12,13]. In [14], analysis on average secrecy capacity (ASC) and secrecy outage probability (SOP) analysis have been done for hybrid satellite-FSO cooperative systems. In [15], ASC and SOP performance metrics have been analysed under Malaga AT but the effect of boresight PEs have been ignored. In [16], analytical expressions of ASC and SOP are derived for a single-input multiple-output (SIMO) simultaneous wireless information and power transfer (SWIPT) based mixed RF/FSO communication system, where the RF link is characterised by Nakagami-m fading and the FSO link follows Malaga distribution. Similarly, in [17], the author's evaluated the PLS performance for a two-way relay based mixed RF/FSO network assuming similar channel models as in [16], but ignored the effects of non-zero boresight. Thereafter, the PLS performance for a mixed RF/FSO system under Gamma-Gamma AT model with PE (only jitter) has been analysed in [18]. It is clear from the above papers and the references therein that the effect of boresight PEs have not been considered when evaluating the PLS of FSO communication systems despite its non-ignorable impact on the system performance. In this regard, the main contributions of this paper are as follows: 1) We derive an exact closed-form expression of ASC under the composite effect of AT and PEs (jitter and boresight) contrary to [15], where only the jitter effect is considered. 2) An exact closed-form expression of SOP under the combined influence of AT and PEs (jitter and boresight) has been evaluated contrary to [15], where only lower bound of SOP is obtained.
3) The derived ASC and SOP expressions are generalised and applicable to both IM/DD and HD techniques. 4) Some useful insights into the FSO secrecy performance are obtained through the asymptotic SOP analysis.

SYSTEM DESCRIPTION
We assume a classical Wyner's wiretap model [19], in which Alice (A) is the Tx, Bob (B) is a legitimate Rx, and Eve (E) is an external eavesdropper. The legitimate (A-B) and eavesdropper (A-E) links experience Malaga distributed flat fading effects [3,7]. PEs are introduced at the Rx due to misalignment between A and B. The received optical signal, after conversion into electrical signal using HD (coherent) or IM/DD (non-coherent) technique at the Rx, is given by where 0 is responsivity of the photodetector (hereinafter assumed unity), s represents the transmitted optical signal, I g = I a I l I p is the FSO communication channel gain that consists of three components: (i) I a denotes Malaga AT, which occurs due to inhomogeneities present along the transmission path that randomly changes the laser beam phase and amplitude at the Rx, (ii) I p represents the PE attenuation considering the combined effects of jitter as well as non-zero boresight caused due to dynamic wind load, building sways, and thermal expansion in the high rise buildings, thereby resulting into misalignment between the Tx and Rx [2,4], (iii) I l indicates path loss which is a deterministic constant described by the Beer-Lambert's law and is a function of distance, visibility, and operating wavelength [4]. Since the investigation of the path loss parameter is not the prime focus of this research and as such it is a deterministic factor. Thus, without loss of generality, I l is a assumed to be unity [2,20]. In (1), w represents additive white Gaussian noise (AWGN) with zero mean and power spectral density N 0 [14], which, without loss of generality, is assumed to be the same for both the links (legitimate and eavesdropper). The closed-form expression of the probability density function (PDF) of Malaga AT is expressed as [7,21]. (2.a) where > 0 is related to the effective number of large scale cells of the scattering process, is a natural number 1 that represents the amount of fading, 2b 0 represents the average power of total scatter components, g = 2b 0 (1 − ) indicates the average power of the scattering component received by off-axis eddies, 0 ⩽ ⩽ 1 is the amount of power coupled to the LOS component, Ω ′ denotes the average power from the coherent component, Ω is the average power of the LOS component, a and b are the deterministic phases of LOS and coupled to LOS scatter term, respectively, K v (.) denotes the modified Bessel function of second kind and order v, and Γ(.) represents Gamma function [22].

PE model with non-zero boresight
The approximation of the Beckmann's distribution into modified Rayleigh distribution is described as [2,4] where U = [U x , U y ] is the radial displacement vector, where U x and U y represent the horizontal and vertical displacement from the Rx plane and are modeled as independent Gaussian ran- , where x and y denotes horizontal and vertical boresights (mean) and 2 x and 2 y represent horizontal and vertical jitters (variance). The beam width z ≈ z, where is divergence angle and z is distance between Tx and Rx, = x 4 x +3 2 y 4 y + 6 x + 6 y 2 ) 1 3 , mod = zeq ∕(2 mod ). The PDF of PE with non-zero boresight is given by [4]

Composite effect of the AT and PE
Using (2) and (4), after applying RV transformation, the PDF of the instantaneous signal-to-noise ratio (SNR) under the combined effect of AT and non-zero boresight PE is given by [7] where x ∈ {B, E }, j = 1 and j = 2 for HD and IM/DD, and jx is electrical SNR established as jx = , E[.] denotes the expectation operator, and G m,n p,q (z| in (5) can be re-written in the following alternative form: where consists of 3 j terms. Using (6), the cumulative distribution function (CDF) of the instantaneous SNR is found as

SOP PERFORMANCE ANALYSIS
In the presence of passive eavesdropper, SOP is an important secrecy performance metric. SOP is defined as the probability that the instantaneous secrecy capacity is less than a given threshold rate, and is expressed as [12] where C th is threshold secrecy capacity and Θ = exp(C th ) ⩾ 1. On substituting (6) and (7) into (16), the SOP is evaluated as Now, using [24,Eq. (2.24.1.3)], (17) can be written as . (18) Remark 2. Although the expression for SOP given by (18) is expressed in terms of an infinite series, it converges quickly for finitely small values of n, that is, n = 8 is sufficient for the convergence of this series. The proof of convergence of this infinite series is shown in Appendix.

ASYMPTOTIC SOP ANALYSIS
To obtain useful insights into the system, we perform asymptotic SOP analysis in this section. Let us carefully observe (18) for high SNR on the main channel, jB , and a fixed eavesdropper channel SNR, jE .

Heterodyne detection, j = 1
On substituting j = 1 in (18), the SOP is written as Using Slater's theorem [27], the Meijer-G function in (19) can be represented in terms of the generalised hypergeometric function as follows: where a = [a 1 , a 2 , … , a 6 ] = [0, n − 2 mod B , n − B , n − k, n, , n], and (⋅) * denotes that the terms a h = a i are to be → 0 and consequently, 6 F 5 (⋅, ⋅, x) → 1. Therefore, the asymptotic SOP for HD, after some simplification, can be approximated as  (21) and observing that the dominant term in the asymptotic SOP expression given by (21) will correspond to the smallest power of 1B , the asymptotic slope of the SOP is given by = min{ 2 mod B , B , k}.

IM/DD detection, j = 2
On substituting j = 2 in (18) and performing a similar asymptotic analysis as mentioned in the previous subsection, the asymptotic slope of the SOP for this case can be derived as = min{ 2 Remark 3. The asymptotic slope of the SOP for HD ( j = 1) or IM/DD ( j = 2) depends only on the AT and PE parameters of the main channel and is independent of the corresponding parameters on the eavesdropper channel for a given 1E and can be expressed as = min{ 2 mod B ∕ j, B ∕ j, k∕ j }.

NUMERICAL RESULTS AND DISCUSSION
In this section, we present the analytical results and their validation using Monte Carlo simulation with the aid of Mathematica and MATLAB platform for the considered FSO system. The link parameter values are chosen from references [7,16,28]. The link parameters used are: link length L=1 km, operational wavelength = 785 nm, and refraction structure parameter,  Figure 2. Moreover, it can be seen from Figures 1 and 2 that HD ( j = 1) technique provides superior ASC performance than IM/DD ( j = 2) technique at the expense of increased cost and complexity. Figure 3a,b displays the SOP performance under the combined effect of strong to weak AT with boresight and jitter. SOP is plotted as a function of jB with jE = 10 dB and 12 dB for C th = 1 and w z ∕a = 10. The SOP performance is better at high jB , whereas it degrades while moving from weak to strong AT. Further for SOP = 0.02 at jE = 10 dB, the required  In Figure 4 Figure 5 illustrates that when A-E link SNR changes from 12 to 10 dB, SOP performance improves for both detection techniques but HD shows better SOP improvement than IM/DD for 1B = 2B ≈ 4 dB to ≈ 23 dB under strong AT; the converse is true for the remaining SNR range. Figures 6 and 7 present the SOP and ASC performance, respectively, under the combined effects of AT and PEs (jitter and boresight), for different values of . It is seen from the figures that the ASC and SOP performance improves with the increase in the values. This is because for higher values of , more amount of scattering power will be coupled to the LOS component, consequently decreasing the turbulence intensity. This observation of improved secrecy performance with increasing is also in line with the results reported in [15].

FIGURE 5
Comparison of the effect of change in electrical SNR on the A-E link (12-10 dB) on SOP performances of HD and IM/DD for equal boresight ( x B ∕a = x E ∕a = 4 and y B ∕a = y E ∕a = 3) and equal jitter ( x ∕a = 1 and y ∕a = 2) on both the links.

CONCLUSIONS
In this work, we developed generalised expressions for the ASC and SOP which are important and useful for system designers for conducting security analyses of FSO communication systems under AT and non-zero boresight PEs. Useful insights about the secrecy performance are obtained through the asymptotic SOP analysis and simulation results under different detection techniques.  (18), we utilise the Cauchy Ratio Test proposed in [29]. On taking the ratio of the (n + 1)th and nth terms given by (A.1) and (A.2), respectively, shown at the top of the page, and further applying the Cauchy Ratio Test [29] to (A.3) shown at the top of the page, we observe that a n = D B D E E E jB