# On the Performance of Dual-Hop Systems over Mixed FSO/mmWave Fading Channels

Free-space optical (FSO) links are considered as a cost-efficient way to fill the backhaul/fronthaul connectivity gap between millimeter wave (mmWave) access networks and optical fiber based central networks. In this paper, we investigate the end-to-end performance of dual-hop mixed FSO/mmWave systems to address this combined use. The FSO link is modeled as a Gamma-Gamma fading channel using both heterodyne detection and indirect modulation/direct detection with pointing error impairments, while the mmWave link experiences the fluctuating two-ray fading. Under the assumption of both amplify-and-forward and decode-and-forward relaying, we derive novel closed-form expressions for the outage probability, average bit error probability (BER), ergodic capacity, effective capacity in terms of bivariate Fox's H-functions. Additionally, we discuss the diversity gain and provide other important engineering insights based on the high signal-to-noise-ratio analysis of the outage probability and the average BER. Finally, all our analytical results are verified using Monte Carlo simulations.

## Authors

• 124 publications
• 23 publications
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• 32 publications
• 177 publications
• ### Shadowed FSO/mmWave Systems with Interference

We investigate the performance of mixed free space optical (FSO)/millime...
05/01/2019 ∙ by Imene Trigui, et al. ∙ 0

• ### All-Optical FSO Relaying Under Mixture-Gamma Fading Channels and Pointing Errors

The performance of all-optical dual-hop relayed free-space optical commu...
10/15/2018 ∙ by Nikolaos I. Miridakis, et al. ∙ 0

• ### Ergodic Capacity of Triple-Hop All-Optical Amplify-and-Forward Relaying over Free-Space Optical Channels

In this paper, we propose a comprehensive research over triple hop all-o...
10/04/2018 ∙ by Mohsen Naseri, et al. ∙ 0

• ### On the Asymptotic Performance Analysis of the k-th Best Link Selection over Non-identical Non-central Chi-square Fading Channels

This paper derives the asymptotic k-th maximum order statistics of a seq...
01/18/2021 ∙ by Athira Subhash, et al. ∙ 0

• ### Error analysis of mixed THz0RF wireless systems

In this letter, we introduce a novel mixed terahertz (THz)-radio frequen...
12/10/2019 ∙ by Alexandros-Apostolos A. Boulogeorgos, et al. ∙ 0

• ### On the Transmission Probabilities in Quantum Key Distribution Systems over FSO Links

In this paper, we investigate the transmission probabilities in three ca...
10/09/2020 ∙ by Hui Zhao, et al. ∙ 0

• ### Performance Analysis of User-centric Virtual Cell Dense Networks over mmWave Channels

This paper analyzes the ergodic capacity of a user-centric virtual cell ...
07/26/2018 ∙ by Jianfeng Shi, et al. ∙ 0

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## I Introduction

It is acknowledged that the dense deployment of small cells is one of the key architecture that enables the coverage of extremely high data rate in the fifth-generation (5G) wireless network. One significant concern in the deployment of such network is the backhauling which connects the massive data traffic from small cells to the core network. Free-space optical (FSO) has been considered as a feasible solution to the emerging backhaul/fronthaul requirements for ultra-dense heterogeneous small cells in 5G networks [2]. An FSO link can be deployed by setting a pair of laser-photodetector transceivers in line of sight between two points and it supports high data rate transmission. Due to its high security level at the unlicensed optical spectrum, FSO can deal with the issue of spectrum crunch in the backhaul link. In addition, the ability of immunity to electromagnetic interference makes FSO link a good solution to offer connectivity between radio frequency (RF) access network and optical fiber based central network. Heterodyne detection and intensity modulation/direct detection (IM/DD) are the two main modes of detection in FSO systems. In heterodyne detection, the received signal is mixed with a coherent signal of a laser beam produced by the local oscillator. The two beams fall on the photodetector by a beam splitter. The signal output from the photodetector contains a component with the difference-frequency between the coherent signal and the received signal, which is called heterodyne frequency. However, in IM/DD, the photodetector detects changes in the light intensity directly without using a local oscillator. Compared with IM/DD, heterodyne detection is more complex but significantly improves the sensitivity of photodetection [3]. Moreover, it is important to mention that on the RF side, the millimeter wave (mmWave) technology is one of the most important techniques for small cells in 5G cellular networks. It has a large spectrum to extend the network capacity massively. Therefore, FSO and RF technologies have been deployed together in the so-called mixed dual-hop FSO/RF systems to combine the advantages of RF access (low cost, flexible coverage) and FSO backhaul (high rate, low latency) [4].

However, fluctuations in both phase and intensity of the received signals caused by atmospheric turbulence are major performance limiting factors in FSO communication [5]

. In addition, FSO communication is vulnerable to weather conditions, such as rain, aerosols, and particularly fog. Moreover, the pointing error caused by buildings sway phenomenon due to thermal expansion, dynamic wind load and weak earthquakes that all result in vibration of the transmitter beam and misalignment between transmitter and receiver may lead to a severe performance degradation over the FSO links. On the other side, RF links are limited by latency problems. From another perspective, relaying technique which can be classified into amplify-and-forward (AF) and decode-and-forward (DF) relaying has been demonstrated as an efficient solution to increase the capacity for wireless communication systems as well as extending cost-efficient coverage. In DF relaying systems, the relay fully decodes the received signal and retransmits the decoded version into the second hop, while AF relays just amplify and forward the incoming signal without performing any sort of decoding, which is less complex in using

relays [6].

### I-B Contribution

Motivated by these studies, we look into the performance of a dual-hop mixed FSO/mmWave system where the FSO and RF links experience Gamma-Gamma and FTR fading channels, respectively. The effect of pointing error on the FSO link is also taken into account. It is necessary to point out that compared to the conference paper [1], which only focuses on the outage probability and average bit error rate of the mixed FSO/mmWave system, in this paper, we provide an extensive performance analysis framework of the mixed FSO/mmWave system, including novel exact closed-form expressions for ergodic capacity and effective capacity. These new results can provide useful insights to design practical dual-hop mixed FSO/mmWave communication systems. More specifically, ergodic capacity is widely employed to assess the maximum long-term achievable rate averaged over ergodic states of the time-varying fading channel [19], [20]. In addition, the effective capacity takes into account the delay constraints imposed by emerging real-time applications, which have different quality of service requirements [21, 22].

The major contributions of this paper can be summarized as follows: 1) employing AF and DF relaying, we derive new closed-form expressions for the outage probability, average BER, ergodic capacity and effective capacity of the considered system. 2) based on 1), the effects of the atmospheric turbulence, pointing errors, relaying techniques and fading figures on the mixed FSO/mmWave system performance are analyzed; 3) in order to get additional insights into the impact of system parameters, we present asymptotic expressions for the outage probability and the average BER at high signal-to-noise ratios (SNRs) to show the achievable diversity gain. Note that our derived results are general and can include the existing results in the literature [23] as special cases, since our adopted RF fading channel in this paper is the most general one.

### I-C Organization

The remainder of this paper is organized as follows: In Section II, we introduce the system and channel models. In Section III, we obtain the end-to-end SNR statistics of AF and DF relaying. In Section IV, we derive closed-form expressions of outage probability, average BER, ergodic capacity, effective capacity followed by the asymptotic expressions at high SNRs. In Section V, some numerical and simulation results are presented to confirm the accuracy of derived expressions. We finally conclude the paper in Section VI.

## Ii System And Channel Models

We consider a dual-hop mixed FSO/mmWave communication system where the source node S communicates with the destination node D through an intermediate relay node R as illustrated in Fig. 1. The FSO link (S-R) is deployed for backhauling/fronthauling combined with the mmWave RF link (i.e., R-D link) for broadband radio access. Subcarrier intensity modulation (SIM) is employed in the source node to generate an optical signal. In the relay node, both heterodyne detection and IM/DD are considered to convert the received optical field to an electrical signal. The FSO link is assumed to follow a Gamma-Gamma fading distribution with pointing error. The probability density function (PDF) of the SNR,

, is given by [24, Eq. (3)]

 fγFSO(γ)=ξ2αβrΓ(α)Γ(β)γG3,01,3⎛⎝αβ(γμr)1r∣∣∣ξ2+1ξ2,α,β⎞⎠, (1)

where

denotes the ratio between the equivalent beam radius at the receiver and the pointing error displacement standard deviation (jitter) at the receiver given as

, with and

represent the equivalent beam radius and the jitter variance at the receiver, respectively

[25]. Note that when , (1) converges to negligible pointing errors case. defines the mode of detection being used (i.e., represents heterodyne detection and represents IM/DD) [13], is the Gamma function as defined in [26, Eq. (8.310)], is the Meijer’s -function as defined in [26, Eq. (9.301)] and refers to the electrical SNR of the FSO link. Particularly, for , , and for , , with the fading parameters and related to the atmospheric turbulence conditions [27], and lower values of and indicate severe atmospheric turbulence conditions. More specifically, when a plane wave propagation in the absence of inner scale is assumed, and can be determined from the Rytov variance as and , where is the Rytov variance, is the refractive-index structure parameter, is the wavelength, and represents the propagation distance [28]. By substituting (1) into and utilizing [29, Eq. (2.54)]

, the cumulative distribution function (CDF) of

can be written as

 FγFSO(γ)=1−ξ2Γ(α)Γ(β) ×H4,02,4((αβ)rγμr∣∣∣(ξ2+1,r),(1,1)(0,1),(ξ2,r),(α,r),(β,r)), (2)

where is the Fox’s -function as defined in [29, Eq. (1.1)]. It is assumed that the RF link experiences the FTR fading and the PDF of is given by [30, Eq. 6]

 fγRF(γ)=mmΓ(m)∞∑j=0Kjdjj!fG(γ;j+1,2σ2), (3)

where

 fG(γ;j+1,2σ2)Δ=γjΓ(j+1)(2σ2)j+1exp(−γ2σ2), (4)

and

 djΔ=j∑k=0(jk)(Δ2)kk∑l=0(kl)Γ(j+m+2l−k)eπ(2l−k)i2 ×((m+K)2−(KΔ)2)−(j+m)2Pk−2lj+m−1⎛⎜ ⎜⎝m+K√(m+K)2−(KΔ)2⎞⎟ ⎟⎠,

where denotes Legendre functions of the first kind [26, Eq. (8.702)]. Moreover, denotes the ratio of the average power of the dominant waves to the scattering multipath, is the fading severity parameter and characterizes the similarity of two dominant waves varying from 0 to 1. In addition, the average SNR of RF link, , is defined as

 ¯γRF=(Eb/EbN0N0)2σ2(1+K), (5)

where is the energy density. Using [31, Eq. (2.9.4), Eq. (2.1.5), and Eq. (2.1.4)], the PDF of the FTR distribution can be represented in terms of the Meijer’s -function as

 fγRF(γ)=mmΓ(m)∞∑j=0Kjdjj!Γ(j+1)γG1,00,1[γ2σ2∣∣∣−j+1]. (6)

The CDF of the FTR distribution can be obtained by using [32, Eq. (07.34.26.0008.01)], , [29, Eq. (2.54)] and then utilizing [32, Eq. (07.34.26.0008.01)] again as

 FγRF(γ)=1−mmΓ(m)∞∑j=0Kjdjj!Γ(j+1)G2,01,2[γ2σ2∣∣∣10,j+1]. (7)

Under the assumption of fixed-gain AF relaying, the end-to-end SNR can be written as [12, Eq. 24]

 γF=γFSOγRFγRF+CR, (8)

where represents a fixed relay gain. The end-to-end SNR for DF relaying scenario can be derived as [12, Eq. (26)]

 γD=min(γFSO,γRF). (9)

## Iii End-To-End SNR Statistics

### Iii-a Fixed-Gain AF Relaying

#### Iii-A1 Exact Result

###### Corollary 1.

The CDF of the end-to-end SNR for a dual-hop mixed FSO/mmWave system using fixed-gain AF relay is

 FγF(γ)=1−ξ2mmrΓ(α)Γ(β)Γ(m)∞∑j=0Kjdjj!Γ(j+1)H0,1:2,0:0,31,0:0,2:3,2 ×⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣(1,1,1r)−−(0,1)(j+1,1)(1−ξ2,1)(1−α,1)(1−β,1)(−ξ2,1)(0,1r)∣∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣∣CR2σ2,1αβ(μrγ)1r⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦. (10)
###### Proof:

Note that the Fox’s -function with two variables in (1) can be calculated straightforwardly in well-known mathematical software, such as MATHEMATICA [33, Eq. (1.1)]. The MATLAB implementation of this function was provided in [34]. As a special case, for the RF link, when , , the FTR fading model reduces to Nakagami- fading model. Using [31, Eq. (2.9.1)] and setting , we can obtain the CDF of mixed Gamma-Gamma/Nakagami- systems using heterodyne detection with pointing errors as previous result [23, Eq. (7)]. In addition, by setting , , and , (10) simplifies to the special case where IM/DD is employed in the FSO link with no pointing errors and the RF link experiences Nakagami- fading. Furthermore, by using [31, Th. 1.7 and Th. 1.11] along with [31, Eqs. (1.5.9) and (1.8.4)], the CDF of the end-to-end SNR can be expressed in the asymptotic high-SNR regime after some algebraic manipulations as

 FγF(γ)≈ξ2mmΓ(α)Γ(β)Γ(m)∞∑j=0Kjdjj!Γ(j+1)4∑i=1ψiμ−θir, (11)

where ,

 (12)
 ψ2Δ= Γ(α−ξ2)Γ(β−ξ2)(γ1rαβ)ξ2 (13)
 ψ3Δ= Γ(β−α)ξ2−α(γ1rαβ)α ×(Γ(j+1−αr)α(CR2σ2)αr+Γ(j+1)α), (14)
 ψ4Δ= Γ(α−β)ξ2−β(γ1rαβ)β ×⎛⎜ ⎜⎝Γ(j+1−βr)β(CR2σ2)βr+Γ(j+1)β⎞⎟ ⎟⎠. (15)

#### Iii-A2 Truncation Error

By truncating (1) up to the first terms, we have

 ∧FγF(γ)=1−ξ2mmrΓ(α)Γ(β)Γ(m)N∑j=0Kjdjj!Γ(j+1)H0,1:2,0:0,31,0:0,2:3,2 ×⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣(1,1,1r)−−(0,1)(j+1,1)(1−ξ2,1)(1−α,1)(1−β,1)(−ξ2,1)(0,1r)∣∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣∣CR2σ2,1αβ(μrγ)1r⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦. (16)

The truncation error of the area under the with respect to the first terms is given by

 ε(N)=Fγ(∞)−∧Fγ(∞). (17)

Table shows the required terms for different channel parameters to demonstrate the convergence of the series in (1). For all considered cases, we only need less than 30 terms to achieve a satisfactory accuracy (e.g., smaller than ).

### Iii-B DF Relaying

Based on (9), the CDF of the end-to-end SNR is given by [12, Eq. (37)]

 FγD(γ) =1−FCγFSO(γ)FCγRF(γ), (18)

where denotes complementary CDF (CCDF) of . By substituting (II) and (7) into (III-B), we obtain the CDF of dual-hop mixed FSO/mmWave systems employing DF relay as

 FγD(γ)=1−mmΓ(m)∞∑j=0Kjdjj!Γ(j+1)H2,01,2[γ2σ2∣∣∣(1,1)(0,1),(j+1,1)] (19)

By using [31, Eqs. (1.5.9) and (1.8.4)] and after some algebraic manipulations, the CDF of the end-to-end SNR can be asymptotically expressed at high SNRs for DF relaying as shown by

 F∞γD(γ)=F∞γFSO(γ)+F∞γRF(γ)−F∞γFSO(γ)F∞γRF(γ), (20)

where

 F∞γFSO(γ) +rξ2Γ(ξ2−α)Γ(β−α)αΓ(α)Γ(β)Γ(ξ2+1−α)((αβ)rμrγ)αr (21)
 F∞γRF(γ)=mmΓ(m)∞∑j=0Kjdjj!(j+1)Γ(j+1)(γ2σ2)j+1. (22)

By truncating (22) up to the first terms, we have

 ∧F∞γRF(γ)=mmΓ(m)N1∑j=0Kjdjj!(j+1)Γ(j+1)(γ2σ2)j+1. (23)

The truncation error of the area under the with respect to the first terms is given by

 ε1(N1)=F∞γRF(∞)−∧F∞γRF(∞). (24)

In order to demonstrate the convergence of the infinite series in (22), Table presents the required truncation terms for different system and channel parameters. It should be noted that we only need less than 10 terms to converge the series for all considered cases and the truncation error is less than .

## Iv Performance Analysis

### Iv-a Fixed-Gain AF Relaying

#### Iv-A1 Outage Probability

we encounter a situation labeled as outage when the instantaneous end-to-end SNR falls below a given threshold , by replacing with in (1), we can easily obtain the outage probability as

 PFout(γth)=Pr[γF<γth]=FγF(γth). (25)

#### Iv-A2 Average Bit-Error Rate

The average BER of a variety of binary schemes and non-binary modulation schemes can be expressed as [12, Eq. (40)]

 ¯Pe=δ2Γ(p)n∑k=1qpk∫∞0γp−1exp(−qkγ)Fγ(γ)dγ, (26)

where , , and denote different modulation schemes. For instance, denotes coherent binary phase shift keying (CBPSK) and denotes differential BPSK (DBPSK).

###### Corollary 2.

The average BER of the dual-hop mixed FSO/mmWave system is given by

 ¯PFe ×⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣(1,1,1r)−−(0,1)(j+1,1)(1−ξ2,1)(1−α,1)(1−β,1)(p,1r)(−ξ2,1)(0,1r)∣∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣∣CR2σ2,(μrqk)1rαβ⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦. (27)
###### Proof:

It is worth to mention that when we set , , , and , the BER in (2) simplifies to [23, Eq. (13)] where the FSO link is operating under heterodyne detection with pointing errors and the RF link experiences Nakagami- fading. Moreover, for , , , , and , (25) reduces to the BER of a mixed Gamma-Gamma/Nakagami- system under IM/DD and no pointing errors. The asymptotic BER can be obtained by substituting (11) into (26) and using [26, Eq. (3.351.3)] after some algebraic manipulations as

 ¯PFe≈δξ2mm2Γ(p)Γ(α)Γ(β)Γ(m)∞∑j=0Kjdjj!Γ(j+1)4∑i=1κiΓ(p+θi), (28)

where

 (29)
 κ2Δ= n∑k=1Γ(α−ξ2)Γ(β−ξ2)⎛⎝(1μrqk)1rαβ⎞⎠ξ2 (30)
 κ3Δ= n∑k=1Γ(β−α)ξ2−α⎛⎝(1μrqk)1rαβ⎞⎠α ×(Γ(j+1−αr)α(CR2σ2)αr+Γ(j+1)α), (31)
 κ4= n∑k=1Γ(α−β)ξ2−β⎛⎝(1μrqk)1rαβ⎞⎠β ×⎛⎜ ⎜⎝Γ(j+1−βr)β(CR2σ2)βr+Γ(j+1)β⎞⎟ ⎟⎠. (32)

By truncating (28) up to the first terms, we have

 ∧¯PFe≈δξ2mm2Γ(p)Γ(α)Γ(β)Γ(m)N2∑j=0Kjdjj!Γ(j+1)4∑i=1κiΓ(p+θi). (33)

The truncation error of the area under the with respect to the first terms is given by

 ε2(N2)=¯PFe−∧¯PFe. (34)

The required terms for different system and channel parameters are presented in Table to demonstrate the convergence of the infinite series in (28). We only need less than 40 terms to achieve a satisfactory accuracy (e.g., smaller than ) for all considered cases. It can be seen from (28)-(IV-A2) that the average BER decreases as the average SNR of both FSO (i.e., ) and RF (i.e., ) links increase, which can be explained from (5) that is an increasing function of with fixed. Moreover, the performance of average BER degrades when the values of and get larger which represent for non-binary modulation schemes. Furthermore, it can be shown that the diversity gain is equal to

 Gd=min(2,ξ2r,αr,βr). (35)

We can observe from (35) that the diversity order is a function of FSO turbulence parameters (i.e., and ), pointing error (i.e., ) and detection mode (i.e., ).

#### Iv-A3 Ergodic Capacity

The ergodic capacity is defined as , where refers to the expectation operator, for heterodyne method (i.e., ) and for IM/DD (i.e., ). By employing part-by-part integration method, ergodic capacity can be expressed in terms of the CCDF of as

 ¯C=cln(2)∫∞0Fcγ(γ)1+cγdγ. (36)

The expression in (36) is exact for the case of heterodyne detection while it is a lower-bound for IM/DD since the transmitted symbols are always positive in IM/DD systems.

###### Corollary 3.

The ergodic capacity of the dual-hop mixed FSO/mmWave systems can be derived as

 ¯CF=ξ2mmln(2)rΓ(α)Γ(β)Γ(m)∞∑j=0Kjdjj!Γ(j+1)H0,1:2,0:1,41,0:0,2:4,3 ×⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣(1,1,1r)−−(0,1)(j+1,1)(1,1r)(1−ξ2,1)(1−α,1)(1−β,1)(1,1r)(−ξ2,1)(0,1r)∣∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣∣CR2σ2,1αβ(μrc)1r⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦. (37)
###### Proof:

For , , , as a special case, (3) reduces to the ergodic capacity of a dual-hop mixed FSO/mmWave system where the RF link experiences Nakagami- fading under pointing error and heterodyne detection, given in [23, Eq. (15)]. We can also set , , and to obtain the special case where the FSO link is under IM/DD for no pointing errors and the RF link experiences Nakagami- fading.

#### Iv-A4 Effective Capacity

The effective capacity is defined as , where with the asymptotic decay rate of the buffer occupancy , the block length , and the system bandwidth [35, 36, 37, 38, 39, 40, 41]. By employing part-by-part integration method, the effective capacity can be expressed in terms of the CCDF of as

 R=−1Alog2(1−A∫∞0(1+γ)−A−1FCγ(γ)dγ). (38)
###### Corollary 4.

The effective capacity of the dual-hop mixed FSO/mmWave systems can be derived as

 RF=−1Alog2(1−ξ2mmrΓ(α)Γ(β)Γ(m)Γ(A)∞∑j=0Kjdjj!Γ(j+1)H0,1:2,0:1,41,0:0,2:4,3 ×⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣(1,1,1r)−−(0,1)(j+1,1)(1−A,1r)(1−ξ2,1)(1−α,1)(1−β,1)(1,1r)(−ξ2,1)(0,1r)∣∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣∣CR2σ2,(μr)1rαβ⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠. (39)
###### Proof:

It can be shown that when we set , , the effective capacity in (4) can be simplified to the special case for Gamma-Gamma/Nakagami- fading channels using heterodyne detection which is given as

 RF=−1Alog2(1−ξ2Γ(α)Γ(β)Γ(m)Γ(A)H0,1:2,0:1,41,0:0,2:4,3 ×⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣(1,1,1)−−(0,1)(m,1)(1−A,1)(1−ξ2,1)(1−α,1)(1−β,1)(1,1)(−ξ2,1)(0,1)∣∣ ∣ ∣ ∣ ∣ ∣ ∣ ∣∣mCR¯γRF,μ1αβ⎤⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥⎦⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠. (40)

### Iv-B DF Relaying

#### Iv-B1 Outage Probability

The outage probability of DF relaying can be obtained by using (III-B) which is given as [42, 43]

 PDout(γth)=Pr(γD<γth)=FγD(γth). (41)

#### Iv-B2 Average BER

Substituting (III-B) into (26), using [44, Eq. (2.25.1/1)] along with [33, Eq. (2.3)] after some algebraic manipulations, the average BER can be obtained in closed-form as

 ¯PeD=nδ2−ξ2mmΓ(α)Γ(β)Γ(m)∞∑j=0Kjdjj!Γ(j+1)n∑k=1H0,1:2,0:4,01,0:1,2:2,4 ×⎡⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢⎣(1−p,1,1)−(1,1)(0,1)(j+1,1)(