Α dual‐hop equivalent structure of a generalised multi‐hop free‐space optics network

Funding information Greece and the European Union (European Social FundESF); Operational Program ‘Human Resources Development, Education and Lifelong Learning’; Strengthening Human Resources Research Potential via Doctorate Research, Grant/Award Number: MIS-5000432 Abstract The performance of free-space optics links strongly depends on scintillation effects caused by the atmospheric turbulence appearance. A widely adopted way to counterbalance the expected deterioration due to scintillation is to employ cooperative and multi-hop arrangements. However, the use of multiple relays increases significantly the complexity of the mathematical models used to describe the overall performance of the system. Taking the latter into account, we approximate a generic multiple relay-assisted free-space optical configuration with a dual-hop link in order to simplify the performance analysis. To this end, we provide new mathematical derivations for the evaluation of the outage probability and the average bit error rate by assuming typical turbulence models.


INTRODUCTION
Free-space-optics (FSO) has attracted increasing research and commercial interest mostly due to the benefits against other wireless technologies, including radiofrequency or millimetre wave systems, in cases where point-to-point connections are required [1][2][3][4]. The most important of them is the high bandwidth access, the operation in the unlicensed spectrum, the low installation and operation cost, the enhanced security provision and the immunity to the electromagnetic interference [5][6][7]. However, the operation of FSO links strongly depends on several impairments, including scintillation caused by atmospheric turbulence effect [4]. In order to counterbalance the expected deterioration, cooperative and multi-hop deployments have been proposed in the open technical literature. However, the employment of multiple relays significantly raises the complexity of the mathematical expressions used to describe the overall performance [8]. To handle this issue, an approximation methodology for multi-hop configurations with amplify and forward (AF) relay nodes was proposed in [9].
In this work, a valid approximation is used, which can be applied in any multi-hop FSO configuration with decode and forward (DF) relays, to produce the equivalent dual-hop link. DF relays, which re-modulate and retransmit the received signals, are often used in FSO transmission since they are more reliable, although they require more processing compared to AF relaying. To this end, we consider intensity modulation/direct detection with on-off keying (OOK) modulation technique, which is the most commonly adopted and straightforward modulation scheme that has been used for experimental or commercial FSO systems [3]. The approximation process leads to a significant simplification of the complicated and widely adopted expressions for the outage probability (OP) and the average bit error rate (ABER) used for the performance evaluation of multi-hop networks.
The remainder of the work is organised as follows. In the section that follows, the channel model is introduced while the approximation methodology is presented and analysed in Section 3. Next, in Section 4, we proceed to the overall performance estimation of the DF-relayed FSO system in both OP and ABER metrics, while the corresponding numerical results

CHANNEL MODEL
In what follows, we emphasise on the DF-relayed FSO link of Figure 1, which is transformed into the dual-hop scheme of Figure 2. In such a configuration, the source, S (node 0), emits optical signals towards several relay nodes, and then every relay node towards the rest relays or the destination D (node L+1), respectively. Hence, the received signal at each hop is given as [10] y x,y = I x,y x x,y + n x,y , where x, y represent the specific hop, that is, from node x towards node y, η is the effective photocurrent conversion ratio, I x,y stands for the normalised received irradiance due to turbulence effects at the corresponding receiver, x x,y is the modulated signal, and n x,y stands for the additive white Gaussian noise with zero mean and variance σ n 2 . Based on the above, the instantaneous signal-to-noise ratio (SNR) at each hop is expressed as [11] x,y = and the average SNR as [11] x,y = where E[I x,y ] stands for the expected normalised irradiance value.
Turbulence is assumed to follow either the gamma, which holds for weak turbulence conditions or the gamma-gamma model, which is appropriate for weak to strong turbulence conditions. The probability density function (PDF) of the gamma model is expressed as [12] f I x, where Γ(⋅) is the standard gamma function [13, eq. (8.310.1)], and x,y stands for the parameter of the gamma distribution determined as while the parameters a x,y and b x,y are calculated as where, 2 x,y represents the Rytov variance, which is given as with k = 2π/λ being the wave number, where λ represents the x,y , D x,y is the receiver's aperture diameter, L x,y is the link length, and C n 2 is a parameter proportional to the atmospheric turbulence strength [12].
After a simple power transformation of Equation (4), the PDF for the SNR is deduced as . (8) Moreover, the PDF for the SNR of the gamma-gamma model is given according to [14] as where K ν (⋅) stands for the ν-th order modified Bessel function of the second kind [13, eq. (8.432.2)].
In what follows, we omit the indices x, y considering the same parameter values in all links.

APPROXIMATION METHOD
By following the methodology of [9], the dual-hop approximation is depicted in Figure 2. More specifically, the approximated normalised PDF for the gamma model described in Equation (8) is derived as while for the gamma-gamma distribution, Equation (9), we get By substituting Equations (10) into (12), the CDF for the gamma distribution is given as Next, by using [13, eq. (3.381.8)], we obtain where (⋅, ⋅) stands for the lower incomplete gamma function [13, eq. (8.310.1)]. For the gamma-gamma model, the CDF is expressed by substituting Equations (11) into (12) as Then, by usingK v ( √ z ) = 0.5G 2,0 0,2 (z∕4| v∕2, −v∕2)from [15], along with the corresponding integral transformation from [16], the CDF expression is provided in terms of the Meijer-G function [13, eq. (9.301)] as

PERFORMANCE ESTIMATION
In this section, the overall performance of the multi-hop network in terms of the OP and ABER, with OOK modulation, is assumed. OP is a significant quantity in order to evaluate the reliability of a communication system and represents the probability that the instantaneous SNR falls below a critical threshold γ th , while the ABER is a crucial metric for the reliability and performance estimation of such communication systems [3].

Outage probability
The OP for a given threshold, γ th , can be estimated as [14,17] P out,r = F ( th ).
Hence, the OP of the first hop, that is, from S towards R L , for gamma and gamma-gamma models, is obtained from Equations (14), (16), and (17) as and respectively. To derive the OP of the second link, that is, from R L towards D, we substitute Equations (8) into (12), and then using Equation (17), we get for the gamma distribution while, for the gamma-gamma, is expressed as [14] P out,D = (ab) Then, the OP for the entire network can be evaluated as [10,18,19]

Average bit error rate
The BER of the first approximated link can be calculated from the following expression as [14] By substituting Equations (23) and (10) or (11) into the following integral we can estimate the ABER expression of the approximated link, that is, from S towards R L for the gamma and the gamma-gamma turbulence, respectively. More specifically, for the gamma distribution, the ABER takes the following form Then, by using erfc( ) and [15], along with the corresponding integral transformation from [16], the ABER expression is derived in terms of the Meijer-G function [13, eq. (9. (17)], the original OP for the entire network is determined as P tot = 1 − [(1 − P out,D )(1 − P out,D P 1 )], with P 1 = 1 − (1 − P out,D )(1 − P 2 )and P 2 = [1 − (1 − P out,D ) 2 ]P out,D . To further test the accuracy of the method, we also provide results for the OP of the system, obtained without using our approximation methodology. Figure 4 demonstrates the OP versus the normalised outage threshold̄∕ th , while Figure 5 presents the ABER versus electrical SNR. We can readily observe that the results for the OP are quite similar either with or without approximation. This is quite promising since the adoption of the described approximation assures the derivation of simplified mathematical expressions, which are useful to researchers who wish to evaluate the performance of complex relayed FSO networks. Additionally, the gamma distribution outperforms in both metrics, the gammagamma one as expected, due to the different C n 2 values.

CONCLUSION
A dual-hop approximation of multi-hop FSO configurations with DF relays over gamma or gamma-gamma turbulence mod-

FIGURE 5
Average bit error rate versus the expected signal-to-noise ratio els was adopted and further analysed in this work. The derived OP and ABER approximated expressions were significantly simplified and the corresponding numerical results revealed the accuracy of the approximation. The work can readily be extended to consider the investigation of AF relays, the adoption of other turbulence models and the evaluation of various performance metrics.