Relay-Assisted Mixed FSO/RF Systems over Málaga-$\mathcal{M}$ and $κ$-$μ$ Shadowed Fading Channels
Imène Trigui Nesrine Cherif Sofiène Affes

	

 Relay-Assisted Mixed FSO/RF Systems over Málaga-ℳ and κ-μ Shadowed Fading Channels
     Imène Trigui, Nesrine Cherif, and Sofiène Affes

INRS-EMT, 800, de la Gauchetière Ouest, Bureau
6900, Montréal, H5A 1K6, Qc, Canada. 

{itrigui, nesrine.cherif, affes}@emt.inrs.ca

    December 30, 2023
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This letter presents a unified analytical framework for the computation of the ergodic capacity and the outage probability of relay-assisted  mixed FSO/RF transmission.  In addition to accounting for different FSO detection techniques, the mathematical model offers a twofold unification of mixed  FSO/RF systems by considering mixed Málaga-ℳ/κ-μ shadowed fading, which includes as special cases nearly all linear turbulence/fading models adopted in the open literature.
		
	

	

		
	

§ INTRODUCTION

	Recently, free-space optical (FSO) communications have gained a significant attention due to their advantages of higher bandwidth in unlicensed spectrum and higher throughput compared to their RF counterparts <cit.>. Hence, the gathering of both FSO and RF technologies arises as a promising solution for securing connectivity between the RF access  and the fiber-optic-based backbone networks. As such, there has been prominent interest in mixed FSO/RF systems where RF transmission is used at one hop and FSO transmission at the other  <cit.>-<cit.>. Most contributions within this  research line  consider restrictive irradiance and channel gain probability density function (PDF) models for the FSO and RF links, respectively. The  most commonly utilized models for the irradiance in FSO links are the lognormal and the Gamma-Gamma (𝒢-𝒢) (<cit.>,<cit.> and references therein). Recently, a new generalized statistical model, the Málaga-ℳ distribution, was proposed in <cit.> to model the irradiance fluctuation of an unbounded optical wavefront  propagating through a turbulent medium under all irradiance conditions. Characterized in <cit.> as a mixture of Generalized-K and  discrete
Binomial distributions, the  Malága-ℳ distribution  unifies most statistical models
exploited so far and is able to better reflect a wider range
of turbulence conditions<cit.>, <cit.>. On the RF side, previous works typically assume either Rayleigh or Nakagami-m fading <cit.>,<cit.>, thereby lacking the flexibility to account for disparate signal
	propagation mechanisms as those characterized  in 5G communications which will accommodate a wide range of usage scenarios with diverse link requirements. To bridge this gap in the literature, the κ-μ shadowed fading model, recently  derived in <cit.>, is an attractive proposition. In addition to offering an excellent fit to the fading observed in a range of real-world applications (e.g. device-to-device, and body-centric fading channels <cit.>), the κ-μ shadowed fading encompasses several RF channel models such as  Nakagami-m, Rayleigh, Rice, κ-μ and shadowed Rician fading distributions. This new channel fading model offers far better and much	more flexible representations of practical fading LOS (line of sight), NLOS (non-LOS), and shadowed channels than the Rayleigh and Nakagami-m distributions.
	Under the assumption of AF relaying and taking into account  the effect
of  pointing errors while considering both heterodyne and intensity modulation/direct (IM/DD) detection techniques,   we derive closed-form expressions
for the ergodic capacity and outage probability of dual-hop
FSO/RF systems over  Málaga-ℳ/κ-μ shadowed channels. We further pursue high signal-to-noise
ratio (SNR) analysis
to derive the diversity order.
	

 

§ CHANNEL AND SYSTEM MODELS

We consider a relay-assisted mixed FSO/RF transmission composed of both Málaga-ℳ with pointing errors and  κ-μ shadowed fading environments. The source communicates with the destination through an intermediate relay, able to activate both heterodyne and IM/DD detection techniques at the reception of the optical beam.

The FSO (S-R) link irradiance is assumed to follow a Málaga-ℳ distribution with pointing errors impairments for which the PDF of the irradiance, I, is given by  <cit.>

  
    f_I(x)=ξ^2A/xΓ (α)∑_k=1^βb_k/Γ(k) G_1,3^3,0[αβ/g β+Ωx/A_0| ξ^2+1 ξ^2,α,k],

where  ξ is the ratio between the equivalent beam radius and the pointing error displacement standard deviation (i.e., jitter) at the relay (for negligible pointing errors ξ→∞), A_0 defines the pointing loss <cit.>,
A=α^α/2[  g β/(g β +Ω)] ^β +α/2g^-1-α/2 and b_k=β-1k-1(g β+Ω)^1-k/2[ (g β +Ω)/αβ] ^α+k/2( Ω/g)^k-1(α/β)^k/2, where α, β, g,  and Ω are the fading parameters related to the atmospheric turbulence conditions  <cit.>. Moreover in (<ref>), G_p, q^m, n [·] and Γ(·) stand for the Meijer-G  <cit.> and the incomplete gamma <cit.> functions, respectively. It is worth highlighting that the ℳ distribution unifies most of the proposed statistical models characterizing  the optical irradiance in  homogeneous and isotropic turbulence <cit.>. Hence both 𝒢-𝒢 and K models are special cases of the Málaga-ℳ distribution, as they mathematically derive from (<ref>)  by setting (g=0, Ω=1) and (g≠0, Ω=0 or β=1), respectively <cit.>.


The RF (R-D) link, experiences the  κ-μ shadowed fading with non-negative real shape parameters κ, μ and m,  for which the PDF of instantaneous SNR, γ_2,  is given by <cit.>


    f_γ_2(x)   =   μ^μ m^m(1+κ)^μ/Γ(μ) γ̅_2 (μκ+m)^m(x/γ̅_2)^μ-1e^-μ(1+κ) x/γ̅_2
          ×  _1 F_1(m,μ;μ^2κ (1+κ)/μκ+mx/γ̅_2),

where _1 F_1(·) is the confluent hypergeometric function <cit.> and γ̅_2=[γ_2]. This fading  model  jointly includes
 large-scale and small-scale propagation effects, by considering
 that only the dominant components (DSCs) are affected by Nakagami-m distributed  shadowing <cit.>. The shadowed κ-μ distribution is an extremely versatile fading model that    includes as special cases  nearly all linear fading models pertaining to LOS and NLOS scenarios,  such as κ-μ (m→∞),  Nakagami-m (μ=m and κ→ 0), Rayleigh (μ=m=1 and  κ→ 0 ),  and Rice (μ=1, κ=K and m→∞),  to name a few  <cit.>.

Assuming AF relaying with channel state information (CSI), then the end-to end SNR can be expressed as<cit.>


    γ=γ_1 γ_2/γ_1+γ_2+1,

where γ_1=(A_0h(g+Ω))^-rμ_rI^r is the instantaneous SNR of the FSO link
(S - R) with r being the parameter that describes the detection technique at the relay (i.e., r = 1
is associated with heterodyne detection and r = 2 is associated
with IM/DD) and, μ_r refers to the electrical SNR of the FSO
hop <cit.> and  h=ξ^2/(ξ^2+1). In particular, for r = 1,   μ_1=μ_heterodyne=[γ_1]=γ̅_1 and for r=2, μ_2=μ_IM/DD=μ_1αξ^2(ξ^2+1)^-2(ξ^2+2)(g+Ω)/((α+1)[2g(g+2Ω)+Ω^2(1+1/β)]) <cit.>.


§ EXACT PERFORMANCE ANALYSIS

In this section, a new mathematical framework investigating the average capacity and the outage probability of the mixed FSO/RF transmission composed of both Málaga-ℳ with pointing errors and shadowed
κ-μ fading environments and accounting for both detection techniques is presented. To the best of the author's knowledge, there are few works that consider these metrics  of mixed FSO/RF systems, yet mostly  considering the mixed  𝒢-𝒢/Nakagami-m fading (<cit.>,<cit.> and references therein). This paper completes and extends the efforts of <cit.>-<cit.>  by unifying the ergodic capacity and the outage probability analysis for any turbulence/fading model under both types of detection techniques.



 §.§ Ergodic Capacity

 Hereafter, we provide capacity formulas for the considered system by using the complementary moment
 generation function CMGF-based approach <cit.> as
 
    CΔ= [ln (1+γ)]/2ln (2)=1/2ln (2)∫_0^∞se^-sM^(c)_γ_1(s)M^(c)_γ_2(s)ds,
	
 where M^(c)_X(s)=∫_0^∞e^-sxF^(c)_X(x) dx stands for the CMGF with  F^(c)_X(x)  denoting the complementary cumulative distribution function (CCDF) of X.

 The ergodic capacity of mixed Málaga-ℳ/κ-μ shadowed fading  FSO transmission system under heterodyne and IM/DD detection techniques with pointing errors taken into account is given for 



  * Integer m, μ, with m ⩾μ  as


    C=ξ^2 A r μ_rB^-r/2 ln(2)Γ(α)∑_k=1^βb_k/Γ(k)∑_l=1^mχ_l/Γ(m) T(θ_2,l,m),


where  B=αβ h (g+Ω)/[(gβ+Ω)], and,


    χ_l = {[ mlθ_2^l -m-μlθ_1^l     for 1⩽ l⩽ m-μ,;            mlθ_2^l        for l> m-μ,;                    ].

 with θ_1=γ̅_̅2̅/μ(1+κ) , and, θ_2=γ̅_̅2̅(μκ+m)/μ m (1+κ). Moreover in (<ref>),

    T(x,y,z)= H_10,43,11^01,14,11[μ_r/B^r; x| (-y,1,1)- | (σ,Σ) (ϕ,Φ)| (1-z,1)(0,1)],

where H[·,·] denotes the Fox-H function (FHF) of two variables <cit.> also known as the bivariate FHF whose Mathematica implementation may be found in <cit.>, whereby (σ,Σ)= (1-r,r),(1-ξ^2-r,r),(1-α-r,r),(1-k-r,r) and (ϕ,Φ)=(0,1),(-ξ^2-r,r),(-r,r).  Moreover, it becomes  for 



  * Integer m, μ, with m  < μ as


    C   =   ξ^2 A r μ_r/2 ln(2)Γ(α)B^r∑_k=1^βb_k/Γ(k)(p,q)≠ (0,0)∑_p=0^m∑_q=0^μ-mmpμ-mq
          θ_2^pθ_1^q(∑_i=1^μ-mΔ_1i/Γ(μ-m-i+1)T(θ_1,q+p,μ-m-i+1)
          -∑_i=1^mΔ_2i/Γ(m-i+1)T(θ_2,q+p,m-i+1)),


where Δ_1i=(-1)^mm+i-2i-1(m/μκ+m)^m(μκ/μκ+m)^-m-i+1 and Δ_2i=(-1)^i-1μ-m+i-2i-1(m/μκ+m)^i-1(μκ/μκ+m)^m-μ-i+1.
 
 	Capitalizing on (<ref>) and recalling the fact that the FSO link's CMGF M^(c)_γ_1(s)=ℒ(F^(c)_γ_1(x)) where ℒ denotes the Laplace transform operator and the FSO link's CCDF is obtained as 

 	 
    F^(c)_γ_1(x)   =   F^(c)_I( A_0 h (g+Ω) ( x/μ_r)^1/r)  
       (a)=   ξ^2A/Γ(α)∑_k=1^βb_k/Γ(k) G_2,4^4,0[B (x/μ_r)^1/r| ξ^2+1,1  0,ξ^2,α,k].

where (a) follows from  integrating (<ref>)   using <cit.>. Then, expressing the Meijer-G function in (<ref>) in terms of Fox-H function by means of <cit.> and resorting to <cit.> with some additional manipulations using <cit.> yield


    M^(c)_γ_1(s)=ξ^2A r  μ_r/Γ(α)B^r∑_k=1^βb_k/Γ(k) H_4,3^1,4[μ_r/B^r s|  (σ,Σ)  (ϕ,Φ)],

where  H_p, q^m, n [·] is the Fox-H function <cit.>.

On the RF side, the CMGF of γ_2 under shadowed  κ-μ fading is given by 


    M^(c)_γ_2(s)=1-M_γ_2(s)/s(a)=1-(θ_1 s+1)^m-μ/(θ_2 s+1)^m /s,

where (a) follows from the recent result in <cit.>. By assuming integer-valued m and μ, the RF link's CMGF can be rewritten after resorting to the transformation Γ(α)(1+z)^-α= H_1,1^1,1 [ z | (1-α,1)  (0,1)] in <cit.> as


    M^(c)_γ_2(s)(a)μ⩽ m=∑_l=1^mχ_l s^l-1/Γ(m) H_1,1^1,1[θ_2s| (1-m,1) (0,1)],

and 

 		
    M^(c)_γ_2(s)   (b)μ> m=   (p,q)≠ (0,0)∑_p=0^m∑_q=0^μ-mmpμ-mqθ_2^pθ_1^qs^p+q-1
          (∑_i=1^μ-mΔ_1i H_1,1^1,1[θ_1 s | (m+i-μ,1) (0,1)]/Γ(μ-m-i+1)-∑_i=1^mΔ_2i H_1,1^1,1[θ_2 s | (i-m,1)(0,1)]/Γ(m-i+1)),

where (a) and (b) follow after applying the binomial expansion and the partial fraction decomposition <cit.>, respectively. Plugging (<ref>), (<ref>) and (<ref>) into (<ref>) and resorting to <cit.> complete the proof.
 
 

 

 §.§ Outage Probability

 The quality of service (QoS) of the considered mixed FSO/RF system  is ensured by keeping the instantaneous end-to-end SNR, γ, above a threshold γ_th. The probability of outage in the mixed FSO/RF  relaying setup is expressed as

 	
    P_out= Pr[γ<γ_th]= Pr[γ_1 γ_2/γ_1+γ_2+1<γ_th].

 Marginalization over γ_1, and letting u=1+γ/γ_th in (<ref>) yield 

  
    P_out(γ_th)=1-γ_th∫_1^∞F^(c)_γ_2(γ_th+1+γ_th/u-1)f_γ_1(uγ_th) du,

  where F^(c)_γ_2  is the CCDF of γ_2 and  f_γ_1 is the PDF of the first-link SNR obtained from deriving (<ref>) with respect to x as

  
    f_γ_1(x)=ξ^2AB^r/Γ(α)μ_r∑_k=1^βb_k/Γ(k) H_1,3^3,0[B^rx/μ_r|  (ξ^2+1-r,r) (ξ^2-r,r),(α-r,r),(k-r,r)].

  Plugging (<ref>) and the RF link’s CCDF expression recently derived in <cit.> for integer  m, μ with  m ⩾μ   into the above integral and making  a Taylor expansion of exponential and power terms, we infer that 

  
    P_out(γ_th)   =   1-ξ^2AB^rγ_the^-γ_th/θ_2/Γ(α)μ_r∑_k=1^βb_k/Γ(k)
          ∑_i=0^m-μ∑_j=0^m-i-1∑_p=0^jjpΥ_i/j!θ_2^jγ_th^j-p (γ_th+1)^p×ℐ,

  with Υ_i=m-μi(m/μκ+m)^i(μκ/μκ+m)^m-μ-i, and ℐ given by 

  
    ℐ   =   ∑_q=0^∞∑_l=0^∞(-1)^q(p+q)_l/q!l!θ_2^q (γ_th+1)^q∫_1^∞ u^-p-q-l
           H_1,3^3,0[B^rγ_th/μ_ru|  (ξ^2+1-r,r)  (ξ^2-r,r),(α-r,r),(k-r,r)]du.

  Substituting  (<ref>) into (<ref>) after resorting to <cit.> yields the outage probability of mixed FSO/RF in Málaga-ℳ/κ-μ shadowed fading (μ⩽ m ) environments  with pointing errors under both detection techniques as


    P_out(γ_th)=1-ξ^2AB^rγ_th/e^γ_th/θ_2Γ(α)μ_r∑_k=1^βb_k/Γ(k)∑_i=0^m-μ∑_j=0^m-i-1Υ_i/θ_2^jΞ(θ_2),


where 


    Ξ(x)=∑_p=0^j∑_q=0^∞∑_l=0^∞(-1)^q(p+q)_ljpγ_th^j-p/j!q!l!x^q(γ_th+1)^-p-q H_2,4^4,0[B^rγ_th/μ_r| (σ_1,Σ_1) (ϕ_1,Φ_1)],

with (a)_n standing for the Pochhammer symbol <cit.>, (σ_1,Σ_1) =(ξ^2+1-r,r),(l+p+q,1), and (ϕ_1,Φ_1)=(l+p+q-1,1),(ξ^2-r,r),(α-r,r),(k-r,r).

Similar to (<ref>) and using <cit.>, the outage probability of mixed FSO Málaga-ℳ/RF shadowed κ-μ (m < μ) is 


    P_out(γ_th)   =   1-ξ^2AB^rγ_th/Γ(α)μ_r∑_k=1^βb_k/Γ(k)
          (e^-γ_th/θ_1∑_i=1^μ-m∑_j=0^μ-m-iΔ_1i/θ_1^jΞ(θ_1)+e^-γ_th/θ_2∑_i=1^m∑_j=0^m-iΔ_2i/θ_2^jΞ(θ_2)).





§ ASYMPTOTIC ANALYSIS

To gain more insights into the effect of turbulence/fading parameters on both the ergodic capacity and the outage probability, we study hereafter their asymptotic behaviors.  To this end, we invoke  the asymptotic expansions of  the Fox-H function <cit.> and the Mellin-Barnes integrals involving the bivariate Fox-H function <cit.>.



 §.§ Asymptotic Ergodic Capacity

We assume that the average SNR of the RF
link γ̅_2 goes to infinity for a fixed and finite valued average
SNR in the FSO link. Then, resorting to the Mellin-Barnes representation of the bivariate FHF <cit.> in  (<ref>), and evaluating the residue  at the poles { -m, -1-l } yield the asymptotic capacity,   when m ≥μ as 


    C^∞   =   ξ^2 A r μ_r/2 ln(2)Γ(α)B^r∑_k=1^β∑_l=1^mb_kχ_l/Γ(k)( H_5,4^2,5[μ_r/B^r θ_2| (-l,1),(σ,Σ)(m-1-l,1),(ϕ,Φ)]/θ_2^1+lΓ(m)
           + θ_2^-m H_5,3^1,5[μ_r/B^r|  (m-l,1),(σ,Σ) (ϕ,Φ)]).

It is worth noting that (<ref>)  is much easier and
faster to calculate than the exact capacity in (<ref>).
Moreover, C^∞ when m<μ follows in the same line of (<ref>) while considering (<ref>).











 §.§ Asymptotic Outage Probability

At high SNR values, the outage probability of the mixed FSO/RF relaying system can
be expressed as P_out≃ (G_c SNR )^-G_d,  where G_c and G_d
 denote  the
coding gain and the diversity order of the system, respectively. Hence, as μ_r →∞ while keeping the low-order terms in (<ref>), i.e. q+l < 1, and   then  applying <cit.>  yield  the asymptotic CDF  when m ≥μ  as 


    P_out^∞   =   1-ξ^2Ae^-γ_th/θ_2/Γ(α)∑_k=1^βb_k/Γ(k)∑_i=0^m-μ∑_j=0^m-i-1∑_p=0^jjpΥ_iγ_th^j-pθ_2^-j/j!(γ_th+1)^-p
          ∑_t=1^41/Φ_1t∏_s ≠ ts=1^4Γ(ϕ_1s-ϕ_1tΦ_1s/Φ_1t)/∏_s=1^2Γ(σ_1s-ϕ_1tΣ_1s/Φ_1t)(B^rγ_th/μ_r)^ϕ_1t/Φ_1t+1.






 Compared
to (<ref>) which is expressed in terms of Fox-H
function, (<ref>) includes only finite summations of elementary
functions.  The diversity gain of the studied system over atmospheric turbulence conditions is inferred after applying e^-γ_th/θ_2γ̅_2≫ 1≈1-γ_th/θ_2 to  (<ref>) as G_d=min{μ, ξ^2/r,α/r,β/r}.

For μ=m, κ→0, g=0 and Ω=1, the CDF in  (<ref>) reduces to P_out^∞ for  𝒢-𝒢/Nakagami-m fading
 channels as


    P_out^∞   =   1-ξ^2e^-mγ_th/γ̅_2/Γ(α)Γ(β)∑_j=0^m-1∑_p=0^j∑_t=1^4jpm^jγ_th^-p(αβ h)^r(ϕ_1t/Φ_1t+1)/j!Φ_1t(γ_th+1)^-p
          ∏_s ≠ ts=1^4Γ(ϕ_1s-ϕ_1tΦ_1s/Φ_1t)/∏_s=1^2Γ(σ_1s-ϕ_1tΣ_1s/Φ_1t)(γ_th/μ_r)^ϕ_1t/Φ_1t+1(γ_th/γ̅_2)^j,

thereby inferring <cit.>, i.e., G_d=min{m, ξ^2/r,α/r,β/r}.
Similar to (<ref>) while considering (<ref>), the asymptotic outage probability can be derived  in
closed form when m< μ. However, the derived expression is omitted due to
space limitations.











§ NUMERICAL RESULTS


Fig.<ref> investigates the impacts of the turbulence-induced fading and pointing errors on the system performance when the RF link is subject to Rician shadowed fading distribution (κ=5, μ=1, m=2). As expected, the ergodic capacity deteriorates by decreasing the pointing error displacement standard deviation, i.e., for smaller ξ, or decreasing the turbulence fading parameter, i.e., smaller α and β, where we associate the strong turbulence to (α,β)=(2.29,2) and the moderate turbulence to (α,β)=(4.2,3). At high SNR, the asymptotic expansion in (<ref>) matches very well its exact counterpart, which confirms the validity of our mathematical analysis for different parameter settings.

Fig.<ref> depicts the outage probability of mixed FSO/RF relay systems in Málaga-ℳ and shadowed κ-μ fading channels for both heterodyne and IM/DD detection at the relay.  Throughout our
numerical experiments, we found out that regardless of the average SNRs and turbulence/fading settings, accurate analytical curves can be obtained by truncating the infinite sums at q=10 and l=5 terms. In the legend, please note that we have identified some particular turbulence and fading distribution cases that simply stem from the general Málaga amd κ-μ shadowed fading scenarios, respectively. The exact match with Monte-Carlo simulation results confirms the precision of the theoretical analysis of section III.B.  Moreover, we notice that the exact and asymptotic expansion in  (<ref>) agree very well at high SNRs.
 	
 

 

§ CONCLUSION

We have presented a unified analytical framework for relay-assisted mixed FSO/RF systems that remarkably accommodates generic turbulence/fading  models including Málaga-ℳ with pointing errors and  shadowed κ-μ  distribution that account for shadowed LOS and NLOS scenarios. The results demonstrate the unification of various FSO turbulent/RF fading scenarios into a single closed-form expression  for the ergodic capacity and the outage probability while accounting for both IM/DD and heterodyne detection techniques at the relay. 

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