Performance Analysis of RIS-Assisted FSO Communications over Fisher–Snedecor F Turbulence Channels

: The Fisher–Snedecor (F-S) F distribution has recently been introduced as a tractable turbulence-induced (TI) fading model that ﬁts well with the experimental data. This paper provides a performance evaluation of a free-space optical (FSO) re-conﬁgurable intelligent surface (RIS)-assisted communications (ACs) link over the F-S F TI fading channels, assuming the intensity modulation– direct detection (IM–DD) technique. In particular, novel and closed-form (C-F) analytical expressions for the probability density function (PDF) and cumulative distribution function (CDF) of the end-to-end signal-to-noise ratio (SNR) in terms of Gaussian hyper-geometric functions are efﬁciently derived. Capitalizing on the obtained results, novel C-F analytical expressions for the moment generating function ( M MGF ), outage probability (OP), average bit error rate (BER) and ergodic channel capacity ( C γ ) of the FSO RIS-ACs system over the F-S F TI fading channels are provided and numerically evaluated under the various TI fading severity conditions. Furthermore, the second-order (S-O) statistical expressions for the level crossing rate (LCR) and average fade duration (AFD) are obtained and thoroughly examined for various FSO RIS-ACs system model parameters.


Introduction
Re-configurable intelligent surface (RIS)-assisted communications (ACs) are envisioned for beyond 5G and 6G wireless systems [1,2]. In a smart propagation environment, RIS blocks are capable of optimizing and improving system performances by controlling incident transmission waves in a directed and programmable way.
RIS-ACs for smart radio environments with suitable RIS applications are considered in [3]. Path-loss models for RIS-ACs supported by experimental and simulation results are provided in [4]. Moreover, a comparison between relay-ACs and RIS-ACs is provided in [5]. In [6], mmWave RIS-ACs systems are considered, whereas visible light communication (VLC) RIS-ACs systems are considered in [7,8].
Free-space optical (FSO) communication is a technology that, due to narrow beam widths, can be distinguished as highly secure, interference immune and energy efficient. Additionally, the FSO system is capable of providing a relatively large bandwidth and can be further distinguished as a license-free and cost effective transmission technology. In turn, the main cause of terrestrial FSO system performance degradation is atmospheric turbulence. The gamma-gamma (G-G) distribution is the most commonly used turbulenceinduced (TI) fading model that is mathematically tractable and experimentally validated for moderate-to-strong TI fading conditions [9,10]. Moreover, the log-normal TI fading model is suitable for weak TI fading conditions, but is mathematically less tractable and often can lead to complex analytical expressions [11], whereas the general Malaga M TI fading model can be used to address weak-to-strong TI fading severity conditions, but is mathematically less tractable if compared with the G-G TI fading model [12].
where X G and Y IG are G and normalized I-G RVs, respectively. Since the normalized I-G RV can be mathematically expressed as Y IG = 1 Y G , the probability density functions (PDFs) of X G and Y G are given as: whose shape parameters are m G 1 and m G 2 , respectively, whereas the mean powers are Ω G 1 and Ω G 2 , respectively. The Γ(·) is the Gamma function ( [39], Equation (8.310.1)). By applying a transformation of RVs, γ x G = X G 2 and γ y G = Y G 2 , the PDFs of the squared G and the normalised squared I-G RVs are, respectively: can be obtained as: where dγ x G dγ F S = γ y G . From (4) to (6), and by applying ( [39], Equation (3.326.2)) and ( [39], Equation (8.384.1)), respectively, the PDF of γ F S can be written as: where β(·, ·) is the Beta function ( [39], Equation (8.380.1)). It can be concluded that p γ F S (γ F S ) in (7) for Ω G 1 =γ 1/2 , Ω G 2 = 1, m G 1 = a and m G 2 = b reduces to the PDF of F-S F TI fading distribution given by ( [13], Equation (20)).

FSO RIS-ACs Link over F-S F TI fading Channels
We considered an FSO RIS-ACs system over F-S F TI fading channels. A simplified block scheme is presented in Figure 1. It was assumed that the RIS module ideally reflects the FSO signal and that the system is not influenced by pointing errors. The output symbol at the receiver y can be expressed as [24]: where x is the transmitted symbol, E s is the symbol's energy and n is AWGN. h 1 and h 2 are channel coefficients from source-to-RIS (S-RIS) and RIS-to-destination (RIS-D), respectively, whereas the quantity µ r e jθ r is deterministic in nature and describes the RIS part of the system under consideration. In detail, µ r ∈ [0, 1] is a coefficient of amplitude reflection and θ r ∈ [0, 2π] is a phase [40]. The PDF of the received signal-to-noise ratio (SNR) for the considered FSO RIS-ACs transmission link can be expressed as ( [24], Equation (17)): where p γ h 1 and p γ h 2 are SNR's PDFs of the links h 1 and h 2 , respectively, which in the case of intensity modulation-direct detection (IM-DD) with the on-off keying (OOK) technique for Fisher-Snedecor F TI fading distribution can be written as ( [13], Equation (20)): whereγ i is the average SNR, whereas a i and b i are Fisher-Snedecor F small-scale and large-scale cells related to TI fading severity conditions, respectively. The scintillation indexes of S-RIS and RIS-D F-S F TI fading can be expressed as ( [13], Equation (10)), respectively: The a i and b i can be written as ( [13], Equation (12a)) and ( [13], Equation (12b)), respectively: where σ 2 lnI a i and σ 2 lnI b i are normalized log-variances of I a i and I b i , respectively. Moreover, I a i and I b i are Gamma ( [13] and Inverse normalized Gamma ( [13], Equation (4)) distributions, respectively. Under the assumption of spherical propagation, σ 2 lnI a i is ( [14], Equation (3)) : where σ SP,i , i = 1, 2 is the spherical scintillation index (SSI) of S-RIS and RIS-D, respectively, and for weak fluctuation conditions, σ SP,i is: where Q l i = 10.89S i /(B i l 2 0 i ), S i is the transmission distance, B i = 2π/λ i is the wavenumber, λ i is the optical wavelength and l 0 i is the inner-scale of the S-RIS and RIS-D Fisher- where σ 2 i represents the Rytov variance. D i is the receiver aperture diameter and C 2 n i is the refractive index for i = 1, 2. The σ 2 can be written as ( [14], Equation (5)): where σ 2 (L 0 i ) are the inner and outer large-scale log-irradiance variances, respectively, that can be expressed as: where

Cumulative Distribution Function (F γ )
The cumulative distribution function (F γ ) of an FSO RIS-ACs system over the F-S F TI fading channels can be obtained by using the following formulae: After substituting (17) in (19), the F γ (γ) of an FSO RIS-ACs system can be written as: where H 1 is: The integral form (I-F) expression H 1 can be solved using the variable substitution, s = a 1 τ 1/2 + (b 1 − 1)γ 1/2 r 1/2 and, then, by applying the binomial formula ( [39], Equation (1.111)). The value of H 1 was calculated and given as: After substituting (22) in (20) and, then, by using ( [39], Equation (3.259.3)), the C-F F γ (γ) expression of the received SNR for the considered FSO RIS-ACs was calculated and given as:

Moment Generating Function (M MGF )
The M MGF of the end-to-end SNR for the considered FSO RIS-ACs system can be derived as ( [24], Equation (28)):

Remark 1.
To date, the statistical distribution functions of the FSO communications based on RIS subject to turbulence effects have been derived. In the context of 5G and 6G, FSO communications coexist with other radiofrequency (RF) systems, e.g., a point-to-point system transmitting in the mmWave frequency domain. Thus, it is possible to combine both frequency domains in order to generate a hybrid FSO and RF system. Although the evaluation of this hybrid system is out of the scope of this paper, it would improve the performance of any system operating independently and its evaluation will be considered in further works.

Performance Analysis of an FSO RIS-ACs Link over F-S F TI fading Channels
The first-order (F-O) performance metrics of the single-input single-output (SISO) FSO RIS-ACs link over F-S F TI fading channels based on end-to-end average SNR for IM-DD with the OOK technique are provided in the following Section.

Outage Probability (P γ )
The OP (P γ ) for a given SNR threshold γ th of an FSO RIS-ACs over F-S F TI fading propagation channels for IM-DD with the OOK technique was calculated as: where F γ was already derived in (23). It is important to note that F γ was valid only for the integer values of a 1 , since F γ was derived as a finite series expression. The numerical analysis of the OP for an FSO RIS-ACs system over Fisher-Snedecor F TI fading channels was provided in the Numerical Results.

Average Bit Error Rate (P BER )
The average bit error rate (P BER ) can be defined as the rate at which errors occur in the considered FSO RIS-ACs transmission system. The P BER of the received SNR for various binary modulation techniques of the FSO RIS-ACs system in F-S F TI fading propagation environment can be calculated using ( [42], Equation (25)): where p and q are the parameters of different modulation types. In particular, P BER (γ) for the non-coherent binary frequency shift keying (NBFSK) (p = 1, q = 1/2), binary frequency shift keying (BFSK) (p = 1/2, q = 1/2), binary phase shift keying (BPSK) (p = 1/2, q = 1) and differential binary phase shift keying (DBPSK) (p = 1, q = 1) can be obtained from (27). According to (23), P BER (γ) can be written as: where H 2 is given as: The C-F P BER (γ) expression can be obtained by solving H 2 in (29) By substituting (30) in (28), the C-F P BER (γ) was efficiently derived, whereas the numerical evaluation and further analysis of P BER (γ) are provided in the Numerical Results.

Channel Capacity (C γ )
The ergodic channel capacity (C γ ) of end-to-end SNR for the considered FSO RIS-ACs system that operates under the IM-DD modulation technique is given by ([24], Equation (31)): By substituting (18) in (31) (31), respectively, the C-F C γ was calculated and given as: The numerical analysis of C γ in terms of different TI fading propagation conditions is provided in the Numerical Results.

Second-Order (S-O) Performance Analysis of FSO RIS-ACs Link over F-S F TI Fading Channels
The important S-O performance metrics of the considered SISO FSO RIS-ACs system over F-S F TI fading channels based on average SNR assuming the IM/DD technique such as the level crossing rate (LCR) and average fade duration (AFD) were further examined. Moreover, S-O statistics can be useful for error control codes design, interleaver design, a throughput and burst error rate analysis in wireless communications [23,43,44].

Level Crossing Rate (N γ )
The level crossing rate (N γ ) is defined as a time rate of change of the TI-faded signal in time-variant TI fading channels. For the predetermined SNR threshold γ th , the N γ (γ th ) can be written as ( [32], Equation (14)): where, p γγ (γ,γ) is the joint distribution of the received SNR for the considered FSO RIS-ACs transmission system, γ and its first derivativeγ. Since we could express the received SNR as γ = γ h 1 γ h 2 , where S-RIS and RIS-D channel coefficients γ h 1 and γ h 2 of the Fisher-Snedecor F TI fading model as already shown in Section 1 can be further expressed as , respectively, the p γγ (γ,γ) can be written as an integral-form (I-F) expression of a joint PDF of i.i.d RVs, γ,γ, γ y G 1 , γ x G 2 and γ y G 2 , as follows from ( [45], Equation (12)): can be further simplified and expressed through independent conditional and individual PDFs as: The conditional distribution p γ|γ y G 1 γ x G 2 γ y G 2 (γ y G 1 γ x G 2 γ y G 2 ) was then transformed into: From (33) to (36), the N γ (γ th ) of the FSO RIS-ACs transmission system in F-S F TI fading propagation environments was expressed as: The parameter σ 2 γ is the variance ofγ. Furthermore,γ is the first derivative of γ = γ h 1 γ h 2 and can be written as:γ whereγ h 1 andγ h 2 are the first derivatives of γ h 1 and γ h 2 , respectively. We assumed thatγ was a zero-mean Gaussian RV whose variance, after some mathematical manipulations, can be expressed as: After substituting (38) in (37) and then p γ x G i and p γ y G i in (37), where p γ x G i and p γ y G i , according to (4) and (5), can be written as: N γ (γ th ) was obtained as: whereγ =γ 1 =γ 2 . H 3 , in (43), was a three-folded integral-form (I-F) expression derived as:

Average Fade Duration (A γ )
The average fade duration (A γ ) is the mean time of the TI-faded signal being below a specified threshold for the received SNR of the considered FSO RIS-ACs system over F-S F TI fading propagation channels and was calculated as: where F γ (γ th ) and N γ (γ th ) are the CDF and LCR for a γ th , given in (23) and (43), respectively. The numerical analysis and observations of S-O statistical results for the SISO FSO RIS-ACs system over F-S F TI fading propagation channels are provided in the Numerical Results.

Numerical Results
The obtained end-to-end SNR statistical results for P γ (γ th ), P BER (γ), C γ , N γ (γ th ) and A γ (γ th ) of an FSO RIS-ACs link over F-S F TI fading propagation channels under various TI fading severity conditions were numerically evaluated and presented in this Section.

Numerical Results for F-O Performance Metrics
The OP versus γ as well as the OP versusγ for weak (a 1 = 5, b 1 = 7.0941, a 2 = 4.5916, b 2 = 7.0941), moderate (a 1 = 2, b 1 = 4.5323, a 2 = 2.3378, b 2 = 4.5323) and strong (a 1 = 1, b 1 = 3.4948, a 2 = 1.4321, b 2 = 3.4948) TI fading severities [14] are presented, respectively, in Figures 2 and 3. Since (23) was the C-F finite series expression, the presented numerical results for the OP were limited to integer values of a 1 . It can be observed that, by shifting from strong-to-moderate or moderate-to-weak TI fading conditions, the OP decreased. In Figure 2, it can be further noticed that, by increasing the average SNR values, the OP decreased, which in turn could provide an additional system performance improvement of the FSO RIS-ACs link over F-S TI fading propagation channels. From Figure 3, it can be observed that a higherγ caused an increase in the OP (e.g., by increasingγ fromγ = 3 toγ = 6).   The P BER (γ) versusγ for a 1 = 6, b 1 = 2.58, a 2 = 5.75, b 2 = 2.58 and a 1 = 3, b 1 = 2.1, a 2 = 2.73 and b 2 = 2.1 TI fading severity values [13], where a 1 was an integer, and for the selected binary modulation schemes such as BFSK, BPSK, NBFSK and DBPSK are presented in Figure 4. It was obvious that less severe TI fading conditions provided a lower P BER (γ) (e.g., by shifting from a 1 = 3, b 1 = 2.1, a 2 = 2.73 and b 2 = 2.1 to a 1 = 6, b 1 = 2.58, a 2 = 5.75 and b 2 = 2.58). It can be further concluded that, for higher dBγ values, TI fading conditions had a stronger impact on the P BER (γ) than the observed modulation schemes.   The C γ of the considered FSO RIS-ACs model over F-S F TI fading propagation channels for weak (a 1 = 4, b 1 = 7.0941, a 2 = 4.5916, b 2 = 7.0941), moderate (a 1 = 2, b 1 = 4.5323, a 2 = 2.3378, b 2 = 4.5323) and strong (a 1 = 1, b 1 = 3.4948, a 2 = 1.4321, b 2 = 3.4948) TI fading severities [14], where a 1 took integer values, is shown in Figure 5. As expected, the C γ could be increased by shifting from severe to less severe TI fading conditions (e.g., by shifting from strong-to-moderate or moderate-to-weak TI fading severities). A similar behaviour for an FSO RIS AC under the IM-DD modulation technique was noticed in [24,27].

Numerical Results for S-O Performance Metrics
The S-O statistical measures of an FSO RIS-ACs model over F-S F TI fading propagation channels are presented in Figures 6 and 7. The variances in (40) were evaluated as σ 2 was given by (11) and was the quasi frequency of the FSO RIS-ACs path ( [30], Equation (15)). Additionally, τ 0 = √ λS U was the turbulence correlation time, λ was the wavelength, S was the distance and U was the average wind speed of the RIS-ACs transmission system. The S-O metrics were evaluated for λ = λ i = 532 nm, U = 1 m/s and S = 980 m [46].
Moreover, the A γ (γ th ) was the least affected by TI fading severity conditions andγ for the threshold values around γ th = 0 dB.

Conclusions
The unified F-O and S-O performance analysis of an FSO RIS-ACs over F-S F TI fading channels under different TI fading severity conditions was investigated. Namely, novel C-F expressions for p γ (γ) and F γ (γ) of the received SNR in terms of Beta and Gausshypergeometric functions were successfully derived. The F γ (γ) was obtained as a finite series expression and, as such, was only valid for integer values of a 1 . Capitalizing on the obtained analytical expressions for p γ (γ) and F γ (γ), the novel C-F expressions of the endto-end SNR for M MGF , P γ (γ th ), P BER (γ) and C γ in terms of the Meijer G function were derived and numerically evaluated for different F-S F TI fading system model parameters. Moreover, the paper provided the mathematical framework for the derivation of S-O statistical measures such as N γ (γ th ) and A γ (γ th ) of an FSO RIS-ACs system over F-S F TI fading channels under different TI fading severity conditions. The obtained results pointed out that a significant performance improvement of SISO FSO RIS-ACs links could be achieved by shifting from more severe to less severe TI fading conditions. Moreover, the system performance improvement in terms of P γ (γ th ) could be further achieved by increasing theγ dB value. The provided analytical and numerical results could be useful for designing SISO FSO RIS-ACs systems over TI fading channels. Further works will consider the experimental validation of the obtained results.

Data Availability Statement:
The authors confirm that the data supporting the results of this work are provided within the paper.

Acknowledgments:
The authors would like to acknowledge the CONEX-Plus project. CONEX-Plus received research funding from UC3M and the European Union's Horizon 2020 programme under the Marie Skłodowska-Curie grant agreement no. 801538. The authors would also like to acknowledge the Spanish National Project IRENE-EARTH (PID2020-115323RB-C33/AEI/10.13039/501100011033) and the Cost Actions CA19111 and CA16220.

Conflicts of Interest:
The authors declare no conflict of interest.

Abbreviations
The following abbreviations are used in this manuscript: