Item Type Article
Authors Li, Sai;Yang, Liang;Zhang, Jiayi;Bithas, Petros S.;Tsiftsis, Theodoros A.;Alouini, Mohamed-Slim
Citation Li, S., Yang, L., Zhang, J., Bithas, P. S., Tsiftsis, T. A., & Alouini, M.- S. (2022). Mixed THz/FSO Relaying Systems: Statistical Analysis and Performance Evaluation. IEEE Transactions on Wireless Communications, 1–1. https://doi.org/10.1109/twc.2022.3188698 Eprint version Post-print
DOI 10.1109/twc.2022.3188698
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Mixed THz/FSO Relaying Systems: Statistical Analysis and Performance Evaluation
Sai Li, Liang Yang, Jiayi Zhang, Senior Member, IEEE, Petros S. Bithas, Senior Member, IEEE, Theodoros A. Tsiftsis, Senior Member, IEEE, and Mohamed-Slim Alouini, Fellow, IEEE
Abstract—In this paper, the performance of a mixed Terahertz/free-space optical (THz/FSO) wireless transmission system is studied, where the joint effects of channel fading and pointing errors are considered for both THz and FSO links. Employing the semi-blind amplify-and-forward protocol, we derive the cumulative distribution function and probability density function of the end-to-end signal-to-noise ratio. By applying the derived statistics, exact expressions for the outage probability (OP), average bit error rate (BER), and average channel capacity are obtained. In order to attain useful physical insights, asymptotic OP and average BER expressions are also presented. Based on them, the diversity gain is determined, which is shown to depend on channel fading and pointing errors of both links. In addition, by taking into account the hardware impairments, the OP of the non-ideal hardware system is derived.
Moreover, the analysis is extended to multi-antenna scenarios and the asymptotic OP is obtained. Finally, illustrative numerical results are plotted and it can be observed that the path loss, channel fading, pointing errors and hardware impairments lead to a considerable degradation in system performance.
Index Terms—Free-space optical (FSO) communications, mixed dual-hop transmission schemes, performance analysis, pointing errors, Terahertz (THz) communications.
I. INTRODUCTION
T
ERAHERTZ (THz) and optical wireless communications have the potential to be one of the key technologies for the successful realization of the future beyond fifth- generation (B5G) communication networks since they provideManuscript received November 16, 2021; revised March 31, 2022 and May 18, 2022; accepted July 2, 2022. This work was supported in part by the Key R&D Projects in Hunan Province under Grant (No. 2022GK2051), the Hunan High-tech Industry Science and Technology Innovation Leading Program Project under Grant (No. 2022GK4004), the Changsha Natural Science Foundation under Grant (No. kq2202172), and in part by the Hunan Natural Science Foundation. This paper is an extended version of work presented at the 13th International Conference on Wireless Communications and Signal Processing (WCSP), Changsha, China, October 2021 [37]. The review of this paper was coordinated by Prof. Miao Wang. (Corresponding author: Liang Yang)
S. Li and L. Yang are with the College of Computer Science and Electronic Engineering, Hunan University, Changsha, 410082, China (e-mail:
[email protected], [email protected]).
J. Zhang is with the School of Electronic and Information Engineer- ing, Beijing Jiaotong University, Beijing, 100044, China (email: zhangjiay- [email protected]).
P. S. Bithas is with the Department of Digital Industry Technologies, National and Kapodistrian University of Athens, Athens, 15772, Greece (e- mail: [email protected]).
Theodoros A. Tsiftsis is with the School of Intelligent Systems Science and Engineering, Jinan University, Zhuhai 519070, China, and also with the Department of Informatics and Telecommunications, University of Thessaly, Lamia 35131, Greece (e-mail: [email protected]).
Mohamed-Slim Alouini is with the CEMSE Division, King Abdullah University of Science, and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia (e-mail: [email protected]).
high available bandwidth, high-speed wireless data transmis- sion rate, and can effectively alleviate the current shortage of radio frequency (RF) communication bands [1]–[4]. The THz frequency band has a short wavelength and can meet the ultra-dense small cell networks requirements, which is beneficial for the extremely high data rate coverage and secure transmission in B5G networks [5]. Moreover, THz communi- cations support both line-of-sight (LoS) and non-LoS (NLoS) propagation conditions and the ability to penetrate dust and fog [6], [7]. On the other hand, the optical frequency band also supports high band availability and transmission rates, with higher confidentiality and stronger anti-interference ability.
Nevertheless, THz and optical communication technologies still have many challenges to overcome, such as higher path attenuation, channel fading, and antenna misalignment [4], [5].
Due to the higher frequency bands in the THz communi- cation system, the huge path loss and molecular absorption may cause limited propagation distance and severe system degradation. In the past, several works have focused their research on accurately describing the impact of total path loss on THz channels [8]–[10]. For example, a path attenuation model for the 275-400 GHz frequency band has been presented in [9], based on which the performance of the THz system was evaluated in [10]. In addition, the multipath fading in THz channels can significantly affect the system performance.
In order to evaluate this effect, recent models to characterize the channel fading of the THz channel were investigated in [11]–[13]. For instance, in [11], the authors adopted Rayleigh or Nakagami-mdistributions to model the fading of the THz channel under NLoS scenarios, while for LoS cases, Rician and Nakagami-m distributions were assumed. In [12], the fluctuating two-ray distribution was used to model the small- scale fading of the THz channel. Very recently, the authors adopted α-µ fading to describe the multipath effect in THz channels [13], since α-µ distribution has a generic form and has been experimentally verified in THz communication systems. Additionally, the misalignment effects between the transmitter and the receiver antennas, also known as pointing errors, have a non-negligible impact on the quality of THz links [14]. Furthermore, hardware imperfections also degrade the performance of THz channels, therefore many efforts have been conducted to tackle this concern [15], [16]. Despite the detailed investigations on various aspects of the THz system performed by many researchers in the past, the degradation effects of the THz channel remains to be further investigated.
In particular, a comprehensive performance analysis of THz links subject to path loss, fading, misalignment effects, and
hardware imperfections, was carried out in [13].
On the other hand, free-space optical (FSO) communication has attracted the researchers’ attention because of its ability to offer an effective solution to the last-mile transmission of broadband networks [17], [18]. However, atmospheric tur- bulence as well as pointing errors may significantly reduce the performance of FSO systems. Regarding the channel behavior, along the years, several distributions have been proposed to describe the statistical behavior of atmospheric turbulence-induced fading in FSO systems [19]–[21], such as Log-Normal, Weibull, double Weibull, Gamma-Gamma, dou- ble generalized Gamma (DGG), M´alaga, and exponentiated Weibull. Among them, the DGG distribution is one of the most accurate models to describe weak-to-strong turbulence conditions [22]. Meanwhile, recent models were proposed to investigate the combined effects of atmospheric turbulence with pointing error impairments in FSO systems. For instance, in [23], assuming that the turbulence-induced fading follows the Gamma-Gamma distribution and taking pointing error effects into consideration, the performance of FSO systems was studied. The performance of a FSO system was analyzed in [24], considering the jointed effects of both DGG turbulence fading and pointing errors.
A. Related Works
Subsequently, due to the rapid development of the relaying technology, the mixed RF/FSO system has been proposed to overcome the connectivity gap between the backbone and the last-mile access network, and some of these works can be found in [25]–[29]. Recently, the data throughput of an un- manned aerial vehicle-assisted FSO/RF system was studied in [29]. Indeed, the RF/THz model can be viewed as an effective solution for alleviating the negative consequences of the limit- ed propagation distance of the THz communications. The main idea is to adopt the relaying techniques for expanding coverage area and improving the system’s performance through dual or multiple shorter hops. Recently, mixed THz/RF relaying systems were proposed by using the decode-and-forward (DF) scheme [30], [31]. The mixed THz/RF relaying concept is envisioned to allow multiple RF links to feed a single high- speed THz link. In [30], assuming the DF protocol, the average bit error rate (BER) of the mixed THz/RF system was evaluated. In [31], the authors derived expressions for the outage probability (OP), ergodic capacity, and average BER of mixed THz/RF systems under the DF protocol. In [32], a fixed-gain amplify-and-forward (AF) RF/THz relaying system has been studied, in which the non-zero boresight pointing errors was considered to model the antenna misalignment.
Additionally, in [33], [34], performance analyses of dual- hop THz communication systems were presented. In [35], the multi-relay THz system was studied, and the system performance under the optimal and random selection strategies was evaluated. In [36], the authors employed the relay-assisted communication to improve the performance of THz frequency nano-networks, and the outage performance under different system parameters was evaluated. Although the results from [25]-[36] are insightful, these works are still insufficient in
some aspects. For instance, the expression is analytically very complex and inconvenient for further analytical investigations when using the DGG distribution to model the FSO channel fading. Furthermore, the impact of hardware impairments on THz-based relaying systems has not been studied in the aforementioned literature.
B. Motivation and Contributions
By combining the advantages of THz and FSO links, the mixed THz/FSO model can provide improved data traffic connectivity between small cells and the core network. The motivation behind this model can be explained as follows.
Compared with FSO, THz communication can better adapt to severe climate conditions, with the drawback of limited propagation distance. On the contrary, the transmission dis- tance of FSO can reach several kilometers, but point-to-point FSO communication on the ground is often affected by terrain, environment, and building obstacles. Therefore, making full use of the advantages of both links, a mixed relaying system can be established to make up for the shortcomings of single- link transmission, with the system model shown in Fig. 1.
More specifically, THz communication is used within a short distance from the source (S) to the relay (R) located in a building, and long-distance FSO communication is carried out from R to the destination (D). The mixed THz/FSO heterogeneous communication scheme can extend the prop- agation range and achieve high-speed, large capacity, and more secure transmissions. Furthermore, since THz and FSO links operate on completely independent frequency bands, any type of interference can be avoided. To the best of authors’
knowledge, the performance of a mixed THz/FSO system has not been investigated in the literature yet. Highly motivated by the aforementioned studies, this work considers a mixed THz/FSO system, where the THz and FSO links obey α- µ and DGG fading, respectively, with both links subject to pointing errors. It is necessary to point out that the conference paper [37] only focuses on the OP and average BER of a mixed THz/FSO system with a fixed-gain AF protocol, where the atmospheric turbulence fading of the FSO channel was modeled by the Gamma-Gamma distribution. In this paper, the analysis is extended by assuming the DGG fading. In addition, a semi-blind fixed-gain AF protocol is employed and a comprehensive performance analysis of a mixed THz/FSO system is provided. The main contributions of this paper are listed as follows:
• For both THz and FSO links, the probability density functions (PDFs) and cumulative distribution functions (CDFs) of the respective received instantaneous signal- to-noise ratios (SNRs) are derived in terms of Meijer’s G-function and Fox’s H-function (FHF), which facilitate further analytical derivations. The SNRs of both links take into account the jointed effects of channel fading, path loss, and pointing errors.
• Considering the semi-blind AF relaying protocol, exact expressions for the CDF, PDF, and the moments of the end-to-end (e2e) SNR are derived. By relying on these statistical metrics, analytical expressions for the OP,
Fig. 1. Mixed THz/FSO transmission system.
average BER, and average channel capacity (ACC) of the system are derived in terms of bivariate Fox’s H-function (BFHF).
• Asymptotic expressions can also be derived in order to obtain additional physical insights. At high SNRs, asymp- totic results for the OP and average BER are carried out.
Based on them, it is observed that the diversity gain of the mixed THz/FSO relaying system is determined by the channel fading parameters and the pointing errors of the two links. Additionally, an upper bound expression of the ACC is provided.
• Insightful discussions are provided. An interesting out- come is that the impact of pointing errors is more impor- tant than the channel fading and propagation distance.
• Considering the effect of hardware imperfections, the OP of the non-ideal hardware system is derived, and the OP of the ideal and non-ideal hardware systems is compared.
• Utilizing the moment generating function (MGF)-based method, we obtain the asymptotic OP of the multi- antenna system. Results show that the diversity gain depends on fading parameters and number of antennas.
C. Organization
The rest of this paper is organized as follows. Section II presents the system and channel models. The statistics of the overall SNR for the considered system are derived in Section III. Section IV obtains the exact expressions of the OP, average BER, and ACC of the considered systems, while the asymptotic OP and average BER are also provided for more insights. In Section V, the exact and asymptotic OPs for the system with hardware impairments are derived. Section VI extends the analysis to multi-antenna scenarios. To verify the accuracy of performance metrics, illustrative numerical results supplemented by Monte Carlo simulations are given in Section VII. Finally, Section VIII concludes this paper. In addition, six appendices provide the proofs of some statistics derived throughout the paper.
II. SYSTEM ANDCHANNELMODELS
As shown in Fig. 1, a mixed THz/FSO uplink transmission system is considered, where S communicates over a THz link with R, and the received signal is converted to an optical one, which is then transmitted through a FSO link to D. It is
assumed that S, R, and D are equipped with a single antenna and semi-blind AF relaying scheme is considered at R. In this section, it is assumed that ideal hardware conditions exist.
In Section V, this assumption will be relaxed. Furthermore, the THz channel is assumed to be subject to α-µ fading that accounts for pointing errors, and the FSO channel is assumed to obey a DGG fading also with pointing error impairments. In the considered dual-hop THz/FSO system, the signal transmission procedure from S to D via R can be divided in two phases. In the first hop, S broadcasts the message to R through a THz link. Thus, the received signal at R, yr, is written as
yr=√
P1hSRs+ω1, (1) wheresrepresents the transmitted symbol with unit energy,P1
stands for the transmit power at S,ω1 represents the additive white Gaussian noise (AWGN) with varianceN01,hSRstands for the channel gain of the THz channel.
In the second hop, the output electrical signal is converted to an optical one. By employing AF relaying scheme, the signal at R is multiplied by a proper amplification factorGand then is forwarded to D via a FSO link. Thus, the received signal at Dyd is expressed as
yd=√
P2GIRDη(√
P1hSRs+ω1) +ω2, (2) where P2 stands for the transmit power at R, η denotes the electrical-to-optical conversion factor, ω2 is the AWGN with variance N02, and IRD represents the channel coefficient of the FSO channel.
By using (2), the e2e instantaneous SNR at D can be deduced as
γo=
|hSR|2P1 N01
η2I2RDP2
N02
η2IRD2 P2
N02 +G2PN201
= γtγf
γf+C, (3) where γt = |hSR|2γt and γf = (ηIRD)2γf represent the instantaneous received SNRs of the THz and FSO links, respectively, with corresponding average SNRsγt=P1/N01 and γf = η2P2/N02, and C = P2/(G2N01) is a constant related toG[38]. More specifically, since the semi-blind AF relaying scheme is considered, the amplification factorGcan be obtained by using statistical channel state information of the first hop [38], that is, G2 = E[
P2 N01(γt+1)
]
, where E[·] denotes the expectation operator. Therefore, the parameter G can be calculated from the statistical characteristics of the THz link, andC is finally derived fromC=P2/(G2N01).
A. Statistical Characteristics of the THz Link
For the THz channel, the channel coefficienthSRis given by hSR=hlhpf, wherehlis the deterministic path loss obtained as [13]
hl=c√ GtGr 4πf d exp
(
−1 2κα(f)d
)
, (4)
whereGt andGr represent the transmit and receive antenna gains, respectively, d denotes the propagation distance of the THz link, c and f are, respectively, the speed of light
and the operating frequency, and κα(f) is the absorption coefficient related to the temperature T, relative humidityϕ, and atmosphere pressure Pa, which can be obtained from [13, Eqs. (8-17)]. Furthermore, the channel coefficienthpf of the THz channel experiences α-µ fading with pointing error impairments. Therefore, hpf encompasses the joint effects of fading and antenna misalignment, whose PDF is given by [13, Eq. (26)]
f|hpf|(x) = ξ12A−ξ
2 1
o1 µξ
21 αxξ21−1 ˆhξf21Γ(µ)
Γ (
ϵ,µxαA−o1α ˆhαf
) , (5)
where ϵ = µ−ξα12, Ao1 and ξ1 denote the parameters of the pointing error of the THz link [23], where Ao1 holds for the constant term and ξ1 stands for the ratio between the equivalent beam radius and the pointing error displacement standard deviation at the receiver, Γ(·)represents the gamma function [39, Eq. (8.310)], and Γ(·,·) denotes the upper incomplete gamma function [39, Eq. (8.350.2)],αandµrefer to the attenuation parameters of the α-µ distribution, ˆhf is the α-root mean value of the fading channel envelope. The α-µ distribution is a generalized model that can simplified to important distributions, such as Nakagami-m, Gamma, Weibull, and Rayleigh [40].
Based on (5), the PDF ofγtis formulated by applying [41, Eq. (8.4.16/2)] as
fγt(γt) =A1A−
ξ2 1 α
2
γt
G2,01,2 [
A2 (γt
γt )α2
1+ξα21
µ,ξα12 ]
, (6) where Gm,np,q [·] refers to the Meijer’s G-function [39, Eq.
(9.301)], A1 = ξ
2 1µ
ξ2 1 αh−ξ
21 l
2ˆhξ
21 f Aξ
21 o1Γ(µ)
, and A2 = µ
(ˆhfhlAo1)α. By utilizing [39, Eq. (9.301)], the CDF of γtcan be derived as
Fγt(γt) =2A1A−
ξ2 1 α
2
α G2,12,3 [
A2 (γt
γt )α2
1,1+ξα12
µ,ξα21,0 ]
. (7) With the help of (6), the amplification factor G is formu- lated by utilizing [42, Eq. (1.112)], [39, Eq. (3.194.3)], and employing [42, Eq. (1.2)] as
G2= 1 N01A1A−
ξ2 1 α
2 H3,12,3
A2
γ
α
t2
(1,α2) ( 1+ξα21,1
)
(µ,1) (ξ12
α,1) ( 1,α2)
. (8)
B. Statistical Characteristics of the FSO Link
For the FSO link, the channel coefficient is given byIRD= IlIpf, where Il is the deterministic path loss and is equal to 1, while Ipf holds for the jointed effects of atmospheric turbulence fading and pointing errors. In particular, the PDF of Ipf can be given in [24, Eq. (8)], however, the expression is analytically very complex and cannot be used for further analytical investigations. Therefore, we obtain tractable results as follows.
Theorem 1: Considering the DGG distributed fading as well as the pointing error, the PDF ofIpf is derived in terms
of FHF by using [20, Eqs. (1)-(4)], [24, Eqs. (6) and (7)], and [42, Eq. (1.2)] as
fIpf(x) =2B1
x H3,01,3 [
B2x
(1+ξ22,1) (m2,λ1
2)(m1,λ1
1)(ξ22,1) ]
, (9) where B1 = 2Γ(mξ22
1)Γ(m2) and B2 = (m1
Ω1
)λ1
1 (
m2
Ω2
)λ1
2 1
Ao2, m1, λ1, Ω1, m2, λ2, and Ω2 hold for the attenuation pa- rameters related to atmospheric turbulence conditions [25], similarly, ξ2 andAo2 indicate parameters related to pointing errors of the FSO link, Hm,np,q [·] stands for the FHF [42, Eq.
(1.2)]1.
Proof:See Appendix A.
Therefore, the PDF of γf is derived as fγf(γf) = B1
γf
× H3,01,3 [
B2 (γf
γf )12
(ξ22+1,1) (
m2,λ1
2
) ( m1,λ1
1
) (ξ22,1)
] , (10) Consequently, the CDF of γf can be obtained as
Fγf(γf) = 2B1
× H3,12,4 [
B2
(γf γf
)12
(1,1)(ξ22+1,1) (
m2,λ1
2
) ( m1,λ1
1
)
(ξ22,1)(0,1) ]
. (11) III. ANALYTICALRESULTS FOR THESTATISTICS OF THE
E2E SNR
In this section, analytical results in terms of the CDF, the PDF, and the moments of the e2e received SNR are provided.
Moreover, by focusing on the high SNR regime, simplified asymptotic results have been extracted.
A. Cumulative Distribution Function
1) Exact Results: From [44, Eq. (14)], the CDF of γo is obtained as
Fγo(γ) =Fγt(γ) +
∫ ∞
0
Fγf
(Cγ x
)
fγt(x+γ)dx
| {z }
I1(γ)
. (12)
Theorem 2: By inserting (6), (7), and (11) into (12) and tak- ing a series of transformations, the CDF can be derived as (13), shown at the top of the next page. whereH0,n1:m2,n2:m3,n3
p1,q1:p2,q2:p3,q3 [·,·] denotes the BFHF [42, Eq. (2.57)]2.
Proof:See Appendix B.
Considerα= 2,λ1= Ω1= 1, andλ2= Ω2= 1, the CDF of γo with a Nakagami-m/Gamma-Gamma fading including pointing errors is written as (14), shown at the top of the next page, where ζ1 = µ
(ˆhfhlAo1)2, and ζ2 = mA1m2
o2 . Furthermore, for µ = 1, hl = 1, and ξ1, ξ2 → ∞, (14) reduces to the Rayleigh/Gamma-Gamma case with non-pointing errors, the result has been presented in the previous literature in [26, Eq.
(3)].
1The MATHEMATICA implementation for computing the FHF has been presented in [43].
2The BFHF can be evaluated by using the MATLAB implementation [45].
Fγo(γ) = 2A1A−
ξ2 1 α
2
α G2,12,3 [
A2
(γ γt
)α2
1,1+ξα12 µ,ξα21,0
]
+ 2A1A−
ξ2 1 α
2 B1H0,1:1,4:0,2 1,0:5,2:2,2
(1;−12,α2)
− (1−m2,λ1
2)(1−m1,λ1
1)(1−ξ22,1)( 0,12)
(1,1) (0,1)(−ξ22,1)
(
1−µ,1)(1−ξα21,1 ) (−ξα21,1) (
0,α2)
γ
1 2
f
B2C12, γ
α 2
t
A2γα2
. (13)
FγS
o(γ) = ξ12 2Γ(µ)G2,12,3
[ ζ1γ
γt
1,1+ξ221 µ,ξ212,0
] +ξ21B1
Γ(µ)
×H0,1:1,4:0,2 1,0:5,2:2,2
(1;−12,1)
−
(1−m2,1)(1−m1,1)(1−ξ22,1)( 0,12)
(1,1) (0,1)(−ξ22,1)
(
1−µ,1)(1−ξ221,1 ) (−ξ212,1
) (0,1)
γ
1 2
f
ζ2C12, γt ζ1γ
. (14)
2) Asymptotic Results: In addition to the exact CDF, we also present the asymptotic analysis of the CDF for the system under consideration. At the high SNR regime, (13) can be reduced asymptotically to simple elementary functions.
Theorem 3: Assuming thatγt, γf → ∞, and applying [46, Eq. (07.34.06.0040.01)] and [47, Eq. (1.8.4)], the asymptotic CDF Fγ∞o(γ) is formulated as
Fγ∞
o(γ) →
γt,γf→∞
2A1Aϵ2Γ(−ϵ) αµΓ(1−ϵ)
(γ γt
)αµ2
+2A1Γ(ϵ) ξ12
(γ γt
)ξ221
+
∑5 i=1
4A1A
ςi−ξ2 1 α
2 B1B2ςi
α ϖi
( Cγ γfγt
)ςi2 ,
(15) where ςi = {m2λ2, m1λ1, ξ22, αµ, ξ21}, ϖ1 =
αΓ(
m1−mλ21λ2)
Γ(µ−m2αλ2)
m2(ξ22−m2λ2)(ξ21−m2λ2) , ϖ2 = αΓ
(
m2−mλ12λ1)
Γ(µ−m1αλ1)
m1(ξ22−m1λ1)(ξ21−m1λ1) , ϖ3 =
αΓ (
m2−λξ222
) Γ
( m1−λξ221
) Γ
( µ−ξα22
)
ξ22(ξ12−ξ22) , ϖ4 =
Γ( m2−αµλ2
)
Γ(m1−αµλ1)Γ(−ϵ) αµ(ξ22−αµ)Γ(1−ϵ) ,ϖ5=
Γ (
m2−λξ212) Γ
( m1−λξ121)
Γ(ϵ) ξ21(ξ22−ξ21) . Proof: See Appendix C.
B. Probability Density Function
From [48, Eq. (25)] and [49, Eq. (51)], the PDF of γo is expressed as
fγo(γ) =
∫ ∞
0
C(x+γ) x2 fγf
(Cγ x
)
fγt(x+γ)dx. (16) Theorem 4: By substituting (6) and (10) into (16) and taking some algebraic operations, the PDF ofγois attained as (17), shown at the top of the next page.
Proof:See Appendix D.
C. Moments
The nth moments of γo is defined as E[γon] =
∫∞
0 γonfγo(γ)dγo, which is given by the following theorem.
Theorem 5: By utilizing (17),E[γon]is derived as
E[γon] = 4A1A−
ξ2 1 +2n
α
2 B1γntΓ(
µ+2nα) (ξ21+ 2n)Γ(n)
×H4,12,4 [
B22C γf
(1−n,1) (1 +ξ22,2) (
m2,λ2
2
) ( m1,λ2
1
)
(ξ22,2)(0,1) ]
. (18) Proof:See Appendix E.
IV. PERFORMANCEANALYSIS
In this section, exact results of the OP, the average BER, and the ACC are obtained. In addition, asymptotic results for the OP and the average BER as well as the upper bound for the ACC are presented.
A. Outage Probability
1) Exact Results: The OP is defined as the probability that the e2e instantaneous SNR falls below a certain thresholdγth. Therefore, the OP is mathematically expressed as
Pout=Fγo(γth). (19)
fγo(γ) =A1A−
ξ2 1 α
2 B1
γ H0,1:0,4:0,2 1,0:4,1:2,2
(1;−12,α2) ( −
1−m2,λ1
2
) (
1−m1,λ1
1
)
(1−ξ22,1)( 1,12) (−ξ22,1)
(
1−µ,1)(1−ξα21,1 ) (−ξα12,1) (
1,α2)
γf12 B2C12, γtα2
A2γα2
. (17)
Pe= A1A−
ξ2 1 α
2
αΓ(p) H2,23,3
A2γ−t α2 qα2
(1−p,α2) (1,1)
( 1+ξα12,1
)
(µ,1) (ξ2
1
α,1 )
(0,1)
+A1A−
ξ2 1 α
2 B1
Γ(p) H0,1:1,4:1,2 1,0:5,2:2,3
(1;−12,α2)
− (1−m2,λ1
2)(1−m1,λ1
1)(1−ξ22,1)( 0,12)
(1,1) (0,1)(−ξ22,1)
(
1−µ,1)(1−ξα21,1 ) (p,α2) (
−ξα21,1) ( 0,α2)
γ
1 2
f
B2C12, γtα2 A2q−α2
. (23)
2) Asymptotic Results: Also, the asymptotic OP is obtained as Pout∞ → Fγ∞
o(γth). From [50], the asymptotic OP can be written asPout∞ →(Gc·γ)−Gd, whereGc denotes the coding gain, Gd refers to the diversity gain, and γ represents the average SNR. By relying on the above definition and (15), the diversity gain of the system can be deduced as
Gd= min {αµ
2 ,ξ12
2, m1λ1, m2λ2, ξ22 }
. (20) Based on (20), one can easily observe that the diversity gain of the system depends on the multipath fading parameters of the THz hop (i.e.,αandµ), atmospheric turbulence parameters of the FSO hop (i.e., m1, λ1, m2, andλ2), and pointing errors of both hops (i.e.,ξ1andξ2). As far as the authors are aware, this remark has not been presented in the literature before.
B. Average Bit Error Rate
1) Exact Results: The average BER for various modulation schemes can be evaluated by [27, Eq. (25)]
Pe= qp 2Γ(p)
∫ ∞
0
γp−1e−qγFγo(γ)dγ, (21) where pandq parameters are determined by the modulation method assumed. For instance, {p = 12, q = 14} and {p =
1
2, q= 1} are used for on-off shift keying (OOK) and binary phase shift keying (BPSK) modulation methods, respectively.
From [44] and [48], the average BER of the considered system can be written as
Pe=Pe,t+ qp 2Γ(p)
∫ ∞
0
γp−1e−qγI1(γ)dγ
| {z }
I2
, (22)
wherePe,t=2Γ(p)qp ∫∞
0 γp−1e−qγFγt(γ)dγ denotes the aver- age BER of the THz link.
Theorem 6: By employing [46, Eq. (07.34.26.0008.01)], [41, Eq. (2.25.2/3)], [39, Eq. (3.326.2)], and together with (13), the following exact expression forPeis deduced in (23), shown at the top of this page.
Proof:See Appendix F.
2) Asymptotic Results: Moreover, in order to get more insights, the asymptotic average BER is also presented. By placing (15) into (21) and employing [39, Eq. (3.326.2)], the asymptotic average BER can be deduced as
P∞e →
γt,γf→∞
A1Aϵ2Γ(−ϵ)Γ( p+αµ2 ) αµΓ(1−ϵ)Γ(p)
( 1 qγt
)αµ2
+
A1Γ(ϵ)Γ (
p+ξ212 )
ξ12Γ(p)
( 1 qγt
)ξ221
+
∑5 i=1
2A1A
ςi−ξ2 1 α
2 B1B2ςiΓ( p+ς2i)
αΓ(p) ϖi
( C qγfγt
)ςi2 .
(24) As a double check, one can observe that the system’s diversity gain is written asGd= min
{αµ
2 ,ξ221, m1λ1, m2λ2, ξ22 }
, which is equal to the result previously obtained with the OP analysis.
C. Average Channel Capacity
1) Exact Results: In addition, the ACC can be defined as [27]
C= 1 2 ln(2)
∫ ∞
0
ln(1 +γ)fγo(γ)dγ. (25) By placing (17) in (25), and using the integral expression ln(1 + γ) = G1,22,2
[ γ
1,1 1,0 ]
[41, Eq. (8.4.6/5)], and then employing [42, Eqs. (2.9) and (2.57)], the ACC can be derived as (26), shown at the top of the next page. Assumingα= 2,
