**Performance Analysis of Asymmetric RF/FSO Dual-hop Relaying Systems for UAV** **Applications**

Jaedon Park, Guisoon Park and Bongsoo Roh
*2-1*

*Agency for Defense Development*
*Daejeon, Korea 305-152*

*jaedon@add.re.kr*

Eunju Lee and Giwan Yoon
*Department of Electrical Engineering*
*Korea Advanced Institute of Science and Technology*

*Daejeon, Korea 305-732*
*lejagnes@kaist.ac.kr*

**Abstract—In this paper, we analyze the performance of****a dual-hop relaying system composed of asymmetric radio**
**frequency (RF) and free-space optics (FSO) (RF/FSO) links.**

**We consider an asymmetric amplify-and-forward (AF) relay**
**which converts the received RF signal into an optical signal**
**using the subcarrier intensity modulation (SIM) scheme. The**
**RF and FSO channels are assumed to experience Rayleigh and**
**Gamma-Gamma fading distributions, respectively. Particularly,**
**we derive the average probability of error as well as ergodic**
**capacity upper bound of the asymmetric RF/FSO dual-hop**
**relaying system, in closed-forms. As a result, the asymmetric**
**RF/FSO relaying system shows slightly worse performance in**
**average probability of error and ergodic capacity upper bound**
**than the RF/RF relaying system in the low SNR. Over the SNR**
**of 20 dB, however, the asymmetric RF/FSO relaying system**
**shows very similar performance in average probability of error**
**and ergodic capacity upper bound to the RF/RF relaying**
**system. The derived mathematical expressions are veriﬁed by**
**exactly matching Monte-Carlo simulation results.**

**Keywords-amplify-and-forward relay; average probability of****error; ergodic capacity upper bound; free-space optics;**

I. INTRODUCTION

Wireless relaying systems have drawn a signiﬁcant at-
tention in radio frequency (RF) wireless communications
due to their strong potential to increase the coverage area
and quality-of-service (QoS) [1]–[5]. The relaying systems
can be classiﬁed into amplify-and-forward (AF) and decode-
and-forward (DF) relays [1]. The DF relaying systems,
also called regenerative systems, decode the received signal
fully and retransmit it to another hop [1]. The AF relaying
systems, also called nonregenerative systems, just amplify
the received signal and forward to another hop with less
complexity than the DF relaying systems [1]. Since the AF
relaying systems do not decode and re-encode the received
signal, they require less power than the DF relaying systems
*in the relaying process [1], [4]. In [1], Hasna et al. studied*
the outage probability and the average probability of error
for the AF relaying systems in Rayleigh fading channels.

*In [3], Karagiannidis et al. studied the outage probability*
and average error probability of the AF multihop relay-
ing systems in Nakagami-n, Nakagami-m and Nakagami-q
fading channels for coherent and non-coherent modulation

*schemes using the moment-based approach. In [2], Zhu et al.*

investigated the upper bound and lower bound of the outage probability of the DF relaying system in Rician fading channels. It is noted that these research works considered the symmetric relay channels where channel fading distributions are the same for the relaying hops.

In many practical scenarios, however, since the com-
munications environments for the relaying hops could be
different, each relay link could have different types of fading
channels, namely asymmetric channels. In [4], Suraweera
*et al. studied the outage probability and average bit error*
probability of the AF relaying systems over asymmetric
*Rayleigh and Rician fading channels. In [5], Gurung et*
*al. analyzed the outage probability and average symbol*
error probability of dual-hop Nakagami-m and Rician fading
channels.

Meanwhile, the free-space optics (FSO) communications systems can provide wider bandwidth and therefore support more users as compared to the RF ones, potentially enabling to solve the problems that the RF communications system may go through due to the expensive and scarce spectrum [6]–[9]. In the implementation of the FSO communica- tions systems, the intensity modulation and direct detection (IM/DD) with on/off keying (OOK) has been widely used for its simplicity. Since the OOK systems, however, have a signiﬁcant performance degradation in the atmospheric turbulence due to the ﬁxed threshold, the subcarrier intensity modulation (SIM) has been widely used for better perfor- mance in the FSO communications [7], [8], [10].

Basically, over the distances of 1 km or longer, the FSO communications system has a signiﬁcant degradation in performance due to the atmospheric turbulence problem [8].

Therefore, the relaying techniques are highly required for
the wideband FSO communications to increase the commu-
*nications distance. Recently, Tsiftsis et al. has studied the*
outage probability of multihop AF and DF relays and the
average bit error rate (BER) (the upper bound) of the dual-
hop DF relays for FSO communications [11]. More recently,
*Kazemlou et al. proposed all-optical relaying techniques to*
improve the error performance and overall distance coverage
2013 IEEE Military Communications Conference

2013 IEEE Military Communications Conference

of FSO communications systems [12], and Aghajanzadeh
*et al. derived the outage probability and quantiﬁed the po-*
tential performance improvements through the derivation of
diversity-multiplexing tradeoff and diversity gain for multi-
hop FSO relaying systems with DF relays [13].

In order to overcome the lack of RF spectrum and the performance degradation of a long FSO channel, the asymmetric RF/FSO relaying system, composed of RF and FSO links with an asymmetric RF/FSO relay which converts the received RF signal into an optical signal using the SIM scheme, could be an efﬁcient solution. Using the asymmetric RF/FSO relaying communications system, it is expected that the relay links can provide an adaptive and effective solution to improve the performance and coverage in real communi- cations networks. In our previous work [14], we have derived the outage probability of an asymmetric RF/FSO dual-hop relaying system in a closed-form. In this work, the authors have further derived the average probability of error of the asymmetric RF/FSO dual-hop relaying system in a closed- form. Also, we have derived the ergodic capacity upper bound of the asymmetric RF/FSO dual-hop relaying system in a closed-form, where the bound is based on Jensen’s inequality [15]–[17].

II. SYSTEM AND CHANNEL MODEL

Consider an asymmetric RF/FSO dual-hop relaying sys-
tem as shown in Fig.1. The source node S is communicating
with the destination node D through a relay node R. There-
*fore, there are two point-to-point propagation links, i.e., S-*
R and R-D links respectively, before the source signal is
arrived at the destination. Here, we assume an asymmetric
*relaying scenario, i.e., the S-R is an RF link and the R-D*
is an FSO link. In the ﬁxed gain AF relay node, an optical
modulator converts the received RF signal into an optical
signal using the SIM scheme [7], [8], [10]. Accordingly, the
S-R link experiences Rayleigh fading distribution which is
frequently used to model the multipath fading with no direct
line-of-sight (LOS) path in RF propagation environments
[1], [18], and the R-D link experiences Gamma-Gamma
fading distribution which is also widely used to model
the atmospheric turbulence in the FSO communications
environments [7]–[9].

Figure 1. A dual-hop relaying system over asymmetric RF/FSO links

The received signal at the relay R can be expressed as

*𝑟*1*= 𝛼*1*𝑠 + 𝑛*1 (1)

where*𝛼*1is the fading amplitude of Rayleigh fading channel
for the S-R link, and*𝑛*1 is an additive white Gaussian noise
(AWGN) with the power spectral density of *𝑁*01. The *𝑠*
indicates the RF signal transmitted from the source S. When
the SIM scheme [7], [8], [10] is employed in the relay, the
retransmitted optical signal at the relay R is

*𝑠**𝑜𝑝𝑡**= 𝐺(1 + 𝜂𝑟*1) (2)
where *𝐺 is the ﬁxed relay gain at the relay R, and 𝜂 is*
the electrical-to-optical conversion coefﬁcient. The received
optical signal at the destination D can be written as

*𝑟*2*= 𝐼𝐺[1 + 𝜂(𝛼*1*𝑠 + 𝑛*1*)] + 𝑛*2 (3)
where*𝐼 is a stationary random variable following Gamma-*
Gamma distribution for the FSO link, and*𝑛*2 is an AWGN
with the power spectral density of *𝑁*02. When the DC
component is ﬁltered out at the destination, the received
signal can be

*𝑟*2*= 𝐼𝐺𝜂(𝛼*1*𝑠 + 𝑛*1*) + 𝑛*2 (4)
Thus, the overall signal-to-noise ratio (SNR) at the desti-
nation can be expressed as

*𝛾 =* *𝐼*^{2}*𝐺*^{2}*𝜂*^{2}*𝛼*^{2}_{1}*𝑃*1

*𝐼*^{2}*𝐺*^{2}*𝜂*^{2}*𝑁*01*+ 𝑁*02 =

*𝛼*^{2}_{1}*𝑃*1

*𝑁*01

*𝜂*^{2}*𝐼*^{2}*𝑃*_{2}
*𝑁*02

*𝜂*^{2}*𝐼*^{2}*𝑃*2

*𝑁*_{02} + * _{𝐺}*2

^{𝑃}*𝑁*

^{2}

_{01}

(5)
where*𝑃*1and*𝑃*2 are the powers transmitted at the source S
and relay R, respectively. If the relay introduces a ﬁxed gain
to the received signal, regardless of the fading information
on the ﬁrst link, we can let*𝐶 = 𝑃*2*/(𝐺*^{2}*𝑁*01), then, (5) can
be rewritten as

*𝛾 =* *𝛾*1*𝛾*2

*𝛾*2*+ 𝐶* (6)

where both *𝛾*1 = ^{𝛼}_{𝑁}^{2}^{1}_{01}^{𝑃}^{1} and *𝛾*2 = ^{𝜂}^{2}_{𝑁}^{𝐼}^{2}_{02}^{𝑃}^{2} are the SNRs
of each hop. The above derivation result is exactly the
same as the dual-hop RF transmissions with Rayleigh fading
distributions [1]. In this work, we assume that the S-R
RF link experiences Rayleigh fading distribution with the
probability density function (PDF) given in [18] as

*𝑓**𝛾*1*(𝛾*1) = 1

*𝛾*1*𝑒*^{−𝛾1}* ^{𝛾1}* (7)

where *𝛾*1 is the average SNR of the S-R link. The R-D
FSO link is assumed to experience Gamma-Gamma fading
distribution with the PDF given in [7]–[9] as

*𝑓**𝛾*_{2}*(𝛾*2) = *(𝛼𝛽)*^{(𝛼+𝛽)/2}*𝛾*_{2}^{(𝛼+𝛽)/4−1}*Γ(𝛼)Γ(𝛽)𝛾*2*(𝛼+𝛽)/4* *𝐾**𝛼−𝛽*

( 2

√
*𝛼𝛽*

√*𝛾*2

*𝛾*2

)

(8)

where *𝛾*2 is the average SNR of the R-D link, *𝐾**𝑎**(⋅) is*
the modiﬁed Bessel function of the second kind of order*𝑎,*
and the parameters*𝛼 and 𝛽 are related to the atmospheric*
turbulence conditions [7]–[9].

III. AVERAGEPROBABILITY OFERROR

In this section, the average probability of error is derived
in a closed-form for the asymmetric RF/FSO dual-hop re-
laying system. The RF and FSO channels are assumed to ex-
perience Rayleigh and Gamma-Gamma fading distributions,
respectively. If we let *𝑃 (𝑒∣𝛾) denote the conditional error*
probability in an AWGN channel, the average probability of
error can be expressed as

*𝑃**𝑒*=

∫ _{∞}

0 *𝑃 (𝑒∣𝛾)𝑓**𝛾**(𝛾)𝑑𝛾,* (9)
where the conditional error probability can be given by [18]

*𝑃 (𝑒∣𝛾) = 𝑄*(√

*𝛿𝛾*)

*,* (10)

where *𝛿 is 2 and 1 for BPSK and QPSK modulations,*
respectively.

Substituting (10) to (9), (9) can be rewritten as [4]

*𝑃**𝑒*= *√*1
*2𝜋*

∫ _{∞}

0 *𝐹**𝛾*

(*𝑡*^{2}
*𝛿*

)

*𝑒*^{−}^{𝑡2}^{2}*𝑑𝑡,* (11)
where*𝐹**𝛾**(𝛾) is the cumulative distribution function (CDF)*
of the random variable*𝛾.*

After the variable change of *𝑥 = 𝑡*^{2}, the average proba-
bility of error can be given by

*𝑃**𝑒*= *√*1
*2𝜋*

∫ _{∞}

0 *𝐹**𝛾**(𝑥*
*𝛿*

)*𝑒*^{−}^{𝑥}^{2} *𝑑𝑥*

2*√𝑥.* (12)
According to our previous work [14], the outage proba-
bility was derived as

*𝑃**𝑜𝑢𝑡* *= 1 − 𝑒*^{−}^{𝛾𝑡ℎ}^{𝛾1}*(𝛼𝛽)*^{(𝛼+𝛽)/2}

*Γ(𝛼)Γ(𝛽)𝛾*2*(𝛼+𝛽)/4**𝐸(𝛾**𝑡ℎ**),* (13)
where the integral function*𝐸(𝛾**𝑡ℎ*) is given by

*𝐸(𝛾**𝑡ℎ*) = 1
*4𝜋*

(*𝐶𝛾**𝑡ℎ*

*𝛾*1

)_{(𝛼+𝛽)/4}

*×*
*𝐺*^{50}_{05}

(

*(𝛼𝛽)*^{2} *𝐶𝛾**𝑡ℎ*

*16𝛾*1*𝛾*2

⏐⏐

⏐⏐

*−*

*𝛼−𝛽*4 *,*^{𝛼−𝛽+2}_{4} *,*^{𝛽−𝛼}_{4} *,*^{𝛽−𝛼+2}_{4} *,−*^{𝛼+𝛽}_{4}

)
*.*
(14)
Since the CDF can be obtained from the outage proba-
bility by the variable change from*𝛾**𝑡ℎ* to*𝛾, 𝐹**𝛾**(𝛾) can be*
expressed as

*𝐹**𝛾**(𝛾) = 1 − 𝑒*^{−}^{𝛾1}^{𝛾}*(𝛼𝛽)*^{(𝛼+𝛽)/2}

*Γ(𝛼)Γ(𝛽)𝛾*2*(𝛼+𝛽)/4**𝐸(𝛾).* (15)

Substituting (15) to (12) with a variable change from *𝛾*
to*𝑥/𝛿, the average probability of error can be expressed as*
*𝑃**𝑒**= 𝐿 − 𝐻*1*(𝛼, 𝛽)𝐻*2*(𝛼, 𝛽),* (16)
where

*𝐿 =* 1
2*√*

*2𝜋*

∫ _{∞}

0

*𝑒*^{−𝑥/2}*√*

*𝑥* *𝑑𝑥* (17)

and

*𝐻*1*(𝛼, 𝛽) =* 1
8*√*

*2𝜋*

*(𝛼𝛽)** ^{(𝛼+𝛽)/2}*(

*𝛿𝛾**𝐶*1*𝛾*2

)*(𝛼+𝛽)/4*

*𝜋Γ(𝛼)Γ(𝛽)* (18)

and

*𝐻*2*(𝛼, 𝛽) =*

∫ _{∞}

0 *𝑥**−(1/2−(𝛼+𝛽)/4)**𝑒** ^{−}*(

_{1}

*𝛿𝛾1*+^{1}_{2})

*𝑥**×*
*𝐺*^{50}05

(

*(𝛼𝛽)*^{2} *𝐶𝑥*
*16𝛿𝛾*1*𝛾*2

⏐⏐

⏐⏐

*−*

*𝛼−𝛽*4 *,*^{𝛼−𝛽+2}_{4} *,*^{𝛽−𝛼}_{4} *,*^{𝛽−𝛼+2}_{4} *,−*^{𝛼+𝛽}_{4}

)
*𝑑𝑥.*

(19) Using the equation (3.361-2) of [19]

∫ _{∞}

0

*𝑒√*^{−𝑞𝑥}*𝑥* *𝑑𝑥 =*

√*𝜋*

*𝑞,* (20)

the *𝐿 in (17) comes to 1/2. Also, using the equation*
(7.813-1) of [19], the integral in (19) can be evaluated as

*𝐻*2*(𝛼, 𝛽) =*
( 1

*𝛿𝛾*1 +1
2

)*−(𝛼+𝛽)/4−1/2*

*×*
*𝐺*^{51}_{15}

⎛

⎝ ^{𝐶(𝛼𝛽)}

2

*16𝛿𝛾*1*𝛾*2

*𝛿𝛾*11 + ^{1}_{2}

⏐⏐

⏐⏐

*1/2−(𝛼+𝛽)/4*

*𝛼−𝛽*4 *,*^{𝛼−𝛽+2}_{4} *,*^{𝛽−𝛼}_{4} *,*^{𝛽−𝛼+2}_{4} *,−*^{𝛼+𝛽}_{4}

⎞

⎠ (21)

IV. ERGODIC CAPACITY UPPER BOUND

In this section, the ergodic capacity upper bound is derived in a closed-form for the asymmetric RF/FSO dual-hop relaying system. The RF and FSO channels are assumed to experience Rayleigh and Gamma-Gamma fading distri- butions, respectively. The ergodic capacity can be written by

*𝐶**𝑒𝑟𝑔* *= 𝐸[𝑙𝑜𝑔*2*(1 + 𝛾)]* (22)
Since*𝑙𝑜𝑔(⋅) is a concave function, using Jensen’s inequal-*
ity [15]–[17], the upper bound of the ergodic capacity can
be expressed as

*𝐶**𝑒𝑟𝑔* *≤ 𝑙𝑜𝑔*2*(1 + 𝐸[𝛾])* (23)
where the mean value *𝐸[𝛾] can be written as*

*𝐸[𝛾] =*

∫ _{∞}

0 *𝛾𝑓(𝛾)𝑑𝛾.* (24)

By differentiating the CDF in (*15) in terms of 𝛾, the pdf*
*𝑓(𝛾) can be expressed as*

*𝑓(𝛾) = −* *(𝛼𝛽)*^{(𝛼+𝛽)/2}*4𝜋Γ(𝛼)Γ(𝛽)𝛾*2*(𝛼+𝛽)/4*

(*𝐶*
*𝛾*1

)_{(𝛼+𝛽)/4}

*× [𝐵*1*(𝛾) + 𝐵*2*(𝛾) + 𝐵*3*(𝛾)], (25)*
where

*𝐵*1*(𝛾) =* *𝛼 + 𝛽*

4 *𝛾*^{(𝛼+𝛽)/4−1}*𝑒*^{−}^{𝛾1}^{𝛾}

*× 𝐺*^{50}_{05}
(

*(𝛼𝛽)*^{2} *𝐶𝛾*
*16𝛾*1*𝛾*2

⏐⏐

⏐⏐

*−*

*𝛼−𝛽*4 *,*^{𝛼−𝛽+2}_{4} *,*^{𝛽−𝛼}_{4} *,*^{𝛽−𝛼+2}_{4} *,−*^{𝛼+𝛽}_{4}

)
*,*
(26)

*𝐵*2*(𝛾) = −𝛾*^{(𝛼+𝛽)/4}*𝛾*1*−1**𝑒*^{−}^{𝛾1}^{𝛾}

*× 𝐺*^{50}05

(

*(𝛼𝛽)*^{2} *𝐶𝛾*
*16𝛾*1*𝛾*2

⏐⏐

⏐⏐

*−*

*𝛼−𝛽*4 *,*^{𝛼−𝛽+2}_{4} *,*^{𝛽−𝛼}_{4} *,*^{𝛽−𝛼+2}_{4} *,−*^{𝛼+𝛽}_{4}

)
*,*
(27)

*𝐵*3*(𝛾) = 𝛾*^{(𝛼+𝛽)/4}*𝑒*^{−}^{𝛾1}^{𝛾}

*×* *𝑑*
*𝑑𝛾𝐺*^{50}_{05}

(

*(𝛼𝛽)*^{2} *𝐶𝛾*
*16𝛾*1*𝛾*2

⏐⏐

⏐⏐

*−*

*𝛼−𝛽*4 *,*^{𝛼−𝛽+2}_{4} *,*^{𝛽−𝛼}_{4} *,*^{𝛽−𝛼+2}_{4} *,−*^{𝛼+𝛽}_{4}

)
*.*
(28)
Substituting (25) to (24), the mean value can be rewritten
as

*𝐸[𝛾] =* *(𝛼𝛽)*^{(𝛼+𝛽)/2}*4𝜋Γ(𝛼)Γ(𝛽)𝛾*2*(𝛼+𝛽)/4*

(*𝐶*
*𝛾*1

)*(𝛼+𝛽)/4*

*× 𝐴(𝛼, 𝛽, 𝛾*1*, 𝛾*2*), (29)*
where

*𝐴(𝛼, 𝛽, 𝛾*1*, 𝛾*2*) = 𝐴*1*(𝛼, 𝛽, 𝛾*1*, 𝛾*2*) + 𝐴*2*(𝛼, 𝛽, 𝛾*1*, 𝛾*2)
*+ 𝐴*3*(𝛼, 𝛽, 𝛾*1*, 𝛾*2*), (30)*

*𝐴*1*(𝛼, 𝛽, 𝛾*1*, 𝛾*2*) = −*

∫ _{∞}

0

*𝛼 + 𝛽*

4 *𝛾*^{(𝛼+𝛽)/4}*𝑒*^{−}^{𝛾1}^{𝛾}

*×𝐺*^{50}_{05}
(

*(𝛼𝛽)*^{2} *𝐶𝛾*
*16𝛾*1*𝛾*2

⏐⏐

⏐⏐

*−*

*𝛼−𝛽*4 *,*^{𝛼−𝛽+2}_{4} *,*^{𝛽−𝛼}_{4} *,*^{𝛽−𝛼+2}_{4} *,−*^{𝛼+𝛽}_{4}

)
*𝑑𝛾,*

(31)

*𝐴*2*(𝛼, 𝛽, 𝛾*1*, 𝛾*2) =

∫ _{∞}

0 *𝛾*^{(𝛼+𝛽)/4+1}*𝛾*1*−1**𝑒*^{−}^{𝛾1}^{𝛾}

*×𝐺*^{50}_{05}
(

*(𝛼𝛽)*^{2} *𝐶𝛾*
*16𝛾*1*𝛾*2

⏐⏐

⏐⏐

*−*

*𝛼−𝛽*4 *,*^{𝛼−𝛽+2}_{4} *,*^{𝛽−𝛼}_{4} *,*^{𝛽−𝛼+2}_{4} *,−*^{𝛼+𝛽}_{4}

)
*𝑑𝛾,*

(32)

*𝐴*3*(𝛼, 𝛽, 𝛾*1*, 𝛾*2*) = −*

∫ _{∞}

0 *𝛾*^{(𝛼+𝛽)/4+1}*𝑒*^{−}^{𝛾1}^{𝛾}

*×* *𝑑*
*𝑑𝛾𝐺*^{50}_{05}

(

*(𝛼𝛽)*^{2} *𝐶𝛾*
*16𝛾*1*𝛾*2

⏐⏐

⏐⏐

*−*

*𝛼−𝛽*4 *,*^{𝛼−𝛽+2}_{4} *,*^{𝛽−𝛼}_{4} *,*^{𝛽−𝛼+2}_{4} *,−*^{𝛼+𝛽}_{4}

)
*𝑑𝛾.*

(33)
Applying the integration by parts to (33), 𝐴3*(𝛼, 𝛽, 𝛾*1*, 𝛾*2)
can be converted to

*𝐴*3*(𝛼, 𝛽, 𝛾*1*, 𝛾*2) =

∫ _{∞}

0

*𝑑*
*𝑑𝛾*

(*𝛾*^{(𝛼+𝛽)/4+1}*𝑒*^{−}^{𝛾1}* ^{𝛾}*)

*×𝐺*^{50}_{05}
(

*(𝛼𝛽)*^{2} *𝐶𝛾*
*16𝛾*1*𝛾*2

⏐⏐

⏐⏐

*−*

*𝛼−𝛽*4 *,*^{𝛼−𝛽+2}_{4} *,*^{𝛽−𝛼}_{4} *,*^{𝛽−𝛼+2}_{4} *,−*^{𝛼+𝛽}_{4}

)
*𝑑𝛾.*

(34)
Differentiating the ﬁrst term in (34), 𝐴3*(𝛼, 𝛽, 𝛾*1*, 𝛾*2)
can be rewritten in terms of *𝐴*1*(𝛼, 𝛽, 𝛾*1*, 𝛾*2) and
*𝐴*2*(𝛼, 𝛽, 𝛾*1*, 𝛾*2) as following

*𝐴*3*(𝛼, 𝛽, 𝛾*1*, 𝛾*2*) = −𝐴*1*(𝛼, 𝛽, 𝛾*1*, 𝛾*2*) − 𝐴*2*(𝛼, 𝛽, 𝛾*1*, 𝛾*2)
+

∫ _{∞}

0 *𝛾*^{(𝛼+𝛽)/4}*𝑒*^{−}^{𝛾1}^{𝛾}

*×𝐺*^{50}_{05}
(

*(𝛼𝛽)*^{2} *𝐶𝛾*
*16𝛾*1*𝛾*2

⏐⏐

⏐⏐

*−*

*𝛼−𝛽*4 *,*^{𝛼−𝛽+2}_{4} *,*^{𝛽−𝛼}_{4} *,*^{𝛽−𝛼+2}_{4} *,−*^{𝛼+𝛽}_{4}

)
*𝑑𝛾.*

(35)
Substituting (35) to (30), 𝐴(𝛼, 𝛽, 𝛾1*, 𝛾*2) can be rewritten
as

*𝐴(𝛼, 𝛽, 𝛾*1*, 𝛾*2) =

∫ *∞*

0 *𝛾*^{(𝛼+𝛽)/4}*𝑒*^{−}^{𝛾1}^{𝛾}

*×𝐺*^{50}05

(

*(𝛼𝛽)*^{2} *𝐶𝛾*
*16𝛾*1*𝛾*2

⏐⏐

⏐⏐

*−*

*𝛼−𝛽*4 *,*^{𝛼−𝛽+2}_{4} *,*^{𝛽−𝛼}_{4} *,*^{𝛽−𝛼+2}_{4} *,−*^{𝛼+𝛽}_{4}

)
*𝑑𝛾.*

(36)
With the help of the equation (7.813-1) of [19],
*𝐴(𝛼, 𝛽, 𝛾*1*, 𝛾*2) can be obtained as

*𝐴(𝛼, 𝛽, 𝛾*1*, 𝛾*2*) = 𝛾*^{(𝛼+𝛽)/4+1}

*× 𝐺*^{51}_{15}
(

*(𝛼𝛽)*^{2} *𝐶*
*16𝛾*2

⏐⏐

⏐⏐

*−*^{𝛼+𝛽}_{4}

*𝛼−𝛽*4 *,*^{𝛼−𝛽+2}_{4} *,*^{𝛽−𝛼}_{4} *,*^{𝛽−𝛼+2}_{4} *,−*^{𝛼+𝛽}_{4}

)
*.*
(37)
Substituting (*37) to (29), the mean value of the 𝛾 can be*
derived as

*𝐸[𝛾] =* *(𝛼𝛽)*^{(𝛼+𝛽)/2}*𝛾*1

*4𝜋Γ(𝛼)Γ(𝛽)*
(*𝐶*

*𝛾*2

)*(𝛼+𝛽)/4*

*× 𝐺*^{51}_{15}
(

*(𝛼𝛽)*^{2} *𝐶*
*16𝛾*2

⏐⏐

⏐⏐

*−*^{𝛼+𝛽}_{4}

*𝛼−𝛽*4 *,*^{𝛼−𝛽+2}_{4} *,*^{𝛽−𝛼}_{4} *,*^{𝛽−𝛼+2}_{4} *,−*^{𝛼+𝛽}_{4}

)
*.*
(38)

Finally, substituting (38) to (23), the ergodic capacity upper bound can be obtained as

*𝐶**𝑒𝑟𝑔**≤ 𝑙𝑜𝑔*2

{

1 + *(𝛼𝛽)*^{(𝛼+𝛽)/2}*𝛾*1

*4𝜋Γ(𝛼)Γ(𝛽)*
(*𝐶*

*𝛾*2

)*(𝛼+𝛽)/4*

*× 𝐺*^{51}15

(

*(𝛼𝛽)*^{2} *𝐶*
*16𝛾*2

⏐⏐

⏐⏐

*−*^{𝛼+𝛽}_{4}

*𝛼−𝛽*4 *,*^{𝛼−𝛽+2}_{4} *,*^{𝛽−𝛼}_{4} *,*^{𝛽−𝛼+2}_{4} *,−*^{𝛼+𝛽}_{4}

) }
*.*
(39)
V. NUMERICAL RESULTS

In this section, we show numerical results of the asym-
metric RF/FSO dual-hop relaying system. The RF link
is modeled as Rayleigh fading channel. The FSO link is
modeled as Gamma-Gamma fading channel [7]–[9], [14],
[20] with the atmospheric turbulence parameters *𝛼 = 4.2, 𝛽*

= 1.4 for a strong regime and*𝛼 = 4.0, 𝛽 = 1.9 for a moderate*
regime [8], [9], [14]. The relay is assumed to convert the
received RF signals to the optical signals using the SIM
scheme [7], [8], [10]. The relay gain factor *𝐶 is ﬁxed to 1*
for simplicity.

0 5 10 15 20 25 30

10^{−4}
10^{−3}
10^{−2}
10^{−1}
10^{0}

Average SNR per Hop (dB)

Average BER

RF(Rayleigh) / FSO(strong) RF(Rayleigh) / FSO(moderate) RF(Rayleigh) / RF(Rayleigh) Monte−Carlo simulation

Figure 2. Average BER of an asymmetric RF/FSO dual-hop relaying
systems.*𝛼 = 4.2 / 𝛽 = 1.4 (a strong FSO regime), 𝛼 = 4.0 / 𝛽 = 1.9 (a*
moderate FSO regime),*𝐶 = 1. BPSK modulation (𝛿 = 2).*

Fig. 2 shows the average BER performance of the asym-
metric RF/FSO dual-hop relaying system given in (16),
where we consider BPSK modulation (*𝛿 = 2) for simplicity.*

The average BER performance of the conventional RF/RF links [1] is also evaluated for comparison. According to the ﬁgure, the RF/FSO links appear to have worse performance than the RF/RF links, and the RF/FSO (Gamma-Gamma fading with a moderate FSO regime) links clearly show bet- ter performance than the RF/FSO (Gamma-Gamma fading with a strong FSO regime) links. The RF/FSO (Gamma- Gamma fading with a strong turbulence regime) links have the SNR loss of 2 dB compared to the RF/RF links in

the low SNR. The RF/FSO (Gamma-Gamma fading with a moderate turbulence regime) links have the SNR loss of 1 dB compared to the RF/RF links in the low SNR.

This is due to the fact that the FSO channel has signiﬁcant performance degradation due to the atmospheric turbulence.

It is observed, however, that the asymmetric RF/FSO relay- ing system (Gamma-Gamma fading with a moderate FSO regime) shows very similar performance in average BER to the RF/RF relaying system over the SNR of 25 dB. It is clearly seen that the analysis results are in a good agreement with the Monte-Carlo simulation results.

0 10 20 30 40 50

0 2 4 6 8 10 12 14 16 18

Average SNR per Hop (dB)

Ergodic Capacity (bits/s/Hz)

[Upper bound] RF(Rayleigh) / FSO(strong) [Upper bound] RF(Rayleigh) / FSO(moderate) [Monte−Carlo sim.] RF(Rayleigh) / FSO(strong) [Monte−Carlo sim.] RF(Rayleigh) / FSO(moderate) [Upper bound] RF(Rayleigh) / RF(Rayleigh)

Figure 3. Ergodic capacity of an asymmetric RF/FSO dual-hop relaying
systems.*𝛼 = 4.2 / 𝛽 = 1.4 (a strong FSO regime), 𝛼 = 4.0 / 𝛽 = 1.9 (a*
moderate FSO regime),*𝐶 = 1.*

Fig. 3 shows our analytical ergodic capacity upper bound performance, given in (39), and the Monte-Carlo Simulated ergodic capacity performance for the asymmetric RF/FSO dual-hop relaying system. It is seen in the ﬁgure that the ergodic capacity upper bound of the RF/FSO (Gamma- Gamma fading with a strong turbulence regime) links is very similar to that of the RF/FSO (Gamma-Gamma fading with a moderate turbulence regime) links. It is also observed that the Monte-Carlo simulation results of the ergodic capacities of the RF/FSO (Gamma-Gamma fading with a strong turbu- lence regime) links and the RF/FSO (Gamma-Gamma fading with a moderate turbulence regime) links are very similar.

It is clearly seen in the ﬁgure that our ergodic capacity upper bound has about 1 bits/s/Hz gap with the ergodic capacity obtained by Monte-Carlo simulation. In this ﬁgure, the ergodic capacity upper bound performance of the con- ventional RF/RF links [1] is also evaluated for comparison.

According to the ﬁgure, the asymmetric RF/FSO relaying system shows slightly worse performance in ergodic capacity upper bound than RF/RF relaying system in the low SNR.

However, the asymmetric RF/FSO relaying system shows very similar performance in ergodic capacity upper bound

to the RF/RF relaying system over the SNR of 20 dB.

VI. CONCLUSION

In this paper, we, for the ﬁrst time, have studied the performance of a dual-hop relaying system composed of asymmetric RF and FSO links. We considered an asym- metric AF relay which converts the received RF signal into an optical signal using the SIM scheme. We have derived the average probability of error and ergodic capacity upper bound of the RF/FSO relaying system in closed- forms. According to the analysis results, the asymmetric RF/FSO relaying system shows slightly worse performance in average probability of error and ergodic capacity upper bound than the RF/RF relaying system in the low SNR.

Over the SNR of 20 dB, however, the asymmetric RF/FSO relaying system shows very similar performance in average probability of error and ergodic capacity upper bound to the RF/RF relaying system.

ACKNOWLEDGMENT

This work was supported by Dual Use Technology Pro- gram

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