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

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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

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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

𝑟1= 𝛼1𝑠 + 𝑛1 (1)

where𝛼1is the fading amplitude of Rayleigh fading channel for the S-R link, and𝑛1 is an additive white Gaussian noise (AWGN) with the power spectral density of 𝑁01. 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

𝑠𝑜𝑝𝑡= 𝐺(1 + 𝜂𝑟1) (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

𝑟2= 𝐼𝐺[1 + 𝜂(𝛼1𝑠 + 𝑛1)] + 𝑛2 (3) where𝐼 is a stationary random variable following Gamma- Gamma distribution for the FSO link, and𝑛2 is an AWGN with the power spectral density of 𝑁02. When the DC component is filtered out at the destination, the received signal can be

𝑟2= 𝐼𝐺𝜂(𝛼1𝑠 + 𝑛1) + 𝑛2 (4) Thus, the overall signal-to-noise ratio (SNR) at the desti- nation can be expressed as

𝛾 = 𝐼2𝐺2𝜂2𝛼21𝑃1

𝐼2𝐺2𝜂2𝑁01+ 𝑁02 =

𝛼21𝑃1

𝑁01

𝜂2𝐼2𝑃2 𝑁02

𝜂2𝐼2𝑃2

𝑁02 + 𝐺2𝑃𝑁201

(5) where𝑃1and𝑃2 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 let𝐶 = 𝑃2/(𝐺2𝑁01), then, (5) can be rewritten as

𝛾 = 𝛾1𝛾2

𝛾2+ 𝐶 (6)

where both 𝛾1 = 𝛼𝑁2101𝑃1 and 𝛾2 = 𝜂2𝑁𝐼202𝑃2 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

𝑓𝛾1(𝛾1) = 1

𝛾1𝑒−𝛾1𝛾1 (7)

where 𝛾1 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

𝑓𝛾2(𝛾2) = (𝛼𝛽)(𝛼+𝛽)/2𝛾2(𝛼+𝛽)/4−1 Γ(𝛼)Γ(𝛽)𝛾2(𝛼+𝛽)/4 𝐾𝛼−𝛽

( 2

𝛼𝛽

𝛾2

𝛾2

)

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where 𝛾2 is the average SNR of the R-D link, 𝐾𝑎(⋅) is the modified Bessel function of the second kind of order𝑎, and the parameters𝛼 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

𝑃𝑒=

0 𝑃 (𝑒∣𝛾)𝑓𝛾(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]

𝑃𝑒= 1 2𝜋

0 𝐹𝛾

(𝑡2 𝛿

)

𝑒𝑡22𝑑𝑡, (11) where𝐹𝛾(𝛾) is the cumulative distribution function (CDF) of the random variable𝛾.

After the variable change of 𝑥 = 𝑡2, the average proba- bility of error can be given by

𝑃𝑒= 1 2𝜋

0 𝐹𝛾(𝑥 𝛿

)𝑒𝑥2 𝑑𝑥

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

𝑃𝑜𝑢𝑡 = 1 − 𝑒𝛾𝑡ℎ𝛾1 (𝛼𝛽)(𝛼+𝛽)/2

Γ(𝛼)Γ(𝛽)𝛾2(𝛼+𝛽)/4𝐸(𝛾𝑡ℎ), (13) where the integral function𝐸(𝛾𝑡ℎ) is given by

𝐸(𝛾𝑡ℎ) = 1 4𝜋

(𝐶𝛾𝑡ℎ

𝛾1

)(𝛼+𝛽)/4

× 𝐺5005

(

(𝛼𝛽)2 𝐶𝛾𝑡ℎ

16𝛾1𝛾2

⏐⏐

⏐⏐

𝛼−𝛽4 ,𝛼−𝛽+24 ,𝛽−𝛼4 ,𝛽−𝛼+24 ,−𝛼+𝛽4

) . (14) Since the CDF can be obtained from the outage proba- bility by the variable change from𝛾𝑡ℎ to𝛾, 𝐹𝛾(𝛾) can be expressed as

𝐹𝛾(𝛾) = 1 − 𝑒𝛾1𝛾 (𝛼𝛽)(𝛼+𝛽)/2

Γ(𝛼)Γ(𝛽)𝛾2(𝛼+𝛽)/4𝐸(𝛾). (15)

Substituting (15) to (12) with a variable change from 𝛾 to𝑥/𝛿, the average probability of error can be expressed as 𝑃𝑒= 𝐿 − 𝐻1(𝛼, 𝛽)𝐻2(𝛼, 𝛽), (16) where

𝐿 = 1 2

2𝜋

0

𝑒−𝑥/2

𝑥 𝑑𝑥 (17)

and

𝐻1(𝛼, 𝛽) = 1 8

2𝜋

(𝛼𝛽)(𝛼+𝛽)/2(

𝛿𝛾𝐶1𝛾2

)(𝛼+𝛽)/4

𝜋Γ(𝛼)Γ(𝛽) (18)

and

𝐻2(𝛼, 𝛽) =

0 𝑥−(1/2−(𝛼+𝛽)/4)𝑒( 1

𝛿𝛾1+12)

𝑥× 𝐺5005

(

(𝛼𝛽)2 𝐶𝑥 16𝛿𝛾1𝛾2

⏐⏐

⏐⏐

𝛼−𝛽4 ,𝛼−𝛽+24 ,𝛽−𝛼4 ,𝛽−𝛼+24 ,−𝛼+𝛽4

) 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

𝐻2(𝛼, 𝛽) = ( 1

𝛿𝛾1 +1 2

)−(𝛼+𝛽)/4−1/2

× 𝐺5115

𝐶(𝛼𝛽)

2

16𝛿𝛾1𝛾2

𝛿𝛾11 + 12

⏐⏐

⏐⏐

1/2−(𝛼+𝛽)/4

𝛼−𝛽4 ,𝛼−𝛽+24 ,𝛽−𝛼4 ,𝛽−𝛼+24 ,−𝛼+𝛽4

⎠ (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

𝐶𝑒𝑟𝑔 = 𝐸[𝑙𝑜𝑔2(1 + 𝛾)] (22) Since𝑙𝑜𝑔(⋅) is a concave function, using Jensen’s inequal- ity [15]–[17], the upper bound of the ergodic capacity can be expressed as

𝐶𝑒𝑟𝑔 ≤ 𝑙𝑜𝑔2(1 + 𝐸[𝛾]) (23) where the mean value 𝐸[𝛾] can be written as

𝐸[𝛾] =

0 𝛾𝑓(𝛾)𝑑𝛾. (24)

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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)𝛾2(𝛼+𝛽)/4

(𝐶 𝛾1

)(𝛼+𝛽)/4

× [𝐵1(𝛾) + 𝐵2(𝛾) + 𝐵3(𝛾)], (25) where

𝐵1(𝛾) = 𝛼 + 𝛽

4 𝛾(𝛼+𝛽)/4−1𝑒𝛾1𝛾

× 𝐺5005 (

(𝛼𝛽)2 𝐶𝛾 16𝛾1𝛾2

⏐⏐

⏐⏐

𝛼−𝛽4 ,𝛼−𝛽+24 ,𝛽−𝛼4 ,𝛽−𝛼+24 ,−𝛼+𝛽4

) , (26)

𝐵2(𝛾) = −𝛾(𝛼+𝛽)/4𝛾1−1𝑒𝛾1𝛾

× 𝐺5005

(

(𝛼𝛽)2 𝐶𝛾 16𝛾1𝛾2

⏐⏐

⏐⏐

𝛼−𝛽4 ,𝛼−𝛽+24 ,𝛽−𝛼4 ,𝛽−𝛼+24 ,−𝛼+𝛽4

) , (27)

𝐵3(𝛾) = 𝛾(𝛼+𝛽)/4𝑒𝛾1𝛾

× 𝑑 𝑑𝛾𝐺5005

(

(𝛼𝛽)2 𝐶𝛾 16𝛾1𝛾2

⏐⏐

⏐⏐

𝛼−𝛽4 ,𝛼−𝛽+24 ,𝛽−𝛼4 ,𝛽−𝛼+24 ,−𝛼+𝛽4

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

𝐸[𝛾] = (𝛼𝛽)(𝛼+𝛽)/2 4𝜋Γ(𝛼)Γ(𝛽)𝛾2(𝛼+𝛽)/4

(𝐶 𝛾1

)(𝛼+𝛽)/4

× 𝐴(𝛼, 𝛽, 𝛾1, 𝛾2), (29) where

𝐴(𝛼, 𝛽, 𝛾1, 𝛾2) = 𝐴1(𝛼, 𝛽, 𝛾1, 𝛾2) + 𝐴2(𝛼, 𝛽, 𝛾1, 𝛾2) + 𝐴3(𝛼, 𝛽, 𝛾1, 𝛾2), (30)

𝐴1(𝛼, 𝛽, 𝛾1, 𝛾2) = −

0

𝛼 + 𝛽

4 𝛾(𝛼+𝛽)/4𝑒𝛾1𝛾

×𝐺5005 (

(𝛼𝛽)2 𝐶𝛾 16𝛾1𝛾2

⏐⏐

⏐⏐

𝛼−𝛽4 ,𝛼−𝛽+24 ,𝛽−𝛼4 ,𝛽−𝛼+24 ,−𝛼+𝛽4

) 𝑑𝛾,

(31)

𝐴2(𝛼, 𝛽, 𝛾1, 𝛾2) =

0 𝛾(𝛼+𝛽)/4+1𝛾1−1𝑒𝛾1𝛾

×𝐺5005 (

(𝛼𝛽)2 𝐶𝛾 16𝛾1𝛾2

⏐⏐

⏐⏐

𝛼−𝛽4 ,𝛼−𝛽+24 ,𝛽−𝛼4 ,𝛽−𝛼+24 ,−𝛼+𝛽4

) 𝑑𝛾,

(32)

𝐴3(𝛼, 𝛽, 𝛾1, 𝛾2) = −

0 𝛾(𝛼+𝛽)/4+1𝑒𝛾1𝛾

× 𝑑 𝑑𝛾𝐺5005

(

(𝛼𝛽)2 𝐶𝛾 16𝛾1𝛾2

⏐⏐

⏐⏐

𝛼−𝛽4 ,𝛼−𝛽+24 ,𝛽−𝛼4 ,𝛽−𝛼+24 ,−𝛼+𝛽4

) 𝑑𝛾.

(33) Applying the integration by parts to (33), 𝐴3(𝛼, 𝛽, 𝛾1, 𝛾2) can be converted to

𝐴3(𝛼, 𝛽, 𝛾1, 𝛾2) =

0

𝑑 𝑑𝛾

(𝛾(𝛼+𝛽)/4+1𝑒𝛾1𝛾)

×𝐺5005 (

(𝛼𝛽)2 𝐶𝛾 16𝛾1𝛾2

⏐⏐

⏐⏐

𝛼−𝛽4 ,𝛼−𝛽+24 ,𝛽−𝛼4 ,𝛽−𝛼+24 ,−𝛼+𝛽4

) 𝑑𝛾.

(34) Differentiating the first term in (34), 𝐴3(𝛼, 𝛽, 𝛾1, 𝛾2) can be rewritten in terms of 𝐴1(𝛼, 𝛽, 𝛾1, 𝛾2) and 𝐴2(𝛼, 𝛽, 𝛾1, 𝛾2) as following

𝐴3(𝛼, 𝛽, 𝛾1, 𝛾2) = −𝐴1(𝛼, 𝛽, 𝛾1, 𝛾2) − 𝐴2(𝛼, 𝛽, 𝛾1, 𝛾2) +

0 𝛾(𝛼+𝛽)/4𝑒𝛾1𝛾

×𝐺5005 (

(𝛼𝛽)2 𝐶𝛾 16𝛾1𝛾2

⏐⏐

⏐⏐

𝛼−𝛽4 ,𝛼−𝛽+24 ,𝛽−𝛼4 ,𝛽−𝛼+24 ,−𝛼+𝛽4

) 𝑑𝛾.

(35) Substituting (35) to (30), 𝐴(𝛼, 𝛽, 𝛾1, 𝛾2) can be rewritten as

𝐴(𝛼, 𝛽, 𝛾1, 𝛾2) =

0 𝛾(𝛼+𝛽)/4𝑒𝛾1𝛾

×𝐺5005

(

(𝛼𝛽)2 𝐶𝛾 16𝛾1𝛾2

⏐⏐

⏐⏐

𝛼−𝛽4 ,𝛼−𝛽+24 ,𝛽−𝛼4 ,𝛽−𝛼+24 ,−𝛼+𝛽4

) 𝑑𝛾.

(36) With the help of the equation (7.813-1) of [19], 𝐴(𝛼, 𝛽, 𝛾1, 𝛾2) can be obtained as

𝐴(𝛼, 𝛽, 𝛾1, 𝛾2) = 𝛾(𝛼+𝛽)/4+1

× 𝐺5115 (

(𝛼𝛽)2 𝐶 16𝛾2

⏐⏐

⏐⏐

𝛼+𝛽4

𝛼−𝛽4 ,𝛼−𝛽+24 ,𝛽−𝛼4 ,𝛽−𝛼+24 ,−𝛼+𝛽4

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

𝐸[𝛾] = (𝛼𝛽)(𝛼+𝛽)/2𝛾1

4𝜋Γ(𝛼)Γ(𝛽) (𝐶

𝛾2

)(𝛼+𝛽)/4

× 𝐺5115 (

(𝛼𝛽)2 𝐶 16𝛾2

⏐⏐

⏐⏐

𝛼+𝛽4

𝛼−𝛽4 ,𝛼−𝛽+24 ,𝛽−𝛼4 ,𝛽−𝛼+24 ,−𝛼+𝛽4

) . (38)

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

𝐶𝑒𝑟𝑔≤ 𝑙𝑜𝑔2

{

1 + (𝛼𝛽)(𝛼+𝛽)/2𝛾1

4𝜋Γ(𝛼)Γ(𝛽) (𝐶

𝛾2

)(𝛼+𝛽)/4

× 𝐺5115

(

(𝛼𝛽)2 𝐶 16𝛾2

⏐⏐

⏐⏐

𝛼+𝛽4

𝛼−𝛽4 ,𝛼−𝛽+24 ,𝛽−𝛼4 ,𝛽−𝛼+24 ,−𝛼+𝛽4

) } . (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 and𝛼 = 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 10−1 100

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

(6)

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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