A hybrid free space optical-millimeter wave cooperative system

Optics Communications 453 (2019) 124400

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Optics Communications journal homepage: www.elsevier.com/locate/optcom

A hybrid free space optical-millimeter wave cooperative system Saud Althunibat a ,∗, Osamah S. Badarneh b , Raed Mesleh b , Khalid Qaraqe c a

Communications Engineering Department, Al-Hussein Bin Talal University, Ma’an, Jordan Electrical and Communications Engineering Department, School of Electrical Engineering and Information Technology, German-Jordanian University, Amman Madaba Street, P.O. Box 35247, Amman 11180, Jordan c Department of Electrical and Computer Engineering, Texas, A&M University at Qatar, Doha, Qatar b

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Keywords: Cooperative communication Gamma–gamma fading Hybrid FSO–mmWave Nakagami fading

ABSTRACT In this paper, a hybrid free space optical (FSO)–millimeter wave (mmWave) cooperative system is proposed. Specifically, a cooperative dual-hop decode-and-forward (DF) relaying system coexists with a direct communication link and two scenarios are considered. In the first scenario, the direct link is used for FSO transmission while the DF dual-hop relaying system is used for mmWave transmission. On the contrary, in the second scenario, the direct link is used for mmWave communication whereas the relaying link is used for FSO communication. In both scenarios, the mmWave link experiences Nakagami-𝑚 fading while the FSO link suffers gamma–gamma turbulence fading. At the destination, the two signals, i.e., the mmWave and the FSO signals, are jointly detected using an optimum maximum likelihood receiver. To investigate the performance of the proposed systems, novel analytical expressions for the overall average bit error rate (BER) and outage probability are derived. The impact of different system and channel parameters on the overall average BER is investigated. Additionally, the performance of the proposed systems is compared with other setups where only direct mmWave or FSO link exists. Monte Carlo simulation results are provided to corroborate the accuracy of the analysis.

1. Introduction The recent spread of cellular systems (smart sensors, mobile phones, base stations, surveillance devices, internet of things, etc.) increased the complexity of processing algorithms and pushed the limits of existing technologies. To cope up with this demand, various technologies were considered to increase the capacity of wireless cellular networks. A promising solution is foreseen in free-space optical (FSO), radio frequency (RF), and cooperative communication along with many other technologies [1–4]. Yet, RF and FSO systems are shown to have complementary properties especially in different weather conditions. As such, several studies have considered hybrid FSO/RF communication systems to benefit from their inherent advantages. Other motivations include variant system requirements by both systems such as line of sight, pointing errors, shadowing impact along with many others. Hence, hybrid FSO/mmWave system is considered in the backhaul cellular networks, where reliable connection is always required and any loss in connection significantly degrades the quality of service for all users. Dual-hop cooperative systems with amplify-and-forward (AF) relay were considered in [5–12], where the first hop is an RF link and the second hop is a FSO link. In [5], the outage probability of mixed

RF/FSO system considering Rayleigh fading in the RF link and Gamma– Gamma (GG) fading over the FSO link was derived. Later, the impact of pointing errors on the average bit error rate (BER) was investigated in [6]. The performance of mixed RF/FSO systems considering different fading channel models was analyzed in [7–9]. The authors in [10], derived closed-form expression for the outage probability over mixed RF/FSO dual-hop communication system, where the RF and FSO links are, respectively, modeled as Rayleigh fading and 𝑀-distribution fading. In [11], the authors derived several performance metrics over a mixed RF/FSO variable-gain AF relaying system. The impact of outdated channel state information on the system performance was studied in [12]. The above studies consider AF relaying system while [13,14] extended the analysis by assuming DF relaying systems. In [13], the authors analyzed the performance of mixed RF/FSO over Nakagami𝑚/GG channels in terms of probability of error and average channel capacity. The multi-user case is considered in [14], where a set of 𝐿 users (sources) transmit RF signals toward a multi-antenna DF relay. The relay re-transmits the detected symbol towards the destination via FSO link over 𝐿 consecutive time slots. A new mixed RF/FSO system is proposed in [15], where a cooperative scenario comprising an RF direct link and a mixed RF/FSO

∗ Corresponding author. E-mail addresses: [email protected] (S. Althunibat), [email protected] (O.S. Badarneh), [email protected] (R. Mesleh), [email protected] (K. Qaraqe).

https://doi.org/10.1016/j.optcom.2019.124400 Received 10 May 2019; Received in revised form 29 July 2019; Accepted 15 August 2019 Available online 17 August 2019 0030-4018/© 2019 Elsevier B.V. All rights reserved.

S. Althunibat, O.S. Badarneh, R. Mesleh et al.

Optics Communications 453 (2019) 124400

the first scenario, the direct link is used for FSO transmission while the dual-hop link is used for mmWave transmission, as depicted in Fig. 1. Yet, mmWave transmission is deployed in the direct link and FSO communication is considered in the dual-hop in the second scenario, as illustrated in Fig. 2. Before transmission, the transmitted symbols over the FSO link are generated using pulse amplitude modulation (PAM) constellation with order 𝑀, while phase shift keying (PSK) with order 𝑀 is used to generate the symbols over the mmWave link. The modulated symbols are then transmitted over the two parallel links where Nakagami-𝑚 and GG turbulence fading channel are respectively assumed for mmWave and FSO links. In the GG channel mode, the normalized irradiance is represented by a product of two Gamma random variables, which represents the small- and large-scale atmospheric turbulence [20–22]. As such, the probability density function (PDF) of the normalized irradiance, denoted by ℎ𝑎 , is given as [23,24] ( √ ) ( ) 2(𝛼𝛽)(𝛼+𝛽)∕2 𝛼+𝛽 −1 𝑓ℎ𝑎 ℎ𝑎 = 𝐾𝛼−𝛽 2 𝛼𝛽ℎ𝑎 , ℎ𝑎 > 0 ℎ𝑎 2 (1) 𝛤 (𝛼) 𝛤 (𝛽)

dual-hop system is assumed in the cooperative AF link. The outage probability in the presence of interference signals was obtained in a closed-form. In addition, the average BER for binary modulation was derived. Two parallel FSO and RF dual-hop system is considered in [16], where multiple relays participate in delivering the data towards the destination. Specifically, two parallel links exist between each relay and the source, and each relay and the destination. The RF link acts as a backup link that is activated once the FSO link is lost. Recently, research efforts have been directed towards considering millimeter wave (mmWave) technology in the hybrid RF/FSO systems to benefit from the promised huge bandwidth in both technologies. Also, atmospheric and weather conditions impact FSO and mmWave links antithetically and combining both in a hybrid system enhances the robustness against such affects. In [17] different practical switching mechanisms for parallel FSO/mmWave links are proposed. Specifically, single threshold and double threshold mechanisms were proposed to allow for a single active link. In single-threshold mechanism, FSO link is activated if its SNR is above a certain threshold. Otherwise, mmWave link is checked and activated if the SNR is above a predefined threshold. Yet, the double-threshold mechanism uses two thresholds for the FSO link, where the lower threshold is used to disconnect the FSO link while the upper threshold is used to re-activate the FSO link. As in the single-threshold mechanism, the RF link is checked once the FSO link is disconnected and used only if its SNR is above a predefined threshold. In the same context, the two parallel links are exploited to enhance the transmission diversity, where a bit-interleaved coded modulation scheme was proposed and analyzed in [18]. The performance of the parallel FSO–mmWave system is analyzed under two different combining rule; namely, selection combining and maximum ratio combining in [19]. Different from existing studies in the literature, hybrid FSO/ mmWave cooperative communication systems are proposed, analyzed and thoroughly discussed in this article. The system model considers a source and a destination communicating over a direct link and through a dual-hop cooperative DF relay. The cooperative link acts as an auxiliary link aiming to enhance the performance of the direct link by combining both inherent advantages of FSO and mmWave systems. The following two scenarios are considered and analyzed: (1) FSO/mmWave system, in which the direct link is a FSO and the cooperative link is mmWave communication, and (2) mmWave/FSO system, where the direct link is mmWave while the indirect link is a FSO. In the cooperative link, a DF relay node decodes the received data from the source and applies error detection techniques. If no error is detected, the relay forwards the received message to the destination. Both FSO and RF data are combined and jointly used to estimate the transmitted source message at the destination. In particular, a novel optimum joint mmWave/FSO maximum likelihood (ML) receiver is designed and analyzed. The overall average and asymptotic BER of the system is derived in closed form assuming Nakagami−𝑚 RF fading and GG FSO turbulent channels. Besides, the outage probability of the direct link both systems assuming fixed energy in the cooperative link is obtained. The derived BER expressions are corroborated through Monte-Carlo simulation results, where a close match is observed over wide range of system and channel parameters. The rest of this paper is organized as follows. The proposed system models are presented in Section 2. Detailed performance analysis are provided in Section 3. Section 4 is dedicated for simulation results and discussion. Finally, conclusions are drawn in Section 5.

where 𝐾𝑥 (⋅) denotes the modified Bessel function of the second kind of order 𝑥, 𝛼 and 𝛽 refer to the number of small and large scale eddies of the scattering environment, respectively. Notice that the calculation of both 𝛼 and 𝛽 depends on the wave number, diameter of the receiver’s aperture, path length and the refractive index 𝐶𝑛2 , as detailed in [18]. The PDF 𝑓ℎ𝑎 can be represented in terms of the Fox-H function using [25, eq. 2.9.19] as ( ) ( ) 𝜂 − | 2,0 𝑓ℎ𝑎 ℎ𝑎 = ℎ𝜇−1 𝐻0,2 𝑐 ℎ𝑎 | ( 𝛽−𝛼 ,1),( 𝛼−𝛽 ,1) , ℎ𝑎 > 0 (2) 𝑎 2 2 | 2 𝜇

2𝑐 . The FSO link is also subject and 𝜂 = 𝛤 (𝛼)𝛤 where 𝑐 = 𝛼𝛽, 𝜇 = 𝛼+𝛽 2 (𝛽) to pointing error where the radial displacement follows a Rayleigh distribution.1 As such, the PDF of the pointing error impairment, denoted by ℎ𝑝 , is given by [27, eq. 2.22]

𝑓ℎ𝑝 (ℎ𝑝 ) =

𝑧2 𝑧2 −1 ℎ𝑝 , 𝐴𝑧2

(3) 𝜔

where 𝐴 is the maximum fraction of the collected power, 𝑧 = 2𝜎𝑧𝑒𝑞 , 𝑧 𝜔𝑧𝑒𝑞 is the equivalent beamwidth, and 𝜎𝑧 is the jitters variance. The combined effect of both turbulence and pointing error, denoted by ℎ, with joint PDF expressed as 𝑓FSO (ℎ) =

∫ℎ𝑝

𝑓ℎ∕ℎ𝑝 (ℎ∕ℎ𝑝 )𝑓ℎ𝑝 (ℎ𝑝 ) ⋅ 𝑑ℎ𝑝 ,

where 𝑓ℎ∕ℎ𝑝 (ℎ∕ℎ𝑝 ) is given as ( ) ℎ 1 𝑓ℎ𝑎 . 𝑓ℎ∕ℎ𝑝 (ℎ∕ℎ𝑝 ) = ℎ𝑝 ℎ𝑝

0 ≤ ℎ𝑝 ≤ 𝐴,

(4)

(5)

Substituting (2) into (5) while using [25, eq. 2.1.3] and [28, eq. 2.25.2.2], (4) can be expressed as ( ) (1−𝜇+𝑧2 ,1) 𝑐 ℎ || 3,0 (6) 𝑓FSO (ℎ) = 𝜁 ℎ𝜇−1 𝐻1,3 2 −𝜇,1),( 𝛽−𝛼 ,1),( 𝛼−𝛽 ,1) , | (𝑧 𝐴| 2 2 2

𝜂𝑧 where 𝜁 = 2𝐴 𝜇. The mmWave channel is modeled in 5G by the 3GPP model for beyond 6 GHz [29], which is very convoluted and mathematically intractable [30]. Thus, approximate channel models have been used in analytical studies in the literature, and the most commonly considered approximation is Nakagami-𝑚 fading channel. Hence, this study follows similar approximation and consider Nakagami-𝑚 channel to approximate the small-scale propagation effects in mmWave links [31], where the envelope PDF is given by

2. System model

𝑓Naka (𝛾) =

Consider a communication system that consists of a source node 𝑆, a relay node 𝑅 and a destination node 𝐷. In this system, the data reaches the destination node through two independent parallel links. The first link is the direct link between 𝑆 and 𝐷, while the second link is a dualhop DF relaying system. In this study, two scenarios are considered. In

2𝑚𝑚 2𝑚−1 −𝑚𝛾 2 𝛾 𝑒 , 𝛤 (𝑚)

(7)

1 Note that the considered pointing error model is zero boresight error model. However, the nonzero boresight error (i.e., Beckmann model) can be treated by the same pdf given in (3) using the approximation provided in [26, eq. 11]

2

S. Althunibat, O.S. Badarneh, R. Mesleh et al.

Optics Communications 453 (2019) 124400

destination, if it is correctly detected. Hence, the received signal at the destination via the cooperative link, 𝑦𝑟𝑑 , is written as {√ 𝑝𝑟𝑑 𝑓 𝑠 + 𝑛, if 𝑠̂ = 𝑠 𝑦𝑟𝑑 = (14) 𝑛, if 𝑠̂ ≠ 𝑠,

and the PDF of its squared envelop follows a Gamma distribution given as 𝑚𝑚 𝑚−1 −𝑚𝛾 𝑓Gam (𝛾) = 𝛾 𝑒 . (8) 𝛤 (𝑚) In the dual-hop cooperative link, a DF relay is used to detect the transmitted symbols from the source. The relay node forwards the received data to the destination only if it is correctly detected. As such, efficient error detection techniques must be employed at the DF relay node [32–34]. Such assumption is widely adopted in the literature and represents an upper bound on the true performance [33–39]. In what follows, the aforementioned scenarios are discussed in more details. Specifically, analytical expressions for the overall average BER are derived. In the analysis, the direct channel between 𝑆-𝐷 is denoted by ℎ, while the channel between 𝑆-𝑅 is denoted by 𝑔 and the channel between 𝑅-𝐷 is denoted by 𝑓 .

with 𝑛 denoting the AWGN at the destination input with zero mean and 𝜎 2 variance, 𝑓 has similar statistics as 𝑔, and 𝑝𝑟𝑑 is the received RF power at the destination given by 𝑝𝑟𝑑 = 𝑝𝑟 𝐿𝑒𝑟𝑑 , where 𝑝𝑟 is the transmit RF power from the relay and 𝐿𝑒𝑟𝑑 is the path loss of the relay–destination mmWave link given as in (12) (with replacing 𝑑𝑠𝑟 by 𝑑𝑟𝑑 ). At the destination, the ML decoder is used to recover the transmitted signal by the source node 𝑆 as, [ ] √ | |2 | |2 𝑞̂ = arg min (15) |𝑦𝑠𝑑 − 𝜀𝜌𝑠𝑑 ℎ𝑥𝑞 | + |𝑦𝑟𝑑 − 𝑝𝑟𝑑 𝑓 𝑠𝑞 | , | | | 𝑞={1∶2𝑀} | where the transmitted pair (𝑥𝑞 , 𝑠𝑞 ) is defined as { (𝑥𝓁 , 𝑠𝓁 ), if 𝑞 ≤ 𝑀 (𝑥𝑞 , 𝑠𝑞 ) = (𝑥𝓁 , 0), if 𝑞 > 𝑀.

2.1. FSO/mmWave system: Direct FSO link with auxiliary cooperative mmwave link In this system model, data transmission over the direct link between 𝑆 and 𝐷 is performed using FSO communication. Yet, the dual-hop cooperative link is accomplished through mmWave transmission. Accordingly, the 𝑀-PAM optical signal, denoted by 𝑥, is transmitted directly to the destination node 𝐷, while the 𝑀-PSK electrical signal, denoted by 𝑠, is transmitted to the destination node 𝐷 through the relay node. The received signal at the destination via the direct link, 𝑦𝑠𝑑 , is expressed as

where 𝑥𝓁 and 𝑠𝓁 (1 ≤ 𝓁 ≤ 𝑀) represent the 𝓁𝑡ℎ symbols of the 𝑀-PAM constellation and the 𝑀-PSK constellation, respectively. Notice that the above formula is written in order to accommodate the case when no signal is transmitted over the 𝑅-𝐷 link. 2.2. mmWave/FSO system: Direct mmWave link with auxiliary FSO cooperative link The mmWave/FSO system employs an opposite scenario where the mmWave signal is now transmitted over the direct link whereas the FSO signal is sent over the cooperative link. Thus, the received signal, at the destination, via the direct link is given as √ 𝑦𝑠𝑑 = 𝑝𝑠𝑑 ℎ𝑠 + 𝑤, (17)

(9)

𝑦𝑠𝑑 = 𝜀𝜌𝑠𝑑 ℎ𝑥 + 𝑤,

where 𝜀 is the optical-to-electrical conversion efficiency, 𝑤 is an additive white Gaussian noise (AWGN) representing shot noise and background noise radiations seen at the destination input, ℎ is the FSO fading channel with the PDF given in (6), and 𝜌𝑠𝑑 denotes the received optical power at the destination from the source which is given as 𝜌𝑠𝑑 = 𝜌𝑠 𝐿𝑜𝑠𝑑 , where 𝜌𝑠 is the optical transmit source power and 𝐿𝑜𝑠𝑑 is the path loss of the source–destination FSO link given as [40] 𝐿𝑜𝑠𝑑 =

𝜋𝑇 2 𝑒−𝜑1 𝑑𝑠𝑑 , 4(𝜙 𝑑𝑠𝑑 )2

with ℎ being a Nakagami−𝑚 fading channel, 𝑤 denotes a complex AWGN with zero mean and 𝜎 2 variance, and 𝑝𝑠𝑑 is the received RF power at the destination given as 𝑝𝑠𝑑 = 𝑝𝑠 𝐿𝑒𝑠𝑑 , where 𝑝𝑠 is the transmitted RF power from the source and 𝐿𝑒𝑠𝑑 is the path loss of the source–destination mmWave link given as in (12) (with replacing 𝑑𝑠𝑟 by 𝑑𝑠𝑑 ) At the relay 𝑅, the received signal is given as

(10)

where 𝑇 is the destination aperture diameter, 𝑑𝑠𝑑 is the distance between the source and the destination, 𝜙 is the transmit beam divergence, and 𝜑1 is a weather dependent coefficient. Similarly, the received signal at the relay 𝑅, 𝑦𝑠𝑟 , can be written as √ 𝑦𝑠𝑟 = 𝑝𝑠𝑟 𝑔𝑠 + 𝑧, (11)

𝑦𝑠𝑟 = 𝜀𝜌𝑠𝑟 𝑔𝑥 + 𝑧,

𝐿𝑒𝑠𝑑 =

(4𝜋𝑑𝑠𝑟 )2 (𝜑𝑜𝑥 𝑑𝑠𝑟 )(𝜑𝑟𝑎𝑖𝑛 𝑑𝑠𝑟 )

,

(18)

where 𝑔 denotes the FSO fading channel, 𝑧 is a real AWGN noise with zero mean and 𝜎 2 variance, and 𝜌𝑠𝑟 is the received optical power at the relay given as 𝜌𝑠𝑟 = 𝜌𝑠 𝐿𝑜𝑠𝑟 , where 𝐿𝑜𝑠𝑟 is the path loss of the source–destination FSO link given as in (10) (with replacing 𝑑𝑠𝑑 by 𝑑𝑠𝑟 ). The relay performs ML detection to retrieve the transmitted symbol 𝑥̂ as

where 𝑠 is the modulated 𝑀-PSK signal, 𝑧 is an AWGN with zero mean and variance 𝜎 2 , 𝑔 is a Nakagami fading channel with the PDF given in (7), and 𝑝𝑠𝑟 is the received RF power at the relay which is given as 𝑝𝑠𝑟 = 𝑝𝑠 𝐿𝑒𝑠𝑟 , with 𝑝𝑠 denoting the transmitted RF power, and 𝐿𝑒𝑠𝑟 is the path loss of the source–relay mmWave link given as [18] 𝐺𝑡 𝐺𝑟 𝜆2𝑒

(16)

2 𝑥̂ = arg min ||𝑦𝑠𝑟 − 𝜀𝜌𝑠𝑟 𝑔𝑥|| , 𝑥∈ 𝑀

(19)

where  𝑀 is the constellation of 𝑀-PAM symbols. Accordingly, the relay will forward 𝑥̂ only if it is identical to 𝑥. As such, the received signal, at the destination, via the second FSO hop is given as { 𝜀𝜌𝑟𝑑 𝑓 𝑥 + 𝑛, if 𝑥̂ = 𝑥 𝑦𝑟𝑑 = , (20) 𝑛, if 𝑥̂ ≠ 𝑥

(12)

where 𝐺𝑡 and 𝐺𝑟 are the gains of the transmit and receive antennas, respectively, 𝜆𝑒 is the wavelength of the mmWave signal, 𝑑𝑠𝑟 is the distance between the source and the relay, 𝜑𝑜𝑥 and 𝜑𝑟𝑎𝑖𝑛 are the attenuations caused by oxygen absorption and rain, respectively. At the relay node, the received signal is decoded using the ML rule as √ | |2 (13) 𝑠̂ = arg min |𝑦𝑠𝑟 − 𝑝𝑠𝑟 𝑔𝑠| , | 𝑠∈ 𝑀 |

where 𝑓 is the FSO fading channel, 𝑛 is a real AWGN noise with zero mean and 𝜎 2 variance, and 𝜌𝑟𝑑 denotes the received optical power at the destination given as 𝜌𝑟𝑑 = 𝜌𝑟 𝐿𝑜𝑟𝑑 , where 𝜌𝑟 is the transmit optical power from the relay and 𝐿𝑜𝑟𝑑 is the path loss of the relay–destination FSO link given as in (10) (with replacing 𝑑𝑠𝑑 by 𝑑𝑟𝑑 ). The detection process at the destination follows the following formula to decode the source signal ] [ √ |2 | |2 | (21) 𝑞̂ = arg min |𝑦𝑠𝑑 − 𝑝𝑠𝑑 ℎ𝑠𝑞 | + |𝑦𝑟𝑑 − 𝜖𝜌𝑟𝑑 𝑓 𝑥𝑞 | , | | | 𝑞∈{1∶2𝑀} |

where  𝑀 is the constellation of the 𝑀-PSK modulation. It is recalled that the relay has the capability of detecting any errors in the received symbols 𝑠̂ and forwards the received symbol, to the 3

S. Althunibat, O.S. Badarneh, R. Mesleh et al.

Optics Communications 453 (2019) 124400

Fig. 1. FSO/mmWave system: Direct FSO link with auxiliary mmWave cooperative link system model.

Fig. 2. mmWave/FSO system: Direct mmWave link with auxiliary FSO cooperative link system model.

where both (𝑠𝑞 , 𝑥𝑞 ) are defined as { (𝑠𝓁 , 𝑥𝓁 ), if 𝑞 ≤ 𝑀 (𝑠𝑞 , 𝑥𝑞 ) = (𝑠𝓁 , 0), if 𝑀 < 𝑞 ≤ 2𝑀,

transmission on the 𝑅 − 𝐷 link) is the error probability on the 𝑆 − 𝑅 link. As such, P𝑞 can be formulated as (𝑅) ) ⎧ 1 ∏𝑀 ( ⎪ 𝑀 𝑝≠𝑞 1 − PEP𝑝𝑞 , P𝑞 = ⎨ (𝑅) ∏𝑀 ⎪1 PEP𝑝𝑞 , ⎩ 𝑀 𝑝≠𝑞

(22)

where 𝑠𝓁 and 𝑥𝓁 are defined earlier below (16).

if 𝑞 ≤ 𝑀

(24)

if 𝑀 < 𝑞 ≤ 2𝑀,

(𝑅)

3. Performance analysis

where PEP𝑝𝑞 is the pairwise error probability between 𝑠𝑝 and 𝑠𝑞 at the (𝐷)

BER𝐷 =

2𝑀 ∑ 𝑞=1

P𝑞

2𝑀 ∑

𝜏𝑝𝑞

𝑝=1

log2 (𝑀)

(𝐷)

PEP𝑝𝑞 ,

3.1. Performance analysis of FSO/mmWave system (𝑅)

3.1.1. Derivation of PEP𝑝𝑞 (𝑅) PEP𝑝𝑞

is defined as the average probability that 𝑠𝑝 is detected at the relay given that 𝑠𝑞 is actually transmitted from the source. Using (11)

(23)

(𝑅)

and (13), the average PEP, PEP𝑝𝑞 , can be formulated as { } √ √ (𝑅) | |2 | |2 PEP𝑝𝑞 = Pr |𝑦𝑠𝑟 − 𝑝𝑠𝑟 𝑔𝑠𝑞 | < |𝑦𝑠𝑟 − 𝑝𝑠𝑟 𝑔𝑠𝑝 | , | | | |

where P𝑞 is the probability that the 𝑞th pair (𝑥𝑞 , 𝑠𝑞 ), as given in (16), is transmitted, 𝜏𝑝𝑞 is the hamming distance between the 𝑞th and 𝑝th (𝐷) PEP𝑝𝑞

transmitted bit blocks, and is the average pairwise error probability defined as the probability that the pair (𝑥𝑝 , 𝑠𝑝 ) is detected at the destination given that the pair (𝑥𝑞 , 𝑠𝑞 ) is actually transmitted. Note that the formula in (23) is valid for both systems (i.e., mmWave/FSO and FSO/mmWave systems), where the differences lie in the computation of both P𝑞 and

(𝑅)

relay. In the following, both probabilities PEP𝑝𝑞 and PEP𝑝𝑞 are derived for each system.

In this section, novel analytical expressions for the overall average and asymptotic BER of the proposed systems are derived. In addition, the outage probability of both communication scenarios assuming fixed power in the cooperative link is obtained. Generally, the average BER at the destination is usually expressed as

(25)

which can be simplified after performing algebraic manipulations as { } )|2 (√ ( ) ) 𝑝𝑠𝑟 | ( (𝑅) PEP𝑝𝑞 = Pr 𝑝𝑠𝑟 𝑔 ∗ 𝑠𝑝 − 𝑠𝑞 𝑧 , (26) |𝑔 𝑠𝑝 − 𝑠𝑞 | < ℜ | 2 | where ℜ(⋅) denotes the real part. (√ ( ) ) Note that, for a given 𝑔, ℜ 𝑝𝑠𝑟 𝑔 ∗ 𝑠𝑝 − 𝑠𝑞 𝑧 is actually a Gaus)|2 𝑝 | ( sian random variable with zero mean and variance 2𝑠𝑟 |𝑔 𝑠𝑝 − 𝑠𝑞 | 𝜎 2 . | | (𝑅) Thus, the conditional PEP on 𝑔, denoted by PEP𝑝𝑞 is expressed in terms of the Q-function as √ √ ⎛ 𝑝 |𝑔𝛥 |2 ⎞ ⎛√ √ 𝑝 |𝑔|2 |𝛥 |2 ⎞ 𝑠𝑟 𝑠 | | | 𝑠| ⎟ , (𝑅) ⎟ = 𝑄 ⎜√ 𝑠𝑟 PEP𝑝𝑞 = 𝑄 ⎜ (27) 2 ⎜ ⎟ ⎜ ⎟ 2𝜎 2𝜎𝑛2 ⎝ ⎠ ⎝ ⎠

(𝐷) PEP𝑝𝑞 .

The probability P𝑞 represents the probability that the transmitted pair (𝑥𝑞 , 𝑠𝑞 ) is sent over the 𝑆 − 𝐷 link and 𝑅 − 𝐷 link, respectively. Based on (16), there are two cases according to the value of 𝑞, and in both cases, P𝑞 depends on the performance of the 𝑆 − 𝑅 link. First, if 𝑞 ≤ 𝑀, the probability that 𝑥𝑞 = 𝑥𝓁 is transmitted from the source is 1 , while the probability that 𝑠𝑞 = 𝑠𝓁 is emitted from the relay towards 𝑀 the destination is actually equal to the complementary error probability on the 𝑆 − 𝑅 link. Second, if 𝑞 > 𝑀, the probability that the source 1 emits 𝑥𝑞 = 𝑥𝓁 is also 𝑀 , while the probability that 𝑠𝑞 = 0 (i.e., no

where 𝛥𝑠 = 𝑠𝑝 − 𝑠𝑞 . As stated earlier, for Nakagami fading channel, |𝑔|2 is a Gamma random variable with the PDF given in (8). Therefore, the 4

S. Althunibat, O.S. Badarneh, R. Mesleh et al.

Optics Communications 453 (2019) 124400

(𝑅) PEP𝑝𝑞 ,

Case I: For the first case, i.e., 𝑞 ≤ 𝑀 or 𝑝 ≤ 𝑀, taking the average of (36) over the PDFs of both links yields

unconditional average PEP, can be computed by averaging (27) over the channel PDF given in (8) as √ ⎛ 𝑝 𝛾|𝛥 |2 ⎞ ∞ (𝑅) 𝑚𝑚 𝑠𝑟 𝑠 ⎟ 𝑚−1 −𝑚𝛾 PEP𝑝𝑞 = 𝑄⎜ 𝛾 𝑒 ⋅ 𝑑𝛾. (28) ⎜ 𝛤 (𝑚) ∫0 2𝜎 2 ⎟ ⎝ ⎠ 𝑝𝑠𝑟 𝛾 |𝛥𝑠 | 2𝜎 2

Defining 𝑡 = (𝑅)

PEP𝑝𝑞 =

𝜆𝑚 𝛤 (𝑚) ∫0

where 𝜆 =

∞

=

2

𝑄

(29)

(𝐷)

2

𝜋∕2

𝑧 ) 𝜋∕2 (− (1∕𝜋) ∫0 𝑒 2 sin2 𝜃

=

𝜋∕2

𝜋(2𝜆 + 1)𝑚+1∕2

𝛤 (𝑚 + 1∕2) 𝛤 (𝑚 + 1)

(𝐷)

PEP𝑝𝑞 =

(31) 𝐼ℎ =

2 𝐹1

=𝜒

𝑚

𝑘=0

where 𝜒 =

1 2

) 2𝜆 , 2𝜆 + 1

√

(34)

PEP(𝐷) 𝑝𝑞

) (41)

𝑓Gam (𝛾) ⋅ 𝑑𝛾.

𝜔

1−𝜇 2

(42)

(43)

(44)

modified

which can be solved using [28, eq. 2.25.1.1] to yield ( ) | 𝜇 1 , ),(0,1),(1−𝜇+𝑧2 ,1) 𝜁 1 𝑐 || (0,1),(1− 4,2 2 2)( ) ( 𝐼ℎ = (√ )𝜇 𝐻4,5 √ | 2 𝛽−𝛼 𝛼−𝛽 2 𝐴 𝜔 || (𝑧 −𝜇,1), 2 ,1 , 2 ,1 ,(0,1),(0,1) 𝜔

(35)

where 𝛥𝑥 = 𝑥𝑝 − 𝑥𝑞 . It can be noted that for (𝑝 > 𝑀 and 𝑞 > 𝑀), the value of 𝛥𝑠 is 0. As such, PEP(𝐷) 𝑝𝑞 can be rewritten as (√ ) 2 2 ⎧ 𝜀2 𝜌𝑠𝑑 2 |ℎ|2 |𝛥𝑥 | +𝑝𝑟𝑑 |𝑓 |2 |𝛥𝑠 | , ⎪𝑄 2𝜎 2 ⎪ = ⎨ (√ ) 2 ⎪ 𝜀2 𝜌𝑠𝑑 2 |ℎ|2 |𝛥𝑥 | , ⎪𝑄 2𝜎 2 ⎩

4𝜎 2 sin2 𝜃

(40)

𝑓FSO (ℎ) ⋅ 𝑑ℎ,

where both Fox-H functions above can be [28, eq. 8.3.2.6] to yield ) (0,1) 𝜁 ∞ 1−𝜇 1,1 ( √ || 𝐼ℎ = 𝜔 2 𝐻1,2 ℎ 𝜔| ( 𝜇−1 , 1 ),(0,1) 2 ∫0 2 2 | ( ) (0,1),(1−𝜇+𝑧2 ,1) 𝑐 ℎ || 3,1 × 𝐻2,4 ⋅ 𝑑ℎ, 2 −𝜇,1),( 𝛽−𝛼 ,1),( 𝛼−𝛽 ,1),(0,1) | (𝑧 𝐴| 2 2

2 2 𝜀2 𝜌𝑠𝑑 2 ||ℎ𝛥𝑥 || + 𝑝𝑟𝑑 ||𝑓 𝛥𝑠 || ⎞ ⎟ ⎟ 2𝜎 2 ⎠ 2 2 𝜀2 𝜌𝑠𝑑 2 |ℎ|2 ||𝛥𝑥 || + 𝑝𝑟𝑑 |𝑓 |2 ||𝛥𝑠 || ⎞ ⎟, 2 ⎟ 2𝜎 ⎠

2 −𝑝𝑟𝑑 𝛾 ||𝛥𝑠 ||

ℎ𝜇−1 exp(−𝜔ℎ2 ) =

Starting from (15) and following similar procedures as in (25)–(27), the conditional PEP(𝐷) 𝑝𝑞 for given ℎ and 𝑔 can be obtained as

⎛ = 𝑄⎜ ⎜ ⎝

(

)

) ( √ | − 1,0 𝐻0,1 ℎ 𝜔| ( 𝜇−1 , 1 ) . 2 2 | 2 Now, substituting (43) into (42) yields 𝜁 ∞ 1−𝜇 1,0 ( √ | 𝜇−1− 1 ) 𝐼ℎ = 𝜔 2 𝐻0,1 ℎ 𝜔| ( , ) | 2 2 2 ∫0 ) ( 2 ,1) | (1−𝜇+𝑧 𝑐 ℎ| 3,0 × 𝐻1,3 ⋅ 𝑑ℎ, 𝛽−𝛼 𝛼−𝛽 2 | 𝐴 | (𝑧 −𝜇,1),( 2 ,1),( 2 ,1)

)

]𝑘 𝑚−1+𝑘 [ 1−𝜒 , 𝑘

∫0

2 −𝜀2 𝜌𝑠𝑑 2 |ℎ|2 ||𝛥𝑥 ||

4𝜎 2 sin2 𝜃

exp

(39)

𝐼ℎ 𝐼𝛾 ⋅ 𝑑𝜃,

2 𝜀2 𝜌 2 |𝛥 | where 𝜔 = 4𝜎𝑠𝑑2 sin 𝑥𝜃 . To solve the integral in (42), the exponential function ) ( ℎ𝜇−1 exp −𝜔ℎ2 is represented in terms of Fox-H function as [25, eq. 2.9.4]

( ) √ 1 1 − 1+2𝜆 .

√

𝑓FSO (ℎ) 𝑓Naka (𝛾) ⋅ 𝑑𝜃 𝑑ℎ 𝑑𝛾.

Now, using (6), the integral 𝐼ℎ can be rewritten as ( ) ∞ ( ) (1−𝜇+𝑧2 ,1) 𝑐 ℎ || 3,0 𝐼ℎ = 𝜁 exp −𝜔ℎ2 ℎ𝜇−1 𝐻1,3 ⋅ 𝑑ℎ, 𝛼−𝛽 𝛽−𝛼 2 | ∫0 𝐴 | (𝑧 −𝜇,1),( 2 ,1),( 2 ,1)

(𝐷)

⎛ = 𝑄⎜ ⎜ ⎝

𝜀2 𝜌𝑠𝑑 2 |ℎ|2 |𝛥𝑥 |2 +𝑝𝑟𝑑 𝛾 |𝛥𝑠 |2 4𝜎 2 sin2 𝜃

𝜋∕2

(

∞

𝐼𝛾 =

3.1.2. Derivation of PEP𝑝𝑞

PEP(𝐷) 𝑝𝑞

−

𝑒

∫0

1 𝜋 ∫0

and

( 1, 𝑚 + 1∕2; 𝑚 + 1;

𝜋∕2

exp

∫0

(32)

where 2 𝐹1 (⋅, ⋅; ⋅; ⋅) denotes the Gauss hypergeometric function. For positive integer 𝑚, (33) reduces to [43, eq. A13] ( 𝑚−1 ∑

∫0

∞

(33)

(𝑅) PEP𝑝𝑞

∞

where

( )𝑚 1 ⋅ 𝑑𝜃, 𝛿

2𝑚−1 𝜆𝑚

∞

After simple mathematical manipulation, (38) can be further simplified to

The latter integral can be solved in a closed-form using [43, eq. A8], where 𝑚 is a positive real number, as (𝑅)

1 𝜋 ∫0

(38)

∞

where the value of 𝛿 can be recovered as )𝑚 𝜋∕2 ( (𝑅) 2𝜆 sin2 𝜃 1 PEP𝑝𝑞 = ⋅ 𝑑𝜃. 𝜋 ∫0 1 + 2𝜆 sin2 𝜃

PEP𝑝𝑞 = √

2 2 𝜀2 𝜌𝑠𝑑 2 |ℎ|2 ||𝛥𝑥 || + 𝑝𝑟𝑑 𝛾 ||𝛥𝑠 || ⎞ ⎟ 𝑓𝐹 𝑆𝑂 (ℎ) 𝑓Gam (𝛾) ⋅ 𝑑ℎ 𝑑𝛾, ⎟ 2𝜎 2 ⎠

PEP𝑝𝑞

𝜆𝑚 𝑡𝑚−1 exp (−𝛿𝑡) ⋅ 𝑑𝑡 𝑑𝜃, (30) = ∫0 𝜋𝛤 (𝑚) ∫0 ( ) where 𝛿 = 𝜆 + 1 2 . The inner integral in (30), with respect to 𝑡, 2 sin 𝜃 can be solved using [42, eq. 3.381.4] yielding 𝜆𝑚 𝜋 ∫0

∫0

√

which can be rewritten as

.

2

(𝑅) PEP𝑝𝑞

(𝑅)

∫0

⎛ 𝑄⎜ ⎜ ⎝

(37)

(√ ) 𝑡 𝑡𝑚−1 exp (−𝜆𝑡) ⋅ 𝑑𝑡,

Using Craig’s formula of the Q-function, 𝑄(𝑧) = 𝑑𝜃 [41], the above integral can be rewritten as

PEP𝑝𝑞 =

∞

, the above integral can be rewritten as

∞

2𝑚𝜎 2 𝑝𝑠𝑟 |𝛥𝑠 |

(𝐷)

PEP𝑝𝑞

The integral 𝐼𝛾 can be rewritten, using (8), as ( ) 2 ∞ −𝑝𝑟𝑑 𝛾 ||𝛥𝑠 || 𝑚𝑚 𝐼𝛾 = exp 𝛾 𝑚−1 𝑒−𝑚𝛾 ⋅ 𝑑𝛾, 𝛤 (𝑚) ∫0 4𝜎 2 sin 𝜃

if 𝑞 ≤ 𝑀 or 𝑝 ≤ 𝑀

using

(45)

(46)

(47)

which can be rewritten using the Taylor series expansion of 𝑒−𝑚𝛾 [42, eq. 1.211.1] as ( ) 2 ∞ ∞ −𝑝𝑟𝑑 𝛾 ||𝛥𝑠 || 𝑚𝑚 ∑ (−𝑚)𝑡 𝐼𝛾 = exp 𝛾 𝑡+𝑚−1 𝑑𝛾. (48) 𝛤 (𝑚) 𝑡! ∫0 4𝜎 2 sin2 𝜃

if 𝑞 > 𝑀 and 𝑝 > 𝑀 (36)

𝑡=0

5

S. Althunibat, O.S. Badarneh, R. Mesleh et al.

Optics Communications 453 (2019) 124400

The later integral can be solved in a closed-form expression using [42, eq. 3.381.4] as 𝐼𝛾 =

∞ ∑

( )𝑡+𝑚 𝛷𝑡 sin2 𝜃 ,

(49)

𝑡=0

where 𝛷𝑡 =

3.1.3. Asymptotic analysis of FSO/mmWave system The asymptotic behavior of a system can be obtained by considering high values of the received power (high SNR values). Starting from (33) or (34) (for non-integer and integer values of 𝑚, respectively), it can be noted that as 𝑝𝑠𝑟 → ∞, the value of 𝜆 → 0. As such, substituting 𝜆 = 0 in

(−1)𝑡 𝑚𝑡+𝑚 𝛤 (𝑡+𝑚) 𝛤 (𝑚)𝑡!

(

)𝑡+𝑚

4𝜎 2

(𝑅)

either (33) or (34), the resultant PEP at the relay is zero, i.e., PEP𝑝𝑞 = 0. Accordingly, 𝑃 𝑞 for 𝑞 > 𝑀, the second term of (24), is also zero, which

.

𝑝𝑟𝑑 |𝛥𝑠 |

2

(𝐷)

(𝐷)

Now, substituting (46) and (49) into (39), the PEP𝑝𝑞 can be rewritten as ∞ 𝜋∕2 ( )𝑡+𝑚 (𝐷) 𝜁 ∑ 1 PEP𝑝𝑞 = 𝛷𝑡 sin2 𝜃 (√ )𝜇 2𝜋 𝑡=0 ∫0 𝜔 ) ( | 𝜇 1 𝑐 | (0,1),(1− 2 , 2 ),(0,1),(1−𝜇+𝑧2 ,1) 4,2 × 𝐻4,5 √ || (𝑧2 −𝜇,1),( 𝛽−𝛼 ,1),( 𝛼−𝛽 ,1),(0,1),(0,1) 𝑑𝜃. 𝐴 𝜔 || 2 2 (50) Letting 𝜔 = rewritten as

𝑎 , 𝑘

where 𝑎 =

𝜀2 𝜌

|𝛥𝑥 | 4𝜎 2

𝑠𝑑

2

2

results in ignoring Case II of PEP𝑝𝑞 .

(𝐷)

To compute the asymptotic PEP𝑝𝑞 given in (52), we start from the PDF given in (1). It is well known that the modified Bessel function of the second kind can be approximated for small arguments as [44] ( )|𝛼−𝛽| √ |𝛼 − 𝛽| 1 𝐾𝛼−𝛽 (2 𝛼𝛽ℎ𝑎 ) ≈ , 𝛼≠𝛽 (58) √ 2 𝛼𝛽ℎ𝑎 As such, the PDF given in (1) can be approximated as 𝑓ℎ𝑎 (ℎ𝑎 ) ≈ 𝑐1 ℎ𝑎 𝛽−1 ,

, and 𝑘 = sin2 𝜃; (50) can be

∞ 1 𝜇 1 1 𝜁 ∑ 𝛷𝑡 = (𝑘)𝑡+𝑚+ 2 − 2 (1 − 𝑘)− 2 × (√ )𝜇 ∫0 4𝜋 𝑡=0 𝑎 ( √ | ) 𝜇 1 , ),(0,1),(1−𝜇+𝑧2 ,1) 𝑐 𝑘 || (0,1),(1− 4,2 2 2)( ( ) 𝑑𝑘, 𝐻4,5 √ | 2 𝛽−𝛼 𝛼−𝛽 𝐴 𝑎 || (𝑧 −𝜇,1), 2 ,1 , 2 ,1 ,(0,1),(0,1) which can be solved using [28, eq. 2.25.2.2] as

, and 𝛼 > 𝛽 for practical GG turbulence where 𝑐1 = 𝛤 (𝛼)𝛤 (𝛽) model [45]. The PDF given in (59) can now be used to obtain the PDF with pointing error included as followed in (3)–(6) to yield

(𝐷) PEP𝑝𝑞

𝑓𝐹 𝑆𝑂 (ℎ) ≈ 𝑐2 ℎ𝛽−1 , (51)

where 𝑐2 = Also, the PDF given in (8) can be approximated as [46]

∞ 𝜁 ∑ 𝛷𝑡 = ( ) × 4𝜋 𝑡=0 √ 𝜇 𝑎 ) ( | 1 𝜇 1 𝜇 1 1 , ),(0,1),(1−𝜇+𝑧2 ,1) 𝑐 || ( 2 −𝑡−𝑚−(2 , 2 ),( 2),0),(0,1),(1− 4,4 2 2 ( ) . (52) 𝐻6,6 √ | 2 𝛽−𝛼 𝛼−𝛽 𝜇 1 𝐴 𝑎 || (𝑧 −𝜇,1), 2 ,1 , 2 ,1 ,(0,1),(0,1),(−𝑡−𝑚− 2 , 2 ) Case II: For the second case in (36), i.e., 𝑃 > 𝑀 and 𝑞 > 𝑀, √ ⎛ 𝜀2 𝜌 2 |ℎ|2 |𝛥 |2 ⎞ ∞ (𝐷) 𝑠𝑑 | 𝑥| ⎟ 𝑓 PEP𝑝𝑞 = 𝑄⎜ (53) (ℎ) ⋅ 𝑑ℎ, ∫0 ⎜ ⎟ 𝐹 𝑆𝑂 2𝜎 2 ⎝ ⎠

𝑓𝐺𝑎𝑚 (𝛾) ≈

1 𝜋 ∫0

(𝐷)

(61)

𝑐2 𝛽 − 𝛽 𝛤 ( )𝜔 2 (62) 2 2 while the resultant integral by substituting (61) in (41) can be solved using [42, eq. 3.381.4] to yield ( )−𝑚 𝑝𝑟𝑑 |𝛥𝑠 |2 𝐼 𝛾 ≈ 𝑚𝑚 (63) (sin 𝜃)2𝑚 4𝜎 2

𝐼ℎ ≈

Substituting both (62) and (63) in (39) yields PEP𝐷

(54)

𝐼ℎ ⋅ 𝑑𝜃.

𝑚𝑚 𝑚−1 𝛾 𝛤 (𝑚)

Now, substituting (60) in (40), 𝐼ℎ can be then formulated as

which can be expressed based on the definition of 𝐼ℎ given in (40) as PEP𝑝𝑞 =

(60)

𝑐 1 𝑧2 . 𝐴𝛽 (𝑧2 −𝛽)

(𝐷) PEP𝑝𝑞

𝜋 2

(59)

(𝛼𝛽)𝛽 𝛤 (𝛼−𝛽)

1 ≈ 𝜋 ∫0

Using (46), (54) can be expressed as

𝜋 2

𝑐2 𝛽 𝛤( ) 2 2

(

𝜖 2 𝜌2𝑠𝑑 |𝛥𝑥 |2

)− 𝛽

(

2

𝑚

4𝜎 2

𝑚

𝑝𝑟𝑑 |𝛥𝑠 |2

)−𝑚

𝛽

(sin 𝜃)2𝑚+ 2 ⋅ 𝑑𝜃

4𝜎 2

(𝐷)

PEP𝑝𝑞 =

𝜁 2𝜋 ∫0

𝜋 2

1 4,2 (√ )𝜇 𝐻4,5 𝜔

(

(64)

) | 𝜇 1 𝑐 | (0,1),(1− 2 , 2 ),(0,1),(1−𝜇+𝑧2 ,1) √ || (𝑧2 −𝜇,1),( 𝛽−𝛼 ,1),( 𝛼−𝛽 ,1),(0,1),(0,1) ⋅ 𝑑𝜃. 𝐴 𝜔 || 2 2

which can be approximated by setting 𝜃 =

PEP𝐷 ≈

(55) 𝜀2 𝜌𝑠𝑑 2 |𝛥𝑥 | Now, substituting 𝜔 = 𝑘𝑎 , where 𝑎 = and 𝑘 = sin2 𝜃, (55) 4𝜎 2 can be rewritten by change of variables (𝜃 to 𝑘) as

𝑐2 𝛤 4

) ( )( 2 2 𝜖 𝜌𝑠𝑑 |𝛥𝑥 |2 𝛽 2 4𝜎 2

(

𝛽 −2

𝑚𝑚

𝜋 2

to lead to

𝑝𝑟𝑑 |𝛥𝑠 |2

)−𝑚 (65)

4𝜎 2

2

Finally, the asymptotic BER value at the destination in FSO/ 1 mmWave system can be obtained by substituting 𝑃 𝑞 = 𝑀 and (65) in (23) to yield

(𝐷)

PEP𝑝𝑞

𝜇

1 𝜇−1 1 𝜁(𝑎)− 2 4,2 = 𝑘 2 (1 − 𝑘)− 2 𝐻4,5 4𝜋 ∫0

(

) 1 | 𝑐 𝑘 2 || (1− 𝜇2 , 21 ),(1−𝜇+𝑧2 ,1) 𝑑𝜃 √ | 2 𝛽−𝛼 𝛼−𝛽 𝐴 𝑎 || (𝑧 −𝜇,1),( 2 ,1),( 2 ,1)

BER𝐷 𝑀 ∑ 𝑀 ∑ 𝑐 1 ≈ 𝜏 2𝛤 𝑀 log2 (𝑀) 𝑞=1 𝑝=1 𝑝𝑞 4

(56)

)− ( )−𝑚 ( )( 2 2 2 𝜖 𝜌𝑠𝑑 |𝛥𝑥 |2 𝑝𝑟𝑑 |𝛥𝑠 |2 𝛽 𝑚 𝑚 . 2 4𝜎 2 4𝜎 2 𝛽

(66)

which can be solved using [28, eq. 2.25.2.2] to obtain the average PEP, (𝐷)

PEP𝑝𝑞 , as 𝜇

(𝐷) PEP𝑝𝑞

𝜁(𝑎)− 2 3,3 = 𝐻5,5 4𝜋

(

) | 𝜇 , 1 ),( 21 ,0),(1− 2 , 12 ),(0,1),(1−𝜇+𝑧2 ,1) 𝑐 || ( 1−𝜇 2 2 . √ | 2 𝛽−𝛼 𝛼−𝛽 𝜇 1 𝐴 𝑎 || (𝑧 −𝜇,1),( 2 ,1),( 2 ,1),(0,1),(− 2 , 2 )

3.1.4. Outage probability of FSO/mmWave system The average electrical SNR at the destination is given by (57) SNR𝐷 = SNR𝑆𝐷 + (1 − 𝑃of f )SNR𝑅𝐷 ,

(67)

where SNR𝑆𝐷 is the average electrical SNR of the 𝑆 − 𝐷 link, SNR𝑅𝐷 is the average electrical SNR of the 𝑅 − 𝐷 link, and 𝑃of f is the probability

Finally, BER𝐷 for FSO/mmWave system is obtained by substituting (52) (or (57)) and (34) into (23). 6

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Optics Communications 453 (2019) 124400

that the relay will not forward a signal to the destination. Notice that 𝑃of f depends mainly on the BER at the 𝑆 − 𝑅 link, and is related to 𝐏𝐄𝐏(𝑅) given in (34). To obtain the outage probability at the destination, the joint PDF of both links should be obtained, which is very sophisticated and no known PDF is available. Therefore, the outage probability at the destination versus the SNR of the 𝑆 −𝐷 link assuming fixed SNR of the dual-hop link is derived hereunder. For FSO/mmWave system, the SNRs of both 𝑆 − 𝐷 and 𝑅 − 𝐷 links can be expressed as SNR𝑆𝐷 =

𝜀2 𝜌2𝑠𝑑 |ℎ|2 𝜎2

where 𝑎 =

(𝐷) (𝐷)

Following the same procedure as considered to obtain PEP𝑝𝑞 , two different cases can be defined depending on the value of 𝑝 and 𝑞 as (𝐷)

Case I: For the case of 𝑝 ≤ 𝑀 or 𝑞 ≤ 𝑀, the PEP𝑝𝑞 is almost identical to (52) as ∞ ∑ 𝛷𝑡′ 𝜁 (√ )𝜇 × 4𝜋 𝑡=0 𝑎′ ( ) | 1 𝜇 1 𝜇 1 1 , ),(0,1),(1−𝜇+𝑧2 ,1) 𝑐 || ( 2 −𝑡−𝑚−(2 , 2 ),( 2),0),(0,1),(1− 4,4 2 2 ( ) 𝐻6,6 , √ | 2 𝛽−𝛼 𝛼−𝛽 𝜇 1 𝐴 𝑎′ || (𝑧 −𝜇,1), 2 ,1 , 2 ,1 ,(0,1),(0,1),(−𝑡−𝑚− 2 , 2 ) )𝑡+𝑚 ( 2 𝑡 𝑡+𝑚 𝛤 (𝑡+𝑚) 𝜀2 𝜌𝑟𝑑 2 |𝛥𝑥 | 4𝜎 2 and 𝑎′ = where 𝛷𝑡′ = (−1) 𝑚𝛤 (𝑚)𝑡! . 2 4𝜎 2 𝑝𝑠𝑑 |𝛥𝑠 | (𝐷)

PEP𝑝𝑞 =

(68)

,

𝑝𝑠𝑑 |𝑔|2

. (69) 𝜎2 The outage probability when the SNR of the mmWave link (𝑅 − 𝐷) is fixed can be expressed as [ ] 𝑃𝑜𝑢𝑡 = Pr SNR𝐷 ≤ SNR𝑡ℎ ] [ = Pr SNR𝑆𝐷 ≤ SNR𝑡ℎ − (1 − 𝑃of f )SNR𝑅𝐷 [ ] = Pr SNR𝑆𝐷 ≤ 𝛾𝑡ℎ , (70)

(𝐷)

3.2.3. Asymptotic analysis of mmWave/FSO system Similar to FSO/mmWave system, in high SNR range, the average received power 𝜌𝑠𝑟 → ∞. As such and by substituting 𝜌𝑠𝑟 = ∞ in (77), (𝑅) the value of PEP𝑝𝑞 is also 0. Thus, from (24), it is clear that 𝑃 𝑞 = 0 for 𝑞 > 𝑀. (𝐷) As for PEP𝑝𝑞 , the second case can be ignored as 𝑃 𝑞 = 0 for 𝑞 > 𝑀.

where 𝐹FSO is the √ Cumulative Density Function (CDF) of the 𝑆 − 𝐷 link 𝜎 2 𝛾𝑡ℎ 𝜀2 𝜌2𝑠𝑑

. As such and using the PDF 𝑓FSO (ℎ) given in

(6), 𝑃out can be obtained as ( ) ℎ𝑡ℎ (1−𝜇+𝑧2 ,1) 𝑐 ℎ || 3,0 𝑃out = 𝜁 ℎ𝜇−1 𝐻1,3 ⋅ 𝑑ℎ 𝛽−𝛼 𝛼−𝛽 2 | ∫0 𝐴 | (𝑧 −𝜇,1),( 2 ,1),( 2 ,1)

(𝐷)

(72)

The H-function can be modified using [28, eq. 8.3.2.6] to yield ( ) ℎ𝑡ℎ (0,1),(1−𝜇+𝑧2 ,1) 𝑐 ℎ || 3,1 𝑃out = 𝜁 ℎ𝜇−1 𝐻2,4 ⋅ 𝑑ℎ, 2 −𝜇,1),( 𝛽−𝛼 ,1),( 𝛼−𝛽 ,1),(0,1) | (𝑧 ∫0 𝐴| 2 2 which can be solved using [28, eq. 2.25.2.2] to yield ) ( 𝑐 ℎ𝑡ℎ || (1−𝜇,1),(0,0),(0,1),(1−𝜇+𝑧2 ,1) 3,3 . 𝑃out = 𝜁 ℎ𝜇𝑡ℎ 𝐻4,5 𝛽−𝛼 𝛼−𝛽 2 𝐴 || (𝑧 −𝜇,1),( 2 ,1),( 2 ,1),(0,1),(−𝜇,1)

Yet, the asymptotic value of PEP𝑝𝑞 in the Case I can be seamlessly obtained using the same procedure considered to obtain (65) in the FSO/mmWave system, which results in

𝑃out ≈ 𝑐2

PEP𝐷 ≈

∫0

𝑀 ∑ 𝑀 ∑ 𝑐 1 ≈ 𝜏 2𝛤 𝑀 log2 (𝑀) 𝑞=1 𝑝=1 𝑝𝑞 4

)− )−𝑚 ( 2 ( )( 2 2 2 𝜀 𝜌𝑟𝑑 |𝛥𝑥 |2 𝑝𝑠𝑑 |𝛥𝑠 |2 𝛽 𝑚 . 𝑚 2 4𝜎 2 4𝜎 2 𝛽

(81) (76) 3.2.4. Outage probability for mmWave/FSO system Similar to the FSO/mmWave system, the outage probability at the destination when the SNR of the 𝑅 − 𝐷 link (FSO) is fixed can be expressed as [ ] 𝑃out = Pr SNR𝑆𝐷 ≤ 𝛾𝑡ℎ , (82)

𝑠𝑑

It is clear from (76) that the diversity order is proportional to 𝛽. 3.2. Performance analysis of mmWave/FSO system The average BER of mmWave/FSO system can also be obtained (𝑅)

(𝐷)

which can be rewritten by substituting the value of SNR𝑆𝐷 to be [ ] 𝜎 2 𝛾𝑡ℎ 𝑃out = Pr ℎ2 ≤ 𝑝𝑠𝑑 ( ) = 𝐹Gam ℎ𝑡ℎ , (83)

using (23). However, the expressions of PEP𝑝𝑞 and PEP𝑝𝑞 are different than the ones derived before since both links are now switched. (𝑅)

3.2.1. Derivation of PEP𝑝𝑞 The 𝑆 − 𝑅 link in mmWave/FSO system is an FSO link. As such, the (𝑅)

(80)

BER𝐷

(75)

The integral is solved to yield ( )𝛽∕2 𝑐2 𝜎 2 𝛾𝑡ℎ 𝑐2 𝛽 𝑃out ≈ ℎ𝑡ℎ ≈ . 𝛽 𝛽 𝜀 2 𝜌2

𝑐2 𝛤 4

Finally, the asymptotic BER formula at the destination in 1 mmWave/FSO system can be obtained by substituting 𝑃 𝑞 = 𝑀 and (80) in (23) to yield

(74)

ℎ𝛽−1 ⋅ 𝑑ℎ.

)− )−𝑚 ( 2 ( )( 2 2 2 𝜀 𝜌𝑟𝑑 |𝛥𝑥 |2 𝑝𝑠𝑑 |𝛥𝑠 |2 𝛽 𝑚𝑚 2 4𝜎 2 4𝜎 2 𝛽

(73)

The asymptotic outage probability can be computed as well by considering the approximated PDF in (60), which is given by ℎ𝑡ℎ

(78)

Case II: For the case of 𝑝 > 𝑀 and 𝑞 > 𝑀, the PEP𝑝𝑞 is almost identical to (34) and given as ( ) 𝑚−1 ∑ 𝑚−1+𝑘 [ ]𝑘 (𝐷) PEP𝑝𝑞 = 𝜒 𝑚 1−𝜒 , (79) 𝑘 𝑘=0 ( ) √ 2 1 and 𝜆′ = 2𝑚𝜎 2 where 𝜒 = 21 1 − 1+2𝜆 ′ 𝑝𝑠𝑑 |𝛥𝑠 | Finally, the overall average BER of the mmWave/FSO system is obtained by substituting (77) and (78) (or (79)) into (23).

with SNR𝑡ℎ denoting the outage SNR threshold, and 𝛾𝑡ℎ = SNR𝑡ℎ − (1 − 𝑃of f )SNR𝑅𝐷 . Now, substituting (68) into (70), 𝑃out can be expressed as √ ] [ 𝜎 2 𝛾𝑡ℎ 𝑃out = Pr ℎ ≤ 𝜀2 𝜌2𝑠𝑑 ( ) = 𝐹FSO ℎ𝑡ℎ , (71)

(FSO), and ℎ𝑡ℎ =

.

3.2.2. Derivation of PEP𝑝𝑞

and SNR𝑅𝐷 =

𝜀2 𝜌𝑠𝑟 2 |𝛥𝑥 | 4𝜎 2

2

(𝐷)

PEP𝑝𝑞 is exactly the same as the case II of the PEP𝑝𝑞 in FSO/mmWave system given in (57) expressed as ( ) 𝜇 | 𝜇 (𝑅) , 1 ),( 21 ,0),(1− 2 , 21 ),(0,1),(1−𝜇+𝑧2 ,1) 𝜁(𝑎)− 2 3,3 𝑐 || ( 1−𝜇 2 2 PEP𝑝𝑞 = 𝐻5,5 , (77) √ | 2 𝛽−𝛼 𝛼−𝛽 𝜇 1 4𝜋 𝐴 𝑎 || (𝑧 −𝜇,1),( 2 ,1),( 2 ,1),(0,1),(− 2 , 2 )

𝜎2 𝛾

where 𝐹Gam is the CDF of the Nakagami fading channel and ℎ𝑡ℎ = 𝑝 𝑡ℎ . 𝑠𝑑 Accordingly, 𝑃out is ( ) 𝑚𝜎 2 𝛾𝑡ℎ 1 𝑃out = 𝛾 𝑚, , (84) 𝛤 (𝑚) 𝑝𝑠𝑑 7

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Optics Communications 453 (2019) 124400

Table 1 Fixed simulation parameters [18,19,40]. FSO parameters 𝜆𝑜 𝜌𝑠 𝑇 𝜙 𝜀

mmWave parameters 1.55 μm 40 mW 20 cm 10 mrad 0.5

𝑓𝑐 𝑝𝑠 𝐺𝑡 𝐺𝑟 𝜑𝑜𝑥

60 GHz 10 mW 44 dBi 44 dBi 15.1 dB/km

Table 2 Parameters for different weather conditions [18]. Weather

𝜑1 [dB/km]

𝜑𝑟𝑎𝑖𝑛 [dB/km]

𝐶𝑛2

Clear air Haze Rain (Moderate)

0.43 4.2 5.6

0 0 5.8

5 × 10−14 1.7 × 10−14 5 × 10−15

where 𝛾(⋅, ⋅) is the lower incomplete gamma function. The asymptotic outage probability can be obtained by approximating the incomplete gamma function in (84) to give ( 2 )𝑚 𝑚𝜎 𝛾𝑡ℎ 1 𝑃out ≈ , (85) 𝛤 (𝑚 + 1) 𝑝𝑠𝑑

Fig. 3. Average BER versus the 𝑆 − 𝐷 distance for the proposed FSO/mmWave system and the FSO–only system at different weather conditions.

It is obvious from (85) that the diversity order is proportional to 𝑚. 4. Simulation results and discussion In this section, we investigate the impact of different system and channel parameters on the overall system performance. Additionally, Monte-Carlo simulation results are provided to substantiate the accuracy of the derived formulas. In all depicted results, the average BER is plotted versus the distance between the source and the destination 𝑑𝑠𝑑 , while considering different values of other parameters. To facilitate the results exploration, the relay 𝑅 has been fixed at the midpoint between 𝑑 𝑆 and 𝐷, i.e., 𝑑𝑠𝑟 = 𝑑𝑟𝑑 = 2𝑠𝑑 . Some simulation parameters that have been fixed for all figures of both systems are listed in Table 1. To emulate different weather conditions, three environments are considered; namely, clear air, haze and rain. The values of 𝜑1 , 𝜑𝑟𝑎𝑖𝑛 , 𝜑𝑜𝑥 and 𝐶𝑛2 for each weather state are listed in Table 2. It should be noted also that the analytical results for the FSO/mmWave and mmWave/FSO systems are obtained by truncating the summations in (52) and (78) to the first 21 terms, where almost no change is noticed when considering more terms.

Fig. 4. The average BER versus the 𝑆 − 𝐷 distance for the proposed FSO/mmWave system and the FSO–only system at different values of the pointing error.

4.1. FSO/mmWave Results the value of 𝑚 will enhance the BER of the dual-hop mmWave link in the proposed system and the overall BER of the system is improved. For example, at 𝑑𝑠𝑑 = 4.5 Km, the obtained BER is about 10−3 , 10−5 , and 10−7 for 𝑚 = 1, 𝑚 = 2 and 𝑚 = 3, respectively. The results in Fig. 6 validates the accuracy of the derived a asymptotic formulas, where the performance of FSO/mmWave system is evaluated under different weather conditions and asymptotic and analytical results are shown to closely match at high SNR values (low S-D distance). The outage probability of the FSO/mmWave system when the SNR 𝜌 of the mmWave cooperative link is fixed versus 𝜎𝑠𝑑2 is shown at Fig. 7 at different turbulence conditions. It can be seen that as the SNR of the FSO link increases, the outage probability decreases as well for all the considered turbulence conditions. However, as expected, weak turbulence conditions result in a lower outage probability. In the figure, analytical and asymptotic results are obtained using (74) and (76), respectively. Additionally, the results show that increasing the turbulence parameter 𝛽 result in a significant performance improvement and especially a higher diversity order as reflected by the steeper slope of the corresponding outage probability curves in the high SNR regime, which confirms the results in (76).

Performance results of FSO/mmWave cooperative system are shown in Figs. 3–6 along with the results for FSO-only system. In FSO-only system, only a direct FSO link is available between the source and the destination and cooperative link exist. In Fig. 3, the BER of FSO/mmWave and FSO-only system at different weather conditions, clear air, haze and rain, is illustrated. As evident, the proposed hybrid FSO/mmWave system substantially enhances the performance of FSO-only system in all weather conditions. The impact of the pointing error on FSO/mmWave and FSO-only systems is studied and results are shown in Fig. 4 while considering clear air weather condition. The value of 𝑧 is set to 0.33, 1.33, 3.33 and infinity (no pointing error case). For both systems, increasing the value of 𝑧 enhances the system performance, where it approaches the no pointing error case. However, it can be observed that the impact of changing the value of 𝑧 has significant impact on the FSO-only as compared to the proposed FSO/mmWave system. This is because the performance of FSO-only system depends mainly on the quality of the FSO link, while FSO/mmWave system relies on two links and the dual-hop mmWave link is not affected by the pointing error. The impact of the Nakagami parameter 𝑚 on the performance of FSO/mmWave system is illustrated in Fig. 5. As expected, increasing 8

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Optics Communications 453 (2019) 124400

Fig. 8. The average BER versus the 𝑆 − 𝐷 distance for the proposed mmWave/FSO system and the mmWave–only system at different weather conditions.

Fig. 5. The average BER versus the 𝑆 − 𝐷 distance for the proposed FSO/mmWave system and the FSO–only system at different values of the Nakagami-𝑚 parameter.

Fig. 9. The average BER versus the 𝑆 − 𝐷 distance for the proposed mmWave/FSO system and the mmWave–only system at different values of the Nakagami-𝑚 parameter.

Fig. 6. Asymptotic and analytic results of the average BER versus the 𝑆 − 𝐷 distance for the proposed FSO/mmWave system at different weather conditions.

4.2. mmWave/FSO results

Fig. 7. The outage probability versus the different turbulence conditions.

𝜌𝑠𝑑 𝜎2

Analytical and simulation results for the mmWave/FSO system are shown in Figs. 8–12. The performance results for the mmWave-only system is also added for the sake of comparison. In the mmWave-only system, there is only a direct mmWave link between the source and the destination. In Fig. 8, the average BER versus the distance between the source and the destination for both the mmWave/FSO and the mmWave-only systems at different weather conditions is depicted. The significant enhancement of the proposed mmWave/FSO system over the mmWave-only system is evident in the figure. The impact of the Nakagami 𝑚 parameter on the performance of both, mmWave/FSO and mmWave-only systems is illustrated in Fig. 9 assuming clear air weather condition. As discussed earlier, increasing the value of 𝑚 enhances the performance of both systems. The pointing error is also studied, and results are shown in Fig. 10. As anticipated, high pointing error has substantial impact of the overall performance of the hybrid system. The derived asymptotic formula for mmWave/FSO system is depicted in Fig. 11, where close match is shown in the figure at arbitrary high SNR values.

the proposed FSO/mmWave system at

9

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Optics Communications 453 (2019) 124400

Fig. 10. The average BER versus the 𝑆 − 𝐷 distance for the proposed mmWave/FSO system and the mmWave–only system at different values of the pointing error.

Fig. 12. The outage probability versus the at different values of 𝑚.

𝑝𝑠𝑑 𝜎2

of the proposed mmWave/FSO system

𝑝

Fig. 13. The outage probability versus the 𝜎𝑠𝑑2 the proposed FSO/mmWave system at different values of 𝛼 and 𝛽 and assuming 𝑧 = 3.33, 𝛾𝑡ℎ = 0.5, 𝐴 = 0.8, and 𝜀 = 0.9.

Fig. 11. Asymptotic and analytic results of the average BER versus the 𝑆 − 𝐷 distance for the proposed mmWave/FSO system at different weather conditions.

5. Conclusions The last results illustrated in Figs. 12 and 13 depict the outage probability versus

𝑝𝑠𝑑 𝜎2

for both systems mmWave/FSO and FSO/mmWave,

Hybrid wireless communication systems using mmWave and FSO technologies while considering cooperative communication are proposed and thoroughly analyzed in this paper. An auxiliary cooperative mmWave link is added to a direct FSO link in a system named FSO/mmWave and shown to significantly enhance the overall BER performance of a direct FSO link. Similarly, an auxiliary FSO link is added to a direct mmWave link resulting in mmWave/FSO system, that is shown to substantially enhance the overall BER performance of a direct mmWave link. Performance analysis of both systems is conducted assuming Gamma–Gamma FSO turbulence fading and Nakagami−𝑚 fading channels. Novel analytical expressions for the overall average and asymptotic BER are derived and corroborated through Monte Carlo simulation results. In addition, outage probability of both systems assuming fixed power of the cooperative link is derived. Future studies will consider the impact of channel and system imperfections on the performance of the proposed systems and investigate the joint PDF of both channels.

respectively. In both scenarios, the cooperative link is assumed to have fixed SNR value. Fig. 12 depicts the impact of varying the Nakagami 𝑚 value on the outage probability, where the higher the value of 𝑚 the lower the outage probability. The results demonstrate also the accuracy of the derived formulas where analytical, asymptotic and simulation results demonstrate close-match over wide range of SNR values. Similarly, Fig. 13 depicts the outage probability for FSO/mmWave system assuming different values of 𝛼 and 𝛽 and considering 𝑧 = 3.33, 𝛾𝑡ℎ = 0.5, 𝐴 = 0.8, and 𝜀 = 0.9. Again, results reveal the accuracy of the conducted analysis. Also, the results demonstrate that increasing the fading parameter 𝑚 leads to a significant performance improvement and especially a higher diversity order as reflected by the steeper slope of the corresponding outage probability curves in the high SNR regime, which confirms the results in (85). 10

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Optics Communications 453 (2019) 124400

Acknowledgment

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